# EES Manual - University of New Hampshire

```EES
Engineering Equation Solver
for Microsoft Windows
Operating Systems
F-Chart Software
4406 Fox Bluff Rd
Middleton, WI 53562
Phone: 608-836-8531
FAX: 608-836-8536
http://www.fchart.com
The authors make no guarantee that the program is free from errors or that the results
produced with it will be free of errors and assume no responsibility or liability for the
accuracy of the program or for the results which may come from its use.
EES was compiled with DELPHI by Borland
Registration Number__________________________
ALL CORRESPONDENCE MUST INCLUDE THE REGISTRATION
NUMBER
v4/7.01
EES
Engineering Equation Solver
for Microsoft Windows
Operating Systems
F-Chart Software
4406 Fox Bluff Rd
Middleton, WI 53562
Phone: 608-836-8531
FAX: 608-836-8536
http://www.fchart.com
Overview ............................................................................................................. 1
Chapter 1: Getting Started ........................................................................... 5
Installing EES on your Computer................................................................. 5
Starting EES ................................................................................................. 5
Background Information............................................................................... 6
An Example Thermodynamics Problem....................................................... 9
Chapter 2: EES Windows........................................................................... 19
General Information ................................................................................... 19
Equations Window ..................................................................................... 21
Formatted Equations Window .................................................................... 23
Solution Window........................................................................................ 25
Arrays Window........................................................................................... 27
Residuals Window...................................................................................... 29
Parametric Table Window .......................................................................... 31
Lookup Table Window ............................................................................... 34
Diagram Window........................................................................................ 36
Plot Windows ............................................................................................. 39
Debug Window........................................................................................... 44
Chapter 4: Built-in Functions ........................................................................ 91
Mathematical Functions ............................................................................. 91
Thermophysical Property Functions ........................................................... 98
Using Lookup Files and the Lookup Table............................................... 104
ii
Chapter 5: EES Modules, Functions and Procedures .......................... 111
EES Functions . ........................................................................................ 112
EES Procedures ........................................................................................ 114
Single-Line If Then Else Statements ........................................................ 116
Multiple-Line If Then Else Statements..................................................... 117
GoTo Statements ...................................................................................... 118
Repeat Until Statements ........................................................................... 118
Error Procedure......................................................................................... 119
Modules .................................................................................................... 120
Library Files.............................................................................................. 123
\$COMMON Directive .............................................................................. 125
The \$INCLUDE directive......................................................................... 126
Chapter 6: Compiled Functions and Procedures ................................... 127
EES Compiled Functions (.DLF Files)..................................................... 127
The PWF Example Compiled Function.................................................... 130
EES Compiled Procedures (.FDL and .DLP Files)................................... 133
Compiled Procedures with the .FDL Format - a FORTRAN Example.... 134
Compiled Procedures with the .DLP Format - a Pascal Example ............ 137
Help for Compiled Routines..................................................................... 139
Chapter 7: Advanced Features ..................................................................... 141
String Variables ........................................................................................ 141
Complex Variables ................................................................................... 142
Array Variables......................................................................................... 146
The DUPLICATE Command ................................................................... 147
Matrix Capabilities ................................................................................... 148
Using the Property Plot............................................................................. 150
Integration and Differential Equations ..................................................... 151
Appendix A: Hints for Using EES .............................................................. 161
Appendix B: Numerical Methods used in EES ....................................... 165
Solution to Algebraic Equations............................................................... 165
Blocking Equation Sets............................................................................. 168
Determination of Minimum or Maximum Values.................................... 170
Numerical Integration ............................................................................... 171
References for Numerical Methods .......................................................... 173
Appendix C: Adding Property Data to EES ............................................. 175
Appendix D: Example Problem Information .......................................... 185
-iii-
__________________________________________________________________________
Overview
__________________________________________________________________________
EES (pronounced ’ease’) is an acronym for Engineering Equation Solver. The basic function
provided by EES is the solution of a set of algebraic equations. EES can also solve
differential equations, equations with complex variables, do optimization, provide linear and
non-linear regression and generate publication-quality plots. Versions of EES have been
developed for Apple Macintosh computers and for the Windows operating systems. This
manual describes the version of EES developed for Microsoft Windows operating systems,
including Windows 3.1, Windows 95, and Windows NT.
There are two major differences between EES and existing numerical equation-solving
programs. First, EES automatically identifies and groups equations which must be solved
simultaneously. This feature simplifies the process for the user and ensures that the solver
will always operate at optimum efficiency. Second, EES provides many built-in
mathematical and thermophysical property functions useful for engineering calculations.
For example, the steam tables are implemented such that any thermodynamic property can
be obtained from a built-in function call in terms of any two other properties. Similar
capability is provided for most organic refrigerants (including some of the new blends),
ammonia, methane, carbon dioxide and many other fluids. Air tables are built-in, as are
psychrometric functions and JANAF table data for many common gases. Transport
properties are also provided for most of these substances.
The library of mathematical and thermophysical property functions in EES is extensive, but
it is not possible to anticipate every user’s need. EES allows the user to enter his or her own
functional relationships in three ways. First, a facility for entering and interpolating tabular
data is provided so that tabular data can be directly used in the solution of the equation set.
Second, the EES language supports user-written functions and procedure similar to those in
Pascal and FORTRAN. EES also provides support for user-written modules, which are selfcontained EES programs that can be accessed by other EES programs. The functions,
procedures, and modules can be saved as library files which are automatically read in when
EES is started. Third, compiled functions and procedures, written in a high-level language
such as Pascal, C or FORTRAN, can be dynamically-linked into EES using the dynamic link
library capability incorporated into the Windows operating system. These three methods of
adding functional relationships provide very powerful means of extending the capabilities of
EES.
1
The motivation for EES rose out of experience in teaching mechanical engineering
thermodynamics and heat transfer. To learn the material in these courses, it is necessary for
the student to work problems. However, much of the time and effort required to solve
problems results from looking up property information and solving the appropriate
equations. Once the student is familiar with the use of property tables, further use of the
tables does not contribute to the student’s grasp of the subject; nor does algebra. The time
and effort required to do problems in the conventional manner may actually detract from
learning of the subject matter by forcing the student to be concerned with the order in which
the equations should be solved (which really does not matter) and by making parametric
studies too laborious. Interesting practical problems that may have implicit solutions, such
as those involving both thermodynamic and heat transfer considerations, are often not
assigned because of their mathematical complexity. EES allows the user to concentrate
more on design by freeing him or her from mundane chores.
EES is particularly useful for design problems in which the effects of one or more
parameters need to be determined. The program provides this capability with its Parametric
Table, which is similar to a spreadsheet. The user identifies the variables which are
independent by entering their values in the table cells. EES will calculate the values of the
dependent variables in the table. The relationship of the variables in the table can then be
displayed in publication-quality plots. EES also provides capability to propagate the
uncertainty of experimental data to provide uncertainty estimates of calculated variables.
With EES, it is no more difficult to do design problems than it is to solve a problem for a
fixed set of independent variables.
EES offers the advantages of a simple set of intuitive commands which a novice can quickly
learn to use for solving any algebraic problems. However, the capabilities of this program
are extensive and useful to an expert as well. The large data bank of thermodynamic and
transport properties built into EES is helpful in solving problems in thermodynamics, fluid
mechanics, and heat transfer. EES can be used for many engineering applications; it is
ideally suited for instruction in mechanical engineering courses and for the practicing
engineer faced with the need for solving practical problems.
The remainder of this manual is organized into seven chapters and five appendices. A new
user should read Chapter 1 which illustrates the solution of a simple problem from start to
finish. Chapter 2 provides specific information on the various functions and controls in each
of the EES windows. Chapter 3 is a reference section that provides detailed information for
each menu command. Chapter 4 describes the built-in mathematical and thermophysical
property functions and the use of the Lookup Table for entering tabular data. Chapter 5
provides instructions for writing EES functions, procedures and modules and saving them in
Library files. Chapter 6 describes how compiled functions and procedures, written as
2
Windows dynamic-link library (DLL) routines, can be integrated with EES. Chapter 7
describes a number of advanced features in EES such as the use of string, complex and array
variables, the solution of simultaneous differential and algebraic equations, and property
plots. Appendix A contains a short list of suggestions. Appendix B describes the numerical
methods used by EES. Appendix C shows how additional property data may be
incorporated into EES. A number of example problems are provided in the Examples
subdirectory included with EES. Appendix D indicates which features are illustrated in the
example problems provided with EES.
3
4
CHAPTER1
__________________________________________________________________________
Getting Started
__________________________________________________________________________
There are two versions of EES: EES and EES32. EES is designed to operate with any of the
Microsoft Windows operating systems. EES32 is a 32-bit version of the program that will
operate only under Windows 95 and NT. This manual is applicable to both versions.
However some of the newer features of the program, e.g., modules (Chapter 5) and complex
variables (Chapter 7) are implemented only in the 32-bit version. If you are using Windows
95 or NT, you should install the 32-bit version of EES.
EES is distributed in a self-installing compressed form in a file called SETUP16.exe (16-bit
version) or SETUP32.exe (32-bit version). To install EES or EES32, execute the
installation program. In Windows 95 or NT, the installation program can be executed by
selecting the Run command from the Start menu and entering A:\SETUP16.exe or
A:\SETUP32.exe.
Here A: is your floppy drive designation. The installation program will provide a series of
prompts which will lead you through the complete installation of the EES program.
Starting EES
The default installation program will create a directory named \EESW (16-bit version) or
\EES32 (32-bit version) in which the EES files are placed. The EES program icon shown
above will identify both the program and EES files. Double-clicking the left mouse button
on the EES program or file icon will start the program. If you double-clicked on an EES
file, that file will be automatically loaded. Otherwise, EES will load the HELLO.EES file
5
Chapter 1
Getting Started
which briefly describes the new features in your version. You can delete or rename the
HELLO.EES file if you do not wish to have it appear when the program is started.
Background Information
EES begins by displaying a dialog window which shows registration information, the
version number and other information. The version number and registration information
will be needed if you request technical support. Click the OK button to dismiss the dialog
window.
Detailed help is available at any point in EES. Pressing the F1 key will bring up a Help
window relating to the foremost window. Clicking the Contents button will present the Help
index shown below. Clicking on an underlined word (shown in green on color monitors)
will provide help relating to that subject.
6
Getting Started
Chapter 1
EES commands are distributed among nine pull-down menus. (A tenth user-defined menu
can be placed to the right of the Help menu. See the discussion of the Load Textbook
command File menu in Chapter 3.) A brief summary of their functions follows. Detailed
descriptions of the commands appear in Chapter 3.
Note the a toolbar is provided below the menu bar. The toolbar contains small buttons
you move the cursor over a button and wait for a few second, a few words will appear to
explain the function of that button. The toolbar can be hidden, if you wish, with a control in
The System menu represented by the EES icon appears above the file menu. The System
menu is not part of EES, but rather a feature of the Windows Operating System. It holds
commands which allow window moving, resizing, and switching to other applications.
and printing.
The Edit menu provides the editing commands to cut, copy, and paste information.
The Search menu provides Find and Replace commands for use in the Equations window.
The Options menu provides commands for setting the guess values and bounds of variables,
the unit system, default information, and program preferences. A command is also
provided for displaying information on built-in and user-supplied functions.
The Calculate menu contains the commands to check, format and solve the equation set.
The Tables menu contains commands to set up and alter the contents of the Parametric and
Lookup Tables and to do linear regression on the data in these tables. The Parametric
Table, similar to a spreadsheet, allows the equation set to be solved repeatedly while
varying the values of one or more variables. The Lookup table holds user-supplied data
which can be interpolated and used in the solution of the equation set.
The Plot menu provides commands to modify an existing plot or prepare a new plot of data
in the Parametric, Lookup, or Array tables. Curve-fitting capability is also provided.
The Windows menu provides a convenient method of bringing any of the EES windows to
the front or to organize the windows.
7
Chapter 1
Getting Started
The basic capability provided by EES is the solution of a set of non-linear algebraic
equations. To demonstrate this capability, start EES and enter this simple example problem
in the Equations window. Note that EES makes no distinction between upper and lower
case letters and the ^ sign (or **) is used to signify raising to a power.
If you wish, you may view the equations in mathematical notation by selecting the Formatted
Equations command from the Windows menu.
Select the Solve command from the Calculate menu. A dialog window will appear indicating
the progress of the solution. When the calculations are completed, the button changes from
Abort to Continue.
Click the Continue button. The solution to this equation set will then be displayed.
8
Getting Started
Chapter 1
An Example Thermodynamics Problem
A simple thermodynamics problem will be set up and solved in this section to illustrate the
property function access and equation solving capability of EES. The problem, typical of
that which may be encountered in an undergraduate thermodynamics course, is as follows.
Refrigerant-134a enters a valve at 700 kPa, 50°C with a velocity of 15 m/s. At the exit of the
valve, the pressure is 300 kPa. The inlet and outlet fluid areas are both 0.0110 m2.
Determine the temperature, mass flow rate and velocity at the valve exit.
State 1
T = 50°C
P = 700
Vel = 15 m/s
State 2
T=?
P = 300 kPa
Vel = ?
To solve this problem, it is necessary to choose a system and then apply mass and energy
balances. The system is the valve. The mass flow is steady, so that the mass balance is:
m1 = m2
(1)
m1 = A1 Vel1 / v1
(2)
m1 = A2 Vel2 / v2
(3)
A1 = A2
(4)
where
m = mass flowrate [kg/s]
A = cross-sectional area [m2]
Vel = velocity [m/s]
v = specific volume [m3/kg]
We know that
The valve is assumed to be well-insulated with no moving parts. The heat and work effects
are both zero. A steady-state energy balance on the valve is:
m& 1 h1 +
Vel12
Vel 2
= m& 2 h2 + 2
2
2
(5)
where h is the specific enthalpy and Vel2/2 is the specific kinetic energy. In SI units,
specific enthalpy normally has units of [kJ/kg] so some units conversions may be needed.
9
Chapter 1
Getting Started
EES provides unit conversion capabilities with the CONVERT function as documented in
Chapter 4.
From relationships between the properties of R134a:
1 6
h = h1T , P 6
v = v 1T , P 6
h = h1T , P 6
v1 = v T1 , P1
(6)
1
1
1
(7)
2
2
2
(8)
2
2
2
(9)
Ordinarily, the terms containing velocity are neglected, primarily because the kinetic energy
effects are usually small and also because these terms make the problem difficult to solve.
However, with EES, the computational difficulty is not a factor. The user can solve the
problem with the kinetic energy terms and judge their importance.
The values of T1, P1, A1, Vel11 and P2 are known. There are nine unknowns: A2, m1 , m2 ,
Vel2, h1, v1, h2, v2, T2. Since there are 9 equations, the solution to the problem is defined. It
is now only necessary to solve the equations. This is where EES can help.
Start EES and select the New command from the File menu. A blank Equations window will
appear. Before entering the equations, however, set the unit system for the built-in
thermophysical properties functions. To view or change the unit system, select Unit System
EES is initially configured to be in SI units with T in °C, P in kPa, and specific property
values in their customary units on a mass basis. These defaults may have been changed
during a previous use. Click on the controls to set the units as shown above. Click the OK
button (or press the Return key) to accept the unit system settings.
10
Getting Started
Chapter 1
The equations can now be entered into the Equations window. Text is entered in the same
manner as for any word processor. Formatting rules are as follows:
1. Upper and lower case letters are not distinguished. EES will (optionally) change the case
of all variables to match the manner in which they first appear.
2. Blank lines and spaces may be entered as desired since they are ignored.
3. Comments must be enclosed within braces { } or within quote marks " ". Comments
may span as many lines as needed. Comments within braces may be nested in which
case only the outermost set of { } are recognized. Comments within quotes will also
be displayed in the Formatted Equations window.
4. Variable names must start with a letter and consist of any keyboard characters except ( ) ‘
| * / + - ^ { } : " or ;. Array variables (Chapter 7) are identified with square braces
around the array index or indices, e.g., X[5,3]. String variables (Chapter 7) are identified
with a \$ as the last character in the variable name. The maximum length of a variable
name is 30 characters.
5. Multiple equations may be entered on one line if they are separated by a semi-colon (;)1.
The maximum line length is 255 characters.
6. The caret symbol ^ or ** is used to indicate raising to a power.
7. The order in which the equations are entered does not matter.
8. The position of knowns and unknowns in the equation does not matter.
After entering the equations for this problem and (optionally) checking the syntax using the
Check/Format command in the Calculate menu, the Equations window will appear as shown.
Comments are normally displayed in blue on a color monitor. Other formatting options are
set with the Preferences command in the Options menu.
1
If a comma is selected as the Decimal Symbol in the Windows Regional Settings Control Panel, EES will recognize
the comma (rather than a decimal point) as a decimal separator, the semicolon (rather than the comma) as an
argument separator, and the vertical bar | (rather than the semicolon) as the equation separator.
11
Chapter 1
Getting Started
Note the use of the Convert function in this example to convert the units of the specific
kinetic energy [m^2/s^2] to the units used for specific enthalpy [kJ/kg]. The Convert
function is most useful in these problems. See Chapter 4 for a detailed description of its use.
The thermodynamic property functions, such as enthalpy and volume require a special
format. The first argument of the function is the substance name, R134a in this case. The
following arguments are the independent variables preceded by a single identifying letter and
an equal sign. Allowable letters are T, P, H, U, S, V, and X, corresponding to temperature,
pressure, specific enthalpy, specific internal energy, specific entropy, specific volume, and
quality. (For psychrometric functions, additional allowable letters are W, R, D, and B,
corresponding to humidity ratio, relative humidity, dewpoint temperature, and wetbulb
temperature.)
An easy way to enter functions, without needing to recall the format, is to use the Function
Information command in the Options menu. This command will bring up the dialog window
shown below. Click on the ‘Thermophysical props’ radio button. The list of built-in
thermophysical property function will appear on the left with the list of substances on the
right. Select the property function by clicking on its name, using the scroll bar, if necessary,
to bring it into view. Select a substance in the same manner. An example of the function
showing the format will appear in the Example rectangle at the bottom. The information in
the rectangle may be changed, if needed. Clicking the Paste button will copy the Example
into the Equations window at the cursor position. Additional information is available by
clicking the Info button.
12
Getting Started
Chapter 1
It is usually a good idea to set the guess values and (possibly) the lower and upper bounds
for the variables before attempting to solve the equations. This is done with the Variable
Information command in the Options menu. Before displaying the Variable Information
dialog, EES checks syntax and compiles newly entered and/or changed equations, and then
solves all equations with one unknown. The Variable Information dialog will then appear.
The Variable Information dialog contains a line for each variable appearing in the Equations
window. By default, each variable has a guess value of 1.0 with lower and upper bounds of
negative and positive infinity. (The lower and upper bounds are shown in italics if EES has
previously calculated the value of the variable. In this case, the Guess value column displays
the calculated value. These italicized values may still be edited, which will force EES to
recalculate the value of that variable.)
The A in the Display options column indicates that EES will automatically determine the
display format for numerical value of the variable when it is displayed in the Solution
window. In this case, EES will select an appropriate number of digits, so the digits column
to the right of the A is disabled. Automatic formatting is the default. Alternative display
options are F (for fixed number of digits to the right of the decimal point) and E (for
exponential format). The display and other defaults can easily be changed with the Default
Information command in the Options menu, discussed in Chapter 3. The third Display options
column controls the hilighting effects such as normal (default), bold, boxed. The units of the
variables can be specified, if desired. The units will be displayed with the variable in the
Solution window and/or in the Parametric Table. EES does not automatically do unit
13
Chapter 1
Getting Started
conversions but it can provide unit conversions using the Convert function (Chapter 4). The
units information entered here is only for display purposes.
With nonlinear equations, it is sometimes necessary to provide reasonable guess values and
bounds in order to determine the desired solution. (It is not necessary for this problem.) The
bounds of some variables are known from the physics of the problem. In the example
problem, the enthalpy at the outlet, h2, should be reasonably close to the value of h1. Set its
guess value to 100 and its lower bound to 0. Set the guess value of the outlet specific
volume, v2, to 0.1 and its lower bound to 0. Scroll the variable information list to bring
Vel2 into view. The lower bound of Vel2 should also be zero.
To solve the equation set, select the Solve command from the Calculate menu. An
information dialog will appear indicating the elapsed time, maximum residual (i.e., the
difference between the left-hand side and right-hand side of an equation) and the maximum
change in the values of the variables since the last iteration. When the calculations are
completed, EES displays the total number of equations in the problem and the number of
blocks. A block is a subset of equations which can be solved independently. EES
automatically blocks the equation set, whenever possible, to improve the calculation
efficiency, as described in Appendix B. When the calculations are completed, the button
will change from Abort to Continue.
By default, the calculations are stopped when 100 iterations have occurred, the elapsed time
exceeds 60 sec, the maximum residual is less than 10-6 or the maximum variable change is
less than 10-9. These defaults can be changed with the Stop Criteria command in the Options
menu. If the maximum residual is larger than the value set for the stopping criteria, the
equations were not correctly solved, possibly because the bounds on one or more variables
constrained the solution. Clicking the Continue button will remove the information dialog
and display the Solution window shown on the next page. The problem is now completed
since the values of T2, m2, and Vel2 are determined.
14
Getting Started
Chapter 1
One of the most useful features of EES is its ability to provide parametric studies. For
example, in this problem, it may be of interest to see how the throttle outlet temperature and
outlet velocity vary with outlet pressure. A series of calculations can be automated and
plotted using the commands in the Tables menu.
Select the New Table command. A dialog will be displayed listing the variables appearing in
the Equations window. In this case, we will construct a table containing the variables P2, T2,
Vel2, and h2. Click on P2 from the variable list on the left. This will cause P2 to be
highlighted and the Add button will become active.
Now click the Add button to move P2 to the list of variables on the right. Repeat for T2, h2,
and Vel2, using the scroll bar to bring the variable into view if necessary. (As a short cut,
you can double-click on the variable name in the list on the left to move it to the list on the
right.). The table setup dialog should now appear as shown above. Click the OK button to
create the table.
The Parametric Table works much like a spreadsheet. You can type numbers directly into
the cells. Numbers which you enter are shown in black and produce the same effect as if
15
Chapter 1
Getting Started
you set the variable to that value with an equation in the Equations window. Delete the P2 =
300 equation currently in the Equations window or enclose it in comment brackets { }. This
equation will not be needed because the value of P2 will be set in the table. Now enter the
values of P2 for which T2 is be determined. Values of 100 to 550 have been chosen for this
example. (The values could also be automatically entered using Alter Values in the Tables
menu or by using the Alter Values control at the upper right of each table column header,
as explained in Chapter 2.) The Parametric Table should now appear as shown below.
Now, select Solve Table from the Calculate menu. The Solve Table dialog window will
appear allowing you to choose the runs for which the calculations will be done.
When the Update Guess Values control is selected, as shown, the solution for the last run
will provide guess values for the following run. Click the OK button. A status window will
be displayed, indicating the progress of the solution. When the calculations are completed,
the values of T2, Vel2, and h2 will be entered into the table. The values calculated by EES
will be displayed in blue, bold or italic type depending on the setting made in the Screen
Display tab of the Preferences dialog window in the Options menu.
16
Getting Started
Chapter 1
The relationship between variables such as P2 and T2 is now apparent, but it can more
clearly be seen with a plot. Select New Plot Window from the Plot menu. The New Plot
Window dialog window shown below will appear. Choose P2 to be the x-axis by clicking
on P2 in the x-axis list. Click on T2 in the y-axis list. Select the scale limits for P2 and T2,
and set the number of divisions for the scale as shown. Grid lines make the plot easier to
read. Click on the Grid Lines control for both the x and y axes. When you click the OK
button, the plot will be constructed and the plot window will appear as shown.
17
Chapter 1
Getting Started
Once created, there are a variety of ways in which the appearance of the plot can be changed
as described in the Plot Windows section of Chapter 2 and in the Plot menu section of
Chapter 3.
This example problem illustrates some of the capabilities of EES. With this example behind
you, you should be able to solve many types of problems. However, EES has many more
capabilities and features, such as curve-fitting, uncertainty analyses, complex variables,
arrays.
18
CHAPTER 2
__________________________________________________________________________
EES Windows
__________________________________________________________________________
General Information
The information concerning a problem is presented in a series of windows. Equations and
comments are entered in the Equations window. After the equations are solved, the values
of the variables are presented in the Solution and Arrays windows. The residuals of the
equations and the calculation order may be viewed in the Residuals window. Additional
windows are provided for the Parametric and Lookup Tables, a diagram and up to 10 plots.
There is also a Debug window. A detailed explanation of the capabilities and information
for each window type is provided in this section. All of the windows can be open (i.e.,
visible) at once. The window in front is the active window and it is identified by its
highlighted (black) title bar. The figure below shows the appearance of the EES windows in
Microsoft Windows 95 and NT 4.0. The appearance may be slightly different in other
Windows versions.
One difference between EES and most other applications is worth mentioning. The Close
control merely hides a window; it does not delete it. Once closed, a window can be
19
Chapter 2
EES Windows
Every window has a number of controls.
1. To move the window to a different location on the screen, move the cursor to a position
on the title bar of the window and then press and hold the left button down while sliding
the mouse to a new location.
2. To hide the window, select the Close command (or press Ctrl-F4) from the control menu
box at the upper left of the window title bar. (Windows 95 and NT 4.0 also provide a
Close icon at the upper right of the title bar.) You can restore a hidden window by
selecting it from the Windows menu.
3. The Maximize box at the upper right of the window title bar causes the window to be
resized so as to fill the entire screen. The Restore box with an up and down arrow will
appear below the Maximize box. Click the Restore box (or select Restore form the
Control menu box) to return the window to its former size.
4. The size of any window can be adjusted using the window size controls at any border of
the window. To change the size of any window, move the cursor to the window border.
The cursor will change to a horizontal or vertical double arrow. Then press and hold the
left button down while moving the mouse to make the window larger or smaller. Scroll
bars will be provided if the window is made too small to accommodate all the
information.
5. Double-clicking the left mouse button on the EES icon at the upper left of the title bar
will hide that window.
6. Use the Cascade command in the Windows menu to move and resize all open windows.
20
EES Windows
Chapter 2
Equations Window
The Equations window operates very much like a word processor. The equations which
EES is to solve are entered in this window. Editing commands, i.e., Cut, Copy, Paste, are
located in the Edit menu and can be applied in the usual manner. Additional information
relevant to the Equations window follows.
1. Blank lines may be used to make the Equations window more legible. Comments are
enclosed in braces {comment} or in quote marks "another comment" and may span
multiple lines. Nested comment fields within braces are permitted. Comments within
quote marks will appear in the Formatted Equations window.
2. Equations may be entered in any order. The order of the equations has no effect on the
solution, since EES will block the equations and reorder them for efficient solution as
described in Appendix B.
3. The order of mathematical operators used in the equations conform to the rules used in
FORTRAN, Basic, C or Pascal. For example, the equation
X= 3+4*5
will result in X having a value of 23. The caret symbol ^ or ** can be used to indicate
raising to a power. Arguments of functions are enclosed in parentheses. EES does not
require a variable to appear by itself on the left-hand side of the equation, as does
FORTRAN and most other programming languages. The above equation could have
been entered as
(X – 3) / 4 = 5
4. Upper and lower case letters are not distinguished. EES will (optionally) change the case
of all variables to match the manner in which they first appear in the Equations window
this change is made only when an equation is first compiled or modified or when
Check/Format command in the Calculate menu is issued.
5. Variable names must start with a letter and consist of any keyboard characters except
(‘|)*/+-^{ } ":;. The maximum variable length is 30 characters. String variables hold
character information and are identified with a \$ as the last character in their names, as in
BASIC. Array variables are identified with square braces around the array index or
indices, e.g., X[5,3]. The quantity within the braces must be a number, except within the
scope of the sum, product or Duplicate commands. As a general rule, variables should not
be given names which correspond to those of built-in functions (e.g., pi, sin, enthalpy).
6. EES has an upper limit of 5000 variables (32-bit version).
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7. Equations are normally entered one per line, terminated by pressing the Return or Enter
keys. Multiple equations may be entered on one line if they are separated by a semicolon2. Long equations are accommodated by the provision of a horizontal scroll bar
which appears if any of the equations is wider than the window. However, each equation
must be less than 255 characters.
8. EES compiles equations into a compact stack-based form. The compiled form is saved
in memory so that an equation needs to be compiled only when it is first used or when it
is changed. Any error detected during the compilation or solution process will result in
an explanatory error message and highlighting of the line in which the problem was
discovered.
9. Equations can be imported or exported from/to other applications by using Cut, Copy and
Paste commands in the Edit menu. The Merge and Load Library commands in the File
menu and the \$INCLUDE directive may also be used to import the equations from an
existing file. The Merge command will import the equations from an EES or text file
and place them in the Equations window at the cursor position. Equations imported with
the \$INCLUDE directive will not appear in the Equations window.
10. Clicking the right mouse button in the Equations window will either insert or remove
curly brace comments around the selected text. If the selected text is already
commented, i.e., begins with a left brace and ends with a right brace, the comments will
be removed - otherwise the braces will be inserted.
11. If EES is configured to operate in complex mode, all variables as assumed to have real
and imaginary components. The complex mode configuration can be changed in the
Preferences Dialog (Options menu) or with the \$Complex On/Off directive.
2
If a comma is selected as the Decimal Symbol in the Windows Regional Settings Control Panel, EES will
recognize the comma (rather than a decimal point) as a decimal separator, the semicolon (rather than the comma)
as an argument separator, and the vertical bar | (rather than the semicolon) as the equation separator.
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EES Windows
Chapter 2
Formatted Equations Window
The Formatted Equations window displays the equations entered in the Equations window in
an easy-to-read mathematical format as shown in the sample windows below.
Note that comments appearing in quotes in the Equations window are displayed in the
Formatted Equations window but comments in braces are not displayed. An examination of
the Formatted Equations Window will reveal a number of EES features to improve the
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display, in addition to the mathematical notation. Array variables, such as B[1] are
(optionally) displayed as subscripted variables. Sums and integrals are represented by their
mathematical signs. If a variable name contains an underscore, the underscore will signify
the beginning of a subscript, as in variable G_2. However, note that although G[2] and G_2
will display in the same manner in the Formatted Equations Window, they are different
variables with different properties. The index of array variables, e.g., G[2], can be used
within the scope of Duplicate statements, or with the Sum and Product functions. In
addition, the calculated value of G[2] can be displayed in the Arrays Window, as described
in more detail in this chapter.
Placing _dot or _bar after a variable name places a dot or bar centered over the name. The
_infinity results in a subscript with the infinity symbol (∞). Variables having a name from
the Greek alphabet are displayed with the equivalent Greek letter. For example, the variable
name beta will display as ß and mu will display as a µ. If the variable name in the Equations
window is entered entirely in capital letters, and if the capital Greek letter is distinct from the
English alphabet, the capital Greek letter will be used. For example, the variable name
GAMMA will be displayed as Γ. A special form is provided for variables beginning with
DELTA. For example, DELTAT displays as ∆T. Capital BETA looks just like a B, so EES
will display the lower case equivalent, i.e., ß.
The formatted equations and comments appearing the Formatted Equations window can be
moved to other positions if you wish. To move an equation or comment, move the cursor to
the item and then press and hold the left mouse button down while sliding the equation or
comment to a new location.
The formatted equations and comments are internally represented as Windows MetaFilePict
items or pictures. You can copy one or more equation pictures from this window to other
applications (such as a word processor or drawing program). To copy an equation, first
select it by clicking the left mouse button anywhere within the equation rectangle. A
selected equation or comment will be displayed in inverse video. You may select additional
equations. Alternatively, the Select Display command in the Edit menu can be used to select
all of the equations and comments which are currently visible in the Formatted Equations
window. Copy the selected equations and comments to the clipboard with the Copy
command. The equations will be unselected after the copy operation. Comments normally
appear in blue text on the Formatted Equations window and they will appear in color when
copied to the Clipboard. If you wish to have the comments displayed in black, hold the Shift
key down while issuing the Copy command.
The text in the Formatted Equations window can not be edited. However, clicking the right
mouse button on an equation in the Formatted Equation window will bring the Equations
window to the front with that equation selected where it can be edited.
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EES Windows
Chapter 2
Solution Window
The Solution window will automatically appear in front of all other windows after the
calculations, initiated with the Solve or Min/Max commands in the Calculate menu, are
completed. The values and units of all variables appearing in the Equations window will be
shown in alphabetical order using as many columns as can be fit across the window.
The format of the variables and their units can be changed using the Variable Info command
in the Options menu, or more simply, directly from the Solution window. Clicking the left
mouse button on a variable selects that variable which is then displayed in inverse video.
Clicking the left mouse button on a selected variable unselects it. Double-clicking the left
mouse button (or clicking the right mouse button) brings up the Format Variable dialog
window. The changes made in the Format Variable dialog are applied to ALL selected
variables. Pressing the Enter key will also bring up the Format Variable dialog window.
The numerical format (style and digits) and the units of the selected variables can be selected
in this dialog window. When configured in Complex mode, an additional formatting option
is provided for displaying the variable in rectangular or polar coordinates. The selected
variables can also be highlighted (with underlining, bold font, foreground (FG) and
background (BG) colors, etc.) or hidden from the Solution window. If a variable is hidden,
it can be made visible again with the Display controls in the Variable Info dialog window.
Additional information pertaining to the operation of the Solution window follows.
1. The Solution window is accessible only after the calculations are completed. The
Solution menu item in the Windows menu will be dimmed when the Solution window is
not accessible.
2. The unit settings made with the Unit System command in the Options menu will be
displayed at the top of the Solution window if any of the built-in thermophysical property
or trigonometric functions are used.
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EES Windows
3. The Solution window will normally be cleared and hidden if any change is made in the
Equations window. However, there is an option in the Preferences dialog of the
Options menu to allow the Solution window to remain visible.
4. The number of columns displayed on the screen can be altered by making the window
larger or smaller.
5. If EES is unable to solve the equation set and terminates with an error, the name of the
Solution window will be changed to Last Iteration Values and the values of the variables
at the last iteration will be displayed in the Solution window.
6. When the Solution window is foremost, the Copy command in the Edit menu will appear
as Copy Solution. The Copy Solution command will copy the selected variables (shown
in inverse video) to the clipboard both as text and as a picture. The text will provide for
each variable (selected or not) a line containing the variable name, its value, and its units.
The picture will show only those variables which are selected in the same format as they
appear in the Solution window. The Select Display command in the Edit menu will
select all variables currently visible in the Solution Window. (If you wish to force a
black and white picture, hold the Shift key down when you issue the Copy Solution
command.) Both the text or the picture can be pasted into another application, such as a
word processor. Most word processors will, by default, paste the text. To paste the
picture instead of the text, select the Paste Special command and select picture.
7. If the Display subscripts and Greek symbols option in the General Display tab of the
Preferences dialog is selected, EES will display subscripts and superscripts of variable
units. For example, m^2 will appear as m2. An underscore character is used to indicate
a subscript so lb_m will appear as lbm.
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EES Windows
Chapter 2
Arrays Window
EES allows the use of array variables. EES array variables have the array index in square
brackets, e.g., X[5] and Y[6,2]. In most ways, array variables are just like ordinary
variables. Each array variable has its own guess value, lower and upper bounds and display
format. However, simple arithmetic operations are supported for array indices so array
variables can be more convenient in some problems as discussed in Chapter 7.
The values of all variables including array variables are normally displayed in the Solution
window after calculations are completed. However, array variables may optionally be
displayed in a separate Arrays window, rather than in the Solution window. This option is
Place array variables in the Arrays window check box in the
controlled with the
Preferences dialog (Options tab) in the Options menu. If this option is selected, an Arrays
window such as that shown below will automatically be produced after calculations are
completed showing all array values used in the problem in alphabetical order with the array
index value in the first column.
The values in the Arrays window may be plotted using the New Plot Window command in the
Plot menu. Part or all of the data in the Arrays window can be copied to another application
by selecting the range of cells of interest followed by use of the Copy command in the Edit
menu. If you wish to include the column name and units along with the numerical
information in each column, hold the Ctrl key down while issuing the Copy command.
The format of values in any column of the Arrays window can be changed by clicking the
left mouse button on the array name at the top of the column. The following dialog window
will appear in which the units, display format and column position can be changed. Note
that you can enter a number in the column number field or use the up/down arrows to change
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Chapter 2
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its value. If the value you enter is greater than the number of columns in the table, the
column will be positioned at the right of the table.
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EES Windows
Chapter 2
Residuals Window
The Residuals window indicates the equation blocking and calculation order used by EES, in
addition to the relative and absolute residual values. The absolute residual of an equation is
the difference between the values on the left and right hand sides of the equation. The
relative residual is the magnitude of the absolute residual divided by the value of left side of
3
the equation. The relative residuals are monitored during iterative calculations to determine
when the equations have been solved to the accuracy specified with the Stopping Criteria
Consider, for example, the following set of six equations and six unknowns.
EES will recognize that these equations can be blocked, i.e., broken into two or more sets, as
described in more detail in Appendix B. The blocking information is displayed in the
Residuals window.
Variables having values which can be determined directly, i.e., without simultaneously
finding the values of other variables, such as G in the example above, are determined first
3
If the value of the left hand side of an equation is zero, the absolute and relative residuals assume the same
value.
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Chapter 2
EES Windows
and assigned to Block 04. Once G is known, H can be determined. The order in which these
individual equations are solved in Block 0 is indicated by the order in which they appear in
the Residuals window. After solving all equations in Block 0, EES will simultaneously
solve the equations in Block 1, then Block 2, and so on until all equations are solved. The
first and third equations in the example above can be solved independently of other
equations to determine X and Y and are thereby placed in Block 1. Similarly, the second
and fourth equations which determine A and B are placed in Block 2. With X, Y, A, and B
now known, Z can be determined, so it appears in Block 3.
The Residuals window will normally be hidden when any change is made in the Equations
window. This automatic hiding can be disabled with the Display Options command in the
It is possible to display the Residuals window in a debugging situation. If the number of
equations is less than the number of unknowns, EES will not be able to solve the equation
set, but the Residuals window can be made visible by selecting it from the Windows menu.
Normally, the block numbers appear in sequential order. When one or more equations are
missing, EES will skip a block number at the point in which it encounters this problem. The
equations in the following blocks should be carefully reviewed to determine whether they
are correctly and completely entered.
The information in the Residuals window is useful in coaxing a stubborn set of equations to
converge. An examination of the residuals will indicate which equations have been solved
by EES and which have not. In this way, the block of equations which EES could not solve
can be identified. Check these equations to be sure that there is a solution. You may need to
change the guess values or bounds for the variables in this block using the Variable Info
Doubling-clicking the left mouse button (or clicking the right mouse button) on an equation
in the Residuals window will cause the Equations window to be brought to the front with the
selected equation highlighted.
The entire contents of the Residuals window will be copied as tab-delimited text to the
Clipboard if the Copy command is issued when the Residuals window is foremost.
4
Variables specified in the Diagram window are identified with a D rather than a block number. See the
Diagram Window section. In Complex mode, each equation is shown twice, once for the real part
identified with (r) and again for the imaginary component labeled with (i)
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EES Windows
Chapter 2
Parametric Table Window
The Parametric Table window contains the Parametric Table which operates somewhat like
a spreadsheet. Numerical values can be entered into any of the cells. Entered values, e.g.,
the values in the P2 column in the above table, are assumed to be independent variables and
are shown in normal type in the font and font size selected with the Preferences command
(Options menu). Entering a value in the Parametric Table produces the same effect as setting
that variable to the value with an equation in the Equations window. Dependent variables
will be determined and displayed in the table in blue, bold type, or italics (depending on the
choice made with the Preferences command) when the Solve Table or Min/Max Table
command in the Calculate menu is issued.
1. A table is generated using the New Parametric Table command in the Tables menu. The
variables which are to appear in the table are selected from a list of variables currently
appearing in the Equations window.
2. Each row of the Parametric Table is a separate calculation. The number of rows is
selected when the table is generated, but may be altered using the Insert/Delete Runs
3. Variables may be added to or deleted from an existing Parametric Table using the
Insert/Delete Vars command in the Tables menu.
4. The initial order in which the columns in the Parametric Table appear is determined by
the order in which the variables in the table were selected in the New Parametric Table
dialog. To change the column number order, click the left mouse button in the column
header cell (but not on the alter values control at the upper right). A dialog window will
appear as shown below in which the column number can be changed be clicking the up
or down arrows to the right of the column number or by directly editing the column
number. The display format, units, and column background color can also be entered or
changed at this point.
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Chapter 2
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5. Values can be automatically entered into the Parametric table using the Alter Values
command in the Tables menu. Alternatively, clicking the mouse on the
control at the
upper right of the column header cell will bring up the dialog window shown below
which provides the same automatic entry somewhat more conveniently.
6. A Sum row which displays the sum of the values in each column may be hidden or made
visible using the 'Include a Sum row in the Parametric table’ control provided in the
Preferences dialog window (Options tab) in the Options menu.
7. A Parametric Table is used to solve differential equations or integrals. See Chapter 7 for
8. The TableValue function returns the value of a table cell at a specified row and column.
9. The TableRun# function returns the row of the table for which calculations are currently
in progress.
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EES Windows
Chapter 2
10. The independent variables in the Parametric Table may differ from one row to the next.
However, when the independent variables are the same in all rows, EES will not have to
recalculate the Jacobian and blocking factor information and can thus do the calculations
more rapidly.
11. Tabular data may be imported or exported from the Parametric Table via the Clipboard
using the Copy and Paste commands in the Edit menu. To copy data from any of the EES
tables, click the mouse in the upper left cell. Hold the Shift key down and click in the
lower right cell, using the scroll bar as needed. The selected cells will be shown in
inverse video. When the Shift key is released, the upper left cell which has the focus
will return to normal display. However, even though it is not displayed in inverse video,
the upper left cell is selected and it will be placed on the clipboard with other cells when
the Copy command is issued. Use the Select All command in the Edit menu to select all
of the cells in the table. The data are placed on the clipboard with a tab between each
number and a carriage return at the end of each row. With this format, the table data will
paste directly into a spreadsheet application. If you wish to include the column name
and units along with the numerical information in each column, hold the Ctrl key down
while issuing the Copy command
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Chapter 2
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Lookup Table Window
The Lookup Table provides a means of using tabular information in the solution of the
equations. A Lookup Table is created using the New Lookup Table command in the Tables
menu. The number of rows and columns in the table are specified when the table is created
and may be altered with Insert/Delete Rows and Insert/Delete Cols commands in the Tables
menu. A Lookup Table may be saved on a disk (separately from the EES file) using the
Save Lookup command in the Tables menu. A .LKT filename extension is used to designate
EES Lookup files. Lookup files can also be saved in ASCII format with a .TXT filename
extension. The Lookup table may then be accessed from other EES programs in either
format.
The Interpolate commands provide linear, quadratic or cubic interpolation or extrapolation
of the data in the Lookup Table. See Chapter 4 for details. In addition, the Lookup,
LookupCol, and LookupRow functions allow data in a Lookup Table to be linearly
interpolated (forwards and backwards) and used in the solution of the equations. The
Lookup Table may either reside in the Lookup Table Window or in a previously-saved
Lookup File with a .LKT filename extension, as explained in more detail in Chapter 4.
A sample Lookup Table is shown above. The column number is displayed in small type at
the upper left of each column header cell. The column number is needed for use with the
Lookup functions. However, the Lookup functions will also accept ‘ColumnName’ in place
of the column number where ‘ColumnName’ is the name of the column shown in the
column heading surrounded by single quote marks.5 The column names are initially
Column1, Column2, etc. but these default names can be changed by clicking the left mouse
button in the header cell which will bring up the following dialog window.
5
EES will also accept #ColumnName in place of the column number.
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EES Windows
Chapter 2
The column title can be changed and units for the values in the column may be specified.
The Format controls allow the data in each column of the table to be displayed in an
appropriate numerical format. A control is provided to change the background color for
each column. The column position may also be changed by either clicking the up or down
arrows to the right of the column number or by editing the number directly.
Data can be imported to or exported from the Lookup Table through the Clipboard in the
same manner as described for the Parametric Table. Data may be automatically entered into
the Lookup table by clicking on the
control at the upper right of the column header cell,
as described for the Parametric table. Hold the Ctrl key down when issuing the copy
command if you wish to copy the column names and units with the other numerical
information on the Clipboard. Data may be interchanged between the Parametric and
Lookup Table windows. In particular, columns of data in the Parametric Table may be
stored in the Lookup Table so that they may be plotted or reused at a later time.
A memory-based Lookup Table can be deleted, if desired, with the Delete Lookup menu item
in the Options menu. Lookup Table files saved with a .LKT or .TXT filename extension can
not be deleted from within EES.
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Chapter 2
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Diagram Window
The Diagram window can be used in two ways. First, it provides a place to display a
diagram (or text) relating to the problem which is being solved. For example, a schematic
diagram of a system identifying state point locations can be displayed in the Diagram
window to help interpret the equations in the Equations window. Second, the Diagram
window can be used for both input and output of information and for report generation.
Shown below is a diagram with a schematic and input and output variable information
superimposed.
The diagram itself is not drawn in EES, but rather in any drawing program which produces
an object drawing such as MSPaint, Microsoft Draw (included in Word for Windows), Corel
Draw, Designer, or Power Point. A scanned image may also be placed in the Diagram
window. Copy the drawing and then paste it into the Diagram window. The diagram will
be saved along with all other problem information.
The diagram can be repositioned in the Diagram window by pressing and holding the left
mouse button down anywhere within the diagram rectangle while sliding the diagram to its
new location. Any text placed on the Diagram window will move with the diagram itself.
The diagram and all associated text will be scaled to fit within the Diagram Window by
double-clicking the left mouse button (or clicking the right mouse button) anywhere within
the Diagram window, except on a text item. The diagram can be made larger or smaller by
first changing the size of the Diagram window and then double-clicking to change the size of
the diagram itself. The aspect ratio of the diagram is not changed as the diagram is resized.
The Add Diagram Text command in the Options menu allows text to be placed anywhere on
the Diagram window. Three different types of text may be selected by the radio buttons at
the upper left of the diagram window. Selecting the Text radio button will cause the window
36
EES Windows
Chapter 2
to appear as shown below, in which the text and its characteristics can be specified. The text
will initially appear in a default position within the Diagram window when the dialog is
dismissed. It can then be dragged to a new position by pressing and holding the left mouse
button down while sliding the text to the desired location. The text or any of its
characteristics can later be changed by double-clicking the left mouse button (or by clicking
the right mouse button) while the cursor is positioned over the text.
Clicking the Input or Output radio buttons changes the dialog window display so that a list
of currently defined variables replaces the text edit box, as shown. Select the variable by
clicking on its name in the list. Both Input and Output variables values are displayed on the
diagram with the option of also displaying the variable name and unit string. An Output
variable displays the value of the selected variable calculated during the previous
calculation. An Input variable will be displayed with the value enclosed in a rectangle. This
value can be edited and it provides the same function as an equation in the Equations
window which sets the variable to a value.
If a string variable (identified with a \$ character as the last character in the variable name) is
selected for an Input variable, EES will provide an option of selecting the variable from a
pull-down list of string constants that you provide. For example, the Diagram window
shown at the start of this section employs a pull-down list of refrigerants. The refrigerant
selected by the user is assigned to a string variable which is then provided as the fluid name
in the thermodynamic functions.
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Chapter 2
EES Windows
When either the Solve or Min/Max commands (Calculate menu) are issued, EES will first
examine the Diagram window to see which variables, if any, are set, provided that the
Diagram window is not hidden. A value which is set in the Diagram window cannot also be
set in the Equations window. After the calculations are completed, the newly-calculated
values of the Output variables will be displayed on the Diagram window. Output values will
display as **** if the value is not currently defined.
The Diagram window input is ignored if the Diagram window is hidden. The Diagram
window can be used with the Parametric table (i.e., the Solve Table command) if the ‘Use
Input from Diagram’ check box is checked in the Solve Table dialog window.
Use the Clear command in the Edit menu to delete an existing diagram and all associated text.
38
EES Windows
Chapter 2
Plot Windows
Variables which appear in the Parametric, Lookup or Array tables may be plotted with the
New Plot Window or Overlay Plot commands in the Plot menu. In addition, plots of the
thermodynamic properties can be generated using the Property Plot command. Up to 10 plot
windows may be constructed, and each may have any number of overlayed plots. There are
many plotting options such as choice of line type and plot symbol, linear/logarithmic
scaling, spline fitting, tick frequency, and grid line control. These options can be set initially
when the plot is first drawn or at a later time using the Plot window controls described
below or the Modify Plot and Modify Axes commands in the Plot menu.
The appearance of the plot can be changed in many ways using the plot menu commands in
the Plot menu and controls in the Plot window. The Plot window controls are as follows.
1. Moving the Plot
The entire plot, including the axis scales and all text items, can be moved to a different
location in the Plot window by holding the mouse button down in any location within the
plot rectangle (but not on a text item) while sliding the mouse to its new location. An
outline of the plot will move with the cursor and the plot will move to the new location
when the button is released.
2. Moving Text
Text items, such as the axis titles and any additional text you have added with the Add
Text command in the Plot menu can be moved to any location within the Plot window by
pressing and holding the left mouse button down while the cursor is on the text item and
dragging it to its new location. A snap-to-grid option for text items is provided in the
Plot Window tab of the Preferences dialog. When this option is selected, the text item
will snap to the nearest position with the specified horizontal and vertical increments.
Snap-to-grid can be overridden by holding the Ctrl key depressed as the text is moved.
3. Moving Lines and Arrows
Lines and arrows can be placed on the plot using the Add Line command in the Plot menu.
The choice of arrowhead on the line is made by double-clicking on the line which will
bring up the following small dialog window. Select the desired type of arrowhead by
clicking on the appropriate control. The line can be rotated or moved to a new location.
To rotate the line, press and hold the left mouse button down while the cursor is
positioned on either end of the line. The line will rotate to follow the cursor movement.
Release the mouse button when the line is correctly positioned. To move the line to a
new location, press and hold the left mouse button down while the cursor is over the
center of the line and then drag the line to its new location and release the mouse button.
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Chapter 2
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4. Resizing the Plot
The size or aspect ratio of the plot can be easily changed by pressing and holding the left
mouse button with the cursor located at the lower right corner of the plot rectangle. The
cursor will change from an arrow to the resize indicator (as shown below) when it passes
over the resize control. The size of the plot will change as you drag the lower right
corner to a new position. When the plot is resized, the size and positions of all text items
and lines are proportionally changed.
5. Changing Text Characteristics
The characteristics (e.g., font, size, style, color, orientation, etc.) of each text item can be
individually changed by double-clicking the left mouse button while the cursor is
positioned within the text rectangle. The Format Text Item dialog window shown below
will appear displaying the text and its current characteristics. The text may be edited in
the text edit field. Subscripts, superscripts, Greek or bold characters may be entered as
follows. First select the text which is to be changed in the text box. Then click the X y
(subscript), Xy (superscript), Σ (Greek), or N (normal font) speed button. Control
40
EES Windows
Chapter 2
characters will be added to the text in the edit field. The text will be displayed as it will
appear on the plot in the box at the top of the window.
EES allows any horizontal text item to be associated with a plot symbol to facilitate
construction of a legend. Clicking in the Legend symbol box will produce a drop-down
list containing a descriptor of each existing plot. If a plot is selected, the line type and
symbol used for that plot will be displayed just to the left of the text item and it will
move when the text item is moved.
6. Modifying the Axis Information
The axis scaling and appearance can be changed by double-clicking the left mouse
button on the abscissa or (left or right) ordinate scales or by selecting the Modify Axes
menu item in the Plot menu. Either action will bring up the Modify Axes dialog window.
The axis for which the changes are to be made is indicated by the radio-button controls
at the upper left. The Minimum, Maximum, and Interval fields initially are the current
values for the axis. These may be changed and the plot will be rescaled and redrawn.
Scale numbers are placed at the position of each interval, as are Grid lines if selected.
Selecting the Zero line causes a vertical (for x-axis) or horizontal (y-axis) line to be
drawn at a value of zero. The No. Ticks/Division is the number of minor ticks, i.e., the
number of tick marks between each interval. If Grid lines is selected, clicking on No.
Ticks/Division will change it to #Grids/Division allowing grid lines to be placed at
points in between the major ticks. If the Show Scale control is selected (as shown), the
scale numbers will be displayed. The characteristics of these numbers are controlled by
the remaining fields on the right-side of the dialog window.
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EES Windows
7. Modifying the Plot Information
The line type, color, plot symbol (or bar type for bar plots), and other information
relating to each plot can be viewed or modified by double-clicking the left mouse button
anywhere within the plot rectangle (but not on text or a line.) The dialog window shown
below will appear. This dialog window can also be made to appear with the Modify Plot
menu item in the Plot menu. All current plots will be listed in the rectangle at the upper
left in the order in which they were constructed. An (R) to the right of the plot name
indicates that the plot uses the right-hand y-axis. Select the plot by clicking on its name.
The Spline fit control, if selected, will cause EES to plot the line using cubic splines to
produce a smooth curve through the data. Automatic Update sets up a direct link
between the plot and the data in the Parametric Table so that the plot will be
automatically redrawn if any change is made to the data in the Parametric Table. The
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EES Windows
Chapter 2
Show error bars control is enabled only if the data being plotted were obtained with the
Uncertainty Propagation Table command in the Calculate menu. Click the Apply
button if you wish to view the changes you have made.
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EES Windows
Debug Window
The Debug window is a diagnostic tool that can be helpful in locating errors in your
equations. Whenever an attempt is made to solve a set of equations in which the number of
equations is not equal to the number of variables, a message box such as that shown below
will appear.
Clicking the Yes button will bring up the Debug window. For example,
This window provides two lists of variables. The first list shows all variables which are
referenced only once in the Equations Window. These variables are possibly spelled wrong
or otherwise not being directly used in the problem, except for informational purposes. The
second list shows the variables which are most likely to be involved in any missing or extra
equations. The information used to construct this second list is determined by examination
of the blocking order of the equations in the Residuals window. You may also find the
information in the Residuals window helpful in identifying the problem with your equation
set.
Clicking the left mouse button on a variable name in the Debug window will bring the
Equations Window to the front with that variable name selected.
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__________________________________________________________________________
__________________________________________________________________________
Open will allow you to access and continue working on any file previously saved with the
Save or Save As… commands.
After the confirmation for unsaved work, the dialog window shown above will appear.
The current directory is indicated in the Folders: field and EES files in that folder
(directory) are shown in the list on the left. To select a file, click on the file name in the
list or enter the filename in the File Name: field. You can open files in another directory
by entering the directory name in File Name: field or by clicking on the folders listed in
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Chapter 3
the Folders list. Clicking on the Drives list displays the available drive designations.
Click on the drive name to select it. Choose the OK button to select the file (or directory)
displayed in the Filename field.
EES can read four types of files which are identified as EES file, Import file, Text file,
and Library file formats. The format is selected with the drop-down list at the bottom
left of the dialog window. EES files with a .EES filename extension are the norm.
Import files with a .XPT extension are files saved by EES with the Export option from a
different operating system, such as the Macintosh. Text files with a .TXT filename
extension contain ASCII text which is read into the Equations window. Library files are
EES files containing one or more functions, procedures or modules which can be
automatically loaded at startup, as described in Chapter 5.
New initiates a new work session. All variables and equations will be cleared. The unit
system will be restored to the settings that would be in effect if the program were
restarted. If an unsaved problem definition exists, you will be asked if you first wish to
Merge allows the equations previously saved in an .EES file to be merged with the current
contents of the Equations window at the cursor position. The Merge dialog window
operates in the same manner as for the Open command. Equations can also be entered
from a text file using the \$INCLUDE directive. EES functions, procedures and modules
Save will save your problem definition with the same file name (which appears after Save in
the File menu and in the title bar of the Equations window) as it was last saved. For a
new work session which has not yet been named, you will be prompted to supply a file
name, just as if the Save As… command were given. All information concerning the
problem definition is saved, including the equations, variable information, the tables, the
plots, and the size and locations of the windows. By default, the file will be saved in the
standard EES file format with a .EES filename extension. If you wish to export the file to
a version of EES on a different operating system, use the Export format in the file type
available in the Save As… command. A check mark will appear to the left of the word
Save in the File menu if the current problem information has been saved on disk. Any
change in the problem information will cause the check mark to disappear.
Save As… provides the same function as the Save command except that it will first prompt
you to supply a filename in the Save File dialog window. The Save As command allows
the problem definition to be saved with another filename or in a form which can be
exported to EES versions on other operating systems. Enter the file name of your choice
in its place. The 32-bit version of EES supports long file names. The file name may
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Chapter 3
include drive and directory information. However, it is not necessary to enter a filename
extension, since EES will supply the extension automatically.
EES recognizes four file types. If EES is displayed in the Type box at the lower left, the
extension in the File Name: field will be set to .EES (the norm) and files having this
extension will be displayed in the filenames list. The Export file type will apply a .XPT
filename extension and save the file in a generic ASCII format which can be transferred to
other operating systems such as the Macintosh. The Text file type will apply a .TXT
filename extension and save only the text in the Equations window in an ASCII file. The
Library type will change the filename extension to be .LIB. Each time EES is started, it
opens all of the .LIB files in the USERLIB\ sub-directory and automatically loads the
functions, procedures, and modules in these files. These functions can be used exactly
like the EES built-in functions. Library files are one of the most powerful features of
EES because the user can easily develop special purpose functions. See Chapter 5 for
Print will print any or all of the EES windows to the printer or to a file on the disk. Each
window has a small check box preceding its name. If the check box is grayed (as it is for
the Arrays window in the Print dialog window shown below) the window is not available
for printing. If an X appears in the box, the window will be printed. To place an X in the
box or to remove an existing X, click the mouse while the cursor is positioned in the box.
If the Page breaks check box is selected, a forced page break will occur so that the printed
output for each window starts on a new page. The printed output will be sent to the
default printer selected with the Printers application in the Windows Control Panel group.
It is possible to direct the output to a file, rather than a printer, with the Connect options
in the Printers applications. See the Windows manual for additional information on
selecting printers. Printing options such as the font, line spacing, and font size are set in
the Preferences dialog window (Options menu). The Preview button will direct a facsimile
of the printed output to the screen.
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Chapter 3
Printer Setup provides a dialog window which allows the printer to be selected along with
printing options, such as the paper size and orientation.
Load Library will bring up the standard open file dialog showing EES Library files (which
have a .LIB file name extension) in the file selection box. Library files contain usersupplied functions, procedures, and/or modules which operate as described in Chapter 5.
Once loaded, these library files remain in memory until EES is closed. Note that when
EES starts, it will preload all of the library and externally-compiled files which are found
in the USERLIB\ sub-directory so the Load Library command is not needed for these
files. The Load Library can also be used to load external functions and procedures with
filename extensions of .DLF, .DLP, and .FDL. See Chapter 6 for additional information.
The \$INCLUDE directive described in Chapter 7 can also be used to load library files.
Load Textbook reads a user-generated Textbook index file with the filename extension
(.TXB) and uses the information in this file to create a Textbook menu at the far right of
the menu bar. A textbook index file can also be loaded automatically, by placing the
textbook index file and associated problem files in a subdirectory within the USERLIB
those developed for use with a text. The Textbook menu section at the end of this chapter
provides instructions for creating and using a Textbook menu.
Make Distributable Program will create a special purpose version of EES which will run
one to five pre-selected problems. The following dialog will appear when this menu
command is chosen. A special version of the EES program with one to five EES
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Chapter 3
problems and all supporting files are placed in a single executable file when you select the
OK button. This executable file can be freely distributed to others.
Quit provides a graceful way to exit the program.
The remaining items in the File menu are recently accessed filenames. Selecting any of these
filenames opens the file. This list can be disabled in the Preferences dialog. It is
recommended that this list be disabled if the program is installed and used on a network.
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Chapter 3
Undo restores the Equations window to the condition it was in before the last editing
operation. The Undo command is available only for the Equations window.
Cut deletes the selected (highlighted) text. The deleted text is placed on the Clipboard where
it can be pasted to another location with the Paste command.
Copy functions in a manner dependent on which window is foremost. Copy will place the
selected text from the Equations window on the Clipboard from which it can be restored
at the cursor position with the Paste command. When the Parametric, Lookup, or Array
Table window is foremost, the Copy command will copy the selected cells (shown in
inverse video.) The data copied from the table are stored on the Clipboard in a standard
format in which numbers within the same row are separated by a tab and each row ends
with a carriage return - line feed. Data in this standard format can be pasted into any
location of the Parametric or Lookup tables or into other applications. (Hold the Ctrl key
down when selecting the Copy command if you wish to also copy the column heading and
units.) Copy will move a selected Plot window or the Diagram window graphics into the
Clipboard from which it can be pasted into other applications. If you hold the Ctrl key
depressed while copying the Diagram window, the text items added with the Add
Diagram Text command will not be included. The copy of the Plot or Diagram window
is stored in MetaFilePict object format. Copy Solution will place the contents of the
Solution window on the Clipboard both as ASCII text with each variable is on a separate
line and as a picture of the formatted Solution window. Use the Paste Special in an
another application, such as a word processor, to select either the text or the picture for
pasting. See the description of the Solution Window for more information. Copy will
place the entire contents of the Residuals window on the Clipboard as text. Tabs separate
the different items on each line of the Residuals windows. The Clipboard contents can be
pasted into a word-processor.
Paste is active for the Equations, Parametric, Lookup, and Diagram windows. Paste moves
the text (or graphics for the Diagram window) previously placed on the Clipboard with
the Cut or Copy commands in EES or in other applications. When Paste is used in the
Parametric or Lookup Table window, the values stored on the clipboard will be copied to
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Chapter 3
the table starting in the cell in which the cursor is currently located. Data can thus be
moved between the Parametric and Lookup tables. Copy and Paste can also be used
with text items placed on the Plot windows.
Clear removes the selected text without placing a copy on the Clipboard. Clear can also be
used to delete the contents of the Diagram window.
Select All will select all of the text in the Equations window, or all of the cells in any of the
three tables, depending on which window is foremost when the command is issued. The
Select All command is normally followed by Copy which places the selected items on the
clipboard. If the Formatted Equations window or Solution window is foremost, the Select
All command appears as Select Display. This command will select all currently visible
items in the window.
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Chapter 3
Find will search the Equations window for the first occurrence of the text entered in the Find
what: field. The search is case-insensitive unless the ‘Match case’ option is selected. If
the ‘Match whole word only’ option is selected, the text will be found only if it is
delimited by spaces or mathematical operators. The Cancel button will change to Done
after the find process is completed.
Replace will search the Equations window for the first occurrence of the text in the Find what:
field and replace it with the text in the Replace with: field. The search options are as
described for the Find command. The Replace All button will replace every occurrence of
the search text with the replacement text. The Cancel button will change to Done after the
find process is completed.
Next will find the next occurrence of the text previously entered with the Find or Replace
command. The search options will be remain in effect if they were set in the Find
command.
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Variable Info will provide a dialog window, as shown below, in which the guess value, lower
and upper bounds, display format, and units of all variables currently appearing in the
Equations window can be viewed or changed. These data are initially set to default
values. The defaults, selected based on the first letter of the variable name, may be set
with the Default Info command.
If the program contains one or more modules (see Chapter 5 for a discussion of modules),
a control will be provided at the top of the dialog to select the module or main program.
The information for the selected module can then be changed or viewed.
Use the scroll bar on the right side of the dialog window to bring the variable information
into view. Note that the height and width of this dialog window can be changed by
selecting an edge and dragging it in the usual Windows manner. All fields, including the
variable name, may be changed as required. If the variable name is changed, EES will
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Chapter 3
change every occurrence of the original variable name in the Equations, Parametric table
and Diagram windows.
The words, -infinity and infinity can be used to indicate unlimited lower and upper
bounds, respectively. The Guess, Lower and Upper value fields will also accept a
variable name, as well as a number. When a variable name is provided, EES uses the
current value of that variable as the guess value or bound.
EES attempts to solve equations having a single unknown before this display appears.
Variables for which the value has been pre-calculated are identified by having the bounds
shown in italics. The pre-calculated value appears in the Guess column. These guess
values and bounds may still be edited which will then cause EES to recalculate the value.
The display format of a variable in the Solutions or Table window is controlled by the
three fields in the Display columns. Clicking in these fields will produce a pop-up menu
for the display style, number of significant digits, and hilighting effects.
The units of the variable (or any other desired information) may be entered in the units
column. Units are used by EES only for display purposes in the Solution and Parametric
Table windows. Note that the display format and units of each variable can also be
changed by clicking on the variable in the Solution Window.
When the OK button is pressed, all changes made to the variable information since the
dialog window appeared are accepted. The Update button replaces the guess value of
each variable with its current value, i.e., the value determined in the last calculation. The
same update feature is provided with the Update Guesses command in the Calculate
menu. The Print button will direct a copy of the information in this table to the selected
printer. The Cancel button will restore all fields to the condition they were in when the
Variable Information dialog was first presented and then remove the dialog.
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Chapter 3
Function Info will bring up the following dialog window.
The four buttons at the top of the dialog window indicate which information is to be
provided. Math Functions and Thermophysical Props refer to the built-in functions for
mathematical and thermophysical property relations, respectively. The User Library
button provides a list of the user functions, procedures, and modules loaded from Library
files. (See Chapter 5 for additional information on Library files). The External Routines
button refers to compiled routines which can be linked to EES as described in Chapter 6.
The functions corresponding to the selected button will be displayed in the Function list
on the left. To select a function, click on the function name in the scrollable list. Click
the Info button at the bottom of the dialog to obtain specific information relating to the
function you have selected.
The units of the thermophysical property function are shown above the function list box.
Thermophysical property functions require specification of a substance. The substances
for which property data are available are shown in the Substance list on the right. Click
on the substance name in the scrollable list to select the substance. ‘Ideal gas’ will appear
above the substance list if the properties of the selected substance are calculated using
ideal gas law approximations. ‘Real substance’ will appear if liquid and vapor states are
determined. Substances represented by their chemical formula (e.g., CO2) are modeled as
an ideal gas and use JANAF table reference values for enthalpy and entropy. Substances
with their name spelled out (e.g., CarbonDioxide) are modeled as real fluids and do not
use JANAF table reference values. Air is an exception to this rule. Air is modeled as an
ideal gas. Psychrometric functions are applicable only to the substance AirH20.
Additional information regarding all built-in functions is provided in Chapter 4.
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Chapter 3
An example of the function with default variables will be shown in the Example rectangle
at the bottom. You can edit this information in the usual manner. If you click the Paste
button, the contents of the Example rectangle will be pasted into the Equations window at
the cursor position.
Unit Conversion Info provides information to support the use of the Convert unit conversion
function. The Convert function has the following format: Convert(’From’, ’To’) where
From and To are character strings identifying the unit type such as ’Btu/hr-ft2-R’ or ’mph’.
(Note that the single quotes marks around the unit identifies are optional.) Many of the
unit identifiers are obvious, but not all. The purpose of this command is to list the unit
identifiers that have been defined, as shown below.
Click on the dimension in the list at the left. All of the units which have been defined
with that dimension are listed in the list on the right. Note that only the defined units
having the selected dimension are listed. For example, if you click on Area, only Acre
and Hectares will be displayed. However, any combination of units having the
dimensions indicated at the top of the right list (e.g., L^2 for Area) can be used in the
Convert function. You can add additional units if needed. They are stored in the
UNITS.TXT file in the main EES directory. Instructions for adding information are
provided at the top of the file. This file may be read as a text file and edited in EES.
Unit System provides a dialog window shown below in which the units of the variables used in
the built-in mathematical and thermophysical property functions may be set. The unit
settings are displayed in the Solution window. The unit system is only needed for the
built-in function calls. EES does not provide automatic unit conversion. The units will
be changed for the remainder of the work session if the OK button is pressed. The
selected units are saved with other problem information when the Save command in the
File menu is issued. These units are then restored with the problem using the Open
command. If you wish to permanently change the default values, press the Store button.
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Chapter 3
Stop Criteria allows the specification of criteria which will terminate the iterative solution.
The criteria are the number of iterations, the maximum relative residual, the maximum
change in a variable value from one iteration to the next and the elapsed time. If any of
these criteria is satisfied, the calculations terminate. All calculations in EES are done in
extended precision with 21 digits of significance. Loss of precision is unlikely to be a
problem even when very small values are set for the maximum residual or variable
change. However, small values for these quantities increase the number of iterations
required for a solution and therefore the computation time. The stopping criteria will be
set as displayed for the remainder of the session by pressing the OK button. The stopping
criteria are saved with other problem information when the Save command in the File
menu is issued and restored using the Open command. To change the default stopping
criteria that EES presents at the start of a new session, press the Store button.
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Chapter 3
Default Info provides a means for specifying the default guess values, bounds, display format
and units of new or existing variables depending on the first letter in the variable name.
There are two ways to use this command. If the problems you do all tend to have the
same nomenclature, it is best to set the default variable information and save it by
pressing the Store button. The Store button will cause the current default settings to be
permanently saved so that these defaults will appear at the start of the program the next
time EES is run.
The Default Variable Information command can also be used to selectively change
information for existing variables. For example, if you change the units for variables
beginning with letter T to [K] and press the OK button, all existing variables beginning
with letter T will take on these new units. No other changes to existing variables will be
made. Each new variable beginning with letter T will then also take on units of [K]. The
OK button sets the current default setting for this problem session only.
Add Diagram Text provides a dialog window in which text of three types may be placed on the
Diagram Window. The three types are plain text, input variables, and output variables.
An input variable provides an edit box in which the value of a variable can be entered.
An output variable displays the most recently calculated value of a selected EES variable
on the Diagram window. See the Diagram Window section of Chapter 2 for details.
Add Diagram Line allows lines or arrows to be placed on the Diagram Window. The
command functions in exactly the same manner as the Add Line command in the Plot
menu except that the line is drawn on the Diagram Window rather than on a Plot window.
Preferences provides six tabs for user choices concerning program options, general display
options, screen display, printer display, plot window, and complex number options.
These options are shown and described below. If the OK button is clicked, the selected
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Chapter 3
preferences remain in effect for the remainder of the work session. The Store button
saves the preferences so that they will be in effect at the start of the program the next time
EES is run.
Allow = in function/procedure equations will suppress the error message that would
normally occur if the assignment symbol (:=) is not used in EES Functions and
Procedures. EES Internal Functions and Procedures (Chapter 5) employ assignment
statements as in FORTRAN and Pascal, rather than equations as used in the main body of
the EES programs. (EES modules use equality statements as in the main body of the EES
program and thus cannot use the := assignment statement syntax.) An assignment
statement sets the variable identified on the left of the statement to the numerical value on
the right. X:=X+1 is a valid assignment statement, but it obviously is not an equality.
The := sign is used to signify assignment, but if this control is selected, EES will also
accept X=X+1.
Show function/procedure/module values will allow the most recent values of local
variables in EES functions, procedures and modules to be displayed in the Solution
window. Module equations will also appear in the Residuals window. The values of
these local variables are ordinarily not of interest but you may wish to know them,
particularly for debugging purposes. Note that only the functions, procedures, and
modules which appear in the Equations window are affected by this setting. The local
values of variables in Functions, procedures, and modules that have been loaded from
library files (See Chapter 5) are not displayed.
Hide Solution Window after change causes the Solution, Arrays, and Residual windows
to be removed from the screen display if a change is made in the Equations window. If
this option is not selected and a change is made in the Equations window, the Solution
window title will change to Last Solution.
Include a Sum row in Parametric Table will result in an extra row being added to the
Parametric table which displays the sum of the values in each column.
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Chapter 3
Place array variables in the Arrays Window instructs EES to display all array variables
in the Arrays window rather than in the Solution window after calculations are completed.
Values in the Arrays window can be plotted and copied just like values in the Parametric
and Lookup tables. See the Arrays Window section of Chapter 2 for additional
information. The existing Arrays window is destroyed if this option is unselected.
Display warning messages will enable or disable warning messages during calculations.
Warnings are issued if thermophysical property correlations are applied outside of their
range of applicability. Warnings can also be turned on or off using the \$Warnings On/Off
directive in the Equations window.
Maintain a list of recent files in the File menu enables or disables a list of up to 8 recent
files at the bottom of the File menu. This list is a convenience that you normally would
want to have. The file names are stored in a file entitled EES.FNL in the EES directory.
However, if EES is placed on a server where multiple users can access the program, it is
best to disable this feature.
Display menu speedbar controls the visibility of the toolbar appearing below the menu
bar. The toolbar will be hidden if this control is unselected.
The first two options under the General tab allow the function names (such as ENTHALPY,
SIN, etc.) and keywords (such as FUNCTION, DUPLICATE, fluid names, etc.) to be
displayed in upper case, lower case, or as typed.
Display uniform case for variable names causes each variable to appear with the upper
and lower case lettering sequence set in the first occurrence of the variable in the
Equations window. If the first occurrence of the variable is changed, the Check/Format
command in the Calculate menu will change all other occurrences.
Display subscripts and Greek symbols affects the appearance of EES variables in the
Solution and Formatted Equations windows. When this option is selected, array variables
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Chapter 3
will be displayed with the array index as a subscript6.. The characters following an
underscore in a variable name will also normally be displayed as a subscript, except for
_dot, _bar, and _infinity. The _dot and _bar designation places a dot or bar centered over
the variable name The _infinity will provide an infinity symbol (∞) subscript. Variables
names which are part of the Greek alphabet (e.g., alpha, beta, gamma, etc.) will be
displayed in symbol font. If the entire variable name is in capital letters, an upper case
Greek symbol will be used; otherwise lowercase will be used.
Changing the Font and Size in the Screen tab will change the appearance of all EES
windows except for the Plot and Diagram windows which have separate controls for text
formatting.
Display equation comments in blue will cause EES to display comments, i.e., text
between {curly braces} or “quote marks” in blue on color monitors. Note that this
option will necessarily reduce the speed in which the Equations window can be
displayed. You may wish to disable this option when working with large files on slow
machines.
Wrap long lines in the Equations window will hide the horizontal scroll bar. Lines
which are too long to be displayed in the Equations window will be broken at an
appropriate point and continued on the following line. A red > symbol will be
displayed in the left margin of continuation lines if Display line-break indicator is
selected.
6
Although array variables, e.g., A[1], and variables containing an underscore, e.g., A_1, will both display
as A1 in the Formatted Equations window, they are different variables and they have different
characteristics. The index of array variables can be used within the scope of Duplicate statements, or
with the Sum and Product functions. In addition, calculated values of array variables can be displayed in
the Arrays window and they can be plotted.
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Chapter 3
Display line-break indicator is applicable only if the Wrap long lines option is
selected. This option controls whether line break characters appear in the left margin
of continuation lines.
Display calculated table values in (same font, italics, blue, bold) controls the
appearance of cells in the Parametric table which have been calculated by EES.
Values entered by the user are always displayed in normal style in the selected font
and font size.
The options in the Printer Tab only affect the appearance of the printed output. Colored
text does not appear clearly on some printers. If the Print text in color option is off, all
text will be converted to black. The Printed output can be viewed using Print Preview in
the Printer dialog.
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Chapter 3
The Plots tab allows the default setting for the plot width and height, the font, font size,
font style, and the major and minor tick sizes for the axis scales to be changed. The plot
width and height are entered in point units. Depending on your equipment and video
setting, a point is either 1/96 or 1/120 inch. Ticks are the short line segments on the axis
scale. Major ticks are placed on the scale at the point where the axis numbers appear
whereas minor ticks occur between the axis numbers. Ticks which are drawn into the plot
rectangle are represented with positive numbers. The plot can be configured for
outdented ticks by specifying negative values for the tick sizes. These default
characteristics are applied whenever a new plot is generated.
The snap to grid will be in effect if the a check appears in the Snap to grid control. The
spacing of the grids in the horizontal and vertical directions is entered with the spin
controls. The number appearing in these controls is the snap grid spacing. For example,
a value of 4 will cause the text to be aligned to the nearest pixel divisible by 4. When this
option is enabled, it is easier to line up text options used in forming a plot legend.
Pressing the Ctrl key while moving text will disable the snap to grid option.
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Chapter 3
The Complex Tab allows the complex algebra capability in EES to be turned on or off.
The complex capability can also be turned on or off with the \$COMPLEX ON/OFF
directive. The imaginary variable name representing the square root of –1 can be
designated to be either i or j depending on the radio button setting.
Pressing the OK button will set the Preferences for this session only. The Store button
will cause the Preferences to be permanently saved so that they will be in effect at the
start of the program the next time EES is run.
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Chapter 3
Check/Format will recompile all equations and apply the formatting options selected with the
Preferences command in the Options menu. The first syntax error found will be indicated
with a message. If no syntax errors are encountered, EES will indicate the number of
equations and variables in the Equations window.
Solve will first check the syntax of the equations in the Equations window. If no errors are
found and if the number of equations is equal to the number of variables, a solution to the
equation set will be attempted. The methods used by EES for solving equations are
described in Appendix B. An information dialog window summarizes the progress of the
elapsed time, number of blocks, the maximum residual (i.e., difference between the left
and right hand sides of an equation), and the maximum change in the value of a variable
since the previous iteration. If the number of equations is not equal to the number of
variables, EES will provide the option of viewing the Debug window which may help
locate a problem in the problem definition. If the Diagram window is being used to enter
one or more variable values, it must be open when the Solve command is issued. See
Chapter 2 for more information concerning the Debug and Diagram windows. I
Solve Table will initiate the calculations using values specified in the Parametric Table. (See
the description of Parametric menu commands on the following pages for information on
the use of the Parametric Table.) The following window will appear.
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Each row of the table is a separate problem. The values of independent variables are
shown in normal type. Blank cells (or cells with bold, blue, or italic type from a previous
Solve Table command) are dependent variables. The values of these variables will be
cleared and the newly calculated values will be entered in the table. If the Update Guess
Values control is selected, the guess values for each run will be set to the calculated
values from the previous run; otherwise each run will be initiated with the guess values
specified with the Variable Info command.
Warnings are issued if a property correlation is applied outside of the range for which it
was developed. If the Stop if warning occurs control is selected, EES will terminate the
table calculations on the run at which the warning occurred. otherwise, EES will continue
the calculations for the remaining rows. A message will be displayed after all of the table
calculations are completed indicating the rows in which a warning occurred.
The “Use input from Diagram” control will be enabled if the Diagram window is open
and has one or more input variables. If this control is selected, EES will accept input
values from the Diagram window, just as if these inputs were specified in the Equations
window.
Min/Max is used to find the minimum or maximum of an undetermined variable in an equation
set for which there is one or more (limited to 10) degrees of freedom. EES will first
check the syntax of the equations in the Equations window. If no errors are found, a
dialog window will appear presenting the undetermined variables in two lists.
Click the Minimize or Maximize button above the left list. The variable which is to be
minimized/maximized is selected by clicking on its name in the list on the left. The
independent variable(s) whose value(s) will be changed in searching for the optimum
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appear in the list on the right. It is necessary to select as many independent variables as
there are degrees of freedom in the Equations window. The number of independent
variables which must be selected is indicated above the right-hand list. To select (or
unselect) a variable, click on its name in the list.
If there is one degree of freedom, EES will minimize/maximize the selected variable
using either a Golden Section search or a recursive quadratic approximation method,
depending on the settings of the buttons at the bottom of the dialog window. (See
Appendix B for information on the optimization algorithms.) The recursive Quadratic
Approximations method is usually faster, but the Golden Section method is more reliable.
Multi-dimensional optimization may be done using either Direct Search or a Variable
Metric algorithm. The Variable Metric method, which uses numerical derivatives, usually
performs much better than the Direct Search method, but it may be confounded if the
optimum is constrained to be on a bound.
EES requires finite lower and upper bounds to be set for each independent variable.
Careful selection of the bounds and the guess value(s) of the independent variables will
improve the likelihood of finding an optimum. You can view or change the bounds and
guess value for each selected independent variable by clicking the Bounds button. This
will bring up an abbreviated version of the Variable Info dialog containing just the selected
independent variables. See the description of the Variable Info command in the Options
The maximum number of times in which the equations are solved (i.e., the number of
function calls) may be specified, along with the relative tolerance. Calculations will stop
if: 1) the relative change in the independent variable(s) between two successive steps is
less than the specified tolerance; or 2) the number of steps exceeds the specified
maximum. EES will also stop the calculations if the equations cannot be solved with
specified value(s) of the independent variables within the tolerance and allowable number
of iterations specified with the Stopping Criteria command in the Options menu.
Min/Max Table provides the same capability as the Min/Max command, except that the
calculations will be repeated for each row in the Parametric Table. (See the description of
Parametric menu commands on the following pages for additional information on the use
of the Parametric Table.) As with the Min/Max command, a dialog window will appear in
which the variable to be maximized or minimized and the independent variable(s) can be
selected. In this case, however, the variable which is to be optimized and all of the
independent variables (whose values will be varied in seeking the optimum) must appear
in the Parametric Table. The start and stop runs in the Parametric Table for which the
calculations will be done may be specified. Values in the Parametric Table which are
shown in normal type are fixed and are treated just as if they were set to that value with an
equation in the Equations window. The variable which is to be optimized and the
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independent variable(s) must be the same for each run. If no errors are encountered, the
optimum is computed and the values of the remaining columns in the table are entered for
each run.
Uncertainty Propagation determines the uncertainty of a selected calculated variable as a
function of the uncertainties of one or more measured values upon which it depends. In
many cases, an important quantity is not directly measured by rather calculated as a
function of one or more variables that are directly measured, i.e., Y = f( X1, X2, ....). The
measured variables, X1, X2, etc. have with a random variability which is referred to as its
uncertainty. In EES, that uncertainty is displayed with a ± symbol, e.g., X1 = 300 ± 2.
The purpose of this command is to calculate how the uncertainties in all of the measured
variables propagate into the value of the calculated quantity, Y. The method for
determining this uncertainty propagation is described in NIST Technical Note 1297
(Taylor B.N. and Kuyatt, C.E., Guidelines for Evaluating and Expressing the Uncertainty
of NIST Measurement Results, National Institute of Standards and Technology Technical
Note 1297, 1994). Assuming the individual measurements are uncorrelated and random,
the uncertainty in the calculated quantity can be estimated as
∂Y ∑ ∂X U
2
UY =
i
2
X
i
where U represents the uncertainty of the variable. After selecting this command, EES
will present two lists of variables. Select the variable for which the uncertainty
propagation is to be determined from the list on the left. Select one or more measured
variables from the list on the right. Note that the variables appearing in the measured
variables list must be constants so that their values are set to a numerical constant with an
equation in the EES Equations window. To specify the uncertainty associated with the
measured variables, click the Set Uncertainties button below the right list. A second
dialog window will appear in which the absolute or relative (fraction of the measured
value) uncertainties for each selected measured variable can be specified. An uncertainty
value for each measured variable must be provided. Click the OK button to set the
uncertainties and close the Specify Uncertainties dialog window. Click the OK button
Uncertainty Propagation dialog to start the calculations.
After the calculations are completed, EES will display an abbreviated Solution Window
containing the calculated and measured variables and their respective uncertainties. The
partial derivative of the calculated variable with respect to each measured variable will
also be displayed.
Uncertainty Propagation Table provides the same function as the Uncertainty Propagation
command, namely the determination of the uncertainty propagation in a calculated
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variable. The difference is that this command allows the uncertainty calculations to be
repeated for one or more measurements by using the Parametric Table. The calculated
and measured variables must all be in the Parametric Table before this command is used.
After selecting the command, the Uncertainty Propagation dialog window will appear in
which the calculated quantity is selected from list of variables in the left list and the
measured variable(s) are selected from the list on the right. The Parametric Table
calculations will proceed after the OK button is selected, just as if the Solve Table
command were applied. The value and uncertainty for the calculated variable and each
measured variable will be displayed in the Parametric Table after the calculations are
completed. The calculated variable can then be plotted with error bars representing the
propagated uncertainty using the New Plot Window command.
Update Guesses replaces the guess value of each variable in the Equations window with the
value determined in the last calculation. This command is accessible after calculations
have been successfully completed. Update Guesses improves the computational
efficiency of an EES calculation since it ensures that a consistent set of guess values is
available for the next calculation. Exactly the same function is provided with the Update
button in the Variable Info dialog window, but the Update Guesses command is more
accessible.
Reset Guesses replaces the guess value of each variable in the Equations window with the
default guess value for that variable. Unless otherwise specified, EES assumes all guess
values are 1.0. You can change the default guess values with the Default Info command in
the Options menu. You should reset the guess values only if you are experiencing
convergence difficulties and you have changed the guess values in an attempt to find a
solution.
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New Parametric Table creates a new Parametric Table after first deleting any existing
Parametric Table. Parametric Tables are used in EES to automate repetitive calculations,
to solve differential equations, and to present data for plotting or curve-fitting. A dialog
window will appear in which information must be entered to create the table, as in this
example.
The number of runs, which corresponds to rows in the table, is entered in the field at the
top. All variables (both independent and dependent) which are to appear in the table are
selected from the alphabetical list of variables on the left. To select a variable, click on its
name in the list, which will cause it to be highlighted. Multiple names can be selected.
Click the Add button to move the highlighted variable names to the list on the right. (As
a shortcut, a variable is automatically added to the list on the right when you double-click
on its name in the list on the left.) The variables in the right-hand list will appear in the
columns of the table in the same order in which they appear in the list. A variable can be
removed from the table list by clicking on its name in the right list and then clicking the
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Remove button or by double-clicking on the variable name. Pressing the OK button will
create the Parametric Table, overwriting any existing table.
The Parametric Table operates somewhat like a spreadsheet. Numerical values can be
entered in any of the cells. Entered values are assumed to be independent variables and
are shown in normal type. Entering a value in a table produces the same effect as setting
that variable to the value in the Equations window. Dependent variables will be
determined and entered into the table in blue, bold, or italic (depending on the choice
made in the Preferences dialog) when the Solve Table, Min/Max Table, or Uncertainty
Propagation Table command is issued. If a variable is set in the table, it can not also be set
in the Equations window; otherwise the problem will be overspecified. Each row of the
table is a separate calculation. The independent variables may differ from one row to the
next. However, for every row, the number of independent variables plus the number of
equations must equal the total number of variables in the problem.
Alter Values provides an automatic way to enter or clear the values of a variable for multiple
runs. There are two other ways of changing the data in the Parametric table. Clicking on
the control at the upper right of each column header cell will bring up a dialog window
which operates just like the Alter Values dialog shown below. Also, you can simply type
the values directly into the Parametric Table.
The runs (i.e., rows) affected are specified at the upper left of the dialog. The variable for
which changes are to be made is selected from the list by clicking on its name. The
column for this variable will be cleared if the Clear Values control is selected. If Set
Values is selected, values for the selected variable will be entered automatically in the
table starting with the value in the First Value field. The list box below the First Value
controls the manner in which successive values in the table are generated. The choices
are Last Value, Increment, and Multiplier. Increment or Multiplier result in successive
values in the table being determined by either adding or multiplying, respectively, the
preceding table value by the value provided in the box. If Last Value is selected (as
shown) the increment will be selected such that the last run has the specified value. The
Apply button will change the Parametric table as specified but control will remain in the
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Alter Table Values dialog window so that additional changes can be made. The OK
button accepts and finalizes all of the changes made to the Parametric table.
The numerical values entered in a table, either directly or through the Alter Values
command, identify independent variables in the equation set; they are shown in normal
type. Independent variables are fixed to a constant for each run, just as if there were an
equation in the Equations window setting the variable to the constant. Dependent
variables are shown in italic, blue or bold depending on the choice made in the
Preferences dialog. These values are automatically entered in the table with the Solve
Table and Min/Max Table commands. If a value is set in a table, it must not also be set in
the Equations window; otherwise an error message will be displayed.
Insert/Delete Runs allows the number of runs in an existing Parametric Table to be changed by
inserting or deleting one or more rows in the table at a specified position.
Insert/Delete Variables allows variables in an existing Parametric Table to be added or
removed. The following dialog window will appear.
The list on the right shows the variables which currently appear in the Parametric Table.
Variables which may be added to the table appear in the list on the left. To add one or
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more variables to the table, click on the variable name(s) causing them to be highlighted.
Click the Add button to move the highlighted variable names to the list on the right. (You
can also add a variable by double-clicking on the variable name.) Variables can be
deleted from the table by selecting them from the list on the right, followed by clicking on
the Remove button.
Variables will appear in columns of the Parametric Table in the same order as they appear
in the list on the right. The order of the variables in this list can be changed by pressing
and holding the left mouse button on a variable name while sliding it up or down to a new
position in the list. The column order of an existing Parametric table can also be changed
by clicking on the column header cell as described in Chapter 2.
Delete Parametric Table will, after a confirmation, delete the Parametric Table and recover the
memory it required.
New Lookup creates a table with the specified number of rows and columns in which tabular
numerical data may be entered. If there is an existing Lookup Table, it will be
overwritten when this command is completed. The tabular data may be automatically
interpolated, differentiated, and used in the solution of the problem using the Interpolate,
Differentiate, Lookup, LookupRow, and LookupCol functions described in Chapter 4.
Only one Lookup Table may appear in the Lookup Table Window at any one time.
However, the data in the Lookup Window may be saved in a Lookup File (with a .LKT
filename extension) and the Lookup Files stored in disk files can also be accessed by the
Interpolate, Differentiate, Lookup, LookupRow, and LookupCol functions. Lookup
tables and files provide a great deal of power to EES by allowing any functional
relationship between variables which can be represented by tabular information to be
entered and used in the solution of the equations.
Open Lookup will read into the Lookup Table Window a Lookup file which was previously
stored with the Save Lookup command or as a text file. A Lookup file is a twodimensional table of data that has been stored in a disk file. A name and display format
for each column of data may also be stored, depending on the file format. Lookup files
can be accessed by the Differentiate, Interpolate, Lookup, LookupCol, and
LookupRow commands. EES recognizes both binary and ASCII forms for Lookup files.
Binary files are identified with a .LKT filename extension. ASCII Lookup files usually
have a .TXT filename extension, although any filename extension (other than .LKT) can
more quickly and it requires smaller file sizes. The ASCII form is easier to edit and it can
be written by spreadsheet or other applications. See Chapter 4, Using Lookup Files and
the Lookup Table, for details.
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Save Lookup copies the data in the Lookup Window into a Lookup file. Lookup files can be
stored either as a binary file with the .LKT filename extension or as an ASCII file with a
.TXT filename extension. The Lookup file can be later read with the Open Lookup Table
command or used directly from the disk in the Interpolate, Differentiate, Lookup,
LookupRow, and LookupCol functions. The information in the Lookup Table Window
is also stored with the problem information when the Save or Save As commands are
given. See Chapter 4, Using Lookup Files and the Lookup Table, for more information
relating to Lookup tables and files.
Insert/Delete Lookup Rows and Insert/Delete Lookup Cols allows one or more rows or columns to
be inserted or deleted at a specified position in an existing Lookup Table.
Delete Lookup will delete the existing Lookup Table and recover the memory it required.
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Linear Regression provides regression capability for the data in the Parametric, Lookup, or
Arrays tables. Note that the Curve Fit command in the Plot menu also provides
regression capability but only for one independent variable. With the Linear Regression
command, the data in any column can be regressed as a function of the data in up to 6
other columns.
The dialog window shown below appears after the command is chosen. Select the table
you wish to operate on from the radio buttons at the upper right and the starting and
stopping rows in that table. Specify the dependent variable by clicking on the variable
name in the list on the left. The independent variable(s) are selected by clicking on the
names in the right list. To de-select an item, click it a second time.
The dependent variable will be represented as a linear polynomial function of the
independent variables. The order of the polynomial is set between 0 and 6 by clicking on
the ’spin button’ up or down arrows. If the cross-terms box is selected then terms
involving the product of the independent variables will be included in the correlation. As
any information relating to the equation form is entered, a representation of the equation
to be fit is displayed in the box at the bottom as shown above.
You may exclude some terms from the regression by clicking on the term. This action
will display a box around the selected term and enable the Exclude button. Click the
Exclude button to remove the term from further consideration. A removed term is
displayed within a crossed-out red box as shown above. If you later wish to include an
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excluded term, click on it. The Exclude button will then be titled Include. Click the
Include button.
When the form of the equation is that which you want to fit, click the Fit button. If the
fitting process is successful, the fitted equation will appear in the display box. The Stats
button will been enabled. Clicking the Stats button will provide a table listing all of the
coefficients, their associated standard errors, and other statistics such as the root-meansquare (rms) error, the bias error, and the R2 value, as shown below. Coefficients which
have been excluded will be represented in the table with stars. The coefficients can be
copied to the clipboard by checking the Copy to clipboard box.
After a successful fitting process, the Fit button in the Linear Regression dialog window
will be changed to Copy and the Cancel button will changed to Done. Either button will
dismiss the dialog window. The Copy button will first copy the fitted equation to the
clipboard. You can then paste this equation into the EES Equations window or into any
other application that accepts text. Note, however, that the Copy process will overwrite
any other information in the clipboard, such as the coefficients copied from the Linear
Regression Coefficients dialog window.
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New Plot Window allows any variable defined in the Parametric, Lookup, or Array Tables to
be plotted as a function of any other variable in that table. There may be up to ten plot
windows. Use the Overlay Plot command if you wish to plot in an existing plot window.
The information needed to produce the plot is specified in the New Plot Window dialog.
All of the information provided in this dialog window can later be changed using Modify
Axes and Modify Plot commands and the Plot Window controls described in Chapter 2.
First, select the table from which you wish to plot using the radio button controls at the
upper right of the window. Tables which are not defined will be shown in grayed type.
The variables to be plotted on the x– and y–axes are selected by clicking on their names in
the x– and y–Axis lists. EES will automatically select appropriate values for the number
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of display digits, the minimum and maximum axis values, and the interval when a
variable is selected. All of these axis formatting variables may be changed.
The two fields to the right of the word Format contain pop-up menus which control the
format of the numbers appearing in the scale for each axis. F and E format the numbers
with a fixed number of decimal places or exponential notation, respectively. The number
in the second field is the number of decimal places (for fixed notation) or significant
figures (for exponential notation).
Grid lines will be drawn if the ‘Grid Lines’ checkbox control is selected. The number of
grid lines and scale numbers is determined by the specified interval value.
The line type, symbol (or bar type), and color of the plot curve can be selected from their
respective drop-down lists. The ‘Spline fit’ control will provide a spline-fit curve through
the plotted points.
When the ‘Automatic update’ control is selected, the plot will be generated using the
current data in the Parametric Table, rather than the data which existed when the plot was
first drawn. The plot will be updated as the data in the table are changed.
If the ‘Add legend item’ is selected, a text item having the name of the Y-axis variable
will be placed at the upper left corner of the plot, preceded by the line and symbol type
used for the plot. The legend item text can be moved, changed, or deleted just as any plot
window text item, as described in the Plot Window section of Chapter 2.
The “Show error bars” control is accessible only if one or both of the variables selected
for plotting have associated uncertainty values specified with the Uncertainty
Propagation Table commands.
Overlay Plot allows a new plot curve to be drawn over existing plots. The use of this
command is identical to that for the New Plot command described above except that it
does not first clear the Plot window. All overlaid plots should share the same x-axis
scaling. If the scale specified for a plot overlay differs from that of the first plot, a control
will be displayed to select the existing scale on the left-hand y-axis or a new scale on the
right-hand y-axis. Following plot overlays may then use either the left or right y-axis
scale, depending on the choice of the axis selection control.
Modify Plot allows the characteristics of existing plot curves to be changed by manipulation of
information in the following dialog window. This command can also be invoked by
double-clicking the mouse button within the plot rectangle.
The plot for which changes are to be made is selected from the list at the upper left. The
plots appear in this list in the order in which they were generated. An (R) following the
plot descriptor indicates that this plot uses the right-hand y-axis scale.
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The line type, symbol, and color of the plot curve can be changed using the drop-down
lists at the lower left. The ‘Spline fit’ and ‘Automatic update’ options may be changed.
(See the New Plot Window command for a description of these options.)
Controls are provided to change the size and characteristics of the plot border and the grid
lines. Click the Apply button after changing the plot line characteristics and before
selected another plot. The plot window will immediately show any changes.
A single plot curve may be selectively deleted (leaving all other plots intact) using the
Delete button. Legend text for the plot is also deleted. The Delete Plot Window command
described below will delete an entire plot window, including all overlays.
Modify Axes allows the appearance of the axes of an existing plot to be changed. This
command can also be invoked by double-clicking the mouse on the axis scale for which
changes are to be made. The dialog window shown below will appear. The axis for
which changes are to be made is selected with the radio-button controls at the upper left.
The current minimum, maximum, and interval values for the selected axis are shown.
These values can be changed and the plot will be redrawn scaled with the new values.
The No. Ticks/Division is the number of minor tick marks in each interval. If selected,
Grid lines are normally placed at each major tick mark. However, the No. Ticks/Division
toggles to No. Grids/Division if you click on this control. Grid lines can be placed at
positions between the major ticks by setting the No. Grids/Division to a value greater than
0.
The display format, font, font size, font style, and color of the scale numbers can be
changed using the drop-down menus appearing on the right of the dialog window. These
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fields will be hidden if the Show Scale checkbox is not selected, in which case a scale
will not be drawn.
Clicking the axis control applies the changes so that they can be viewed in the Plot
window. The OK button accepts the changes and exits the dialog.. The Cancel button
will restore the plot to the condition it was in before this command was issued.
Add Text allows a line of text to be placed in the current plot window. A text item will be
allocated with the string ‘New Text’ and with the characteristics of the last allocated text.
The Format Text dialog shown below will then appear in which the text can be edited and
formatted.
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The font, font size, font style, and color of the text can be selected from the drop-down
lists appearing below the text field. The orientation of the text may be horizontal or
rotated 90° counterclockwise. The latter orientation is useful for labeling the y-axes. The
Opaque control displays the text with a white background so that the background does not
interfere with the text. Frame text places a border around the text.
The formatting capabilities for text are extensive and they are facilitated with the four
speed buttons for superscript (Xy), subscript (Xy), symbol font (Σ), and normal font (N).
To create a subscript, for example, select the text which is to be subscripted and then click
the subscript button. The formatted text as it will appear in the plot window is shown at
the top. Note that EES puts characters into the string to generate the subscript and other
formatting options. The backslash (\ ) character is a special character used for formatting
and it cannot be displayed in the text. You can enter a backslash to cancel a formatting
option.
EES allows any horizontal text item to be associated with a legend symbol. The Legend
drop-down list will display a description of each existing plot. If a plot is selected, the
line type and symbol used for that plot will be displayed just to the left of the text item
and it will move when the text item is moved. If the plot is deleted, the text item will also
be deleted.
All text items can be moved by dragging them with the mouse as described in the Plot
Windows section of Chapter 2. The characteristics of any text item, including the axis
labels and plot title, can be altered by double-clicking the mouse within the text rectangle
which will bring up the Format Text dialog window.
Add Line allows a line or arrow to be placed anywhere within the Plot window. After this
command is issued, the cursor will appear as a cross. Press and hold the mouse button
down at the position where you wish the line to begin. Hold the mouse button down
while moving the mouse to the desired end position of the line and then release it.
You can move, resize or rotate the line after it is created. To move the line, press and
hold the mouse button down anywhere near the middle of the line while dragging it to its
new position. To rotate the line and/or change its length, press and hold the mouse button
on either end. Move the end to its new position and release the mouse button. Initially, a
line will be created with the default characteristics of the previous line or with default
characteristics. If you wish to change the characteristics, double-click on the line. A
dialog window will appear in which the line and arrowhead characteristics may be
selected.
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Delete Plot Window will delete the entire contents of the selected Plot Window. Use the Delete
button in the Modify Plot dialog window if you wish to delete only one of several overlayed
plots.
Property Plot creates a new plot window with thermodynamic property data for a selected
substance. Once created, additional property data or thermodynamic cycle state points
can be superimposed on the plot using the Overlay Plot command. Also, the plot
characteristics and axis scales can be modified in the usual manner with the Modify Axes
and Modify Plot commands.
Select the substance from the list at the left. The substance type (real fluid or ideal gas) is
shown above the list. The general rule is that the substance is modeled as a real fluid if its
name is spelled out (e.g., Oxygen) and as an ideal gas if its name is a chemical formula
(e.g., O2). Air is the exception to this rule.
For all substances except AIRH2O (psychrometric air-water mixtures), there are four
buttons which allow a specification of Temperature-entropy (T-s), Temperature-volume
(T-v), Pressure-volume (P-v) or Pressure-enthalpy (P-h) coordinate systems. The
AIRH2O substance provides a field in which the total pressure can be specified.
At the right are controls which allow specification of four isobars or isotherms. A line of
constant pressure or temperature with the value shown in the box will be superimposed on
the plot. Suggested values are provided. If you do not wish to display the isobar or
isotherm, click on the check box preceding the value.
Curve Fit will find the best fit of a smooth curve through a previously plotted set of data points
using unweighted least squares. (The Curve fit dialog provides a fit with a single
independent variable. The Linear Regression command in the table menu allows a
variable to be fitted with as many as 6 independent variables.) The dialog window shown
below will appear.
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Chose the data to be fitted from the list of plots at the left. Note that data can be plotted
from the Parametric Table, the Lookup Table, or the Arrays Table with the New Plot or
Overlay Plot commands. Select the form of the curve fit by clicking the appropriate radio
button. A sample of the equation form will appear in blue in the box at the bottom of the
dialog window. The first four buttons correspond to commonly used equation forms for
which linear least squares can be used to determine the unknown coefficients. The
Enter/edit equation button allows you to enter any equation form or to edit a previously
entered equation. The equation you enter may be linear or non-linear in the unknown
parameters. You will prompted to supply guess values bounds for the unknown
parameters.
Click the Fit button (or press the Enter key). The fitted equation will be displayed in the
box at the bottom of the dialog window. A Stats button will appear. Clicking the Stats
button will display the following statistical information relating to the curve fit.
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Std. Error is the standard error of the curve-fitted parameter value; rms is the root mean
square error of the fit; bias is the bias error of the fit. R^2 is the ratio of the sum of
squares due to regression to the sum of square about the mean of the data.
The Fit button will now have changed into the Plot button. Click the Plot button if you
wish to have the curve fit equation overlayed on your plot. If the Plot Legend check box
is selected, a legend containing the equation will be created and displayed on the plot.
The curve fit equation will be copied to the clipboard if the To Clipboard checkbox is
selected when either the Plot or Cancel button is selected.
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Equations causes the Equations window to become the active window by bringing it to the
front of all other windows and making it visible if it were hidden previously.
Formatted Equations first checks the syntax of the equations and then brings the Formatted
Equations window to the front displaying the contents of the Equations window in
mathematical format.
Solution, Arrays and Residuals cause the Solution, Arrays, and Residual windows, respectively,
to be moved to the front of all other windows. These windows are normally viewed after
the Solve, Min/Max, or Uncertainty Propagation command has been successfully completed.
Any change made to the Equations window will remove these windows from the screen if
the Hide Solution after Change Option in the Preferences dialog (Options tab) is selected.
If EES is unable to solve the equation set and terminates with an error, the name of the
Solution window will be changed to Last Iteration Values and the values of the variables
at the last iteration will be displayed in the Solution window; the residuals for the last
iteration will be displayed in the Residuals window.
Plot Windows will display a submenu with items for each of the 10 possible plot windows.
Selecting a plot window will bring that window to the front of all other windows. The
Plot menu commands, e.g., Modify Plot and Modify Axes, will operate without asking for a
plot window designation when one of the plot windows is foremost. The menu item will
be grayed if the corresponding plot window is not defined. The graphics in any of the
plot windows can be copied to the Clipboard by selecting Copy from the Edit menu.
Parametric Table and Lookup Table bring the Parametric and Lookup Table windows,
respectively, to the front of all other windows and make it the active window. The
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Chapter 3
Parametric and Lookup Table windows may be hidden by choosing close from the
Windows control menu or by pressing Ctrl-F4.
Diagram will bring the Diagram to the front of all other windows. A diagram is entered into
EES from a drawing program. Copy the diagram in the drawing program, switch to EES
and bring the Diagram window to the front. Then use the Paste command to move the
diagram into the window. Note that values can be entered from the Diagram window
only if it is currently displayed on the screen.
Debug Window will be generally be disabled as shown. This window is available only after
the Solve command is attempted with the number of equations not equal to the number of
variables. In this case, an option is presented to display the Debug Window and this
menu item becomes enabled. Any change made to the Equations window will clear the
Debug window and disable this menu item.
Tile arranges all open windows to fill the screen so that a portion of each can be viewed.
Cascade arranges the currently visible windows so that the title bar of each is shown.
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Help Index will activate the Help processor which provides specific information on the use of
EES. The Help processor will open to the EES Information index which lists the subjects
for which help is available. Clicking on the subject opens the Help window to
information for that subject. Help can also be accessed by pressing the F1 key which will
bring up help information specific to the window or dialog which is foremost. The online help provides most of the information contained in this manual.
Using Help shows information provided by the Windows Help processor on how to use the
features in the Help program.
About EES will bring up the EES header window. This window registration information and
indicates the version of your EES program and the amount of available memory. This
information will be needed in any correspondence with F-Chart Software.
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Chapter 3
The Textbook menu is perhaps misnamed. It is really a user-defined menu that is designed to
allow easy access to EES files. However, it has been used to provide a convenient means to
access EES problems associated with a textbook, and thus its name. This menu can be created
either by opening a textbook index file with the Load Textbook command (File Menu) or by
placing the textbook index file in the USERLIB subdirectory.
A textbook index file is an ASCII file identified with the filename extension (.TXB). When
EES reads a textbook index file, it creates the Textbook menu at the far right of the menu bar, as
shown above. The format of the textbook index file is quite simple. The menu shown above
was created with the following textbook index file.
1
Textbook information line
Textbook information line
Textbook information line
Reserved
Descriptive problem name1
Descriptive problem name2
Descriptive problem name3
1
2
3
| FileName1.EES | HelpFile1.HLP | NO.BMP
| FileName2.EES | HelpFile2.HLP | NO.BMP
| FileName3.EES | HelpFile3.HLP | NO.BMP
Descriptive problem name1 | FileName1.EES | HelpFile1.HLP | NO.BMP
Descriptive problem name2 | FileName2.EES | HelpFile2.HLP | NO.BMP
Descriptive problem name3 | FileName3.EES | NO.HLP | NO.BMP
etc | FileName1.EES | HelpFile1.HLP | NO.BMP
The first line in the file is the menu title. This title is the name of the menu which will appear in
the menu bar to the right of the Help menu. The following line is a version number used
internally by EES. EES currently ignores this line but a 1 should be provided, as shown. The
following three lines provide information about the textbook or problem set. This information
will be displayed whenever any menu item is selected from the Textbook menu. A fourth line
containing the word "reserved" is provided for possible future use. EES currently ignores the
information on this line, but it must be provided. The following lines contain a menu item name
(preceded by an identifying > character) and then one or more problem descriptions. The menu
item name is the name that will appear in the Textbook menu list.
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Chapter 3
Each problem description line contains four pieces of information, separated by a | character.
The first item is a descriptive name for the problem, which may be up to 128 characters. The
second item is the filename for the EES program file. This filename may be partially or fully
qualified with directory information, e.g., C:\myBook\Chapter1\Problem1.EES. However, in
most cases, directory information is not necessary and it should not be included. The third item
is an optional help file which is to be associated with this file. The help file can be an ASCII
text file or a Windows HLP file produced by a Help formatting program. By convention, the
help files have a .HLP filename extension. However, EES will accept any filename extension
for either type of help file. If no help file is available, enter NO.HLP for this field. The final
item is the filename for a figure associated with the problem. EES does not currently use the
figure, but a figure name must be provided. Use NO.BMP as a placeholder.
The textbook index file should be placed in the same location (i.e., floppy disk, subdirectory, or
folder) with all of the referenced EES program and help files. When the user selects a command
from the Textbook menu, a dialog window will appear showing a list of the descriptive names of
the problems for that menu item. The user can then select a name from the list and the file
associated with that problem will be opened.
89
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CHAPTER 4
__________________________________________________________________________
Built-in Functions
__________________________________________________________________________
EES has a large library of built-in mathematical functions. Many of these (e.g., Bessel,
hyperbolic, error functions, etc.) are particularly useful for engineering applications. EES also
provides a function to convert units and functions that help manipulate complex numbers. The
major feature that distinguishes EES from other equation solving programs, however, is its
extensive library of built-in functions for thermophysical properties. Thermodynamic and
transport properties of steam, R22, R134a, R407C, air, ammonia, carbon dioxide, and many
others are implemented in a manner such that any independent set can be used to determine the
remaining unknown properties. EES also provides a Lookup Table which allows tabular data
to be entered and used in the solution of the equations set. The first two sections of this chapter
provide reference information for the built–in mathematical and thermophysical functions. The
third section provides information on the use of the Lookup Table. Much of the information
provided in this chapter can also be obtained from within the program using the Info button in
Function Info dialog.
Mathematical Functions
The mathematical functions built into EES are listed below in alphabetical order. (The
functions which operate on the Lookup Table are described in the Using the Lookup Table
section at the end of this chapter.) All of the functions (except pi and tableRun#) require one
or more arguments which must be enclosed in parentheses and separated with commas. The
argument may be a numerical value, a variable name, or an algebraic expression involving
values and variables.
abs(X) returns the absolute value of the argument. In complex mode, abs returns the
angle(X), angleDeg(X) and angleRad(X) all return the angle (also called amplitude or
argument) of complex variable X. Representing X as X_r + i*X_i, this function returns
arctan(X_i/X_r). Angle will return the angle in either degrees or radians depending on the
Trig Function setting in the Unit System dialog. AngleDeg will always return the angle in
degrees and AngleRad will always return the angle in radians. All three functions return the
angle in the correct quadrant of the complex plane. Note that the Angle functions are used
to extract the angle of a complex number variable or expression but they cannot be used for
assigning the angle of a complex number. For example, the equation Angle(X)=4 will
produce an error.
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arcCos(X) returns the angle which has a cosine equal to the value of the argument. The units
of the angle (degrees or radians) will depend on the unit choice made for trigonometric
functions with the Unit System command.
arcCosh(X) returns the value which has a hyperbolic cosine equal to the value of the argument.
arcSin(X) returns the angle which has a sine equal to the value of the argument. The units of
the angle (degrees or radians) will depend on the unit choice made for trigonometric
functions with the Unit System command.
arcSinh(X) returns the value which has a hyperbolic sine equal to the value of the argument.
arcTan(X) returns the angle which has a tangent equal to the value of the argument. The units
of the angle (degrees or radians) will depend on the unit choice made for trigonometric
functions with the Unit System command.
arcTanh(X) returns the value which has a hyperbolic tangent equal to the value of the
argument.
bessel_I0(X) returns the value of zeroth-order Modified Bessel function of the first kind for
argument value X where -3.75≤ X <infinity.
bessel_I1(X) returns the value of first-order Modified Bessel function of the first kind for
argument value X where -3.75≤ X<infinity.
bessel_J0(X) returns the value of zeroth-order Bessel function of the first kind for argument
value X where -3≤ X <infinity.
bessel_J1(X) returns the value of first-order Bessel function of the first kind for argument
value X where -3≤ X <infinity.
bessel_K0(X) returns the value of zeroth-order Modified Bessel function of the second kind for
argument value X where 0≤ X <infinity.
bessel_K1(X) returns the value of first-order Modified Bessel function of the second kind for
argument value X where 0≤ X <infinity.
bessel_Y0(X) returns the value of zeroth-order Bessel function of the second kind for argument
value X where 0< X <infinity.
bessel_Y1(X) returns the value of first-order Bessel function of the second kind for argument
value X where 0< X <infinity.
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cis(X) is a complex mode function that returns cos(X)+i*sin(X). The required units (degrees or
radians) of the angle is controlled by the unit choice made for trigonometric functions with
the Unit System command. However, you can append deg or rad to the angle to override the
Unit System setting. For example, V=3*cis(20deg) will set the value of complex variable
V to have a magnitude of 3 and an angle of 20 degrees, regardless of the Unit System
setting.
conj(X) returns the complex conjugate of a complex variable X. Representing X as X_r +
i*X_i, this function returns X_r - i*X_i. Note the this function returns a complex result.
The EES equation, Y=conj(X),will set the real part of Y (Y_r) to the real part of X and the
imaginary part of Y to negative of the imaginary part (Y_i).
convert(’From’, ’To’) returns the conversion factor needed to convert units from the unit
designation specified in the ‘From’ string to that specified in the ‘To’ string. The single
quote marks are optional. For example, FI = convert(ft^2, in^2) will set FI to a value of
144 because 1 square foot is 144 square inches. Combination of units and multiple unit
terms may be entered. In a combination of units, such as Btu/hr-ft^2-R, the individual units
are separated with dash (i.e., minus) or division signs. Only one division symbol may be
used in any one term. All units to the right of the division symbol are assumed to be in the
denominator (i.e., raised to a negative power.) The ^ symbol is optional so ft2 and ft^2 are
equivalent. The convert function will accept multiple unit terms if each term is enclosed
within parentheses. Terms are separated with an optional * symbol or with a / symbol, as in
the example below.
P =15* Convert((lbm/ft3)*(ft)/(s^2/ft), kPa)
The defined unit symbols can be displayed with the Unit Conversion Info command in the
Options menu. If you find that a unit you need is not defined, you can enter it by editing
the UNITS.TXT file in the EES directory.
cos(X) will return the cosine of the angle provided as the argument. The required units
(degrees or radians) of the angle is controlled by the unit choice made for trigonometric
functions with the Unit System command.
cosh(X) will return the hyperbolic cosine of the value provided as the argument.
differentiate('Filename', 'ColName1', 'ColName2', ColName2=Value) returns the derivative
determined from two columns of tabular data based on cubic interpolation. See the Using
Lookup files and the Lookup Table section of this chapter for more information and
examples.
erf(X) returns the Gaussian Error function of X.
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erfc(X) returns the complement of the Gaussian Error function of X which is 1-erf(X).
exp(X) will return the value e raised to the power of the argument X.
if (A, B, X, Y, Z) allows conditional assignment statements. If A<B; the function will return a
value equal to the value supplied for X; if A=B, the function will return the value of Y; if
A>B, the function will return the value of Z. In some problems, use of the if function may
cause numerical oscillation. It is preferable to use the if then else, repeat until and goto
statements in a function or procedure for conditional assignments. See Chapter 5 for
imag(X) returns the imaginary part of a complex variable X. Representing X as X_r + i*X_i,
this function returns X_i. Note that the Imag function is used to extract the imaginary part
of a complex number variable or expression. It cannot be used for assigning the imaginary
part of a complex number. For example, the equation Imag(X)=4 will produce an error.
Instead you should just enter X=4*i which will set X to 0 + 4*i. If you wish to only set the
imaginary part of X, you can enter X_i=4.0
integral(Integrand, VarName) or integral(Integrand, VarName, Start, Stop, Step) returns the
integral of the expression represented by Integrand with respect to the variable VarName,
i.e., ∫(Integrand) d(VarName). There are two basic forms of the integral function which
differ in their reliance on the Parametric table. If the values of Start, Stop, and Step are not
provided, the integral function is used only in conjunction with the Parametric Table. In
this case, VarName must be a legal variable name which has values defined in one of the
columns of the Parametric table and Integrand can be a variable or any algebraic expression
involving VarName and other variables or values. If Start, Stop, and (optionally) Step are
provided, EES will numerically integrate all equations involving variable VarName, setting
the value of VarName to values between Start and Stop as appropriate. If Step is not
provided, EES will internally chose a step size using an automatic stepsize adjustment
algorithm. The integral function can be used to solve initial value differential equations.
See Chapter 7 for additional information.
interpolate(’Filename’, ’ColName1’, ’ColName2’, ColName2=Value) returns an interpolated or
extrapolated value from tabular data in the Lookup table, a Lookup file (if filename is
provided), or the Parametric table using cubic interpolation. See the Using Lookup files
and the Lookup Table section of this chapter for more information and examples.
interpolate1(’Filename’, ’ColName1’, ’ColName2’, ColName2=Value) provides the same
function as the interpolate command except that it uses linear interpolation.
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Chapter 4
interpolate2(’Filename’, ’ColName1’, ’ColName2’, ColName2=Value) provides the same
function as the interpolate command except that it uses quadratic interpolation.
lookup(‘Filename’, Row, Column) returns the value in the Lookup Table or Lookup file at the
specified row and column. Filename is optional. See Using Lookup files and the Lookup
lookupCol(‘Filename’, Row, Value) uses the data in the specified row of the Lookup Table or
Lookup file to determine the column which corresponds to the value supplied as the second
argument. Filename is optional. See the Using Lookup files and the Lookup Table
lookupCol(‘Filename’, Row, Value) uses the data in the specified row of the Lookup Table or
Lookup file to determine the column which corresponds to the value supplied as the second
argument. Filename is optional. See the Using Lookup files and the Lookup Table section
ln(X) will return the natural logarithm of the argument.
log10(X) will return the base 10 logarithm of the argument.
magnitude(X) returns the magnitude (also called modulus or absolute value) of a complex
variable X. In complex mode, the abs function also returns the magnitude. Representing X
as X_r + i*X_i, this function returns sqrt(X_r^2+X_i^2). Note that the magnitude function
is used to extract the magnitude of a complex number variable or expression. It cannot be
used for assigning the magnitude of a complex number. For example, the equation
magnitude(X)=4 will produce an error.
max(X1, X2, X3, …) will return the value of the largest of its arguments. The number of
arguments must be greater or equal to 1.
min(X1, X2, X3, …) will return the value of the smallest of its arguments. The number of
arguments must be greater or equal to 1.
pi is a reserved variable name which has the value of 3.1415927.
product(Arg, Series_info) returns the product of a series of terms. Arg can be any algebraic
expression. Series_info provides the name of the product index variable and the lower and
upper limits which must be integers or variables which have been previously set to integer
values. product (j, j=1,4) will return 1*2*3*4 or 24, which is 4 factorial. The product
function is most useful when used with array variables, e.g., X[j]. For example, the product
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Chapter 4
Built-in Functions
of the square of all 10 elements in the vector X can be obtained as product (X[j]*X[j],
j=1,10).
real(X) returns the real part of a complex variable X. Representing X as X_r + i*X_i, this
function returns X_r. Note that the real function is used to extract the real part of a
complex number variable or expression. It cannot be used for assigning the real part of a
complex number. For example, the equation real(X)=4 will produce an error. Instead you
should just enter X=4 which will set X to 4 + i*0. If you wish to only set the real part of
variable X, you can enter X_r=4.
round(X) will return a value equal to the nearest integer value of the argument.
sin(X) will return the sine of the angle provided as the argument. The required units (degrees
or radians) of the angle is controlled by the unit choice made for trigonometric functions
with the Unit System command.
sinh(X) will return the hyperbolic sine of the value provided as the argument.
sqrt(X) will return the square root of the value provided as the argument which must be greater
than or equal to zero.
step(X) will return a value of 1 if the argument is greater than or equal to zero; otherwise the
Step function will return zero. The step function can be used to provide conditional
assignments, similar to the if function. The step and if functions are provided to maintain
compatibility with earlier versions. Conditional assignments are more easily and clearly
implemented with the IF THEN ELSE statement in functions or procedures as described in
Chapter 5.
sum(Arg, Series_info) returns the sum of a series of terms, i.e., ΣArg. Arg can be any algebraic
expression. Series_info provides the name of the summation index variable and the lower
and upper limits. These limits must be integers or variables which have been previously set
to integer values. The function is best explained by examples. sum(j, j=1,4) will return
1+2+3+4 or 10. The sum function is most useful when used with array variables, e.g., X[j].
For example, the scalar product of two vectors, X and Y, each with 10 elements can be
obtained as sum(X[j]*Y[j], j=1,10). See Chapter 7 for information on how the sum
function can be used with array variables to manipulate matrices.
tableRun# returns the Parametric Table run number, i.e., the current row in the Parametric
Table or zero, if the Parametric Table is not being used in the calculations. This function
should only be used with the Solve Table or Min/Max Table command in the Calculate menu.
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Chapter 4
tableValue(Row, Column) or tableValue(Row, ‘VariableName’) returns the value stored in a
specified row and column of the Parametric Table. The column number may be either
entered directly as an integer number or indirectly by supplying the variable name for the
desired column, enclosed by the single quotes, e.g., TableValue(6,’ABC’)7. An error
message will be generated if the row or column (or corresponding variable name) does not
exist in the Parametric Table or if the referenced cell does not have a value. The
tableValue function is useful in the solution of some 'marching-solution' type problems in
which the current value of a variable depends on its value in previous calculations.
tan(X) will return the tangent of the angle provided as the argument. The required units
(degrees or radians) of the angle is controlled by the unit choice made for trigonometric
functions with the Unit System command.
tanh(X) will return the hyperbolic tangent of the value provided as the argument.
trunc(X) will return a value equal to the integer value corresponding to the argument rounded
toward zero.
UnitSystem(‘Unittype’) is a function which allows an EES program to know what unit settings
have been selected with the Unit System command. This function takes one argument which
must be placed within single quote marks. Legal arguments are 'SI', 'Eng', 'Mass', 'Molar',
'Deg', 'Rad', 'kPa', 'bar', 'psia', 'atm', 'C', 'K', 'F', and 'R'. The function returns either 1 (for
true) or 0 (for false). As an example, the following assignment statement in an EES
function or procedure,
g:=unitsystem('SI') + 32.2*unitsystem('Eng')
will set g equal to 1 if the user has selected the SI unit system and g equal to 32.2 if the user
has selected the English unit system.
7
For compatibility with earlier versions, EES will also accept the #symbol preceding the variable name in place of
enclosing it within single quotes, e.g., TableValue(6, #ABC).
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Thermophysical Property Functions
The first argument of all built-in thermophysical property functions is the name of the
substance. The substance names recognized by EES are8:
Substance Names for Thermophysical Property Functions
Air
AirH2O
Ammonia
C2H6
C3H8
C4H10
CarbonDioxide
CH4
CO
CO2
Oxygen
Propane
R11
R12
R13
R14
R22
R32
R113
R114
CH2
H2O
Helium
Methane
n-Butane
N2
Nitrogen
NO
NO2
O2
R123
R134a
R141b
R500
R502
R600a
SO2
Steam (Water)
Steam_NBS
Additional substances that are provided with the 32-bit version of EES in external files in the
USERLIB subdirectory are R113, R125, R143a, R404A, R407C, R410A, and R600a. These
substances are provided with a user-supplied property file having a .MHE filename extension,
as described in Appendix C. Also provided in the USERLIB subdirectory are the external
routines providing thermodynamic property data for lithium bromide-water mixtures (H_LIBR,
T_LIBR, V_LIBR, Q_LIBR, P_LIBR, X_LIBR), ammonia-water mixtures (NH3H2O) and
specific heat, enthalpy and entropy for hundreds of additional substances with JANAF table
references (JANAF). Documentation for these routines is provided through the Function Info
command in the Options menu. Click the External routines button at the top right and then
select the external routine name from the list. Clicking the Info button at the bottom of the
dialog will provide the documentation.
It may appear from the above list that some substances, e.g., N2 and Nitrogen, CO2 and
CarbonDioxide, H2O and Steam (or Water), are duplicated, but this is not true. Whenever a
chemical symbol notation (e.g., N2, CO2, CH4 etc.) is used, the substance is modeled as an
ideal gas and the enthalpy and entropy values are based on JANAF table references. The
JANAF table reference for enthalpy is based on the elements having an enthalpy value of 0 at
298K (537R). The entropy of these substances is based on the Third Law of Thermodynamics.
Whenever the substance name is spelled out (e.g., Steam (or Water), Nitrogen, R12,
CarbonDioxide, Methane, etc.) the substance is modeled as a real fluid with subcooled,
saturated, and superheated phases. Exceptions to this rule occur for Air and AirH2O, both of
8
Your version of EES may have additional fluids. Property data may be added as described in Appendix D.
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Chapter 4
which are modeled as ideal gases. AirH2O is the notation for air-water vapor mixtures, i.e.,
psychrometrics. Property information for up to 150 additional fluids can be added by the user
as explained in Appendix C.
The property keywords Water and Steam are treated identically. Either keyword provides
access to approximate water property functions based on empirical correlations which have
been developed for rapid calculations. The Steam/Water property correlations assume the fluid
is incompressible in the subcooled region; This assumption is not accurate for pressures above
350 atm and for states near the critical point. The Steam_NBS keyword uses property
correlations are extremely accurate at any condition. However, they require considerably more
computing effort than the Steam/Water relations.
Many of the thermodynamic functions can take alternate sets of arguments. For example, the
enthalpy function for steam can be accessed with temperature and pressure as arguments;
alternatively, the same function could be accessed with entropy and quality as arguments. In
general, any independent set of arguments can be supplied for thermodynamic functions. The
transport functions (conductivity and viscosity), however, require temperature as the argument
(for ideal gas substances) or temperature and pressure (for real fluids).
All arguments in thermophysical property functions, aside from the substance name, are
identified by a single case-insensitive letter followed by an equal sign. The value or algebraic
expression representing the value of the argument follows the equal sign. The letters which are
recognized in function arguments and their meaning are as follows:
Property Indicators
for Use in Thermophysical Functions
B
D
H
P
R
S
= Wetbulb Temperature
= Dewpoint Temperature
= Specific Enthalpy
= Pressure
= Relative Humidity
= Specific Entropy
T = Temperature
U = Specific Internal Energy
V = Specific Volume
W = Humidity Ratio
X = Quality
Arguments must be separated with commas and may be in any order, provided that the
substance name is first, as in the examples shown below. EES will display the function name
in the format selected for Functions in the Display Options dialog window. The substance name
is an EES keyword and it will be displayed in the format selected for Keywords in the Display
Options dialog window
EES does not require the argument to a function to have a known value. For example:
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h1 = enthalpy(STEAM, T=T1, P=P1)
will return the value of h1 corresponding to known temperature and pressure, T1 and P1. If,
however, the value of h1 is known, but T1 is unknown, the same equation will return the
appropriate value of the temperature. Alternatively, the temperature could be found by:
T1 = temperature(STEAM, h = h1, P=P1)
The latter method is preferable in that the iterative calculations implemented for steam are less
likely to have convergence difficulty.
The built-in thermophysical property functions are listed below in alphabetical order. The
units, which depend on the choices made with the Unit System command in the Options menu,
are shown in brackets. One or more examples, showing the allowable formats for the function,
are also given.
Conductivity [W/m-K, Btu/hr-ft-R] returns the thermal conductivity of the specified substance.
For substances modeled as an ideal gas, the conductivity function takes temperature as its
only argument in addition to the substance name. Real fluids require temperature and
pressure as arguments. Steam, Water and Steam_NBS will accept specific volume as an
alternative to pressure. For AIRH2O (moist air), the temperature, pressure, and humidity
ratio (or relative humidity) must be supplied.
Examples:k1 = conductivity (AIR, T=200)
k2 = conductivity (AMMONIA, T=100, P=200)
k3 = conductivity (STEAM_NBS, T=100, v=0.345)
k4 = conductivity (AIRH2O, T=80, P=14.7,R=0.5)
Density [kg/m3, kgmole/m3, lb/ft3, lbmole/ft3] returns the density of a specified substance.
Two arguments are required for all pure substances; three are needed for moist air.
Example: d1 = Density(AIR, T=300, P=100)
d2 = Density (Steam, h=850, P=400)
d3 = Density (AirH2O, T=70, R=0.5, P=14.7)
DewPoint [°F, °C, R, K] returns the dewpoint temperature for air-water gas mixtures. This
function can be used only with AIRH2O as the substance name. Three arguments follow
the substance name in any order: temperature, total pressure, and relative humidity (or
humidity ratio or wetbulb temperature).
Example: D1 = dewpoint(AIRH2O, T=70, P=14.7, w=0.010)
D2 = dewpoint(AIRH2O, T=70, P=14.7, R=0.5)
D3 = dewpoint(AIRH2O, T=70, P=14.7, B=50)
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Enthalpy [kJ/kg, kJ/kgmole, Btu/lb Btu/lbmole] returns the specific enthalpy of a specified
substance. The exact form of the enthalpy function depends on the substance and
independent variable(s) selected. Substances which obey the ideal gas law, such as air,
require a single argument, (temperature or internal energy), in addition to the substance
name whereas real fluid substances, e.g., STEAM and CARBONDIOXIDE, will always
require two independent variables. For AIRH2O, three arguments are required.
Example: h1 = enthalpy(AIR, T=300)
h2 = enthalpy(STEAM, T=900, P=300)
h3 = enthalpy(AIRH2O, T=70, P=14.7, R=0.50)
Entropy [kJ/kg-K, kJ/kgmole-K, Btu/lb-R, Btu/lbmole-R] returns the specific entropy of a
specified substance. For all pure substances, the entropy function always requires two
arguments, in addition to the substance name. For AIRH2O, three arguments are required.
Example: s1 = entropy(O2, T=400, P=100)
s2 = entropy(AIRH2O, T=70, P=14.7, R=0.50)
HumRat [dimensionless] returns the humidity ratio (defined as the mass of water vapor per
mass of dry air) for air-water gas mixtures. This function is applicable only to the
substance AIRH2O. The function requires three arguments which must include pressure
and any two remaining independent variables such as temperature, relative humidity,
enthalpy, or dew point.
Example: w1 = humRat(AIRH2O, T=70, P=14.7, R=0.50)
w2 = humRat(AIRH2O, T=70, P=14.7, h=25)
IntEnergy [kJ/kg, kJ/kgmole, Btu/lb, Btu/lbmole] returns the specific internal energy of a
specified substance. The exact form of the IntEnergy function depends on the substance
and independent variable(s) selected. Substances which obey the ideal gas law, such as air,
require a single argument (temperature or enthalpy) whereas real fluid pure substances, like
steam, will always require two arguments in addition to the substance name. AIRH2O
Example: u1 = intEnergy(AIR, T=300)
u2 = intEnergy(STEAM, T=1320, P=300)
u3 = intEnergy(AIRH2O, T=70, P=14.7, R=0.50)
MolarMass returns the molar mass (often called molecular weight) of the fluid provided as the
parameter.
Example: M_CO2 =MolarMass(CarbonDioxide)
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Pressure [kPa, bar, psia, atm] returns the pressure of a specified substance. The argument list
for the pressure function always requires the substance name followed by two arguments,
each item separated by commas. The pressure function is not implemented for AIRH2O;
however, an unknown pressure can still be determined using any of the functions which are
applicable to moist air and which take pressure as an argument.
Example: P1 = pressure(STEAM, h=1450, T=900)
P_Crit [kPa, bar, psia, atm] returns the critical pressure of the specified fluid. The fluid may
be a fluid name or a string variable. Critical property information is not available for ideal
gas substances.
Example: Pc=P_Crit(R134a) "returns the critical pressure of R134a"
Quality [dimensionless] returns the quality (vapor mass fraction) for substances modeled as
real fluids such as WATER and R12. Two independent arguments are required.
Temperature and pressure are not independent for saturated states. If the state of the
substance is found to be subcooled, the quality is returned as –100. If it is superheated, 100
is returned.
Example: x1 = quality(R12, h=50, T=80)
Relhum [dimensionless] returns the relative humidity as a fractional number for air-water gas
mixtures. There are three arguments to this function, in addition to the substance name,
AIRH2O. The three arguments are temperature, total pressure and any two remaining
independent variables such as temperature, wetbulb, enthalpy, dew point, or humidity ratio.
Example: R1 = relhum(AIRH2O, T=70, P=14.7, w=0.01)
R2 = relhum(AIRH2O, T=70, P=14.7, h=25)
R3 = relhum(AIRH2O, T=70, P=14.7, B=55)
Specheat [kJ/kg-K, kJ/kgmole-K, Btu/lb-R, Btu/lbmole-R] returns the constant pressure
specific heat of the specified substance. For pure substances which obey the ideal gas law,
the specific heat function has temperature as its only other argument in addition to the
substance name. The temperature and pressure must both be provided as arguments for
substances modeled as real fluids. The specific heat of the liquid or vapor may be returned,
depending on the temperature and pressure values provided.
Example: Cp1 = specheat(AIR, T=350)
Cp2 = specheat (AMMONIA, T=100, P=30)
Temperature [°C, K, °F, R] returns the temperature of the substance. The exact form of the
function depends on the substance and argument(s) selected. Substances which are
assumed to obey the ideal gas law, such as air, may require one or two arguments whereas
pure real fluid substances, like STEAM, will always require two arguments.
Example: T1 = temperature(AIR, h=300)
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T2 = temperature(AIR, s=1.75, P=100)
T_Crit [°C, K, °F, R] returns the critical temperature of the specified fluid. Critical property
information is not available for ideal gas substances.
Example: Tc=T_Crit(R134a)
Volume [m3/kg, m3/kgmole, ft3/lb, ft3/lbmole] returns the specific volume of a specified
substance. Two arguments are required for all pure substances; three are needed for moist
air.
Example: v1 = Volume(AIR, T=300, P=100)
v2 = Volume(Steam, h=850, P=400)
v3 = Volume(AirH2O, T=70, R=0.5, P=14,7)
V_Crit [m3/kg, m3/kgmole, ft3/lb, ft3/lbmole] returns the critical specific volume of the
specified fluid. Critical property information is not available for ideal gas substances.
Example: vc=v_Crit(R134a) "returns the critical volume of R134a"
Wetbulb [°C, K, °F, R] returns the wetbulb temperature for air-water gas mixtures. This
function is applicable only to the substance AIRH2O. There are three arguments to this
function, in addition to the substance name. The three arguments are temperature (or
enthalpy), total pressure, and relative humidity (or humidity ratio or dewpoint).
Example: B1 = wetbulb(AIRH2O, T=70, P=14.7, w=0.01)
B2 = wetbulb(AIRH2O, h=25, P=14.7, w=0.01)
B3 = wetbulb (AIRH2O, h=25, P=14.7, D=30)
Viscosity [N-sec/m2, lbm/ft-hr] returns the dynamic viscosity of the specified substance. For
substances modeled as an ideal gas, the viscosity function takes temperature as its only
argument in addition to the substance name. Temperature and pressure arguments are
required for real fluids. STEAM and STEAM_NBS will accept temperature and specific
volume. Relative humidity or humidity ratio must be supplied for AIRH2O.
Example: v1 = viscosity(AIR, T=300)
v2 = viscosity(R12, T=40, P=30)
v3 = viscosity(STEAM_NBS, T=100, v=0.335)
v4 = viscosity(AIRH2O, T=80, P=14.7, R=0.5)
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Using Lookup Files and the Lookup Table
A Lookup file is a two-dimensional set of data with a specified number of rows and columns.
Lookup files provide a means to enter functional relationships with tabular data and to use
these relationships in the solution of the equations. Lookup files can be stored in a disk file.
Alternatively, a single Lookup file, called the Lookup Table, can exist in the Lookup Table
Window. The six menu commands which pertain to the Lookup Table Window appear at the
bottom of the Options menu and are summarized here.
New Lookup creates a new empty Lookup Table with a specified number of rows and columns
in the Lookup Table Window. If a Lookup Table already exists in the Lookup Table
window, it will be overwritten.
Open Lookup reads a Lookup file from the disk into the Lookup Table Window. If a Lookup
Table already exists in the Lookup Table window, it will be overwritten. There are two
lookup file formats identified with filename extensions of .LKT and .TXT. Either type can
be opened with the Open Lookup command. The .LKT format is a binary format unique
to EES that is created with the Save Lookup command. The .TXT format is an ASCII text
file that can be created with the Save Lookup command in EES or with any other text
processing application.
Binary Lookup files (.LKT)
Binary Lookup files store all of the information that appears in a Lookup Table window in a
binary file on disk, including the data, and the column name, units, and display format for
each file type. A binary (.LKT) Lookup file is created using the Save Lookup Table
command. Once created, the Lookup file can be opened into the Lookup Table window
using the Open Lookup Table command. Binary files require less disk storage space and
the can be opened and saved more quickly by EES. However, they cannot be created,
edited or viewed by any application other than EES.
ASCII Lookup files (.TXT)
There are several variations for the ASCII Lookup file format. In the basic form, the first
line of the file contains the number of rows and columns in the table. Each following line
provides the data for one row with the value for each column separated by one or more
spaces or a tab character. The basic form does not provide a means of specifying the
names, units, or display format for the data. EES assigns the names "COLUMN1’,
’COLUMN2’, etc. and these column names should be used when the file is used with the
Interpolate or Differentiate commands. Automatic formatting is used to display the data
if the file is read into the Lookup Table with the Open Lookup Table command. The
following example shows the ASCII data needed for a Lookup file with five rows and three
columns.
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5 3
1
11
111
2
22
222
3
33
333
4
44
444
5
55
555
If a negative number is provided in the file for the number of rows, EES will determine the
number of rows of data in the file. If the number of columns is a negative number, EES
will expect to find the format specification (e.g., A3, F3 or E4) followed by one space and
then the column heading and units for each column. The units are enclosed in square
brackets. The following lines contain the data for each row, separated by one or more
spaces or a tab. The example below would create a table with 2 rows and 3 columns. The
columns would be formatted with E4, F0, and F3 format specifications and the column
names will be ColA, ColB, and ColC.
2 -3
E4 ColA [Btu]
F0 ColB
F3 Col
1.23E-12
2
2.34E-11
4
4.56
7.89
In addition to being read into the Lookup Table, Lookup files in either the binary or ASCII
formats can be accessed directly with Interpolate, Differentiate, Lookup, LookupCol,
and LookupRow functions. These functions are documented below.
Save Lookup saves the Lookup Table in the Lookup Table Window as a Lookup file on the disk.
Lookup files can be accessed with the Lookup functions described below. The Lookup file
can be saved as a binary file with a .LKT filename extension or as an ASCII text file. The
ASCII format allows the data to be exported to another application. Note that the content of
the Lookup Table Window is also saved with other problem information when the Save
command is issued. It is not necessary to save the Lookup Table separately unless it is to be
used by more than one EES program or the program requires more than one Lookup table.
Insert/Delete Rows allows the number of rows in the Lookup Table to be changed.
Insert/Delete Cols allows the number columns in the Lookup Table to be changed.
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Delete Lookup deletes the Lookup Table and recovers the memory it required.
Data in the Lookup Table can be accessed with the Differentiate, Interpolate, Lookup,
LookupRow, and LookupCol functions. These functions may either operate on the data in the
Lookup Table Window or in a Lookup file on disk. In the latter case, the filename must be
provided as the first argument of the function. The file name can be supplied as a string
constant (surrounded with single quotes) or as a string variable, identified by the \$ character at
the end of the variable name. Chapter 7 provides details on the use of string variables. The
filename extension can either be .LKT (for binary lookup files) or .TXT (for ASCII lookup
files). If a filename extension is not provided, EES will assume that the file format to be binary
(i.e., EES will automatically append the .LKT extension).
Differentiate(’Filename’, ’ColName1’, ’ColName2’, ColName2=Value) returns the derivative
determined from two columns of tabular data based on cubic interpolation. These data can
be in the Lookup table, a Lookup file, or the Parametric table. ‘FileName’ is an optional
parameter which, if supplied, must be the name of an existing Lookup file having either a
.LKT or .TXT filename extension previously stored with the Save Lookup Table command.
The filename can be entered either a string constant (within single quotes) or as a string
variable that has been assigned to the name of the file. If the ‘FileName’ parameter is not
supplied, the Differentiate function will be applied on the existing Lookup Table.
ColName1 and ColName2 are the column header names. The single quotes enclosing the
column header names are optional. These columns names can also be supplied with string
variables. The final parameter is of the form ColName2=Value where the text to the left of
the equal sign can be either of the column header names (ColName1 or ColName2)
specified with the two previous parameters. Value is a numerical value or expression. EES
will return an estimate of the derivative d(ColName1)/d(ColName2) at a point fixed by the
specified value of either ColName1 or ColName2. The Differentiate function can also be
made to operate on data in the Parametric table if Filename is set to ‘Parametric’. In this
case, the values in the Parametric table must already exist, such as those entered from the
keyboard. Values which are to be calculated when the Solve Table command is issued
cannot be used with the Differentiate command.
Examples: dXdY=Differentiate('X', 'Y', Y=2.34) {returns the derivative dX/dY at
a value of Y=2.34 using data in the Lookup Table. Quotes are optional}
Y=Differentiate('C:myFile',T,X,T=100) {returns the derivative dT/dX
at a value of T=100 using data from Lookup file myFile.LKT on drive C}
Interpolate(’Filename', 'ColName1', 'ColName2', ColName2=Value) returns an interpolated or
extrapolated value from tabular data in the Lookup table, a Lookup file, or the Parametric
table using cubic interpolation. ‘FileName’ is an optional parameter. It can be supplied as
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a string constant (within single quotes) or as a string variable. If this parameter is not
supplied, the interpolate function will applied on the existing Lookup Table. If ‘FileName’
is supplied, it must be the name of an existing Lookup file having either a .LKT or .TXT
filename extension. ColName1 and ColName2 are the column header names. These single
quotes enclosing the column header names are optional. The column header names can
alternatively be entered as string variables. The final parameter is of the form
ColName2=Value where the text to the left of the equal sign can be either of the column
header names (ColName1 or ColName2) specified with the two previous parameters.
Value is a numerical value or expression. EES will return the interpolated value from the
data in column ColName1 corresponding to the specified value of ColName2. If the value
of ColName1 is supplied, EES will return the interpolated value of ColName2. If Filename
is 'Parametric', the interpolate command will be applied to the existing Parametric table. In
this case, the values in the Parametric table must already exist, such as those entered from
the keyboard. Values which are to be calculated when the Solve Table command is issued
cannot be used with the interpolate command.
Examples: Z=interpolate(‘Col1’, ‘Col2’, Col1=2.3) {returns a value from column
Col2 of the Lookup table corresponding to a value in Col1 equal to 2.3 using
cubic interpolation. Note that the quotes are optional.}
X= Interpolate(C:\myData,X,Y,Y=4.5) {returns a value from column X
in the lookup table called myData.LKT on drive C: corresponding to a value in
column Y equal to 4.5 using cubic interpolation.}
interpolate1(’Filename’, 'ColName1', 'ColName2', ColName2=Value) provides the same
function as the interpolate command except that it uses linear interpolation.
interpolate2(’Filename’, 'ColName1', 'ColName2', ColName2=Value) provides the same
function as the interpolate command except that it uses quadratic interpolation.
Lookup(‘Filename’, Row, Column) returns the value in the Lookup Table or Lookup file at the
specified row and column. Note that the column can be specified either by providing a
numerical value (or expression) for the column number or by providing the column name as
a string constant (enclosed in single quotes) or as a string variable. An older format in
which the column name is preceded with the # symbol is still supported. ‘Filename’ is
optional. If a filename is provided, EES will first check to ensure that the specified Lookup
file exists and then load the Lookup file data into memory. The row and column arguments
need not be integers. The value returned will be interpolated between rows and columns as
needed. For example, Lookup(2.5, 3) will return a value which is midway between the
values on the second and third rows in the third column. If the specified row or column is
less than 1, the value in the first row or column will be returned. Similarly, if the row or
column is greater than the number of rows and columns in the lookup table, the value in the
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last row or column will be returned. The Lookup function can be used with the
LookupCol and LookupRow functions to provide interpolated values of user-supplied
tabular information. However, the Interpolate commands are usually more convenient for
this purpose.
Examples:
X=Lookup(1,2) { Set X to the value in row 1, column 2 of the Lookup table}
X=Lookup(1,’X’) { Set X to the value in row 1 of the column in the Lookup table
which is named X. }
X=Lookup(‘C:\abc\ CopperK,R,’T’) {Set X to the value in row R and
the column which is named T in Lookup file C:\ abc\ CopperK.LKT}
LookupCol(‘Filename’, Row, Value) uses the data in the specified row of the Lookup Table or
Lookup file to determine the column which corresponds to the value supplied as the second
argument. The column value returned may not be an integer. Interpolation between
columns will be provided as needed. The purpose of the LookupCol function is to provide
a means of relating tabular information in different rows of the Lookup Table or Lookup
file.
Examples:
C=LookupCol(2, 100)
{Set C to the column number in row 2 of the Lookup
table which has a value of 100}
C=LookupCol(‘C:\abc\ CopperK’, R, X) {Set C to the column number in row R
of \Lookup file C:\abc\ CopperK.LKT having the value X}
LookupRow(‘Filename’, Column, Value) uses the data in the specified column of the Lookup
Table or Lookup file to determine the row corresponding to the value supplied as the
second argument. Note that the column can be indicated either by supplying its numerical
value or by providing the column name as a string constant (enclosed in single quotes) or as
a string variable. An older format in which the column name is preceded by the # symbol is
also accepted. The row value returned may not be an integer. Interpolation between rows
will be provided as needed. The purpose of the LookupRow function is to provide a means
of relating tabular information in different columns of the Lookup Table.
Examples:
R=LookupRow(2, 100) {Set R to the row number in column 2 of the Lookup
table which has a value of 100}
R=LookupRow(‘C:\abc\ CopperK’, C, X) {Set R to the row number in column C
Lookup file C:\abc\ CopperK.LKT which has the value X}
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When a new Lookup Table is created, the columns are initially named Column1, Column2,
etc. These default names and the table display format can be changed by clicking the left
mouse button in the header cell as indicated in the Lookup Window section of Chapter 2.
Information can be copied to or from the Lookup Table via the Clipboard. In this way, data
may be transferred between the Lookup Table and the Parametric Table or between other
applications such as a spreadsheet program. To select a rectangular group of cells in the Table,
click the left mouse in the upper left cell. Hold the Shift key down and then click in the lower
right cell. Selected cells will be displayed in inverse video. Use the Select All command in
the Edit menu to select all of the cells in the Lookup table. Next, use the Copy command in the
Edit menu to copy a selected range of table cells to the Clipboard. Hold the Ctrl key depressed
if you wish to copy the column header name and units, in addition to the selected data. Data
may be copied from the Clipboard by clicking the upper-left cell into which the data are to be
pasted, followed by the Paste command. The data in the Clipboard will be pasted into the
Lookup Table, starting from the selected cell.
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____________________________________________________________________________
EES Functions, Procedures and Modules
____________________________________________________________________________
Most high-level programming languages allow the user to write subprograms. EES also offers
this capability in a variety of ways. An EES subprogram is a function, procedure, or module
written within EES. A function is a subprogram that accepts one or more inputs and returns a
single result. A procedure can return one or more results. A module is similar to a procedure
in that it can return one or more results. However, it differs from a procedure in that it employs
equalities rather than assignment statements, as explained below. EES can access both internal
subprograms that have been written within EES and external subprograms written in Pascal, C,
C++, FORTRAN, or any compiled language. The development of external subprograms is
described in Chapter 6. Both internal and external subprograms can be stored in the USERLIB\
subdirectory from which they are automatically loaded when EES is started.
EES subprograms offers a number of advantages. First, they make it easier to formulate the
solution for a complicated system by breaking the problem up into a number of smaller parts.
Programs which rely on subprograms are easier to understand. Second, subprograms can be
saved in a library file and reused in other EES programs. Third, EES functions and procedures
(but not modules) allow use of if then else, repeat until and goto statements. The statements
appearing in functions and procedures differ from those in the main body of EES in that they
are assignment statements, similar to those used in most high-level programming languages,
rather than equality statements. They are executed in the order in which they appear. Modules
employ equality statements just as those used in the main body of an EES program. EES
reorders the equality statements as needed to efficiently solve the equations. The combination
of both statement types offers a great deal of flexibility in the manner in which a problem can
be formulated in EES.
There are several ways to access subprograms. The Merge command in the File menu can be
used to import EES subprograms from one EES file into another. In addition, EES also allows
subprograms to be saved in a library file. Library files are EES files containing one or more
functions, procedures, and/or modules which have been saved with a .lib filename extension
using the Save As command. Subprograms stored in library files that reside in the USERLIB\
subdirectory are automatically and transparently loaded when EES starts. Library files can also
Functions, procedures and modules in library files act just like EES internal functions. They
can even provide help when requested. The steps necessary for creating library files is
described at the end of this chapter.
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EES Functions
EES provides the capability for the user to write functions directly in the Equations window
using the EES equation processor. EES functions are similar to those in Pascal. The rules for
these functions are as follows:
1. The user functions must appear at the top of the Equations window, before any modules or
any equations in the main body of the EES program.
2. User functions begin with the keyword FUNCTION. The function name and arguments,
enclosed in parentheses and separated by commas, follow on the same line.
3. The function is terminated by the keyword END.
4. The equations appearing in EES functions and procedures are fundamentally different from
those appearing in the main body of EES. The equations in functions and procedures are
more properly called assignment statements, similar to those used in FORTRAN and
Pascal. An assignment statement sets the variable identified on the left of the statement to
the numerical value on the right. X:=X+1 is a valid assignment statement but it obviously
cannot be an equality, as assumed for all equations in the main body of EES. The := sign
(rather than the = sign) is used to signify assignment. However, EES will accept an equal
Allow = in Functions/Procedures control in the
Display Options dialog window (Options menu) is selected.
5. EES normally processes the assignment statements in a function or procedure in the order
they appear. However, If Then Else, Repeat Until and goto statements may be used in
functions and procedures to alter the calculation order. The format of these logic control
statements is described below.
6. Functions are called simply by using their name in an equation. The arguments must follow
the name, enclosed in parentheses. The function must be called with the same number of
arguments appearing in the FUNCTION statement.
7. Equations in user functions may call any of the built-in functions. In addition, they may call
any previously-defined user function or procedure or any functions or procedures previously
loaded as Library files. Recursive functions which call themselves are, however, not
allowed. Functions may not call modules.
8. All variables used in the function body are local to the function except those variables
defined in the scope of the \$COMMON directive. The function returns the value to which
its name is assigned.
9. Functions always operate in real mode regardless of the setting for complex algebra.
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Functions can be used to implement an analytical relationship between two or more variables.
For example, the specific availability of a flowing stream, often called ψ, is
ψ = (h - ho) - To (s - so) + V2/2 + g z
where
h and s are specific enthalpy and entropy, respectively
ho and so are specific enthalpy and entropy at the ‘dead’ state condition, To and Po
V is the velocity
g is gravitational acceleration
z is elevation, relative to a selected zero point
Once the temperature and pressure of the dead state are selected, ho and so are constants. A
user function for the availability of steam, with To=530 R and Po=1 atm, could be implemented
by placing the following statements at the top of the Equations window. A reference to psi(T1,
P1, V1, Z1) from an equation would return the specific availability of steam in Btu/lbm for the
FUNCTION psi(T, P, V, Z)
To := 530
ho := 38.05 “Btu/lbm specific enthalpy at dead state conditions”
so := 0.0745 “Btu/lbm-R specific entropy at dead state conditions”
h := enthalpy(STEAM, T=T, P=P)
s := entropy(STEAM, T=T, P=P)
g = 32.17 “ft/s^2 gravitational acceleration
psi := (h-ho)- To * (s – so) + (V^2 / 2 + g * Z) * Convert(ft^2/s^2, Btu/lbm)
END
Functions can also be used to change the name of any built-in function and/or to shorten the
argument list. For example, the following function changes the name of humrat, the built-in
function for humidity ratio, to w, eliminates the need to specify the substance AIRH2O as an
argument, and sets the total pressure to 100 kPa in each case.
FUNCTION w(T,RH)
w := humrat(AIRH2O, T=T, P=100, R=RH);
END
The two example functions both employ EES internal property functions and as a result, they
depend on the EES unit settings to be properly set. Using the UnitSystem function (Chapter 4)
and the IF THEN ELSE statements documented below, it is possible to write general functions
that will operate correctly with any unit settings.
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EES Procedures
EES procedures are very much like EES functions, except that they allow multiple outputs.
The format of a Procedure is:
PROCEDURE test(A,B,C : X,Y)
...
...
X :=...
Y :=...
END
Procedures must be placed at the top of the Equations window, before any of the modules or
equations in the main body of an EES program. The procedure name, TEST, in the example
above, can be any valid EES variable name. The argument list consists of a list of inputs and a
list of outputs separated by a colon. In the example above, A, B, and C are inputs and X and Y
are outputs. Each procedure must have at least one input and one output. Each output variable
must be defined by an equation with the output variables name on the left of the assignment
sign. An END statement closes the procedure.
To use the procedure, place a CALL statement anywhere within your equations. The CALL
statement appears as
...
CALL test(1,2,3 : X,Y)
...
The numbers of inputs and outputs in the CALL statement argument list must exactly match the
PROCEDURE declaration statement. The arguments may be constants, string variables,
numerical variables, or algebraic expressions. Additional arguments can be passed between the
main body of an EES program and a procedure using the \$COMMON directive. EES will
evaluate the outputs using the input variables supplied in the argument list. Procedures may
also call other functions and procedures, provided that they are defined previously. Procedures
may not call modules.
The equations within a procedure differ from ordinary EES equations in modules or in the main
body of an EES program. First, all variables except for the inputs and outputs are local to the
procedure. Second, the equations are really assignment statements, rather than equalities, and
to make this distinction clear, the assignment symbol (:=) is used in place of the equal sign.
You may override this convention by enabling the
Allow = in Functions/Procedures
control in the Preferences dialog window (Options menu). Third, if then else, repeat until and
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goto statements may be used. The format of these flow control statements is described in the
next section.
Implicit equations can not be directly solved in a procedure or function, as they are in modules
and in the main equation body. Using the If Then Else, Repeat Until and goto statements, it is
possible to program your own iterative loop. However, it is also possible to have EES solve
implicit equations within a procedure. For example, consider the following two non-linear
equations.
X^3 + Y^2 = 66
X/Y = 1.23456
To solve for X and Y in a procedure, subtract the right-hand side from the left hand side of each
equation and set them to residuals, R1 and R2, respectively. Now use EES to solve for X and
Y such that the residuals are 0. Here’s a program which does this. However, it should be noted
that implicit equations could be solved more directly and efficiently using a module, as
described later in this chapter.
PROCEDURE Solve(X,Y:R1,R2)
R1:=X^3+Y^2-66
R2:=X/Y-1.23456
END
CALL Solve(X,Y:0,0)
{X = 3.834, Y = 3.106 when executed}
Procedures offer a number of advantages for the EES user. Commonly-used procedures may be
saved separately and merged into the Equations window with the Merge command in the File
menu. Alternatively, the procedure could be saved as a library file so that it is loaded
automatically when EES is started. Procedures can be selectively loaded with the Load
Library command in the Options menu or with the \$INCLUDE directive. For example, the
equations describing a turbine can be entered once and saved. Each time a turbine calculation
is needed, the CALL Turbine statement can be used to determine the turbine work and outlet
state variables.
EES supports both internal and externally-compiled procedures. Internal procedures are
entered directly at the top of the Equations window, as described in this section. Compiled
procedures are written in a high-level language such as C, Pascal, or FORTRAN and called
from EES. The CALL statement for both types of procedures is identical. See Chapter 6 for a
detailed description of writing and using compiled functions and procedures.
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Single-Line If Then Else Statements
EES functions and procedures support several types of conditional statements. These
conditional statements can not be used in modules or in the main body of an EES program.
The most common conditional is the If Then Else statement. Both single-line and multiple-line
formats are allowed for If Then Else statements. The single-line format has the following form.
If (Conditional Test ) Then Statement 1 Else Statement 2
The conditional test yields a true or false result. The format is very similar to that used in
Pascal. Recognized operators are =, <, >, <=, >=, and <> (for not equal). The parenthesis
around the conditional test are optional. Note that string variables (see Chapter 7) can be used
in the condition test. The Then keyword and Statement 1 are required. Statement 1 can be
either an assignment or a GoTo statement. The Else keyword and Statement 2 are optional. In
the single-line format, the entire If Then Else statement must be placed on one line with 255 or
fewer characters. The following example function uses If Then Else statements to return the
minimum of its three arguments.9
Function MIN3(x,y,z) { returns smallest of the three values}
If (x<y) Then m:=x Else m:=y
If (m>z) Then m:=z
MIN3:=m
End
Y = MIN3(5,4,6)
{ Y will be set to 4 when this statement executes}
The AND and OR logical operators can also be used in the conditional test of an If Then Else
statement. EES processes the logical operations from left to right unless parentheses are
supplied to change the parsing order. Note that the parentheses around the (x>0) and (y<>3)
are required in the following example to override the left to right logical processing and
produce the desired logical effect.
If (x>y) or ((x<0) and (y<>3)) Then z:=x/y Else z:=x
9
Note the the built-in MIN function accepts any number of arguments so this function would not be needed.
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Multiple-Line If Then Else Statements
The multiple-line If Then Else statement allows a group of statements to be executed
conditionally. This conditional statement can be used in functions and procedures, but not in
modules or the main body of an EES program. The format is as follows:
If (Conditional Test) Then
Statement
Statement
...
Else
Statement
Statement
...
EndIf
The If keyword, the conditional test, and Then keyword must be on the same line. The
parentheses around the conditional test are optional. The statements which are to be executed if
the conditional test is true appear on following lines. These statements may include additional
If Then Else statements so as to have nested conditionals. An Else (or EndIf) keyword
terminates this first group of statements. The Else keyword should appear on a line by itself,
followed by the statements which execute if the conditional test is false. The EndIf keyword,
which terminates the multiple-line If Then Else statement, is required and it must appear on a
line by itself. The format is illustrated in the following example. Indentation is used to make
the logic flow more clear. However EES ignores the blank spaces. Also upper and lower case
are treated equally.
Function IFTest(X, Y)
If (X<Y) and (Y<>0) Then
A:=X/Y
B:=X*Y
If (X<0) Then
{ nested If statement}
A:=-A; B:=-B
EndIf
Else
A:=X*Y
B:=X/Y
EndIf
IFTest:=A+B
End
G=IFTest(-3,4) { G will be set to 12.75 when this statement executes}
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GoTo Statements
EES will normally process the assignment statements in a function or procedure in the order
they appear starting with the first statement. However, the flow control can be altered using
GoTo statements. The format of a GoTo statement is simply
GoTo #
where # is a statement label number which must be an integer number between 1 and 30000.
Statement labels precede an assignment statement separated with a colon (:). The GoTo
statement must be used with If Then Else statements to be useful. The following function
illustrates the use of GoTo and If Then Else statements in the calculation of the factorial of a
value supplied as the argument.
Function FACTORIAL(N)
F:=1
i:=1
10: i:=i+1
F:=F*i
If (i<N) Then GoTo 10
FACTORIAL:=F
End
Y= FACTORIAL(5) { Y will be set to 120 when this statement executes}
Repeat Until Statements
Looping within functions and procedures can be implemented with If Then Else and GoTo
statements described above, but it is generally more convenient and readable to use a Repeat
Until construct. The Repeat Until statement has the following format. Note that Repeat Until
statements can only be used in functions and procedures.
Repeat
Statement
Statement
...
Until (Conditional Test)
The conditional test yields a true or false result using one of the following operators: =, <, >,
<=, >=, and <> (for not equal). The format is identical to that used in Pascal. Here is the same
Factorial example presented in the previous section implemented with a Repeat Until construct.
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Function Factorial(N)
F:=1
Repeat
F:=F*N
N:=N-1;
Until (N=1)
Factorial:=F
End
Y= FACTORIAL(5) { Y will be set to 120 when this statement executes}
Error Procedure
The Error procedure allows the user to halt calculations if a value supplied to a function or
procedure is out of range. The format of the Error procedure is
Call Error(‘error message’,X)
or
Call Error(X)
where ‘error message’ is an optional character string enclosed within single quotes and X is the
value of the parameter which caused the error. If the error message string is not provided, EES
will generate the following error message when it executes the ERROR procedure.
Calculations have been halted because a parameter is out of range. The
value of the parameter is XXX.
The value of X supplied to the Error procedure replaces XXX. If an error string is provided,
EES will display that string, inserting the value of X in place of the characters XXX. If a
formatting option, such as F1 or E4 follows the XXX, as in the example below, the value of X
will be accordingly formatted, otherwise a default format will be applied. The ERROR
procedure will most likely be used with an IF - THEN - ELSE statement as in the following
example.
Function abc(X,Y)
if (x<=0) then CALL ERROR('X must be greater than 0.
supplied.', X)
abc:=Y/X
end
A value of XXXE4 was
g:=abc(-3,4)
When this function is called, the following message will be displayed and calculations will
stop:
X must be greater than 0. A value of -3.000E0 was supplied.
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Modules
Modules can be considered to be stand-alone EES subprograms that can be called from the
main EES program. The format of a Module is similar to that for an internal Procedure. The
Module is supplied with inputs and it calculates outputs. The formal format of the Module
statement uses a colon in the argument list. The number of arguments to the provided to the
left of the colon is the number of degrees of freedom in the module, or stated another way, the
number of values that must be supplied to have the number of equations equal to the number of
unknowns within the module. An example of a Module statement is
MODULE Testme(A, B : X, Y)
In this case, EES understands that there are two inputs (A and B) and two outputs (X and Y).
However, EES Modules employ equalities rather than assignment statements as used in
Procedures. In most cases, it does not matter what variables are specified as inputs as long as
the appropriate number are specified. As a consequence, the colon which separates inputs from
outputs is irrelevant and it can be replaced with a comma (or semi-colon for the European
numerical format). The following form for the Module statement is equivalent to the format
shown above.
MODULE Testme(A, B, X, Y)
A module is accessed with a CALL statement. For example, the following statement would
access the Testme module.
CALL Testme(77,1.5, X,Y)
Note that if a colon is used to separate the inputs and output in the MODULE statement, it must
also be used in the CALL statement. Similarly if a colon is not used in the MODULE
statement, it should not be used in the CALL statement.
When EES encounters a CALL statement, it transparently grafts the equations in the module to
the equations in the main program. The steps necessary for this process are as follows. First,
every variable in the module, including the inputs and outputs in the MODULE statement, is
renamed with a unique qualifier that EES can recognize. Then EES adds one equation for each
input and output which sets the value of the parameter in the calling program to the value of the
value in the module. Finally, all of the equations in the module, with their renamed variables,
are merged into the EES program at the point at which it is called. If the module is called a
second time, the process is repeated, but with a different qualifier for the variable names in the
module. The net effect is that a copy of all of the equations in the module are merged into the
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main EES program each time a CALL statement is encountered. EES currently allows up to
6000 equations so rather large problems can be developed. EES then uses its efficient blocking
techniques to reorganize the equations for optimal solution. As a result of this reorganization,
the equations in the module may not necessarily be called in sequence. In fact, this is rarely the
case. You can view the equations in the order that EES has rearranged them in the Residuals
window. Equations from a module are identified with the module name followed by a
backslash and then the call index number. For example, the following equation in the
Residuals window
Turbine\2: h2=h1+Q/m
would indicate that the equation h2=h1+Q/m originated from the second call to the Turbine
module.
The local values of variables in modules are normally not displayed in the Solution window.
However, you can view these local solutions for modules which appear in your equations
window by selecting the ’Show function/procedure/module variables’ control in the Options tab
of the Preferences dialog.
Variables values are normally passed to the module through the argument list. However, the
\$COMMON directive (Chapter 7) may be used to set the values of variables that are defined in
the main program.
All variables defined within the module assume the same guess value, lower and upper bounds,
and formatting information. The variable information is accessible with the Variable Info
command. The local variables in a module are always real regardless of the Complex Numbers
setting.
The Module is terminated with an END statement. A CALL statement is used to call the
module, just as for the procedure. The important difference between a procedure and a module
is that the module is composed of equality statements whereas the procedure is composed of
assignment statements. Consequently, a module cannot support logic constructs such as IF
THEN ELSE, but it can provide iterative solutions to implicit equations, when needed, and
order independent equation input, just as in the main part of EES. Here is an example.
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When Solve command is issued, EES will merge the equations in the Testme module twice into
the main set of equations then solve the entire set. The Solution window will then appear as
Note that the local variables for each call to the Testme module will be shown only if the
‘Show function/procedure/module values’ control in the Options tab of the Preferences dialog
is selected.
Modules may be stored as library files, just like internal functions and procedures. (Library
files are described in the following section.) The library files can be automatically loaded if
they are placed in the USERLIB subdirectory. Alternatively, a \$INCLUDE directive can be
used to transparently load a library file. Help can be included within the module using the same
syntax as used in procedures or it can be supplied with a separate ASCII or Windows help file
having the same filename as the library file with a .HLP filename extension.
Modules can significantly increase the capabilities of your EES programming.
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Library Files
EES allows files containing one or more functions, procedures or modules (subprograms) to be
saved as Library files. A Library file has a .LIB filename extension. When EES starts, it will
automatically load all of the functions, procedures, and modules in the library files which reside
in the USERLIB\ subdirectory. Library files can also be loaded manually with the Load Library
command in the File menu and with the \$INCLUDE directive. Library subprograms will not
displayed in the Equations window. They are used just like EES built-in functions. To create a
Library file, enter one or more functions, procedures, and/or modules into the Equations
window. Compile the equations using Check, Solve or Solve Table. Then save the file with a
.LIB filename extension using the Save As command.
Subprograms in library files can provide help information in the Function Info dialog window,
just like the built-in functions. The help text is placed in the file as a comment in braces. The
first character after the opening comment brace is a \$, followed by the name of the function,
procedure or module followed by a carriage return. The lines following, up to the closing
comment brace, are the help text that will be displayed when the user selects the Info button in
the Function Info dialog window as shown in the example below.
The Library file concept is among the most powerful features of EES because it allows the user
to easily write customized subprograms for personal use or for use by others. The following
example uses a library file to provide a fourth-order Runge-Kutta numerical integration
function in EES. The Runge-Kutta algorithm is used to numerically solve a differential
equation of the form:
dY = f (X ,Y)
dX
where f(X,Y) is any function involving the dependent variable Y and independent variable X. Y
must have a known initial value, Y0, corresponding to the initial value of X.
The Runge-Kutta algorithm has been implemented as a general purpose library function called
RK4. RK4 requires 4 parameters: the initial value of X (LowX), the final value of X (HighX),
the step size (StepX), and the value of Y at X=LowX (Y0). The function returns the value of Y
at X=HighX. The RK4 function calls another function, fRK4(X,Y), to provide the value of
dY/dX for given X and Y values. A dummy fRK4 function is provided in the RK4.LIB file as
a placeholder. In an actual application, the user overrides the dummy fRK4 function by
entering another fRK4 function in the EES Equations window. The RK4 and fRK4 functions
have been saved in a library file called RK4.LIB in the USERLIB\ sub-directory. EES will load
these functions when it starts. If you were to open the RK4.LIB file in EES, you would see the
following statements. Note how the functions provide help text as a comment with the
\$filename key. Help can also be provided in an ASCII file having a .HLP filename extension.
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FUNCTION fRK4(X,Y)
{\$fRK4
fRK4 is a user-supplied function to evaluate dY/dX. This
function is used with the RK4 function to solve differential
equations with the Runge-Kutta method. Enter a fRK4(X,Y)
function in the Equations window to evaluate dY/dY for your
problem. See the RK4 function for additional information.}
fRK4:=(Y+X)^2
END
FUNCTION RK4(LowX,HighX,StepX,Y0)
{ \$RK4
RK4 is a general purpose function which solves a first-order
differential equation of the form dY/dX=fRK4(X,Y) using the
Runge-Kutta 4th order algorithm. The RK4 function calls function
fRK4(X,Y) supplied by the user to evaluate dY/dX at specified values
of X and Y. The user must supply the fRK4 function.
RK4 requires four input parameters. LowX is the initial
value of independent variable X. HighX is the final value
of independent variable X and StepX is the step size. Y0 is
the value of Y when X is equal to LowX.}
X := LowX
Y := Y0;
Tol := 0.1*StepX
10:
IF (X>HighX-Tol) THEN GOTO 20
k1 := fRK4(X,Y)*StepX
k2 := StepX*fRK4(X+0.5*StepX,Y+0.5*k1)
k3 := StepX*fRK4(X+0.5*StepX,Y+0.5*k2)
k4 := StepX*fRK4(X+StepX,Y+k3)
Y := Y+k1/6+(k2+k3)/3+k4/6
X := X+StepX
GOTO 10;
20:
RK4:=Y
END
2
X 2dx using the RK4 function.
Suppose you wish to numerically solve the equation
0
You provide a function fRK4 to evaluate the integrand, which is X2 in this case. Your function
overrides the fRK4 function in the RK4 library file. Assuming RK4 was in the USERLIB\ subdirectory when EES was started, all that would be needed is:
FUNCTION fRK4(X,Y)
fRK4:=X^2
END
V=RK4(0,2,0.1,0)
When you solve this problem, EES will display V=2.667 in the Solution window.
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\$COMMON Directive
The \$COMMON directive provides a means for passing information from the main program to
internal functions, procedures, and modules. Use of \$COMMON provides an alternative to
passing values as arguments. This directive is similar in concept to the COMMON statement
in FORTRAN. It differs in that information flow is one-way. Variable values can be passed
from the main program to the function or procedure. However, the function or procedure may
not assign or alter these values.
The \$COMMON directive must directly follow the FUNCTION, PROCEDURE, or MODULE
declaration on a line by itself. Variables appearing in the \$COMMON statement are separated
with commas, as in the following example.
FUNCTION TESTCOMMON(X)
\$COMMON B,C,D {variables B,C, and D are from the main program}
TESTCOMMON:=X+B+C+D
END
B=4; C=5; D=6
G=TESTCOMMON(3)
\$COMMON should only be used with functions, procedures, and modules appearing in the
Equations window. It should not be used with library functions.
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\$INCLUDE Directive
The \$INCLUDE directive provides an automatic method for loading a library file or ASCII text
file containing EES equations. The format is:
\$INCLUDE FILENAME
FILENAME is the filename including the filename extension which can be one of .TXT, .LIB,
.FDL, .DLF, or .DLP. The filename should also include the complete path name, e.g.,
C:\EESW\myDefn.TXT. However, if a path name is not provided, EES will look in the current
directory. If EES is unable to find the file, it will provide the opportunity to browse so that you
can locate it. The \$INCLUDE statement must be on a line by itself, starting in column 1. It is
best to place the \$INCLUDE directives at the top of the Equations Window to ensure that the
directive is processed before compilation of the equations is initiated.
.TXT Files
If the filename extension is .TXT, EES expects FILENAME.TXT to be an ASCII text file
containing EES equations. Syntax errors can not be identified in the library file so care should
be taken to ensure the equations are correct before saving the library file. EES will include
these equations with others in the Equations window during compilation. However, the
equations and the variables associated with these equations will be hidden. Nested use of the
\$INCLUDE directive is not supported so that the text file must not include any \$INCLUDE
statements.
Equations can also be entered from a file with the Merge command in the File menu. The
difference between the Merge command and the \$INCLUDE directive is that equations entered
with the Merge command are placed directly into the Equations window as if they were typed
and they remain visible. The text entered with the \$INCLUDE directive will be hidden. Note
that the speed of editing in the Equations window is reduced as the size of the text in the
Equations window increases. Use of the \$INCLUDE directive for very large problems can
eliminate this problem on slower machines.
Library Files
If the filename extension is .LIB, .FDL, .DLF, or .DLP, EES will expect the file to be a library
file of a type corresponding to the filename extension. EES internal functions, procedures. and
modules are recognized with the .LIB extension, whereas external functions use a .DLF
extension and external procedures use either the .FDL or .DLP extension. EES will
can also be loaded automatically when EES is started by placing them in the USERLIB
subdirectory or by applying the Load Library command.
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__________________________________________________________________________
Compiled Functions and Procedures
__________________________________________________________________________
EES has an extensive library of built-in functions, but it is not possible to anticipate the needs
of all users. A remarkable feature of EES is that the user can add (and later remove) functions
and procedures written in any compiled language, such as Pascal, C, C++ or FORTRAN.
These compiled routines may have any number of arguments. Functions return a single value
whereas procedures may return multiple values. The compiled routines are used in exactly the
same manner as internal EES subprograms. This capability gives EES unlimited flexibility and
it is among its most powerful features.
Compiled functions and procedures are written as dynamic link library (DLL) routines under
the Windows operating system10. Compiled functions are identified with a .DLF. There are
two formats for compiled procedures identified by .DLP and .FDL filename extensions. When
EES is started, it examines the files in the EES USERLIB\ subdirectory. Any files having a
.DLF, .DLP, or .FDL filename extension are assumed to be compiled functions or procedures
command in the File menu or with the \$INCLUDE directive. The function name to be
referenced in EES equations is the filename (without the extension).
Compiled functions and procedures can (optionally) be setup to work with the Function Info
command (Options menu) so that they provide an example and detailed help when it is
requested. The following sections of this chapter provide detailed information and examples of
compiled functions and procedures.
EES Compiled Functions (.DLF files)
Compiled functions can be written in C, C++, Pascal, or any language which can produce a
dynamic link library (DLL). The function statement header, however, must have a specific
format. To avoid having to set a fixed upper limit on the number of inputs, the input
information to a compiled function is implemented as a linked list. The linked list record or
structure consists of an extended precision value and a pointer to the next input. The last input
10
Note that there are both 16 and 32-bit DLLs in the Windows Operating System and they are not interchangeable.
The 16-bit version of EES can only use 16-bit DLLs and the 32-bit version can only use 32-bit DLLs. The 32-bit
version of EES can be used with Windows 95 or NT, but not with Window 3.1. The 16-bit version will run under
any Windows operating system, but it executes more slowly than the 32-bit version. Instructions for preparing both
DLL types are provided in this chapter. If you are not sure as to whether you are using the 16-bit or 32-bit version
of EES, start the program and examine the startup screen. The 32-bit version of EES will display (32-bit) on the
same line as the version number and in the main window title.
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points to nil. Some compiled languages, such as FORTRAN 77 do not support pointers so
.DLF compiled functions cannot be written in these languages The .FDL format described in
this chapter should be used in this case.
The compiled function should check that the number of inputs supplied in the linked list is
equal to the number the function expects. (The PWF function example in the next section
shows how this checking can be done.) Although the values of the inputs may be changed in
the function, these changes are local and will be disregarded by EES. Only the function result
will be used by EES. A skeleton listing of a compiled function written in Borland’s Delphi 1.0
(16-bit) or Delphi 3.0 (32-bit) is as follows:
library XTRNFUNC;
{\$N+}
type
ParamRecPtr = ^ParamRec;
ParamRec = record { defines structure of the linked list of inputs }
Value: extended;
next: ParamRecPtr;
end;
function FuncName (var S:string; Mode:integer; Inputs:ParamRecPtr): extended; export; stdCall;
begin
...
FuncName:=Value; { Funcname must be extended precision }
end;
11
exports FuncName;
begin
end.
The major concern is the function header. To be recognized by EES, the function name, called
FuncName in the above example, must be the same as the filename. The function statement
has three arguments.
S is a standard 255-character Pascal string. The first character contains the actual length of the
string. S can be used for both input and output. If the first parameter provided in the EES
function is a string (within single quotes), EES will pass this string to the external routine. If an
error is encountered, S should be set in the external routine to an appropriate error message. If
the length of S is not zero, EES will terminate calculations and display S as an error message.
11
The stdCall keyword is required for Delphi 3.0 (32-bit). It is not needed for Delphi 1.0 (16-bit)
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Mode is an integer set by EES. If Mode=–1, then EES is requesting that the function return in
S an example of the function call. If Mode>=0, then the function should simply return the
function value. Currently, EES does not use the return value of Mode.
Inputs is a pointer to the head of a linked list of input values supplied by EES. Each input
consists of a value (extended precision) and a pointer to the next input, as indicated by the
ParamRec structure. The function may have one or more inputs. The next field of the last
input will be a null pointer (nil). The function should count the inputs to be sure that the
number supplied is as expected and issue an error message in S if this is not the case.
A skeleton listing of a compiled function written in Borland's C++ follows:
#include <windows.h>
#include <stdlib.h>
#include <string.h>
#define EXAMPLE=-1;
// Use extern "C" to prevent C++ name mangling
extern "C"
{
long double far pascal _stdcall _export FUNCNAME
(char* S, int Mode, struct ParamRec *FirstInput);
}
int far pascal LibMain //DLL entry point for initiation
(HINSTANCE hInstance, WORD wDataSeg, WORD cbHeapSize, LPSTR lpstrCmdLine)
{
if (cbHeapSize) UnlockData(0);
return TRUE;
}
int far pascal WEP(int nParam) //Windows exit procedure - not needed in Borland C++
{
return TRUE;
}
struct ParamRec
{
long double
value;
struct ParamRec
*next;
};
long double far _export _stdcall pascal FUNCNAME(char* S, int Mode, struct ParamRec *FirstInput)
{
….
….
return (v);
}
Note that the pascal keyword must be provided to ensure that the calling parameters are in an
order that will be understood by EES.
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The PWF Compiled Function
EES does not have any internal economic functions. An economic function called the present
12
worth factor (PWF) has been added as a compiled function. PWF is the present worth of a
series of N future payments which inflate at rate i per period accounting for the time value of
money with a market discount rate per period of d. The equation for PWF is
Σ
N
PWF(N,i,d) =
j=1
1+i
j–1
1+d
j
=
1
1+i
d –i 1–
1+d
N
1+i
N
if i ≠ d
if i = d
where
N is the number of periods (e.g., years)
i is the interest rate per period, expressed as a fraction
d is the market discount rate per period, expressed as a fraction
A compiled function, called PWF, has been written to do this economic calculation. This
function is stored in the PWF.DLF file on the EES disk. EES treats this compiled function just
like any of its internal functions. Shown on the following pages is the complete listing for the
PWF compiled function written in Borland’s Delphi 3.0 (32-bit version).
Four other compiled functions are included with EES. These functions implement a
generalized equation of state using Redlich-Kwong-Soave equation13.
Compressibility(Tr, Pr, w) returns the compressibility of a gas, i.e., the ratio of the specific
volume of the gas to the specific volume of an ideal gas at the same conditions. Tr is the
reduced temperature, Pr is the reduced pressure, and w is the acentric factor. The third
parameter is optional.
EnthalpyDep(Tr, Pr, w) and EntropyDep(Tr, Pr, w) return the dimensionless enthalpy and
entropy departures, respectively. The dimensionless enthalpy departure is defined as (h[ideal]h)/(R Tc). h[ideal]-h is the difference in enthalpy between an ideal gas and a real gas at the
same temperature and pressure, R is the gas constant and Tc is the critical temperature. The
dimensionless enthalpy departure is similarly defined as (s[ideal]-s)/(R).
FugCoef(Tr, Pr, w) returns the dimensionless fugacity coefficient (fugacity / pressure).
Do you need a function which EES doesn't include? Write your own. It's EESY!
12
Duffie, J.A. and Beckman, W.A., Solar Engineering of Thermal Processes, 2nd edition, J. Wiley and Sons, 1992,
Chapter 11.
13
G. Soave, Chem. Eng. Science, Vol. 27, pp. 1197-1203, 1972
130
Compiled Functions and Procedures
Chapter 6
Listing of the PWF Compiled Function in Borland’s Delphi 3.0
library PWFP;
uses
SysUtils, Classes;
{\$N+}
const doExample = -1;
type
ParamRecPtr=^ParamRec;
ParamRec=record
Value:extended;
next:ParamRecPtr;
end;
function CountValues (P: ParamRecPtr): integer;
var N: integer;
begin
N := 0;
while (P <> nil) do begin
N := N + 1;
P := P^.next
end;
CountValues := N;
end; {CountValues}
function PWF(var S:Shortstring; Mode:integer; Inputs:ParamRecPtr):extended; export; stdcall;
var P: ParamRecPtr; V: extended;
function CountValues (P: ParamRecPtr): integer;
var
N: integer;
begin
N := 0;
while (P <> nil) do begin
N := N + 1;
P := P^.next
end;
CountValues := N;
end; {CountValues}
function PWFCalc: extended;
var
Periods, NArgs: integer;
interest, discount: extended;
begin
PWFCalc:=0; {in case of error exit}
S := ’’;
P := Inputs;
Periods := round(P^.value);
if (Periods < 1) then begin
S := ’The number of periods for the PWF function must be >0.’;
exit;
end;
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Compiled Functions and Procedures
P := P^.next;
interest := P^.value;
if (interest >= 1) or (interest < 0) then begin
S := ’The interest rate is a fraction and must be between 0 and 1.’;
exit;
end;
P := P^.next;
discount := P^.value;
if (discount >= 1) or (discount < 0) then begin
S := ’The discount rate is a fraction and must be between 0 and 1.’;
exit;
end;
if (interest <> discount) then
PWFCalc := 1 / (discount - interest) * (1 - exp(Periods * ln((1 + interest) / (1 + discount))))
else
PWFCalc := Periods / (1 + interest);
end; {PWF}
begin
PWF:=1;
if (Mode = doExample) then begin
S := ’PWF(Periods,Interest,Discount)’;
exit;
end;
if (CountValues(Inputs)<>3) then
S := ’Wrong number of arguments for PWF function.’
else begin
PWF:=PWFCalc;
end;
end; {PWF}
exports PWF;
begin
{no initiation code needed}
end.
When this Pascal code is compiled with the Borland Delphi 3.0, a dynamically-linked library
routine is created. The compiler automatically generates a .DLL filename extension for the
compiled code. EES must distinguish compiled functions from compiled procedures. It does
this by the filename extension. Compiled functions must have a .DLF filename extension.
Rename the compiled file so that it has a .DLF extension.
Access the external PWF function by a statement of the following form in your EES program.
P = PWF(Periods,Interest,Discount)
132
Compiled Functions and Procedures
Chapter 6
EES Compiled Procedures (.FDL and .DLP files)
EES compiled procedures are very similar to EES compiled functions. In either case, the user
supplies the function or procedure in compiled form as a Windows dynamically linked library
routine. The major difference between functions and procedures is that procedures can return
one or more values, whereas a function returns a single value. Procedures are useful, for
example, for thermodynamic property evaluations where multiple properties (e.g., volume,
enthalpy, entropy, etc.) are to be determined given one set of independent variables (e.g.,
temperature and pressure).
External procedures are written as dynamic link libraries (DLL’s). Be careful to provide 16-bit
DLLs for the 16-bit version of EES and 32-bit versions for the 32-bit version. There are two
formats for external procedures and they are identified to EES by their filename extension. The
two formats differ in the manner in which EES exchanges information with the external
routine. The .FDL format passes inputs and outputs in double precision floating point arrays
which may contain up to 50 elements. The .DLP format passes inputs and outputs as linked
lists (as in the .DLF functions) so there is no limit to the number of inputs and outputs. EES
identifies the format by the filename extension which must be .FDL or .DLP. External
procedures using arrays to hold the inputs and outputs must have a .FDL filename extension.
This is the usual case when providing DLLs written in FORTRAN. C, C++, and Pascal
procedures can use either format.
Compiled procedures are accessed from EES with the CALL statement which has the following
format,
CALL procname(‘text’, A, B : X, Y, Z)
where
procname is the name of the procedure
‘text’ is an (optional) text string that will be passed to the procedure. This text can either be
a string constant enclosed in single quote marks or a string variable. (See Chapter 7).
A and B are inputs. There may be one or more inputs, separated by commas, appearing to
the left of the colon. Inputs may be numerical constants, EES variable names, or algebraic
expressions. String variables cannot be supplied as inputs.
X, Y, and Z are outputs determined by the procedure. There should be one or more outputs
to the right of the colon, separated by commas. Outputs must be EES numerical variable
names. String variables cannot be provided for outputs.
Note that the CALL statement used to access compiled functions is identical in format to the
CALL statement used for internal EES Procedures.
The following two sections describe the .FDL and .DLP external procedure formats and
provide a simple example which should serve as a model.
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Chapter 6
Compiled Functions and Procedures
Compiled Procedures with the .FDL Format - a FORTRAN Example
The .FDL format is illustrated by the following FORTRAN subroutine fragments. The code
differs slightly depending on whether 16-bit or 32-bit libraries are being compiled and which
compiler is being used.
16-bit .FDL library using the Microsoft FORTRAN 5.1 compiler
SUBROUTINE MYPROC(S,MODE,NINPUTS,INPUTS,NOUTPUTS,OUTPUTS)
INTEGER*2 MODE, NINPUTS, NOUTPUTS
REAL*8 INPUTS(50), OUTPUTS(50)
CHARACTER*255 S
…
OUTPUTS(1)=…
…
RETURN
END
32-bit .FDL library using the Digital Visual FORTRAN 5.0 compiler
SUBROUTINE MYPROC(S,MODE,NINPUTS,INPUTS,NOUTPUTS,OUTPUTS)
!DEC\$ATTRIBUTES ALIAS:'MYPROC' :: MYPROC
!DEC\$ATTRIBUTES DLLEXPORT :: MYPROC
INTEGER(4) MODE, NINPUTS, NOUTPUTS
REAL(8) INPUTS(50), OUTPUTS(50)
CHARACTER(255) S
…
OUTPUTS(1)=…
…
RETURN
END
S is a null-terminated C-style character string containing 255 characters. If the first parameter
in the EES Call statement is a text string (within single quotes), EES will pass this string to the
external program in S. When EES calls the subroutine with MODE =-1, it is asking for an
example of the calling sequence of this procedure from EES to be placed in S so that it can be
displayed in the Function Info Dialog window. S is also used to return user-supplied error
messages if necessary. If an error is detected in the subroutine, MODE should be set to a value
greater than 0 to signal EES to terminate calculations. If S is defined, it will be displayed in the
EES error message. In normal operation, MODE =0 and S need not be defined.
and NOUTPUTS are the number of inputs and outputs provided by EES. The routine
should check to see if these agree with the expected number of inputs and outputs and return an
error condition (MODE>0) is this is not the case. INPUTS and OUTPUTS are arrays of 50 double
precision (REAL*8) values. EES will supply the values in the INPUTS array. Results calculated
by the subroutine are placed in the OUTPUTS.
NINPUTS
134
Compiled Functions and Procedures
Chapter 6
The external program must be compiled and linked as a dynamic link library (DLL) routine.
The compiling procedure differs among different languages and compilers. To compile and
link a FORTRAN external procedure called MYPROC as a DLL in the Microsoft FORTRAN 5.1
environment to produce the MYPROC.FDL 16-bit external EES procedure, you would enter:
fl /c /Aw /Gw MYPROC.FOR
The compiler and linker options are defined in the Microsoft FORTRAN 5.1 manuals. A
definition file, MYPROC.DEF, is required by the linker. This file has the following format.
LIBRARY MYPROC
DESCRIPTION MYPROC TEST FDL’
EXETYPE WINDOWS 3.0
HEAPSIZE 1024
EXPORTS MYPROC @1
WEP
@2 RESIDENTNAME
Creating a 32-bit DLL with Digital Visual Fortran 5.0 is most easily done within the Microsoft
Developer Studio environment. A new project workspace is selected as a Dynamic Link
Library. The FORTRAN sources file(s) are inserted into the workspace and compiled with the
standard options. Note that the two !DEC\$ATTRIBUTES directives should be included in the
main program, as noted above. The output filename in the Link settings should be set to
MYPROC.FDL where MYPROC is the name that will be used in the EES Call statement.
Alternatively, the default filename MYPROC.DLL should be changed to MYPROC.FDL after
building the dynamic link library project.
The simple FORTRAN program listed below provides the product, dividend, sum, and
difference of two input values. This program should provide a model for writing external EES
procedures in FORTRAN.
135
Chapter 6
Compiled Functions and Procedures
Listing of the FORTRAN MDASF Program
C.
C.
SUBROUTINE MDASF(S,MODE,NINPUTS,INPUTS,NOUTPUTS,OUTPUTS)
The following two lines are specific to Microsoft Power Station 4.0
!MS\$ATTRIBUTES ALIAS:’MDASF’ :: MDASF
!MS\$ATTRIBUTES DLLEXPORT :: MDASF
Replace INTEGER(4) with INTEGER*2 for a 16 bit DLL in the following line
INTEGER(4) MODE, NINPUTS, NOUTPUTS
REAL(8) INPUTS(25), OUTPUTS(25)
CHARACTER(255) S
C.
C.
100
C.
C.
C.
C.
C.
C.
200
300
900
C.
IF (MODE.EQ.-1) GOTO 900
IF (NINPUTS.NE.2) GOTO 100
IF (NOUTPUTS.NE.4) GOTO 200
DO CALCULATIONS
X=INPUTS(1)
Y=INPUTS(2)
IF (ABS(Y).LE.1E-9) GOTO 300
OUTPUTS(1)=X*Y
OUTPUTS(2)=X/Y
OUTPUTS(3)=X+Y
OUTPUTS(4)=X-Y
MODE=0
S=’’C
RETURN
CONTINUE
ERROR: THE NUMBER OF INPUTS ISN’T WHAT THIS SUBROUTINE EXPECTS
NOTE: SET MODE>0 IF AN ERROR IS DETECTED. IF S IS EQUAL TO A
NULL STRING, THEN EES WILL DISPLAY THE MODE NUMBER IN AN ERROR
MESSAGE. IF S IS DEFINED, EES WILL DISPLAY THE STRING IN THE
ERROR MESSAGE. THE C AT THE END OF THE STRING INDICATES C-STYLE
S=’MDASF REQUIRES 2 INPUTS’C
MODE=1
RETURN
CONTINUE
S=’MDASF EXPECTS TO PROVIDE 4 OUTPUTS’C
MODE=2
RETURN
CONTINUE
S=’DIVISION BY ZERO IN MDASF’C
MODE=3
RETURN
CONTINUE
PROVIDE AN EXAMPLE OF THE CALLING FORMAT WHEN MODE=-1
S=’CALL MDASF(X,Y:A,B,C,D)’C
RETURN
END
136
Compiled Functions and Procedures
Chapter 6
Compiled Procedures with the .DLP Format - a Pascal Example
The .FDL format described in the previous section was illustrated with FORTRAN, but it can
be implemented in any compiled language. The .DLP calling format described in this section
uses linked lists for inputs and outputs, and thereby is not suitable for use with FORTRAN.
There is essentially no difference in efficiency between the two formats. Both are provided for
backward compatibility and complete flexibility.
Compiled procedures using the .DLP format are very similar to compiled functions (.DLF files)
described previously. The only difference is that a procedure must have, in addition to a linked
list of input values, a linked list of output values. The calling sequence for a compiled Pascal
procedure with the .DLP format has the following form
procedure procname (var S: string; Mode: integer; Inputs, Outputs: ParamRecPtr);
S, Mode, and Inputs are identical to their counterparts for the EES compiled functions. Outputs
is a linked list of extended values which provides the results of the calculations to EES in the
order in which they appear in the CALL statement.
Shown on the following page is a complete listing of an EES compiled procedure, called
MDAS (an acronym for MyDearAuntSally) which provides the product, dividend, sum, and
difference of two input values. (This is the same program used in the .FDL example.) The
code checks to make sure that the number of inputs and outputs provided in the CALL
statement are what the routine expects before it does the calculations and sets S to an error
message if this is not the case.
Example Compiled Procedure (.DLP) in Borland’s Delphi 2.0
library MDASP;
uses
SysUtils, Classes;
{\$N+}
const Example = -1;
type
ParamRecPtr=^ParamRec;
ParamRec=record
Value:extended;
next:ParamRecPtr;
end;
function CountValues (P: ParamRecPtr): integer;
var N: integer;
begin
N := 0;
while (P <> nil) do begin
N := N + 1;
137
Chapter 6
Compiled Functions and Procedures
P := P^.next
end;
CountValues := N;
end; {CountValues}
procedure MDAS(var S:Shortstring; Mode:integer;
Inputs,Outputs:ParamRecPtr); export; stdcall;
procedure MyDearAuntSally;
var
P1, P2: extended;
P: ParamRecPtr;
begin
P := Inputs;
P1 := P^.Value;
P := P^.next;
P2 := P^.value;
P := Outputs;
P^.Value := P1 * P2;
P := P^.next;
P^.Value := P1 / P2;
P := P^.next;
P^.Value := P1 + P2;
P := P^.next;
P^.Value := P1 - P2;
end; {doCall}
begin {MDAS}
if (Mode = -1) then
S := ’CALL MDAS(In1,In2:Out1,Out2,Out3,Out4)’
else begin
if (CountValues(Inputs) <> 2) then begin
S := ’Wrong number of inputs for MDAS.’;
exit;
end;
if (CountValues(Outputs) <> 4) then begin
S := ’Wrong number of outputs for MDAS.’;
exit;
end;
MyDearAuntSally;
S:=’’;
end;
end; {MDAS}
exports
MDAS;
begin
{no initiation code needed}
end.
138
Compiled Functions and Procedures
Chapter 6
Help for Compiled Functions and Procedures
The Function Info dialog (Options menu) has a INFO button which, when used, provides help
explaining the use of the selected function. When the user clicks the INFO button, EES will
look for a file with the name of the compiled routine and a .HLP extension. This file can either
be an ASCII text file or a Windows .HLP file. This help file will be displayed if the file is
found in the directory in which the external library file resides; otherwise a message will
appear which states the help is not available for this item.
If an ASCII file is provided, it should be formatted so that each paragraph ends with a carriage
return. Long lines which do not fit within the Help window will be broken and word-wrapped
as needed. Blank lines and spaces can be used to make the text more clear.
Note that the Windows .HLP allows figures and formatting options to be used in the .HLP file
and so it is a better way of providing help. The Window .HLP file can be composed within any
word processor which produces a RTF file or more conveniently, using any of the commercial
Help generating programs.
139
.
140
CHAPTER 7
__________________________________________________________________________
__________________________________________________________________________
The advanced features in EES allow the program to work with string, complex, and array
variables and solve simultaneous algebraic and differential equations. The commands and
functions which implement these features are described in this chapter and illustrated with
examples.
String Variables
EES provides both numerical and string variables types. A string variable holds character
string information. A string variable is identified to EES with a variable name that ends with
the \$ character, as in the BASIC language. The variable name must begin with a letter and
consist of 30 or fewer characters, including the \$ character.
String variables can be set to string constants. A string constant is a set of up to 255 characters
enclosed within single quote marks, e.g.
A\$=’carbon dioxide’
String variables can be set equal to other string variables, e.g.,
B\$=A\$
String variables may be passed as arguments to internal functions, procedures and modules or
to external functions and procedures as described in Chapter 6.
In general, string variables can be used in EES equations anywhere in which character
information is provided. For example, the name of a fluid provided to a thermophysical
property function may be a string variable, e.g.,
h=enthalpy(R\$,T=T,P=P)
String variables may be used with the Parametric table. In the example below, a Parametric
table is used to tabulate the values of specific enthalpy for four refrigerants. Note that the
single quote marks which are normally used when specifying a string constant should not be
used when entering the strings in the Parametric table.
141
Chapter 7
A string variable may be used to hold the name of a Lookup file or the name of a column for
use with the Interpolate or Lookup commands, e.g.,
m=Interpolate(File\$,Col1\$,Col2\$,Col1\$=x)
k=Lookup(File\$,Row,Col\$)
String variables may be supplied to the Equations Window using the Diagram Window, either
with an edit box or with a pull-down list of alternatives. See the Diagram Window section in
Complex Variables
EES will solve equations involving complex variables of the form a+b*i if the ‘Do Complex
Algebra’ control in the Complex tab of the Preferences dialog is checked. The imaginary
number operator may be set to either i or j in the Preferences dialog although i is used in the
following discussion.
When set in complex mode, every (non-string) EES variable is represented internally as two
variables corresponding to the real and imaginary components of a complex number. The real
part is internally identified by appending _r to the variable name. The imaginary part has an
appended _i. (You should not use _r or _i at the end of a variable name unless you are
specifically referring to the real or imaginary component.) If, for example, you enter the
equation
X=Y
EES will automatically create variables X_r, X_i, Y_r and Y_i corresponding to the real and
imaginary components of the variables. You will normally not have to refer to the renamed
variables, although they will appear with these names in the Variable Info and New Parametric
Table dialogs, as well as in column headers of the Parametric Table. However, you can set the
142
Chapter 7
value of a real or imaginary part of a complex number by entering the real or imaginary variable
name on the left of an equation in the Equations window. For example, the following equation
will set the imaginary part of variable omega to 0.
omega_i=0
Any later attempt to set the imaginary part of omega will cause EES to display an error
message. If, for example, you also enter the equation, omega = 3 in the Equations window,
EES will present an error message when you attempt to solve because the omega=3 equation
sets both the real and imaginary parts of omega and the imaginary part would have already been
set. You could, of course, enter omega_r=3.
Complex numbers can be entered in either rectangular or polar form. In rectangular form, the
complex number is entered using the imaginary number operator (i or j) with a multiplication
symbol (*) separating the imaginary number operator from variables or constants. A complex
constant can be entered in polar form by entering the magnitude (also called absolute value) of
the number and the angle separated with the < symbol. The < character will be displayed in the
Solution and Formatted Equations windows as ∠. The angle can be entered in either degrees or
radians. If no designation is provided, the angle is assumed to be in the same units as indicated
for trigonometric functions in the Unit System dialog. However, you can ensure that the angle
you enter is in degrees or radians regardless of the unit system setting by appending deg or rad
to the number (with no spaces). For example, the value of Y can be set to the same complex
constant in any one of the following three ways.
Y=2 + 3 * i
Y=3.606 < 56.31deg
The use of deg or rad to indicate the units of the angle is strongly recommended for three
reasons. First, the value of the constant you enter will not be changed by the unit system
setting. Second, you can enter complex variables in polar form with the angle in degrees yet do
all calculations with the unit setting in radians which is more efficient. Third, if you designate
the angle to be in degrees, the degree sign will be shown with the angle in the Formatted
Equations window. The output display of the number in the Solution Window can be set for
degrees or radians (in polar notation) independent of how the number is entered. The display is
changed by clicking the right mouse button on the value and selecting the display options from
the Format Variable dialog.
143
Chapter 7
Internally, EES creates two equations for each equation that is entered in the Equations
window. One equation is used to equate the real parts of the variables whereas the second
equation equates the imaginary parts. The actual set of equations used in complex mode is
most clearly seen by viewing the Residuals window which displays the residual and blocking
order for each equation. Equations used for the real and imaginary parts are identified with (r)
and (i), respectively.
When configured in complex mode, some EES functions, such as Min and Max, are not
accessible. However, most of the built-in functions (including the thermophysical property
functions) have been modified to work with complex numbers. For example, the sin, cos, ln,
exp, and tanh functions will accept and return appropriate complex numbers. User-written
functions, procedures, and external routines can be used but they will accept and return real
numbers only. (Modules are currently not supported in complex mode.) Only the real part of a
complex variable will be placed in the argument list of internal or external functions,
procedures, and modules.
There are a few built-in functions that operate only in complex number mode. They are Real,
Imag, Magnitude, Angle, AngleDeg, AngleRad, and Conj. These functions all take one
(complex number) argument and set the real part of the complex variable to the selected value.
One limitation of the manner in which EES implements complex numbers is that EES can only
return one solution, although two or more solutions may exist. There is, however, a simple way
to coax EES into providing multiple solutions. Consider the problem of determining the five
roots to the complex equation z5 + 9 + 9 i = 0. Entering this equation into the Equations
window and solving with the default guess values will produce a solution of z=1.176+1.176 i.
Setting the Solution window display to polar (degree) coordinates will produce the following
solution.
This is a correct solution to the equation, but there are four others. To find a different solution
(without changing guess values), divide the equation by the difference between z and the value
of this root. The Formatted Equations window will appear as shown.
144
Chapter 7
Now, a second solution will be found.
This process can be repeated to find the third, fourth, and fifth roots.
The Formatted Equations window and solution for this last step are shown below.
145
Chapter 7
Array Variables
EES recognizes an array variable by placing the array index within square brackets, e.g. X[5].
Multi-dimensional array variables may also be used with the indices separated by commas, e.g.,
Z[1,2,3]. The special requirements pertaining to array variables are:
1. An array index may be an integer number, an EES variable which has been previously set to
a constant value, the TableRun# function, or an algebraic expression involving these
quantities with operators +, –, *, and /. Index arithmetic is done left to right with no
operator precedence. For example, X[2*3+1] is a valid array variable which EES will
transform into X[7]. X[1+2*3] will be transformed into X[9]. The index variable for the
DUPLICATE command or the sum or product functions can also be used in an expression
for the array index as shown below.
2. Valid index values range between –32760 and +32760, including zero.
3. The right bracket must be the last character in the variable name.
4. The total length of the variable name, including the brackets and the integer value of the
index, must not exceed 30 characters.
EES treats array variables in a very different manner than FORTRAN or Pascal. In EES, each
array variable, such as X[99], is a unique variable name. As such, X[99] appears to EES just
like any other variable such as ZZZ. The guess value and bounds (along with other
information) may be specified for X[99] with the Variable Info command, just as for any other
variable. It is legal (but not good practice) to have EES variables names of X, X[1], X[2,3] all
within the same equation set. The fact that X[99] appears in the Equations window does not
cause EES to reserve memory for 99 elements. Memory is allocated only for the variables
which appear in the equations.
Array variables can be useful in several ways. They provide a means of grouping variables of
similar type. For example, the temperatures at each state in a system can be written as T[1],
T[2], etc. Array variables can be plotted. For example, the temperature and entropy of each
state in a thermodynamic cycle can be overlaid on a T-s property diagram. See Property Plot
in the Plot Menu section of Chapter 3 for details. Finally, array variables can be used with the
DUPLICATE command and the sum and product functions to provide matrix capability and
thereby significantly reduce the amount of typing needed to specify some problems.
146
Chapter 7
The DUPLICATE Command
The DUPLICATE command provides a shorthand way of entering equations into EES. The
equations which are to be duplicated are enclosed between the DUPLICATE and END
command words. DUPLICATE is useful only when used with array variables. For example,
the following statements:
N=5
X[1]=1
DUPLICATE j=2,N
X[j]=X[j–1]+j
END
are equivalent to:
X[1]=1
X[2]=X[1]+2
X[3]=X[2]+3
X[4]=X[3]+4
X[5]=X[4]+5
Note that, within the scope of the DUPLICATE command, the DUPLICATE index variable (j
in the example above) can be used in an algebraic expression for the array index. The
DUPLICATE index is not an EES variable, but rather just a temporary placeholder for the
integers applied in the DUPLICATE command.
The special format requirements pertaining to the DUPLICATE command are as follows:
1. The DUPLICATE command must be on its own line in the Equations window or separated
from other equations with a semicolon.
2. The lower and upper limits specified for the index variable in the DUPLICATE command
must be integers, EES variables previously assigned to constant values, or the TableRun#
function.
3. DUPLICATE commands may be nested within one another as deep as desired. However,
each DUPLICATE command must use a different index variable name and each must be
terminated with an END command. The lower or upper limit of an internal DUPLICATE
may be the index value of an external DUPLICATE, e.g.,
DUPLICATE i=1,5; DUPLICATE j=i,6; X[i,j] = i*j; END; END
4. The END command terminates the last opened DUPLICATE command.
147
Chapter 7
Matrix Capabilities
Many engineering problems can be formulated into a linear system of algebraic equations of the
form
[A] [X] = [B]
where [A] is a square matrix of coefficients, and [X] and [B] are vectors. Ordinarily, the matrix
equation is solved to determine the elements in the vector [X] for known [A] and [B]. In this
case,
-1
[X] = [A] [B]
EES can directly solve the equations represented by [A] [X] = [B] by entering each equation
directly into the Equations window in any format or order. However, a more elegant and
convenient method for solving these equations in EES is to make use of the matrix capability.
EES can solve matrix equations, formulated with array variables, by using the DUPLICATE
command and the sum function. For example, consider the following radiation heat transfer
problem in which [A] and [B] are given below, and the radiosity vector, [X] is to be
14
determined.
10
−1
! −1
−1
A = −1 3.33
"#
−1
#
2 \$#
−1
940584"#
B = 4725
#
! 0 \$#
The equations required in EES to solve this problem are as follows:
The calculated elements in the X array will appear in the Arrays window.
14
Incropera, F.P. and DeWitt, D.P., Fundamentals of Heat and Mass Transfer, 2nd edition, John Wiley and
Sons, 1985, Chapter 13
148
Chapter 7
Note that it was not necessary to determine the inverse of [A] to obtain the solution. EES
calculates the inverse matrix internally, as needed to solve these and any other simultaneous
equations. However, the inverse matrix [A]-1 can be determined by setting the matrix product
[A] [A]-1 equal to the identity matrix in the following manner.
The inverse matrix Ainv will appear in columns of the Arrays window.
The two examples above provide a general procedure for determining the product of a matrix
and a vector or the product of two matrices. Using the DUPLICATE command along with
array variables in EES is no more efficient than the alternative of entering each equation
separately with non-array variable names; however, the matrix capabilities in EES can
significantly reduce the amount of typing required to enter the problem and, more importantly,
make the equations easier to follow.
149
Chapter 7
Using the Property Plot
The Property Plot menu item in the Plot menu generates T-s, T-v, P-v, or P-h diagrams for any of
the fluids in the EES data base. A psychrometric chart is generated if substance AirH2O is
selected. The property plot is placed in one of the EES plot windows.
Additional property data or thermodynamic state point information can be superimposed on the
property plots using the Overlay Plot command in the Plot menu. This is most conveniently done
if array variables are used for the thermodynamic variables. Another benefit of using array
variables is that the state property data then appear in the Arrays Table in a convenient tabular
form.
The P-h plot below shows the state points for simple refrigeration cycle operating between an
evaporator temperature of 10°C and a condensing temperature of 48°C with a compressor
isentropic efficiency of 0.70. The plot was prepared by first producing a P-h plot for R12 with
isotherms at 10°C and 48°C with the Property Plot command and then overlaying the P[i] and
h[i] arrays (plotting from the Arrays table) for the four state points in a refrigeration cycle
analysis. The equations can be found in file REFRIG.EES in the Examples subdirectory.
150
Chapter 7
Integration and Differential Equations
The INTEGRAL function is used to evaluate an integral and in the solution of differential
equations. The format of the Integral function is:
t2
f dt = Integral
(f,t)
t1
There are two basic forms of the integral function which differ in their reliance on the
Parametric table.
Table-based Integral function
The table-based Integral function uses the Parametric table to provide the limits and step size of
the integration variable. The format of this function is Integral(f, t). This form of the Integral
function can be used only in conjunction with the Parametric Table. The integration variable, t,
must be a legal variable name which has values defined in one of the columns of the Parametric
table. The limits on the integration variable, t1 and t2 are the first and last values specified for
variable t. The integrand, f, can be a variable or any implicit or explicit algebraic expression
involving variables, values, and the integration variable, t.
Equation-based Integral function
The Equation-based integral function serves the same purpose as the table-based integral
function but it does not require the use of the Parametric table. The format for the equationbased integral function is
F = INTEGRAL(f, t, t1, t2, tStep)
or
F = INTEGRAL(f, t, t1, t2) {automatic step size}
t1 and t2 are the lower and upper limits of the integration variable. These limits may be
specified with a constant or an EES expression. However, the limits cannot be a function of the
integration variable t or any other variable which changes during the course of the integration.
tStep is the increment EES will use for the integration variable while numerically evaluating
the integral between the specified limits. tStep cannot vary during the course of the integration.
Note that specification of tStep is optional. If tStep is not provided, EES will select the stepsize
using an automatic stepsize adjustment algorithm.
EES uses a second-order predictor-corrector algorithm for evaluating the integral. This
algorithm is designed for solving combined algebraic and differential equations which result
when the integrand is a complex function of other variables. The algorithm is especially suited
to stiff equations.
151
Chapter 7
EES uses the Integral function to solve initial value differential equations. Any first-order
differential equation can be transformed into an appropriate form by integrating both sides. For
example, the differential equation, dy/dx = f(x,y) can be equivalently written as
y = y0 + ∫ f(x,y) dx
where y0 is the initial value of y. This equation can be solved using either the table-based or
equation-based forms of the Integral function. The table-based form would be entered into the
EES Equations Window as
y = y0 + INTEGRAL(fxy, x)
where fxy is an EES variable or expression. To solve the equation it is necessary to create a
Parametric table containing a column for variable x. Values of x are entered into the
Parametric table with the value in first row corresponding to the lower limit of x and the value
in the last row corresponding to the upper limit. The step size is determined by the difference
between the value of x in successive rows and need not be a fixed value. The integral is
evaluated when the Solve Table command is applied.
The equation-based form would appear in the EES Equations window as
y = y0 + INTEGRAL(fxy, x, low, high)
y0 and fxy are as defined above. Low and high are the lower and upper limits for x. A
Parametric table is not required for use with this form of the Integral command. Since the
stepsize is not specified, EES will use automatic stepsize selection.
Solving First-Order Initial Value Differential Equations
Initial-value differential equations can be solved in a number of ways with EES. Chapter 5
describes a Library function included with EES in the USERLIB subdirectory which
implements a 4th order Runge-Kutta algorithm. This method can only be used if the derivative
can be expressed explicitly as a function of the dependent and independent variables. This
section demonstrates two ways of solving simultaneous algebraic and differential equations
using the Integral function or the TableValue function in conjunction with the Parametric
Table.
Method 1: Solving Differential Equations with the Integral Function
Consider the problem of determining the time–temperature history of a sphere of radius r=5
mm, initially at a uniform temperature of 400°C. The sphere is exposed to 20°C air with a
convection coefficient of h=10 W/m2-K. The thermophysical properties of the sphere material
are:
152
Chapter 7
ρ = density = 3000 kg/m3
k = thermal conductivity = 20 W/m-K
c = specific heat = 1000 J/kg-K
Calculation of the Biot number will indicate that the sphere can be treated as a lumped system,
and therefore it may be assumed to be at a uniform temperature at any instant of time.15 The
relation between the sphere temperature and time is given by an energy balance on the sphere,
which results in the following differential equation
– h A (T – T∞) = ρ c V d T
dt
where
h
is the convective heat transfer coefficient
T
is the uniform temperature of the sphere at any time
T∞
is the temperature of the air stream = 20°C
A
is the surface area of the sphere = 4 π r2
V
is the volume of the sphere = 4/3 π r3
t
is time
This differential equation has the following analytic solution which can be used to check the
accuracy of the numerical solution provided by EES.
T – T∞
= exp – h A t
ρcV
Ti – T∞
To solve the differential equation numerically in EES, enter the following equations.
Next, a Parametric Table is generated with the New Table command in the Parametrics menu.
(Note that, if the start and stop times were provided as the third and fourth parameters for the
15
Incropera, F.P. and DeWitt, D.P., Fundamentals of Heat and Mass Transfer, 2nd edition, John Wiley and
Sons, 1985, Chapter 5
153
Chapter 7
Integral function, it would not be necessary to use the Parametric table. However,
intermediate results for times between the start and stop times would not be available for
plotting.)
Select T, Time, and Texact as the three variables to include in the table. Enter 11 runs which
will allow a time–temperature history for 1000 seconds starting at 0 with 100 second intervals.
The New Table dialog window should now appear as shown below.
Click the OK button. It is next necessary to enter the values of Time in the table for which the
temperature T is to be calculated. A timestep of 100 seconds has been chosen. With a fixed
timestep, the values of Time can be most easily entered by clicking on the
control at the
upper right of the Time column header cell. Enter 0 for the First Value. Set the drop-down list
control to ‘Last Value’ and enter 1000 for the last value as shown.
The value of Time from 0 to 1000 will be automatically entered into the table when you click
the OK button and displayed in normal type. Now select Solve Table from the Calculate menu to
calculate the numerical and analytical values of temperature corresponding to each value of
Time in the table. When the calculations are completed, the Parametric Table window will
display the solutions. Calculated values are shown in bold. (The format of calculated values in
154
Chapter 7
the Parametric Table can be set using the Preferences command in the Options menu.) The plot
shows that the numerically determined temperature agrees closely with the exact analytic
solution.
155
Chapter 7
Method 2: Solving Differential Equations with the TableValue Function
In this section, we solve the same first-order differential equation described in Method 1.
– h A (T – T∞) = ρ c V d T
dt
The differential is approximated as
new
old
d T∼T – T
dt
∆t
T new is the temperature at the current time which is to be calculated. T old is the temperature at
the previous time which can be found in the previous row of the Parametric table using the
TableValue function. The TableValue function returns the value in the Parametric Table at a
specified row and column, as described in Chapter 4. With this function, it is possible to access
the values of variables calculated in previous runs during the Solve Table calculations. ∆t is
the timestep which, in the equations shown below, is the difference of the current and previous
values of the variable Time.
Both an explicit method (Euler’s method) and an implicit method (Crank-Nicolson) are used to
solve this first-order differential equation and compared with the exact solution. In the Euler
method, only previous temperatures are used to evaluate the right-hand side of the differential
equation. In the Crank-Nicolson method, the average of the previous and current temperatures
is used. The Crank-Nicolson method is implicit because the current temperature is not as yet
determined. The implicit method is no more difficult to implement since EES is designed to
solve implicit equations. Shown below is a listing of all of the equations needed to solve this
problem.
Most of the equations are identical to those used for Method 1. T_Euler is the temperature
calculated by Euler’s method. T_CN is the temperature calculated by the Crank-Nicolson
method. (In the Formatted Equations and Solution windows, these variables will display as
TEuler and TCN, respectively.) To proceed, a Parametric table must be defined, as in Method 1.
The values of T_Euler, T_CN and T_exact in the first row of the table, corresponding to
Time=0, are the initial conditions and their values (400°C) must be entered. Then, the Solve
Table command is used to complete the table, with calculations starting at Run 2. Note the use
of the TableRun# function to return the row number in the Parametric Table for which
calculations are currently being done, and the TableValue function which returns the value at a
specified row and column in the Parametric Table.
156
Chapter 7
Shown below is the completed table with the numerical and analytical solutions. Calculated
values are shown in italics. It is evident that Euler’s method does not provide as accurate a
solution as that obtained with the Integral function (Method 1) or the Crank-Nicolson method.
Improved accuracy could be obtained by reducing the time step, but this would require
additional computational effort and storage space (which is not significant here).
Solving Second and Higher Order Differential Equations
Higher order differential equations can also be solved by repeated use of the Integral function.
Shown below is an EES program which solves a second-order differential equation to calculate
the velocity and position of a freely falling object, subject to aerodynamic drag. The Solution
Window appears after the Solve command (F2) is issued.
157
Chapter 7
Multiple-Variable Integration
Multiple integration is provided by nesting calls to the Integral function. Up to six levels can
be nested. The following example performs a numerical double integration.
158
Chapter 7
Combined Use of the Table and Equation-Based Integral Functions
An advantage of the table-based form of the Integral function over the Equation-based form is
that intermediate values are provided in the Parametric table so that the trajectory of the
integrated quantity may be observed and plotted. A possible disadvantage is that the integrated
variable must appear in the table for each step of the integration. It is possible to combine the
equation-based form of the Integral function with the Parametric table so that only some of the
intermediate results are reported in the Parametric table, as demonstrated in the following
example.
159
160
Appendix A
_____________________________________________________________________________
Hints for Using EES
_____________________________________________________________________________
1. The Variable Information command in the Options menu produces an alphabetical list of all
variables appearing in the Equations window. Check this list to make sure that you have not
misspelled a variable name.
2. The Residuals window provides an indication of the accuracy in which each non-trivial
equation in the Equations window was solved and the order in which the equations are
solved. An examination of the residuals indicates which equations were not solved when
EES indicates that a solution could not be found.
3. If your equations do not converge, it may be that the guess values are poor. In this case, the
problem can often be solved by entering equations which set guess values for one or more of
the unknown variables and modifying the equations as needed to ensure an equal number of
variables and equations. If a solution is then obtained, use Update Guesses in the Calculate
menu to set the guess values of all variables to their current values. Then return the
Equations window to its original form and solve again.
4. If EES is unable to solve your set of nonlinear equations, try exchanging some independent
and dependent variables to produce an equation set which is easier to solve. For example,
EES may not be able to solve the following heat exchanger equations to determine NTU
with the default guess values and bounds.
Eff=.9
Cmax = 432
Cmin = 251
eff = (1 - exp(-NTU * (1 - (Cmin/Cmax)))) / (1 - (Cmin/Cmax) * exp(-NTU *
(1 - (Cmin/Cmax))))
However, the equations would be easily solved if the value of NTU were specified in place
of Eff.
NTU = 5
Cmax = 432
Cmin = 251
eff = (1 - exp(-NTU * (1 - (Cmin/Cmax)))) / (1 - (Cmin/Cmax) * exp(-NTU *
(1 - (Cmin/Cmax))))
A few trials will indicate that NTU must be between 3 and 5 for Eff = 0.9. Setting a guess
value for NTU of 4 allows EES to quickly determine the final value of 3.729.
161
Appendix A
Hints for Using EES
5. A sure way to solve difficult problems with EES is to add an additional variable so that the
problem has one more degree of freedom. Then, use the Parametric Table to vary the values
of one of the implicit variables in order to find the solution in which the additional variable
has a value of zero. For example, consider the following radiation calculation in which the
value of T is to be determined. The first three equations must be solved simultaneously and
they are non-linear because T is raised to the fourth power. EES may have trouble
determining the solution, depending upon the guess values.
QL = AL*Sigma*(T^4 - TL^4)
QB = AH*Sigma*(TH^4 - T^4)
QL = QB
Sigma=0.1718E-8
AL=.5; AH=1; TL=300; TH=1000
Alternatively, add a variable, Delta, such that
QL = AL*Sigma*(T^4 - TL^4)
QB = AH*Sigma*(TH^4 - T^4)+Delta
QL = QB
Sigma=0.1718E-8
AL=.5; AH=1; TL=300; TH=1000
Now, set up a Parametric Table containing variables T and Delta. Use the Alter Values
command to set a range of values of T and Solve Table to calculate the corresponding values
of Delta. The value(s) of T for which Delta is zero constitute a solution to the equation set.
The New Plot Window command is handy for visualizing the relationship between T and
Delta. If the values of Delta do not cross zero, there is no solution to the equation set for the
range of T values investigated. This is perhaps the most useful method of solving a difficult
set of non-linear equations.
6. The Store button in the Default Info dialog can be particularly convenient if you have a
customary set of nomenclature for your variable names. For example, if variables beginning
with letter T often designate temperatures, set the bounds, display format and units for letter
T and then Store the default information. You can later use the Load button to restore this
set of default variable information. EES will then always set this information for you in
following problems.
7. The arrow keys help some users move about more quickly in the Equations, Parametric and
Lookup Tables. In the Equations window, the up and down arrows move the cursor up and
down one line; left and right arrow move the cursor left and right one character. Home and
End move to the start and end of the current line. In the tables, the arrow keys move to the
next cell in the direction of the arrow. The Return and Tab keys produce the same effects as
the down arrow and right arrow keys, respectively.
162
Hints for Using EES
Appendix A
8. Use the Tab key in the Equations window to set off the equations for improved readability.
9. Except for the Steam_NBS substance, the EES property correlations are not specifically
applicable in the compressed (sub-cooled) liquid range. Instead, it is assumed that
subcooled liquid is incompressible and properties are taken to be those for saturated liquid.
Thus, in the subcooled region, v(T,P)=v(T,Psat), u(T,P)=u(T,Psat) and s(T,P)=s(T,Psat). To
calculate the ideal work of a pump, for example, recall that h2-h1 = -Wpump = ∫v dP =
v*(P2-P1), since for an incompressible substance, v is independent of P.
10. The Arrays window can be quite useful for organizing the property information in a
thermodynamics problem having multiple states. Use array variables, such as T[1], P[1],
and h[1] (rather than T1, P1, and h1) for the properties at each state. The state properties
will appear in a neat table in the Arrays window, rather than jumbled together in the
Solutions window. Be sure the Use Arrays window option in the Display Options dialog has
been selected.
11. Considerable effort has been expended to design EES so that it does not unexpectedly quit
under any circumstances. However, this may still happen. In this case, EES will save your
work in a file called EESERROR before it terminates. You can restart EES and load the
EESERROR file so none of your work is lost.
12. Use the \$INCLUDE directive to load commonly used constants, unit conversions, or other
equations into the Equations window. You won’t see them, but they’re there, available for
use. You can also load library files with the \$INCLUDE directive.
13. If you write an EES Library Function which calls any of the built-in thermodynamic or
trigonometric functions, use the UnitSystem command to determine the current unit system
settings. Then you can use If Then Else statements to ensure that the arguments provided to
the thermodynamic or trigonometric functions have the correct values.
14. The background color option for columns in the Parametric Table is useful if you are
preparing an EES program in which others will be entering data into the columns, as in a
spreadsheet. Set the background color for the columns in which data are to be entered, to
distinquish them from the columns in which calculated results will appear.
15. If you are working in Complex Mode, use the \$COMPLEX ON directive at the top of the
Equations window. Its more convenient than changing the Complex Mode setting in the
Preferences dialog.
163
164
Appendix B
_____________________________________________________________________________
Numerical Methods Used in EES
_____________________________________________________________________________
EES uses a variant of Newton’s method [1-4] to solve systems of non-linear algebraic equations.
The Jacobian matrix needed in Newton’s method is evaluated numerically at each iteration.
Sparse matrix techniques [5-7] are employed to improve calculation efficiency and permit rather
large problems to be solved in the limited memory of a microcomputer. The efficiency and
convergence properties of the solution method are further improved by the step-size alteration
and implementation of the Tarjan [8] blocking algorithm which breaks the problems into a
number of smaller problems which are easier to solve. Several algorithms are implemented to
determine the minimum or maximum value of a specified variable [9,10]. Presented below is a
summary of these methods, intended to provide users with a better understanding of the
processes EES uses in obtaining its solutions.
Solution to Algebraic Equations
Consider the following equation in one unknown:
x3 - 3.5 x2 + 2 x = 10
To apply Newton’s method to the solution of this equation, it is best to rewrite the equation in
terms of a residual, ε, where
ε = x3 - 3.5 x2 + 2 x - 10
The function described by this equation is shown in Figure 1. There is only one real solution
(i.e., value of x for which ε = 0) in the range illustrated at x = 3.69193.
20
Improved
guess
10
Initial guess
ε
0
-10
-20
-2
-1
0
1
2
3
4
x
Figure 1: Residual of x3 - 3.5 x2 + 2 x = 10 as a function of x
165
5
Appendix B
Numerical Methods used in EES
Newton’s method requires an estimate of the total derivative of the residual, J. For this
equation, the derivative is:
J = dε = 3 x2 – 7 x + 2
dx
To solve the equation, Newton’s method proceeds as follows:
1. An initial guess is made for x (e.g., 3).
2. The value of ε is evaluated using the guess value for x. With x = 3, ε = -8.5.
3. The derivative J is evaluated. With x = 3, J = 8.
4. The change to the guess value for x, i.e., ∆x, is calculated by solving J ∆x = ε. In this
example, ∆x is -1.0625.
5. A (usually) better value for x is then obtained as x - ∆x. In the example, the improved
value for x is 4.0625 (which results in ε = 7.4084).
Steps 2 to 5 are repeated until the absolute value of ε or ∆x becomes smaller than the specified
tolerances in the Stop Criteria dialog. The method, when it converges, converges quite quickly.
However, a bad initial guess can cause the method to diverge or to converge quite slowly. Try,
for example, an initial guess of 2 and see what happens.
Newton’s method can be extended for solving simultaneous non-linear equations. In this case,
the concept of "derivative" generalizes into the concept of "Jacobian matrix." Consider the
following two simultaneous equations in two unknowns:
x21 + x22 – 18 = 0
x1 - x2 = 0
The equations can be rewritten in terms of residuals ε1 and ε2:
ε1 = x21 + x22 – 18 = 0
ε2 = x 1 - x 2 = 0
The Jacobian for this matrix is a 2 by 2 matrix. The first row contains the derivatives of the first
equation with respect to each variable. In the example above, the derivative of ε1 with respect
to x2 is 2 x2. The Jacobian matrix for this example is:
J=
2⋅x1 2⋅x2
1 –1
Newton’s method as stated above applies to both linear and nonlinear sets of equations. If the
equations are linear, convergence is assured in one iteration, even if a "wrong" initial guess was
made. Non-linear equations require iterative calculations. Consider the following initial guess:
x= 2
2
166
Numerical Methods used in EES
Appendix B
ε = – 10
0
The values of ε and J for this initial guess are:
J=
4 4
1 –1
Improved values for the x vector are obtained by solving the following matrix problem
involving the Jacobian and the residual vector.
4 4
1 –1
∆x1
= –10
∆x2
0
Solving this linear equation results in:
∆x1
= – 1.25
– 1.25
∆x2
Improved estimates of the x1 and x2 are obtained by subtracting ∆x1 and ∆x2, respectively, from
the guess values.
x1
3.25
x2 = 3.25
The correct solution to this problem is x1 = x2 = 3.0. The calculated values of x1 and x2 are
closer to the correct solution than were the guess values. The calculations are now repeated
using the most recently calculated values of x1 and x2 as the guess values. This process is
repeated until convergence is obtained.
The Jacobian matrix plays a key role in the solution of algebraic equations. The Jacobian matrix
can be obtained symbolically or numerically. Symbolic evaluation of the Jacobian is more
accurate, but requires more processing. Accuracy of the Jacobian, however, does not necessarily
lead to more accuracy in the solution, only to (sometimes) fewer iterations. EES evaluates the
Jacobian numerically. Because EES does all calculations with 96 bit precision (about 20
decimal places), numeric evaluation of the Jacobian rarely results in convergence problems from
loss of precision.
In most equation sets, many of the elements of the Jacobian matrix are zero. A matrix with
many zero elements is called a sparse matrix. Special ordering and processing techniques make
handling of sparse matrices quite efficient. In fact, without sparse matrix techniques the number
of simultaneous equations which could be solved by EES would be substantially less than 6000,
the current limit implemented in EES. Further references on sparsity and on how to handle
sparse matrices are available in [5, 6]. A collection of routines that are designed to handle very
large sparse matrices are described in [7].
Newton’s method does not always work, particularly if a "bad" initial guess for the x vector is
supplied. The solution obtained after applying the correction ∆x to the previous x vector should
be more correct (i.e., result in a smaller maximum residual) than the solution obtained before the
167
Appendix B
Numerical Methods used in EES
correction. EES always checks for this condition. If this is not true, EES will halve the step ∆x
and evaluate the residuals again. If this does not improve the solution, the step is halved again
(up to 20 times). If the resultant solution is still not better than the solution prior to the
correction, EES will reevaluate the Jacobian and try again until one of the stopping criteria
forces the calculations to stop. Step halving is very helpful when a bad initial guess is provided.
Figure 2 illustrates the process for the solution of the single equation in the first example,
starting from a guess of x=2.5. In this case, step halving works quite well.
60
40
Initial step
ε
20
Initial guess
Reduced step
0
Half
step
Full
step
-20
0
1
2
3
4
5
6
x
Figure 2: Use step-halving for improving the convergence
Blocking Equation Sets
Even though you may have what looks like a set of simultaneous equations, it is often possible
to solve these equations in groups (sometimes one at the time) rather than all together as one set.
Solving equations in groups makes Newton’s method work more reliably. For this reason, EES
organizes the equations into groups (or blocks) before solving. Consider, for example, the
following set of equations:
x1 + 2 x2 + 3 x3 = 11
5 x3 = 10
3 x2 + 2 x3 = 7
168
Numerical Methods used in EES
Appendix B
These equations can be solved as one simultaneous set. However, they can be more easily
solved if they are reordered and blocked. It is better to re-order first. EES automatically
recognizes that equation 2 can be solved directly for x3. Once this is done, equation 3 can be
solved for x2. Finally, equation 1 can be solved for x1. This results in three blocks of equations,
each with one equation and one variable which are directly solved. Because the equations in
this example are linear and they can be totally uncoupled, the process looks trivial. Things can
get a little more interesting if the blocks are a little less obvious. Consider the following
example with 8 linear equations in 8 unknowns:
+ x8
x3
x5
- x6
- x6
+ x4
x1
x7
- x7
x2
x3
+ x8
+ x8
- x5
x4
x1
+ x6
+ x7
= 11
=7
= -8
= -1
= 10
= 6
= 4
= 14
These equations and variables can be re-numbered and blocked. Each block is solved in turn.
In the case above, blocking allows the equations to be solved in 6 blocks as follows:
Block 1: Equation 7
x4 = 4
Block 2: Equation 2
x7 = 7
Block 3: Equations 4 and 8
x1 + x4 - x6 = -1
x1 + x6 + x7 = 14
Block 4: Equation 3
x5 - x6 - x7
= -8
From here
and:
x1 = 1
x6 = 6
From here:
x5 = 5
Block 5: Equations 1 and 6
x3 + x8 = 11
x3 - x5 + x8 = 6
From here:
and:
x3 = 3
x8 = 8
Block 6: Equation 5:
x2 + x8 = 10
From here:
x2 = 2
The first two blocks contain a single equation with a single variable. These blocks simply
define constants. EES will recognize that equations that depend from the start on a single
variable are in reality parameter or constant definitions. These parameters are determined before
any solution of the remaining equations takes place. No lower and upper limits on the guesses
169
Appendix B
Numerical Methods used in EES
are needed for parameters, since the values of these parameters are determined immediately.
The solution of the remaining equations is now very simple, although it did not appear trivial at
the beginning of the process.
Grouping of equations is useful when the equations are linear, but it is not essential. When the
equations are nonlinear, grouping of equations is nearly indispensable, otherwise later groups of
equations begin iterating with totally incorrect values of earlier variables. The result is often
divergence. EES is able to recognize groups of equations prior to solution by inspecting the
Jacobian matrix using the Tarjan [8] algorithm. See reference [6] for more details on this
algorithm.
Determination of Minimum or Maximum Values
EES has the capability to find the minimum or maximum (i.e., optimum) value of a variable
when there is one to ten degrees of freedom (i.e., the number of variables minus the number of
equations). For problems with a single degree of freedom, EES can use either of two basic
algorithms to find a minimum or maximum: a recursive quadratic approximation known as
Brent’s method or a Golden Section search [9]. The user specifies the method, the variable to be
optimized and an independent variable whose value will be manipulated between specified
lower and upper bounds. When there are two or more degrees of freedom, EES uses Brent’s
method repeatedly to determine the minimum or maximum along a particular direction. The
direction is determined by a direct search algorithm known as Powell’s method or by the
The recursive quadratic approximation algorithm proceeds by determining the value of the
variable which is to be optimized for three different values of the independent variable. A
quadratic function is fit through these three points. Then the quadratic function is differentiated
analytically to locate an estimate of the extremum point. If the relationship between the variable
which is being optimized and the independent variable is truly quadratic, the optimum is found
directly. If this is not the case, the algorithm will use the newly obtained estimate of the
optimum point and two (of the three) points which are closest to it to repeat the quadratic fit.
This process is continued until the convergence criteria set for the minimization/maximization
process are satisfied.
170
Numerical Methods used in EES
Appendix B
The Golden Section search method is a region-elimination method in which the lower and upper
bounds for the independent variable specified by the user are moved closer to each other with
each iteration. The region between the bounds is broken into two sections, as shown in Figure
3. The value of the dependent variable is determined in each section. The bounds for the
section which contains the smaller (for minimization) or larger (for maximization) dependent
variable replace the interval bounds for the next iteration. Each iteration reduces the distance
between the two bounds by a factor of (1-τ) where τ =0.61803 is known as the Golden Section
ratio.
S ecti o n 1
τ
(1 − τ )
S ect io n 2
Figure 3: Region Elimination using the Golden Section Method
Numerical Integration
EES integrates functions and solves differential equations using a variant of the trapezoid rule
along with a predictor-corrector algorithm. In explaining this method, it is helpful to compare
the numerical scheme with the manner one would use to graphically determine the value of an
integral.
Consider the problem of graphically estimating the integral of the function
f = 5 -5 X + 10 X2
for X between 0 and 1. In graphical integration, a plot of f versus X would be prepared. The
abscissa of the plot would be divided into a number of sections as shown below. The area under
the curve in each section is estimated as the area of a rectangle with its base equal to the width
of the section and its height equal to the average ordinate value in the section. For example, the
ordinate values at 0 and 0.2 in the plot below are 5 and 4.4, respectively. The area of the first
section is then 0.2 * (5+4.4)/2 or 0.94. The estimate integral value between 0 and 1 is the sum
of the areas of the 5 sections. The accuracy of this method improves as the number of sections
is increased.
171
Appendix B
Numerical Methods used in EES
10.0
f = 5.5 X + 10 X2
8.0
6.0
4.0
2.0
0.0
0. 0
0. 2
0. 4
0. 6
0. 8
1. 0
X
Figure 4: Numerical approximation of an integral
Integration in EES takes place in a manner quite analogous to the graphical representation. The
abscissa variable, X, in the example above, is placed in the Parametric Table. The values of X
entered into the table identify the width of each section. EES does not require each section to
have the same width. The numerical value of the function, f, which is to be integrated, is
evaluated at each value of X and supplied to EES through the Integrate function, e.g.,
Integral(f,X).
In some situations, such as in the solution of differential equations, the value of f may not be
explicitly known at a particular value of X. The value of f may depend upon the solution to
non-linear algebraic equations which have not yet converged. Further, the value of f may
depend upon the value of the integral up to that point. In this case, iteration is needed. EES will
repeatedly evaluate the section area using the latest estimate of f at the current value of X until
convergence is obtained. The procedure in which the estimate of the integral made on the first
calculation is corrected with later information is referred to as a predictor-corrector algorithm.
172
Numerical Methods used in EES
Appendix B
References
1. A. W. Al-Khafaji and J. R. Tooley, Numerical Methods in Engineering Practice, Holt,
Rinehart and Winston, 1986, pp. 190 & ff.
2. C. F. Gerald and P. O. Wheatley, Applied Numerical Analysis, Addison-Wesley 1984, pp.
135 & ff.
3. J. H. Ferziger, Numerical Methods for Engineering Application, Wiley-Interscience 1981,
Appendix B.
4. F. S. Acton, Numerical Methods that Usually Work, Harper and Row 1970.
5. I. S. Duff, A. M. Erisman and J. K. Reid, Direct Methods for Sparse Matrices, 1986 Oxford
Science Publications, Clarendon Press.
6. S. Pissanetsky, "Sparse Matrix Technology," Academic Press 1984.
7. F. L. Alvarado, "The Sparse Matrix Manipulation System," Report ECE-89-1, Department
of Electrical and Computer Engineering, The University of Wisconsin, Madison, Wisconsin,
January 1989.
8. Tarjan, R. "Depth-First Search and Linear Graph Algorithms," SIAM J. Comput. 1, 146160, (1972)
9. Powell’s Method of Successive Quadratic Approximations, Reklaitis, Ravindran and
Radsdell, Engineering Optimization, John Wiley, New York, (1983)
10. W. H. Press, B. P. Flannery and S. A. Teukolsky, and Vetterling, W.T., Numerical Recipes
in Pascal, Cambridge University Press, Chapter 10, (1989)
173
174
Appendix C
_____________________________________________________________________________
_____________________________________________________________________________
Background Information
EES uses an equation of state approach rather than internal tabular data to calculate the
properties of fluids. For some substances and conditions, the ideal gas law is applicable. EES
employs a naming convention to distinguish ideal gas and real fluid substances. Substances
which are represented by their chemical symbol (e.g., N2) are modeled with the ideal gas law
whereas substances for which the name is spelled out (e.g., Nitrogen) are considered to be real
fluids. (Air and AirH2O are exceptions to this naming convention.)
Ideal gas substances rely on JANAF table data [Stull, 1971] to provide the enthalpy of formation
and absolute entropy at a reference state of 298 K, 1 atm. Specific heat correlations for these
gases and the ideal gas law are used to calculate the thermodynamic properties at conditions
other than the reference state. A number of ideal gas substances are built into EES. The
external JANAF program provides thermodynamic property information for hundreds of
USERLIB folder, as explained below.
The Martin-Hou [1955] equation of state (or variations of it) is used for all real fluids except
water. (Several equations of state are provided for water, the most accurate and computationally
intensive being the equation of state published by Harr, Gallagher, and Kell [1984]. Ice
properties rely upon correlations developed by Hyland and Wexler [1983].) Thermodynamic
property relations are used to determine enthalpy, internal energy and entropy values based upon
the equation of state and additional correlations for liquid density, vapor pressure, and zeropressure specific heat as a function of temperature. A modification to the Martin-Hou equation
of state proposed by Bivens et al. [1996] allows this equation equation of state to be applied for
mixtures, such as the R400 refrigerant blends.
Viscosity and thermal conductivity of liquids and low-pressure gases are correlated as
polynomials in temperature. Temperature alone determines the transport properties for ideal
gases. For real fluids, the effect of pressure on the gas transport properties is estimated using
correlations from Reid et al. [1977].
175
Appendix C
EES has been designed to allow additional (ideal and real) fluids to be added to the property
data base. The user must supply the necessary parameters for the thermodynamic and transport
property correlations. The parameters are placed in an ASCII text file which must be located in
the EES\USERLIB subdirectory. EES will load all fluid files found in the EES\USERLIB
subdirectory at startup. The additional fluids will appear in every way identical to the built-in
fluids. The following sections describe the format required for the property data files.
Ideal Gas files
Ideal gas files must have a .IDG filename extension. An equation of state is not needed since it
is assumed that the fluid obeys the ideal gas equation of state. However, particular attention
must be paid to the reference states if the gas is to be used in calculations involving chemical
reactions. The enthalpy of formation and Third-law entropy values at 298 K and 1 bar (or 1
atm) must be supplied. An example file providing the parameters for CO2 is provided below.
The properties of ideal gas fluid can be entered by adapting the file format to the new fluid.
SAMPLE TESTCO2.IDG File
TestCO2
44.01
100.0
250
1500
-3.7357 0
30.529 0.5
-4.1034 1.0
0.02420 2.0
0 0
0 0
0 0
0 0
0 0
0 0
298.15
100
-393520
213.685
0
0
200
in K}
1000
in K}
-8.09519E-7
6.039533E-8
-2.8249E-11
9.84378E-15
{Molar mass of fluid
{Tn Normalizing value in K}
{Lower temperature limit of Cp correlation in K}
{Upper temperature limit of Cp correlation in K}
{a0, b0 Cp=sum(a[i]*(T/Tn)^b[i], i=0,9 in kJ/kgmole-K}
{a1, b1}
{a2, b2}
{a3, b3}
{a4, b4}
{a5, b5}
{a6, b6}
{a7, b7}
{a8, b8}
{a9, b9}
{TRef in K}
{Pref in kPa}
{hform - enthalpy of formation in kJ/kgmole at TRef}
{s0 - Third law entropy in kJ/kgmole-K at Tref and PRef}
{reserved - set to 0}
{reserved - set to 0}
{Lower temperature limit of gas phase viscosity correlation
{Upper temperature limit of gas phase viscosity correlation
{v0 Viscosity = sum(v[i]*T^(i-1)) for i=0 to 5 in Pa/m^2}
{v1}
{v2}
{v3}
176
Appendix C
-1.4732E-18 {v4}
0
{v5}
200
{Lower temperature limit of gas phase thermal conductivity
correlation in K}
1000 {Upper temperature limit of gas phase thermal conductivity
correlation in K}
-1.1582E-3 {t0 Thermal Conductivity = sum(t[i]*T^(i-1)) for i=0 to 5
in W/m-K}
3.9174E-5
{t1}
8.2396E-8
{t2}
-5.3105E-11 {t3}
3.1368E-16 {t4}
0
{t5}
0
{Terminator - set to 0}
Real Fluid Files
A pure real fluid is identified with a .MHE (for Martin-Hou Equation) filename extension. A
sample file named XFLUID.MHE is listed on the following pages illustrating the required file
format. (The sample file contains the parameters used for n-butane.) The file consists of 75
lines. The first line provides the name of the fluid which EES will recognize in the property
function statements. For example, the first line in the sample file contains UserFluid. The
enthalpy for this substance would then be obtained as follows.
h=Enthalpy(UserFluid,T=T1, P=P1)
The fluid name will appear in alphabetical order with other fluid names in the Function
Information dialog window. The following 74 lines each contain one number. A comment
follows on the same line (after one or more spaces) to identify the number. The forms of all of
the correlations except the pressure-volume-temperature relation are indicated in the
XFLUID.MHE file. Pressure, volume and temperature are related by the Martin-Hou equation
of state in the following form. A method for obtaining the coefficients is described by Martin
and Hou, [1955].
Martin-Hou Equation of State (parameters in lines 18-36)
A + B2T + C2e– βT/Tc A3 + B3T + C3e– βT/Tc
P= RT + 2
+
2
3
v–b
v–b
v–b
A + B4T + C4e– βT/Tc A5 + B5T + C5e– βT/Tc A6 + B6T + C6e– βT/Tc
+ 4
+
+
4
5
eαv(1 + C′eαv)
v–b
v–b
where
P [=] psia, T [=] R, and v [=] ft3/lbm
You may need to curve fit tabular property data or data obtained form a correlation in a different
form to obtain the appropriate parameters. Most of the correlations are linear with respect to the
parameters so that they can be determined by linear regression. A parameter set which improves
177
Appendix C
upon the fit resulting from the Martin and Hou method can be determined by non-linear
regression. EES can be used to do these regressions.
SAMPLE XFLUID.MHE File for pure fluids
UserFluid
58.1
0
12.84149
{ molecular weight}
{ not used}
{ a} Liquid
Density=a+b*Tz^(1/3)+c*Tz^(2/3)+d*Tz+e*Tz^(4/3)+f*sqrt(Tz)+g*(Tz)^2}
33.02582
{ b}
where Tz=(1-T/Tc) and Liquid
Density[=]lbm/ft3
-2.53317
{ c}
-0.07982
{ d}
9.89109
{ e}
0
{ f}
0
{ g}
-6481.15338
{ a}
Vapor pressure fit: lnP=a/T+b+cT+d(1T/Tc)^1.5+eT^2
15.31880
{ b}
where T[=]R and P[=]psia
-0.0006874
{ c}
4.28739
{ d}
0
{ e}
0
{ not used}
0.184697
{ Gas constant in psia-ft3/lbm-R}
1.5259e-2
{ b}
Constants for Martin-Hou EOS/English_units
-20.589
{ A2}
9.6163e-3
{ B2}
-314.538
{ C2}
0.935527
{ A3}
-3.4550e-4
{ B3}
19.0974
{ C3}
-1.9478e-2
{ A4}
0
{ B4}
0
{ C4}
0
{ A5}
2.9368e-7
{ B5}
-5.1463e-3
{ C5}
0
{ A6}
0
{ B6}
0
{ C6}
5.475
{ Beta}
0
{ alpha}
0
{ C’}
-7.39053E-3
{ a}
Cv(0 pressure) = a + b T + c T^2 + d T^3 +
e/T^2
6.4925e-4
{ b}
where T[=]R and Cv[=]Btu/lb-R
9.0466e-8
{ c}
-1.1273e-10
{ d}
178
5.2005e3
124.19551
0.0956305
550.6
765.3
0.07064
0
0
2
260
535
-3.790619e6
5.42356586e4
-7.09216279e1
5.33070354e-2
115
235
2.79677345e3
-2.05162697e1
5.3698529e-2
-4.88512807e-5
2
change}
250
535
7.5931e-3
-6.3846e-5
3.95367e-7
-2.9508e-10
115
235
2.776919161e-1
-8.45278149e-4
1.57860101e-6
-1.8381151e-9
0
Appendix C
{
{
{
{
{
{
{
{
{
{
{
{
{
{
{
{
{
{
{
{
{
{
e}
href offset}
sref offset}
Pc [=] psia}
Tc [=] R}
vc [=] ft3/lbm}
not used}
not used}
Viscosity correlation type: set to 2: do not change}
Lower limit of gas viscosity correlation in K}
Upper limit of gas viscosity correlation in K}
A}
GasViscosity*1E12=A+B*T+C*T^2+D*T^3
B}
where T[=]K and GasViscosity[=]N-s/m2
C}
D}
Lower limit of liquid viscosity correlation in K}
Upper limit of liquid viscosity correlation in K}
A}
Liquid Viscosity*1E6=A+B*T+C*T^2+D*T^3
B}
where T[=]K and Liquid Viscosity[=]N-s/m2
C}
D}
Conductivity correlation type: set to 2: do not
{
{
{
{
{
{
{
{
{
{
{
{
{
Lower limit of gas conductivity correlation in K}
Upper limit of gas conductivity correlation in K}
A}
GasConductivity=A+B*T+C*T^2+D*T^3
B}
where T[=]K and GasConductivity[=]W/m-K
C}
D}
Lower limit of liquid conductivity correlation in K}
Upper limit of liquid conductivity correlation in K}
A}
LiquidConductivity=A+B*T+C*T^2+D*T^3
B}
where T[=]K and LiquidConductivity[=]W/m-K
C}
D}
not used: terminator}
Fluid Properties for Blends
The Martin-Hou equation of state can be adapted for mixtures as proposed by Bivens et. al. The
major modifications needed to make this pure component equation of state applicable to blends
is to provide separate correlations for the bubble and dew point vapor pressures and a
correlation for the enthalpy of vaporization, since the equation of state can not provide this
information. Shown below is a listing of the R410A.MHE file that is used to provide property
data for R410A, along with an explanation of each line in the file.
R410A
179
Appendix C
72.584
{molecular weight Bivens and Yokozeki}
400
{Indicator for blend}
30.5148
{a} Liquid density = a+b*Tz^(1/3)+c*Tz^(2/3)+d*Tz
60.5637
{b}
+e*Tz^(4/3)+f*sqrt(Tz)+g*(Tz)^2}
-5.39377
{c} where Tz=(1-T/Tc) and Liquid Density[=]lbm/ft3
55.5360815
{d}
-21.88425
{e}
0
{f}
0
{g}
-5.9789E+03 -5.9940E+03 {a} Bubble and Dew Pt Vapor pressure fit:
24.06932 24.04507
{b}
lnP=a/T+b+cT+d(1-T/Tc)^1.5+eT^2
-2.1192E-02 -2.1084E-02 {c}
where T[=]R and P[=]psia fit
-5.5841E-01 -4.4382E-01 {d}
1.3718E-05 1.3668E-05
{e}
0 0
{not used}
0.1478
{Gas constant in psia-ft3/lbm-R}
0.006976
{b} Constants for Martin-Hou EOS/English_units from Bivens
-6.40764E+00
{A2}
3.40372E-03
{B2}
-2.34220E+02
{C2}
1.41972E-01
{A3}
4.84456E-06
{B3}
9.13546E+00
{C3}
-4.13400E-03
{A4}
0
{B4}
0
{C4}
-9.54645E-05
{A5}
1.17310E-07
{B5}
2.45370E-02
{C5}
0
{A6}
0
{B6}
0
{C6}
5.75
{Beta}
0
{alpha}
0
{C’}
0.036582
{a}
Cv(0 pressure) = a + b T + c T^2 + d T^3 + e/T^2
2.808787E-4
{b}
where T[=]R and Cv[=]Btu/lb-R from Bivens
-7.264730E-8
{c}
2.6612670E-12 {d}
0
{e}
65.831547
{href offset}
-0.082942
{sref offset}
714.5
{Pc [=] psia}
621.5
{Tc [=] R}
0.03276
{vc [=] ft3/lbm}
0
{not used}
7
{# of coefficients which follow - used for blends}
1
{DeltaH Correlation type}
0.5541498
{Xo}
87.50197
{A} DeltaH_vap=A+B*X+C*X^2+D*X^3+E*X^4 Bivens
180
Appendix C
185.3407
{B}
where X =(1-T/Tc)^.333-X0, T in R and enthalpy in Btu/lb
13.75282
{C}
0
{D}
0
{E}
2
{Viscosity correlation type: set to 2: do not change}
200
{Lower limit of gas viscosity correlation in K}
500
{Upper limit of gas viscosity correlation in K}
-1.300419E6
{A}
GasViscosity*1E12=A+B*T+C*T^2+D*T^3
5.39552e4
{B}
where T[=]K and GasViscosity[=]N-s/m2
-1.550729e1
{C}
0
{D}
-999
{Lower limit of liquid viscosity correlation in K}
-999
{Upper limit of liquid viscosity correlation in K}
0
{A}
Liquid Viscosity*1E6=A+B*T+C*T^2+D*T^3
0
{B}
where T[=]K and Liquid Viscosity[=]N-s/m2
0
{C}
0
{D}
2
{Conductivity correlation type: set to 2: do not change}
200
{Lower limit of gas conductivity correlation in K}
500
{Upper limit of gas conductivity correlation in K}
-8.643088e-3
{A}
GasConductivity=A+B*T+C*T^2+D*T^3
7.652083e-5
{B}
where T[=]K and GasConductivity[=]W/m-K
2.144608e-9
{C}
0
{D}
-999
{Lower limit of liquid conductivity correlation in K}
-999
{Upper limit of liquid conductivity correlation in K}
0
{A}
LiquidConductivity=A+B*T+C*T^2+D*T^3
0
{B}
where T[=]K and LiquidConductivity[=]W/m-K
0
{C}
0
{D}
0 {terminator}
{The forms of the correlations and in some cases the coefficients have been
adapted from D.B. Bivens and A. Yokozeki, "Thermodynamics and Performance
Potential of R-410a," 1996 Intl. Conference on Ozone Protection Technologies
Oct, 21-23, Washington, DC.}
181
Appendix C
References
ASHRAE Handbook of Fundamentals, (1989, 1993, 1997), American Society of Heating,
Refrigerating and Air Conditioning Engineers, Atlanta, GA
ASHRAE, Thermophysical Properties of Refrigerants, American Society of Heating,
Refrigerating, and Air-Conditioning Engineers, Atlanta, GA, (1976)
D.B. Bivens and A. Yokozeki, "Thermodynamics and Performance Potential of R-410a," 1996
Intl. Conference on Ozone Protection Technologies Oct, 21-23, Washington, DC.
Downing, R.C. and Knight, B.W., "Computer Program for Calculating Properties for the
"FREON" Refrigerants," DuPont Technical Bulletin RT-52, (1971); Downing, R.C.,
"Refrigerant Equations", ASHRAE Transactions, Paper No. 2313, Vol. 80, pt.2, pp. 158-169,
(1974)
Gallagher, J., McLinden, M, Morrison, G., and Huber, M., REFPROP - NIST Thermodynamic
Properties of Refrigerants and Refrigerant Mixtures, Versions 4, 5, and 6, NIST Standard
Reference Database 23, NIST, Gaithersburg. MD 20899, (1989)
Harr, L. Gallagher, J.S., and Kell, G.S (Hemisphere, 1984).
Hemisphere Publishing Company, Washington, (1984)
NBS/NRC Steam Tables,
Howell, J.R., and Buckius, R.O., Fundamentals of Engineering Thermodynamics, McGrawHill, New York, (1987)
Hyland and Wexler, “Formulations for the Thermodynamic Properties of the Saturated Phases
of H2O from 173.15 K to 473.15 K, ASHRAE Transactions, Part 2A,Paper 2793 (RP-216),
(1983)
Keenan, J.H., Chao, J., and Kaye, J., Gas Tables, Second Edition, John Wiley, New York,
(1980)
Keenan, J.H. et al., Steam Tables, John Wiley, New York, (1969)
Irvine, T.F. Jr., and Liley, P.E., Steam and Gas Tables with Computer Equations, Academic
Press Inc., (1984)
Martin, J.J. and Hou, Y.C., ”Development of an Equation of State for Gases,” A.I.Ch.E Journal,
1:142, (1955)
McLinden, M.O. et al., "Measurement and Formulation of the Thermodynamic Properties of
Refrigerants 134a and 123, ASHRAE Trans., Vol. 95, No. 2, (1989)
Reid, R.C.Prausnitz, J.M. and Sherwood, T.K., The Properties of Gases and Liquids,
McGraw-Hill, 3rd edition, (1977)
Shankland, I.R., Basu, R.S., and Wilson, D.P., "Thermal Conductivity and Viscosity of a New
Stratospherically Sate Refrigerant - 1,1,1,2 Tetrafluoroethane (R-134a), published in CFCs:
182
Appendix C
Time of Transition, American Society of Heating, Refrigeration and Air-Conditioning
Engineers, Inc., (1989)
Shankland, I.R., "Transport Properties of CFC Alternatives", AIChE Spring Meeting,
Symposium on Global Climate Change and Refrigerant Properties, Orlando, FL, March, (1990)
Stull, D.R., and Prophet, H., JANAF Thermochemical Tables, Second Edition, U.S. National
Bureau of Standards, Washington, (1971)
Van Wylen, G.J., and Sonntag, R.E., Fundamentals of Classical Thermodynamics, Third
Edition, John Wiley, New York, (1986)
Wilson, D.P. and Basu, R.S., "Thermodynamic Properties of a New Stratospherically Safe
Working Fluid - Refrigerant 134a", paper presented at the ASHRAE meeting, Ottawa, Ontario,
Canada, June, (1988), published in CFCs: Time of Transition, American Society of Heating,
Refrigeration and Air-Conditioning Engineers, Inc., (1989)
183
184
Appendix D
_______________________________________________________________________
Example Problem Information
_______________________________________________________________________
The EXAMPLES subdirectory within the EES directory contains many worked-out example
problems. Each example problem illustrates one or more EES features, as indicated in the
information below.
Feature
Arrays
Complex numbers
Curve-fitting
Diagram window
Differential equations
Differentiate function
DUPLICATE command
Formatted Equations
Functions, user-written
Greek symbols
Integration
Interpolate function
JANAF table
LOOKUP table
Minimize or maximize
Modules
Overlay Plot
Parametric table
Plotting
Procedures, user-written
Procedures, external
Properties, thermodynamic
Property Plot
Psychrometric functions
Regression
Subscripted variables
SUM function
Systems of equations
TABLEVALUE
Transport properties
Unit conversion
EES Example Files
MATRIX.EES, MATRIX2.EES, RANKINE.EES,
REFRIG.EES, REGEN.EES
COMPLEXROOTS.EES
HEATEX.EES
COPPER.EES
DIAGRMW.EES
DRAG.EES, RK4_TEST.EES, SUBSTEPS.EES,
DIFEQN1.EES, DIFEQN1.EES
COPPER.EES
MATRIX.EES, MATRIX2.EES, NLINRG.EES, REGEN.EES
HEATEX.EES, DRAG.EES
CONVECT.EES, MOODY.EES, RK4_TEST.EES
HEATEX.EES, NLINRG.EES
DBL_INTEG.EES, DIFEQN1.EES, DIFEQN2.EES,
SUBSTEPS.EES, DRAG.EES, RK4_TEST.EES
COPPER.EES
FLAMET.EES
NLINRG.EES, COPPER.EES
MAXPOWER.EES, NLINRG.EES, RANKINE.EES
MOODY.EES
CH1EX.EES RANKINE.EES
CAPVST.EES, CH1EX.EES, DIFEQN1.EES, FLAMET.EES,
SUBSTEPS.EES
CAPVST.EES, DIFEQN2
REGEN.EES
ABSORP.EES
REFRIG.EES, CATVST.EES, CH1EX.EES, FLAMET.EES,
REGEN.EES
RANKINE.EES, REFRIG.EES
SUPERMKT.EES
NLINRG.EES
MATRIX.EES, MATRIX2.EES, HEATEX.EES
MATRIX.EES, MATRIX2.EES, NLINRG.EES
HEATEX.EES, CH1EX.EES
DIFEQN2.EES
CONVECT.EES
DRAG.EES
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