COMPUTATIONAL FLUID DYNAMICS ANALYSIS OF FLOW

COMPUTATIONAL FLUID DYNAMICS ANALYSIS OF FLOW
COMPUTATIONAL FLUID DYNAMICS ANALYSIS OF FLOW
THROUGH HIGH SPEED TURBINE USING FLUENT
A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE
REQUIREMENTS FOR THE DEGREE OF
Bachelor of Technology
In
Mechanical Engineering
BY
BIDHAN KUMAR PRADHAN
Under the Guidance of
PROF. SUNIL KUMAR SARANGI
National Institute of Technology
Rourkela
CERTIFICATE
This is to certify that the thesis entitled “CFD ANALYSIS OF FLOW THROUGH HIGH SPEED
TURBINE USING FLUENT” submitted by Bidhan Kumar Pradhan in partial fulfillment of the
requirements for the award of Bachelor of technology Degree in Mechanical Engineering at the
National Institute of Technology, Rourkela (Deemed University) is an authentic work carried out
by him under my supervision and guidance.
To the best of my knowledge, the matter embodied in the thesis has not been submitted to any
other University / Institute for the award of any Degree or Diploma.
Prof. Sunil Kumar. Sarangi
Date
National Institute of Technology
Rourkela-769008
ACKNOWLEDGEMENT
I would like to express my deep sense of gratitude and respect to my mentor Prof. Sunil
Kumar Sarangi, for his excellent guidance and suggestions. He has been a motivating factor for
me. I feel myself extremely lucky to get a chance to work under the guidance of such a dynamic
personality.
I am thankful to Prof P. Rath for building up my concepts in Computational Fluid Dynamics.
Besides I would like to thank Mr. Banjara and Mr. Subrato Ghosh for their help.
Bidhan Kumar Pradhan
Roll No- 10503054
8th Semester B. TECH
Mechanical Engineering Department
National Institute of Technology, Rourkela
LIST OF CONTENTS
S.NO
TOPIC
PAGE NO
1
ABSTRACT
4
2
INTRODUCTION
5
3
HISTORY REVIEW
8
4
THEORY
14
3.1
GENERAL
DESCRIPTION
OF
A
15
CRYOGENIC TURBINE EXPANDER
5
3.2
DESIGN AND OVERALL GEOMETRY
16
3.3
PARAMATERS OF TURBINE WHEEL
21
OBJECTIVE AND ORGANISATION OF THE
23
THESIS
6
4.1
WHAT IS CFD?
24
4.2
DISCRETIZATION METHODS IN CFD
26
4.3
HOW IS THE WORKING DONE IN CFD
27
GAMBIT DESIGNING OF THE MODEL
30
5.1
OVERVIEW OF GAMBIT
31
5.2
MODELLING OF THE COMPONENTS
31
5.3
ASSEMBLYING
40
5.4
MESHING AND DEFINING
41
BOUNDARY CONDITIONS
7
FLUENT ANALYSIS
42
6.1
OVERVIEW OF FLUENT
43
6.2
ANALYSIS
44
8
RESULTS
48
9
CONCLUSION
53
10
REFERENCE
55
1
LIST OF FIGURES
FIGURE
PARTICULAR
PAGE NO
1
SCHEMATIC OF A CRYOGENIC TURBOEXPANDER
15
2
SECTION OF THE TURBINE DISPLAYING ITS
16
NO
COMPONENTS
3
MAJOR DIMENSIONS OF NOZZLE
19
4
STATE POINTS OF TURBO EXPANDER
21
5
BLADE PROFILE GENERATED IN GAMBIT
34
6
MESHED MODEL OF BLADE PROFILE
34
7
BLADE PASSAGE GENERATED IN GAMBIT
35
8
MODEL REPRESENTING BLADE PASSAGE
35
9
COORDINATES OF A SINGLE VANE AND ITS
36
REPRESENTATION
10
NOZZLE ARRANGEMENT REPRESENTATION
37
11
NOZZLE ARRANGEMENT
37
12
DIFFUSER NOMECLATURE
38
13
DIFFUSER GENERATED IN GAMBIT
40
14
BLADE AND NOZZLE ASSEMBLY
40
15
MESHED MODEL OF THE ASSEMBLY
41
16
VELOCITY CONTOURS FOR LAMINAR FLOW IN
49
TURBINE
17
VELOCITY CONTOURS FOR TURBULENT FLOW IN
50
TURBINE
18
TEMPERATURE VARIATION ALONG MERIDONAL
51
STREAMLENGTH IN TURBINE WHEEL
19
PRESSURE VARIATION ALONG MERIDONAL
51
STREAMLENGTH IN TURBINE WHEEL
20
VARIATION OF VELOCITY ALONG THE MERIDONAL
STREAMLENGTH IN TURBINE
2
52
LIST OF TABLES
TABLE
PARTICULARS
PAGE NO
1
OPERATING CONDITIONS OF TURBINE
22
2
COORDINATES FOR GENERATION OF BLADE
32
NO
PROFILE-1
3
COORDINATES FOR GENERATION OF BLADE
33
PROFILE-2
4
COORDINATES FOR GENERATION OF DIFFUSER
3
39
ABSTRACT
This project deals with the computational fluid dynamics analysis of flow in high speed
turbine. This involves with the three dimensional analysis of flow through of a high
turbine having radial inlet and axial outlet. The software used for this purpose are
GAMBIT and FLUENT. The 3 D model of the parts of the turbine are made by GAMBIT
and analysis are to be carried out by FLUENT. The models are first generated using the
data and then are meshed and then various velocity and pressure contours are to be
drawn and graphed in this paper to analyze the flow through the cryogenic turbine.
Various graphs indicating the variation of velocity, pressure and temperature along the
stream length of the turbine are given.
Keywords: Radial Inlet, Axial Outlet, Gambit, Fluent, Stream length
4
CHAPTER 1
INTRODUCTION
5
INTRODUCTIONOxygen, Nitrogen, Helium, Argon etc are industrially important gases.
Nature has provided us an abundant supply of these gases in atmosphere and under
the earth crust. Oxygen and Nitrogen are available from atmosphere and helium, argon
etc are available from the earth crust. The main aim is to harness these gases and use
it for important purpose. The production and utilization of these gases form the major
part of economy. This can be said as a indicator of technological improvement. These
gases have various uses. Oxygen is used for steel manufacturing, rocket propulsion
and medical applications. Argon is used in TIG welding and high temperature furnaces.
Helium finds its use in superconductivity, nuclear reactors etc. Hydrogen is used as fuel
in rocket propulsion systems. Nitrogen is a major input to the fertilizer industry. It is also
used in cryosurgery and semiconductor industry. Nitrogen is used as a blanket gas in
most chemical processes, and serves as the basic raw material in production of
ammonia based fertilizers and chemicals. High purity nitrogen is used as a carrier gas in
the electronic industry; and liquid nitrogen provides the most effective cooling medium
for many low temperature processes – from shrink fitting to cryosurgery.
These important gases are first trapped from the various sources and a
low temperature process known as air separation is used to separate them from each
other. This air separation is carried out using expanding turbines. Air separation using
turbo expanders has several benefits over high pressure (Linde) process. These
benefits include low capital cost, better product mix and high operational flexibility.
While room temperature processes, based on adsorption and membrane separation,
are finding increasing application, particularly for low purity products, cryogenic
distillation still remains the predominant method of producing bulk industrial gases. The
cryogenic distillation process, operating at temperatures below 100K, offers several
advantages over its room temperature counterparts. This process is economical in large
scale, delivers both gaseous and liquid products, produces argon and rare gases (such
as neon, krypton and xenon), and can respond to variation in demand in product mix.
In petrochemical industries, turbo expanders are used for separation
of propane and heavier hydrocarbons from natural gas stream. Turbo expanders
generate low temperature necessary for recovery of ethane and do it less expensively.
6
Expansion turbines are also used for power recovery applications as in refrigeration and
high pressure wellhead gas, in power cycles using geothermal heat, in energy recovery
in pressure let down, in Organic Rankine Cycle used in cryogenic process plants in
order to achieve total utility consumption and in paper and other industries for waste gas
energy recovery.
The expansion turbines can operate continuously for years and are
more reliable than the other forms of reciprocating expanders used widely earlier. This
is made possible by use of gas lubricated bearings, which use the process gas as the
lubricant. While larger machines use axial flow geometry, cryogenic turbines universally
adopt mixed flow, radial inlet and axial discharge, configuration. Multi-staging is difficult
to achieve with radial or mixed flow geometry. Therefore, cryogenic turbines always
adopt single stage expansion, irrespective of the expansion ratio. In addition to their role
in producing liquid cryogens, turbo-expanders provide refrigeration in a variety of other
applications, at both cryogenic and normal temperatures. Closed cycle cryo-coolers
based on the Reversed Brayton cycle are used in cooling of radiation detectors and
superconducting magnets.
7
CHAPTER 2
HISTORY REVIEW
8
HISTORY REVIEW-
The concept that a turbine can be used as a refrigerant machine was
first introduced by Lord Rayleigh. In a letter of 26th June 1898 to Nature, he suggested
the use of turbine instead of a piston expander for air liquefaction because of practical
difficulties being encountered with the low temperature reciprocating machines. In this
letter, Rayleigh emphasized the most important function of and cryogenic expander,
which is to production of the cold, rather than the power produced. This followed a
series of early patents on cryogenic expansion turbine. In 1898 The British engineer
Edgar C Thrupp patented a simple liquefying system using an expansion turbine.
Thrupp’s expander was a double flow machine entering the center and dividing into two
oppositely flowing streams. Each end of the rotor consists of 7 discs on each of which
were from two to four row of blades parallel with the rotor axis. Airflow was from the
center outward through the moving blades on each disk and intervening fixed blades on
the turbine casing. The casing was so shaped internally as to bring air discharge from
periphery of each rotor disk back to hub of the succeeding disk for further expansion.
Contemporaneously with Thrupp, an American engineer Joseph E
Johnson patented an apparatus for liquefying gases. A fraction of air to be liquefied was
to be condensed in the turbine nozzle and fall to the bottom of the liquefaction chamber
for collection, and run off upon exhausting from the turbine. A refrigerative expansion
turbine with a tangential inward flow pattern was patented by the Americans Charles F
and Orrin J Crommett in 1914. Gas was to be admitted to the turbine wheel by a pair of
nozzles, but it was specified that any desired numbers of nozzle could be used. The
turbine blades were curved to present slightly concave faces to the jet from the nozzle.
These blades were comparatively short, not exceeding very close to the rotor hub.
In 1922, the American engineer and teacher Harvey N Davis had
patented an expansion turbine of unusual thermodynamic concept. This turbine was
intended to have several nozzle blocks each receiving a stream of gas from different
temperature level of high pressure side of the main heat exchanger of a liquefaction
apparatus. Davis pointed out that if the supply pressure were sufficiently high all
9
streams would be expanded into the two phase region and so although achieving
varying degrees of wetness, would reach the same terminal temperature.
Successful commercial application of an expansion turbine for gas
liquefaction does not have been made until the early 1930’s. This was done in Linde
Works in Germany. The turbine used was an axial flow single stage impulse machine.
Later in the year 1936 it was replaced by an inward radial flow turbine based on a
patent by an Italian inventor, Guido Zerkowitz. One feature was a reversing chamber
fitted inside the turbine wheel to give a second admission of the gas to the moving
blades. In this way, velocity compounding could be achieved with a consequent
reduction in the wheel speed. The patent specification set forth may details of turbine
construction to keep refrigerate and piping losses minimum. The shaft bearings were to
be entirely outside the turbine housing being within the casing of the machine driven
and so removed from the cold zone.
Peter Kapitza, a well known Russian in cryogenics in the year 1939
came out with a break through paper. It contains of two useful conclusions:
1. In this Kapitza compared the thermodynamics of the liquefiers operating on high
and low pressure cycles and concluded that a low pressure liquefier is better
than high pressure liquefier.
2. Secondly Kapitza undertook to show by analysis and experimental results that an
inward radial flow turbine would preferable to an axial impulse machine.
After the works of Kapitza one of the first well documented air liquefaction
turbines to be built and operated was that designed by Elliot company and constructed
by Sharples company which was done in 1942 and the machine was described as
Swearingen. The turbine was a radial inflow reaction type with designed speed of
22000rpm. The turbine was supported by ball bearings.
Work on the small gas bearing turbo expander commenced in the early
fifties by Sixsmith at Reading University on a machine for a small air liquefaction plant.
In 1958, the United Kingdom Atomic Energy Authority developed a radial inward flow
turbine for a nitrogen production plant. During 1958 to 1961 Stratos Division of Fairchild
Aircraft Co. built blower loaded turbo expanders, mostly for air separation service. Voth
et. developed a high speed turbine expander as a part of a cold moderator refrigerator
10
for the Argonne National Laboratory (ANL). The first commercial turbine using helium
was operated in 1964 in a refrigerator that produced 73 W at 3 K for the Rutherford
helium bubble chamber. A high speed turbo alternator was developed by General
Electric Company, New York in 1968, which ran on a practical gas bearing system
capable of operating at cryogenic temperature with low loss.
National Bureau of Standards at Boulder, Colorado developed a turbine of
shaft diameter of 8 mm. The turbine operated at a speed of 600,000 rpm at 30 K inlet
temperature. In 1974, Sulzer Brothers, Switzerland developed a turbo expander for
cryogenic plants with self acting gas bearings. In 1981, Cryostar, Switzerland started a
development program together with a magnetic bearing manufacturer to develop a
cryogenic turbo expander incorporating active magnetic bearing in both radial and axial
direction. In 1984, the prototype turbo expander of medium size underwent extensive
experimental testing in a nitrogen liquefier. Izumi at Hitachi, Ltd., Japan developed a
micro turbo expander for a small helium refrigerator based on Claude cycle. The turbo
expander consisted of a radial inward flow reaction turbine and a centrifugal brake fan
on the lower and upper ends of a shaft supported by self acting gas bearings. The
diameter of the turbine wheel was 6mm and the shaft diameter was 8mm. The rotational
speeds of the 1st and 2nd stage turbo expander were 816,000 and 519,000 rpm
respectively.
A simple method sufficient for the design of a high efficiency expansion
turbine is outlined by Kun. A study was initiated in 1979 to survey operating plants and
generates the cost factors relating to turbine developed by Kun & Sentz. Sixsmith in
collaboration with Goddard Space Flight Centre of NASA, developed miniature turbines
for Brayton Cycle cryo coolers. They have developed of a turbine, 1.5 mm in diameter
rotating at a speed of approximately one million rpm. Yang developed a two stage
miniature expansion turbine made for 1.5 L/hr helium liquefier at the Cryogenic
Engineering Laboratory of the Chinese Academy of Sciences. The turbines rotated at
more than 500,000 rpm. The design of a small, high speed turbo expander was taken
up by the National Bureau of Standards (NBS) USA. The first expander operated at
600,000 rpm in externally pressurized gas bearings. The turbo expander developed by
11
Kate was with variable flow capacity mechanism (an adjustable turbine), which had the
capacity of controlling the refrigerating power by using the variable nozzle vane height.
A wet type helium turbo expander with expected adiabatic efficiency of
70% was developed by the Naka Fusion Research Centre affiliated to the Japan Atomic
Energy Institute. The turbo expander consists of a 40 mm shaft, 59 mm impeller
diameter and self acting gas journal and thrust bearings. Ino developed a high
expansion ratio radial inflow turbine for a helium liquefier of 100 L/hr capacity for use
with a 70 MW superconductive generator. Davydenkov developed a new turbo
expander with foil bearings for a cryogenic helium plants in Moscow, Russia. The
maximum rotational speed of the rotor was 240,000 rpm with the shaft diameter of 16
mm. The turbo expander third stage was designed and manufactured in 1991, for the
gas expansion machine regime, by “Cryogenmash”. Each stage of the turbo expander
design was similar, differing from each other by dimensions only produced by
“Heliummash”.
The ACD Company incorporated gas lubricated hydrodynamic foil
bearings into a TC–3000 turbo expander. Several Cryogenic Industries has been
involved with this technology for many years including Mafi-Trench. Agahi have
explained the design process of the turbo expander utilizing modern technology, such
as Computational Fluid Dynamic software, Computer Numerical Control Technology
and Holographic Techniques to further improve an already impressive turbo expander
efficiency performance. Improvements in analytical techniques, bearing technology and
design features have made turbo expanders to be designed and operated at more
favorable conditions such as higher rotational speeds. A Sulzer dry turbo expander,
Creare wet turbo expander and IHI centrifugal cold compressor were installed and
operated for about 8000 hrs in the Fermi National Accelerator Laboratory, USA. This
Accelerator Division/Cryogenics department is responsible for the maintenance and
operation of both the Central Helium Liquefier (CHL) and the system of 24 satellite
refrigerators which provide 4.5 K refrigeration to the magnets of the Tevatron
Synchrotron. Theses expanders have achieved 70% efficiency and are well integrated
with the existing system. Sixsmith at Creare Inc., USA developed a small wet turbine for
a helium liquefier set up at the particle accelerator of Fermi National laboratory. The
12
expander shaft was supported in pressurized gas bearings and had a 4.76 mm turbine
rotor at the cold end and a 12.7 mm brake compressor at the warm end. The expander
had a design speed of 384,000 rpm and a design cooling capacity of 444 Watts. Xiong
at the institute of cryogenic Engineering, China developed a cryogenic turbo expander
with a rotor of 103 mm long and weighing 0.9 N, which had a working speed up to
230,000 rpm. The turbo expander was experimented with two types of gas lubricated
foil journal bearings. The L’Air liquid company of France has been manufacturing
cryogenic expansion turbines for 30 years and more than 350 turbo expanders are
operating worldwide, installed on both industrial plants and research institutes. These
turbines are characterized by the use of hydrostatic gas bearings, providing unique
reliability with a measured Mean Time between failures of 45,000 hours. Atlas Copco
has manufactured turbo expanders with active magnetic bearings as an alternative to
conventional oil bearing system for many applications.
India has been lagging behind the rest of the world in this field of
research and development. Still, significant progress has been made during the past
two decades. In CMERI Durgapur, Jadeja developed an inward flow radial turbine
supported on gas bearings for cryogenic plants. The device gave stable rotation at
about 40,000 rpm. The programme was, however, discontinued before any significant
progress could be achieved. Another programme at IIT Kharagpur developed a turbo
expander unit by using aerostatic thrust and journal bearings which had a working
speed up to 80,000 rpm. Recently Cryogenic Technology Division, BARC developed
Helium refrigerator capable of producing 1 kW at 20K temperature
13
CHAPTER 3
THEORY
14
THEORY
3.1 GENERAL DESCRIPTION OF A CRYOGENIC TURBINE EXPANDER-
A cryogenic turbo expander consists of the following components Nozzle
Turbine Wheel
Diffuser
Shaft
Brake Compressor
Radial Bearing
Thrust Bearing
Housing
Plumbing an Instrumentation
FIGURE 1: SCHEMATIC OF A CRYOGENIC TURBOEXPANDER
15
3.2 DESIGN AND OVERALL GEOMETRY-
FIGURE 2: SECTION OF THE TURBINE DISPLAYING ITS COMPONENTS
A cryogenic turbo-expander is a complex equipment whose design
depends on the working fluid, the flow rate and the thermodynamic states at inlet and
exit. The high-pressure process gas enters the turbine through piping, into the plenum
of the cold end housing and from there, radially into the nozzle ring (1). A tangential
velocity is imparted to the fluid, which eventually provides the torque to the rotor. The
fluid accelerates through the converging passages of the nozzles. Pressure energy is
transformed into kinetic energy, leading to a reduction in static temperature. The high
velocity fluid streams impinge on the rotor blades, imparting force to the rotor creating
torque. The nozzles and the rotor blades are so aligned as to eliminate sudden changes
in flow direction and consequent loss of energy. The turbine wheel is of radial or mixed
flow geometry, i.e. the flow enters the wheel radially and exits axially. While larger units
are generally shrouded, smaller wheels are open, the turbine housing acting as the
shroud. The blade passage has a profile of a three dimensional converging duct,
changing from purely radial to an axial tangential direction. Work is extracted as the
16
process gas undergoes expansion with corresponding drop in static temperature. The
diffuser is a diverging passage, and acts as a recompressor that converts most of the
kinetic energy of the gas leaving the rotor to potential energy, in the form of a gain in
pressure. Thus the pressure at the outlet of the rotor is lower than the discharge
pressure of the turbine system. The expansion ratio in the rotor is thereby increased
with a corresponding gain in efficiency and rate of cold production. A loading device is
necessary to extract the work output of the turbine. This device, in principle, can be an
electrical generator, an eddy current brake, an oil drum, or a centrifugal compressor.
The turbine wheel is mounted on a rotating shaft at one end. The
torque produced by the expanding gas is transmitted by the shaft to a braking device
which can be an oil drum, an electrical generator or a compressor. In the given design,
a compressor has been chosen for ease of manufacture and dynamic balancing of the
rotor. The shaft is supported on a pair of journal bearings and a pair of thrust bearings,
the thrust bearings being placed on opposite sides of a collar built on the shaft. The
wheel, the shaft (with the thrust collar built on it) and the brake compressor constitute
the rotor. The rotor is surrounded by bearings, turbine and compressor shrouds, and a
housing holding them in place. In addition, there are a set of small but critical parts,
such as seals, fasteners and spacers. The various components are described below-
ROTORThe rotor is mounted vertically. The rotor consists of the shaft with a
collar integrally machined on it to provide thrust bearing surfaces, the turbine wheel and
the brake compressor mounted on opposite ends. The impellers are mounted at the
extreme ends of the shaft while the bearings are in the middle.
NOZZLEThe nozzles expand the inlet gas isentropically to high velocity and
direct the flow on to the wheel at the correct angle to ensue smooth, impact free
incidence on the wheel blades. A set of static nozzles must be provided around the
turbine wheel to generate the required inlet velocity and swirl. The flow is subsonic, the
absolute Mach number being around 0.95. Filippi has derived the effect of nozzle
17
geometry on stage efficiency by a comparative discussion of three nozzle styles: fixed
nozzles, adjustable nozzles with a centre pivot and adjustable nozzles with a trailing
edge pivot. At design point operation, fixed nozzles yield the best overall efficiency.
Nozzles should be located at the optimal radial location from the wheel to minimize
vaneless space loss and the effect of nozzle wakes on impeller performance. Fixed
nozzle shapes can be optimized by rounding the noses of nozzle vanes and are
directionally oriented for minimal incidence angle loss.
The throat of the nozzle has an important influence on turbine
performance and must be sized to pass the required mass flow rate at design
conditions. Converging–diverging nozzles, giving supersonic flow are not generally
recommended for radial turbines. The exit flow angle and exit velocity from nozzle are
determined by the angular momentum required at rotor inlet and by the continuity
equation. The throat velocity should be similar to the stator exit velocity and this
determines the throat area by continuity. Turbine nozzles designed for subsonic and
slightly supersonic flow are drilled and reamed for straight holes inclined at proper
nozzle outlet angle. In small turbines, there is little space for drilling holes; therefore two
dimensional passages of appropriate geometry are milled on a nozzle ring. The nozzle
inlet is rounded off to reduce frictional losses.
An important forcing mechanism leading to fatigue of the wheel is the
nozzle excitation frequency. As the wheel blades pass under the jets emanating from
the stationary nozzles, there is periodic excitation of the wheel. The number of blades in
the nozzle and that in the wheel should be mutually prime in order to raise this
excitation frequency well beyond the operating speed and to reduce the overall
magnitude of the peak force. The number of vanes as 17 in the nozzle for 7 in the wheel
has been chosen.
18
FIGURE 3: MAJOR DIMENSIONS OF NOZZLE
DIFFUSERThe diffuser acts as a compressor, converting most of the kinetic energy
in the gas leaving the rotor to potential energy in the form of pressure rise. The
expansion ratio in the rotor is thereby increased with a corresponding gain in efficiency.
The efficiency of a diffuser may be defined as the fraction of the inlet kinetic energy that
gets converted to gain in static pressure. The Reynolds number based on the inlet
diameter normally remains around 105. The efficiency of a conical diffuser with regular
inlet conditions is about 90% and is obtained for a semi cone angle of around 5° to 6°.
According to Shepherd, the optimum semi cone angle lies in the range of 3°-5° [24]. A
higher cone angle leads to a shorter diffuser and hence lower frictional loss, but
enhances the chance of flow separation.
The diffuser can be seen as an assembly of three separate sections
operating in series – a converging section or shroud, a short parallel section and finally
the diverging section where the pressure recovery takes place. The converging portion
of the diffuser acts as a casing to the turbine. A clearance is provided to cover the
tolerances of form, position and profile of the wheel, diffuser and the assembly. In
addition, it covers the radial deflection of the wheel due to centrifugal stresses. The
differential contraction between the wheel and the diffuser at low temperature usually
acts to enhance this clearance.
19
SHAFTThe force acting on the turbine shaft due to the revolution of its mass
center and around its geometrical center constitutes the major inertia force. A restoring
force equivalent to a spring force for small displacements, and viscous forces between
the gas and the shaft surface, act as spring and damper to the rotating system. The film
stiffness depends on the relative position of the shaft with respect to the bearing and is
symmetrical with the center-to-center vector.
BRAKE COMPRESSORThe power developed in the expanders may be absorbed by a geared
generator, oil pump, viscous oil brake or blower wheel. Where relatively large amounts
of power are involved, the generator provides the most effective means of recovery.
Induction motors running at slightly above their synchronous speed have been
successfully used for this service. This does not permit speed variation which may be
desirable during plant start up or part load operation. A popular loading device at lower
power levels is the centrifugal compressor. Because of its simplicity and ease of control
the centrifugal compressor is ideally suited for the loading of small turbines. It has the
additional advantage that it can operate at high speeds. For small turbines whose work
output exceeds the capacity of a centrifugal gas compressor, an electrical or oil brake
may be used. The electrical device may be an eddy current brake or permanent magnet
alternator, the latter having the advantage that heat is generated in an external load.
The power generated by the turbine is absorbed by means of a centrifugal blower which
acts as a brake. The helium gas in the brake circuit is circulated by the blower through a
water cooled heat exchanger and a throttle valve. The throttle valve is used to adjust the
load on the blower and the corresponding speed of the shaft. The blower is overdesigned so that when the throttle is fully open the shaft speed is less than the optimum
value. The heat exchanger removes the heat energy equivalent of the shaft work
generated by the turbine from the system. Thus the turbine removes heat from the
process gas and transfers it to the cooling water.
The
turbo expander brake assembly is designed in the form of a
centrifugal wheel of diameter 11.5 mm with a control valve at the inlet which provides for
20
variation in rotor speed within 20%. The heat of friction is removed by the flow of
lubricant through the static gas bearings thereby ensuring constant temperature of the
parts supporting the rotor.
Various other components that form the turbo expander include
bearings(thrust bearing, journal bearing etc) and seals. These also form an integral part
of the turbo expander. The bearings are meant for proper support to the rotors. Seals
are meant to minimize the heat leakage between warm and cold ends due to flow of gas
along the shaft.
3.3 PARAMATERS OF TURBINE WHEEL-
The various thermodynamic parameters that at inlet and the exit are
listed below. The turbine assembly include the nozzle, turbine wheel and diffuser. The
phrase “inlet state” depicts the total or stagnation condition at the inlet, whereas “exit
states” refer to the static conditions at the exit of the diffuser (state ex). The various that
are enlisted below include the pressure, temperature, density, enthalpy, entropy. The
inlet, actual exit along with ideal (isentropic) exit state is tabulated below. This table
indicates the operating conditions of the turbine.
FIGURE 4: STATE POINTS OF TURBO EXPANDER
21
TABLE 1: OPERATING CONDITIONS OF TURBINE
22
CHAPTER 4
OBJECTIVE AND
ORGANISATION OF
THE THESIS
23
OBJECTIVE AND ORGANISATION OF THE THESISThis thesis deals with the study of flow of a cryogenic turbo expander.
Computational fluid analysis is carried out to determine the velocity profile and the
temperature profile. Computational fluid analysis is carried using two software- Gambit
2.3 and Fluent 6.3
Gambit is used to build the model and mesh it and Fluent is used to carry
out the velocity, temperature and pressure analysis. This total analysis is known as
Computational fluid dynamics analysis. Before doing the analysis it is important to have
an overview of what fluent is and how does it work.
4.1 WHAT IS CFD?
CFD or computational fluid dynamics is predicting what will happen,
quantitatively, when fluids flow, often with the complications of simultaneous flow of
heat, mass transfer (eg perspiration, dissolution), phase change (eg melting, freezing,
boiling), chemical reaction (eg combustion, rusting), mechanical movement (eg of
pistons, fans, rudders), stresses in and displacement of immersed or surrounding solids.
Computational fluid dynamics (CFD) is one of the branches of fluid mechanics that uses
numerical methods and algorithms to solve and analyze problems that involve fluid
flows. Computers are used to perform the millions of calculations required to simulate
the interaction of fluids and gases with the complex surfaces used in engineering. Even
with simplified equations and high-speed supercomputers, only approximate solutions
can be achieved in many cases. Ongoing research, however, may yield software that
improves the accuracy and speed of complex simulation scenarios such as transonic or
turbulent flows. Initial validation of such software is often performed using a wind tunnel
with the final validation coming in flight test.
The most fundamental consideration in CFD is how one treats a
continuous fluid in a discretized fashion on a computer. One method is to discretize the
spatial domain into small cells to form a volume mesh or grid, and then apply a suitable
24
algorithm to solve the equations of motion (Euler equations for inviscid and NavierStokes equations for viscous flow). In addition, such a mesh can be either irregular (for
instance consisting of triangles in 2D, or pyramidal solids in 3D) or regular; the
distinguishing characteristic of the former is that each cell must be stored separately in
memory. Where shocks or discontinuities are present, high resolution schemes such as
Total Variation Diminishing (TVD), Flux Corrected Transport (FCT), Essentially Non
Oscillatory (ENO), or MUSCL schemes are needed to avoid spurious oscillations (Gibbs
phenomenon) in the solution. If one chooses not to proceed with a mesh-based method,
a number of alternatives exist, notably Smoothed particle hydrodynamics (SPH), a
Lagrangian method of solving fluid problems, Spectral methods, a technique where the
equations are projected onto basis functions like the spherical harmonics and
Chebyshev polynomials, Lattice Boltzmann methods (LBM), which simulate an
equivalent mesoscopic system on a Cartesian grid, instead of solving the macroscopic
system (or the real microscopic physics). It is possible to directly solve the NavierStokes equations for laminar flows and for turbulent flows when all of the relevant length
scales can be resolved by the grid (a direct numerical simulation). In general however,
the range of length scales appropriate to the problem is larger than even today's
massively parallel computers can model. In these cases, turbulent flow simulations
require the introduction of a turbulence model. Large eddy simulations (LES) and the
Reynolds-averaged Navier-Stokes equations (RANS) formulation, with the k-ε model or
the Reynolds stress model, are two techniques for dealing with these scales. In many
instances, other equations are solved simultaneously with the Navier-Stokes equations.
These other equations can include those describing species concentration (mass
transfer), chemical reactions, heat transfer, etc. More advanced codes allow the
simulation of more complex cases involving multi-phase flows (e.g. liquid/gas, solid/gas,
liquid/solid), non-Newtonian fluids (such as blood), or chemically reacting flows (such as
combustion).
25
4.2 DISCRETIZATION METHODS IN CFD
The stability of the chosen discretization is generally established
numerically rather than analytically as with simple linear problems. Special care must
also be taken to ensure that the discretization handles discontinuous solutions
gracefully. The Euler equations and Navier-Stokes equations both admit shocks, and
contact surfaces.
Some of the discretization methods being used are:
•
Finite volume method (FVM). This is the "classical" or standard approach used
most often in commercial software and research codes. The governing equations
are solved on discrete control volumes. FVM recasts the PDE's (Partial
Differential Equations) of the N-S equation in the conservative form and then
discretize this equation. This guarantees the conservation of fluxes through a
particular control volume. Though the overall solution will be conservative in
nature there is no guarantee that it is the actual solution. Moreover this method is
sensitive to distorted elements which can prevent convergence if such elements
are in critical flow regions. This integration approach yields a method that is
inherently conservative (i.e. quantities such as density remain physically
meaningful
•
Finite element method (FEM). This method is popular for structural analysis of
solids, but is also applicable to fluids. The FEM formulation requires, however,
special care to ensure a conservative solution. The FEM formulation has been
adapted for use with the Navier-Stokes equations. Although in FEM conservation
has to be taken care of, it is much more stable than the FVM approach.
Subsequently it is the new direction in which CFD is moving. Generally
stability/robustness of the solution is better in FEM though for some cases it
might take more memory than FVM methods.
•
Finite difference method. This method has historical importance and is simple to
program. It is currently only used in few specialized codes. Modern finite
26
difference codes make use of an embedded boundary for handling complex
geometries making these codes highly efficient and accurate. Other ways to
handle geometries are using overlapping-grids, where the solution is interpolated
across each grid.
•
Boundary element method. The boundary occupied by the fluid is divided into
surface mesh.
•
High-resolution schemes are used where shocks or discontinuities are present.
To capture sharp changes in the solution requires the use of second or higher
order numerical schemes that do not introduce spurious oscillations. This usually
necessitates the application of flux limiters to ensure that the solution is total
variation diminishing.
4.3 HOW IS THE WORKING DONE IN CFD
Working in CFD is done by writing down the CFD codes. CFD codes
are structured around the numerical algorithms that can be tackle fluid problems. In
order to provide easy access to their solving power all commercial CFD packages
include sophisticated user interfaces input problem parameters and to examine the
results. Hence all codes contain three main elements:
1. Pre-processing.
2. Solver
3. Post - processing.
PRE-PROCESSING
Preprocessor consists of input of a flow problem by means of an operator friendly
interface and subsequent transformation of this input into form of suitable for the use by
the solver.
The user activities at the Pre-processing stage involve:
27
1) Definition of the geometry of the region: The computational domain. Grid generation
is the subdivision of the domain into a number of smaller, no overlapping sub domains
(or control volumes or elements Selection of physical or chemical phenomena that need
to be modeled).
2) Definition of fluid properties: Specification of appropriate boundary conditions at cells,
which coincide with or touch the boundary. The solution of a flow problem (velocity,
pressure, temperature etc.) is defined at nodes inside each cell. The accuracy of CFD
solutions is governed by number of cells in the grid. In general, the larger numbers of
cells better the solution accuracy. Both the accuracy of the solution & its cost in terms of
necessary computer hardware & calculation time are dependent on the fineness of the
grid. Efforts are underway to develop CFD codes with a (self) adaptive meshing
capability. Ultimately such programs will automatically refine the grid in areas of rapid
variation.
SOLVER
These are three distinct streams of numerical solutions techniques: finite difference,
finite volume& finite element methods. In outline the numerical methods that form the
basis of solver performs the following steps:
1) The approximation of unknown flow variables are by means of simple functions
2) Discretization by substitution of the approximation into the governing flow equations
& subsequent mathematical manipulations.
POST-PROCESSING
As in the pre-processing huge amount of development work has recently has taken
place in the post processing field. Owing to increased popularity of engineering work
stations, many of which has outstanding graphics capabilities, the leading CFD are now
equipped with versatile data visualization tools.
These include:
28
1) Domain geometry & Grid display
2) Vector plots
3) Line & shaded contour plots
4) 2D & 3D surface plots
5) Particle tracking
6) View manipulation (translation, rotation, scaling etc.)
29
CHAPTER 5
GAMBIT
DESIGNING OF THE
MODEL
30
GAMBIT DESIGNING OF THE MODEL5.1 OVERVIEW OF GAMBIT
The model development is carried out on gambit. It is the pre
processing process where the model development is done and meshing of model is
followed for further analysis. GAMBIT is a software package designed to help analysts
and designers build and mesh models for computational fluid dynamics (CFD) and other
scientific applications. GAMBIT receives user input by means of its graphical user
interface (GUI). The GAMBIT GUI makes the basic steps of building, meshing, and
assigning zone types to a model simple and intuitive, yet it is versatile enough to
accommodate a wide range of modeling applications.
The various components of the turbine are designed using Gambit. The
various components include blade, blade passage, nozzle and diffuser. The
components are individually made and all are assembled. The profiles are generated
with the help of coordinates available which have been generated. A single blade profile
is made and then a group of seven blades are arranged to make the blade passage.
After the formation of the blade passage the nozzle arrangement is done followed by
the designing of the diffuser using the coordinate. All these components are finally
assembled and meshing is done.
5.2 MODELLING OF THE COMPONENTSMODELLING THE BLADE PROFILEThe hub and the tip streamlines are available in the table below. A
ruled surface is created by joining the hub and tip streamlines with a set of tie lines. The
surface so generated is considered as the mean surface within a blade. The suction and
pressure surfaces of two adjacent channels are computed by translating the mean
surface in the positive and negative Ф directions through half the blade thickness.
Coordinates of all the blade surfaces are computed by further rotating the pair of
surfaces over an angle 2π / Z, i.e. 51.43 degrees for Z = 7. Non Uniform Rational B
Splines are used to develop the solid surface.
31
TABLE 2: COORDINATES FOR GENERATION OF BLADE PROFILE-1
z= axial length
r= radius
phi= angle of deflection measured in clockwise direction
32
TABLE 3: COORDINATES FOR GENERATION OF BLADE PROFILE -2
33
FIGURE 5: BLADE PROFILE GENERATED IN GAMBIT
FIGURE 6: MESHED MODEL OF BLADE PROFILE
34
BLADE PASSAGE MODELLINGA single blade profile which created is aligned to create the blade passage. Seven
blades are aligned about a circle to form the blade passage profile.
FIGURE 7: MODEL REPRESENTING BLADE PASSAGE
35
FIGURE 8: BLADE PASSAGE GENERATED IN GAMBIT
NOZZLE AND ITS ARRANGEMENTThe number blades in the nozzle and the blades passage should be
mutually prime. The number blades taken in blade passage are 7 and the number of
blades in the nozzle is taken to be 17. The nozzle profile is done as per the autocad
profile and the extrusions of the vanes are done to develop the nozzle arrangement 3D
profile.
FIGURE 9: COORDINATES OF A SINGLE VANE AND ITS REPRESENTATION
36
FIGURE 10: NOZZLE ARRANGEMENT REPRESENTATION
FIGURE 11: NOZZLE ARRANGEMENT
37
MODELLING THE DIFFUSERThe coordinates are provided for the development of the 2D model of
the diffuser. The model is then rotated about 360 degrees to get the 3D profile. For
design purposes, the diffuser can be seen as an assembly of three separate sections
operating in series – a converging section or shroud, a short parallel section and finally
the diverging section. The converging portion of the diffuser acts as a casing to the
turbine. The straight portion of the diffuser helps in reducing the non-uniformity of flow,
and in the diverging section, the pressure recovery takes place. The geometrical
specifications of the diffuser have chosen somewhat arbitrarily. Diameter of diffuser inlet
is equal to diameter of the turbine inlet. Diameter of throat of diffuser is depending on
the shroud clearance. The recommended clearance is 2% of the exit radius, which is
approximately 0.2 mm for wheel. The differential contraction between the wheel and the
diffuser at low temperature usually acts to enhance this clearance. The profile of the
convergent section has been obtained by offsetting the turbine tip profile by o.2 mm
radially. For diameter of diffuser exhaust, suggested exit velocity of the diffuser should
be maintained near about 20 m/s with a half cone angle of 5.50.
DinD = Inlet Diameter Diffuser, DthD = Throat Diameter Diffuser, DexD = Exit Diameter
Diffuser, LcD = Length Convergent Section, LdD = Length Divergent Section
FIGURE 12: DIFFUSER NOMECLATURE
38
TABLE 4: COORDINATES FOR GENERATION OF DIFFUSER
Z(in mm)
X(in mm)
Z(in mm)
X(in mm)
Z(in mm)
X(in mm)
0
26.632
2.4
20.3325
7.4
18.6645
0.1
26.1098
2.5
20.2364
7.6
18.6604
0.2
25.5823
2.6
20.1404
7.8
18.6552
0.3
25.0383
2.7
20.0444
8
18.6486
0.4
24.5761
2.8
19.9646
8.2
18.6421
0.5
24.1616
2.9
19.8848
8.4
18.5836
0.6
23.7879
3
19.805
8.6
18.5204
0.7
23.4373
4.5
19.0485
8.8
18.492
0.8
23.1446
4.6
19.015
9
18.492
0.9
22.8519
4.7
18.9888
9.2
18.492
1
22.6023
4.8
18.9625
9.4
18.492
1.1
22.3606
4.9
18.9363
9.6
18.492
1.2
22.1343
5
18.9097
9.8
18.492
1.3
21.9335
5.2
18.8699
10
18.492
1.4
21.7328
5.4
18.8299
10.5
18.492
1.5
21.5547
5.6
18.7993
11
18.492
1.6
21.3874
5.8
18.7701
11.5
18.492
1.7
21.2202
6
18.7474
12
18.492
1.8
21.0742
6.2
18.7266
12.5
18.492
1.9
20.9349
6.4
18.7096
13
18.492
2
20.7956
6.6
18.6966
13.5
18.492
2.1
20.672
6.8
18.6845
14
18.492
2.2
20.5561
7
18.6769
14.7
18.492
2.3
20.4402
7.2
18.6693
39
FIGURE 13: DIFFUSER GENERATED IN GAMBIT
5.3 ASSEMBLYINGThe blades passage, nozzle and diffuser are assembled to give the turbine
assembly.
FIGURE 14: BLADE AND NOZZLE ASSEMBLY
40
5.4 MESHING AND DEFINING BOUNDARY CONDITIONS The assembly is meshed using tetrahedral elements of T-grid scheme type.
The boundary conditions are defined as given. Nozzle inlet is taken as mass flow inlet
and diffuser is taken as the pressure outlet. Here mixing planes are also defined at the
interface of nozzle outlet and blade passage inlet as pressure outlet and inlet
respectively and the interfaces at the blade passage outlet and diffuser inlet as pressure
and inlet respectively. The nozzle, blades and diffuser flow path are defined as fluid and
the nozzles and the blades are taken as solid. File is then saved for analysis in fluent.
FIGURE 15: MESHED MODEL OF THE ASSEMBLY
41
CHAPTER 6
FLUENT ANALYSIS
42
FLUENT ANALYSIS6.1 OVERVIEW OF FLUENTFLUENT is the software used for modeling fluid flow and heat transfer in
complex geometries. It provides complete mesh flexibility, including the ability to solve
your flow problems using unstructured meshes that can be generated about complex
geometries with relative ease. It is written in the C computer language and makes full
use of the flexibility and power offered by the language. Consequently, true dynamic
memory allocation, efficient data structures, and flexible solver control are all possible.
All functions required to compute a solution and display the results are accessible in
FLUENT through an interactive, menu-driven interface. The basic procedural steps for
solving a problem in FLUENT include:
1) Define the modeling goals.
2) Create the model geometry and grid.
3) Set up the solver and physical models.
4) Compute and monitor the solution.
5) Examine and save the results
6) Consider revisions to the numerical or physical model parameters, if necessary
FLUENT can model flow involving moving reference frames and moving cell zones,
using several different approaches, and flow in moving and deforming domains
(dynamic meshes). Solving flows in moving reference frames requires the use of
moving cell zones. Problems that can be addressed include flow in a (single) rotating
frame and flow in multiple rotating and/or translating reference frames. However in case
of turbo machinery fluent uses multiple reference frame model, mixing plane model and
sliding mesh model.
43
6.2 ANALYSISThe analysis is carried in fluent by importing the meshed file saved in
gambit. The steps that are followed are given below which include all the conditions and
the boundaries values for the problem statement.
Checking of mesh and ScalingThe fluent solver is opened where 3D is selected and then the importing of the meshed
file is done. The meshed file is then undergoes a checking where 51620 number of
grids are found. After this grid check is done following which smoothing and swapping of
grid is done. For this skewness method is selected and minimum skewness is set as 0.8
and the iterations are set as 8. The model is then smoothed and swapped where the
number of faces comes to be 2696.
Following this the scaling is done. Scale is scaled to mm. Grid created was changed to
mm. After this defining of various parameters are done.
Solver and Material Selection and Operating Condition DefiningThe solver is defined first. Solver is taken as pressure based and formulation as implicit,
space as 3D and time as steady. Velocity formulation as absolute and gradient options
as Green-Gauss Cell based are taken. Energy equation is taken into consideration. The
viscous medium is also taken. First the analysis is carried using laminar flow and then
the k-epsilon is considered. 2 results are to be found out.
The selection of material is done. Material selected is nitrogen gas. The properties of
nitrogen is taken as followsDensity = 1.138 kg/m3
Cp (specific heat capacity) = 1040.67 J/kg K
Thermal conductivity = 0.0242 W/m K
44
Viscosity = 1.663 x 10-5 kg/m s
The analysis is carried out under operating conditions of 101325 Pascal. Gravity is not
taken into consideration.
Mixing Plane SelectionThe mixing plane model was set up. Two mixing planes were needed, one at the
interface between the pressure outlet of the Upstream nozzle outlet region and the
pressure inlet at the adjacent face of the Blades passage Region. It was defined as
radial mixing plane geometry. Similarly, the second mixing plane was defined at the
pressure outlet of Blades passage and the pressure inlet to the Downstream Diffuser
inlet region. It was defined as Axial mixing plane geometry. Under relaxation value was
set as 1.
Boundary Conditions-
Nozzle InletMass flow inlet was taken for the nozzle inlet and the value of mass flow
rate was taken as 0.0606kg/s. Initial gauge pressure was taken as 500000 Pascal.
Temperature was taken as 120K.
Mixing PlanesMixing planes were defined as defined earlier in gambit.
Blades and Flow PathFor blades solid material taken was Aluminium and rotation axis origin
was taken as (0,0,0) and rotation axis direction was set as (0,0,-1) and motion type
selected was moving mesh. The rotational velocity was taken as 10400rad/s.
Translational velocity was set as (0,0,0).
45
OutletThe diffuser was set as pressure outlet and the temperature was set to
80K and the pressure was set to 0 Pascal.
The similar analysis was also carried out for turbulent flow, using k-epsilon method.
Here turbulence energy and dissipation rate are taken into consideration.
Controls Set UpThe solutions controls are set as listed below.
The under relaxation factor was set as givenPressure-0.2
Density-1
Body forces-1
Momentum-0.5
Energy-1
Pressure Velocity Coupling was taken as SIMPLE
Discretization Equation are selected as givenPressure- PRESTO (Pressure Staggering Option)
Momentum- Second Order Upwind
Energy- Second Order Upwind (For turbulent flow Power Law was taken into
consideration)
46
InitializationSolution initialization is done. Initial values of velocity are taken as zero along x, y and z
direction. Temperature is taken as 300K.
Residual Monitorization is done and convergence criteria are set up. The convergence
criteria of various parameters are listed below.
Continuity- 0.001
X-Velocity- 0.001
Y-Velocity- 0.001
Z-Velocity- 0.001
Energy- 1e-06
The number of iterations is then set up and iterations starts. The iteration continues till
the convergence is reached.
47
CHAPTER 7
RESULTS
48
RESULTSThe velocity profiles are drawn for both the laminar and turbulent (k-epsilon) flow. In
case of laminar flow the velocity increases from the nozzle inlet (blue region) to reach a
maximum at the nozzle outlet (red region). In case of the turbulent flow higher velocity
range are to be seen. The velocity profile is quiet random in case of turbulent flow.
However it is to be noted that the convergence is reached in turbulent flow much
quicker in comparison to laminar flow.
FIGURE 16: VELOCITY CONTOURS FOR LAMINAR FLOW IN TURBINE
49
FIGURE 17: VELOCITY CONTOURS FOR TURBULENT FLOW IN TURBINE
50
Temperature
98
96
94
92
90
88
86
84
82
80
78
T
e
m
p
i
e
n
r
a
K
t
u
r
e
0
2
4
6
8
10
12
Meridonal Streamlength in mm
FIGURE 18: TEMPERATURE VARIATION ALONG MERIDONAL STREAMLENGTH
IN TURBINE WHEEL
Pressure
3
P
R
E
S
S B
U A
R R
E
2.5
2
1.5
1
0.5
0
I
N
0
2
4
6
8
10
12
MERIDONAL STREAMLENGTH IN MM
FIGURE 19: PRESSURE VARIATION ALONG MERIDONAL STREAMLENGTH IN
TURBINE WHEEL
51
FIGURE 20: VARIATION OF VELOCITY ALONG THE MERIDONAL
STREAMLENGTH IN TURBINE
Average Velocity magnitude has been found out. The average velocity is found out
using
= 86m/s for laminar flow
52
CHAPTER 8
CONCLUSION
53
CONCLUSION-
Cryogenic turbine has great utility in industrial collection of gases. They also provide a
wonderful medium for refrigeration. The modeling of the various parts of the turbine is
done using Gambit and the computational fluid dynamics analysis is done using fluent
software. The velocity contours and graphs indicating the variation of temperature,
pressure and velocity along the meridonal streamlength is given. However the results
are not accurate as the leakage are not taken into consideration. Moreover the original
material used to construct the turbine is also not taken into consideration.
54
REFERENCES:
1)
Patankar Suhas V Numerical Heat Transfer and Fluid Flow
2)
Ghosh, P and Sarangi, S Thesis on “Analytical and
Experimental Studies on
Turboexpander” , IIT Kharagpur (2002)
3)
Arkharov, A., Marfenina, I. and Mikulin, Ye. (Trans. Kuznetsov, B.V.) Theory
and Design of Cryogenic Systems, Mir Publishers, Moscow
4)
Barron, R. F. Cryogenic Systems Oxford University
Press(1985)
5)
Blackford, J. E., Halford P. and Tantam, D. H. Expander and pumps in G G
Haselden (Ed) Cryogenic Fundamentals Academic Press London (1971)
6)
Flynn, T.M. Cryogenic Engineering, Marcel-Dekkar Inc.
7)
Rama S. R. Gorla and Aijaz A. Khan Turbomachinery Design and Theory
8)
Logan Jr, E. Turbomachinery: Basic Theory and Applications Marcel Dekker
Inc.
55
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