Determination of thermal transpiration effect for biomolecular gases with capacitance manometer

Determination of thermal transpiration effect for biomolecular gases with capacitance manometer
Master of science programme in engineering physics
Determination of thermal transpiration effect for
biomolecular gases with capacitance manometer
Martin Viktor Johansson
December, 2015
Department of physics
Physikalisch-Technische Bundesanstalt Braunschweig and Berlin
Faculty of science and technology
Division 6 Ionizing radiation
Umeå University
Department 6.6 Radiation effects
Umeå, Sweden
Braunschweig, Germany
Examinator: Ludvig Lizana
Supervisor: Dr. Woon Yong Baek
Abstract
Capacitance manometer with sensors maintained at temperatures above the temperature of the vacuum vessel may read a higher gas pressure than the true value. This
arises due to a transport process of molecules induced by molecule-surface collisions
called thermal transpiration effect. Thermal transpiration effect depends on the pressure, the temperature gradient, gas, geometry and surface properties of the interconnecting pipe between the capacitance manometer and the vacuum vessel. To determine
the height of the thermal transpiration effect for the biomolecular gas tetrahydrofuran,
an experimental setup has been built. Its suitability to measure the thermal transpiration effect has been tested. Measurements of thermal transpiration effects for nitrogen
and tetrahydrofuran have been analyzed with the semi-empirical Takaishi-Sensui equation. The coefficients of the Takaishi-Sensui equation can be used to determine the
magnitude of the thermal transpiration effect for different temperature gradients, diameters of the interconnecting pipe and pressures.
Acknowledgements
I wish to give special thanks to Alexander Arndt for his patience and support in the
many discussions we have had and for sharing his expertise of experimental physics. I
also wish to thank Dr.Woon Yong Baek for his trust to give me responsibility for this
project and for his guidance, and Dr.Hans Rabus whom has been a great support
throughout the project. I would also like to thank Hartmut Lucius for his help in the
assembly of the vacuum experiment and the technical personal of
dep. 6.6 Radiation effects, PTB Braunschweig. I also wish to thank all of my
colleagues at dep. 6.6 for making this work a very enjoyable experience.
I would also like thank my family, especially Sara, for their support.
Chapter 1
Introduction
1.0.1
Introduction
In a vacuum system with a temperature gradient, thermal transpiration effect causes a
greater pressure in the region of higher temperature than in the cooler region of the system. This effect depends mainly on the rarefaction of the gas, the temperature difference between the warm and cold region, and on the molecule-surface scattering [14,33].
Rarefaction of a gas is quantified by the Knudsen number Kn which depends on the
characteristic dimension of the vessel, collisional diameter of the molecules, temperature and pressure. Capacitance manometer is often used for measurement of the gas
pressure in high and intermediate vacuum. It has high accuracy, fast dynamic response
and it is not easily contaminated. It is based on the measurement of the induced mechanical deflection of a thin metal diaphragm caused by the pressure difference between
the vacuum vessel and a reference side. The deflection leads to a change in capacitance
between the diaphragm and a fixed plate in the reference side which causes a shift in
the electrical capacity of the gauge.
The capacitance manometer measures absolute pressure when the reference side is
in high vacuum. When no temperature gradient between the capacitance manometer
and the vacuum vessel is present, the pressure readings are independent of gas type.
However, the capacitance manometer is usually heated internally in order to increase
long term stability, especially with respect to the variation in ambient temperature
and contamination. This internal heating leads to a temperature difference between
the capacitance manometer and the vacuum vessel. If the pressure is below 1 mbar,
1
thermal transpiration effect causes a higher pressure in the head of the capacitance
manometer than in the vacuum vessel [7, 8, 31].
There are several semi-empirical equations used to correct the pressure readings for this
effect [10, 40]. One of the most commonly used equation is the semi-empirical formula
of Takaishi-Sensui [35] which may accurately (within 0, 5 % Ref. [40] for Nitrogen)
correct the readings for thermal transpiration effect if the gas specific coefficients are
known. These coefficients are known for the great part of common gases but not for
biomolecular gases such as tetrahydrofuran C4 H8 O. Tetrahydrofuran is used as the
model molecule for the deoxyribose in DNA. The quantification of radiation induced
damages requires the interaction cross sections between charged particle and the DNA
constituents. These cross sections are commonly determined in a scattering experiment
using a gas target. In scattering experiments, the gas pressure usually lies in the region of 5 × 10−2 mbar - 1 × 10−3 mbar where thermal transpiration effect is highly gas
dependent and plays an important role for the pressure measurements. When the gas
specific coefficients in the semi-empirical formula of Takaishi-Sensui are unknown, they
may be estimated using intermolecular collisional diameter of the gas [35]. This correction scheme is recommended when calibrating pressure gagues when the gas specific
coefficients are unknown [12], and commonly used when measuring the gas pressure
of tetrahydrofuran, for example in [22, 41]. However, it has recently been shown that
this correction scheme fails for large molecules and gives a unphysical scaling with
molecular size [10]. Alternative semi-empirical equations of Sětina [32] or Miller [20]
have not been verified for large molecules [10].
The aim of this work is to determine the gas specific coefficients of Takaishi-Sensui
formula for the biomolecular gas tetrahydrofuran. An experimental setup have been
built and tested, including the generation of software for data acquisition and temperature regulation. Two methods for the measurement of thermal transpiration effect
have been evaluated. In the first method, the temperature of the vacuum vessel is
varied with constant pressure. The second method is based on the comparison of readings of two capacitance manometers with different internal temperatures over different
pressures. Measurements of the pressure ratio for nitrogen, argon and tetrahydrofuran
have been carried out. The gas specific coefficients to the Takaishi-Sensui formula
for thermal transpiration effect of tetrahydrofuran relative to the calculated thermal
transpiration effect of nitrogen has been determined.
2
1.0.2
Context of the work
This is a project within the framework of the master of science program in engineering
physics held by the department of physics of Umeå university. The project has been
carried out at the department 6.6 Radiation effects , at the national metrology institute
of Germany, Physikalisch-Technische Bundesanstalt Braunschweig and Berlin (PTB),
in Braunschweig.
3
Chapter 2
Theory
This chapter begins with the introduction to some basic concept of vacuum
physics such as rarefaction and gas molecule-surface scattering which is of
importance to understand thermal transpiration effect. Then, a description of thermal transpiration effect and the vacuum technology used in the
experimental setup is given.
2.1
2.1.1
Vacuum physics
Gas in a vacuum vessel
For most purposes in vacuum physics, gas molecules are considered as spheres with
diameter D in first order. If the gas in a vacuum vessel behaves similar to the model of
an ideal gas, the molecules are considered as independent spheres colliding elastically
with negligible size in comparison to the distance between them [5]. For an ideal gas,
the number density n and the pressure p is related by the ideal gas law
p = nkB T,
(2.1)
where kB is Boltzmann constant, T is the temperature in Kelvin. The SI unit for
pressure p is Pascal, Pa, which is defined as 1 Newton per square meter N m−2 . In this
work the unit
1 mbar = 100 Pa
4
is used for pressure.
Rarefaction
The behaviour of gas depends on the rarefaction. The rarefaction determines the relative frequency of molecule-molecule collisions and molecule-surface collisions. When
molecule-molecule collision is the governing transport processes, the gas behaves like
a fluid. This is called continuum regime. In the so called molecular regime, the governing transport process is molecule-surface collisions. The regime between these two
extremes, is called Knudsen regime. Rarefaction is quantified by the unit less Knudsen
number. The Knudsen number is defined as the quotient of the mean free path λ,
which is the average distance covered by a gas molecule between successive collisions,
and the characteristic length d of the vessel containing the gas.
In the case of a pipe, the characteristic length is the diameter of the pipe so that
the Knudsen number is defined as
Kn :=
λ
d
The mean free path may be expressed as [5]
1
λ= √
,
2πD2 n
where D is the molecular diameter and n is the number density. Using the ideal gas
law, Eq.2.1, the Knudsen number can then be written as
Kn :=
λ
kB T
=√
.
d
2dπD2 p
(2.2)
In the continuum regime Kn is greater than 1, while in the molecular regime it is
lower than 0, 01. A gas is said to be a rarefied gas when Kn is greater than one.
In this case, surface-molecule collisions significantly influence the transport process of
gas molecules. For example, air at room temperature under atmospheric pressure is a
rarefied gas when the characteristic length is in order of or less than micro meters. Air
at room temperature in a pipe of diameter 1 cm is rarefied if the pressure is below 1
mbar.
5
Gas-surface scattering
In a rarefied gas, the transport of gas molecules is governed by molecule-surface collisions. How this scattering occurs depends on the surface smoothness, the material
as well as temperature of the surface, the mass and the structure of the molecule. If
molecules are reflected with the same angle of incidence upon the collision with the surface, by reversing the direction of the perpendicular component of the velocity to the
surface and leaving the parallel component unchanged, the scattering is called specular, see Fig. 2.1(a). If molecules lose the information about the incoming direction and
velocity and are scattered into all directions with an equal probability with a velocity
depending on the surface temperature the scattering is called diffuse, this is depicted
in Fig. 2.1(b).
a)
b)
dS
dS
Figure 2.1: Left picture a) shows specular scattering. The ball represents a molecule
hitting the surface element and leaving it with the reflection angle and speed same
as the incidence angle and speed. The picture b) depicts diffuse scattering. Here,
the large solid arrows represent the average outgoing momentum of molecules to left
and right side of dS. The small arrows the tangential and normal components of the
average outgoing momentum. The arrow along the dotted line represents the total
average outgoing momentum. Molecules leave the surface in any direction with an
equal probability. The speed depends only on the temperature of dS.
Diffuse scattering
In the case of collisions with metallic surface the scattering is mainly diffuse. In diffuse scattering, molecules attach on the surface, this is called adsorption, and are later
released from the surface in a process called desorption. There are several mechanisms
for adsorption. The molecule may be physically adsorbed by establishing a bond with
6
the atoms of the metal due to fluctuating electric dipole forces. Another mechanism
is chemisorption that occurs when the gas molecule transfers or shares electrons in
the bonding with the atoms of the solid. The letter bond is typically much stronger
than in the case of physically adsorbed molecules [5]. Thermal energy causes the adsorbed gas molecules and the atoms in the metal to be in continuous vibratory motion.
Thermal energy is randomly shared back and forth between the atoms in the metal
and adsorbed gas molecules by their vibrating motion. The molecules may be released
from the surface when the adsorbed molecule, by statistical chance, acquire enough
thermal energy to break the bond. Desorbed molecules leave the surface randomly in
any direction, with kinetic energy corresponding to the temperature on the place of
emission. Molecules may be adsorbed in many layers on the metal surface. They may
be transported from the surface deeper into the metal by the process of diffusion. As
the metallic surface is microscopically rough, molecules which are not adsorbed still
may scatter in a diffuse way.
7
2.1.2
Thermal transpiration effect
In this section thermal transpiration effect will be first qualitatively described, then the magnitude of the effect in the limit of continuum and
a) Incoming
b) Outgoing
HOT
COLD HOT
dS
COLD
dS
c)
In =
+
=
C.o.M: In + Out = 0
Out =
Figure 2.2: The mechanism causing the thermal transpiration effect is depicted. The
left side of dS is hot, the right side of dS is cold. The upper left picture a) shows
incoming particles hitting dS. The large solid arrows on each side of dS represent the
average incoming momentum of the molecules hitting dS from the hot and the cold
side, the small arrows the tangential and normal components of the average incoming
momentum. Assuming that there is on average equal number of molecules hitting dS
from all directions, the mean incoming momentum from the hot side will have a larger
tangential average momentum. This may be seen in the dotted line representing the
total average incoming momentum which is the sum of the incoming average momentum from both sides illustrated in the first line in c). Therefore, from conservation of
momentum, the second line in c), the total average outgoing momentum must be in
the direction toward the hot side of dS as depicted in b). Assuming diffuse reflection,
the emission velocity is equal in all directions as it only depends on the temperature of
dS. Therefore, more molecules must be emitted toward the hot side so that the total
average momentum have a larger tangential component toward the hot side of dS.
8
molecular regime followed by the discussion of semi-empirical equations of
Takaishi-Sensui, Sětina and Miller which may be used to calculate thermal
transpiration effect in Knudsen regime.
Qualitative description
Thermal transpiration effect in a rarefied gas occurs in a system with a temperature
gradient when the boundary scattering is diffuse. The mechanism of the effect is
explained with the assumption that there are equally many incoming molecules per
unit time hitting a small piece of the surface dS from all directions. In a rarefied gas
the molecules keep their kinetic energy from their origin of emission, as the moleculemolecule collisions may be neglected relative to the frequency of the molecule-surface
collisions. Therefore the incoming molecules from the hot region will have a larger
average tangential momentum in the direction toward the cold side which can be seen
in Fig.2.2(a). From conservation of momentum there is an total average tangential
momentum of the outgoing molecules in the direction of the hot region as can be seen
in Fig.2.2(b) and Fig.2.2(c). Because the outgoing molecules are in thermal equilibrium
with dS, the emission velocity only depends on the temperature of dS and is isotrop.
Therefore more molecules must be emitted in the direction of the hot side [34]. This
flow of molecules toward the warmer region is called thermal transpiration effect, or
thermal creep as the molecules will creep toward the hot region along the walls.
Thermal transpiration effect in molecular
A model for deriving maximum value of thermal transpiration effect in the molecular
regime is by considering two gas filled vessels at fixed, different, temperatures. The
vessels are connected by a pipe of length l, and diameter d, as illustrated in Fig. 2.3.
The hot vessel represents the capacitance manometer of internal temperature T2 with
its interconnecting pipe to the cold vacuum vessel at temperature T1 .
Suppose that the vessels are in continuum regime and initially not connected.
If the particle densities in the vessels are the same, the pressure will be greater in
the hot vessel. When connecting the vessels there will initially be a flow of molecules
from the hot vessel to the cold vessel induced by molecule-molecule collisions. In
stationary state the pressure will even out throughout the two vessels and the pressure
9
p2 , T2
HOT
T
d
p1 , T1
COLD
l
Figure 2.3: Two gas filled vessels connected by a pipe of diameter d and of length l,
under the influence of a temperature gradient between the vessels. The pressure and
temperature of the cold vessel is T1 , p1 and of the hot vessel T2 , p2 . The circle shows an
example of a velocity field of a rarefied gas, the arrows along the walls in the direction
of the hot vessel is due to thermal transpiration effect and the ones toward the cold
vessel are due to inter-molecular collisions.
ratio (when T2 > T1 )
R :=
p2
p1
will be unity. If T2 < T1 the pressure ratio is defined as R := pp12 . In rarefaction below
continuum, thermal transpiration effect induces a flow of molecules from the cold to
the hot region in the vacuum system. The flow induced by thermal transpiration effect
opposes the the flow induced by molecule-molecule collisions.
In stationary state the flows are in balance and the pressure will be is higher in the
hot region than in the cold region. The velocity field in Fig. 2.3 illustrates the flows,
the flow in the middle of the velocity field toward the cold region is due to moleculemolecule collisions and the field along the walls in the direction of the hot vessel is due
to thermal transpiration effect.
The magnitude of thermal transpiration effect will increase as the rarefaction increases and reaches its maximum value in the molecular regime called Knudsen value
[17]. In the molecular regime at stationary state it may be derived as follows. Let
10
the cold vessel be denoted by number 1 and the hot vessel number 2. The molecule
1
flux from cold vessel is dN
, where N1 is the number of molecules per area in the cold
dt
vessel. This flux is proportional to the mean absolute velocity of the molecules and
the particle number density
dN1
∼ n(T1 , p1 ) hv(T1 )i .
dt
As inter-molecular collision occurs seldom in comparison to surface-molecule collisions
they may be neglected and the flux from respective sides may be considered independent from each other. Assuming the diameter of the vessel is large in comparison to
the diameter of the connecting pipe the molecular velocities are mainly determined by
the temperature of the vessels. Then the net flux of molecules traveling from cold to
hot side is
dN1 dN2
dN
=
−
.
dt
dt
dt
In stationary state, the net flux is zero,
dN
=0
dt
then
n(T1 , p2 ) hv(T1 )i = n(T2 , p2 ) hv(T2 )i ,
here, it is assumed that the entrances of the pipes are of the same size so that the
constant of proportionality cancels. From the ideal gas law Eq.2.1
p2
p1
hv(T1 )i =
hv(T2 )i .
kB T1
kB T2
From simple kinetic theory, the average kinetic energy of a molecule is given by
1
3
m hvi2 = kB T.
2
2
It means that the average velocity is proportional to square root of temperature hvi ∼
√
T . Then the pressure ration may be expressed as
p2
RK :=
=
p1
11
r
T2
T1
(2.3)
This maximum value of the pressure ratio due to thermal transpiration effect is the
Knudsen value and depends only on the temperatures of the respective vessels. When
the surface interaction is not completely diffuse but has a degree of specular scattering,
the transmission probability will not be the same from both sides of the pipe and an
lower Knudsen value is expected [33].
2.1.3
Knudsen regime
Thermal transpiration effect dependencies
Thermal transpiration effect mainly depends on the temperature difference and the
rarefaction is quantified by the Knudsen number Eq.2.2, which depends on the pressure, temperature, interconnecting pipe diameter and collisional molecular diameter.
The main gas dependency in the collisional molecular diameter is included in the
Knudsen number but it also depends weakly on the microscopic molecule-surface scattering which may change the degree of diffuse scattering. The microscopic moleculesurface interaction depends on the mass and molecular bindings of the molecule, the
surface microscopic smoothness material and the temperature where the interaction
occurs [9]. The semi-empirical equations for thermal transpiration effect assumes that
the molecule-surface scattering is completely diffuse which is often not the case and
therefore typically a slightly lower value than the ideal Knudsen limit is achieved. To
account for this T2 may be interpreted as effective temperature of the capacitance
manometer. Thermal transpiration effect also depends on the geometry of the interconnecting pipe as the geometry changes the Knudsen number which is defined for a
pipe. The actual geometry may be complex in practice, the semi-empirical equations
assume that the geometry is a pipe of diameter d. To consider the non-ideal geometry
and molecule-surface scattering the diameter parameter in the semi-empirical equations may be considered as an effective diameter deff instead of the actual diameter of
the interconnecting pipe [14]. Thermal transpiration effects depends on the length of
the interconnecting pipe relative to the diameter. When the interconnecting pipe is
very short relative to it’s diameter, as in the case when the vessels are connected by
an aperture,
If the scattering is completely diffuse thermal transpiration effect is independent
of the length of the interconnecting pipe and the pressure ratio assumes the Knudsen
12
value in the molecular regime. However, if the scattering is not completely diffuse
but it has a degree of specular scattering, the Knudsen value of the pressure ratio
in the molecular regime is only obtained when the length of the interconnecting pipe
is so short relative to the diameter, so short that it may be considered an aperture.
When it is long relative to the diameter and the scattering is not completely diffuse,
the pressure ratio in the molecular regime assumes a lower value than the Knudsen
value depending on the length of the pipe relative to the diameter [33]. Thermal
transpiration effect depends on the shape of the temperature gradient [6]. In the
typical case of the capacitance manometer with internal heating connected with a pipe
to a vacuum system, the shape of the gradient is simply a linear slope from T2 to T1 .
All semi-empirical equations assumes this shape of the temperature gradient.
2.1.4
Empirical equation
There are several empirical equations for describing thermal transpiration effect. A
summary of the most common semi-empirical equations can be found [10], [40], most
of the equations are based on an approximate solution of the differential equation
1 dT
dp
= Θ(Kn)
p
2 T
where Θ(Kn) is an inverted S-shaped function ranging from 0 to 1. As Kn depends
on pressure and temperature it is difficult to find an analytic solution.The three most
accurate expression for the pressure ratio are Takaishi-Sensui equation [35] (1963),
Miller equation [20] (1963) and Sětina equation [32] (1999), which have been shown
to be more accurate than 0, 5% under ambient conditions for nitrogen [40] and for
argon [10]. All of the equations have the form
−1
R−1
= θ(x); θ(x) = αx2 + βx + f (x)
RK − 1
(2.4)
with different f (x) and x. Takaishi-Sensui calls the form of expressing thermal transpiration effect by RR−1
Degree of thermal transpriation effect [35] , where R and RK
K −1
are pressure ratios defined in Eq. 2.1.2 and Eq. 2.3. θ(x) have been termed degree
of thermal transpiration [35], x is a variable proportional to p2 . α, β are empirical
13
parameters and f (x) is a slow varying function which tends to 1 as x → 0 [40]. The
pressure ratio of thermal transpiration effect may be expressed as
R=
2.1.5
αx2 + βx + f (x) − 1 +
αx2 + βx + f (x)
q
T2
T1
.
Takaishi-Sensui equation
The most widely used, and recommended semi-empirical equation for calibration of
capacitance manometers for thermal transpiration effect [12], is Takaishi-Sensui equation [35]. This equation has the function and parameter 1
√
f (x) := γ x + 1;
x :=
1 p2 d
1, 33 T̄
in Eq. 2.4 where p2 is pressure reading of the capacitance manometer in mbar, d the
diameter of the interconnecting pipe in meter and T̄ is the mean temperature, T̄ :=
T1 +T2
and T1 , T2 the temperatures of the vacuum vessel and capacitance manometer
2
in Kelvin with T2 > T1 . Takaishi-Sensui equation is a modification of Liang’s equation
[19]. Liang’s equation have a function f (x) = 1 in Eq. 2.4, the Takaishi-Sensui equation
have been shown to have better agreement with experimental data [31]. Inserting
Eq. 2.1.7 in Eq. 2.4 the pressure ration have the more familiar form
q
√
T2
2
αx
+
βx
+
γ
x
+
T1
p2
√
=
R=
2
p1
αx + βx + γ x + 1
(2.5)
when T2 > T1 . The numerical value of A∗ , B ∗ , C ∗ (α, β, γ with our notation following
[10]) in the original paper [35] may be used directly and the stated units replaced with
the units used in out definition of the coefficients. If the coefficients are not known
for the measurement gas the recommended procedure [12, 14, 15, 31, 40] is the use of
the molecular diameter in the paper and the expressions for the coefficients from the
Note that in the original paper [35] the definition of x incorrectly omits the power 1, 33−1 of the
factor 1, 33 [12].
1
14
original paper [35] as
4
α = 1.4 × 10 exp 0, 507D/Å
β = 5, 6 exp 0, 607D/Å
γ = 110Å/D − 14
mm Pa
K
mm Pa
K
mm Pa
K
−2
−1
− 12
.
If the molecular diameter is not known it may be obtained from viscous data from
5
η̄ =
16D2
r
mkB T̄
π
where m is the molecular mass, kB Boltzmann constant, T̄ is the mean temperature [35].
However, the calculated coefficient does not accurately reproduce the calibration curve
especially when the molecular diameter is large in comparison to nitrogen and has a
non-physical scaling of γ as it is inverse proportional to the molecular diameter and
turns negative above molecular diameter 7, 9 Å [10]. Yasumoto (1980) [38] suggested
an alternative way to calculate the coefficients,
−9
α = 2.2 × 10
(D/pm)
mm Pa
K
−1
4
β = 0, 75...6
mm Pa
K
γ = (0, 024D/pm − 4, 8)
mm Pa
K
−2
− 12
which has a physical scaling of γ. Although as stated by Yasumoto the diameter
dependency only partially reproduce the experimental values, and is especially limited
for large molecules [10].
15
2.1.6
Miller equation
Miller (1963) [20] stated a universal semi-empirical equation, the coefficients are no
longer dependent on the type of gas but the gas dependency of the molecular diameter
is in the parameter. Miller’s equation is parameterized with the inverse of the Knudsen
number,
d
1 + γx
; xM := Kn−1 =
f (x) :=
1 + δx
λ
used in Eq. 2.4 , where δ is an empirical coefficient and Kn−1 inverse of Knudsen
number. The numerical values of the coefficients α, β, γ and δ , were obtained by best
fit of the experimental data of measurements of H2 , He, Ne, Ar, Kr, Xe,
γ=
α=
1
100
β=
245
1000
5
(0, 024D/pm − 4.8)
2
δ = 2.
In the form of the pressure ratio
α Kn−2 + β Kn +
R=
1+γ Kn−1
1+δ Kn−1
α Kn−2 + β Kn +
−1+
q
T2
T1
1+γ Kn−1
1+δ Kn−1
.
2.1.7
Sětina equation
From pressure ratio measurements of four gases He, Ne, Ar and N2 Sětina (1999) [32]
introduced a normalized pressure scale
xS :=
p2
η̄ v¯Th
; p∗ :=
∗
p
d
q
B T̄
used as x in Eq. 2.4 , where v¯Th = 8kmπ
is the thermal molecular velocity. Turning
the form of Takaishi-Sensui’s equation into a universal equation we note that this is
16
also a parametrization of the Knudsen number by
xS =
5
5
(Kn)−1 = xM
4
4
from Eq.2.2. Then the pressure ratio is
R=
2.1.8
√
q
+ βxS + γ xS + TT21
.
√
αx2S + βxS + γ xS + 1
αx2S
(2.6)
Measurement gas
Tetrahydofuran
In the study of radiation damage in biomolecular systems it is difficult to produce
a pure enough nucleic acid, and as most nucleobases are solids at room temperature
they are not convenient to use as gas target in scattering experiments. Therefore
prototype molecules for the building blocks of the nucleic acids are used. One such is
tetrahydrofuran (THF) C4 H8 O which represents an analogue for the sugar rings that
constitute the phosphate- deoxyribose backbone structure in the nucleic acids, see
the similarity in Fig.1 in [22].Tetrahydrofuran is a liquid at room temperature under
atmospheric pressure with boiling point of 66 ◦C . The molecular diameter is D = 6, 3
Å in [18, 22].
2.2
Vacuum technology
2.2.1
Producing in vacuum
To achieve high vacuum first the air molecules have to be evacuated from the vacuum
system. Depending on the rarefaction of the gas different pumps must be used. The
first stage pump is most effective in the continuum regime and the second stage in the
molecular regime. For the first stage pump a rotary displacement pump is used, it let
gas flow into a exhaust which is a enlarging volume that then is sealed off and then
the gas is compressed and then let out in atmospheric pressure. This principle works
well in continuum regime as the gas behaves as a fluid and flows into the enlarging
volume as the gas the pressure is lower pressure in the expanding volume than in the
17
vacuum system. To further lower the pressure in the vacuum system and increase the
rarefaction below continuum a second stage pump must be used, such as a molecular
dispersion pump. One molecular dispersion pump is the turbo molecular pump, which
works on the principle that the gas molecules are given momentum in the direction of
the exhaust by repeated collision with a moving solid surface.
There are several sources which continuously increase the number of molecules in the
vacuum vessel. Leaks let in air into the vacuum vessel, typically due to untight connections of the vacuum equipment. Water vapour absorbed into the materials, fingerprints,
slow down the evacuation process. Outgassing is desorption of molecules from the interior walls of the container. Desorption of molecules typically starts to contribute to
the gas load below 1 × 10−1 mbar . As pumping continues and the pressure decreases
contribution of degassing becomes stronger and sets the limit for which pressure is
achievable in the vacuum system. Other sources are chemical processes, heating of
materials causes them to release gas or chemical reactions, or back-stream of oil from
the pumps. When measuring the pressure of a test gas all these sources balance with
the pumping system. [5].
18
Environment
Leaks
Vapor
Vacuum vessel
Test gas
Pump
Outgassing
Figure 2.4: Principle of a vacuum vessel with pumping system. Leaks, inlet of test
gas, outgassing and release of vapor contributes to the gas load.
2.3
Pressure measurement
This sections begins with a brief history of pressure measurements with the
origin of some unit and introduces the terminology used in pressure measurements, followed by the principles of the pressure measurement gauges
used in the experiment.
Introduction to pressure measurement, units, terms
There are several used units of measuring pressure such as Torr, mHg, mbar, Pa. Evangelista Torricelli (1608-1647) was the first person who measured atmospheric pressure
with his invention of the mercury barometer, a glass tube in a bowl of mercury. The
mercury in the glass tube sinks down and produces vacuum in the evacuated space
inside the glass tube. The pressure inside the glass tube is balanced with atmospheric
19
pressure on the mercury in the bowl. This device measures pressure relative to zero
pressure, this is called absolute pressure. As the height of the mercury depends on the
force per unit area exerted on the mercury in the cup it direct measures pressure. At
atmospheric pressure the height of the column is 760 millimeters . This pressure unit is
mmHg, millimeters of mercury, it is called Torr after Torricelli. The SI (Systeme International) unit of pressure is newton per square meter, defined as pascal Pa. The unit
expressing pressure in terms of atmospheric pressure is called standard atmospheric
pressure
1 atm = 760 mmHg = 760 Torr = 101, 3 Pa,
pressure below atmospheric pressure is defined as vacuum. Pressure with reference to
atmospheric pressure, and not relative to zero as the mercury barometer, is called gauge
pressure. When the reference is in the same system as pressure is being measured the
pressure is called differential pressure.
2.3.1
Indirect gauges
Gauges measuring some property which depends on the gas pressure is called indirect
gauges, such as Pirani gauge 2 . This gauge has a filament with a constant power
running through it, so that when the gas pressure is constant, the temperature of the
filament is constant. When the gas pressure increases there will be an heat loss from
the filament causing a decrease in temperature. Then the gas pressure can be indirectly
measured either by measuring the temperature of the filament, or by measuring the
change in resistance in the filament caused by the change in temperature. Today the
most common Pirani gauges measure the required power to a bridge circuit such that
the resistance (temperature) remain constant as the gas pressure changes. However, the
temperature will be almost constant in the full continuum and full molecular regime as
the gas pressure changes. Lower pressure can be measured using hot cathode ionization
gauge. This gauge relates the current produced by ionizing gas molecules with a hot
cathode filament. This current is proportional to the number density of particles, which
is related to pressure by the ideal gas law. Bayard-Alpert gauge, is a combination of a
Pirani and a hot cathode ionization gauge, where the upper part of the span is measured
by the Pirani and lower by the ionization gauge. Indirect gauges are however highly
2
Marcello Pirani (1880- 1968), laboratory head at Siemens & Halske AGs bulb factory in Berlin
20
gas dependent. The thermal conductivity in Pirani gauges varies with the mass of the
gas molecules, the heavier the gas molecules the lower the thermal conductivity. In
ionization gauges, for example, smaller molecules have tighter bindings, and therefore
require more energy to ionize [11]. Pirani and Ionization gauges are most commonly
used for monitoring the quality of vacuum, for accurate pressure measurements the
capacitance manometer is commonly used. In comparison to the Pirani and ionization
gauge which measure indirect pressure, the capacitance manometer measure direct
pressure by deflect of diaphragm. For example a Pirani typically have 2-25 % the
accuracy of its readings while in the same range a capacitance manometer may have
2.5 × 10−1 % [3].
2.3.2
Capacitance manometer
The capacitance manometer measures direct pressure and it has relatively high accuracy, repeatability, resolution, stability and fast response and is relatively resistant to
corrosive gases. Therefore it is often used as a transfer standard [16]. The capacitance
manometer is an direct pressure measurement device as it measures the deflection of
an elastic diaphragm Fig. 2.5 a). It may sense a deflection of the diaphragm by 0, 4
nm. The deflection is induced by a pressure differences between the measurement
side Fig. 2.5 b) and reference side Fig. 2.5 c) of the diaphragm. When the reference
side behind is evacuated the capacitance manometer measures absolute pressure. The
pressure in the reference side Fig. 2.5 c) is typically in the order of 1 × 10−7 mbar,
the vacuum in the reference side is supported by a chemical getter pump Fig. 2.5
g) [13]. When the pressure is greater on the measurement side than on the reference
side the diaphragm is bent toward the electrodes Fig. 2.5 e). This deformation causes
the change of the capacitance of the two electrodes, which are metallic films situated
on a ceramic disc Fig. 2.5 e). Both electrodes will sense a change in capacitance by a
different amount, this difference is sensed with a bridge circuit driven by a sine-wave
oscillator and amplified by the buffer , see Fig. 2.6 and converted to a DC voltage signal
output [13]. When the diaphragm is flat, the bridge is balanced and gives 0 voltage
output. When the diaphragm deformed, the bridge circuit becomes unbalanced and
gives a voltage output proportional to the deformation. This signal is then linearized
with signal conditioning. The measurement side, the enclosure and the diaphragm is
21
Electrode Structure
(Metal on Ceramic)
g) Chemical
Getter Pump
a) Diaphragm
Electronics
Baffle
f) Span pot
e) Linearity pot
d) Zero pot
Electrical
Connector
b) Measurement (Px) Side,
Connected to System
c) Reference (Pr) Side,
Constant Temperature
High Vacuum
Oven (Heated-Temperature Controlled
Units Only)
Figure 2.5: Principle sketch of an absolute capacitance manometer from MKS. The
left side is connected to the vacuum vessel, behind the baffle is the diaphragm dividing
the measurement side and the reference side. The vacuum in the reference side is
supported by a Getter pump. Behind the diaphragm is a ceramic disk on which there
are metal pattern in the form of a circular bull’s eye surrounded by a ring. All of
which are encapsulated by an electrically heated oven. On the outside at the end of
the capacitance manometer there are three pots, two of which are sealed. Only the zero
pot, Fig. 2.5 d), is adjustable without opening the sealing. This figure is a modification
of the principle sketch Fig.7 in [11]
made of corrosion-resistant materials to prevent contamination of the instrument and
increase stability and accuracy. Typically either as ceramics and glass, or INCONEL,
a nickel / chromium alloy. INCONEL capacitance manometers have been shown to be
22
Oscillator
Bridge
Reference
Buffer,
Scaling,
Amplification
DC signal
Output
Figure 2.6: Sketch of the electronics of the capacitance manometer. The inner and
outer electrodes sense a change in capacitance by a different amount, this difference
is sensed with a bridge circuit driven by a sine-wave oscillator and amplified by the
buffer and converted to a DC voltage signal output [13]. This figure is a from Fig.7
in [11]
more resistant to contamination than previous ceramic design, and exhibits less shift
in offset when cycling to atmospheric pressure [13]. When the diaphragm becomes
contaminated, the offset can change. After long time of exposure the instrument may
be so contaminated such that the offset is not possible to set to zero. The output
voltage is proportional to the deformation of the diaphragm, for the output voltage to
be proportional to pressure the capacitance manometer is calibrated. Calibration is
made using three potentiometers, span Fig. 2.5 f), linearity Fig. 2.5 e) and zero Fig. 2.5
d). When the span is off, the range will be smaller or larger. When the linearity is not
properly set, the zero and F.S (full scale) may be at the right place, but in between
there may be error due to non-linearity.
The span Fig. 2.5 f) and linearity Fig. 2.5 e) potentiometers are sealed, they may
only be when calibrated of the instrument is preformed. The zero potentimeter Fig. 2.5
d) may be set by the user. To increase accuracy and improve repeatability capacitance
manometers may have internal heating element with an internal temperature typically
in the range of 35 ◦C - 200 ◦C . The internal heating decreases sensitivity to the change
in ambient temperature. The drift in zero may be reduced with a higher temperature
than 45 ◦C to prevent contamination depending on which gas is to be measured [2].
The capacitance manometer measures direct pressure. However, in a rarefied gas if the
23
capacitance manometer has internal heating due to the thermal transpiration effect
the pressure will be higher in the capacitance manometer than in the vacuum vessel,
first pointed out by J.Philip Bromberg [8]. With 45 ◦C internal heating the error
due to thermal transpiration effect may be up to 4 %, with 200 ◦C it may be up
to 27 % . Even without internal heating thermal transpiration is considered when
preforming calibration as due to heating from the electronics the error induced by
thermal transpiration effect is up to 0.5 % .
24
Chapter 3
Method
This chapter covers the experimental setup, software algorithm and two
measurement methods for thermal transpiration effect.
3.1
3.1.1
Experimental setup
Vacuum system
The experimental setup consists of a vacuum vessel with a pumping system capable
of bringing down the pressure to 1 × 10−8 mbar. The pressure of the vacuum vessel is
measured with three capacitance manometers, two of which with a span of 1 mbar 1 × 10−4 mbar and one with 1 × 10−1 mbar - 1 × 10−5 mbar, and one BA Ionization
gauge with a measuring range 1 × 103 mbar - 5 × 10−10 mbar. Temperature is measured with 8 thermocouples and all measurements are controlled by a PC, which also
regulates the heating bands wired around the vacuum vessel. The pumping system is
connected through a small aperture to the vacuum vessel and is in balance with the
inlet of the gas so that a stable pressure in the vacuum vessel is established.
26
(O) CM (200
631B
(N) CG:
CMR 364
(Q) MSM
DN63CF/DN40CF
DN40CF
C):
DN16CF
DN40CF
DN40CF
o
DN16CF
DN40CF
DN40CF
DN40CF
DN63CF/DN40CF
(M) BAG :
PBR 260
DN40CF/DN16CF
(J) 304L
DN16CF
(K) DN40CF
(L)
DN63CF/DN40CF
DN63CF/DN40CF
(H)
(P) CM (100
CMR628
o
C):
DN40CF/DN16CF
(D) PG: TPR 280
DN16CF
DN40CF
(I)
DN16CF
N2
DN16CF
THF
(C)
(A) RVP:
Trivac16B
(B)
(E) TMP: TURBOVAC50
(F)(G)
Figure 3.1: Description of the vacuum system according to DIN-28401 ( [19] DIN28401. Vakuumtechnik Bildzeichen Ubersicht. Beuth, Berlin, 2008). The letters in
parenthesis referee to the equipment in table 3.1 .
3.1.2
Pumping system
The vacuum vessel has a two stage pumping system consisting of a fore-vacuum pump,
which is a rotary vane pump (A) [28] and a turbo molecular pump (E) [27]. The rotary
vane pump and the rest of the pumping system are equipped with a valve (C) which is
used shutting the RVP to prevent back-suction of oil into the vacuum vessel. The forevacuum pump may bring down the pressure in the pumping system and vacuum vessel
to the order of 10−2 mbar . The fore-vacuum pump is most effective in continuum
regime as the pumping principle is to suck out the gas. To bring down the pressure
further the turbo molecular pump (E) [27] is turned on, which hits gas molecules in
the direction of the outlet. When the gas load due to leaks is negligible the pressure
is typically in the order of 10−6 mbar, then the gas load is due to outgassing and
water vapour. When baking the vacuum vessel, heating it up with heating bands, the
27
.
Figure 3.2: Photo of the experimental setup with nitrogen as test gas. On the left side
of the experiment the three CM can be seen. The experiment is wrapped in heating
bands, and the temperature is measured with
28 TC (the green wires). On the right side
of the experiment is the BAG ionization gauge, and in the middle of and the upper
part of the photo is the MS. Underneath the experiment the TMP and the RVP can
be seen.
Index
Item
Manufacturer
Model
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
Rotary Vane Vacuum Pump
Exhauast Filter with Lubricant Feedback
Shutoff Vacuum Valve
Compact Pirani Gauge
Turbomolecular Pump
Electronic Frequency Converter
Air Cooling Unit for TURBOVAC
Twisting Valve
Aperature
6-Way Cross
All-Metal Dosing Valve
Cylinder Pressure Regulator
Compact FullRange BA Gauge
Ceramic Capacitance Gauge
High Temperature Capacitance Manometer
High Temperature Capacitance Manometer
Control Valve
Oerlikon Leybold V.
Oerlikon Leybold V.
BALZERS
PFEIFFER V.
Oerlikon Leybold V.
Oerlikon Leybold V.
Oerlikon Leybold V.
VAT
Trivac B - D 16 B
AR
EVA 025 HX
TPR 280
TURBOVAC 50
TURBOTRONIK NT 10
Air cooler for TURBOVAC 50
UHV Gate Valve 10846-CE01
PFEIFFER V.
Oerlikon Leybold V.
MESSER
PFEIFFER V.
PFEIFFER V.
MKS I.
MKS I.
PFEIFFER VACUUM
Stainless Steel 304L
UD 040 HR
FM61
PBR 260
CMR 364
631B01MBFP
628
EVR 116
Table 3.1: Equipment in Fig. 3.1. The company names are shortened, V. stands for
Vacuum and I. stand for Instruments.
pressure further lowered to 10−8 mbar - 10−9 mbar. Pressure in the pumping system
in between the RVP and TMP is monitored by a Pirani gauge (D) [26]. The Pirani
gauge is connected to a electronic frequency converter (F) [24] which controls the turbo
molecular pump. The Pirani gauge make ensures that the pressure is not too high for
the TMP to operate, and to prevent overheating it is equipped with a air cooling unit
(G) [25]. The vacuum vessel and the pumping system is divided with a twisting valve
(H), which may be used to evacuate the system. To establish a constant flow rate and
stable pressure the vacuum vessel is connected by a small cooper aperture (I) with a
diameter of 2 mm. The chamber of the vacuum vessel is a 6-way cross (J). Gas flow
rate into the vacuum vessel is adjusted with a dosing valve (K) [23]. The gas bottle
has cylinder pressure regulator (L) to reduce the outlet pressure of the bottle. The gas
bottle is nitrogen, with a purification of 99.999%. Tetrahydrofuran is liquid in room
temperature and under atmospheric pressure with boiling point of 66 ◦C. The vapor
pressure may be measured when first lowering pressure in the connecting pipe between
the pipe to the dosing valve and the bottle, see Fig. 3.1. This is done by letting the
fore vacuum pump be connected to the connecting pipe of the gas bottle and vacuum
vessel. Once the vapour pressure has been decreased, it may be let into the vacuum
29
vessel. The vapour pressure is sensitive to pressure and temperature. Therefore, to
decrease heat loss due to convection of surrounding air and thereby increase stability
in the gas inlet the connecting pipes between the bottle and the dosing valve have been
wrapped in foil.
3.1.3
Pressure
The vacuum vessel is equipped with four pressure measurement devices, Bayard Alpert
gauge (M) [30] with a range of 1 × 103 mbar - 5 × 10−10 mbar, a Ceramic Capacitance
Manometer (N) [29] and two Capacitance Manometers with Internal Heating from
MKS with sensor temperature of 100 ◦C (O) and 200 ◦C degrees (P) [21]. The
capacitance manometer with 100 ◦C internal temperature (O) has Full Scale of 0.1
mbar measuring 4 decades, the one with 200 ◦C degrees (P) has Full Scale of 1 mbar
and also measure 4 decades. The diameter of the interconnecting pipe is d = 11.049 mm
and the respective uncertainties are listed in Table 6.1 in appendix. The measurements
are read by the instrument controllers, a the data is fed to the PC.
3.1.4
Heating
The vacuum vessel is wired with 6 heating tapes (h) [36]. The electricity to the heating
tapes is controlled either to be on or off by a USB-hub (d). The USB-hub is controlled
by the PC (e). Temperature is measured by 8 thermocouples (f), the measurements
are read by USB Thermocouple data logger (g) and fed to the PC. To increase the
stability of the temperature, as a lot of heat is lost due convection of surrounding air,
the heating bands have been wrapped in foil and the experiment enclosed in a plastic
box. To prevent heat loss due to diffusion ceramic feet have been placed under vacuum
vessel.
30
Pirini
(f)
(i) MSM
(g)
Thermocouple
data logger
BA ion
(b)
Ethernet
(d) USB Switch
USB
Pfeiffer
controller
CM
USB
RS-232
Electrical socket
(e) PC
CDM (100
o
C)
CDM (200
o
C)
RS-232
(a)
(h) Heating bands
MKS
controller
Figure 3.3: 6 Heating tapes (E) [36] attached to vacuum vessel controlled via a USBhub (D), which is controlled by the PC. Temperature is measured by 8 thermocouples
(A), the measurements are read by USB Thermocouple data loggers (B) and fed to
the PC.
Index
Item
Manufacturer
Model
Connection
a
b
c
d
e
f
g
h
i
Controller
Controller
MKS
PFEIFFER
PR4000B F2V2
MaxiGauge TPG 256A
RS232
RS232
USB Electricity Switch
Computer
Thermocouple
Thermocouple data logger
Heating Tapes
Interface for EVR 116
CLEWARE
Intel
Pico Technology
Pico Technology
THERMOCOAX ISOPAD
PFEIFFER VACUUM
USB-Switch 4
Pentium 4
GTF 300GS
USB TC-08
IT-S45 9903009
IF 200
USB
Table 3.2: Table of equipment in Fig. 3.3.
3.2
Software
The PC (e) acquires the measurement data and regulates the temperature of the heating bands. It reads the controllers (a),(b) and the thermocouple USB-hub (g). The
controllers are connected by RS-232 connections. The program saves all the measurements and the settings used for the experiment and controls the USB electricity switch
(d) either to give power to the heating bands or not (h).
3.2.1
Measurement and temperature regulation program
The measurement and temperature regulation program is written in LabVIEW (2012
Version 12.0.1f2). The program runs through three stages. INITIATE, RUN and SHUT
DOWN. In the first stage, all variables are defined and the communication between all
the devices is opened. When the INITIATE stage is completed, the program moves
on to the RUN stage. In the RUN stage there are five loops running independently
of each other , see Fig. 6.3. The loops run independently which makes the program
robust toward failure of any device. If there is an error in any of the instruments, for
example if the contact is lost or a spike in electricity supply, the error is contained
within the loop and the others keep going. This is particularly important to prevent
overheating if the loop controlling the heating bands would freeze.
START
Read UI parameters
Open all USB and RS-232
communication
Define all variables
RUN
Regulate temperature
Read Pressure
Measure Count down
Read Temperature
Save measurements
Turn off heating
Close all USB and RS-232
communication
STOP
Figure 3.4: Flow scheme of the measurement and temperature regulation program
using standard flow scheme symbols except the ellipsoid with arrow which represents
a loop. The loops are inside the subprogram Run and if there is an error in any loop
the other ones keep running.
As the measurements of pressure and temperature occurs in different loops they
need be synchronized. The algorithm of synchronizing the readings is shown in Fig. 3.5.
The Measure count down synchronizes the time of the readings and the frequency of
the data logging.
When the STOP & SAVE button is pressed on the user interface, see Fig. 3.6,
all five loops stop, and the program proceeds to the last stage. In the SHUT DOWN
stage, the communications of all devices are closed.
In the settings window of the program , see Fig. 3.7 the thermocouples used for
calculating the regulation temperature and calibration of the regulation temperature
may be adjusted. Here the heating may be scheduled.
P : array with all sensors pressuer
START RUN
T : array with all sensors temperature
Temperature
too hgih or too
low?
Heating bands off
Heating bands on
Read
temperature
Read
pressure
Add readings to P
Add readings to P
No
No
Ready to save
pressuer?
Yes
Time to save
reading?
Yes
Ready to save
pressuer?
No
Yes
Add mean of P
and of T to file
STOP RUN
Figure 3.5: Flow scheme of the synchronization of the loops within the program run
Run Fig. 6.3
Figure 3.6: User interface of measurement and temperature regulation program. There
is a big pressure vs time in the middle, and a small temperature vs time in the right
upper corner. The STOP & SAVE button in the middle saves the files.
Figure 3.7: User interface of the measurement and temperature regulation program,
this is the settings page. In the upper left corner is the input data for the calibration
of input value of temperature and actual output temperature. In the bottom left, the
green dots associate thermocouples to heating bands. In the middle there is a scheme
of temperatures for a certain amount of hours. In the upper right is the settings for
the communication ports used by the pressure reading of the controllers. Here is also
the input of the speed of the loops.
3.3
Measuring thermal transpiration effect
In this section two ways of measuring thermal transpiration effect will be
described. One method is based on varying the pressure, and the other is
carried out varying the temperature of the vacuum vessel.
3.3.1
Relative measurement method, variation of the system
pressure
In the measurement method variation of the the pressure of the vacuum
vessel, one needs to confirm that there is no pressure dependent bias in
the measurand and that the only non-linearity in the measurand is due to
thermal transpiration effect. This is typically done by setting the temperature of the instruments to the same temperature by turning on or off the
internal heating [7, 10, 31] so that there is no thermal transpiration effect,
then by quantifying the error and correcting the measurand for any pressure dependent bias. However, this is not possible with the pressure gauges
used in this experimental setup as the internal heating cannot be turned on
or off. Instead, the measurand is first corrected for thermal transpiration
effect by using Takaishi-Sensui equation with nitrogen as test gas, which
is the standard procedure when calibrating gauges [39] as the TakaishiSensui equation has shown to be accurate [40] for nitrogen. After removing
thermal transpiration effect from the pressure ration using Takaishi-Sensui
equation, it is then possible to correct for any pressure dependent bias.
After this has been done, thermal transpiration effect may be measured for
any other gas.
Instruments
Three capacitance manometers are used, all of which have different internal temperature. The capacitance manometer from Pfeiffer, the measurement of which is denoted
p1 , have no internal heating element. However it does have a slightly elevated internal temperature due to heating of electronics. It is estimated less than 3 ◦C degrees
above ambient temperature. It has a helicoidal baffle instead of a diaphragm as the
gauges from MKS, however this does not affect the overall magnitude of thermal transpiration effect as shown in Monte-Carlo simulation [37]. The capacitance manometers
from MKS instruments are capacitance diaphragm manometers with internal heating
of 100 ◦C and 200 ◦C degrees. The pressure signal of the capacitance manometer with
internal heating of 100 ◦C degrees is denoted p2 and the one with 200 ◦C degrees p3 .
Measurement method
The unmodified signal of the measurand, the pressure ratio is denoted as R100 := pp21 ,
R200 := pp32 , R300 := pp31 , when subscript is neglected the argument holds for all measurands.
The deviation of R00 is due to thermal transpiration effect, calibration and uncertainty
of the instruments. To determine, and correct, for the deviation due to the calibration.
Influence of thermal transpiration effect is removed by multiplying the inverse of the
pressure ratio calculated using Takaishi-Sensui equation or Sětinas equation, denoted
RCALC with nitrogen,
R0 := R00 /RCALC .
R3′′
R2′′
p3 , T 3
CDM
R1′′
p2 , T 2
p1 , T 1
CDM
CDM
p0 , T 0
38
Vacuum vessel
Then the deviation from unity, or zero when plotting the relative sensitivity defined as R0 − 1, is due to the calibration and is quantified using 7 degree polynomial fitted to the relative sensitivity rCl P = c0 + c1 log (p/mbar) + c2 log (p/mbar)2 +
c3 log (p/mbar)3 + c4 log (p/mbar)4
+ c5 log (p/mbar)5 + c6 log (p/mbar)6 + c7 log (p/mbar)7 . When the polynomial P is
fitted to the relative sensitivity using Takaishi-Sensui equation, R00 /RCALC − 1, the
correction polynomial is denoted P TS , or P S when Sětinas equation is used. Then the
relative sensitivity with the polynomial correction R0 /P − 1 should be close to zero
and scattering of data due to uncertainty. Then the pressure ratio due to thermal
transpiration effect relative to Takaishi-Sensui equation of nitrogen is
RTS := R00 / P TS+1 ,
and may be used to measure any gas, similarly when using Sětinas equation.
High vacuum
Background pressure of at least 1 decade lower than measure range has to be obtained,
to ensure that the gas measured is the major contributor to the gas load. This is done
with by using the mass spectrometer to see what was contributing to the gas load in
the vacuum vessel. Detecting air indicates leaks, detection of water is due to water
vapour on the interior surfaces, detection of oil is either from back suction from the
vacuum pumps or from finger prints. Leaks were found by spraying helium on the
connections outside the vacuum vessel and monitor any increase in the detection rate.
If there are no leaks, the main contribution to the pressure was from water vapour.
The outgassing of water vapour and oil is steadily being pumped out, this process
is significantly shortened by heating vacuum vessel. Background pressure with the
twisting valve open has been reached down to ultra-high vacuum 1 × 10−8 mbar , and
1 × 10−6 mbar with the twisting valve closed.
39
Varying pressure
At pressure below the reading level of the instruments, the offset was set to zero. After
zeroing the Twisting vale was closed and when the background pressure was stable,
the pressure was slowly increased till reading level was reached by opening the valve
between the gas bottle and the dosing vale, as the dosing valve is not completely tight.
The pressure was was further increased by opening the dosing vale. Once the pressure
has been stabilized in the vacuum vessel, the first point may be measured. The time
for the instruments and the pressure in the vacuum vessel to stabilize should be at
least 20 minutes per measurement point. After measuring one point, the pressure was
successfully increased by further opening of the dosing valve till full scale was reached,
then it was successfully decreased until the dosing valve is completely closed. One
cycle takes therefore at least one day.
3.3.2
Absolute measurement method, variation of the system
temperature
Instruments
For this method only one capacitance manometer is used with internal temperature at
T0 = 200 ◦C , and the BAG ionization gauge to monitor the vacuum vessel pressure.
Measurement method
By varying the temperature of the vacuum vessel T1 the influence of thermal transpiration effect will vary. The particle density and the temperature of the capacitance
manometer with internal heating T2 is fixed. When the vacuum vessel is heated to the
same temperature as the capacitance manometer with internal heating so that T1 = T2 ,
can be seen in Fig. 3.8 a), there is no thermal transpiration effect and the pressure in
the vacuum vessel and the pressure reading of the capacitance manometer will be the
same, which is designated as p0 . Let p∗ be a successive pressure reading at vacuum
vessel temperature of T1 lower than T2 , see Fig. 3.8 b).
The difference between p0 and p∗ is due to thermal transpiration effect, temperature
coefficient of the capacitance manometer and change in particle density. Temperature
coefficient arises due to the stress of the gauge head caused by thermal expansion or
40
a)
p0 , T 0
b)
CDM
p* , T0
CDM
p0 , T 0
Vacuum vessel
p, T
Vacuum vessel
Figure 3.8: The capacitance manometer with internal heating measures p∗ with sensor
temperature at T2 , with vacuum vessel temperature T1 < T2 .
contraction as the temperature changes. If p is the pressure reading corrected for this
effect, the quotient
p
p0
is due to thermal transpiration effect.
Experimental method
After the system has been cleaned and the heating calibrated the vacuum vessel temperature is set to 180 ◦C . When the temperature and instruments stabilize at the new
temperature, this typically takes half a day, the offset of the instruments is set to zero.
To increase the pressure to desired value p0 , first the twisting valve is closed and after
the pressure have stabilized it is increased further by slowly opening the dosing valve.
The temperature of the vacuum vessel is then stepwise decreased and the pressure is
measured once the temperature and instruments have stabilized. When the temperature of the vacuum vessel have reached room temperature, the temperature is stepwise
increase back to 180 ◦C . This process was repeated for different initial pressures.
41
Chapter 4
Result
Results include measurements of thermal transpiration effect using the methods of
variation of the system pressure and system temperature. The gases measured include
nitrogen, argon and tetrahydrofuran vapour.
4.1
Variation of the system pressure
In this method the pressure is varied and the system temperature and the
temperature of the gauges is kept constant. The gauges are calibrated with
nitrogen as test gas. Thermal transpiration effect of tetrahydofuran and
argon is measured relative to calibration curve of nitrogen calculated using
Takaishi-Sensui and Sětina’s equation.
Typical data of thermal transpiration effect measured by varying inlet pressure is
shown in Fig. 4.1. As the pressure increases the effect decreases because the rarefaction
decreases. The mean value, or time average, is taken when the pressure have reached
stationary state. This mean value is used in calculation of the pressure ratio, denoted
in Fig. 4.1 as R300 = p3 /p1 . The time for the pressure to stabilize depends mainly on
how much the pressure inlet is increased, at which pressure it is stabilizing and how
fast the inlet is opened. Ambient temperature during the experiment was measured to
be T0 = 26 ◦C and stable within 1 ◦C .
The deviation of the unmodified pressure ratio R00 from unity in Fig. 4.2 is due
to thermal transpiration effect, calibration and uncertainty of the instruments. The
42
Typical data of tetrahydrofuran (THF) by method of varying pressure
6x10
-2
300
p1
p3
-2
p / mbar
T1
4x10
295
T = 473,15 K
-2
3
T / Kelvin
5x10
290
3x10
-2
285
0
10
20
30
40
50
60
t / min
Figure 4.1: Typical measurement data from the method of varying pressure. The
left axis is the measured pressure over time, the red curve is the measurement of the
capacitance manometer with internal heating of 200 ◦C , p3 , and the black curve the
reference capacitance manometer at room temperature measuring p1 . The right axis is
the temperature, the top curve is the measured temperature of the vacuum vessel over
time. The temperature in this this figure is slightly higher than room temperature as
the heating of the system is on.
influence of thermal transpiration effect in R00 is removed by multiplying the inverse
of the pressure ratio calculated RCALC using Takaishi-Sensui or Sětina equations. The
remaining deviation from unity is due to calibration and uncertainty of the instruments.
In Fig. 4.2 a),b) and c) shows the unmodified pressure ratios of nitrogen and the
calculated calibration curves using Takaishi-Sensui with nominal parameters (red solid
curve), with effective diameter (red dashed curve) and Sětina and Millers equation
with nominal parameters. The relative sensitivity, R00 /RCALC − 1, are the black dots
in Fig. 4.3 a) , Fig. 4.4 a) , Fig. 4.5 a) using Takaishi-Sensui equation and a), b), c)
in Fig. 4.6, using Sětina equation. The spread in the data of the relative sensitivity
reflects the uncertainty and the deviation from unity a systematic pressure dependent
43
bias. The bias is quantified using a fitting a 7 degree polynomial, P , (red thick curve)
to the relative sensitivity (black dots) with the corresponding coefficients in Tab. 4.1.
The red dots shows corrected data R0 / (P + 1) − 1, setting the relative sensitivity close
to zero.
The correction polynomial align the unmodified pressure ratio R = R00 / (P + 1) to
the calculated pressure ratio curve, as shown in Fig. 4.3 b) , Fig. 4.4 b) , Fig. 4.5 b)
using P TS and d), e), f) in Fig. 4.6 using P S . The correction polynomial may be used
to correct the unmodified pressure ratio for other gases, when measuring the pressure
ratio for other gases as in Fig. 4.3 c) , Fig. 4.4 c) , Fig. 4.5 c) using P TS and a), b), c)
in Fig. 4.7 using P S for tetrahydrofuran, and for argon in Fig. 4.8 using P TS and P S .
In Fig. 4.3 c) the red dots shows the corrected pressure ratio relative to TakaisiSensui equation R3TS , to this data the coefficients of Takaishi-Sensui equation are fitted
(black curve) with coefficients in Tab. 4.2.
For R2TS , Fig. 4.4 c), the coefficients fitted to R3TS has similar shape to the trend of
the measurements but is slightly shifted. Using effective diameter and the coefficients
obtained from the fitting of R3TS for R2TS the curve agrees well with the measurements
(dotted black curve). For R1TS , Fig. 4.5 c), both curves with the fitted coefficients of
R3TS with nominal parameters and effective diameter from R2TS is shown. In Fig. 4.7
RS of tetrahydrofuran is measured and the calculated calibration curve of Sětina is
plotted, for R3S ,R2S the calculated curve shows too slow increasing slope but agrees
quite well with R1S . Measurements of argon in Fig. 4.8 showing both RTS and RS and
the calculated calibration curves of Takaishi-Sensui and Sětina equation are shown.
The difference between RTS and RS is relative little compared to the unmodified pressure ratio. For R1S and R2S the calibration curve of Sětina shows the most similar
measurements to the calculated curve.
4.2
Measurement uncertainty analysis
The instrumental uncertainty at different pressure of the MKS capacitance manometer
with 200 ◦C internal heating reading p3 and Pfeiffer capacitance manometer reading
p1 , the instrumental uncertainty can be found in Tab. 6.1.
The pressure regime have very large instrumental uncertainties in comparison to
the reading. The largest contribution to the uncertainty is the temperature coefficient
44
c0
c1
c2
c3
c4
c5
c6
c7
P1TS
P2TS
P3TS
P1S
P2S
P3S
-0.59438
-2.30696
-3.41449
-2.31353
-0.52401
0.16626
0.10454
0.01434
3.1539
15.32216
30.88018
33.66948
21.53195
8.07731
1.64398
0.13995
-0.00258
0.06712
0.35202
0.90491
1.20288
0.779
0.23679
0.02714
-0.09112
0.25794
1.9692
3.69777
3.29262
1.54428
0.36806
0.03507
3.27062
16.20394
33.28549
36.93293
23.92249
9.03805
1.84371
0.15677
-0.00254
0.09341
0.47247
1.11124
1.27495
0.7323
0.20436
0.02207
Table 4.1: Coefficients of the 7 degree polynomial used for the in Fig. 4.3 a) P3TS ,
Fig. 4.4 a) P2TS , Fig. 4.5 a) P1TS and a), b), c) in Fig. 4.6 P3S , P2S , P1S .
a
b
c
Molecular diameter / m
Nitrogen
Argon
1.20E+06
1.00E+03
14
3.75E-10
1.08E+06
8.08E+01
15
3.42E-10
THF
THF-Fit, rel. TS
5.00E+06
1.00E+01
1
6.30E-10
Table 4.2: Coefficients used in the calculation of the calibration curve using TakaishiSensui equation, nitrogen and argon from literature [35], the coefficients of THF is
from Fig.4.3. The molecular diameters used in the calculation of the calibration curve
for nitrogen and argon [4] and for tetrahydrofuran [18, 22]
Pressure reading at
1,00E+00
1,00E-01
1,00E-02
1,00E-03
1,00E-04
mbar
Measurement uncertainty
Resolution
Zero temperature coefficient
Span temperature coefficient
2,00E-03
3,00E-05
1,50E-04
1,00E-04
2,00E-04
3,00E-05
1,50E-04
1,00E-05
2,00E-05
3,00E-05
1,50E-04
1,00E-06
2,00E-06
3,00E-05
1,50E-04
1,00E-07
2,00E-07
3,00E-05
1,50E-04
1,00E-08
mbar
mbar
mbar
mbar
Total uncertainty
2,28E-03
3,90E-04
2,01E-04
1,82E-04
1,80E-04
mbar
Total uncertainty
0,23
0,39
2,01
18,21
180,21
% of reading
Table 4.3: Typical instrumental uncertainty calculated at four decade of pressure with
uncertainties from Table.6.1 of the Pfeiffer ceramic capacitance manometer. Measurement uncertainty includes non-linearity, hysteresis and non-repeatability. 1 K of
ambient temperature changed is assumed.
45
induced uncertainty, particularly of the temperature coefficient of the offset.
Pressure reading at
1,00E+00
1,00E-01
1,00E-02
1,00E-03
1,00E-04
mbar
Measurement uncertainty
Resolution
Zero temperature coefficient
Span temperature coefficient
5,00E-03
1,70E-05
1,60E-04
1,00E-04
5,00E-04
1,70E-05
1,60E-04
1,00E-04
5,00E-05
1,70E-05
1,60E-04
1,00E-04
5,00E-06
1,70E-05
1,60E-04
1,00E-04
5,00E-07
1,70E-05
1,60E-04
1,00E-04
mbar
mbar
mbar
mbar
Total uncertainty
5,28E-03
7,77E-04
3,27E-04
2,82E-04
2,78E-04
mbar
Total uncertainty
0,53
0,78
3,27
28,20
277,50
% of reading
Table 4.4: Typical instrumental uncertainty calculated at four decade of pressure with
uncertainties from Table.6.1 of the MKS high temperature capacitance manometer.
Measurement uncertainty includes non-linearity, hysteresis and non-repeatability. 1 K
of ambient temperature changed is assumed.
The error bars used in all the figures are calculated using the instrumental uncertainty from provided by the manufacturer and statistical deviation, see appendix
Uncertainty calculation. The change in room temperature during each measurement
point was measured with thermocouples.
4.3
Method of varying system temperature
The temperature of the vacuum vessel is initially raised to 183 ◦C and p0 is measured.
At this temperature, thermal transpiration effect is neglected in-comparison to the
magnitude of the effect when the vacuum vessel is at room temperature. After p0 hasF
been measured the temperature is stepwise decreased, as can be seen in Fig. 4.9.
In Fig. 4.10, the initial p0 = 1, 003 mbar (red dots) and thermal transpiration
effect is close to unity. When the initial pressure is lower, at p0 = 7.57 × 10−3 mbar
(blue dots) thermal transpiration effect can be seen to increase in magnitude as the
in temperature gradient increases. The pressure ratio due to thermal transpiration
effect at lowest temperature, with initial pressure p0 = 7.57 × 10−3 mbar , agrees with
measurements made with the method of varying pressure within the uncertainty.
46
1.3
a)
1.3
THF
TS
T1 = 299,5 K
T1 = 299,5 K
T2 = 373,15 K 1.2
T2 = 373,15 K
TS eff.diam.
M
S
R1′′
1.2
d)
Nitrogen
1.1
1.1
1.0
1.0
10
-3
10
-2
10
p2 / mbar
-1
-3
10
b)
10
-2
10
p2 / mbar
e)
Nitrogen
THF
T1 = 373,15 K
1.2
T1 = 373,15 K
1.2
R2′′
T2 = 473,15 K
T2 = 473,15 K
1.1
1.1
1.0
1.0
10
-3
10
-2
10
-1
10
p3 / mbar
c)
Nitrogen
p3 / mbar
THF
T2 = 473,15 K
R3′′
p3 / mbar
-1
-1
T2 = 473,15 K
1.0
10
10
T1 = 299,5 K
1.0
-2
-2
1.4
1.2
10
10
T1 = 299,5 K
1.2
-3
-3
f)
1.4
10
-1
10
0
10
-3
10
-2
10
-1
10
0
p3 / mbar
Figure 4.2: Unmodifed pressure ratio of nitrogen on the left column a), b), c) and of
tetrahydrofuran on the right column d), e), f), for pressure ratio R100 ,R200 and R300 . The
different colors of the points indicates that the measurement set was done on different
occasions. The red solid curve (TS) is calculated using Takaishi-Sensui equation with
nominal parameters, the dotted red (TS eff.diam) with effective diameter fitted to
the nitrogen measurements. Green (M) and blue (S) curve shows Millers and Sětina
equations with nominal parameters.
47
a)
0.15
b)
PTS
R3′ − 1
R3′ /(P TS+ 1)−1
Nitrogen
T1 = 299, 5 K
T2 = 473, 15 K
0.10
R3′′
Nitrogen
T1 = 299, 5 K
T2 = 473, 15 K
1.4
RTS
3
TS
R3
R3′ − 1
0.05
1.2
0.00
-0.05
1.0
-0.10
10
-3
c)
10
-2
10
-1
10
0
10
-3
10
p3 / mbar
-2
10
-1
10
0
p3 / mbar
THF
T1 = 299, 5 K
T2 = 473, 15 K
1.4
R3′′
RTS
3
RTS-FIT
R3
1.3
1.2
1.1
1.0
10
-3
10
-2
10
-1
10
0
p3 / mbar
Figure 4.3: The upper left graph a) shows the relative sensitivity after removing thermal transpiration effect using Takaishi-Sensui equation with nominal parameters (black
dots). The red solid curve is a 7 degree polynomial fit, and the red dots shows the
relative sensitivity after the multiplying the black dots with the inverse of the polynomial. b) shows the unmodified pressure ratio (black dots) of nitrogen, and the red is
after correction with the polynomial in a). Similarly to b), c) shows the unmodified
pressure ratio and the corrected pressure ratio but with tetrahydrofuran. The black
curve in c) is Takaishi-Sensui eqation with fitted coefficients Tab. 4.2
48
0.15
a)
Nitrogen
T1= 373,15 K
T2= 473,15 K
0.10
1.3
PTS
R2′ − 1
R2′ /(P TS+ 1)−1
R2′′
Nitrogen
T1= 373,15 K
T2= 473,15 K
RTS
2
TS
1.2
R2
0.05
R2′ − 1
b)
0.00
1.1
-0.05
-0.10
1.0
-0.15
10
c)
-2
10
-1
10
p3 / mbar
-3
10
-2
10
-1
p3 / mbar
THF
T1= 373,15 K
T2= 473,15 K
R2′′
TS
R2
RTS-Fit
R2
1.2
1.1
1.0
10
-3
10
-2
10
-1
p3 / mbar
Figure 4.4: Similarly as Fig. 4.3, but with pressure ratio R2 . Here c) shows the TakaishiSensui equation with fitted coefficients from Fig. 4.3 with nominal parameters (black
solid curve) and with effective diameter (dashed black curve).
49
a)
0.15
Nitrogen
T1= 299.5 K
T2= 473,15 K
0.10
1.3
PTS
R1′ − 1
R1′ /(P TS+ 1)−1
b)
R1′′
Nitrogen
T1= 299.5 K
T2= 473,15 K
RTS
1
TS
1.2
R1
R1′ − 1
0.05
0.00
1.1
-0.05
-0.10
1.0
-0.15
-2
1.3
c)
10
p2 / mbar
10
-1
10
-3
-2
10
p2 / mbar
10
THF
T1= 299,5 K
T2= 473,15 K
R1′′
RTS
1
RTS-Fit
R1
1.2
1.1
1.0
10
-3
10
-2
p2 / mbar
Figure 4.5: Similarly as Fig. 4.4, but with pressure ratio R1 .
50
-1
10
-1
a)
d)
0.15
1.3
S
P
R1′ − 1
R1′ /(P S + 1)−1
0.10
1.2
R1′′
R1S
S
R1
R1′ − 1
0.05
0.00
1.1
-0.05
Nitrogen
T1= 299.5 K
T2= 373,15 K
-0.10
-0.15
10 -2
p2 / mbar
0.15
Nitrogen
T1= 299.5 K
T2= 373,15 K
1.0
10 -1
10 -3
b)
1.2
S
10 -1
e)
Nitrogen
T1= 373,15 K
T2= 473,15 K
P
R2′ − 1
R2′ /(P S + 1)−1
0.10
10 -2
p2 / mbar
R2′′
R2S
S
R2
R2′ − 1
0.05
0.00
1.1
-0.05
Nitrogen
T1= 373,15 K
T2= 473,15 K
-0.10
-0.15
0.15
c)
10 -2
p3 / mbar
1.0
10 -1
10 -3
f)
PS
R3′ − 1
R3′ /(P S + 1)−1
0.10
10 -2
p3 / mbar
10 -1
Nitrogen
T1= 299.5 K
T2= 473,15 K
1.4
R3′′
R3S
S
R3
R2′ − 1
0.05
0.00
1.2
-0.05
Nitrogen
T1= 299.5 K
T2= 473,15 K
-0.10
-0.15
10 -3
10 -2
p3 / mbar
10 -1
1.0
10 0
10 -3
10 -2
p3 / mbar
10 -1
10 0
Figure 4.6: The graphs on the left side, a), b) and c) shows the relative sensitivity of
the pressure ratio after multiplying with the inverse of the calibration curve calculated
using Sětinas equation with nominal parameters R10 − 1 (black dots). The red solid
curve PS is a 7 degree polynomial fitted to the black dots, the blue dots is the result
after multiplying with the pressure ratio R10 − with inverse of PS . The right side,
d), e) and f) shows the unmodified pressure ratio (black dots) and the pressure ratio
00
RS = PRS +1 after correction applying the correction polynomial.
51
a)
THF
T1= 299,5 K
T2= 473,15 K
1.3
R1′′
R1S
S
R1
1.2
1.1
1.0
10
-3
10 -2
p2 / mbar
b)
10
-1
THF
T1= 373,15 K
T2= 473,15 K
R2S
S
R2
1.2
R2′′
1.1
1.0
10
-3
10 -2
p3 / mbar
10
-1
c)
THF
T1= 299,5 K
T2= 473,15 K
1.4
R3′′
R3S
S
R3
1.3
1.2
1.1
1.0
10
-3
10
-2
p3 / mbar
10
-1
10
0
Figure 4.7: Pressure ratio of the unmodified pressure ratio (black dots) R00 of tetrahydrofuran and the pressure ratio with polynomial correction using Sětina equation for
nitrogen in Fig. 4.6 RS . The blue curve is the calculated curve using Sětinas equation
(S).
52
a)
R1′′
R1′′
Argon
T1= 299,5 K
T2= 473,15 K
1.2
R1S
R1TS
S
TS
1.1
1.0
10
b)
-2
10
p2 / mbar
-1
R3′′
Argon
T1= 373,15 K
T2= 473,15 K
R2TS
S
TS
R2′′
1.1
R2S
1.0
10
-3
10 -2
p3 / mbar
10
c)
1.3
Argon
T1= 299,5 K
T2= 473,15 K
1.2
R3′′
-1
R3′′
R3S
R3TS
S
TS
1.1
1.0
10
-3
10
-2
10
-1
10
0
p3 / mbar
Figure 4.8: The three figures shows the pressure ratio of argon, the black dots the
unmodified pressure ratio, the red and blue the pressure ratio relative to the correction
of nitrogen measurements using Takaishi-Sensui equation RTS and Sětina equation RTS .
53
The blue (S) and the red (TS) are the calculated
curve using Sětina and Takaishi-Sensui
equation with nominal parameters.
T / Kelvin
p / mbar
t/h
Figure 4.9: Typical data set with pressure on the left axis and temperature on the
right against time. The most upper (blue) curve is the vacuum vessel temperature.
Variation of the vacuum vessel temperature induces a change in pressure. The first
curve from bottom is pressure measurements of BA full scale, the second is that of the
ceramic capacitance manometer and the third curve is that of the high temperature
capacitance manometer.
54
p = 7, 57 x 10
-3
mbar
0
1, 2
p = 1 ,003 mbar
p / p
0
1, 1
1, 0
320
340
360
380
400
420
440
460
T / Kelvin
Figure 4.10: Pressure ratio over temperature of thermal transpiration effect using
nitrogen gas. The blue and red dots show different initial pressures, the red dots have
high enough initial pressure to have no thermal transpiration effect, the deviation from
unity of blue dots is due to thermal transpiration effect.
55
Chapter 5
Discussion
This chapter covers a discussion of the result, methods, uncertainties and the suitability
of the experimental setup the measurement methods of measuring thermal transpiration effect.
5.1
Method of varying pressure
The unmodified pressure ratios of nitrogen in Fig. 4.2 a), b) and c), the calibration
curves of Sětina (blue) and Miller (green) fits slightly better to the measurements than
to the calibration curve of Takaishi-Sensui equation (solid red). The slope of the curve
of Takaishi-Sensui is steeper and is more similar to the slope of the measurements than
the slope of Sětina and Millers equation. This may be because the measurements in
the experiment which Takaishi-Sensui equation is based on, has a larger temperature
gradient than the temperature gradient used in the experiment which Sětinas equation
is based on. By using effective diameter with Takaishi-Sensui equation (dashed red),
that is, letting the diameter be fitting parameter, the curve is translated toward the
measurements and fits very well to the measurements. The difference between Sětina
and Millers equation is very little for nitrogen as well as for tetrahydrofuran and argon.
None of the calculated curves for tetrahydrofuran in Fig. 4.2 d), e) and f) shows good
agreement with the measurements. The calculated curve of Takaishi-Sensui shows a
much too late increase in slope, this confirms that Takaishi-Sensui equation does not
scale well with large molecules as pointed out in [10]. The curves of Sětina and Miller
57
shows better agreement with the measurements but have a too slow increasing slope
in comparison to the measurements. The difference in color of the points in Fig. 4.2
represents different data sets. The repeatability of the nitrogen measurements is good,
as almost all points are within uncertainty. The data sets of the measurements of
tetahydrofuran shows a larger variation, especially in R100 and R200 (red dots). In R100
the red dots are lower than the trend and in R200 larger than the trend, this suggests
that the common data set which they are based on, the denominator of R200 and the
numerator of R100 , p2 is lower than the others which may be due to the offset being
slightly lower.
In Fig. 4.4 the fit of Takaishi-Sensui equation to the corrected tetrahydrofuran data
(black solid line) to R3 is shown, incomparision to the fitted curve to R2 the (dotted
black line) in Fig. 4.3 the slope of the curves are similar but a slight translation was
needed by using effective diameter.
The curve using the coefficients fitted of Takaishi-Sensui equation to the measurements of R3 in Fig. 4.3, shows a very similar slope to the measurements of R2 In Fig. 4.4.
Only a slight translation is needed, this may be done by using effective diameter. The
effective diameter of R3 and R2 is likely to be different as the instruments are from different manufacturers and the geometry of the interconnecting pipe differs. In Fig. 4.5
the fits obtained from Fig. 4.3 and Fig. 4.4 to the tetrahydrofuran data are shown,
no separate fit to this data has been made as the data is too scattered. In Fig. 4.6
Sětina equation is used as reference from the nitrogen measurements (blue curve in
Fig. 4.2). The result of tetrahydrofuran using the correction relative to Sětina is show
in Fig. 4.7, the calculated calibration curve using Sětinas equation (blue curve) shows a
too slow increasing slope in comparison to the trend of the measurements of R2S and R3S
in Fig. 4.7 b) and c). For R1S , Fig. 4.7 a), the calculated curve using Sětinas equation
agrees quite well. Fig. 4.8 shows measurements of argon (black points), the corrected
measurements using Takaishi-Sensui equation (red points) and Sětina equation (blue
points) compared with the calculated curves using known coefficients. Ideally, the red
dots should be on the red curve and the blue dots on the blue curve, however, only
in Fig. 4.8 a) and b) the measured R1S and R2S and the blue points shows quite good
agreement with blue curve.
58
5.1.1
Application
The molecular diameter of tetrahydrofuran is 6, 3 Å according to [18, 22]. Current
application of thermal transpiration corrections as in [41] for example, have a temperature gradient of about 100 ◦C so their situation is similar to Fig. 4.2 d). They are
using Takaishi-Sensui equation and calculate the coefficients, which correspond to the
red solid curve in Fig. 4.2 d). This is about 6% difference between the measurements
in this work and what would be used to correct for thermal transpiration effect for
tetrahydrofuran.
5.1.2
Uncertainty
Temperature coefficient The calculated uncertainty tables Tab. 4.3 and Tab. 4.4
shows how important it is to have a stable ambient temperature. As in the lower and
middle span of measurement ambient temperature variation less than a degree causes
large uncertainty. This uncertainty depends on how much the ambient temperature
varies and how fast it changes. To achieve high accuracy it is important to have a
experimental environment with small and slow varying ambient temperature changes.
5.1.3
How it should be done
In the relative measurement method the instruments first needs to be calibrated to
insure that the deviation from unity in the pressure ratio is due to thermal transpiration effect. In this work, this was done by using the calibration curves for nitrogen
using Takaishi-Sensui equation and Sětinas equation. Therefore the measurements of
thermal transpiration effect is relative to the calculated calibration curve of thermal
transpiration of nitrogen. It is better to use capacitance manometers where the internal heating could be turned on or off it would be possible to calibrate the instruments
by direct comparison such as in [7, 10, 31]
5.2
Method of varying temperature
In the method of varying temperature thermal transpiration effect can be seen, with
initial pressure 7.57 × 10−3 mbar . The magnitude agrees measurements with varying
59
pressure of varying pressure within the uncertainty bars. The curve in Fig. 4.10 with
initial pressure 1, 003 mbar has pressure ratio close to unity as the temperature varies.
This is expected as the rarefaction is in the continuum regime. This curve also shows
that the correction due to temperature coefficient of the offset, and the correction of
the change in particle density in the system measured by ionization gauge is good.
There is only two good data sets when preforming the experiment by varying the
temperature. This is because the rapid heating and cooling of the vacuum vessel soon
causes the connecting surfaces of the vacuum pipes to become oblique causing major
leaks. Therefore after a few measurements with this method, the vacuum vessel has
to be open up, and the connecting surfaces have to be grinded even. Another setback
of this method is that each measurement point takes hours, for the temperature to
properly stabilize through out the system. The biggest setback is that is difficult to
determine T1 , the vacuum vessel temperature, as the temperature is not uniformly
distributed. The temperature is not uniform, and cannot be exactly determined, as
this experiment was using heating bands as heat source. What to think about when
using this method is, other means of heating the experiment, such as surrounding
the experiment, or all the pipes leading to the gauge, inside a heated enclosure with
regulated air temperature, for example as one in Ref. [1]. A setback of this method is
also that the system may not be heated to 200 ◦C as the temperature regulation inside
the capacitance manometer requires a temperature difference, else the heater controller
will respond system temperature instead of the sensor temperature [2]. For further
accuracy, temperature coefficient induced errors of the span must also be determined.
5.2.1
Temperature regulation
The regulation of temperature is stable, within 1 ◦C , at the point where it is regulating.
This is when the regulation value is taken as the mean value of the temperature of
the point of regulation and of the heating band close to this point. This works well
as the heat capacity of the system is much lower than of the heating band. However
the heating is not uniform through out the vacuum vessel. This is because the heating
bands have to be wired around the system and it is therefore not possible to have a
uniform temperature distribution.
60
5.2.2
How it should be done
With the method of varying temperature, using a gauge with external electronics which
gives low temperature coefficient such as 615A from MKS which may be heated up to
300 ◦C and measures 6 decades of pressure. Instead of heating bands, which gives very
non-uniform temperature, a temperature regulated enclosure should be used, such as
in the heating box in [1].
5.3
Conclusion
Two experimental methods to measure thermal transpiration effect have been evaluated. The method of varying temperature can measure thermal transpiration effect but
heating with heating bands is not suitable for this purpose, and the temperature has
to be varied very slowly in between measurement points not to produce leaks. With
the method of varying pressure, the coefficients of Takaishi-Sensui equation for thermal
transpiration effect of tetrahydrofuran relative to Takaishi-Sensui equation using nitrogen have been determined. Comparing the measurements of thermal transpiration
effect and the calculated curve using Takaishi-Sensui equation, with molecular diameter of 6, 3 Å, the calculated slope of the calculated curve starts to increase at too low
pressures and increase too slow comparing for all three measurands. The calculated
curve using Sětina equation shows a too slow increasing slope in comparison to the
trend of the measurements of thermal transpiration effect of tetrahydrofuran relative
to Sětina equation using nitrogen for R2S and R3S , but agrees quite well with R1S .
61
Chapter 6
Appendix
6.1
Uncertainty calculation
Source
p2 MKS 100
p3 MKS 200
p1 PFEIFFER
Measurment uncertainty
Resolution
Zero temperature coefficient
Span temperature coefficient
Span temperature coefficient
Full scale
0,5%
0,001%
0,002%
0,02%
—
1.33E-01
0,5%
0,0017%
0,016%
0,01%
—
1.33E+00
0,2%
0,003%
0,015%
—
0,01 %
1.00E+00
Reading
F.S
F.S Kelvin
F.S Kelvin
Reading Kelvin
mbar
Table 6.1: Instrumental uncertainties provided by the manufacturers.
p2 is the time average of pressure measurements by MKS capacitance manometer,
and p1 of the reference gauge Pffeifer capacitance manometer. T1 is the time average of
temperature measurements of the vacuum system. σp1 , σp2 , σT1 are the standard deviations of the time averages. T2 is internal temperature of MKS capacitance manometer.
for the pressure ratio R := pp12 , the uncertainty is
s
δR =
where
p2
− 2 ∆p1
p1
2
+
1
∆p2
p1
2
,
σp
∆p1 = √ 1 + Instrument Error(p1 , ∆TEnviroment , F.S),
N
62
Where N is the number of measurement points, σp1 is the standard deviation of
σ
the mean value p1 , the term √pN1 is called standard error of p1 . Similarly
σp
∆p2 = √ 2 + Instrument Error(p2 , ∆TEnviroment , F.S).
N
For the degree of thermal transpiration effect y :=
p2
1− p1
q
1− T2
,
1
s
δy =
∂y
∆p1
∂p1
2
+
where
∂y
∆p2
∂p2
2
+
∂y
∆T1
∂T1
2
+
∂y
∆T2
∂T2
2
p2
∂y
p1
q
=
∂p1
1 − TT21
−1
∂y
p1
q
=
∂p2
1 − TT21
r
∂y
p2 1 T2
=− 1−
∂T1
p1 2 T1
∂y
=
∂T2
p2
1
√
1−
p1 2 T1 T2
r
1−
r
1−
T2
T1
T2
T1
!−2
!−2
σT
∆T1 = √ 1 + Instrument Error
N
∆T2 is the uncertainty in the internal temperature of the MKS capacitance manometer.
2p2 d
1
X := 1,33
, where d is the diameter of the interconnecting pipe of the MKS
(T1 +T2 )
capacitance manometer in mm, and p2 the pressure reading in mbar .
s
δX =
∂X
∆p2
∂p2
2
+
2 2 2
∂X
∂X
∂X
∆d +
∆T1 +
∆T2
∂d
∂T1
∂T2
∂X
1
2d
=
∂p2
1, 33 T1 + T2
63
1
2p2
∂X
=
∂d
1, 33 T1 + T2
∂X
1
2p2 d
=−
∂T1
1, 33 (T1 + T2 )2
∂X
1
2p2 d
=−
∂T2
1, 33 (T1 + T2 )2
where ∆d is the uncertainty in the diameter of the MKS capacitance manometer
interconnecting pipe.
6.2
6.2.1
Experimental procedures
Calibrate the temperature regulation
• Close gas inlet and open the main shaft, to clean the system while calibrating
the temperature.
• Attach the thermocouples such that one is on the point where temperature is
chosen, and at the heating band.
• Set the temperature regulation value as a mean of the two temperatures measurements.
• Heat up the system at various temperatures.
• Plot the heating data with input on Y-axis and finial temperature of heating
point on the X-axis.
• Perform a linear regression analysis, put in the coefficients in settings of the
measurement and temperature regulation program.
Starting up the vacuum system
• Main shaft open.
• Fore-vacuum pump valve open.
64
• Start up the Pffifer controller.
• Start the fore-vacuum pump. Let it pump until the pressure is in the order of
10−2 − 10−3 mbar.
• Start the electricity, and press START on the turbo molecular pump controller.
• Start up the MKS controller.
• In the Pffifer controller, put Ionization gauge to ON.
6.2.2
Shutting down the experiment
• Turn off the ionization gauge on the Pffiffer controller.
• Turn off the MKS controller.
• Close the valve to the fore-vacuum pump, in order to prevent back stream of oil
into the system.
• Shut off the fore-vacuum pump.
• Shut off the turbo-molecular pump.
• Wait 20 min in order for the ionization gauge to cool down, and the turbomolecular pump to slow down.
• Increase the inlet slightly by opening the inlet valve.
• Let the pressure raise slowly, and successfully increase the inlet until the system
reaches atmospheric pressure.
6.2.3
Varying pressure
• Set the system at the some temperature.
• After half a day the temperature will be completely stable.
• Close main shaft.
65
• After another two days for the system to equilibrate.
• Raise the pressure to just below reading point.
• Zero all gauges.
• Raising the pressure stepwise till about 1.02 mbar and then back down 3.0 × 10−3
mbar, for one measurement cycle. Using small steps yield shorter equilibrating
time upon raising or lowering the pressure.
6.2.4
Measuring temperature coefficient induced change in
offset
• Set the system temperature to as close as 180o C.
• When the temperature have stabilized, 3 hours. Set offset to zero.
• Close main shaft, make sure the pressure does not begin to rais, this would
indicate leaks.
• Set a schedule for stepwise decreasing the temperature.
• Run the schedule, each measurement point needs enough time for the system to
stabilize, 3 hours.
• As the pressure readings are below minimum reading pressure, the change in each
pressure measurement point is due to the temperature coefficient of the offset.
• Varying temperature
6.2.5
Measure thermal transpiration effect by varying temperature
• Set the system temperature to as close as 180o C, this system temperature is T0 .
• When the temperature have stabilized, 3 hours. Set offset to zero.
• Close main shaft, make sure the pressure does not begin to raise, this would
indicate leaks.
66
• Turn the inlet valve until desired system pressure is reached.
• When the pressure has stabilized, measure the pressure, this measurement is p∗0
1
.
• Set the same schedule for step wise decreasing the temperature as when calibrating the temperature coefficient offset, the pressure reading at system temperature
T is p∗ .
• Run the schedule, each measurement point needs enough time for the system to
stabilize, 3 hour.
6.2.6
Data analysis of varying temperature
• For each system temperature system temperature reading T , with pressure gauge
reading p∗ .
• Subtract the variation due to temperature coefficient of zero.p∗ −pTemperature coeffeicient
for temperature T .
• Subtract the difference in ionization gauge, difference in p∗ − ∆pion , ∆pion =
Pion, 0 − Pion .
6.3
6.3.1
Equipment
Vacuum pumps
Rotary vane pump
The constructional principal is displayed in figure below. A metallic rotor driven by an
electric engine, rotating about 1500 rpm, is placed off center in the cylindrical pumping
chamber with only a few millimetres between the rotor and the chamber. The rotor
have diametrical sliding vanes, which are pushed out by spring and centrifugal forces to
maintain contact to the in contact with the chamber. The vanes are made of lightweight
plastic to keep the vibration due to unbalanced load minimal. The exhaust flap,
1
See experiment theory section
67
Figure 6.1: Principial sketch of a rotary vane pump
opening when partial pressure exceeds somewhat greater pressure than atmospheric
pressure, is below the oil level to keep the outgoing gas sealed. Oil is also drawn into
the pumping chamber by gravity and movement of the pars, the oil keep it running
smoothly, seal small gaps and provide cooling by moving away heat generated. The
Figure 6.2: Cycle of a rotary vane pump, induction, isolation, compression, exhaust
process is illustrated in the picture above, induction, isolation, compression, exhaust.
First, the gas is induced into the chamber when it occupies the expanding volume.
By further rotation the gas isolated until the vane reaches the exhaust valve. At the
exhaust valve the gas is compressed by the decreasing volume resulting in an increase
of pressure and temperature. Rotary vane pump bring down the pressure to about
10−2 − 10−3 mbar. pumps.
Turbo-molecular pump
The turbo molecular pump works similar to a turbine with spinning blades, with
the major difference that the blades are not design to press fluids but rather to hit
molecules. A turbo molecular pump have many layers of spinning rotors, and by each
68
the particles acquire more momentum and the gas is getting more compressed until it
reaches the outlet.
69
6.4
Software
INITIATE
Give all variables,
parameters and
constans their
inital values.
RUN
Measure Count down
SHUT DOWN
Close communication
between the PC and
all USB and serial
RS232 devices.
Read Pressure
Open communication
between the PC and
all USB and serial
RS232 devices.
Read Temperature
Save measurements
Regulate temperature
Figure 6.3: The top arrow inducate the order for the brackets. The first stage is
INITATE, the second RUN, third SHUT DOWN. The elliptic shapes with a arrow on
top represent for loops. The loops run until the STOP % SAVE button is pressed on
the user interface.
70
RUN
Measure count down
Count down for Measurement speed ms
Count down ended ->
Measure -> True
Read all
pressure
sensors
Add a row
of values
to Pressure Array
Read Temperature
If
If
Measure = True
and Ready to read pressure = True
and Got pressure value = False
Ready to read -> False
Got value -> True
Clear Pressure Array
Else
Keep reading
Read all
temperature
sensors
Add a row
of values
to Pressure Array
Read Pressure
If
Measure = True
and Ready to read temperature= True
and Got temperature value = False
Ready to read -> False
Got value -> True
Clear Temperature Array
Else
Keep reading
Measure = True
and Got temperature value = True
and Got pressure value = True
Pressure array -> Mean value, standard deviation
Temperature array -> Mean value, standard deviation
Add the values to Data file
Measure -> Fals
Got pressure value -> Fals
Got temperature value -> Fals
Pressure ready to read -> True
Temperature ready to read -> True
Else
Do nothing
Save measurements
Regulate temperature
Figure 6.4: Detailed algorithm of syncronization of temperature and pressure data
logginig. The arrows indicate the order of the brackets, the elliptic shapes with arrow
71
on top represent for loops.
RUN
Measure Count down
Read Pressure
Read Temperature
Save measurements
Regulate temperature
For Each USB 4-Switch slot.
‘1.Connect termocouple readings to USB 4-Switch,
read Temperature array. Take the values from
Temperature array which are connected to the slot,
produce a mean value T of those readings.
2. If
T < T-limit
Turn the USB 4-Switch slot ON.
else
Turn the USB 4-Switch slot OFF.
Continue to next USB 4-Switch slot.
Figure 6.5: Detailed algorithm of temperature regulation. The arrows indicate the
order of the brackets, the elliptic shapes with arrow on top represent for loops.
72
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