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This is to certify that the

thesis entitled

STEADY STATE INVERSE THERMAL ANALYSIS IN
SUPERCOOLED ACCELERATOR CAVITIES

presented by
AHMAD AI ZAZ

has been accepted towards fulfillment
of the requirements for ‘

MASTERS degree in MECHANICAL ENGINEERING

 

MW

Major professor

0-7639 MS U i: an Affirmative Action/Equal Opportunity Institution

 

 

 

 

LIBRARY

Michigan State

University

 

 

PLACE IN RETURN BOX to rem
TO AVOID FINES return

MAY BE RECALLED with earlier

ove this checkout from your record.
on or before date due.
due date if requested.

 

DATE DUE

DATE DUE

DATE DUE

 

 

n 252008
qfiiuhg

 

 

 

 

 

 

 

 

 

 

 

 

 

6/01 cJClRC/DatoDue.p65-p.15

 

STEADY STATE INVERSE THERMAL ANALYSIS
IN SUPERCOOLED ACCELERATOR CAVITIES
BY

AHMAD AIZAZ

A THESIS

Submitted to
Michigan State University
In partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE

Department of Mechanical Engineering

2003

ABSTRACT
STEADY STATE INVERSE THERMAL ANALYSIS
IN SUPERCOOLED ACCELERATOR CAVITIES
BY
AHMAD AIZAZ

A heat conduction problem in Super Conducting Radio Frequency (SRF) cavities has
been studied. The wall thickness is l-4mm niobium and a ‘point heating’ source on the
inner surface of the evacuated niobium cavity is assumed. Liquid helium acts as a coolant
on the outer surface of the cavity to make the niobium electrically super conducting
below its critical temperature. Using finite difference techniques, a computer program has
been developed to solve the direct steady-state heat conduction equation. To simulate the
actual thermal diagnostic setup at the Cyclotron Laboratory as closely as possible, the
geometry for this computer program is 3-dimensional with insulated thermal sensors
installed on the outer surface of the cavity. The insulated enclosures have different
thermal properties from those of the niobium.

The results obtained through the program are useful for understanding the
maximum sensor spacing, which allows detection of a 1 degree temperature rise at the
heated surface (inner side) of the cavity.

Code verification is done with results obtained through the exact solutions generated by
COND3D, developed for Sandia National Laboratory, and through the Kokopelli Finite
Element Method program developed for Los Alamos National Laboratory. The results are

discussed in detail with suggestions for future research.

ACKNOWLEDGEMENTS

I would like to take this opportunity to thank James Calthorp, Steve Bricker and Scott
Hitchcock for their invaluable practical help in performing thermometric experiments at
the National Superconducting Cyclotron Laboratory (NSCL), at Michigan State
University (MSU). I would also like to thank Dr. Terry Grimm and Dr. Walter Hartung
for many helpful discussions and explanations of the physics of thermometry in RF
cavities.

My special thanks go to Dr. Robert L. McMasters whose patient guidance gave
this work direction and purpose, and to my wife, Asma, whose continuous support kept
the work moving along. This thesis could not have been finished without these people.

Finally, I wish to acknowledge the financial support of the National

Superconducting Laboratory throughout my graduate studies.

iii

TABLE OF CONTENTS

 

 

 

 

 

 

LIST OF FIGURES - VIII
LIST OF TABLES X
CHAPTER 1 1
INTRODUCTION 1
1.1 SUPERCONDUCTING CAVITIES ............................................................................. 1
1.2 THERMAL BREAKDOWN ....................................................................................... 1
1.3 PAST RESEARCH .................................................................................................. 2
CHAPTER 2 _ 4
NUMERICAL SIMULATIONS 4
2.1 MODEL DEFINITION ................................................................................................... 4
2.2 ASSUMPTIONS 1N MODEL ........................................................................................... 6
2.3 GOVERNING DIFFERENTIAL EQUATION ...................................................................... 7

2. 3.] Boundary Conditions on the Niobium Surface .................................................. 8
2.3.2 Boundary Conditions on the Sensor .................................................................. 9

2.4 DISCRETIZATION OF THE DIFFERENTIAL EQUATION ................................................... 9
2.5 IMPLEMENTATION OF BOUNDARY CONDITIONS ....................................................... 11
2.5.] On Heated Face ............................................................................................... 11
2.5.2 Convective Boundary Condition ...................................................................... 13
2.5.3 Boundary Condition at the Niobium-Sensor Interface .................................... 13

iv

2.5.4 Other Sides of Boundary Conditions 0n Sensor Housing ............................... I4

 

 

 

2.5.5 Boundary Conditions on Face 3 and 4 ............................................................ 15
2.5.6 Boundary Conditions on Face 5 and 6 ............................................................ 15

2.6 PROGRAM CHARACTERISTICS .................................................................................. 15
2.6.1 Convergence .................................................................................................... 15
2.6.2 Post Processing ................................................................................................ I 7
2.6.3 Program Verification ....................................................................................... 18
2.6.4 Program Validation ......................................................................................... 18
2.6.5 Verification through Inverse Heat Conduction Calculations .......................... 21
CHAPTER 3 26
THERMAL PROPERTIES OBTAINED FROM THE LITERATURE ................... 26
3.1 LITERATURE SEARCH ............................................................................................... 26
3.1.1 Thermal Conductivity of Niobium .................................................................... 26
3.1.2 Thermal Conductivities of Common Cryogenic Materials .............................. 29
3.1.3 Convection Heat Transfer Coefi‘icient For Liquid Helium .............................. 31
CHAPTER 4 -_ 35
EXPERIMENTAL STUDIES - 35
4.1 THERMAL SENSOR DESIGN AND MANUFACTURE ..................................................... 35
4.1.1 Sensor Selection ............................................................................................... 36
4.1.2 Sensor Manufacture ......................................................................................... 36

4.2 SENSOR DESIGN ISSUES ........................................................................................... 38
4.2.1 Cooling Eflects on Sensor ................................................................................ 38

4.2.2 Cooling of Sensor Through G-IO Housing ...................................................... 39

 

4.2.3 Cooling Effect On The Sensor Through Manganin Wire ................................. 39
4.2.4 How much is infinite length? ........................................................................... 40
4.2.5 Ohmic Self Heating of the Sensor .................................................................... 41
4.3 SENSOR INSTALLATION/ MOUNTING SYSTEM ......................................................... 42
4.3.1 The Tree ........................................................................................................... 42
4.3.2 Ribbon Cables .................................................................................................. 43
4.4 SENSOR MEASUREMENTS TECHNIQUE ..................................................................... 46
4.4.1 Four-Lead Measurement ................................................................................. 46
4.4.2 Measurement of Two-Lead Sensors ................................................................. 46
4.5 DESIGN AND MANUFACTURE OF VACUUM FEED-THROUGH. .................................. 48
4.6 DESIGN CRITERIA FOR DATA SIGNAL CABLES. ....................................................... 50
4.6.1 Grounding And Shielding Sensor Leads .......................................................... 50
4.7 SIGNAL PROCESSING ................................................................................................ 50
4.8 DATA LOGGING ....................................................................................................... 51
4.9 SENSOR CALIBRATION. ............................................................................................ 51
4.10 SENSOR SENSITIVITY AND MEASUREMENT RESOLUTION. ...................................... 54
APPENDICES 56
5.1 PROGRAM DESCRIPTION .......................................................................................... 57
5.1.1 User Graphical Interface (UGI) ...................................................................... 57
5.1.2 Input Text Boxes of U0] .................................................................................. 57
5.1.3 Option Boxes of the UGI .................................................................................. 58
5.1.4 Program Listing ............................................................................................... 59

vi

5.2 PROPERTIES OF LIQUID HELIUM .............................................................................. 73

5.2.1 Background ...................................................................................................... 73
5.2.2 Isotopes of Helium ........................................................................................... 73
5.2.3 Liquid Helium .................................................................................................. 73
5.2.4 General Properties of Helium .......................................................................... 73
5.2.5 Unique Properties of Liquid Helium ................................................................ 74
5.3 TEMPERATURE MEASUREMENTS .............................................................................. 75
5.3.1 Defining Temperature Measurement [5] ......................................................... 75
5.3.2 Uncertainty Conversions Using Dimensionless Sensitivities .......................... 76
5.3.3 Resolution ........................................................................................................ 77
5.3.4 Precision And Accuracy ................................................................................... 78
5.3.5 Sources 0f Measurement Uncertainty ............................................................. 79
5.3.6 Calibration Uncertainty ................................................................................... 82
5.3. 7 Combining Measurement Uncertainties .......................................................... 82
5.4 SUPER-CONDUCTIVITY ............................................................................................ 85
REFERENCES 86

 

vii

LIST OF FIGURES
FIGURE 1 REGION OF INTEREST ON AN ELLIPTICAL SUPER-CONDUCTING CAVITY Is
MODELED AS A RECTANGULAR PARALLELEPIPED SURFACE WITH POINT OF HEAT
SOURCE ON THE INTERIOR SURFACE AND COOLED BY THE LIQUID HELIUM ON THE
EXTERIOR SURFACE ...................................................................................................... 4
FIGURE 2 RECTANGULAR PARALLELEPIPED SURFACE HAS POINT OF HEAT SOURCE ON
FACE 1 AND SENSORS INSTALLED ON FACE 2 WHERE LIQUID HELIUM IS TO COOL DOWN
THE WHOLE CAVITY To THE DESIRED OPERATING TEMPERATURE. X-AXIS IS CHOSEN IN
THE DIRECTION OF THICKNESS OF THE NIOBIUM METAL. .............................................. 5
FIGURE 3 PROGRAM OUTPUT SHOWING MOST HEATED SPOTS AT THE FOUR CORNERS
UNDER THE SENSOR LOCATIONS ................................................................................. 18
FIGURE 4 EXPERIMENTAL SETUP To VALIDATE THE COMPUTATIONAL MODEL. SIX
SENSORS PLACED 1 CM APART OVER A 4 MM THICK NIOBIUM PLATE WITH HEATER
INSTALLED ON THE BOTTOM SURFACE ........................................................................ 20
FIGURE 4A SENSOR EFFICIENCY ((1)) SHOWN As A FUNCTION OF RADIAL DISTANCE (R).
THE RELATIONSHIP OBTAINED FROM NUMERICAL COMPUTATION IS COMPARED WITH
THE ONE OBTAINED FROM THE EXPERIMENTAL SETUP AT ROOM TEMPERATURE ......... 21
FIGURE 5 THERMAL CONDUCTIVITY OF NIOBIUM AT Low TEMPERATURES .................... 28
FIGURE 6 THERMAL CONDUCTIVITY OF G- 10 AND OTHER MATERIALS COMMONLY USED
IN CRYOGENIC ENVIRONMENT .................................................................................... 30
FIGURE 7 VARIATION OF HEAT TRANSFER COEFFICIENT WITH TEMPERATURE FOR Two

VALUES OF HEAT FLUX AT TWO DIFFERENT PRESSURES .............................................. 32

viii

FIGURE 8 VARIATION OF HEAT TRANSFER COEFFICIENT WITH HEAT FLUX FOR THE
SPECIFIC CASE OF TWO BULK HELIUM TEMPERATURES, I.E. FOR 1.8 K AND 2.0 K ...... 33
FIGURE 9 SENSOR EFFICIENCY AS A FUNCTION OF SENSOR SPACING .............................. 34
FIGURE 10 DRAWING DETAILS OF SENSOR HOUSING .................................................... 37
FIGURE 1 1 SENSOR INSTALLATION TREE ..................................................................... 44
FIGURE 12 SENSOR INSTALLATION TREE (2) ................................................................ 45
FIGURE 13 FOUR LEAD SENSOR MEASUREMENT TECHNIQUE ........................................ 47
FIGURE 14 VACUUM FEED-THROUGH HAVING TEN AMP-CPC CONNECTORS, TWENTY-
FOUR PINS EACH ......................................................................................................... 49
FIGURE 15 CALIBRATION CURVE FOR CARBON SENSOR RESISTANCE AS A FUNCTION OF
TEMPERATURE BASED ON A GERMANIUM SENSOR ...................................................... 53
FIGURE 16 RESIDUAL OF POLYNOMIAL FIT To THE CALIBRATION DATA ...................... 54

ix

LIST OF TABLES
TABLE 1 SHOWING DIFFERENCE BETWEEN ACTUAL TEMPERATURE RISE AND ESTIMATED
TEMPERATURE RISE AT SENSOR LOCATION BY USING ESTIMATED PARAMETERS. ........ 24
TABLE 2 THERMAL ANALYSIS OF OHMIC SELF-HEATING. HERE, DELTA I IS THE RISE IN
SENSOR TEMPERATURE DUE TO SELF HEATING. .......................................................... 42
TABLE 3 SENSOR SENSITIVITY AND DISPLAY RESOLUTION OF THE SENSOR MONITOR 55

TABLE 4 GENERAL PROPERTIES OF LIQUID HELIUM ...................................................... 74

CHAPTER 1
INTRODUCTION
1.1 Superconducting Cavities
Superconductivity has become an important technology for particle accelerators.
Radio frequency superconducting cavities (SRF) have been operating routinely for
many years in a variety of accelerators for high-energy physics, low-energy to
medium-energy nuclear physics research, and free-electron lasers. A key component
of the modern particle accelerator is the device that imparts energy to the charged
particles. This is the electromagnetic cavity resonating at a microwave frequency at
very low temperatures Where the material of the cavity is electrically super-
conducting. For more information on super-conductivity refer to appendix 5.4 to this
report.
1.2 Thermal Breakdown
Thermal breakdown, commonly known as “Quench”, is a phenomenon that occurs in
the accelerating cavity due to a rise in the temperature of the cavity, in a small localized
region known as a “defect”. With the increase in If power, these sub-millimeter-sized
regions that have If losses substantially higher than the surface resistance of an ideal

superconductor, grows and become normal conducting regions. Hence, the result is
sudden loss of power in the accelerating field Em. The two most critical design features
of a super-conducting accelerating cavity are its average accelerating field, Em, and the
quality factor Q0, which is the intrinsic Q of the resonant cavity defined as the. ratio of

the energy stored (U) in the cavity to the energy lost (PC) in one If period. This thermal

breakdown (quench) is a mechanism that ultimately limits field strengths, Ease, in

superconducting cavities whose microwave performance is not affected by
multipacton'ng (a resonant process in which a large number of electrons build up within a
small region of the cavity surface) or field emission (emission of electrons from high
electric field regions of the cavity). [1]. It is now a well-known fact that this effect takes
place within a localized, point-like region of the cavity inner surface rather than over a
large fraction of the super-conducting areas. At these localized regions, therefore, the

temperature continues to rise with the increase in RF power input and could eventually

cross the critical temperature (TC), which may result in a quench condition [1].

1.3 Past Research

Thermal breakdown in superconducting RF cavities is one of the limiting factors in

achieving high values of average accelerating field (Eacc). Thermal breakdown originates

at sub-millimeter—size regions that have RF losses substantially higher than the surface
resistance of an ideal superconductor.

K.R. Krafft et al. have studied the mechanism of thermal-magnetic breakdown in
superconducting cavities by systematically investigating the two most important thermal
transport phenomena in the cavity-cooling bath system: the thermal conductivity of the
metal and heat transport across the metal to liquid helium interface. They concluded that
the thermal transport in super-conducting niobium cavities is determined by the interplay
of the temperature dependant functions of the surface resistance, thermal conductivity,
and the metal to liquid helium thermal boundary resistance [2].

H. Padamsee et al. have investigated the behavior of thermal breakdown through the
construction of a phenomenological model by considering it as a purely thermal effect.

They found that the factors influencing the production of heat at the defect are the defect

size and resistance. Also, the surface resistance of the metal determines the additional
power dissipated in the neighboring super-conducting niobium. They developed l-D
computer simulations to model thermal-magnetic breakdown by incorporating heat
production as well as heat transportation factors. Essentially, the program calculates the
temperature of the defect and vicinity for increasing RF field levels until the thermal
breakdown takes place [1].

R. Romijn, W.Weingarten and H.Piel have presented calibration measurement for the
temperature mapping system in use at European Organization for Nuclear Research
(CERN) for super-conducting acceleration cavities immersed in sub-cooled liquid
helium. They concluded, from their experiment, that the resistor thermometer response on
a heat source at the cavity interior surface could be described by turbulent convection
heat flow [4].

HP. Kramer et al have discussed the heat transfer from technical copper to flowing
He II for different heat fluxes and different flow velocities. They argued that since the
theory of Kapitza conductance can at best account for the heat transfer from specially
treated ultra-clean surfaces, heat transfer from “dirty” technical surfaces still has to be
described empirically. He found that the heat transfer coefficient of liquid helium H does

not depend on the flow velocity [3].

W
NUMERICAL SIMULATIONS

2.1 Model Definition

The SRF cavity is modeled as a rectangular parallelepiped 3D surface, as shown in

Figure 1

EXAMPLE REGION OF
-A/INIEREST

 

 

 

 

INTERIUR HEATED
SURFACE

—EXTERIUR SURFACE
COOLED BY LIQUID
HEI IUM

 

DETAILED VIEW OF
REGION OF INTEREST

 

 

 

 

Figure 1 Region of interest on an elliptical super-conducting cavity
is modeled as a rectangular parallelepiped surface with
point of heat source on the interior surface and cooled by

the liquid helium on the exterior surface

This section of the cavity is chosen to be representative of the whole cavity. A more

elaborate View of this three dimensional surface with sensors installed on one face,

known as cold surface, is shown in Figure 2

 

 

 

 

 

 

 

 

 

 

 

 

  
   
 

 

 

  

 

 

 

   

 

 

 

 

   

 

 

 

 

Face 4
Interface b/w Nb and
sensor housing
4.: Face 5
.1:
Face 1 if: Face 2 With
——p Sensor Housing
iii
E
II
I!
Face 6 p L!
y | Face 3
I
——>
x
Figure 2 Rectangular parallelepiped surface has point of heat source

on face 1 and sensors installed on face 2 where liquid
helium is to cool down the whole cavity to the desired
operating temperature. x-axis is chosen in the direction of

thickness of the niobium metal.

2.2 Assumptions in Model

In order to describe the heat transfer behavior in the niobium cavities, mathematical
modeling is done to relate it with the actual physical phenomena. This mathematical
model is based upon certain underlying assumptions, which are discussed below:

Steady State Heat Conduction
Since thermal breakdown in SRF cavities is a phenomenon that limits the Eacc of

the cavity, accurately computing the steady state temperature distribution is of

prime importance. Also from a typical order of magnitude analysis, the transient

. . at . .
time constant, known as Fourier number (F0 = 2;), IS quite large, even for small

C

times of several milliseconds, and steady state conditions are reached very

cmz

 

),L

t“-

rapidly. In this relation, a is the thermal diffusivity of the niobium in (
sec

is the Characteristic length i.e. thickness of the niobium in (cm) and, t is time in
(sec). Moreover, since the present study is focused upon finding the optimum
sensor spacing as a function of detectable temperature rise above ambient, a
steady state solution to the problem is more desirable.

Since there is no heat being generated inside the niobium, the equation for
transport of heat across the niobium surface can be simplified. Point heating has
been assumed on the inner surface of the cavity. Since no finite heating source
can be described as point heating, a single grid point of heating is assumed to

model the point heating behavior.

Since the operating temperature of the cavities is below the critical
temperature of the niobium (TSTF) where Tc 5 9.2 ”k for niobium the Meissner

effect [12] dictates that the magnetic field does not affect heat conduction in the
niobium. Transport of heat out of the cavity through radiation is considered to be
negligible as compared with convective heat transfer through liquid helium.

Though the insulated sensor housings are made with several materials, the
main composition of the sensor that has any significant role in transportation of
heat is the casing of the sensor. This is made of 6-10 material, a kind of epoxy
filled fiberglass.

Due to the low thermal conductivity of the 0-10 sensor housing, the contact
resistance with the niobium is neglected. Moreover, this contact resistance is
further reduced by applying a thin film of varnish (IM-7031) that has low
electrical conductivity but high thermal conductivity at low temperatures. The use
of this varnish, thus allows us to neglect the interface resistance without
significant loss of accuracy.

2.3 Governing Differential Quation
Based upon the assumptions described above, the heat transport equation in the SRF
cavity material is described as:

WT+NT+NTz
8X2 8Y2 322

 

 

2.1

This differential equation in three dimensions, is commonly known as the ‘Laplace

Equation’ and is categorized as elliptical in nature.

The boundary conditions on each of the six faces are described with respect to the
faces identified in Figure 2. Since there are two distinct material surfaces in the problem,
i.e. niobium plate and the material of the sensor, the boundary conditions are described
accordingly.

2.3.1 Boundary Conditions on the Niobium Surface

(a) Face 1; at x=0. (Heated Page)
There are two boundary conditions (a mixed boundary condition) specified on

Face 1.

qu.z(Pt.heat) = User specified

ql y.z(exciptPt.heat) = 0

Heat flux is zero on all grid points except on one single grid point (input by
the user) where flux q is specified in Watts / mmz.
(b) Face 2: at x: L ( cooled surface)

This face also has two types of boundary conditions i.e. mixed boundary
conditions: -

(i) Surface with no sensor

=h AT

No sensor conv, He

 

4

(ii) Surface with sensors
(JIM, = q
TNb

 

Sensor

T

(131.“ = Sensor(i.j.k)

Here L is the thickness of the niobium plate.

(c) Faces 3 & 4:at y=O,W

ar_

——0
3y

Here, W is the width of the niobium plate

(d) Faces 5 & 6; at z=O,H
8: ._. 0
dz

Here, ‘H’ is the height of the niobium plate
2.3.2 BoundarlConditions on the Sensor
Since the whole sensor is assumed to be immersed in the liquid helium, a
convective boundary condition, as given below, is valid for the five sides of the

sensor.

sensor

AT

conv,He

 

 

q

The boundary condition on the sixth surface of the sensor i.e. at the interface with
the niobium surface is same as described for the niobium surface and is repeated here

for completeness.

51M; FL 2 qSensor X=L
TNb(i,j.k) = TSensor(i.j,k)
Here ‘L’ is the thickness of the niobium plate.
2.4 Discretization of the Differential Equation
The discretization of the differential equation is done using a finite difference

technique. The second order differentials in the equation are discretized at a point by

using the central difference with respect to the two neighboring grid points. Hence the

scheme is second order accurate.

Tm.” — 2T. + T

 

 

 

r.j.k i+l.j.k + Ti.j-I.k " 2Ti.j.k + Ti.j+l.k +
2 2
Ax Ay 22
T1,}.k—l "' 2Ti.j.k + Ti,j.k+l __ 0
Az2

This discretized equation, which is now a simple algebraic equation, can be solved
through SOR (Successive Over Relaxation) iterative technique.

Solving for T

13M , we get

1
Ti,j,k = 7[a(Ti-I.j.k + Ti+l.j.k ) + fl(Ti.j—I.k + Ti.j+l,k ) +
2.3
7(TI.).k—I + Ti.j.k+l)]

Where,

x = 2(Ax2Ay2 + AJCZAZZ + AzZAyz)

a = AyzAz2
,3 = Az2Ax2
y = AyzAx2

This equation is a Gauss-Seidel method representation of the Laplace equation.

Introducing the convergence acceleration parameter a), this equation converts to an SOR

representation of the three-dimensional Laplace equation.

1
n+l _ n +1
Ti,j,k — Ti,j.k + w[7{a(Tifl,j,k + Ti:l,j,k ) +

2.4
I n l n
fl<Tiflthk + Ti,nj+l.k ) + y(Ti.;k—I + Ti.j.k+l )} — Ti.nj.k]

10

Here the superscripts”n+1” is the current iteration number and ”n” is the

previously computed iteration number.

“to” varies between 1 and 2 and is determined through numerical experimentation to

enhance the convergence. For (0:1, the equation reduces to equation (2.3), the

Gauss-Seidel iterative method.
2.5 Implementation of Boundary Conditions

At the boundaries, since one of the two end surface grid points is not defined in the
central differencing scheme, a hypothetical point is assumed to exist beyond the surface
boundary. The relation for the hypothetical point is then derived from discretization of
the known boundary condition, which is then substituted into the equation (2.4) for that
point. It is also realized that except at the interface boundary of niobium plate with the
sensor, all boundary equations developed through this technique are similar to the ones
that can be obtained by applying finite control volume approach. In the following sub
paragraph, this principle is applied to implement the corresponding boundary conditions
and one boundary equation is derived from both these methods to illustrate this similarity.

2.5.1 On Hezged Facg

At Face 1 on the point of heating, the boundary condition is specified as:

8T

=—k __
‘1 Nb ax,

. . n+1 . . . . , , .
Here the grid pornt TH, 13;, Is the hypothetical pornt, as ‘I-l doesn t exrst on the

surface. To find the relation for this point, the above boundary condition can be

expressed algebraically with central difference as

11

k n+ n
q = _-2—XXL(7;HI _Ti—l)

2qAx
+1
or 7‘:ng = —+Ti:I,j.k

Nb

Hence the complete boundary equation becomes

 

2"“ + 2m... > +
Nb 2.5

n+1 n+l
fl(Ti,j—1,k + Ti,nj+l,k ) + y(Ti,j,k—l + Till/(+1)} _ Till/c]

n+ n 1
Ti,j,lr : Tr,” + w[7{a(

Now applying the finite control volume approach to the same boundary point, we

get:-

AxAz AxAz
KNb (Z—Ay-XTIJHJI '— Ti.j.k ) + KNb (Hy—X13141: " Ti.j,k)

AxAy AxAy

KNb (EXTIJJHI — TIJJI ) + KNb (Exam—I _ TIJJI)
A AZ

K~.<—i—x—><T...... —T.-.,-,. > + was: = 0

After simplification to solve for Ti.j.k , we get

2qAx

 

1 n
Ti,j,k = 7{a( + 2Ti+l,j,k)+

Nb
n+1 n n+1 n
r6(Ti,j—l,k + Ti,j+l.k ) + y(Ti,j,k—l + Ti,j.k+l)}
Here a , fl and y are the same as used in equation (2.4).

Hence it is seen that this equation is exactly the same as equation 2.5 once

assembled in SOR technique. This shows that the two methods of applying the

boundary conditions results in similar boundary equations.

Similarly, for the boundary points where heat flux is zero, the relation for a

hypothetical point is developed as:

12

=0=—k _
q NbaX

_ . n+l _ n
which results Into 71-1,” — Tm,”

2.5.2 Convective Boundary Condition

The convective boundary condition is given by working on the similar principle:

3T
q : ”ko — = hHe(TNb -T°°) ,
ax

Here TM is the temperature at the niobium-helium interface and To, is the known
bulk temperature of the liquid helium. When central differenced, the relation

becomes:

n n 2Axh ,
T- = T4" ——i"'—<T.. —T..>
ko
2.5.3 Boundary Condition at the Niobium-Sensor Interface
On the interface boundary between the niobium surface and the 6-10 material of

the sensor housing, as shown in figure 2, the boundary condition can be expressed

more easily with the help of a finite control volume approach and is expressed as:

 

I.j.k

AyAz AyAz
Ax (km; + kG—IO )T' _ —Kx—(koTi—I.j.k + kG—10Ti+1,j.k) +

AxAz AXAZ
17(ko + kG-IO )TiJJc — FUCM, + kG-IO)(Ti.j—1J€ + T131411) +

 

 

AyAx AyAx
AZ (km; + [(0—10 )T - A (km, + k0_lo )(Ti.j,k—l + Ti.j,k+l) : 0

i,j.k
Z

Solving for T,‘ 1.1: and simplifying the equation we get,

13

 

 

,, 1 2k ,. 2k _
T131311: ._. _[a{( Nb )Ti-I.lj,k +( G 10 )
’1 ko + [(6-10 ko + kG—IO 25
TIZIJJ: } + fl(Ti.nj+-ll.k + Ti,"j+1,k ) + ),(Tlfzjtllt—l + TIE-1+0]

Here 61,13 and y are the same as used in equation (2.4).

If a close look at this equation is made, it becomes evident that this equation is
quite similar to equation (2.4) except for the coefficients of the x-direction terms,
which now have the influence of the thermal conductivities of the two materials. This
can also be seen if we simplify this equation with the thermal conductivities of the
two materials is made equal, the equation becomes the same as equation 2.4 for one
homogeneous material. Hence a convenient and Simple logic is found to implement
this boundary condition along with the other boundary conditions into the computer
program.

2.5.4 Other Sides of Boundary Conditions On Sensor Housing

Since we are dealing with a three dimensional case, the sensor housing must have
five other surfaces in addition to the one mentioned in the case of Face 2. If the sensor
is not on any of the side edges then all other five faces shall have same type of

convective boundary condition as described above in 2.5.2 and is given as:
q = hue (T — Tm)
But if any sensor is modeled on the edge or comer of the niobium surface, then
this side of the sensor shall have the insulated boundary condition. So, if the sensor is
aligned with the top or bottom surface, then the valid boundary condition imposed as:

(a) 8_T = 0 and,

3y

If the sensor is aligned with the edges of face 5 or face 6, then

14

3T
b =0
( ) I)?

2.5.5 Boundary Conditions on Face 3 and 4
These faces are modeled as insulated surfaces, and the boundary condition is
given as:

E=O,then

n+l
Ti,j—I,k= —T-n “.1 k for face 3 and,

n+1 n+1
T ”14.1 k :Tj —lk for face4.

2.5.6 Boundary Conditions on Face 5 and 6
These faces are also modeled as insulated surfaces, since no heat transfer takes
place across them. So the boundary condition is given as: -

£=0,then

az

n+1 n
71311—1 =72)“, for faceSand

TMl =T"+,,_ 1 for face 6

1 j k+l
2.6 Program Characteristics
2.6.1 Convergence
Numerical formulation of Laplace equation can be represented in a system of
linear algebraic equations. These algebraic equations can be written in the matrix
form with U representing the solution vector.

A U=E

15

The solution of this system of equations exists and is unique if and only if A is
non-singular, in which case

U=A" E.

It may be shown that the following conditions are sufficient for the Gauss-Seidel
methods of iteration to converge [13]:

(a) The n x n matrix A is irreducible.

(b) 3,, >0

n
< . . . . . .
(c) Elaijl _ aii , wrth stnct Inequality holding for at least one value of ‘r’.
i=1

jar-i

It is observed that the system of algebraic equations arising from the application
of equation 2.4 along with the boundary conditions described above in section 2.5
satisfy the above conditions of convergence.

Defining the value of epsilon sets the convergence criterion for the program. The
program scans the whole grid to find the absolute largest ‘difference value’ on the
entire grid between the previous value and the newly computed value at the
corresponding grid point. It then compares this largest ‘difference value’ with the user
defined epsilon value. If this largest difference value over the whole computational
domain is greater than the epsilon, the program continues for the next iteration
otherwise, it displays the indication of convergence. The value of epsilon is selected
by the user depending upon the number of significant digits to which the accuracy is
desired in the solution. Smaller values of epsilon would require a greater number of
iterations to converge and hence require more computing time. Also, it is important to

note that, if the solution already contains very small numbers, then the value of

16

epsilon must be selected such that it is significantly lower (depending upon accuracy
desired to the number of significant digits) than the smallest value expected in the
solution domain. This can be done through obtaining an initial-guess solution by test
running the program on a coarse grid with smaller domain size. From this guess-
solution, the smallest value in the solution is used to Figure out the value of epsilon

that shall provide the desired accuracy in the actual solution. The default value of

epsilon used in the program is 10*.
2.6.2 Post Processing

As shown in Figure 3, the highest temperature locations on the cooled surface are
the four comers where sensors are installed because of the relatively reduced cooling
effects of the liquid helium. Due to the relatively high thermal conductivity of
niobium at low temperatures, heat generated by a point source is conducted away
diffusively within the metal and a very low temperature rise is expected at the outer
surface of cavity. Moreover, the rise in temperature on the cooled surface to be
detected by the sensors is greatly influenced by the excellent cooling properties of
superfluid helium. Figure 8 shows the relation between the sensor efficiency and the
sensor spacing for different thickness sizes of the niobium plate. Sensor efficiency is
a function of distance between the sensors (sensor spacing) and can be defined as the
ratio of temperature sensed by the sensor on the cooled surface and the maximum
temperature on the inner surface of the niobium cavity. The program is repeated for
each thickness of niobium plate i.e. 1mm and 4 mm and for each value of selected
bulk temperature of liquid helium i.e. 2 K and 4 K, for varying sensor spacing in each

C2186.

l7

 

 

 

 

 

Figure 3 Program output showing most heated spots at the four

comers under the sensor locations

2.6.3 Proggam Verification

Code verification is done with the results obtained through the exact solutions
generated by COND3D, developed for Sandia National Laboratory [15] and through
the Kokopelli Finite Element Method program developed for Los Alamos National
Laboratory [16]. As shown in Figure 8, the results obtained from all three programs
are in good agreement with each other and the error is within 1% of the exact
solution.

2.6.4 Program Validation
The validity of the model is assured by comparing the results obtained
from the program with the experimental data. Since no past experiment with
similar boundary conditions is known to exist, an independent experiment was

performed at room temperature. Six thermocouples as shown in Figure 4, were

18

used to record the temperature of the cold surface of a 4mm thick niobium plate.
A thin foil heater of 12.7 mm diameter was used to act as the point heating source
for the problem. The bottom surface, the hot surface, was then insulated using two
G-lO sheets of different thicknesses. The thin sheet (approx 0.5-1.0mm) was
flexible enough to bear against the bottom surface of the niobium plate with the
heater sandwiched between the two. Then, the thick (approx 20-30 mm) sheet of
6-10 was placed under the thin sheet to further insulate the bottom of the niobium
surface. An insulation ‘duct’ tape was used to bond the three plates together
which also provided insulation to the side walls of the niobium plate to match the
boundary condition as close to the model as possible. The experimental data thus
obtained is then compared with the data obtained from the program. As shown in
Figure 4a, the two results are in agreement with each other. The standard

deviation in difference of the two results is 0.009427°C

l9

 

 

\

 

Sensor

Locations

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

/
Heater on the other

 

x-Axis

 

side of the Plate+Sensor

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1 Dlv=1cm

 

Thickness=0.4cm

I

I

 

 

Figure 4

Experimental setup to validate the computational model.

Six sensors placed 1 cm apart over a 4 mm thick niobium

plate with heater installed on the bottom surface.

20

.0
O)
01

I

I

I

I

I

Experimental

Computed

o
'01 .0
01 or
J

Fitted

0.45 - - ;

Sensor Efficiency (O)
O
U"!

.0
A
I

o = -0.0474Ln(r) + 0.4894 ‘
0.35 . T . . , A ,
o 1 2 3 4 5 6 7 8

Radial Distance (cm)

 

 

Figure 4a Sensor efficiency ((1)) shown as a function of radial distance
(r). The relationship obtained from numerical computation
is compared with the one obtained from the experimental

setup at room temperature

2.6.5 Verification through Inverse Heat Conduction Calculations

To verify the results obtained from the program, yet another weapon in the
arsenal is the inverse heat conduction calculations through parameter estimation.
Here the parameters are the unknown temperature at the niobium surface right
under the heater and the ‘x’ and ‘y’ coordinates of the heater. Since in this
experiment we know the actual parameters before their estimation, the accuracy
of the results obtained from this technique may provide a good level of

confidence; especially, once it is applied during the actual operation of the cavity

21

where these parameters would not be known a priori. The equation employed in
this method, and the result thus obtained, are described briefly below.
From the definition of sensor efficiency i.e. the ratio of temperature at the sensor

to the maximum temperature on the heated surface , we know that
(15(r): T( r) / Tmax
where, (D(r) is the sensor efficiency and is function of distance ‘r’ which itself is given

as

 

r = {Ur-X.)2 +(y-y.~)2
here, (x, y) is the unknown location of the heat source and (xbyi) is the known location
of the sensor ‘i’ over the plate
Ti, is the temperature at a distance ‘r’ measured by the sensor ‘i’, and
Tm is the unknown maximum temperature on the heated surface of the plate i.e. at

the point of source of heating.

In matrix form this equation can be written as
[11-]: Tm...[<I><r.->]
In this equation, as described above, Tmax, and the coordinates (xi,y,-) of the

heater (which determines the value of r,- ) are the unknowns. Now if we have, let us

say six temperature sensors installed on the top (cold) surface of the plate to detect
the heat produced by the heater on the bottom (hot) surface of the plate, then each
sensor shall have its efficiency curve as a function of its location ‘r’ away from the

point of heating. Such a curve for each sensor is obtained from the direct computation

22

through the 3-D steady state program output. Thus from the above equation, we shall

have following six set of equations,
T1: Tmax ¢( 71)

T2: Tmax $0.2)

T6: Tmax $06)

Now, in this set of six equations, we have just three unknowns. So we have an
over-defined set of equations whose unique solution doesn’t exist. However, through
parameter estimation software written by Dr Robert L. McMasters, unknown variables

with optimum best (least square sense) values can be obtained through a non—linear least

squares technique. These values (xi, yi and Tmax) can then be obtained using the

functional form of (PM) through the direct solution.

Thus the close agreement of the two results as shown in Table 1 validates the
potential of estimation of parameters through inverse heat conduction methods. The
values of estimated parameters are then used to calculate the temperature rise detected by
each sensor. The difference between the measured temperature rise from the sensor and
the calculated temperature rise is quite small with standard deviation of only 0.045

Kelvin.

23

 

Verification of Estimated Parameters through Inverse Technique

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

x y "“ng1 Temp Estimate; Temp Error
1 0 3.5 3.43799008 -0.062
0 2 3.1 3.104853539 0.0048
3 0 3 3039895125 0.0399
0 4 2.9 2.899929045 -7.095E-05
5 0 2.8 2.863678044 0.0637
0 6 2.8 2.77476351 4 -0.0252
Std.Dev (K) 0.0449
Parameter
Estimated Experimental
Max
Temp 6.89735 6.9
Rise (K)
x (cm) 0.222 0
y (cm) -0.278 0
Tabla; Showing difference between actual temperature rise and estimated

temperature rise at sensor location by using estimated parameters;

24

 

2.7 Results and Discussion

The result shown in Figure 9 shows the variation of sensor efficiency with the
sensor spacing. At low sensor spacing for a particular sensor, high sensor efficiency can
be achieved. However, it has been previously stated [9] that a signal of 60 uK can be
resolved with a higher scanning time. So, from this result, we can assume a signal of 60
uK from which we can find a corresponding sensor spacing. For example, sensors on the
cooled surface of a 1 mm thick niobium plate and at bath temperature of 2 K can be
placed at a spacing 4 cm distance apart and will be able to detect a lK-temperature rise
anywhere in the material. Similarly, at 4 k the sensors on a 4 mm thick niobium plate can
be placed at a distance of almost 13 cm apart. The essential requirements to apply this
technique during the actual operation of the cavity have also been carried out and are

described in the subsequent chapters.

25

CHAPTER 3

THERMAL PROPERTIES OBTAINED FROM THE LITERATURE
3.1 Literature Search

A significant amount of effort has been expended to extract thermal property data for
niobium metal and liquid helium at extremely low temperatures. Since very little
information is available about materials at such low temperatures in standard data books,
much of the information is obtained through research papers published in various journals
and other publications.

3.1.1 Thermal Conductivity of Niobium

As shown in Figure 5 [3], the thermal conductivity of niobium metal depends on

the RR value of niobium metal, which is a measure of metal purity. As the graph
shows, there is a sharp decrease in the curve below TC=9.2 K. However, the higher the
RRR value, the higher the thermal conductivity. As discussed in chapter 3 of

reference [9], the reason for these features is that electrons are the dominant carriers

of heat. Though, the phonons (lattice vibrations) also play a role in heat conduction,
but this is significant only at T<4 K. Below Tc, the thermal conductivity drops
sharply as more electrons condense into Cooper pairs. Because the depairing energy

is not available from random thermal motion, the pairs are not scattered by the lattice

vibrations and therefore cannot conduct heat from one part of the niobium to another.
At high temperatures (4 k < T < TC ), a significant, though small, fraction of electrons

is not frozen into Cooper pairs and can carry heat effectively, provided that the
electron-impurity scattering is low. As electrons condense into Cooper pairs,

electron-phonon scattering also decreases. Below about 4 K, the thermal conductivity

26

from phonons dominates and begins to increase, leading to the phonon peak near 2 K.
With decreasing temperature, the number of phonons decreases oc T3. Ultimately the
value of the phonon conductivity maximum is limited by phonon scattering from
lattice imperfections, of which grain boundary density is the most important. If the
crystal grains of niobium are very large, because of annealing at high temperature,
one observes a large phonon peak, as shown in the thermal conductivity behavior of

the sample with RRR=250, which was annealed at 1400 °C [9].

27

 

 

 

 

1000 _ I I fl ' l I’ I T ‘
W/(mK) : o RRR = 400 33
500 " o A ..
_ . RRR = 250 .. f- _
3. . RRR = 90 "' .‘ ‘
g 200 - c .‘ .
3 ' a
'0 O
5 n ‘
0 100 - "A A
Te : .o " 3
g g a 4 . ‘ i “
o " fl 0 ‘ I "
5 50.. ' “ I I ‘-
l. . ~‘ 9.. I.
I I "
‘3“:
20 - A . -I
I . . I...
10 ib-J L L L 1 L I r 1
TIK]
Figure 5 Thermal conductivity of niobium at low temperatures

28

3.1.2 Thermal Conductivities of Common Cryogenic Materials

The thermal conductivity values of some commonly used materials are given in
Figure 6 for a wide range of temperatures from around 1 °C to 300 °C on a
logarithmic scale [5]. Thermal conductivity of G-lO, an epoxy filled fiberglass
material, is also shown in this Figure which is a poor conductor of heat at low

temperatures .

29

thermal conductivity (WIm-K)

 

‘n—r'r rTTm
.1

d

I

.J

-1

i

.l

l A_.l 1 ill]

1 L1 iturl

 

 

E
i
i .
l: '5
E i
k . «1
10%— '3
. i
i ;
g
l- :i
l J
i _. I
0.1 E “:i
g . 2
iris)” /‘«,,o :
as?“
001 ’E (‘6’ , a
/
E ".

/

000111; ”~42. ...L__L_.L_J_..l 1 LI
1

\

i
._. ”LE—J ._._L__L L_,L .L_L.L....-_r._ ._L.J

 

3 5 7 10 30 so 70 100 200 300
temperature (keivin)
Fi ure 6 Thermal conductivity of G-lO and other materials

commonly used in cryogenic environment

3.1.3 Convection Heat Transfer Coefficient For Liquid Helium

Unlike thermal conductivity, the heat transport across a solid-liquid helium
interface is not well understood [9]. Kramer et al. have found the experimental value
of the convection heat transfer coefficient of liquid helium for a temperature range of
1.5 °K to 2.3 °K as shown in Figure 7 and 8 [3]. Figure 7 shows the variation of heat
transfer coefficient with temperature. An important conclusion drawn from this
Figure is that the heat transfer coefficient of helium H is the same for system
pressures 0.1 and 0.25 Mpa. Also, while crossing the lambda transition temperature of
helium i.e. the temperature at which phase change take place between normal liquid
helium and the superfluid helium (See appendix 5.2 for more details), the heat

transfer coefficient drops to about 30 % of its previous value.

31

 

2

h [wlcm K]

 

 

 

 

_ o '
r 0.1 MPa 0.25 MPa ' ' ' .H
' o
0'2 i x + 0.21 wlcm2 . .4
2
l O
o 0.85 wlcm +5.,” X+i
0.0 ‘ l ‘ ‘ ‘ ‘ - l - I - A A n _
‘5 16 17 1.8 19 20 2,1 22 23
T [Kl
Figure 7 Variation of heat transfer coefficient with temperature for

two values of heat flux at two different pressures

32

2

h [wicm K]

.0
a)

'0'.»
WWW.
1

.0
co

 

O
\J
—-u v“: v r'

0.5

0.4

0.3 -
0.0

Figure 8

Y

A_._ l

Tfiz1.8 K

_ I A Q A ._.-r 1.; Ln.

 

I
l
,
.
.
I
..

b

0.4 0,8 2 1.2‘
q [Wicm l

d

0)
1"
0

Variation of heat transfer coefficient with heat flux for the
specific case of two bulk helium temperatures, i.e. for 1.8 K

and 2.0 K

33

 

Sensor Efficiency

Sensor Efficiency

Three solutions included at each condition: (bare plate), (insulated Sensors), and Exact

 

 

 

Solution
1
§\\
0.1 y \ 4mm Thick Niobium at 4K.
4mm Thick Niobium at 2K.
0.01 1
1mm Thick Niobium at 4K.
0.001 1 1mm Thick Niobium at 2K.
0.0001 4
0.00001 « ‘
0.000001 . . . 4 T . * .
0 2 4 6 8 10 12 14 16 18
Sensor Spaclng (cm)
Figure 9 Sensor efficiency as a function of sensor spacing

34

 

20

CHAPTER 4

EXPERIMENTAL STUDIES
4.1 Thermal Sensor Design and Manufacture

Since the operation of the SRF cavity is around 2 Kelvin, at which the liquid helium
is a super-fluid (liquid helium at temperatures below its phase transition i.e. 2.17 K,
doesn’t change to solid but remains a liquid known as super fluid. For more information
refer to appendix 5.2 to this report), it is important to have special design features, which
may enable the sensors to monitor the temperature rise above the bulk temperature of the
liquid helium. Previous studies at Cornell University [9] have demonstrated that a signal
of 60 11K above ambient can be resolved through a careful design of carbon resistor
temperature sensors. However, in the Cornell study, it is maintained that, because of the
excellent cooling characteristics of the superfluid helium in which the thermometers are
immersed, the efficiency of the sensors is greatly reduced from 100 % of the wall
temperature to about 20-30%. Here, again sensor efficiency is defined as the ratio of the
detected temperature increase to the actual rise in wall temperature. Consequently, they
used a large number of fixed thermometers to cover the whole cavity. Typically, for a 1.5
GHz single cell cavity almost 700 thermometers would be required [9]. The focus of the
present study, however, is to estimate the appropriate sensor spacing required to detect a
temperature rise of 1 °K above ambient anywhere on the inner surface of niobium cavity.

The basic design features of the sensors used in this research are essentially the same
as those developed at Cornell University [6] and manufactured in the same way.
However, the sensor installation procedure on the single cell cavity is somewhat modified

as described in the subsequent paragraphs.

35

4.1.1 Sensor Selection
A sensor selection criterion is based upon the following considerations: -

0 High sensitivity especially at low temperatures.

Insulated from liquid helium.

o Insensitive to magnetic or ionization fields.

Simple in design and low cost.

Though the selection of 1/8 watt, 100 ohm Allen Bradley carbon resistor and its
housing (capsule) design is based upon the one already used at Cornell University [6],
the present study tries to quantify the sensor selection properties through thermal
analysis.

4.1.2 Sensor Manufacture
The sensor manufacturing procedure used in this research is also based on the

guidelines used at Cornell University and involves, typically, the following processes:

0 Manufacturing of sensor housing with G-lO material as shown in Figure 10.

0 Spot welding of resistor leads with 36 AWG ‘Manganin wire’.

0 Epoxy filling of the housing with resistor held in middle of the housing.

0 Observing the epoxy curing time.

0 Grinding the epoxy from the top of the housing to expose the heat sensitive

carbon element of the resistor.

36

 

 

i

 

 

 

 

.400 .350 13.032 THRU
l # 67 DRILL
, m” (2) HOLES

 

(l

 

 

 

.OSO—J _,.. «In—.073

 

 

 

 

 

:ffiqn— I
[I513 l
J +._ ‘ i

 

 

Figure 10 Drawing details of sensor housing

37

4.2 Sensor Design Issues
The sensor selection and its manufacturing has been done on the basis of past
experiences of Cornell University, however, the order of magnitude analysis to estimate
the relative heat transfer from such a sensor into the surrounding liquid helium is
provided in the subsequent paragraphs.
4.2.1 Cooling Effects on Sen_sgr_'

Cooling of the carbon element inside the sensor housing can be attributed to the
two main sources i.e. cooling through the sides of the housing and cooling through
the manganin wires. The estimates for the cooling of the sensor are based on a 1 K
rise in temperature above the ambient temperature, which is a much larger
temperature rise than any actually encountered, providing a conservative estimate of

heat loss.

Manganin
Wire \‘

 

 

 

 

 

 

 

 

 

 

(3-10 Housing / 1

Carbon Element

38

4.2.2 Cooling of Sensor Through G-10 Housinfig
From Fourier's Law of heat conduction, we know that

AT
= 494...,
Q Ax

Where, Q is the heat transfer from the sensor to the liquid helium
AT is 1 K (i.e. one degree rise in temperature) above ambient.
‘K’ is the thermal conductivity of the G—lO material at 4 K
‘Ax’ is the wall thickness of the G-lO material
‘A’ is the surface area of the sidewall of (31-10 exposed to liquid helium.
So, the heat transfer from one surface, Q = 5.29><10‘4 Watts
Hence, the total heat transfer from all the 5 surfaces of the sensor that are exposed
to the liquid helium is, Q = 2.43x10'3 Watts.
4.2.3 Cooling Effect On The Sensor Through Manganin Wire
For simplicity and applying a more conservative approach, this problem is divided
into two parts:
(a) Assuming the whole portion of wire outside the G-lO housing is at the same
temperature as that of liquid helium. Hence, the effective temperature difference
(i.e. 1 degree difference in temperature) takes place on the length of wire between
the sensor and the edge of (31-10 housing.

For AWG-36 wire having diameter of 0.36 mm,

Q = 2.227x10‘5 Watts (2 Wires)

Q = 4.454x10" Watts (4 Wires)

39

(b) Heat transfer from the length of the wire inside the liquid helium, can be taken
as a typical case of heat transfer through fins. The solution is available in
standard textbooks on heat transfer. Interestingly, the total heat transfer through

the wire of infinite length is

Q = 5.162E" Watts for 2 Wires, and

Q = 1.032E‘3 Watts for 4 Wires.

4.2.4 How much is infinite length?
For the conditions stated in part (b), the infinite length i.e. the length of fin at end
of which the heat transfer is effectively zero and temperature of the fin is the same as

the temperature of the fluid, is computed from the following fin equation [13]:

Q = JhPKAc (To —T,,)tanh[ ’12: L]

For infinite length tanh[ hP L] = 1
KA

Or tanh( hP L] E .99
V KA
L2 ’KAr
hP

Where k is the thermal conductivity of the wire.

 

 

 

 

(2.65)

‘A’ is the cross sectional area of the wire.
‘h’ is the convective heat transfer coefficient of liquid helium.
‘P’ is the circumferential perimeter of the wire.
Substituting in the values for these parameters, we get, L 2 0.363 mm
This effective length of wire, in which any significant heat transfer is taking

place, is a small number. This is because, beside the extremely small cross sectional

4O

area of the manganin wire, liquid helium has a very high convective heat transfer
coefficient as compared to the low thermal conductivity of the manganin wire at 4 K.
Hence, nearly all the heat coming out of the wire from the edge of the G—lO housing,
is convected away within this length of wire. The rest of the length of wire has nearly
no effect on cooling the sensor.

In view of the two comparisons, we can observe that the cooling of the sensor
through the manganin wires is not significant as compared with the cooling of the
sensor from the G-lO sidewalls.

4.2.5 Ohmic Self Heating of the Senfl

For resistance temperature detector (RTD) type sensors, ohmic self-heating is a
known problem which occurs especially when using negative RTDs (resistor whose
resistance increases as temperature goes down). The monitoring instrument used in
the experiment is from LakeShore® Inc. model 218 and can display a maximum
resistance of 7.2 K52. Since the instrument provides a constant current of 10 uamps,
the maximum resistance of the sensor the instrument can display limits maximum
power dissipated inside the resistor. This power dissipation can be computed from the
following relation:

P = I 2 R
Where ‘P’ is the dissipated power
‘I’ is the current of the instrument, and
‘R’ is the resistance of the sensor.

Once the power is known, rise in sensor temperature due to this dissipated power

can be calculated from the relation:

41

AT 2 Q/(h A),
Where, Q is the dissipated power,
‘h’ is the convective heat transfer coefficient of liquid helium, and
‘A’ is the surface area of the sensor.
From this relation, the estimates for rise in temperature of sensor due to self-

heating are given in table 2.

 

 

 

 

 

 

 

 

 

Helium Temp (K) Current Source Power ( [1 Watts) AT ( ”K )
(,U Amps)
4.0 10 .18 .59
2 10 .75 .98
Table 2 Thermal analysis of ohmic self-heating. Here, delta t is the rise in

sensor temperature due to self heating=
4.3 Sensor Instzflttion / Mounting System

4.3.1 The Tree

The installation procedure used in this experiment is fundamentally different than
the one used at Cornell. Instead of using pogo sticks on the sensors and using G-lO
board to map these sensors onto the contour of the cavity from the top iris to the
equator and to the bottom iris, a simple but more versatile design is used. This design,
as shown in Figure 11, has been specifically selected so that it can be applied to a
wide variety of shapes of cavities as well as on multi-cell cavities without any major
modification in the basic design. The sensors on this design pattern, called a Tree
pattern, are equally spaced in both radial and transverse directions. The ‘Tree’ can be

mounted on any one of twelve cavity support bars. Since these support bars are

42

equally spaced at 30 degrees, a maximum of 12 such trees can be mounted for one
side (either the top or bottom) surface of the cavity. Each tree, then has two radial
rows of sensors, called the branches, with each having 4 to 5 sensors depending upon
the desired spacing between the sensors. As shown in Figure 12, manganin wires
from each sensor on the branch of a tree can be attached to their respective ‘D-type’
connectors installed for each branch on the tree.
4.3.2 Ribbon Cages

Normal ribbon cables take the temperature signal of the sensor from the D-
connectors to the ‘Feed Through’ on top of the cryogenic Dewar as shown in Figure

14.

43

Sensor Support Tree Cavity SllPPOrt

Rods

Sensor Support Tabs

7 Space for
‘ Ribbon
connector

Sensor Under
the Tab

 

Manganin
. . . Wires
Figure 11 Sensor installatron tree

Sensor Tree

 
 
   

Ribbon Cable
and Connector

Figure 12 Sensor installation tree (2)

45

4.4 Sensor Measurements Technique
4.4.1 Four-Lead Measurement

The four-lead measurement technique eliminates the effect of lead resistance on
the measurement. If it is not accounted for, lead resistance causes errors in the
temperature measurements.

As shown in Figure 13, a two lead sensor is shown in a four lead measurement
technique. In this configuration, the current leads and voltage leads run separately to
the sensor. With separate leads, there is little current in the voltage leads so their
resistance does not enter into the measurement, whereas resistance in the current
leads will not change the current as supplied by the constant current source.

4.4.2 Measurement of Two-Lead Sensors

Due to limited space in a congested cryogenic environment, sometimes it is not
possible to have the luxury of having four wire measurement sensors as described
above. However, it is possible to use a four wire measurement technique on a sensor
with two leads coming out of the cryogenic dewar. Four-lead sensor measurement in
this situation, can be adopted by attaching the plus voltage to plus current and minus
voltage to minus current leads at the back of the vacuum feed through on top of the
cryogenic dewar. The error in such a resistive measurement is the resistance of the
lead wire run with current and voltage together inside the cryogenic dewar. This
resistance may have to be traded off using the two lead measurements for a greater

number of sensors inside the dewar.

45

Urcufi Inade

 

the instrument Current
x Leeds
Rurrent Source —] /
l , l t

 

 

 

l ‘ Temperature
Is a R‘“ E} Sensor

1 R.

l \\ ! \\;_____\Hfitoge

L‘ _J Leeds
iNMfltmeter

 

 

 

 

 

Figure 13 Four lead sensor measurement technique

47

4.5 Design And Manufacture Of Vacuum Feed-Through.

Since the number of wires going into the cryogenic Dewar is always limited by the
capacity of the feed-through as well as the capacity of the pipe and passages from where
the bundle of wires has to pass, a good design for a Vacuum Feed —-Through is always
important. Many off-the-shelf vacuum feed-throughs are available from different
vendors, but none was considered sufficient to meet the specified requirements. For
example, a commonly used six-way multiple connector confiat® type vacuum feed—
through is not only expensive but also has a limited capacity of only 100 wires.
Moreover, it requires special connectors on the vacuum side wire connections, which
themselves may incur extra cost. It was for this reason an ingeniously designed feed-
through is used (Manufactured by Mr. Steve Bricker of N SCL) as shown in Figure 14.

This feed-through has a capacity of 240 wires in a conflat type vacuum seal. It has a
total of 10 connectors of 24-pin standard ‘CPC AMP’ type for the airside connections and

a D-type connector with ribbon cables on the vacuum side connections.

48

 

Figure 14 Vacuum feed-through having ten amp—cpc connectors,

twenty—four pins each

49

4.6 Design Criteria For Data Signal Cables.

The data signal cables used outside the cooling system can be much different from
those used inside. Between the instrument and the feed-through, heat leak is not a
problem. Error and noise pickup is minimized by the use of the following criteria for the
selection of data signal cables. The cable shall be

0 Twisted pair (to cancel out inductance noise)

0 Braided or foil shielded (to reduce noise pickup from external sources or

environment)

0 Larger conductor i.e. 22 to 28 AWG stranded copper wire. (flexible but

maintaining low resistance)

4.6.1 Grounding And Shielding Sensor Leads

Since the sensor input measurements are not isolated from ground, sensor lead
cables are not grounded with the outside chassis of the instrument [5]. Shielding the
data signal cables is important to keep external noise from entering the measurement.

A shield is most effective when it is near the measurement potential. The instrument

used in this experiment is Model 218 from LakeShore® Inc. and offers a shield that

stays close to the measurement potential. Therefore, the shield pin in the connector of

the data signal cable is connected to the input connector shield pin of the model 218

instrument. However, the other end of this shield pin in the data signal cable is not

connected to ground or to the instrument chassis or in the cooling system.
4.7 S'gnal Processing
Signal processing is done within the instrument model 218. The instrument has two

groups of eight input channels. Each group must have the same input type. The

50

instrument has 8 (one for each input channel) constant current sources to provide an
excitation current of 10 u Amperes for each sensor. It employs four-lead differential
measurement for each input channel and the measurement of this experiment from the
RTD sensor is displayed in ohms. The range the instrument can display is 0 to 7500
ohms.

The instrument has two AID converters to provide 16 new readings per second.
Therefore the display rate is twice each second. Though the measurement resolution of
the unit is 50 m but the display resolution is limited to 100 m9.

4.8 Data Logging

Data logging is done on a standard IBM compatible PC with the help of a serial
interface connection with the instrument model 218. A program has been under use at
NSCL (developed by Alberto Rodriguez) to log data from each sensor onto the hard disk
of the computer. A file can be generated either in Microsoft Excel or HTML file format
to save the data. The program has a user selectable data logging rate which can log the
data as fast as approximately 237 ms, which is faster than the data acquisition rate of the
temperature monitor.

4.9 Sensor Calibration.

The accuracy of a sensor relates to how closely the measurement of resistance can be
converted to temperature relative to some recognized temperature scale. Understanding
how the accuracy of temperature sensors is specified begins with the definition of the
response curve, i.e. Resistance vs. Temperature for the carbon resistor sensor. Though

some sensors follow a known standard response within a given tolerance, carbon resistor

51

sensors must be calibrated to determine their response curve. Therefore, the carbon
resistor sensors are not interchangeable or at least require calibration after installation.

We know that the accuracy of a sensor is determined by the stability of the sensor and
the system in which it is calibrated. Calibration of carbon resistor sensors, in this
experiment, is done with respect to a standard Germanium resistor sensor that is pre-
calibrated from factory standard settings (ITS-90) and can be traced back to NTST.

In the calibration procedure, data is collected for the variation in resistance of the
carbon sensors with respect to the change in temperature recorded by the Ge sensor as
shown in Figure 15. Since the fitted curve is very close to the experimental curve,
therefore no significant variation in the two curves can be seen in Figure 15. However,
Figure 16 shows the residuals present in the polynomial fit to the calibration curve. The
fourth order polynomial

T = A+ BR" + CR’2 + DR‘3 + ER‘4
is then fitted to the experimental data.
The parameters A, B, C, D, and E are computed to fit the experimental data and hence
yield a standard deviation of 0.003789 Kelvin.

Now, once the relation is known, other pertinent relation for data analysis can be
obtained very easily. For example the sensitivity of the carbon sensor can be obtained by
differentiating the above equation as: -

d—T- = —BR‘2 - 2CR‘3 — 3BR" — 4ER'5
dR

The relation used at Cornel University is given as:

T: l

 

Aln(R) + + C

 

ln(R)

52

This relation requires only 3 unknown parameters to be found, but is not trivial to
solve since it is a non-linear relation, especially once sensitivity calculations require the

differential of this relation.

 

Carbon Resistor Calibration

6000

 

5000 - ~
4000 -

3000 «—
2000 ~

 

1000 ~ ~ - __ _

Resistance From Carbon
(Ohms)

 

 

 

0 1 2 3 4 5
Temperature From Ge (Kelvin)

 

 

 

Figure 15 Calibration curve for carbon sensor resistance as a function

of temperature based on a germanium sensor

53

 

 

 

 

 

lb
or

 

Resistance Residuals (Ohms)
,{3 L. J. , _. _..
O 01 O 01 O 01 O 01

1 2 3 4 5
Temperature (Kelvin)

Figure 16 Residual of polynomial fit to the calibration data

4.10 Sensor Sen_sitivityand Mgsurement Resolution.

A typical calculation is performed on sample data, as shown in Table 3, which is
obtained from the calibration curve of a carbon resistor. The comparison of sensitivity
between the Germanium and the Carbon resistor shows that, at low temperatures, the
Carbon resistor is more sensitive than the Ge resistor especially once the temperature
goes down from 4 °K to 2 °K. The chart also indicates that the Carbon sensor at 2 K, with
the given monitor display resolution, is sensitive enough to resolve a heat signal of 9.4

01K.

54

 

 

 

 

 

 

 

 

 

 

 

 

 

Sensor Type Germanium Carbon
Temperature Coefficient Negative Negative
Monitor Display Resolution 100 m9 100 m9

Sensor Sensitivity at 4 K -1 16.98Q/K -57 2 (UK

2 K -930.04 Q/k -10537.8 Q/k
Monitor Display 4 K 854 [K 174 [K
Resolution w.r.t. 100 m9 2 K 107.52 “K 9.4 ”K
W Sensor sensitivity and display resolution of the sensor monitor

55

APPENDICES

56

5.1 Program Description
5.1.1 User Graphigrl Interface (UGI)

To simulate the thermal model of the super conducting cavity into a computer
program, ‘Visual Basic’ as a programming language is used. The advantage of this
language over other high level languages is that Visual Basic, besides being as fast as
others, has a system by which a graphical interface can be generated much more
easily. This added advantage of a better graphical interface results in providing a
professional appearance to the program and provides a more ‘user friendly’
environment. The first window, where the user interacts with the program, commonly
known as user graphical interface or simply the UGI, has some text boxes along with
some option buttons for the input by the user. Two command buttons allow the user
to either execute the program with a ‘Start’ button, or abort the program with a
‘Cancel’ button.

5.1.2 Input Text Boxes of UGI

Input text boxes of UGI are the places where the user can describe the dimensions
of the model. These dimensions include the configuration of the niobium plate i.e.
thickness (x overall dimension), height (y overall dimension) and, length (2 overall
dimension) and the configuration of the sensors installed on the cold surface of the
niobium plate i.e. thickness of the sensor (height), length and width of the sensor.
Moreover, flexibility in the design for the placement of sensors is also incorporated.
This is done by allowing the user to define the distance, as per design requirement,

between the base of the niobium plate and the bottom surface of the first sensor. This

57

distance is zero in case the two surfaces i.e. base of the niobium plate and the bottom
surface of the sensor, are in-line with each other.

Depending upon the number of grid points required by the user in the numerical
problem, values of delta x, delta y and, delta 2 can also be prescribed in this UGI.

The UGI not only facilitates the user to define material properties required by the
program such as thermal conductivities and heat transfer coefficient of the materials
involved, but also allows for prescribing the location and the amount of point heating
flux on to the heated surface of niobium plate, so that physically correct boundary
conditions can be implemented in the program

Flexibility of choosing the desired accuracy for the convergence of the solution is

also built into the program by allowing the user to prescribe the value of epsilon in

one of the input text boxes. The default value is 10"").

During the program execution, iteration numbers are displayed with an increment
of 20 iterations, to indicate the processing of the program.
5.1.3 Option Boxes of the UGI

Option for applying SOR to the scheme is left at the discretion of the user by
providing an option button. This button is checked for SOR by default and if
unchecked the scheme will become an ordinary Gauss-Seidel iterative procedure. If
the SOR option is utilized, value of omega=l.8 is used to accelerate the convergence
of the program. It is estimated that, in using this option, the program can reduce the
number of iterations to almost one fifth of the time required by the ordinary Gauss-

Seidel iterative method with no loss of accuracy.

58

Since the sensors are rectangular in shape, their orientation with respect to the
niobium plate coordinates can also be prescribed. The user can optionally choose the
orientation of the sensors by placing the width of the sensor to be either aligned with
the y-direction or with the z-direction.

A word of note is also displayed on the GUI, at the bottom right comer of the
screen; to remind the user for selecting consistent units while designing the input
values for the problem. The results would be meaningless if units for different

dimensions are not kept consistent.

5.1.4 Program Listing
The computer program code developed in Visual Basic® is given below to

implement the mathematical model described in chapter 2.

59

Option Explicit
'3-D STEADY STATE HEAT CONDUCTION PROBLEM IN A PARRALLELPIPED
‘DOMAIN WITH POINT HEATING ON HEATED SURFACE AND SENSORS

‘INSTALLED ON THE SURFACE COOLED BY LIQUID HELIUM.

Dim Lx As Single 'Overall dimension in the x direction
Dim Ly As Single 'Overall dimension in the y direction
Dim Lz As Single 'Overall dimension in the z direction
Dim omega As Double

Dim Dx As Single rDelta x
Dim Dy As Single 'Delta y
Dim Dz As Single 'Delta 2

Dim strMyDataString As String
Dim strInputDataString As String
Dim blnsensel As Boolean

Dim seldec As Integer

Dim txtll As String

 

Dim txt12 As String

Dim tloop As Integer

Dim consttl As Double ' Defined as = dt/(rho*cp*dx) for left b.c (congant flux)
Dim constt2 As Double ‘Defined as = 2*alpha*dx*h/k for right b.c (convective)

 

Dim KNb, KglO, Hf, Tf 'constants of thermal conductivity, convective heat coefficient
‘of fluid, Fluid temperature T(infinity),(Rho, Spht not requried)

 

Dim Alpha, Beta, Gama, Nue 'Computed Constant Coefficients in all three dimensions
Dim Index As Integer

Dim q As Double

Dim YinQ, ZinQ, sor As Integer

Dim imx As Integer 'Mx limit if no sensor found in x-dir

Dim imax As Integer 'Mx limit if sensor found in x-dir

Dim jmax As Integer

Dim krnax As Integer

Dim postopO As Single
Dim poslft() As Single

' Dimensions of sensors

Dim Hs, Ls, Ws As Double

Dim Ytotsen, Ztotsen As Integer

Dim Zbase, Ybase, ytop, ztop As Double

Dim Zgap, ygap As Double

60

Dim qygp, ngp As Integer

Dim YSize, ZSize, intfac As Double

Dim Nygps, Nygpg, Nygpb, Nzgps, Nzgpg, Nzgpb, YsenNUM, ZsenNUM As Integer
Dim zsentbl() As Integer

Dim ysentbl() As Integer

Dim Ysensr, Zsensr, flag As Boolean

Dim ZsLL, ZsUL, YsLL, YsUl As Integer

Dim itr As Integer

Dim Errmx As Double
Dim Error As Double
Dim Epsilon As Double

Dim wid As Single
Dim Hht As Single

'Dynamic Array of Temperature distribution according to the user i/p
Dim Tmp() As Double

Private Sub Command1_Click()
Dim k, j As Integer

txtll = ""

txt12 = ""

Text9.Enabled = False

Inputblock 'Gets the input from the user form 1

strMyDataString = ""

jmax=Ly/Dy+l
kmax=Lz/Dz+1

qygp = YinQ / Dy + 1
ngp = ZinQ / Dz +1

If (Check1.Value = 1) Then ' To enable SOR techniqumlculations
Dim sigma, pi As Double
pi = 22 I 7

omega = 1.8

61

Else
omega 2 1
End If
Text30.Text = Str(omega)

' To define the orientation of sensors installed on the cooled surface.
If (Option1.Value = True) Then

YSize = Ws

ZSize = Ls
Else

YSize = Ls

ZSize = Ws
End If

Nygps = YSize / Dy + 1 '# of Y-dir grid points in a sensor
Nzgps = ZSize / Dz + l '# of Z-dir grid points in a sensor
Nygpg = ygap / Dy - l '# of Y-dir grid points in a Gap
Nzgpg = Zgap / Dz - 1 '# of Z-dir grid points in a Gap
Nygpb = Ybase / Dy '# of Y-dir grid points in a Base
Nzgpb = Zbase / Dz ' # of Z-dir grid points in a Bag;

 

Ytotsen = jmax / Nygps + 1 'Guess value of total sensors in y-dir
Ztotsen = kmax / Nzgps + 1 'Guess value of total sensors in z-dir

ReDim ysentbl(l To Ytotsen, 1 To 3)
ReDim zsentbl(l To Ztotsen, 1 To 3)

' Create sensor locaLtion table; one for y-dir
Dim templ, temp2 As Integer
Dim ii As Integer
ysentbl(l, l) = l
ysentbl(l, 2) = N ygpb + l
ysentbl(l, 3) = ysentbl(l, 2) + N ygps - 1
For ii = 2 To Ytotsen
templ = ysentbl(ii - 1, 3) + Nygpg + l
temp2 = templ + Nygps - 1
If (templ < jmax And temp2 <= jmax) Then
ysentbl(ii, l) = ii ' Sets the Y-index # for the sensor
ysentbl(ii, 2) = templ ' Sets the lower limit of a particular sensor
ysentbl(ii, 3) = temp2 ' Sets the upper limit of a particular sensor
Ytotsen = ii
Else
GoTo 41
End If
Next ii

62

' Crgrte sensor locfiation table; one for z-dir
41 zsentbl(l, l) = l
zsentbl(l, 2) = Nzgpb + l
zsentbl(l, 3) = zsentbl(l, 2) + Nzgps - 1
For ii = 2 To Ztotsen
templ = zsentbl(ii - l, 3) + Nzgpg + 1
temp2 = templ + Nzgps - 1
If (templ < kmax And temp2 <= kmax) Then
zsentbl(ii, l) : ii ' Sets the Z-index # for the sensor
zsentbl(ii, 2) =temp1 ' Sets the lower limit of a particular sensor
zsentbl(ii, 3) = temp2 ' Sets the upper limit of a particular sensor
Ztotsen = ii
Else
GoTo 51
End If
Next ii

51 ytop = jmax - ysentbl(Ytotsen, 3)
ztop = kmax - zsentbl(Ztotsen, 3)

imx=(Lx+Hs)/Dx+l

Alpha = (Dy * Dz) ." 2

Beta = (Dx * Dz) " 2

Gama = (Dx * Dy) " 2

Nue = 2 * (Alpha + Beta + Gama)
intfac = Lx / Dx + 1

ReDim Tmp(l To imx, 1 To jmax, 1 To kmax) As Double

initializ ' Set an initial eduacated guess at liquid helium temp.
tlevel
'5 Unload Form2

End Sub
Sub Domenlmt()
Dim i As Integer

For i = 2 To nez
kmin(i) = kmax(i - 1) +1
kmax(i) = kmin(i) + kmax(l) - 1
Next i
jmin(1) = l
jmax(l) = Ly / ney

63

For i = 2 To ney
jmin(i) = jmax(i - 1) + l
jmax(i) = jmin(i) + jmax(l) - 1
Next i
tmx = (ToT / Dt) +1
imx = (Lx / Dx) + 1

End Sub

Sub Inputblock()

Lx = Val(Text1.Text)
Ly = Val(Text2.Text)
L2 = Val(Text3.Text)

Dx = Val(Text5.Text)
Dy = Val(Text6.Text)
Dz = Val(Text7.Text)

KNb = Val(Text16.Text)
Kg10 = Val(Text15.Text)

Ls = Val(Text4.Text)

Ws = Val(Text8.Text)

Hs = Val(Text9.Text)
Zbase = Val(Text12.Text)
Ybase = Val(Text23.Text)
Zgap = Val(Text19.Text)
ygap = Val(Text24.Text)

Epsilon = Val(Text28.Text)

q = Val(Text25.Text)
YinQ = Val(Text26.Text)
ZinQ = Val(Text27.Text)

Hf = Val(Text17.Text)
Tf = Val(Text18.Text)

End Sub

Private Sub tlevel()

Dim i, k, j, tind As Integer
Dim cnt As Integer

itr=l
me=1

64

While (Errmx > Epsilon)

xyswpnge ' 3D Laplace equation solving scheme.i.e. sweep on
'xy face with paging in z-direction
cnt = itr/ 20
1f (itr = cnt * 20) Then
Text29.Text = Str(itr)
Textl3.Text = Str(Errmx)
Forrnl .Refresh
End If

itr = itr + l
Wend

Text29.Text = Str(itr)
Text30.Text = Str(omega)

' Output Format Section of the Program
5 Forj= l Tojmax

Fork=1Tokmax

strMyDataString = strMyDataString & Str(Format(Tmp(intfac, j, k),
"##.00000000")) & vbTab

Next k
strMyDataString = strMyDataString & vbCr
Next j

' Calling Output file for the storage of the computed data
pstproc

End Sub

Private Sub initializ()

' This provides temperature initiaiconditions
Dim i As Integer

Dim j As Integer

Dim k As Integer

' Note!

' A RECTANGULAR DOMAIN (INCLUDING THE SENSORS IN IT) IS
‘ASSUMED FOR APPLYING UNIFORM INITIAL GUESS VALUES. BUT THE
‘POINTS OUTSIDE THE ACTUAL DOMAIN SHALL NOT BE COMPUTQ

' IN THE MAIN PROGRAM.

65

Fori = 1 To imx
For j = 1 To jmax
For k = 1 To kmax
Tmp(i, j, k) = Tf ' Initial Guess temp equal to fluid temp Tf
Next k
Next j
Next i

End Sub

Public Sub Ysenloc(jk)

Dim ii As Integer
Ysensr = False
For ii = 1 To Ytotsen
If (jk >= ysentbl(ii, 2) And jk <= ysentbl(ii, 3)) Then
Ysensr = True
YsenNUM = ii
YsLL = ysentbl(ii, 2)
YsUl = ysentbl(ii, 3)
GoTo 5
End If
Next ii

5 End Sub
Public Sub Zsenloc(jk)

Dim ii As Integer
Zsensr = False
For ii = 1 To Ztotsen
If (jk >= zsentbl(ii, 2) And jk <= zsentbl(ii, 3)) Then
Zsensr = True
ZsenNUM = ii
ZsLL = zsentbl(ii, 2)
ZsUL = zsentbl(ii, 3)
GoTo 5
End If
Next ii

5 End Sub
Public Sub xyswpnge()

Dim i, j, k, SC As Integer
Dim AMn, AMx, BMn, BMx, CMn, CMx, tempo As Double

66

Dim kk As Double
Dim constt, constl, const2, const3, const2y, const32 As Double
Errmx = 0

flag = False
For k = 1 To kmax
Call Zsenloc(k)
For j = 1 To jmax
Call Ysenloc(j)
If (Ysensr = True And Zsensr = True) Then
flag = True
Else
flag = False
End If

If (flag = False) Then
imax = Lx IDx + l
kk = KNb

Else
imax = imx
kk = KglO

End If

constt=6*Hf/ ll lkk

constl = l + constt * Dx
const2 = l + constt * Dy
const2y = 1 - constt * Dy
const3 = 1 + constt * Dz
const32 = 1 - constt * Dz

For i = 1 To imax
IMPLEMENTATION OF X-DIRECTION BOUNDARY CONDITIONS
' Inside the computational domain when not on anyof the boundary
If (i <> 1 And i <> intfac Andi <> imax) Then
AMn = Tmp(i - l,j, k)
' On the Heated surface excludingpoint heating
ElseIf (i = 1 And (j 0 qygp Or k <> ngp)) Then
AMn = Tmp(i + l,j, k)‘

' On the Heated surface at the point of heating

67

ElseIf (i = 1 And j = qygp And k = ngp) Then AMn = 2 * Dx * q / KNb + Tmp(i
+ l,j, k)

' At the interfape of Niobium and the Sensor
ElseIf (imax <> intfac And i = intfac) Then AMn = 2 * KNb / (KNb + Kg10) *
Tmp(i - l,j, k)

' At the cooled surfac_a
ElseIf(i =imax) Then AMn = l/constl *(2/11 * Tmp(i - 3,j, k) - 9/11*
Tmp(i - 2, j, k))

End If

' Inside the computational domain when not on any of the boundary
If (i < imax And i <> intfac) Then AMx = Tmp(i + 1, j, k)

' At the cooled surfrg
If (i = imax) Then AMx =1/const1 *(18/11* Tmp(i - l,j, k) + constt * Dx *

Tf)

' At the interface of Niobium and the Sensor
If (flag = True Andi = intfac) Then AMx = 2 * KglO / (KNb + KglO) * Tmp(i +
l, j, k)

' Implementation of Y and Z Direction boundary conditiona
If (i <= intfac) Then ' when i is in Niobium computational dormip
If (i = 1) Then
BMn = Tmp(i,j + 1, k)
BMx = Tmp(i,j + l, k)

ElseIf (j = jmax) Then
BMn = Tmp(i,j - l, k)
BMx = Tmp(i,j - l, k)

Else
BMn = Tmp(i,j - l, k)
BMx = Tmp(i,j + 1, k)

End If
If (k = 1) Then
CMn = Tmp(i,j, k +1)
CMx = Tmp(i,j, k +1)
ElseIf (k = kmax) Then

CMn = Tmp(i,j, k - 1)
CMx = Tmp(i,j, k - l)

68

Else
CMn = Tmp(i,j, k - l)
CMx = Tmp(i,j, k +1)
End If

Else ' when i is in sensor computational domain

' For Y-Direction Bouniary Conditions
If (j <> YsLL And j 0 YsUl And j <> 1 And j 0 jmax) Then
BMn = Tmp(i,j-1, k)
BMx = Tmp(i,j + l, k)
ElseIf (j = 1) Then
BMn = Tmp(i,j + l, k)
BMx = Tmp(i,j+1, k)
Elself (j = YsLL And j <> 1) Then
BMn = l /const2y *(18/11 * Tmp(i,j-+1, k) - 9/11 * Tmp(i,j + 2, k))
BMx = l lconst2y * (2/ 11 * Tmp(i,j + 3, k) - constt * Dy * Tf)
Elself (j = YsUl And j <> jmax) Then
BMn = l /const2 *(2/11 * Tmp(i,j - 3, k) -9/ 11 * Tmp(i,j — 2, k))
BMx = l/const2 *(18/11 * Tmp(i,j - l, k) + constt * Dy * Tf)
ElseIf (j = jmax) Then
BMn = Tmp(i,j - l, k)
BMx = Tmp(i,j - 1, k)
End If

' Similarly for conditions in Z-Direction
If (k <> ZsLL And k <> ZsUL And k <> 1 And k <> kmax) Then
CMn = Tmp(i,j, k - l)
CMx = Tmp(i,j, k +1)
ElseIf (k = 1) Then
CMn = Tmp(i,j, k + l)
CMx = Tmp(i,j, k + 1)
ElseIf (k = ZsLL And k 0 1) Then
CMn = 1 lconst3z *(18/11 * Tmp(i,j, k+ 1) - 9/11 * Tmp(i,j, k+ 2))
CMx = l /const3z * (2/ 11 * Tmp(i,j, k + 3) - constt * Dz * Tf)
ElseIf (k = ZsUL And k 0 kmax) Then
CMn = l /const3 *(2/11*Tmp(i,j, k-3)-9/11 * Tmp(i,j,k- 2))
CMx = l lconst3 *(18/11* Tmp(i,j, k -1)+ constt * Dz * Tf)
ElseIf (k = kmax) Then
CMn = Tmp(i,j, k — 1)
CMx = Tmp(i,j, k - 1)
End If

End If

tempo = Tmp(i, j, k)

69

Tmp(i, j, k) = Tmp(i, j, k) + omega * ((l / Nue * (Alpha * (AMn + AMx) + Beta *
(BMn + BMx) + Gama * (CMn + CMx))) - Tmp(i, j, k))

' Monitor the Absolute Error at each grid point
Error = Abs(Tmp(i, j, k) - tempo)

' Trap the maximum error in each itration
If (Error > Ernnx) Then Errrnx = Error

Next i
Next j
Next k

End Sub

Private Sub Command2_Click()
End
End Sub

Private Sub Form_Load()
Text9.Enabled = False

End Sub

Sub pstproc()
'this program :-
'1. checks to see if there is a folder on C drive called "C:\DataResult "
'2. If finds it then Ok, if not then creates a folder.
'3. Then checks to see if there is a file named C:\DataResult\MyDataFile-l.xls
'4. If found then cheks to see if there is a file named C:\DataResult\MyDataFile-2.xls
'5. Keeps trying till it finds a name that is not in use in that folder.
'6. Creates a file at the location found in step 5.
'7. Everytime run this program it will create a new file.

Dim objFSO As New Scripting.FileSystemObject
Dim objTextStream As Scripting.TextStream
Dim strCompletePath As String

Dim strFolderName As String

Dim strFileName As String
Dim strFileExtension As String

strFolderName = "C:\DataResult\"

7O

strFileName = "MyDataFile"
strFileExtension = ".xls"

Dim blnFileNameFound As Boolean
Dim intFileNameCount As Integer

strCompletePath = strFolderName & strFileName

' first Check to see if the folder exists if it doesnt then create one
If objFSO.FolderExists(strFolderName) = False Then

'folder doesnt exist so create one

objFSO.CreateFolder strFolderName
End If

'check to see if folder wcrezflea
If objFSO.FolderExists(strFolderName) = False Then
'folder doesnt exist so create one
MsgBox "Failed to create a folder at : " & strFolderName
GoTo DestroyObjects
End If

blnFileNameFound = False
Do While blnFileNameFound = False ' Means keep looping till grew file we is
‘decided
intFileNameCount = intFileNameCount + l
strCompletePath = strFolderName & strFileName & "-" & intFileNameCount &
strFileExtension
If objFSO.FileExists(strCompletePath) = False Then
' File name is unique exit
blnFileNameFound = True
End If
Loop

If Len(strMyDataString) > 0 Then
Set objTextStream = objFS0.0penTextFile(strCompletePath, ForWriting, True,
TristateUseDefault)
objTextStream.Write strMyDataStrin g
objTextStream.Close
Else
MsgBox "There was no data provided so file was not created"
End If

If objFSO.FileExists(strCompletePath) = True Then

MsgBox "Data file was successfully created at " & strCompletePath
Else

71

MsgBox "Data file was NOT created"
End If

DestroyObjects:
' Set flobiects to nothing otherwise they will my in the memory
Set objFSO = Nothing
Set objTextStream = Nothing
End Sub

72

5.2 Properties Of Litrpid Helium

5.2.1 Backggound

Janssen first discovered traces of helium during the solar eclipse of 1868 in the
solar spectrum of which he detected a new line. Kaerlingh Onnes in his laboratory
first achieved liquification of helium in 1908 by using liquid hydrogen pre-cooling in
a J oule-Thomson liquefier.
5.2.2 Isotopes of Helium

Helium has two isotopes, known as helium-3 and helium-4. The helium-4 is a
more common isotope, which consists of two electrons surrounding a nucleus of two
neutrons and two protons. The helium-3 atom consists of two electrons surrounding a
nucleus of two protons plus one neutron.
5.2.3 Liquid Helium

As described above, helium-4 is the most common of the two isotopes. Ordinary

helium gas consists of about 1.3x10'4 percent helium-3, so usually when reference is
made to liquid helium or helium, generally helium-4 is being referred to. It has a
molecular weight 4.0026.
5.2.4 General Properties of Helium

Liquid helium~4 is odorless and colorless and sometimes difficult to see in a

container. Table 4 lists some of its properties [10].

73

 

Boiling point temperature

4.224 K

 

Density at normal boiling point
Liquid

Vapor

124.96 kg/m3 or 0.03122 My cm3

16.89 kg/m3 or 0.00422 Wycm3

 

Heat of vaporization at normal boiling

point

10.73 kJ/kg

 

Solidification point at normal atmospheric

pressure

Doesn’t exist even if temperature is

reduced to absolute zero

 

 

Index of refraction n1-

 

1.02

 

Table 4

5.2.5 Unigue Promrties of Liquid Helium

General properties of liquid helium

Liquid helium has number of unique properties, which are not present in many

other liquids. One of the first properties of liquid helium that attracted great attention

is the absence of solid-liquid-vapor triple point. The most amazing properties,

however, are those shown by liquid helium at temperatures below 2.17 k. As the

liquid is cooled below this temperature, instead of solidifying, it changes to a new

liquid phase hence marking another distinct transitional line on the phase diagram.

This transitional line separates the two phases of liquid helium. Liquid helium above

this transitional line is known as helium-I and liquid helium below this transitional

74

 

line is known as helium-II. Properties of helium-II are remarkable as it does not obey
normal laws. Helium-H expands on cooling: its conductivity for heat is enormous and
is as high as 86,500 W/m K — much higher than that of pure copper at room
temperature [1]. In fact, measurement of true values of thermal conductivity below
the transition point, known as lambda point, is very hard to make. As an estimate, k
for He H below the transition point is about 80,000 W/m-K between 1.4 and 1.75 K
and it might go as high as 340,000 W/m-K at 1.92 K. These values are the highest
conductivities known of copper, silver, and diamond [11].
5.3 Temperature Measurements

5.3.1 Defining Temperature Measfurement [5]

Temperature is not a directly measurable quantity, but rather inferred from other
measurements, which can be made directly. All measurements have an associated
uncertainty, u, which depends on the statistical nature of the measurement. The
choice of resolution, precision or, accuracy as the important statistical feature of the
measurement can be important. Good resolution is easiest to achieve and good
accuracy is hardest to achieve. Selecting an inappropriate statistical measure either
adds cost to or decreases the value of a measurement. As an example, resolution is the
most important statistical measure if small temperature changes must be measured,
but the absolute temperature either is not important or can be measured
independently. Precision is most important when successive measurements under
identical conditions must return the same value, but the true value of the quantity is
not required. An example might be a temperature control system, which must run

successive samples through identical temperature cycles, but the actual temperatures

75

are less important than the repeatability of the cycles. Accuracy is most important
when absolute temperature must be known.

It is always true that the accuracy is worse than the precision, and precision is
worse than the resolution of a measurement.

No temperature sensor operates by itself. The expected resolution, precision and
accuracy of a temperature measurement depend not only on the properties of the
temperature sensor, but also on the measurement system and the sensor’s thermal
environment.

5.3.2 Uncertainty Conversions Using Dimensionless Sen_sitivities

A sensor produces some output signal such as a voltage, frequency, resistance or
capacitance, which is measured by the instrumentation.

Uncertainties in the measured quantity are not in units of temperature and must be

converted. In the case of a temperature sensor with an output Resistance R, the

temperature “T uncertainty is related to the resistance uncertainty u R by the

dimensionless formula

m/ uV
34;- R = R 5.1

_ dR _
T (T/Rx /dT) ST
Where 5, = (T/R)(dR/dT) is defined as the dimensionless temperature

sensitivity. The dimensionless sensitivity is also equal to d(log R)/d(log T), the slope
of the resistance verses temperature curve on a log-log plot. There are advantages to
this form of the equation. First, the dependence of the resolution upon the excitation
is now explicit rather than implicit in the sensitivity. Second, the dimensionless nature

of equation 5.1 allows its application to thermometers based on other temperature

76

dependant properties (capacitance, voltage, or pressure) by replacing R with C, V, or
P as appropriate. This allows for easier comparison of calculation of resolutions based
on different temperature-dependent properties.

A dimensionless sensitivity in the 0.1 to 10 range is usually best for property
measurements over a wide range, although other factors such as physical size or
sensitivity to environmental conditions can be much more important. A large
dimensionless sensitivity allows good relative temperature resolution, but the
temperature range becomes limited if the value of the property measured becomes too
large or small to be determined accurately.

5.3.3 Resolution
Resolution is given the symbol r and has the same units as the quantity measured.

The smallest distinguishable change in a temperature measurement is termed the

temperature resolution, 7'7 . The temperature resolution is often determined by some

limiting factor in the measurement system. The resolution of a digital system is
typically determined by the smallest unit change, which can be displayed and is
usually a fixed value for a given range. In the simple case of a temperature readout

displaying three digits to the right of the decimal, the temperature resolution is 0.001

times the temperature unit (e.g. K or "C ).

As an example, consider an instrumentation system consisting of an Ohmmeter
with a 5-digit display capable of displaying 0 to 99999 counts. If the smallest range is
0-10 mg , then the best resistance resolution, Q is 9.9999 mSZ/99999=0.l #92. A

resistance 10 mg or larger must be read on another scale with poorer resolution. The

resulting temperature resolution can be calculated using

77

5.2

rT

_ r R
01%”)

where I} is the temperature resolution, Q is the resistance resolution, and dR/dT

is the sensitivity of the sensor. Note that the sensitivity (1de is a function of both

temperature and excitation while “R is fixed by the instrument design for a given

range. The relative resolution is defined as the temperature resolution divided by the
absolute temperature and can be calculated using dimensionless quantities as

1;: 6%) _(%) 5.3

T (T/Rx‘1%T)_ 57

Hence, in general, the sensor with the best resolution will depend on the
measurement system and considerations other than the resolution can be very
important in selection of a sensor.

5.3.4 Precision And Accuracy

The probable precision or accuracy of a temperature measurement is never better
than the resolution of and typically depends on several additional factors. Estimating
the precision or accuracy of a measurement involves the following steps

0 Identify the relevant sources of measurement uncertainty.

0 Change the units of all uncertainties to temperature, and

0 Combine all of the uncertainties using the root sum of squares method as
described later.

Examples of source of measurement uncertainties affecting the accuracy, but not

the precision of a measurement include offset voltages and calibration uncertainties.

78

5.3.5 Sources Of Measurement Uncertainty
The following sections discuss specific sources of temperature measurement
uncertainty and how to estimate their magnitudes.
(a) Instrumentation Measurement Uncertainty. The manufacturer normally
specifies the accuracy of the instrument measuring the output of a temperature
sensor. The accuracy of a digital meter is usually specified as a percentage of
reading plus a number of counts of the least significant digit. The percent
accuracy is calculated as a function of the actual reading, not full scale as done for
analog meters. For example, a digital accuracy specification of i(0.05%+l

count) for 4-1/2 digit meter reading 3.000 mg on a 20 m9 range equals a

resistance uncertainty “R of i(0.0015+0.001) mSZ=i2.5 ,u 52. Equation 1 is

again used to translate to temperature uncertainty.

(b) Sensor Self-Heating. Any difference between the temperature of the

 

sensor and environment the sensor is intended to measure produces a temperature
measurement error or uncertainty. Dissipation of power in the temperature sensor
will cause its temperature to rise above that of the surrounding environment.
Power dissipation in the sensor is also necessary to make a temperature
measurement. Minimization of the temperature measurement uncertainty thus
requires balancing the uncertainties due to self-heating and output signal
measurement. Attempting to correct for ‘self heating’ errors by calculation or
extrapolation is not considered good practice. An estimate of the ‘self heating’

errors should be included in the total uncertainty calculation instead.

79

(c) Thermal (Johnson) Noise Thermal energy produces random motions of the

charged particles within a body, giving rise to electrical noise. The minimum root

mean square (rms) noise power available is given by P" = 4kTAf,l , where k is the

Boltzmann constant and Afn is the noise bandwidth. Peak-to-peak noise is

approximately five times greater than the rrns noise. Metallic resistors approach
this fundamental minimum, but other materials produce somewhat greater thermal
noise. The noise bandwidth is not necessarily the same as the signal bandwidth,
but is approximately equal to the smallest of the following [Low Level
Measurements, Keithley Instruments, Inc., Cleveland, Ohio, USA (1993).]

o n/2times the upper 3 db frequency limit of the analog DC measuring
circuitry, given as approximately 1/(4 Re); Cm) where Reflis the effective

resistance across the measuring instrument (including the instrument’s input

impedance in parallel with the sensor resistance and wiring) and Cm is the

total capacitance shunting the input;

0 0.55/t, where tris the instrument’s 10%-90% rise time;

o 1 Hz if an analog panel meter is used for readout; or

0 One half the conversion rate (readings per second) of an integrating digital
voltmeter.

(d) Thermoelectric And Zero Offset Volargg Voltages develop in electrical

conductors with temperature gradients when no current is allowed to flow

(Seeback effect). Thermoelectric voltages appear when dissimilar metals are

80

joined and joints are held at different temperatures. Typical thermoelectric
voltages in cryogenic measurement systems are on the order of microvolts.

A zero offset is the signal value measured with no input to the measuring
instrument. The zero offset can drift with time or temperature and is usually
included in the instrument specifications. Thermoelectric voltages and zero offsets
can be eliminated from voltage measurements on ohmic resistors by reversal of
the excitation current and use of the formula:

V: (V+ - V_ )/2

Where V+ and V_ are the voltages with respectively positive and negative
excitation currents. Alternating current (AC) excitation can also be used with
ohmic sensors to eliminate zero offsets. The sum of the thermoelectric voltages
and zero offset can be calculated as

V0=(V+ + V_)/2
Note that the resolution of Vois practically limited by the resolution of the
measured system. The value of V0 can be expected to vary little in a static system,
but may change during a thermal transient under study. The value of V0 should be
rechecked as often as practical. The offset voltage V0 is best measured by
reversing the current through a resistor. Measurement of V0 with zero excitation
current is also possible, but large resistances can produce excessive time constants
for discharge of any capacitances in the circuit, requiring long waiting times

before V0 can be measured accurately.

81

(e) Environmental Effects Temperature sensors can be affected by changes in
the environment. Examples include magnetic fields, ionizing radiation, or changes
in the pressure, humidity, or chemistry of environment.
(f) Ground Loops. Improper grounding of instruments or grounding at
multiple points can allow current flows which result in small voltage offsets. One
common problem is the grounding of cable shields at both ends. The current flow
through ground loops is not necessarily constant, resulting in a fluctuating
voltage.
(g) Electromagnetic Noise Electromagnetic pickup is a source of additional
noise. Alternating current noise is a serious problem in sensors with nonlinear
current-voltage characteristics. Measurement of the AC noise across the terminals
of the reading instrument can give a quick indication of the magnitude of this
noise source (thermal noise will be included in this measurement). Twisting wire
pairs, using coaxial cables, adding shielding, or shortening the wires can reduce
electromagnetic pickup.
5.3.6_C_alibration Uncertainty
Commercially calibrated sensors should have calibrations traceable to
international standards. The calibration uncertainty typically increases by a factor of 3
to 10 between successive devices used to transfer a calibration. An algorithm for
selecting the number and spacing of calibration points is given by Nara, et al [6].
5.3.7 Combining Measurement Uncerflinties
The expected uncertainty of a measurement is expressed in statistical terms. As

stated in the guide to the expression of uncertainty in measurement: [7].

82

“The exact values of the contributions to the error of the measurement arising
from the dispersion of the observations, the unavoidable imperfect nature of the
corrections, and incomplete knowledge are unknown and unknowable, whereas
the uncertainties associated with these random and systematic effects can be
evaluated ........ the uncertainty of a result of a measurement is not necessarily an
indication of the likelihood that the measurement result is near the value of the
measurement; it is simply an estimate of the likelihood of neamess to the best
value that is consistent with presently available knowledge.”

The uncertainty, u, has the same units as the quantity measured. The

combined uncertainty “c arising from several independent uncertainty sources

discussed above can be estimated by assuming a statistical distribution of
uncertainties, in which case the uncertainties are summed in quadrature according

to

 

uc = Juf +u;2 +...+u,.2 +...+uff 54

Both random and systematic uncertainties are treated in the same way. The
justification for this practice is given in Annex E of Reference [7]. Note that both
sides of equation 4.4 can be divided by the measurement quantity to express the
measurement uncertainty in relative terms. Finding statistical data suitable for
addition by quadrature can be a problem; instrument and sensor specification
sometimes give maximum or typical values for uncertainties. Two approaches
may be taken to dealing with maximum uncertainty specifications. The
conservative approach is to use the specification limit value in the combined

uncertainty calculation. The less conservative approach is to assume a statistical

83

distribution within the specification limits and assume the limit is roughly three
standard deviations, in which case one third of the specification limit is used in
uncertainty calculations. Practical recommendations and procedures for problems
related to the estimation of measurement uncertainties are discussed in greater

detail by Rabinovich [8].

84

5.4 Super-Conductivity

Super-conductivity was first observed by Kamerlingh Onnes in 1911. It is a state
when electrical resistivity of a material suddenly drops to zero at a temperature known as
the critical temperature. This state is found in many materials and it occurs when the
material is sufficiently cooled, often to temperatures in the liquid helium range.

In the simple free electron picture for the behavior of a metal, electrons in the outer
shells of atoms in a metal are easily detached from the core and wander away from the
parent ions that define the solid lattice and are responsible for the excellent electrical and
thermal conductivity of metals.

At temperatures above the critical temperature, these atoms are not organized and
remain disordered. However, the electrons are ordered at temperatures below the
transition temperature. Thus the transformation from the normal state to the super-
conducting state may be considered to be a phase change involving the electronic state
while not affecting the crystal structure of the material [12].

It has been stated that the super-conducting state is an ordered state. Super-
conductivity can be considered as arising from the coupling between pairs of electrons
and vibrating atoms. In the super-conducting state, pairs of electrons develop an attractive
interaction by simultaneously interacting with vibrating atoms. When one electron of the
electron pair is scattered, the other responds so that there is no net effect on the motion of
the pair. If the thermal energy of the atom is sufficient it will disrupt the coupling. Thus
super-conductivity is only observed at very low temperatures where the thermal energy of

the atom will be insufficient to destroy the electron pair coupling.

85

REFERENCES

1. H. Padamsee, “Calculations For Breakdown Induced by Large Defects in
Superconducting niobium cavities”, IEEE transactions on magnetics, vol-19,
No.3, 1322(1983)

2. K.R. Krafft, “An investigation of Thermal Transport in Superconducting Cavities
Made of High Thermal Conductivity niobium”, IEEE transactions on magnetics,
vol. Mag-l9 pg 1326

3. HP. Kramer, “Heat Transfer to Forced Flow Helium 11”, Proceedings of 12‘h
ICEC Southampton UK 1988, page 299.

4. R. Romijn, W. Weingarten and H. Piel, “Calibration of the Scanning
Thermometer Resistor System for a Superconducting Accelerating Cavity”, IEEE
transactions on magnetics, vol. Mag-19, pg 1318

5. “Temperature Measurement and Control” Lake Shore Cryotronics, Inc.

6. . k.Nara, H.kato and M.Okaji, “Derivation of optimized calibration procedures for
practical thermometers”, Cryogenics 35 (1995) 291-295

7. “ISO Guide to the Expression of Uncertainty in Measurement”, first edition,
ISO/T AG 4/WG 3, International Organization for Standardization, Geneva,
Switzerland (1992)

8. S. Rabinovich, “Measurement Errors”, American Inst. of Phs.(1993)

9. Hasan Padamsee, Jens Knobloch, and Tom Hays “RF Superconductivity For
Accelerators”, John Wiley & Sons, INC, 1998.

10. Timmerhaus D. and Thomas M.F1ynn, Cryogenic Process Engineering.

11. Lienhard H.John and Chap. 6 by Roger Eichhom, “A heat transfer textbook”

86

12. Irving Granet, “Modern Material Science”, page 408. Reston Publishing
Company Inc. A PRENTICE-HALL COMPANY.

13. Frank P. Incropera, David P. DeWitt “Introduction to Heat Transfer” John Wiley
& Sons, 1990.

14. Brice Camahan and James O. Wilkes, “Numerical Methods, Optimization
Techniques and Process Simulation For Engineers”; p.279 (1967)

15. J. Beck, R. McMasters, A Haji-Sheikh, "COND3D, Computer Code for
Computing Exact Solutions for Transient Heat Conduction in Three-Dimensional
Parallelepipeds", Under Contact to Sandia National Laboratories, Albuquerque,
NM, 2002.

16. D. Tullock and E. Montalbano, "Kokopelli 96.2 Finite Element Program", Los

Alamos National Laboratory, Los Alamos, NM, 1996.

87