Layered heat flux gauges for aeroentry application

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Authors Oishi, Tomomi

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LAYERED HEAT FLUX GAUGES FOR AEROENTRY APPLICATION

by

Tomomi Oishi

A Dissertation Submitted to the Faculty of the

DEPARTMENT OF AEROSPACE AND MECHANICAL ENGINEERING

In Partial Fulfillment of the Requirements For the Degree of

DOCTOR OF PHILOSOPHY WITH A MAJOR IN AEROSPACE ENGINEERING

In the Graduate College

THE UNIVERSITY OF ARIZONA

2 0 0 3

UMI Number: 3089995

UMI UMI Microform 3089995

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9

THE UNIVERSITY OF ARIZONA ® GRADUATE COLLEGE

As members of the Final Examination Committee, we certify that we have

read the dissertation prepared by Tomomi Oishi

entitled Layered Heat Flux Gauges for Aeroentry Application

and recommend that it be accepted as fulfilling the dissertation

requirement for the Degree of Doctor of Philosophy

K R- ""'-J ̂r/o J

Cholik Chan Date

Matthias Gottmann

Date

Date

Final approval and acceptance of this dissertation is contingent upon the candidate's submission of the final copy of the dissertation to the Graduate College.

I hereby certify that I have read this dissertation prepared under my direction and recommend that it be accepted as fulfilling the dissertation requirement.

Dissertation Director ^ Sridhar Date

3

STATEMENT BY AUTHOR

This dissertation has been submitted in partial fulfillment of requirements for an advanced degree at The University of Arizona and is deposited in the University Library to be made available to borrowers under the rules of the Library.

Brief quotations from this dissertation are allowable without special permission, pro­vided that accurate acknowledgment of source is made. Requests for permission for extended quotation from or reproduction of this manuscript in whole or in parts may be granted by the head of the major department or the Dean of the Graduate College when in his or her judgement the proposed use of the material is in the interests of scholarship. In all other instances, however, permission must be obtained from the author.

SIGNED

4

ACKNOWLEDGMENTS

I thank the following people for their role in helping me finish this dissertation.

Dr. K.R. Sridhar for the opportunity, his guidance and advice, and his patience with

me and the research progress.

Dr. Cholik Chan for his ideas and suggestion in math oriented topics.

Dr. Matthias Gottmann for his advice and ideas in broad view point.

Dr. Srini Raghavan for being my minor advisor.

Dr. Kenneth A. Jackson for being my committee member.

Dien Nguyen for making me ceramic tapes and giving advice about ceramics.

NASA Ames Research Center for JRI grants that supported this work.

Primex Aerospace company for donating the arcjet chamber parts.

Nicole Meckel for helping me assemble and modify the arcjet chamber.

Robert Ricks for his advice in electronics.

Dr. Jochen Marschall for conducting his independent experiment of the flux gauge.

Dave Lyle for his advice related to machining and experiment.

Mark Densmore, Anna Gradillas, and Richard Center for helping me setup testing apparatus.

Gege Tao and Christie lacomini for their thoughts and ideas in academic topics and for going over this dissertation.

Darren Hickey for proofreading this dissertation.

Ian Russel for proofreading my preliminary proposal.

Jeff Blanchard, Koorosh Araghi, Bob Byron, Chris Lewicki, Roger Foerstner, Mar­cus Fromm, Dietmar Welberg, Patric Szabo, Sophie Lelu de Brach, Nick Phillips and others who worked at the Space Technologies Lab for showing me useful skills.

5

TABLE OF CONTENTS

LIST OF FIGURES 9

ABSTRACT 11

1 INTRODUCTION 12

1.1 Motivation 12

1.2 Heat Flux Component 14

1.3 Heat Flux Measurement 15

1.3.1 Temporal Type Heat Flux Gauge 15

1.3.2 Spatial Type Heat Flux Gauge 17

1.4 Past Work by Others 19

1.5 Layered Heat Flux Gauge for Aeroentry Application 20

1.6 Gauge Intrusiveness 21

1.7 Objectives 21

1.8 Overview 22

2 ANALYSIS OF LAYERED HEAT FLUX GAUGE 23

2.1 Analysis Outline 23

2.2 Gauge Model 23

2.3 One-Dimensional Transient Composite Slab Solution 25

2.4 Gauge Time Response 26

2.4.1 Simple Difference Method 26

2.4.2 Delay Time 26

6

TABLE OF CONTENTS - Continued

2.4.3 Time Shift Method 28

2.4.4 Quadratic Method 30

2.4.5 Inverse Method 32

2.5 Effect of Noisy Temperature Data 34

2.6 Intrusiveness of the Gauge 37

2.7 One-Dimensional Intrusiveness Analysis 39

2.8 Radial Heat Leak 43

2.9 Two-Dimensional Steady State Analysis 44

2.9.1 Two-Dimensional Model 44

2.9.2 Two-Dimensional Steady State Solution 45

2.9.2.1 Analytical Solution of Each Region 45

2.9.2.2 Interface Matching Condition 47

2.9.2.3 Determining Coefficients 49

2.9.3 Contour Plot 49

2.9.4 Non-Monotonic Temperature Behavior 51

2.9.5 Two-Dimensional Effect 54

2.9.6 Buffer Zone 55

2.10 Simple Mounted Gauge 57

2.11 Gauge Output 61

3 DESIGN, FABRICATION AND MEASUREMENT OF GAUGES 62

7

TABLE OF CONTENTS - Continued

3.1 Gauge Design 62

3.1.1 Gauge Material 62

3.1.2 Gauge Dimensions 63

3.1.3 Gauge Schematic 64

3.2 Temperature Sensors 64

3.3 Fabrication Process 66

3.4 Electronics 67

4 TESTING APPARATUS AND EXPERIMENT 69

4.1 Qualitative Testing Apparatus 69

4.2 Arcjet Chamber 71

4.3 Calibration 73

4.3.1 Calibration Process in General 73

4.3.2 Conduction Calibration Furnace 74

4.4 Experimental Result 77

5 CONCLUSIONS 82

5.1 Analytical and Numerical Simulation 82

5.2 Gauge Design and Fabrication 83

5.3 Experiment 83

5.4 Future Work 84

5.4.1 Calibration 84

8

TABLE OF CONTENTS - Continued

5.4.2 Gauge Mounting and Connection 84

5.4.3 Temperature Sensor Option 85

5.4.4 Transient Intrusive Analysis 85

5.5 Summary 86

SYMBOLS 87

REFERENCES 90

9

LIST OF FIGURES

Figure 1-1 : illustration of heat flux component at the surface 14

Figure 1-2 : simple models of temporal temperature heat flux gauges 16

Figure 1-3 : simple models of spatial temperature heat flux gauges 18

Figure 2-1 : schematic of layered heat flux gauge 24

Figure 2-2 : time response of alumina gauge for different backing 27

Figure 2-3 : gauge response for different numerical methods 29

Figure 2-4 : quadratic temperature distribution within the gauge 31

Figure 2-5 : gauge response simulation to noisy data 36

Figure 2-6 : illustration of gauge intrusiveness and heat "leak" 38

Figure 2-7 : relative error in heat flux 40

Figure 2-8 : relative error bound for gauges more conductive than backing 42

Figure 2-9 : axi-symmetric model 45

Figure 2-10 : temperature contour plot 50

Figure 2-11; surface temperature distribution 52

Figure 2-12 : illustration of one-dimensional solutions 53

Figure 2-13 : two-dimensional effect 55

Figure 2-14 ; two-dimensional effect as function of temperature sensor radius 56

Figure 2-15 : axi-symmetric model of simple mounted gauge 57

Figure 2-16 : relative error in heat flux of simple mounted gauge 59

Figure 2-17 : two-dimensional effect on simple mounted gauge 60

Figure 3-1 : schematic and photo of heat flux gauge 65

10

LIST OF FIGURES - Continued

Figure 3-2 : schematic of gauge reader electronics 68

Figure 4-1 ; schematics of qualitative gauge testing apparatus 70

Figure 4-2 : schematics of arcjet testing apparatus 71

Figure 4-3 : photos of arcjet testing apparatus and arcjet 72

Figure 4-4 : conduction calibration furnace illustration 75

Figure 4-5 : gauge response on fire brick in high temperature apparatus 78

Figure 4-6 : gauge response on cold plate in high temperature apparatus 79

Figure 4-7 : gauge response in arcjet apparatus at arc current of 10 Amp 80

11

ABSTRACT

A layered heat flux gauge, which can withstand a high temperature environment for

applications such as for use on thermal protection shields on aeroentry vehicles, is ana­

lyzed, designed, fabricated, and tested. The heat flux gauge consists of two resistance

temperature detectors on the top and bottom faces of a thin ceramic substrate. The heat

flux is calculated from temperature measurements of the two temperature detectors. An

analytical model is used to simulate the gauge response. Several numerical methods to

calculate the heat flux are investigated to improve the time response of the gauge. The

error due to gauge intrusiveness and the validity of one-dimensional heat transfer within

the gauge is studied by solving a steady state two-dimensional composite problem using

a semi-analytical approach. Gauge fabrication techniques and measurement devices are

discussed. Testing apparatus, including a "close-to-entry" condition apparatus using an

arcjet at low pressure and a conduction calibration furnace, are explained. Experimental

data showing qualitative gauge response is presented.

12

1 INTRODUCTION

1.1 Motivation

There are many space missions which require vehicles to return to the Earth's surface.

These vehicles must be reliable for reentry, and the reliability of the vehicles for any

human missions must be extremely high to ensure the safety of the vehicle's occupants.

Other space missions demand vehicles to enter a planet's atmosphere for landing or for

aeromaneuvering which can be aerobraking or aerocapturing [1], Aerobraking uses atmo­

spheric drag to slow down the vehicle to change the orbital parameters without the use of

fuel. Aerocapturing is a term for using drag to slow down the vehicle from a hyperbolic

orbit to an elliptical orbit so that the vehicle is "captured" by a planet.

Since the finding of possible fossilized life forms in a meteorite originating from Mars,

the scientific and public interest in Mars has grown [2]. The success of the Mars Path­

finder mission increased public interest even further [3]. There are many Mars missions

planned to pave the way to human Mars missions, which involve Mars landing. Earth

landing and possible aeromaneuvering. These are the missions likely to benefit most from

improvements in vehicle designs for reliability and maneuvering and eventual cost reduc­

tion because of their large mission scale.

A space vehicle encounters a harsh environment during aeroentry or aeromaneuvering

[4-8]. Most of its kinetic and potential energy is converted into thermal energy. The space

vehicle is exposed to a large amount of thermal energy and must endure very high heat

flux. Such a space vehicle has a thermal protection shield. The design of the shield has

13

evolved over the years and the design differs by the nature of the mission [1], The shield

can be heat sinking, insulating, or ablative. The heat sinking technique uses a large ther­

mal mass of the shield to absorb the heat load. The insulating shield uses a thermally resis­

tive material to reduce the conducted heat. The ablating shield evaporates gradually where

most of the heat load is used up for the evaporation. The design of a thermal protection

shield is generally considered half science and half art. Numerical simulation and scaled

model experimental data are used to design the protection shield. However, uncertainty in

the available data is large, causing thermal protection shields to be designed with very

large safety margins. In order to improve the accuracy of the numerical simulation and

experimental procedure, a comparison to actual flight data is important. Among many

parameters of interest, one of the most important pieces of information is the heat flux into

a protection shield. Improved numerical and experimental data will minimize the guess

work and help reduce the mass of the vehicle. In addition, they will aid in the design of a

highly maneuverable vehicle without compromising safety.

Heat flux into a thermal protection shield can be measured in different ways. Regard­

less of the method, gauges must deal with the harsh environment of aeroentry or aeroma-

neuvering. Some gauges are designed to survive the high temperature, other gauges are

protected away from the harsh environment, and some utilize the ablating nature of the

environment. The accuracy of the gauge depends on these choices and each heat flux mea­

surement technique has its own pitfalls, which can degrade the gauge performance.

14

1.2 Heat Flux Component

The heat flux to which a space vehicle is exposed during aeroentry is illustrated in Fig­

ure 1-1, showing different components of heat flux at the surface of the thermal protection

shield. The arrows, for illustration purposes, indicate if the flux component is coming into

irradiation

surface

chemical emission reflection convection reaction

-I- + +

conduction

thermal protection shield

Figure 1-1 : illustration of heat flux component at the surface

or going out from the surface. The size of the arrow indicates the magnitude of the flux

component. In some cases, some heat flux components play no role, or the direction of a

flux component can be opposite to what is shown. Different surface coatings or different

materials for the thermal protection shield can increase or decrease a heat flux component.

The energy balance at the surface yields that the heat flux components add up to zero

because the surface has no thickness and no thermal mass to store energy. The heat flux

15

gauge discussed in tliis dissertation measures the heat flux conducted into the thermal pro­

tection shield. The conducted component of the heat flux is equal to the magnitude of the

sum of the flux components above the surface, .

1.3 Heat Flux Measurement

Heat flux measurements are categorized into two main groups, although there are

other techniques to measure heat flux [9], One technique relies on temporal temperature

measurements and the other uses spatial temperature measurements. Temporal type heat

flux gauges use temperature recorded as a function of time and process the temperature

history to compute the heat flux. The spatial type heat flux gauges use instantaneous tem­

perature measurement at different locations and calculate the flux at the instant.

1.3.1 Temporal Type Heat Flux Gauge

Three examples of heat flux gauges using the temporal temperature method are illus­

trated in cross sectional view in Figure 1-2. The slug calorimeter assumes constant tem­

perature within the gauge and perfect insulation from the thermal protection shield or the

backing material. The surface heat flux is given from the time derivative of the slug tem­

perature as

Qur = (1-1)

where p is the density of the gauge material, L is the thickness of the slug gauge, is

the specific heat of the gauge material, and T is the rate of change in temperature with

16

Heat Flux

slug calorimeter thin film gauge Thermocouple gauge

thermal mass surface temperature sensor

L

temperature sensor

thermal protection shield or backing material

Figure 1-2 : simple models of temporal temperature heat flux gauges

respect to time t. The thin film gauge assumes negligible thermal mass of the gauge and

semi-infinite length below the gauge. Then the heat flux is given by

' ^ Wt 2Jp J (1-2)

where is the product of the conductivity, the density and the specific heat of the

backing material and 5 is the dummy variable of the integration [10]. Eq. (1-2) can be per­

formed numerically or solved by using an electronic circuit analogy. The thermocouple

gauge is used to calculate the heat flux from the measured temperature by solving the

inverse heat conduction problem of the backing material.

17

In general, the temporal temperature method requires solving the conduction heat

problem or the inverse problem of the model. Therefore, the accuracy of modeling is very

important. Modeling, in general, requires information from the surroundings as boundary

conditions. In the case of the slug meter, the boundary condition is simplified to a per­

fectly insulated boundary. In the case of the thin film gauge, the duration of the experi­

ment is limited to a short period of time such that the backing material behaves as an

infinite slab. For the thermocouple heat flux gauge, the boundary condition of the bottom

of the backing material must be known or assumed to solve the inverse problem.

1.3.2 Spatial Type Heat Flux Gauge

There are many designs for spatial type hear flux gauges. Two examples are illustrated

in Figure 1-3 in cross sectional view. A radial flux gauge is a circular disk which has ring

shaped thermal diffusion barriers of different thickness on top of it [11]. The surface heat

flux creates a radial temperature distribution because of the different thickness of the ther­

mal diffusion barrier. With some assumptions, the flux is computed by using

where k is the thermal conductivity of the thermal diffusion barrier, is the temperature

of the inner sensor, is the temperature of the outer sensor and is the thickness of

the upper diffusion ring. A layered heat flux gauge consists of two temperature sensors

sandwiching a thermal diffusion barrier. Assuming one-dimensional heat flow, a linear

18

Heat Flux

radial flux

symmetn

T T out in

temperature sensor

thermal diffusion barrier

layered gauge

T . top

bo t tom

thermal protection shield or backing material

Figure 1-3 : simple models of spatial temperature heat flux gauges

temperature distribution within the gauge, negligible temperature sensor thermal mass,

and constant material properties, the heat flux, q-^, is given by

T - T top bo t tom

L, (1-4)

where /:, is the thermal conductivity of the thermal diffusion barrier, T is temperature,

and Lj is the thickness of the gauge. The subscript top denotes the top of the gauge and

bottom denotes the bottom of the gauge. The assumptions are justified by use of a thin and

wide thermal diffusion barrier and significantly thinner temperature sensors than the diffu­

sion barrier. However, the minimum thickness of the diffusion barrier is limited by the fact

19

that measuring the temperature difference becomes more difficult for a thinner gauge

because its magnitude for a fixed value of heat flux decreases as the thickness decreases.

1.4 Past Work by Others

In the past, thermocouples imbedded in the thermal protection shield have provided

data for the calculation of heat flux [12-14]. An inverse problem of the thermal protection

shield is then solved from the thermocouple temperature data. However, the inverse prob­

lem is susceptible to noise and the thermal model has a significant influence on the solu­

tion.

There is a technique to deduce heat flux, which is unique to ablating shields. An ablat­

ing shield evaporates away when exposed to a heat load. The technique uses a recession

gauge which measures the rate of ablation. The rate of ablation is related to the heat flux

via the evaporation mechanism. Strictly speaking, the heat flux obtained from the rate of

ablation differs from the conductive component shown in Figure 1-1 because the surface is

receding. Several thermocouple junctions have been used as discrete recession gauges.

Other recession gauges which can produce the continuous ablation rate have been used

recently [15], Recession gauges only work with an ablating shield. The accuracy of the

heat flux calculated from the ablation rate depends on the mechanism of ablation which

has large uncertainty.

20

1.5 Layered Heat Flux Gauge for Aeroentry Application

For ablating shields, the recession gauge is the logical choice for the application. For

insulating type thermal protection shields, a heat flux gauge is needed. The question is

which gauge type is suitable for the application. The necessity to know the properties of

the surrounding material is disadvantageous for gauges using the temporal temperature

method because it implies that the gauge is not universal. The gauge is only accurate in the

particular setting for which it is designed. Every time a gauge is moved from one setting to

another, its applicability needs to be questioned. When a boundary condition is changed,

the process used to calculate the heat flux needs to be changed. The process to obtain the

heat flux can be an algebraic equation, a numerical program, or an electronic circuit. The

change is necessary because the actual situation differs from the model used to design the

heat flux gauge. The change to the process may involve using different property values of

the backing material, modifying the computational routine, or using different equations.

Another disadvantage of the temporal temperature method is that the process used to cal­

culate the heat flux from the temperature data is, in most cases, more complicated than the

spatial temperature method.

For aeroentry applications, heat flux gauges utilizing the spatial temperature method

are suitable because of the accuracy, simplicity, and flexibility of the gauge. Among the

spatial type heat flux gauges, the layered flux gauge is chosen for its simple fabrication

process. For the layered gauge, the substrate functions as the diffusion barrier. It only

needs to have temperature sensors on the top and bottom.

21

1.6 Gauge Intrusiveness

With a good design and calibration, a heat flux gauge can measure heat flux through it

fairly accurately. However, regardless of the type, a heat flux gauge is generally

intrusive [16&17]. The presence of the gauge changes the heat conduction path in the

backing material unless the gauge has the same physical properties as the backing mate­

rial. For instance, the surface absorptivity and emissivity of the gauge may differ from

those of the backing material, which change the radiative heat flux components. The use

of a coating may mitigate the problem of mismatched radiative properties. Additionally,

the conductivities of the gauge and the backing are important. These issues need to be

taken into consideration when designing a heat flux gauge.

1.7 Objectives

The objective of this research is to analyze, design, manufacture and test layered heat

flux gauges which can survive in the high temperature environment encountered during

aeroentry or aeromaneuvering with better accuracy than gauges used in the past. The

gauge design is analyzed using numerical and analytical solutions. Based on these analy­

ses, actual gauges are manufactured. Issues of using the actual gauges are studied includ­

ing calibration.

22

1.8 Overview

The layered gauge is discussed in more detail in Chapter 2. Design and fabrication of

the gauge and measurement electronics are discussed in Chapter 3. The experimental

apparatus and data are presented in Chapter 4. Conclusions are given in Chapter 5.

23

2 ANALYSIS OF LAYERED HEAT FLUX GAUGE

2.1 Analysis Outline

A layered heat flux gauge, which is categorized as a spatial type heat flux gauge, is

chosen for an aeroentry application for its simplicity. The performance of the gauge is

studied in detail using numerical simulation for different criteria. The gauge is exposed to

a constant heat flux and its time response is studied using an analytical solution. The study

shows that the time response of the gauge can be quite slow in some cases. Means to

improve the time response of the gauge is sought by modifying Eq. (1-4). With the modifi­

cation, the categorization of the gauge becomes a hybrid of the spatial and temporal

method. The effect of noise in temperature data on different flux calculation methods is

studied by adding artificial random noise to simulated temperature data. Intrusiveness of

the gauge is quantified by using one-dimensional solutions. The validity of a one-dimen-

sional heat flux assumption is investigated by solving a two-dimensional heat model and

comparing it to the one-dimensional solution.

2.2 Gauge Model

A model of a layered heat flux gauge with a backing material is illustrated in Figure 2-

1, where x is a coordinate normal to the surface, L is the thickness of a layer, k is the

thermal conductivity of the material, a is the thermal diffusivity of the material. The sub­

script I denotes the property of the gauge, and the subscript 2 denotes the property of the

backing. The heat flux through the surface is denoted as ^. The temperature at the bot-

24

heat flux

top

/ ' 1

L

r A, u, y

i I

Li X

1 f

k-2 ^2

bottom

top temperature sensor

thermal diffusion barrier

bottom temperature sensor

backing material

'^hack - 0

Figure 2-1 : schematic of layered heat flux gauge

tom of the backing, , is assumed to be zero. The gauge consists of two temperature

sensors attached to either side of a thermal diffusion barrier. Assuming that the heat flow

is one-dimensional along the x-axis and the thermal mass of the temperature sensor is

negligible, the heat flux is given by

_ , d T ^bottom •

top ^ d x

(2-1)

bottom

where is the heat flux through the top surface and '^he heat flux through the

bottom of the gauge.

25

2.3 One-Dimensional Transient Composite Slab Solution

Assuming material properties are constant, the gauge response to a constant heat flux

can be simulated using an analytical one-dimensional transient composite slab solution.

The initial temperature throughout the gauge and the backing is at and a constant

heat flux of is applied at the top at time t = 0 . Temperatures of the top and bottom

sensors are calculated analytically.

The composite slab solution is the superposition of a homogeneous and a particular

solution. The particular solution, T^(x), is given by

Because a constant heat flux is assumed as the top surface boundary condition, is a

constant. Via eigenfunction expansion, the homogeneous solution, Tf^(x, t), is given by

for x < L

for x > L (2-2)

for x < L

for x > L

(2-3)

Z-,j(x) is the eigenfunction and is given by

26

cos(p,jX/^)

-cos((3„L]/7^) . ^=^si

sin(P„^2/V^2)

P„(x- ( L i+ L 2 ) )

for i = 1

\

for i = 2 (2-4)

J

(3,J is the eigenvalue and obtained by solving the transcendental equation of a form

sm p„'-i

sm V y v v ^ 2 y k ] J a ^

cos cos V y

= 0

V ^ ^ 2 7

(2-5)

2.4 Gauge Time Response

2,4.1 Simple Difference Method

Heat flux through the surface of the gauge can be calculated by using a "simple differ­

ence method". Assuming a linear temperature distribution within the gauge and treating

the material properties as constant, both heat fluxes become the same and are given by

^ = lb •>ottom

T - T _ ^ top bottom (2-6)

2.4.2 Delay Time

To investigate the validity of the liner temperature distribution within the gauge, the

system response to a constant heat flux is simulated using the analytical solution shown in

Eq. (2-2) through Eq. (2-5). From the simulated top and bottom gauge temperature, the

heat flux is calculated by using the simple difference method, Eq. (2-6). The diffusion bar­

rier is assumed to be made of alumina. Alumina is chosen for the gauge material from the

fabrication point of view. The actual gauge design and fabrication are discussed in

27

Chapter 3. Typical gauge responses are shown in Figure 2-2 in terms of percents of ideal

response for three different backing materials. These materials are used to illustrate the

110

Ideal 100

Alumina 90

Plaster

70

o 60 OJ >

(D X 3

0 0.5 1 1.5 2 2.5 3 3.5 4

time (sec)

Figure 2-2 : time response of alumina gauge for different backing

cases of different conductivity and diffusivity ratios between the gauge and the backing

material. The thermal properties used for the calculation are values at room temperature.

Dimensional values are used to give an idea of physical quantity. A gauge thickness of

500 |im and a backing thickness of 2 cm are used for the calculation. The data acquisition

rate is assumed to be 500 Hz. The ideal response of the gauge is a step function. Figure 2-

2 shows that there is a transition time to reach the actual heat flux value. The delay time,

Xq 95, is defined here as the time the gauge takes to reach 99 % of the applied heat flux.

28

For example, the delay time of the case with copper backing is 1.1 sec. In the case with a

plaster backing, it takes on the order of minutes for the measured flux to reach the input

heat flux value. The duration of aeroentry or aeromaneuvering differs from one mission to

another mission, but it is in the order of minutes. In order to obtain reasonable data, the

heat flux gauge should have time response of a second. Some researchers may be inter­

ested in evan smaller time-scale phenomena. Gauges with a delay time of tens of seconds

or minutes are too slow to measure heat flux with adequate time resolution.

2.4.3 Time Shift Method

The previous section shows that the time response of the gauge can be slow when

using the simple difference method. Some means to improve the time response of the

gauge without changing the physical design are sought. One way to reduce the delay time

for a given gauge design is to view the heat flow as a heat wave traveling through the dif­

fusion barrier. It takes some time for the heat wave to travel through the barrier and to

reach the bottom temperature sensor. Therefore, the bottom temperature sensor responds

to the input heat flux with some delay while the top temperature sensor responds instanta­

neously. It is proposed to exploit this delay and use time shifted temperature data for the

calculation of the heat flux. The bottom temperature reading is time shifted backward by a

constant value and the temperature difference is then computed. Eq. (2-6) becomes

- '^bottom

f ( i

' t - C — a, V V ' yy

',n(0 = k, (2-7)

29

2 where c is a constant and Lj /QL^ is the time constant of the gauge. Numerical simulation

shows that a value of 0.5 is suitable for c in many different material combinations and

thickness ratios. Figure 2-3 shows the alumina gauge response in alumina backing for dif­

ferent methods of calculating heat flux. The parameters used to produce the plot are the

110

100 'aaratic

90 time shift simple difference

80

-a

60

40

30

20

0 20 40 60 80 100 120 140 160 180 200

time (msec)

Figure 2-3 : gauge response for different numerical methods

same as those used to produce Figure 2-2. The values from Eq. (2-6), labeled as simple

difference, makes up the slowest rising curve. The values of Eq. (2-7), which is the time

shift method, are plotted as the second slowest rising curve. The improvement in time

response is very clear. In terms of delay time, Tq ^ci for Eq. (2-6) is 10.2 sec and for

Eq. (2-7) is 34 msec. The delay time is reduced by orders of magnitude.

30

2.4.4 Quadratic Method

The time shift method provides a significant reduction in Xq gg over the simple differ­

ence method. However, the constant c is determined in an ad-hoc way. Implementing Eq.

(2-7) is difficult to accomplish in real situations. Measuring the two temperatures and sub­

tracting them later is less accurate than directly measuring the temperature difference.

Another method is sought to improve the delay time problem of the simple difference

method. Since the linear approximation of the temperature distribution within the gauge is

unsatisfactory, the distribution is assumed to be quadratic and given by

where is the average of the top and bottom temperature and Cj- is the deviation of

the quadratic curve from the linear curve at the mid point. The quadratic temperature dis­

tribution within the gauge is illustrated in Figure 2-4. Three plots on the left show the con­

tribution of each term in Eq. (2-8). The plot to the right shows the superposition of the

three terms. The quadratic temperature assumption gives heat flux as

bottom mean (2-8)

bottom

bottom (2-9)

bottom ^bottom

bottom

31

mean

T — T Top bottom

T — T bottom Top

L,/2

L,/2

X

T — T Top bottom

Figure 2-4 : quadratic temperature distribution within the gauge

By integrating an energy equation within the gauge, Cj is given by

(2-10)

The dot denotes the time derivative. Eq. (2-10) is an ordinary differential equation and can

be implemented by the use of an electronics analogy or evaluated numerically [10],

Numerically, Eq. (2-10) can be implemented as

(Cr),-f3At

I + 12 ^ + 0^ ^

( L j / t t ] )

{ T m e a n ) + { \ - 2 ^ ) { C j ) . - ( 1 - e ) ( C r ) . _ j ( 2 - 1 1 )

32

where 0 is the discretization parameter. The subscript i denotes discrete values at time t - .

Evaluation of Tmean is dependent on the availability of {T,nean)i+ l • For a real time com­

putation, {T,nean)i + 1 is not available and Tmean is calculated from older values.

In Figure 2-3, the heat flux values computed from the quadratic method, Eq. (2-9) and

Eq. (2-11), are plotted as the fastest rising curve with iggg of 16 msec. Kinks at the begin­

ning of the plot are due to Eq. (2-11). The accuracy of Eq. (2-11) depends on the time step

size, Ar and how quickly Cj changes. The kinks can be made smaller by increasing the

data acquisition rate. Compared to the simple difference method, the improvement is sig­

nificant. Compared to the time shifting method, the value of Tq is reduced by a factor of

2. Although the quadratic method is more involved than the simple difference method, it is

much simpler than solving the heat conduction equation within the gauge, which is dis­

cussed in Chapter 2.4.5.

2.4.5 Inverse Method

Theoretically, the heat flux is calculated from the temperature measurements most

accurately by using the inverse method. The heat flux at the surface creates the tempera­

ture distribution in the system consisting of the gauge and the backing. The inverse

method reconstructs the heat flux at the surface from temperature measurements by using

the least square method and sensitivity coefficients. Sensitivity coefficients describe the

relationship between a cause and an effect. The surface heat flux is the cause and the tem­

perature evolution in the system is the effect. The sensitivity coefficients are obtained by

solving the direct problem, which is the heat conduction model with specified boundary

conditions and an initial condition. Once the sensitivity coefficients are obtained, the heat

flux can be determined by using the least square method.

For the layered heat flux gauge, the inverse method can be used to calculate the heat

flux. However, strictly speaking, the process is not the inverse method because of the well

positioned location of the temperature sensors. When the gauge alone is considered, the

temperatures at the top and bottom boundaries are measured and are used as the boundary

conditions for a heat transfer model of the gauge. When the system as the gauge and the

backing is considered, the top temperature measurement provides one boundary condition

and the other condition is given from the assumption of the fixed temperature at the bot­

tom of the backing. Considering either the gauge alone or the system ends up with a heat

transfer model with two boundary conditions and an initial condition. Because of the

known boundary conditions, the process does not require using the least square method. It

is the same as solving the direct problem. Although the process itself is not the inverse

method, the idea of calculating the heat flux from the temperature measurement by solving

the entire heat transfer model is termed "the inverse method" in this writing.

The heat conduction model can be solved analytically by using a series solution or

numerically by using the finite difference method. However, the computations for these

operations involve much more than computing Eq. (2-6) or Eq. (2-9) and Eq. (2-11), espe­

cially in the case of the finite difference method. The entire temperature distribution in

each time step must be computed, even though the temperature slope at the top of the

gauge in each time step gives the surface heat flux. Solving the heat conduction problem

requires the initial temperature distribution within the gauge. This is inconvenient in some

cases when the heat flux at the end of a lengthy experiment is the only point of interest.

The computation still needs to be started at the time the experiment is started because this

is the only time an accurate temperature distribution is known or reasonably assumed. A

guessed temperature distribution can be used to start the computation at the middle of the

experiment but it diminishes the accuracy and then the inverse method loses its advantage

over other methods.

In addition to the complicated process of calculating the heat flux, the inverse method

has another disadvantage. Although the inverse method is most accurate in an ideal situa­

tion, it is very susceptible to noise in general. Depending on the location of the tempera­

ture sensor, the noise in temperature measurements can be amplified many times through

the inverse method.

2.5 Effect of Noisy Temperature Data

Theoretically, the inverse method is the most accurate way to calculate the surface heat

flux from measured temperature. However, it is computationally expensive and suscepti­

ble to noise. The quadratic method is shown to be better than the simple or the time shift­

ing method in terms of time response with a penalty of having some more complexity in

the computational process. Although it is much simpler than the inverse method, the qua­

dratic method also might be susceptible to noise. The quadratic method involves taking

time derivatives of temperature data. The evaluation of time derivatives of real data is

sometimes difficult because of measurement noise. The response of the gauge is simulated

35

to compare the quadratic and the inverse method using temperature data with artificially

introduced noise.

A model of the alumina gauge on an alumina backing exposed to a constant heat flux

is solved analytically using Eq. (2-2) through Eq. (2-5) to obtain the top and bottom tem­

peratures of the gauge. The random noise is then added to the temperature data and Eq. (2-

9) and Eq. (2-11) are used to determine the heat flux for the quadratic method. The heat

flux from the inverse method is computed by solving the entire heat transfer model in the

gauge with the top and bottom temperature as the boundary conditions. The finite differ­

ence method is used to solve the model of the gauge for the inverse method. The result is

plotted in Figure 2-5. The system parameters are the same as those used for Figure 2-2.

The noise introduced is normally distributed values with a standard deviation of 0.25 K,

which is equivalent to 2.3 % of the steady state temperature difference of the gauge for the

specified heat flux. The nodal spacing used to approximate the spatial derivative at the

surface for the inverse method is 1 /50 of the gauge thickness.

The solid line is the heat flux calculated without the noise and the points represent the

heat flux estimated from noisy data. Figure 2-5 (a) shows the heat flux estimate of the qua­

dratic method using Eq. (2-9) and Eq. (2-11). Figure 2-5 (b) shows the heat flux estimate

from the inverse method. For a small time period, the solid line of (b) overshoots for the

inverse method. The width and height of the overshoot depend on whether the analytical

or the finite different method is used to solve the model, the nodal spacing to approximate

the spatial derivative, and the data acquisition rate. Comparison of the heat flux from the

noisy temperature in (a) and (b) shows that the scattering of the data is smaller for the qua-

36

120

100

80

X =5

S <D J=

T3 (O

a V

60

f

\ . a * * • • ''

^r-» % -•, • •• '• • .i* ••^••^4 • • - • • • • •

noiseless

with noise quadratic method (a)

120

100

80

60

0 20 40 60 80 100 120 140 160 180 200

time (msec)

El I ' 9 ' # ' ' ' ' ' • • t

* ••• - ••• %•> • •. • * • -• * * - • » •**->

noiseless

with noise inverse method

(b)

20 40 60 80 100 120 140 160 180 200

time (msec)

Figure 2-5 : gauge response simulation to noisy data

dratic method. In terms of standard deviation, the quadratic method has 4.1 % error and

the inverse method has 8.1 % error. This appears to be due to the fact that the inverse

method is relying mostly on the top surface temperature measurement while the quadratic

method uses the top and bottom temperature data equally. The numerical simulation of the

noisy temperature data shows that the quadratic method is not only simple but it is less

affected by noisy data compared to the inverse method without much penalty in the time

response.

2.6 Intrusiveness of the Gauge

The delay time is one of the parameters to consider for the performance of the gauge.

Another issue to consider is the intrusiveness of the gauge. The presence of the gauge

alters the heat transfer of the system. The gauge surface needs to be kept as close to the

original surface as possible. The gauge should not disturb the fluid flow which governs the

convective heat transfer. The radiative properties of the gauge should be close to that of

the backing material so that it reacts in the same way for irradiation and emits the same.

Cataliticity of the gauge should be the same as the backing material so that reaction rates

are the same if any chemical reactions are happening at the surface. These surface require­

ments are assumed to be satisfied with careful mounting of the gauge and surface coatings.

Besides the surface properties, the thermal properties of the diffusion barrier need to

be considered. Unless the diffusion barrier has the same properties as the backing material,

the presence of the gauge changes the heat conduction path of the system being measured.

This situation is illustrated in Figure 2-6. The steady state temperature is sketched as the

function of .x at the gauge center axis for a system with the gauge on the left and the tem­

perature of the system without the gauge on the right. The bottom face of the backing is

assumed to be kept at a constant temperature. The surface heat flux is assumed to be sum­

marized with

1 i „ = (2-12)

where h is the heat transfer coefficient, is the ambient gas tempe r ature, and T. r is s u r j

the surface temperature. For the system with the gauge, is equal to . The gauge

38

Q i . = h { T ^ - T

'^back - 0

Figure 2-6 : illustration of gauge intrusiveness and heat "leak'

causes a second kink in the temperature distribution, and changes the surface temperature

and slope in the temperature plot. Because of this, the surface heat flux of the system with­

out the gauge is different from the flux value of the system with the gauge even if every­

thing else is the same.

The introduction of the gauge creates two-dimensional heat transfer. The gauge tem­

perature is colder or hotter than the temperature of the system without the gauge at the

same depth. This implies that there is heat transfer in the radial or lateral direction in addi­

tion to the axial or the surface normal direction. This radial heat transfer is termed "leak"

in this dissertation and relates to the deviation of the actual heat flow from the one-dimen­

sional assumption.

39

2.7 One-Dimensional Intrusiveness Analysis

For the moment, the two-dimensional nature of the temperature distribution due to the

gauge presence is neglected for the sake of simplicity. The intrusiveness of the gauge can

be quantified by comparing the one-dimensional steady state heat flux of the system with

and without the gauge. Assuming no contact resistance between the gauge and the back­

ing, the one-dimensional steady state heat flux is given by

= kl'; H L ,

where k- is the material thermal conductivity which can be either k-^ or Substituting

k2 for k- in Eq. (2-13) gives the heat flux of the system without the gauge while substitut­

ing k^ for kj gives the heat flux of the system with the gauge. Now the relative error in the

heat flux value is obtained by subtracting Eq. (2-13) with k- = /cj from Eq. (2-13) with

kj = k^ then dividing by Eq. (2-13) with = /cj • Assuming the heat transfer coefficient,

/z, is a constant, the relative heat flux error, is given by

1 _ ^ ^ ^ w i t h ~ ̂ w i t h o u t ^ ^ 1 -^ \ D - - K / L \ k L ^ ^ ^ ^ w i t h o u t ' ^ 2 2

/ i ( L | - I - L j ) V J ^ \ J k ^ L |

where q is the one-dimensional heat flux of the system and the subscripts w i t h denotes

the system with the gauge and without denotes the system without the gauge. Eq. (2-14)

is plotted in Figure 2-7 for h{L^ + L2)/k2 values of oo, 1, and 0.2, which is related to the

40

Biot number, h{L^/k^ + £2/^:2) [18]. A space vehicle trajectory and a shield design nar­

row the value of h(L^ + L2)/^2 • trajectory determines what is expected for h and the

shield design fixes the value of (Lj +L2)/k2 - A h{L^ +L2)/k2 value of 0° is the limit-

/KL| +L2) h { L ^ + L 2 ) h { L ^ + ̂ 2 )

— = 0.001 -0.01

L . / L

Figure 2-7 : relative error in heat flux

ing case where the surface temperature is equal to the ambient temperature. A ^2/^1

value of 1, which is the case of the gauge made of the same material as the backing, gives

a zero percent error shown as the horizontal line in the plots. Values less than 1 for ^2/^1

describe the curves with positive error. This is the case when the gauge is more conductive

than the backing. The total resistance between the ambient and the bottom of the backing

temperature becomes smaller because of the gauge. Thus, the flux through the gauge is

larger than that of the system without the gauge. Similarly, k2/k^ values greater than 1

give the curves with negative error. In the case of values less than 1, the magnitude

of error decreases drastically with increasing length ratio, Lj/Lj . In the case of

values greater than 1, the magnitude of the error is large though bounded within 100 %.

Additionally, the dependencies on the length ratio are small. Since increasing the length

ratio does not reduce the error significandy for greater than 1, this is unfavorable

situation. For the gauges to be non-intrusive, gauges with a k2/k-^ value of 1 is ideal.

When using a diffusion barrier with a k2/k^ of 1 is difficult or impractical, it is important

to choose a diffusion barrier with a material property of k2/k^ less than 1 and make the

gauge as thin as possible to make the gauge non-intrusive.

In Figure 2-7, the difference between the curves with /t j /Zc i values of 0.001 and 0.01

is very small and indistinguishable on that scale. This implies that a ^2^^! value of 0

bounds the error for the gauges with values of less than 1. Eq. (2-14) is plotted as a

function of /z(L, + £2)7^2 Lj/Lj in Figure 2-8 for a k2/k^ value of 0 showing the

relative error bound. The intrusiveness analysis relies on many assumptions and may be

too simplistic but Figure 2-8 can be used as an estimate of gauge intrusiveness. For exam­

ple, assume a mission trajectory and shield design fixes /i(L] +L2)/k2 value of 1, then

the intrusiveness caused by a gauge with a thickness corresponding to L2/LJ of 20 is less

than 2.6 %.

There are a few assumptions that need to be addressed for the one-dimensional intru­

siveness analysis. One is the constant heat transfer coefficient. Even if the heat transfer

mechanism at the surface is only convection, the heat transfer coefficient is expected to

42

10

9

8

7

6

0.3 5

4

3

2 0.03

1 0.01

Figure 2-8 : relative error bound for gauges more conductive than backing

change in some degree with the surface temperature. Eq. (2-12) does not involve an

explicit radiation term, but the radiation component can be lumped into the heat transfer

coefficient as a cubic function of temperature. Depending on the surface temperature of

the gauge, the change in the heat transfer coefficient may be large due to this non-linear

radiative contribution.

The conductivity of the material is assumed to be constant. However, the temperature

of the system can be any value between the ambient and the bottom of the backing temper­

ature. In some cases, the conductivity can vary by an order of magnitude. Even if this is

the case, the above analysis should be applicable by using the appropriate average conduc­

tivities to calculate k2/k^ .

The contact resistance between the gauge and the backing has a significant effect on

the one-dimensional intrusiveness analysis. It is ignored in the previous analysis. A bond­

ing agent reduces the contact resistance but it introduces an extra layer. However, when

the bonding layer is made thin enough so that the thermal resistance is negligible, its effect

can be ignored.

2.8 Radial Heat Leak

The gauge intrusiveness analysis with the one-dimensional assumption shows that the

temperature distribution with the gauge may differ from the one without the gauge. This

implies that there is a radial heat flow as illustrated in Figure 2-6. For a gauge made of

material more conductive than the backing, the temperature along x at a given depth, in

most cases, is higher for the system with the gauge than the one without the gauge. The

radial heat flow, or "leak", happens at the side of the gauge and in the backing below the

gauge. The leak tends to even out the hot and cold spots, which makes the gauge less

intrusive than the one-dimensional case. However, the radial heat flow is another source of

error because it is unaccounted in the one-dimensional analysis and needs to be investi­

gated.

The one-dimensional heat flow in the gauge can be justified by using a gauge with a

high aspect ratio of the radius over thickness. In one extreme, when the entire body is cov­

ered with the gauge, the one-dimensional assumption holds provided that the heat transfer

44

coefficient is location independent. Covering the entire body is impractical. The assump­

tion of a location independent heat transfer coefficient in a local sense is reasonable, but it

is unrealistic for the entire body. In order to determine how high the aspect ratio needs to

be to justify the one-dimensional assumption, the two-dimensional heat transfer model

needs to be analyzed.

2.9 Two-Dimensional Steady State Analysis

2.9.1 Two-Dimensional Model

The steady state heat transfer of the model in Figure 2-6 is two-dimensional when the

axis of symmetry is assumed at the center of the gauge. The deviation of the actual gauge

response from the one-dimensional assumption can be evaluated by solving the two-

dimensional model. The axi-symmetric model with boundary conditions, length dimen­

sions and material property is illustrated in Figure 2-9 where R is the radius of the gauge.

The temperature sensors are assumed to be thin enough to be neglected. A perfect thermal

contact is assumed between the gauge and the backing. A semi-analytical approach is used

to compute the two-dimensional composite steady state temperature distribution of the

model. The reasons for the semi-analytical approach are the following:

1) An analytical solution can incorporate an infinite domain naturally.

2) There is no need to compute the internal temperatures.

3) Once the model is solved for fixed thickness ratio and material, computing the

effect of changing the gauge radius is less involved.

45

in

interface

Region I

back

Figure 2-9 : axi-symmetric model

2.9.2 Two-Dimensional Steady State Solution

2.9.2.1 Analytical Solution of Each Region

The semi-analytical approach separates the model into two regions. Region I is the

region with r<R and Region II is the region with r> R. The non-homogeneous bound­

ary condition is removed by superposition of homogeneous and particular solutions. The

particular solution of each region, T^j(x), is the one dimensional solution given by

46

= _ i l l '" ' ' A:- ^ 2 V ^i ^ A - ^ i v J j = I I ^rid k - = k 2 for Region II

(2-15) r^[ for x<L]

k - = for Region I /c- = for Region II [k2 for x> Lj

where <7-,^ is given by Eq. (2-13). The homogenous solution of each region,r^^(x, r ) , is

obtained via eigenfunction expansion for a given interface temperature, F{x), at r = i? in

a form of

5 . 9 t - f r ) = 1 f o r x < L |

{, = 2 fo, n = 1

+ L-) ' { F { x ) - r ^ / x ) ) Z ^ . i „ ( x ) J x + ' V U ) - T ^ j { x ) ) X j 2 , i x ) d x

0 Li \

9^^-,j(r) is given by

[ '^o(Y;,/) for J = I 9?,„(r) = (2-17) ' Uo(Y;,/) for J =

where is the modified Bessel function of order 0 of the first kind and K^iyj^r) is

the modified Bessel function of order 0 of the second kind. X^„;(x) is the eigenfunction

and is given by

47

for i = 2

(2-18)

is the eigenvalue and obtained by solving the transcendental equation given by

Since the eigenvalues are independent of the gauge radius, it is not necessary to repeat

computing the eigenvalues when studying the effect of the gauge radius on the solution,

once the heat transfer coefficient, material combination, and thickness ratio are fixed.

Computing the eigenvalues takes about half the entire computational time. Naturally, a lot

of computational time can be saved by not calculating the eigenvalues every time.

The analytical solution of each region is specified in the previous section as a function

of the interface temperature, F{x). The closure to the process is to determine the interface

temperature such that it satisfies the interface condition. The solution from each region

combines seamlessly when the r component of the flux in Region I matches the one in

Region II at the interface r = R. This matching process is done using the approximate

analytic methods of residuals [19]. The residual, E(x), is defined here as

^(tan(Y.„L2) + ^tan(y^.„L])j + ^-tan(y^.„L,)tan(y.„L2) = 0 (2-19)

2.9.2.2 Interface Matching Condition

/ = 1 for X < L

i = 2 for x > L (2-20)

Two forms of functions are used as the trial function for the interface temperature. One

is a finite sum of the eigenfunctions in Region I multipUed with undetermined coefficients

given as

where is the undetermined coefficient and N is the number of undetermined coeffi­

cients. Another is a piece-wise quadratic with undetermined coefficients as nodal values.

Regardless of the form of the trial function, the undetermined coefficients will be deter­

mined by making the residuals close to zero. Using the eigenfunction as the trial function

works well for the length ratio, Lj/Lj, less than 5. However, for higher values of Lj/Lj,

it encounters a problem. The number of eigenvalues is related to the spatial resolution of

the homogenous solution. To keep a reasonable spatial resolution to study the gauge, the

number of eigenvalues increases proportionally with Lj/Lj and so does the number of

undetermined coefficients. The number of undetermined coefficients fixes the size of a

matrix which needs to be inverted. At high values of Lj/L, , the size of the matrix

becomes too big to be computed quickly. A piece-wise quadratic is used to remedy the

problem. The most drastic deviation from the one-dimensional solution happens close to

the gauge; whereas smaller deviations occur away from the gauge. By using coarser nodal

spacing at larger values of x away from the gauge than the nodal spacing at x close to the

N

F ( x ) = D „ X „ „ ( x ) (2-21)

n = 1

49

gauge, the number of undetermined coefficients necessary to resolve the spatial detail is

not influenced by Lj/Lj. Reasonably accurate solutions are obtained with the number of

coefficients in tens or at most a few hundred when the piece-wise quadratic is used.

2.9.2.3 Determining Coefficients

The undetermined coefficients are determined such that the residuals are close to zero

at any x location. To accomplish this, the Galerkin method is used [19]. In general, keep­

ing the residuals close to zero is accomplished by equating the weighted integral of the

residuals along the interface to zero as

\ ' ^ ^ ' W { x ) E { x ) d x = 0 (2-22) •'o

where W { x ) is the weighting function. Different methods use different weighting func­

tions. The Galerkin method uses the basis function of the trial function as the weighting

function. When the sum of eigenfunctions is used for the interface temperature, the

weighting for the Galerkin method is the eigenfunction itself. When the piece-wise qua­

dratic is used as the interface temperature, the Galerkin method requires the weighting to

be a piece-wise linear or quadratic function.

2.9.3 Contour Plot

The integration and the differentiation in Eq. (2-16) through Eq. (2-22) are performed

analytically. Calculation of the eigenvalues and the inversion of the final matrix is done

numerically. The whole process to obtain the homogeneous solution along with the partic­

50

ular solution is programed in MATLAB. Because the main objective of the two-dimen­

sional analysis is to quantify the deviation of two-dimensional solution from the one-

dimensional solution, it is not necessary to compute the temperature in the whole domain.

But if it is desired to visualize the result qualitatively, the temperature contour of the sys­

tem can be plotted as isotherms by evaluating the temperature using Eq. (2-16) and super-

positioning it with the particular solution. As an example. Figure 2-10 shows a

temperature contour of the system with "^^lue of 2, value of 0.1, hL^/k^

value of 0.1, and R/L^ value of 10. The temperature increment between isothermal lines

R / L ,

L,

^2

h L , = 0.1

r/L,

Figure 2-10 : temperature contour plot

is uniform except for the top three isothermal lines. The top three lines are further divided

51

into fifteen lines to resolve the temperature contour within the gauge. The small triangles

indicate the isothermal lines with the coarser increment to distinguish them from the finer

ones. From the kinks in the isothermal lines, the presence of the gauge is almost seen in

Figure 2-10. The gauge occupies the region of r/Lj < 10 and x/Lj < 1 . Within the

gauge, it is apparent that there is two-dimensional effect, especially at the r location close

to R, because isothermal lines are not horizontal.

2.9.4 Non-Monotonic Temperature Behavior

Another thing worth mentioning about Figure 2-10 is the surface temperature. By

inspecting the contour plot, a qualitative trend of the surface temperature is revealed.

When following the surface temperature in the direction of increasing r starting at r = 0,

the surface temperature cools down first, then heats up close to the interface, and finally

cools down again. The surface temperature is plotted in Figure 2-11. Intuitively, it may be

expected that the one-dimensional solution in each region bounds the surface temperature

of the two-dimensional solution. In other words, the surface temperature is expected to

increase or decrease monotonically. However, Figure 2-11 shows that the surface tempera­

ture of the two-dimensional solution at r = 0 is slightly lower than the one-dimensional

solution in Region I and the deviation increases as r increases to the interface. Close to

the interface, the temperature of the two-dimensional solution starts increasing and

exceeds the one-dimensional solution of Region II. Then the temperature reaches a peak

and approaches the one-dimensional solution of Region II as r increases. The occurrence

and the degree of the non-monotonic temperature behavior as function of r depend on the

52

R / L ,

T — 0.71 T oo

0.7

0.69

0.68

0.67

0.66

1-D solution

2-D solution

h L , = 0.1

0 r / L ,

10 15

Figure 2-11 : surface temperature distribution

parameters. When the surface temperature does not show the non-monotonic trend, the

temperature as function of r location at the gauge depth shows the non-monotonic behav­

ior.

The explanation of the non-monotonic temperature trend as function of r is given as

the following. The one-dimensional solution in each region as function of ;c is illustrated

in Figure 2-12. The one-dimensional temperature in Region I is the thick black Hne. The

one-dimensional temperature in Region II is the thin line. Depending on parameters, the

surface boundary condition changes but the boundary condition of Region I is bounded by

two extreme cases. One is the fixed surface temperature condition and the other is the

53

I: fixed flux

I: fixed temperature

L T

Figure 2-12 : illustration of one-dimensional solutions

fixed surface heat flux condition. These bounds are shown with thick grey lines in Figure

2-12. The thick black line can be somewhere between the grey lines depending on param­

eters. On the average along x, assume the temperature of Region I is hotter than that of

Region II. In Region I far away from the interface, Region I sees Region II as a heat sink.

This explains why the temperature of the two-dimensional solution is lower than the one

o f o n e - d i m e n s i o n a l a t r = 0 a n d w h y t h e s u r f a c e t e m p e r a t u r e g o e s f u r t h e r d o w n a s r

increase. As r gets close to the interface, it starts "seeing" the details of Region II and

"senses" that the surface temperature of Region II is higher than that of Region I. This

explains why the surface temperature goes up close to the interface. The reason why it

exceeds the one-dimensional surface temperature of Region II and causes a peak is

because Region II sees Region I as a heat source in the average sense. This means that the

54

heat flows outward in the r direction in Region II far away from the interface. The peak

and the gradual decrease in the surface temperature outside of the interface are consistent

with this outward heat flow. When Region I is colder than Region II in average, the non­

monotonic behavior in temperature is observed at the gauge depth.

2.9.5 Two-Dimensional Effect

Once the steady state two-dimensional temperature distribution is calculated, the two-

dimensional effect, E2D, can be quantified as the deviation of the two-dimensional solu­

tion from the one-dimensional solution and is given by

£20 = (2-23) ^ i n

where is the one-dimensional heat flux given in Eq. (2-13) with kj = . For the

actual gauge fabrication, a Resistance Temperature Detector, or RTD, is used for the tem­

perature sensor. Because of the RTDs, the measured temperature at the top is the surface

area average of the gauge top temperature. The measured bottom temperature is the area

average of the gauge bottom temperature. is calculated for different combinations of

Lj/Lj, , and h{L-^ +L2)/^2- The results are summarized in Figure 2-13, which

contains subplots of constant k2/k^ in columns and constant /i(Lj +L2)/k2 in rows.

Each subplot has 5 different curves corresponding Lj/L, values of 5, 10, 20, 40, and 80.

The arrows in the subplots indicate the direction of increasing Lj/Lj.

Figure 2-13 can be used to quantify the two-dimensional effect for a given heat flux

gauge design which contributes to measurement error, to design gauges for a given error

55

/ i ( L i + L 2 )

\ \ 20 40 60 80 100 20 40 60 80 100 20 40 60 80 100 20 40 60 80 100

\ 20 40 60 80 100 20 40 60 80 100 20 40 60 80 100 20 40 60 80 100

\ 20 40 60 80 100 20 40 60 80 100 20 40 60 80 100 20 40 60 80 100

10

^2 — -V- —

/ 40-- ' /

/ i / . / / . 20 40 60 80 100 20 40 60 80 100

R / L ,

20 40 60 80 100 20 40 60 80 100

Figure 2-13 : two-dimensional effect

specification, or to obtain a correction factor to "remove" the error due to the two-dimen-

sional effect.

2.9.6 Buffer Zone

Figure 2-13 shows that two dimensional effects are significant for gauges with

i?/L) <5 and becomes almost negligible at R/L^ - 100 in most cases. Even if the

gauge geometry is fixed for some other consideration, the two-dimensional effect can still

56

be reduced by using the outer gauge area as a "buffer" zone. Instead of covering the entire

gauge with the temperature sensor, the temperature sensor only occupies an area of radius

smaller than the gauge radius R . Denoting the temperature sensor radius as , the effect

of the buffer zone is shown in Figure 2-14 for ^2^^!

of 0.01. The plot shows that the two dimensional effect can be reduced up to 50 %

20

= 0.01

Q (N

0 10 20 30 40 60 70 50 80 90 100 i?/L]

Figure 2-14 : two-dimensional effect as function of temperature sensor radius

by a "buffer zone" as small as the length of Lj . This is expected from a contour plot, such

as Figure 2-10, because the radial heat flow is most significant close to the interface.

57

2.10 Simple Mounted Gauge

Up to this point, the gauge is assumed to be mounted in a recess in the backing such

that the gauge surface is flush with the backing surface. The flush mounted gauge mini­

mizes the disturbance to the flow, which governs convective heat transfer. The general

practice of using layered heat flux gauges is to mount the gauge on top of the backing

without the recess. This configuration is called "simple mounted" in this dissertation to

distinguish it from flush mounted gauges and is illustrated in Figure 2-15. The translated

= h { T ^ - T ) i n

i n

i n

interface

Region I Region II

back

Figure 2-15 : axi-symmetric model of simple mounted gauge

coordinate of x by Lj for Region II, denoted as x ' , is introduced to emphasize that the top

surfaces of Region I and II do not coincide. The gauge is exposed to a heat flux at the top

and side face. It is questionable if the heat transfer coefficient is the same for the top and

58

side of the gauge and the top surface of the backing, but it is assumed to be constant for the

sake of simplicity.

With the same assumptions used for Eq. (2-14), the gauge intrusiveness of the simple

mounted gauge can be quantified from the one-dimensional solution as

where the subscript s i m p l e denotes the values for the simple mounted gauge,

is plotted in Figure 2-16. It should be noted that the vertical scale of Figure 2-16 is one

tenth of that of Figure 2-7 and is the same as that of Figure 2-8. For the case with small

values of k2/k^, is almost negligible. As long as stays on the order of

one or less, t)e kept small with a decent value of Lj/Lj.

The two-dimensional effect for the simple mounted gauge is also studied. The same

semi-analytical approach, as that used for the flush mounted gauge, is used to obtain the

two-dimensional simple mounted gauge temperature distribution. The only differences are

how to define the residuals and some modification in equations to account for the trans­

lated coordinate of Region II. At the section where x<L^ , the r component of the flux

m u s t s a t i s f y t h e h e a t t r a n s f e r e q u a t i o n a t t h e s i d e b o u n d a r y . A l o n g t h e i n t e r f a c e a t j c > L j ,

the r component of the flux from each region must be the same, which is the same condi­

tion as the flush mounted. This is expressed in terms of residuals, Egimple^^)' shown in

the following equation.

E 1 D - s i m p l e s i m p l e

(2-24)

59

/i(Z> J + ^2) /i(Lj + L ^ ) 0.2

0 10

- — = 0.0

L./L

Figure 2-16 : relative error in heat flux of simple mounted gauge

^ s i m p l e ^ ^ ^

d T ^ l i x , r )

' d r

d T , ^ j ( x , r )

h { T ^ - { T ^ j { x , R ) + T p , { x , R ) ) ) for x < L ,

d r - k .

d T , h - s i m p l e - I I { x , r )

r = R d r

(2-25)

for X > Lj r = R

The homogeneous solution of Region II of the simple mounted gauge is denoted as

T . h-simpie-ii^^''')' which accounts for the difference in thickness of the backing and for the

fact that the surface of Region II is located at x = Lj. For the flush mounted gauge, the

thickness of backing material in Region II is L^+Lj while it is Lj for the simple

mounted gauge. The equations of the flush mounted gauge can still be used for the simple

60

mounted gauge by substituting x' for x and for Lj+Lj when calculating

T h-simple-ii^^^ r). The two-dimensional effect of simple mounted gauge is defined by Eq.

(2-23), which is the same equation used for the flush mounted gauge. The two-dimen-

sional effect of the simple mounted gauge is summarized in Figure 2-17. In some cases,

h { L , + L , )

/ 20 40 60 80 100 20 40 60 80 100 20 40 60 80 100

10

20 40 60 80 100

20 40 80 100 20 40 60 80 100 20 40 60 80 100 20 40 60 80 100

20 40 60 80 100 20 40 60 80 100 20 40 60 80 100 20 40 60 80 100

\

^2

/ / 40 ^ / /

20 40 60 80 100 20 40 60 80 100

R / L

20 40 60 80 100 20 40 60 80 100

Figure 2-17 : two-dimensional effect on simple mounted gauge

even a gauge with small values of R/L^ has negligible two-dimensional effect. These

cases do not mean that there is no radial heat flow. There still is non-negligible radial heat

61

flow, but it affects the top and bottom gauge temperature equally and its effect is cancelled

out through the subtraction inside the integral in Eq. (2-23).

2.11 Gauge Output

Another thing to consider when designing a gauge is the gauge output. For all other

parameters being constant, a thicker gauge has a larger temperature difference between the

top and bottom temperature sensors. A larger temperature difference is easier to measure.

Therefore, in terms of the gauge output, a thicker gauge is better than a thinner gauge

which is actually an opposite result from the other design considerations mentioned previ­

ously.

62

3 DESIGN, FABRICATION AND MEASUREMENT OF GAUGES

3.1 Gauge Design

Several important performance characteristics of a heat flux gauge are numerically

simulated or analyzed in Chapter 2. The result of this analysis shows that dimensional and

non-dimensional parameters determine the characteristics of the gauge.

The design specifications for the gauge differ from one mission to the next and even

from one location to another location on a space vehicle. As a gauge design exercise, a

physical gauge is designed from assumed specification values. Before discussing dimen­

sions of the gauge design, material selection is discussed.

3.1.1 Gauge Material

The one common trait of gauges in aeroentry applications is that they are exposed to

high temperature. The gauge materials must tolerate high temperature. The choice of a dif­

fusion barrier material depends on mounting methods. A flush mounted gauge should

have similar conductivity as the backing while a simple mounted gauge needs to be more

conductive than the backing to reduce the intrusiveness. In general, insulating materials

are fragile and not suited for a thin substrate. For these considerations, alumina is chosen

for the diffusion barrier or the gauge substrate. Alumina has a high melting point and a

conductivity of which falls in the lower range of metals. It is mechanically suited as a sub­

strate. Though alumina is not the best for the flush mounting, the intrusiveness error can

be reduced by making the gauge thin.

63

Resistance Temperature Detectors, or RTDs, are used as the temperature sensors on

the top and bottom of the gauge. Platinum is chosen as an RTD material because it is a

standard in temperature measurement and because it is chemically stable even at high tem­

peratures.

By choosing these materials for the gauge fabrication, the gauge can function at tem­

peratures up to 1000 °C.

3.1.2 Gauge Dimensions

Specification values for the gauge are assumed from the Stardust mission [20]. The

gauge needs to measure a heat flux up to 75 W/cm^. A time resolution of 1 Hz is assumed.

The measurement accuracy of the gauge is 5 %. The thermal protection shield thickness is

assumed to be 13 mm.

Further assuming that the temperature measurement process gives an accuracy of

0.3 °C, the diffusion barrier needs to produce a temperature difference of 30 °C to have an

accuracy of 1 % of full scale output. At a temperature of 500 "C, the conductivity of alu­

mina is 10W/(m-K) and the diffusivity is 2.2-10"^ m^/K. Using Eq. (2-6), the diffusion

barrier thickness needs to be 400 |im or thicker. 400 |a,m thick alumina is thick enough to

be handled easily. The thickness of 400 |im gives the time constant of 0.07 sec. As long as

the quadratic method is used, the gauge responds quickly enough to provide the time reso­

lution of 1 Hz. The thickness of 400 jim gives a Lj/L, value of 30. At the value

of 30, the relative error due to the gauge intrusiveness is less than 3 %.

64

The gauge diameter is chosen to be 15.5 mm, which gives a R / Lj value of 20. Ideally,

a higher value of R/L^ is desired to reduce the two-dimensional effect. This diameter is

chosen to keep the gauge size reasonably small for usefulness and experimental consider­

ations. When increasing the diameter of the gauge, testing the gauge at a uniform heat flux

becomes a challenge. At the R/L^ value of 20, the two-dimensional effect is less than

4 % except for the cases of a flush mounted gauge with high /i(Lj + L2)//c2 values. By

making the radius of the temperature sensors smaller than the gauge radius, this problem

can be reduced. Choosing the temperature sensor radius to be smaller than the gauge

radius by 1 mm, the two-dimensional effect is reduced by a factor of 2. This gauge design

satisfies the specification for most applications.

3.1.3 Gauge Schematic

The schematic of the gauge is shown in Figure 3-1. The top and bottom RTDs have a

common junction through a via and another via is used to make the lead wire connection

to the top RTD from the bottom side. A total of five lead wires are attached to the gauge.

The outer pair is used for current excitation and the remaining three wires are used for

voltage measurements. A photo of the gauge is shown in Figure 3-1.

3.2 Temperature Sensors

For temperature sensors, RTDs are chosen for simplicity. RTD utilizes the fact that the

resistivity of a material is a function of temperature. The manufacturing process of RTDs

is much less complicated than that of thermopiles. A thermopile is an array of thermocou-

65

15.5 mm in diameter

400 |im

Platinum lead wires

via

Platinum RTD (10 [i-m)

alumina disk as thermal diffusion barrier

^ Platinum RTD (10 |im)

bottom view top view

m

Figure 3-1 : schematic and photo of heat flux gauge

pies connected in series. The voltage output of the thermopile is a multiple of the number

of junctions. With many junctions, the thermopile output can be comparable to that of an

RTD or better [21-29]. The most attractive part of using thermopiles is that the output

voltage is self generated, which means that the thermopiles require no excitation voltage

or current. It also eliminates the necessity to have dedicated electronics to handle two RTD

66

signals for subtractions. The other advantage of the thermopile is that its measurement is

local because the physical size of thermocouple junction can be much smaller than that of

an RTD. The implication of this is that the junctions can be positioned close to the axis of

symmetry of the gauge. By doing this, the two-dimensional effect is reduced because of

the buffer zone, which is mentioned in Section 2.9.6.

Disadvantages of the thermopile are that an additional thermocouple is needed and

that the fabrication process is more complicated than that for an RTD. Because the ther­

mopile only measures the temperature difference, the additional thermocouple is needed

to measure the temperature of the gauge to account for the temperature dependent proper­

ties. Thermocouples are made of two materials and in most cases each material is an alloy

or mixture of pure metals. Depending on the fabrication technique, keeping the mixture

ratio consistent within a batch becomes a big issue. The thermopile requires the same

number of connections from the top to the bottom of the gauge as the number of junctions.

Requiring to have many reliable top to bottom connections is a major fabrication issue

associated with thermopiles. Because of these disadvantages, the RTD is chosen as the

temperature sensor of the heat flux gauge.

3.3 Fabrication Process

The gauge fabrication steps go through many iterations to make the RTD adhere better

to the alumina, to make the steps simpler, or to make the dimensions more precise. In-

house made pre-fired alumina tape is CNC machined for the gauge perimeter and via

holes. The tape is fired in a high temperature furnace. Off-the-shelf platinum paste is

67

printed on the fired alumina substrate using a screen printer. After a second firing in the

furnace, the platinuna paste becomes RTDs. Platinum lead wires are attached to the gauge

terminals with the platinum paste and fired a third time in the furnace to cure the paste.

3.4 Electronics

There are different ways to excite and measure an RTD. In this research, constant cur­

rent excitation and subtractions are used to measure the gauge output. A dedicated elec­

tronic circuit is designed and fabricated in-house. A schematic of the electronics is shown

in Figure 3-2. The RTDs are excited by the constant current source. The voltage drop of

each RTD is amplified with operational amps and recorded to provide the resistance of

each RTD. From the resistance, the temperature of each RTD is calculated. If the resis­

tance difference is computed from these temperature measurements after analog to digital

conversion, accuracy can be lost. Therefore, the difference of the two voltages is boosted

by another operational amplifier to provide the resistance difference. For very simple

gauge models, only the temperature difference is needed to compute the heat flux. How­

ever, in most cases, the temperature of the gauge must be measured to account for temper­

ature dependent effects.

68

electronics enclosure

Voltage of bottom RTD Voltage of top RTD

Voltage difference of RTDs

Figure 3-2 : schematic of gauge reader electronics

69

4 TESTING APPARATUS AND EXPERIMENT

Several testing apparatus including an arcjet chamber and a calibration furnace are

setup to test the flux gauge. Some of the apparatus are used to observe the qualitative

response of the gauge. The arcjet chamber is assembled to subject the gauge to close-to-

aeroentry conditions. The calibration furnace is constructed to calibrate the flux gauge

against a material of published conductivity data.

4.1 Qualitative Testing Apparatus

Several testing apparatus with relatively simple setups are constructed to test the

gauge in a qualitative sense for gauge performance confirmation and troubleshooting. The

schematics of these testing apparatus are shown in Figure 4-1. The gauge is flush mounted

on a fire brick as a substitute for a thermal protection shield or simple mounted on a cold

plate which is water cooled. Mounting the gauge on the cold plate enables high gauge out­

put, which is useful for troubleshooting the gauge and electronics.

The convective testing apparatus uses a temperature-controlled heat gun. Flow of hot

air impinges upon the gauge. A mechanical chopper can be used to interrupt the flow and

give the periodic heating of impinging hot air followed by natural convective cooling. The

radiative testing apparatus uses a 500 W halogen lump as a heat source. The mechanical

chopper can be used to block the radiation. With these apparatus, the gauge temperature

stays relatively cold even with the fire brick as the backing. The maximum air temperature

that the heat gun can maintain is 350 °C. The gauge reaches a steady state temperature of

70

convective radiative high temperature

heat gun halogen lump

r'cT

t t t t t mechanical

chopper

blank

swap

gauge block:

furnace

flsuh mounted gauge on fire brick

o r

simple moutned gauge on cold plate

Figure 4-1 : schematics of qualitative gauge testing apparatus

200 °C with this condition. With the radiation testing apparatus, the gauge heats up to

150 °C. Testing the gauge at a high temperature is important for aeroentry applications.

To expose the gauge to higher temperature, a high temperature testing apparatus is

constructed. A tube heater in the furnace is used as a heat source. The gauge block is posi­

tioned on the opening on the top of the furnace. By exchanging the gauge block with a

blank block repeatedly, the gauge is exposed to the hot and cold temperature periodically.

71

4.2 Arcjet Chamber

An arcjet chamber, donated by Primex Aerospace company, is assembled and modi­

fied to be used as an arcjet testing apparatus for the heat flux gauges. It is used to mimic

aeroentry conditions. The setup schematic is shown in Figure 4-2. Inert gas is fed into the

Power Supply

mass flow controller

vacuum pump

test

platfon

measurement port

mass ( flow flow

1 * control meter 1 ^ ^ / valve

Ar

Nn

chilled water supply

chilled water return

p ) :pressure transducer

Figure 4-2 : schematics of arcjet testing apparatus

upstream of a nozzle which acts as an anode. Along the center axis of the nozzle upstream

of the throat, there is a cathode rod electrically insulated from the nozzle. The power sup­

ply provides the electric current and creates an arc between the cathode and anode which

heats up the gas. The gas expands through the nozzle and exits at super sonic speed. Oxy-

72

gen, carbon dioxide, or other gases can be injected into the arcjet from a ring shaped injec­

tor to simulate the composition of given atmospheres. The pressure in the chamber is

reduced with a vacuum pump which has a pumping speed of 150 cfm. The chamber and

the test platform in the chamber are water cooled. The arcjet runs at up to 5 kW.

A photograph of the arcjet testing apparatus is shown in Figure 4-3. The vertical tube

standing at the center with access and measurement ports is the chamber. The left of the

chamber, out of the picture, is the power supply which is capable of providing 100 Amp

DC current.

Figure 4-3 : photos of arcjet testing apparatus and arcjet

73

A photograph of the arcjet and the test platform is also shown in Figure 4-3. The col­

umn at the center is the arcjet which spreads horizontally after striking the test platform.

On top of the test platform is a glowing hot ceramic tube carrying thermocouple wires.

4.3 Calibration

4.3.1 Calibration Process in General

For a gauge to be useful, it must be calibrated. However, the calibration of heat flux

gauges is not straightforward. There is no national standard for the process

involved [30&31]. In most cases, the calibration is performed by comparing the gauge

output to known input heat fluxes, utilizing radiative, convective or conductive heat

transfer [32-39]. Depending on the design of the gauge, the calibration done with convec­

tion or conduction may differ noticeably from that with radiation [40&41].

Calibrating the gauges with a radiative heat source is the most commonly used method

because it is relatively easy to setup. The problem in calibrating with a radiative heat

source is the uncertainty in the absorptivity and the emissivity of the gauge. The magni­

tude of the incident radiation can be accurately estimated but the amount absorbed and

emitted by the gauge depends on the absorptivity and the emissivity. The uncertainty in

these radiative properties is large, sometimes on the order of tens of percent. Determining

the absorbed flux may be accomplished by using a reference heat flux gauge. A reference

gauge is positioned in the place of the gauge being calibrated and is exposed to a fixed

intensity of heat source. The reference gauge outputs the magnitude of the absorbed heat

flux. Now, the gauge being calibrated replaces the reference gauge and is exposed to the

74

same heat source, whose absorbed magnitude is supposed to be known from the reference.

However, the use of a reference gauge is hkely to alter the heat transfer of the system and

introduce other errors unless the reference gauge and the gauge being calibrated have the

same design. Performing convective calibration of a heat flux gauge is difficult to do at

high temperatures. At high temperatures, it is difficult to suppress radiative components.

Conduction calibration should give the least complication when calibrating high tempera­

ture heat flux gauges.

The calibration methods mentioned above, regardless of the heat source, require

knowledge of the magnitude of the heat flux going through the gauge. There are several

other methods which require no knowledge of the magnitude of the heat flux. One such

method compares the gauge response to a response of a reference material, whose physical

properties are well studied [42]. The result gives the gauge properties as ratios to the refer­

ence material. Another method records the gauge response to periodic excitation with dif­

ferent frequencies. The data can be processed to calculate the gauge thermal

properties [43-54].

4.3.2 Conduction Calibration Furnace

As mentioned in Section 4.3.1, the calibration process has many pitfalls. The uncer­

tainty in the flux flowing through the gauge is the main problem. It is proposed to measure

the flux by another layered heat flux gauge in series as a reference. One of the diffusion

barriers has thermocouples on the top and bottom faces, which functions as the reference.

In this way, the flux through the gauge being calibrated is equal to the flux through the ref­

75

erence. By using a material with published conductivity data for the reference diffusion

barrier, the flux through the reference is known. Figure 4-4 illustrates the conduction cali­

bration furnace setup. It consists of a plate heater at the bottom, water cooled cold plate at

thermocouple junction

cold plate

adjustable diffusion barrier

insulation

reference

gauge and spacer

heater

Figure 4-4 : conduction calibration furnace illustration

the top, a spacer and the gauge being calibrated, and thermal diffusion barriers between

the heater and the cold plate, all surrounded by insulation. The plate heater and the cold

plate create one-dimensional heat flow. The spacer is a ring shaped disk made from the

same stock of the alumina tape of which the gauge is made. The gauge is positioned in the

hole of the spacer. Together, they act as one layer of diffusion barrier. The thickness or the

material of the adjustable diffusion barrier can be changed to provide a certain heat flux at

a certain gauge temperature.

Similar furnaces have been setup with added symmetry about the plate heater and with

guard heaters to minimize radial heat flow in order to obtain the thermal conductivity of

material from the heater power. In the conduction calibration furnace, the complexity of

guard heaters is removed in favor of simplicity and the heater power is used as sanity

check of the heat flux value. One functionality of the guard heaters is to provide one-

dimensional heat flow, but the same task is accomplished by using a much larger heater

than the gauge being calibrated. Using insulation of much smaller conductivity as com­

pared to the diffusion barrier material reduces the radial heat loss.

The conduction calibration furnace's plate heater and diffusion barrier diameters are

five times as large as the gauge diameter. The heater is made in-house to function up to

1000 °C and to minimize the radial heat loss. Originally, it is designed to be made of cop­

per with machined out grooves where nickel-chromium wire is imbedded. Copper has a

high thermal conductivity but oxidizes easily. A ceramic coating is used to protect it from

oxidizing and to function as an electrical insulator. However, the coating did not function

as a oxidizing barrier. Titanium is used instead of copper because it can have a fairly stable

electrically insulating thin oxide film. The heater is capable of power output up to 1700 W,

which is equivalent to 50 W/cm^.

77

4.4 Experimental Result

The high temperature testing apparatus is used to test the quaUtative response of the

gauge. The furnace temperature is kept at 950 °C. A gauge is flush mounted to a 6 cm

thick fire brick (Thermal Ceramics, K-25). The gauge block is exposed to the furnace tem­

perature for 10 minutes and to the room temperature for 10 minutes cyclically. The qua­

dratic method, Eq. (2-9) and Eq. (2-11), is used to calculate the flux from measured

temperature data. The published conductivity data of alumina and resistivity data of Plati­

num are used for the calculation [18&55]. The data acquisition rate is 1 Hz. The gauge

output is shown in Figure 4-5. Qualitatively, the plot shows what is expected. Highest in­

flux and out-flux are seen at the beginning of each exposure cycles. The flux value

decreases gradually which can be explained by the decrease in the difference between the

gauge and ambient temperature. Several problems can be seen in the plot. One is the shift­

ing of zero or offset problem. When the gauge is exposed to room temperature long

enough, the gauge can be assumed to be at an isothermal condition. When the gauge is at

isothermal, it should give the output of zero. The electronics is adjusted such that the out­

put is zero at the beginning of the experiment. After many thermal cycles, the output is

checked and usually has a non-zero value. It is seen in Figure 4-5 as a horizontal line at the

end of experiment with a heat flux value of 0.6 W/cm . The second problem is that the

gauge output has noticeable time lag. The 9 th peak at 311 minutes is magnified and

shown at the upper corner with 24 seconds worth of data. The heat flux value from the

simple difference method is also calculated for a comparison purpose. In the magnified

scale, the difference in the calculation method is clearly seen. According to the simple heat

78

quadratic

simple difference

-el 1 1 1 1 I I ^ 150 200 250 300 350 400 450 500

time (min)

Figure 4-5 : gauge response on fire brick in high temperature apparatus

transfer equation given by Eq. (2-11), the gauge should see the highest flux in magnitude

at the beginning of each exposure. The gauge output with the quadratic method should

have time resolution much better than 1 Hz. However, the result shows that the gauge

takes 5 seconds to reach the peak.

The result for a gauge simple mounted on a water cooled cold plate are shown in Fig­

ure 4-6. The gauge is exposed to 5 minutes of hot temperature and 5 minutes of cold tem­

perature. The plot is very asymmetric upside down compared to that of the gauge on the

fire brick. The magnitude of the heat flux values are high and stay relatively high for the

79

S

X s c "S

12

10

-2

magniHed I I

¥

quadratic

/•'/simple difference

19.15 49.2 49 25 49.3 49 35 49.4

_J \ I l_ _1 l_

0 10 20 30 40 50 60 70 80 90 100

time (min)

Figure 4-6 : gauge response on cold plate in high temperature apparatus

hot cycle. This can be explained because the gauge does not heat up to high temperature

and stays close to the cold plate temperature. The zero offset of the gauge after the experi­

ment is relatively small. The window starting at 49 minutes and ending at 49.4 min is

magnified and shown at the upper right corner. Again, there is a delay in the gauge

response. It takes 18 seconds to reach the peak even for the quadratic method.

The gauge response on the fire brick in the arcjet apparatus is shown in Figure 4-7.

The applied arc current is 10 Amp and the gas flow rate is 6.6 slm of Argon. The arcjet is

started at 2.1 min after the data recording is started and terminated at 6.65 min. The mea-

80

magnified 12

quadratic

10

simple difference 8

6

stop quadratic 4

j/ simple difference 2

0

start

•2

0 5 20 10 15 25

time (min)

Figure 4-7 : gauge response in arcjet apparatus at arc current of 10 Amp

sured heat flux is almost constant while the arcjet is on. This can be explained from the

very high gas temperature of 2500 °C. After the arcjet is stopped, the gauge should output

negative values. However, the gauge output stays positive. The zero offset after the expo­

sure is almost as large as the output. The cause of these problems is unknown and needs to

be investigated. The magnified plot of a window starting at 2.1 min and ending at 2.5 min

is shown at the lower right corner.

Calibration using the conduction calibration furnace is conducted starting at a low

value of heat flux. Unfortunately, the in-house made heater failed due to internal shorting.

81

There are four layers of electrical insulation between the heating element and a metal part,

yet the heating element still shorted out through the metal. Applied heat fluxes are as small

as the measurement error of the gauge. The experiment provides no useful calibration

data.

82

5 CONCLUSIONS

5.1 Analytical and Numerical Simulation

The performance of the gauge is simulated in detail using numerical simulation and

analytical solutions for different criteria. It is found that the time response of the gauge can

be slow in some material and thickness combinations when the simple difference method

is used to calculate heat flux. The time response of a gauge can be improved by modifying

the simple difference method. Such modification results in the quadratic method, which

puts the categorization of the gauge as a hybrid of the spatial and temporal method. The

quadratic method improves the time response without having the complexity of solving

the entire heat model. The quadratic method is proven to be less effected by noise com­

pared to the "inverse method".

Intrusiveness of the gauge is quantified by using one-dimensional solutions. The valid­

ity of a one-dimensional heat flux assumption is investigated by solving a two-dimen­

sional heat model and comparing it to the one-dimensional solution. Two-dimensional

effects can be kept smaller than 5 % by having a gauge with its radius 20 times as large as

its thickness in most cases. Different mounting methods have a large influence on the

gauge intrusiveness. Flush mounting minimizes the disturbance to the fluid flow and is

suited for gauges with the same conductivity as the backing material. Simple mounting

has disadvantage of disturbing the flow. Intrusiveness of a simple mounted gauge can be

reduced by using a material more conductive than the backing for the gauge diffusion bar­

rier.

83

5.2 Gauge Design and Fabrication

Based on the simulations and analyses, a physical gauge is designed and fabricated.

The gauge is made of 400 )a.m thick alumina with a diameter of 15.5 mm. Alumina is cho­

sen for its thermal conductivity and mechanical property. The thickness is determined to

provide measurable gauge output, while keeping the error due to intrusiveness and the

time constant of the gauge small. Platinum RTDs are screen printed on the top and bottom

of the gauge as temperature sensors. The gauge is designed to measure a heat flux up to

75 W/cm with an accuracy of 5 % and the time resolution of 1 Hz. A dedicated electron­

ics circuit to operate the gauge is designed and constructed.

5.3 Experiment

The convective, radiative and high temperature testing apparatus are setup to test the

gauge qualitatively. The convective and radiative apparatus are easy to setup and to use,

but do not provide high enough heat fluxes to expose the gauge in designed full range.

Experimental data with the high temperature testing apparatus is presented as a typical

gauge response. The arcjet chamber is assembled to expose the gauge to a condition close

to aeroentry. The arcjet experiment provides a mixed result revealing the zero off set prob­

lem. The conduction calibration furnace is setup to calibrate the gauge against a material

with published thermal conductivity data. Because of a problem with the heater in the cal­

ibration furnace, no useful calibration data is obtained.

84

5.4 Future Work

Some issues of the gauge analyses and experiments still remain. Some of them must be

resolved to provide a useful gauge for aeroentry applications. Others can be investigated

for improvements.

5.4.1 Calibration

For the gauge to be useful in a real application, it needs to be calibrated. The heater

failure of the conduction calibration furnace is a small technical problem. With a different

design for the heater, which uses a solid ceramic disk as a substrate for a heating element,

the calibration furnace functions as it is designed for. With a functioning apparatus, cali­

bration of the gauge should be finished.

5.4.2 Gauge Mounting and Connection

The gauge will be mounted on the thermal protection shield of space vehicles. Mount­

ing the gauge and connecting the lead wires to the on-board electronics are non-trivial

matters. The safety and reliability of the shield can not be compromised during mounting

and connecting of the gauges. Yet the mounting and connection process should be simple

enough to reduce the installation work hours and to minimize any possible mistakes. In

addition, for better measurement accuracy of the gauge, some care needs to be taken dur­

ing installation.

85

5.4.3 Temperature Sensor Option

For the temperature sensors, an RTD is chosen over a thermopile because of the simple

fabrication process of RTDs. However, dealing with RTDs in experiments is found to be

very cumbersome because of the complexity in obtaining its output. A dedicated electron­

ics circuit is used to operate the gauge and the circuit needs to be balanced for each gauge.

Balancing the circuit is necessary because each RTD has a different resistance. It was orig­

inally thought that the complexity of handling RTDs in experiments is a small penalty to

pay compared to the savings in the fabrication process. However, experience with the

gauge shows that user friendliness is more important. If the process of balancing needs to

be eliminated to make the gauge more user friendly, a more elaborate fabrication process

is needed to produce RTDs with an exact resistance value. Though difficulties in fabricat­

ing a reliable thermopile are unknown, they may be similar to producing an RTD with a

fixed resistance value. If that is the case, using thermopiles is an attractive option.

Screen printing, or thick film technology, is used to produce RTDs. Whether continu­

ing to work with RTDs or trying a thermopile, another production technique may improve

fabrication processes. Thin film technologies, such as vacuum deposition, can print small

features on the order of sub microns with a thickness on the order of nanometers. This may

help produce more precise RTDs or consistent thermocouples for the thermopile.

5.4.4 Transient Intrusive Analysis

The intrusive analyses done in Chapter 2 is based on the steady state assumption. For

the heat flux with low frequency content, this analysis is reasonably accurate. However,

86

the high frequency content of the heat flux is unlikely to "reach" the bottom of the thermal

protection shield. The effect of the gauge on these high frequency contents of heat flux, or

transient gauge intrusiveness, may be quantified by using an analytical solution of the

transient composite slab problem.

5.5 Summary

As a summary, this research presented here accomplished the following objectives.

First, gauge performances are characterized as a function of gauge geometries and materi­

als. Secondly, different testing apparatus is constructed and used to test the gauge. Finally,

a proof of concept gauge is demonstrated.

It is hoped that this research will aid in the optimization, fabrication and testing of lay­

ered heat flux gauges for aeroentry applications.

87

SYMBOLS

a thermal diffusivity of material

a J thermal diffusivity of heat flux gauge material

a2 thermal diffusivity of backing material

eigenvalues of transient composite slab problem

eigenvalues of two-dimensional steady state composite problem

0 discretization parameter to compute finite difference

p density of material

Tq gg delay time

A,J coefficients for homogeneous part of transient composite slab problem

Bj^ coefficients for homogeneous part of two-dimensional steady state

composite problem

specific heat at constant pressure

Cj deviation of temperature at midpoint between quadratic and linear

curve in quadratic method

c constant for time shift method

undetermined coefficient

relative error due to gauge intrusiveness for flush mounted gauge

^\D-simple relative error due to gauge intrusiveness for simple mounted gauge

£"20 two-dimensional effect for flush mounted gauge

E2D _ simple two-dimensional effect for simple mounted gauge

E { x ) residuals of flush mounted gauge

E^impiei^) residuals of simple mounted gauge

F ( x ) interface temperature between Region I and Region II

h heat transfer coefficient

/o(r) modified Bessel function of order 0 of first kind

KQ{r) modified Bessel function of order 0 of second kind

k thermal conductivity of material

thermal conductivity of gauge or backing

/c. thermal conductivity of gauge

^2 thermal conductivity of baking or thermal protection shield

product of thermal conductivity, density and specific heat for backing

L thickness of slug calorimeter

thickness of layered gauge

L2 thickness of backing or thermal protection shield

L d thickness of top diffusion ring for radial heat flux gauge

N number of undermined coefficients

N n norm of eigenfunction, X - ^ { x )

^ j n norm of eigenfunction, X j - ^ ^ { x )

b o t t o m heat flux through bottom of gauge

surface heat flux

^ t o p heat flux through top of gauge

^ w i t l i heat flux of system with gauge

^ w i t h o u t heat flux of system without gauge

% M ) Bessel functions

R radius of gauge

R s radius of temperature sensor

r coordinate in radial direction

s dummy variable of integration

T rate of change in temperature with respect to time

T 00 ambient temperature

T b a c k temperature at bottom of backing or thermal protection shield

T ^ { x , t ) homogeneous temperature solution of transient composite slab problem

ThM, r) homogenous part of two-dimensional steady state temperature solution

h - s i m p l e - I I ^ ^ ' r ) homogenous part of two-dimensional steady state temperature solution

in Region II for simple mounted gauge

89

temperature at inner sensor of radial heat flux gauge

average of top and bottom gauge temperature

temperature at outer sensor of radial heat flux gauge

T^{x) steady state temperature solution of composite slab

Tpj{x) one-dimensional steady state temperature solution of composite slab

temperature at surface

t time

t - i-th discrete time node

W { x ) weighting function

X coordinate inward normal to surface

j:' translated coordinate of x by Lj

eigenfunction of transient composite slab problem

eigenfunction of two-dimensional steady state composite problem

90

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