the thermal behavior of spherically curved to …

THE THERMAL BEHAVIOR OF SPHERICALLY CURVED

SOLAR COLLECTOR MIRROR PANELS EXPOSED

TO CONCENTRATED SOLAR RADIATION

by

VIJAY K. AGARWAL, B.E.

A THESIS

IN

MECHANICAL ENGINEERING

Submitted to the Graduate Faculty of Texas Tech University in Partial Fulfillment of the Requirements for

the Degree of

MASTER OF SCIENCE

IN

MECHANICAL ENGINEERING

December, 1980

- - y

'7^/ ACKNOWLEDGEMENTS

I am deeply indebted to Dr. Herbert J. Carper whose guidance,

constant encouragement and help made this work possible. I am also

grateful to Dr. J. R. Dunn, Dr. R. J. Pederson and Dr. J. H. Strickland

from all of whom I received substantial help.

Thanks are also due to Mr. Paul Davenport and Mr. Norman Jackson

of the Mechanical Engineering Lab for help with experimental setups.

I must also express my gratitude to all the members of Crosbyton

Solar Power Project team, especially Dr. J. D. Reichert, Col. Travis

Simpson and Mr. Bobby Green.

Special thanks goes to Mrs. Dunree Norris for typing this manu

script.

11

TABLE OF CONTENTS

ACKNOWLEDGEMENTS ii

ABSTRACT v

LIST OF TABLES vi

LIST OF FIGURES vii

NOMENCLATURE viii

CHAPTER 1. INTRODUCTION 1

1.1 Background 1

1.2 The Hot Spot Problem 3

1.3 Objectives 8

1.4 Plan of Work 9

CHAPTER 2. COl-IPUTATIONAL MODEL 11

2.1 Background 11

2.2 Basis of Computational Model 13

2.3 Assumptions and Simplifications 13

2.4 The Finite Difference Formulation and

the Stability Criteria 14

2.5 The Computer Program 16

2.6 Certain Characteristics of the Model 17

CHAPTER 3. EXPERIMENTAL DATA FOR A HOT SPOT ON A METAL

PLATE AND COMPARISON V7ITH PREDICTED RESULTS 20

3.1 Introduction 20

3.2 Hot Spot Characteristics 21

3.3 Thermal Response of a Metal Plate to a

Hot Spot 31 3.4 Correction for Thermocouple Errors 43

111

.n

TABLE OF CONTENTS (continued)

3.5 Comparison of Experimental Data with Predicted Results 45

CHAPTER 4. EXPERIMENTAL DATA FOR ADVS B0I7L AND COMPARISON

WITH PREDICTED RESULTS 60

4.1 General Setup 60

4.2 Hot Spot Characteristics 62

4.3 Corrections for Thermocouple Errors 64

4.4 Temperature Distribution 66 4.5 Comparison of Experimental Data with Pre

dicted Results for ADVS Mirror Panel 88

CHAPTER 5. PARAMETRIC STUDIES 113

5.1 Introduction 113

5.2 Effect of Mirror Thickness Variation 113

5.3 Effect of Mirror Glass Density Variation 115

5.4 Effect of Thermal Conductivity Variation 115

5.5 Effect of Specific Heat Variation 118

5.6 Effect of Absorptivity Variation 118

5.7 Effect of Emissivity Variation 121

5.8 Effect of Spot Irradiation and Size Variation .. 121

5.9 Effect of Spot Speed Variation 121

5.10 Effect of Variation of Convective Heat

Transfer Coefficient 126

CHAPTER 6. CONCLUSIONS AND RECOMMENDATIONS 128

LIST OF REFERENCES 130

APPENDIX 1. Listing and a Sample Output for Program TEMPDIST ... 131

APPENDIX 2. Reflectivity and Absorptivity for a Mirror Panel ... 162

iv

ABSTRACT

Under certain conditions the mirror panels in 'Fixed Mirror

Distributed Focus* (FMDF) type of solar collectors are subject to a

moving heat source (hot spot) formed by concentrated solar radiation.

A two-dimensional finite difference scheme was devised to predict the

temperature distribution in the panels under these conditions. The

predicted results were compared with experimental data and, show good

agreement. The prediction model is therefore considered adequate for

use in design and selection of materials for the panels. Some para

metric studies based on the above model show that severity of the

problem can be reduced by choosing mirror materials having higher

thermal capacity and lower absorptivity.

LIST OF TABLES

2.1 Effect of Time Increment Variation and Q.„<,

Averaging on Program Output 19

3.1 Experimental Data for Single-Mirror Hot Spot 26

3.2 Experimental Conditions 36

3.3 Input Parameters for Prediction of Thermal

Response of Aluminum Target #2 48 3.4 Predicted and Measured Temperatures, °F, for

Aluminum Target #2 56-58

4.1 Experimental Conditions and Data on Hot Spot

in ADVS Bowl 63

4.2 Thermocouple Temperature Corrections 67

4.3 Input Parameters for Prediction of Temperature

Profile on an ADVS Panel 90 4.4 Predicted and Measured Temperature, °F, for

ADVS Panel 103-105

5.1 Effect of Mirror Thickness Variation 114

5.2 Effect of Mirror Density Variation 116

5.3 Effect of Thermal Conductivity Variation 117

5.4 Effect of Specific Heat Variation 119

5.5 Effect of Absorptivity Variation 120

5.6 Effect of Emissivity Variation 122

5.7 Effect of Spot Irradiation Variation 123

5.8 Effect of Spot Size Variation 124

5.9 Effect of Spot Speed Variation 125

5.10 Effect of Variation of Convection Coefficient 127

VI

LIST OF FIGURES

1.1 FMDF Solar Thermal Electric Power Plant Scheme 2

1.2 ADVS Mirror Panel 4

1.3 Hot Spot Formation on Hemispherical Collector 5

1.4 Mirror Hot Spot Concentration as a Function of

Angular Distance 7

3.1a Schematic Plan of Experimental Setup 22

3. lb Target //I 22

3.2 Cutaway Schematic of Calorimeter 23

3.3 Typical Calorimeter Trace for a Single-Mirror

Hot Spot 24

3.4a-d Comparison of Distribution Functions 27-30

3.5 Irradiation Function for a General Point 32

3.6 Schematic Plan of Experimental Setup 33

3.7a Nodes and Thermocouple Locations on Target #2 34

3.7b Thermocouple Attachment Method to Aluminum

Target #2 34

3.8 Measured Temperatures, °F, at Thermocouple Nodes 37-42

3.9 Two Wire Thermocouple Model 44

3.10 Predicted and Measured Temperatures, "F 49-54

4.1a Nodes and Thermocouple Locations for Mirror Panel .... 61

4. lb Thermocouple Attachment to Mirror Panel 61

4.2 Irradiation Distribution in ADVS Hot Spot 65

4.3 Thermocouple Temperature Corrections 69

4.4 Measured and Corrected Node Temperatures, °F 70-87

4.5 Predicted and Corrected Measured Node Temperatures,

*'F 102

4.6a-f Predicted and Measured Temperature Prof i les 107-112 v i i

NOMENCLATURE

a Half of horizontal axis of spot ellipse (ft)

2 A Area (ft )

b Half of v e r t i c a l ax is of spot e l l i p s e ( f t )

c Specif ic heat (Btu/lb-°F)

2 h Convective heat t ransfe r coeff ic ient (Btu/ft •hr»°F)

i , j Node numbers

2 I Di rec t normal in so la t ion (Btu/f t -hr)

k Thermal conduct ivi ty (Btu/ft»hr»°F)

L C h a r a c t e r i s t i c length ( f t )

Q,Q(x) & 2

Q(x,y) Local irradiation in the spot (Btu/ft -hr)

^ABS Energy absorbed due to hot spot (Btu/ft -hr)

2 Q Maximum irradiation in the spot (Btu/ft -hr)

2

Q Minimum irradiation in the spot (at the edges) (Btu/ft 'hr)

AQ Net heat gain for the finite difference element (Btu)

R Thermal resistance ("F-hr/Btu)

t Temperature ^F)

T Absolute Temperature ("R)

v Spot speed (in./min or ft/hr)

x,y Coordinates of a point Cft)

X,, Maximum x coordinate in hot spot area (ft)

Ax,Ay Incremental changes in x and y directions (ft)

z Plate thickness in finite difference (ft)

Vlll

NOMENCLATURE (continued)

Greek Symbols

2 a Thermal diffusivity (ft/hr)

e

6

P

0

T

Sub

Emissivity

Angle

Densi

'. (degrees)

ty (Ib/ft^)

Stefan Boltzmann

Time

scripts

(brs)

.2., .o„A,

a Ambient air

f Fluid

s Surface

te Thermocouple

w Wire

IX

CHAPTER 1

INTRODUCTION

1.1. Background

One of the concepts being evaluated for solar-thermal electric

power generation is the so called "Fixed Mirror Distributed Focus" con

cept. It consists of a fixed mirror in the shape of a hemispherical

bowl that concentrates sunlight along a line joining its center of curv

ature to the sun (the "solar vector axis"). The receiver (solar boiler)

is located along this axis and tracks the sun diurnally and seasonally.

The focus is distributed in the sense that the concentrated sunlight

is distributed along the solar vector axis rather than being concen

trated at a point as is the case with parabolic collectors. Figure 1.1

shows schematically the general arrangement of the bowl, the receiver,

and the character of the reflected rays that enter the bowl.

When in operation, subcooled water at approximately 1Q0°F is pumped

into the receiver at the end nearest the bowl surface, and exits as

steam at approximately lOOO^F and 1000 psi at the top of the receiver.

Flexible couplings and high-pressure flexible hoses located in the

vicinity of the pivot point are used to bring the water to and remove

the steam from the receiver.

Under contract to the United States Department of Energy, Texas

Tech University was responsible for the design, construction, operation,

and evaluation of a 65-ft diameter prototype called the Analog Design

Verification System (ADVS). The operation and evaluation of the ADVS

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by Texas Tech began at the test site in Crosbyton, Texas, in January,

1980. The bowl in this unit has a radius of curvature of 37.5 feet

and an aperature or rim diameter of 65 feet.

1.2. The Hot Spot Problem

The reflecting surface of the bowl is in actual practice made up

by arranging numerous mirror panels in a space filling pattern. In the

ADVS at Crosbyton, the bowl is composed of 430 individual panels. Each

of these panels is approximately a square of 39 in. x 39 in. , and Fig

ure 1.2 shows the salient features of their construction.

As shown in Figure 1.3, when the receiver is not in focus, part

of the sunlight is concentrated on a relatively small area on the sur

face of the bowl itself, resulting in the so called mirror hot spot.

This spot is located directly beneath the point where the receiver would

normally be located, and moves across the surface with a speed of the

order of 2 in. per minute.

The size, shape, and irradiation distribution characteristics of

the hot spot are expected to be sensitive to manufacturing accuracies,

alignment, and reflectivity characteristics of the mirrors. While

the reflectivity of the mirrors is well known, the effects of manufact

uring errors and alignment on the character of the hot spot are diffi

cult to predict with any degree of accuracy. Moreover, until the ADVS

was constructed, it was not known what mirror manufacturing and align

ment accuracies could be achieved in actual practice.

In a previous study [1], complex optical computer codes were devel-

83

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Figure 1.3

Hot Spot Formation on Hemispherical Collector

oped to predict the characteristics of the hot spot. With reasonable

assumptions for manufacturing errors these calculations predicted that

optical concentration in the hot spot could reach 100 suns in some

cases. A typical result of this study is shown in Figure 1.4 where

optical concentration at the bowl surface has been plotted as function

of the angle T|> , measured from the center of curvature for two differ

ent values of mirror reflectivity. (See Figure 1.3 for explanation of

angle }p^,)

Results of Figure 1.4 are for a bowl which is a complete hemis

phere. The bowl of the ADVS at Crosbyton is a 120-degree segment of a

sphere, and even for this size concentrations of the order 32 suns were

predicted. As will be seen in Chapter 4, actually recorded concentra

tions were nearly 20 percent higher than this. Also, it is not incon

ceivable that future, bigger FI4DF systems might have an angular aperture

of greater than 120* and higher hot spot concentrations.

It was realized that the absorption of some of this energy by the

mirrors would result in elevated temperatures and thermal gradients

within the glass. Thus the following concerns regarding the surviv

ability of mirrors become immediately apparent:

(i) All materials used in the construction of mirrors must

obviously be able to withstand the highest temperature

resulting from the hot spot.

(ii) The accompanying thermal stresses coupled with any residual

stresses from the forming process, could lead to a mechani

cal failure of the glass.

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8

Prior to the construction of the mirrors, rough estimates were

made of the maximum mirror glass temperature to be expected. These

estimates showed that a maximum temperature of approximately 250°F

could be expected, and selection of the epoxy and other materials used

in the panel fabrication was based on this estimate. Due to cost and

time schedule constraints, the stress problem was not dealt with ana

lytically, particularly since the inputs to the stress problem (hot

spot characteristics and residual stress in the mirror glass due to the

cold forming process) were not considered to be known with any degree

of accuracy. As will be seen later on, however, these estimates con

siderably underpredicted the maximum temperature in the mirrors.

Early in the performance evaluation of the ADVS at Crosbyton,

mirrors did in fact begin to develop cracks as a result of the hot spot.

On several occasions, the author and others observed these cracks

initiate within the hot spot area. However, not all mirror panels

exposed to the track of the hot spot have cracked, which indicates to

some extent the statistical nature of the failure phenomenon.

1.3. Objectives

Since it has now been determined that the hot spot problem must

be dealt with in the design of future systems employing the FMDF con

cept, the present study was undertaken. The objectives of the present

study were:

A. To formulate a computational model to predict temperature

distribution in the mirror glass under the influence of

the hot spot.

B. To determine the characteristics (the size and incident

radiative power distribution) of the hot spot. These would

obviously need to go in as inputs to any computational model

for thermal response of the mirror glass.

C. To test the computational model against some experimental

observations.

The accomplishment of these objectives will provide valuable informa

tion that can be used in the selection of materials and thermal stress

analysis for future FMDF systems.

1.4. Plan of Work

The main steps in the work described in the following pages were

accomplished in the order listed below:

(i) A finite difference computational model for predicting

thermal response of a panel was developed starting from

the first principles,

(ii) Experimental data for hot spot irradiation distribution

and resulting temperature profiles were obtained in the

laboratory using a single ADVS panel as a source mirror

to form a hot spot on an aluminum plate target. The

irradiation distribution was used as an input to the

computational model, and the predicted temperature profiles

were compared with measured ones.

(iil) Data on hot spot irradiation distribution and corresponding

temperatures in a panel installed in the ADVS bowl were

then obtained, and a comparison between predicted and

10

measured values was made as before,

(iv) Reasonably good agreement between predicted and measured

results being obtained, the computer model was then used

to study the effect of variation of some important proper

ties and other parameters.

CHAPTER 2

COMPUTATIONAL MODEL

2.1. Background

The problem of moving sources of heat was first encountered in

connection with grinding, cutting, and welding of metals. Jaeger [6]

presented an analytical treatment for infinite and semi-infinite bodies

subject to a moving uniform-strength heat source in the shape of'an

infinitely long band of finite width or a rectangle. He assumed a

steady state, i.e., he assumed that at the time of consideration, the

motion had been going on for infinite time. He obtained the tempera

ture profiles for a band or rectangle source by integrating the temp

erature profile due to an "instantaneous point source" over infinite

time and applicable space dimensions. Jaeger did not consider any

surface losses. Des Ruisseaux and Zerkle [7] extended Jaeger*s work

to include the effect of convective cooling on the entire surface of

the solid, but all the other limitations still remained.

Rosenthal [8] presented another analytical treatment for a dlmen-

sionless point source of infinite temperature. He cited experimental

evidence to assert that from the point of view of the moving source,

the problem can be treated as quasi-stationary, i.e., "if the solid

is long enough as compared to the extent of the heat source, the temp

erature distribution around the heat source soon becomes constant.

In other words an observer stationed at the point source fails to notice

any change in the temperature around him as the source moves on."

11

12

This allows him to solve a homogeneous, steady form of the conduction

equation in which the effect of the heat source is accounted for by

imposing a condition that the temperature gradient at the source loca

tion equal the source strength divided by thermal conductivity.

Dusinberre [9] presented an outline for a finite difference scheme based

on Rosenthal's analysis but the stability criteria of this model when

applied to our problem of mirror panels in the ADVS would restrict

the time increment to less than half a second and the space increment

to less than 0.035 in. Considering that each mirror panel is 39 in. x

39 in. and it takes the hot spot over 20 minutes to cross it, the above

restrictions become unacceptable.

There are several basic problems that make all of the above treat

ments unsuitable for the problem of the ADVS hot spot. First of all,

the extent of the heat source (the hot spot) in the ADVS is a 16 in.

diameter circle on a mirror panel 39 in. x 39 in. in size, whereas in all

the analytical models above, the extent of the solid is assumed to be

large compared to the size of heat source. Secondly, none of the models

discussed above includes the effect of surface losses due to radiation.

For the problem under study however, radiation was expected to be the

significant mode of heat transfer as in the worst case the convective

losses are limited to natural convection in still air. Finally, all

the schemes involve a source of uniform strength whereas the hot spot

is very far from being such a source. Therefore a somewhat different

approach was used to develop a computational model for the problem at

hand.

13

2.2. Basis of Computational Model

The basic principle utilized in obtaining a computational model

is simply an energy balance for each small element of material in an

explicit finite difference scheme. The heat input from the source is

lumped along with all the other heat transfer terms (due to conduction,

convection, radiation) for the finite element to calculate a net heat

gain by the element. This gain, when equated to the change in internal

energy of the element, gives the change in its temperature.

2.3. Assumptions and Simplifications

Besides the usual assumptions of uniform and temperature-independ

ent properties, the geometry of the mirror panels allows some further

simplifications.

The mirrors used in the ADVS at Crosbyton have a thickness of 3/32

inches and a radius of curvature of 37.5 feet. For the full-scale system

planned later on, the radius of curvature will be over 100 feet while

the thickness of glass will be of the same order as before. Therefore,

in first approximation, the following assumptions seem reasonable.

(i) The temperature gradients across thickness may be neglected.

(ii) For this geometry, the mirrors may be considered flat.

These two simplifications lead to a two-dimensional thin flat plate

model.

14

2.4. The Finite Difference Formulation and the Stability Criteria

k Consider a node (i,j) in the interior of the plate. Let t denote

k+1 temperatures at time T , and t denote temperatures at time T + AT .

AY \

1-1.3

. i,j+l

Ax

Ay •i+l.j

— V

Ax

i.J-1 Thickness = z

Then the heat input to the element in a time interval AT is

fkAyz . k «. k k v

kAxz (^k - 2 t ^ , + t ^ , , ) + hAxAy(t - t ^ ) Ay 1,3+1 i»3 i»3-J- ^ ^»3

+ aeAxAy(T^ - T ^ ) + Q 'ABS

AxAy AT

(2.1)

where T's denote absolute temperatures and Q.„c- i^ the heat absorbed

due to the hot spot and which will in general be a function of the

location of node i,j with respect to center of the hot spot. This

function is as yet to be determined. Since net heat gain must equal

the increase in internal energy of the element we have

k+1 ^ k AQ = p c AX Ay 2 i,j " i,j ^ (2.2)

15

For simplicity, if we take Ax = Ay then a combination of (2.1) and

(2.2) gives

^k+1 OAT , k ^ k k k ^»3 / 2 1-1,j 1+1,3 1,3+1 i.3-1

1.3 4aAT h . deaAT ,^ . k ^ .2 ^-k^"^^--T^(â^î,j>(â

(2.3)

, haAT . OEOAT , k,, 2 k 2. ^-TTâ^-^T- â^\ + î,j >(â + ' i,j

^ ^ A B S ^ kz

On the right hand side of Equation (2.3) all terms except the colt

efficient of t. . are inherently positive. Making the coefficient of

k t. . positive will ensure stability. This gives a stability criterion

aAT ^ 1 A 2 - A 2 , V 9 > (3.4a)

or

AT <

Ax

(3.4b)

Because of the presence of the radiation term, the right hand side of

the stability condition involves the nodal temperature. But in most

cases an estimate of maximum expected temperature is available or can

be made and this can be used to establish the stability criterion. In

the case of the present study this estimate was available from the ex

perimental measurements. Denoting this estimate by T^^^, the stability

16

condition can be wr i t t en as

AT <

Ax

It is worth noting that because of the presence of another inherently

positive term involving T. .in Equation 2.3, the stability criterion 1,3

given by Equation C2.5) errs on the side of safety.

2.5. The Computer Program-

A listing and a sample output of the computer program used for

calculation of temperature profile is included in Appendix 1« As men

tioned above, it is based on an explicit two-dimensional finite differ

ence formulation. The given plate surface is divided into a rectangular

mesh with node 1,1 located at the lower left corner of the plate. Main

steps in the program are as follows,

(i) At each time step, calculate location of the center of

hot spot,

(ii) Then for each node in succession calculate:

(a) Its distance from center of spot

(b) Heat input from hot spot. This is done in a separate

subroutine incorporating the hot spot irradiation

distribution function

(c) All conduction, convection and radiation terms based

on old temperature of the given node and its surround

ing nodes

17

(d) Net heat gain for the element, AQ

(e) New temperature using the fact that

_ net heat gain new old thermal capacity of the element

(iii) Go back to step (i) and repeat for next time increment.

Besides the parameters defining the element geometry, material

properties and heat transfer coefficients, the main input to the

program is the characteristics of the hot spot. These are:

- The maximum irradiation in the spot

- The semi-major and minor axis. (The spot is assumed to be an

ellipse)

- Speed and direction of motion of the spot with respect to posi

tive direction of x axis.

- Orientation of spot major axis with respect to the positive

direction of x axis.

The above parameters need to be determined either experimentally or

analytically before the temperature distribution can be calculated.

2.6. Certain Characteristics of the Model • - J

To calculate the heat gain of an element from the moving source

in a small time increment, two methods were tried. In one it was assumed

that the source remains stationary at its position at the beginning of

the time interval. In the other an average was taken of the heat rates

18

due to spot positions at the beginning and end of the incremental time.

This is of course equivalent to a one-step integration over the time

increment and can be further refined if necessary, l^en the results

of these two methods were compared for a typical case with reasonable

assumptions for hot spot characteristics and other properties for the

mirror panel, the differences were found to be slight (See Table 2.1)

especially a few (about 5) minutes after the start when the peak temp

erature starts levelling off. Thus, either method could be used but the

second method was retained because it is physically closer to the actual

conditions.

Any explicit finite difference scheme is expected to improve in

accuracy as the size of time increment is reduced. Results for three

increment values of 20 seconds, 10 seconds, and 5 seconds were studied

for a typical case for the mirror panel for both the methods mentioned

above and are partially listed in Table 2.1. Again it seems that the

differences are small and for the case of the mirror; an increment value

of 20 seconds will give reasonably good results. Of course due to higher

thermal diffusivity a much smaller value is necessary if a metal plate

were to be used, as was done in early checking of the model as described

in the next chapter.

19

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CHAPTER 3

EXPERIMENTAL DATA FOR A HOT SPOT ON A METAL PLATE AND COMPARISON WITH PREDICTED RESULTS

3.1. Introduction

The experimental work in this phase was done in and on the roof

of the Mechanical Engineering Laboratory at Texas Tech University. A

single spherically curved mirror panel from ADVS bowl was used as a

source mirror to generate the hot spot on a thin flat aluminum plate.

The laboratory experiment was undertaken first because

1. It is difficult to work in the bowl^ and it was desirable to

have some verification of computational model before going

there.

2. It is easier to control and monitor experimental conditions

in the laboratory.

3. The measurements can be made more accurately because better

thermal contact can be obtained between thermocouples and

plate ensuring good temperature measurement and the hot spot

can also be controlled to some extent allowing greater

accuracy in determining its characteristics.

The work was done in two parts; tne first to determine the spot

characteristics and the second to obtain resulting temperatures, A

brief description of both follows.

20

21

3.2. Hot Spot Characteristics

The experimental setup used in this part is shown schematically

in Figure 3.1a. The mirror used was similar to one of the mirror

panels in the bowl of the ADVS at Crosbyton. It had a radius of

3

curvature of 37.5 ft and an aperture of 37 in. x 36T- in. The target

was formed from 1/8-in. thick aluminum alloy (ALCOA 5083-H-343) plate

20 in. X 20 in. in size. In the center of this target was mounted a

Hycal Engg. Model C-1300-A asymptotic response, water cooled calori

meter, shown in Figure 3.2, which has an electrical output directly

proportional to heat flux incident on it. The front surface of the

target V7as painted black using Pyromark 2500 paint and marked as

shown in Figure 3.1b to enable estimation of the size of the hot spot

image, as well as its speed across the target. The procedure used to

measure irradiation distribution in the image is described below.

Sunlight was focused on the target near, but not on, the calori

meter sensor. The image so formed was found to be elliptical in shape

with major axis in the range of 4 in. to 7 in. and minor axis 3.5 in.

to 6 in. As the sun moves across the sky, the image moves across the

target. By properly adjusting the image location on the target, it

can be ensured that as the image moves across the target, the calori

meter sensor traces a diagonal through the ellipse, A strip chart

trace of the calorimeter output will then represent the irradiation

distribution along this diagonal. Several such traces were obtained,

both for central as well as off-center chords, and a typical trace

is shown in Figure 3.3. Other data recorded besides this were

22

Hycal Engg. Mode!^ 'C-1300A Calorimeter^

Cooling water

I I

Target //I 20 in. X 20 in. Al. alloy (ALCOA 5083-H-343)

^ jMirror. Radius of 'Curvature '37.5 ft Aperture 37 in. X 36.75 in.

Two-Way Switch

HP-660 Strip

Chart JSLecDxdar

Figure 3.1a

Eppley Pyrheliometer

Schematic Plan of Experimental Setup

Figure 3.1b

Target //I

Circles (Etched white on black surface) at 1 in* radial interval Innermost circle 1 in. radius

1 in. dia. calorimeter face with 1/8 in, di^, sensor in the center

23

Constantan / Disc

Cold Junction of Differential TC

Hot Junction of Differential TC

y////////A, (Section) • / / /

r / / / /

\ / / , / /

Heat Sink J T

V/a i! Y//\ /

/ / / A

-Copper Heat Sink

Cooling Channel Nega t ive Lead P o s i t i v e Lead

Figure 3.2

Cutaway Schematic of Calorimeter

24

~ '"!!rr!j.~ririi~^ii;~i~

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25

the direct normal insolation, I^ , image size (a x b) and speed across

the target, distance between mirror and target, and the ambient

temperature and wind speed. Some of these data are listed in Table 3.1

which also lists Qj^^. the maximum irradiation in the hot spot.

The irradiation distribution recorded above was non-dimensionalized

using half the length of its extent (X .y ^^^ ^^ maximum irradiation

Functional representations in the form of (1) a polynomial and

(2) an exponential or Gaussian distribution were tried. A least

squares curve fit showed the best polynomial representation to be of

the form

- ^ = 1 - 4(-^)2 + 6 ( - ^ ) ^ - 3 ( ^ ) ^ (3.1) ^MAX ' AX ^ lAX ^lAX

For an exponential fit^ a modified normal distribution of the jform

- ^ ^ ^ ^ = EXP[-h(^)^] (3.2) ^MAX MAX

was tried. Figures 3.4a to 3.4d compare both these distributions with

the measured distribution for some typical cases. Figures 3.4a to

3.4c are for cases where the measured data are for a diagonal close

to the center of the image, and Figure 3.4d is for an off-center case.

(The data set numbers in these figures refer to the data set numbers

in Table 3.1). The polynomial fit seems to break down at x/X^^^

values greater than 7, but the exponential fit seems to do reasonably

well in the entire range for all cases. It therefore seems reasonable

26

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to assume that the non-dimensionalized distribution function along

lines like AB, CD, and GH shown in Figure 3,5 has the same form. Then

the irradiation at a point like P in Figure 3,5 will be given by

Q(x»y) ^ f/ X . ^f y . ,3 ON

Q ^^X > V(x)^ ^ ^ MAX MAX

where function f has the form of Equation (3.2), and where y(x) is the

maximum ordinate corresponding to a given x, and for an ellipse of

semi-axes equal to a and b is given by (with origin located at its

center)

y(x) = bjl - ^ (3.4)

^r - = tlAX (3.5)

3.3. Thermal Response of a Metal Plate to a Hot Spot

The experimental setup for' determining the thermal respons-e of

a metal plate under the influence of a hot spot formed by a single

mirror is shown in Figure 3.6. Another aluminum target CTarget y/21

was constructed and was instrumented with thermocouples to be-monitored

by a sixteen channel Data Acquisition System (DAS). Tbe second

^ - «f 1/R in aluminum alloy (ALCOA 5083-H-343) target was also made out of i/o in. ctiumxi ^ v..

plate 20 in. x 20 in. in size. Its front surface was painted black using

Pyromark 2500 paint, and on this was etched a 1 in. x 1 in. square grid

pattern. The rear surface was sand blasted and to it were attached

15 Chromel-Alumel (type K) thermocouples in a pattern shown in Figure

3.7a. The thermocouple beads were embedded in little pits drilled

32

Distribution along CD

Distribution along AB, Qg = Q ^ ^ f (r )

Figure 3.5

Irradiation Function for a General Point

33

Target #2

DATS

ACQUISITION

SYSTEM

Esterline-Angus Model PD-2064

/

15, Type K. ThermocoL^les

Target //I

Calorimeter Hycal C-1300____j-r

Mirror

Strip Chart Recordet

Two-way Switch

Pyrheliometer

Figure 3.6

Schematic Plan of Experimental Setup

34

91 ^x

15

in •Lyj

9 8

7

6

5

1

—

—

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—

—

—

—

—

-A

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i r' t

(Typical)

10 13 15 17 19 21

Figure 3.7a

Nodes and Thermocouple Locations on Target //2

Thermocouple bead

F r o n t of t a r g e t

Omega h igh - t empera tu r e , h i g h - c o n d u c t i v i t y thermocouple cement

1/8"

Figure 3.7b

Thermocouple Attachment Method to Aluminum Target P2

35

into the back of the plate and cemented there using high-conductivity

thermocouple cement. Figure 3.7b illustrates the method. The thermo

couples were connected to the DAS which is capable of scanning all the

thermocouples practically simultaneously at specified intervals and

recording the temperature. The DAS has 16 channels, one of which acts

as reference channel and records ambient temperature. The experimental

procedure now was to

(i) Record direct normal isolation I using the pyrheliometer

(ii) Focus sunlight on target //I to obtain an irradiation

distribution or at least the peak irradiation (Qw.v)

(ill) Focus sunlight on target #2, and let the image track

across it and record temperatures indicated by the 15

thermocouples at specific time intervals

(iv) Record size of spot in step (iii), as well as its loca

tion on the target at different intervals to estimate

its speed across the target.

Table 3.2 and Figure 3.8 are records of one set of data obtained

using such a procedure. Table 3.2 is a list of experimental conditions,

while Figure 3.8 contains "node pictures" of the measured temperature

distribution on the plate at one minute time intervals beginning from

one minute after start of the experiment. In these pictures the

thermocouple nodes are placed in geometric relationship to each other

and temperatures recorded at all nodes at the given time instant are

shown. Spot position and direction of travel are also indicated in

some of these tables by an asterisk and an arrow respectively. The

36

Table 3.2

Experimental Conditions

Date: 7/5/80, Time 18:30 - 18:45 hrs.

Direct normal insolation I = 230 Btu/ft • hr

o

Maximum irradiation in-spot, Q^..^ = 79120 Btu/ft • hjc

spot size (a X b): 4.25 in. x 4.75 in.; Shape: Ellipse

Spot focused on target at 18:31 hrs.

Spot removed from target at 18:44 hrs.

Ambient temperature: lOO^F Plate temperature at Start: 120°F

Spot Travel:

Location at 18:32 hrs; x = 13.0 in., y = 6 in.

Location at 18:41 hrs; x = 8.0 in., y = 13 in.

Speed = 0.896 in per min.

Direction of travel: at an angle of 119.75" with respect to positive

direction of x axis.

Orientation;

Spot appeared to be oriented parallel to x axis.

37

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spot speed and direction indicated in Table 3.2 were estimated by

observing the spot location on the target at different times during

the experiment.

3.4. Correction for Thermocouple Errors

Whenever thermocouples are used to measure the temperature of a

body or surface, some energy is conducted away from the body by the

thermocouple wires. This results in a local depression in body or

surface temperature where thermocouples are located.Thermocouples will

then indicate this lower temperature rather than the true temperature

that would have prevailed in their absence.

3.4.1. Correction Models

Two different models available in the literature for estimation

of this error were tried. The first one by Sparrow and Hennecke [3],

[4] is based on an analysis that treats the effect of heat transfer

due to thermocouple wires as being equivalent to the creation of a

local heat sink on the surface. Based on this analysis, the results

for dimensionless temperature error ———^ are presented graphically

s f in reference [3] as a function of two other dimensionless groups.

/kA/R tanh (kAR) \ ^^^ ^s^l ^^^^ subscript s refers to the solid

^^l^s ^s surface, tc to the thermocouple, and f to the ambient fluid. The

second group above is of course the Blot number of the solid and kA

is the equivalent total conductance of the two wires of the thermo

couple given by

44

kA = (k , + k^)Aw wl w2

(3.6)

where k ^ and k _ represent conductivities of the wires and Aw is wl w2

the area of cross-section of a single wire. The two wires of the

thermocouple including their insulation are modeled by a single wire

of radius r with an outer insulation radius of r„, and the relation

ships between r. , r„ and the actual dimensions of thermocouple cross-

section are shown in Figure 3.9.

" 1

' 2

=

=

r w

h •

+

4

/ 2

^2 T

Wire 1, radius r^ conductivity

Wire 2, radius r^ conductivity k

-Outer Insulation

Inner Insulation

Figure 3.9

Two-Wire Thermocouple Model

R is then the thermal resistance of thermocouple wires given by

R =

ln(r /r ) _1 + ^_A_ (3.7)

2 Trr h 2 irk

where h is the coefficient of convection between thermocouple and

ambient fluid, k^ is thermal conductivity of thermocouple insulation

and L is the thermocouple length.

A somewhat different formulation for an estimate of thermocouple

CHAPTER 4

EXPERII-IENTAL DATA FOR ADVS BOWL AND COMPARISON

WITH PREDICTED RESULTS

4.1. General Setup

For this phase of the experiment, the calorimeter was mounted in

an aluminum panel of the same size as one of the mirror panels in the

ADVS bowl, and 15 thermocouples were attached to the rear surface of

the glass in a mirror panel, identical to one of the bowl panels.

These two panels were then used to replace two adjacent panels in the

bowl. The hot spot was obtained by moving the receiver of the ADVS

out of focus. The hot spot was allowed to track across the calorimeter

and the instrumented mirror panel, and the irradiation distribution and

temperatures were recorded as before. The thermocouples were attached

at various locations in a s in. x J$ in. grid as shown in Figure 4.1a.

The thermocouples were attached to the glass surface using the

method shown in Figure 4.1b. A 1/8-in. diameter hole was drilled

through the steel back of the mirror panel and through the 2-in. thick

paper honeycomb. The epoxy coating on the back of the glass was

removed by scraping as far as practical without destroying the mirror

coating or endangering the glass. Each thermocouple bead was coated

generously with a commercially available silicone "heat sink" compound

to enhance thermal contact between the mirror glass and the bead. To

seal the 1/8-in. diameter hole in the steel back of the panel, and to

hold the bead in contact with the mirror glass, an RTV compound was

applied to the thermocouple sleeve and steel back of the panel as

60

46

previously described experiment.

The experimental data were obtained in an open outdoor environ

ment. Each experiment had a duration of nearly half an hour. Thus

the ambient conditions like wind speed and temperature can at best be

estimated averages. The experiment was conducted in fairly calm

conditions (wind speed did at no time exceed 10 mph) and it was

thought that use of convection coefficient for natural convection

should be reasonable. Fortunately, as will be shown in Chapter 5,

for the magnitude of coefficients involved in such a case, the temper

ature distribution has only a slight dependence on the convection

coefficient. Because of this, simplified relations for free convec

tion in air taken from reference [10] were used. As can be seen,

these relationships also take into account the fact that the convective

heat transfer coefficient is a function of the temperature difference

between the surface and the ambient fluid, AT.

(i) Laminar boundary layer on vertical surface

(10^ < Gr^Pr^ < 10^)

0.25

ft"'hr«°F = o-(^)'-ei7

(ii) Laminar boundary layer, on horizontal surface

(10^ < Gr^Pr^ < 10^)

h = 0.27(^)''''-^^^^^ (3.10) ^ ^' ft'^-hr-^F

(iii) Turbulent boundary layer, on vertical surface

47

(Gr^Pr^ > 10^)

h = 0.19(AT)°-333 Btt^__ ^3^^^^

f t -hr-^F

( iv) Turbulent boundary l a y e r , on h o r i z o n t a l surface

(Gr^Pr^ > 10^)

h = 0.22(AT)^-^^-^ ^ ^ " (3.12) f t ' h r ' ^ F

In all the above equations L is the characteristic horizontal or

vertical dimension of the surface in feet, and the subscript f denotes

that the Grashof and Prandtl numbers are evaluated at film temperature,

which is taken to be average of surface temperature and ambient fluid

temperature. Since surface temperature is unknown to begin with,

estimated average values were used for calculation of Gr and Pr. For

the case of the aluminum plate, Equation (3.9) was found to be appli

cable. Estimates of radiation properties were made using mainly

references [11] and [1.2], but it must be noted that these can only be

approximate averages. Emmissivity values of .9 and .3, and absorptivity

values of .9 and .4, were estimated for the front and back surfaces of

the plate, respectively.

3.5.2. Comparison of Predicted and Experimental Results

Table 3.3 lists the main input parameters for the computer model.

Figure 3.10 again consists of "node pictures" at one minute time

intervals. Shown here are the predicted temperatures for all thermo

couple nodes and nodes adjacent to them. For comparison, the

48

Table 3.3

Input Parameters for Prediction of Thermal Response of Aluminum Target //2

Plate size: 20 in. x 20 in., thickness: 0.125 in.

Finite element size: 0.5 in. x 0.5 in.

Time increment: 2 seconds

Thermal conductivity: 67.7 Btu/ffhr«**F

Specific heat: 0.22 Btu/lb' 'F

Density: 167 Ib/ft"

Emissivity: front surface: 0.9; backsurface: 0.3

Absorptivity: front surface: 0.9; backsurface: 0.4

Ambient temperature: 100**F

Plate temperature at start of calculation: 120°F

0 25 2 Convection coefficient: 0.256(AT) * Btu/ft -hr- F at front, back,

and edges

Spot Characteristics

Size (semi-axes): 2.12 in. x 2.37 in.

Location a t s t a r t of ca lcu la t ion : x = 13.5 i n . , y = 5 in .

Speed: 0.896 i n . per min.

Direction: 119.75° w . r . t . pos, d i rec t ion of x-axis

Orientation: 0° w . r . t . pos, d i rec t ion of x-axis

2 Maximum i r r ad i a t i on = 79120 Btu/f t -hr

Minimum i r r ad ia t ion (at edge of spot) = 0

2 Direct normal insola t ion: 230 Btu/f t 'hr

49

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55

measured temperature at each thermocouple node is shown in parenthesis

besides the predicted one.

Table 3.4 summarizes part of the information shown in Figure 3.10

and indicates percentages by which the predicted temperatures at

thermocouple nodes deviate from measured values. The last column of

this table lists root-mean-square deviation at each node. The

following observations can be made from Table 3.4.

Of the 15 nodes, at all except five (13,9; 14,8; 14,10; 15,7;

16,10), the agreement between measured and predicted values is always

within ±20 percent, and in most cases the values are within ±15 per

cent. In fact, if we look at the root-mean-square deviation for each

node for the entire time, it is seen that agreement at all the nodes

except 13,9 is on the average better than 20 percent, and for 10 out

of 15 nodes it is better than 15 percent. The consistently high

deviations at nodes like 13,9 and 16,10 probably indicate a poor bond

between thermocouple beads and the plate surface. Considering the

difficulties of measuring surface temperature, and the uncertainties

involved in property values that go as input to the computational

model, the above agreement can be considered reasonably good. Where

there is disagreement, the calculated temperatures are usually higher

than the measured temperatures which one could expect since any

thermocouple inaccuracies (and some are inevitable in a situation like

this) would tend to indicate measured temperatures that are lower than

actual. Even if the computational model does slightly overpredict

temperatures, it would be an error on the safe side if in an actual

56

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59

situation this calculated temperature profile were to be used for

evaluation of thermal stresses and/or material selection.

CHAPTER 4

EXPERII-IENTAL DATA FOR ADVS BOWL AND COMPARISON

WITH PREDICTED RESULTS

4.1. General Setup

For this phase of the experiment, the calorimeter was mounted in

an aluminum panel of the same size as one of the mirror panels in the

ADVS bowl, and 15 thermocouples were attached to the rear surface of

the glass in a mirror panel, identical to one of the bowl panels.

These two panels were then used to replace two adjacent panels in the

bowl. The hot spot was obtained by moving the receiver of the ADVS

out of focus. The hot spot was allowed to track across the calorimeter

and the instrumented mirror panel, and the irradiation distribution and

temperatures were recorded as before. The thermocouples were attached

at various locations in a in. x J$ in. grid as shown in Figure 4.1a.

The thermocouples were attached to the glass surface using the

method shown in Figure 4.1b. A 1/8-in. diameter hole was drilled

through the steel back of the mirror panel and through the 2-in. thick

paper honeycomb. The epoxy coating on the back of the glass was

removed by scraping as far as practical without destroying the mirror

coating or endangering the glass. Each thermocouple bead was coated

generously with a commercially available silicone "heat sink" compound

to enhance thermal contact between the mirror glass and the bead. To

seal the 1/8-in. diameter hole in the steel back of the panel, and to

hold the bead in contact with the mirror glass, an RTV compound was

applied to the thermocouple sleeve and steel back of the panel as

60

61

79

1 46

45

44

/ o 42

41

40

3y

38

3 /

36

35

34

1

;

-

1

1 t

i

I ,

r ,,

— - 1

;

J

\ •7

-4

L

^ L

1 1

1 .....jl k"-

^ > .

^ 7

in.

f.

(Typical)

>-

i 1

^ 7

i 1

L i

'

c

I

-:

79 > ' 1

^ - • •

J M

if i n . >f ( T y p i P P l )

- 1 T 36 37 38 39 40 41 42 43 44

Figure 4.1 a

Nodes and Thermocouple Locations for Mirror Panel

79

Steel back

Paper honeyco

. Thermocouple leads _ .---l/8 in. Inconel Sleeve

Silicone "heat sink' compound

Epoxy coating on back of mirror

3/32-iii. thick glass pie bead

Figure 4.1 b

Therm.ocouple Attachment to Mirror Panel

62

shown in Figure 4.1b. The experimental procedure used was essentially

the same as used in the laboratory. But here the difficulties

anticipated with this phase of experiment were encountered in full

measure. Only one run could be made each day since the time at which

the hot spot would traverse the instrumented panels was fixed by

their position in the bowl and by the sun*s path in the sky. For

each test about one hour of completely clear sun was needed, but oyer

the period available for experiment weather permitted only one good

test though nearly a dozen attempts were made. Table 4.1 summarizes

the experimental conditions, including data on the hot spot, and

Figure 4.4 (included and discussed in Section 4.4) presents the tempera

ture data obtained as a result of the experiment which is discussed below,

4.2. Hot Spot Characteristics

Unlike the hot spot formed by a single mirror, the hot spot in

the bowl is much less intense and comparatively large in extent. As

against a concentration of nearly 345 suns, obtained with a single

mirror, the bowl hot spot was found to have a peak concentration of

less than 40 suns. Compared to a diameter of 5 to 6 inches for the

single mirror spot, the high-intensity extent of the bowl hot spot

was nearly 16 In. in diameter, and the intensity at the "edge" of the

spot does not fall to zero but has a value in the range of 5 to 6 suns.

Despite all this the intensity distribution in the spot can still

be fairly well described by an exponential function of type described

in Equation 3.2 with only a slight modification that the base for the

intensity is now Q^^,. the minimum intensity at the "edge", rather than

63

Table 4.1

Experimental Conditions and Data on Hot Spot

in ADVS Bowl

Date: 8/25/80, Time: 14:30 - 15:30 hrs.

Ambient temperature: 100**F

2 Direct normal insolation I^ = 262 Btu/ft -hr

Maximum irradiation in spot: 10024 Btu/ft' 'hr

Minimum irradiation in spot: 1559 Btu/ft^'hr

Spot size: 16 in. diameter circle

Spot speed: 2 in. per minute

Direction of motion 1 356.67** angle V7,r.t. pos. direction of x-axis

Orientation: 0" (is immaterial for a circle)

Spot edge moved on to mirror at 14:44 hrs. and spot center moved on

to mirror at 14:48 hrs.

Time zero = 14:44 hrs.

64

z e r o . With t h i s , we o b t a i n fo r t h e d i s t r i b u t i o n a long a c e n t r a l

d i a g o n a l , a f u n c t i o n of t h e form

Q. Q(x) ^MIN , 3v ? Q - = -Q + E X P [ - J , ( - ^ ) 2 ] ( 4 . 1 )

and fo r a g e n e r a l p o i n t ( x , y ) i n an e l l i p s e w i t h t h e o r i g i n a t i t s

c e n t e r and semi a x e s a and b

- f ^ = ^ + EXP[->,(3 |)2]*EXP(-^ A . ) 2 ] ( 4 . 2 )

^MAX ^MAX ^^^^

where y(x) i s given by Equation 3.4 as before. Figure 4.2 shows a

plot of the i r r a d i a t i o n d i s t r ibu t ion t race obtained experimentally as

well as a p lo t of Equation C4-1) with 0 ^ ^ equal to 61 „ . As can be MIN DN

seen there is good agreement between Equation (4,1) and experimentally

obtained results. Also shown for comparison is the theoretically

estimated distribution (reference [2]) applicable to this case, and

it is seen that the peak of this estimate is nearly 15 percent lower

than the measured value.

4.3. Corrections for Thermocouple Errors

Before presenting the measured temperatures it is appropriate

to discuss the method of correcting thermocouple readings. Calculations

for thermocouple error were made using both models described in Section

3.4 and, unlike the case of the aluminum plate, very large corrections

were predicted by both models for this case. This is essentially a

consequence of low thermal conductivity of glass. Because of low

65

• ti •H

U 0) 4J

ti 0) o S o M

MH

0) O C td ^

Dis

OI •

» *

0) M 9 bO

•H PC4

4J O P.

i n

4J

o W to >

§ c •H

(3 O

•H •P 3

.o •H M 4J, 00

•H Q

rra

dia

tio

n

66

diffusivity of glass,energy from hotter areas of the surface is

unable to reach thermocouple location fast enough, and as a result

the temperature at that location drops signficantly below that of the

surrounding surface.

Sparrow's procedure predicts lower values than that of Chapman

but only slightly so, the difference for indicated temperature of up

to 250°F being less than 7 percent. Results of both are tabulated as

Table 4.2 and are presented graphically in Figure 4.3. Corrections to

the temperatures indicated by the thermocouples attached to the mirror

were made using the formulation by Sparrow [3] as the other model [5]

involves a greater degree of approximation. As mentioned before, hov;ever

and illustrated by Table 4.2 and Figure 4.3, the two predictions are

fairly close.

4.4. Temperature Distribution

Figure 4.4 shows the temperatures recorded at various thermo

couple nodes with the corrected value enclosed in parentheses along

side. The "node pictures" are given at one minute time intervals

beginning from 4 minutes after start of experiment. Temperatures

could be obtained at only 13 of the 15 thermocouples. The highest

recorded uncorrected temperature was 255''F at node 38,38 some 14

minutes after the start of the test. The corrected measured temperature

at this point is 324°F. At this time the spot center would be in the

vicinity of node 41,39 or nearly 1.5 inches away. Thus, as expected,

the highest temperature is at a location near but slightly behind the

67

Table 4.2

Thermocouple Temperature Correct ions

(a) Based on Sparrow [ 3 ] , the c o r r e c t i o n equation i s :

t^ - 0.32t^ tg - t ~ r-^ = 0.32 OR t = "^^ ^ , , t - t s 0.68

(b) Based on Chapman [5], the equation for correction is

t - t t - 0 37t - ^ — £ £ = 0 . 3 7 OR t = '- „ : f t„ - t_ s 0.63

Measured (t ) and Corrected (t ) Temperatures, °F

for an Ambient Fluid Temperature (t^) = 110°F

•tc t^ (a)

% (b)

% difference based on t^

tc

100

95.3

94.1

-1.2

120

124.7

125.9

1.0

140

154.1

157.6

2.5

160

183.5

189.4

3.7

180

212.9

221.1

4.6

200

242.4

252.9

5.25

220

271.8

284.6

5.8

240

301.2

316.3

6.3

260

330.6

348,1

6.7

Property Values and Dimensions Used for Both Cases

Chromel Alumel Thermocouple:

Wire dia. 2*r , = 0.0125" w

Outside dimensions L^ = 0.056", L^ = 0.033" [ref. Fig. (3.9)]

Total length L = 5 ft = 60 in.

Thermal conductivities

Chromel, k =10 Btu/ft-hr'^F w

Alumel k = 2 5 B t u / f t - h r - ^ F ^2

In su l a t i on ( f i b e r g l a s s ) k^ = 0.439 Btu/ffhr-**F

used average k = 17.5 for case (b) above

68

Table 4.2 (continued)

Property values (continued)

Thermal conductivity of air k = -0156 Btu/ft'hr-^F

Thermal conductivity of mirror k = -439 Btu/ft'hr'"F s

Convective Heat Transfer Coefficients

2 Between mirror surface and a i r h = 0 Btu/ft •hr '°F

s 2

Between thermocouples and air h = 1 Btu/ft 'hr'^F

TEMPERRTURE CORRECTION (FOR TYPE K THERMOCOUPLES ON (SLPSSy

69

o

0)

u U Cd u <o a 6 <0 H

(U 4-1 O (U

O U

Indicated Temperature,

.1 L

100 120 140 160 180 200 220 240 260 280 300

Figure 4.3

Thermocouple Temperature Corrections

70

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88

center of the spot. As mentioned earlier, cracks were seen to initiate

in the hot spot area during these experiments.

4.5. Comparison of Experimental Data with Predicted Results for ADVS

Mirror Panel

4.5.1. Input Parameters to Computer Model

All the comments made in Section 3.5 about difficulties of obtain

ing good estimates of property values and other input parameters are

also applicable here. An additional problem in this case was that

exact specifications for the glass used in the mirror panels were not

available even from the manufacturer. The absorptivity of the mirror

panels is a function of the mirror reflectivity, glass thickness,

coefficient of extinction and also varies with angle of incidence. To

calculate absorptivity,an expression for reflectivity and transmis-

sivity of the mirror panels v;as derived using a ray tracing technique

similar to the one described in reference [13]. Due to the geometry

of the ADVS bowl, none of the light rays entering the hot spot have

an incidence angle of less than 60". Assuming that the silvered

surface reflects 98 percent of the light incident on it, and using

an extinction coefficient of 0.3 cm"" for glass, the absorptivity of

3/32-in. thick roirrorsis found to be in the range of 0.16 to 0.15 for

incidence angles 60** to 80°. A value of 0.16 was used in the calcu

lations for the temperature profile. For details of method of

calculation, see Appendix 2.

For the convective heat transfer coefficient, Equation(3.12)was

89

found to be applicable for this case. The above and other values

of input parameters used in the computational model are summarized

in Table 4.3.

4.5.2. Comparison

Figure 4.5 displays node temperatures at one-minute time intervals

beginning with 10 minutes after start of experiment. Predicted

temperatures at all thermocouple nodes and nodes adjacent to them are

shown, and also shown alongside in parenthesis are the corrected

measured tenperatures at the corresponding thermocouple nodes.

Table 4.4 presents part of the above information in a different

format, displaying variation of temperature with time at different

thermocouple nodes. It also lists the deviation between predicted

and measured temperatures at each node at each time, and in the last

column the root-mean-square deviation for each node is presented.

It must be pointed out here that the measured temperatures in

this case are not as reliable as in the case of laboratory experiment.

Presence of any amount of mirror epoxy coating in the bond between

glass and thermocouple would tend to make indicated temperatures lower

than what they would be otherwise. Also, if this epoxy has the effect

of modifying the effective thermal conductivity of glass surface, the

temperature corrections applied to the measured temperatures would

also be affected. Still, the corrected measured temperatures have been

used for conparison since they are the best available estimates. But

it is well to keep in mind that those could be somewhat lower than

90

Table 4.3

Input Parameters for Prediction of Temperature

Profile on an ADVS Panel

Finite element size: 0.5 in. x 0.5 in. (Mirror = 39 in. x 39 in.)

Time step = 20 seconds

k = 0.439 Btu/ffhr«*'F, c = 0.192 Btu/lb-°F, p = 169.0 Ib/ft^

Emissivity = 0.9, Absorptivity = 0.16

Ambient temperature = 110°F, Starting temperature = 120°F

0 '^'\'^ 9

Convection coefficient with air, h = 0.2 (AT) Btu/ft -hr' F

Convection coefficient at edge and back = 0.

Spot Characteristics

Size (semi-axes) 8 in. x 8 in., speed 2 in./min.

Starting location x = -8.0 in., y = 19.7 in.

Path: 356.67** v/.r.t. pos, x-axis, orientation = 0,0° 2

Maximum irradiation = 10024 Btu/ft 'hr

2 Minimum irradiation at edge of spot = 1559 Btu/ft "hr

Direct insolation = 262.0 3tu/ft 'hr

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actual temperatures.

Figure 4.5 and Table 4.4 both show reasonably good agreement

between predicted and measured temperatures. A look at the last column

of Table 4.4 gives a rough but quick idea of average deviation, and

from this it is seen that except at node 36,36, the agreement at all

nodes is better than ±25 percent. In fact, the agreement is better

than ±20 percent at 11 out of 13 nodes and better than ±15 percent at

9 out of 13 nodes. Consistently high deviation at node 36,36 is again

an indication of a malfunctioning thermocouple.

Considering the difficulties in measuring the surface temperatures,

and the approximations that are part of input parameters, these

agreements are considered good.

Another comparison of the data is made in Figure 4.6a to 4.6f. Each

figure has 3 sets of curves for a given instant in time, and the two

curves in each set show predicted and measured temperature profiles

along a vertical (x = constant) line on the mirror. Figure 4.6

further illustrates what has already been shown; that the model tends

to be conservative in that it tends to overpredict the magnitudes of the

temperatures. The additional information shown by Figure 4.6 is that

the model is very nearly correctly predicting the temperature gradients

in the mirror glass. This information will be extremely useful as

input to the problem of thermal stress analysis.

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CHAPTER 5

PARAMETRIC STUDIES

5.1. Introduction

The computer program described in Chapter 2 was used to study

the effect of the variation of different material properties and

other parameters on the temperature profiles. The basic model used

in each case was the one used in Chapter 2, Table 2.1, and in each

case only one property value was varied.

The indicators used for comparison are the highest temperature,

its location on the mirror, the spot location at that instant, and the

difference between maximum and minimum temperatures on the mirror, at

any given instant. Tables 5.1 to 5.10 list all these parameters at

10 one minute time intervals for ten different property variations.

The time in all these tables, is referenced to the instant the center of

the spot crosses the left edge of the mirror. The maximum temperature

difference in these comparisons is used as an indicator of temperature

gradients than can be expected.

5.2. Effect of Mirror Thickness Variation

Table 5.1 shows this effect for four different thicknesses, namely

1/16 in., 3/32 in., 1/8 in., and 3/16 in. This probably covers the

range of glass thickness ever likely to be considered for mirror

panels. The effect of increasing glass thickness is, of course, to

increase the thermal capacity of the mirrors, and the results in

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115

Table 5.1 exhibit the trends expected due to this effect. Thus

increasing thickness causes lower peak temperatures as well as lower

temperature gradients. Also, the location of the hottest point lags

further and further behind the hot spot location as thickness is

increased. All these effects seem very desirable, but the decision

to reduce thermal gradients, and hence thermal stresses, by increasing

mirror glass thickness will, of course, have to depend on consideration

of its effect on costs of mirror panels, supporting structures, and

the increased forming stresses that are likely. Increase in the

thickness of only those panels that lie in the path of the hot spot

may be worth considering.

5.3. Effects of Mirror Glass Density Variation.

Increase of density also has the effect of increasing mirror's

thermal capacity. The results of this comparison shown in Table 5.2

are qualitatively similar to the results of mirror thickness variation

and the same comments apply.

5.4. Effect of Thermal Conductivity Variation.

This effect is shown in Table 5.3, and it can be seen that the

effect here is very small. Even a 25 percent change in conductivity

does not cause any appreciable change in peak temperatures, their

location, or temperature gradients.

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5.5. Effect of Specific Heat Variation

As sho\m in Table 5.4, the effects are similar to those of

changing mirror thickness or density. But if a glass with the same

density but higher specific heat could be found, all the advantages

of increased thermal capacity could be realized without increasing

structure weight. Tables 5.1, 5.2, and 5.4 all demonstrate that a

50 percent increase in thermal capacity will result in lowering of

peak temperature rises as well as temperature gradients by nearly

25 percent, which is a considerable gain especially if it could be

obtained without incurring much additional expenditure.

5.6. Effect of Absorptivity Variation

The fraction of incident energy in the hot spot that is absorbed

by the mirrors is directly proportional to their absorptivity and^as

can be seen from Table 5.5, this has a very significant effect. A

25 percent reduction in absorptivity reduces both the peak temperature

rise and temperature gradients by about same factor. Lower absorptivity

can be obtained by using glass that has a low coefficient of extinction

and by improving reflectivity of the silvered surface of mirror. Thus,

if the coefficient of extinction is lowered from 0.3 cm to 0.04 cm

even with same silvered surface reflectance of 0.98, the absorptivity

goes down from nearly 0.16 to nearly 0.02, i.e., in nearly the same

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121

5.7. Effect of Emissivity Variation

This effect is not very significant as can be seen from Table 5.6

Also not much can be done to increase the emissive characteristics of

glass surfaces.

5.8. Effect of Spot Irradiation and Size Variation

These effects need to be discussed together since they are closely

related. Thus, a small spot will be more intense and vice-versa. It

can be shown that for the type of irradiation distribution seen in the

hot spot, the total power in the spot is proportional to the product

of its area and peak irradiation. Thus, for same total power, a spot 25

percent smaller in area will have a 25 percent higher peak irradiation

At the present time, sufficient data are not available to show that

for a given bowl-receiver configuration, the total power in the hot

spot is the same for different sizes. But if this is so, then reading

Tables 5.7 and 5.8 together it can be seen that a smaller, more

intense spot would lead to higher peak temperatures and temperature

gradients.

5.9. Effect of Spot Speed Variation

As is to be expected. Table 5.9 shows that a slower moving spot

causes higher peak temperatures and temperature gradients. The spot

speed is of course a function of sun's path in the sky, which varies

with time of year, and the radius of bowl.

122

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126

5.10. Effect of Variation of Convective Heat Transfer Coefficient

As has been mentioned earlier, the convection effects for the

present case are essentially limited to natural or free convection.

But it can be seen in Table 5.10 that even with convection losses

being proportional to the 1.33 power of temperature difference, the

effect is not very large. A 50 percent change in the convection

coefficient changes the peak temperature rises and temperature

gradients by less than 5 percent.

127

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r-i s f s«x

^ o r H

CHAPTER 6

CONCLUSIONS AND RECOMMENDATIONS

The main purpose of this study was to develop a computational

model for prediction of temperatures in the mirror panels of FMDF

systems. While this objective can be said to have been achieved in

fair measure, it is suggested that some more work on collection of

actual on-site experimental data be done. \^en construction of

future FMDF systems is undertaken, provisions should be made, at least

in the first few, to have some precisely manufactured instrumented

test panels with a large number of thermocouples. These panels should

be used to collect data on hot spot characteristics and temperature

profiles at different locations in the bowl at different times of the

year.

Before the construction of the present ADVS began, the highest

expected glass temperature was estimated to be 250*'F and material I

selection was based on this. In the present inyestigation. However, '

temperatures of the order of 320°F were measured, and the computational

model predicts temperatures of the order of 350''F. The problem there

fore is more severe than anticipated and demands greater attention

than hitherto given. An immediate first step that can be recommended

is to undertake a thermal stress analysis of the mirrors based on

temperature distributions predicted by the model developed here.

Also, as shown in the chapter on parametric studies, the severity

of the problem can be reduced by suitable material selection. At

128

129

least the possibility of using glass with lower absorbing character

istics is worth investigating.

LIST OF REFERENCES

1. "Interim Technical Report; Crosbyton Solar Power Project Phase 1 " Vol. 2, p. C-122.

2. Personal Communication with Dr. J.D. Reichert, Project Director, Crosbyton Solar Power Project, Texas Tech University.

3. Sparrow, E.M., "Error Estimates in Temperature Measurement," in Measurements in Heat Transfer, Ed., E.R.G. Eckert and R.J. Goldstein, 2nd Ed., McGraw-Hill, 1976.

4. Hennecke, D.K. and Sparrow, E.M., "Local Heat Sink on a Convec-tively Cooled Surface - Application to Temperature Measurement Error," Int., J. of Heat Transfer, Vol. 13, pp. 287-304,

5. Chapman, Alan J., Heat Transfer, 3rd Ed., pp. 552-554, Macmillan, 1974.

6. Jaeger, J.C., "Moving Sources of Heat and the Temperature at Sliding Contacts," Proc. of the Royal Society of New South Wales, Vol. 76, pp. 203-224, 1942.

7. Des Ruisseaux, N.R. and Zerkle, R.D., "Temperature in Semi-Infinite and Cylindrical Bodies Subjected to Moving Heat Sources and Surface Cooling," Trans. ASME, pp. 456-464, August 1970.

8. Rosenthal, D., "Theory of Moving Sources of Heat and its Application to Metal Treatments," Trans. ASME, pp. 849-866, November 1946.

9. Dusinberre, G.M., Heat Transfer Calculations by Finite Differences, Chapter 8, pp. 121-126, International Textbook Co., 1961.

10. Chapman, Alan J., Heat Transfer, 3rd Ed., p. 385, Macmillan, 1974.

11. Gubareff, G.G. et al., Thermal Radiation Properties Survey, 2nd Ed., Honeywell Research Center, 1960.

12. Edwards, D.K. and Catton, I., "Radiation Characteristics of Rough and Oxidized Metals," in Advances in Thermophysical Properties at Extreme Temperatures and Pressures, pp. 189-199, ASME, 1965.

13. Kreith, Frank and Kreider, J.F., Principles of Solar Engineering, Tab. 3.16, p. 174, Hemisphere Publishing Co., 1978.

130

APPENDIX 1

Listing and a Sample Output

for Program TEMPDIST

131

132

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APPENDIX 2

Reflectivity and Absorptivity

of a Mirror Panel

162

163

Appendix 2

R e f l e c t i v i t y and A b s o r p t i v J t v ^ p ^ a mirror pane l .

Notat ion

a = Total absorptivity

a' = Fraction of incident energy absorbed during each pass through

the plate thickness

p = Total reflectivity

p' = Fraction of incident radiation reflected by a single unsilvered

surface

T = Total transmittivity

0J. = Angle of refraction

i = Angle of incidence

K = Extinction coefficient

L = t/cos 6r

n = Index of refraction

N = Suffix denoting normal component of radiation

P = Suffix denoting parallel component of radiation

r = Fraction of incident energy reflected at silvered surface

t = Mirror plate thickness

Optical Laws

Angle of refraction 0^ is given by

• ft _ si ^ i ^^"^ "r - n (A2.1)

164

S i n g l e s u r f a c e r e f l e c t a n c e s a r e g i v e n by:

a . For 0° < i < 90°

s i n ^ ( i - 0 j . )

^'N==~r2~777 (A2.2) s m ( i + 0 - )

t a n ^ ( i - 0 j . )

P ' p = 2 7 ~ T T (A2.3) ^ t a n " ( i + 0 ^ )

b . For normal i n c i d e n c e ( i = 0")

2

(n+1)

c . For g r a z i n g i n c i d e n c e ( i = 90")

« i ^1 ( n - 1 ) " N = P p = -, ~2 <A2-'i)

P ' N = P ' P - 1 (A2.5)

For n o n - p o l a r i z e d l i g h t , add t h e two components to g i v e

P ' = | C P ' J ^ + P 'p ) (A2.6)

P = 7 C P N + Pp) (A2.7)

'' =i^''N+ V ^^^'^^ For homogeneous substances, absorptivity for a single pass is given

by

a' = 1 - e"^ (A2.9)

Total Reflectivity and Transmittivity

Since by definition

energy reflected energy incident

and

165

T- = energy transmitted energy incident

Then from Figure 2.1, for each component (normal and parallel)

P = P' + d-p') V r + (l-p')2aVp' + aVp.2(,.p,)2 ^

= p» + (l-p')Vr[l + a^rp' + a V p ' + ...]

- D' I (l-p')Vr ~ ^ , 2 , (A2.10)

1 - a rp* and

T = a(l-p')(l-r) + a\p'(l-p')(l-r) + a\^p'^(l-p') (1-r) + ...

= a(l-p')(l-r)[l + a^rp' + aVp'^ + ...]

= a(l-pM(l-r) 1 2 ,

1 - a rp' (A2.11) In all of above

a = 1 - a' (A2.12)

Calculation Procedure for Absorptivity

For each value of incidence angle i

- Calculate Q^. using equation (A2.1)

- Calculate p* & p' using equation (A2.2-A2.5)

- Calculate a' using equation (A2.9)

- Calculate p & p using equation (A2.10)

- Calculate p using equation (A2.7)

- Calculate T„ & T_, using equation (A2.11) N P

- Calculate T using equation (A2.8)

- Calculate a using

a = 1 - (p+T) (A2.13)

166

Reflected rays

a^r(l-p') a*rV(l-p')^ a^^p'^l-p')^

a\p'(l-p')(X-r) ^ " a-P')P' a-r)

Silvered surface. Reflects fraction r of incident energy, CAssumed to be same for both components.)

a = 1 - a'

Figure A2.1

Ray Trace Diagram for Mirror Panel

167

Some typ ica l r e s u l t s using the above procedure are

for n = 1.526, K = 0.3 cm'-"-, r = .98

i

60°

70°

80°

a

0.155

0.157

0.149

T

0.018

0.0175

0.016

the thermal behavior of spherically curved to …

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