Astm stp 544 1980

Contents Introduction 1 Definitions and Thermal Modelling What Property Do We Measure-^^TM Subcommittee CI 6.30 5 Measurement Philosophy of Subcommittee CI 6-30 5 Heat Transfer 7 Th

HEAT TRANSMISSION

MEASUREMENTS IN

THERMAL INSULATIONS

A symposium sponsored by ASTM Committee C-16 on Thermal and Cryogenic Insulating Materials AMERICAN SOCIETY FOR

TESTING AND MATERIALS Philadelphia, Pa., 16-17 April 1973

ASTM SPECIAL TECHNICAL PUBLICATION 544

R P Tye, symposium chairman

List price $30.75 04-544000-10

AMERICAN SOCIETY FOR TESTING AND MATERIALS

1916 Race Street, Philadelphia, Pa 19103

Library of Congress Catalog Card number: 73-87351

NOTE The Society is not responsible, as a body, for the statements and opinions advanced in this publication

Printed in Baltimore, Maryland

June, 1974 Printed in Philadelphia, Pennsylvania

Second Printing, January, 1980

Foreword

The Symposium on Contributions of Basic Heat Transmission Measurements

to the Design and Behavior of Thermal Insulation Systems was held at the

American Society for Testing and Materials Headquarters in Philadelphia, Pa.,

on 16-17 April 1973 The symposium was sponsored by ASTM Committee C-16

on Thermal and Cryogenic Insulating Materials R P Tye, Dynatech R/D

Company, presided as the symposium chairman

ASTM Publications

Thermal Insulating Covers for NPS Piping, Vessel Lagging and Dished Head

Segments ASTM Recommended Practice for Prefabrication and Field

Fabri-cation o f - C 450 adjunct (1965), $4.25, 12-304500-00

Manual on the Use of Thermocouples in Temperature Measurement, STP 470A

(1974), $17.50 04-470010-40

Contents

Introduction 1

Definitions and Thermal Modelling

What Property Do We Measure-^^TM Subcommittee CI 6.30 5

Measurement Philosophy of Subcommittee CI 6-30 5

Heat Transfer 7

The Necessity of Multiple Measurements 9

Recommendations for Future Changes 11

Establishing Steady-State Thermal Conditions in Flat Slab

Specimens-C.J.Shirtliffe 13

Basic Problem 15

Common Factors in Models 15

Model Descriptions and Solutions 15

Simplification of Solutions 16

Truncation of Solutions 17

Inversion of the Solutions 22

Comparison of Settling Times 23

Accuracy of Equations 24

Conclusions 24

Mechanisms of Heat Transfer in Permeable Insulation and Their

Investi-gation in a Special Guarded Hot Plate—C G Bankvall 34

Measurement of Heat Transfer 35

The Guarded Hot Plate 35

Heat Transfer Mechanisms in Fibrous Insulation 40

The Natural Convective Heat Transfer 43

Summary 48

Water Vapor Diffusion and Frost in Porous Materials—^ A uracher 49

Diffusion in Porous Frost-Containing Materials 50

Diffusion on Simple, Frost-Containing Pore Models 51

Diffusion in Frost-Containing Sphere Packings 61

Discussion 66

Conclusion 67

Radiative Contribution to the Thermal Conductivity of Fibrous

Insula-tions-i? M F Linford, R J Schmitt, and T A Hughes 68

Predicting Spacecraft Multilayer Insulation Performance from Heat

Trans-fer Measurements—I D Stimpson and W A Hagemeyer 85

System-Level MLI Blanket Results 87

Types of Calorimeters Used 89

The JPL Test Program 90

Conclusions 92

Techniques

Design Criteria for Guarded Hot Plate Apparatus-F De Ponte and P Di

Filippo 97

The Guarded Hot Plate 99

The Cold Plate 109

Conclusions 116

Suitable Steady-State Methods for Measurement of Effective Thermal

Conductivity in Rigid Insulations-IV T Engelke 118

Comparative Rod Apparatus 120

Radial Inflow Apparatus 126

New Development in Design of Equipment for Measuring Thermal

Conductivity and Heat Flow-is Brendeng and P E Frivik 147

Nomenclature 148

Steady-State Measurements 149

Test Results 156

Transient State Measurements 164

Robinson Line-Heat-Source Guarded Hot Plate Apparatus-M H Hahn,

H E Robinson, and D R Flynn 167

Mathematical Analysis of Line-Heat-Source Guarded Hot Plate 169

Design of Proposed Apparatus 185

Conclusion 191

Calibrated Hot Box: An Effective Means for Measuring Thermal

Conduc-tance in Large Wall Sections—X R Mumaw 193

Description of Test Apparatus 194

Construction of Test Apparatus 195

Hot Side Construction Details 196

Cold Side Construction Details 198

Specimen Frame Construction 199

Air Infiltration Test Capability 199

Obtaining Proper Test Results-The Data System 200

Hot Side Chamber Calibration 201

Testing Procedure 202

Discussion of Testing Results 203

Conclusions and Recommendations 211

Results and Applications

Improving the Thermal Performance of the Ordinary Concrete Block—

H N Knickle and Edgar Ducharme 215

Procedure 216

Experimental Work 218

Economic Analysis 220

Conclusions 221

Some Recent Experimental Data on Glass Fiber Insulating Materials and

Their Use for a Reliable Design of Insulations at Low

Temperatures-D Fournier and S Klarsfeld 223

Theoretical Data 224

Measurements Facilities 227

Materials Investigated and Test Procedure 230

Experimental Results 231

Some Applications of Both Theoretical Results and Experimental Data

to Design Actual Insulations at Low Temperatures 235

Thermal Conductivity of Evacuated Glass Beads: Line Source

Measure-ments in a Large Volume Bead Bed Between 225 and 300 K—M G

Langseth, F E Ruccia, and A E Wechsler 256

Nomenclature 257

Bead Tank Conductivity Measurements Using a Line Source 259

Heat Flow Probe and Line Source Probe Comparisons 270

Conclusions 273

High Performance Thermal Insulation for an Implantable Artificial

Heart-Z) R Stoner, R C Svedberg, J W H Chi, and T Vojnovich 275

Thermal Test Apparatus 277

Fabrication of Insulation Systems 280

Experimental Results 281

Discussion 285

Study of Thermophysical Properties of Constructional Materials in a

Temperature Range from 10 to 400 Yi—A V Luikov, A G

Shash-kov, L L Vasiliev, S A Tanaeva, Yu P BolshaShash-kov, and L S

Domorod 290

Nomenclature 290

Experimental Procedure 292

Analysis and Measurement of the Heat Transmission of Multi-Component

Insulation Panels for Thermal Protection of Cryogenic Liquid

Storage Vessels-/ G Bourne and R P Tye 297

Materials and Systems Evaluated 299

Introduction

During the past decade there has been an ever-increasing utilization of

thermal insulation materials and systems Furthermore, the impact of the world

energy crisis has fostered additional expansion in the prediction and use of

thermal insulation which will not diminish in the coming years Applications

have become more exotic, conditions of temperature and environment more

extreme, and the consequent insulation systems and their means of evaluation

are now more sophisticated As a result, new methods for measurement of

thermal performance must be developed and existing methods improved in order

to keep abreast of this continued use of thermal insulation

Insulating materials are generally inhomogeneous, because heat transfer in

them can take place through a number of separate and interacting mechanisms

By means of more reliable measurements of heat transmission, we become more

aware of these mechanisms and how the performance of certain materials and

systems depend less upon solid conduction than upon other processes such as

radiation, convection, and mass transfer With more confidence in the results, we

can better understand heat transmission behavior and, consequently, develop

better and more economical materials and systems

In the United States, ASTM Committee C-16 on Thermal and Cryogenic

Insulating Materials is responsible for the promulgation of standards concerning

thermal and cryogenic insulation materials, systems, and test methods Within

this committee, Subcommittee CI6.30 on Thermal Conductance is directly

responsible for test methods relating to heat transmission characteristics The

subcommittee has kept abreast of developments in the field by continuously

revising and upgrading the relevant standard test methods under their

jurisdic-tion and by communicating, where possible, with their counterparts on similar

national committees In addition, they have foreseen future requirements by

developing new or extending existing standards to fulfill the potential needs The

purpose of these test methods is to uphold the realistic philosophy by evaluating

an insulation under operating conditions rather than by measuring a

physically-defined property which may have no meaning for these materials and systems

Seven years ago, Committee C-16 sponsored a similar technical meeting where

the topic related specifically to heat transmission measurements at cryogenic

temperatures We have arrived at a point where significant developments in the

evaluation of heat transmission have taken place; therefore, the committee

1

2 HEAT TRANSMISSION MEASUREMENTS

decided that a further international meeting among workers in this field was

justified so current technologies and ideas could be discussed and subsequently

applied to future worldwide activities The goal of this symposium was to

provide a forum which would extend our horizons, cover all types of insulations

at all operating temperatures, and illustrate that better measurement and

performance characteristics can lead to further improvements in materials and

systems

The international group of papers in this volume covers representative

subjects in the areas of fundamental studies of heat transmission processes,

experimental techniques, both large and small scale, and the measurement and

analysis of particular materials or systems for specific applications The wide

variety of subjects discussed, especially the Subcommittee C 16.30 position paper

which outlines their future philosophy, should stimulate further activities The

international representation of authors produces a further cross-fertilization of

ideas which ultimately promotes greater international cooperation One

particu-lar area concerns that of the well characterized reference materials of low

thermal conductivity being made available in the future The Appendix briefly

outlines how Subcommittee C16.30 has started the work to solve the problem

In conclusion, I wish to thank all of the authors for their efforts in making

the symposium a success The paper by 0 B Tsevetkov, "Experimental

Determinations of the Thermal Conductivity of Fluids by Coaxial-Cylinder

Apparatus," was received too late for inclusion in this pubhcation and will

appear in the July 1974 issue of the Journal of Testing and Evaluation I trust

that we have discovered new areas of concentration resulting in more numerous

future meetings

Manager, Testing Services, Dynatech R/D Co., Cambridge, Mass

symposium chairman

Definitions and Thermal Modelling

W h a t Property Do W e Measure

REFERENCE: ASTM Subcommittee C16.30, "What Property Do We

Measure?" Heat Transmission Measurements in Thermal Insulations, ASTM

STP 544, American Society for Testing and Materials, 1974, pp 5-12

ABSTRACT: ASTM Subcommittee C16.30 on Thermal Properties of

Com-mittee C-16 on Thermal and Cryogenic Insulating Materials discusses in this

paper its philosophy of the measurement of heat transfer properties of

insulations and its concern about the present manner in which certain heat

transfer properties are used or possibly misused, in describing thermal

insulation performance Recommendations for changes in the standard test

methods under the jurisdiction of the subcommittee to avoid the present

problems are given

KEY WORDS: heat transfer, thermal insulation, thermal conductivity,

thermodynamic properties, thermal resistance, heat transmission

Within Subcommittee CI6.30 on Thermal Properties of ASTM Committee

C-16 (Thermal and Cryogenic Insulating Materials) there has been concern

regarding the proper application of terminology used to describe the heat

transfer properties of insulating materials, especially regarding the interpretation

of measurements made using the standard test methods under their jurisdiction

It is the intent of the subcommittee to set forth in this paper some technical

background and terminology relating to heat transfer in insulations, and also to

make some recommendations for changes in the manner in which terminology is

used and test results are reported in the standard test methods involved

Measurement Philosophy of Subcommittee CI6.30

A thermal insulation is typically used to limit the amount of heat transferred

between its two surfaces when a temperature difference (possibly large) is

maintained between those two surfaces The actual details of heat transfer

within the interior of the insulation involves a complicated combination of solid

conduction, gaseous conduction, and sometimes convection and radiation

Possibly, even a mass transfer of some sort may be involved From a practical

point of view, one does not usually look at the effect of each of these

mechanisms in detail, but rather one empirically determines for a particular

material the total amount of heat that flows from one of its surfaces to the other

when a given temperature is maintained on each surface In almost all cases of

practical importance, heat flow and temperature measurements are carried out

under steady-state conditions over some range of temperature differences and

thicknesses between the two surfaces

The standard test methods under the jurisdiction of Subcommittee CI6.30

related to heat transfer properties of insulating materials attempt to specify test

conditions near those of "real world" conditions Measurements are usually

made in a test situation which establishes a one-dimensional heat flow in a

slab-like specimen with parallel faces, or in a cylindrical geometry used for pipe

insulations, or in apparatus designed for testing large, multicomponent built-up

systems, such as wall sections Test situations provided by the various methods

are aimed at providing a measure of the performance of the insulating material

or system under conditions that duplicate or approximate as closely as possible

its actual enduse thermal environment Then, the empirically developed

knowledge of heat flows as they are related to temperature differences and

actual specimen geometries describe the thermal performance of a material in a

useful manner for the designer who must select an insulation with the proper

thermal characteristics for a given application

The goal of Subcommittee CI6.30 is that all of the standard test methods

under its jurisdiction directly measure applicable characteristic properties of

insulation At present, the existing standard test methods have been or are

presently being made appropriate and sound as prescriptions of measurement of

heat transfer properties, but their present use of terminology could be

misleading under some circumstances The methods have, in fact, been

misapplied in practice in some instances, and have been interpreted as being

unnecessarily limiting in others

It is worth noting the contrast between our measurement philosophy and the

philosophy of standards developers in some other countries Ours is aimed at

determining the thermal performance of a material by measurement at or near

the conditions of actual use, while their approach is aimed at measuring inherent

or "theoretical" properties of the materials Their test methods employ thermal

conditions which are best for determining "theoretical properties," but which

may be far from typical thermal conditions encountered in applications Actual

use thermal performance is then calculated from the measured theoretical

property An example of such a property would be thermal conductivity which

is measured by a test using very small temperature differences These are

necessary to provide a close experimental approximation to a true theoretical

temperature gradient, (the infinitesimal limit of the ratio Ar/thickness) and a

reasonably well defined mean temperature While our approach may not yield a

thermal conductivity value as close to the "theoretical" value as that obtained

from a test method such as the one just described, we get a better idea of how

the material will actually perform in a given application, because that

performance is measured directly

ASTM SUBCOMMITTEE C16.30 ON MEASUREMENT OF PROPERTIES 7

Heat Transfer

We will now examine a set of thermal transfer properties and also discuss the

differences between properties attributable to a specimen taken from a sample

of a material and properties attributable to the material as a whole

Heat transfer in insulations, even though it is a very complex process, stUl

often obeys to a good approximation Fourier's law of conduction, which relates

the heat flux at a point in a body to the temperature gradient at that point Heat

flux is the time rate of thermal energy transfer per unit area through some

imaginary surface and the temperature gradient is the same as was just

mentioned

Definition of Thermal Conductance

From ASTM Definitions of Terms Relating to Thermal Insulating Materials

(C 168-67), we take the following definition:

thermal conductance, C-a property of a particular body or assembly measured

by the ratio of steady state heat flux in common between two definite surfaces to the difference between the average temperatures of the two

surfaces

NOTE 1—The average temperature of a surface is one that adequately

approximates that obtained by integrating the temperatures over the entire

surface

NOTE 2—The value of the thermal conductance is peculiar to the specific

geometric configuration of the particular body or assembly

NOTE 3—Terms ending in "-ance" generally designate properties of a

particular object and thus may depend not only on its component elements,

but also on its size, shape, or surface conditions Strictly speaking, the terms

"conductance" and "resistance" apply to an object having a particular

and individual total or whole area of cross section through which heat flows

However, in general practice and usage it is convenient to refer to unit area

conductance where the unit area is considered to be representative of the

whole area of cross section "Conductance (or Resistance) per Unit Area" could be used, but in ordinary usage, this is shortened to "Conductance" or

"Resistance" with the unit area concept understood To avoid confusion, in

those cases where whole or total area conductance or resistance is meant, it

should be so designated, or simply called "areal conductance (or resistance)."

Examples might include bodies with concentric bounding surfaces (pipe insulation), nonparallel bounding surfaces (wedge-shaped bodies), or bodies

that are not homogeneous in a direction perpendicular to the temperature

gradient

We can therefore write the following expression involving the thermal

conductance of a specimen, C

q = C-^T (1)

and similarly for the areal thermal conductance of a specimen, C'

Q = C' • /^T (2)

where AT" is the temperature difference between the surfaces of the specimen,

and q and Q are the heat flux and total heat flow through the specimen,

respectively

Definition of Thermal Resistance

Again from ASTM C 168-67 we have the following definition:

thermal resistance, /?-a property of a particular body or assembly measured by

the ratio of the difference between the average temperatures of two

surfaces to the steady state heat flux in common through them

There are two additional notes which are part of this definition but not germane

in this discussion and will not be included here

From the definition we can write the following expression involving the

thermal resistance of a specimen, i?

AT = R-q (3)

and similarly for the areal thermal resistance R' of a specimen

AT = R'-Q (4)

where AT, q, and Q have the same definitions as before

Definition of Thermal Conductivity

We take one more definition from ASTM C 168-67 for thermal conductivity:

thermal conductivity, k-a property of a homogeneous body measured by the

ratio of steady state heat flux (time rate of heat flow per unit area) to the

temperature gradient (temperature difference per unit length of heat flow

path) in the direction perpendicular to the area

NOTE 1-A body is considered homogeneous when the value of k is

unaffected by variations in specimen thickness or area within the range

normally used

NOTE 2—A thermal conductivity value must be identified with respect to:

(a) Mean temperature, since it varies with temperature

(b) Direction and orientation of heat flow, since some materials are not

isotropic with respect to thermal conductivity

ASTM SUBCOMMITTEE C16.30 ON MEASUREMENT OF PROPERTIES 9

Note the important difference, in the meaning of the term temperature

gradient as defined here, and as defined earlier in this paper

From this definition, we can write two final equations, one for the thermal

conductivity, k, of a material

q = k- AT/D (5)

and the thermal resistivity of a material, r

AT=r-D-q (6)

where in addition to the quantities as defined before,/) is the distance between

the same two isothermal surfaces which determine AT

Heat Transfer Properties as Proportionality Constants

The equations as written show the properties as proportionality constants

This assumption of proportionality is, in fact, basic to the definitions as written,

but it is really only an assumption In every real case, it is assumed that there is

some range of temperature, specimen area and thickness, and even time for

which these equations are valid The actual extent of the range of validity in

each of these parameters can only be determined by a series of measurements

involving different mean temperatures, temperature differences, areas,

thick-nesses, and at different times

Foam plastic insulations with a cellular structure containing a gas other than

air are one example of a material whose properties can change over time In such

a material the thermal properties will vary as the gases diffuse in or out of the

pores of the material, or interact with the solid part of the material or both,

changing its properties Another example is the apparent thickness dependence

of thermal conductivity which may show up in a series of measurements In

insulations this is most likely due to radiation or the coupling between radiation

and conduction, or both, and it becomes evident either in very thin specimens or

at high temperatures or both

The Necessity of Multiple Measurements

In the case where there are variations of a property from one specimen to

another, verified by measurement, then one can only speak of the property as

characterizing that particular specimen or, at best, the sample and not as a

property characterizing the type of material from which the specimen or the

sample was made The only way a given property can be said to be characteristic

of a material and not only of a given specimen is through a series of

measurements of the desired property on a number of specimens from several

samples of the material For example, if the heat flux through only a small unit

portion of the total area of the material and corresponding local surface

temperatures are measured, then only the thermal conductance property of that

measured part of the material is known with certainty Asserting that the heat

transfer performance is the same for parts of the material different from the

actually measured area involves the assumption that the material possess a

uniformity of the thermal conductance property over its total area, or that the

average value over the tested area applies to other areas as well There is,

therefore, an inherent danger in ascribing a thermal conductance value which is

obtained from a single measurement on a single specimen to all materials which

are the same type as the one from which the specimen originally came

Highly important to the concept of thermal conductivity is the operational

definition of "homogeneous" which requires the measured value of the thermal

conductivity to remain constant with variations of specimen thickness or area,

within the range of normal use In addition, the thermal conductivity must be

independent, to within some tolerance, of the temperature gradient in the

material (this condition depends on the definition of gradient used) It may be

generally dependent on the mean temperature of the material, although the

usefulness of the thermal conductivity concept becomes questionable when the

dependence on mean temperature becomes large

These last statements lead to the following key point: there are dangers

involved in ascribing the property of thermal conductivity to a material as a

result of a single measurement obtained according to a standard test method on a

single specimen regardless of how good the test method is All methods prescribe

the measurement of heat fluxes and surface temperature by some means, for

some physical configuration of a specimen, from which a resultant thermal

conductivity is calculated There is no requirement in these methods, however,

that this calculated property must be shown to be independent of area,

thickness, and temperature gradient Therefore, there is no assurance without

such a verification that the calculated property is in fact a true thermal

conductivity

It should also be noted that strictly speaking, one should not use the term

conductivity to describe a property of a material which involves modes of heat

transfer other than conduction, even though the definitions in ASTM C 168-67

do not rule out such usage It is implicitly understood that in such a case the

measurement would only be valid for a particular range of thicknesses and

temperature differences—a point all too often forgotten

A similar situation with regard to usage exists for the other terms that have

been mentioned There is no requirement in the standard test methods that a

calculated thermal resistivity, conductance, or resistance must in fact be shown

to be characteristic of the material as a whole, rather than just the given

specimen that was measured

We see, therefore, that multiple measurements are necessary for two different

ASTM SUBCOMMITTEE C16.30 ON MEASUREMENT OF PROPERTIES 11

reasons, either or both of which can be important in a given situation First,

when performed under the same test conditions they are necessary to determine

whether or not the measured heat transfer property of the individual specimens

is also ascribable to the material from which they were fabricated Second,

multiple measurements on the specimen at different test conditions are

necessary to determine whether or not a "theoretical value" of thermal

conductivity (or resistivity) does exist (that is, is the property actually

independent of the geometry and thermal gradient of the specimen)

Recommendations for Future Changes

In order to avoid possible problems involved in the use of a calculated

property derived from temperature and heat flow measurements in the present

standard test methods, the preceding considerations lead to some

recommenda-tions by the subcommittee for future changes Before stating them, however, it

must be stressed that the changes are basically ones of employing the correct

terminology for the thermal properties actually being measured, and are not

intended to imply that operational changes be made in the test methods

themselves, or that there are errors in presently reported values The test

methods always measure the thermal resistance or conductance of a specimen

for a particular set of test conditions and, in fact, these can be sufficient to

characterize insulating materials well enough for their selection for the proper

end use; thermal conductivity or resistivity are usually not necessary If a

comparison is desired between different specimens, then it is necessary that

property measurements be made for the same thickness and temperature

differences and that these be typical of the applications considered for the

materials

We come finally to the following recommendations: First, the subcommittee

recommends that in those standard test methods where heat flux and surface

temperatures are measured, the thermal property calculated should be thermal

conductance or resistance of the specimen A standard set or sets of test

conditions involving thickness, temperature difference, and boundary plate

emittance should be specified by the test method Sufficient sampling should be

done to determine whether the conductance or resistance of the specimen is

representative of the whole In those cases where additional measurements on

additional specimens confirm that the calculated property is independent of the

sample from which the specimen is selected and also the area of the specimen,

then it may be called the thermal conductance or resistance of the material The

test methods should specify the minimum number of specimens required to

identify the properties of a material In general, as the test area of a specimen

becomes larger, fewer specimens would require testing

Second, only in those cases where additional measurements are made and

they confirm that the calculated ratio of heat flux to temperature gradient is

independent of the specimen, area, thickness, and temperature gradient to

within prescribed and expHcitly stated Hmits (which must be within the

uncertainty of the test), should the properties of thermal conductivity and

resistivity be ascribed to the material This will usually be possible for those

materials which transfer heat only by the processes of gaseous and solid

conduction When this condition is satisfied by a material where, in addition to

the conduction heat transfer mode, a significant amount of heat is also

transferred by convection or radiation, or a coupled process which involves all

three modes, then the term "apparent thermal conductivity (resistivity)" or

"ef-fective thermal conductivity (resistivity)" should be used In such cases the

con-ditions of measurement should also be specified along with the effective thermal

conductivity (resistivity) value Care must be exercised in such a way that this

property only be applied where conditions are reasonably similar to those of the

test

In conclusion the subcommittee would like to emphasize that the changes

proposed do not mean to imply that past results obtained in accordance with

standard test methods are invalid or that the test methods are in error Rather,

the way results are presently reported, and particularly the terminology used,

could lead to the incorrect application and interpretation of heat transfer

property data These potential problems could be avoided by uniformly using

correct thermal property terminology and reporting test results in a more

complete fashion

Acknowledgment

The Chairman of ASTM Subcommittee CI6.30 thanks the members whose

contributions made this paper a reality Explicit acknowledgment is made to

W L Carroll (National Bureau of Standards, Washington, D C.) and C J

Shirtliffe (National Research Council, Ottawa, Canada) for compiling the ideas

and writing and editing the manuscript

C J Shirtliffe1

Establishing Steady-State Thermal

Conditions in Flat Slab Specimens

REFERENCE: Shirtliffe, C J., "Establishing Steady-State Thermal Conditions

in Flat Slab Specimens," Heat Transmission Measurements in Thermal

Insulations, ASTM STP 544, American Society for Testing and Materials,

1974, pp 13-33

ABSTRACT: The thermal response of flat slab specimens subjected to four

sets of thermal boundary conditions is examined The conditions are typical of

thermophysical test apparatus, and the four sets of conditions are similar

Initial temperatures are uniform and the cold face temperature is stepped to

the required value There are four different hot face conditions: (1) constant

temperature, (2) constant heat flow, (3) zero heat flow followed by constant

temperature, and (4) zero heat flow followed by constant heat flow

Solutions are given for the heat transfer problems, and the equations are

truncated, then inverted, to yield simple, approximate expressions for

response time The precision of the approximate equations is shown to be

adequate for prediction purposes; the four sets of conditions are ranked in

order of increasing response time; and the occurrence of optimum situations is

noted

KEY WORDS: thermal insulation, heat transmission, heat transfer, heat flow

meters, flat slab, temperature, analytic functions, one-dimensional flow,

reaction time, thermodynamic properties, thermophysical properties, time

dependence, transient heat flow

The objectives of this present study are twofold: to determine theoretically

the fastest way to achieve steady-state heat flow conditions in a flat specimen,

using simple, realistic operating conditions; and to determine the theoretical

settling time for specimens subjected to conditions similar to those in the four

most common operating modes of guarded hot plate and heat meter apparatus

The initial temperature is always assumed to be uniform, and only the

temperature level is allowed to vary Although the ultimate in initial

condition-1 Research officer, Building Services Section, National Research Council of Canada,

Ottawa, Canada K1A 0R6

ing is to establish the correct Hnear temperature gradient, it is seldom practicable

to precondition samples so precisely The surface at the lower temperature was

always assumed to be at constant temperature since this is typical of most

apparatus where one surface is a temperature controlled heat sink

The temperature of a body is known to approach steady-state conditions

asymptotically Thus, the time to reach steady state depends on what criteria are

used to determine when steady state is reached Temperature and heat flow may

approach steady conditions at different rates The most useful criterion

is the error in the parameter to be determined, in this case, thermal

con-ductivity This was the approach used by Shirtliffe and Orr^ in an earlier

paper, when they considered two types of boundary conditions Results were

given for a limited range of initial temperatures and are now extended and

presented in a more convenient form The solutions for two other sets of

boundary conditions are included

The following approach was used to obtain simplified equations for settling

time: (1) the series form of the solution to the heat conduction problem was

derived, (2) the solution was substituted into the equation for thermal

conductivity yielding an expression for the deviation from the correct value, and

(3) the series in the expression were truncated and then inverted This yielded

relatively simple expressions for settling time in terms of the parameters of the

problem and the error in measuring thermal conductivity

Optimum starting conditions were identified for three cases As the simple

expressions for settling time did not hold where these optimum conditions

existed, special expressions were derived for these cases Results were compared

and the boundary conditions ranked according to the speed of reaching steady

state

Simplified equations derived for the four idealized cases are useful in

predicting lower limits for settling time in actual apparatus A solution of a

model for the full apparatus is necessary for an upper limit Lower limits can be

used in establishing guidelines in standard test methods It should be noted that

no existing thermal property test method attempts to establish test duration on

a theoretical basis

The equations may also be used in establishing the design of an apparatus

The advantages and disadvantages of each case are made apparent by this means

In addition, the equations can serve as a guide in experimental determination of

settling time for different specimens in a particular apparatus They are useful in

establishing optimum conditioning temperatures and optimum turn-on

tempera-ture, which can be particularly useful in quality control applications where rapid

measurements are required The equations can be used, as well, in determining

which of the many configurations of heat meter apparatus is fastest

^ Shirtliffe, C J and Orr, H W., Proceedings, Seventh Conference on Thermal

Conductivity, National Bureau of Standards, Special Publication 302, 1967, pp 229-240

SHIRTLIFFE ON STEADY-STATE THERMAL CONDITIONS 15

Basic Problem

Four heat transfer problems have been solved and the solutions rearranged to

yield an estimate of thermal settUng time Each problem represents a model of

either a typical thermophysical test apparatus or a different mode of operation

for the same apparatus

The four cases are described in mathematical terms in the Appendix, where

the solutions are given The models are of a one-dimensional, flat slab specimen

All parameters are nondimensionalized; the space variable is divided by the

specimen thickness, and time is multiplied by the thermal diffusivity and divided

by the thickness squared The temperature scale is referenced to the cold surface

temperature and scaled to produce a unit temperature difference across the

specimen Heat flow is divided by the steady-state heat flow

The range of nondimensional variables is as follows:

space: Ar = 0 to 1

time: T = 0 to «>

temperature: d = - 5 0 to +50

= 0 at cold surface for steady state

= 1 at hot surface for steady state

heat flow: d O/bX = -«> to +«>

= 1 at steady state, for A^ = 0 to 1

Common Factors in Models

There are a number of factors common for all cases; specimens are always

preconditioned to a constant temperature either above or below the cold plate

temperature The nondimensional initial temperature, W, can vary widely, but a

range of —50 to +50 covers most cases of practical interest

Step changes in surface temperature are assumed in the analysis In a real

apparatus, the cold face of the specimen is often placed against a liquid heat

exchanger that can either extract or supply heat It cannot, however, supply the

infinite heat flows required to produce step changes in the surface temperature

The heat capacity of the cooling system helps to provide high heat flows and

give a reasonable approximation of step changes

Heat flow is always assumed to be one dimensional in the models Modern

apparatus usually have automatic temperature control of one or more guards, so

that this assumption is reasonable

Model Descriptions and Solutions

Case (7) Constant Temperature-The first model is of an apparatus with heat

exchanger plates on both the hot and cold surfaces, which are at constant

temperature after time zero

Case (2) Constant Heat Flow-lhe second model is of an apparatus with a

liquid heat exchanger on the cold surface and an electric heater on the hot

surface The other face of the heater is perfectly insulated, preventing heat loss

This model is typical of one half of the guarded hot plate apparatus except that

in the model the heater has no thermal capacitance The power to the heater is

turned on at T = 0 and held constant thereafter

Case (J) Zero Heat Flow Followed by Constant Tempera tare-The third and

fourth models are of the same apparatus, but control of the power to heater is

different and the initial specimen temperature, W, is always above the

steady-state hot surface temperature, that is, W^ I In the third model the

power to the heater on the hot surface is turned on only when the hot surface of

the specimen has cooled to the correct value, 6(1, r*) = 1 From this point on,

power is controlled to maintain the hot surface temperature constant The

power on, or starting time, is termed r* and is determined from the solution for

the case where heat flux is zero at A" = 1 The equation must be solved by trial

and error

The solution for the rest of the problem, that is, when r is greater than T* and

the hot surface is at a constant temperature, was found by superposition of

simple solutions

Case (4) Zero Heat Flow Followed by Constant Heat Flow-Jhe fourth

model is similar to the third, that is, heat flow across the hot surface is zero until

that surface cools to a prescribed temperature, S, when constant power is

suppHed to the heater Temperature S is any value between the cold surface

temperature, 0, and the initial temperature, W

Time r*, as for Case (3), is found by solving essentially the same equation by

trial and error The solution to the heat transfer problem for T > T* is derived

using the final temperature distribution of the initial phase as the initial

distribution for the second phase

The solutions are all in the form of three terms since they are derived by

superposition The first term is the steady-state solution, normally one in a

nondimensionalized problem; the second is the transient solution for the

boundary value problem, that is, the effect of boundary conditions; the third is

the transient solution for the initial value problem, namely, the effect of the

initial temperature distribution The last two terms are in the form of infinite

series in order to satisfy the conditions at all positions and times The two series

simplify considerably at hot surface, X = 1 At large times many of the terms are

negligible The solutions are listed in the Appendix, and are given in the section

on inversion of the solutions in simplified form

Simplincation of Solutions

The solutions given in the Appendix are not in a particularly convenient

form They express temperature or heat flow in terms of infinite series involving

X, T, W, and S The techniques previously used by Shirtliffe and Orr^ may be

used to define a settling time, which is defined as the time required to reach

SHIRTLIFFE ON STEADY-STATE THERMAL CONDITIONS 17

conditions where measurement of the thermal conductivity of the specimens can

be made to a desired degree of precision

The error due to neglect of the remaining transient effect is termed e and is

expressed as a fractional error in all formulas The results are presented in such a

way that e can be specified by the user according to the application, and the

required settling time can then be determined It is assumed that in most

thermophysical applications an error value of less than 10 percent will be

desired

The derivations of the equations for e in terms of temperature and heat flow

are given in the Appendix The equations are essentially identical to those given

by Shirtliffe and Orr.^

At Z = 1, for temperature specified

d0_

for heat flux specified,

e / ( l + e ) = l - 0 ( l , r ) (lb)

The solutions for each heat transfer problem may be substituted in the

appropriate expression for error, and the resulting solution plotted for a range of

each variable, W, r, and S The solution could then be "graphically inverted" to

yield a solution for settling time T This is basically the tack taken by Shirtliffe

and Orr^ for the first and second cases Figures 1 through 5 are such solutions

for all cases, plotted in a more convenient manner

Approximate solutions have been derived for all the models, but in equation

form rather than graphical form These will be shown to have adequate precision

for the intended purpose They will also be shown to be more useful in

comparing the setthng times for the four models

Truncation of Solutions

The solutions for the heat transfer problems given in the Appendix are all in

infinite series form rather than in the equally common, error-function form The

series form was selected because it is a simpler form for truncation and for

estimating the truncation error

The Appendix gives the truncated form of the solution for e The series are

truncated after the second order term(s) Truncation error is small as long as e is

small, since a small e only occurs at a relatively large time, T These equations

can be further simplified because they still contain r* and C (n) which are in

series form

Lines I ndicate Approx Equations

FIG I-Settling time for constant temperature mode

I W-2/ir I

FIG 2-Settling time for constant heat flow mode

SHIRTLIFFE ON STEADY-STATE THERMAL CONDITIONS 19

3.0

2.5

T I I I I I I~P1 Points Indicate Correct Values Lines Indicate Simple Equations

The series solution from which r* was obtained had also been truncated

Figure 6 indicates that for W greater than 1.2, a single term is adequate The

simphfied equation is

The equation for the error, e, in the third case contains constants C(l) and

C(2) in the first and second terms, respectively Terms that contain these

constants are the effects of the initial temperature distribution on e The series

for the constants C{n) have been truncated at the first term The truncated series

from which r* was found was then used in those truncated series to obtain an

expression for C(l) and C(2) containing only W and S These multiple

truncations are shown to be justified by the results given in the section on

accuracy of equations and by Figs 3 to 5

The equations for C(l) and C(2) are:

C(\)=-TTA/(12W) (3a)

SHIRTLIFFE ON STEADY-STATE THERMAL CONDITIONS 21

FIG S—Settling time for slab-all cases, e = 1 percent and T*

The four equations, after truncation of the series and substitution of 1, 2, and

3,for e are:

Case 1, constant temperature

e = (2-4H/) exp (-TT^ T) + 2 exp i-4n^ T) (4)

Case 2, constant heat flow

e/(l + e) = (4/;:) ((2/7r)-IV) exp (-77^r/4)

+ (4/(37r)) ((2/(37r)) + W) exp (-97r^ r/4) (5)

Case 3, zero heat flow followed by constant temperature

e = -(2/3) (4H'/7r)'* exp (-7r^T)+ (62/15) (4Hy7r)* * exp (-4rr^T) (6)

Case 4, zero heat flow followed by constant heat flow

e/(l + e) = (4/7r) (W/S) [((S/TT^) - S ) exp (-^27/4)

+ (1/3) ((8/(37:^)) (4W/iTrS)f + S) exp (-97rV/4)] (7)

The first terms in Eqs 4, 5, and 7 equal zero when W= 1/2, W = 2/n, and

S = 8/n^, respectively The second terms then express the relation between the

variables The second terms are negligible in comparison with the first, except

very near these optimum points

Inversion of the Solutions

Equations 4 through 7 are truncated at the first term, then inverted to

produce equations for T in terms of the other variables These do not hold at or

near the points where the first terms of the original series equal zero When the

first terms are zero, the second terms can be used to give equations for r In the

relatively small regions where the two terms are of comparable size a graphic

solution must be used

The complete set of equations follows:

A t Z = l ,

Case 1, constant temperature

r «» (I/TT^) In [i2^W)le], W¥^ 1/2, e¥=0 (8) and

T'«(l/(47r^))ln[2/e],W/=l/2,e^tO (8a) Case 2, constant heat flow

r«(4/(97r^))ln [2(1 +e)/e] +(4/7r^)ln H/+0.0434

W / > 1 2 , e ¥ = 0 o r - l , 5 = 8/7r^ (11a)

SHIRTLIFFE ON STEADY-STATE THERMAL CONDITIONS 23

The 1 + e could be set equal to 1 in Eqs 9,9a, 11, and 1 la if desired, because

it contributes less than 0.04 to T for e = 0.1 and less for smaller errors The e in

Eq 3 is normally negative so that the sign cancels the minus sign The sign of e

and the sign of the other terms in the square brackets are consistent and always

lead to a positive value Equation 4 should reduce to Eq 2 when S=W, but this

is not quite the case because of the approximation used for T*

Comparison of Settling Times

The equations for settling time, though not exact, reveal the most important

features of the response: the occurrence of optimum conditions, and relative

values of response time away from these optimum points

The optimum points occur very near the singularities, that is, the undefined

points in the logarithmic function For constant temperature this occurs at an

initial temperature W = 0.5, and for constant heat flow at fV = 2/7r««0.637

These have been confirmed by calculations using the full series

The third and fourth cases have no optimum initial temperature, W There is

an optimum S at S/n^ for the fourth case Calculations with the full series

indicate that it is actually between 0.811 and 0.812

The occurrence and significance of optimums for W has already been pointed

out^ and discussed Estimated values were 0.52 and 0.64 If specimens are

conditioned to the optimum temperatures, the response time can be reduced

very substantially

The occurrence of the optimum value for S is also significant This is the

optimum value to which the hot surface should be allowed to cool before power

to the heater is turned on As with the second case, constant heat flow, the

correct power must be applied The difficulty in achieving this and the

consequence of making an incorrect estimate have been discussed.*

Away from the optimum Ws, the constant temperature case can be seen to

have a settling time approximately one fourth that of the constant heat flow

case This is evident from the l/n^ and 4/;:* constant multipliers in Eqs 8 and 9

and from Figs 6 and 7 The settling of the third case is longer than that for

constant temperature but shorter than that for constant heat flux (Figs 6 and

7) The comparison can only be made for H'> 1.2 because of the approximation

for T*

Comparison of the fourth case is more difficult It is clearly faster than the

constant heat flux case and slower than the constant temperature case It is

significantly slower than the third case when 5 = 1.0, but faster when S has a

value close to 0.811 (Figs 6 and 7)

In order of increasing settling times the cases are:

(1) constant temperature,

(3) zero heat flow followed by constant temperature,

(4) zero heat flow followed by constant heat flow, and

(2) constant heat flow

FIG 1-Settling time for slab-all cases, e = O.J percent

In Case (4), when S is between 0.788 and 0.835 and e = 1 percent, the order

is changed to (1), (4), (3), (2) The same order holds when e = 0.1 percent and S

is in an even narrower range centered about 0.811

At the optimum values of W and S the order is (1), (2), (4), (3), and this

holds for a very small region on either side of the optimum values

Accuracy of Equations

The accuracy of the approximate equations for predicting settling time is

outlined in Table 1 The equations were determined by extensive calculations

using the full series Their accuracy depends on range of e, error due to the

transient, and nearness to optimum values These conditions are also listed in

Table 1

The accuracy of the equations is more than adequate for estimating the

settling time of the test specimens because the thermal properties are seldom

known to better than 10 percent The simplified equations are not so precise for

the last two cases The more precise equations are therefore listed The error in

the fit of either set of simplified equations to the data decreases rapidly as the

distance from the optimum S increases

Conclusions

Factors governing the settling time of flat slab specimens are thermal

properties, thickness, initial temperature, maximum allowable error, and, in

SHIRTLIFFE ON STEADY-STATE THERMAL CONDITIONS 25

TABLE 1 Sumimry of errors in estimating settling time with approximate equations

Case

Constant temperature

Constant heat flow

Zero heat flow followed by

10 1.0 0.1

10

1 0.1

20

10

1 0.1

10

1 0.1

F o r e Less Than, %

transient, is less than the value given, and when the initial temperatures, W, or the turnon

temperature, S, is as indicated

certain cases, the hot surface temperature at which the boundary conditions are

appHed

Approximate equations for the thermal response of a specimen can be used to

determine settling time Accuracy is adequate for prediction purposes

The four hot surface conditions have been ranlced in order of increasing

settling time For other than near the special points they are:

(1) constant temperature,

(2) zero heat flow followed by constant temperature,

(3) zero heat flow followed by constant heat flow, and

(4) constant heat flow

Optimum operating conditions have been determined For constant

tempera-ture and constant heat flow conditions the optimum initial temperatempera-tures, on a

scale of 0 to 1 from cold to hot surface, are 0.5 and 2/?: For zero heat flow

followed by constant heat flow the optimum hot surface temperature at which

to supply the heat flux is 0.811 to 0.812

The ranking of the hot surface conditions changes when the responses for

optimum conditions are compared Referring to the numbers above, these are

(1), (2), (4), and (3) Figures 5,6, and 7 give a fuller comparison

The results are in a form that can be utilized to improve the design and

operation of thermal test equipment They are also in the correct form for

incorporation in standard test methods for thermal transference properties

Acknowledgments

The author gratefully acknowledges the programming assistance of G

Arsenault of the Division of Building Research, National Research Council of

Canada, who prepared the programs, and the useful criticism and guidance of

D G Stephenson of the same organization

This paper is a contribution from the Division of Building Research, National

Research Council of Canada, and is published with the approval of the Director

X = space variable running from 0 to Z,, and

t = time, a = the thermal diffusivity

The nondimensional heat flux is written as

q dd

qi dX

where

q = heat flux per unit area,

qi = -X [T(L,°°) - T(0,'=°)]/L, steady-state heat flux

SHIRTLIFFE ON STEADY-STATE THERMAL CONDITIONS 27

X = thermal conductivity, and

°° = indicates the steady-state value

(1) constant temperature: 0( 1, T) = 1, (16)

' " ( 1 7 , (2) constant heat flux: - ,,

aX (3) with W>\, zero heat flux till 0( 1, T*) = 1, then

constant temperature 0( 1, T) = 1, r > T* , and (18)

(4) with W>\, zero heat flux till d{\,T*) = S, 0 <S <W then

= l,T>T* (19) l,r

In every case the steady-state temperature difference and steady-state heat

flux are equal to unity

constant heat flux:

r-r-oX

Solutions to Basic Problems

Solutions have been determined using equations derived from basic models

and the "superposition principle" given by Churchill.^ The solutions are in terms

of the variable not specified at Z = 1, that is, in terms of heat flux for the

constant temperature boundary condition and in terms of temperature for the

constant heat flux boundary condition The solutions are given in general form,

then f or X = 1

General form of solutions

(a) Constant temperature, BC

bx X.T f^l = 1 + 2 y " ( - 1 ) " c o s ( 4 „ Z ) e x p ( - ^ ^ T)

+ 4^^ ^ cos (B„ X) exp ( - 5 ^ r) (20) n=l

Churchill, R V., Operational Mathematics, 2nd ed., McGraw-Hill, New York, 1958

where

A„ = mr By, = {2n~\)-n, exp is the exponential function

{b) Constant heat flux, BC

£ ; ( - l ) " - l ( l / 5 „ ) e x p ( - 5 ^ ^ * 4 ) = 5 / 4 H ' (26)

«=1 Note—If T* = 0, 14 reduces to 10

SHIRTLIFFE ON STEADY-STATE THERMAL CONDITIONS 29

Solutions for X = 1

(a) Constant temperature, BC

3 ^

dX 1,T «=1 n=\ = 1 + 2 X 1 e x p ( - ^ ^ T ) + 4H/ £ ] ( - l ) " e x p ( - f i ^ T ) (27) (b) Constant heat flux, BC

0 0

0(1,r) = 1 - X ) mi/Bl) - 4 H / ( ( - l ) " - V 5 « ) ] e x p ( - f i ^ r / 4 ) (28)

«=1 (c) Zero heat flux until 9( 1, r*) = 1 followed by constant temperature, r > T*

ax l,r n=l = l + £ [ 2 + ( 3 2 W 7 r ^ ) ^ ^ C ( n ) ] e x p ( - ^ ^ ( r - T * ) ) (29)

where

C(/i) = XL ( - 1 ) ' " " 1 {(TijBmMiBmh? ' 4«^) e x p ( - f i ^ r * / 4 ) (30)

m=l and

Bf„ = ( 2 w - l ) 7 r

7* is found by solving

0 0

^ ( - l ) « ^ l ( l / 5 „ ) e x p ( - 5 ^ r * / 4 ) = ^ / ( 4 H ' ) (31) n=l

where B„ = (In — 1 )?T and A = \

(d) Zero heat flux until 0 ( 1 , T*) = S, followed by constant heat flux, T> T*

Calculation of Errors

Errors in the determination of thermal conductivity, if measurements are

made before steady-state conditions have been attained, are derived as follows

Constant Temperature, BC, and Case (3)

Error at time T and position X is expressed as

Constant Heat Flux, BC, and Case {4)

Error at time r and position X is expressed as

[ e ( X , ° ° ) - ^ ( 0 , ° ° ) ]

^ " ^ ^ = ' = [eiX,r) - 6iOj)]

X 9(X,T)

SHIRTLIFFE ON STEADY-STATE THERMAL CONDITIONS 31

Constant Heat Flux BC

Truncating Eq 17 at two terms, rearranging, and substituting in Eq 24, gives

-=^ = 0(1,7) - 1 « -(4/7:) ((2/7r)-H/) exp ( - 7 7 ^ / 4 )

1+e

+ (4/(37r^)) ((2/(37r)) + W) exp(-97r^T/4) (38)

then e is found using Eq 25

Zero Heat Flow Until 9 ( 1 , T*) = 1, Followed by Constant Temperature, for T

N o t e - I f T* = 0 then C( 1) = —7r/8 and, when substituted into the first term of

Eq 28, reduces to the first term of Eq 26

Zero Heat Flow Until 9(1, T*) = S, Followed by Constant Heat Flux, for T>T*

Truncating Eq 21 at two terms, rearranging, and substituting in Eq 24 gives

^ = 0(1,T) - 1 = - [(8/7r^) exp(7rV*/4) e+1

- (4H'/7r)] exp(-7r^7/4) - [(8/9?:^) exp(97rV*/4) + 4H'/(37r)l exp(-977^7/4) (40)

e is found using Eq 2 5

Simplification of 7* and C

The Series in Eq 20 from Which T* is Found for Eqs 39 and 40 Can Be

Truncated at One Term

e x p ( - 7 r V * / 4 ) * 7 r y l / ( 4 H ' ) (41)

or, solving for 7*

T * « ( 4 / 7 r 2 ) ] n ( 4 j ^ / ( ^ ^ ) ) (42)

where In is the natural logarithm function

Note—The equation holds for If > 1.2 (see Fig 6)

Truncating Eq 19, the Series for C(n), at One Term

C ( l ) « - ( l / 3 ) e x p ( - 7 r V * / 4 ) (43)

and substituting Eq 30 into Eq 32, yields

C(l) ^-nAKnW) (43a)

where A is either 1 or S, as the case requires

Similarly,

C ( 2 ) « » ( l / 1 5 ) e x p ( - 7 r ^ T * / 4 ) (44) and substituting Eq 30 into Eq 3 3 , yields

inverted to give

T « ( l / ( 4 7 r ^ ) ) l n ( 2 / e ) , H'= 1/2,6=5^0 (45a)

Note—An optimum exists at W = 1/2, where T is reduced by a factor of

approximately 4

Constant Heat Flow, BC, X = 1

Neglecting the second term in Eq 27, inverting gives

Substituting for r* and C from Eqs 31 and 33 with A = 1 gives a simpler

expression which holds for W^ 1.2

SHIRTLIFFE ON STEADY-STATE THERMAL CONDITIONS 33

T «(1/rr^) In [(4^/77)" (-2/(3e))],H'# 0,6^^0 (48)

Note—No optimum W exists

Zero Heat Flow Until 6(1, r*) = S, Followed by Constant Heat Flux, X = 1, for

Neglecting the second term in Eq 26 and inverting

r«(4/7r^)ln [(4/7r)((l +e)/e) ([(2/?:) exp ( T r V * ^ ] - ^ ) ] (49)

Substituting for T* and C from Eqs 31 and 33 with A =S gives a simpler

expression which holds for PV > 1.2

T «(4/7:^) In [(4/7:) {W/S) ((8/7r^)-5) (1 + e)/e] (50) Note—No optimum value for W exists, but there is an optimum S at

S = S/TT^ » 0.811 Calculations with the full series confirm a value between

«»(4/(97r^)) In [ 2.(1 + e)/e ] + (4/7:^) In If + 0.043 (51a)

Note—At optimum S, T is reduced by a factor of approximately 9

Equation 39 no longer reduced to Eq 35 when S = W, due to the error in

finding T* near W = 1.0