NATIONAL BUREAU OF STANDARDS REPORT

8663

MATHEMATICAL' ANALYSIS OF TEMPERATURE RISE

IN THE

HEAT CONDUCTION REGION OF AN UNDERGROUND SHELTER

by

F. J. Jerome Drapeau and Neil V. Baggette

Prepared for

Office of Civil Defense

Department of the Army

under

Contract No. OCD -OS -62-44?

Subtask 1211A

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NATIONAL BUREAU OF STANDARDS REPORT

NBS PROJECT NBS REPORT

42103-12-4210436 April 1965 8663

MATHEMATICAL ANALYSIS OF TEMPERATURE RISE

IN THE

HEAT CONDUCTION REGION OF AN

UNDERGROUND SHELTER

by

F. J„ Jerome Drapeau and Neil V. Baggette

Mechanical Systems Section

Building Research Division

Prepared for

Office of Civil Defense

Department of the Army

under

Contract No. 0CD-0S-62-44

Sub task 121 1A

This report has been reviewed in the Office of Civil

Defense and approved for issuance. Approval does not

signify that the contents necessarily reflect the views

and policies of the Office of Civil Defense.

NATIONAL BUREAU OF STA

for use within the Government. B and review. For this reason, the whole or in part, is not authoriz Bureau of Standards, Washingtor

the Report has been specifically p

IMPORTANT NOTICE

Approved for public release by the

Director of the National Institute of

Standards and Technology (NIST)

on October 9, 2015.

accounting documents intended

ubjected to additional evaluation

isting of this Report, either in Office of the Director, National he Government agency for which

Dies for its own use.

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

NATIONAL BUREAU OF STANDARDS REPORT

MBS PROJECT NBS REPORT

42103-12-4210436 April 1965 8663

MATHEMATICAL ANALYSIS OF TEMPERATURE RISE

IN THE

HEAT CONDUCTION REGION OF AN

UNDERGROUND SHELTER

by

F. J Jerome Drapeau and Neil V. Baggette

Mechanical Systems Section

Building Research Division

Prepared for

Office of Divil Defense

Department of the Army

under

Contract No. OCD-OS-62-44

Subtask 121LA

This report has been reviewed in the Office of Civil

Defense and approved for publication. Approval does

not signify that the contents necessarily reflect the

views and policies of the Office of Civil Defense.

IMPORTANT NOTICE NATIONAL BUREAU OF STANDARDS REPORTS are usually preliminary or progress accounting documents intended

for use within the Government. Before material in the reports is formally published it is subjected to additional evaluation and review. For this reason, the publication, reprinting, reproduction, or open-literature listing of this Report, either in whole or in part, is not authorized unless permission is obtained in writing from the Office of the Director, National Bureau of Standards, Washington 25, D.C. Such permission is not needed, however, by the Government agency for which

the Report has been specifically prepared if that agency wishes to reproduce additional copies for its own use.

y. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

TABLE OF CONTENTS

PAGE

ABSTRACT 1

1. Introduction 2

1.1 Objectives 2

1.2 Scope 3

1.3 Types of Solutions 5

2. Summary of Problems 6

3 . Evaluation of Experimental Parameter Data 9

3.1 Average Initial Temperature 10

3.2 Properties of Concrete 11

3.3 Properties of Earth 12

3.4 Heat Flux 14

b . Solution I 15

One-dimensional, homogeneous, semi-infinite

region* uniform initial temperature, constant

heat flux.

4.1 Statement of Problem 15

4.2 Mathematical Summary 15

^.3 Assumptions 16

4.4 Results 17

4.5 Discussion 17

5. Solution II " 26

uniform initial temperature, constant heat flux.

5

*1 Statement of Problem 26

5.2 Mathematical Summary 26

5.3 Assumptions 27

5.4 Results 29

5.5 Discussion 29

i
TABLE OF CONTENTS (Cont.)

6 . Solution III

Infinite cylindrical cavity, homogeneous,

infinite region, uniform initial temperature,

constant heat flux.

6.1 Statement of Problem

6.2 Mathematical Summary

6.3 Convergence

6.4 Assumptions

6.5 Results

6.6 Discussion

?. Solution IV

One-dimensional, composite, semi-infinite

region, uniform initial temperature, constant

heat flux.

7-1 Statement of Problem

7.2

Mathematical Summary

7-3 Convergence

7.4 Assumptions

7.5 Results

7-6 Discussion

8 . Solution V

One-dimensional, composite, semi-infinite

region, prescribed initial temperature,

constant heat flux.

8.1 Statement of Problem

8.2 Mathematical Summary

8.3 Convergence

8.4 Assumptions

8.5 Results

8.6 Discussion

PAGE

32

32

32

33

34

35

35'

38

38

38

40

40

41

48

49

49

49

53

55

58

62

11
TABLE OF CONTENTS (Cont.)

PAGE

9 . Solution VI 63

One-dimensional, composite, semi-infinite

region, prescribed initial temperature,

constant heat flux.

9.1 Statement of Problem 63

9.2 Mathematical Summary 63

9.3 Convergence 69

9*9 Assumptions 69

9-5 Results 72

9.6 Discussion 72

10 . General Discussion 75

10.1 Geometry 75

10.2 Composition of Region 79

10.3 Initial Temperature 82

10.4 Thermal Properties 85

10.5 Shelter Irregularities 87

10.6 Shelter Environment 88

11 . Conclusions 90

APPENDIX A Initial Temperature Distribution 92

APPENDIX B heat Flux 96

APPENDIX C Numerical Example 1-02 APPENDIX D References 105

iii
LIST OF FIGURES

FIGURE

4.4.1

4.4.2

4.4.3

4.4.4

4.4.5

4.4.6

5.4.1

5.4.2

6.5.1

6.5.2

7.5.1

TITLE

Solution I - NBS Test 3

Predicted Surface Temperatures

for North Wall Region

Solution I - NBS Test 3

Predicted Surface Temperatures

for West Wall Region

Solution I - NBS Test 4

Predicted Surface Temperatures

for North Wall Region

Solution I - NBS Test 4

Predicted Surface Temperatures

for West Wall Region

Solution I - NBS Test 5

Predicted Surface Temperatures

for North Wall Region

Solution I - NBS Test 5

Predicted Surface Temperatures

for West Wall Region

Solution II - NBS Test 3

Predicted Surface Temperatures

for Equivalent Spherical Radius a Solution II - NBS Test 5

Predicted Surface Temperatures

for Equivalent Spherical Radius a Solution III - NBS Test 3

Predicted Surface Temperatures

for Equivalent Cylindrical Radius a Solution III - NBS Test 5

Predicted Surface Temperatures

for Equivalent Cylindrical Radius a Solution IV - NBS Test 3

Predicted Surface Temperatures

for North Wall Region

PAGE

18

19

20

21

22

23

30

31

36

37

42
LIST OP FIGURES (Cont.)

FIGURE

7.5.2

7.5.3

7 . 5.4

8.4.1

8.5.1

8.5.2

8.5.3

9.4.1

9.5.1

9.5.2

TITLE

Solution IV - NBS Test 3

Predicted Surface Temperatures

for West Wall Region

Solution IV - NBS Test 5

Predicted Surface Temperatures

for North Wall Region

Solution IV - NBS Test 5

Predicted Surface Temperatures

for West Wall Region

Solution V - NBS Test 3

Initial Temperature Distribution

in Heat Conduction Region

Solution V - NBS T st 3

Predicted Surface T mperatures

for North Wall Region, Various

Concrete and Earth Thermal Properties

Solution V - NBS Test 3

Predicted Surface Temperatures for

West Wall Region, Various Concrete

and Earth Thermal Properties

Solution V - NBS Test 5

Predicted Surface Temperatures

for West Wall Region

Solution VI - NBS Test 3

Initial Temperature Distribution

in Heat Conduction Region

Solution VI - NBS Test 3

Predicted Surface T mperatures for

North Wall Region, Various Concrete

and Earth Thermal Properties

Solution VI - NBS Test 3

Predicted Surface Temperatures

for West Wall Region, Various

Concrete and Earth Thermal Properties

PAGE

43

44

45

56

59

60

61

70

73

74

v
LIST OF FIGURES (Cont.)

FIGURE TITLE

10 . 1.1

10 . 1.2

10.1.3

10 . 2.1

10.3.1

10.3.2

A. 1

A.

2

A. 3

B. l

B . 2

Comparison of Solutions I, II, & III

NBS Test 3

- Predicted Surface

Temperatures, Earth Thermal Properties

Comparison of Solutions I, II, & III

NBS Test 4 - Predicted Surface

Temperatures, Earth Thermal Properties

Comparison of Solutions I, II, & III

NBS Test 5

- Predicted Surface

Temperatures, Earth Thermal Properties

Solution I vs. Solution IV

NBS Test 3 - Predicted Surface

Temperatures for West Wall Region

Comparison of Solutions IV, V, & VI

NBS Test 3

- Predicted Surface

Temperatures for North Wall Region

Comparison of Solutions IV, V, & VI

NBS Test 3

- Predicted Surface

Temperatures for West Wall Region

NBS Test 3. Initial Temperature

Distribution in Heat-Conduction Region

on Sept. 13, 1959

NBS Test 4. Initial Temperature

Distribution in Heat Conduction Region

on Oct. 6 , 1959

NBS Test 5. Initial Temperature

Distribution in Heat Conduction Region

on March 25, I960

Solution IA. NBS Test 3

- Predicted

Surface Temperatures for North Wall

Region K = 0.75 = 0.026

Solution IA. NBS Test 3

- Predicted

Surface Temperatures for We 3 t Wall

Region K = O .75 = 0.026

vi

PAGE

76

77

78

80

83

84

93

94

95

100

101
LIST OF TABLES

TABLE TITLE PAGE

1.1.1 Schedule of NBS Test Conditions

for 0CDM-NBS-6Q-1 2

3-1.1 Initial Temperatures 11

4.4.1 Interior Surface Temperatures

Predicted by Solution I, K = 1.15 24

7-5.1 Interior Surface Temperatures

Predicted by Solution IV

Ki = 1.15, K 3 = 0.75 46

7.5.2 Interior Surface Temperatures

Predicted by Solution IV

Kx = 1.4-0, Ks = 1.04 47

8.4.0 Values of Curve-Fitting Constants

for Solution V 57

9-4.0 Values of Curve-Fitting Constants

for Solution VI 71

B. 1 Cumulative Average Values of Heat

Flux, F0 99

vii
LIST OF SYMBOLS

a curve-fitting constant; also equivalent

radius, feet

b, m curve-fitting constants

c curve-fitting constant; also specific heat,

Btu/lb- °F

F

o heat flux, Btu/hr-ft s

k /H

\ a2

K thermal conductivity, Btu/hr-ft-°F

Kq

( qr) ,

K

T (qa ) modified Bessel functions of the second kind

f thickness of concrete, feet

n summation index

P Laplace transform variable

\f a

r radial distance, feet

t time, hours

T Fourier time, dimensionless

u temperature in the medium, °F

u transformed temperature

u

o average initial temperature in concrete

and earth, °F

X linear distance, feet

viii
LIST OF SYMBOLS (Cont.)

a thermal diffusivity, ft 3 /hr

P a „ i

a + 1

e temperature rise in the medium, °F

e transformed temperature rise

p density, lb/ft3

a K

Al

erfc (x) complementary error function

i

nerfc (x) integral error function of order n

exp(x) e x

D(x)

2 »X

e“x

J e dX , Dawson's integral

o

the subscripts 1 and 2 denote concrete

and earth, respectively.

ix

MATHEMATICAL ANALYSIS OF TEMPERATURE RISE

IN THE HEAT CONDUCTION REGION

OF AN UNDERGROUND SHELTER

by

F. J. Jerome Drapeau and Neil V. Baggette

ABSTRACT

Six analytic solutions of the heat conduction equation

to predict interior surface temperatures in an underground

concrete shelter are presented In this report. Four solutions employ one-dimensional, semi-infinite geometry and

are applied to the wall and floor heat conduction regions

of the small NBS test shelter. These solutions are not

applicable to the roof region. Two solutions use spherical

and cylindrical geometry, respectively, and are applied to

the entire shelter. These solutions are not applicable to

individual surfaces. The one-dimensional solutions were

used primarily to examine the importance of the initial

temperature distribution, the composite nature of the heat

c nduction regions, and in conjunction with the other two

solutions, the geometry of the model. Surface temperature

predictions for selected surfaces, together with the

observed performance, are presented for the one-dimensional

solutions in tabular and/or graphical form, covering a range

of assumed thermal properties. Similar results are presented for the spherical and cylindrical solutions in

graphical form, for two sets of thermal properties and two

values of equivalent shelter radius. The results show that

the one-dimensional solutions give high predictions, the

spherical solution low predictions, and the cylindrical

solution generally good agreement with observed performance

of the NBS test shelter.

- 1 -
- 2 -

1. INTRODUCTION

1.1 Objectives

In February 1959, a study was begun to determine the

thermal characteristics of a family-size underground concrete shelter with simulated occupancy.

The first phase of this study was experimental, with

the objective of obtaining performance data on the environmental factors of temperature, humidity, ventilation, and

heat exchange in an underground installation of this type

during fourteen-day periods of occupancy In summer and

winter. This objective was accomplished and the results

[l] 1 reported in March 1961.

The schedule of test conditions from which the NBS

performance data were obtained is presented in the following table. This report will compare analytical predictions

of shelter surface temperatures with experimental data from

NBS Tests 3? and 5 only.

TABLE 1.1.1

SCHEDULE OF NBS TEST CONDITIONS [l]

Ventilation Air

NBS Test

Number

Date at

Start

Duration

of Test

Internal

Heat Input

Flow

Rate

Avg. Dry

Bulb Temp.

Dew Point

Temp

.

mo-da-yr days Btu/hr CFM 9 F

9 F

1 8-13-59 7 110 42 85 69

2 8-28-59 7 2500 0 — --

3 9-13-59 14 2500 42 85 69

4 10-6-59 14 2500 18 85 69

5 3-25-60 14 2500 18 35 33

Figures In brackets indicate literature references at

the end of this report.

i
-3-

The experimental phase indicated that theoretical

approaches were desirable in order to provide methods for

predicting environmental conditions in a shelter of any

size or shape under various ambient conditions, and thus

to reduce the need for exhaustive testing under an empirical approach. Tests in a few representative shelters

could then be used to verify the applicability of those

methods. One such approach was to develop mathematical

equations which would predict air temperature and relative

humidity inside shelters characterized by the following

known parameters:

a) initial earth temperatures

b) thermal properties of earth and wall

materials

c) construction detail

d) shape and size

e) weather conditions

f) ventilation rate

g) internal heat sources

This objective would also include a comparison of the

theoretically predicted results with the performance data

obtained in the experimental phase, and with any other

data on shelters, when available.

When these equations were adequately substantiated by

experiment for sufficient ranges of the above-listed parameters. they would be used to develop a design manual,

including extrapolated ranges.

1.2 Scope

The scope of this report was confined to the presentation of mathematical solutions to the problem of transient

heat conduction in a certain class of shelter models, and

a comparison of predicted shelter temperatures with observed temperatures in the NBS test series [1]. The class
- 4-

of models considered were those characterized by a single

space variable (linear distance* spherical radius, or

cylindrical radius)* and a constant heat flux applied at

the interior boundary of the heat conduction region.

This report* then* deals with only part of the whole

problem--the heat conduction region* and all solutions

begin at the interior boundary. They do not include the

parameters of heat generation and ventilating air inside

the shelter* the mechanisms of transfer from these sources

to the surfaces by convection* radiation* condensation*

and evaporation* or weather conditions above the shelter.

However* experience with prototype shelters has shown that

wall surface temperature is a good index of effective air

temperature* especially in small shelters* where they are

about equal.

This division of the problem was possible because the

rate of heal^flow to all of the interior surfaces as a function of tim^ determined experimentally during each test on

the shelter studied at the National Bureau of Standards.

This method had the obvious and important advantage of

isolating existing sources of error to those lying in the

heat conduction region. Poor agreement between observed

and predicted results could not be attributed to poor

assumptions associated with the parameters of heat generation or the coefficients of heat transfer from air to

surface.

In summary* therefore* this first attempt at obtaining

mathematical solutions to an inherently complicated problem

was simplified by dealing with available data measured at

the boundary* instead of undetermined heat exchange processes inside the shelter. It provided a means of studying

some of the heat conduction aspects of this problem* such

as the importance of a prescribed initial temperature distribution in the region and the effects of the thermal and

physical properties of the materials comprising the region.

It also provided a means of determining the applicability

of the linear heat flow theory (i.e. temperature and position Independent coefficients), to a problem of this kind.
-5-

1.3 Types of Solutions

Pour solutions contained in this report were based

on the one-dimensional model. In this model, the heat

conduction region exterior to the wall and floor boundaries of the rectangular parallelepiped cavity was represented by a layer of infinite thickness and infinite

surface area. Each region was considered to be either

a single homogeneous layer of earth or a composite of

concrete and earth. The heat flux was presumed to be

applied uniformly over the entire surface so that heat

flowed in only one direction, normal to each plane surface. The finite (roof) region is not treated in this

report

.

An infinitely thick layer, such as described above,

is called a semi-infinite region. This denotes a region

bounded on one side only, o £ x <®in the homogeneous case,

or - i

<. x < o and o s x <« in the composite case.

One solution was based on the spherical model in which

the heat conduction region was represented by an infinite,

homogeneous mass of earth bounded internally by a spherical

cavity of radius a. The heat flux was presumed to be

applied uniformly over the entire surface so that heat

flowed radially in all directions.

In the cylindrical solution, the heat conduction

region was also represented by an infinite, homogeneous

earth medium, bounded internally by an infinitely long

cylindrical cavity of radius a. The heat' flux was presumed

to be applied uniformly over the entire surface so that

heat flowed only radially to the cylinder.

The assumption of uniform heat flux application over

the entire surface of each model permitted the mathematical

treatment of the Cartesian (x, y, z) model, the spherical

(r, ft, 0 ) model, and the cylindrical (r, 9, z) model as

simple models of only one (x or r) space variable.

Each mathematical model represents a simplification

of the actual physical shelter, and the effects of this

simplification will be discussed later in the report.
-6 -

2, SUMMARY OF PROBLEMS

Six mathematical solutions to study temperature

predictions in an underground shelter, as determined by

heat conduction into the surrounding wall and floor regions,

are presented in this report. The boundary conditions and

assumptions for each of the heat transfer models are summarized for each solution. The initial temperature distribution and the heat conduction model are shown diagrammat ically

for each summary.

SOLUTION I — One-dimensional, homogeneous, semi-infinite

heat conduction region; uniform initial temperature; constant

heat flux across the surface.

This solution was used as a first approximation to predict NBS shelter performance.

SOLUTION II — Spherical cavity bounded by an infinite,

homogeneous heat conduction region; uniform initial temperature; constant heat flux across the surface.

This solution was used as a second approximation to

predict performance.

oo
- 7

-

SOLUTION III — Infinitely long cylindrical cavity

bounded by an infinite, homogeneous heat conduction region;

uniform initial temperature; constant heat flux across the

surface

.

This solution was used as a third approximation to predict performance.

SOLUTION IV — One-dimensional, composite, semi-infinite

heat conduction region; uniform Initial temperature;

constant heat flux across the surface.

This solution was used to observe the effect of the

composite region assumption.

U

©,= 0

t = 0

©2=0

u,

-1 0

-*»- ©O 0 X

©o

TrvrT
-8 -

SOLUTION V — One-dimensional, composite, semi-infinite

heat conduction region; prescribed initial temperature;

constant heat flux across the surface.

This solution was derived on the basis of a prescribed

initial temperature distribution, and used to observe the

effect of such a distribution.

SOLUTION VI — One-dimensional, composite, semi-infinite

heat conduction region; prescribed initial temperature;

constant heat flux across the surface.

This solution is similar to the preceding solution,

differing only in the mathematical expression for initial

temperature distribution in the earth conduction region.

- ©o

0 \ X CO
-9-

3. EVALUATION OF EXPERIMENTAL

PARAMETER DATA

The purpose of this analysis was not only to develop

mathematical solutions, but also to extract results from

those solutions for comparison with experimental

observations. For this reason, it was necessary to

establish numerical values for the several parameters

appearing in each solution. This section deals with

the selection of these values for each of the following

parameters, namely

1) average initial temperature

2) properties of concrete

3) properties of earth

4) heat flux.
- 10-

3.1 Average Initial Temperature

Solutions I through IV were based on the assumption that

the initial temperature distribution throughout each region

was constant (or zero for mathematical convenience). Therefore., a single value of temperature had to be used for

computing numerical results.

Since the Initial temperature distribution for all

regions deviated to some extent from a constant value over

the domain in question (see Appendix A), the average Initial

temperature u

Q

was calculated for each composite region on

the basis of the following heat balance;

XiPiCi(u0-u0l ) = xs p ? c 2 (u02 -u0 ) (1)

where the subscripts 1 and 2 denote concrete and earth.

uQ1 and u02 were calculated by graphically evaluating

the Integrals

ucu _

1

xi u(x)dx

—X -i

( 2 )

and uo?

1 f

Xs

u(x)dx (3)

using the Interface as the reference point. The functions

u(x) are the initial temperature profiles shown in Appendix A.

The values of average initial temperature obtained by

this method are listed in Table 3.1.1.

It will be noted In the graphs for Solutions I and

IV that the origins of the observed temperature curves

show© in Sections 4 through 7 are not necessarily the

same as those of the predicted curves. This is because

the Initial temperature of the observed curve Is the

actual value at the Interior surface (see Appendix A),

whereas the initial temperature of any predicted curve

is the average value shown In Table 3.1.1.
- 11 -

TABLE 3.1.1

COMPUTED INITIAL TEMPERATURES

Average Region Temperature un , °F

NBS

Test

Number

Heat Conduction Region

North West South East Roof Floor

1 70.8 68.7 69 .

6

70.1 72.2 64.2

2 71.2 69.3 69.5 71.0 72.7 64.5

3 71.7 69.7 68.3 71.0 67.5 64.8

4 68.7 67.7 67.9 68.0 66.8 64.7

5 43.9 45.4 45.0 44.0 42.8 50.5

3.2 Properties of Concrete

The density of the shelter concrete was taken to be

145 lb/ft 3

. This value was based on the total weight of

cement and aggregates, approximately 63,200 pounds, an

estimated water of hydration content of 2,400 pounds, and

the purchased volume of 16.75 cubic yards.

The specific heat of mass concrete varies but

little with the type of aggregate and richness of mix.

and for practical purposes is usually taken as a constant

for a given concrete at about 0.22 Btu/lb-°F [2, p.585!»

A mean value of thermal conductivity was selected from

among several references [3,4, 5,6] as 1.15 Btu/hr-f t-°F,

based on an estimated moisture content and a physical

composition fitting that of the shelter concrete described

above.

The thermal diffusivity was determined from the

K, ,

. relation eg = — ? the subscript (1) denoting concrete,

Pi c i
- 12 -

and the computed value was 0,036 ft 3 /hr. These values of

thermal properties were thought to be representative of

those in the NBS test shelter, and will henceforth be

referred to as the "preferred'1 concrete values.

The references consulted on this subject listed

widely divergent thermal property values for concrete in

the 140-155 lb/ft 3 density range. The values of specific

heat ranged from 0.156 to O. 23 O, thermal conductivities

from 0.50 to 1 - 35 , and calculated thermal diffusivities

from 0.016 to 0.046. As a result, minimum and maximum

values were also used for analytical predictions.

3.3 Properties of Earth

During the excavation process, representative samples

of earth were removed for density and moisture content

determinations. Three distinct earth strata were found

and the average values of density and moisture content

were determined to be approximately 100 lb/ft 3 and 17

percent

.

A value of 0.17 Btu/lb-°F was selected as a representative average specific heat for the three earth strata

in the dry state [7, P - 33 3

Since specific heat is known

to vary with moisture content, the selected value was

corrected for 17 percent moisture, giving a value of 0.29

Btu/lb-°F for specific heat of the moist earth [7, p. 89 ].

A determination of the thermal diffusivity was made

from experimental data obtained during the NBS test

series. The solution of the heat conduction equation for

a one-dimensional , homogeneous, semi-infinite region with

a simple periodic boundary condition at the surface yields

a convenient expression for determining the thermal diffusivity, based on the time lag between the maxima of the

ground surface temperature wave and the corresponding wave

at any depth below the surface. (This phase-lag method is

more accurate than the temperature-amplitude method because

the values of amplitude are difficult to measure accurately [8]).
-13-

The expression is

a

-£*L

4irt 2 ’ where P = period, hours

x = depth of temperature measurement, feet

t = time lag between maxima, hours

Although temperatures were measured at several depths

in undisturbed earth near the shelter, only two were useful for this determination - those made at the six-inch

and one-foot depths . Measurements made at the two-foot

depth exhibited no clearly defined wave form, if any at

all. The amplitude of the daily cycle decreases rapidly

with depth and is almost completely damped out at the

two-foot depth [ 9 1 ® For this reason, thermal

diffusivity determinations in deeper ground are extremely

difficult on a diurnal cycle basis, and must be made on

averages of annual cycles.

Ambient temperatures were used because ground surface

temperatures were not recorded. However f ambient and

ground surface temperature maxima occur in phase, for

practical purposes. It was observed that the temperature

maxima at the six-inch and one-foot depths, during three

early consecutive days of NBS Test 3 , lagged the ambient

maxima by about 4 1/4 and 8 1/2 hours, respectively.

These phase-lags each indicate a thermal diffusivity of

0.026 ft a /hr.

A similar determination was based on data obtained

during three late consecutive days of NBS Test 5, with

exactly the same phase-lags and results. No precipitation had occurred for at least six weeks prior to the

start of or during NBS Test 3? whereas two feet of snow

had completely melted a few days before th© start of NBS

Test 5*
-14-

The thermal conductivity was determined from the

same relation K 2 = p 2 c2 as , the subscript ( 2 ) denoting

earth, and the computed value was 0,75 Btu/hr-ft- °P.

These values of thermal properties are likely to

be representative of at least the upper two-foot layer

of earth surrounding the NBS experimental shelter, and

possibly deeper. They will henceforth be referred to

as the preferred earth values. However, in order to

account for the likelihood of other values at greater

depths, and changes occurring in the upper layer during

the intervening times, minimum and maximum values were

again selected to form a range which would account for

reasonable variations.

3.4 Heat Flux

Although the term "constant heat flux" is used

throughout the discussion of Solutions I to VI, inclusive, the actual value of the heat flux F0 was used for

each plotted value of time. The heat flux used for time

t was a cumulative average of heat flow meter readings,

averaged over the interval from o to t . This procedure

was intended to approximate, to some extent, the effect

of the observed exponential decay of the heat flux.

This is discussed more fully in Appendix B which also

contains a table listing the cumulative average values

used in each solution.

The west and north wall regions were selected for

analysis and graphical presentation of the results because

they were considered to be most and least closely approximated by the one-dimensional model. The reasons for these

selections will be discussed in greater detail in Section

10.5.
-15-

4 . SOLUTION I

4.1 Statement of Problem

Semi-infinite homogeneous region. The heat flux F0 is

constant across the surface x = o. The initial temperature

is uniform.

4.2 Mathematical Summary

The differential equation to be satisfied by the

temperature rise q is

a9 a

2 - e = * >° , t> o (1 ) with the following initial and boundary conditions:

X > o t = o 0 = 0 (2)

X = 0 t > o 1 S

07!

a> II o (3)

ax X—> oo t > o 0 O (4)

The solution of the Laplace transformed equation is

- F o , , 0 _ exp (_qX) qKp

and the inverse transformation yields the solution for

temperature rise

(5)

e =

2F0 r at

12

- x

2 I T i

exp ^

»

tt Aat _

x x

” erfc

2 2yol }

( 6 )

For an initial temperature u0 , the temperature at any

distance x in the region is given by

u = uQ+ 9 (7)

At the interior surface x = o, Eq . *+.2(6) reduces to

'o at I 5

K; t it J ( 8 )
- 16 -

4.3 Assumptions

This solution [10, p. 75] was used as a first approximation for predicting shelter performance for all heat conduction regions except the roof region.

The assumptions used in this approximation are listed

and discussed below.

1 ) geometry - one-dimensional model

The NBS shelter deviated from the one-dimensional

model discussed in Section 1.3 in that the shelter

surfaces were finite in area with the result that

the solution, being one -dimensional, could not

account for heat lost to the corner regions.

Except for the floor, deviation from the onedimensional model was also caused by the additional boundary at the ground surface.

2 ) homogeneous heat conduction region

Since the region was considered to be homogeneous,

only one thermal conductivity and one thermal

dlffusivity appear in the solution, i.e. both

the concrete and earth media are assumed to have

identical thermal properties. These properties

are assumed to be invariant with position and

changes in temperature and moisture content.

For more detail, see Section 10.4.

3 ) uniform initial temperature

Figures 1 through 5 of Appendix A show that the

initial temperature distribution in each region

was not uniform for all tests. The assumed initial

condition 4.2(2) most nearly fitted the wall regions

of NBS Tests 1 and 5. but did not fit, for example,

the wall regions of fes Test 3 . In order to obtain numerical results, the temperature u was

computed on the basis of the average initial

temperatures u0 listed in Table 3 . 1 . 1 .
-17-

4.4 Results

Figures 4.4.1 and .2 show interior surface temperatures versus time, predicted hy Solution I, for the

north and west wall conduction regions of NBS Test 3,

together with the observed performance. Figures 4.4.3

through .6 show similar temperature-time curves for NBS

Tests 4 and 5«

Curves 1, 2, and 3 show predicted performance based

on an estimated range of earth thermal properties, including the preferred values. Curve 4 is based on the

preferred concrete properties, values which would be

reasonable if the concrete predominated in the heat

storage.

Table 4.4.1 shows predicted surface temperatures

for the five semi-infinite regions of NBS Tests 3? 4,

and 5* evaluated at elapsed times of one and two weeks,

together with the observed temperatures and the resulting

deviations. The predicted temperatures were computed

using the preferred concrete properties, the values producing the best general agreement with the observed

performance

.

4.5 Discussion

Inspection of Figures 4.4.1 through 4.4.6 shows

that the lower thermal properties gave better agreement

for the north wall region, and that the higher properties

were better for the west wall region. The behavior of

the other three surfaces, in most cases, was similar to

that of the west wall region in this respect. This can

be seen from Table 4.4.1 which shows that in 11 out of

15 cases, use of the preferred concrete thermal properties

produced agreement to within 1.5°F at the end of a twoweek period.
- 18 -

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ELAPSED

TIME

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- 19-

ELAPSED

TIME

-DAYS
SOLUTION

I

-

NBS

TEST

4

PREDICTED

SURFACE

TEMPERATURES

FOR

NORTH

WALL

REGION

- 20 -

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- 21 -

ELAPSED

TIME

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SOLUTION INBS

TEST 5

PREDICTED

SURFACE

TEMPERATURES - 22 -

ELAPSED

TIME —

DAYS
SOLUTION INBS

TEST 5

PREDICTED

SURFACE

TEMPERATURES - 23 -

ELAPSED

TIMEDAYS
INTERIOR

SURFACE

TEMPERATURES

PREDICTED

BY

SOLUTION

I

- 24-

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-25-

A comparison of analytical predictions with experimental data from NBS Test 4 is included in this section

to show the effect on the observed curve of an adverse

ambient temperature change which occurred about halfway

through the test and continued for its duration. This

can be seen in Figures 4.4.3 and 4.4.4 where the value

of observed temperature at fourteen days is actually

lower than the value at ten days, producing an observed

performance impossible to duplicate with models of this

kind. For this reason, it will be discarded for discussion purposes in the succeeding solutions.

Further inspection of the figures for NBS Tests 3

and 5 shows that the predicted curves have a basically

different shape from the observed curve, which levels

off more rapidly with increasing time. The continuing

rise of the predicted curves is due to the one-dimensional

nature of the solution in which the heat flow is confined

to only one direction in space. The shelter itself could

lose heat in three dimensions, with the additional heat

sinks emanating from the corners becoming more important

with increasing time. This resulted in an observed curve

which leveled off rapidly, appearing to approach a constant temperature.

These observations suggest a spherical model, which

conducts heat in three dimensions. This approximation

should accordingly produce a temperature-time curve of

more reasonable shape, since the solution is bounded in

time

.
-26-

5. SOLUTION II

5-1 Statement of Problem

Infinite homogeneous medium bounded internally by a sphere of radius a. The heat flux F0 is constant across

the surface r = a. The initial temperature is uniform.

5.2 Mathematical Summary

The differential equation to be satisfied by the

temperature rise 9 is

se _

aL

~ a

1 a ( r2

ae V

r

2 V / .

, r ^ a t > o ( 1 )

with the

The

following

solution

initial and boundary conditions:

r ^ a , t = 0 , 9 = o (2) r = a , t > 0 , -K - - Fo (3)

r — oo t > 0 , 0 O (4)

for temperature rise is

9 erfc

r - a

2/cxt

( 5 )

For an initial temperature u Q , the temperature at any

distance r in the region is given by

u = u0 + 9

( 6 )
"27~

At the interior surface r = a, 5-2(5) reduces to

5*3 Assumptions

This solution is available on p. 248 [10] and was

used as a second approximation for predicting overall

shelter performance.

The assumptions used in this approximation are

listed and discussed below.

1) geometry - spherical model

The NBS shelter deviated from the spherical

model in that part of the system was actually

finite, bounded externally by the ground

surface. It also differed in the obvious

matter of shape, which would alter the local

heat flow pattern.

Since the solution depends only on radial distance and time, as discussed in Section 1.3?

it can make no distinction among the six

surfaces. Thus, it provides only a single

temperature prediction at a given radius in

all directions.

( 7 )

at

where T = “TF
- 28 -

a) equivalent spherical radius

Since the assumed shape in this model

is a sphere, it was necessary to

choose a sphere of such radius as to

have heat transfer characteristics

similar to those of the actual shelter.

This was done in two elementary ways,

as follows:

i) by requiring the sphere to have

the same surface area as the

actual shelter, thus maintaining

the same total heat loss rate.

On this basis, the equivalent

radius is 5.7 feet.

ii) by requiring the sphere to have

the same volume to surface area

ratio as the actual shelter,

thus taking into account, to

some extent, the difference in

shape. On this basis, the

equivalent radius is 4.0 feet.

2) homogeneous heat conduction region

This assumption was discussed in Section 4.3.

3) uniform initial temperature

Since this model is spherically symmetric in

every respect, it follows that only one value

of initial temperature can be used. Therefore, the six average region temperatures u0, listed in Table 3-1* 1? were arithmetically

averaged to obtain a single overall initial

temperature

.
-29-

5.4 Results

Figure 5.4.1 compares the observed interior surface

temperatures with predicted values using Solution II,

for the two values of equivalent spherical radius described in the preceding section and for two sets of earth

thermal properties, including the preferred values. Figure 5*4.2 shows similar comparisons for NBS Test 5.

The observed temperatures shown in these figures are

arithmetic averages of the smoothed experimental data for

all surfaces, including the roof.

The values of heat flux F0 used in computing these

results were arithmetic averages of the cumulative averages, including the roof, listed in Table B.l of Appendix

B.

5-5 Discussion

Inspection of Figures 5*4.1 and 5*4.2 shows that

better agreement was achieved between observed and predicted values by using the estimated minimum values of

earth properties rather than the preferred values. It

can also be seen that the equivalent surface area radius

of 5*7 feet gave closer agreement than the volume to

surface area radius. For the minimum values, the 5.7 foot

radius gave agreement to within 1°F or less at t = 336 hours

for each test, whereas the 4.0 foot radius provided agreement to about 2°F.

Of the two choices for equivalent spherical radius

described, the spherical cavity which provided an interior

surface area (a = 5*7 feet) equal to that of the parallelepiped shelter would be expected to give better agreement

between the predicted and observed surfaces temperatures

because the test shelter had a low aspect ratio and thus

the total amount of earth serving as a heat sink was more

nearly comparable in the two cavities.

Further inspection of Figures 5.4.1 and 5.4.2 shows

that the shapes of the predicted curves for the spherical

model leveled off more rapidly than those for Solution I,

as would be expected. The effect of the ground surface

boundary can b^ interpreted as a reduction in the volume

of earth available to the shelter for heat conduction, but may not,

in all seasons of the year, represent a reduction in the total heat

absorption capacity of the region.
- 30 -

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- 31 -

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ELAPSED

TIME

-

DAYS
-32-

6. SOLUTION III

6.1 Statement of Problem

Infinite homogeneous region bounded internally by an

infinitely long cylinder of radius a. The heat flux F0

is constant across the surface r = a. The initial temperature is uniform.

6.2 Mathematical Summary

The differential equation to be satisfied by the

temperature rise 0 is

£§ = a

at

re +

i ae Sr

2 r

r > a , t > o

( 1 )

with the following initial and boundary conditions:

a , t = o , 0 = 0 (2)

a , t > o , -K r

= F° (3)

t > o ,

OD i o (4)

the Laplace transformed equation is

,Mq_ qKpKj (qa) (5)

An asymptotic series expansion of Eq. 6.2(5) yields the

solution for temperature rise for small values of time

9 =

2F,

F oat ll / . „ r - a (3r + a) r ..f _ „ t - a

L — 1 \ ierfc iw - c dt i 1 erfc ^sr

9a2 + 6ar + 33r"

32a

2 r

2^2 L at J i

3 erfc

2/at

75as + 27a

2 r + 33ar2 + 249r3

s r - a \

[ at ] i erfc + . • 128a • J 3?”

(6)
-33-

For an initial temperature Ug, the temperature at

any distance r in the region is given by

u = uQ + 9 (7)

At the interior surface r = a, 6.2(6) reduces to

e = - t£ + t - -i- ^ + . . .1(8)

K ^ */tT 4 4/tF 32 J

where T = .

6.3 Convergence

The first two terms of this series solution, Eq.

6.2(6), were available on p. 339 [10], and were tried as

a third approximation for predicting overall shelter

performance. However, the contribution from the second

term was found to be considerable for moderate values

of the Fourier number T, suggesting that one or more

higher order terms might contribute significantly.

Two additional terms were obtained by expanding the

modified Bessel functions of the second kind,, which appear

in Eq . 6.2(5)* into asymptotic series, performing the

indicated division, and taking the inverse transform of

each term.

The expression in Eq . 6.2(8) converges for small

values of T, but will diverge for large values. The change

from convergence to divergence appears to occur at approximately T = 1. For values of T in the neighborhood of 1,

convergence may be quite slow, so that many terms of the

series would be needed to provide reasonable accuracy. In

this case, four terms of the series were sufficient to produce accuracy of the order of hundredths of a degree Fahrenheit .

A large time solution p. 339 [10] , would be required

to produce convergence for values T>1.
-34-

6.4 Assumptions

The assumptions used in this approximation are

listed and discussed below.

1) geometry - cylindrical model

The NBS shelter deviated even more from this

model than it did from the previous spherical

model. In addition to the reasons given in

paragraph (l) of Section 5-3* the cylinder is

assumed to be infinite in length, as well as

circular in shape. However, as stated in

Section 5.5 j this model was introduced for

purely mathematical reasons.

Like the previous solution, this solution

provides only a single temperature at a given

radius, in all directions normal to the central axis of the cylinder.

a) equivalent cylindrical radius

The cylindrical radius chosen to approximate the shelter was obtained by assuming

that the shelter was a cylindrical cavity

whose length (x) was equal to the longest

internal dimension of the shelter and

whose radius was such that the curved area,

2Trax, was equal to the total shelter surface area. This method yielded an equiva*

lent radius of 6.0 feet. Only the curved

area was used to represent the total surface area because no heat flows axially in

the cylindrical model.

2) homogeneous heat conduction region

This assumption was discussed in Section 4.3.

3) uniform Initial temperature

The same average initial temperature used in

the previous solution was also used here, for

the same reasons of symmetry.
35-

6.5 Results

Figures 6.5.1 and 6.5.2 show interior surface temperatures versus time, predicted by Solution III, for an

equivalent cylindrical radius, a = 6.0 ft. These curves

together with the observed performance, were plotted for

NBS Tests 3 and 5 for two sets of thermal properties,

including the preferred earth values.

The observed temperatures shown on each figure are

the same ones used in Section 5.4, and the values of

heat flux F0 used in the computations are the same as

those used in the previous solution.

6.6 Discussion

Inspection of Figures 6.5.1 and 6.5.2 shows that

the preferred earth thermal properties gave better

agreement between predicted and observed temperatures

for an equivalent cylindrical radius of 6.0 feet. This

combination gave agreement to within less than 0.5°F

for both tests, for elapsed times of both 168 and 336

hours. The shape and position of the predicted curves

are an improvement over those produced by Solutions I

and II.

At this point, further improvement dealing with

models characterized by one space variable could probably be achieved by treating the composite case, or the

nonuniform initial temperature case. These refinements

would introduce considerable complexity into the mathematics; however, treating the one-dimensional model will

avoid some of the complexity, and will give some indication of the magnitude of the effects. Since the composite nature of the heat conduction region is a fundamental

property applicable to all concrete shelters, it will be

treated in the following section.
- 36 -

lf)

<£)

id»

Ll.

ELAPSED

TIME

-

DAYS
- 37 -

ELAPSED

TIME

-

DAYS
- 38 -

7. SOLUTION IV

7.1 Statement of Problem

Semi-infinite composite region. The heat flux F0 is

constant across the surface x = -1 . The initial temperature is uniform.

7.2 Mathematical Summary

The differential equations to be satisfied by the

temperature rise @ are

with the following initial and boundary conditions:

( 1 )

and

at a*3 = as o < x < e© , t > o

( 2 )

£ < X < o , t = o, 0 =0 ( 3 )

o < x < ©o , t = o

( 4 )

( 5 ) X i , t > o

X = o t > o , 9 (6)

X = o , t > o

( 7 )

X ©o , t > o » 2 o (8)
-39-

The solutions of the Laplace transformed equations are

g =

F° f exP [ - q 1 ( l + x) 3 - B exp [ - q, (

l

- x)]

j

q a K t p

t 1 + 8 exp (-2q t £) J

and 02 = / exp [- q,(/ + xk)] 1 q^pCcr + 1) l 1 + 3 exp( - 2q 1 4) J ( 10 )

The inverse transformations yield the solution for

temperature at any point in the concrete medium

Uo + ~ F 0T^Y (-l) n

B n

{ ierfc - ^/AK

i U 1

and the solution for temperature at any point in the earth

medium

U; = Uq + H (1-b)F 0T^ Y(-1)V ierfc

K . L 1

( 12 ) n=o 2T

where T = .

At the interior surface x = - i , Eq . 7.2(ll) becomes

00

U, = U0 + IL F 0T^ Y(-1)V ; ierfc \ - B ierfc ^

}

( 13 ) K, l T 5 Tf

>

1 n=o

At the concrete-earth interface x =o , either 7.2(ll)

or (12) may be used, giving

u un + — (1-0)FOT^ 7 (-l)n

3n ierfc

K, ° L> n=o

(2n+l)

T“

2T 2 (14)
-40-

7.3 Convergence

Since o < ierfc(x) <; l//rr for all values of x, the

convergence of (-0) n is sufficient to insure convergence

of Eqs. 7.2(ll) and ( 12 ). Therefore the series converges

whenever | 0 | < 1 . This condition is always satisfied, as

can be seen from an examination of the definition;

whence - 1 < 3 < 1 for o < a <

7.4 Assumptions

Similar solutions were published in references [ll]

and [12], using a different origin for the coordinate

system. This solution, Eqs . 7.2(11) and (12), was derived

with the origin located at the Interface for mathematical

simplicity.

The assumptions used in this approximation are listed

and discussed below.

1) geometry — one-dimensional model

This assumption is the same as for Solution I

and was discussed in Section 4-. 3 . It will be

listed in the following solutions without

further comment.

2) composite heat conduction region

The heat conduction region was a composite of

8 inches of concrete in perfect thermal contact

with an infinite earth medium, so that the

solution contains the thermal properties of

both media. This assumption will be listed

in the following solutions without further

comment

.

3 - and

a + 1
-41-

3 ) uniform initial temperature

This assumption is the same as for Solution I

and was discussed in Section 4.3.

7.5 Results

Figures 7.5*1 through .4 show interior surface

temperatures versus time* computed from Eq. 7*2(13),

for the north and west wall conduction regions of NBS

Tests 3 and 5, together with the observed performance.

Curves 1 and 3 show predicted performance based on

the preferred concrete thermal properties and two sets

of earth properties, including the preferred values.

Curves 2 and 4 show performance based on estimated

’'maximum" concrete properties and two sets of earth

properties, namely, estimated maximum and preferred values.

Table 7.5*1 shows predicted surface temperatures

for the five semi-infinite regions of NBS Tests 3? 4,

and 5 , evaluated at elapsed times of one and two weeks,

together with the observed temperatures and the resulting deviations. The computations were based on

the preferred concrete and earth thermal properties.

Table 7*5*2 is similar to 7*5*1, except that the

predicted surface temperatures were based on the maximum sets of thermal properties.

A numerical example for this solution is shown in

Appendix C.
- 42 -

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TIMEDAYS
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TIMEDAYS
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INTERIOR

SURFACE

TEMPERATURES

PREDICTED

BY

SOLUTION

IV

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CD> X ftCD un-4- o h on 10- unco LTMO- -=t vo oj oj un

CO CO CT\H O t

00 oo oo cnvo unoo o o to- on

cn

uoo

14- [0-COCO to- 10- 10- 10- E- [o- un unco vo un

rH vo

-4- on o o

o

1 CO

CDj.

TO

(D -Po unm-oovoo oje-[poje- cvi o enco on

to [ft cno ft o unocvcnoo ,-icn-4 0co

cd e-e-oooooo tftoon-ifttft ununvovoun

ft

a., o’ o’

-P c

ii ii CO o

ft ft

r—1 CV (D -p r-iunununvo O-4-voooj oj cm oj unvo

a a ft CO •••••

m £ ft ftOOOO -4" ft OJ OJ O onOOJOO

cn CD > III 1 1 1 1 I II 1 to- Eh to (D

ft Q

o’ CD 3

o O

ii CO TO

ft CD b ft CO > on ft oo cnoo 4-ftOjft4- oftooovoo

3 VD ft o -4- CO 1—1 <d tft-oo ovoo un oocnoavun ojuovovo^j4- o co t— t— — tft Eft E-oo [ft [ft un un un un un

• • ft ,-Q

i—1 t—1 o o

ft

ii ii ftCD TO H CV -p CD K K a -P OJ vo cn-4" -=4 4" E-VO ft OJ CO Cn ft -4" -4" M O ..... ..... .....

ft vo c— ov cnvo -4”[fttft!ftun ao4-cnvoun

to ftft.ft.ft.ft. ftftftftft ^runununun

CD

ft

PL,

CD CDXlUC ft ft £2 ft £ O fti-P-PftO ft -p -P p O -P-P-P-PO

CO ft co 0 co o ft co 0 co o ft ra 0 CO O

Eh ft O CD O CO ft O CD O CO ft O <D O CO ft

ft ^SWUli, gSWWfc S^cenWtiH

TO 3

CD co

COft

CO ft

rH CO w cd on -4* un

EH

57.5
48-

7.6 Discussion

Inspection of Figures 7.5-1 and .3 shows that the

preferred thermal properties of concrete and earth

(Curve 1) gave better agreement for the north wall

region. This region, as previously pointed out, was

selected as the least typical in the sense of conforming least to the one -dimensional model. The maximum

values of thermal properties (Curve 4) gave better

agreement for the west wall region, as well as for the

other three regions, in most cases. These findings are

similar to those pointed out by Solution I, and can be

seen from Table 7-5*2 which shows that In 10 out of 15

cases, the maximum values produced agreement to within

1.5°F at the end of a two-week period. The preferred

sets, on the other hand (Table 7 - 5 - 1), produced agreement to within the same amount in only 2 out of 15

cases. This lack of agreement for the preferred sets

of values is due to the one-dimensional nature of the

solution which has already been seen to cause high

predictions. However, it should be recalled that the

purpose of this solution is to examine the importance

of the composite assumption.

Curves 2 and 3* based on approximately equal fractional increases in concrete values on one hand and

earth values on the other, indicate, when compared

with Curve 1, that the two media play about equal roles

in conducting heat away from the shelter for two-week

periods. However, since the earth is generally less

conductive than the concrete, and becomes increasingly

utilized for heat storage as time increases, it would

ultimately be expected to control the heat flow.

This tendency can be seen from further examination

of Curves 1, 2, and 3 in Figs. 7-5*2 and .4 which shows

that increasing the concrete properties lowers the temperature more than increasing the earth properties in

the first week, but less in the second week. However,

the lower temperatures at the end of two weeks (Curve 3)

are not significantly lower than Curve 2. and the reason

that the greater mass of earth involved is not a dominant

factor by this time will b^\ discussed in Section 10.2.

This solution can be further improved, at the expense

of some increase in mathematical complexity, by considering the non-uniform initial temperature distribution -which

actually existed. (See Appendix A).
-49-

8. SOLUTION V

8.1 Statement of Problem

Semi-infinite composite region. The heat flux F 0 is

constant across the surface x = - f . Each medium has

initial temperature u = f(x).

8.2 Mathematical Summary

The differential equations to be satisfied by the

temperature u are

and

Su. 3

a u,

a. :. x

L. at *x2

9, < x <r o , t > o

^u3 = as ^ u ? , o < x < oo , t > o St rjx 2

( 1 )

(2)

with the following initial and boundary conditions:

-£<x<o,t = o,Uj=ax+b (3)

o < x oo , t = o , u? = c+ (b-c) exp (-mx) (4)

x = - l , t > o , -K1 = F 0 ( 5 ) ^x

x = o,t>o,ut =u2 (6)

X = o , t > o = K. fix

dua Sx

(7)

X oo 9 t > o * u- c ( 8 )
-50-

The solutions of the Laplace transformed equations

are

_ _ f F0 + aK

1 f exp [-q x (x + x) ] - R exp [-q^u - x) ]

'I l q i K 1 p

J ( 1+8 exp (-2q 1 ^)^

f (b - c) K 2 (qs - m) \ f exp [-q,(2£ + x) ] + exp (q : x)

( q1 K 1 (a + 1) (p-ctgm2

) ^ ^ 1 + B exp ( -2q 1 l) J

+ / q 1 mc-b)-a

| | exp [ ~g T ( 2jg + x)] + exp (q,x)

^ ax + b

l'q 1 p( (7 + l)''^ 1 + 8 exp ( -2q 1 1) J +

p

(9)

and

r F 0 + aK 1

-i r exp [ -g 1 U + xk) ] - p exp [ -g 1 (l + xk)] ~i

L q,K x p

J l + p exp (-2q1 i) J

q, a (c-b)-a

'I r exp (-q^xk) + exp [-

q

a (2£ + xk)]

q x p (cr + 1) H 1+8 exp (-2q1 l) J

r

( b - c)(q 1 K 1 a - mK2 )

. , exp (-q T xk) + exp [-q l (2/ + xk)] + { “ --—

i

4 —— —

qi K ,(a + l)(p - asm

s

)

J L 1 + g exp (-2q i) }

(b - c)

p - a-m* | exp (-mx) - exp (-q : xk)

|

+

{

(b-c) exp (-q : xk) + c p }

( 10 )
51 -

The inverse transformations yield the solution for

temperature at any point in the concrete medium

u, =

21

K,

(Fq + aK 1 )T® £(-B)

n ierfc hlrd - Bierfo

n=o

2T ! 2T‘

V.U - ai( 1-8)T^ £(-$)n ierfc iUhrrd + ierfc h_^x/' 1

2T

n=o

(b-c)

U+0 )

CD I(-«

n

n=o

2T s

2(n+l) + x/l ^ 2n - x/j?

erfc _________ + erfc -l 2Tc 2Te

+ (b-c)

( 1+0 )

00 erfc

n=o

2(nf} + x/ * + h T* | exp

2T^ k

to

+ 2( n+1)

^ + TM

k k + erfc T i 1 exp

l^f- k

J

„ £m mx

+ 2n — - —

k k } + ax + b

and the solution for temperature at any point in the earth

medium

u. = P (F0 + aKjJTpf (-e)n(l-B) ierfc

2T? n=o

CO - a£(l-B)T^£ ( -8) n

n=o

ierfc

2(n+1) + ^ + ierfc 20 + ^

1

2T' 2T’

(ID
-52-

where

-(b-c) i^V(-B)n

2

V (_3) [" erfc 2(n+1 ) ++ erfc gj +

^_j i n n=o

i

2T ? 2T2

+ (b-c) (1+P)

I

(- B ) n {

erfc

n=o

2(n+l) + kx/A Urn

2t?

+ •— T' k

exp

Am\2

T

+ 2(n+l) — + mx

k + erfc 2n+kx/A , Am T — 2

.

T— —

L 2T^ k

i n exp

Am— I T

•k

» Am + 2n + mx

b-c f „ + —— ^erfc

2 L Am

T

|r kx/A

k 2T^

f Am \

2 exp - T-®

_ L\ k ./ J

- erfc Am rpi kx/A

k 2t4

exp(?> *» |+ (b-c) erfc + c 2T^

(12)

rp CL^ "t

f_2

At the interior surface x = - a > Eq. 8.2(11) becomes

= — (F 0 + aK, ) T

? y (-P) n

K, n=o

ierfc - 3 ierfc flti

Te Ts -

00 00 - 2aA(l-g)T^Y ( -B) n ierfc - (l+f3)(b-c) y (-g) n erfc

L* 2Tt 2T? n=o n=o

+ (b-c)(l+fl)

y

(-3)n erfc

2n+l Am i

+ -— T

s

n=o

L2T£ k

/ Am>

expj M—- ) T

( 2n+l)

Am

- aA + b

( 13 )
-53-

At the concrete-earth interface x = o, either 8.2(ll)

or (12) may be used, giving

u =

21

uu (F + aK,)T 5 Y (-B) n(l-B) ierfc U

2T*

n=o

- ai(l-B)T^Y (-B) n n=o

n+1 n

ierfc — + ierfc __t! J

(b-c)

(1+S) n n+1 n

erfc ~X~ + erfc T n=o

+ (b-c)

(1+B)

erfc

n=o

—7- + —

t? k

exp ifp + 2( n+1) i2

k / k

+ erfc

l t

" + *? T*

& k

exp + 2n *=

k }

+ b ( 14)

8.3 Convergence

The first two series in Eq. 8.2(11) are of the same form

as that of Eq . 7.2(ll), and converge for the reasons given

in Section 7-3-

The third series contains complementary error functions

which are also bounded

o < erfc(x) £ 1

for all values of x. Therefore this series also converges

whenever | g | <r 1 .
-54-

The remaining series converges for all values of T,

whenever a certain condition among the parameters 3 , m,

l , a:

j, and aR is satisfied. This condition can be obtained

by use of the ratio test for convergence. The absolute

value of the ratio of the (n+l)^b term to the nth term of

the series is

pn+1

erfc]

-

. L

2(n+l)-x/ i ^ q|

2T# k

T

J|

|excp|" Y*fj T + 2(n+l)^~ . EEJ

3n erfc

r2n-x^

+

_ 2T ® k 1 | exI{(f

J

T

This reduces to

( 1)

3

erfcj^

2n-x/

l

, T~

2T s — +

k

41

r2n-x/i, , lm il

erfc -i-T*

< L 2T^ k J

( 2)

The value of this expression must be less than one, in the

limit of large n, for convergence. The bracketed term

approaches one for large n, and thus the ratio of terms

reduces to

3 exp

( 3 ) The condition for convergence, then, is that (3) be kept

less than one. The smaller the value of T, the faster is

the convergence, and for T corresponding to about 14 days,

only a few terms are required.

Eq. 8.2(12) is similar to (ll), with two additional

terms which are not summed. These terms vanish for large T.
-55-

8.4 Assumptions

A solution of this type was not available in any of

several apposite references examined, and was derived,

as in the previous solution, with the origin of the coordinate

system located at the interface.

The assumptions used in this approximation are listed

and discussed below.

1) geometry — one-dimensional model

2) composite heat conduction region

3) non-uniform initial temperature

In the previous solutions, the initial

temperature distribution throughout each

region, e.g., - l < x <r 00, was assumed to

be uniform, and an appropriate average

value was used to represent the observed

distribution shown in Appendix A. The

figures in this appendix show that the

initial temperature distribution was

not uniform in most cases, especially

at the beginning of NBS Test 3. Even

though an initial condition such as that

of Test 3 would not be likely to exist

in an actual situation, especially one

in which the shelter had been unoccupied

for a long period of time, the kind of

unusual distribution which did exist

afforded an opportunity to observe its

maximum effect and significance.

In this solution, the initial temperature

was approximated in the concrete medium

by a linearly increasing function,

u, = ax + b, Eq. 8.2(3)* and in the earth

medium by the exponential function

u2 = c + (b-c) exp (-mx), Eq. 8.2(4).

See Fig. 8.4.1 and Table 8.4.0.
- 56 -

do‘n - 3yniva3dW3i

DISTANCE

IN

REGION

-

FEET
-57-

TABLE 8.4.0

VALUES OF CURVE-FITTING CONSTANTS

FOR SOLUTION V

Curve-Fitting Constants

Test Surface a b c m

3 No rth 2.40 66 .

6

78.0 0.50

West 1.65 65-2 74.0 0.60

South 1.00 61.7 76.0 0.50

East 1.00 65.0 77-0 0.55

Floo r 1.40 63.8 66.0 0.60

4 North 0.30 66.1 73.0 0.40

West 0.00 66.0 70.0 0.40

South 0.00 66.0 71.0 0.40

East 0.00 65.8 72.5 0.30

Floor -1.00 64.7 64.7 0.00

5 No rth 0.00 45.2 41.5 0.30

West 0.00 46.0 43.5 0.30

South 0.00 46.0 42.0 0.27

East 0.00 45.6 40.5 0.30

Floor 1.00 48.3 54.0 0.33
-58-

This figure partly reproduces Fig. 3 of

Appendix A* showing the initial temperature distribution in the north and west

wall heat conduction regions* selected as

representative regions for detailed

analysis in this report.

8.5 Results

Figure 8.5.1 shows interior surface temperatures

versus time* predicted by Solution V* for the north wall

conduction region of NBS Test 3 , together with the observed performance.

Curves 1* 2 and 4 in Figure 8.5.1 were based on the

preferred earth thermal properties while varying the concrete thermal properties to cover the range of values mentioned in Section 3.2.

Curves 1* 3 and 5 in Figure 8.5.1 were based on the

preferred concrete thermal properties while varying the

earth properties to cover an estimated range of values that

might occur at the NBS test site.

Figure 8.5.2 shows similar temperature-time curves for

the west wall conduction region of NBS Test 3.

Figure 8.5.3 shows a temperature-time curve for the

west wall conduction region of NBS Test 5. The figure

shows predicted performance* together with the observations*

based on the preferred thermal properties only.

The values of the curve-fitting constants used in the

computations are listed in Table 8.4.0.

The results of this solution were obtained on the IBM

7090 digital computer at the rate of 0.35 minutes per run.

Each run consisted of computing temperatures at four positions in the concrete medium and eight positions in the

earth medium* for the eight values of time shown in the

figures* a total of 96 calculations.
SOLUTION

3E-N3S

TEST 3

PREDICTED

SURFACE

TEMPERATURES

FOR

NORTH

WALL

REGION - 59 -

in

coi a* L

ELAPSED

TIMEDAYS
SOLUTION

3E

-NBS

TEST

3

PREDICTED

SURFACE

TEMPERATURES

FOR

WEST

WALL

REGION

- 60 -

C\3 ~ q 00 iD

00 qo h- Is- S»

do ‘ n - 3amvM3dW3i aovdans

ELAPSED

TIME

-

DAYS
- 61 -

ELAPSED

TIME

-DAYS
- 62-

8.6 Discussion

Comparisons of Curves 1 and 2 in both Figures 8.5.1

and 8.5.2 show that an increase of 14 percent in the thermal

diffusivity of the concrete decreased the inside surface

temperature 0.5-P or less throughout the l4~day period.

On the other hand* comparisons of curves 1 and 4 in Figures

8.5.1 and 8.5.2 show that a decrease of a little over 50

percent in the thermal diffusivity of the concrete from the

preferred value Increased the temperature about 1°F for the

north wall surface and about 2°F for the west wall surface.

Comparisons of Curves 3 and 5 in Figures 8.5.1 and

8.5.2 show that an increase in thermal diffusivity of the

earth from 0.02 to 0.03 ft2 /hr caused a change of less than

0.5°F at any time in the surface temperature of the north

wall and a maximum change of about 1°F in the surface temperature of the west wall after 14 days. Thus it appears

that the thermal properties of the concrete in the composite

model affects the surface temperature of the walls more

significantly than the thermal properties of the earth.

Comparison of Figure 8.5.3 with Curve 1 of Figure 7.5.4

for Test 5 shows much similarity between the two* especially

after the first week* indicating that when the initial temperature is reasonably uniform (See Fig. 5 of Appendix A)*

the predictions of Solution V approach those of Solution IV.

It was pointed out in Section 7.6 that Curves 2 and 3

of* e.g.* Figure 7.5.2* cross at about 7 days* showing the

tendency of the earth region to succeed the concrete as the

dominant medium. In Solution V* however* Curves 2 and 3 of

Figures 8.5.1 and 8.5.2 do not cross* even after 14 days*

Indicating that the concrete remained dominant throughout

this period. The reason is that in Solution V the concrete

was assumed to be initially colder than the earth* as shown

in Figure 8.4.1* and therefore dominated the heat flow for

a longer time than usual. This medel serves to illustrate

the fundamental interdependence of the composite nature of

the region and the nonuniform Initial temperature* which

could only be seen in a solution containing both assumptions.
-63-

9 . SOLUTION VI

9.1 Statement of Problem

Semi-infinite composite region. The heat flux FQ is

constant across the surface x = - /, . Each medium has initial

temperature u = f(x).

9.2 Mathematical Summary

The differential equations to be satisfied by the temperature u are

Bu 1 B

su — 1 =

°S _ » - £ < x < 0 , t > 0 (1)

at Bx2

Bu2 B

2u2 — = a 0 < X < ©O , t > 0

g Bt dx2

’ (2)

with the following initial and boundary conditions:

A X < 0 t — 0 a, = ax + b ( 3 )

0 < X < 00 t = 0 us = b + cx exp (-mx) ( 4)

Bu, X = - l t > 0 ” K

I Bx

= F 0 ( 5 )

X = 0 t > 0 = ^2 ( 6 )

X = 0 t > 0 K, II

09 *1 ( 7 ) 3x Bx

X -> CO t > 0 U2 -> b (8)
- 64-

The solutions of the Laplace transformed equations are

F 0 + aKj ^ f exp [ -g 1 (l+x)] - 0 exp [ -g t (l-x)] |

1 + 8 exp (-2q t i)

^ u

- =

L q,KlP

J L

r a

-J r exp [ -q

x ( 21 + x) ] + exp (q 1 x)

1 q, p ( a+1) J 1 '1 + R exp (-2q 1 /) }

+

{

cK«

q T K 7 (cr+1) (p- a2 m

s

) H

exp [-qj(2£ + x)] + exp (qjx)

1 + 8 exp (-2^/)

}

r 2ag cmKg (q 8 -m) ^

r exp [ -q 1 (2/ + x)] + exp (q,x) 1 ax + b

l q^K : (o+l) (p - a2m

y

) y

J l 1 + p exp (-2qt X) -> + ( 9 )

and

u® =

F 0 + aKj 'i r [1-3] ©xp [-q t (i + xk) ] = {“}{ q,KlP 1+0 exp ( -2q x

!,) }

- {

a

j

r exp [-q, (21 + xk)] + exp (-q^xk) 1 q,P (o+l) ^

1+fl exp (-2^/)

i +

1

cK.

q 1 K 1 (o+l) (p - a2 m

a

) }{

exp [-q a (21 + xk) j + exp (-q 1 xk)

1+R exp (-2q i £) }

-{

20gCmKs (q 3 -m) -> r exp [-q 1 (21 + xk) ] + exp (-q l xk)

q t K, (o+l)(p ——} { a2 m

2

) a ^ ^ 1+0 exp ( -2q,£) }

^ 2a 2 cm^ +

J {

- exp(-mx) } +

( p-a 2ra2

)

cxexp( -mx) b

p - a2

rrr p

( 10 )
- 65 -

The inverse transformations yield the solution for

temperature at any point in the concrete medium

u i =

2

1

Ki

n=o

[F0 +aKj T

? V - pierfc

i 2T

2

(2n+l)-x/ j,-|

aje(l-P)Ts y

(-3)n 2(n+l)+x/

l

2n-x/

l

ierfc i + ierfc i 2T^ 2T^

n=o

c ( 1+P) n _ m 2 l

( ' e)

{[

2 'V

k )

T + 2(n+1) k

+

k >

rfc

/m mx r 2( n+l)+x/ l

n=o 2T ?I + T^|

exp[(jjT + + “]

}

£ ( 1+P)

m 2

n=o

mx

k J

erfc

- 2n-x/l

- 2T* ]

+

-^1+$)T*

2, ( -B)n

|'expT- / 2(n+l)+x/

i

L V 2T1 ') ] + exp[-

2n-x/;

2T‘

1

+ ax + b (ID
- 66 -

and the solution for temperature at any point in the

earth medium

us =%F0+aKUT

n=o

n, „ (2n+l)+kx/i

( l-(3)ierfc i 2T ?

- a;(l-3)T

1. CD

( n. y

( - B)

p

n=o

ierfc 2(n+l)+kx/l

+ ierfc^ilWi"|

2T 2 2T^ j

C (1+g)

m 2

n=o

exp^-^J T + 2(n+l)~ + mxj

j

s. R^

m 2

! y (_e)

r-{[2(|2)

2 T + 2n^ + nx]erfc[^*|^ + ^T*]

n=o 2T '

exp£^^JT + 2n—i + mxl

n=o

err//mY mx-i r/m i kx//-i r//m\2 1

mlL J

T +

2 J

erfc

Lk"^

5 + Tyir^LUJ T + mc

]

[(

;'£m\2 mx-, r £m i kx/i

TjT . Yjerfc^ - yyr

]

e)/(p) T - tnxj 1- + b

where T = . I

*

(12)
At the interior surface x = - i, Eq. 9.2(ll)

> = |Av«K, 1T* f

(-S)

n[ierf=U . Sierfo^]

n=o

n 2n+l

- 2a£(l-3)T s ) (-3) ierfc—

U 2 T~

n=o

r2n+l Jem 1- ~

T

c

I k j n=o

e^[(r) T + (2n+1)*r] J

+ g^(Wy(-e)

nK.g|)

2 ]}-a, + b

n=o

becomes

(13)
- 68 -

At the concrete-earth interface x = o, either

9.2(ll) or (12) may be used, giving

O 4 2. u = —rF0+aKj 1 T 2 y

( -3)n

( l-3)ierfc

n=o

2n+l

2T^

i

00

. - a/( 1-0)T* y (-P)nfierfc£ii

n=o T

rierfo 11 x + ierfc—

n

r ir±e> v, a \ nir-/—

m 2 * f’*]

exp[(j^YT + 2(n+l)-^l}

c (1+P)

m 2 Z(-

n=o

. n r r \ r n

p) | l

\

k P + ^

k >rf=[7? + rT1

+ £

(. 3 ) n=o

n exp

r /n+lN

- J

+ b (14)
-69-

9. 3 Convergence

Equations 9.2(ll) and (12) converge subject to the

same condition discussed in Section 8 . 3 , namely

3 ( 1 )

9*^ Assumptions

This solution was not available in the references

examined, and was derived like the previous one. It differs

from Solution V only in the form of the initial temperature.

The assumptions are:

1) geometry — one-dimensional model

2 ) composite heat conduction region

3 ) non-uniform initial temperature

As stated in Section 8 . 6 , the region beyond 4 feet

was expected to contribute to the surface temperatures, resulting in the need for a better

approximation to the initial profile in this

region. In this solution, the initial temperature was approximated in the concrete medium by

a linear function u x = ax + b, Eq. 9-2(3),

as in Solution V, and in the earth medium by the

function u 2 = b + cx exp (-mx), Eq . 9-2(4), shown

in Pig. 9-4.1 (See Table 9-*+.0). This figure

is completely analogous to Fig. 8.4.1. It can be

seen that the function u a has a maximum corresponding to that suggested by the observed curve, and

subsequently decays exponentially to the value b

at infinite x. This is probably closer to the

actual situation for a much greater distance into

the earth, and should improve the solution for

times greater than one week.
- 70 -

DISTANCE

IN

REGION

-FEET
-71-

table 9.4; o

VALUES OF CURVE-FITTING CONSTANTS

FOR SOLUTION VI

Curve-Fitting Constants

Test Surface a b c m

3 North 2.40 66 .

6

5.23 0.20

West 1.65 65.2 5.96 0.30

South 1.00 61.7 6.85 0.20

East 1.00 65.0 5.79 0.20

Floor 1.40 63.8 2.99 0.50

4 North 0.30 66.1 2.09 0.10

West 0.00 66.0 1.46 0.15

South 0.00 66.0 1.57 0.10

East 0.00 65.8 1.83 0.10

Floor -1.00 64.7 0.00 0.00

5 North 0.00 45.2 -1.25 0.15

West 0.00 46.0 -0.60 0.15

South 0.00 46.0 -1.15 0.15

East 0.00 45.6 -1.39 0.10

Floor 1.00 48.3 -1.87 0.10
-72-

9.5 Results

Figures 9.5.1 and 9*5.2 are exactly analogous to the

corresponding figures of Solution V. The curve for NBS

Test 5 is omitted, since it is almost identical with that

in Figure 8.5.3.

These results were also obtained with the aid of the

digital computer, at approximately the same running time.

The values of the curve-fitting constants used in the

computations are listed in Table 9.5.0.

9.6 Discussion

Inspection of Figures 9.5.1 and 9.5.2 shows the same

general results concerning thermal properties found in

Solution V, except that Curves 2 and 3 of Figure 9.5.2 do

cross near the end of the test, indicating that the earth

has begun to take over a little earlier than in Solution

V. Since Solutions V and VI were derived on assumptions

identical except for the initial temperature function u

wide variations in results were not expected.

Solution VT represents the last investigation of

one-dimensional, semi-infinite region, constant heat flux

shelter models. A general discussion and comparison of

all the solutions presented in this report follows in the

next section.
- 73 -

ELAPSED

TIME

-

DAYS
- 74 -
- 75 -

The

cance of

for four

1 )

2 )

3 )

4 ) and also

3)

6 )

10. GENERAL DISCUSSION

purpose of this section Is to discuss the signifithe assumptions made in the Individual solutions

basic parameters, namely:

geometry

composition of region

initial temperature

thermal properties,

the questions of

shelter irregularities

treatment of the complete problem.

10.1 Geometry

The first three solutions provide a means of comparing

the effects of the basic geometries considered, since they

differ only in this respect. Figures 10.1.1, .2, and .3

show surface temperatures versus time predicted by Solutions I, II, and III for NBS Tests 3 , and 5* based on

the preferred earth thermal properties. The equivalent

radii used in Solutions II and III were determined on the

basis of equal surface areas. Since Solution I, unlike

the other two, represents a one-dimensional model and is

applicable only to the individual surfaces, it could still

be included in this comparison by assuming all six surfaces

to be In one plane and using the same overall average

values of heat flux and initial temperature.

It can be seen from these figures that Solution III,

based on the cylindrical approximation, produces the most

accurate predictions for two-week periods. As stated in

Section 6.3* this solution might still be best for threeweek periods and even longer with increasing size. In the

case of large shelters, because of their shape and shallow

depth, spherical approximations would not be applicable.

With increasing size, the one-dimensional solutions become

Increasingly better.
- 76 -

6

Os

ELAPSED

TIME

-DAYS
- 77 -

C\J

o

GO

<0

stCVJ

C\J

6

ia*

Ll

d 0

4 n-3dniVd3dl^31 30VddflS

o

ELAPSED

TIME

-

DAYS
- 78 -

roO

S d*

iZ

ELAPSED

TIME

-

DAYS
-79-

10.2 Composite Region

In order to examine the significance of the composite

assumption, it is necessary to compare Solutions I and IV.

This comparison, however. Is complicated by the question

of which set of thermal properties in Solution I corresponds to which two sets in Solution IV. Figure 10.2.1 provides a comparison on the basis of the preferred concrete

and earth thermal properties. The west wall region of NBS

Test 3 was chosen as a typical case. This figure is a reproduction of Curves 2, 3 > and 4 of Figure 4.4.2, and

Curve 1 of Figure 7.5.2.

Inspection of Figure 10.2.1 shows Solution IV, Curve

1, lying between two cases of Solution I, Curves 2 and 4.

The composite assumption lowers the predicted temperature

toward Curve 4 for small values of time, when concrete

dominates, and raises the prediction toward Curve 2 for

large values of time, when the earth properties are of more

significance. This effect is more clearly observed by comparing Solution IV with Solution I, Curve 3 , which is based

on an intermediate set of thermal properties.

It can be seen that some intermediate combination of

thermal conductivity and dlffusivity will translate Curve

3 into a position of optimum agreement with Curve 1, but

never into complete agreement, since the times under consideration are neither small enough to approximate the

whole region by concrete nor large enough to approximate

It by earth. This result may or may not be true for the

cylindrical and spherical models. While it seems likely

that the effect of the concrete would be less in these

models, because of the increasing mass of earth per unit

surface area within a given distance from the surface, any

differences could only be determined by solving the appropriate composite models.

It was first pointed out In Section 7.6 that in the

one-dimensional geometry, the earth does not dominate the

heat flow sufficiently, after two weeks, for the whole

region to be treated as earth. This fact can be made

plausible by calculating the relative amounts of heat

stored in the concrete and earth, according to Solution

IV. The heat stored in each medium Is simply the integral

of the temperature rise over the medium. Thus, the heat

stored in the concrete is:
- 80 -

CVJ 0

1 d>

L

ELAPSED

TIME

-

DAYS
- 81 -

o

Q*(t) PiCx e 1 (x,t)dx

-l

( 1 )

and the heat stored In the earth

Q 2 (t)

00

O

( 2 )

The temperature rise, e, and e a , can be obtained from Eqs.

7.2(11) and (12), giving

00 Q T

( t) = F0t V ( „Q) n

| 49i2 erfc + 4is erfc IL - 4{R+l)i2 erfc n

'

r"^

1 n=Q T

2

' (3)

and Q s (t) = FQt( 3+1) Y(-0)

n

4i2 erfc n+|

W w

Since F0

t is the total heat per unit area lost at time t,

it follows that the fraction stored in the concrete is the

coefficient of F0 t in Eq. (3)* giving 24 percent when

evaluated at t = 336 hours. It may be noted that this

agrees fairly well with an estimate made in reference [1]

,

based on measured temperatures in NBS Test 3- That estimate

established the fraction of heat stored in all the concrete,

including the shielding wall, at about 26 percent of the

total heat stored within 4 feet of the shelter.
-82-

10.3 Initial Temperature

Figure 10.3.1 shows surface temperatures versus time

predicted by Solutions IV, V, and VI for the north wall

region of NBS Test 3* based on the preferred sets of thermal

properties. Figure 10.3.2 shows similar curves for the west

wall region. These figures facilitate a comparison between

observed performance and three one-dimensional solutions

which differ only in the form of the initial temperature.

Figures 10.3.1 and 10.3.2 show the effect of the particular temperature distribution which existed at the start

of Test 3 on the predicted surface temperatures. The assumed

temperature at the shelter surface for Solutions V and VI

were as much as 5.6°F lower than the average value used in

Solution IV, resulting in generally lower predictions from

these latter solutions for the first 6 to 8 days of the test.

However, the predicted surface temperature at the end of

14 days was higher from Solution VI than from Solution IV

and still higher from Solution V.

Solution V was derived with the simplest exponential

approximation to the initial temperature on the assumption

that the region beyond 4 feet would have little effect on

the surface temperature up to 14 days. The magnitude of

the difference in the predicted temperatures at 14 days’

time from Solutions IV and V, however, suggested that the

effect of the earth temperature beyond 4 feet might be significant. It is probable that the initial temperature more

than 4 feet from the shelter surface for Test 3 decreased

as indicated in Figure 9. 4 , 1 . [l]. The effect of this latter

assumption regarding the initial earth temperature beyond

4 feet is seen, from a comparison of Solutions V and VI, to

be a decrease in shelter surface temperature of a little

more than 0.5°F after 14 days.

Solutions V and VI represent successive refinements

in the assumptions regarding the initial earth temperature.

It appears that the effect of these refinements is not large

after the first week of occupancy. Since a more nonuniform

initial temperature distribution than that observed for Test

3 is not likely to occur in undisturbed earth, the initial

temperature can be considered uniform for design purposes,

unless the first week of occupancy is of comparable importance to the second week.
- 83 -

C\J

o

00

iD

00

o

PO

o

I. O'

L

ELAPSED

TIME

-

DAYS
- 84 -

CVJ roo

s O'

IL.

ELAPSED

TIME

-

DAYS
-85-

Since the composite nature of the region and the nonuniform initial temperature have been taken into account

in Solutions V and VI, the steepness of the curves in

Figures 10. 3.1 and 10.3.2 after the first week is due

chiefly to the one-dimensional geometry.

10.4 Thermal Properties

It was observed in Sections 4.6 and 7.6, in discussing

one-dimensional Solutions I and IV, that higher thermal

properties for the conduction region produced better agreement between predicted and observed temperatures in most

cases. The north wall region was an exception to this trend.

Since heat flows in one direction only in a one-dimensional

model, artificially high thermal properties were required

to produce reasonable temperature predictions.

It seems probable that the NBS family shelter, without

widely different width, length and height, would be best

represented by a spherical cavity, especially if it were

located deep in the earth. However, the comparison of

observed and predicted surface temperatures by Solution II,

(See. Fig. 5.4.1 and 5.4.2), indicate that the heat conduction region around the actual shelter was a less effective

heat sink than that assumed for the spherical model during

both the summer and winter tests. This was probably due to

the effect of the relatively thin layer of earth over the

shelter and the exposure to atmospheric conditions at the

earth's surface and also to the difference in shape of the

cavities. Thus, artificially low thermal properties were

required for Solution II to best approximate the observed

shelter surface temperatures.

Since the cylindrical model provided a larger heat sink

per unit shelter surface area than the one-dimensional model,

and a smaller heat sink per unit shelter surface area than

the spherical model, it tended to produce the best agreement

between predicted and observed surface temperature with the

preferred, or most probable, thermal properties of the heat

conduction medium.

In NBS Tests 3 and 4, significant absorption by the

concrete walls of existing condensation would have raised

the thermal properties; however, this possibility was minimized by the application of two coats of rubber-base paint.

In NBS Test 5, due to the melted snow, the thermal properties

of the earth could have been higher than the preferred set,

but this was not indicated by the diffusivity determination

discussed in Section 3*3.
- 86 -

The models used for this analysis implicitly assumed

that the heat conduction in the concrete and earth was linear

in nature, i.e., the thermal properties did not vary with

temperature, so the classical linear heat flow theory was

applicable. While it may be that the temperature dependence

of the thermal properties could be neglected for the gradients encountered in these tests, these gradients nevertheless

may have caused some moisture migration in the soil. This

migration would have acted both to transfer sensible and

latent heat directly, and to change the thermal properties,

which are much more sensitive to moisture changes than to

temperature changes [7]. The result is a problem of combined heat and moisture flow which involves coupled, nonlinear, partial differential equations, amenable to solution

by numerical analysis [13]

Although a few measurements of moisture content of the

earth were made during the course of the five NBS tests,

the data did not indicate whether significant moisture

transfer occurred. It probably was not sufficient to cause

significant changes in the thermal properties of the earth.
10.5 Shelter Irregularities

Throughout this analysis, both heat conduction media

were assumed to be homogeneous, and in the case of the

earth. Infinite in extent. These conditions were not fully

met by the NBS test shelter and site, due to certain

irregularities In the walls and earth. The deviations for

each region are listed as follows:

1) north wall region — a concrete road was located

about ten feet from the wall, which might

actually have acted as a heat source, storing

heat from the sun. This additional heat source

would tend to make the surface temperature rise

faster, requiring lower thermal properties to

make predictions agree with observations (cf.

Sections 4.6 and 7.6). The hatchway exit,

adjacent to this wall, might have affected the

distribution of heat flux over the wall. The

shielding wall might also have affected the heat

flow distribution, by interfering with the radiation component of the internal heat exchange.

This wall also contained the metal exhaust pipe,

which would tend to conduct some heat directly

to the air. It is for these reasons that this

region was designated the least typical.

2) west wall region — the south half of this wall was

probably exposed to more heat than the north half,

due to the arrangement of simulated occupants and

shielding wall, but since it faced uphill on a site

of 12 percent grade, it was exposed to more earth

than any other wall. This region was designated

the most typical.

3) south wall region — the ventilating air supply

system used during the NBS tests was located in

the shielding wall and aimed at the south wall.

Furthermore, three simulated occupants were located

near this wall. This arrangement possibly affected

the heat flux distribution. A spare ventilating

air supply and exhaust system, installed in this

wall, would tend again to conduct some heat directly

to the air.
- 88 -

4) east wall region — this wall faced downhill and

thus was exposed to less earth than any other wall.

In fact, the original grade line was about one

foot below the top edge of this wall. It was

consequently the one most affected by weather conditions. Also, the hatchway portion of this wall

received less heat than the rest.

5) floor region — this region had a truly infinite

exposure of earth. However, the hatchway portion

of the floor received less heat, and the arrangement of the simulated occupants and the supply

air stream might have affected the heat flux

distribution.

10.6 Shelter Environment

In this report, the conduction problem has been treated

as separate from the rest of the shelter-environment system,

using the known heat flux at the surfaces. While this procedure is not invalid, it is Incomplete in the sense of

omitting that part of the heat transfer process between the

Internal heat sources and the walls. A more complete treatment would take into account these sources, namely, ventilation and occupants. In such a problem, it would not be

possible to treat each surface separately, as was done in

the one-dimensional solutions, because the heat losses to

all the surfaces are interrelated through the common sources

In that case, the simplest approach would be to use a single

symmetrical model such as the cylinder or sphere, and

according to Section 10.1, the cylinder is the better choice

A cylindrical solution with heat source boundary conditions is available [ 143, and has been expanded and applied

to the NBS shelter. However, the composite problem has not

been found in a literature search, and the composite assumption may be important as seen in Section 10.2.
-89 -

When the conduction problem Is treated separately,

the roof region can be accurately represented by a onedimensional finite layer, with boundary conditions at both

ceiling and ground surfaces. (Three such solutions have

been developed, but are not included in this report.)

Because of the proximity of the ground surface, the heat

flow from the four walls is chiefly two-dimensional, except near the floor. The floor is exposed to the most

earth of all the surfaces, and heat flow in that region

is truly three-dimensional. These facts suggest that the

overall effect might be two-dimensional, adding weight to

the choice of cylindrical geometry as the best simple

model

.

It should be noted that none of the one-dimensional

solutions in this report are really applicable to the

floor region, because of its three-dimensional behavior.

However, since the floor region was approximately semiinfinite, the predicted temperatures were included in the

tables for the sake of completeness.

A variety of more elaborate models might be considered

which would explicitly take into account the asymmetry

caused by the ground surface. An example might be a closed

hemispherical cavity convex downward located at a finite

depth within a semi-infinite region. This model would

probably exhibit a more desirable heat flow pattern, approximately one-dimensional in the roof region, two-dimensional

at the sides, and three-dimensional at the bottom. Models

of this type are, of course, mathematically complex.
-90-

11. CONCLUSIONS

On the basis of the analyses covered by this report,

it is concluded that a cylindrical model provides the best

predictions of the surface temperatures in a small underground shelter with limited earth cover, probably because

it most nearly compensates for the effect of the finite

region over the cavity. It is probable that a spherical

model would be better if the small shelter were considerably

deeper in the earth. The one-dimensional model was less

effective in predicting temperatures in such a small shelter,

but would probably be an adequate representation for larger

shelters in which corner effects were relatively of less

importance

.

The composite nature of a concrete shelter surrounded

by earth was shown to have a small effect on shelter surface

temperatures. The higher thermal properties of concrete as

compared to earth caused the shelter surface temperatures

to be a little lower than if the entire conduction region

was comprised of earth. This effect was slightly greater

during the first week of occupancy than during the second.

A heat conduction region comprised entirely of concrete

might reduce surface temperatures as much as 3°F in a small

shelter as compared to a shelter with only moist earth as a heat sink. This disparity would be even greater if the earth

medium was essentially dry.

The initial distribution of temperature from the shelter

surface to a distance of 4 feet into the surrounding medium

was found to make an appreciable difference in the surface

temperature during the first week of occupancy, but the effect was of little significance after two weeks.

The solutions described in this report do not take into

account the effect of moisture migration on heat conduction,

but these effects were believed to be small in the test shelter.

Further efforts toward fulfilling the overall objectives

of this analytical study should begin with models that take

into account heat generation and ventilation of the shelter,

the finite roof cover, and the composite nature of the shelter surroundings

.
91-

appendix A

INITIAL TEMPERATURE DISTRIBUTION

IN

SHELTER CONCRETE

AND

SURROUNDING EARTH MEDIUM

FOR

FIVE NBS SHELTER TESTS

CONDUCTED BETWEEN

August 13, 1959 and April 8, i960
Tt.MPfc.NA

I

UNfc

92

NBS TEST 3

INITIAL TEMPERATURE DISTRIBUTION IN
TEMPERATURE

-°F

93

NBS TEST 4

INITIAL TEMPERATURE DISTRIBUTION IN HEAT CONDUCTION REGION ON OCT. 6 , 1959

DISTANCE IN CONCRETE

INCHES

DISTANCE IN EARTH FROM EXTERIOR

SURFACE OF CONCRETE , INCHES

FI©. A 2
94

MBS TEST '5 INITIAL TEMPERATURE DISTRIBUTION IN HEAT CONDUCTION REGION ON MAR. 25, I960
-95-

APPENDIX B

Heat Flux

The mathematical solutions in this report are based

on a constant ( time -independent ) heat flux, applied uniformly over the entire bounded surfaces of each mathematical model. These are conditions which would appear

more readily applicable to a sealed-up shelter, and which

would fail to describe the case of ventilated shelters,

whose heat flux would not be expected to be either constant or uniformly applied. Thus, in order to deal

properly with the constant flux parameter appearing in

each solution, it was necessary to extract numerical

values from the experimental data based on methods considered to be compatible with the constant flux assumption.

Three methods were considered and each is discussed below,

one of which required the derivation of a new solution.

Method 1 Constant Heat Flux

Each solution based on this assumption required that

a single value of heat flux be used for all values of

time. This value could reasonably be selected as the

overall average value, shown in Table B.l at 336 hours,

and which is close to the experimental value observed

during approximately the seventh day of each test, for

any surface. However, the observed performance diverges

from this value for ocher times.

Method 2 Cumulative Average Heat Flux

This is the method used throughout the report. The

values of heat flux were cumulative averages, values which

are essentially time-dependent, averaged over the interval

from o to t, and considered to be constant in that

interval. The smaller intervals gave a better overall

approximation than Method 1 because the values thus obtained were distributed such as to incorporate the observed exponential decay feature. Even though this

method was obviously a deviation from the models, it was

nevertheless considered to be a reasonable compromise
" 96”

between the mathematical assumption and the experimental

observations. The predicted curve determined on this basis

is then actually a curve drawn through points obtained by

solving each solution eight separate times, each time varying the heat flux in accordance with the cumulative average

values listed in Table B.l.

Method 3 Prescribed Heat Flux

This method would have required the derivation of

each solution on the basis of an actual time-dependent

heat flux boundary condition, but, as previously mentioned,

this was considered to be beyond the scope of this report.

However, in order to determine the effects of the mathematical simplifications involved in Method 1, and of the

" compromise " made in Method 2, Solution I was reworked by

replacing Eq. 4.2(3)* the constant flux condition, by

a prescribed flux condition approximating the experimental situation of exponentially decaying heat flux.

This solution is identified as Solution IA.

The temperature rise 0 at the interior surface x

is then

( 1 )

( 2 )

where D(//mt) represents Dawson's integral^ equal in this

case to the imaginary part of the complex error function.

Both functions are tabulated in [l 5]*
-97-

Figures B.l and B.2 illustrate predicted surface

temperatures obtained from Methods 1, 2, and 3* together

with the observed data, for the north and west wall regions

of NBS Test 3. In both cases, the prescribed flux solution.

Solution IA, yielded results in closer agreement with the

observed temperatures at fourteen days . Method 1 is generally better for the first seven days because the constant

flux value used was lower than or about equal to the observed values for those times. At the end of fourteen

days, the deviations in temperature between Methods 2 and

3 were less than 1°F for the west wall and practically

negligible for the north wall. For times larger than 336

hours, results from Methods 2 and 3 will tend to merge,

that trend being already visible in Figure B.l, so that

the greatest deviation is probably encountered at times in

the neighborhood of fourteen days or less.

A similar improvement using Eq.(l) could be made to

the other solutions in this report, but the likelihood of

uncovering deviations much wider than those illustrated

by this solution appears doubtful.
Time

24

48

72

96

120

168

240

336

Time

24

48

72

96

120

168

240

336

Time

24

48

72

96

120

168

240

336

-98-

TABLE B.l

CUMULATIVE AVERAGE VALUES OF

HEAT FLUX FQ

Test 3

No rth West South East Roof

2.06 4.41 8.32 4.78 3.60

2.16 4.02 7.09 4.44 3.34

2.15 3.70 6.21 4.13 3.18

2.02 3.56 5-57 3.90 2.97

1.93 3.41 5.10 3.71 2.86

1.85 3.25 4.50 3.46 2.80

1.76 3.04 3.99 3.22 2.70

1.65 2.84 3.53 3.01 2.36

Test 4

North West South East Roof

2.65 4.78 5.49 4.71 3.96

2.63 4.61 4.97 4.42 3.45

2.53 4.42 4.64 4.18 3.15

2.48 4.34 4.40 4.02 2.93

2.45 4.28 4.26 3.90 2.77

2.31 4.10 3-98 3.74 2.41

2.07 3.87 3.63 3.42 2.27

1.83 3.69 3.35 3.18 2.36

Test 5

No rth West South East Roof

1.81 4.19 6.38 5.74 4.57

1.82 3.99 6 . 08 5.35 4.27

1.85 3.93 5.96 5.26 4.16

1.89 3.89 5 . 88 5.18 4.01

1.90 3.84 5.79 5.10 3.80

2.02 3.91 5.80 5.09 3.44

2.09 3.90 5.74 4.97 2.99

2.06 3.86 5.65 4.76 2.80

Floor

5.24

5.17

5.12

5.02

4.90

4.66

4.50

4.24

Floor

4.70

4.51

4.40

4.35

4,32

4.23

4.04

3.90

Floor

1.73

1.74

1.76

1.78

1.81

1.95

2.08

2.19
SOLUTION

IA

-

NBS

TEST

3

PREDICTED

SURFACE

TEMPERATURES

FOR

NORTH

WALL

REGION

K

=

0.75

CC

=

0.026

- 99 -

00

•*ro

CO

LlJ >

1

<r3

O 3

O O

Q

LlJ go

cru

co

LlI cr

CL

CD

I CO

OJ K> Q

O

X

hLl!

00

— CO

(X

•o

o

00

COh3

r-

•sf- do‘n- 3dfUVd3dlAI31 30VJdDS

o

00

NELAPSED

TIME

-DAYS
SOLUTION

IA

-

NBS

TEST

3

PREDICTED

SURFACE

TEMPERATURES

FOR

WEST

WALL

REGION

K

=0.75

&

=

0.026

- 100 -
- 101 -

APPENDIX C

Numerical Example

The example worked out below for Solution IV is the

type of computation necessary to obtain numerical results

from any of the first four solutions contained in this

report

.

To compute the interior surface temperature u a

> at x = -/

,

for the north wall of NBS Test 3 at 336 hours, Eq. 7.2(13)

is manipulated as follows.

Rewriting the summation portion of the equation in a slightly more convenient form yields

The factor of 2 was brought inside the summation sign because

the integral error function of the first order is usually

tabulated in this form. Numerical values for this function,

as well as for those of higher order appearing in Solution III,

are tabulated in [15] and in Appendix II of [10].

Expanding summation (l) to produce sufficient convergence

for the desired accuracy for the particular values of time,

wall thickness, and thermal properties involved will require,

in this example, the first four terms, and a convenient

arrangement for listing these terms is shown below. At

n=o, 1, 2, 3 . .

.

,

the summation reduces to the following:

n

( 1 )

n = o

n = 1
- 102-

n = 2, 202 ierfc -2 1 „ 20

3 ierfc

^ T"5 ^ _3

-

n = 3 , -203 ierfc ! —y 1 + negligible terra

T

2

Eq. 7.2(13) can be rewritten* combining the above

terms* to read

u i = uo + — f

2 ierfc (o)

Ki

1 26

[

2 ierf0

(^)]

+ 202

[

2 ierfc

(fi)]

2P ^2 ierfc

^ + • • • j

- (2)

The following substitutions can now be made in the above

expression to obtain the required numerical result:

Prom Table 3 • 1- 1?

From Table B. 1,

From Table 7.5.1*

Thickness of concrete.

u0 = 71.7

F0 = 1.65

K l = 1.15

a x = 0.036

a = 0.767

l = 0.667

Therefore* = 27 ® 216

and B = — = -0.1316

cr+1
-103-

Substituting in (2), we get

ua = 71.7 +

( I.65 )( O.667)

1.15

+ 0.03^6(0.5235) + 0.0046(0.3321) + ...

Uj = 71.7 + 6.76 = 78.46 or 78.5°F

,

the predicted value shown in Table 7-5.1.

It should be noted that Solutions V and VI are too

cumbersome for the type of manipulation shown in the above

example, especially when many computations are required.

Even when convergence is fairly rapid, the use of a digital

computer is almost a necessity. As indicated in Sections

8.5 and 9*5* these two solutions were programmed in

Fortran language for use on the IBM 7090 computer, and

these programs are available from the Mechanical Systems

Section of NBS.
APPENDIX D

References

Achenbach, P.R., Drapeau, F.J.J., and Phillips, C.W.

Studies of Environmental Factors in a Family-Size

Underground Shelter. Mechanical Systems Section,

Building Research Division, National Bureau of

Standards, Washington, D. C., March 1961.

Proceedings of the American Concrete Institute,

Volume 3^9 May-June 1938*

American Society of Heating, Refrigerating, and AirConditioning Engineers, Guide and Data Book, 1961.

Marks, L.S., Mechanical Engineers' Handbook, Fifth

Edition, 1951.

Thermal Properties of Concrete, American Society of

Heating and Ventilating Engineers, Research Bulletin

No. 2, Volume 53 9 September 1947.

Ingersoll, L.R., Zobel, O.J., and Ingersoll, A.C.,

Heat Conduction, University of Wisconsin Press,

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c*