NBS BUILDING SCIENCE SERIES 156

Annual Variation

of Temperature Field and Heat Transfer

TA—T-jr Heated Ground Surfaces

435

.U58 Dn-Gradc Floor Heat Loss Calculation)

to. 156

1983

Q 2 TMENT OF COMMERCE • NATIONAL BUREAU OF STANDARDS "»«T»*
NATIONAL BUREAU OF STANDARDS

The National Bureau of Standards' was established by an act ot Congress on March 3, 1901. The Bureau's overall goal is to strengthen and advance the Nation's science and technology

and facilitate their effective application for public benefit. To this end, the Bureau conducts

research and provides: (1) a basis for the Nation's physical measurement system, (2) scientific and technological services for industry and government, (3) a technical basis for equity in trade, and (4) technical services to promote public safety. The Bureau's technical work is per- formed by the National Measurement Laboratory, the National Engineering Laboratory, and

the Institute for Computer Sciences and Technology.

THE NATIONAL MEASUREMENT LABORATORY provides the national system of

physical and chemical and materials measurement; coordinates the system with measurement

systems of other nations and furnishes essential services leading to accurate and uniform

physical and chemical measurement throughout the Nation's scientific community, industry,

and commerce; conducts materials research leading to improved methods of measurement,

standards, and data on the properties of materials needed by industry, commerce, educational

institutions, and Government; provides advisory and research services to other Government

agencies; develops, produces, and distributes Standard Reference Materials; and provides

calibration services. The Laboratory consists of the following centers:

Absolute Physical Quantities^ — Radiation Research — Chemical Physics —

Analytical Chemistry — Materials Science

THE NATIONAL ENGINEERING LABORATORY provides technology and technical ser- vices to the public and private sectors to address national needs and to solve national problems; conducts research in engineering and applied science in support of these efforts; builds and maintains competence in the necessary disciplines required to carry out this research and technical service; develops engineering data and measurement capabilities;

provides engineering measurement traceability services; develops test methods and proposes

engineering standards and code changes; develops and proposes new engineering practices;

and develops and improves mechanisms to transfer results of its research to the ultimate user. The Laboratory consists of the following centers:

Applied Mathematics — Electronics and Electrical Engineering^ — Manufacturing

Engineering — Building Technology — Fire Research — Chemical Engineering^

THE INSTITUTE FOR COMPUTER SCIENCES AND TECHNOLOGY conducts

research and provides scientific and technical services to aid Federal agencies in the selection, acquisition, application, and use of computer technology to improve effectiveness and

economy in Government operations in accordance with Public Law 89-306 (40 U.S.C. 759), relevant Executive Orders, and other directives; carries out this mission by managing the Federal Information Processing Standards Program, developing Federal ADP standards

guidelines, and managing Federal participation in ADP voluntary standardization activities; provides scientific and technological advisory services and assistance to Federal agencies; and

provides the technical foundation for computer-related policies of the Federal Government.

The Institute consists of the following centers:

Programming Science and Technology — Computer Systems Engineering.

'Headquarters and Laboratories at Gaithersburg, MD, unless otherwise noted; mailing address Washington, DC 20234. 'Some divisions within the center are located at Boulder, CO 80303.
PsIBS BUILDING SCIENCE SERIES 156

Annual Variation

of Temperature Field and Heat Transfer

Under Heated Ground Surfaces

(Slab-on-Grade Floor Heat Loss Calculation)

SEP 1 5 1985

T. Kusuda, O. Piet, and J. W. Bean

Center for Building Technology

National Bureau of Standards

Washington, DC 20234

U.S. DEPARTMENT OF COMMERCE, Mjilcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

Issued June 1983
Library of Congress Catalog Card Number: 83-600539

National Bureau of Standards Building Science Series 156

tl. Bur. Stand. (U.S.), Bldg. Sci. Ser. 156, 67 pages (June 1983)

CODEN: BSSNBV

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1983

For sale by the Superintendent of Documents, U.S. Government Printing Office Washington, DC 20402

Price (Add 25 percent for other than U.S. mailing)
Annual Variation of Temperature Field

and Heat Transfer Under Heated Ground Surfaces

(Slab-on-Grade Floor Heat Loss Calculation)

by T. Kusuda, 0. Piet-'^,

Center for Building

National Bureau of

and J. W. Bean

Technology

Standards

Abstract

Seasonal sub-surface ground temperature profiles and surface heat transfer were

determined for the condition whereby one and more than one region of the

earth's surface temperature were disturbed. The analysis was conducted by

numerical integration using a closed form solution based on the Green's

function. Monthly profiles of earth temperature isotherms under a house of

20' X 20' (6.1m X 6.1m) floor area and under a ground of six houses near a

wooded area are presented. The heat losses calculated from this approach for

square slabs of various sizes were compared with those derived from the recent

analytical solution of Delsante et al» resulting in good agreement.

The Delsante solution, which is based upon a Fourier Transform technique, is

then extended to generate the frequency domain thermal response factors suitable

for the periodic heat transfer calculation for multi-layer slab floors on grade.

In the appendix, this thermal response factor method was used to generate annual

cycles of monthly heat loss from several slab floor constructions shown in the

1981 ASHRAE Handbook of Fundamentals. The maximum values of these monthly slab

floor heat losses agree relatively well with the ASHRAE design values.

Key words: ASHRAE design values; building heat transfer; Delsante method;

earth temperature; slab-on-grade heat transfer; soil temperature.

*/ Guest worker from Ecole Nationale des Fonts et Chaussees, France.

iii

Contents

Page

Nomenclature and Introduction 1

Lachenbruch Solution 5

HEATPATCH (Computer Program) 6

Delsante Solution 16

Steady State Solution - I; Equations (23) and (24) 28

Steady State Solution - II; Equation (24) 29

Steady State Solution - III; Equation (25) 30

Steady State Solution - IV; Equation (26) 34

A Simplified Procedure (PC-1) 35

Comparison Between the HEATPATCH and PC-1 35

Daily Thermal Cycles 41

Conduction Transfer Functions for Composite Floor 50

Suimnary and Discussion 56

References 57

Appendix 59

List of Figures

Figure 1. Surface grid design for the HEATPATCH calculation: example of

six houses near forest.

Figure 2. One quandrant of the surface temperature grid used for the square

patch heat transfer analysis.

Figure 3. Annual average earth temperature profile under a house with

slab-on-grade floor.

Figure 4. Monthly earth temperature profile under a house with slab-on-grade

floor.

V
Figure 5. Annual average earth temperature profile under six houses and a forest (site plan, see fig. 1).

Figure 6, Monthly earth temperature profile under six houses and a forest

(site plan, see fig. 1).

Figure 7, Temperature profile in the perimeter zone.

Figure 8. Absolute value of the Delsante's Perimeter Function I(Z).

Figure 9. Heat transfer from one-sided (semi- infinite) floor slab.

Figure 10. Two-dimensional slab heat transfer function.

Figure 11. Rectangular slab of dimension 2a x 2b with perimeter width of 2e.

Figure 12. Thermal resistance of square slabs with respect to slab sizes.

Figure 13. Thermal resistance calculated by for slabs of different

aspect ratios.

Figure 14. Correction factors for the non-square slab heat loss calculations.

Figure 15. Comparison of annual average floor heat flux values calculated by

the HEATPATCH method and by the Delsante formula.

Figure 16. Annual heat flux cycle from 20'x20' (6.1m x 6.1m) slab floor with

different perimeter widths.

Figure 17. Annual heat flux cycle from AO'xAO' (12.1m x 12.1m) slab floor with

different perimeter widths.

Figure 18. Annual cycle of heat flux from square slab floor of different sizes

Figure 19. Annual heat loss cycle from 40'x40' (12.1m x 12.1m) slab floor for

different soil thermal dif f usivities

Figure 20. Comparison between the diurnal and annual heat flux cycles for a slab-on-grade floor under the constant indoor temperature condition

Figure 21. Comparison between the diurnal and annual heat flux cycles for

slab-on-grade floor when the indoor temperature fluctuates.

Figure 22. Diurnal cycle of hourly temperature and heat transfer from lO'xlO'

patch.

Figure 23. Heat Flux distribution over the floor lO'xlO' (0.31m x 0.31m)

segment of which undergoes a cyclic heating.

Figure 24. Diurnal floor heat flux cycle due to solar heat gain.

Figure 25. Diurnal heat loss cycle for 40'x40' (12.1m x 12.1m) slab floor.

vi
Annual Variation of Temperature Field

and Heat Transfer Under Heated Ground Surfaces

(Slab-on-Grade Floor Heat Loss Calculation)

Nomenclature and Introduction

Nomenclature : The following symbols are used unless otherwise noted.

a, a^ : half width of slab floor, m

A : an element of a transfer function matrix

b, b" : half length of a slab floor, m

B : an element of a transfer function matrix

c : specific heat, kJ/kg'K

2 C : thermal capacitance C = pc£, kj/m 'K

C : an element of a transfer function matrix

'VD : an element of a transfer function matrix

g : Fourier transform of the surface temperature

H : a variable defined in equation (29) in the text

i : complex number index i =

L : Green's function integral

I : perimeter function defined in equation (18)

k : thermal conductivity of soil, W/m'K

K. : integrated Bessel function

1

Z : thickness of solid, m

N : number of temperature cycle harmonics

P : perimeter of the slab, m

q : heat flow, W

r : distance between the surface point and a point in the earth, m

R : distance between the two surface points, m

2

S : area of the slab, m

t : time, hr

T : temperature, °C, or K

T^ : amplitude of the annual temperature cycle, K

Tg : average temperature of the slab above the annual

average outdoor temperature, °C

1
amplitude of the annual indoor temperature cycle, K

annual average earth temperature, °C

period of the periodic cycle, hr

Fourier transform variable

Fourier transform variable

coordinate system for the sub-surface point, m

integration variables used in equation (2)

frequency domain thermal response factors

2 thermal diffusivity of earth, m /day

Euler constant

defined in the text . . . equation (18)

half thickness of the perimeter zone, m

defined in the text = */—

f a

defined in the text = r

angular frequency = 2t\/T^, rad/hr

surface temperature distribution in the disturbed segment

heat flux function W/K

phase angle, radian

solid angle, steradian

angle, radian

density kg/m

equation (8)

Subscripts

floor slab

earth, ground or soil

inside the building

i-th harmonics

k-th layer

outside

°°

: undisturbed condition

1 : inside condition

2 : outside condition

Others

Variables with denote complex variables,

Variables with — denote average values.

2
Introduction : The subsurface temperature of the earth undergoing a seasonal

cycle depends upon the surface temperature condition, the thermal diffusivity

of soil, and the distance from the surface. When the ground surface is exposed

to open atmosphere without surface obstructions such as buildings, pavement,

forest, etc., the subsurface temperature may be determined by the well known

equation for the natural and undisturbed earth temperature [l]

T = T + E A.e sin (oo.t + Z * 2a + C,-) (1) CO Q n 1 ^X*^ ^'

^ i=l

where T^: Undisturbed earth temperature

T : Annual average surface temperature

Aj|^: Amplitude of the surface temperature variation

with angular frequency a)_^

Z : Depth from the surface

a : Thermal diffusivity of the earth (soil)

Phase angle or time delay

0)^: Angular frequency of the i-th harmonic of the

surface temperature variation

N : Total number of harmonics, usually N=l for annual cycle

This equation is based upon the solution for a semi-infinite heat conduction

system and shows that the surface temperature fluctuation will quickly diminish

at depths greater than two feet (Z=2' or 0.61m) unless the periods of harmonics

are extremely large (more than 24 hours)

When the segment of interest o£ the earth's surface is covered by forest,

building or pavement, this equation is no longer applicable because the surface

temperature of that segment experiences a different annual temperature cycle

than the undisturbed and open region. The influence of a temperature disturbance

on the subsurface temperature profile depends upon the size and shape of the

disturbance, the temperature characteristics (both with respect to time and
space) of the disturbance, and soil thermal diffusivity. For a long term

temperature disturbance such as that due to the erection of a building, paving ,

or vegetation, the earth temperature calculations for the annual cycle are of

the most interest. From the ecological point of view, it is important to know

the influence of the disturbance which will be felt far outside of the disturbed

area. In the study of heating buildings, it is also important to know the magnitude of the floor heat loss to the ground as a function of the season. The

latter can be determined by the gradient of the subsurface temperature near the

surface. The heat transfer from the disturbed surface region is strongly affected

by the manner in which the surface temperature changes from the disturbed region

to the undisturbed region (from the inside of the house to the outside) . If this

change is abrupt, an extremely large heat loss at the boundary of the disturbed

region is expected. In spite of the noteworthy work of numerous researchers

[2, 3, ^, 5, 6, T, 8, 9 and lO], comprehensive analysis of seasonal fluctuation

of the earth temperature and heat transfer for the heated ground surface have

not been available. This is because of the complexity of the three-dimensional

and transient heat conduction problem that characterizes the ground heat transfer

analysis.

The purpose of this paper is to describe sub-surface temperature variations under

disturbed ground surface (s) obtained by computer-based simulation. The computer

program used is based on the Green's function technique introduced by Lachenbruch

[h] . Heat loss from the slab-on-grade floor was determined and compared against

the results obtained from the analytical solutions of Delsante [ll]. Delsante

solutions are exact for the steady-state heat flow condition and approximate

for the periodic solution. This paper also describes a simplified slab-on-grade

heat loss calculation procedure, which is an extension of the Delsante solution

and is suitable for the multi-layered floor slab on earth.

4
Lachenbruch Solution

According to Carslaw and Jaeger [l] the basic equation describing the underground temperature affected by the surface temperature disturbance is

dt' (2)

o S

where (})(x ',y ',t = ground surface temperature at z = 0

a = thermal diffusivity of earth,

2 2 2 2 r = (x-x') + (y-y") + z

Lachenbruch [h] applied this formula for the case of a heated slab with a

Surface temperature cycle having an angular frequency of co such that

(t)(x%y',t') = Tg(x',y') + T^(x^yO sin wt' (3)

over an area S and

{t)(x%y%t') = sin oot^ (4)

outside the area S.

The Lachenbruch solution for this boundary condition can be expressed as

T(x,y,z,t) = i- / /

[T^ + (T - T )L] dQ (5)

2tt y y B c A

Q.

where Q, is the solid angle subtended by the surface segment S above the subsurface

point at x,y,z, for which the temperature is calculated, and where

—A

^ L = e {sin (wt^-X") + X[cos (wt^-X^ + sin (wt^-X^]} (7)

and

]^ " = -r . / 2a

r (8)

5
HEATPATCH (Computer Program)

A computer program was developed to determine the soil temperature profile under

the ground surface using the Lachenbruch integral over an elementary rectangle

surface segment. The program is called HEATPATCH and can simulate the existence

of several disturbed regions at the ground surface, each disturbance having a

seasonal cycle of its own different from that of the cycle of the naturally

exposed outdoor surface.

HEATPATCH assumes that the ground surface may be broken into a rectangular grid;

each of the disturbed areas is prescribed by the respective x and y coordinates of its boundary. Within a given boundary, surface temperature can be

varied as a spatial function. Figure 1 illustrates a typical surface grid

design where six heated patches measuring 20' (6.1m) square are located next

to a forest. The subsurface soil temperature can then be computed by superimposing temperature solutions under rectangular segments of the grid. Assuming that

Tg(x,y) and T (x,y) are constant over a finite rectangular segment AxAy, approxic mate integration may be performed based on equation (5),

, M N

T(x,y,z,t) Z E {Tg + (Tc . . - T )L } Af2 (9)

where each of the variables with subscripts i and j is evaluated at a finite

difference grid point x^^ and y^ , as illustrated in Fig. 2 for a quarter section

of a square symmetric region containing two different temperatures. The Afi in

equation (9) may be determined by integration of equation (6) over a rectangular

element of sides Ax and Ay, resulting in

. Ax Ay,

A^^ = G(x-x , y-y , y-^ 2^)

0/ - - -'^x Ay, - G(x-x , y-y , —^ ^) (10)

- .Ax -Ay\

- G(x-x , y-y , y~ ^ 2

+ G(x-x , y-y , 2 ^
TO

0)3o

CT3 U

3CUH<H<Wu • O 4-1

OJO

•H 4-1 \.

0) t-i » OJ ^ c

•H00 (U

CO

(u :3 o o

14-1

>-i -A

(U3

OO

in in 7
SURFACE GRID

1.25 2.50 3.75 5 6 7 8 9 10 11 12 14

18 19 20 21 2223 2425 28

2e

Figure 2. One quandrant of the surface temperature grid used for the square

patch heat transfer analysis.

8
where

_l (x+a) (y+b)

G(x,y,a,b) = tan r-r (11)

^ + (x+a)^ + (y+b)^

In order to study the subsurface temperature distribution in the form of isotherms, the earth temperature can then be calculated for any set of specified

points below the earth surface, such as a cross-section along the centerline

of the slab, along a slab edge or along a diagonal.

The HEATPATCH program may be used to examine various surface and temperature

configurations. A number of variations are possible:

1. The size and shape of the disturbed surface segments (or heat patches)

can be varied.

2. The number of the heated patches can be varied.

3. Each of the patches can assume independent mean temperature and amplitude.

4. Within a given patch, the surface temperature can be varied from point

to point.

5. Vertical cross-section isotherms under a specified line across the

ground surface can be studied.

6. Heat flux under specified surface segments can be calculated and averaged.

Figures 3 and 4 illustrate results using HEATPATCH to show the annual average

and month by month soil temperature distributions, respectively, under an isolated

square region representing the slab floor of a house. For this case it is assumed

that the 20'x20' (6.1m x 6.1m) slab was over earth having a thermal diffusivity

2 2 of 0.025ft /hr (0.056m /day) and undergoing an annual cyclical temperature variation with an average of T = 70°F (21.1°C) and an amplitude of T = 5.0°F (2.78 K)

B ^ The undisturbed or outdoor surface undergoes an annual cycle having an average

temperature of T = 56°F (13.3°C) and an amplitude of T = 20.6°F (11.44 K) . Fig-

§ A

ures 5 and 6 show the annual and monthly temperature variation under six houses

adjacent to a forest, the site plot of which is shown in figure 1. The

9
_o

01

u

CD

O

I—

I—

Q

O

01

cc

O

CD

4-1

COo

OJ

3

oa

0) u

cc

l-i (Ua

B

4-1 U

u o

03 O

OJ rH

•4-1 0)bO 0) CO -a

)-i cdu

> bO

CO I C

rH O

CO I d CO

01u

•H

P4

T

S OT SI

ij 'Hid3a

10
DISTRNCE, FEET MONTH 5

eOUNOflRT CONDITION NUMBER 1

DISTANCE, FEET nONTH 11

BOUNDfiRT CONDITION NUHBER 1

DISTRNCE, FEET CISTflNCE, FEET MONTH 6 MONTH 12

eOUNDflRr CONDITION NUMBER I 80UNDflRY CONDITION NUMBER 1

igure 4. Monthly earth temperature profile under a house with slab-on-gra

floor

11
house temperature and outdoor temperature cycles are the same as in

figure 4. The forest temperature was assumed to undergo an annual cycle with

the same annual temperature as the undisturbed exposed surface but with a much

smaller amplitude of 10°F (5.55K). The profile similar to figures 3 through 6

would also be determined for a plane normal to the surface and passing through

any line on the surface.

In HEATPATCH, the surface heat flow is approximated by the numerical differentiation formula such that

3T 1

q = ~ k[T(x,y,0,t) - T(x,y,£,t) ]/£ (12)

z=0

where k = thermal conductivity of soil

and £ = an arbitrary thickness of the subsurface layer.

The depth parameter £ in the numerical differentiation by equation (12) depends

upon the type of temperature distribution over the disturbed region. If there

is a very abrupt temperature transition such as in the case of a heated slab floor

during the winter, as shown in figures 4 and 6 the temperature near the perimeter

of the slab undergoes a rapid change in directions normal to the depth. In this

case, I must be chosen small enough to account for this large temperature gradient. In the extreme case, if the surface temperature transition from the slab

to the exposed outdoor surface is a step function, the slab floor heat flux at

the floor edge becomes infinitely large. In actual situations, however, slab

surface temperature would be expected to change rather gradually from the indoor

condition to the outdoor condition, as shown in figure 7 for the perimeter zone

defined in figure 2.

The two different types of transition temperature profiles shown in figure 7

were studied: the first is a linear change, while the second is a smooth

transition representing continuous derivatives. It was found that these two

12
I- o

S OT SI OZ SZ 0£

13
OlSTFiNCE, FEET MONTH 1

[STANCE, EEET MONTH 7

OISTRNCE. EEET MONTH 2

DISrflNCE, FEET MONTH 8

DISTANCE, FEET MONTH 3

72 5

72, 5

OISTRNCE, FEET MONTH 9

OISTRNCE, FEET MONTH -i

DiSTRNCE, FEET MONTH 10

OISTRNCE, FEET MONTH 5

67. 5

46.2

67. S 67.5

DISTANCE, FEET MONTH I 1

OISTRNCE. FEET MONTH 6

'^onno 65 7 iiii^iiiiV

DISTANCE, FECT nONTH 12 Figure 6. Monthly earth temperature profile under six houses and a forest

(site plan, see fig. 1).

lA
15
different transitional temperature profiles yield virtually identical results.

Only the linear transition is therefore used in the HEATPATCH analyses.

The choice of I for the numerical differentiation of equation (12) has to be

made in relation to the perimeter thickness 2e shown in figure 7. In order

to examine the relationship between I and 2e , comparative analyses were made

between the heat flow data calculated by HEATPATCH and exact solution developed

by Delsante for heated square slabs, details of which are given in the following

section

Delsante Solution

Recently Delsante solved the slab-on-grade heat loss problem by determining the

Fourier transform solution for the heat conduction differential equation such

as follows [9] '•

00 00

T(x,y,z,t).^ , / / e-i(>'^+^y'> e'^^^+^^^^^iu, v)dn iv (13)

-oo —oo

oo oo

_oo —oo

where

(14)

By differentiating this temperature equation with respect to depth, he obtained

a closed form analytical expression for the integrated average floor heat loss

as follows:

% f f dTl f f — -i (ux+vy) g(u,v)dudv

z=0 —oo —00

For an infinitely long slab of width 2b having the surface temperature profile

shown in figure 7,

'

16
q = —^— i -nXe 7 - K. (2A£) + K. {2Xk) - K, (2Ab) + K, (2>b)

4 1^ I3 ±1 13

(16)

+ K.^13 [2X(b+e)] - K. [2A(b+e)]} + 2bAkT

where and are generalized indoor/outdoor temperature cycles expressed in

complex variables. and are integrals of modified Bessel functions K^,

such that

(x) = /

(t)dt

r J r-1

X

r

00

J r-1

(x) = / (t)dt

and (17)

/

K (x) = / K (t)dt

1^ / o X

Since these Bessel functions are extremely difficult to evaluate, a perimeter

heat loss function I(z) is defined by Delsante as follows:

I(z) = [x - K. (z) + K. (z)]/zTT (18)

^1 3

Moreover, the authors of this paper developed an asymptotic approximation of

I(z) as follows:

I(z) = 1 [-Inz +6 + ^ z]

z^ %

+ ~ [Inz - ( 6 + 7/3)] 12

, z 61, [Inz - ( 5 + ^)] 960 " ^ " ' 30

^6 ry, + 80640 - 2lo)^

+ 9289728 ^^^^ " + ^ ^19)

17
where 6 = ln2 - y

Y in the above expression is the Euler constant (0.5772).

Substituting equation (18) into (16) and rearranging the terms in a dimensionless form, one obtains

* ^ 2k(T^-T^) = ^(2Ae) + ^ i(2Ab) - ^

I(2Ab + 2Ae)

or by letting 6^^ = 2Ae, = 2A(b+£), and 63 = 2Ab

2k(f^ = I(Ql> - I('^2) + ^

(1(^3) - 1(^2^) (20)

tinder the steady-state condition, A 0 and

^^-mf^'H '"X^) (21)

Equation (18), however, becomes divergent for large |z| as shown by the dotted

I I curve in Figure 8. For |z| ^3 it can be approximated by

I(z) z \ , 4z

which yields a smooth monotonieally decreasing function as indicated in figure

8. I(z) in fact represents a dimensionless slab heat transfer from a one-sided

(or semi-infinite) and infinitely long slab for very thin perimeter width 2e.

The use of I(z) also permits the evaluation of the effect of higher harmonics

upon the floor heat loss. To do this a complex plane representation of I(z) is

shown in figure 9 for various cyclic periods T" when the soil thermal diffusivity

? 2 is 0.025 ft /hr (0.56 m /hr) and the perimeter thickness is 1 ft (0.3m). As the

cyclic period decreases from the annual cycle to the hourly cycle, both imaginary

and real components of I approach 0. The absolute value of the perimeter heat

loss I(z) for a given periodic cycle is the vector connecting the point at a

given T' on the curve in figure 9 and the origin. The angle of the vector indicates the phase relationship between the temperature and the heat flow cycles.

This figure clearly shows that the daily outdoor temperature cycle (T"=24 hrs)

has a relatively small impact on floor heat transfer compared to the annual cyclej

18
19
20
Figure 10 on the other hand shows equation (20) on a complex plane for a 20'

(6.1m) wide and infinitely long slab floor with a fixed soil thermal diffusivity

and various perimeter thicknesses. Unlike figure 9 for T(z) this figure indicates that the imaginary component of the heat flux disappears as oj approaches

0 or A -> 0, leaving only the real and steady state component, which is expressed

by equation (21)

The above solution is applicable only to an extremely long floor slab, which

is without the corner effect of the rectangular slab. Delsante also evaluated

equation (15) for a rectangular slab such as shown in figure 11. In this calculation he assumed that the rectangular slab was divided into five zones; one

core zone and four perimeter zones. While the slab surface temperature T(x,y,o,t)

is assumed constant at an indoor temperature throughout the core zone A, it

varies linearly from the indoor temperature to the outdoor temperature in the

four perimeter zones, I through IV. Only the steady state solution has been

available so far, which amounts to the evaluation of the following convolution

integral

oo CO

q = 2k f f J (a-u)^+(b-y)^ ^ T(u,v)dudv (22)

i\ I I ^ (a-u) (b-v)

_oo _oo

This equation was derived from equation (15) by setting X -> 0,

Using the surface temperature profile T(x,y,o,t) shown in figure 3, this

integral was evaluated. The results are shown in equations (23), (24), (25)

and (26) in four different ways as (j)^ through 4)^.

While (f)-j^ is an original Delsante solution from reference [9] which is calculated

only for the area of zone A of figure 11, 4)^ was developed by the authors to

include the heat loss from perimeter zones I, II, III and IV of the slab. In

other words the use of for the slab heat loss calculation excludes the heat

conducted from the temperature transition or perimeter zone to the earth.

21


The authors of this paper contend, however, that the perimeter zone heat loss

should be a part of the total floor heat loss. This contention is especially

valid since the perimeter zone is usually much wider then the wall thickness;

detailed discussion for this subject will be given in a later chapter. The

solution ^ special case of (f)^ applicable only when a>>e. The solution

(j)^ is a special case of (p ^ when a = b (a square slab). Using these cf) functions,

the thermal resistance of slab-on-grade floors of different sizes and shapes can

be derived and depicted as shown in figures 12 and 13. In figure 12 it is shown

that the error by using the simplified expression <f)^, in lieu of the exact

formulation (j) , is small unless the slab size is small. Since (J)^ is extremely

complicated, it was used to generate correction factors for non-square slab heat

transfer to be applied to the values determined for square slabs by the much

simpler formulation (}) . The correction factors correlated with respect to the

4

P 2 hydraulic diameter A/ (^) agreed well with that determined by Muncey [6]. Figure

14 shows Muncey 's correction factor which is to be applied to the thermal resistances of various non-square slab floors.

24
PERIMETER THICKNESS, 2e, m

1 2

PERIMETER THICKNESS, 26, ft

Figure 12. Thermal resistance of square slabs with respect to slab sizes

25
PERIMETER THICKNESS, 2e, m

0.2 0.4 0.6

8 -

CO

w 6

00

Z 4 -

CO

00

0.8

• lO'x^ (3.05m>ixoo

o 10'x20' |3.05mx6.1m]

10'x30' |3.05mx9.15m|

£^ 10'x50' |3.05mxl5.2m|

A 10'xlO' |3.05mx3.05m)

i

1.5

1.0

E

00

0.5

00

0.5 1.0 1.5 2.0

PERIMETER THICKNESS, 2e, ft

2.5 3.0

Figure 13. Thermal resistance calculated by tf-^ for slabs of different

aspect ratios.

26
Applicable shapes for

slabs on grade

AREA

( 1/4 PERIMETER)^

Figure lA . Correction factors for the non-square slab heat loss calculations.

27
Steady State Solution - I Equation (23) and (24)

b a

* 9T(x,y,z)

q = -k / / 3Z

D a // dxdy

-b -a z=0

9V

^ib V?;? *T(a,b)

"*" signifies the

convolution integral)

oc cc

2k I V

(a-u)V(b-v)^ , TT j j ' (a-u)(b-v) v)dudv = A (T -T ) ^1^ 1

2^

—cc — oc

> = 2k

i — G(a,b,e) (23)

where

G(a,b,e.)= (2+-^) -y/(a+£ ) 2+(b+£ )

-^2(1+ 1^) Va2+(a+2£)2

-^(1-f^) Vb2+(b+202

-(^)Vr2^^

[1+^ sin h--L (-1)] + a2+b2 . AT . , -1

+ Iziyfi + sin h ^ (-1)]

-V2 (i^)^ [sin h-1 -.in (^) :

+ (a-b)2-(b-te )2 sin

28
+ (a-b)^-(a-te)^ . , -1 / a4e sm h ^ '

+ £(2bz£l 3in h-1 b

£ \a

+ b(2a-b) ^.^ ^-1

- sin h-

- (bild) sin h-

+ (2a-fe ) sin h

+ (2b+£)sin h'

(a-te)

(b^)

(?)

+V2 |! sin h-l(^)

+ V^^ sin h-1

Steady State Solution - II

Equation (24)

29
Steady State Solution - III

Equation (25)

b+2£ a+2e

/ /

q = -k / / 9T(x,y,z) dxdy

-b-2£ -a-2£ z=0

oo oo

2k

J J

^

(a:-u)"+(b>v) 2 2 la'-u) (h'-Y) T(u,v),dudv = ^3(1^

-00 —00

where a^ = a+Ts = b+2£

= 2k

where

+ 2 (a*) [sin h-1 (^)-si. h"! (f^)]

+ 2(b-te)[ sin h

+ 2e [sin h"^ (—\ + sin (~^-^ sin h ^(1)]

y b+£ 1 \ a-te

. [Sin -(f''-e)sln h"! + 2a

where

a = 1- 1 sin h-^1) + + l-^^n - X/+2X-1 (Xo 2+2X^-1)' 2

1 in /Xn+l-\/2\

30
2

2

H-^-j. = (sin \ {l)-yj2)e +(^^

-^siv h-^^S^^+ 2b

where

6 = 1 - JL sin h ^(1) + __ilVl +

^ ^^o

2 ^2 X^'^+2X/-1 (X^'2+2X^'-1)2

- V2 ln(Xo^+l-V2)

and

^III = ^11+^ ^'"^ (~ ~"^) ^ '^''(^^) " ^ V8

(X^2+2X^-1) (X-^2+2x^-l)^

where.

a' = b; then J^-j = l^yj^ " sin h ^ (1)) £ = Jm

J.^ = 2^(b:^) raa^^+(a^-bOL_^j if a' b

where

^11 a"-£ lnXr> -

2 -

a'-b^

a^

a'-b^

InX^ + In "X2-I Xi+1

X2+I

"

+

2(i-x,) 2(1-Xtl)

X.2+2X,-1 X^^+2X^-l

'2 2

+ In

]

\ 31
- 2 ^2

L-j- = -(a-e) lnX2+a InX.^

2(a'-bO^

- 3V2;in

4

and

> H

X-,+H-y/2 X.+1-a/2

X

1 X. +

X-^^+2X^-l X2^+2X2-1

- In

(X-^+l X^-l ^ X^ X2+1 )

2-6X^ 2-6X2

(X-l2+2x^_1)2 (X2^+2X2-1)^

- N ' VP) + 1

IV III J^^^ + z sin h~lf^ ^1 + 2a'2 In

- 3V2 In

/Y^+l-V2\

4 \Y-^+l+yl2

+

6Y-^-2

(Y]^^+2Y^-1)^

where

Y = /£- - 1 U - 1\ + 1

^Xii

" 2(a^-bO |b^M-j.-^+(b^-aOM^|- if ^ b

if a' = h\ then J^^^ = |^2 - sin ^ h(l)|e

32
-(b'-e)^lnY2+b' ^InY2(b'-a')2

- In

Y2+1

21 + \ 2_i ov 1 2~

Y^^+2Y^-1 Y2^+2Y2-1

2-6Y. 2-6Y, +

(Y^ 2+2Y,-l)'^ (Y 2+2Y„-l)2

1 1 ^2 2

^

M = b'-e lnY„ - inY,

II 2 . 1

b -a b^-a-

\Y2+1 Y^-1/

+

2(1-Y2) _ 2(l-y.)

Y2 +2Y2-I (Y^^+2Y;l~^^

+ y[2 ln\\+1-^2

^ Y^+l+y2

+l+-^2 Y^+l-yJl

where

^1

= + 1

a - E

b^- £ +

a - c b^- e + 1
Steady State Solution IV

Equation (26)

A special case of solution where = b

4 = 7^'(%^-^«i^ + (26)

H^' = 2i^/2a'-2

+ e

|^2a'-2 V(a'-£)^^e^~+ (a'-e) [sin h ^ " ^ "^d)]}

[sin (^^^ ~ ^^"^ 1^ ^(1)]

= [sinh-l(l)- yj2h ^(r-^) --^"'(t^)

, 2a ^ + a

where a = i - sin h ^(1) 2-X, +

X +2X -1

o o

2-6X +

(X^2+2x^-l)^ ^

-km

X = S o a' - £ + + 1

-1 -1 / a - e

I-j-j-]-" = [-^2 - sin h ""(1) + sin h

1+1-

2a

where B = In

X-L 6X-L-2

X^2+2X^-1 (X;l^+2X^-1)2

and X, = +

1 £

34
A Simplified Procedure (PC-1)

For a large slab (a=b>>e), a combination of periodic and steady state components

of Delsante's floor heat loss equation yields

q = Re {k P(T;L-T7)I(2Ae)+k^AST, } + * (T -Tj (27)

where Re = real part of a complex variable

k = soil thermal conductivity

g

P = slab perimeter length

S = slab area

T^,T2 = complex temperature function representing periodic components

T^j^,!^ = average or steady part of the temperature component

This equation was used in a pocket calculator procedure called PC-1 to determine

the annual cycle of the square slab floor heat loss calculated by the Delsante

method. It is important to point out that the PC-1 calculation is based on the

assumption that the periodic response of the perimeter heat transfer along each

edge of the slab will not be affected by other edges (both adjacent and opposing

edges)

.

Comparison between the HEATPATCH and PC-1

Table 1 and figure 15 show the result of comparative calculations between the

HEATPATCH and Delsante (PC-1) for the steady-state heat loss from two square

slabs, one 20 foot (6.1m) and another 40 foot (]-2.1m) squares, as a function

of perimeter zone thickness, 2e. In this comparison, an I of six inches (0.15m)

was used for equation (12). Figures 16 and 17 show similar comparisons for the

monthly floor heat flux.

35
PERIMETER WIDTH, 2e, m

0.5 1.0

PERIMETER WIDTH, 2e, ft

Figure 15. Comparison of annual average floor heat flux values calculated by

the HEATPATCH method and by the Delsante formula.

36
3U1/M 'xnij iV3H avis imm aihinow

lO

E

E

E

r-oo

CO o LO CO O unoo

un

I

cm"

- o

00

4-1 •H

o

o

.—

t

XI

ra t—

(

w

e

r-H

6

o

CNIXo

B • o m

l-M 4-1 T3

OJ -H rH >

a

CJ 0)

4-1 X OJ 3 e

r-l -H

14-1 i-l

4-1 a

TO0) 4J x: c

0)

.-H ^1

CO (U 3 <+-4 c ^

OJuM

•H

Ij-,

CM

U'^^/m 'xmj ivbh avis bovubav aihinow

37
;Ui/M 'XnU 1V3H 9V1S 39Vd3AV AIHINOM

CO

W-JM/nia 'Xnid 1V3H avis BOVHBAV AlHiNOW

/ 38
20 ft

square

slab

TABLE 1

Steady State Heat Loss

(Btu/hr)

HEATPATCH

Delsante

(PC-1)

a = b 2e

(ft) (m) (ft) (m) (Btu/hr) (W) (Btu/hr) (W)

1 90 J-/ Z U . -L J9 8/17 Q91 7 Z X zu D . ± J. U . JU Z joZ 0 ^; n Q ZbUo 1

20 6.1 2 0.61 1944 570 2036 597

20 6.1 5.7 1.73 944 111 1074 315

/ 10 3.05 0.5 0.15 1128 331 1304 382

1 10 3.05 1 0.30 922 270 1018 298

]10 3.05 1-3/4 0.53 734 215 769 225

(10 3.05 2.85 0.87 500 147 537 157

lio 3.05 4.3 1.31 319 93 335 98

These computations were made using a annual average temperature of 56.5°F (13.5°C)

and an amplitude of 20.6°F (11. 4K) for the outside condition, while the average

and amplitude temperature were 70°F (21.1°C) and 5°F (2.8K), respectively, for the

slab surface. Thermal conductivity of 1 Btu/hr* ft -^F (1.728 W/m*K) and thermal

2 2 diffusivity of 0.025 ft /hr (0.056 m /day) were assumed for soil. The agreement

between the Delsante and HEATPATCH results is good except where the perimeter

thickness is very large or very small. One of the reasons for the large discrepancies between the HEATPATCH and the Delsante is that the perimeter zone was

considered a part of the floor for the HEATPATCH calculation.

Generally, as the perimeter thickness increases, the annual average heat loss

tends to decrease and the annual amplitude of the heat loss also decreases. A most

interesting aspect of the result is that heat loss between May through August is

scarcely affected by the perimeter thickness. This is understandable because the

temperature profiles during that period are practically parallel to the ground

39
surface even near the edge, so that heat flow Is practically normal (even over

the perimeter zone) to the floor surface across the entire floor (see figures 4

and 6)

.

As pointed out before, the formula used in the PC-1 calculation for the transient or periodic component of the floor heat loss is not as exact as for the

steady-state case and is based upon the assumption that the three-dimensional

corner effect is extremely small. Thus, it is only valid for very thin perimeter widths. On the other hand, HEATPATCH calculations yield errors due to

its arbitrary nature for selecting the depth parameter, £, for the numerical

differentiation. Theoretically, the smaller the value of £, the closer it will

be to the exact solution. There is, however, a problem in making the value of

I too small or smaller than the surface grid size Ax and Ay. As can be seen

from equation (2), the integrand L contains a variable r which is the distance

between the earth temperature point and the differential surface segment AxAy.

For the numerical calculations dealing with the finite magnitudes of Ax and Ay,

this r represents the distance between the earth temperature point (x,y,£,t)

and the centroid of the surface segment (x^,y^,o,t) expressed by the following

equation

^ = '^£^ + (x-xO^ + (y-yO^

Approximation of the original integral equation by a finite difference Integral

(equation 9) will be valid as long as r Is considerably larger than Ax and Ay.

For the small value of z = Z this criteria is difficult to meet, unless Ax and

Ay are made extremely small throughout the surface region. A smaller Ax and Ay

for the entire region, on the other hand, results in excessive computer

time. For normal slab-on-grade heat transfer calculations, with a floor size of

more than 20'x20' (6.1m x 6.1m), the smallest practical Ax and Ay is in the order

of perimeter thickness 2e and the depth parameter £ of six inches (0.15m) seems

to be adequate.

40
Figures 16 and 17 show that the floor heat loss becomes smaller as the perimeter

thickness gets larger. This is, of course, partly due to the fact that the

average temperature difference between the slab surface and the outdoors decreases.

The physical significance of the perimeter thickness 2£ is somewhat unclear. It

has been considered as the wall thickness by Muncey [6] as well as by Delsante [9]

on the assumption that the floor temperature is constant from one end of the floor

to another. Under actual conditions, however, the measured floor temperature is

not uniformly constant; it gradually changes from the center of the floor toward

the wall. In other words, the temperature transition zone is considerably wider

than the wall thickness. (For this reason the HEATPATCH calculation included heat

loss under the perimeter zone since it is by definition the temperature transition

zone.) Unfortunately, the magnitude of 2e is not well known and it varies depending upon several factors such as the conductivity of the floor slab, perimeter

insulation, foundation, and wall construction, etc. Moreover, in many cases, the

type of heating system or location of heating equipment on the floor significantly

affects the floor temperature distribution. An additional point is that, in reality,

the floor surface temperature is seldom maintained at a constant value when the

room air temperature is controlled by the thermostat. Limited experience with NBS

thermal mass test houses on 20 ft (6.1m) square slab floors indicates that Iz or

the width of the temperature transition zone, is sometimes as much as two ft (0.61m).

While figure 18 shows the effect of floor sizes on the monthly cycles of the heat

loss figure 19 indicates a somewhat strange effect of the thermal diffusivity

upon the floor heat loss. It is important, however, to point out that figure 19

was obtained for the constant thermal conductivity and in reality the thermal diffusivity is directly proportional to the soil thermal conductivity.

Daily Thermal Cycles

While the previous discussions are concerned with the monthly temperature and

heat flow of the annual cycle, identical equations can be used to solve for the

41
OO CO ^

C>4
4.0

0 I I I I I I I I I I

I I JFMAMJJASOND

MONTH

Figure 19. Annual heat loss cycle from 40'x40' (12.1m x 12.1m) slab floor for

different soil thermal dif f usivities

.

43
daily cycle by simply changing the value of co from 2tt/8760 to 2tt/24. Figures

20 and 21 compare the diurnal floor heat flux cycle with the annual cycle under

identical conditions for temperature data, slab size and soil thermal properties. Figure 20 is for the constant indoor temperature condition and shows

that the diurnal outdoor temperature cycle affects the diurnal floor heat flux

to much less of a degree than the annual temperature cycle affects the annual

heat flux. Figure 21, on the other hand, shows that an indoor temperature fluco tuation of 5 F (2.8K) significantly affects the diurnal floor heat transfer but

has very little effect on the annual cycle of the monthly heat loss.

The daily cycle floor heat loss is important for the consideration of solar heat

absorption by the floor. To study such a situation, calculations were performed

for a lO'xlO' (3.05m x 3.05m) section of a large floor, which is cyclically

o heated in such a manner that it experiences a daily maximum temperature of 130 F

(54.4°C) and a minimum of 70°F (21.1°C), while the rest of the floor is maintained

at 70°F (21.1°C). Figure 22 shows the results of HEATPATCH calculations for the

average heat flux from the heated section and its temperature profile under a

steady periodic condition. The results using PC-1 agree very closely with the

HEATPATCH results when Iz is assumed to equal 0.5 ft (0.15m). Figure 23 shows

the distribution of floor heat flux along the floor indicating large heat flux

change around the perimeter of the heated section. During the time that the

floor was absorbing the heat, the region immediately outside the heated section

was releasing the heat that was conducted into the heat section. This type

of calculation should be extended to include higher harmonics to simulate

the more complex diurnal surface temperature cycle representative of solar

radiation incident upon the floor surface. Figure 24 is a result of such calculations for the 20'x20' (6.1m x 6.1m) floor during a typical winter day while

44
UJ/UBM 'Xmj iV3H MOOU 39VH3AV

CM

ex

CM

C

0

•H

4-J •H H t3 o

0

0

C/J 0

1—

t

o

^>^ O CO A»-i r-b

U-(

4-1 4-» CO H

0

rC 0

t3

1

1 C

CO •H DC 4-1

CD CO

4-1

C/} C C

CU 0

0

t-HCO QJ C

>-i 4-1 3

•HT3 OJ -d

0) c

p

>-l c 0

(U 0

(U rH

15 4-1 u

0) -n

CO c u

0 M

CO 1 •H c

u 0

CO 1 e CO 0 iH 0 CO 0

<N

diu

3

bO

•H

zU 'Ji|/nia 'xniJ iV3H dOOU 39Vd3AV

45
^lu/UBM 'Xmj iV3H 39Vd3AV

o

cn Cfi (U iH 4-1 o CO3 o +Ju

X

r-l rH 14-1

0)

V4 CO U

CO rH cu

CO e

c

c 4-1 CO

t-l T3 O

C o

CO C

t—

1

•H

COC 0) d 4-J •HTD c

0)

0) x:

4-1 C o

0) o

0) r-l

M-l

4-1 QJ QJ T3

CO

>-i o bO

1 •H C

S-i o

CO 1

CO o iH u

CU

•H

CO CO CM OO to CM

zU 'i^/m 'Xmj iV3H dOOU 33Vd3AV

46
3o 'auniVU3dM31 HOVJHnS HOiVd

47
DISTANCE FROM CENTER, m

4 3 2 1 0 1 2 3 4

15 10 5 0 5 10 15

DISTANCE FROM CENTER, ft

Figure 23. Heat flux distribution over the floor lO'xlO' (0.31in x O.Slm)

segment of which undergoes a cyclic heating.

48
1 1 1 \ r 70 F (21 rc)

I I I I I 1 1 1 4 8 12 16 20 24

HOUR

Figure 24. Diurnal floor heat flux cycle due to solar heat gain.
room temperature was maintained at 70°F (21.1°C) during the day and 65 F (18.3°C)

during the night. The figure shows the relationship among solar heat gain, floor

heat loss and the heat given off by the warm floor to room during the night hours.

Figure 25 shows the effect of perimeter thickness upon the diurnal floor heat

loss for a 40'x40' (12.1m x 12.1m) floor, which is significantly greater than

the annual heat loss shown in figure 13.

Conduction Transfer Functions for Composite Floor

Strictly speaking, what has been shown in the previous sections is applicable

only to a bare earth floor in a house and when T^ and T^ used in equations 16,

20, and 21 are essentially the surface temperature of the earth floor and outdoor

earth surface temperature. In reality, however, the floor slab is a multilayered structure consisting of floor covering, concrete slab, insulation, subgrade

gravel, etc. Moreover, the surface film heat transfer coefficients for inside

floor surface, must be considered. According to Carslaw and Jaeger [l], the relationship between surface heat flux and temperature is given by the following matrix

equation:

contain periodic components.

If the floor system is a homogeneous plane slab of thickness I, thermal conductivity k, density p, and specific heat c.

(28)

where all the quantities with denote complex variables which indicate they

A cosh H

B -R sinh H

H

(29)

C

-H sinh H

R

D cosh H

50
4000

3000 -

T

\ r T—

r

Daily cycles

26 = 0.5 ft (0.15 m)

26 = 1.0 ft. (0.3 m)

GO

CO

CO

2000

1000

0

26-2.0 ft. (0.6 m)

26 = 4.0 ft. (1.2 m)

2ax2b - 40' X 40' (12.1 m x 12.1 m)

Tg = 56.5T (13.6°C)

Ta=10.3°F(5.7 K)

Tb-70°F (21.1°C)

Tc-0°F (0 K)

k = 1.0 Btu/hr ft °F (1.726 W/m K)

a = 0.025 ftVhr (0.056 mVday)

J \ \ L I I I I I L

1100

1000

900

800

700

^ 600

500

400

300

- 200

- 100

CO

CO

s

0 2 4 6 8 10 12 14 16 18 20 22 24

ELAPSED TIME, HOURS

Figure 25. Diurnal heat loss cycle for 40'x40' (12. Im x 12.1m) slab floor.

51
O; , ;

'

' where H = / iooRC

I R = : thermal resistance

C = pc£ : thermal capactance

and i =

If the system is purely resistive (or when c=0) , such as the surface film,

having the heat transfer coefficient h, '

A = 1

B = -1/h

C = 0

D = 1

If the floor system is a multi-layer composite, one can write

CD/ D^/ \C2 D^/

(30)

% % % %

where A^, B^, C^, D^, (k=l, 2 ... n) are evaluated at each layer and postmultiplied. Using this convention, the conduction system eauatiQn for the room

air to outside air floor/earth may be written as

C:) d (^: y (y (31)

/ \ a - /

g

% %

where A„, B , C^, and D„ are for the multi-layered finite thickness floor and

F' F F F

a- % % %

A„, B , C , and D are for the semi- infinite earth region around the building.

o 6 & &

For the floor heat loss calculations, only B and D are needed, values of which

can be determined by modifying equation (27) in the form consistent with equation

(28) as follows:

% c -1

Bg =1 * [kg I(2eA)] ' (32)

52
The frequency domain conduction transfer functions X and Y can be defined as

ll = Tl - ^ To (34)

where X = - (35)

Ap.B + B„D

F g F

_ -1

Y - '\, % % (36)

A„B + Bt?D F g F

g

Finally the floor heat loss can be calculated by summing up the contribution of

all the harmonics as follows:

Z X(a3^) T^(w^) - Y(co^) T^(oJ^) (37)

i-1 )

where Re indicates that q is the real part of the complex number in the bracket.

Of course when co =0, this represents the steady-state and X and Y will be replaced

i

% o, _ _

by Likewise, T^{u}) and T-^(^) will be replaced by and T^.

Typical values for the frequency response factors have been calculated and are

shown in Table 2-A and 2-B for 20'x20' (6.1m x 6.1m), 30'x40' (9.1m x 12.2m) and

40'x50' (12.2m x 15.2m) floors. In these sample calculations a soil thermal dif2 9 fusivity of 0.025 ft /hr (0.056 m^^/day) and thermal conductivity of 0.5 Btu/hr-ffF

(0.072 W/m*K) were assumed. The floor is assumed to be bare for Table 2-A, while

Table 2-B is for a floor consisting of 6" concrete and 6" gravel above the soil.

These tables show that the Y components of the response factors are practically

zero for all the frequency levels higher than the daily cycle, indicating that

only the annual cyclic outdoor temperature changes affect the floor heat transfer. The X factors are, however, extremely large for the high frequency component indicating that the indoor temperature fluctuation due to the thermostat

cycling, solar beam penetration and/or lighting schedule would have a large impact

on the short time floor heat transfer. Also interesting is that the X factors are

virtually unaffected by a change in floor size and shape, but are influenced by

the floor layer structure.

53
Table 2-A

Frequency domain thermal response factors for slab floor

Surface Resistance

Floor Construction Bare Soil

20' X 20' slab X Y

3 hour cycle

6

" 12 " 24 " Annual cycle

Steady state

1.2981 - 1(0.6508)

1.1400 - 1(1.0206)

0.6461 - 1(1.3068)

0.05767 - 1(1.1533)

-0.07655 - 1(0.0208)

0.09366

-0.00078 - 1(0.00155)

-0.00243 - 1(0.00273)

-0.00629 - 1(0.00336)

-0.01109 - 1(0.00151)

-0.04507 + 1(0.01182)

0.09366

30' X 40' slab X Y

3 hour cycle

6

" 12 " 24 " Annual cycle

Steady state

1.2976 - 1(0.6508)

1.1388 - 1(1.0197)

0.6455 - 1(1.3030)

0.06136 - 1(1.1479)

-0.05667 - 1(0.0256)

0.0656

-0.00045 - 1(0.00090)

-0.0014 - 1(0.00159)

-0.00366 - 1(0.00195)

-0.00644 - 1(0.00087)

-0.02587 + 1(0.00666)

0.0656

40' X 50' slab X Y

3 hour cycle

6

12 " 24 " Annual cycle

Steady state

1.2975 - 1(0.6507)

1.1384 - id. 0194)

0.6453 - 1(1.3019)

0.06253 - 1(1.1461)

0.05045 - i(0.0270)

0.0547

-0.00035 - 1(0.00070)

-0.00109 - 1(0.00122)

-0.00282 - 1(0.00150)

-0.00496 - 1(0.00067)

-0.01986 + 1(0.00509)

0.0547

54
Table 2-B

Frequency domain thermal response factors for slab floor

Surface Resistance

Floor Construction 6" Concrete

20' X 20' slab X Y

3 hour cycle

6

" 12 " 24 " Annual cycle

Steady state

0.9679 + 1(0.0918)

0.9264 + 1(0.1169)

0.8779 + 1(0.1530)

0.7976 + 1(0.1981)

-0.0942 - 1(0.0087)

0.0758

0

0

-0.00011 - 1(0.00026)

0.00116 - 1(0.00117)

-0.05610 + 1(0.01301)

0.0758

30' X 40' slab X Y

3 hour cycle

6

" 12 " 24 " Annual cycle

Steady state

.96.89 + 1(0.0920)

.9272 + 1(0.1171)

.8786 + 1(0.1533)

.7983 + 1(0.1983)

-0.0652 - 1(0.0135)

0.0564

0

0

0

0.00069 - 1(0.00068)

-0.03040 + 1(0.00653)

0.0564

40' X 50' slab X Y

3 hour cycle

6

" 12 " 24 " Annual cycle

Steady state

.9689 + 1(0.0920)

.9282 + 1(0.1171)

.8786 + 1(0.1533)

.7983 + 1(0.1983)

-.0568 - 1(0.0147)

0.0481

0

0

0

0.00053 - 1(0.00052)

-.02292 + 1(0.00481)

0.0481

55
Summary and Discussion

Using the Lachenbruch Green's function technique a computer 'program called

HEATPATCH was developed to determine underground temperature profiles under

disturbed ground surface areas of various sizes and shapes. This program can

also be used to compute the heat loss or heat transfer from the disturbed regions

to the undisturbed ground. The heat transfer calculation is based upon the numerical differentiation of the soil temperature near the surface. Comparison

with Delsante's exact solution shows that the temperature gradient through a

six inch thickness of soil layer is adequate for the determination of the floor

heat loss except for the case of a very thin perimeter z.one. A simplified pocket

computer program was developed using Delsante's solution which could be used as

a part of the annual energy calculation if it is translated into the seasonal

average subfloor temperature at six inches below the floor surface. Using Delsante's

equation, spectral sensitivity of the floor heat loss due to shorter periodic

cycles such as the diurnal cycle was also Investigated. The diurnal floor heat

loss as affected by the daily outdoor temperature cycle is very small. This

implies that floor heat loss is basically one-dimensional or the heat flow is

normal to the floor surface from center to edge as far as the hour-by-hour simulation of the building heat transfer process is concerned. This justifies the

use of the one-dimensional thermal response factor approach to determine the heat

storage effect of the floor for the hour-by-hour energy calculations. A frequency

domain thermal response factor concept was developed by combining the slab floor

composite with the surrounding earth. Sample response factors for typical floor

construction were obtained and presented.

It is important to recognize the limitation of the HEATPATCH and Delsante solution. Both solutions do not address edge insulation. Finite difference (FDM)

and/or finite element (FEM) type calculations such as shown in references [T] arid

56
[9] are required to account correctly for the effect of edge insulation. An

examination of Wang's results [ 9 ], however. , indicates that the heat loss from

the edge insulated slab is very closely approximated by the annual average heat

loss that ignores the periodic component of the monthly perimeter heat loss of

equation (27). Moreover, the HEATPATCH technique should still be useful in

expediting the FDM/FEM calculation by being able to provide accurate seasonal

boundary temperature conditions under annual cyclic conditions. The HEATPATCH

calculation principle also can be extended for basement wall and floor analyses,

but will require more complexity in keeping track of the disturbed surfaces,

which will be five instead of one (for the slab-on-grade floor problem).

References

1. H. S. Carslaw and J. C. Jaeger, "Conduction of Heat in Solids", Oxford at

the Clarendon Press, 1946, pp 353-357.

2. R. S. Dill, W. C. Robinson, and H. E. Robinson, "Measurements of Heat

Losses from Slab Floors," NBS Building Material and Structures Report

EMS 103 , March 1943.

3. H. D. Bareither, A. N. Fleming, and B. E. Alberty, "Temperature and Heat

Loss Characteristics of Concrete Floors Laid on the Ground," University

Of Illinois Small Homes Council Technical Report , PB 93920, 1948.

4. A. H. Lachenbruch, "Three Dimensional Heat Conduction in Permafrost Beneath

Heated Buildings," Geological Survey Bulletin 1052-B , U.S. Government

Printing Office, Washington, D.C., 1957.

5. B. Adamson, "Soil Temperature Under Houses Without Basements," Byggf orskningen

Handlinger, Nr 46 Transactions, 1964.

6. R. W. R. Muncey and J, W. Spencer, "Heat Flow into the Ground Under a House,"

Energy Conservation in Heating, Cooling, and Ventilating Buildings , Hemisphere

Publishing Corporation, Washington, D.C., Vol. 2, pp 649-660, 1978.

7. H. Akasaka, "Calculation Methods of the Heat Loss Through a Floor and Basement Walls," Transactions of the Society of Heating, Air-Conditiong , and

Sanitary Engineers of Japan , No. 7, pp 21-35, June 1978.

8. C. W, Ambrose, "Modeling Losses from Slab Floors," Building and Environment,

Vol. 16, No. 4, pp 251-258, 1981.

9. F. S. Wang, "Mathematical Modeling and Computer Simulation of Insulation

Systems in Below Grade Applications", ASHRAE SP 28, Proceedings of ASHRAE/

DoE Conference on Thermal Performance of the Exterior Envelopes of Buildings, Orlando, Florida, December 1979.

57
10. T. Kusuda, M. Mizuno, and J.. W. Bean, "Seasonal Heat Loss Calculation for

Slab-on-Grade Floors," NBSIR 81-2420, National Bureau of Standards, March 1982.

11. A. F. Delsante, A. N, Stokes and P. J. Walsh, "Application of Fourier Transform to Periodic Heat Flow Into the Ground Under a Building," to be published

in the International Journal of Heat and Mass Transfer, December 1982.

58
Appendix

Compatibility with the current ASHRAE data :

The 1981 ASHRAE Handbook of Fundamentals lists in its Table 4 (page 25.9) data

to be used for evaluating design heat loss from three different types of floor/

wall constructions. The ASHRAE data are presented in terms of the factor,

which is the heat loss per unit length of floor perimeter per degree F difference

between the in- and outdoor temperatures. When the edges of the floor were insulated with R = 5.4 hr- ft^.^F/Btu (0.95 m^-K/W) material, the value of F2

is virtually constant around at 0.5 Btu/hr • f

t • °F . For the uninsulated slab

floor, the value of Fo for Madison (WI) , for example, varied from 0.62

Btu/hr-ft^-°F (1.07 W/km) to 1.15 Btu/hr-ft-°F (1.99 W/m-K) depending upon

the wall/floor construction.

The ASHRAE data were generated by a finite element computer program of Wang [1]

under three climatic conditions; Madison (WI) , Columbus, (OH) and Atlanta (OA)

for the soil thermal conductivity of 0.8 Btu/hr-ff°F (1.382 W/m-K) and indoor

temperature of 70°F (21.1°C). Table A shows the conditions used for the generation of ASHRAE data and the comparison between the ASHRAE F2 and those calculated

by PC-1 for uninsulated floors. Since reference [1] does not mention the soil

thermal diffusivity and floor size, it is assumed in this comparative analysis

that

a = 0.025 ft^/h (0.056 m^/day)

a = b = 20 ft (6.1m)

and the value of F^ was computed for the January floor heat loss data, which is near maximum.

The agreement between the ASHRAE F2 and that obtained by PC-1 is relatively good

for the block walls except that the trend with respect to degree-day is reversed.

The ASHRAE heat loss coefficient increases with the warmer climate contrary to

the value obtained by PC-1. The discrepancy for the metal stud wall can be made

small if the effective perimeter width 2e is assumed 2 inches (0.051m) instead

of 6 inches (0.15m).

The insulated floor heat loss factor of 0.5 Btu/hr'ff°F (0.86 W/K-m) corresponds

very closely to that derived from the annual average floor heat loss calculated

by the procedure described herein. Table B compares the average floor heat flux

data obtained from ASHRAE F^ values and those determined as the annual average

floor heat loss by PC-1. Again, the agreement is relatively good between the

data obtained by two different methods for the cold climatic condition. A large

discrepancy between these two results for Atlanta is very puzzling.

Reference

[1] F. S. Wang, "Mathematical Modeling and Computer Simulation of Insulation

Systems in Below Grade Application", ASHRAE SP 28, ASHRAE/DoE Conference

on Thermal Performance of the Exterior Envelopes of Buildings, Orlando,

Florida, December 1979.

59
Table A

Heat Loss Coefficient F2 of Slab

Floor Construction in Btu/h • f

• f t (W/K-m)

2 P(T-L-T2)

Construction

and 2e in

ft (m)

Locations Madison Columbus Atlanta

degree day

at 65°F

7433 5350 2950

T^, F ( C) 46.1 ( / . 0) 52.5 (11.4) 60.8 (16.0)

B, °F (K) 27.5 (15.2) 22.9 (12.7) 18.3 (10.2)

(T-L-T2), °F (K) 51.4 (28.5) 40.4 (22.4) 27.5 (15.3)

8-inch block

wall, brick

facing

2e=1.08 ft

(0. jjm;

F2 (PC-1)

F^ (ASHRAE)

0.70 (1.20)

0.62 (1.07)

0.69 (1.19)

0.68 (1.17)

0.64 (1.10)

0.72 (1.24)

4-inch block

wall, brick

facing

2e=0.75 ft

(0.23m)

F2 (PC-1)

F2 (ASHRAE)

0.79 (1.36)

0.80 (1.38)

0.76 (1.31)

0.84 (1.45)

0.73 (1.26)

0.92 (1.61)

Metal stud

wall, stucco

2e=0.5 ft

(0.15m)

F2 (PC-1)

F2 (ASHRAE)

0.90 (1.55)

1.15 (1.99)

0.88 (1.51)

1.20 (2.07)

0.83 (1.43)

1.34 (2.32)

60
Table B Floor Heat Flux of Slab in

Btu/hr.ft2(W/m2)

i= P-F2(Ti-T2)

Floor

Construction

and 2c

in ft (m)

Locations Madison Columbus Atlanta

degree days

at 65°F

7433 5350 2950

T , °F (°C) m

46.1 (7.8) 52.5 (11.4) 60.8 (16.0)

B, °F (K) 27.5 (15.2) 22.9 (12.7) 18.3 (10.2)

Tj-T^, °F (K) 51.4 (28.5) 40.4 (22.4) 27.5 (15.3)

8-inch block

wall, brick

facing

2£=1.08 ft

(0.33m)

PC-1

From ASHRAE

^2 edge

insulated

floor

2.30 (7.25)

2.47 (7.81)

1.69 (5.32)

2.0 (6.38)

0.89 (2.80)

1.54 (4.87)

4-inch block

wall, brick

facing

2e=0.75 ft

(0.23m)

PC-1

From ASHRAE

F2 edge

insulated

floor

2.53 (7.97)

2.41 (7.62)

1.86 (5.86)

1.97 (6.22)

0.98 (3.09)

1.49 (4.71)

Metal stud

wall, stucco

2e=0.5 ft

(0.15m)

PC-1

From ASHRAE

F2 insulated

floor

2.78 (8.76)

2.62 (8.28)

2.04 (6.43)

2.14 (6.76)

1.07 (3.37)

1.60 (5.06)

* , -

61
NBS-n4A (REV. 2-6C)

1. PUBLICATION OR 2. P.erforming Organ. Report No. 3. Publication Date BIBLIOGRAPHIC DATA

" REPORT NO.

SHEET (See instructions) NBS BSS 156 Jimp. 198-^

4. TITLE AND SUBTITLE

Annual Variation of Temperature Field and Heat Transfer Under Heated Ground Surfaces

(Slab-on-Grade Floor Heat Loss Calculation)

5. AUTHOR(S)

T, Kusuda, 0. Piet, and J. W. Bean

6. PERFORMING ORGANIZATION (If joint or other than NBS. see instructions) 7. Contract/Grant No.

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE 8. Type of Report & Period Covered

WASHINGTON, D.C. 20234 Final

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City. State. ZIP)

Same as Item 6, above.

10. SUPPLEMENTARY NOTES

I

Library of Congress Catalog Card Number: 83-600539

I I Document describes a computer program; SF-185, FIPS Software Summary, is attached.

11. ABSTRACT (A 200-word or less factual summary of most si gnifi cant information. If document includes a significant

bi bliography or literature survey, mention it herej

Seasonal sub-surface ground temperature profiles and surface heat transfer were

determined for the condition whereby one and more than one region of the earth's surface temperature were disturbed. The analysis was conducted by numerical integration

using a closed form solution based on the Green's function. Monthly profiles of earth

temperature isotherms under a house of 20' x 20' (6.1m x 6.1m) floor area and under

a ground of six houses near a wooded area are presented. The heat losses calculated

from this approach for square slabs of various sizes were compared with those derived

from the recent analytical solution of Delsante et al. resulting in good agreement.

The Delsante solution, which is based upon a Fourier Transform technique, is then ex- tended to generate the frequency domain thermal response factors suitable for the

periodic heat transfer calculation for multi-layer slab floors on grade.

In the appendix, this thermal response factor method was used to generate annual cycles

of monthly heat loss from several slab floor constructions shown in the 1981 ASHRAE

Handbook of Fundamentals. The maximum values of these monthly slab floor heat losses

agree relatively well with the ASHRAE design values.

12. KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolons)

ASHRAE design values; building heat transfer; Delsante method; earth temperature;

slab-on-grade heat transfer; soil temperature.

13. AVAILABILITY

[jjj Unl imited

I I For Official Distribution. Do Not Release to NTIS

I X I Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402.

Q^] Order From National Technical Information Service (NTIS), Springfield, VA. 22161

14. NO. OF

PRINTED PAGES

67

15. Price

USCOMM-DC 6043-P80
NBS TECHNICAL PUBLICATIONS

PERIODICALS

JOURNAL OF RESEARCH—The Journal of Research of the National Bureau of Standards reports NBS research and develop- ment in those disciplines of the physical and engineering sciences in which the Bureau is active. These include physics, chemistry,

engineering, mathematics, and computer sciences. Papers cover a broad range of subjects, with major emphasis on measurement

methodology and the basic technology underlying standardization. Also included from time to time are survey articles on topics

closely related to the Bureau's technical and scientific programs. As a special service to subscribers each issue contains complete

citations to all recent Bureau publications in both NBS and non- NBS media. Issued six times a year. Annual subscription: domestic

$18; foreign $22.50. Single copy, $5.50 domestic; $6.90 foreign.

NONPERIODICALS

Monographs—Major contributions to the technical literature on

various subjects related to the Bureau's scientific and technical activities. Handbooks—Recommended codes of engineering and industrial practice (including safety codes) developed in cooperation with in- terested industries, professional organizations, and regulatory

bodies. Special Publications—Include proceedings of conferences sponsored by NBS, NBS annual reports, and other special publications

appropriate to this grouping such as wall charts, pocket cards, and

bibliographies. Applied Mathematics Series—Mathematical tables, manuals, and

studies of special interest to physicists, engineers, chemists,

biologists, mathematicians, computer programmers, and others engaged in scientific and technical work.

National Standard Reference Data Series—Provides quantitative data on the physical and chemical properties of materials, compiled from the world's literature and critically evaluated. Developed under a worldwide program coordinated by NBS under

the authority of the National Standard Data Act (Public Law

90-396).

NOTE; The principal publication outlet for the foregoing data is the Journal of Physical and Chemical Reference Data (JPCRD)

published quarterly for NBS by the American Chemical Society

(ACS) and the American Institute of Physics (AIP). Subscriptions,

reprints, and supplements available from ACS, 1 155 Sixteenth St., NW, Washington, DC 20056.

Building Science Series—Disseminates technical i*" '"ormation developed at the Bureau on buildmg materials, components,

systems, and whole structures. The series presents research results,

test methods, and performance criteria related to the structural and

environmental functions and the durability and safely characteristics of building elements and systems.

Technical Notes—Studies or reports which are complete in themselves but restrictive in their treatment of a subject. Analogous to monographs but not so comprehensive in scope or definitive in treatment of the subject area. Often serve as a vehicle for final reports of work performed at NBS under the sponsorship of other government agencies.

Voluntary Product Standards—Developed under procedures

published by the Department of Commerce in Part 10. Title 15, of

the Code of Federal Regulations. The standards establish nationally recognized requirements for products, and provide all concerned interests with a basis for common understanding of the

characteristics of the products. NBS administers this program as a supplement to the activities of the private sector standardizing

organizations. Consumer Information Series—Practical information, based on NBS research and experience, covering areas of interest to the con- sumer. Easily understandable language and illustrations provide

useful background knowledge for shopping in today's tech- nological marketplace. Order the above NBS publications from: Superintendent of Documents, Government Printing Office, Washington, DC 20402. Order the following A' 55 publications—FIPS and NBSlR s—from

the National Technical Information Service, Springfield. VA 22161.

Federal Information Processing Standards Publications (FIPS

PUB)—Publications in this series collectively constitute the Federal Information Processing Standards Register. The Register

serves as the official source of information in the Federal Govern- ment regarding standards issued by NBS pursuant to the Federal Property and Administrative Services Act of 1949 as amended.

Public Law 89-306 (79 Stat. 1127), and as implemented by Executive Order 11717(38 FR 12315, dated May II, I973)and Parl6

of Title 15 CFR (Code of Federal Regulations).

NBS Interagency Reports (NBSIR)—A special series of interim or

final reports on work performed by NBS for outside sponsors

(both government and non-government). In general, initial dis- tribution is handled by the sponsor; public distribution is by the National Technical Information Service , Springfield, VA 22161,

in paper copy or microfiche form.
U.S. Department of Commerce

National Bureau of Standards

Washington, D.C. 20234

Official Business

Penalty for Private Use $300

POSTAGE AND FEES PAID

U S DEPARTMENT OF COMMERCE

COM-215

THIRD CLASS BULK RATE