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NBSIR 81-2321

Single-Room Heat Balance for

Building Heat Transfer

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Engineering Laboratory

Center for Building Technology

Washington, DC 20234

November 1 981

IGU

.U56

81-2321

1981

c. 2

Sponsored by:

Department of Energy

Passive and Hybrid Solar Energy Division

Office of Solar Heat Technologies

Washington, DC 20585
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NBSIR 81-2321 ^

SINGLE-ROOM HEAT BALANCE FOR . ^ BUILDING HEAT TRANSFER

B. A. Peavy

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Engineering Laboratory

Center for Building Technology

Washington, DC 20234

November 1981

Sponsored by:

Department of Energy

Passive and Hybrid Solar Energy Division

Office of Solar Heat Technologies

Washington, DC 20585

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

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director
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TABLE OF CONTENTS

Page

1. INTRODUCTION 1

2. RADIATION AND RADIOSITY SHAPE FACTORS 2

3. CONVECTION HEAT TRANSFER BETWEEN ROOM SURFACES AND AIR 4

4. HEAT BALANCE 5

5. CONCLUSIONS 8

References 9

Appendix A A-1

ill
Nomenclature

A == surface areas

B == constants

s

== specific heat of air

C == cloud cover

H'!' H® ^

= i,t o,t

= inside and outside coefficients of heat transfer for

building construction m, at time t

u®

^ci

= convection coefficient of heat transfer for building

construction m

Ha == coefficient of heat transfer between room air and simulated

room mass

pin -= factors of past heat flux history for building construction

cm

Hi,t = sum of resultant radiative internal heat gains (flux) and

transmitted solar energy to Inside surface of building

construction, m

fpin rpin = inside and outside surface temperature for building

construction m, (wall, ceiling, floor, window or door), m = 2,3 .... n total number of room surfaces

0? 0® = heat flux at inside and outside surface of building

construction, m at time t

Qf

== sum of resultant convective Internal heat gains at time t.

^in *7 in »^n,k»^n,k

'= kth order conduction transfer function for building

construction, m, n = 1,2,3...

•^m

= absorptance of surface m

I- := incident solar radiation on surface m

== long-wave radiation factor

F = ^ m, n

= radiosity shape factor, seeing surface m, to receiving

surface n

Tb.t = outdoor air temperature at time, t

Ta,t '= room air temperature at time t

Tc.t = surface temperature of simulated room mass

''o.t

== mass flow of Infiltration air to room

''a.t

•= mass flow of supply air to room
Single-Room Heat Balance for Building Heat Transfer

by

B. A. Peavy

ABSTRACT

A single-room heat balance has been developed to provide a more

precise computational tool. The primary purpose for this tool is

to evaluate the effects of approximations presently used in computer

programs on the determination for building heating and cooling

loads. Specific algorithms to be incorporated in the room heat

balance concern radiosity shape factors, temperature difference

dependent convection heat transfer coefficients, simulated room

mass, and an iterative methodology for solution of room temperatures.

Ke3^ords: building heating/cooling loads; heat balance for a single

room; heat transfer; radiosity shape factors.

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1.

INTRODUCTION

Fundamental to the determination of heating and cooling loads in buildings is

the formulation of a heat balance for a single room within a building. Items

to be considered as elements of the heat balance are discussed briefly:

1. Conduction heat transfer in building constructions such as through the solid interior surfaces of ceilings, floors, walls,

windows, doors, etc. as affected by temperature changes at their

exterior surfaces.

2. Radiation heat transfer by emitted and reflected energy among the room surfaces.

3. Convection heat transfer between room air and the room surfaces.

4. Distribution and magnitude of transmitted solar radiation passing

through fenestration areas.

5. Heat generation within the room and the resultant convection and

radiation heat transfer.

6. Heat transfer to room mass such as furniture, furnishings, etc.

This is to be considered with changes in room air temperature

only.

7. Convection heat and mass transfer from sources and/or forces

acting exterior to the room such as Infiltration, exfiltration,

circulating air, and inter- and intra-room convective air motion.

A fairly thorough discussion of the above items is found in Kusuda [1], and

the heat balance equation involving most of the above items has been incorporated in the computer program NBSLD [2]. In this paper, algorithms will be

developed for items 2 and 3, heat balance equations will be derived using the seven items listed above, and a method will be proposed for solving the

resulting set of equations using an iterative technique.

1
2. RADIATION AND RADIOSITY SHAPE FACTORS

The algorithms for calculating radiation shape factors are found in [2] and

include angle-factor algebra and solutions for determining shape factors

between a plane rectangular surface of arbitrary dimension and another parallel

or perpendicular rectangular plane surface.

Radiosity is defined as the total radiant flux leaving the surface of a system

and includes both energy emitted and energy reflected from a system. Radiosity

shape factors for room surfaces are computed using Hottel's method [3] for a gray enclosure, and are determined from the radiation shape factors and the areas and emittances of the separate surfaces. Radiosity shape factors will be

used in the same manner for room heat balances as radiation shape factors are presently used in NBSLD. One difference is that the room surfaces will be

able to "see" themselves due to reflection. Another difference will be that

the emittance of the seeing surface will be incorporated in the shape factors.

Hottel's method is preferred for the transient numerical analysis of a complete

system, particularly where the surface temperatures are unknown.

A subroutine has been developed for calculating radiation and radiosity shape

factors between inside rectangular surfaces of a room. It was developed for

accepting data in a form similar to Data Sheets 12, 13 and 14 (with slight

modifications) of the computer program NBSLD [2]. For computing radiation

shape factors, the exact location of a window, door or secondary wall must

be defined, in addition to the respective areas and emittances. This is different from present practice, where only the area is defined for windows, doors

and secondary walls.

At present, the program is designed to calculate factors for rooms with walls

containing different building constructions or surface coverings. The program

presently does not include more than one surface for either the floor or ceiling. Consequently, floors with two or more surface coverings such as wood,

tile and rug combinations are modeled as one surface^ and skylights in a ceiling are not modeled. In its present form, the program can accept data for 30

rectangular surfaces placed either parallel or perpendicular to each other,

including celling, floor, 4 primary walls and a collection of up to 24 windows,

doors and secondary wall sections located on the four walls. A simulated room

layout is shown in figure 1, which illustrates the sequence of surfaces and

necessary measurements needed for calculation of radiation shape factors.

Appendix A is a listing of the subroutine needed for determining the radiation

and radiosity shape factors for use with NBSLD, as well as the necessary additional information needed for Data Sheets 13 and 14 of reference 2. The integration of the radiosity shape factor into the room heat balance will be

discussed in section 4.

2
SECONDARY WALL

( 10)

NORTH WALL (5j

WINDOW

( 11 )

WEST WALL (4)

FLOOR

( 2)

EAST WALL (6]

WINDOW

( 12)

DOOR |8)

SOUTH WALL (3)

CEILING

SAMPLE ROOM LAYOUT

Figure 1. Sample room layout.

3
3. CONVECTION HEAT TRANSFER BETWEEN ROOM SURFACES AND AIR

Generally, the heat transfer between room surface and the air involves motion

in the air due to difference in density and the action of gravity or natural

convection. Natural convection heat transfer coefficients for air are defined

[4] by the simplified relationships

= 0.19 (AT)^*^^, horizontal heat flow to vertical plates (la)

= 0.22 (AT)^*^^, heat flow up to or from horizontal plates (lb)

0 33 = 0.11 (AT) , heat flow down to or from horizontal plates (Ic)

where AT is assumed to be the absolute temperature difference between the air

and the surface considered. These relationships make the computation more com- plicated because in the heat balance the temperatures of the surfaces and the

air are unknowns and need to be determined. In the iterative technique this can be easily handled by assuming surface and air temperatures from the previous time period for the first iteration of present time step. In the present

version of NBSLD, the convection heat transfer coefficients are assumed to be

constants for the three directions of heat flow.

4
4. HEAT BALANCE

At exterior facing surfaces the heat balance is given by

+

pm + QS.t -

Qs

= 0 ( 2 )

where

a) Incident solar radiation

b) Convection heat transfer from the outdoor air

^A o,t'b,t o,t

c) Conduction heat flow at outdoor surface of building construction

k

^o,t ^l,k ^i,t ^l,k ^o,t ^

^n %,t-n

n=l

I y 'pin _ »7ni rpin \ ^^n,k "^ijt-n+l ^^n,k '^o,t-n+l'^

n=2

d) Long-wave radiation to the sky

Qs = 23 (10-C).

Solving for the outside surface temperature at time t gives the relationship

(3)

+

^T,k>

, B, T? , + B,

O, t 1 1, t 2

(H“,t + n,k> »2 = “It +

"S.t Tb.t + 2ll(10-C) + 2 R“

n=l

_i_ y «pin ^ yin 'ptn \ '^n,k ^i,t-n+l ^n,k '^o,t-n+l'

n=2

The coefficients and B 2 are known for each time, t.

Equation (3) is solved for the outside surface temperature of a wall exposed to

outdoor weather conditions. If the wall divides two rooms, then a proper heat

balance is necessary at the surface of the other room, for which another definition for B]^ and B£ must be determined. Similarly, proper heat balances must

be performed for attic spaces and crawl spaces where appropriate definitions

may be derived for the constants B^ and B£. It is not the purpose of this

paper to propose algorithms for defining heat transfer in attic and crawl

5
spaces. A model will be proposed in section 5 which will adequately define the

external conditions as they affect the room heat balance, particularly in ref- erence to comparison of various determination methods (such as found in NBSLD,

BLAST, AND DOE-2).

The heat balance at a room surface is

+

^‘T.k -

’'T.k ^ «S.k

n"2 w

- ^S.k TS,t-n+l) + S + S?

t ° n"l

where

»i,t n"l

(5)

Solving for the surface temperature T^

^ in (4) gives

B3 + B4

D _ um 4. ym _ n ^l,t ^

^l,k ^l,k

( 6 )

a 4 _ vin a X mm T 1 y® t® — T® T® 1 ° ^l,k °2 ^ ®i,t \ ^i,t-n+l ^n,k ^,t-n+y

- \ 'S «l.t-n

n"l

The term is known for each time, t, and eoefflclent suit b@ coapulid

before each Iteration.

The heat balance for the room air can be expresiid by the following

relationship

- Ta.t) +

+ M»(Te,e - Ta,{) + Q( 0 (7)

At the surface of the room maii

- Ta.t) + ^

- 2y| I • 0 (8)

n*l

where the temperature at the surface of the simulated room mass Is
^c,t ^a,t ^6 (9)

^5 = »a ^l,k "l,k

- 2Yf,k

B

6 = 2 (^S,k + zS,k - 2YS^k)Tc,t-n+l

n=2

The temperature of the room air becomes

^7^a,t I . + Bo1 n 1 , t o

B

7 = HiAn + (^o.t '^a.t^-p ' —-5

A H

^ ^ (Be »a)

( 10 )

»8 -

(''o.t ’^b.t +''a,C Ts,t:><^p + «f +

a a D B5

( 11 )

Constants B5, B5, By and Bg are computed once for each time, t.

Using (6) and (10), the surface temperatures and the room air temperature may

be solved for by iteration where for the first iteration these temperatures are assumed to be the same as for the previous time, t-1, for a unit time interval.

After each iteration, H? must be redetermined due to changes in T?

^ and

, and changes in H^^’from the relationships (la), (lb), or (Ic). * The total

number of iterations is to be determined by establishing conditions for the con- vergence of the iterative process. When adequate convergence is attained for

T?,t and then TTM^^ (3), T^^^ (9), Q? and

^ (2c) are to be determined. Q? * is the sum of the second, third, fourtfi and fifth terms of (4). 1 t

7
5. CONCLUSIONS

The heat transfer algorithms presented in sections 2 and 3 of this report

are intended to be used in part or whole for the purpose of determining heat

exchange within a room. Although they represent an ideal case, they are probably the most exact of methods presented previously and could be used to

compare other methods.

For comparing different analytical techniques dealing with heat exchange in a room, the effects of external and internal heat balance algorithms on the

results should be distinguished. To perform such comparisons, a simplistic

model is suggested, which simulates four wall surfaces, a flat roof and a floor

raised from the ground level so that all six envelope surfaces are exposed to

the environment. This model, which would be similar to vacation homes in some coastal regions, offers a somewhat realistic condition by which the use of

available algorithms can be justified. With this model, comparisons of the

method proposed in this paper can be made to other methods, particularly for

geometrical and orientation considerations for the placement of windows, doors,

etc., effect of internal mass, effect of variation in the surface coefficients

of heat transfer, and effect of surface emlttance and reflectance (radiosity).

The Iterative solution for the surface and air temperature offers a method

by which the more exact algorithms may be used whereby a computation time

savings is evident when compared to other methods of solution. A computer

program using this feature is possible. Use of this feature on NBSLD was not

attempted because this was not within the scope of this study.

8
References

1. T. Kusuda, Fundamentals of Building Heat Transfer, J. Research, Natl. Bur.

Stds., Vol. 82, No. 2, 1977.

2. T. Kusuda, NBSLD, the Computer Program for Heating and Cooling Loads in

Buildings. BSS 69, Natl. Bur. Stds., July 1976, pp 48-59a, 79-81a, 94-99a,

45-51C.

3. J. A. Wiebelt, Engineering Radiation Heat Transfer. Holt, Rinehart and

Winston, 1966.

4. Handbook of Fundamentals, ASHRAE, New York, 1977.

9

Appendix A

Subroutine ROOMZ - to compute the radiation shape factors and the radiosity

shape factors for a room in the shape of a rectangular parallelepiped. The

walls of the room may have up to 24 rectangular shapes on them which may represent windows, doors and/or secondary walls. The first 3 inputs (Lines 13, 24,

and 58 of listing) are identical to Data Sheets 12 and 13 of [2]. At line 62

of the listing, 5 numbers are input to the subroutine, if other than the

primary walls, ceiling, or floor are to be considered; namely,

1. Width of the window door or secondary wall

2. Height of above

3. Distance from left corner of wall to closest side

4. Height from floor to bottom of rectangle

5. Emittance of surface (needed for all surfaces).

With the above information, the radiation shape factors are calculated using

the algorithms found in [2], pages 48a-58a. At line 280, the radiation shape

factors have been computed and stored in the array F (M,N). Radiosity shape

factors are then computed and stored in the array SF(M,N) (line 307).

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235

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CONTINUE

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A-
NBS»n4A (REV. 2»8C) U.S. DEPT. OF COMM,

BIBLIOGRAPHIC DATA

SHEET fSee instructions)

1. PUBLICATION OR

REPORT NO.

2. Performing Organ. Report NoJ 3.

4. TITLE AND SUBTITLE

SINGLE-ROOM HEAT BALANCE FOR BUILDING HEAT TRANSFER

Publication Date

November 1981

5. AUTHOR(S)

Bradley A. Peavy

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

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

NBS and Department of Energy

Passive and Hybrid Solar Energy Division

Office of Solar Heat Technologies

Washington, Df. 20‘58‘j

10. SUPPLEMENTARY NOTES

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

11.

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

bibliography or literature survey, mention it here) A single—room heat balance has been developed to provide a more precise computational

tool. The primary purpose for this tool is to evaluate the effects of approximations

presently used in computer programs on the determination for building heating and

cooling loads. Specific algorithms to be incorporated in the room heat balance

concern radiosity shape factors, temperature difference dependent convection heat

transfer coefficients, simulated room mass, and an iterative methodology for solution

of room temperatures.

12. KEY WORDS (S/'x to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolon s) building heating/cooling loads; heat balance for a single room; heat transfer;

radiosity shape factors

13. AVAILABILITY 14. NO. OF

Unlimited

1 1 For Official Distribution. Do Not Release to NTIS

PRINTED PAGES

22

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

20402. 15. Price

QQ Order From National Technical Information Service (NTIS), Springfield, VA. 22161 $5.00

USCOMM-DC 6043-P80

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