single-room heat balance for building heat transfer
TRANSCRIPT
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NBSIR 81-2321
Single-Room Heat Balance for
Building Heat Transfer
U.S. DEPARTMENT OF COMMERCENational 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 EnergyPassive and Hybrid Solar Energy Division
Office of Solar Heat TechnologiesWashington, DC 20585
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WATIOWAL burbau09 0TANDA1UC
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DfC 8 1981
NBSIR 81-2321^
SINGLE-ROOM HEAT BALANCE FOR . ^
BUILDING HEAT TRANSFER
B. A. Peavy
U.S. DEPARTMENT OF COMMERCENational 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 buildingconstruction m
Ha == coefficient of heat transfer between room air and simulatedroom mass
pin -= factors of past heat flux history for building construction
cmHi,t = sum of resultant radiative internal heat gains (flux) and
transmitted solar energy to Inside surface of buildingconstruction, m
fpin rpin= inside and outside surface temperature for buildingconstruction 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 buildingconstruction, 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 receivingsurface 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 moreprecise computational tool. The primary purpose for this tool is
to evaluate the effects of approximations presently used in computerprograms on the determination for building heating and coolingloads. Specific algorithms to be incorporated in the room heatbalance concern radiosity shape factors, temperature differencedependent convection heat transfer coefficients, simulated roommass, and an iterative methodology for solution of room temperatures.
Ke3^ords: building heating/cooling loads; heat balance for a singleroom; 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 passingthrough fenestration areas.
5. Heat generation within the room and the resultant convection andradiation heat transfer.
6. Heat transfer to room mass such as furniture, furnishings, etc.
This is to be considered with changes in room air temperatureonly.
7. Convection heat and mass transfer from sources and/or forcesacting 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], andthe heat balance equation involving most of the above items has been incorpo-rated in the computer program NBSLD [2]. In this paper, algorithms will bedeveloped for items 2 and 3, heat balance equations will be derived using theseven items listed above, and a method will be proposed for solving theresulting 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] andinclude angle-factor algebra and solutions for determining shape factorsbetween a plane rectangular surface of arbitrary dimension and another parallelor perpendicular rectangular plane surface.
Radiosity is defined as the total radiant flux leaving the surface of a systemand includes both energy emitted and energy reflected from a system. Radiosityshape factors for room surfaces are computed using Hottel's method [3] for a
gray enclosure, and are determined from the radiation shape factors and theareas and emittances of the separate surfaces. Radiosity shape factors will be
used in the same manner for room heat balances as radiation shape factors arepresently used in NBSLD. One difference is that the room surfaces will be
able to "see" themselves due to reflection. Another difference will be thatthe emittance of the seeing surface will be incorporated in the shape factors.Hottel's method is preferred for the transient numerical analysis of a completesystem, particularly where the surface temperatures are unknown.
A subroutine has been developed for calculating radiation and radiosity shapefactors between inside rectangular surfaces of a room. It was developed foraccepting data in a form similar to Data Sheets 12, 13 and 14 (with slightmodifications) of the computer program NBSLD [2]. For computing radiationshape factors, the exact location of a window, door or secondary wall mustbe defined, in addition to the respective areas and emittances. This is dif-ferent 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 wallscontaining different building constructions or surface coverings. The programpresently does not include more than one surface for either the floor or ceil-ing. Consequently, floors with two or more surface coverings such as wood,
tile and rug combinations are modeled as one surface^ and skylights in a ceil-ing 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 roomlayout is shown in figure 1, which illustrates the sequence of surfaces andnecessary measurements needed for calculation of radiation shape factors.
Appendix A is a listing of the subroutine needed for determining the radiationand radiosity shape factors for use with NBSLD, as well as the necessary addi-tional information needed for Data Sheets 13 and 14 of reference 2. The inte-gration 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 motionin the air due to difference in density and the action of gravity or naturalconvection. 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 airand the surface considered. These relationships make the computation more com-plicated because in the heat balance the temperatures of the surfaces and theair are unknowns and need to be determined. In the iterative technique this
can be easily handled by assuming surface and air temperatures from the previ-ous time period for the first iteration of present time step. In the presentversion 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 B2 are known for each time, t.
Equation (3) is solved for the outside surface temperature of a wall exposed tooutdoor weather conditions. If the wall divides two rooms, then a proper heatbalance is necessary at the surface of the other room, for which another defi-nition for B]^ and B£ must be determined. Similarly, proper heat balances mustbe performed for attic spaces and crawl spaces where appropriate definitionsmay be derived for the constants B^ and B£. It is not the purpose of thispaper 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,tn"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-nn"l
The term is known for each time, t, and eoefflclent suit b@ coapulidbefore each Iteration.
The heat balance for the room air can be expresiid by the followingrelationship
- 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
B6
= 2 (^S,k + zS,k - 2YS^k)Tc,t-n+ln=2
The temperature of the room air becomes
^7^a,t I . + Bo1 n 1 ,
t o
B7
= 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 maybe solved for by iteration where for the first iteration these temperatures areassumed 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 T™^^ (3), T^^^ (9), Q? and^
(2c) are to be deter-mined. 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 reportare intended to be used in part or whole for the purpose of determining heatexchange within a room. Although they represent an ideal case, they are prob-ably 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 theresults should be distinguished. To perform such comparisons, a simplisticmodel is suggested, which simulates four wall surfaces, a flat roof and a floorraised 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 somecoastal regions, offers a somewhat realistic condition by which the use of
available algorithms can be justified. With this model, comparisons of themethod proposed in this paper can be made to other methods, particularly forgeometrical and orientation considerations for the placement of windows, doors,etc., effect of internal mass, effect of variation in the surface coefficientsof heat transfer, and effect of surface emlttance and reflectance (radiosity).
The Iterative solution for the surface and air temperature offers a methodby which the more exact algorithms may be used whereby a computation timesavings is evident when compared to other methods of solution. A computerprogram using this feature is possible. Use of this feature on NBSLD was notattempted 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 andWinston, 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 repre-sent 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 wall2. Height of above3. Distance from left corner of wall to closest side4. Height from floor to bottom of rectangle5. Emittance of surface (needed for all surfaces).
With the above information, the radiation shape factors are calculated usingthe algorithms found in [2], pages 48a-58a. At line 280, the radiation shapefactors have been computed and stored in the array F (M,N). Radiosity shapefactors are then computed and stored in the array SF(M,N) (line 307).
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7
NBS»n4A (REV. 2»8C)
U.S. DEPT. OF COMM,
BIBLIOGRAPHIC DATASHEET fSee instructions)
1. PUBLICATION ORREPORT 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 standardsDEPARTMENT 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 EnergyPassive and Hybrid Solar Energy DivisionOffice of Solar Heat TechnologiesWashington, 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 significantbibliography 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 1For 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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