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Analytic model for passively-heated solar houses—1. Theory
So/arEnemy Vol.27,No. 4,pp. 331-M2. 1981
0038-092X/8111~0331-1250"2.~/0
Printed in Great Britain.
© 1981Perpmon Press Ltd.
ANALYTIC MODEL FOR PASSIVELY-HEATED SOLAR HOUSES--1. THEORY? J. M. GORDON
and Y. ZARMI
Applied Solar Calculations Unit, Jacob Blaustein Institute for Desert Research, Ben-Gurion University of the Negev, Sede Boqer Campus, Israel
(Received 22 July 1980; accepted 2 April 1981) Abstract--A simple analytic method for the prediction of the long-term thermal performance of passively-heated solar houses is presented. The treatment includes a new coarse method for "energy bookkeeping"and the use of a distribution function which represents the frequency of occurrence of different values of the solar load ratio. Due to its generality, this formalism is applicable to any passive heating element. As specificexamples, the cases of direct gain and water wall houses are treated in detail. Relative to the parameterizationof computer simulation results, this method offers the user a design tool that can be used to predict, in closed form, the thermal effect on the house of differentbuildingand climaticparametersand is not restricted to a "reference" building.Agreementwith published numericalsimulationresults in satisfactory.The presentation is divided into two separate papers: a users guide for the reader who may not be interested in the details of derivation and validation,and the present paper, in which the theory is presented in detail.
One advantage of the proposed methodology is its simplicity. It provides closed-form analytic expressions for the same performance variables that have, until now, been obtained via large-scale computer simulations. Consequently, one can easily calculate on a hand calculator the effect of variations in building and climatic parameters such as thermal storage mass, thermal resistances, storage absorbance, insolation, etc. Another advantage of this approach is that it provides insight into the basic physical ingredients which govern the performance of passive solar systems. Thermal performance is expressed in terms of building and climatic variables whose physical meaning is clear. In Section 2 we note a few of the fundamental physical considerations upon which the theory is based. In Section 3 we present our energy bookkeeping method and the general formalism for seasonal calculations. In Section 4 we discuss the particular details of the distribution function we employ to describe the frequency of occurrence of various daily solar load ratio values. In Section 5 this formalism is applied to direct gain houses. In Section 6 our results for direct gain houses are presented and compared with the corresponding results from published numerical simulations[I,2]. In Section 7 the method is applied to water wall houses, and in Section 8 the water wall results are compared with the corresponding computer simulation results [3]. The ability of the model to predict the dependence of the performance of passively-heated solar houses on various parameters of the building is also presented. Our conclusions are presented in Section 9. In the Appendix, we show the effect of introducing a time-dependent daily insolation and to what extent it limits the validity of our treatment.
I. INTRODUCTION
The prediction of the seasonal thermal performance of passively-heated solar houses is an essential element in the design of these structures. Much of the successful work in this area has emerged from numerical solutions of the large number of equations which govern heat transfer and storage through multiple thermal nodes. A primary example is the work of the Los Aiamos team in predicting solar heating fraction as a function of the solar load ratio [1-5]. There is, however, a need for simple analytic models which capture the essential physics of the problem, afford the user the flexibility of not being restricted to a "reference" house, and can be applied with equal ease to a variety of passively-heated solar building types. Such models may be particularly well suited to the prediction of seasonal performance and its dependence on various parameters of the building since the effects of hourly temperature and insolation variations can be averaged out over the course of an entire heating season. In this paper a new methodology is proposed for a simple treatment of seasonal thermal performance of passively-heated solar houses. The particular cases of direct gain and water wall houses are treated in detail. A new method for "energy bookkeeping" is presented-one which is suited to monthly or annual calculations as opposed to hourly performance. The effect of variations in total daily insolation is accounted for by weighting different values of the solar load ratio by the frequency of their occurrence. Since most users of this methodology may not be interested in the details of its derivation and validation, we divide our presentation into two separate papers: a users guide [6], with little explanation of the origins of the formulae, and the nresent Darter which describes
2. FUNDAMENTALCONSIDERATIONS (a) Definitions The object of the calculation is to express a solar heating fraction, SHF, of a house in terms of the solar load ratio, SLR, as well as building and climatic parameters. SHF is defined as the fraction of a house
"~Research supported by the Center for Absorption in Science, The Ministry for ImmigrantAbsorption, State of Israel. 331
J.M. Go~u~oNand Y. Z ~ I
332
heating load which is supplied by solar energy. SLR is defined as the ratio of the insolation transmitted into the house, Qr, divided by the house heating load, L. The insolation transmitted through the vertical south-facing giazings can be readily calculated from the more commonly available data of insolation on a horizontal surface [1]. Over a given period of time, t, (which can be the entire heating season or a sizable fraction thereof, e.g. a month), SHF is calculated as SHF;
QdL, = filL,)
fo'
q,(t') dr',
(1)
where L, is the total heating load over time, t, Q. is the useful solar gain collected over time, t, and q.(t) is the useful solar gain per unit time. For the sake of specificity, the period of time under consideration throughout the remainder of this paper will be the entire heating season. By useful solar gain, we mean solar energy gain which does not result in heating room air above a specified comfort temperature, To. The house is assumed to have a back-up heating system which insures that room air temperature never falls below To. Let us stress that the heating load should be calculated with respect to the temperature below which the back-up system is activated (To). It is also assumed that whenever the room air temperature exceeds Tc ventilation is employed to maintain room air temperature at To If the heating load is calculated as specified above, then the effect on SHF of activating ventilative cooling of a temperature slightly above T~ is negligible [7]. (b) Dependence on SLR Before any detailed calculations are performed, let us indicate what can be said about the behavior of SHF as a function of SLR. Were there no overheating at any time during the heating season, then all collected energy would be useful energy, and SHF would be proportional to SLR, SHF-- Co SLR.
(2)
The coefficient, co, is a function of climatic as well as building parameters. Equation (2) is general, although the value of Co and its dependence on building parameters vary with the type of passive heating element. Clearly, for sufficiently large SLR values, SHF becomes unity. Since overheating may occur during part of the heating season, not all collected energy is useful energy (energy is dumped), and SHF averaged over the heating season is less than the value predicted by the linear form of equation (2). That is, there is a non-linear dependence of SHF on SLR in a "transition" region characterized by intermediate SLR values. To summarize, SHF is linear in SLR at low SLR values, nonlinear in a "transition" region of intermediate SLR values, and unity at large SLR values. A calculation of SHF in this transition region depends on a proper accounting of energy that is not useful (dumped). This entails accurate "energy bookkeeping'*
for determining what fraction of the time overheating occurs and what magnitude of collected energy is dumped. 3. ENERGY B O O ~ G
We divide the heating season into periods during which solar gain provides all of the heating load (no back-up required) and periods during which solar gain provides only part of the heating load. For our purposes of energy bookkeeping, we use a relatively coarse division of time into days and nights (rather than the hourly calculations usually used in numerical simulations). We then further divide the calculation into days (or nights) during which solar gain (or the storage thereof) provides all of the load, and days (or nights) during which solar gain provides only part of the load. We assume that the fraction of the average daily load that is represented by the daytime load, Ld, and the nighttime load, L., does not vary appreciably over the heating season. We can then express SHF as a weighted average over daytime and nighttime periods during which back-up heating is either required or not required, SHF = (llN)([df) day-~-'.~.o 1 back-up
+(llN)(£d£)d.~._,a (QdL~) bm:k,.up
+(l/N)(F~dFO ~,,o, l +(llN)(£dF~) .~.~,,,h (QdL.) no brock-up
Ix~k-up
(3) where N is the number of days in the heating season, and £d and/.~ are the average daytime and nighttime loads, respectively. The useful energy provided by the passive solar element, denoted by Q. in eqn (3), is a combination of both energy supplied by solar on the day it is collected and "residual" energy which has remained in storage from previous days. We then denote by SHFd(°) and SHF. <°) the SHF value for daytime and nighttime periods, respectively, during which back-up is required, namely SHF~°)= Q.(day)/L,t SHF. (°)= Q.(night)/L.. In order to compute the summations of eqn (3), we need the frequency of occurrence of different values of SLR during the heating season. We therefore introduce a normalized distribution function, p(SLR), which gives the probability of occurrence of any SLR value. SHF then becomes
SHF=([df)[f;L~.. ~#(x) dx + - " F =(°) foSLR"dp(x)SH d (x ) dx J I(l-d/~)
Analytic model for passively-heated solar houses--l. Theory where SLR=i..a and SLRm~,,. are the minimun values of SLR such that no back-up heating is required during daytime and nighttime, respectively, and the dummy variable x denotes SLR. SLR=,,.a and SLRm,.,. are the points at which SHF) °) and SHF. ~°) become unity, SHFd~°)(SLRmin.,~)= 1
(5}
SHF.~°>(SLR=i...) = 1.
0
distribution (assuming a thermostat set temperature of 18.3eC (65°F)). This distribution is a peaked, symmetric function with a statistically insignificant tail for large SLR values. Therefore, for the purpose of presenting a closed-form analytic solution, we use a parabolic distribution function which is peaked about the average SLR value, SLR, which vanishes at SLR -+& and which is normalized,
0 < SLR < SLR - 8
p(SLR) = 3(82- (SLR- SLR)2) SLR - 8 < SLR < SLR + 8 483 0
(6)
SLR > SLR + &
Let us again note that, as shown below, in the determination of SHF~°~ and SLR=i., residual energy that remains in storage from previous days, as well as energy collected on a given day, are taken into account.
4. TXEmswamUTlONFUNCTION In order to calculate SHF, we need to know the function which describes the frequency of occurrence of different values of the ratio of daily insolation to load (SLR) over the course of a heating season. It should be noted that most commonly considered passive solar heating elements are vertical, so that the insolation considered is that on a vertical surface. It is interesting to note that although insolation on the horizontal varies significantly during the heating season and gives rise to a skewed distribution [8, 9], the insolation on the vertical is much more uniform throughout the heating season and yields a far more symmetric distribution [8, 9]. With typical meteorological year data for the coastal region of Israel [9], we have determined the actual SLR
For the particular case cited above [9], 8 = (0.8) SLR (see Fig. 1). In principle, the p(SLR) employed in eqn (4) should be the actual measured distribution function particular to one's location. Inspection of some available data [8] indicates that 8/SLR typically varies from 0.6 to 1.0. It will be shown below that the functions SHF~~°~and SHF, ~°~are proportional to SLR, i.e. SHFa~°'= c / S L R
Hence, eqn (5) yields SLRmi..d = l/co'~
Given this result, the distribution function of eqn (6L and eqn (7) are used in eqn (4) to yield SHF = (L,I£) SHF '"~ + (L./£) SHF"""'
0.8 0.6 0-4
0-5
(8)
SLRmi... = 1/co".
i
o
(7)
SHF. ~°)= Co" SLR.
I.O i
°it
333
I "O
1"5
5LR Fig. 1. The parabolic distribution function p(SLR).
2.0
(9)
334
J. M, GORDONand Y. ZARMI
with SLR=.,., > SLR+ (5
Cod " ~
SHF a-' --- 1 + [(I - Co"S ~ + c°aS)3(ll6(coaS)' Coa - S - ' ~ - 3coaS)lj
SLI~ + 8 -> SLR,.,..d -> S L R - 8
(1o)
SLR=i.., < SLI~ -
l
and
SHI~i. h,
coSR
f I+
(1 - c o " S L R
+ Co"8)3(I - co"SLR - 3Co~8)
1
16(Co"8) 3
s. ~nacArlon ro ~ DmEcr GAINSYSTEM A direct gain house is one with a large south-facing glazing and thermal storage mass in the interior. We shall assume that the building envelope is sufficiently well insulated and that storage is located so that heat transfer from storage to ambient can be neglected, and that the other thermal mass of the building is negligible relative to the storage mass. We treat a direct gain house with a two-node model, the two nodes being the storage S and room air A. As all we need to complete the calculation for SHF (via eqn 4) are SHFJ °) and SHF~ (°) (namely, the SHF values for periods during which back-up is required), we consider the heat balance equations for the storage. During daytime, this equation is dTsd/dt d
= (1-F)Q,~ -
UsA~'Td s
L
- F)Qr( I - e-'"l') ] u~a:
(mC)s dTs"ldt ~ = - UsA(Ts" - Tc),
(15)
whose solution for Ts" is ~Ts a'At " -m. , T , " ( t ' ) - Tc = ~ ( d") - Tc)e
(16)
where Tsa(AI") is the value of Ts at the beginning of nighttime, and superscript n denotes nighttime. We use eqn (13) in eqn (16) to obtain the solution for Ts'(tn),
T,)e-""'|
T~),
where (mC)s is the heat content of the storage; Ts is the storage temperature; t denotes time; superscript d denotes daytime; At" is the length of daytime; F is the fraction of transmitted insolation Q-r which is not absorbed by storage and hence directly heats room air; and Us^ is the heat transfer coefficient from storage to room air. We have assumed above that insolation can effectively be treated as constant throughout the day (the justification for this is presented in the Appendix). The solution of eqn (12) for Tsd(t d) is
f(l
(ll).
The heat balance equation for the storage at night when back-up is required is
+ (Ts(O)-
(12)
Tsa(t d) - T~ =
SLR + 8 --- SLR=~..,, -> SLR - 8 SLR.~... < SLR - 8
I
(mC)s
SLR.,..o > SLR + 8
e-r,t,. J
(17)
The energy collected by room air during daytime (Qjr)) and night-time (Q,,")) on a given twenty-four hour day is Qd"t = FQr + foa'~ UsA(Ts . ( t d) -
Q."' =
0 ¢n
Tc)dt d
Us,~(Ts"(t ~) - To) dt",
(18) (19)
where FQ-r in eqn (18) represents the fraction of transmitted radiation absorbed directly into room air during daytime. Using eqns (13) and (17) in eqns (18) and (19), respectively, we obtain
J
+ (Ts(0)- T~)e -'"I"
(13) + UsA(Ts(0)- Tc)r(l - e-A'd/;)
where
(20) "r =
(mC)sT Usa
(14)
and is the relaxation time for heat transfer from storage to room air; and Ts(O) is the value of Ts ~ at the beginning of the day.
and Q . ) _ (I - F)Qr.r(l - e -A'"I")(1 - e -~m') At" + UsA(Ts(O)- T~)T e~'"/"(1
e -~m')
(21)
Analytic model for passively-heated solar houses--l. Theory Both eqns (20) and (21) have terms which represent the fraction of QT which is put to use the same day, and the fraction of "residual" energy (which has remained in storage from previous days) which is also put to use that same day. Since our calculation is a monthly or seasonal one, we are interested in how the total useful QT is distributed between daytime and nighttime over the course of a month or a season. Equations (20) and (21) indicate how the residual energy will, on the average, be distributed between daytime and nighttime. Residual energy QR is simply that fraction of QT which is not put to use on that same day, hence
335
20% of transmitted solar flux being absorbed directly into room air). We specify our "reference" location which, in conjunction with the house's properties, has the following seasonal average properties: 8/SLR = 0.8 £/£a = 3
At ~ = 10 hr. For these parameters, eqns (26)-(27) and (8) yield Coa = I/SLR,,,.a = 1.935
QR
(I -
F)QT¢ e-A'"(l
e A,a,,)
Consequently, the average useful daytime (nighttime) energy gain Q a ( Q , ) is
Qa Q~- r l L -
r.
- F),r(1
L
- e-"''')] ]
- e-~"'r/'/'rX1
Ata(l -
e -'''`+'''")")
'
JJ
(23) (7. = QT - Qa.
Equation (23) provides us with a solution for the SHF for days during which back-up is required, namely, SHFat°>-- QdLa
=
(f-,If-,a)(Q,IL)
=
Coa SLR
(24)
and SHF, (°)= Q.IL,, = (/ZII~,,,XQ./C)
Co" = I/SLRmi,., = 0.533.
(22)
At d
= co" SLR,
(25)
With eqns (9)-(11), we calculate SHF as a function of SLR, which is presented in Fig. 2. On the same plot we also present the corresponding seasonal SHF for a reference house with and without night insulation as calculated by the numerical simulations [1]. The results of Ref. [1] represent a best fit to results from a variety of climatic conditions, while our calculation is for a location of.specified characteristics. In our model, the variable ElLa decreases with the addition of night insulation. To show the effect of varying f . / f . a (one manifestation of which could be the addition of night insulation, another of which could simply be different climatic conditions), we also plot in Fig. 2 our prediction for the same house with f-I£a equal to 2 and 4. As expected, smaller values of L/La (a larger fraction of the heating load occurring during daytime) increase SHF.
where
Cod=(c,/c,a)r1-r(l- F),(1-e-:")(,- e-:''-)'ri Atd(l - e-U,,d+A,.)1,) jj L L (26)
co"= (C1£.)[ (1 - F)'r(1 - e - " ' " ) ( l
- e-"'"l')] Aid( 1 _ e_,,,~+~,,)/,) j.
(27)
The fact that SHF~°) is proportional to SLR is a key result. It enables us to perform the integrations described in Section 3 and to obtain a closed-form analytic expression for SHF. With these results, eqns (24)--(27) and eqn (6) are used in eqn (4) to yield the closed-form expressions for SHF as described in eqns (10)-(11). 6. RESULTS FOR DIRECT GAIN HOUSES
To offer a quantitative comparison with the results of existing computer simulations, we first present an exemplary calculation for a direct gain house whose parameters are taken to be those of the "reference" house considered in Ref. [1], namely, ¢ = (mC)slUs^
--
15 hr,
and 1 - F = 0.64 (a mass surface absorbance of 0.8 and
In Fig. 3, we present the effect of varying the daytime length At a (with all other parameters fixed at their reference values). We note that as At a decreases, SHF increases. This is a consequence of the fact that the same amount of insolation is absorbed over a shorter daytime period (in a house with a time constant for heat transfer from storage to room air of 15 hr), and put to use at night when the load is greater. In Fig. 4, we display the dependence of SHF on the relaxation time ~- for several fixed values of SLR. Increasing ~" corresponds to either increasing the storage thermal mass or decreasing the heat transfer coefficient from storage to room air, or both (at fixed F). It should be noted that increasing storage thermal mass by increasing its area rather than its thickness also affects the parameter F. SHF increases with increasing ~" and approaches an asymptotic value at large ~'. Our results are consistent with the findings of Ref. [2]. In Fig. 5, we show the effect on SHF of varying F at fixed values of SLR. At sufficiently low values of SLR, SHF is independent of F and is equal to SLR. At larger, more commonly encountered values of SLR, SHF is a decreasing function of F, which is in agreement with the findings of Ref. [10]. This is due to the fact that large F values result in larger fractions of daily insolation being absorbed directly by room air during daytime and less energy being stored for higher load nighttime periods. We also note that at sufficiently large values of SLR, the
336
J.M. GOR~ONand Y. ZAP.MI
SHF ~.O
b 0"6
0"6
0-4
0.2
0
o
t.o
2:o
3-o
4:0
SLR Fig. 2. Dependence of SHF on L/L~ and comparison _with numerical simulation results (SHF vs SLRY---Direct Gain; a, our model with reference conditions (including/.,/~ = 3); b. same with [,/[.a = 2; c, same with f./f.a = 4: d, numerical simulation results' without night insulation; and e, numerical simulation results Lwith night insulation.
SHF ,,
1-0
0.8
0'6
0"4
0.2
0 0
1~
2"0 SLR
3"0
4"0
Fig. 3. Dependence of SHF on Atd--DireCt Gain; a, our model with reference conditions (including At d = l0 hr); b, same with At J = 12 hr; and c, same with &t ~ = 8 hr.
SHF |.0 f 0.8 0.6 f
0.4 0.2
0 5.00
tO.O0
~5,00
-L-
20.00
25"00 (hr)
Fig, 4. SHF vs the relaxation time ~ for fixed SLR--Direct Gain; a, SLR = 2.00; b, SLR -- 1.00; and c, SLR = 0.50.
Analytic model for passively-heated solar houses--l. Theory
337
SHF
1.0,
O'B
'
0.6,
0.4.
0.2.
0 0
0"2 5
0.5
0.7 5
1.0
F
Fig. 5. SHF vs fraction F of transmitted insolation not absorbed by storage, for fixed SLR--Direct Gain; a, SLR = 2.00; b, SLR= 1.00; c, SLR = 0.50; and d, SLR = 0.10. limiting value of SHF at F = 1.0 is simply L d L This is a result of the fact that if no solar energy is being stored for nighttime use (F = 1.0), then sufficiently large SLR values will provide 100% of the daytime load only. In Fig. 6, we present the effect of varying the relative width of the parabolic distribution function, 8/SLR. We note that SHF is not a sensitive function of ~/SLR.
the two nodes being the water storage S and room air A. We consider the heat balance equation for the storage. During daytime when back-up is required, this equation is (mC)s dTs d Idt d = (c~QrlAt a ) - U d (Ts a - T~a) - UsA(Tsd - To),
where a is the fraction of transmitted insolation QT which is absorbed by storage; U~a is the daytime heat transfer coefficient from storage to ambient; and UsA is the heat transfer coefficient from storage to room air. One further approximation to be made here is that the relatively fast rate of heat transfer which results from natural convection within the water wall allows us to treat the water wall as an element of negligible thermal resistance. The convective heat transfer coefficient within a water wall can in fact be calculated from the
7. APPLICATION TO THE WATER WALL
The house considered here has a glazed, south-facing water storage wall. The absorbing surface area of the water wall is taken to be equal to the glazing area. The house's other thermal mass is assumed to be negligible relative to the water wall. The use of night insulation is optional and will be treated in detail below. We treat the water wall house with a two-node model, SHF | °O
O'B
0.6.
O.4
0.2
0
o
~.o
(28)
z:o
3.0
4-0
SLR
Fig. 6. Dependence on SHF on 8/SLR--Direct Gain: a. our model with referenc_l£conditions (including 8/SLR = 0.8); b, same with 8/$LR ffi0.6; and c, same with 8/SLR = 1.0.
338
J. M. GOaDONand Y. ZARMI
theory of the natural convection of a fluid between two parallel plates[l 1]. For the temperature differences typically encountered in such systems, the thermal resistance to heat flow offered by the water wall is of the order of 0-5% of the thermal resistance to heat flow from the inner surface of the water wall to room air and will hence be treated as a negligible contribution to the overall thermal resistance to heat flow from storage to room air, l/UsA. The solution of eqn (28) for Ts`.(t ~) is I"sd(t `.)-
day is O,m =
foAidUs^(Ts`.(t`.) -
T~) dt`.
(35)
Q m = fo ~lln Us^(Ts"(t ") - T~) dt ~.
(36)
Similarly, the energy lost from storage to ambient during daytime (E2 ~) and nighttime (E. m) for the same twenty-four day is
Tc =
r<,qT(1-e-'"")] Cat (U.A+U..)J d
'
`.
rvz°(r - r ° ` . ) ( l - e - ' " " ) ] L
~
'
d
u~A+u~°
+ (Ts(0)- T.)e -'"/'a
J
EaOl= fo ~lld g ~ a ( T s d ( t a ) - T . a ) d t a
(37)
E. m = fo Ai n U ~.(Ts"(t")- T.") dt".
(38)
(29) Using eqns (29) and (34) in eqns (35)-(38), we obtain
where
(30)
r ~ = (mC)~I(U~A + U L )
and is the relaxation time for daytime heat transfer from storage to both room air and ambient; and T.(0) is the value of Tsd at the beginning of the day. The heat balance equation for the storage at night when back-up is required is
o,,,= =@is.., (USA+ U~.)L
jj
gain
+ UsAr"(Ts(O)- Tc)(l - e -'
residual
- Li
(39) wall load
Q(I)_ OtQTUS^T"(I -- e-~"O"")(1 -
e -''"/"")
At`.(Us^ + U g.)
(mC)s dTs"Idt" = - U].(Ts" - T.")- UsA(Ts" - T,)
(31)
At"
gain
+ UsAz"(Ts(O) - T~)e-"e"r'"(1 - e-'<'"t"")
residual whose solution for Ts" is -
Ts"(t") - T~ =
E,,,,:
/'lff dlAtd'~ T, "le-t~l'rn itlall 1-- c/
r" = (mC)s/(UsA + U ~0).
03)
We have distinguished between daytime and nighttime heat transfer coefficients from storage to ambient (U go and U~.) in order to allow for the effect of night insulation should it be employed. We use eqn (29) in eqn (32) to obtain the solution for Ts"(t"), -
e-~"*'")e -'''°
. + d at (UsA Us<,)
L _ r v
L
]
J
- T.`.)(1 - e-"'d'"")e - ' ' ' ' ' ]
U.,, + u L
[ U ,~.(r: ~ - r.")(l - e-'""") ] ~. --;,-~,L UsA Us. J + (Ts(0) - Tc)e-At~/'d)e-t'/~'.
r, r,`.(,-e-"'")]]
(USA+ U ~ . ) [ ' - L
At"
jj
gain
+ U~.r`.(Ts(O)- T~)(I - e -a'~/"d)
residual
+Li
(41) wall load
where Tsa(At a) is the value of Ts at the beginning of nighttime; superscript n denotes nighttime; and
[<~(1
(40)
02)
+tiS
Ts"(t")- T< =
L,
wall load
_ U s°(T~• 7,,. )(I-e-'"1""). ] UsA- "~ "¢~" " Us°
J
-
(34)
The energy collected by room air during daytime (Qdm) and nighttime (Q2") on a given twenty-four hour
E m =
e',QrU~.r"(1-e-~i`.l'`.)(1-e -~l'"/'~) At"(UsA + U ~.)
gain
+ U~°~"(Ts(O)- TDe-"'"#~"I(1- e - a ' ' l ' ' )
residual + L3
(42) wall load
Equations (39)-(42) each have three types of terms. Terms proportional to aQT represent the fraction of the absorbed solar gain, aQr, which is transfered either to room air (Qd"> and Q m) or to ambient (Eam and E. m) on the same day. Terms proportional to Ts(0)-Tc represent the fraction of "residual" energy (which has remained in storage from previous days) which is also transfered to either room air or ambient that same day, Terms L~. L2 and L, represent the contributions of the water wall to the heating load on that same day, and will not enter any further into our calculations. Hence their exact form is omitted for the sake of brevity.
Analytic model for passively-heated solar houses--1. Theory
Since our calculation is a monthly or seasonal one, we are interested in how the total useful solar gain is distributed between daytime and nighttime over the course of a month or a season. Equations (39)-(42) indicate how the transfer of residual energy to either room air or ambient will, on the average, be distributed between daytime and nighttime. Residual energy Q~ is simply that fraction of aQr which is not transfered to either room air or ambient on that same day. From eqns (39)-(42) we obtain OtQ'l~
u - e --AtdJTd~.je - A t n l ' r n . / I
QTus P _ P d(1- e - ' " ' ) ( l [
e-"'"")l]
Ata(l_e_,a,,,.,+t,,./..)
At d = 10 hr.
With eqns (9)-(11), we calculate SHF as a function of SLR, which is presented in Fig. 7. On the same plot we also present the corresponding seasonal SHF for a reference house with and without night insulation, as calculated by the numerical simulations [3]. The results of Rd. [3] represent a best fit to results from a variety of climatic conditions, while out calculation is for a location of specified characteristics. The effect on SHF of the addition of night insulation is incorporated into our model via the nighttime storageto-ambient heat transfer coefficient U ~o, as distinct from its daytime value U do. The addition of night insulation also decreases f-./Ld and hence should also affect SHF via this parameter. At the relatively long relaxation times (¢d and C) which characterize the reference house, however, the effect on SHF of even a marked decrease in fJf-.a (e.g. from 3 to 2) is negligible (see Fig. 8). Hence the effect on SHF of night insulation, as displayed in Fig. 7 is primarily one of different storage-to-ambient heat loss coefficients for daytime and night-time. In Fig. 8 we show the effect of varying f-/f.a (one manifestation of which could be different climatic conditions) with all other parameters fixed at their reference values, for a house without night insulation. As noted above, the long relaxation times associated with the reference house give rise to a negligible effect for decreasing L//~d. Larger values of L//~d (a larger fraction of the heating load occurring during nighttime) give rise to small decreases in SHF. In Fig. 9, we present the effect of varying the daytime length At d (for a house with night insulation). We note that SHF increases with decreasing At d. This is a consequence of the fact that the same amount of insolation is absorbed over a shorter daytime period (in a reference house with a relatively long time constant for heat transfer from storage to room air) and put to use at night when the load is greater. In Fig. 10, we display the dependence of SHF on the water wall thickness d (for a house with night insulation). Thicker walls represent increased storage
(46)
SHF.'°'= q.lf.. = (£1f..)( O.l£)= co" SLR
(47)
and
where aUsA [1 cod = (LIEd) UsA + U L
~
~
~
_ e-,,,",-"111
"JJ
Co~ = l/SLRmi..a -- [0.848 without night insulation 10.996 with night insulation
(45)
SHFd'°~= Qdf.a = ([.I[.d)(QalL) = co d SLR
_
r a = 30 hr; r" = 30 hr (without night insulation); r" = 40.9 hr (with night insulation); and
Co" = I/SLRmi.,. = [0.576 without night insulation t0.684 with night insulation.
Equations (44) and (45) provide us with a solution for the SHF for days during which back-up is required, namely,
re(1
L/Ld = 3
jj (44)
otQrUsArd(l - e-",a¢-a)( 1 _ e-,,,./,,)
Q" =/~td(UsA + U ~,,)(l - e,e,,d¢.a+,,,.J..))'
-L' "
~5/SLR = 0.8
(43)
Consequently, the average useful daytime (night-time) energy gain to room air Qd(Q,) is
'UsA+ Us.a tl
sulation); and U~°=0.57W/(m2K) (with night insulation). As in Section 6, our "reference" location has the following seasonal average properties:
For these parameters, we have
d
QR = ~
Qd=
339
(48)
r[_,lf_, ~ aUsArd [ (l-e-a'~j'd)(l-e-a'"l"")] Co" = , , .I Ata(UsA + U ~,~) I - e -.,,a/.,a+,,;i,t.,.~ j (49)
Again, we note the key result that SHF~°) is proportional to SLR, and that the closed-form expressions for SHF are eqns (9)-(11).
8. RESULTSFOR WATERWALLHOUSF£
To offer a quantitative comparison with the results of existing computer simulations, we first present an exemplary calculation for a water wall house whose parameters are taken to be those of the "reference" house considered in Ref. [3] namely, a=l.0 water wall thickness d = 0.22 m; (mC)s = 9.20 x lOs J/(m" of glazing-K); UsA = 5.68 W/(m2K); U Io = 2.84 W/(mZK); U ~. = 2.84 W/(m2K) (without night in-
340
J. M. GORDON and Y. Z~u~[
SHF
SHF
1'0'
d
0"8'
0"8
0.6~
0-6
f 04,
0"4
0"2
0.2 0
0
O
1~0
2~0
3:0
4~O
SIR Fig. 7. Dependence of SHF on the use of night insulation and comparison with numerical simulation results (SHF vs SLR)-Water Wall; a, our model with reference conditions aM no night insulation; b, same with night insulation; c, numerical simulation results [3] without night insulation; and d, numerical simulation results [3] with night insulation.
SHF
"°t
0~2
10
20
30
4;0
SLR Fig. 8. Dependence of SHF on [lid (house without night insulation).-Water Wall; a, our model with reference conditions (including /.~/[, = 3); b, s~me with [If.d = 2; and c, same with [.If-,, = 4.
0.6 0.8 1"0 ira) d Fig. I0. SHF vs the water wall thickness d for fixed SLR (house with night insulation); a, SHF vs d at SLR = 2.00; b, SHF vs d at SLR = 1.00; and c, SHF vs d at SLR = 0.50. 0
0;2
0;4
thermal mass and hence longer relaxation times for heat transfer from storage to room air. SHF increases with increasing wall thickness and approaches an asymptotic value for sufficiently thick walls. Our results are consistent with the findings of Refs. [4, 5] that the SHF for a water wall (a storage wall of very high effective thermal conductivity) increases with thermal mass ((mC)s) and exhibits no optimal thickness at which SHF is maximized. Were the small non-zero thermal resistance of a water wall taken into account, an optimal thickness would be obtained. This optimal thickness would be so large, however, as not to be of any practical interest. In Fig. 11, we show the effect on SHF of varying a at fixed values of SLR (for a house with night insulation). At sufficiently low values of SLR, SHF is linear in a for all values of a. At large values of SLR, SHF is linear in a for small a, with a more gradual increase with a at large OL
In Fig. 12, we present the effect of varying the relative width of the parabolic distribution function 81SLR (for a house with night insulation). We note that, as in the case of direct gain systems, SHF is not a sensitive function of ~ISLR. 9. CONCLUSIONS
We have presented a simple analytic method with which the monthly or seasonal thermal performance of
SHF" t.O
SHF
00
10
0.6
0"8
04
b
0"6
0.2 0
a
04 0
1:0
2:0
3:0
4~0
SLR Fig. 9. Dependence of SHF on At d (house with night insulation)--Water Wall; a, our model with reference conditions (including At d= lOhr); b, same with Aid= 12hr; and c, same with Atd = 8 hr.
0"2 0 Fig. I I. SHF vs a for fixed SLR (house with night insulation)-Water Wall; a, SLR = 2.00; b, SLR = 1.00; and c, SLR = 0.50.
Analytic model for passively-heated solar houses--l. Theory SHF
1 °0'
0.8
0.6 0.4 0-2 0
120
2"0
.'320
4:0
SLR FiB. 12. Dependence of SHF on 6/SLR (house with night insulation)--Wate__grWall', a, our model withnreference conditions (including 8/SLR = 0.8); b, same with 8/SLR = 0.6; and c, same with ~/SLR = 1.0. passively-heated solar houses can be calculated in closed form. The formalism presented above can be applied to any type of passively-heated solar house, which is one of its primary advantages. In this paper, the specific cases of direct gain and water wall houses have been treated in detail. This method has also been applied to the case of the massive storage wall house--a case sufficiently complex so as to merit a separate paper [12]. This method provides a useful tool for easy calculation of the dependence of thermal performance on building and climatic parameters. Unlike using the parameterizations of computer simulation results, it affords the user the flexibility of not being limited to a "reference" house. Whereas the treatment of different types of passive heating elements can require the writing of new computer simulation programs, our method requires identification of the basic factors governing heat balance and the solution of the appropriate equations. There may be two major limitations to our model. First, we have ignored the true time dependence of insolation during the course of the day. The fact that this effect is small for commonly considered direct gain houses is shown in the Appendix. Second, the coarse division in time used in our energy bookkeeping can result in over-estimates of SHF. This can occur whenever ventilation must be activated to avoid overheating during only a fraction of daytime or nighttime in the actual situation, while such overheating would not occur on the coarse time scale employed in our model. For relatively large storage thermal mass, this error occurs at large SLR values, for which SHF varies sufficiently slowly with SLR that the error incurred is small. For small storage thermal mass, the SLR values at which this error occurs are lower, which can result in slightly larger overestimates of SHF. Significant errors are introduced, however, only at such small values of thermal mass so as not to be of practical interest. REFERENCES
1. W. O. Wray, J. D. Balcomb and R. D. McFarland, A SemiEmpirical Method for estimating the Performance of Direct
341
Gain Passive Solar Heated Buildings. Los Alamos Scientific Laboratory Publication LA-UR-79-117(1979). 2. W. O. Wray and 2. D. Balcomb, Trombe Wall vs Direct Gain: A Comparative Analysis of Passive Solar Heating Systems. Los Alamos ScientificLaboratory Publication LA-UR-79-116 (1979). 3. J. D. Balcomb and R. D. McFarland, A Simple Empirical Method for Estimating the Performance of a Passive Solar Heated Building of the Thermal Storage Wall Type. Los Klamos Scientific Laboratory Publication LA-UR-78-1159 (1978). 4. R. D. McFarland L D. Balcomb, The Effect of Design Parameter Changes on the Performance of Thermal Storage Wall Passive Systems. Los Alamos Scientific Laboratory Publication LA-UR-79-423(1979). 5. J. E. Perry, Jr., Mathematical Modelling of the Performance of Passive Solar Heating Systems. Los Alamos Scientific Laboratory Publication LA-UR-77-2345(1977). 6. J. M. Gordon and Y. Zarmi, Analytic model for passively heated solar bouses--II. Users Guide. Solar Energy 27, 343--347(1981). 7. R. W. Jones, private communication, Los Aiamos Scientific Laboratory (Oct. 1980). 8. P. Berdahl, D. Grether, M. Martin and M. Wahlig, California Solar Data Manual. Lawrence Berkeley Laboratory (1978). 9. S. Schweitzer, A representative 'average' weather year for Israel's coastal plain. Report No. 3-78. Agricultural Research Organization, Bet-Dagan, Israel (1978). 10. W. O. Wray and J. D. Balcomb, Sensitivity of direct gain space heating performance to fundamental parameter variations. Solar Energy 23, 421 (1979). 11. H. Tabor, Radiation, convection and conduction coefficients in solar collectors. Bull. Res. Council Israel 6C, 155 (1958). 12. J. M. Gordon and Y. Zarmi, Massive storage walls as passive solar heating elements: an analytic model. Solar Energy 27, 349-355 (1981). 13. B. Y. H. Liu and R. C. Jordan, The interrelationship and characteristic distribution of direct, diffuse and total solar radiation. Solar Energy 4(3), 1 (1960).
/dJlpEh'DIX We consider here the effect on our results of using a timedependent insolation during daytime. The only change in our formalism is the replacement of the constant insolation rate Or/At# used in the daytime eqn (12) by a time-dependent one. The gross features of the time dependence of the insolation can be represented [13] by a function of the form qo(COSaJ- COSws),
where oJ, is the sunrise (sunset) hour angle, ~o~= ~rAtd/T (T = At a + At" = 24 hr), and
co= co,-fit, with fi=2w/T. Equation (12) now becomes (mC)s dTsaldt ~ = (I - F)qdcos ~o- cos ~o~)- UsA(Tsd - To). (AI)
The solution to eqn (A1) is straightforward, yielding
Ts" T, (]-
=
)qofrr
UsA
LEE
l+fz~ ~
J ÷c°s
rcos ,o - fir sin to'] _ cos ,o,]. +L I+N ~T~ J + (Tsd(0)- T~)e-'~/L
(A2)
Repeating the procedure of section 5, we recalculate Qjl), Q(]) and QR. The partition of the residual energy QR between daytime and nighttime is still the same as in section 5, yielding the final partition of Qr between daytime and nighttime as
342
J. M. GORDON and Y. ZARMI For commonly used values of ~',however, the differencebetween the two resultsis negligible.For example, for a at d of I0 hr and a value of ,ras small as 5 hi',eqn (A3) yields for Q, a value which is smaller than that obtained by eqn (23) by 6.4%. With increasing ~-,agreement is better,and for large values of ~-both expressions for Q,, approach the same limit,
- (I - F)Qr fl~:: Q,- 2(i + flzl.-') • " fsmco,-fl~'coso,,+(sm,o,
+
1 flrcos,o,)e- ~ ' / " 1F-e
-At,l,
I
Qd = Q r - Q . .
These expressions for Q, and Qa are obviously differentfrom the corresponding ones calculated in section 5 (see eqn 23). The main differenceappears at low values of ,r,where eqn (23) yields 0,, = (I - F)QTv/At d (A4) while the more exact result, eqn (A3), yields
0,. =
(1
-
F) ^ [
sin w~
,
~ [ 2 ( s i n ~ , - m, cos ~,,)] a2~'''
(A5)
Q.
, (I -
F)QrAt"IT.
large v
Also, the limit of Q. for short daytime periods (At ~ small) is the same for both calculations, Q. ~
(i - F)~r. &t
~mall
0038-092X/8111~0331-1250"2.~/0
Printed in Great Britain.
© 1981Perpmon Press Ltd.
ANALYTIC MODEL FOR PASSIVELY-HEATED SOLAR HOUSES--1. THEORY? J. M. GORDON
and Y. ZARMI
Applied Solar Calculations Unit, Jacob Blaustein Institute for Desert Research, Ben-Gurion University of the Negev, Sede Boqer Campus, Israel
(Received 22 July 1980; accepted 2 April 1981) Abstract--A simple analytic method for the prediction of the long-term thermal performance of passively-heated solar houses is presented. The treatment includes a new coarse method for "energy bookkeeping"and the use of a distribution function which represents the frequency of occurrence of different values of the solar load ratio. Due to its generality, this formalism is applicable to any passive heating element. As specificexamples, the cases of direct gain and water wall houses are treated in detail. Relative to the parameterizationof computer simulation results, this method offers the user a design tool that can be used to predict, in closed form, the thermal effect on the house of differentbuildingand climaticparametersand is not restricted to a "reference" building.Agreementwith published numericalsimulationresults in satisfactory.The presentation is divided into two separate papers: a users guide for the reader who may not be interested in the details of derivation and validation,and the present paper, in which the theory is presented in detail.
One advantage of the proposed methodology is its simplicity. It provides closed-form analytic expressions for the same performance variables that have, until now, been obtained via large-scale computer simulations. Consequently, one can easily calculate on a hand calculator the effect of variations in building and climatic parameters such as thermal storage mass, thermal resistances, storage absorbance, insolation, etc. Another advantage of this approach is that it provides insight into the basic physical ingredients which govern the performance of passive solar systems. Thermal performance is expressed in terms of building and climatic variables whose physical meaning is clear. In Section 2 we note a few of the fundamental physical considerations upon which the theory is based. In Section 3 we present our energy bookkeeping method and the general formalism for seasonal calculations. In Section 4 we discuss the particular details of the distribution function we employ to describe the frequency of occurrence of various daily solar load ratio values. In Section 5 this formalism is applied to direct gain houses. In Section 6 our results for direct gain houses are presented and compared with the corresponding results from published numerical simulations[I,2]. In Section 7 the method is applied to water wall houses, and in Section 8 the water wall results are compared with the corresponding computer simulation results [3]. The ability of the model to predict the dependence of the performance of passively-heated solar houses on various parameters of the building is also presented. Our conclusions are presented in Section 9. In the Appendix, we show the effect of introducing a time-dependent daily insolation and to what extent it limits the validity of our treatment.
I. INTRODUCTION
The prediction of the seasonal thermal performance of passively-heated solar houses is an essential element in the design of these structures. Much of the successful work in this area has emerged from numerical solutions of the large number of equations which govern heat transfer and storage through multiple thermal nodes. A primary example is the work of the Los Aiamos team in predicting solar heating fraction as a function of the solar load ratio [1-5]. There is, however, a need for simple analytic models which capture the essential physics of the problem, afford the user the flexibility of not being restricted to a "reference" house, and can be applied with equal ease to a variety of passively-heated solar building types. Such models may be particularly well suited to the prediction of seasonal performance and its dependence on various parameters of the building since the effects of hourly temperature and insolation variations can be averaged out over the course of an entire heating season. In this paper a new methodology is proposed for a simple treatment of seasonal thermal performance of passively-heated solar houses. The particular cases of direct gain and water wall houses are treated in detail. A new method for "energy bookkeeping" is presented-one which is suited to monthly or annual calculations as opposed to hourly performance. The effect of variations in total daily insolation is accounted for by weighting different values of the solar load ratio by the frequency of their occurrence. Since most users of this methodology may not be interested in the details of its derivation and validation, we divide our presentation into two separate papers: a users guide [6], with little explanation of the origins of the formulae, and the nresent Darter which describes
2. FUNDAMENTALCONSIDERATIONS (a) Definitions The object of the calculation is to express a solar heating fraction, SHF, of a house in terms of the solar load ratio, SLR, as well as building and climatic parameters. SHF is defined as the fraction of a house
"~Research supported by the Center for Absorption in Science, The Ministry for ImmigrantAbsorption, State of Israel. 331
J.M. Go~u~oNand Y. Z ~ I
332
heating load which is supplied by solar energy. SLR is defined as the ratio of the insolation transmitted into the house, Qr, divided by the house heating load, L. The insolation transmitted through the vertical south-facing giazings can be readily calculated from the more commonly available data of insolation on a horizontal surface [1]. Over a given period of time, t, (which can be the entire heating season or a sizable fraction thereof, e.g. a month), SHF is calculated as SHF;
QdL, = filL,)
fo'
q,(t') dr',
(1)
where L, is the total heating load over time, t, Q. is the useful solar gain collected over time, t, and q.(t) is the useful solar gain per unit time. For the sake of specificity, the period of time under consideration throughout the remainder of this paper will be the entire heating season. By useful solar gain, we mean solar energy gain which does not result in heating room air above a specified comfort temperature, To. The house is assumed to have a back-up heating system which insures that room air temperature never falls below To. Let us stress that the heating load should be calculated with respect to the temperature below which the back-up system is activated (To). It is also assumed that whenever the room air temperature exceeds Tc ventilation is employed to maintain room air temperature at To If the heating load is calculated as specified above, then the effect on SHF of activating ventilative cooling of a temperature slightly above T~ is negligible [7]. (b) Dependence on SLR Before any detailed calculations are performed, let us indicate what can be said about the behavior of SHF as a function of SLR. Were there no overheating at any time during the heating season, then all collected energy would be useful energy, and SHF would be proportional to SLR, SHF-- Co SLR.
(2)
The coefficient, co, is a function of climatic as well as building parameters. Equation (2) is general, although the value of Co and its dependence on building parameters vary with the type of passive heating element. Clearly, for sufficiently large SLR values, SHF becomes unity. Since overheating may occur during part of the heating season, not all collected energy is useful energy (energy is dumped), and SHF averaged over the heating season is less than the value predicted by the linear form of equation (2). That is, there is a non-linear dependence of SHF on SLR in a "transition" region characterized by intermediate SLR values. To summarize, SHF is linear in SLR at low SLR values, nonlinear in a "transition" region of intermediate SLR values, and unity at large SLR values. A calculation of SHF in this transition region depends on a proper accounting of energy that is not useful (dumped). This entails accurate "energy bookkeeping'*
for determining what fraction of the time overheating occurs and what magnitude of collected energy is dumped. 3. ENERGY B O O ~ G
We divide the heating season into periods during which solar gain provides all of the heating load (no back-up required) and periods during which solar gain provides only part of the heating load. For our purposes of energy bookkeeping, we use a relatively coarse division of time into days and nights (rather than the hourly calculations usually used in numerical simulations). We then further divide the calculation into days (or nights) during which solar gain (or the storage thereof) provides all of the load, and days (or nights) during which solar gain provides only part of the load. We assume that the fraction of the average daily load that is represented by the daytime load, Ld, and the nighttime load, L., does not vary appreciably over the heating season. We can then express SHF as a weighted average over daytime and nighttime periods during which back-up heating is either required or not required, SHF = (llN)([df) day-~-'.~.o 1 back-up
+(llN)(£d£)d.~._,a (QdL~) bm:k,.up
+(l/N)(F~dFO ~,,o, l +(llN)(£dF~) .~.~,,,h (QdL.) no brock-up
Ix~k-up
(3) where N is the number of days in the heating season, and £d and/.~ are the average daytime and nighttime loads, respectively. The useful energy provided by the passive solar element, denoted by Q. in eqn (3), is a combination of both energy supplied by solar on the day it is collected and "residual" energy which has remained in storage from previous days. We then denote by SHFd(°) and SHF. <°) the SHF value for daytime and nighttime periods, respectively, during which back-up is required, namely SHF~°)= Q.(day)/L,t SHF. (°)= Q.(night)/L.. In order to compute the summations of eqn (3), we need the frequency of occurrence of different values of SLR during the heating season. We therefore introduce a normalized distribution function, p(SLR), which gives the probability of occurrence of any SLR value. SHF then becomes
SHF=([df)[f;L~.. ~#(x) dx + - " F =(°) foSLR"dp(x)SH d (x ) dx J I(l-d/~)
Analytic model for passively-heated solar houses--l. Theory where SLR=i..a and SLRm~,,. are the minimun values of SLR such that no back-up heating is required during daytime and nighttime, respectively, and the dummy variable x denotes SLR. SLR=,,.a and SLRm,.,. are the points at which SHF) °) and SHF. ~°) become unity, SHFd~°)(SLRmin.,~)= 1
(5}
SHF.~°>(SLR=i...) = 1.
0
distribution (assuming a thermostat set temperature of 18.3eC (65°F)). This distribution is a peaked, symmetric function with a statistically insignificant tail for large SLR values. Therefore, for the purpose of presenting a closed-form analytic solution, we use a parabolic distribution function which is peaked about the average SLR value, SLR, which vanishes at SLR -+& and which is normalized,
0 < SLR < SLR - 8
p(SLR) = 3(82- (SLR- SLR)2) SLR - 8 < SLR < SLR + 8 483 0
(6)
SLR > SLR + &
Let us again note that, as shown below, in the determination of SHF~°~ and SLR=i., residual energy that remains in storage from previous days, as well as energy collected on a given day, are taken into account.
4. TXEmswamUTlONFUNCTION In order to calculate SHF, we need to know the function which describes the frequency of occurrence of different values of the ratio of daily insolation to load (SLR) over the course of a heating season. It should be noted that most commonly considered passive solar heating elements are vertical, so that the insolation considered is that on a vertical surface. It is interesting to note that although insolation on the horizontal varies significantly during the heating season and gives rise to a skewed distribution [8, 9], the insolation on the vertical is much more uniform throughout the heating season and yields a far more symmetric distribution [8, 9]. With typical meteorological year data for the coastal region of Israel [9], we have determined the actual SLR
For the particular case cited above [9], 8 = (0.8) SLR (see Fig. 1). In principle, the p(SLR) employed in eqn (4) should be the actual measured distribution function particular to one's location. Inspection of some available data [8] indicates that 8/SLR typically varies from 0.6 to 1.0. It will be shown below that the functions SHF~~°~and SHF, ~°~are proportional to SLR, i.e. SHFa~°'= c / S L R
Hence, eqn (5) yields SLRmi..d = l/co'~
Given this result, the distribution function of eqn (6L and eqn (7) are used in eqn (4) to yield SHF = (L,I£) SHF '"~ + (L./£) SHF"""'
0.8 0.6 0-4
0-5
(8)
SLRmi... = 1/co".
i
o
(7)
SHF. ~°)= Co" SLR.
I.O i
°it
333
I "O
1"5
5LR Fig. 1. The parabolic distribution function p(SLR).
2.0
(9)
334
J. M, GORDONand Y. ZARMI
with SLR=.,., > SLR+ (5
Cod " ~
SHF a-' --- 1 + [(I - Co"S ~ + c°aS)3(ll6(coaS)' Coa - S - ' ~ - 3coaS)lj
SLI~ + 8 -> SLR,.,..d -> S L R - 8
(1o)
SLR=i.., < SLI~ -
l
and
SHI~i. h,
coSR
f I+
(1 - c o " S L R
+ Co"8)3(I - co"SLR - 3Co~8)
1
16(Co"8) 3
s. ~nacArlon ro ~ DmEcr GAINSYSTEM A direct gain house is one with a large south-facing glazing and thermal storage mass in the interior. We shall assume that the building envelope is sufficiently well insulated and that storage is located so that heat transfer from storage to ambient can be neglected, and that the other thermal mass of the building is negligible relative to the storage mass. We treat a direct gain house with a two-node model, the two nodes being the storage S and room air A. As all we need to complete the calculation for SHF (via eqn 4) are SHFJ °) and SHF~ (°) (namely, the SHF values for periods during which back-up is required), we consider the heat balance equations for the storage. During daytime, this equation is dTsd/dt d
= (1-F)Q,~ -
UsA~'Td s
L
- F)Qr( I - e-'"l') ] u~a:
(mC)s dTs"ldt ~ = - UsA(Ts" - Tc),
(15)
whose solution for Ts" is ~Ts a'At " -m. , T , " ( t ' ) - Tc = ~ ( d") - Tc)e
(16)
where Tsa(AI") is the value of Ts at the beginning of nighttime, and superscript n denotes nighttime. We use eqn (13) in eqn (16) to obtain the solution for Ts'(tn),
T,)e-""'|
T~),
where (mC)s is the heat content of the storage; Ts is the storage temperature; t denotes time; superscript d denotes daytime; At" is the length of daytime; F is the fraction of transmitted insolation Q-r which is not absorbed by storage and hence directly heats room air; and Us^ is the heat transfer coefficient from storage to room air. We have assumed above that insolation can effectively be treated as constant throughout the day (the justification for this is presented in the Appendix). The solution of eqn (12) for Tsd(t d) is
f(l
(ll).
The heat balance equation for the storage at night when back-up is required is
+ (Ts(O)-
(12)
Tsa(t d) - T~ =
SLR + 8 --- SLR=~..,, -> SLR - 8 SLR.~... < SLR - 8
I
(mC)s
SLR.,..o > SLR + 8
e-r,t,. J
(17)
The energy collected by room air during daytime (Qjr)) and night-time (Q,,")) on a given twenty-four hour day is Qd"t = FQr + foa'~ UsA(Ts . ( t d) -
Q."' =
0 ¢n
Tc)dt d
Us,~(Ts"(t ~) - To) dt",
(18) (19)
where FQ-r in eqn (18) represents the fraction of transmitted radiation absorbed directly into room air during daytime. Using eqns (13) and (17) in eqns (18) and (19), respectively, we obtain
J
+ (Ts(0)- T~)e -'"I"
(13) + UsA(Ts(0)- Tc)r(l - e-A'd/;)
where
(20) "r =
(mC)sT Usa
(14)
and is the relaxation time for heat transfer from storage to room air; and Ts(O) is the value of Ts ~ at the beginning of the day.
and Q . ) _ (I - F)Qr.r(l - e -A'"I")(1 - e -~m') At" + UsA(Ts(O)- T~)T e~'"/"(1
e -~m')
(21)
Analytic model for passively-heated solar houses--l. Theory Both eqns (20) and (21) have terms which represent the fraction of QT which is put to use the same day, and the fraction of "residual" energy (which has remained in storage from previous days) which is also put to use that same day. Since our calculation is a monthly or seasonal one, we are interested in how the total useful QT is distributed between daytime and nighttime over the course of a month or a season. Equations (20) and (21) indicate how the residual energy will, on the average, be distributed between daytime and nighttime. Residual energy QR is simply that fraction of QT which is not put to use on that same day, hence
335
20% of transmitted solar flux being absorbed directly into room air). We specify our "reference" location which, in conjunction with the house's properties, has the following seasonal average properties: 8/SLR = 0.8 £/£a = 3
At ~ = 10 hr. For these parameters, eqns (26)-(27) and (8) yield Coa = I/SLR,,,.a = 1.935
QR
(I -
F)QT¢ e-A'"(l
e A,a,,)
Consequently, the average useful daytime (nighttime) energy gain Q a ( Q , ) is
Qa Q~- r l L -
r.
- F),r(1
L
- e-"''')] ]
- e-~"'r/'/'rX1
Ata(l -
e -'''`+'''")")
'
JJ
(23) (7. = QT - Qa.
Equation (23) provides us with a solution for the SHF for days during which back-up is required, namely, SHFat°>-- QdLa
=
(f-,If-,a)(Q,IL)
=
Coa SLR
(24)
and SHF, (°)= Q.IL,, = (/ZII~,,,XQ./C)
Co" = I/SLRmi,., = 0.533.
(22)
At d
= co" SLR,
(25)
With eqns (9)-(11), we calculate SHF as a function of SLR, which is presented in Fig. 2. On the same plot we also present the corresponding seasonal SHF for a reference house with and without night insulation as calculated by the numerical simulations [1]. The results of Ref. [1] represent a best fit to results from a variety of climatic conditions, while our calculation is for a location of.specified characteristics. In our model, the variable ElLa decreases with the addition of night insulation. To show the effect of varying f . / f . a (one manifestation of which could be the addition of night insulation, another of which could simply be different climatic conditions), we also plot in Fig. 2 our prediction for the same house with f-I£a equal to 2 and 4. As expected, smaller values of L/La (a larger fraction of the heating load occurring during daytime) increase SHF.
where
Cod=(c,/c,a)r1-r(l- F),(1-e-:")(,- e-:''-)'ri Atd(l - e-U,,d+A,.)1,) jj L L (26)
co"= (C1£.)[ (1 - F)'r(1 - e - " ' " ) ( l
- e-"'"l')] Aid( 1 _ e_,,,~+~,,)/,) j.
(27)
The fact that SHF~°) is proportional to SLR is a key result. It enables us to perform the integrations described in Section 3 and to obtain a closed-form analytic expression for SHF. With these results, eqns (24)--(27) and eqn (6) are used in eqn (4) to yield the closed-form expressions for SHF as described in eqns (10)-(11). 6. RESULTS FOR DIRECT GAIN HOUSES
To offer a quantitative comparison with the results of existing computer simulations, we first present an exemplary calculation for a direct gain house whose parameters are taken to be those of the "reference" house considered in Ref. [1], namely, ¢ = (mC)slUs^
--
15 hr,
and 1 - F = 0.64 (a mass surface absorbance of 0.8 and
In Fig. 3, we present the effect of varying the daytime length At a (with all other parameters fixed at their reference values). We note that as At a decreases, SHF increases. This is a consequence of the fact that the same amount of insolation is absorbed over a shorter daytime period (in a house with a time constant for heat transfer from storage to room air of 15 hr), and put to use at night when the load is greater. In Fig. 4, we display the dependence of SHF on the relaxation time ~- for several fixed values of SLR. Increasing ~" corresponds to either increasing the storage thermal mass or decreasing the heat transfer coefficient from storage to room air, or both (at fixed F). It should be noted that increasing storage thermal mass by increasing its area rather than its thickness also affects the parameter F. SHF increases with increasing ~" and approaches an asymptotic value at large ~'. Our results are consistent with the findings of Ref. [2]. In Fig. 5, we show the effect on SHF of varying F at fixed values of SLR. At sufficiently low values of SLR, SHF is independent of F and is equal to SLR. At larger, more commonly encountered values of SLR, SHF is a decreasing function of F, which is in agreement with the findings of Ref. [10]. This is due to the fact that large F values result in larger fractions of daily insolation being absorbed directly by room air during daytime and less energy being stored for higher load nighttime periods. We also note that at sufficiently large values of SLR, the
336
J.M. GOR~ONand Y. ZAP.MI
SHF ~.O
b 0"6
0"6
0-4
0.2
0
o
t.o
2:o
3-o
4:0
SLR Fig. 2. Dependence of SHF on L/L~ and comparison _with numerical simulation results (SHF vs SLRY---Direct Gain; a, our model with reference conditions (including/.,/~ = 3); b. same with [,/[.a = 2; c, same with f./f.a = 4: d, numerical simulation results' without night insulation; and e, numerical simulation results Lwith night insulation.
SHF ,,
1-0
0.8
0'6
0"4
0.2
0 0
1~
2"0 SLR
3"0
4"0
Fig. 3. Dependence of SHF on Atd--DireCt Gain; a, our model with reference conditions (including At d = l0 hr); b, same with At J = 12 hr; and c, same with &t ~ = 8 hr.
SHF |.0 f 0.8 0.6 f
0.4 0.2
0 5.00
tO.O0
~5,00
-L-
20.00
25"00 (hr)
Fig, 4. SHF vs the relaxation time ~ for fixed SLR--Direct Gain; a, SLR = 2.00; b, SLR -- 1.00; and c, SLR = 0.50.
Analytic model for passively-heated solar houses--l. Theory
337
SHF
1.0,
O'B
'
0.6,
0.4.
0.2.
0 0
0"2 5
0.5
0.7 5
1.0
F
Fig. 5. SHF vs fraction F of transmitted insolation not absorbed by storage, for fixed SLR--Direct Gain; a, SLR = 2.00; b, SLR= 1.00; c, SLR = 0.50; and d, SLR = 0.10. limiting value of SHF at F = 1.0 is simply L d L This is a result of the fact that if no solar energy is being stored for nighttime use (F = 1.0), then sufficiently large SLR values will provide 100% of the daytime load only. In Fig. 6, we present the effect of varying the relative width of the parabolic distribution function, 8/SLR. We note that SHF is not a sensitive function of ~/SLR.
the two nodes being the water storage S and room air A. We consider the heat balance equation for the storage. During daytime when back-up is required, this equation is (mC)s dTs d Idt d = (c~QrlAt a ) - U d (Ts a - T~a) - UsA(Tsd - To),
where a is the fraction of transmitted insolation QT which is absorbed by storage; U~a is the daytime heat transfer coefficient from storage to ambient; and UsA is the heat transfer coefficient from storage to room air. One further approximation to be made here is that the relatively fast rate of heat transfer which results from natural convection within the water wall allows us to treat the water wall as an element of negligible thermal resistance. The convective heat transfer coefficient within a water wall can in fact be calculated from the
7. APPLICATION TO THE WATER WALL
The house considered here has a glazed, south-facing water storage wall. The absorbing surface area of the water wall is taken to be equal to the glazing area. The house's other thermal mass is assumed to be negligible relative to the water wall. The use of night insulation is optional and will be treated in detail below. We treat the water wall house with a two-node model, SHF | °O
O'B
0.6.
O.4
0.2
0
o
~.o
(28)
z:o
3.0
4-0
SLR
Fig. 6. Dependence on SHF on 8/SLR--Direct Gain: a. our model with referenc_l£conditions (including 8/SLR = 0.8); b, same with 8/$LR ffi0.6; and c, same with 8/SLR = 1.0.
338
J. M. GOaDONand Y. ZARMI
theory of the natural convection of a fluid between two parallel plates[l 1]. For the temperature differences typically encountered in such systems, the thermal resistance to heat flow offered by the water wall is of the order of 0-5% of the thermal resistance to heat flow from the inner surface of the water wall to room air and will hence be treated as a negligible contribution to the overall thermal resistance to heat flow from storage to room air, l/UsA. The solution of eqn (28) for Ts`.(t ~) is I"sd(t `.)-
day is O,m =
foAidUs^(Ts`.(t`.) -
T~) dt`.
(35)
Q m = fo ~lln Us^(Ts"(t ") - T~) dt ~.
(36)
Similarly, the energy lost from storage to ambient during daytime (E2 ~) and nighttime (E. m) for the same twenty-four day is
Tc =
r<,qT(1-e-'"")] Cat (U.A+U..)J d
'
`.
rvz°(r - r ° ` . ) ( l - e - ' " " ) ] L
~
'
d
u~A+u~°
+ (Ts(0)- T.)e -'"/'a
J
EaOl= fo ~lld g ~ a ( T s d ( t a ) - T . a ) d t a
(37)
E. m = fo Ai n U ~.(Ts"(t")- T.") dt".
(38)
(29) Using eqns (29) and (34) in eqns (35)-(38), we obtain
where
(30)
r ~ = (mC)~I(U~A + U L )
and is the relaxation time for daytime heat transfer from storage to both room air and ambient; and T.(0) is the value of Tsd at the beginning of the day. The heat balance equation for the storage at night when back-up is required is
o,,,= =@is.., (USA+ U~.)L
jj
gain
+ UsAr"(Ts(O)- Tc)(l - e -'
residual
- Li
(39) wall load
Q(I)_ OtQTUS^T"(I -- e-~"O"")(1 -
e -''"/"")
At`.(Us^ + U g.)
(mC)s dTs"Idt" = - U].(Ts" - T.")- UsA(Ts" - T,)
(31)
At"
gain
+ UsAz"(Ts(O) - T~)e-"e"r'"(1 - e-'<'"t"")
residual whose solution for Ts" is -
Ts"(t") - T~ =
E,,,,:
/'lff dlAtd'~ T, "le-t~l'rn itlall 1-- c/
r" = (mC)s/(UsA + U ~0).
03)
We have distinguished between daytime and nighttime heat transfer coefficients from storage to ambient (U go and U~.) in order to allow for the effect of night insulation should it be employed. We use eqn (29) in eqn (32) to obtain the solution for Ts"(t"), -
e-~"*'")e -'''°
. + d at (UsA Us<,)
L _ r v
L
]
J
- T.`.)(1 - e-"'d'"")e - ' ' ' ' ' ]
U.,, + u L
[ U ,~.(r: ~ - r.")(l - e-'""") ] ~. --;,-~,L UsA Us. J + (Ts(0) - Tc)e-At~/'d)e-t'/~'.
r, r,`.(,-e-"'")]]
(USA+ U ~ . ) [ ' - L
At"
jj
gain
+ U~.r`.(Ts(O)- T~)(I - e -a'~/"d)
residual
+Li
(41) wall load
where Tsa(At a) is the value of Ts at the beginning of nighttime; superscript n denotes nighttime; and
[<~(1
(40)
02)
+tiS
Ts"(t")- T< =
L,
wall load
_ U s°(T~• 7,,. )(I-e-'"1""). ] UsA- "~ "¢~" " Us°
J
-
(34)
The energy collected by room air during daytime (Qdm) and nighttime (Q2") on a given twenty-four hour
E m =
e',QrU~.r"(1-e-~i`.l'`.)(1-e -~l'"/'~) At"(UsA + U ~.)
gain
+ U~°~"(Ts(O)- TDe-"'"#~"I(1- e - a ' ' l ' ' )
residual + L3
(42) wall load
Equations (39)-(42) each have three types of terms. Terms proportional to aQT represent the fraction of the absorbed solar gain, aQr, which is transfered either to room air (Qd"> and Q m) or to ambient (Eam and E. m) on the same day. Terms proportional to Ts(0)-Tc represent the fraction of "residual" energy (which has remained in storage from previous days) which is also transfered to either room air or ambient that same day, Terms L~. L2 and L, represent the contributions of the water wall to the heating load on that same day, and will not enter any further into our calculations. Hence their exact form is omitted for the sake of brevity.
Analytic model for passively-heated solar houses--1. Theory
Since our calculation is a monthly or seasonal one, we are interested in how the total useful solar gain is distributed between daytime and nighttime over the course of a month or a season. Equations (39)-(42) indicate how the transfer of residual energy to either room air or ambient will, on the average, be distributed between daytime and nighttime. Residual energy Q~ is simply that fraction of aQr which is not transfered to either room air or ambient on that same day. From eqns (39)-(42) we obtain OtQ'l~
u - e --AtdJTd~.je - A t n l ' r n . / I
QTus P _ P d(1- e - ' " ' ) ( l [
e-"'"")l]
Ata(l_e_,a,,,.,+t,,./..)
At d = 10 hr.
With eqns (9)-(11), we calculate SHF as a function of SLR, which is presented in Fig. 7. On the same plot we also present the corresponding seasonal SHF for a reference house with and without night insulation, as calculated by the numerical simulations [3]. The results of Rd. [3] represent a best fit to results from a variety of climatic conditions, while out calculation is for a location of specified characteristics. The effect on SHF of the addition of night insulation is incorporated into our model via the nighttime storageto-ambient heat transfer coefficient U ~o, as distinct from its daytime value U do. The addition of night insulation also decreases f-./Ld and hence should also affect SHF via this parameter. At the relatively long relaxation times (¢d and C) which characterize the reference house, however, the effect on SHF of even a marked decrease in fJf-.a (e.g. from 3 to 2) is negligible (see Fig. 8). Hence the effect on SHF of night insulation, as displayed in Fig. 7 is primarily one of different storage-to-ambient heat loss coefficients for daytime and night-time. In Fig. 8 we show the effect of varying f-/f.a (one manifestation of which could be different climatic conditions) with all other parameters fixed at their reference values, for a house without night insulation. As noted above, the long relaxation times associated with the reference house give rise to a negligible effect for decreasing L//~d. Larger values of L//~d (a larger fraction of the heating load occurring during nighttime) give rise to small decreases in SHF. In Fig. 9, we present the effect of varying the daytime length At d (for a house with night insulation). We note that SHF increases with decreasing At d. This is a consequence of the fact that the same amount of insolation is absorbed over a shorter daytime period (in a reference house with a relatively long time constant for heat transfer from storage to room air) and put to use at night when the load is greater. In Fig. 10, we display the dependence of SHF on the water wall thickness d (for a house with night insulation). Thicker walls represent increased storage
(46)
SHF.'°'= q.lf.. = (£1f..)( O.l£)= co" SLR
(47)
and
where aUsA [1 cod = (LIEd) UsA + U L
~
~
~
_ e-,,,",-"111
"JJ
Co~ = l/SLRmi..a -- [0.848 without night insulation 10.996 with night insulation
(45)
SHFd'°~= Qdf.a = ([.I[.d)(QalL) = co d SLR
_
r a = 30 hr; r" = 30 hr (without night insulation); r" = 40.9 hr (with night insulation); and
Co" = I/SLRmi.,. = [0.576 without night insulation t0.684 with night insulation.
Equations (44) and (45) provide us with a solution for the SHF for days during which back-up is required, namely,
re(1
L/Ld = 3
jj (44)
otQrUsArd(l - e-",a¢-a)( 1 _ e-,,,./,,)
Q" =/~td(UsA + U ~,,)(l - e,e,,d¢.a+,,,.J..))'
-L' "
~5/SLR = 0.8
(43)
Consequently, the average useful daytime (night-time) energy gain to room air Qd(Q,) is
'UsA+ Us.a tl
sulation); and U~°=0.57W/(m2K) (with night insulation). As in Section 6, our "reference" location has the following seasonal average properties:
For these parameters, we have
d
QR = ~
Qd=
339
(48)
r[_,lf_, ~ aUsArd [ (l-e-a'~j'd)(l-e-a'"l"")] Co" = , , .I Ata(UsA + U ~,~) I - e -.,,a/.,a+,,;i,t.,.~ j (49)
Again, we note the key result that SHF~°) is proportional to SLR, and that the closed-form expressions for SHF are eqns (9)-(11).
8. RESULTSFOR WATERWALLHOUSF£
To offer a quantitative comparison with the results of existing computer simulations, we first present an exemplary calculation for a water wall house whose parameters are taken to be those of the "reference" house considered in Ref. [3] namely, a=l.0 water wall thickness d = 0.22 m; (mC)s = 9.20 x lOs J/(m" of glazing-K); UsA = 5.68 W/(m2K); U Io = 2.84 W/(mZK); U ~. = 2.84 W/(m2K) (without night in-
340
J. M. GORDON and Y. Z~u~[
SHF
SHF
1'0'
d
0"8'
0"8
0.6~
0-6
f 04,
0"4
0"2
0.2 0
0
O
1~0
2~0
3:0
4~O
SIR Fig. 7. Dependence of SHF on the use of night insulation and comparison with numerical simulation results (SHF vs SLR)-Water Wall; a, our model with reference conditions aM no night insulation; b, same with night insulation; c, numerical simulation results [3] without night insulation; and d, numerical simulation results [3] with night insulation.
SHF
"°t
0~2
10
20
30
4;0
SLR Fig. 8. Dependence of SHF on [lid (house without night insulation).-Water Wall; a, our model with reference conditions (including /.~/[, = 3); b, s~me with [If.d = 2; and c, same with [.If-,, = 4.
0.6 0.8 1"0 ira) d Fig. I0. SHF vs the water wall thickness d for fixed SLR (house with night insulation); a, SHF vs d at SLR = 2.00; b, SHF vs d at SLR = 1.00; and c, SHF vs d at SLR = 0.50. 0
0;2
0;4
thermal mass and hence longer relaxation times for heat transfer from storage to room air. SHF increases with increasing wall thickness and approaches an asymptotic value for sufficiently thick walls. Our results are consistent with the findings of Refs. [4, 5] that the SHF for a water wall (a storage wall of very high effective thermal conductivity) increases with thermal mass ((mC)s) and exhibits no optimal thickness at which SHF is maximized. Were the small non-zero thermal resistance of a water wall taken into account, an optimal thickness would be obtained. This optimal thickness would be so large, however, as not to be of any practical interest. In Fig. 11, we show the effect on SHF of varying a at fixed values of SLR (for a house with night insulation). At sufficiently low values of SLR, SHF is linear in a for all values of a. At large values of SLR, SHF is linear in a for small a, with a more gradual increase with a at large OL
In Fig. 12, we present the effect of varying the relative width of the parabolic distribution function 81SLR (for a house with night insulation). We note that, as in the case of direct gain systems, SHF is not a sensitive function of ~ISLR. 9. CONCLUSIONS
We have presented a simple analytic method with which the monthly or seasonal thermal performance of
SHF" t.O
SHF
00
10
0.6
0"8
04
b
0"6
0.2 0
a
04 0
1:0
2:0
3:0
4~0
SLR Fig. 9. Dependence of SHF on At d (house with night insulation)--Water Wall; a, our model with reference conditions (including At d= lOhr); b, same with Aid= 12hr; and c, same with Atd = 8 hr.
0"2 0 Fig. I I. SHF vs a for fixed SLR (house with night insulation)-Water Wall; a, SLR = 2.00; b, SLR = 1.00; and c, SLR = 0.50.
Analytic model for passively-heated solar houses--l. Theory SHF
1 °0'
0.8
0.6 0.4 0-2 0
120
2"0
.'320
4:0
SLR FiB. 12. Dependence of SHF on 6/SLR (house with night insulation)--Wate__grWall', a, our model withnreference conditions (including 8/SLR = 0.8); b, same with 8/SLR = 0.6; and c, same with ~/SLR = 1.0. passively-heated solar houses can be calculated in closed form. The formalism presented above can be applied to any type of passively-heated solar house, which is one of its primary advantages. In this paper, the specific cases of direct gain and water wall houses have been treated in detail. This method has also been applied to the case of the massive storage wall house--a case sufficiently complex so as to merit a separate paper [12]. This method provides a useful tool for easy calculation of the dependence of thermal performance on building and climatic parameters. Unlike using the parameterizations of computer simulation results, it affords the user the flexibility of not being limited to a "reference" house. Whereas the treatment of different types of passive heating elements can require the writing of new computer simulation programs, our method requires identification of the basic factors governing heat balance and the solution of the appropriate equations. There may be two major limitations to our model. First, we have ignored the true time dependence of insolation during the course of the day. The fact that this effect is small for commonly considered direct gain houses is shown in the Appendix. Second, the coarse division in time used in our energy bookkeeping can result in over-estimates of SHF. This can occur whenever ventilation must be activated to avoid overheating during only a fraction of daytime or nighttime in the actual situation, while such overheating would not occur on the coarse time scale employed in our model. For relatively large storage thermal mass, this error occurs at large SLR values, for which SHF varies sufficiently slowly with SLR that the error incurred is small. For small storage thermal mass, the SLR values at which this error occurs are lower, which can result in slightly larger overestimates of SHF. Significant errors are introduced, however, only at such small values of thermal mass so as not to be of practical interest. REFERENCES
1. W. O. Wray, J. D. Balcomb and R. D. McFarland, A SemiEmpirical Method for estimating the Performance of Direct
341
Gain Passive Solar Heated Buildings. Los Alamos Scientific Laboratory Publication LA-UR-79-117(1979). 2. W. O. Wray and 2. D. Balcomb, Trombe Wall vs Direct Gain: A Comparative Analysis of Passive Solar Heating Systems. Los Alamos ScientificLaboratory Publication LA-UR-79-116 (1979). 3. J. D. Balcomb and R. D. McFarland, A Simple Empirical Method for Estimating the Performance of a Passive Solar Heated Building of the Thermal Storage Wall Type. Los Klamos Scientific Laboratory Publication LA-UR-78-1159 (1978). 4. R. D. McFarland L D. Balcomb, The Effect of Design Parameter Changes on the Performance of Thermal Storage Wall Passive Systems. Los Alamos Scientific Laboratory Publication LA-UR-79-423(1979). 5. J. E. Perry, Jr., Mathematical Modelling of the Performance of Passive Solar Heating Systems. Los Alamos Scientific Laboratory Publication LA-UR-77-2345(1977). 6. J. M. Gordon and Y. Zarmi, Analytic model for passively heated solar bouses--II. Users Guide. Solar Energy 27, 343--347(1981). 7. R. W. Jones, private communication, Los Aiamos Scientific Laboratory (Oct. 1980). 8. P. Berdahl, D. Grether, M. Martin and M. Wahlig, California Solar Data Manual. Lawrence Berkeley Laboratory (1978). 9. S. Schweitzer, A representative 'average' weather year for Israel's coastal plain. Report No. 3-78. Agricultural Research Organization, Bet-Dagan, Israel (1978). 10. W. O. Wray and J. D. Balcomb, Sensitivity of direct gain space heating performance to fundamental parameter variations. Solar Energy 23, 421 (1979). 11. H. Tabor, Radiation, convection and conduction coefficients in solar collectors. Bull. Res. Council Israel 6C, 155 (1958). 12. J. M. Gordon and Y. Zarmi, Massive storage walls as passive solar heating elements: an analytic model. Solar Energy 27, 349-355 (1981). 13. B. Y. H. Liu and R. C. Jordan, The interrelationship and characteristic distribution of direct, diffuse and total solar radiation. Solar Energy 4(3), 1 (1960).
/dJlpEh'DIX We consider here the effect on our results of using a timedependent insolation during daytime. The only change in our formalism is the replacement of the constant insolation rate Or/At# used in the daytime eqn (12) by a time-dependent one. The gross features of the time dependence of the insolation can be represented [13] by a function of the form qo(COSaJ- COSws),
where oJ, is the sunrise (sunset) hour angle, ~o~= ~rAtd/T (T = At a + At" = 24 hr), and
co= co,-fit, with fi=2w/T. Equation (12) now becomes (mC)s dTsaldt ~ = (I - F)qdcos ~o- cos ~o~)- UsA(Tsd - To). (AI)
The solution to eqn (A1) is straightforward, yielding
Ts" T, (]-
=
)qofrr
UsA
LEE
l+fz~ ~
J ÷c°s
rcos ,o - fir sin to'] _ cos ,o,]. +L I+N ~T~ J + (Tsd(0)- T~)e-'~/L
(A2)
Repeating the procedure of section 5, we recalculate Qjl), Q(]) and QR. The partition of the residual energy QR between daytime and nighttime is still the same as in section 5, yielding the final partition of Qr between daytime and nighttime as
342
J. M. GORDON and Y. ZARMI For commonly used values of ~',however, the differencebetween the two resultsis negligible.For example, for a at d of I0 hr and a value of ,ras small as 5 hi',eqn (A3) yields for Q, a value which is smaller than that obtained by eqn (23) by 6.4%. With increasing ~-,agreement is better,and for large values of ~-both expressions for Q,, approach the same limit,
- (I - F)Qr fl~:: Q,- 2(i + flzl.-') • " fsmco,-fl~'coso,,+(sm,o,
+
1 flrcos,o,)e- ~ ' / " 1F-e
-At,l,
I
Qd = Q r - Q . .
These expressions for Q, and Qa are obviously differentfrom the corresponding ones calculated in section 5 (see eqn 23). The main differenceappears at low values of ,r,where eqn (23) yields 0,, = (I - F)QTv/At d (A4) while the more exact result, eqn (A3), yields
0,. =
(1
-
F) ^ [
sin w~
,
~ [ 2 ( s i n ~ , - m, cos ~,,)] a2~'''
(A5)
Q.
, (I -
F)QrAt"IT.
large v
Also, the limit of Q. for short daytime periods (At ~ small) is the same for both calculations, Q. ~
(i - F)~r. &t
~mall