Evaluation of complex heat transfer coefficients for passive heating concepts

Building and Environment, Vol. 22, No. 4, pp. 259-268, 1987. Printed in Great Britain.

0360-1323/87$3.00+0.00 © 1987Pergamon Journals Ltd.

Evaluation of Complex Heat Transfer Coefficients for Passive Heating Concepts N. K. B A N S A L * M. S. S O D H A * S A N T RAM1" S. P. S I N G H * Passive heatin9 concepts namely Trombe wall, Water wall and Trans wall have been analysed to obtain overall heat transfer coefficients for average values and for time-dependent variations. The numerical values have been obtained and tabulated for various wall thicknesses.

rh~ n r So S,

mass flow rate of air (kg/sec) number of harmonics (integer) reflectance (dimensionless) average solar insolation (W/m 2) nth harmonic of solar insolation in its Fourier series expansion (W/m 2) T temperature of the wall and the trap (°C) t time (sec) Ta ambient temperature (°C) Ta0 average value of ambient temperature (°C) T~, nth harmonic in the Fourier series expansion of ambient temperature (°C) T~ room temperature (°C) T~a solair temperature (°C) Tw~ temperature of first water column (°C) Tw2 temperature of second water colunm (°C) X coordinate (m) co 21r/T ~., constant (°C) 2~ constant (°C) rg transmittance of glazing (dimensionless) % absorptance of the blackened surface of the wall (dimensionless) p density (Kg/m 3) (Ebi--Ebi ~)/Eb emissive power of a black body source with Eb0 = 0 and Eb, = Eb (dimensionless) 2 wavelength ~m) #j extinction coefficient of trap material for solar radiation in the wavelength region between j and j - 1 vj fractional absorption of solar radiation in water in the wavelength region between j and j - 1 (dimensionless) ~/j extinction coefficient of water for solar radiation in the wavelength region between j and j - 1. tr, phase of the nth harmonic of heat transfer coefficient (radians)

NOMENCLATURE A Ao ao At a, A~ Aw2 C C, Cw d~t d~2 ho ht h'~ h2

area (m 2) constant (°C) average solair temperature (°C) constant (°C/m) nth harmonic in the Fourier expansion of solair temperature (°C) area of first water column (m 2) area of second water column (m 2) specific heat (J/Kg °C) specific heat of air (J/Kg °C) specific heat of water (J/Kg °C) width of first water column facing glazing (m) width of second water column facing the room (m) heat transfer coefficient from front surface of the wall to ambient (W/m 2 °C) heat transfer coefficient from blackened surface of the wall to the enclosed air (t~V/m 2 °C) heat transfer coefficient from blackened surface of plate to water (W/m 2°C) heat transfer coefficient from enclosed air to glazing (W/m ~ °C)

h'2 heat transfer coefficient from water to the concrete layer (W/m 2 °C) h3 heat transfer coefficient from glazing to ambient (W/m ~ °C)

h4 heat transfer coefficient from interior surface of the wall to the room (W/m 2 °C) h~ radiative heat transfer coefficient from blackened surface of the wail to the glazing (W/m 2 °C) hR heat transfer coefficient from concrete wall to room in the case of water wall ('W/m 2 °C) htt heat transfer coefficient from front surface of the trap to first water column ( W / m 2 °C) ht2 heat transfer coefficient from trap to second water column (W/m 2 °C) h~. heat transfer coefficient from first water column to the ambient (W/m 2 °C) K thermal conductivity (W/ink) L thickness of the wall and trap material (m) M~l mass of water in first water column (kg) M~2 mass of water in second water column (kg)

1. I N T R O D U C T I O N PASSIVE heating concepts for building have been a subject of several studies in recent years [1-6]. Using numerical simulation analysis [7,8], the year long performance of storage walls has been well characterized. Simple empirical methods [9,10] have also been presented for estimating the annual solar heating performance of a building using either thermal storage wall or direct gain concept. The periodic technique based on Fourier series rep-

* Centre of Energy Studies, Indian Institute of Technology, Hauz Khas, New Delhi-110016, India. t Department of Nonconventional Energy Sources Ministry of Energy. 259

N . K . Bansal et al.

260

resentation of temperature and heat flux has been extensively applied [11,12] to rate solar passive heating concepts. This rating has often been expressed in terms of the heat flux coming into a room assumed to be maintained at a constant temperature. In this approach each time either the geometry or the location of the building changes, one has to perform a fresh set of detailed numerical calculations. In this paper, the passive heating concepts, namely Trombe wall with or without vents, Water wall and Trans wall, have been characterized in terms of overall complex heat transfer coefficients, appropriate for periodic variations. The heat transfer coefficients, which are independent of the meteorological parameters and the area of the wall, can be used to quickly evaluate the effect of passive concepts on building response; the heat flux in an air-conditioned room (with constant temperature of room air) can be expressed as a simple function on these coefficients and Fourier coefficients of the solair temperature at the outer interface of the wall. For some concepts in which the heat collection and the subsequent heat transfer mechanism is different than an opaque wall, an equivalent solair temperature has to be redefined to take advantage of the overall definition of heat transfer coefficient for the calculation of heat flux.

determined by the application of appropriate boundary conditions which can be written as

- KOq~Tx=oVX

= ho[T,. - T(x, t)lx=0]

(3)

and

K dT -- ~Xx=L=h4[T(x,t)lx=L-T,].

(4)

In equation (3), the solair temperature is an equivalent environmental temperature which takes into account the mechanism of radiation absorption and subsequent heat transfer from the hot surface. The right hand side in equation (4) represents the heat flux, Q, coming into the room. Substituting for T from equation (l) in equations (3) and (4) and solving the resultant algebraic equations one obtains the expressions for the unknown constants and the heat flux. The expression for the heat flux, Q, can be expressed in the form

Q. = Uo(ao- TR)+ Re ~ U,a.e ~'°',

(5)

n=l

where a0 and a, are Fourier coefficients of the time variation of solair temperature, i.e.,

Ts~ = ao + Re ~ a. e ~"°''

2. ANALYSIS

(6)

n=l

and Uo and U. defined as the average overall heat transfer coefficients and the complex heat transfer coefficient for the harmonic are given by the expressions.

2.1. Trombe wall without vents Corresponding to the configuration given in Fig. la, the heat transfer through the glazed mass wall can be assumed to be one dimensional governed by the following heat conduction equation, the periodic solution of which is written as

L ,]-i

Vo =

E,

(7)

and

T ( x , t ) = A o + A I X + R e ~ (2.e#"X+2~,e-~"x)C "~', (1)

u.=

n=l

where

Khoh4fl. [(hoh4 + K2fl.2) sinh (fl.L) + (ho + h,)K~, cosh (fl.L)]'

~ncopc~~/2 fl.=(l+0(2K J "

(2)

(8)

The unknown constants A0, A~, 2. and 2~, are to be

The overall heat transfer coefficient from the collecting

Room TR

~cnGcl~ Ell

:(a) Troml~ woll ~

(b) Troml~wall with ventS. I:-=_~-~_-=-=-~

Water

ZI Rl

Room TR

RoomrR(2o'c) \

_

~-'---_"?~=-_-S - U X=0

(c) Water wall.

: Flux

X=L

(d) Trans wall

Fig. I. Schematics of the systems.

Wa ,f

Evaluation o f Complex Heat Transfer Coefficients f o r Passive Heating Concepts surface to the ambient is in equation (7). In fact, this again consists of 3 parts, viz the conductance of the air gap between the hot surface and the glazing, the conduction effect of the glazing and the heat loss from the outermost surface of the glazing. As with a fiat plate collector, an overall heat transfer coefficient is assigned to single or double glazed surfaces [11,12]. 2.2. Trombe wall with vents For Trombe wall with vents (Fig. 1b), the heat transfer from the hot surface consists of two parts (i) heat losses to the ambient through glazing and (ii) heat transfer to the moving air in the air gap, in addition to heat conduction through the wall. The heat flux, therefore, into the room consists of two parts i.e. 0 = 0~ + 0~,

(9)

(2k = hR[T(x, t)l~=L - TR]

(10)

where

and ~haca

C = ~-

[0ou~- T~].

(11)

0o,t is the temperature of hot air, entering the room through the upper vent. After rearrangement, the expression for heat flux can be put in the form,

U0 and U, assume the following expressions

(15)

v0= and

U, = hR[F1 e#'L +F2 e-#.L].

(16)

In equation (15) h~, is the heat transfer coefficient from water to the concrete surface in contact with water, where F1 and F2 are given in Appendix B. 2.4. Trans wall Trans wall, a translucent storage wall introduced by Fuchs and McClelland [9] and shown conceptually in Fig. ld, has been studied earlier by Sodha e t al. [13]. Since solar radiation is absorbed throughout the translucent media, the governing heat conduction equation for the temperature distribution is O2T Ox2

1 0T t3T K ax = p c ~

(17)

where I(x,t), the solar intensity at any point inside the translucent material is given by the expression [14],

I(x,t) = %(1-r)S(t) j= 1

Ebj-- Ebj- i

E~

e-U/"

(18)

with I(x,t) given by equation (18), the solution of equation (18) is obtained as,

0 = Uo [GI(G2So+ T~o)- Te.]

+R~ ~ U.(G.S~+ To.)e~'°~'. (12)

261

T(x, t) = A o + A I x

Sora(1-s) ~ Ebj--Ebj_l K #jEb j=l

n=l

x e - ' x vJ e-"'d"'+Re,=l ~ {2, @"~+2~, e -~"x

As with equation (5), Uo and (.7.may be termed as overall average and associated nth harmonic heat transfer coefficients respectively, and GI(G2S0 + Tao) and (G.S.+T~.) as the modified solair temperatures, expressions for various parameters being given in Appendix A. 2.3. Water wall The water wall consists of water filled in drums as the storage media (Fig. lc). The water wall may have a concrete structure at its back surface facing the room. Following the analysis given in Nayak et al. [11], the expression for the heat flux, when there is no concrete wall, can be written in the same form as equation (5) with, Uo=

~oo+ h~ + ~

S.zg(1-s) ~

-K

j__ ,

b9

ebj--Ebj-1

~.

,j--

Eb

• vj e-~ ~," e-Ur~} e~°'.

(19)

With appropriate boundary conditions given as

K OT ffxxx=0 = h,l[T(x, t)l~=0- T~t(t)],

(20)

_ K OT ax x=,. = h,2[T(x, t)l~=L-- T~2(t)],

(21)

M~IC,o OT~I -- h,l[T(x, t)l~=o- T~t(t)] A,ol at

(13) +xg(1-s)S(t)

{5

j~l

and

h°h'lh4 ] F, U, = Lhoh I +h0h4 + h'lh4 + (h:~"~-+h4)(MwCwinco/ A)-.J" (14) In equation (13) h o is the overall heat transfer coefficient from the heat collection surface to the ambient, h" is the coefficient of heat transfer from hot surface to water and h4 is the overall heat transfer coefficient from water to the room air. If the water wall is followed by a thin concrete wall,

t

1 - ~ vj e-~J dw,

)

-- h~. [Tw, ( t ) - T.(t)]

(22)

and

M~2C~, OT~2 A~ 2 Ot

h,2[r(x, t) lx~L- Tw2(t)] 5

+xg(1 --s)S(t) ~ vj e-¢jaw,{1 --vj e-¢~d-2} j=l Ebj--Ebj-I

Eb

e -u~z -h4[Tw2(t) -- TR]. (23)

262

N. K. Bansal et al. Table 1. Values of different heat transfer coefficients(HTC) and thermophysical parameters HTC (W/m3 °C) h0 h~ h'l h2 h'2 h3 h4 hr hR h, t hi2 hwo Concrete Trap Air Water

Specificheat (J/kg °C)

K (W/m K)

Density (kg/m3)

Other

6.0 3.2 206.5 2.9 206.5 21.96 8.29 4.9 8.29 200 200 6.0

rh~ s TR rq %

795.5 1500 1008 4200

0.72 0.2

After involved but simple algebra, the flux entering into the room can be expressed in the form, O = h4S'[G', (G'2So -k Tao) - TR] + hahwaS"[G'nSn - Tan]

(24) where h4S' ( = Uo) and h4h~,aS" ( = Un) are the overall average and nth harmonic complex heat transfer coefficients and the equivalent solair temperatures for Trans wall have the following Fourier coefficients ao = G'l (G'2So + T,o) an = (G'~S, - T.n )

The expressions for G1, G:, Gn, S' and S" are given in Appendix C. 3. RESULTS AND DISCUSSIONS Various values of heat transfer coefficients and thermophysical properties of the materials used in the calculation of overall heat transfer coefficients are given in Table 1. The values of extinction coefficients and absorption coefficients for Transwall are given in Table 2. With these values of the wall properties, the numerical values of the average and other harmonic for overall heattransfer coefficients for various passive heating concepts discussed in this paper are given in Tables 3-7. The nth harmonic heat transfer coefficient Un is complex as Un = A, exp (ian), An and trn being given in the tables. On

0.104 kg/sec 0.04 20°C 0.9 0.9

1858 1190

the average, the overall U-value for the Trombe wall with vents is higher than other passive components i.e. Trombe wall without vents, Water wall and Trans wall. Also there is little change in its value with increasing wall thickness for the case of vents, essentially because the convective heat transfer starts dominating and the phase lag attenuation goes on reducing as the wall thickness increases. Average heat flux into a room of constant temperature environment depends on the equivalent average solair temperature and the overall heat transfer coefficient. These are given in Tables 8 and 9 for different heat capacities. On average it is seen that though the solair temperature for the water wall and Trombe wall is identical, the water wall has larger U-value. This results in the transfer of greater heat flux into the room. Looking only at the average values, the Trombe wall with vents gives the best performance ; however the timedependent behaviour shows that the storage capacity of this type of Trombe wall is negligible. The time-dependent behaviour of the heat flux (a measure of the overall performance) is shown in Figs 2 and 3 corresponding to the solair temperature variations for Trombe wall, Water wall and Trans wall given in Figs 4 and 5. It is observed that the solair for the Trombe wall without vents and the water wall has maximum amplitude ; this, along with the appropriate damping effect of various harmonics of heat transfer coefficients, yields positive flux for the water wall throughout 24 hr.

Table 2. Value of extinction and absorption coefficients in trap (translucent) material for solar radiation in different wavelength regions Wavelength region

Ebj-- Ebj- j/Eb

#

v

0 ~<2 ~<0.36 0.36 ~<2 ~< 1.06 1.06 ~<2 ~< 1.3 1.3 ~<2 ~< 1.6 1.6 ~<2 ~<0.0

0.08137 0.66880 0.08620 0.06120 0.10244

0.0 0.725 3.82 9.45 0.0

0.237 0.193 0.167 0.179 0.224

0.032 0.450 3.00 35.0 255.0

Evaluation o f C o m p l e x H e a t Transfer Coefficients f o r Passive Heatin 9 Concepts

263

Table 3. Amplitude and phase of the heat transfer coefficient of Trombe wall without vent n

L(m) 0.10 0.15

An tr, tr.

0.20

an

0.25

a,

0.30

or,

0

1

2

3

4

5

6

2.35 0 2.02 0 1.77 0 1.58 0 1.42 0

1.88 5,4502 1.191 4.9225 0.74 4.4600 0.47 4.02730 0.3034 3.5999

1.202 4.8701 0.613 4.2186 0.326 3.6116 0.177 3.0043 0.0959 2.3939

0.82 4.4914 0.37 3.7359 0.174 2.9925 0.083 2.2448 0.039 1.4968

0,596 4.2034 0,244 3.3433 0.103 2.4814 0.043 1.6176 0.018 0.75416

0.453 3.9633 0.17 3.0033 0.065 2.0380 0.025 1.0724 0,0094 0,10721

0.356 3.7528 0,123 2.6997 0.426 1.6417 0.015 0.5843 0.00513 5.8100

Table 4. Trombe wall with vents (rh/A = 0.104 kg/sec) n

L(m) 0.05 0.10 0.15

A, a. A, trn A. tr,

0.20

A,

0.25

a. A. a,

0

1

2

3

4

5

6

4.65 0 4.51 0 4.43 0 4.35 0 4.30 0

4.49 6.0885 3,8048 5.9686 3.24 5.9535 2.9264 6.0059 2,805 6.0683

3.982 5.9606 3.014 5.9287 2.65 6.0265 2.5985 6.1115 2.648 6.1538

3.50 5.9031 2.64 5.9775 2.497 6.1005 2.5521 6.1592 2.62 6.172

3.14 5.8907 2.48 6.0328 2.456 6.1445 2.5385 6.1760 2,586 6.1741

2,88 5.9016 2.39 6.0783 2.445 6.1686 2.5238 6.1812 2.55 6.1742

2.70 5.9229 2.36 6,1131 2.441 6.1817 2.5055 6.1827 2.517 6,1751

Table 5, Water wall without a concrete slab n

M~ (kg) 50 100 150 200 250

A, tr. A, tr, A. tr. An a. A. trn

0

1

2

3

4

5

6

3.41 0 3.41 0 3.41 0 3.41 0 3.41 0

2.316 5.45779 1.431 5,14478 1.0042 5.9088 0,7678 4.9392 0.620 4,8949

1,431 5,14478 0.7678 4.93918 0,519 4.86503 0.3914 4.8272 0.314 4.80443

1.004 5.01088 0.5192 4.86503 0.348 4.81469 0.2619 4,7892 0.209 4.7738

0.768 4.93918 0.3914 4,82726 0.2619 4.78915 0,1966 4.7700 0.157 4.7585

0,620 4.89494 0.3138 4,80443 0.2097 4.77384 0.1574 4.7585 0.126 4.74929

0.520 4.86503 0.2620 4.78915 0.175 4.76362 0.131 4.7508 0.105 4.74315

Table 6. Water wall with a concrete layer of thickness 0.1 m n

Mw (kg) 50 100 150 200 250

A, or, An tr, A, a. A, tr, An tr.

0

1

2

3

4

5

6

2.26 0 2.26 0 2.26 0 2.26 0 2.26 0

0.8678 4.68986 0.5382 4,49300 0.3853 4.4060 0.2990 4.3578 0.24404 4.3272

0.4114 4.10009 0.2391 3.9524 0,1675 3.8925 0.1286 3,8603 0.1044 3.8402

0.2380 3.69385 0.1338 3.5660 0.0927 3.5163 0.0708 3.49004 0.0573 3.4738

0.1519 3.37274 0.0836 3.2590 0,05748 3.2161 0.0437 3,19364 0.0353 3.1798

0.1033 3.10431 0.0560 3.0021 0.0383 2.9642 0.0291 2.9445 0.02345 2.9325

0.0735 2.87136 0.03944 2.7784 0,2689 2,7444 0.0204 2.72687 0,01642 2.7161

264

N . K . Bansal e t al. Table 7. Amplitude and phase of the heat transfer coefficient of trans wall with Mw~ = Mw2 = 100kg and

d~] =dw2=O.lm n

L(m)

0.005 0.01 0.02 0.03 0.04 0.05

A. a. A. a. A. a. A. a. A. a. A~ a.

0

I

2

3.1 0 2.87 0 2.51 0 2.23 0 2.01 0 1.82 0

0.639 1.3507 0.517 1.1338 0.352 0.8585 0.2569 0.6751 0.198 0.5216 0.158 0.3750

0.262 0.9169 0.1824 0.6686 0.107 0.4010 0.074 0.2046 0.0542 0.00857 0.0416 6.0800

4. C O N C L U S I O N S Explicit expressions for the average overall heat transfer coefficients a n d the associated complex heat transfer coefficients for various h a r m o n i c s have been o b t a i n e d for the T r o m b e wall, W a t e r wall a n d T r a n s wall passive heating concepts. The p e r f o r m a n c e o f buildings incorp o r a t i n g any o f these c a n be easily evaluated by using the values o f overall heat transfer coefficients t a b u l a t e d in this paper. The expressions for a n equivalent solair t e m p e r a t u r e for the T r o m b e wall with vents, T r a n s wall a n d W a t e r wall are also o b t a i n e d in the process.

3

0.138 0.6747 0.089 0.4469 0.0497 0.1901 0.033 6.2479 0.0239 5.9972 0.0177 5.7174

20( 16(

4

5

6

0.0845 0.5234 0.0521 0.3173 0.0283 0.0575 0.0186 6.0754 0.013 5.7681 0.0093 5.4266

0.0564 0.4212 0.034 0.2304 0.0182 6.2398 0.0117 5.9320 0.00798 5.5705 0.0054 5.1757

0.0401 0.3478 0.024 0.1665 0.0126 6.1559 0.008 5.8046 0.0053 5.3931 0.0035 4.9525

Flux through I Simpletromb¢ wall II Trombcwclll with natural convention

m Water wail 121: N Trans wall

/

~

/'/A i tO-

I

0 -,60"

~

"

nerT,~"

(hr)~

Fig. 3. Hourly variations of the flux entering into the room through different systems for heat capacity equal to 443.2 kJ/°C.

260~i Solair for (L=O.15m) Tromb¢ wall without vents and water wail 200 11 Tromb¢wall with / vents ] 1601~ Trans wall /

t~ E

1

12o

x it

(hr)

Fig. 2. Hourly variations of the flux entering into the room through different systems for heat capacity equal to 221.6 kJ/°C.

Fig. 4. Hourly variations of the solair temperatures for different systems for heat capacity equal to 221.6 kJ/°C.

T,o (°C)

44.4 44.4 44.4

Uo (W/m 2 °C)

2.02

1.42 1.10

4.43 4.26 4.17

Uo (W/m 2 °C) 33.0 31.8 30.7

T,o (°C) 57.6 50.2 44.6

Qo (W/m 2) 3.42 3.42 3.42

Uo (W/m 2 °C)

49.3 34.6 26.8

Qo (W/m 2) 4.43 4.26 4.17

Uo (W/m: °C) 33.0 31.8 30.7

T,o (°C) 57.6 50.2 44.6

Qo OV/m ')

Tromb¢ wail with vents

=

3.42 3.42 3.42

44.4 44.4 44.4

T,o (°C)

Water wall

83.45 83.45 83.45

Qo (W/m 2)

44.4 44.4 44.4

T~o (°C)

Water wall

83.45 83.45 83.45

Qo (W/m 2)

1.82 1.82 1.82

Uo (W/m 2 °C)

1.82 1.82 1.82

Uo (W/m 2 °C)

0.3 m; aw2 = 0.10 m ; trap thickness = 0.05 m.

Uo (W/re' °C)

* (I) aw~ = 0.1 m ; aw2 = 0.0 m; trap thickness = 0.05 m. (II) awl = 0.15 m ; aw2 = 0.10 m ; trap thickness = 0.05. (III) awl

0.15 0.30 0.45

49.3 34.6 26.8

Qo (W/m 2)

Trombe wail with vents (natural convection)

Table 9. Average solair, heat transfer coefficients and flux for different passive heating concepts for different wall thicknesses

44.4 44.4 44.4

T,o (°C)

Trombe wall without vents

2.02 1.42 1.10

221.6 443.2 664.8

Total thickness of the wall (L) (m)

Uo (W/m 2 °C)

Trombe wall without vents

Total heat capacity (n ~,Cp/mr) of the wail (kJ/°C)

Table 8. Average solair, heat transfer coefficients and flux for different passive heating concepts for different heat capacities (kJ/°C)

30.0 34.0 35.3

T,o (°C)

Trans wall*

30.0 33.0 34.0

T,o (°c)

Trans wall

23.7 25.5 28.0

Qo (W/m r)

18.2 23.7 25.5

Qo (W/m 2)

to O~ L~

¢'a

¢b

g~

g~

~'

q~

N . K . Bansal et al.

266

A v

E

g,

".__

Fig. 5. Hourly variations of the solair temperature for different systems for heat capacity equal to 443.2 kJ/°C.

REFERENCES 1. J.D. Balcomb and R. D. McFarland, A simple empirical method for estimating the performance of a passive solar heated building of a thermal storage wall type. Proc. 2nd National Passive Solar Conference, 2, 377 (1978). 2. J.D. Balcomb, R. W. Jones, R. D. McFarland and W. O. Wray, Performance analysis of passive solar heated buildings of the solar load ratio method. LA-UR-82-670 (1982). 3. W.A. Beckmann, S. A. Klein and J. A. Duffle, Solar Heating Design by the F-chart Method. Wiley Interscience, New York (1977). 4. D.B. Goldstein, Modelling passive solar buildings with hand calculations. Conf. 4 (1980). 5. F.D. Heidt, Physical principles of passive solar heating systems. Solar World Congress, 14-19 August, Perth, Australia (1983). 6. M.S. Sodha, J. K. Nayak, N. K. Bansal and I. C. Goyal, Thermal performance of a solarium with a removable insulation. Bid# Envir. 17, 23-32 (1982). 7. J. D. Balcomb, R. D. McFarland and S. W. Moore, Simulation analysis of passive solar heated buildings--comparison with test room results, LA-UR-77-939, Los Alamos Scientific Laboratory, University of California, Los Alamos, New Mexico (1977). 8. R . D . McFarland, PASOLE. A general simulation program for passive solar energy. Los Alamos National Laboratory Report LA-7433-MS (1978). 9. R. Fuchs and J. F. McClelland, Passive solar heating of buildings using a transwall structure. Sol. Energy, 23, 123 (1979). 10. M. Mazaria, M. S. Baker and F. C. Wessling, An analytical model for passive solar heated buildings. PAM 10 (1977). 11. J.K. Nayak, M. S. Sodha and N. K. Bansal, Analysis of passive heating concepts. Sol. Energy, 30, 51 69. 12. J.A. Duffle and W. A. Beckman, Solar Energy Thermal Processes. John Wiley, New York (1970). 13. M.S. Sodha, N. K. Bansal and Sant Ram, Periodic analysis of a transwall : a passive heating concept. Appl. Eneryy, 14, 3348 (1983). 14. E. Lumsdaine, Transient solution and criteria for achieving maximum fluid temperature in solar energy applications. Sol. Energy, 13, 3 19 (1970).

APPENDIX

A

Uo = h 4 P I e P 1 4 - PIT

PI9 = W ( S 2 - - S I P ~ P t 3 )

(A6)

PI8 = - S1WP~oP~3

(A7)

e,7 = W[S3+S,(1-P,2P,3)]

(A8) (A9)

(A1)

G~ = (P~9 +h4P,2P~4P~6)/Uo

(A2)

G2 = (Pto+P,2Pt4P,sh4)/(P~9+P~2Pt4P~6)

(A3)

Pl6 = - P l l / P I 2

U,, = ZI~ +h4(Zu e&L+Z7 e t~,/

(A4)

P~s

G n = [ Z i o + h 4 ( Z 0 e&L+z6 e &L)]/U.

(A5)

PH = K / &

= --Pto/Pt2

(A10) (All)

Evaluation of Complex Heat Transfer Coefficients for Passive Heating Concepts K

P,3 = L + h~

(A12)

R6 = (h2+ Kfl.)- Lh"+h2+M~Cjn~o l

P l 2 = - h4 (P9 - PT)/H

(A 13)

Pit = - h 4 P s / H

(A14)

R7 =

P, o = -- h4%zg/H

"(A 15)

Rs = (hR + Kfl~) e #.L

P9 = e6(hl +h~P2)

(A16)

R9 =

Ps = h ~Ps + h~(e3 + P2P5)

(A17)

P6

W 1 -Gh2P2

Gh2P3 P5 = 1-Gh2P2

P4

G(h, + h 2 P l ) 1 -

P3

h3 h2 +h3 + hy

(h~-K/L)

(B9)

(A 18)

F, = Rio

(A19)

F2 =

R7R8 Rio

APPENDIX C S' = S,5 G', = S,3/$15

(A22)

h~

(A23)

S" : Y 3 6

h2+h3+h r

G'~

(A24)

H = h4K+ P7 ( K + h4Z ) S, = hi +h2P, +h2P2P4

(A25)

S: = h 2(P2P5 + e3)

(A26)

$3 = h: (P2 P6 - 1) - h 1

(A27)

$2 = h4

(A28)

ht2 $3 = h~

(A29)

$4

G = W = A-( h ,-+ h 2 )

1-W hi+h2

-- ~

1-exp

-)_]

Z I I = W ( Z 6 -r Z 8 )

(A30)

Zlo = WS, (Z7 + Zg) + $2 W

(A31)

Z4Z7 Z 9 = -- - -

T

$5 =

z(1 - y)h,2 K'hs

z(l --y) hs

h4 $6 = h~ $7 = S3h,o

Z4Z6

Z s = ---Z3 Z7 =

S,EE(3) S"h~a

S, = z(1--y)/K

(A32)

Z3

(A33) $8

Z3P8

(A34)

Z5

(ht i - B6) B2

S 9 = (htl +Ssht2) S,o = K - S s h 5

Z6 = Z3zg%

Z5

(A35)

Z5 = Z2Z3 - ZIZ4

(A36)

Z4 = (h4 - Kfl,,) e - # J

(A37)

Z 3 = (h 4 +Kfl.) e a.z

(A38)

Z2 = (P7 + Kfl.)

(A39)

Z, = (e7-Kfl.)

(A40)

S , I = Sghlo.+ S l o S 7 Si2 = SloS 6

Sm~

S13 ~ - -

Blm

$12 514 = ~Bl S15 = I-.[-SI4

SS =

APPENDIX B /1"I h. =

~(1

-~)

h5 = h , 2 L + K

l ~ -1 +

(s,)

h6 = htl + h~a h 7 = ht2+h 4

h2 (B2)

h8 = ht2L

R 2 = he + h2 + M~,Cwinco

(B3)

h6 h9 = ht~

R 3 = h 2 --Kfl,,

(B4)

R l = h, + h2 + M w C j n o 9

h,

R4 = h2 +Kfl~

(B5) [_

h~

l

R5 = (h2-- Kiln)-- l h e + h2 + MwCwinto_]

(B6)

(Bll)

RTR9

G': = E9/S 13 P,

(B8)

(B10)

e -#:

R,o = (hn+Kfl.) e a,L R 6 - ( h R - K f l . ) e -#,L R5

(A21)

h2 P2 = h2 +ha + h r

(B7)

h~h: h~ + h 2 + M~C~inco

(A20)

Gh 2P2

267

E, = z~

:= 1

Ebj--Ebj --

/dj Eb

. . e-U/~vj e-":~'

E 2 = ~ Ebj --Ebj- ' e -#]L vj e-~.,dwl j=l

Eb

(B12) (BI3)

268

N . K . Bansal et al.

E3-

e-~, z v2 e-qf(d.,+d.2) j=l

E4

= ~ Ebj--Ebj_,

#jEb

j=,

htly3

Ylo

Eb

vj e-~/a~,

YlY9

Y,~ = h t , ( 1 - Y 2 ~ k Y,/

5

E 5 = ~ vj e-~fl~,

El2 = - -

j=]

YJ

z# I ( 1 - s ) h 2 E ~ E6 = ~ ~ +(I-s)(E2-E3)

Yl3 = Y12 --Y5

(1 --s)h,2l 4 K

YI4 -- YJl Y9

ht2(1 - E s ) ]

+

U,

-J

g7=~,(-Ebj~bJ-I-)vje-~/d~,

Y15 =

Y3YJ l YlY9

YI6 = YloYll

j=l E8 = St h~lE4 + S S E 7 + SI Ssht2E1 - S s S S E 2 + S1 oE6 - $9S1 E4 + $9S2 - $9S2E5

E8 E9 = B~

YI7 =

Y2Y ~I Y9

El8 = Y13 --YI6 Y14 YI9 = - Yl8

_ ~

EE(J) -

j=l

f

#2 ~/f12 ~--

. Ebj- Ebj_

. rh,~

• Je- ~ [~(1

Eb

I. YJ e_/~/.dwI

+y37)-#y37-~

h,, K

Yls--(h,l/y,) Yl8

Y2o

Y~7 Y21 = - Yl8 1

X (736 -{-Y38)"4-~jY38} - - (1 -- vj e-~J d.z)

Y22 = - Y18

f Ebj--et,j-i . vje-,i,a.,} x ~Y36+Y35" Eb e -u.'~

Y23 = Y7 - - Yl

h7

Y6Y2

Y24

htlY6 -

Yl

B, =h8 B2-

Y25 = Y24--YloY23

ht2h9

h8

y23ht2

Y26

Y9

B 3 = ht2h 9 - h s B 2

Y6 Y27 = - Y,

B4 = hsBt--ht2 B4 B5 = B~3

Y28

t

B 8 = B 7 -B6B5

Y32 = --Y22Y25

Y3 = (hi2 + K~n) e x p (~.L) Y4 = (ht2 + K~.) e x p ( - ~ . L ) mw i C~in~o

Y5 = h,~ + h ~ + -

A~t

Y6 = ht2 exp (~.L) Y7 = ht2 e x p ( - ~ . L )

Y2Y3 Y,

Y9

Y31 = Y26--Y21Y25

Y2 = h,1 + K~.

Y9 = Y4

Y29 = 223 Y30 = Y28--Y27

Y, = h,, - Kot°

Y8 = ht2 + h4 +

Y3Y23 --

YJY9

B 6 = htjh9+KB2 B 7 = htlB 5-KB

Y8

inmC m~2 Aw2

Y33 = Y19Y25 --Y29 Y34 = Y30--Y20Y2~ 1

Y35 = - Y3J Y32 Y36 = - Y31 Y33 Y37 = - Y3t Y34 Y38 = - Y3~