Applied Energy 13 (1983) 157 164

Technical Note

Effects of Fancy-structured Walls or Roofs on the Heat Losses from Buildings

N O M E N C L A T U R E

C ~ C 1 =

C 2 =

d = h o =

h i =

h s =

K = l =

Q f = Qr =

Qs = Qt = Sf = S r =

S t : -

specific heat of the building material (kJ/kg °C) constant eqn (9) constant eqn (9) length of the fin (m) heat transfer coefficient between the exposed plane wall and the atmosphere (kJ/h m z °C) heat transfer coefficient for inner surface (kJ/h m 2 ° C )

heat transfer coefficient between fin and atmosphere (kJ /hm 2 °C) thermal conductivity of the wall and fin (kJ/h m °C) thickness of the wall (m) heat loss through bot tom of the fin (k J/h) heat loss through plane wall of area S r (k J/h) heat loss through side faces of the fin (k J/h) total heat loss to the atmosphere (k J/h) area of the fin bot tom (m 2) area of the plane wall (m z) area of side faces of the fin (m 2) total area of wall (m:)

t = time coordinate (s) x = coordinate axis for wall (m)

x + = coordinate axis for fin (m) Yo = half the width of the fin (m) V = wind velocity (m/s)

0 A = atmospheric temperature (°C)

157

Applied Energy 0306-2619/83/0013-0157/$03.00 © Applied Science Publishers Ltd, England, 1983. Pr inted in Grea t Bri tain

158 Alok Thakur, J. K. Sharma, Ram Chandra

OR = room temperature (°C) 0 r = temperature in the wall at a point x (°C) 0 r = temperature in the fin at a point x + (°C) p - -densi ty of the building material (kg/m 3)

I N T R O D U C T I O N

Recent energy crises have stimulated a renewed interest in design and fabrication of passive solar heated/cooled houses and buildings. One can clearly distinguish designs of houses built in desert areas from those in cold mounta inous areas. In hot climatic zones proper designing requires the reduction of the incoming heat flux. This may be achieved by using insulated walls and roofs, and introducing air gaps 1 - 3 (single or double), maintaining a water film on the roof, 4 or having flowing water on the roof.5 - 7 However, in colder zones (e.g. Europe) the situation is reversed and efforts are made to reduce the outgoing heat flux to ensure economical heating of the house. In both situations efforts are directed towards thermal load reduction of the air-conditioning system.

In this paper the effects of structure and design of exposed walls and roofs on the economical air-conditioning of buildings are studied.

T H E O R Y

Consider the proposed model (Fig. 1) for investigating the effect of fancy structured walls/roofs on incoming or outgoing heat fluxes for a building. It has been assumed that the ambient temperature is less than room temperature and the fancy structure on the wall is exposed to the atmosphere. The heat losses to the atmosphere may be divided into three categories (i) rate of heat loss Qr from plane surface, excluding base area of the fancy structure, of the wall; (ii) rate of heat loss Qr from the end of the fancy structure; and (iii) rate of heat loss Qs from the surface of the fancy structure. For the sake of convenience, rectangular fins have been chosen as the fancy structure considered, and S t, S r, Sf and S s are the total area of the wall, area of the wall excluding the base of the fin, area of cross- section of the fin and surface area of the fin respectively. Thus

St = Sr -~- St- (1)

Heat losses f rom buildings 159

e R

%

d - -

/ • ,-I x "=d

X:O ×~[ x*=o

Fig. 1. Sketch of fancy wall element considered.

and the total rate of heat loss to the atmosphere is

Qt = Q, + Qr + Qs (2)

The one-dimensional heat conduction equation for the plane wall bounded by two parallel planes x -- 0 and x = l is described by

~ 2 0 r ~ 0 r K ~x z = pC ~ (3)

The steady-state solution of eqn (3) can be expressed, with the help of the following two boundary conditions at the outer and inner faces of the wall, as

- K aOr = h i [ 0 R - O r (X = 0 ) ] ( 4 )

Ix=o

- K 0 0 , = h o [ 0 r ( x = l ) - 0A] (5)

Hence the steady-state temperature distribution in the wall is

h°hi(OA - - OR) (X + 1 O r = K ~ h ~ ( h o l ; K) \ h-i) "t- 0R (6)

and heat loss to the atmosphere will be

hohi(O R -- 0A) Or = gSr Kh ° + hi(hol + K) (7)

160 Alok Thakur, J. K. Sharma, Ram Chandra

Equation (7) determines the rate of conductive heat losses to the atmosphere through the fin-less portion of the wall.

To evaluate the rate of heat loss through the fin, one can write the differential equation for the rectangular fin s as:

d20s h s dx+2 K y ° 0s=0 (8)

where 0s is the excess temperature given by Of - 0A, and the convective heat transfer coefficient hs is expressed as 5-7 + 3.8V, V being the wind velocity (in m/s). The solution ofeqn (8) may be expressed in the following form

Of = 0A + C, exp [ ( h s / K Y o ) ° S x + ] + C 2 exp [ - ( h s / K Y o ) ° S x + ] (9)

where C~ and C 2 are arbitrary constants to be evaluated by using the following set of boundary conditions:

and

Of(x + = O) = Or(x = l ) (10)

- K o x + OOr x + - a = h ° [ O r ( x + = d ) - O A ] (11)

In deciding the boundary condition eqn (11) to be applied, it has been assumed that the temperature at the end of the fin is higher than the ambient temperature and this is physically true for fins of very short length. However, for a long fin, the temperature of the end of the fin is equal to the ambient temperature, and in this case 8 eqn (11) reduces to

OOf x + = 0 (12) (gx + :~

Thus using eqns (9)-(11) values of C 1 and C 2 a r e

{0r(X = l ) -- O A } { K fl + ho} exp (r id)

C , = (K f l + ho) exp (fld) + (K f l - ho) exp( - rid) (13)

{ K 2 f l 2 - h 2 } { O , ( x = l ) - 0A} C2 - - (K2fl 2 - - h 2) + ( K f l + ho) 2 exp (2fld)

(14)

Heat losses J~'om buildings 161

where

= ( h s / K y o ) O . 5

The total rate of heat loss through the fin surface and end may be assumed to be the heat conducted into the fin. Thus

~0f x + (15) Qs + Q f = - K S f ~ x ~ = o

o r

Qs + Q f = K S f f l ( C 1 - C 2 )

Equation (7) together with eqn (15) determines the total rate of heat loss to the atmosphere Qt.

R E S U L T S A N D D I S C U S S I O N

To obtain a numerical appreciation of the effect of the fancy structure on the incoming or outgoing heat fluxes for a building, the following set of parameters has been chosen:

0k = 22 °C

0A= 10°C

h i = 22-08 kJ/h m 2 °C

h o = h s = 17.38 kJ/h m 2 °C (referring to a 3-0m/s wind velocity)

S t = 4 m 2 (i.e. 2 x 2 m 2)

K = 2.6 kJ/h m °C)

l = 0 . 5 m

Figure 2 illustrates the variation of the total rate of heat loss to the atmosphere through the fancy wall: Curves I, II and III represents one-, two- and four-fin structured walls respectively. It is evident from the curves that the total heat loss increases with fin length and attains a maximum. This is the expected result since at zero fin length the total heat loss corresponds to the heat loss through a wall bounded by the two parallel planes x = 0 and x = 1 and is a minimum. As the fin length increases the total surface area exposed to the atmosphere increases and thus the rate of convective heat transfer also increases. The straight

........

9

190

180

170

160 j ] 0 0 0 8

Fig. 2.

I ] I

, I , I , I , I 0.16 0.2& 0.32 0 ./-,0

F I N - L E N G T H ( m )

Variat ion o f total heat loss to the a tmosphere wi th fin length; Y0 = 0.01 m.

A

a;:

s

165

160

155

150 ~1 I I 1 1 I I I I I I 0.0"- 0 . 0 8 0.12 0. !6 0 .20 0.2 L 0 . 2 8 0.3Z 0.36 0.,'- 0

FIN - W I D T H ( m }

Fig. 3. Variat ion o f total heat loss to the a tmosphere wi th fin width.

Heat losses from buildings 163

portions of the curves correspond to the case where 0r = 0A and there is no heat transfer from the fin end.

Figure 3 depicts the variations of the total rate of heat loss to the atmosphere through the finned wall with width of the fin. Curves I, II, III and IV correspond to the plane, the single, the double and the four-finned wall respectively for a fin length d = 0.01 m. An increase in the fin width results in a decrease in the rate of heat transfer through the plane area of the wall and an increase in the heat transfer through fin end. However, the surface area (excluding the bo t tom cross-sectional area) of the fin remains almost constant because the fin width is much smaller than the fin lateral dimensions and thus the heat transfer is also constant. The initial increase in heat transfer through the finned wall over the plane wall is due to additive effect of the heat transfer through the side faces of the fin. However, as the thickness increases further, the heat transfer through the side faces remains constant but there is a sharp decrease in the heat transfer through the fin-less port ion of the wall.

C O N C L U S I O N S

For cold climates, fancy structured walls (which are exposed to the atmosphere) are undesirable because they result in enhanced rates of heat losses.

R E F E R E N C E S

1. R. C. Sonderegger, Harmonic analysis of building thermal response applied to the optimal location of insulation within the walls, Energy Bld., 1 (1977), pp. 131-40.

2. M. S. Sodha, S. C. Kaushik, G. N. Tiwari, I. C. Goyal, M. A. S. Malik and A. K. Khatry, Optimum distribution of insulation outside and inside the roof, Bldg Envir. (1978), pp. 47-52.

3. M.S. Sodha, A. K. Seth and S. C. Kaushik, Periodic heat transfer in a hollow concrete slab: optimum placement of air gap, Appl. Energy, 6 (1979), pp. 113 25.

4. M. S. Sodha, A. K. Khatry and M. A. S. Malik, Reduction of heat flux by a water film over the roof, Solar Energy, 20 (1978).

5. C. H. M. Van Bavel and E. J. Sadler, Analysis of the film of Roof Solar Greenhouse, IIIrd Annual Conf. of Solar Energy for heating greenhouses and greenhouse/residence combinations, Fort Collins, Colorado, 1978, pp. 60 7.

164 Alok Thakur, J. K. Sharma, Ram Chandra

6. K.D. Mannan and L. S. Cheema, Year round studies on natural cooling and heating of greenhouses in Northern India, ISES Congress, Atlanta, USA, 1979, Abstract p. 191.

7. M.S. Sodha, Alok Srivastava, Ashvini Kumar and G. N. Tiwari, Heating and cooling of building by flow of water over the roof. Appl. Energy, 7 (1980), pp. 229~,2.

8. M. Jakob, Heat Transfer, John Wiley & Sons, Inc., N.Y., 1962.

Alok Thakur, J. K. Sharma and Ram Chandra Centre o f Energy Studies, Indian Institute o f Technology, Hauz Khas, New Delhi-- l lO 016 (India)