Periodic heat flux through extended structures and projections of buildings

Building and Environment, Vol. 20, No. 1, pp. 43~6, 1985, Printed in Great Britain.

0360-1323/85 $3.00 +0.00 © 1985 Pergamon Press Ltd.

Periodic Heat Flux Through Extended Structures and Projections of Buildings A. K. S. T H A K U R * An explicit expression for periodic heat flux through extended structures and projections has been obtained in terms of Fourier series. Numerical results show that it is imperative to include hourly variation of solair and ambient temperatures for realistic thermal load calculation of buildinos with projections. It is interesting to note that projections may enhance or reduce thermalfluctuations in the incomin# heat flux depending upon their transverse dimensions.

NOMENCLATURE

which are situated within sub-Saharan belt and experience one of the hottest climates. These mud houses provide a quite comfortable environment even on the hottest day. Modern architecture is extremely complex involving a large number of extended structures and projections (columns and girders) which are necessary to support massive structures like cold-storage warehouses, commercial buildings, etc. Heat transfer through extended surfaces has been studied in the past [1-3]. However, these studies were restricted to steady-state environmental conditions which deviate significantly from the actual conditions met in the thermal analysis of buildings since one or more faces of a building are exposed to the sun and thus subjected to variable solar insolation. Hence, the earlier analyses, based on steady-state conditions, are not valid in this context and need modification. In this paper the effects of inside extended structures on periodic heat flux entering the building have been considered. To maintain the generality of the analysis we assumed that the base (roof in the present case) as well as the tip of the structures are subjected to periodic temperature conditions, i.e. variable solar insolation and variable ambient temperature respectively.

ao am bo bm A C Cp d h

mean value of solair temperature (°C) = amexp(-ia,m) mean value of ambient temperature (°C) = b'exp(--iObm) cross-section area of projection (m 2) perimeter of projection (m) specific heat of projection material (kJ kg- 1 K- 1) projection length (m) heat transfer coefficient for projection surface (W m -2 K -a) ha heat transfer coefficient for base, x + = 0 surface ( W m - 2 K -a ) h 2 heat transfer coefficient for base, x ÷ = I surface (W m -2 K - l) ht heat transfer coefficient for projection tip (W m -2 K -a)

i,/:i

k 1 m AR

S(t) t T T~ T^ e' e 00

thermal conductivity of projection (W m - a K-a) base (roof) thickness (m) harmonic number difference between longwave radiations incident on the surface from sky and surroun.dings, and the radiations emitted by a blackbody at atmospheric temperature T~, (W m- 2) solar insolation (Wm -2) time coordinate (h) temperature distribution inside the base (°C) solair temperature (°C) ambient temperature (°C) absorptivity of the sunlit surface longwave emittance of the sunlit surface average excess temperature (°C) 2n/24, period for daily variation.

THEORY Consider a straight projection of uniform cross sectional area A at the inner surface of the roof(Fig. 1). For a more general approach outer as well as inner surfaces of the roof are assumed to have periodic temperature conditions, i.e. solair and ambient temperatures respectively. If p and Cp are the density and specific heat of the projection material then the thermal energy stored in it would be pCp ~O/~t; 0 being the difference between temperatures of projections and surroundings. Thus, the differential equation governing heat flow in the projection may be expressed as

INTRODUCTION S E V E R A L methods are being evolved for thermal modelling of residential and commercial buildings for conservation of non-renewable energy and best utilization of renewable energy : solar energy. The concept of passive solar heating/cooling is surprisingly not new but inherited from earlier times when due consideration of prevailing climatic and environmental conditions was given in constructing houses from a combination of mud and wood. Such houses are still being built in rural areas of the northern part of Nigeria especially in K a n o and Sokoto

~20 Ox 2

pCp O0 k

Ot

hC kA

0= 0

(1)

where excess temperature 0 is expressed in terms of Fourier series o0 0 = 0 o + Real ~ 0 m exp (im~ot). (2)

* Department of Physics, Indian Institute of Technology, Hauz Khas, New Delhi 110016, India.

m=l

43

A. K. S. Thakur

44

where += •

32 =--'hC kA'

x*=[

rA

f12 = (~2+ im~opCp k

and arbitrary constants A', B', C' and D' are evaluated on the basis of heat flux passing through the base of the projection of thickness l and appropriate boundary conditions. The temperature distribution inside the base is expressed by the following differential equation

r~

x=d

(~2T k s~T2

Fig. 1. Schematics of the analysis.

8T pCp ~ -

=

(9)

satisfying the boundary conditions In writing (1) we assume that the projection and its base are of the same materials and that their physical properties do not exhibit spatial or temperature variation. If the temperature at the tip of projection is higher than its surroundings then the most appropriate boundary conditions satisfying (1) are {o}x=o = Tx+ = , - T,,

- k ~c~T~

(o~+L+=o

= h,{T~- T~+ =o}

(10)

and

-- [~x+jx+= =h2{Tx+=,-TA}

(11)

(3)

where

and

h2=l

(4)

(Ox;~=.

~-1 + ~ + ~ - t -

(lla)

and Ts is the solair temperature given by where ambient temperature is Fourier analysed and expressed as

T~ = let'S(t) + h, T'A -- e" AR]/h,

(12)

which is Fourier analysed over a period of 24 h, thus TA =

b o + Real ~ b,, exp (ime)t).

(5) T~ = ao+Real ~ am exp (ima~t).

m=l

(13)

m--1

Substituting 0 from (2) in (1) and separating the timeindependent and time-dependent terms, we have

hC 0o = 0 kA

0200

~x 2

T-

(6)

Only six harmonics are sufficient for reasonably good convergence. Equation (9) is analysed for temperature distribution inside the base by employing boundary conditions, (10) and (11),

{hx(k+h21)ao+h2kbo} + (bo-ao)hlh2 x+ {h,(k+hfl)+h2k} {h2k+h,(k+h21)} °0[{ hla"(ka+h2) exp(ctl)+h2br"(kct-hl) +Real ~__1 ( k c t + h , ) ( ~ i ~ ~ - - h ~ 2 ) e x p ( - M )

}

h2bm(kct+hO+hlam(ka-h2) exp (-at) } x exp ( -- ax +) + [(ka + hlXke + h2) exp (~l) + (hi - kct)(k~t- h2) exp ( - ct/) (14)

x exp (ctx+)] exp (imt~t)

with

and

8x 2 -

+~

0m=0.

(7)

Solutions of (6) and (7) are straightforward and thus the general solution of (1) is given by 0 = A' exp ( - f i x ) + B' exp (fix)

~2 -- im~pCp k The total amount of incoming heat flux through the projection surfaces and the tip may be assumed to be the heat conducted into the projection through its base, thus

+Real ~ {C' exp (-fix) m

1

+ D' exp (fix)} exp (imcot)

(8)

Q = -

(~x ;x=o

Periodic Heat Flux Through Extended Structures and Projections of Buildings

45

200

1 127

15o

CJ

IOO

137

i

117

50

107 -

2 4

8

12

16

20

Fig. 2. Hourly variation of temperature : (1) solair; (2) ambient.

or

Q = - kA{(B'-- A')5

~ (D'-C')exp (imcot)}.

(16)

+Real.=1

RESULTS AND DISCUSSION To gain a numerical appreciation of the analysis, we choose the following parameters relevant to some typical concrete roof and projection -1,

Cp=0.84kJkg-lK

8

24

t(h)

k=l.55Wm-lK

4

12 T(h)

16

20

24

Fig. 3. Hourly variation of incoming heat flux through projection surfaces with roof thickness for A = 0.9 m 2, d = 0.6 m, C = 6.6m : (1) l = 0.3; (2) l = 0.4; (3) l = 0.5 m. through its surfaces is shown in Fig. 4. It is obvious that heat flux increases with projection perimeter. Further, the difference between the maxima and minima of heat flux increases with increasing perimeter. It may be noted that the heat flux attains a maximum for very small projection lengths (x < 8 cm) [31 and thus the heat flux variation with projection length has no practical importance because structural columns and girders normally have lengths greater than 8 cm.

145 I-

-1,

p = 2200 kg m - 3

Further, the solair and ambient temperatures over a period of 24 h for some typical hot day (Fig. 2) have been Fourier analysed and presented in Tables 1 and 2. Figure 3 illustrates the daily variation of incoming heat flux through surfaces of the projection with the thickness of the base (roof). It is evident from various curves that the incoming heat flux decreases sharply with increasing base thickness. Further, the difference between maxima and minima of incoming heat flux reduces with base thickness and approaches a constant after certain value of base thickness; in the present case this thickness corresponds to 40cm. The effect of projection perimeter on incoming heat flux

95

I

I

I

I

I

I

4

8

12

16

2O

24

T(h)

Fig. 4. Hourly variation of incoming heat flux through projection surfaces with perimeter for d = 0.6 m, l = 0.3 m : (1) C = 6.6 m ; (2) C = 6.4 m; (3) C = 6.2 m.

Table 1. Fourier representation of solair temperature m a~, (°C) gain (rad.)

0

1

2

3

4

5

6

86.9096 --

80.9464 3.6968

31.7143 1.0511

11.7172 0.8039

5.4195 4.8155

2.7488 0.6866

3.0279 4.0130

Table 2. Fourier representation of ambient temperature m bm (°C) (rbm(rad.)

0

1

2

3

4

5

6

38.0250 --

6.5270 4.2962

1.4076 1.7424

0.8864 1.8559

0.1897 6.7313

0.3365 5.7927

0.2424 5.1903

46

A. K, S. Thakur

480 \\\

////

7

\\\

increases with the projection perimeter. It is interesting to note that additional thermal fluctuations in the incoming heat flux introduced by thin projections slowly dies out as projection perimeter increases. A further increase in perimeter results in a gradual decrease in the difference between the maximum and minimum of total incoming heat flux and thus thick projections help in thermal load levelling of the building. Straight lines depict steady-state heat flux entering the building.

\\\

////

CONCLUSION 80

I 4

I 8

12

16

I 20

I 24

T(h}

Fig. 5. Hourly variation of total incoming heat flux entering the buildingwith perimeter for d = 0.6 m, l = 0.3 m : (1) plane surface; (2) C = 6.2 m ;(3) C = 6.4 m ; (4) C = 6.6 m ;(5-8) steady-state heat flux corresponding to curves 1-4 respectively.

Hourly variation of total incoming heat flux is shown in Fig. 5. An increase in the perimeter results in an increase in the heat flux through the projection surfaces but reduces the rate of heat transfer through the plane area of the roof. However, the decrease in the heat flux through the plane surface is smaller than the increase in flux through the projection surfaces and thus effectively total heat flux

The above results show that it is imperative to include hourly variation of solair and ambient temperature for realistic thermal analysis of buildings with projections. In practice the incoming heat flux is independent of projection lengths greater than 8 cm but depends significantly upon perimeter. Projections may reduce or enhance thermal load levelling in buildings depending upon their transverse dimensions. The present study seems to have applications in more complex extended structures especially in steel framed buildings and cold storage warehouses where a steady internal temperature is required in spite of severe variations in atmospheric temperature. Acknowledgements--The author is grateful to Drs R. Raman, A. Dang and A. Sharma and Mr. A. Mehta of IIT Delhi, India, for their assistance in the course of deriving the numerical results.

REFERENCES 1. C.F. Kayan and R. G. Gates, Temperature distribution in fins and other projections, including those of building structures by several procedures. Trans. Am. Soc. mech. Enors 80, 1599-1608 (1958). 2. M. Jakob, Heat Transfer. John Wiley, New York (1962). 3. A. Thakur, J. K. Sharma and R. Chandra, Effect of fancy structured wall or roof on heat losses from buildings. Appl. Eneroy 13, 157-164 (1983).