IL NUOVO CIMENTO

VOL. 18 D, N. 1

Gennaio 1996

Geometrical characteristic and heat transfer in buildings' protruding structure (*) U. LUCIA Dipartimento di Energetica - 1-50139 Firenze, Italy

(ricevuto il 27 Marzo 1995; revisionato 1'8 Agosto 1995; approvato il 26 Settembre 1995)

Summary. - - A non-equilibrium thermodynamical and geometrical analysis of the buildings' protruding structures is developed. The geometrical conditions for the buildings' protruding structures to obtain a decreasing in the buildings lost heat are obtained. PACS 44.10 - Heat conduction (models, phenomenological description). PACS 44.90 - Other topics in heat transfer, thermal and thermodynamic processes.

1. - I n t r o d u c t i o n

The energy needs in the industrialized countries have been demonstrated to increase continuously [1,2]. Cogeneration, energy saving and rationality represent some answers to the energy problem [2], for their ambient relapse too. Solar Architecture represents a contribution to the development of ( [3, 4] and, during the last fifteen years, the scientific and technical literature on the subject of passive solar architecture has been constantly growing [3-7]. The interest in more thermal comfort buildings is always growing too [8, 9], and this subject is one of the research developments both about a better use of the energy and about the more <,ecological buildings,, for the man [9]. Following the works about solar energy[3-5], here we want to analyse the non-radiative heat transfer in buildings, examining the building structures in their geometrical and thermal properties, with regards to the buildings' protruding structures as balconies, cornices, etc. 2. - H e a t t r a n s f e r in buildings' p r o t r u d i n g s t r u c t u r e s

The buildings' protruding structures are solids with small transversal section which protrudes from the building, with temperature T and are in a fluid, the air, with different temperature Ta (fig. 1). (*) The author of this paper has agreed to not receive the proofs for correction.

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Fig. 1. - a) Buildings; b) buildings representation,

Here we want to obtain t h e distribution function of the temperature T ( x ) in the buildings' protruding structure. To do that we consider that there exists a difference of temperature between the air and the building, and this temperature gradient defines the direction n of the heat flux. So we define the origin of the geometrical frame of reference at the base of the building's protruding structure and, because of the geometrical symmetry of this structure, in the middle of its high, as in fig. 2. Using the definition of the position r = r n , where r is the distance between the position considered and the origin of the geometrical frame of reference, we can use a scalar analysis, in r variable, for this problem. Now, after having defined the geometrical frame of reference, we must examine ,

Tb

0

Fig. 2. - Builfling's protruding structure.

//

/

D

43

GEOMETRICAL CHARACTERISTIC AND HEAT TRANSFER ETC.

Tb T~

0

Fig. 3. - Considered building's protruding structure. the power sources; considering an element of volume of the buildings' protruding structure as in fig. 3, they are 1) The incoming conduction differential thermal power in the r position dQ~o.dr, given by the Fourier law[9-15]: dT dQco.dr = - k d A - - , dr

(1)

where k is the thermal conductivity and d A is the area of the buildings' protruding structure transversal elementary section. 2) The outgoing conduction differential thermal power in r + dr position, dQeondr + dr :

(2)

dQco,dr +dr =

[

dT] d (_kdA dT)dr. -kdA -~r r § =_kdA dT d--~ + d--~ d--~

3) The outgoing convection differential thermal power through the surface between the positions r and r + dr, dQconv, given by the Newton law[9-11,16,17]

(3)

dQeonv =

hdp(T -

Ta)dr,

where h is the convective thermal coefficient between the buildings' protruding structure and the fluid, dp is the perimeter of the buildings' protruding structure transversal elementary section. From (1)-(3) we can obtain the power balance equation (4)

-kdA dT--dr=-kdA

+--dr dT --drd ( -kdA dT)dr d r + hdp(t - Ta)dr

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from which we can obtain the first-order linear differential equation (5)

d2T(r) - ~ 2 [ T ( r ) - Ta],

dr 2 where we have defined the constant ~ = V ~ / k A , considering that d p / d A = p / A , because of geometrical similarity. Referring to fig. 3 the boundary conditions for eq. (5) are

{T(_ r0)i Tb,

(6)

dT

_

r=R

h [T(R) - Ta] k

where Tb is the building temperature, and R is the boundary surface coordinate in the n direction. Now, solving eq. (5)[18] and considering its boundary conditions (6), we can obtain (7)

T(r) = Ta - (Tb - T~)

cosh [~(R - r)]

1 + (hiCk) tgh [$(R - r)]

cosh (~R)

1 + (h/~k) tgh (~R)

This is the distribution function for the temperature in the buildings' protruding structure. From eq. (7) and applying the Fourier equation (1), we can obtain the lost heat per unit of time: (8)

Q~o~t = - k A

dT dr

I

= ~ ( T b r= o

- T~)

tgh (~R) + h / $ k

1 + (h/~k) t g h ( ~ R )

3. - Physical interpretation of the mathematical results First we must notice that the lost heat is negligible if the buildings' protruding structure end is thermal isolated or if it receives conduction heat. If this is the case, then h is zero[10,11,13,16,17] and we can obtain that the distribution function becomes (9)

T(r) = T~ - (TB - T~)

cosh [~(R - r)] cosh (~R)

and the lost heat

(10)

Qlost -- h ~ / - h ~ (Tb - T~) t g h (~R).

Moreover the convective heat transfer cannot establish if T ( R ) = T~. In the end if the convective heat establishes, we can write (8) as (11)

h (~k/h) tgh (~R) + 1 Q~o~t= $Ak(Tb - Ta) ~-~ 1 + ( h / ~ k ) t g h ( ~ R ) = Q b r ,

GEOMETRICALCHARACTERISTICAND HEAT TRANSFERETC.

F 1.7 1.5 1.3 1.1 i 0.9 00m e_' = = 0.7 ~m= 0.5 0.01

45 m=l

r ,

d

9

.

,.....

, ....

F -'-'-/

y

. ......

m=lO.

.0 - - ~. -. - 0 2.01

4.01

.0

..

r .

" ....

A '

6.01

.

.

.

. -

m=O.1

1 8.01

10.01

~k/h Fig. 4. - Numerical results in 0-10 ~k/h range with 0.001-1000 m (= hL/k) range. where

Qb = hA(Tb - Ta),

(12) (13)

F-

Q~o~t _ (~k/h)tgh(~R) + 1 Qb 1 + (h/~k) tgh (~R)

Now we can notice that Qb is the heat lost by the building without considering the effect of the buildings' protruding structure: this effect is expressed by the term F. So we can identify three different cases: 1) ~ = h/k, then p/A = h/k and F = 1: the buildings' protruding structures have no effect on the building lost heat; 2) ~ > h/k, then p/A > h/k and F > 1: the buildings' protruding structures have the effect of increasing the building lost heat; 3) ~ < h/k, then p/A < h/k and F < 1: the buildings' protruding structures have the effect of decreasing the building lost heat. The last case is the better condition for the buildings; in fact the lost heat decreases because of the presence of the buildings' protruding structures. In this condition the building releases less heat to ambient fluid if Tb > T~ (in cold months), and absorbs less heat ff Tb < T~ (in hot months) (fig. 4). This is an important result, obtained by a thermodynamical and geometrical analysis of the building lost heat, because it represents a criterion of buildings designing to obtain good thermal comfort conditions. 4. - C o n c l u s i o n s

We have analysed the non-radiative buildings heat transfer by a non-equililibrium thermodynamics and geometrical method of investigation. We have obtained the geometrical conditions for the buildings' protruding st~actures to get the decreasing in the building lost heat and, as a consequence, good thermal comfort conditions.

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REFERENCES [1] PEDROCCHI E., Previsioni di fabbisogno energetico per l'Italia (La Termodinamica) 1993. [2] CARNEVALEE. and DE LUCIA M. and LUCIA U., Scenari energetici ed economici per la cogenerazione nelle cartiere, A.T.I., Gruppi Combinati e Cogenerativi, Atti del VII Convegno Nazionale, 20-21 Ottobre 1993. [3] DUFFIE J./~ and BECKMANW. A., Solar Engineering of Thermal Processes (John Wiley & Sons, New York) 1980. [4] MASSOBRIO/~ and SERTORIO L., Nuovo Cimento B, 106 (1991) 69. [5] MnSSOBRIOA. and SERTORI0 L., Nuovo Cimento C, 15 (1992) 317. [6] SWINBANKW. C., Q. J. R. Meteorol. Soc., 89 (1963) 339. [7] BLISS R. W., Solar Energy, 5 (1961) 103. [8] FANGER P. 0., Thermal Comfort (McGraw-Hill Book Co., New York) 1973. [9] COCCHIA., Elementi di Termofisica (Esculapio, Bologna) 1990. [10] ECZ_ERTE. R. G. and DRAKER. M., Analysis of Heat and Mass Transfer (McGraw-Hill Book Company, New York) 1972. [11] KESTIN J., A Course in Thermodynamics, Vol. I-II (Hemisphere Publishing Corporation, McGraw-Hill Book Company, New York) 1979. [12] (~ZISIKM. N., Heat Conduction (John Wiley & Sons, New York) 1980. [13] ()ZISlZ M. N., Basic Heat Transfer (McGraw-Hill Book Company, New York) 1977. [14] KREITH F., Principi di trasmissione del calore (Liguori Editore, Napoli) 1974. [15] GYARMATII., Non-Equilibrium Thermodynamics (Springer-Verlag, Berlin) 1970. [16] BEJAN A., Advanced Engineering Thermodynamics (John Wiley & Sons, New York) 1988. [17] BE JAN A., Entropy Generation through Heat and Fluid Flow (John Wiley & Sons, New York) 1982. [18] BENDER C. M. and ORSZAGS. A., Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill International Editions, New York) 1987.