- Email: [email protected]
Periodic heat transfer through a hollow concrete slab: Optimum placement of the air gap
AppliedEm'rg)6 (1980) I 13-23
PERIODIC HEAT TRANSFER THROUGH A HOLLOW CONCRETE SLAB: OPTIMUM PLACEMENT OF THE AIR GAP
M. S. SODHA, A. K. SETH and S. C. KAUSHIK
Centre oJ Energy Studies, Indian Institute oj Technology, New Detffi, 110029 (India)
SUMMARY
This paper describes an investigation oJ the periodic heat trans]er and optimum placement o] the air gap in a hollow concrete slab subjected to solar radiation and atmospheric air on one side and in contact with air at a fixed temperature on the other. The heat conduction equation has been solved using the appropriate boundaJ3' conditions at the interlaces. It isJound that the presence o/an air gap eonsiderabO rethu'es the heat flux through the slab,jurther,jor a given total thickness oJconcrete the best load levelling is achieved when the thickness o/the outer layer of the eoncrete is h,ast, consistent with structural considerations.
NOMENCLATURE
a
b C
C1 ('2
hi h.
KI K2
Q
AR
s(t)
Thickness of the upper wall (in). Thickness of the lower wall (in). Thermal conductance of the air gap (BTU/ft 2 h °F). Specific heat of material of the upper wall (BTU/lb °F). Specific heat of material of the lower wall (BTU/Ib F ) . Inside heat transfer coefficient (BTU/ft 2 h °F). Outside heat transfer coefficient (BTU/ft 2 h °F). Thermal conductivity of material of the upper wall (BTU/ft h °F). Thermal conductivity of material of the lower wall (BTU/ft h °F). Heat flux entering system (BTU/ft 2 h). Difference between the long-wave radiation, incident on the surface from the sky and surroundings, and the radiation emitted by a black body at atmospheric temperature (BTU/ft 2 h). Intensity of solar radiation (BTU/ft z h). 113
Apph'ed Energy 0306-2619/80/0006-0113/$02"25 :(( Applied Science Publishers Ltd, England, 1980 Printed in Great Britain
114 T t x, y
M. S. SODHA, A. K. SETH, S. C. KAUSHIK
Temperature distribution in the upper wall (°F). Time (h). Axes normal to the walls.
Greek symbols ~0
Pl P2 0A
Os (D g
Absorptivity of the material of the upper wall. Density of material of the upper wall (lb/ft3). Density of material of the lower wall (lb/ft3). Atmospheric air temperature (°F). Solair temperature (°F). 27t/period (h- ~). Long-wave emissivity of the material of the upper wall. INTRODUCTION
The capacity of an air-conditioning plant--and hence its capital cost---is determined by the maximum and minimum heat flux into the building: hence load levelling (reduction of peak and enhancement of the valley) should be one of the goals of the thermal design of a building. Since hollow concrete slabs are favourite building materials for buildings of low height in harsh climates, this paper is devoted to a study of periodic heat fluxes through such slabs. Although tables z for conductance of hollow concrete slabs have been compiled and the thermal performance of hollow concrete blocks by an electrical analogue method has been studied by Rao and Chandra, 2 optimum placement of the air gap for best load levelling has not been considered. This paper presents a straightforward analysis for the periodic heat transfer through, and optimum placement of, an air gap in a stratified hollow concrete slab subjected to periodic solar radiation and atmospheric air on one side and in contact with room air at constant temperature on the other. The solair temperature has been expanded as a Fourier series in time and the onedimensional heat conduction equation has been solved under appropriate boundary conditions at the two interfaces. Numerical calculations for the heat flux through a horizontal slab (on the roof) using the hourly data of solar radiation and atmospheric temperature for 21 July, 1975 in Kuwait have been presented. It is found that the presence of an air gap (even ] in) considerably reduces the heat flux through the slab compared with a solid slab (zero air gap). It is further seen that for a given total thickness of concrete the best load levelling is achieved when the thickness of the outer layer of the concrete is least, consistent with structural considerations. ANALYSIS
Figure 1 illustrates the geometry of the problem. The equations governing the temperature distributions in the two walls are:
PERIODIC HEAT TRANSFER THROUGH A HOLLOW CONCRETE SLAB
1 15
SOLAR RADIATIONS(t)ANDATMOSPHERIC AIR AT TEMPERATUREOA(t) ×--o
\ \ \ N cO,,NC,
x:a
AIR GAP, THERMAL CONDUCTANCEC
ROOM AIR AT CONSTANT TEMPERATURE TR Fig. 1. Hollow concrete slab.
gO
K 1 ~20
(t
t)1C1 aA2
0 < x < a
(1)
and gT
K 2 ~2T
Pt
p2(~2 c y -
.)
{'3
Let us assume the periodic solutions as: J
0 = 0o(x) + 2
0,,(x) exp ( i r a . t )
(3}
T,,(y) exp (im,)t)
(4)
n = I
and: J
T = T.(y) + E n = ]
where n is an integer. The a p p r o p r i a t e b o u n d a r y c o n d i t i o n s are:
K~ -
= ~oS(t) + h,,(O.4It) - 0(x = 01 - c A R (Lv),,=o
(5(a))
= h,,(Os - O(x = 0)) where: 0s(t ) = OA(t ) +
:%S(t)
tt,,
--
cAR It,,
(6(a))
i s , c o m m o n l y k n o w n as the solair t e m p e r a t u r e which in general can be e x p a n d e d a s
116
M. S, SODHA, A. K. SETH, S. C. KAUSHIK
0s(t) = a o + ~ ?1 =
a,,exp(ino)t)
(6(b))
|
where: a,, = A,, exp ( - i~b,,) Further, we have:
_ K1 90.,
= _ K2 cT_ =,,
~T
-K2~r=
(5(b))
9) r=o
= C(O(x = a) - TO'= 0))
o
(5(c))
dT] - K2~fy ,.=b = hi(T()'= b) - Te,)
(5(d))
C is defined in accordance with the A S H R A E H a n d b o o k . Equation (5(d)) also determines the heat flux, Q. per unit area which is transferred to the r o o m air at temperature, T e. Substituting for 0 and T from eqns (3) and (4) in eqns (1) and (2), the solutions for the time independent and time dependent parts of 0 and T are, respectively, given by: 0o = A o + A l x
(7(a))
T o = B o + BLy
(7(b))
O.(x) = 2° exp ( + ict,,x) + "~'nexp ( - i%x)
(7(c))
T.(y) = / a . exp( + i/~.y) +/~'. exp( - i[3,,y)
(7(d))
where: ct.=\-
nu)p i C l ) J/2 ~i
_
(.~op2c2) '/2
ft.=\-
2 ~ z ,/
(l-i)
(8(a))
(1 - i )
(8(b))
The real constants (A o, A 1, Bo and B ~) and the complex constants (£.,).~, #. and/~'.) are determined from the appropriate boundary conditions given by eqns (5); hence:
E+o ~C
- Kl
0
- Ca
+ C
0
+h i
+ K2 -
-
K2
( K 2 + bhi)
J
i olioiol Al = Bo
BI
iTR
(9(a))
PERIODIC HEAT TRANSFER THROUGH A HOLLOW C()NCRETE SLAB
117
and :
h; - iK'otn ho + iKl~,, 0 %,exp(+i%a) -Kla. exp(-i~.a ) - K2fi,, C e x p ( +icq,a) -Cexp(-&.a) C - iKefi , 0 0 (ki + iK2fl,)exp(ifl,,h)
IKl
+ K2fl.
)"n
C + iK2fi,,
t~n
(h~ - iK2fl.)exp{-ifl,,b)
t~'.
=
(9(b)) Thus we can write for the t e m p e r a t u r e dislributions in the walls:
O=Ao+Alx+Re[
E{2nexp(+iGlx)+)/,exp(-i~,,x)]exp(in~ot);
lO(a))
n=l
T=Bo+B~y+Re[
Ell4, exp(+i/~,,y)+tJ'~expl-iflny)',exp(incot) 1 n-
lO(b))
1
where Re denotes the real part of the quantity. Substituting: .;-,1 = 2,,,, exp ( + iqS., )
1 lta))
i(o;)
1 l(b))
2'. = 2'no exp( +
/Jn = tq., exp( + i G )
( 11 (c))
#'. = U'ooexp( + ia~,)
(1 l(d))
in eqns (I0) one obtains: 0 = Ao + A lx + ~
[':'nOe x p ( +Hl/2~oX)COS (tl(ot + tll,/2~o X + ~),,)
n=l
+ Go exp ( -
nl/2%x) cos (ne)t
- n ' - % x + ~b;,)] (12(a))
and:
T = Bo + B l ) ' +
~ rt=l
[#no exp(
+nl/2[3o)')cos(noJt
+ nl/2[~oy-k ty,,)
+ fnoexp(--nl/2floy)Cos(n(ot--n
1,2
rio)'+
a'.)]
(12(b))
118
M. S. SODHA, A. K. SETH, S. C. KAUSHIK
where: ( 0 1 ( D C I ~ 1/2
~o
=
(p2(D(~T2~ 1/2
\ 2K, J
and
[7o =
The heat flux entering the room is given by: Q(t) = hi[B 0 + Bib - TR+
Z
.Ipnoexp(+n~"2[Job)cos(ne)t + nl"2//ob + ~,)
n--l
+ ff~°exp(-nl"2]t°b)c°s(n~°t - nu2~°b + ~")}1
(13)
NUMERICAL RESULTS AND DISCUSSION
To obtain a numerical appreciation of the results we have computed the amount of heat flux through a horizontal slab (on the roof) as a function of time using the hourly data of solar radiation and atmospheric temperature for a typical hot summer day, i.e. 21 July, 1975, in Kuwait. The Fourier analysis for the solair temperature for 21 July, 1975 for concrete is given in Table 1. It is evident that the first six harmonics are enough to obtain good convergence in most of the calculations. The following parameters have been used for the calculations: TABLE 1 FOURIER ANALYSISOF PERIODIC SOLAIR TEMPERATURE FOR A SUMMER DAY 21 JULY, 1975 IN KUWAIT n A n (°F) ~0,, R a d i a n
1
2
3
4
5
6
7
113.1990
34.5608
8.0878
2.7436
1.4208
0.9746
0.3813
3.9810
1-1082
1-5828
6.1722
6.2306
5.709
--
TR = 70°F ho = 6"0 BTU/ft z h °F
(wind velocity = 15 mph)
h i = 1.2 BTU/ft z h °F For concrete ~o = 0"9 ~:= 1"0 p = 1161b/ft 3 C = 0-156 BTU/Ib °F K = 5-0/12 BTU/ft h °F AR = 20 BTU/h ft 2 L = a + b = 2 i n , 4 in o r 6
in
PERIODI( HEAT TRANSFER ]HROUGH
A HOLLOW ('ON('RE]E
119
SLAB
The values of the air gap conductance for different sizes of air gap are given in Table 2. Figure 2 illustrates the effect of varying the size of the air gap on the periodic heat flux. It is seen that the variation of the air gap from 3in to 4in does not have any significant effect on the heat flux. However, the introduction ()fan air gap (even ~ in) does lead to a considerably smaller heat flux than in thecase of the solid slab (zero air gap). Figure 3 shows the variation of the Q ..... and Q,m,, with the total thickness (L) of the concrete for hollow (a = b) and solid slabs. It is seen that the O .... decreases with the total thickness more rapidly for a solid slab than for a hollow slab of the same concrete thickness; on the other hand, OmJ,, is almost independent of the total thickness of the concrete. Figures 4(a) and (b) show the daily variation of the heat flux into the room lor different values of a,L corresponding to a 1.5 in wide air gap and L = 4 in and 6 in. T A BI_.E 2 AIR (IAP (ONDt (JAN( 1 IOR HORIZfIN]AL STRIA( I AND I)O%ND,'ARI) HIAI t L()\x, , (SU MMIR) 7 h i( ,tn(,.~..+
(,
(in)
( B I L,It } h t . )
0.75 1-5 4-0
1-07
1-18 l-Ol
o=b :3" "---o-'--o-- Solm r temperoture - ...... Solid slab
3E--
I
+
=
Air gap : . 7 5
:~.o
?
/ /
160
/ / \ ',,, /%
/
,
\\
/ ,,' j / - % \ , \ ' %
;. 100
6t 0
i 3
"-""5 6
1 9
1 12 I (HOURS) - - - ~
I 15
I 18
I 21
BO 24
Fig. 2. Dependence of periodic heat flux through hollow concrete slab on air gap; 1,11 a n d 111 refer to air gap thicknesses of 0.75 in, 1.5 in and 4 in, respectively; the dotted curve refers to single slab L = a + b = 6in; the solair temperature is denoted by - - O - - O O--.
120
M. S. SODHA, A. K. SETH, S. C. KAUSH1K 58
54
4e
A
mox (solid stab)
-o
E~
.£ E
Qrnox(hO|lOW slob)
~
~min(sotid ~
l
4
~
I
stab)
Qmin[holtow slob)
8
L (inches)
Fig. 3.
Variation of Q,,a~ and Q,,,. with total thickness of concrete for hollow (a = b) and solid slabs.
respectively. The curve with circles along it shows hourly variation of the solair temperature (on the right-hand-side scale). It is seen that there is a finite phase shift (between the maxima of the incoming heat flux and incident solair temperature) of the order of 2-4 h, depending on the total thickness of the concrete. One can also observe that the phase shift of the maximum solair temperature is larger in comparison with that of the minimum heat flux with respect to the minimum solair temperature, the difference being of the order of 1 h, Figures 5(a) and (b) show the variation of the maximum and minimum heat fluxes with the thickness distribution of concrete outside the air gap, riz. alL for L = 4 in and 6 in, respectively. It is seen that for a given total thickness of concrete the greatest load levelling (i.e. highest value o f Qmin and lowest value of Qmax) is obtained when a/L -~ O, i.e. as the thickness of the outside layer of concrete tends to zero: this is of course not possible and one has to be satisfied with the least value, consistent with structural constraints.
180 I olL:O ]I Ig
:.2S : -SO
I£
: "75
2
:1.0
/UR GAP=I.5~ L=4" . 160
Solakr t emoeratu
?
T 120
.0
100
'[
I
0
3
--
~
~
t (HOURS)
Fig. 4(a).
I
'~
I
I~
i ,8
,s
1 2,
~
180
'~
Hourly variation of periodic heat flux through hollow concrete slab corresponding to different values of a/L (with air gap = 1.5 in and L = 4 in).
x o/Leo A,R GAP=I.S~°6 2,
1T g[ l~ ~"
20
=.166 :.500 = .833 = 1.000
So,air temperature 7
ZVl v y / ~
J/-H
- 160 150
18 ,,~ 16 120
•o
12
110
IQ
100
Go
4
B
6;
3
6
9
I 12
I 15
# 1B
l 21
I 0 2/.
t(HOURS)
Fig. 4(b).
P e r i o d i c h e a t flux t h r o u g h h o l l o w c o n c r e t e s l a b c o r r e s p o n d i n g to different values a i r g a p = 1.5in a n d L = 6 i n ) .
ofa/L (with
"d
-El
m
2
E
S
6.5
7S
23
2s
27
Fig. 5(a).
({1}
f
"
I .75
~
"
I 100
=1.5"
AIRGAP :.7 5~.
4.0"
1.S"
Qmaxand Q=~. on a/L for L = 4in.
I .S 0
o/,
I
.2 S
Dependence of
L=M'
=.
"0
¢:
~o
"Z
E
x o
.¢ t~
I .333
L=6
#
I .500
I -666
I -833
l 1.00
:4.0"
: 1.5"
AIR GAP-0.7S"
40
=1.5"
Dependence of Qm=~ and Qm,~ on a/L for L = 6in.
I .166
Fig. 5(b).
6.61 0
2~
(b)
>
3:
>
3: >
©
t~
PERIODIC HEAT TRANSFER THROUGH A HOLLOW CONCRETE SLAB
123
A l t h o u g h we have m a d e these c a l c u l a t i o n s for a h o r i z o n t a l r o o f on a p a r t i c u l a r d a y in K u w a i t we expect the results to be valid in general. Extensive c a l c u l a t i o n s have revealed that the m a x i m u m a n d m i n i m u m values o f the heat flux are only slightly d e p e n d e n t on the m a g n i t u d e a n d phase o f the second h a r m o n i c o f solair t e m p e r a t u r e . Hence the present c o n c l u s i o n s a b o u t o p t i m a l c o n d i t i o n s for best load levelling, based on a specific set o f m e t e o r o l o g i c a l p a r a m e t e r s , will be a p p l i c a b l e to a wide range o f climatic c o n d i t i o n s .
CONCLUSIONS F o r a given total thickness o f concrete, best load levelling ( a p p r o p r i a t e to m i n i m u m c a p a c i t y o r cost o f the a i r - c o n d i t i o n i n g plant) is achieved when the thickness o f the outside layer has the least value, consistent with structural c o n s i d e r a t i o n s .
REFERENCES I. ANON., ASHRAE Handbook o.[ Fundamentals, American Society of Heating, Refrigeration and Air-
conditioning Engineers, Inc., New York, 1974. 2. K. P. RAOand PRAKASHCnANDRA,A study of thermal performance of concrete hollow blocks by an electric analogue method, BuiMing Science, 5 (1970), p. 31. 3. A. K. KHATP,Y, M. S. SODHAand M. A. S. MALIK,Periodic variation of ground temperature with depth and time, J. Solar Energy, 20 (1978), p. 425.
PERIODIC HEAT TRANSFER THROUGH A HOLLOW CONCRETE SLAB: OPTIMUM PLACEMENT OF THE AIR GAP
M. S. SODHA, A. K. SETH and S. C. KAUSHIK
Centre oJ Energy Studies, Indian Institute oj Technology, New Detffi, 110029 (India)
SUMMARY
This paper describes an investigation oJ the periodic heat trans]er and optimum placement o] the air gap in a hollow concrete slab subjected to solar radiation and atmospheric air on one side and in contact with air at a fixed temperature on the other. The heat conduction equation has been solved using the appropriate boundaJ3' conditions at the interlaces. It isJound that the presence o/an air gap eonsiderabO rethu'es the heat flux through the slab,jurther,jor a given total thickness oJconcrete the best load levelling is achieved when the thickness o/the outer layer of the eoncrete is h,ast, consistent with structural considerations.
NOMENCLATURE
a
b C
C1 ('2
hi h.
KI K2
Q
AR
s(t)
Thickness of the upper wall (in). Thickness of the lower wall (in). Thermal conductance of the air gap (BTU/ft 2 h °F). Specific heat of material of the upper wall (BTU/lb °F). Specific heat of material of the lower wall (BTU/Ib F ) . Inside heat transfer coefficient (BTU/ft 2 h °F). Outside heat transfer coefficient (BTU/ft 2 h °F). Thermal conductivity of material of the upper wall (BTU/ft h °F). Thermal conductivity of material of the lower wall (BTU/ft h °F). Heat flux entering system (BTU/ft 2 h). Difference between the long-wave radiation, incident on the surface from the sky and surroundings, and the radiation emitted by a black body at atmospheric temperature (BTU/ft 2 h). Intensity of solar radiation (BTU/ft z h). 113
Apph'ed Energy 0306-2619/80/0006-0113/$02"25 :(( Applied Science Publishers Ltd, England, 1980 Printed in Great Britain
114 T t x, y
M. S. SODHA, A. K. SETH, S. C. KAUSHIK
Temperature distribution in the upper wall (°F). Time (h). Axes normal to the walls.
Greek symbols ~0
Pl P2 0A
Os (D g
Absorptivity of the material of the upper wall. Density of material of the upper wall (lb/ft3). Density of material of the lower wall (lb/ft3). Atmospheric air temperature (°F). Solair temperature (°F). 27t/period (h- ~). Long-wave emissivity of the material of the upper wall. INTRODUCTION
The capacity of an air-conditioning plant--and hence its capital cost---is determined by the maximum and minimum heat flux into the building: hence load levelling (reduction of peak and enhancement of the valley) should be one of the goals of the thermal design of a building. Since hollow concrete slabs are favourite building materials for buildings of low height in harsh climates, this paper is devoted to a study of periodic heat fluxes through such slabs. Although tables z for conductance of hollow concrete slabs have been compiled and the thermal performance of hollow concrete blocks by an electrical analogue method has been studied by Rao and Chandra, 2 optimum placement of the air gap for best load levelling has not been considered. This paper presents a straightforward analysis for the periodic heat transfer through, and optimum placement of, an air gap in a stratified hollow concrete slab subjected to periodic solar radiation and atmospheric air on one side and in contact with room air at constant temperature on the other. The solair temperature has been expanded as a Fourier series in time and the onedimensional heat conduction equation has been solved under appropriate boundary conditions at the two interfaces. Numerical calculations for the heat flux through a horizontal slab (on the roof) using the hourly data of solar radiation and atmospheric temperature for 21 July, 1975 in Kuwait have been presented. It is found that the presence of an air gap (even ] in) considerably reduces the heat flux through the slab compared with a solid slab (zero air gap). It is further seen that for a given total thickness of concrete the best load levelling is achieved when the thickness of the outer layer of the concrete is least, consistent with structural considerations. ANALYSIS
Figure 1 illustrates the geometry of the problem. The equations governing the temperature distributions in the two walls are:
PERIODIC HEAT TRANSFER THROUGH A HOLLOW CONCRETE SLAB
1 15
SOLAR RADIATIONS(t)ANDATMOSPHERIC AIR AT TEMPERATUREOA(t) ×--o
\ \ \ N cO,,NC,
x:a
AIR GAP, THERMAL CONDUCTANCEC
ROOM AIR AT CONSTANT TEMPERATURE TR Fig. 1. Hollow concrete slab.
gO
K 1 ~20
(t
t)1C1 aA2
0 < x < a
(1)
and gT
K 2 ~2T
Pt
p2(~2 c y -
.)
{'3
Let us assume the periodic solutions as: J
0 = 0o(x) + 2
0,,(x) exp ( i r a . t )
(3}
T,,(y) exp (im,)t)
(4)
n = I
and: J
T = T.(y) + E n = ]
where n is an integer. The a p p r o p r i a t e b o u n d a r y c o n d i t i o n s are:
K~ -
= ~oS(t) + h,,(O.4It) - 0(x = 01 - c A R (Lv),,=o
(5(a))
= h,,(Os - O(x = 0)) where: 0s(t ) = OA(t ) +
:%S(t)
tt,,
--
cAR It,,
(6(a))
i s , c o m m o n l y k n o w n as the solair t e m p e r a t u r e which in general can be e x p a n d e d a s
116
M. S, SODHA, A. K. SETH, S. C. KAUSHIK
0s(t) = a o + ~ ?1 =
a,,exp(ino)t)
(6(b))
|
where: a,, = A,, exp ( - i~b,,) Further, we have:
_ K1 90.,
= _ K2 cT_ =,,
~T
-K2~r=
(5(b))
9) r=o
= C(O(x = a) - TO'= 0))
o
(5(c))
dT] - K2~fy ,.=b = hi(T()'= b) - Te,)
(5(d))
C is defined in accordance with the A S H R A E H a n d b o o k . Equation (5(d)) also determines the heat flux, Q. per unit area which is transferred to the r o o m air at temperature, T e. Substituting for 0 and T from eqns (3) and (4) in eqns (1) and (2), the solutions for the time independent and time dependent parts of 0 and T are, respectively, given by: 0o = A o + A l x
(7(a))
T o = B o + BLy
(7(b))
O.(x) = 2° exp ( + ict,,x) + "~'nexp ( - i%x)
(7(c))
T.(y) = / a . exp( + i/~.y) +/~'. exp( - i[3,,y)
(7(d))
where: ct.=\-
nu)p i C l ) J/2 ~i
_
(.~op2c2) '/2
ft.=\-
2 ~ z ,/
(l-i)
(8(a))
(1 - i )
(8(b))
The real constants (A o, A 1, Bo and B ~) and the complex constants (£.,).~, #. and/~'.) are determined from the appropriate boundary conditions given by eqns (5); hence:
E+o ~C
- Kl
0
- Ca
+ C
0
+h i
+ K2 -
-
K2
( K 2 + bhi)
J
i olioiol Al = Bo
BI
iTR
(9(a))
PERIODIC HEAT TRANSFER THROUGH A HOLLOW C()NCRETE SLAB
117
and :
h; - iK'otn ho + iKl~,, 0 %,exp(+i%a) -Kla. exp(-i~.a ) - K2fi,, C e x p ( +icq,a) -Cexp(-&.a) C - iKefi , 0 0 (ki + iK2fl,)exp(ifl,,h)
IKl
+ K2fl.
)"n
C + iK2fi,,
t~n
(h~ - iK2fl.)exp{-ifl,,b)
t~'.
=
(9(b)) Thus we can write for the t e m p e r a t u r e dislributions in the walls:
O=Ao+Alx+Re[
E{2nexp(+iGlx)+)/,exp(-i~,,x)]exp(in~ot);
lO(a))
n=l
T=Bo+B~y+Re[
Ell4, exp(+i/~,,y)+tJ'~expl-iflny)',exp(incot) 1 n-
lO(b))
1
where Re denotes the real part of the quantity. Substituting: .;-,1 = 2,,,, exp ( + iqS., )
1 lta))
i(o;)
1 l(b))
2'. = 2'no exp( +
/Jn = tq., exp( + i G )
( 11 (c))
#'. = U'ooexp( + ia~,)
(1 l(d))
in eqns (I0) one obtains: 0 = Ao + A lx + ~
[':'nOe x p ( +Hl/2~oX)COS (tl(ot + tll,/2~o X + ~),,)
n=l
+ Go exp ( -
nl/2%x) cos (ne)t
- n ' - % x + ~b;,)] (12(a))
and:
T = Bo + B l ) ' +
~ rt=l
[#no exp(
+nl/2[3o)')cos(noJt
+ nl/2[~oy-k ty,,)
+ fnoexp(--nl/2floy)Cos(n(ot--n
1,2
rio)'+
a'.)]
(12(b))
118
M. S. SODHA, A. K. SETH, S. C. KAUSHIK
where: ( 0 1 ( D C I ~ 1/2
~o
=
(p2(D(~T2~ 1/2
\ 2K, J
and
[7o =
The heat flux entering the room is given by: Q(t) = hi[B 0 + Bib - TR+
Z
.Ipnoexp(+n~"2[Job)cos(ne)t + nl"2//ob + ~,)
n--l
+ ff~°exp(-nl"2]t°b)c°s(n~°t - nu2~°b + ~")}1
(13)
NUMERICAL RESULTS AND DISCUSSION
To obtain a numerical appreciation of the results we have computed the amount of heat flux through a horizontal slab (on the roof) as a function of time using the hourly data of solar radiation and atmospheric temperature for a typical hot summer day, i.e. 21 July, 1975, in Kuwait. The Fourier analysis for the solair temperature for 21 July, 1975 for concrete is given in Table 1. It is evident that the first six harmonics are enough to obtain good convergence in most of the calculations. The following parameters have been used for the calculations: TABLE 1 FOURIER ANALYSISOF PERIODIC SOLAIR TEMPERATURE FOR A SUMMER DAY 21 JULY, 1975 IN KUWAIT n A n (°F) ~0,, R a d i a n
1
2
3
4
5
6
7
113.1990
34.5608
8.0878
2.7436
1.4208
0.9746
0.3813
3.9810
1-1082
1-5828
6.1722
6.2306
5.709
--
TR = 70°F ho = 6"0 BTU/ft z h °F
(wind velocity = 15 mph)
h i = 1.2 BTU/ft z h °F For concrete ~o = 0"9 ~:= 1"0 p = 1161b/ft 3 C = 0-156 BTU/Ib °F K = 5-0/12 BTU/ft h °F AR = 20 BTU/h ft 2 L = a + b = 2 i n , 4 in o r 6
in
PERIODI( HEAT TRANSFER ]HROUGH
A HOLLOW ('ON('RE]E
119
SLAB
The values of the air gap conductance for different sizes of air gap are given in Table 2. Figure 2 illustrates the effect of varying the size of the air gap on the periodic heat flux. It is seen that the variation of the air gap from 3in to 4in does not have any significant effect on the heat flux. However, the introduction ()fan air gap (even ~ in) does lead to a considerably smaller heat flux than in thecase of the solid slab (zero air gap). Figure 3 shows the variation of the Q ..... and Q,m,, with the total thickness (L) of the concrete for hollow (a = b) and solid slabs. It is seen that the O .... decreases with the total thickness more rapidly for a solid slab than for a hollow slab of the same concrete thickness; on the other hand, OmJ,, is almost independent of the total thickness of the concrete. Figures 4(a) and (b) show the daily variation of the heat flux into the room lor different values of a,L corresponding to a 1.5 in wide air gap and L = 4 in and 6 in. T A BI_.E 2 AIR (IAP (ONDt (JAN( 1 IOR HORIZfIN]AL STRIA( I AND I)O%ND,'ARI) HIAI t L()\x, , (SU MMIR) 7 h i( ,tn(,.~..+
(,
(in)
( B I L,It } h t . )
0.75 1-5 4-0
1-07
1-18 l-Ol
o=b :3" "---o-'--o-- Solm r temperoture - ...... Solid slab
3E--
I
+
=
Air gap : . 7 5
:~.o
?
/ /
160
/ / \ ',,, /%
/
,
\\
/ ,,' j / - % \ , \ ' %
;. 100
6t 0
i 3
"-""5 6
1 9
1 12 I (HOURS) - - - ~
I 15
I 18
I 21
BO 24
Fig. 2. Dependence of periodic heat flux through hollow concrete slab on air gap; 1,11 a n d 111 refer to air gap thicknesses of 0.75 in, 1.5 in and 4 in, respectively; the dotted curve refers to single slab L = a + b = 6in; the solair temperature is denoted by - - O - - O O--.
120
M. S. SODHA, A. K. SETH, S. C. KAUSH1K 58
54
4e
A
mox (solid stab)
-o
E~
.£ E
Qrnox(hO|lOW slob)
~
~min(sotid ~
l
4
~
I
stab)
Qmin[holtow slob)
8
L (inches)
Fig. 3.
Variation of Q,,a~ and Q,,,. with total thickness of concrete for hollow (a = b) and solid slabs.
respectively. The curve with circles along it shows hourly variation of the solair temperature (on the right-hand-side scale). It is seen that there is a finite phase shift (between the maxima of the incoming heat flux and incident solair temperature) of the order of 2-4 h, depending on the total thickness of the concrete. One can also observe that the phase shift of the maximum solair temperature is larger in comparison with that of the minimum heat flux with respect to the minimum solair temperature, the difference being of the order of 1 h, Figures 5(a) and (b) show the variation of the maximum and minimum heat fluxes with the thickness distribution of concrete outside the air gap, riz. alL for L = 4 in and 6 in, respectively. It is seen that for a given total thickness of concrete the greatest load levelling (i.e. highest value o f Qmin and lowest value of Qmax) is obtained when a/L -~ O, i.e. as the thickness of the outside layer of concrete tends to zero: this is of course not possible and one has to be satisfied with the least value, consistent with structural constraints.
180 I olL:O ]I Ig
:.2S : -SO
I£
: "75
2
:1.0
/UR GAP=I.5~ L=4" . 160
Solakr t emoeratu
?
T 120
.0
100
'[
I
0
3
--
~
~
t (HOURS)
Fig. 4(a).
I
'~
I
I~
i ,8
,s
1 2,
~
180
'~
Hourly variation of periodic heat flux through hollow concrete slab corresponding to different values of a/L (with air gap = 1.5 in and L = 4 in).
x o/Leo A,R GAP=I.S~°6 2,
1T g[ l~ ~"
20
=.166 :.500 = .833 = 1.000
So,air temperature 7
ZVl v y / ~
J/-H
- 160 150
18 ,,~ 16 120
•o
12
110
IQ
100
Go
4
B
6;
3
6
9
I 12
I 15
# 1B
l 21
I 0 2/.
t(HOURS)
Fig. 4(b).
P e r i o d i c h e a t flux t h r o u g h h o l l o w c o n c r e t e s l a b c o r r e s p o n d i n g to different values a i r g a p = 1.5in a n d L = 6 i n ) .
ofa/L (with
"d
-El
m
2
E
S
6.5
7S
23
2s
27
Fig. 5(a).
({1}
f
"
I .75
~
"
I 100
=1.5"
AIRGAP :.7 5~.
4.0"
1.S"
Qmaxand Q=~. on a/L for L = 4in.
I .S 0
o/,
I
.2 S
Dependence of
L=M'
=.
"0
¢:
~o
"Z
E
x o
.¢ t~
I .333
L=6
#
I .500
I -666
I -833
l 1.00
:4.0"
: 1.5"
AIR GAP-0.7S"
40
=1.5"
Dependence of Qm=~ and Qm,~ on a/L for L = 6in.
I .166
Fig. 5(b).
6.61 0
2~
(b)
>
3:
>
3: >
©
t~
PERIODIC HEAT TRANSFER THROUGH A HOLLOW CONCRETE SLAB
123
A l t h o u g h we have m a d e these c a l c u l a t i o n s for a h o r i z o n t a l r o o f on a p a r t i c u l a r d a y in K u w a i t we expect the results to be valid in general. Extensive c a l c u l a t i o n s have revealed that the m a x i m u m a n d m i n i m u m values o f the heat flux are only slightly d e p e n d e n t on the m a g n i t u d e a n d phase o f the second h a r m o n i c o f solair t e m p e r a t u r e . Hence the present c o n c l u s i o n s a b o u t o p t i m a l c o n d i t i o n s for best load levelling, based on a specific set o f m e t e o r o l o g i c a l p a r a m e t e r s , will be a p p l i c a b l e to a wide range o f climatic c o n d i t i o n s .
CONCLUSIONS F o r a given total thickness o f concrete, best load levelling ( a p p r o p r i a t e to m i n i m u m c a p a c i t y o r cost o f the a i r - c o n d i t i o n i n g plant) is achieved when the thickness o f the outside layer has the least value, consistent with structural c o n s i d e r a t i o n s .
REFERENCES I. ANON., ASHRAE Handbook o.[ Fundamentals, American Society of Heating, Refrigeration and Air-
conditioning Engineers, Inc., New York, 1974. 2. K. P. RAOand PRAKASHCnANDRA,A study of thermal performance of concrete hollow blocks by an electric analogue method, BuiMing Science, 5 (1970), p. 31. 3. A. K. KHATP,Y, M. S. SODHAand M. A. S. MALIK,Periodic variation of ground temperature with depth and time, J. Solar Energy, 20 (1978), p. 425.