- Email: [email protected]
Effects of Wall's thermophysical properties on time lag and decrement factor
ELSEVIER
Energy
and Buildings
28 (1998)
159-166
Effects of Wall’s thermophysical properties on time lag and decrement factor H. Asan *, Y.S. San&tar Department
of Mechanical
Engineeting, Received
Karadeniz
Technical
University,
61080 Trabzon,
Turkey
7 June 1997; accepted 24 June 1997
Abstract In this study, the effects of thermophysical properties andthicknessof a wall of a building on time lag and decrement factor have been investigated. For this purpose, one dimensional transient heat conduction equation was solved using Crank-Nicolson scheme under convection boundaryconditions.To the outersurface of the wall, periodic boundary conditions were applied. A very general code which can take care of composite walls under any kind of boundary condition was developed. Single and combined effects of the thickness and thermophysical properties on the time lag and decrement factor were investigated. It was found that thermophysical properties have a very profound effect on the time lag and decrement factor. The computations were repeated for different building materials and the results are discussed. 0 1998 Elsevier Science S.A. All rights reserved. Keywords:
Time Lag; Decrement
factor;
Thermophysical
properties
1. Introduction The adobe houseconstruction is very common in many areasof the world. In the USA, it is developedby the Indians in the southwestregions.In Turkey, adobehouseconstruction is seen in eastern and southern part of the country. In the Black Sea region, northern coast of the country, traditional humsesalso‘nave waMs mahe of &one an6 ah&e. It is we% known that adobehouseconstruction yield a natural air condirtiron@ eI’&z~.Tne roomsare coo1at mid&q an& warm a1 night. The houseconstructionsmentionedabove are nothing but passive so&buildings. For passivesolarbuildings, heatingthe building is possible via direct heatgain aridl’or thermal storagemelho& andthere have been many researcheson this area [I--3). Although direct heat gain method is simple and inexpensive, it suffers from iage %emperaWeswingsbesidesstrong directiOna day lighting [ 41. In addition, direct heat gain method can be effected very fast from outside temperature fluctuations which results to a bed comfort level for indoors [ 5-71. For thermal storagebuildings on the other hand, walls and floors are used as heat storageelements,and stored energy in the walls and floors during day period can be usedfor heating during nights. * Corresponding
author.
0378-7788/98/$ - see front matter 0 PUSO378-7788(98)00007-3
1998 Elsevier
At the cross-sectionof the outer wall of a building, there are different temperatureprofiles during any instant of l-day period. These profiles are functions of inside temperature, outside temperature and thermophysical properties of the wall. Since the outsidetemperaturechangesperiodically during one day period, there will be new temperatureprofiles at any instant of time of the day. During this transient process, aheatvKwe ~QWSt!ArQu~~~e wa!LframQuts.ideta kw.idean!! the amplitudesof thesewaves show the temperaturemagnitudes, and ax. wauele%g
Science S.A. All rights reserved
160
H. Asart, Y.S. Sancaktar/Energy
I
I
and Buildings
wall
28 (1998)
159-166
I
I
,T,&X
outdoor
._....... __........--.e....-
Ti
Twin Fig. I. The schematic
representation
of time lag r#~and decrement
time lags and decrement factors can be obtained. The stored energy during day period can be used during night period when the outside temperature is low. In some dry regions, inside temperature is too high for normal comfort level. Walls with high time lags and small decrement factors, gives comfortable inside temperatures even if the outside is very hot [ 91. By designing special walls in which decrement factors are very low and time lags are high, the propagation of the big fluctuations of outside temperatures to inside can be prevented and almost constant inside temperatures can be obtained which results to a good comfort level [ 10-l 31. In this study mainly, to determine the effects of the thickness and thermophysical properties of a wall on time lag and decrement factor, a detailed computational study was made. The computations were repeated for different wall materials and the results are compared to each other. The results of this study are useful for designing more effective passive solar buildings and related other areas.
factor5
Cf=A,=,,IA,,)
layer 1
interior
layer
layer
2
3
layer ”
exterior
Ti
TO
hi
h, ‘#TX,
TX
*X
0-1----+-L
Fig. 2. The schematics
of the problem
geometry.
80
2. Method
Inside -
In this study, the wall under investigation is assumed to be only in x direction and time-dependent. The problem geometry is shown in Fig. 2. One dimensional, transient heat conduction equation for this problem is as follows:
k$ =pc,$
Surface -
T*mpmtun
Analybcalsdutii
( Thdkek! [IS] ) Numerical solutim (PnuamdY)
-
(1)
where k is the thermal conductivity, p is the density and C, is the heat capacity of the wall material. To solve this problem, two boundary conditions and one initial condition are needed. On both sides of wall, convection boundary conditions are present. At the inner surface, the boundary condition is: =hi[T~=o(t)-Til,
(2)
*=O
whereason the outer surface of the wall, the boundary condition can be written as:
0
"
Fig. 3. Comparison Threlkeld [ 151.
"
n"
4
of present
8
I "
I 12
Time, hr computations
16
with
20
analytical
24
solution
of
H. Asan, Y.S. Sancaktar/Energy
and Buildings
Here, again hi is the wall inner surface heat transfer coefficient, h, the wall outer surface heat transfer coefficient, TX= o is the wall inner surface temperature, TxxL is the wall outer surface temperature,T, is the room air temperatureand rs;;,(t) is the ‘sol-air temperature’. This temperature includes the effects of the solar radiation combined with outside air temperatureand changesperiodically. The sol-air temperatureis assumedto show sinusoidalvariations during 24-h period. At the sametime, since time lag and decrement factor are dependent on wall material only not on the climatological data [ 141, a very generalequation for sol-air temperatureis taken as follows:
ITmax-Tminl
2
+T
(4)
In,".
159-166
161
ature of a room, T,, is taken to be constant. As seenfrom Eq. (4), 7’?,,(t) changesin between T,,,,, and Tmin during 24-h period. The problem now is reducedto one-dimensionalheat conduction which has periodic boundary condition on the outer surface, the sol-air temperature boundary condition, and normal convection boundary condition on the inner surface. The analytical solution of this problem for one layer is given in Ref. [ 151. Here, the algorithm is developed to take careof II layer. For this purpose,finite-difference formulation of Eq. ( 1) isobtainedand Crank-Nicolson methodis applied. The input values of code are: number of layers, the thickness of each layer, density of each layer, specific heat and conductivity of each layer and heat generationsof each layer if any. To interpret the graphicsandthe resultsbetter, Tmin= 0°C and T,,,,, = 1“C were selectedin Eq. (4) and indoor temperature is selectedas0.X accordingly. The outputsof thecode were time lag, decrement factor, wall inner surface temperature and the temperatureof any location at any time of the day. To test the correctnessof the code developed,computed inner surfacetemperatureprofile is comparedwith harmonic analysisresultsof Threlkeld [ 1.51.As seenfrom Fig. 3, computed results of present study match pretty well with the
(3)
+
28 (1998)
As an initial condition, the steady-steady solution of the problem at t = 0 is taken. In the computations,insidetemper-
24 20 16 12 6 4 0 1 E+4
Cc)
Fig. 4. (a) Variation decrement factorfwith
of time dependent heat capacity.
wall inner surface temperature
1 E+5
1 E+6
1E+6
1E+7
C (J/Km*)
(b)
1 E+4
lE+5
lE+7
C (J/Km*) with heat capacity.
(b) Variation
of time lag C#Jwith heat capacity.
(c) Variation
of
162
H. Asan, Y.S. Sancaktar
/Energy
and Buildings
28 (I998)
159-166
24
-
20
-;
,lnlL
0.01
0.1
1
10
100
1000
k(WlmK)
0.6 0.5
(cl Fig. 5. (a) Variation of time dependent wall inner surface temperature Variation of decrement factorfkth thermal conductivity.
k (WImK) with thermal
harmonic analysis results of Threlkeld. The details of the analytical solution for one layer with real climatological data can be found in the work of Threlkeld [ 151.
3. Results and diicussion To see the effects of heat capacity, thermal conductivity, and the thickness of wall on the time lags and decrement factor, first the code was applied to a single layer wall. For 0.2-m-thick wall, the thermal conductivity is taken to be constant and the heat capacity is varied from 1.0 X lo4 to 1.OX 10’ J/K m2. Secondly, the thickness and the heat capacity are fixed and the thermal conductivity is varied from 0.01 to 1000 W/m K. Lastly, the thermal conductivity and the heat capacity are fixed to a certain constant and the thickness is varied from 0.001 to 1 m. For all three cases, time-dependent wall inner surface temperature, time lags and decrement factors are computed and plotted in Figs. 4-6. In Fig. 4, a wall inner surface temperature is given as a function of time and heat capacity. Here, the thickness of the wall and thermal conductivity are fixed to a certain value. As seen from the
conductivity,
(b) Variation
of time lag C$with thermal
conductivity.
(c)
figure, for increasing heat capacity, wall inner surface temperature goes to a constant value. This was expected because if the heat capacity is too high, the stored heat energy in the wall can sustain almost constant inner wall temperature. From Fig. 4a, time lags and decrement factors are also apparent. But to see the effects of heat capacity on time lags and decrement factors more clearly, time lag vs. heat capacity and decrement factor vs. heat capacity are plotted in Fig. 4b and 4c, respectively. As seen from Fig. 4b, there is an exponential relationship between time lag and heat capacity. As the heat capacity goes to its maximum value, time lag exponentially goes to infinity. On the other hand, as the heat capacity goes to zero, time lag goes to zero also. In Fig. 4c, the relationship between decrement factor and heat capacity is given. Here, an inverse exponential relationship between decrement factor and heat capacity is present. As heat capacity goes to its maximum value, decrement factor goes to zero and as heat capacity goes to zero, decrement factor takes certain constant value. Fig. 5 gives the time dependent wall inner surface temperature in the case of varying thermal conductivity and constant heat capacity. As seen from Fig. 5, for small values of thermal
I63
24
16
0.001
0.01
0.1
1
1 (ml
&I
0.8 0.7 0.6 0.5 0.4 0.3 0.2
0.1 0.0 0.001
Cc)
0.01
0.1
1
l(m)
Fig. 6. (a) Variation of time dependent wall inner surfacetemmrature with thickness of the wall. (b) Variation of time lag QIwith thickness of the wall, (c) Variation of decrement facrorfwith thickness of the wall.
conductivity (insulation materials), almost constant wall inner wall temperature is obtained. Again, in Fig. Sb and 5c, the relationship between time Iag vs. thermal conductivity and decrement factor vs. thermal conductivity are plotted. As seen from Fig. 5b, an inverse exponential relationship exists between time lag vs. thermal conductivity. For small values of thermal conductivity, for k = 0.05, time lag take the value of 24 h. Thermal conductivity smaller than 0.01 results to very high time lags. As thermal conductivity increases, the time lag gets smaller and after certain value of thermal conductivity, k> 100, time lag takes a certain constant value around 2 h. As seen from Fig. SC, there exists a direct relationship between thermal conductivity and decrement factor. Small values of thermal conductivity, k < 0.0 1I results to zero decrement factor. As k increases after this value, the decrement factor also increases. After k > IO, the increase of decrement factor slows down as shown in Fig. 5~. Fig. ha gives the time dependent wall inner surface temperature in the case of varying wall thickness and constant L-3nacity and constant thermal conductivity. As seenfrom -.+ps of wall thickness, almost constant -Wined. When the wall -. --nnaeates
inside without any changes in its shape. As seen from Fig, 6b, there is an exponential relationship between time lag and wall thickness. This relationship is similar to that of heat capacity-time lag relationship. This is not surprising because as the wall thickness gets thicker its heat storage capability increases, which is like increasing heat capacity. As the wall thickness goes to its maximum value, time lag exponentially goes to infinity. On the other hand, as the wall thickness goes to zero, time lag goes to zero also. In Fig. 6c, the relationship between decrement factor and the wall thickness is given. Here, as the wall thickness goes to its maximum value, decrement factor goes to zero and as the wall thickness goes to below a certain value, decrement factor takes certain constant value.
In Fig. 7a, the combined effects of heat capacity and thermal conductivity on time lag is given. As seen from Fig. 7a, low thermal conductivity and large heat capacity results to high time lags. Fig. 7b is nothing but the top view of Fig. 7a. Here, computed time lags for different materials are presented. As seen from Fig. 7b, since metals like steel ant aluminum have high heat capacities, it is expected that the should have high time lags. But since they have high therrr conductivity which is inversely proportional with time 1,
164
H. Asan, KS. Sancaktar/Energy
and Buildings
159-166
logk(WlmK)
logk(WlmK) Fig. 7. (a) Combined effects of heat capacity time lag. (b) Computed time lags for different
28 (1998)
and thermal conductivity building materials.
on
the desired time lags are not high enough. On the other hand, materials with low thermal conductivity like formaldehyde and polyurethane have low heat capacity at the same time and results to low time lags. As seen from Fig. 7b, especially asbestos, rubber and, to some extent, asphalt andgraniteresult to considerably high time lags because these materials have low thermal conductivity and high heat capacity. The combined effects of heat capacity and thermal conductivity on decrement factor is given in Fig. 8a. As seen from Fig. 8a, heat capacity has very mild effect on decrement factor and it is determined mostly by thermal conductivity. As mentioned before, materials with high thermal conductivity results to high decrement factor. Again Fig. 8b is the top view of Fig. 8a. Here, computed decrement factors for different materials are shown. As seen from Fig. 8b, formaldehyde and polyurethane result to very small decrement factors because they have very low thermal conductivity. But as remembered before, since these materials have small heat capacities they have small time lags also. Comparing Fig. 7b with Fig. 8b, it is seen that asbestos, rubber and, to some
Fig. 8. (a) Combined effects of heat capacity and thermal conductivity on decrement factor. (b) Computed decrement factors for different building materials.
extent, asphalt and granite result to high time lags and small decrement factors. In Fig. 9a, the cross-section of three-layer wall without insulation and in Fig. 9b, the cross-section of five-layer wall (with insulation) are given. In Fig. lOa, the transient temperature profiles of any location of the wall without insulation is given. Here, x = 0 represents the wall inner surface and x= 14 cm represents the wall outer surface. From Fig. lOa, time lags and time factors are also visible. Fig. lob shows the decrease of amplitude of the temperature profile with respect to x. In Fig. 1 la, the transient temperature profiles of any location of the wall with insulation is given. Here, also time lags and decrement factors are seen from the graph. Comparing Fig. 1Oawith Fig. 1la, it is apparent that wall with insulation results to high time lag and small decrement factor even if both walls have the same total thickness. As seen from Fig. lla, between wall inner surface and insulation location, almost constant temperature profile is obtained (small decrement factor). Again in Fig. 1lb, the decrease of the amplitude of the temperature profile with respect to x is given. The work is going on to optimize the insulation thickness and
H. Asan, Y.S. Sancaktar
three-layer wall (without
/ Energy
and Buildings
28 (I 998) 159-166
165
insulation)
outdoor
,
2
10
,
I 2 I[cml
(4 five-layer wall (with insulation) outdoor
2
,
4
,
2
/
4
! 2 I[cml
I 0.02
-
/ 0.04
I 0.06
(b)
(b)
g
I 0.08
r
/ 0.10
I 0.12
r
‘I 0.14
x(m)
Fig. IO. (a) The transient temperature profiles of any location of the wall without insulation. (b) The decrease of the amplitude of the temperature profile with x for wall without insulation.
Fig. 9. (a) The cross-section of three-layer wall (wall without (b) The cross-section of five-layer wall (wall with insulation).
insulation position from the viewpoint of lags and small decrement factors. Finally, puted time lags and decrement factors for materials are given. Here, for all materials, is used.
1 0.00
insulation).
getting large time in Table 1, comdifferent building 2.5cm thickness
4. Conclusions In this study, to determine the effects of the thickness and thermophysical properties of a wall on time lag and decrement factor, a detailed computational study was made. Single and combined effects of the thickness and thermophysical properties on the time lag and decrement factor were investigated. It was found that thermophysical properties have a very profound effect on the time lag and decrement factor. The computations were repeated for different building materials and the results are discussed. The results of this study are useful for designing more effective passive solar buildings and other related areas.
A 1 I 0.00 0.02
(b)
I,
0.04
I
0.06
1
0.08
I
0.10
/
0.12
.I, 1 0.14
x(m)
Fig. 11. (a) The transient temperature profiles of any location of the wall with insulation. (b) The decrease of the amplitude of the temperature profile with x for wall with insulation.
166
H. Asan, Y.S. Sancaktarl
Table 1 Compound Building
decrement material
Cement sheet Concrete block Brick block Gypsum plastering Granite (red) block Marble (white) block Sandstone block Clay sheet Asphalt sheet Steel slab Aluminum slab Cork board Wood block Plastic board Rubber board P.V.C. board Asbestos sheet Formaldehyde board Thermalite board Fibreboard Siporex board Polyurethane board Light plaster Dense plaster
factors and true lags for different
building
Energy
[21 131
141
L-51
[61
[71
28 (1998)
159-166
materials
P Wm3)
Cp (J/kg K)
k (W/m
700 1400 1800 1200 2650 2500 2200 1900 2300 7800 2700 160 800 1050 1600 1379 2500 30 753 300 550 30 600 1300
1050 loo0 840 837 900 880 712 837 1700 502 880 1888 2093 837 200 1004 1050 1674 837 1000 1004 837 1000 1000
0.36 0.5 1 0.62 0.42 2.90 2.00 1.83 0.85 1.20 50.00 210.00 0.04 0.16 0.50 0.30 0.16 0.16 0.03 0.19 0.06 0.12 0.03 0.16 0.50
References Cl1 C.
and Buildings
Carter, I. DeWilliers, Principles of Passive Solar Building Design, Pergamon, NY, 1987. J.D. Balcomb et al., Passive Solar Buildings, MIT Press, MA, 1992. B. Givoni, Characteristics, design implications, and applicability of passive solar heating systems for buildings, Solar Energy 47 (6) (1991) 425435. A.K. Athienitis, H.F. Sullivan, K.G.T. Hollands, Analytical model, sensitivity analysis, and algorithm for temperature swings in direct gain rooms, Solar Energy 36 (4) ( 1986) 303-3 12. J. Maloney, T. Wan, B. Chen, J. Thorp, Thermal network predictions of the daily temperature fluctuations in a direct gain room, Solar Energy29(3) (1982) 207-223. M.S. Sodha, J.K. Nayak, N.K. Bansal, I.C. Goyal, Thermal performance of a solarium with removable insulation, Building Environ. 17 (1) (1982) 23-32. G. Athanassouli, A model to the thermal transient state of an opaque
K)
C (kJ/K 735 1400 1512 1004 2385 2200 1566 1590 3910 3916 2376 302 1674 879 3200 1385 2625 50 630 300 552 25 600 1300
m’)
4 (h)
f
0.26 0.44 0.46 0.28 0.59 0.56 0.40 0.45 1.03 0.89 0.55 0.32 0.79 0.27 1.17 0.65 1.23 0.06 0.28 0.24 0.26 0.03 0.28 0.4 1
0.544 0.588 0.609 0.564 0.701 0.689 0.688 0.639 0.647 0.179 0.733 0.174 0.403 0.587 0.50 1 0.406 0.396 0.139 0.439 0.234 0.355 0.139 0.408 0.586
wall due to solar radiation absorption, Solar Energy 41 ( 1) ( 1988) 71-80. [81 R.J. Duffin, A passive wall design to minimize building temperature swings, Solar Energy 33 (3-4) (1984) 337-342. composites for efficient thermal storage [91 T.R. Knowles, Proportioning walls, Solar Energy 31 (3) (1983) 319-326. [lOI J. M. Jordan, Y. Zarmi, Massive storage walls as passive solar heating elements: an analytic model, Vol. 27, No. 4. 1981, pp. 349-355. [Ill R.J. Duffin, G. Knowles, Use of layered walls to reduce building temperature swings, Solar Energy 33 (6) ( 1984) 543-549. [I21 R.J. Duffin, G. Knowles, A simple design method for the trombe wall, Solar Energy 34 ( 1) ( 1985) 69-72. control of buildings by adobe [I31 R.J. Duffin, G. Knowles, Temperature wall design, Solar Energy 27 (3) ( 1981) 241-249. [I41 P.J. Bums, K. Han, C.B. Winn, Dynamic effects of bang-bang control on the thermal performance of walls of various constructions, Solar Energy46 (3) (1991) 129-138. Engineering, Prentice-Hall, 1151 J. L. Threlkeld, Thermal Environmental NJ, 1970.
Energy
and Buildings
28 (1998)
159-166
Effects of Wall’s thermophysical properties on time lag and decrement factor H. Asan *, Y.S. San&tar Department
of Mechanical
Engineeting, Received
Karadeniz
Technical
University,
61080 Trabzon,
Turkey
7 June 1997; accepted 24 June 1997
Abstract In this study, the effects of thermophysical properties andthicknessof a wall of a building on time lag and decrement factor have been investigated. For this purpose, one dimensional transient heat conduction equation was solved using Crank-Nicolson scheme under convection boundaryconditions.To the outersurface of the wall, periodic boundary conditions were applied. A very general code which can take care of composite walls under any kind of boundary condition was developed. Single and combined effects of the thickness and thermophysical properties on the time lag and decrement factor were investigated. It was found that thermophysical properties have a very profound effect on the time lag and decrement factor. The computations were repeated for different building materials and the results are discussed. 0 1998 Elsevier Science S.A. All rights reserved. Keywords:
Time Lag; Decrement
factor;
Thermophysical
properties
1. Introduction The adobe houseconstruction is very common in many areasof the world. In the USA, it is developedby the Indians in the southwestregions.In Turkey, adobehouseconstruction is seen in eastern and southern part of the country. In the Black Sea region, northern coast of the country, traditional humsesalso‘nave waMs mahe of &one an6 ah&e. It is we% known that adobehouseconstruction yield a natural air condirtiron@ eI’&z~.Tne roomsare coo1at mid&q an& warm a1 night. The houseconstructionsmentionedabove are nothing but passive so&buildings. For passivesolarbuildings, heatingthe building is possible via direct heatgain aridl’or thermal storagemelho& andthere have been many researcheson this area [I--3). Although direct heat gain method is simple and inexpensive, it suffers from iage %emperaWeswingsbesidesstrong directiOna day lighting [ 41. In addition, direct heat gain method can be effected very fast from outside temperature fluctuations which results to a bed comfort level for indoors [ 5-71. For thermal storagebuildings on the other hand, walls and floors are used as heat storageelements,and stored energy in the walls and floors during day period can be usedfor heating during nights. * Corresponding
author.
0378-7788/98/$ - see front matter 0 PUSO378-7788(98)00007-3
1998 Elsevier
At the cross-sectionof the outer wall of a building, there are different temperatureprofiles during any instant of l-day period. These profiles are functions of inside temperature, outside temperature and thermophysical properties of the wall. Since the outsidetemperaturechangesperiodically during one day period, there will be new temperatureprofiles at any instant of time of the day. During this transient process, aheatvKwe ~QWSt!ArQu~~~e wa!LframQuts.ideta kw.idean!! the amplitudesof thesewaves show the temperaturemagnitudes, and ax. wauele%g
Science S.A. All rights reserved
160
H. Asart, Y.S. Sancaktar/Energy
I
I
and Buildings
wall
28 (1998)
159-166
I
I
,T,&X
outdoor
._....... __........--.e....-
Ti
Twin Fig. I. The schematic
representation
of time lag r#~and decrement
time lags and decrement factors can be obtained. The stored energy during day period can be used during night period when the outside temperature is low. In some dry regions, inside temperature is too high for normal comfort level. Walls with high time lags and small decrement factors, gives comfortable inside temperatures even if the outside is very hot [ 91. By designing special walls in which decrement factors are very low and time lags are high, the propagation of the big fluctuations of outside temperatures to inside can be prevented and almost constant inside temperatures can be obtained which results to a good comfort level [ 10-l 31. In this study mainly, to determine the effects of the thickness and thermophysical properties of a wall on time lag and decrement factor, a detailed computational study was made. The computations were repeated for different wall materials and the results are compared to each other. The results of this study are useful for designing more effective passive solar buildings and related other areas.
factor5
Cf=A,=,,IA,,)
layer 1
interior
layer
layer
2
3
layer ”
exterior
Ti
TO
hi
h, ‘#TX,
TX
*X
0-1----+-L
Fig. 2. The schematics
of the problem
geometry.
80
2. Method
Inside -
In this study, the wall under investigation is assumed to be only in x direction and time-dependent. The problem geometry is shown in Fig. 2. One dimensional, transient heat conduction equation for this problem is as follows:
k$ =pc,$
Surface -
T*mpmtun
Analybcalsdutii
( Thdkek! [IS] ) Numerical solutim (PnuamdY)
-
(1)
where k is the thermal conductivity, p is the density and C, is the heat capacity of the wall material. To solve this problem, two boundary conditions and one initial condition are needed. On both sides of wall, convection boundary conditions are present. At the inner surface, the boundary condition is: =hi[T~=o(t)-Til,
(2)
*=O
whereason the outer surface of the wall, the boundary condition can be written as:
0
"
Fig. 3. Comparison Threlkeld [ 151.
"
n"
4
of present
8
I "
I 12
Time, hr computations
16
with
20
analytical
24
solution
of
H. Asan, Y.S. Sancaktar/Energy
and Buildings
Here, again hi is the wall inner surface heat transfer coefficient, h, the wall outer surface heat transfer coefficient, TX= o is the wall inner surface temperature, TxxL is the wall outer surface temperature,T, is the room air temperatureand rs;;,(t) is the ‘sol-air temperature’. This temperature includes the effects of the solar radiation combined with outside air temperatureand changesperiodically. The sol-air temperatureis assumedto show sinusoidalvariations during 24-h period. At the sametime, since time lag and decrement factor are dependent on wall material only not on the climatological data [ 141, a very generalequation for sol-air temperatureis taken as follows:
ITmax-Tminl
2
+T
(4)
In,".
159-166
161
ature of a room, T,, is taken to be constant. As seenfrom Eq. (4), 7’?,,(t) changesin between T,,,,, and Tmin during 24-h period. The problem now is reducedto one-dimensionalheat conduction which has periodic boundary condition on the outer surface, the sol-air temperature boundary condition, and normal convection boundary condition on the inner surface. The analytical solution of this problem for one layer is given in Ref. [ 151. Here, the algorithm is developed to take careof II layer. For this purpose,finite-difference formulation of Eq. ( 1) isobtainedand Crank-Nicolson methodis applied. The input values of code are: number of layers, the thickness of each layer, density of each layer, specific heat and conductivity of each layer and heat generationsof each layer if any. To interpret the graphicsandthe resultsbetter, Tmin= 0°C and T,,,,, = 1“C were selectedin Eq. (4) and indoor temperature is selectedas0.X accordingly. The outputsof thecode were time lag, decrement factor, wall inner surface temperature and the temperatureof any location at any time of the day. To test the correctnessof the code developed,computed inner surfacetemperatureprofile is comparedwith harmonic analysisresultsof Threlkeld [ 1.51.As seenfrom Fig. 3, computed results of present study match pretty well with the
(3)
+
28 (1998)
As an initial condition, the steady-steady solution of the problem at t = 0 is taken. In the computations,insidetemper-
24 20 16 12 6 4 0 1 E+4
Cc)
Fig. 4. (a) Variation decrement factorfwith
of time dependent heat capacity.
wall inner surface temperature
1 E+5
1 E+6
1E+6
1E+7
C (J/Km*)
(b)
1 E+4
lE+5
lE+7
C (J/Km*) with heat capacity.
(b) Variation
of time lag C#Jwith heat capacity.
(c) Variation
of
162
H. Asan, Y.S. Sancaktar
/Energy
and Buildings
28 (I998)
159-166
24
-
20
-;
,lnlL
0.01
0.1
1
10
100
1000
k(WlmK)
0.6 0.5
(cl Fig. 5. (a) Variation of time dependent wall inner surface temperature Variation of decrement factorfkth thermal conductivity.
k (WImK) with thermal
harmonic analysis results of Threlkeld. The details of the analytical solution for one layer with real climatological data can be found in the work of Threlkeld [ 151.
3. Results and diicussion To see the effects of heat capacity, thermal conductivity, and the thickness of wall on the time lags and decrement factor, first the code was applied to a single layer wall. For 0.2-m-thick wall, the thermal conductivity is taken to be constant and the heat capacity is varied from 1.0 X lo4 to 1.OX 10’ J/K m2. Secondly, the thickness and the heat capacity are fixed and the thermal conductivity is varied from 0.01 to 1000 W/m K. Lastly, the thermal conductivity and the heat capacity are fixed to a certain constant and the thickness is varied from 0.001 to 1 m. For all three cases, time-dependent wall inner surface temperature, time lags and decrement factors are computed and plotted in Figs. 4-6. In Fig. 4, a wall inner surface temperature is given as a function of time and heat capacity. Here, the thickness of the wall and thermal conductivity are fixed to a certain value. As seen from the
conductivity,
(b) Variation
of time lag C$with thermal
conductivity.
(c)
figure, for increasing heat capacity, wall inner surface temperature goes to a constant value. This was expected because if the heat capacity is too high, the stored heat energy in the wall can sustain almost constant inner wall temperature. From Fig. 4a, time lags and decrement factors are also apparent. But to see the effects of heat capacity on time lags and decrement factors more clearly, time lag vs. heat capacity and decrement factor vs. heat capacity are plotted in Fig. 4b and 4c, respectively. As seen from Fig. 4b, there is an exponential relationship between time lag and heat capacity. As the heat capacity goes to its maximum value, time lag exponentially goes to infinity. On the other hand, as the heat capacity goes to zero, time lag goes to zero also. In Fig. 4c, the relationship between decrement factor and heat capacity is given. Here, an inverse exponential relationship between decrement factor and heat capacity is present. As heat capacity goes to its maximum value, decrement factor goes to zero and as heat capacity goes to zero, decrement factor takes certain constant value. Fig. 5 gives the time dependent wall inner surface temperature in the case of varying thermal conductivity and constant heat capacity. As seen from Fig. 5, for small values of thermal
I63
24
16
0.001
0.01
0.1
1
1 (ml
&I
0.8 0.7 0.6 0.5 0.4 0.3 0.2
0.1 0.0 0.001
Cc)
0.01
0.1
1
l(m)
Fig. 6. (a) Variation of time dependent wall inner surfacetemmrature with thickness of the wall. (b) Variation of time lag QIwith thickness of the wall, (c) Variation of decrement facrorfwith thickness of the wall.
conductivity (insulation materials), almost constant wall inner wall temperature is obtained. Again, in Fig. Sb and 5c, the relationship between time Iag vs. thermal conductivity and decrement factor vs. thermal conductivity are plotted. As seen from Fig. 5b, an inverse exponential relationship exists between time lag vs. thermal conductivity. For small values of thermal conductivity, for k = 0.05, time lag take the value of 24 h. Thermal conductivity smaller than 0.01 results to very high time lags. As thermal conductivity increases, the time lag gets smaller and after certain value of thermal conductivity, k> 100, time lag takes a certain constant value around 2 h. As seen from Fig. SC, there exists a direct relationship between thermal conductivity and decrement factor. Small values of thermal conductivity, k < 0.0 1I results to zero decrement factor. As k increases after this value, the decrement factor also increases. After k > IO, the increase of decrement factor slows down as shown in Fig. 5~. Fig. ha gives the time dependent wall inner surface temperature in the case of varying wall thickness and constant L-3nacity and constant thermal conductivity. As seenfrom -.+ps of wall thickness, almost constant -Wined. When the wall -. --nnaeates
inside without any changes in its shape. As seen from Fig, 6b, there is an exponential relationship between time lag and wall thickness. This relationship is similar to that of heat capacity-time lag relationship. This is not surprising because as the wall thickness gets thicker its heat storage capability increases, which is like increasing heat capacity. As the wall thickness goes to its maximum value, time lag exponentially goes to infinity. On the other hand, as the wall thickness goes to zero, time lag goes to zero also. In Fig. 6c, the relationship between decrement factor and the wall thickness is given. Here, as the wall thickness goes to its maximum value, decrement factor goes to zero and as the wall thickness goes to below a certain value, decrement factor takes certain constant value.
In Fig. 7a, the combined effects of heat capacity and thermal conductivity on time lag is given. As seen from Fig. 7a, low thermal conductivity and large heat capacity results to high time lags. Fig. 7b is nothing but the top view of Fig. 7a. Here, computed time lags for different materials are presented. As seen from Fig. 7b, since metals like steel ant aluminum have high heat capacities, it is expected that the should have high time lags. But since they have high therrr conductivity which is inversely proportional with time 1,
164
H. Asan, KS. Sancaktar/Energy
and Buildings
159-166
logk(WlmK)
logk(WlmK) Fig. 7. (a) Combined effects of heat capacity time lag. (b) Computed time lags for different
28 (1998)
and thermal conductivity building materials.
on
the desired time lags are not high enough. On the other hand, materials with low thermal conductivity like formaldehyde and polyurethane have low heat capacity at the same time and results to low time lags. As seen from Fig. 7b, especially asbestos, rubber and, to some extent, asphalt andgraniteresult to considerably high time lags because these materials have low thermal conductivity and high heat capacity. The combined effects of heat capacity and thermal conductivity on decrement factor is given in Fig. 8a. As seen from Fig. 8a, heat capacity has very mild effect on decrement factor and it is determined mostly by thermal conductivity. As mentioned before, materials with high thermal conductivity results to high decrement factor. Again Fig. 8b is the top view of Fig. 8a. Here, computed decrement factors for different materials are shown. As seen from Fig. 8b, formaldehyde and polyurethane result to very small decrement factors because they have very low thermal conductivity. But as remembered before, since these materials have small heat capacities they have small time lags also. Comparing Fig. 7b with Fig. 8b, it is seen that asbestos, rubber and, to some
Fig. 8. (a) Combined effects of heat capacity and thermal conductivity on decrement factor. (b) Computed decrement factors for different building materials.
extent, asphalt and granite result to high time lags and small decrement factors. In Fig. 9a, the cross-section of three-layer wall without insulation and in Fig. 9b, the cross-section of five-layer wall (with insulation) are given. In Fig. lOa, the transient temperature profiles of any location of the wall without insulation is given. Here, x = 0 represents the wall inner surface and x= 14 cm represents the wall outer surface. From Fig. lOa, time lags and time factors are also visible. Fig. lob shows the decrease of amplitude of the temperature profile with respect to x. In Fig. 1 la, the transient temperature profiles of any location of the wall with insulation is given. Here, also time lags and decrement factors are seen from the graph. Comparing Fig. 1Oawith Fig. 1la, it is apparent that wall with insulation results to high time lag and small decrement factor even if both walls have the same total thickness. As seen from Fig. lla, between wall inner surface and insulation location, almost constant temperature profile is obtained (small decrement factor). Again in Fig. 1lb, the decrease of the amplitude of the temperature profile with respect to x is given. The work is going on to optimize the insulation thickness and
H. Asan, Y.S. Sancaktar
three-layer wall (without
/ Energy
and Buildings
28 (I 998) 159-166
165
insulation)
outdoor
,
2
10
,
I 2 I[cml
(4 five-layer wall (with insulation) outdoor
2
,
4
,
2
/
4
! 2 I[cml
I 0.02
-
/ 0.04
I 0.06
(b)
(b)
g
I 0.08
r
/ 0.10
I 0.12
r
‘I 0.14
x(m)
Fig. IO. (a) The transient temperature profiles of any location of the wall without insulation. (b) The decrease of the amplitude of the temperature profile with x for wall without insulation.
Fig. 9. (a) The cross-section of three-layer wall (wall without (b) The cross-section of five-layer wall (wall with insulation).
insulation position from the viewpoint of lags and small decrement factors. Finally, puted time lags and decrement factors for materials are given. Here, for all materials, is used.
1 0.00
insulation).
getting large time in Table 1, comdifferent building 2.5cm thickness
4. Conclusions In this study, to determine the effects of the thickness and thermophysical properties of a wall on time lag and decrement factor, a detailed computational study was made. Single and combined effects of the thickness and thermophysical properties on the time lag and decrement factor were investigated. It was found that thermophysical properties have a very profound effect on the time lag and decrement factor. The computations were repeated for different building materials and the results are discussed. The results of this study are useful for designing more effective passive solar buildings and other related areas.
A 1 I 0.00 0.02
(b)
I,
0.04
I
0.06
1
0.08
I
0.10
/
0.12
.I, 1 0.14
x(m)
Fig. 11. (a) The transient temperature profiles of any location of the wall with insulation. (b) The decrease of the amplitude of the temperature profile with x for wall with insulation.
166
H. Asan, Y.S. Sancaktarl
Table 1 Compound Building
decrement material
Cement sheet Concrete block Brick block Gypsum plastering Granite (red) block Marble (white) block Sandstone block Clay sheet Asphalt sheet Steel slab Aluminum slab Cork board Wood block Plastic board Rubber board P.V.C. board Asbestos sheet Formaldehyde board Thermalite board Fibreboard Siporex board Polyurethane board Light plaster Dense plaster
factors and true lags for different
building
Energy
[21 131
141
L-51
[61
[71
28 (1998)
159-166
materials
P Wm3)
Cp (J/kg K)
k (W/m
700 1400 1800 1200 2650 2500 2200 1900 2300 7800 2700 160 800 1050 1600 1379 2500 30 753 300 550 30 600 1300
1050 loo0 840 837 900 880 712 837 1700 502 880 1888 2093 837 200 1004 1050 1674 837 1000 1004 837 1000 1000
0.36 0.5 1 0.62 0.42 2.90 2.00 1.83 0.85 1.20 50.00 210.00 0.04 0.16 0.50 0.30 0.16 0.16 0.03 0.19 0.06 0.12 0.03 0.16 0.50
References Cl1 C.
and Buildings
Carter, I. DeWilliers, Principles of Passive Solar Building Design, Pergamon, NY, 1987. J.D. Balcomb et al., Passive Solar Buildings, MIT Press, MA, 1992. B. Givoni, Characteristics, design implications, and applicability of passive solar heating systems for buildings, Solar Energy 47 (6) (1991) 425435. A.K. Athienitis, H.F. Sullivan, K.G.T. Hollands, Analytical model, sensitivity analysis, and algorithm for temperature swings in direct gain rooms, Solar Energy 36 (4) ( 1986) 303-3 12. J. Maloney, T. Wan, B. Chen, J. Thorp, Thermal network predictions of the daily temperature fluctuations in a direct gain room, Solar Energy29(3) (1982) 207-223. M.S. Sodha, J.K. Nayak, N.K. Bansal, I.C. Goyal, Thermal performance of a solarium with removable insulation, Building Environ. 17 (1) (1982) 23-32. G. Athanassouli, A model to the thermal transient state of an opaque
K)
C (kJ/K 735 1400 1512 1004 2385 2200 1566 1590 3910 3916 2376 302 1674 879 3200 1385 2625 50 630 300 552 25 600 1300
m’)
4 (h)
f
0.26 0.44 0.46 0.28 0.59 0.56 0.40 0.45 1.03 0.89 0.55 0.32 0.79 0.27 1.17 0.65 1.23 0.06 0.28 0.24 0.26 0.03 0.28 0.4 1
0.544 0.588 0.609 0.564 0.701 0.689 0.688 0.639 0.647 0.179 0.733 0.174 0.403 0.587 0.50 1 0.406 0.396 0.139 0.439 0.234 0.355 0.139 0.408 0.586
wall due to solar radiation absorption, Solar Energy 41 ( 1) ( 1988) 71-80. [81 R.J. Duffin, A passive wall design to minimize building temperature swings, Solar Energy 33 (3-4) (1984) 337-342. composites for efficient thermal storage [91 T.R. Knowles, Proportioning walls, Solar Energy 31 (3) (1983) 319-326. [lOI J. M. Jordan, Y. Zarmi, Massive storage walls as passive solar heating elements: an analytic model, Vol. 27, No. 4. 1981, pp. 349-355. [Ill R.J. Duffin, G. Knowles, Use of layered walls to reduce building temperature swings, Solar Energy 33 (6) ( 1984) 543-549. [I21 R.J. Duffin, G. Knowles, A simple design method for the trombe wall, Solar Energy 34 ( 1) ( 1985) 69-72. control of buildings by adobe [I31 R.J. Duffin, G. Knowles, Temperature wall design, Solar Energy 27 (3) ( 1981) 241-249. [I41 P.J. Bums, K. Han, C.B. Winn, Dynamic effects of bang-bang control on the thermal performance of walls of various constructions, Solar Energy46 (3) (1991) 129-138. Engineering, Prentice-Hall, 1151 J. L. Threlkeld, Thermal Environmental NJ, 1970.