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Transient heat conduction in a composite slab by a time-varying incident heat flux combined with convective and radiative cooling
INT. ~ . HEAT M A S S T R A N S F E R 0735-1933/86 $3.00 + .00 Vol. 13, pp. 515-522, 1986 ©Pergamon Journals Ltd. Printed i n t h e U n i t e d States
TRANSIENT HEAT CONDUCTION IN A COMPOSITE SLAB BY A TIME-VARYING INCIDENT HEAT FLUX COMBINED WITH CONVECTIVE AND RADIATIVE COOLING
Bengt SundSn Department of Applied Thermo and Fluid Dynamics, Chalmers University of Technology, 41296 GSteborg, Sweden
(C~udnicated by D.B. Spalding) ABSTRACT Numerical calculations based on finite difference approximations are carried out to assess the thermal response of a composite slab due to a time-varying incident heat flux. The slab, which consists of several layers of various material, is cooled by combined convection and radiation. The slab is rotating and the front and rear surfaces receive the incident heat flux in an alternating fashion. Temperature distributions versus time and space are presented.
Introduction Transient heat conduction in a solid with complicated thermal boundary conditions has received little attention in the literature. In [I], where transient heating and cooling of a plate by combined convection and radiation were analysed,a review of studies dealing with a heat con~1~ting solid exposed to combined convection and radiation was presented. Such problems become highly nonlinear and
analytical solutions to these problems are diffi-
cult to find and sometimes not even exist
if not more or less rough approxi-
mations are used. Recently, Davies [2] presented an investigation where transient conduction in a plate heated by convection and cooled by radiation was treated. An one-dimensional approximate solution method was employed. The present study is concerned with transient heat conduction in a composite slab heated by a time-varying incident heat flux and cooled by combined convection and radiation. The problem is more complicated than those already presented in the literature. The problem originates from a real world application and unfortunately some information is confidential and can thus not be given. The purpose of this paper is to predict the temperature distribution 515
516
B. Sunden
Vol. 13, No. 5
in the slab as function of time and to predict the peak temperatures. Due to the complexity of the problem a numerical solution by finite difference approximations is presented.
Statement of the proble m This investigation is concerned with the temperature distribution in a composite slab as shown in Figure I.
iO0
1 2 3 l,l~il6
_
I
7
__
9
!
L1 L2 L3 Ll. L~L6~Lv Le L%
FIG. I The composite slab. Initially, the slab has a uniform temperature equal to the ambient temperature but suddenly the surfaces are exposed to an incident time-varying heat flux. The slab exchanges heat with the surrounding environment by convection and radiation. The intensity of the incident heat flux versus time
is depicted in Fi-
gure 2. However, this intensity is modulated because the slab is rotating (30 rpm). This means that the intensity has to be multiplied by the cosine between the surface normal and the direction of the incident heat flux. The rotation of the slab is started in such a manner that the maximum intensity at the front surface is achieved at t = I second.
Vol. 13, No. 5
TRANSIENT H E A T ~ O N I N A C E M P ( ~ I ' I ~ S ~ A B
517
1.8
~
8.5
G~AX= 8.311E~8~ TO'I'.~= O.836E:~eO,J/ttm~
0.8
8
5
18
T:i'E: S6::.
15
20
25
FIG. 2 Incident heat flux versus time. The main purpose of this investigation is to predict the thermal response of the composite slab during a time period of 25 seconds. The followingass~nptions are made in the analysis: a) The heat conduction is one-dimensional. b) The slab is opaque to thermal radiation and has no internal heat sources or heat sinks. c) The physical properties of each layer of the slab are uniform and independent of temperature. Values are given in Table I. d) The convective heat transfer coefficient is independent of the surface temperatures. e) The ambient air temperature and the background radiation temperature are independent of time (25 and 10 °C, respectively). f) The ambient air is transparent to thermal radiation.
518
B. Sunden
Vol.
13, No. 5
TABLE I. Physical Data o f bhe Composite Slab. Layer no
Thickness L.z mm
Thermal conductivity k.z W/mK
Density 3 Specific heat Pi kg/m epi J/kgK
I
0.85
0.6
1500
1100
2
1.2
0.4
1700
1200
3
0.85
o.6
1500
1100
4
0.85
0.6
1500
11oo
5
0.2
0.2
1200
1000
6
0.85
0.6
1500
1100
7
0.85
o.6
]500
1]oo
8
0.85
0.6
1500
1100
9
0.9
0.7
1900
1000
Basic equations and boundary conditions Within each layer of the slab, the one-dimensional duction equation yields
3T at
--=
a.
form of the heat con-
(see [3])
32T z ax 2
(I)
where a. is the thermal diffusivity of layer i. l Since the thermal conductivities of the layers are different the interfaces between the layers are treated separately and the conditions of continuity in heat flux and temperature are applied. Thus at every interface we have
k (3T)i
=
i ~x
k
(~T]i+1 i+l"~x ~
Ti
=
where the superscripts
(2)
T i+I
(3)
indicate Which layer the variable belongs to.
For the surfaces exposed to the incident radiation a heat balance yields: absorbed incident heat flux convective cooling or in mathematical
terms:
-
-
thermal radiation loss
outward conduction
=
0
-
Vol. 13, No. 5
aQA
TRANSI~qrHEATOOI~YJCrI(~ IN A C O M P ( ~ I T E SLAB
-
4 oaA(T~ - Tol)
-
hA(T s - To2)
+
~T kA (~x)s =
0
519
(4)
In equation (4), Q is the modulated incident heat flux, Tol and To2 the temperatures of the radiation background and the surrounding air, respectively. h is the convective heat transfer coefficient and k is the thermal conductivity of the layers exposed to the surroundings. The absorbtlvity a for the incident heat flux and the emissivity a for the thermal radiation was determined from measurements carried out by the company delivering the slab. The values used in the calculations were: a = 0.6 and c = 0.8.
Numerical solution procedure Equation (I) is solved numerically by a finite difference technique with respect to the initial and boundary conditions outlined previously. This technique is second-order in the space coordinate but only first order in time. A fully implicit iterative solution algorithm with an over-relaxation procedure is employed. The proper step sizes in space and time were estimated by carrying out several test calculations. 271 non-uniformly distributed grid points were used in the slab. The time steps in the final calculations were either 0.05 or 0.1 seconds and the resolution in time and the accuracy in the time history were found to be sufficient.
Results and discussion Figure 3 depicts the surface temperatures and one interface temperature (between layers I and 2) with convective cooling corresponding to h = 40 W/m2K. The peak temperature appears on the front surface after approximately 1.2 seconds. Both the front and rear surfaces behave in a damped periodic manner after the initial exposition to the incident heat flux. The interface temperature showed has a slightly periodic variation after the initial heating up phase. Figure 4 shows the temperature distribution in the slab for various instants of time. After approximately 20 seconds the temperature is uniform in the slab. In Figure h some discontinuities in the temperature gradient appear. These originate from the fact that the thermal conductivity varies from layer to layer. Figures 5 and 6 show similar results as in Figures 3 and 4 but with convective cooling corresponding to h = 200 W/m2K. The peak temperature is reduced
approximately 13 degrees and the uniform temperature after 20 seconds
520
B. Sunden
i
Vol.
FRONT SURFACE TEMPEPATURE
2
INTERFACE
3
REAR SURFACE TEMPERATURE
TEMPERATURE
O
I
8
e
I
I
I
I
5
,
i
,
i
I t
i
,
,
I
,
,
16 15 TIME ~ .
FIG. 3 Surface and interface temperatures h = 40 W/m2K.
i
2
IS8 t.)
I
TIME =
,
,
I
,
,
,
28
versus
,
2S
time.
1 S
2s
3
3s
4
4s
5
5s
6
i0 s
7
15 s
I88
Se
0
! I:'08 IN MKI~I~_
FIG. 4 Temperature distributions at various h = 40 W/m2K.
instants
of time.
13, No. 5
Vol. 13, NO. 5
TRANSIENT HE~T ~ O N
i
IS8
IN A O3MPOSITE SLAB
FRONT SURFACE TEMPERATURE
2
INTERFACE TEMPERATURE
3
REAR SURFACE TEMPERATURE
O
f
lu
@ @
18
S
16
2B
TIME ~ ,
FIo. 5 Surface and interface temperatures versus time. h = 200 W/m2K.
2@@
t@@
i
ll@@
~- TIME =
] S
2
2s 3s
L}
/.I s
.5
5s
6
i0 s
7 8
20 s
15 s
5@
@ @
I~
IN I~IIRIAL
I
FIQ. 6 Temperature distributions at various instants of time. h = 200 W/m2K.
521
522
B. Sunden
Vol. 13, No. 5
is 5 degrees less than that in Figure 4. Otherwise the results are very similar. It should be noted that the calculated results have been obtained with the initial data, physical data etc. as specified in the preceding sections. All these data are important and changes in them may influence the results. the predicted peak temperature it. So for instance,
is too high, there are various means to reduce
the thermooptieal
characteristics
can be altered and ab-
lation layers on the surfaces may be useful.
Nomenclature A
surface area
a. 1
thermal diffusivity of layer i
c
specific heat of layer i
Pi
h
convective heat transfer coefficient
k. 1 L. 1
thermal conductivity of layer i thickness
of layer i
Q
incident heat flux
(W/m 2)
~ax
maximum value of the incident heat flux (W/m 2)
t
time
T
temperature
T
surface temperature S
TO 1
background radiation temperature
TO2
ambient air temperature
x
coordinate in space
Greek symbols absorbtivity of the surfaces
for incident
heat flux £
If
emissivity of the surfaces
Pi
density of layer i
c
Stefan - Boltzmann constant
for thermal radiation
References I. A.L. Crosbie and R. Viskanta,
Int. J. Heat Mass Transfer
11, 305 (1968)
2. T.W. Davies, Appl. Math. Modelling ~, 337 (1985) 3. E.R.G. Eckert and R.M. Drake Jr., Analysis of heat and mass transfer, McGraw-Hill, New York (1972)
TRANSIENT HEAT CONDUCTION IN A COMPOSITE SLAB BY A TIME-VARYING INCIDENT HEAT FLUX COMBINED WITH CONVECTIVE AND RADIATIVE COOLING
Bengt SundSn Department of Applied Thermo and Fluid Dynamics, Chalmers University of Technology, 41296 GSteborg, Sweden
(C~udnicated by D.B. Spalding) ABSTRACT Numerical calculations based on finite difference approximations are carried out to assess the thermal response of a composite slab due to a time-varying incident heat flux. The slab, which consists of several layers of various material, is cooled by combined convection and radiation. The slab is rotating and the front and rear surfaces receive the incident heat flux in an alternating fashion. Temperature distributions versus time and space are presented.
Introduction Transient heat conduction in a solid with complicated thermal boundary conditions has received little attention in the literature. In [I], where transient heating and cooling of a plate by combined convection and radiation were analysed,a review of studies dealing with a heat con~1~ting solid exposed to combined convection and radiation was presented. Such problems become highly nonlinear and
analytical solutions to these problems are diffi-
cult to find and sometimes not even exist
if not more or less rough approxi-
mations are used. Recently, Davies [2] presented an investigation where transient conduction in a plate heated by convection and cooled by radiation was treated. An one-dimensional approximate solution method was employed. The present study is concerned with transient heat conduction in a composite slab heated by a time-varying incident heat flux and cooled by combined convection and radiation. The problem is more complicated than those already presented in the literature. The problem originates from a real world application and unfortunately some information is confidential and can thus not be given. The purpose of this paper is to predict the temperature distribution 515
516
B. Sunden
Vol. 13, No. 5
in the slab as function of time and to predict the peak temperatures. Due to the complexity of the problem a numerical solution by finite difference approximations is presented.
Statement of the proble m This investigation is concerned with the temperature distribution in a composite slab as shown in Figure I.
iO0
1 2 3 l,l~il6
_
I
7
__
9
!
L1 L2 L3 Ll. L~L6~Lv Le L%
FIG. I The composite slab. Initially, the slab has a uniform temperature equal to the ambient temperature but suddenly the surfaces are exposed to an incident time-varying heat flux. The slab exchanges heat with the surrounding environment by convection and radiation. The intensity of the incident heat flux versus time
is depicted in Fi-
gure 2. However, this intensity is modulated because the slab is rotating (30 rpm). This means that the intensity has to be multiplied by the cosine between the surface normal and the direction of the incident heat flux. The rotation of the slab is started in such a manner that the maximum intensity at the front surface is achieved at t = I second.
Vol. 13, No. 5
TRANSIENT H E A T ~ O N I N A C E M P ( ~ I ' I ~ S ~ A B
517
1.8
~
8.5
G~AX= 8.311E~8~ TO'I'.~= O.836E:~eO,J/ttm~
0.8
8
5
18
T:i'E: S6::.
15
20
25
FIG. 2 Incident heat flux versus time. The main purpose of this investigation is to predict the thermal response of the composite slab during a time period of 25 seconds. The followingass~nptions are made in the analysis: a) The heat conduction is one-dimensional. b) The slab is opaque to thermal radiation and has no internal heat sources or heat sinks. c) The physical properties of each layer of the slab are uniform and independent of temperature. Values are given in Table I. d) The convective heat transfer coefficient is independent of the surface temperatures. e) The ambient air temperature and the background radiation temperature are independent of time (25 and 10 °C, respectively). f) The ambient air is transparent to thermal radiation.
518
B. Sunden
Vol.
13, No. 5
TABLE I. Physical Data o f bhe Composite Slab. Layer no
Thickness L.z mm
Thermal conductivity k.z W/mK
Density 3 Specific heat Pi kg/m epi J/kgK
I
0.85
0.6
1500
1100
2
1.2
0.4
1700
1200
3
0.85
o.6
1500
1100
4
0.85
0.6
1500
11oo
5
0.2
0.2
1200
1000
6
0.85
0.6
1500
1100
7
0.85
o.6
]500
1]oo
8
0.85
0.6
1500
1100
9
0.9
0.7
1900
1000
Basic equations and boundary conditions Within each layer of the slab, the one-dimensional duction equation yields
3T at
--=
a.
form of the heat con-
(see [3])
32T z ax 2
(I)
where a. is the thermal diffusivity of layer i. l Since the thermal conductivities of the layers are different the interfaces between the layers are treated separately and the conditions of continuity in heat flux and temperature are applied. Thus at every interface we have
k (3T)i
=
i ~x
k
(~T]i+1 i+l"~x ~
Ti
=
where the superscripts
(2)
T i+I
(3)
indicate Which layer the variable belongs to.
For the surfaces exposed to the incident radiation a heat balance yields: absorbed incident heat flux convective cooling or in mathematical
terms:
-
-
thermal radiation loss
outward conduction
=
0
-
Vol. 13, No. 5
aQA
TRANSI~qrHEATOOI~YJCrI(~ IN A C O M P ( ~ I T E SLAB
-
4 oaA(T~ - Tol)
-
hA(T s - To2)
+
~T kA (~x)s =
0
519
(4)
In equation (4), Q is the modulated incident heat flux, Tol and To2 the temperatures of the radiation background and the surrounding air, respectively. h is the convective heat transfer coefficient and k is the thermal conductivity of the layers exposed to the surroundings. The absorbtlvity a for the incident heat flux and the emissivity a for the thermal radiation was determined from measurements carried out by the company delivering the slab. The values used in the calculations were: a = 0.6 and c = 0.8.
Numerical solution procedure Equation (I) is solved numerically by a finite difference technique with respect to the initial and boundary conditions outlined previously. This technique is second-order in the space coordinate but only first order in time. A fully implicit iterative solution algorithm with an over-relaxation procedure is employed. The proper step sizes in space and time were estimated by carrying out several test calculations. 271 non-uniformly distributed grid points were used in the slab. The time steps in the final calculations were either 0.05 or 0.1 seconds and the resolution in time and the accuracy in the time history were found to be sufficient.
Results and discussion Figure 3 depicts the surface temperatures and one interface temperature (between layers I and 2) with convective cooling corresponding to h = 40 W/m2K. The peak temperature appears on the front surface after approximately 1.2 seconds. Both the front and rear surfaces behave in a damped periodic manner after the initial exposition to the incident heat flux. The interface temperature showed has a slightly periodic variation after the initial heating up phase. Figure 4 shows the temperature distribution in the slab for various instants of time. After approximately 20 seconds the temperature is uniform in the slab. In Figure h some discontinuities in the temperature gradient appear. These originate from the fact that the thermal conductivity varies from layer to layer. Figures 5 and 6 show similar results as in Figures 3 and 4 but with convective cooling corresponding to h = 200 W/m2K. The peak temperature is reduced
approximately 13 degrees and the uniform temperature after 20 seconds
520
B. Sunden
i
Vol.
FRONT SURFACE TEMPEPATURE
2
INTERFACE
3
REAR SURFACE TEMPERATURE
TEMPERATURE
O
I
8
e
I
I
I
I
5
,
i
,
i
I t
i
,
,
I
,
,
16 15 TIME ~ .
FIG. 3 Surface and interface temperatures h = 40 W/m2K.
i
2
IS8 t.)
I
TIME =
,
,
I
,
,
,
28
versus
,
2S
time.
1 S
2s
3
3s
4
4s
5
5s
6
i0 s
7
15 s
I88
Se
0
! I:'08 IN MKI~I~_
FIG. 4 Temperature distributions at various h = 40 W/m2K.
instants
of time.
13, No. 5
Vol. 13, NO. 5
TRANSIENT HE~T ~ O N
i
IS8
IN A O3MPOSITE SLAB
FRONT SURFACE TEMPERATURE
2
INTERFACE TEMPERATURE
3
REAR SURFACE TEMPERATURE
O
f
lu
@ @
18
S
16
2B
TIME ~ ,
FIo. 5 Surface and interface temperatures versus time. h = 200 W/m2K.
2@@
t@@
i
ll@@
~- TIME =
] S
2
2s 3s
L}
/.I s
.5
5s
6
i0 s
7 8
20 s
15 s
5@
@ @
I~
IN I~IIRIAL
I
FIQ. 6 Temperature distributions at various instants of time. h = 200 W/m2K.
521
522
B. Sunden
Vol. 13, No. 5
is 5 degrees less than that in Figure 4. Otherwise the results are very similar. It should be noted that the calculated results have been obtained with the initial data, physical data etc. as specified in the preceding sections. All these data are important and changes in them may influence the results. the predicted peak temperature it. So for instance,
is too high, there are various means to reduce
the thermooptieal
characteristics
can be altered and ab-
lation layers on the surfaces may be useful.
Nomenclature A
surface area
a. 1
thermal diffusivity of layer i
c
specific heat of layer i
Pi
h
convective heat transfer coefficient
k. 1 L. 1
thermal conductivity of layer i thickness
of layer i
Q
incident heat flux
(W/m 2)
~ax
maximum value of the incident heat flux (W/m 2)
t
time
T
temperature
T
surface temperature S
TO 1
background radiation temperature
TO2
ambient air temperature
x
coordinate in space
Greek symbols absorbtivity of the surfaces
for incident
heat flux £
If
emissivity of the surfaces
Pi
density of layer i
c
Stefan - Boltzmann constant
for thermal radiation
References I. A.L. Crosbie and R. Viskanta,
Int. J. Heat Mass Transfer
11, 305 (1968)
2. T.W. Davies, Appl. Math. Modelling ~, 337 (1985) 3. E.R.G. Eckert and R.M. Drake Jr., Analysis of heat and mass transfer, McGraw-Hill, New York (1972)