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Transient conduction in a plate with counteracting convection and thermal radiation at the boundaries
Transient conduction in a plate with counteracting convection and thermal radiation at the boundaries T. W. Davies Department of Chemical Engineering, Exeter University, Exeter EX4 4QF, UK (Received February 1985; revised May 1985)
During the flash dehydroxylation of powdered kaolinite it is desirable that a rapidly propagating thermal wave penetrates the cold powder particles in a way that raises the particle interior to the reaction temperature of 600°C without the particle exterior being heated beyond 1000°C. In a production unit this is achieved by performing the heat treatment in a device where particles are heated by convection from hot gas and are subject to heat loss by thermal radiation to cool walls. This paper concerns the fundamental heat transfer problem of the process, decoupled from the thermal effects of the dehydroxylation reaction. Using a plate as the approximation for the particle shape a semi-analytical solution for the plate temperature distribution is obtained as a function of the five dimensionless process parameters: Biot number, radiation number, wall/ gas and particle/gas temperature ratios and mode of convection. Accuracy is demonstrated by comparison with an existing numerical solution for the limiting case of pure radiative heating of a plate initially at absolute zero. Key words: mathematical models, transient conduction, flash heating, heat transfer
The flash heating of a powdered solid is a technique attracting growing interest in a number of mineral processing industries. One particular process is that of the flash dehydroxylation of kaolinite. Kaolinite, A1203.2SIO2.2H20, undergoes an endothermic loss of hydroxyl water at around 600°C. The product of this reaction is meta-kaolin, its physical form depending on the speed of the heating process. Although the fundamental size of crystalline kaolinite particles is submicron, in practice agglomerations of particles a few microns thick and in the form of flakes are the usual result of the powdering process. If a fast moving thermal wave is propagated through such particles it is possible to release steam in the particle interior faster than it can diffuse to the particle edges. Under these circumstances internal voids are created and the product has a low bulk density. If the voids are around the wavelength of light the product opacity is also high because of increased scattering. Such a product is attractive as a filler in the paint and paper industries. If the temperature of the
0307-904X/85/05337-04/$03.00 © 1985Butterworth& Co. (Publishers)Ltd.
meta-kaolin exceeds 1000°C a transformation to mullite begins with an associated and undesirable increase in bulk density and abrasiveness. The heat transfer requirements of the process are therefore that particles should experience a void-generating fastmoving isotherm of 600°C but should not be heated beyond about 1000°C. The process parameters which are available to achieve this are the reactor temperature distribution and the residence time distribution. The design of such reactors is proprietorial but involves passage of cold particles through a hot gas front near a wall under a cyclonic action. This gives some degree of variation in the heat treatment of large and small particles. A laboratoryscale furnace designed to replicate the heat transfer conditions in an industrial flash calciner has been described by Davies. l The heat transfer process may be modelled on the assumption that the plate-like particles are suddenly immersed in a hot flowing fluid and have a view of cold walls.
Appl. Math. Modelling, 1985, Vol. 9, October 337
Transient conduction in a plate: T. W. Davies They are heated by convection and as their temperature rises they begin to lose heat by thermal radiation. This nonlinear boundary condition renders the problem insoluble by analytical methods. Numerical solution is possible but numerical methods do not lend themselves easily to an extension of the problem to include a moving reaction front and a variation in physical properties. The heat transfer analysis described in this paper was developed with the aim of linking the process variables of gas, wall and initial particle temperatures and residence time with the thermal history of the particle. Problem def'mition It is required to predict the transient temperature distribution in a plate of thickness 21, which is initially at a uniform temperature Ti throughout. At time t = 0 the plate surfaces are suddenly subjected to forced or free convective heating from a fluid at temperature Ta and to radiative cooling to surroundings at temperature Tr, i.e. Ta> Ti> Tr. The problem of a plate of thickness 2L subjected to identical heat transfer at both faces is mathematically similar to that of a plate of thickness L which has one adiabatic face and one heat transfer face. The assumptions are that: • The heat conduction in the plate is one-dimensional. • The solid is homogeneous, isotropic and has constant physical properties. • The fluid temperature Ta, the wall temperature Tr, the heat transfer coefficient h, and the surface emissivity e, are constant. In dimensionless form the problem may be defined as follows:
00 Or
020 0X 2
0~
r>0
(1)
0 (5", O) == 1 00 ~-~(1, r) = 0 00 ----(0, r)=B(1--/3)n--R(~--h4)
aX
(2) (3)
1<~n<~4/3
(4)
where/3 is a function of r. Equation (4) is the nonlinear boundary condition for convective heating with counteracting radiative cooling. The objective of the analysis was to obtain a solution of equation (1) for any given values of the process parameters B. R, X and n. Typical ranges of these parameters which may occur in practice are 0.1 < B < 10, 0.1 < R < 10, 0~< ?~< 1, 1 <~n<~413.A solution of equation (1) with the boundary condition equation (4), is not available in analytical form. Nonlinear diffusion problems of this kind are, however, amenable to treatment by the heat balance integral technique described by Goodman. = This technique permits a semi-analytical solution using a route analogous to that used in the integral analysis of boundary layer flows. For the case at hand a thermal wave propagates into the plate from both heated faces at a rate which depends upon the boundary fluxes. The leading edge of each wave is defined in a manner analogous to that used in defining the edge of a boundary layer. (In an 'exact' analysis the thermal disturbance originating at the plate surface would be felt throughout the plate instantaneously.) At the beginning of
338 Appl. Math. Modelling, 1985, Vol. 9, October
the heat transfer process the waves will travel through the plate as they would through a semi-infinite solid. When they meet at the centre plane (the penetration time) the semi-infinite solid behaviour ceases and the boundary condition at the leading edge changes. The heat balance integral solution of this problem consequently has two components, the pre-penetration or semi-infinite solid solution and the post-penetration or finite solid solution. The solutions are matched at the penetration time. The preopenetration solution The reduction of equation (1) to an algebraic expression for the temperature distribution within the plate proceeds as follows. At time t when the conduction front has penetrated a distance x = 8, a heat balance on the thermal wave is obtained by integration of equation (1): 8*
0
This equation may be transformed into an ordinary differential equation for/3(0 by the application of Leibnitz's rule and the substitution of a suitable approximation for 0 (r, X) as follows: 6"
0 clX--Ox=n* -~-¢ -.
=\~'Y/x=6o-v..
OX/x=o (6)
0
Langford 3 has discussed the choices available for the approximation of the temperature profile across the thermal wave. A second-order polynomial was chosen for the present analysis. Assuming therefore that:
O = a o + a l X + a 2 X 2, a ( r ) ,
i=0,1,2
(7)
with the boundary conditions: X=8*
a0 --=0 aX
0=7
X=0
(8a)
(8b)
0 =/3
These boundary conditions may be used with equation (7) to give: ao =/3 as . . . .
(9a) al
(9b)
25*
2 (~, -
8* = - -
~)
(9c)
al
Substitution of equation (7) into equation (6) gives:
8, ~
~r(aoS* x
+ a~ --2
dr
which simplifies to:
8,3k
+a2-3--J--(ao+alS*+a~8*' )
=6*--
=0
Transient conduction in a plate: T. 14/. Davie~
dr
2 dr
3 dr=--
where 13" may be estimated from equation (13) using the Newton-Raphson me~hod. The dimensionless time, R2r *, at which the value of 13" occurs may be calculated from equation (12), since at the penetration time the pre-penetration solution is matched with the new solution for the post-penetration period.
a-x x=o
Substituting for ao, 8" and noting that:
(~X)x=o = a l
Post-penetration
gives:
2(7--13)d13f 2(7--13)2 dal q- 8 (7--13) da2 = --al ax
dr
a]
3
dr
a]
dr
But:
solution
The heat balance integral (equation (6)) is simplified in this period because the conduction front is no longer propagating, dS*/dr is therefore zero and it is assumed that the new temperature distribution is of the form: b(r);
O=bo+blX+b2X2; da2
~2(7--13)a,(dal/dr)+a](d13/dr)~
= --[
x=
so the heat balance becomes: 4 (7 -- 13) d13 3
2 (7 -- 13)2 dal + --+1=0 dr 3 a] dT"
a]
1 f(7--13)2]
(7--13)
.q
1(~--13)2 2~(7--~) a~
3J
a]
d13
dO --=0 dX
b2 =
---2
-
(10)
(11)
The heat balance integral for the pre-penetration period is thus:
which may be rearranged to give an expression relating the dimensionless surface temperature and the dimensionless time in the post-penetration period:
id13 ~
li db' ~
=
-
fdr
(15)
7"•
For the particular case of convective heating and counteracting radiative cooling substitution of the expression for b ~gives:
(134 _ x4)
- ~
(B/R)(1 -- 13)n _ (134 X4)
R2r = R2r * + R
#
(v - 13) d13
--3. ((aiR)(1 -
O*
(I 2)
R 2 f B(1 -- 13")"/R -- (1 -- ~*')~ + - - lng - - - ~
13)" - (134_ x , ) } 2
7
The integral in equation (12) cannot be evaluated analytically but may easily be evaluated numerically using a standard quadrature procedure. The dimensionless heating time corresponding to any chosen value of the dimensionless surface temperature may then be calculated from equation (12) and the complete temperature profile estimated from equations (7) and (9).
Penetration
--b,=a(l--13)"--R(~4--X ')
so that equation (6) reduces to:
where a~ embodies the heat transfer mechanism at the surface. For the present problem equation (4) thus gives:
2[
X= 1
be = 13
3,
a l = R (134 -- k 4) --B(1 -- 13)n
0 =t3
bx
d13
giving the following general form of the heat balance integral for a one-dimensional thermal wave travelling through a semi4nfinite solid:
dI"=3
0
Application of these boundary conditions to equation (14) gives:
which may be further simplified using the relation: (1 _13)2
(14)
subject to the boundary conditions:
j
4(1 _13)2
i=0,1,2
time
The semi-infinite solid solution may be used up to the time at which the conduction front reaches the plate centreplane. This point in time may be estimated from equation (9c) with 6" = 1 and al given by equation (4). Thus, using the following notation: when 6" = 1,13 = 13", the expression for 13" is: a ( l - 13.)" - R (13" - X4) + 2 ( 7 - 13") = 0
03)
2 t B(l--13)"IR (I--13') J
(16)
The complete temperature profile in the post-penetration period is then available through equation (14). Results No analytical solution of the nonlinear conduction problem considered in this paper is available. The only published numerical solution of a problem in this class which may be used to demonstrate the accuracy of the present solutions is that by Crosbie and Viskanta.4Figure 8 of their paper includes a curve for the dimensionless surface temperature history of a plate initiaUy at absolute zero throughout which is suddenly exposed to radiative heating. Although this is a simpler case than those considered here, it is a highly nonlinear problem and as such presents a fair test of the present approximate solution technique. Table 1
Appl. Math. Modelling, 1985, Vol. 9, October 339
Transient conduction in a plate: T. W. Davies 1.o B----(i)
Nomenclature
equation (18)
(ii) equation ( 4 )
ai
/
/
0.8
f
bi
/ / ¢~ 0 . 6
0.4
0.2 10 -3
I 10-2
I I 10 "1 10 ° Rzl; o r B21:
I 10 t
I
102
Figure 1 Dimensionless surface temperature history of a semiinfinite solid subject (i) to convective heating and (ii) to convective heating with counteracting radiative cooling: B ----- 1, R = 1, "7 = 0.2, 2~ = 0.2, n = 4/3
Table I
Comparison of numerical and integral predictions of the dimensionless surface temperature history of a plate, initially at absolute zero, heated by thermal radiation
R=T
R2T
r/
(ref 4)
(equations (10), (15) and (17))
0.1 0.2 0.3 0.4 0.5 O.6 0.7 0.8
0.008 0.032 0.069 0.104 0.20 0.31 0.48 0.70 1.02
0.007 0.027 0.061 0.110 0.18 O.32
0.9
B /~ h k L n R t t* T
Ta Ti Tr Ts x X t~ /~ 7 6 6* e 0 ~. o r r*
0.49
0.72 1.11
time-dependent coefficients, pre-penetration temperature profile time-dependent coefficients, post-penetration temperature profile Blot number, hL/k heat transfer coefficient (=h(Ts -- Ta) n-l) pseudo heat transfer coefficient thermal conductivity of solid plate half-thickness exponent in convective flux function (1 ~
Discussion and shows a comparison of the surface temperature history taken from Figure 8 of reference 4 and that calculated using the integral solution with a similar boundary condition, i.e.:
al = - R ( 1 --74 )
(17)
where R is redefined as (oeLTra/k). For many engineering purposes the accuracy of the integral solution is seen to be reasonable. Figure 1 shows an example of the use of equation (12) to predict the surface temperature history of a plate (in the pre-penetration period) subjected to convective heating and radiative cooling. The process parameters used were R = 1, B = 1,7 = 0.2, ?~= 0.2, n = 4]3 (i.e. turbulent natural convection on an upward facing horizontal plate). This was part of the solution, with different process parameters, which was required for the optimization of the flash heating process. Included in Figure 1 is the surface temperature history of a plate heating up by convection, as predicted using the integral solution with the boundary condition: a, = --B(1 --/3)n
340
A p p l . Math. Modelling, 1985, V o l . 9, October
(18)
conclusions
An approximate solution of one-dimensional transient conduction in a solid with a nonlinear boundary condition of practical importance, has been presented in a form which allows the effects of the main process parameters to be studied for the purposes of process optimization. The accuracy of the solution appears to be adequate for these purposes, particularly so when one considers the uncertainties in the values of the physical properties involved in most industrial processes. The solution has the merit .of simplicity and numerical results may be obtained by means of a standard numerical integration procedure. References i Davies,T. W. 'Equipment for the study of the flash heating of particle suspensions', High Temp. Tech., 1984, 2, 141 2 Goodman, T. R. 'Advancesin heat transfer, vol. 1', Academic Press, New York, 1964, 51 3 Langford, D. 'The heat balance integral', Int. J. Heat Mass Transfer, 1973, 16, 2424 4 Crosbie, A. L. and Viskanta, R. 'Transient heating or cooling of a plate by combined convection and radiation', Int. J. Heat Mass Transfer, 1968, 11,305
During the flash dehydroxylation of powdered kaolinite it is desirable that a rapidly propagating thermal wave penetrates the cold powder particles in a way that raises the particle interior to the reaction temperature of 600°C without the particle exterior being heated beyond 1000°C. In a production unit this is achieved by performing the heat treatment in a device where particles are heated by convection from hot gas and are subject to heat loss by thermal radiation to cool walls. This paper concerns the fundamental heat transfer problem of the process, decoupled from the thermal effects of the dehydroxylation reaction. Using a plate as the approximation for the particle shape a semi-analytical solution for the plate temperature distribution is obtained as a function of the five dimensionless process parameters: Biot number, radiation number, wall/ gas and particle/gas temperature ratios and mode of convection. Accuracy is demonstrated by comparison with an existing numerical solution for the limiting case of pure radiative heating of a plate initially at absolute zero. Key words: mathematical models, transient conduction, flash heating, heat transfer
The flash heating of a powdered solid is a technique attracting growing interest in a number of mineral processing industries. One particular process is that of the flash dehydroxylation of kaolinite. Kaolinite, A1203.2SIO2.2H20, undergoes an endothermic loss of hydroxyl water at around 600°C. The product of this reaction is meta-kaolin, its physical form depending on the speed of the heating process. Although the fundamental size of crystalline kaolinite particles is submicron, in practice agglomerations of particles a few microns thick and in the form of flakes are the usual result of the powdering process. If a fast moving thermal wave is propagated through such particles it is possible to release steam in the particle interior faster than it can diffuse to the particle edges. Under these circumstances internal voids are created and the product has a low bulk density. If the voids are around the wavelength of light the product opacity is also high because of increased scattering. Such a product is attractive as a filler in the paint and paper industries. If the temperature of the
0307-904X/85/05337-04/$03.00 © 1985Butterworth& Co. (Publishers)Ltd.
meta-kaolin exceeds 1000°C a transformation to mullite begins with an associated and undesirable increase in bulk density and abrasiveness. The heat transfer requirements of the process are therefore that particles should experience a void-generating fastmoving isotherm of 600°C but should not be heated beyond about 1000°C. The process parameters which are available to achieve this are the reactor temperature distribution and the residence time distribution. The design of such reactors is proprietorial but involves passage of cold particles through a hot gas front near a wall under a cyclonic action. This gives some degree of variation in the heat treatment of large and small particles. A laboratoryscale furnace designed to replicate the heat transfer conditions in an industrial flash calciner has been described by Davies. l The heat transfer process may be modelled on the assumption that the plate-like particles are suddenly immersed in a hot flowing fluid and have a view of cold walls.
Appl. Math. Modelling, 1985, Vol. 9, October 337
Transient conduction in a plate: T. W. Davies They are heated by convection and as their temperature rises they begin to lose heat by thermal radiation. This nonlinear boundary condition renders the problem insoluble by analytical methods. Numerical solution is possible but numerical methods do not lend themselves easily to an extension of the problem to include a moving reaction front and a variation in physical properties. The heat transfer analysis described in this paper was developed with the aim of linking the process variables of gas, wall and initial particle temperatures and residence time with the thermal history of the particle. Problem def'mition It is required to predict the transient temperature distribution in a plate of thickness 21, which is initially at a uniform temperature Ti throughout. At time t = 0 the plate surfaces are suddenly subjected to forced or free convective heating from a fluid at temperature Ta and to radiative cooling to surroundings at temperature Tr, i.e. Ta> Ti> Tr. The problem of a plate of thickness 2L subjected to identical heat transfer at both faces is mathematically similar to that of a plate of thickness L which has one adiabatic face and one heat transfer face. The assumptions are that: • The heat conduction in the plate is one-dimensional. • The solid is homogeneous, isotropic and has constant physical properties. • The fluid temperature Ta, the wall temperature Tr, the heat transfer coefficient h, and the surface emissivity e, are constant. In dimensionless form the problem may be defined as follows:
00 Or
020 0X 2
0~
r>0
(1)
0 (5", O) == 1 00 ~-~(1, r) = 0 00 ----(0, r)=B(1--/3)n--R(~--h4)
aX
(2) (3)
1<~n<~4/3
(4)
where/3 is a function of r. Equation (4) is the nonlinear boundary condition for convective heating with counteracting radiative cooling. The objective of the analysis was to obtain a solution of equation (1) for any given values of the process parameters B. R, X and n. Typical ranges of these parameters which may occur in practice are 0.1 < B < 10, 0.1 < R < 10, 0~< ?~< 1, 1 <~n<~413.A solution of equation (1) with the boundary condition equation (4), is not available in analytical form. Nonlinear diffusion problems of this kind are, however, amenable to treatment by the heat balance integral technique described by Goodman. = This technique permits a semi-analytical solution using a route analogous to that used in the integral analysis of boundary layer flows. For the case at hand a thermal wave propagates into the plate from both heated faces at a rate which depends upon the boundary fluxes. The leading edge of each wave is defined in a manner analogous to that used in defining the edge of a boundary layer. (In an 'exact' analysis the thermal disturbance originating at the plate surface would be felt throughout the plate instantaneously.) At the beginning of
338 Appl. Math. Modelling, 1985, Vol. 9, October
the heat transfer process the waves will travel through the plate as they would through a semi-infinite solid. When they meet at the centre plane (the penetration time) the semi-infinite solid behaviour ceases and the boundary condition at the leading edge changes. The heat balance integral solution of this problem consequently has two components, the pre-penetration or semi-infinite solid solution and the post-penetration or finite solid solution. The solutions are matched at the penetration time. The preopenetration solution The reduction of equation (1) to an algebraic expression for the temperature distribution within the plate proceeds as follows. At time t when the conduction front has penetrated a distance x = 8, a heat balance on the thermal wave is obtained by integration of equation (1): 8*
0
This equation may be transformed into an ordinary differential equation for/3(0 by the application of Leibnitz's rule and the substitution of a suitable approximation for 0 (r, X) as follows: 6"
0 clX--Ox=n* -~-¢ -.
=\~'Y/x=6o-v..
OX/x=o (6)
0
Langford 3 has discussed the choices available for the approximation of the temperature profile across the thermal wave. A second-order polynomial was chosen for the present analysis. Assuming therefore that:
O = a o + a l X + a 2 X 2, a ( r ) ,
i=0,1,2
(7)
with the boundary conditions: X=8*
a0 --=0 aX
0=7
X=0
(8a)
(8b)
0 =/3
These boundary conditions may be used with equation (7) to give: ao =/3 as . . . .
(9a) al
(9b)
25*
2 (~, -
8* = - -
~)
(9c)
al
Substitution of equation (7) into equation (6) gives:
8, ~
~r(aoS* x
+ a~ --2
dr
which simplifies to:
8,3k
+a2-3--J--(ao+alS*+a~8*' )
=6*--
=0
Transient conduction in a plate: T. 14/. Davie~
dr
2 dr
3 dr=--
where 13" may be estimated from equation (13) using the Newton-Raphson me~hod. The dimensionless time, R2r *, at which the value of 13" occurs may be calculated from equation (12), since at the penetration time the pre-penetration solution is matched with the new solution for the post-penetration period.
a-x x=o
Substituting for ao, 8" and noting that:
(~X)x=o = a l
Post-penetration
gives:
2(7--13)d13f 2(7--13)2 dal q- 8 (7--13) da2 = --al ax
dr
a]
3
dr
a]
dr
But:
solution
The heat balance integral (equation (6)) is simplified in this period because the conduction front is no longer propagating, dS*/dr is therefore zero and it is assumed that the new temperature distribution is of the form: b(r);
O=bo+blX+b2X2; da2
~2(7--13)a,(dal/dr)+a](d13/dr)~
= --[
x=
so the heat balance becomes: 4 (7 -- 13) d13 3
2 (7 -- 13)2 dal + --+1=0 dr 3 a] dT"
a]
1 f(7--13)2]
(7--13)
.q
1(~--13)2 2~(7--~) a~
3J
a]
d13
dO --=0 dX
b2 =
---2
-
(10)
(11)
The heat balance integral for the pre-penetration period is thus:
which may be rearranged to give an expression relating the dimensionless surface temperature and the dimensionless time in the post-penetration period:
id13 ~
li db' ~
=
-
fdr
(15)
7"•
For the particular case of convective heating and counteracting radiative cooling substitution of the expression for b ~gives:
(134 _ x4)
- ~
(B/R)(1 -- 13)n _ (134 X4)
R2r = R2r * + R
#
(v - 13) d13
--3. ((aiR)(1 -
O*
(I 2)
R 2 f B(1 -- 13")"/R -- (1 -- ~*')~ + - - lng - - - ~
13)" - (134_ x , ) } 2
7
The integral in equation (12) cannot be evaluated analytically but may easily be evaluated numerically using a standard quadrature procedure. The dimensionless heating time corresponding to any chosen value of the dimensionless surface temperature may then be calculated from equation (12) and the complete temperature profile estimated from equations (7) and (9).
Penetration
--b,=a(l--13)"--R(~4--X ')
so that equation (6) reduces to:
where a~ embodies the heat transfer mechanism at the surface. For the present problem equation (4) thus gives:
2[
X= 1
be = 13
3,
a l = R (134 -- k 4) --B(1 -- 13)n
0 =t3
bx
d13
giving the following general form of the heat balance integral for a one-dimensional thermal wave travelling through a semi4nfinite solid:
dI"=3
0
Application of these boundary conditions to equation (14) gives:
which may be further simplified using the relation: (1 _13)2
(14)
subject to the boundary conditions:
j
4(1 _13)2
i=0,1,2
time
The semi-infinite solid solution may be used up to the time at which the conduction front reaches the plate centreplane. This point in time may be estimated from equation (9c) with 6" = 1 and al given by equation (4). Thus, using the following notation: when 6" = 1,13 = 13", the expression for 13" is: a ( l - 13.)" - R (13" - X4) + 2 ( 7 - 13") = 0
03)
2 t B(l--13)"IR (I--13') J
(16)
The complete temperature profile in the post-penetration period is then available through equation (14). Results No analytical solution of the nonlinear conduction problem considered in this paper is available. The only published numerical solution of a problem in this class which may be used to demonstrate the accuracy of the present solutions is that by Crosbie and Viskanta.4Figure 8 of their paper includes a curve for the dimensionless surface temperature history of a plate initiaUy at absolute zero throughout which is suddenly exposed to radiative heating. Although this is a simpler case than those considered here, it is a highly nonlinear problem and as such presents a fair test of the present approximate solution technique. Table 1
Appl. Math. Modelling, 1985, Vol. 9, October 339
Transient conduction in a plate: T. W. Davies 1.o B----(i)
Nomenclature
equation (18)
(ii) equation ( 4 )
ai
/
/
0.8
f
bi
/ / ¢~ 0 . 6
0.4
0.2 10 -3
I 10-2
I I 10 "1 10 ° Rzl; o r B21:
I 10 t
I
102
Figure 1 Dimensionless surface temperature history of a semiinfinite solid subject (i) to convective heating and (ii) to convective heating with counteracting radiative cooling: B ----- 1, R = 1, "7 = 0.2, 2~ = 0.2, n = 4/3
Table I
Comparison of numerical and integral predictions of the dimensionless surface temperature history of a plate, initially at absolute zero, heated by thermal radiation
R=T
R2T
r/
(ref 4)
(equations (10), (15) and (17))
0.1 0.2 0.3 0.4 0.5 O.6 0.7 0.8
0.008 0.032 0.069 0.104 0.20 0.31 0.48 0.70 1.02
0.007 0.027 0.061 0.110 0.18 O.32
0.9
B /~ h k L n R t t* T
Ta Ti Tr Ts x X t~ /~ 7 6 6* e 0 ~. o r r*
0.49
0.72 1.11
time-dependent coefficients, pre-penetration temperature profile time-dependent coefficients, post-penetration temperature profile Blot number, hL/k heat transfer coefficient (=h(Ts -- Ta) n-l) pseudo heat transfer coefficient thermal conductivity of solid plate half-thickness exponent in convective flux function (1 ~
Discussion and shows a comparison of the surface temperature history taken from Figure 8 of reference 4 and that calculated using the integral solution with a similar boundary condition, i.e.:
al = - R ( 1 --74 )
(17)
where R is redefined as (oeLTra/k). For many engineering purposes the accuracy of the integral solution is seen to be reasonable. Figure 1 shows an example of the use of equation (12) to predict the surface temperature history of a plate (in the pre-penetration period) subjected to convective heating and radiative cooling. The process parameters used were R = 1, B = 1,7 = 0.2, ?~= 0.2, n = 4]3 (i.e. turbulent natural convection on an upward facing horizontal plate). This was part of the solution, with different process parameters, which was required for the optimization of the flash heating process. Included in Figure 1 is the surface temperature history of a plate heating up by convection, as predicted using the integral solution with the boundary condition: a, = --B(1 --/3)n
340
A p p l . Math. Modelling, 1985, V o l . 9, October
(18)
conclusions
An approximate solution of one-dimensional transient conduction in a solid with a nonlinear boundary condition of practical importance, has been presented in a form which allows the effects of the main process parameters to be studied for the purposes of process optimization. The accuracy of the solution appears to be adequate for these purposes, particularly so when one considers the uncertainties in the values of the physical properties involved in most industrial processes. The solution has the merit .of simplicity and numerical results may be obtained by means of a standard numerical integration procedure. References i Davies,T. W. 'Equipment for the study of the flash heating of particle suspensions', High Temp. Tech., 1984, 2, 141 2 Goodman, T. R. 'Advancesin heat transfer, vol. 1', Academic Press, New York, 1964, 51 3 Langford, D. 'The heat balance integral', Int. J. Heat Mass Transfer, 1973, 16, 2424 4 Crosbie, A. L. and Viskanta, R. 'Transient heating or cooling of a plate by combined convection and radiation', Int. J. Heat Mass Transfer, 1968, 11,305