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Transient temperature distribution in a multilayer medium subject to radiative surface cooling
Transient temperature distribution in a multilayer medium subject to radiative surface cooling W. Y. D. Yuen Research
and Technology
Centre,
BHP
Steel,
Port
Kembla,
New
South
Wales,
Australia
In the hot processing offlat products, it is important to estimate accurately the temperature distribution in the workpieces with eficient algorithms to complement the process control system. Normally, this can be achieved by standard numerical means, such as the finite-dtflerence method, after a model for the conduction of heat in the workpieces subject to the appropriate boundary conditions is formulated. However, when the workpieces are stacked together or when they are wound into the form of a coil, the numerical computation becomes more involved. The thermal resistance at the contact interfaces, which arises from imperfect contact due to oxidation of the workpieces or air gaps, plays an important role in restricting the heat flow in the workpieces and must be included in the formulation of the model. In this paper, the heat balance integral method is applied to solve the above problem, with special emphasis on the modelling of the thermal resistance at the contact interfaces. The resulting solution procedure is simple and the numerical scheme is eficient. Good accuracy, s@cient for general engineering applications, is obtained. This study may be applied to general heat conduction problems in a multilayer medium subject to nonlinear boundary conditions, and in particular, finds application in the hot rolling process. Keywords: conduction,
contact resistance, heat balance integral, hot rolling
1. Introduction In the hot processing of flat products, precise control of the workpiece temperature is important in achieving products of desired properties. Associated with this objective is the need to estimate accurately the transient temperature distribution in the workpieces when they are stacked together or wound up in the form of a coil. Examples are the hot charging operation’ in which slabs produced from the continuous casting process are stacked and transferred to the hot strip mill for further processing, and the hot rolling process in which the workpiece is wound into a coil at the coil box or downcoiler.2 In these cases, the heat flow in the workpiece, which is induced by the surface heat losses on the top and bottom of the stack or outside and bore of the coil, can be regarded as unidirectional as a first approximation. When the elapsed time under consideration is not overly long such that the thermal effect has not penetrated through the entire stack of the workpieces or all the wraps in the coil, a semi-infinite medium can be assumed. However, because of the existence of air gaps
Address reprint requests to Dr. Yuen at the Research and Technology Centre, BHP Steel, P.O. Box 77, Port Kembla, New South Wales, 2505, Australia Received 1993
0
7 September
1992; revised
1994 Butterworth-Heinemann
17 June
1993; accepted
28 July
between the workpieces/wraps of the coil (arising from a low contact pressure) and because of the possibility of oxidation of the workpiece surfaces (due to the elevated temperature involved during processing), imperfect contact between adjacent workpieces or wraps of the coil results. Hence, a suitable model for this process is to consider the heat transfer, induced by surface heat losses, in a multilayer semi-infinite medium with thermal resistance at the interface of the layers. Again, because of the high temperature involved, the surface heat losses are dominated by radiation. The heat balance integral method is a powerful technique, suitable for solving nonlinear diffusion problems and/or those with nonlinear boundary conditions. Goodman3 provided a comprehensive review of the method as applied to heat transfer. Other applications of the technique have been reported by Landford, who proposed a criterion for the estimation of the accuracy of the solutions obtained from the various approximated temperature profiles; by Sfeir,5 who examined the twodimensional steady-state conduction in a fin; and by Davies6 who examined the transient one-dimensional heat transfer in a plate with convective and radiative boundary conditions as applied to the flash heating process. In this study, the heat balance method is applied to examine the conduction in a multilayer medium with thermal resistance at the contact interfaces. Although only radiative surface cooling is considered below, the formulation for other boundary conditions (such as con-
Appl.
Math.
Modelling,
1994,
Vol. 18, February
93
Temperature distribution in a multilayer medium
with radiative surface cooling: W. Y. D. Yuen
vection or with a prescribed heat flux) is a trivial extension of the approach discussed herein.
2. Formulation
and solution
Consider a semi-infinite medium consisting of multiple layers and having an initially uniform temperature, subject to surface heat losses that are primarily dominated by radiative effects. The transient temperature distribution in the medium is examined below with the assumptions that 1. the layers are homogeneous and have the same physical properties and thickness; 2. the thermal properties of the layers are constant for the temperature range being considered; and 3. the imperfect contact between the adjacent layers, which may arise from an air gap or contamination (such as an oxide layer or lubricant), can be modelled by a thermal resistance with neither heat capacity nor physical dimension. In the following analysis, the temperature drop of the medium from its initial temperature T, will be nondimensionalized by the initial temperature T,, the linear dimension by the thickness of each layer h, and the elapsed time by h2/cx,where CI is the thermal diffusivity of the medium. Hence, the thermal conduction equation application for the bulk of the medium in its nondimensional form is
de a28 aT ap
-=-
condition
Assume, at a certain elapsed time r, that the penetration has reached the nth layer (from the but not beyond the (n+ 1)th layer (thus n I 6 and that the temperature in each layer can be imated by a third-order polynomial of the form
thermal surface) I n+ l), approx-
Yo{ao + bo(6 - 5) + co@ - O2 + (6 - O”> fornI456
1
yi{ai + b,(n - i + 1 - 5) + c,(n - i + 1 - o2 + (n - i + 1 - r)“} for (n - i) I 4 I (n - i + l), i = 1,2,. . . , n (10)
0 cct r=-
h2 and
Here T z T(x, t) is the temperature in the medium, t the time variable, and x the spatial coordinate measured from the surface of the medium as shown in Figure 1. The heat balance integral technique3 is now applied to equation (l), giving ae(0, r)
(6)
a<
where the coefficients yi, ai, bi, and ci are time-dependent functions. In general, equation (10) does not satisfy equation (l), but the coefficients yi, ai, b,, and ci of equation (10) will be determined such that they will satisfy the surface boundary condition, the conditions at the interfaces of the layers and those at the penetration depth, as well as equation (6), which relates the heat flux at the surface to the rate of change of the heat content “stored” in the body, an average condition that has been deduced from the exact form of equation (1). By differentiating equation (8) with respect to r (and using equation (1)) an additional relation is obtained:
a2w, 4
o
(11)
-=
at2 It is obvious,
from
equations
(8), (9), and
(ll),
that
a, = b, = co = 0. At a contact interface, say between the ith and (i + 1)th layer, continuity of heat flux requires
where d (7)
edg s0
Appl.
(9)
5 are the nondimensionalized
8=1_g
dz
(8)
and
(2)
where 8 = t!J(&z), T, and variables, given by
94
e(s, T) = 0
8=
M, 0) = 0
1=
Here, 6 E 6(~) is the thermalpenetration depth and equation (6) has been derived using the properties associated with 6 deduced from its definition:
(1)
with the initial
dl _=-_
Figure 1. The thermal model: a multilayer medium subject to surface heat loss.
ae(i+, at
Math.
Modelling,
1994,
Vol. 18, February
T)
ae(i-, 7) a<
e(i+, 5) R
e(i-, 2)
(12)
Temperature
distribution in a multilayer medium
where
RJ!2
(13)
h
Here R, is the contact resistance at the interfaces and k the thermal conductivity of the layers. Two more boundary conditions are required at each interface in order to determine uniquely the coefficients of equation (10). They are known as the derived boundary conditions3 in contrast with the natural boundary conditions of equation (12). Although a logical approach to obtain these is by differentiating equation (12) with respect to 7, here they are derived by assuming the second and third derivatives of the temperature with respect to 5 on both sides of the interface to be equal, thus
7) 8’8(i-, 7) ap = ap
a20(i+,
(14)
and
a3e(i+,7) a30(i-,~) a(3 = ap
Yi =
for i = 1,2,. . . , n
YO
ae(O, 7) _ (1 - e(O, 7114
~i+l = ~i + bi(l + R) + Ci(l + 2R) + 1 + 3R
(17)
bi+l = bi + 2ci + 3
(18)
ci+1=ci+3
(19)
a, = (6 - n)2(6 - n + 3R)
(20)
b, = 3(6 - n)2
(21)
c1 = 3(6 - n)
(22)
On summing the series resulting from the recursive relations of equations (17)-(19), it can be shown that, for i= 1,2 ,..., n, oi = ~‘(c#J+ 3R) + (i - 1){342(1 + R) - 34 + l} +
ji(i - 1)(2#(1 + R) - l}
+ $i(i -
1)(2i - l)(l + R)
bi = 3~$~ + 64(i - 1) + 3(i - 1)2 ci = 3(+ + i - 1)
(27)
L=-
k &ahT;
Here E is the emissivity, G the Stefan-Boltzmann constant, T, the ambient temperature (K), and L can be regarded as a normalized surface thermal resistance. For situations where radiative effects dominate, the term involving T, can be neglected because then T,/T, < 1. On substitution of equation (10) into equation (27), a fourth-order polynomial in y. is obtained: {(a,, + b, + c, + l)y, - l}” - (b, + 2c, + 3)Ly, = 0 (29) The next step is to relate the elapsed time 7 to the thermal penetration depth 6. From equation (7) (with equation (lo)), the following is obtained after summing the resulting series: 2 = yo{i($ + n)4 + n(n + l)R[$z(n - 1) + 4(n - I) +
%“I>
(30)
Further, the following differential 6 can be deduced from equation dz
Iv, +
dd
3~0(4 + 4'
p=
equation (6):
relating
7
and
YoV2
(31)
where 6yo{L(4 + n) + 2(I - &)“C(4 + n)2 + R(n2 - n + 2#n)]}
VI = -
3(4 + n)2y, L + 4e,(1 -
v2 = (4 +
(32)
es)3
n)3 + n(n + l)(n - 1 + 3d)R
(33)
and
8, = e(0, 7) = yo{(4 +
n)3 + nR[g2n
- l)(n - 1)
+ 3&n - 1) + 342-J)
(34)
The right-hand side of equation (31) consists of the only unknown 6; thus 7 may be readily computed in terms of 6 by numerical integration. Of interest also are the average temperature drop [ of each layer, which may be deduced from equation (lo), where for the ith layer,
(23) (24)
iit71
=
et,
7)
dt
si i-l
(25)
where 4=6-n
vn)4
where
with
and
-
L
at
(16)
and for i = 1,2, . . . , n - 1,
W. Y. D. Yuen
Having expressed yi, ai, bi, and ci in terms of y. and 6, a relation between y. and 6 (through a,, b,, and c,) is now derived based on the radiative surface boundary condition:
(15)
Again, the above conditions (with those of equation (12)) would satisfy only the average condition of equation (6) but not the exact form of equation (1). It will be shown in Section 4 that the resulting solution procedure using these boundary conditions is much simpler than those using the more “logical” boundary conditions, yet it produces largely similar results. Expressions for the coefficients a,, bi, ci, and yi can now be deduced from equations (lo), (12), (14), and (15), giving:
wirh radiative surface cooling:
YO(an+l-i
+
@“+I-i
$yo(a - n)4
(26)
Appl. Math.
Modelling,
+
hn+l~i
+
a)
for i = 1,2, _. . , n fori=n+ 1
=
1994,
Vol. 18, February
(35)
95
Temperature
distribution
in a multilayer
medium
and the cumulative heat loss Q from the medium, may be obtained by numerical integration:
Qa
3 s’oy062 d7
which
for 6 5 1
i&yo(b,+2c,+3)dz
q-K=
with radiative surface cooling:
(36)
ford>1
It should be noted that for the special case of the absence of the thermal resistance at the interface (R = 0), equation (12) takes a degenerated form. The first part of the equation still holds while the last part becomes e(i-, z) = 0(i+, z), i.e., continuity of temperature across the interface is preserved because perfect contact is assumed. All subsequent solutions derived are still valid with R = 0. In fact, for this degenerated case, the solution of the problem becomes one of applying the heat balance integral method to a homogeneous medium by dividing it into hypothetical layers, the temperature profile in each of which is approximated by a third-order polynomial. This provides improved accuracy over the single polynomial normally adopted in the application of this method, but the resulting solution is naturally more complicated.
3. Numerical implementation
W. Y. D. Yuen
3. calculate 2, vl, and v2 from equations (30), (32), and (33) respectively after the surface temperature drop, 8,, is determined from equation (34); 4. determine the elapsed time z by numerical integration of equation (31); and finally 5. compute the temperature distribution 13from equation (10) with ai, b,, and ci determined from equations (23x25). Some numerical results are illustrated below. The variations of the penetration depth and surface temperature with the elapsed time for various values of the thermal resistance are shown in Figure 2. It can be seen that the penetration of the thermal effects decreases sharply and the corresponding surface temperature drop increases as the interface resistance R increases. Hence, an increase in the interface resistance offers the advantage of preserving the bulk temperature of the medium, especially during long elapsed time, while causing the temperature drop near the surface to increase. This is illustrated in Figure 3 in which the temperature distribu-
and results 0.1
Based on the above analysis, a computer program has been written to calculate the temperature distribution 0 in the medium and the elapsed time z for a specified interface thermal resistance R, surface thermal resistance L, and penetration depth 6. The computational procedure is as follows:
0.06 o 0.06 0.04 0.02 0.0
1. Calculate a,, b,, and c, from equations (23H25); 2. solve the fourth-order polynomial equation (29) for y0 (a Newton-Raphson iterative technique has been used);
1
0
2
4
3
(a)
5
6
7
5
6
7
R=O.
0.1 0 0.05
\ 1
3
1
’
‘20 ’
1 4
5
1
‘50 ’
0.0
0
1
2
4
3
6
F
T
(b) (a)
0.0
0
Penetration
depth
1
2
3
R=l.
(1).
I
1
I
I
I
(8) versus time
4
I
I
5
6
F
T
(c) (b)
Surf. temp.
drop (8,)
versus time (7).
Figure 2. perature
96
Variations of the penetration depth drop with the elapsed time (L = 15).
Appl.
Math.
Modelling,
1994,
and surface
Figure 3. tem-
Vol. 18, February
values
(L=15).
R=20.
Temperature distribution 0 near the surface for various of the elapsed time T and interface thermal resistance R
Temperature
disrriburion
in a mu/Mayer
tion is plotted for three values of the interface resistance. The temperature jump at the interface is, indeed, high for large values of R and for long elapsed time r. At r = 6, six layers of the medium will have a temperature drop exceeding 1% of its initial (absolute) value when R = 0. The same condition is detected in only four layers when R = 1, and in two layers when R is increased to 20. The average temperature drop of the layers near the surface is plotted against the elapsed time in Figure 4 for several values of R. The “sacrificial effect” of the outermost layer in conserving the thermal energy in the rest of the medium when the interface resistance becomes large is evident. The average temperature drop in the first layer is found to increase with increasing R while all other layers display a reverse trend. It is also interesting to note that the strip temperature drop in these other layers increases at a faster rate (with time) with a higher interface resistance and will eventually surpass that with a lower interface resistance when the elapsed time is sufficiently long (for instance, the temperature drop for R = 1 overtakes that for R = 0 in the second layer at r = 4.4 as shown in Figure 4(a)). When the interface
0.20
-
0.15
-
0.10
-
Layer No. 1 Layer No. 2
-.-
0
2
1
3 7 (a)
0.05
I
0.04
-
0.03
-
0.02
-
-.
I -
,
Layers
1
,
4
5
6
1 & 2.
,
,
1
,
,
Loyer No. 3 Layer No. 4
medium
with radiative surface cooling: I
,
1
,
I
Approx.:
1
0
2
3
,
W. Y. D. Yuen I
q=r/L
4
5
6
T
Figure 5. elapsed
Variation of the cumulative surface heat loss
q with
the
time 5 (L = 15).
resistance is high such that the heat flow between the first and second layers can be neglected, the average temperature drop of the first layer ii can be approximated from a heat balance consideration, in which case it can be shown that ii = q, where q is given by equation (36). An examination of the results for this approximation, as indicated in Figure 4(a), reveals an excellent agreement with the full theory when R is large. The cumulative heat loss from the surface of the medium, plotted in Figure 5, indicates that a medium with a high interface thermal resistance will lose less heat than that with a low interface resistance, especially when the elapsed time is long. It may be argued, when the surface temperature drop is small, that the rate of heat loss from the surface would be fairly constant and the cumulative heat loss will then be linearly proportional to the elapsed time; thus, q = t/L. This level of approximation, as shown in Figure 5, proves to be grossly unsatisfactory even when the elapsed time is small. A constant surface thermal resistance L has been used in the above illustrations (L = 15). The effects of variations in the surface thermal resistance are illustrated in Figure 6. It can be seen that an increase in the surface thermal resistance L would reduce the thermal penetration into the medium. In addition, the temperature drop in the layers is also substantially reduced. 4. Effect of alternative boundary conditions
G
As mentioned in Section 2, two extra relations in addition to the boundary conditions of equation (12), which relate the heat fluxes on both sides of the interface, are required to determine the coefficients of equation (10). Instead of those used in Section 2 (equations (14) and (15)) an alternative and more logical approach is to deduce these relations by differentiating equation (12) with respect to r, and using equation (l), giving: (b)
d38(i+, 2) = d30(i-, 7) = A d20(i+, T) d28(i-, 7) at3 R at2 - at2 x3
Layers 3 & 4.
Figure 4. Variation of the average temperature drop [of the layers with the elapsed time T (L = 15).
(37)
Appl.
Math.
Modelling,
1994,
Vol. 18, February
97
Temperature
-
distribution
-.-
in a multilayer
medium
with radiative surface cooling:
W. Y. D. Yuen
R=7_0
6-
0.08
or 0
’
’ 1
’
’ 2
’
’ 3
’
’ 4
’
’ 5
m 0.06 ’
6
t (a)
Penetration
depth
(d) versus time (7).
(a)
I
I
0
1
I
I
2
R=l.
4
3
4
5
6 B.C. of eqn(37) B.C. of eqns( 14)&(15)
T (b)
Ave. temp.
drop (0
vs. time (T) CR=ZOl.
Figure 6. Variations of the penetration depth and average temperature drop of the layers with the elapsed time for various values of the surface thermal resistance L.
i
Following the same procedure and using the above relations in place of equations (14) and (15), it can be shown that the coefficients ai and bi are given by the same expressions as derived before (equations (17), (18), (20), and (21)), while ci is given by ci+l
=
Ci +
3(l + R,
(b)
(38)
R=20.
Figure 7. Effect of the choice of the boundary conditions at the interfaces on the temperature distribution 8.
with ci = 3(6 - n + R)
(39)
and y0 can, again, be determined from equation (29). An inherent complication with this formulation, however, is that the temperature distribution in the medium varies appreciably as the penetration depth 6 “crosses” an interface; i.e., for the nth interface, the temperature distribution at 6 = n- will be significantly different from that at 6 = n+. This arises because the approximation by cubic polynomials of the temperature distribution (equation (10)) would not satisfy all boundary conditions of equations (12) and (37) without introducing a discontinuity as 6 crosses an interface. As an illustration, the temperature distributions at 6 = n- and 6 = n+ for the first four layers are shown, together with the results obtained earlier (using the boundary conditions of equations (14) and (15)), in Figure 7. It can be seen that the latter results do not exhibit variations as 6 crosses an interface and always lie between the limits given by the former ones. With the boundary conditions of equation (37), an additional elapsed time with further heat loss at the surface will be experienced as 6 crosses an interface because the temperature distribution in the medium has changed. This must be taken into account and added to
98
-.-.-
Appl.
Math.
Modelling,
1994,
Vol. 18, February
the accumulated values deduced previously. From equation (6), the time taken for 6 to cross the nth interface is
s c5=n+
z=
-
dA (40)
d=n- (No, z)lX
and the total heat loss during this elapsed 6=nf dA = &_+ - II,=,4= s 6=n-
time is (41)
The progressive change in temperature distribution as 6 varies from n- to n+ must be deduced to evaluate the integral in equation (40). It is stipulated, from physical reasoning, that the temperatures vary in a manner corresponding to the thermal resistance at the nth interface being increased from zero to its full value R as 6 crosses the nth interface (6 = n- to 6 = n’). Thus, the solution procedure developed earlier is applicable (using the boundary condition of equation (37)), with equations (20) and (39) replaced by, respectively, a, = (6 - n)‘(6 - n + 3R’)
(42)
Temperature
distribution
in a multilayer
and
medium
with radiative surface cooling:
0.30
cr = 3(6 - n + R’)
(43)
where R’ changes from 0 to R as 6 crosses the interface. Based on this algorithm, the variation of the penetration depth 6, surface temperature drop 8,, average temperature drop at the outermost layer cl, and the cumulative heat loss 4 with time r are plotted in Figures 8 and 9, superimposed with the earlier results (according to the boundary conditions of equations (14) and (15)) for comparison. It can be seen that the differences resulting from the two approaches are small (except for the penetration depth). This is not unexpected because the governing heat transfer mechanism of equation (6) has been satisfied in both approaches. It is noted that the simpler conditions of equations (14) and (15) do not require special considerations as 6 crosses an interface, and the solution procedure is much simpler (it can be easily shown from equations (lo), (23t(25), and (27) that the are temperature distributions at 6 = n- and 6=nt
10 8
-.-
6 0 4 2 0
0
3
2
1
4
5
0.25
-
0.15
-
Penetration
’
/
I
1
I
0”
o.oy 0 1
’
1
’
2
’
’
’
3
’
4
’
’
5
’
’
6
T (b)
0
Surface
1
(c)
temp.
2
Ave. temp.
drop (8,)
vs. time
3
4
drop ({,)
I
,
I
I
(
I
,
I
I
I
B.C. of
0.25
0
eqn(37) B.C. of eqns(14)&(15)
R=“\&
0.20
Cs 0.15
0.10
I
0
I 1
I
I
2
I
I
3
1
I
4
5
1
-.-I
6
T
Figure 9. interfaces
Effect of the choice of the boundary on the cumulative surface heat loss.
conditions
at the
identical in this case). Hence, they have been used in the formulation given in Section 2. It should be noted that for the special case of R = 0, the comments made at the end of Section 2 apply here. In addition, the special consideration when the thermal front crosses an interface discussed above is no longer required as it can be shown that the temperature distribution for 6 = n- is identical to that for 6 = n+ in this instance.
5. Conclusion
depth (6) versus time (1).
’
,
6
T (a)
I
W. Y. D. Yuen
(1).
5
The transient temperature distribution in a multilayer semi-infinite medium with radiative cooling on its surface has been examined using the heat balance integral method. A thermal resistance at the layer interface has been included in the model. It has been shown that two different forms of the derived boundary conditions produce largely similar results although one form requires substantially more computational effort than the other. Hence a judicious choice of the derived boundary conditions can reduce the numerical complexity of the solution. The application of the heat balance integral method to this problem does not require the nonlinear boundary condition to be linearized, and the numerical procedure is simple: It only requires the numerical solutions of a fourth-order polynomial and a first-order differential equation. The effect of the thermal resistance at the layer interface on the temperature distribution has been discussed. It has been observed that a high thermal resistance would help conserve the thermal energy in the medium during a long elapsed time, while the surface layer would, as a result, cool more rapidly. This study finds applications in the hot rolling process.
6
vs. time (T).
Figure 8. Effects of the choice of the boundary conditions at the interfaces on the penetration depth, surface temperature drop, and average temperature drop of the first layer.
Acknowledgment The author wishes to thank the management of BHP Steel, Sheet and Coil Products Division, for permission to publish the information contained in this paper.
Appl.
Math.
Modelling,
1994,
Vol. 18, February
99
Temperature
distribution
in a multilayer
medium
with radiative surface cooling:
Nomenclature
&
a,(z),hi(z),ci(z) coefficients associated with the temperature distribution of the (n + 1 - i)th layer, equation (10) h thickness of each layer k thermal conductivity of the medium L nondimensionalized surface thermal resistance the layer (from the surface) that the n(r) thermal penetration has just passed through nondimensionalized heat loss from the 4(r) surface of the medium heat loss from the surface of the medium Q(t) R nondimensionalized thermal resistance in between layers contact thermal resistance in between RC layers t elapsed time temperature distribution in the medium T(x, r) (K) initial temperature of the medium (K) T, ambient temperature (K) T, X distance from the surface of the medium c1 thermal diffusivity of the medium coefficient associated with the temperaYitz) ture distribution of the (n + 1 - i)th layer, equation (10) nondimensionalized penetration depth 6(r)
100
Appl.
Math.
Modelling,
1994,
Vol. 18, February
ii@)
A vt,
v2
W. Y. 0. Yuen
emissivity at the surface of the medium nondimensionalized average temperature drop of the ith layer nondimensionalized temperature drop nondimensionalized surface temperature drop, 19(0,r) term defined in equation (7) terms defined in equations (32) and (33) respectively nondimensionalized x-coordinate Stefan-Boltzmann constant nondimensionalized elapsed time 6-n
References 1 2
3
Yabuuchi, K. Progress of the iron and steel technologies in Japan in the past decade. Trans. ISIJ 1985, 25, 733-738 Hewitt, E. C. Developments in rolling mill technology. The Steel Industry in the Eighties, Book No. 267, The Metals Society, London, 1980, pp. 86103 Goodman, T. R. Application of integral methods to transient nonlinear heat transfer. Advances in Heat Transfer, Vol. 1. Academic Press, 1964, pp. 51-122 Langford, D. The Heat Balance Integral Method. In?. J. Heat Mass Transfer 1973, 16, 24242428 Sfeir, A. A. The Heat Balance Integral in Steady-State Conduction. ASME J. Heat Transfer 1976, 98, 46C-470 Davies, T. W. Transient conduction in a plate with counteracting convection and thermal radiation at the boundaries. Appl. Math. ModeRing 1985, 9, 337-340
and Technology
Centre,
BHP
Steel,
Port
Kembla,
New
South
Wales,
Australia
In the hot processing offlat products, it is important to estimate accurately the temperature distribution in the workpieces with eficient algorithms to complement the process control system. Normally, this can be achieved by standard numerical means, such as the finite-dtflerence method, after a model for the conduction of heat in the workpieces subject to the appropriate boundary conditions is formulated. However, when the workpieces are stacked together or when they are wound into the form of a coil, the numerical computation becomes more involved. The thermal resistance at the contact interfaces, which arises from imperfect contact due to oxidation of the workpieces or air gaps, plays an important role in restricting the heat flow in the workpieces and must be included in the formulation of the model. In this paper, the heat balance integral method is applied to solve the above problem, with special emphasis on the modelling of the thermal resistance at the contact interfaces. The resulting solution procedure is simple and the numerical scheme is eficient. Good accuracy, s@cient for general engineering applications, is obtained. This study may be applied to general heat conduction problems in a multilayer medium subject to nonlinear boundary conditions, and in particular, finds application in the hot rolling process. Keywords: conduction,
contact resistance, heat balance integral, hot rolling
1. Introduction In the hot processing of flat products, precise control of the workpiece temperature is important in achieving products of desired properties. Associated with this objective is the need to estimate accurately the transient temperature distribution in the workpieces when they are stacked together or wound up in the form of a coil. Examples are the hot charging operation’ in which slabs produced from the continuous casting process are stacked and transferred to the hot strip mill for further processing, and the hot rolling process in which the workpiece is wound into a coil at the coil box or downcoiler.2 In these cases, the heat flow in the workpiece, which is induced by the surface heat losses on the top and bottom of the stack or outside and bore of the coil, can be regarded as unidirectional as a first approximation. When the elapsed time under consideration is not overly long such that the thermal effect has not penetrated through the entire stack of the workpieces or all the wraps in the coil, a semi-infinite medium can be assumed. However, because of the existence of air gaps
Address reprint requests to Dr. Yuen at the Research and Technology Centre, BHP Steel, P.O. Box 77, Port Kembla, New South Wales, 2505, Australia Received 1993
0
7 September
1992; revised
1994 Butterworth-Heinemann
17 June
1993; accepted
28 July
between the workpieces/wraps of the coil (arising from a low contact pressure) and because of the possibility of oxidation of the workpiece surfaces (due to the elevated temperature involved during processing), imperfect contact between adjacent workpieces or wraps of the coil results. Hence, a suitable model for this process is to consider the heat transfer, induced by surface heat losses, in a multilayer semi-infinite medium with thermal resistance at the interface of the layers. Again, because of the high temperature involved, the surface heat losses are dominated by radiation. The heat balance integral method is a powerful technique, suitable for solving nonlinear diffusion problems and/or those with nonlinear boundary conditions. Goodman3 provided a comprehensive review of the method as applied to heat transfer. Other applications of the technique have been reported by Landford, who proposed a criterion for the estimation of the accuracy of the solutions obtained from the various approximated temperature profiles; by Sfeir,5 who examined the twodimensional steady-state conduction in a fin; and by Davies6 who examined the transient one-dimensional heat transfer in a plate with convective and radiative boundary conditions as applied to the flash heating process. In this study, the heat balance method is applied to examine the conduction in a multilayer medium with thermal resistance at the contact interfaces. Although only radiative surface cooling is considered below, the formulation for other boundary conditions (such as con-
Appl.
Math.
Modelling,
1994,
Vol. 18, February
93
Temperature distribution in a multilayer medium
with radiative surface cooling: W. Y. D. Yuen
vection or with a prescribed heat flux) is a trivial extension of the approach discussed herein.
2. Formulation
and solution
Consider a semi-infinite medium consisting of multiple layers and having an initially uniform temperature, subject to surface heat losses that are primarily dominated by radiative effects. The transient temperature distribution in the medium is examined below with the assumptions that 1. the layers are homogeneous and have the same physical properties and thickness; 2. the thermal properties of the layers are constant for the temperature range being considered; and 3. the imperfect contact between the adjacent layers, which may arise from an air gap or contamination (such as an oxide layer or lubricant), can be modelled by a thermal resistance with neither heat capacity nor physical dimension. In the following analysis, the temperature drop of the medium from its initial temperature T, will be nondimensionalized by the initial temperature T,, the linear dimension by the thickness of each layer h, and the elapsed time by h2/cx,where CI is the thermal diffusivity of the medium. Hence, the thermal conduction equation application for the bulk of the medium in its nondimensional form is
de a28 aT ap
-=-
condition
Assume, at a certain elapsed time r, that the penetration has reached the nth layer (from the but not beyond the (n+ 1)th layer (thus n I 6 and that the temperature in each layer can be imated by a third-order polynomial of the form
thermal surface) I n+ l), approx-
Yo{ao + bo(6 - 5) + co@ - O2 + (6 - O”> fornI456
1
yi{ai + b,(n - i + 1 - 5) + c,(n - i + 1 - o2 + (n - i + 1 - r)“} for (n - i) I 4 I (n - i + l), i = 1,2,. . . , n (10)
0 cct r=-
h2 and
Here T z T(x, t) is the temperature in the medium, t the time variable, and x the spatial coordinate measured from the surface of the medium as shown in Figure 1. The heat balance integral technique3 is now applied to equation (l), giving ae(0, r)
(6)
a<
where the coefficients yi, ai, bi, and ci are time-dependent functions. In general, equation (10) does not satisfy equation (l), but the coefficients yi, ai, b,, and ci of equation (10) will be determined such that they will satisfy the surface boundary condition, the conditions at the interfaces of the layers and those at the penetration depth, as well as equation (6), which relates the heat flux at the surface to the rate of change of the heat content “stored” in the body, an average condition that has been deduced from the exact form of equation (1). By differentiating equation (8) with respect to r (and using equation (1)) an additional relation is obtained:
a2w, 4
o
(11)
-=
at2 It is obvious,
from
equations
(8), (9), and
(ll),
that
a, = b, = co = 0. At a contact interface, say between the ith and (i + 1)th layer, continuity of heat flux requires
where d (7)
edg s0
Appl.
(9)
5 are the nondimensionalized
8=1_g
dz
(8)
and
(2)
where 8 = t!J(&z), T, and variables, given by
94
e(s, T) = 0
8=
M, 0) = 0
1=
Here, 6 E 6(~) is the thermalpenetration depth and equation (6) has been derived using the properties associated with 6 deduced from its definition:
(1)
with the initial
dl _=-_
Figure 1. The thermal model: a multilayer medium subject to surface heat loss.
ae(i+, at
Math.
Modelling,
1994,
Vol. 18, February
T)
ae(i-, 7) a<
e(i+, 5) R
e(i-, 2)
(12)
Temperature
distribution in a multilayer medium
where
RJ!2
(13)
h
Here R, is the contact resistance at the interfaces and k the thermal conductivity of the layers. Two more boundary conditions are required at each interface in order to determine uniquely the coefficients of equation (10). They are known as the derived boundary conditions3 in contrast with the natural boundary conditions of equation (12). Although a logical approach to obtain these is by differentiating equation (12) with respect to 7, here they are derived by assuming the second and third derivatives of the temperature with respect to 5 on both sides of the interface to be equal, thus
7) 8’8(i-, 7) ap = ap
a20(i+,
(14)
and
a3e(i+,7) a30(i-,~) a(3 = ap
Yi =
for i = 1,2,. . . , n
YO
ae(O, 7) _ (1 - e(O, 7114
~i+l = ~i + bi(l + R) + Ci(l + 2R) + 1 + 3R
(17)
bi+l = bi + 2ci + 3
(18)
ci+1=ci+3
(19)
a, = (6 - n)2(6 - n + 3R)
(20)
b, = 3(6 - n)2
(21)
c1 = 3(6 - n)
(22)
On summing the series resulting from the recursive relations of equations (17)-(19), it can be shown that, for i= 1,2 ,..., n, oi = ~‘(c#J+ 3R) + (i - 1){342(1 + R) - 34 + l} +
ji(i - 1)(2#(1 + R) - l}
+ $i(i -
1)(2i - l)(l + R)
bi = 3~$~ + 64(i - 1) + 3(i - 1)2 ci = 3(+ + i - 1)
(27)
L=-
k &ahT;
Here E is the emissivity, G the Stefan-Boltzmann constant, T, the ambient temperature (K), and L can be regarded as a normalized surface thermal resistance. For situations where radiative effects dominate, the term involving T, can be neglected because then T,/T, < 1. On substitution of equation (10) into equation (27), a fourth-order polynomial in y. is obtained: {(a,, + b, + c, + l)y, - l}” - (b, + 2c, + 3)Ly, = 0 (29) The next step is to relate the elapsed time 7 to the thermal penetration depth 6. From equation (7) (with equation (lo)), the following is obtained after summing the resulting series: 2 = yo{i($ + n)4 + n(n + l)R[$z(n - 1) + 4(n - I) +
%“I>
(30)
Further, the following differential 6 can be deduced from equation dz
Iv, +
dd
3~0(4 + 4'
p=
equation (6):
relating
7
and
YoV2
(31)
where 6yo{L(4 + n) + 2(I - &)“C(4 + n)2 + R(n2 - n + 2#n)]}
VI = -
3(4 + n)2y, L + 4e,(1 -
v2 = (4 +
(32)
es)3
n)3 + n(n + l)(n - 1 + 3d)R
(33)
and
8, = e(0, 7) = yo{(4 +
n)3 + nR[g2n
- l)(n - 1)
+ 3&n - 1) + 342-J)
(34)
The right-hand side of equation (31) consists of the only unknown 6; thus 7 may be readily computed in terms of 6 by numerical integration. Of interest also are the average temperature drop [ of each layer, which may be deduced from equation (lo), where for the ith layer,
(23) (24)
iit71
=
et,
7)
dt
si i-l
(25)
where 4=6-n
vn)4
where
with
and
-
L
at
(16)
and for i = 1,2, . . . , n - 1,
W. Y. D. Yuen
Having expressed yi, ai, bi, and ci in terms of y. and 6, a relation between y. and 6 (through a,, b,, and c,) is now derived based on the radiative surface boundary condition:
(15)
Again, the above conditions (with those of equation (12)) would satisfy only the average condition of equation (6) but not the exact form of equation (1). It will be shown in Section 4 that the resulting solution procedure using these boundary conditions is much simpler than those using the more “logical” boundary conditions, yet it produces largely similar results. Expressions for the coefficients a,, bi, ci, and yi can now be deduced from equations (lo), (12), (14), and (15), giving:
wirh radiative surface cooling:
YO(an+l-i
+
@“+I-i
$yo(a - n)4
(26)
Appl. Math.
Modelling,
+
hn+l~i
+
a)
for i = 1,2, _. . , n fori=n+ 1
=
1994,
Vol. 18, February
(35)
95
Temperature
distribution
in a multilayer
medium
and the cumulative heat loss Q from the medium, may be obtained by numerical integration:
Qa
3 s’oy062 d7
which
for 6 5 1
i&yo(b,+2c,+3)dz
q-K=
with radiative surface cooling:
(36)
ford>1
It should be noted that for the special case of the absence of the thermal resistance at the interface (R = 0), equation (12) takes a degenerated form. The first part of the equation still holds while the last part becomes e(i-, z) = 0(i+, z), i.e., continuity of temperature across the interface is preserved because perfect contact is assumed. All subsequent solutions derived are still valid with R = 0. In fact, for this degenerated case, the solution of the problem becomes one of applying the heat balance integral method to a homogeneous medium by dividing it into hypothetical layers, the temperature profile in each of which is approximated by a third-order polynomial. This provides improved accuracy over the single polynomial normally adopted in the application of this method, but the resulting solution is naturally more complicated.
3. Numerical implementation
W. Y. D. Yuen
3. calculate 2, vl, and v2 from equations (30), (32), and (33) respectively after the surface temperature drop, 8,, is determined from equation (34); 4. determine the elapsed time z by numerical integration of equation (31); and finally 5. compute the temperature distribution 13from equation (10) with ai, b,, and ci determined from equations (23x25). Some numerical results are illustrated below. The variations of the penetration depth and surface temperature with the elapsed time for various values of the thermal resistance are shown in Figure 2. It can be seen that the penetration of the thermal effects decreases sharply and the corresponding surface temperature drop increases as the interface resistance R increases. Hence, an increase in the interface resistance offers the advantage of preserving the bulk temperature of the medium, especially during long elapsed time, while causing the temperature drop near the surface to increase. This is illustrated in Figure 3 in which the temperature distribu-
and results 0.1
Based on the above analysis, a computer program has been written to calculate the temperature distribution 0 in the medium and the elapsed time z for a specified interface thermal resistance R, surface thermal resistance L, and penetration depth 6. The computational procedure is as follows:
0.06 o 0.06 0.04 0.02 0.0
1. Calculate a,, b,, and c, from equations (23H25); 2. solve the fourth-order polynomial equation (29) for y0 (a Newton-Raphson iterative technique has been used);
1
0
2
4
3
(a)
5
6
7
5
6
7
R=O.
0.1 0 0.05
\ 1
3
1
’
‘20 ’
1 4
5
1
‘50 ’
0.0
0
1
2
4
3
6
F
T
(b) (a)
0.0
0
Penetration
depth
1
2
3
R=l.
(1).
I
1
I
I
I
(8) versus time
4
I
I
5
6
F
T
(c) (b)
Surf. temp.
drop (8,)
versus time (7).
Figure 2. perature
96
Variations of the penetration depth drop with the elapsed time (L = 15).
Appl.
Math.
Modelling,
1994,
and surface
Figure 3. tem-
Vol. 18, February
values
(L=15).
R=20.
Temperature distribution 0 near the surface for various of the elapsed time T and interface thermal resistance R
Temperature
disrriburion
in a mu/Mayer
tion is plotted for three values of the interface resistance. The temperature jump at the interface is, indeed, high for large values of R and for long elapsed time r. At r = 6, six layers of the medium will have a temperature drop exceeding 1% of its initial (absolute) value when R = 0. The same condition is detected in only four layers when R = 1, and in two layers when R is increased to 20. The average temperature drop of the layers near the surface is plotted against the elapsed time in Figure 4 for several values of R. The “sacrificial effect” of the outermost layer in conserving the thermal energy in the rest of the medium when the interface resistance becomes large is evident. The average temperature drop in the first layer is found to increase with increasing R while all other layers display a reverse trend. It is also interesting to note that the strip temperature drop in these other layers increases at a faster rate (with time) with a higher interface resistance and will eventually surpass that with a lower interface resistance when the elapsed time is sufficiently long (for instance, the temperature drop for R = 1 overtakes that for R = 0 in the second layer at r = 4.4 as shown in Figure 4(a)). When the interface
0.20
-
0.15
-
0.10
-
Layer No. 1 Layer No. 2
-.-
0
2
1
3 7 (a)
0.05
I
0.04
-
0.03
-
0.02
-
-.
I -
,
Layers
1
,
4
5
6
1 & 2.
,
,
1
,
,
Loyer No. 3 Layer No. 4
medium
with radiative surface cooling: I
,
1
,
I
Approx.:
1
0
2
3
,
W. Y. D. Yuen I
q=r/L
4
5
6
T
Figure 5. elapsed
Variation of the cumulative surface heat loss
q with
the
time 5 (L = 15).
resistance is high such that the heat flow between the first and second layers can be neglected, the average temperature drop of the first layer ii can be approximated from a heat balance consideration, in which case it can be shown that ii = q, where q is given by equation (36). An examination of the results for this approximation, as indicated in Figure 4(a), reveals an excellent agreement with the full theory when R is large. The cumulative heat loss from the surface of the medium, plotted in Figure 5, indicates that a medium with a high interface thermal resistance will lose less heat than that with a low interface resistance, especially when the elapsed time is long. It may be argued, when the surface temperature drop is small, that the rate of heat loss from the surface would be fairly constant and the cumulative heat loss will then be linearly proportional to the elapsed time; thus, q = t/L. This level of approximation, as shown in Figure 5, proves to be grossly unsatisfactory even when the elapsed time is small. A constant surface thermal resistance L has been used in the above illustrations (L = 15). The effects of variations in the surface thermal resistance are illustrated in Figure 6. It can be seen that an increase in the surface thermal resistance L would reduce the thermal penetration into the medium. In addition, the temperature drop in the layers is also substantially reduced. 4. Effect of alternative boundary conditions
G
As mentioned in Section 2, two extra relations in addition to the boundary conditions of equation (12), which relate the heat fluxes on both sides of the interface, are required to determine the coefficients of equation (10). Instead of those used in Section 2 (equations (14) and (15)) an alternative and more logical approach is to deduce these relations by differentiating equation (12) with respect to r, and using equation (l), giving: (b)
d38(i+, 2) = d30(i-, 7) = A d20(i+, T) d28(i-, 7) at3 R at2 - at2 x3
Layers 3 & 4.
Figure 4. Variation of the average temperature drop [of the layers with the elapsed time T (L = 15).
(37)
Appl.
Math.
Modelling,
1994,
Vol. 18, February
97
Temperature
-
distribution
-.-
in a multilayer
medium
with radiative surface cooling:
W. Y. D. Yuen
R=7_0
6-
0.08
or 0
’
’ 1
’
’ 2
’
’ 3
’
’ 4
’
’ 5
m 0.06 ’
6
t (a)
Penetration
depth
(d) versus time (7).
(a)
I
I
0
1
I
I
2
R=l.
4
3
4
5
6 B.C. of eqn(37) B.C. of eqns( 14)&(15)
T (b)
Ave. temp.
drop (0
vs. time (T) CR=ZOl.
Figure 6. Variations of the penetration depth and average temperature drop of the layers with the elapsed time for various values of the surface thermal resistance L.
i
Following the same procedure and using the above relations in place of equations (14) and (15), it can be shown that the coefficients ai and bi are given by the same expressions as derived before (equations (17), (18), (20), and (21)), while ci is given by ci+l
=
Ci +
3(l + R,
(b)
(38)
R=20.
Figure 7. Effect of the choice of the boundary conditions at the interfaces on the temperature distribution 8.
with ci = 3(6 - n + R)
(39)
and y0 can, again, be determined from equation (29). An inherent complication with this formulation, however, is that the temperature distribution in the medium varies appreciably as the penetration depth 6 “crosses” an interface; i.e., for the nth interface, the temperature distribution at 6 = n- will be significantly different from that at 6 = n+. This arises because the approximation by cubic polynomials of the temperature distribution (equation (10)) would not satisfy all boundary conditions of equations (12) and (37) without introducing a discontinuity as 6 crosses an interface. As an illustration, the temperature distributions at 6 = n- and 6 = n+ for the first four layers are shown, together with the results obtained earlier (using the boundary conditions of equations (14) and (15)), in Figure 7. It can be seen that the latter results do not exhibit variations as 6 crosses an interface and always lie between the limits given by the former ones. With the boundary conditions of equation (37), an additional elapsed time with further heat loss at the surface will be experienced as 6 crosses an interface because the temperature distribution in the medium has changed. This must be taken into account and added to
98
-.-.-
Appl.
Math.
Modelling,
1994,
Vol. 18, February
the accumulated values deduced previously. From equation (6), the time taken for 6 to cross the nth interface is
s c5=n+
z=
-
dA (40)
d=n- (No, z)lX
and the total heat loss during this elapsed 6=nf dA = &_+ - II,=,4= s 6=n-
time is (41)
The progressive change in temperature distribution as 6 varies from n- to n+ must be deduced to evaluate the integral in equation (40). It is stipulated, from physical reasoning, that the temperatures vary in a manner corresponding to the thermal resistance at the nth interface being increased from zero to its full value R as 6 crosses the nth interface (6 = n- to 6 = n’). Thus, the solution procedure developed earlier is applicable (using the boundary condition of equation (37)), with equations (20) and (39) replaced by, respectively, a, = (6 - n)‘(6 - n + 3R’)
(42)
Temperature
distribution
in a multilayer
and
medium
with radiative surface cooling:
0.30
cr = 3(6 - n + R’)
(43)
where R’ changes from 0 to R as 6 crosses the interface. Based on this algorithm, the variation of the penetration depth 6, surface temperature drop 8,, average temperature drop at the outermost layer cl, and the cumulative heat loss 4 with time r are plotted in Figures 8 and 9, superimposed with the earlier results (according to the boundary conditions of equations (14) and (15)) for comparison. It can be seen that the differences resulting from the two approaches are small (except for the penetration depth). This is not unexpected because the governing heat transfer mechanism of equation (6) has been satisfied in both approaches. It is noted that the simpler conditions of equations (14) and (15) do not require special considerations as 6 crosses an interface, and the solution procedure is much simpler (it can be easily shown from equations (lo), (23t(25), and (27) that the are temperature distributions at 6 = n- and 6=nt
10 8
-.-
6 0 4 2 0
0
3
2
1
4
5
0.25
-
0.15
-
Penetration
’
/
I
1
I
0”
o.oy 0 1
’
1
’
2
’
’
’
3
’
4
’
’
5
’
’
6
T (b)
0
Surface
1
(c)
temp.
2
Ave. temp.
drop (8,)
vs. time
3
4
drop ({,)
I
,
I
I
(
I
,
I
I
I
B.C. of
0.25
0
eqn(37) B.C. of eqns(14)&(15)
R=“\&
0.20
Cs 0.15
0.10
I
0
I 1
I
I
2
I
I
3
1
I
4
5
1
-.-I
6
T
Figure 9. interfaces
Effect of the choice of the boundary on the cumulative surface heat loss.
conditions
at the
identical in this case). Hence, they have been used in the formulation given in Section 2. It should be noted that for the special case of R = 0, the comments made at the end of Section 2 apply here. In addition, the special consideration when the thermal front crosses an interface discussed above is no longer required as it can be shown that the temperature distribution for 6 = n- is identical to that for 6 = n+ in this instance.
5. Conclusion
depth (6) versus time (1).
’
,
6
T (a)
I
W. Y. D. Yuen
(1).
5
The transient temperature distribution in a multilayer semi-infinite medium with radiative cooling on its surface has been examined using the heat balance integral method. A thermal resistance at the layer interface has been included in the model. It has been shown that two different forms of the derived boundary conditions produce largely similar results although one form requires substantially more computational effort than the other. Hence a judicious choice of the derived boundary conditions can reduce the numerical complexity of the solution. The application of the heat balance integral method to this problem does not require the nonlinear boundary condition to be linearized, and the numerical procedure is simple: It only requires the numerical solutions of a fourth-order polynomial and a first-order differential equation. The effect of the thermal resistance at the layer interface on the temperature distribution has been discussed. It has been observed that a high thermal resistance would help conserve the thermal energy in the medium during a long elapsed time, while the surface layer would, as a result, cool more rapidly. This study finds applications in the hot rolling process.
6
vs. time (T).
Figure 8. Effects of the choice of the boundary conditions at the interfaces on the penetration depth, surface temperature drop, and average temperature drop of the first layer.
Acknowledgment The author wishes to thank the management of BHP Steel, Sheet and Coil Products Division, for permission to publish the information contained in this paper.
Appl.
Math.
Modelling,
1994,
Vol. 18, February
99
Temperature
distribution
in a multilayer
medium
with radiative surface cooling:
Nomenclature
&
a,(z),hi(z),ci(z) coefficients associated with the temperature distribution of the (n + 1 - i)th layer, equation (10) h thickness of each layer k thermal conductivity of the medium L nondimensionalized surface thermal resistance the layer (from the surface) that the n(r) thermal penetration has just passed through nondimensionalized heat loss from the 4(r) surface of the medium heat loss from the surface of the medium Q(t) R nondimensionalized thermal resistance in between layers contact thermal resistance in between RC layers t elapsed time temperature distribution in the medium T(x, r) (K) initial temperature of the medium (K) T, ambient temperature (K) T, X distance from the surface of the medium c1 thermal diffusivity of the medium coefficient associated with the temperaYitz) ture distribution of the (n + 1 - i)th layer, equation (10) nondimensionalized penetration depth 6(r)
100
Appl.
Math.
Modelling,
1994,
Vol. 18, February
ii@)
A vt,
v2
W. Y. 0. Yuen
emissivity at the surface of the medium nondimensionalized average temperature drop of the ith layer nondimensionalized temperature drop nondimensionalized surface temperature drop, 19(0,r) term defined in equation (7) terms defined in equations (32) and (33) respectively nondimensionalized x-coordinate Stefan-Boltzmann constant nondimensionalized elapsed time 6-n
References 1 2
3
Yabuuchi, K. Progress of the iron and steel technologies in Japan in the past decade. Trans. ISIJ 1985, 25, 733-738 Hewitt, E. C. Developments in rolling mill technology. The Steel Industry in the Eighties, Book No. 267, The Metals Society, London, 1980, pp. 86103 Goodman, T. R. Application of integral methods to transient nonlinear heat transfer. Advances in Heat Transfer, Vol. 1. Academic Press, 1964, pp. 51-122 Langford, D. The Heat Balance Integral Method. In?. J. Heat Mass Transfer 1973, 16, 24242428 Sfeir, A. A. The Heat Balance Integral in Steady-State Conduction. ASME J. Heat Transfer 1976, 98, 46C-470 Davies, T. W. Transient conduction in a plate with counteracting convection and thermal radiation at the boundaries. Appl. Math. ModeRing 1985, 9, 337-340