We have demonstrated that given the inlet temperature, the procedure of using the heat transfer coefficient or the cooling region lengths to control the surface temperature is well-posed and the iterative procedure converges rapidly. An analysis on a slightly simpler problem using a Neumann condition with a primary cooling region also showed that the problem is indeed well-posed

Huang, H.

English

Mathematics in Industry Information Service Eprints Archive

Temperature Control by Laminar Flows
A. Fitt*, Z. Guot, H. Huang+, L. Jiang§, J. Ockendon~, Y. Tanll, J. Wylie**
Written by H. Huang
where T and v are the (non-dimensional) temperature and the speed of the steel bar, respectively.
K. is the thermal diffusivity, x and z are the coordinates nondimensionalized by the thickness and
the total length of the primary laminar flow cooling region. A secondary (feed-back controlled)
flow cooling region is located between the nondimensionalized length z = 1 and z = d where
d« 1. Temperature is measured at the inlet (z = 0) and exit (z = 1+ d) which are denoted by
FT (feed temperature) and CT (control temperature), respectively. The boundary conditions
are
{
-aLT, 0 < z < L,
0, L < z < 1,
Txlx=o = 0, Txlx=l = -alT, 1 < z < 1+ 1,
0, 1+ 1< z < 1+ d,
where O'.L, ai, Land 1are the heat transfer coefficients and lengths for the primary and secondary
laminar flow cooling regions, respectively and T! is the temperature at the inlet (FT position).
When v, K., O'.L and al are constants, the solution of this initial-boundary-value problem can
be obtained using Fourier series. For example, when L = 1 and 1= d = 0, we have
T = Lan exp( -A~Z) COS(An,8X)
n
where ,8 = JK./v and An are the eigenvalues determined by
T! = Lan COS(An,8X).
n
When L < 1 and d =f: 0, the exact solution can also be obtained in principle by matching
the Fourier expansions for the various regions 0 < z < L, L < z < 1, 1 < z < 1 + 1 and
1+ 1< z < 1+ d.
* Southampton
t Baa Steel
+York
§ Tongji
, Oxford
IIFudan
** City University of HK
To illustrate the basic idea, we consider first the following control problems: Given the inlet
temperature Tf, find the correct value of 0'.£ (given v and K with L = 1 and 1= d = 0) such
that the temperature at the surface of bar at the exit (z = 1+ d and x = 1) equals to the target
temperature T;. In other words, we want to solve the following problem
Tc = Lan exp( -A~) COS(An,B).
n
In principle, we can consider a Newton type of iteration using the following procedure: we start
with an initial guess for the value of O'.~), we compute the eigenvalues corresponding A~O). We
can then find the value TJO), and using (4), we can find an improvement for 0'.£ using various
method. For example, for Newton's method, we have
0'.(1) _ 0'.(0) F(O'.~))
£ - £ - FI(O'.~))·
Repeat the process until convergence is achieved, i.e., limk-+oo 1000t)- at-I) I = 0 where at) and
at-I) are the two successive values of the above iteration procedure.
However, this procedure may not converge. Therefore, we use a simpler bisection method in
this study since the optimization problem (4) is equivalent to finding the root of Tc - T; = 0
when we vary 0'.£. We start with two initial guesses O'.~) and O'.~) and find the approximated
value using the recursive formula
(k-l) + (k-2)
(k) _ 0'.£ 0'.£
0'.£ - 2
To illustrate the procedure, we first considered the each Fourier modes separately due to the
linearity of the problem. Another way of looking at it will be to think of Tf is one of the
eigenfunctions
Tf = COS(.BAnX).
The first six eigenvalues computed using Maple for ,B = 1 and 0'.£ = 1 are 0.8603, 3.4256,
6.4373, 9.5233, 12.6453, 15.7713. The procedure converges for all the eigen-functions.
We then consider the case where the inlet temperature profile is uniform. Since the eigenfunc-
tions are nearly orthogonal, we compute the first six coefficients in the eigenfunction expansion
as
f01 COS(,BAnx )dx
an ~ Ifo COS(,BAnx)2dx
and the values are 1.1191, -0.1517, 0.0466, -0.0217, 0.0124, -0.0080.
Simulations have been carried using a Matlab code which does the following:
Figure 1: Convergence history for the test case: (a) temperature at the exit Te; and (b) the heat
transfer coefficient aL. The solid lines are the exact values and circles with line are computational
values.
1. Compute the target value Te= Lan exp( -A~) COS(An);
2. Pick up two initial values for aL as a£) = 0.2 and a~) = 3.4, we can compute the
eigenvalues associated with these two values A~k), the corresponding exit temperature
TJk), and the values Fk = TJk) - Te, for k = 1 and 2;
3. We then use the bisection method to solve the equation F(aL) = 0 with a~) = (a~-l) +
a~-2)) /2 until convergence is reached.
For example, when we set the tolerance Io::~) - a~-l)1 < 10-3, the iteration converges in a few
iterations, as shown in Figure 1.
The case we considered is useful for illustrating the control procedure. It can also be used to
find the value ofthe heat transfer coefficient since aL is often not known. However, the cases of
more practical relevance are to find the length of the laminar cooling regions (l and/or L) so that
the target temperature at the exit is achieved. Since the values of physical parameters v and K
may fluctuate, we need to adjust the value of L, more often the length of secondary laminar flow
region l based on the measured temperature Te. In the following, we consider two cases: (1).
Controlling primary laminar flow region length L with l = d = 0 and aL = 1; (2). Controlling
the secondary laminar flow region length l with L = 0.7, O::L = 1, d = 0.1 and O::l = 0.9. In both
cases, we use the same procedure as the one described earlier with aL replaced by L or l.
The temperature dis:ributions in the z-direction for the two cases can be obtained by solving
the forward problem using a finite difference method described below. Its values on the surface
and at the center of the bar are plotted in Figure 2.
5.1 Case 1: controlling primary region (l = d = 0)
We now consider the case where O::L is given with no secondary cooling but the length of the
primary laminar flow region can be adjusted so that the exit temperature Te equals to the target
,,,,,,,,,,,,,,
,,,,,,,,,,,,,,,,,,,,,,,
value. In principle, we can use the exact solution represented by the Fourier series in the two
regions (z < L and z > L). However, it may be more efficient to use numerical method to find
an approximated solution for the forward problem.
We use the method of lines to solve the heat equation (1) by discretizing in the x directions
only, i.e., replacing Txx by the central difference formula and we obtain
for i = 2, ...N - 1. For i = 1 (x = 0) and i = N (x = 1), we use the boundary conditions and
the equations are
where the {} denotes the derivative with respect to z. The resulting systems of ordinary differen-
tial equations for the grid values Ti and the initial condition is THO) = Tf, the inlet temperature.
In Figure 2, we have plotted the convergence history of the control process by setting the
objective temperature Tc. Again, we start with two initial guesses, L = 0 and L = 1, the correct
value for the cooling region is found after few iterations.
5.2 Case 2: controlling secondary cooling region
In the manufacturing process, controlling the secondary cooling region length is used more often
as a feed-back controlling device in response to possible variations in. v (speed of moving steel
bar) or K, (heat conductivity due to difference in steel composition, etc.). Here we consider the
case where the primary cooling region length is set to be 1 and the heat transfer coefficients
(}:L = 1 and (}:l = 0.9. We set the target temperature to be the one corresponding to v = 1,
K, = 1, d = 0.1 and 1 = 0.05. We assume that 1 is unknown and use the iterative procedure to
find its value.
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Figure 3: Convergence history for Case 1: (a) for temperature at the exit Tc and (b) for the
length of the primary cooling region L. The solid lines are the exact values and circles with line
are computational values.
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Figure 4: Convergence history for Case 2: (a) for temperature at the exit Tc and (b) for the
length of the secondary cooling region l. The solid lines are the exact values and circles with
line are computational values.
In Figure 4, we have plotted the convergence history of the control process by setting the
objective temperature Tc• Again, we start with two initial guesses, l = a and l = d, corresponds
to two cases with no cooling and cooling applied to the entire region. The correct value for the
cooling region (l = 0.05) is found after few iterations.
We have demonstrated that given the inlet temperature, the procedure of using the heat transfer
coefficient CY.L or the cooling region lengths (L or l). to control the surface temperature Tc is
well-posed and the iterative procedure converges rapidly. An analysis by Dr. John Ockendon,
also included in this report, on a slightly simpler problem using Neumann condition Txlx=l = -CY.
with a primary cooling region showed that the problem is indeed well-posed.


Temperature control by laminar flows

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