Example4.10 rev 1.mw
Example 4.10 Steady State Heat Conduction in a Plate
Consider steady state conduction in a semi-infinite rectangular strip. The governing equation in dimensionless form is
u(x,0) = u(∞,y)=1 (4.1.17)
(0,y) = u(∞,z) =0
The following transformation is used to combine the variable:
η =
(4.1.18)
The variable u in the new coordinate η is represented by U. The governing equation (ordinary differential equation) for U is obtained by converting the spatial derivatives (x and y) in equation (4.1.17) (partial differential equation) to derivatives in the η coordinate. The boundary conditions for U are:
U(0)=0
(4.1.19)
U(∞)=0
Example 4.10 is solved i n Maple below:
The governing equation is entered here:
| > |
eq:=diff(u(x,y),x$2)+diff(u(x,y),y$2); |
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(1) |
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eq:=changevar(u(x,y)=U(eta(x,y)),eq): |
| > |
eq1:=(simplify(subs(eta(x,y)=y/x,eq))): |
| > |
eq1:=subs(y=eta*x,eq1): |
| > |
eq1:=simplify(eq1*x^2): |
The governing equation in the combined variable is:
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eq2:=convert(eq1,diff); |
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(2) |
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(3) |
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(4) |
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U:=rhs(dsolve({eq2,bc1,bc2},U(eta))); |
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(5) |
The dimensionless temperature U is given by:
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(6) |
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plot(U,eta=0..10,thickness=5,title="Figure Exp. 4.1.21.",axes=boxed); |
The dimensionless temperature in the original coordinate is:
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u:=expand(subs(eta=y/x,U)); |
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(7) |
The solution obtained is plotted:
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plot3d(u,x=0..50,y=0..50,axes=boxed,title="Figure Exp. 4.1.22.",orientation=[120,60],labels=[x,y,"u"]); |
The dimensionless heat flux at y = 0 is given by:
| > |
flux:=subs(y=0,-diff(u,y)); |
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(8) |