ME 353 Heat Transfer 1

M.M. Yovanovich

CARTPOISS1.MWS

Solution of Poisson equation in plane wall with Dirichlet boundary conditions.

The plane wall has thickness and constant thermal conductivity . The volumetric heat sources are constant and uniform at strength . The two surface temperatures are maintained at temperatures: and .

Obtain the temperature distribution, the heat transfer rates out of the wall at and , and the location of the maximum temperature.

> restart:

Poisson Equation and Its Solution.

> ode:= diff(T(x), x$2) = - P/k;

> sol:= dsolve(ode, T(x));

Dirichlet Boundary Conditions.

> bc1:= subs(x=0, rhs(sol)) = T[s1];

> bc2:= subs(x=L, rhs(sol)) = T[s2];

Solve for constants of integration.

> consts:= solve({bc1,bc2}, {_C1,_C2});

Assign the constants of integration to get the temperature distribution.

> assign(consts):

> expand(sol);

Heat transfer rates at the two boundaries.

> Q[0]:= expand(k*A*subs(x=0, diff(rhs(sol),x)));

> Q[L]:= expand(- k*A*subs(x=L, diff(rhs(sol),x)));

Check the result by addition of the two relations.

> Q[total]:= Q[0] + Q[L];

The total heat transfer rate from both boundaries is equal to the heat generation rate.

Location of the temperature maximum.

> x[max]:= expand(solve(diff(rhs(sol), x) = 0, x));

Verification of the location of temperature maximum.

When the two boundary temperatures are equal, the maximum temperature will occur in the midplane of the wall. Also the temperature gradient should be zero.

> subs(T[s1]=T[s2], x[max]);

> simplify(subs(x=x[max], diff(rhs(sol),x)));

The result is correct.