Example4.3 rev 1.mw

Example 4.3 Heat Conduction with Flux Boundary Conditions

In the previous two examples, the temperature (dependent variable) at x = 0 was specified. The same technique can be applied for eh case where the derivative of the dependent variable is known at the boundary x = 0 (flux boundary conditions). Consider the transient heat conduction problem in a slab.[4] The governing equation in dimensionless form is

=α

u(x,0) = 0

(0,t) = - k and u(∞,t) is defined (4.1.3)

The flux boundary condition has to be considered while taking the Laplace transform. Equation (4.1.3) is solved in Maple below:

> | restart:with(linalg):with(inttrans):with(plots):

> | eq:=diff(u(x,t),t)=alpha*diff(u(x,t),x$2);

| (1)

> | u(x,0):=0;

| (2)

> | bc1:=diff(u(x,t),x)=-k;

| (3)

> | bc2:=u(infinity,t)=defined;

| (4)

> | eqs:=laplace(eq,t,s):

> | eqs:=subs(laplace(u(x,t),t,s)=U(x),eqs);

| (5)

> | bc1:=laplace(bc1,t,s);

| (6)

The boundary condition in the Laplace domain is:

> | bc1:=subs(laplace(u(x,t),t,s)=U(x),bc1);

| (7)

> | bc1:=subs(x=0,convert(bc1,D));

| (8)

> | U(x):=rhs(dsolve({eqs,bc1},U(x)));

| (9)

> | eqc:=coeff(U(x),exp(1/alpha^(1/2)*s^(1/2)*x));

| (10)

> | _C2:=solve(eqc,_C2);

| (11)

> | U(x):=eval(U(x));

| (12)

> | u:=invlaplace(U(x),s,t);

| (13)

The dimensionless temperature profile is given by:

> | u:=convert(u,erfc);

| (14)

> | pars:={alpha=0.001,k=1};

| (15)

Plots are made for particular values of parameters:

> | plot3d(subs(pars,u),x=1..0,t=300..0,axes=boxed,title="Figure Exp. 4.1.7.",labels=[x,t,"u"],orientation=[-60,60]);

> | plot([subs(pars,t=1,u),subs(pars,t=10,u),subs(pars,t=100,u),subs(pars,t=200,u)],x=0..1,axes=boxed,title="Figure Exp. 4.1.8.",thickness=5,labels=[x,"u"],legend=["t=1","t=10","t=100","t=200"]);

> |