Example4.2 rev 1.mw

Example 4.2 Heat Conduction with Transient Boundary Conditions

Heat condition with a constant boundary condition at x =0 was considered in example 4.1. The same technique can be applied for time dependent boundary conditions. Consider the transient heat conduction problem in a slab.[4] The governing equation is:

=α

u(x,0) = 1

u(0,t) = and u(∞,t) is defined (4.1.2)

> | 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:=u(0,t)=k/t^(1/2);

| (3)

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

| (4)

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

The governing equation in the Laplace domain is:

> | 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(0,t),t,s)=U(0),bc1);

| (7)

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

| (8)

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

| (9)

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

| (10)

The dimensionless temperature in the Laplace domain is:

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

| (11)

Next, the solution is inverted to the time domain:

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

| (12)

Maple is not able to invert the solution in the Laplace domain. This can be solved by using dummy variables for x and α and defining them to be positive:

> | U(x):=subs(x=x1,alpha=alpha1,U(x));

| (13)

> | assume(x1>0,alpha1>0);

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

| (14)

> | u:=subs(x1=x,alpha1=alpha,u);

| (15)

Hence, the final solution is:

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

| (16)

> | plot3d(subs(pars,u),x=1..0,t=300..0,axes=boxed,title="Figure Exp. 4.1.5.",labels=[x,t,"u"],orientation=[45,45]);

> | 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. 4.1.6.",thickness=5,labels=[x,"u"],legend=["t=1","t=10","t=100","t=200"]);

> | animate(subs(pars,u),x=0..1,t=1..100,thickness=5,title="Figure Exp. 4.1.6.",axes=boxed,labels=[x,"u"]);

You can make an animation using the command illustrated in example 4.1.

> |