Transient Conduction in Halfspace: Neumann Problem

M.M. Yovanovich

HSNEUMANN.MWS

One-dimensional conduction in halfspace. The equation is in the halfspace for time .

The initial condition is uniform temperature throughout the halfspace, . The boundary conditions are (i) as

, , and (ii) at the free surface , the surface flux is prescribed such that , where is a constant flux incident on the surface. Laplace transform methods are used to find the solution which is

.

> restart:

> case1:= (q0=1e4, k=20, alpha=20e-6, Ti = 300);

> T:= Ti + 2*q0/k*sqrt(alpha*t)*exp(-x^2/(4*alpha*t))- (q0*x)/k*erfc(x/(2*sqrt(alpha*t)));

Instantaneous surface temperature rise.

> T0:= expand(simplify(subs(x=0,T)));

Temperature and surface temperature rise for a given problem.

> T1:= subs(case1, T);

> T01:= evalf(subs(case1, T0),6);

Given the time, the temperature and the surface temperature rise can be calculated.

> evalf(subs(t=200, x=30/1000, T1),6);

> evalf(subs(t=200, T01),6);

Given the temperature and the position, the time can be calculated. A numerical solution is required.

> fsolve(subs(x=30/1000, T1) = 348, t);

>