Transient Conduction in Halfspace: Dirichlet Problem

M.M. Yovanovich

HSDIRICHLET.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 temperature is maintained at a constant value which may be greater than the initial temperature excess, or smaller than the initial temperature excess. Laplace transform methods give the solution directly which can be expressed as where .

> restart:

Heating of the halfspace.

> T:= Ti + (T0-Ti)*erfc(x/(2*sqrt(alpha*t)));

Instantaneous heat flux at arbitrary location .

> qx:= -k*diff(T,x);

Instantaneous surface heat flux.

> q0:= simplify(subs(x=0,qx));

Average surface heat flux.

> q0ave:= Int(q0,t=0..t1)/t1;

> q0ave:= value(%);

Alternative relation.

> q0ave:= 2/sqrt(Pi)*k*(T0-Ti)/sqrt(alpha*t1);

> case1:= (T0=350, Ti=300, x=10/1000, alpha=20e-6, k= 80, t=100);

> T1:= evalf(subs(case1, T));

> q01:= evalf(subs(case1, q0),6);

> q0ave1:= evalf(subs(t1 = t, case1, q0ave),6);

Plot of temperature distribution.

> case11:= (T0=350, Ti=300, alpha=20e-6, k= 80, t=100);

> T11:= subs(case11, T);

> plot(T11,x=0..200/1000);

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