Resistance in Rectangle: General Solution

M.M. Yovanovich

RESRECTGEN.MWS

The dimensionless resistance in a rectangle of width , height , and thermal conductivity . The left boundary at is adiabatic, and the right boundary at is convectively cooled where . The upper boundary at is convectively cooled where . The lower boundary at has a uniform heat flux over the interval from to , and the remainder of that boundary is adiabatic. The dimensionless resistance which is based on the average temperature of the heat source area, is a function of four dimensionless parameters: . The dimensionless resistance is obtained from the series: where , and the eigenvalues are the positive, real roots of the characteristic equation: .

There are several special cases which arise from the general solution.

> restart:

> nmax:= 40:

General solution.

> pars:= (beta=1/2, epsilon=0.2, Bi=1e-5, Bie=100);

> term:= (2/epsilon^2)*sin(d*epsilon)^2/(d^2*phi*(d + sin(d)*cos(d)));

> phi:= (d*beta*tanh(d*beta) + Bie)/(d*beta+ Bie*tanh(d*beta));

> ce:= d*sin(d)=Bi*cos(d);

> dvals:= [seq(fsolve(subs(pars, ce), d=(n-1)*Pi..(n-1/2)*Pi), n=1..nmax)]:

> terms:= [seq(evalf(subs(pars, d=dvals[j], term)),j=1..nmax)]:

> terms[1]; terms[nmax];

> psi:= add(terms[j], j=1..nmax);

Summary of input and output parameters.

> pars:= pars; nmax:= nmax; kR:= psi;

> psi1:=(epsilon,nmax)->evalf(add(2/Pi^3/epsilon^2*sin(n*Pi*epsilon)^2/n^3,n=1..nmax));

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