General Fin Solution

M.M. Yovanovich

FINGENSOL.MWS

General fin solution.

Constant cross-section extended surface (fin) with area , perimeter , length , and thickness . The thermal conductivity is . The heat transfer coefficient along the lateral boundary is , and the contact conductance at the fin base is , and the heat transfer coefficient at the fin tip is . The fin base temperature is , and the surrounding fluid temperature is .

The temperature excess is defined as , and the fin parameter is defined as .

There are three dimensionless parameters defined as which must be less than to ensure that the temperature is one-dimensional, i.e. , which cannot be zero, and which lies in the range .

> restart:

System parameters.

> syspar:= (A, P, L, k, h, h[c], h[e], T[b], T[f]);

Dimensionless parameters.

> BiotNos:= [Bi = h*t/k, Bi[c] = h[c]*L/k, Bi[e] = h[e]*L/k];

Governing differential equation and its solution.

> ode:= Diff(theta(x),x$2) - m^2*theta = 0;

> sol:= C[1]*cosh(m*x) + C[2]*sinh(m*x);

Robin boundary conditions at the fin base and at the fin tip .

> bc1:= Diff(theta(x),x)[x=0] = -h[c]/k*(theta[b] - theta(0));

> bc2:= Diff(theta(x),x)[x=L] = - h[e]/k*theta(L);

Constants of integration.

> C[1]:= theta[b]/(1 + (m*L*phi)/Bi[c]);

> C[2]:= - theta[b]*phi/(1 + (m*L*phi)/Bi[c]);

> phi:= (m*L*tanh(m*L) + Bi[e])/(m*L + Bi[e]*tanh(m*L));

Heat transfer through the fin base.

> Q[base]:= - k*A[base]*Diff(theta(x),x)[x=0];

> Q[fin]:= - k*A[base]*m*C[2];

Fin resistance.

> R[fin]:= theta[b]/Q[fin];

General expressions have been presented for fin equation, fin solution, fin heat transfer rate and the fin resistance. These general solution can be used to find many special cases such as

1. Perfect contact at the base where and therefore . This parameter will not appear in the temperature distribution, the fin heat transfer rate and the fin resistance.

2. Perfect contact with , and adiabatic fin tip where , therefore . Here the fin parameter .

3. Perfect contact with , and perfect contact at fin tip where , therefore . Here the fin parameter .

4. Perfect contact with , infinitely long fin, i.e. , and .