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A Rectangular Cooling Fin*

*Maple 6

1998 Waterloo Maple Inc.

NOTE: This worksheet demonstrates how Maple can be used in the design of a rectangular cooling fin.

Introduction

In this worksheet we determine a formula for the height of a rectangular cooling fin to give the maximum heat flow.

[Maple OLE 2.0 Object]

> restart;

Definition of the Problem

We consider a rectangular cooling fin with constant volume V, length l and thickness d . Cooling takes place by free and forced convection. The heat transfer coefficient is alpha , in [W/(m^2*K)] and the thermal conductivity, lambda , is in [W/(m*K)] . We neglect radiation and assume the surface temperature of the heat source is constant.

Solution

The temperature distribution in the fin, is given by:

> deq := diff(T(x),x,x)-2*alpha/lambda/d*T(x) = 0

We simplify this by substituting m^2 for 2*alpha/lambda .

> deq2 := algsubs(2*alpha/lambda = m^2,deq)

deq2 := -(-diff(T(x),`$`(x,2))*d+T(x)*m^2)/d = 0

This differential equation is solved subject to certain boundary conditions: The first is that the temperature at the base of the fin is equal to dT, the difference between the temperature of the heat source and the ambient fluid. The second condition is that there is no heat loss through the top of the fin.

> sol := dsolve({deq2, T(0) = dT, D(T)(h) = 0},T(x));

sol := T(x) = dT*exp(-m*h/(sqrt(d)))*exp(m*x/(sqrt(...

We then determine the heat flow, using the flow through the base of the fin.

> Q := simplify(-lambda*l*d*eval(subs(x = 0,diff(rhs(...

Q := lambda*l*sqrt(d)*dT*m*(-exp(-m*h/(sqrt(d)))+ex...

> Q := simplify(-lambda*l*d*(2*dT*m/(sqrt(d)*(exp(m*h/(sqrt(d)))^2+1))-(dT+dT*exp(m*h/(sqrt(d)))^2)*m/((exp(m*h/(sqrt(d)))^2+1)*sqrt(d))));

Q := lambda*l*sqrt(d)*dT*m*(-1+exp(2*m*h/(sqrt(d)))...

We eliminate d using the formula V = d*l*h , giving Q as a function of h .

> s1 := subs(sqrt(d) = sqrt(V/l)/sqrt(h),1/sqrt(d) = ...

s1 := lambda*l*sqrt(V/l)*dT*m*(-1+exp(2*h^(3/2)*m/(...

The optimal value of Q is a maximum, ie when its derivative with respect to h is zero.

> s2 := simplify(diff(s1,h))

s2 := 1/2*lambda*l*dT*m*(sqrt(V/l)-sqrt(V/l)*exp(4*...

We substitute t for m*h^(3/2)/sqrt(V/l) and determine the root of this equation.

> s3 := simplify(algsubs(m*h^(3/2)/sqrt(V/l) = t,s2))...

s3 := -1/2*lambda*dT*t*V*(-1+exp(4*t)-12*exp(2*t)*t...

We only need consider the numerator and remove lambda , V , h and dT from the equation by equating them to 1:

> s4:=numer(subs(lambda=1, V=1, h=1, dT=1, s3));

s4 := -t*(-1+exp(4*t)-12*exp(2*t)*t)

We find the root as follows, using positive values of t.

> tsoln:=fsolve(s4, t=0..infinity, avoid={t=0});

tsoln := 1.419223190

To get the optimal value for the height of the fin under the conditions of constant volume and length, we substitute back.

> gl := simplify(subs(m = sqrt(2*alpha/lambda),V = l*...

gl := h*sqrt(2)*sqrt(alpha*lambda)/(lambda*sqrt(d))...

Solving this for h we get:

> solve(gl, h);

1.003542342*lambda*sqrt(d)/(sqrt(alpha*lambda))

Conclusion

The mathematical model delivers a simple formula for the optimal height at constant volume, dependant on the parameters of the fin.

Reference

Frank P. Incropera and David P. De Witt, Fundamentals of Heat and Mass Transfer, John Wiley & Sons.

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