ME 353 Heat Transfer 1

M.M. Yovanovich

EX3P7.MWS

Example 3.7 of 4th edition of Incropera and DeWitt.

A long cylindrical tube of inner radius and outer radius is insulated at the outer surface and cooled at the inner surface by a fluid of temperature and a heat transfer coefficient . Uniform volumetric heat generation occurs in the tube wall.

1. Obtain the general solution of the temperature distribution in the tube wall.

> restart:

Assumptions are

1. Steady-state

2. One-dimensional conduction, therefore

3. Constant properties

4. Uniform volumetric heat generation

5. Outer surface is adiabatic

6. Inner surface convectively cooled

7. Let the surface temperature at the inner surface be

8. Assume unit length of tube, i.e.

> ode:= expand(1/r*diff(r*diff(T(r),r),r)) = - P/k;

> sol:= dsolve(ode, T(r));

> dersol:= diff(rhs(sol), r);

Boundary conditions.

The outer surface is adaibatic, therefore the temperature gradient is zero by Fourier's Rate Equation.

The maximum temperature occurs at the adiabatic surface. Incropera and DeWitt denote the temperature at to be . They used this as a boundary condition.

The inner surface is convectively cooled. The inner surface temperature is denoted as . It will be specified at the inner surface, then it will be found by means of a heat balance after the solution has been found.

> bc1:= subs(r=r1, rhs(sol)) = Ts1;

> bc2:= subs(r=r2, dersol) = 0;

Solve for constants of integration.

> consts:= solve({bc1,bc2}, {_C1,_C2});

Assign the constants to get the temperature distribution.

> assign(consts):

> Temp:= expand(rhs(sol));

Alternative expression for the temperature distribution.

> Temp2:= Ts1 - P/k*(r^2-r1^2)/4 + P/(2*k)*r2^2*ln(r/r1);

This relation is different from the one given in the text. The text expression is based on the temperature of the adiabatic surface.

Heat transfer rate from inner surface into fluid.

> Qgen:= P*V; V:= Pi*(r2^2 - r1^2)*L;

> Qconv:= h*Ainner*(Ts1 - Tf); Ainner:= 2*Pi*r1*L;

Find relation for temperature of inner surface by heat balance.

> Ts1:= expand(solve(Qgen - Qconv = 0, Ts1));

Find relation for maximum temperature which occurs at the adiabatic surface.

> Tmax:= expand(subs(r = r2, Temp));

Alternative expression for the maximum temperature.

> Tmax2:= Tf + P/(4*k)*(r1^2-r2^2) + P*r2^2/(2*k)*ln(r2/r1) + P/(2*h)*(r2^2/r1 - r1);

Incropera and DeWitt did not report this relation.