1-D steady state heat conduction equation (Cartisian, Cylindrical and Sperical)

[gabbagabbahey]

There is no need for a power series here...see my last post ^^^

 

[Schmoozer]

Ok so I'm on the last one and have:
$$T(r)=1+\frac{\dot{q}}{4k}(R^2-r^2)+C_1(\frac{1}{R}-\frac{1}{r})$$

Now $$T(x)=\sum_{n=0}^{\infty}b_n x^n$$ ?

 

[gabbagabbahey]

I think you should have:

$T(r)=1+\frac{\dot{q}}{6k}(R^2-r^2)+C_1(\frac{1}{R}-\frac{1}{r})$

Again, you would expect the temperature to be finite everywhere inside the sphere; so you can set C_1=0 since the 1/r blows up at r=0.

PS. don't forget to answer part (b) of the question for each case ;0)

 

[Schmoozer]

b. T(r)=1 when q_dot=0 for all three cases right?

Edit: Except the first one where T(r)=1+x

 

[alpha754293]

Homework Statement

The one dimensional steady-state heat conduction equation in a medium with constant conductivity (k) with a constant volumetric heat generation in three different coordinate systems (fuel rods in a nuclear power plant) is given as:

$$\frac{d^2 T}{dx^2}=-\frac{\dot{q}}{k}$$ T(x=0)=1 T(x=1)=2 Cartisian

$$\frac{1}{r}\frac{d}{dr}(r\frac{dT}{dr})=-\frac{\dot{q}}{k}$$ T(r=R)=1 Cylindrical

$$\frac{1}{r^2}\frac{d}{dr}(r^2\frac{dT}{dr})=-\frac{\dot{q}}{k}$$ T(r=R)=1 Sperical

a. Find an expression for the temperature distribution in a solid for each case
b. What is a temperature distribution if the heat generation is zero?

Homework Equations

The Attempt at a Solution

$$T(x)=\sum_{n=0}^{\infty}b_n x^n$$

$$T'(x)=\sum_{n=0}^{\infty}nb_n x^n^-^1$$

$$T''(x)=\sum_{n=0}^{\infty}n(n-1)b_n x^n^-^2$$ so n=0, n=1

$$b_{0}=0$$

$$b_{1}=0$$

$$b_{2}=2x$$

$$b_{3}=3x^2$$

Am I at least on the right track?

LOL. Navaz's class? MECH522? Awesome!