Transient heat conduction of a semi-infinite solid

[mudweez0009]

Homework Statement

This is a problem regarding transient heat conduction in an undefined semi-infinite solid, initially at a temperature T₀ whose surface temperature is suddenly raised to a new constant level at T_s.
I also supplied the problem as an attachment for ease in explaining the problem.
We want to:

1. Derive the Energy Equation
2. State initial and boundary conditions
3. Define two new variables
   1. η=Cxt^m
      
   2. θ=(T-T₀)/(T_s-T₀)
4. Determine C and m (the constants?) of η and θ from Part 3
5. Solve for η and θ themselves.

I understand and completed Parts 1 and 2 of this problem. Unfortunately I don't understand how to begin Part 3... I assume its a method of Separation of Variables?

Homework Equations

There are a decent amount of equations for transient heat conduction, but I don't think Parts 3, 4, and 5 relate to any specific equation. I think its a method (I forget the name) of where you define new equations so that you can re-define the variables, then plug into the original Energy equation (shown in Part 1).

The Attempt at a Solution

Here is what I've tried so far. I am getting very lost on what variables to solve for, which ones to take derivatives of, etc... I would appreciate it if someone could help me get started if I'm wrong, or point me in the right direction if I am on the right track. Thanks in advance.

 

[Chestermiller]

This is not a separation of variables problem. The approach they are trying to lead you through is called a similarity transformation. They are trying to help you transform the differential equation from a partial differential equation to an ordinary differential equation by means of a change of independent variable. The original independent variables are combined in such a way that, when you apply the transformation, they form a single independent variable. In part 3, what they want you to try is to transform the problem from a PDE to an ODE by expressing the dimensionless temperature θ uniquely in terms of the combined variable η of the form:

θ = θ(η)

If you can do this without any t's or x's remaining in the transformed equation, then you can solve the resulting ODE for the dimensionless temperature profile in terms of η. You need to choose a value of m to get rid of the t in the transformed equation. And you need to choose a value of C to work the equation into the form that they have provided. That's what they want you to do in part 4.

Chet

 

[mudweez0009]

Okay so I get that we need to get rid of the "x's" because the final equation we want in Part 3 does not have an x. However, it does have a "t". So I'm not sure I understand the reason why I would guess an "m" that would eliminate the "t".

Is the initial approach I took correct?
Do I need to do anything with θ? I'm not sure how the final equation has dθ/dη and d²θ/dη² in it, when θ is only in terms of T, not η.

 

[Chestermiller]

Your equation for $\frac{\partial ^2 θ}{\partial x^2}$ is incomplete. You need to apply the chain rule here too. After you do this, you get:

$$\frac{\partial θ}{\partial t}=\frac{dθ}{dη}mCxt^{m-1}=\alpha C^2t^{2m}\frac{d^2θ}{dη^2}$$

But, $η=Cxt^m$

So you can substitute $x=\frac{η}{Ct^m}$.

Substitute that, and see what this leaves you with.

Chet

 

[mudweez0009]

Sorry for the late reply, I was actually in the midst of studying for the PE Exam, which I took on Friday, so I didnt have time to get to this until now. I did however get through Parts1 through 4, and am fairly confident in my answer. I am stuck on solving Part 5, where you have to solve d²θ/dη²+2η(dθ/dη)=0, to obtain θ as a functino of η.
I believe this is somewhat of a common diff. eq. problem, but my diff. eq. is rusty to say the least!

 

[Chestermiller]

Hint: Substitute u=dθ/dη

Chet