# Difficult Separable Integration Problem

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1. Jul 20, 2016

### argpirate

1. The problem statement, all variables and given/known data
Q=-1*K(T)*(H*W)*(dT/dx)+((I^2)(p)(dx)/(H*W))

K(T)=(197.29-.06333333(T+273))
H=0.01905
W=0.06604
I=700
p=10*10^-6
Q=some constant

Please separate and differentiate to solve for Q using variables of T and x.

Boundaries:
T: Upper=T1 (constant)
Lower=T0 (constant)

x: Upper=L (constant)
Lower=0 (obv. constant)

2. Relevant equations
a=dT/dx ----> a*dx=dT ----> integrate ax|=T|

3. The attempt at a solution

I plugged in all the values and tried to make common denominator to move dx to the Q side. But I could never get around getting rid of the dx in the numerator on the right side of the plus symbol in the original equation. Also, i wasnt sure whether to double integrate with boundaries for both integrals (was sort of weird)... Please help. Been working on this for a long time and cant figure out a way to manipulate it. Main issue is the two dx's and only one dT, so straight up integration wont work bc you would be integrating a dx when there is no dT left.

2. Jul 20, 2016

### Delta²

Let me see if I get this right, your equation looks like $Q=c_1K(T)\frac{dT}{dx}+c_2dx$ where $c_1,c_2$ constants right? If yes , then you can ignore the $c_2dx$ term like it doesn't exist.

The reason is that if you take the $\lim_{dx\rightarrow 0}$ in both sides of the equation you ll end up with an equation that will be

$Q=c_1K(T(x))T'(x)$ which is fairly easy to solve.

3. Jul 21, 2016

### argpirate

Where do you see somewhere where T(x). There is no function for that. Can only use the information given.

4. Jul 21, 2016

### Delta²

It has to be a function of x, otherwise $\frac{dT}{dx}$ is zero hence the whole equation is Q=0.

5. Jul 21, 2016

### argpirate

Okay thanks. I finished this problem before I posted it on here. I was just looking for a method that involved separation and integration. In my original solution (which was successful), I had to make a T(x) equation. I was trying to look for a method that didn't involve this.

6. Aug 5, 2016

### haruspex

Is that a correct statement of the problem as given to you? That isolated dx makes looks wrong, and I don't see where "separable integration" comes in. If you mean separation of variables, that is usually in the context of a differential equation involving one dependent variable and two or more independent. Delta2 proposes to ignore the dx, but I suspect a typo.