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Difficult Separable Integration Problem

  1. Jul 20, 2016 #1
    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. jcsd
  3. Jul 20, 2016 #2
    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.
     
  4. Jul 21, 2016 #3
    Where do you see somewhere where T(x). There is no function for that. Can only use the information given.
     
  5. Jul 21, 2016 #4
    It has to be a function of x, otherwise ##\frac{dT}{dx}## is zero hence the whole equation is Q=0.
     
  6. Jul 21, 2016 #5
    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.
     
  7. Aug 5, 2016 #6

    haruspex

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    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.
     
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