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Modeling heat ODE

  1. Jan 14, 2016 #1
    • Member warned about posting without the homework template
    Im stuck on this question, can someone please help me?

    u(t) = input power [W].
    x(t) = temperature in plate [Celsius]
    v = 0, temperature of surroundings [Celsius]
    C = 400, heat capacity for plate [J/ Celsius]
    g = 2, heat transfer plate / air [W / Celsius]

    Question is something like this:
    You're playing with the heat plate (Kitchen). The plate might be considered like a flat heat element, that radiates heat to the surroundings.
    Find a differential equation for x(t).

    Nb: the change of temperature in the room migtht be neglected due to
    air circulation

    Should be on the form like: x'(t) = ax + bu

    I assume i need to use the ΔE = W + Q, but i kind of have no clue how to even start.
    I checked if there was a solutions manual but, it was partly broken. The only experience i have with modelling something with heat is the heat equation. But that's a partial differential equation, and only one-dimentional, and i 'm not sure how to handle two functions in one equation?
     
  2. jcsd
  3. Jan 14, 2016 #2
    I know should try to come up with a possible solution, but I’ve been stuck with this all day, and I can't find any similar problems to start with either.
     
  4. Jan 14, 2016 #3
    This is a 1D steady state heat conduction problem with heat generation within the body and a convective heat loss boundary condition at one of the surfaces. You need to perform a differential heat balance on a portion of the plate between x and x + Δx.
     
  5. Jan 16, 2016 #4
  6. Jan 17, 2016 #5
    Ok, I think I got it.

    C dx = u dt - g(x-v)dt
    x'(t) = - (g / C)x + (1 / C)u

    Just an additional question: Is there some better way to type in equation, like Latex or similiar?
     
  7. Jan 17, 2016 #6
    Do you see the words "LaTex / BBcode Guides" under the reply window?
     
  8. Jan 17, 2016 #7
    Ah, found it !

    So the answer would be [itex] \dot{x} = - \frac{g}{C}x + \frac{1}{C}u [/itex]
     
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