# Modeling heat ODE

1. Jan 14, 2016

### CyberneticsInside

• Member warned about posting without the homework template

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. Jan 14, 2016

### CyberneticsInside

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.

3. Jan 14, 2016

### Staff: Mentor

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.

4. Jan 16, 2016

### CyberneticsInside

Ok, thanks.

5. Jan 17, 2016

### CyberneticsInside

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?

6. Jan 17, 2016

### Staff: Mentor

Do you see the words "LaTex / BBcode Guides" under the reply window?

7. Jan 17, 2016

### CyberneticsInside

Ah, found it !

So the answer would be $\dot{x} = - \frac{g}{C}x + \frac{1}{C}u$