I have an assignment that ask me to calc the Thermal Capacity($C_{t}$) of an Oven over the Transient state. I am not asking for the solution, just a help to understand the following:
They give us this formulae:
$C_{t}\frac{\Delta T}{\Delta t} = P - C_{p}(T-T_{e}) $
Clearing $C_{t}$ we get:
$C_{t} = (P - C_{p}(T-T_{e})) \frac{\Delta t}{\Delta T} $
We have to solve it in Ada Programming. These are the variables with the corresponding units.
$C_{t}$($\frac{J}{K}$) = Thermal Capacity
$C_{p}$($\frac{W}{K}$) = Thermic Loss Coefficient (incorrect translation?)
$P(W)$ = Oven Power
$T(K)$ = Temperature in function of time
$T_{e}(K)$ = Ambient Temperature
$t(s)$ = time
I have all the values but Ct, I know that $\Delta$ means variation, but the thing is that I do not understand is, what exactly means $\Delta$T and $\Delta$t in this problem and how to calc it in a computer?
Would it be an aproximation?
Okay, that is what I expected, so this would be the aproximation, where the Sum is a loop?
$C_{t} = \sum_{i}^{n}\left [ (P-C_{p}(T_{i}-T_{e})) \frac{ t_{i} - t_{i-1} }{T_{i-1} - T_{i}} \right ]$
where the $\sum$ is an loop?
$\Delta t$ is the time step you use for your integration and $\Delta T$ is the change in temperature during that time step. The physics equation has $\frac {dT}{dt}$ in that place. When you model the system in a computer you cannot take infinitesimal time steps so we use the $\Delta$ symbolism. You need to choose the time step. Smaller steps will reduce the truncation error as you are ignoring higher order effects. The price of this is added computation (more steps to cover the same time) and perhaps numeric inaccuracy when the changes in a time step become too small for the accuracy of your variables.