How Do You Calculate Cooking Intensity from Cooling in an Oven Cycle?

[Bacat]

I have a block of food in an oven and I want to calculate the "cooking intensity" of the block during a controlled oven cycle. The cycle of the oven is that it heats at 5 degrees (C) per minute, holds the temperature at 300C for 1 hour, and then shuts off the oven to cool to room temperature. I define as "cooking" any temperature higher than 23C (room temperature). I use P for cooking intensity.

I am doing fine on the ramp-up and the hold. And I can use the heat equation to find the temperature of cooling at any time. But how do I integrate the total heat from the cooling function? I seem to be stuck on this point. Here's my work so far:

Ramping Up

For a constant heating rate r, this is just finding the area of a triangle. In this case, r = 5.

$$P=\int_0^t{rxdx}$$

Holding

In this case, T = 300.

$$P=\int_0^t{TdT}$$

Cooling

Newton's Cooling Law (using k to temporarily ignore A, m, c, and R):

$$T(t) = T_a + (T_0 - T_a)*e^{-kt}$$

In this case:
$$T_a = 23$$

Let $$k = 0.0035$$

I want to find P from T(t). Can I just integrate like this?

$$P = \int_0^t{T(t)dT}$$

Don't I need to take the derivative of T(t) first and add that under the integral?

And before anyone asks: no, this isn't homework. Really.

 

[Bacat]

Should I ask the question in a different way? Or did Memorial Day just derail the forum? :)

 

[Bacat]

Please help if you know the answer.

 

[russ_watters]

The concept here seems pretty odd to me because you're just calculating a temperature profile of the oven and not considering the heating of the food, so this doesn't have a whole lot to do with "cooking". The reality is that you always have a Newton's Law of cooling/heating scenario going on inside the oven between the oven and the food.

Anyway, being an engineer, I'd integrate numerically with Excel, so I can't help you do it with calculus...