# Calculating cooking intensity

1. May 28, 2010

### 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.

2. Jun 1, 2010

### Bacat

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

3. Jun 6, 2010