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.(adsbygoogle = window.adsbygoogle || []).push({});

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.

[tex]P=\int_0^t{rxdx}[/tex]

Holding

In this case, T = 300.

[tex]P=\int_0^t{TdT}[/tex]

Cooling

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

[tex]T(t) = T_a + (T_0 - T_a)*e^{-kt}[/tex]

In this case:

[tex]T_a = 23[/tex]

Let [tex]k = 0.0035[/tex]

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

[tex]P = \int_0^t{T(t)dT}[/tex]

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.

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# Calculating cooking intensity

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