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Calculating cooking intensity

  1. May 28, 2010 #1
    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.

    [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.
     
  2. jcsd
  3. Jun 1, 2010 #2
    Should I ask the question in a different way? Or did Memorial Day just derail the forum? :)
     
  4. Jun 6, 2010 #3
    Please help if you know the answer.
     
  5. Jun 6, 2010 #4

    russ_watters

    User Avatar

    Staff: Mentor

    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...
     
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