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...

 

[PJC01]

I would approach this problem by first defining what exactly we mean by "cooking intensity." From the given information, it seems that cooking intensity is related to the temperature of the block of food during the cooking process. Therefore, we can define cooking intensity as the average temperature of the block of food during the cooking cycle.

Next, we need to consider the different stages of the oven cycle and how they contribute to the average temperature of the block of food. During the ramp-up stage, the temperature of the block of food increases at a constant rate. We can use the heat equation to calculate the temperature at any given time during this stage. However, since we are interested in the average temperature, we will need to integrate the temperature function over the entire ramp-up period.

During the holding stage, the temperature of the block of food remains constant at 300C. Therefore, the contribution of this stage to the overall cooking intensity is simply 300C multiplied by the duration of the holding time.

Finally, during the cooling stage, the temperature of the block of food decreases according to Newton's Cooling Law. To calculate the average temperature during this stage, we will need to integrate the temperature function over the entire cooling period.

Once we have calculated the contributions of each stage, we can add them together to get the total cooking intensity. This can be represented mathematically as:

P = (ramp-up contribution) + (holding contribution) + (cooling contribution)

P = \int_0^{t_{ramp}}T(t)dt + 300C(t_{hold}) + \int_{t_{ramp}+t_{hold}}^{t_{total}}T(t)dt

where t_{ramp}, t_{hold}, and t_{total} represent the duration of the ramp-up, holding, and total cooking time, respectively.

To answer the specific question about integrating the cooling function, yes, you will need to take the derivative of T(t) and add it under the integral in order to integrate the function. This is because the integral represents the area under the curve, and the derivative represents the slope of the curve at any given point. Therefore, in order to find the total area under the curve, we need to integrate the derivative of the function.

In conclusion, calculating cooking intensity involves considering the contributions of each stage of the cooking cycle and integrating the temperature function over each stage. By doing so, we can find the average