# Calculating new temperature of an object when air temperature changes

I'm trying to calculate the new temperature of an object when the air temperature around it changes, given a period of time.

Basically I get periodic readings from an air temperature sensor in a refrigerator. In some cases these readings are every 5 minutes, in others every 1 minute, so the time between readings is variable.

For each reading I get, I'd like to also calculate the approximate temperature of food at its core; something like a chicken for example (I know that part is vague, but if there is a variable I can tweak then that is fine).

The result should be a "damped" version of the actual air temperature, as obviously any objects will slowly change temperature to eventually meet the air temperature.

Initially there used to be "food simulant" put around the sensor, so the temperature would automatically be damped, but this is no longer the case.

I do not know much about thermodynamics. I'm not sure if I can just add a percentage of the temperature change to the previous damped value, or if I need a calculation based on the last few air temperature readings, or what.

I guess I'm looking for a result a bit like:

10:00 2 degrees (air), 2 degrees (product)
10:05 2.5 degrees (air), 2.1 degrees (product)
10:10 2.5 degrees (air), 2.2 degrees (product)
10:20 2.7 degrees (air), 2.5 degrees (product)


I could do something really cheap like averaging the readings over the last 30 minutes but I don't think that will cut it!

I'd appreciate any help - thanks very much.

• You can't, at least not without knowing the thermal conductivity(aka rate of heat transfer) of the air-object interface and the thermal capacity (aka joules per gram per kelvin) of the object; to get the core temperature you need the heat transfer rate of the solid as well. Aug 27, 2014 at 11:31
• However, with a controlled experiment you can estimate the conductivity: Start with a known air and object temperature, respectively. Then take readings of both at regular intervals. The average change in the temperature difference divided by the time interval yields the conductivity. With Fourier's law you can predict the object's temperature. Aug 27, 2014 at 11:51
• Thanks Carl. It does not have to be perfectly precise. It is for food safety in restaurants (and similar). They just need to roughly know when the temperature of the products have gone out of range. They can't really have separate readings for each food type, so it is based on something like a chicken. I think they even got roughly the same response by putting some "hot melt" on the sensor. So basically, I'm trying to roughly simulate a block of protein in a fridge.
– Mark
Aug 27, 2014 at 11:53
• Thanks Aziraphale. Does the rate of change vary with how near or far it is to the air temperature (not physically, numerically)? Or how long it has been exposed to that temperature?
– Mark
Aug 27, 2014 at 11:55
• Before going to a lot of trouble, you might want to check with the regulatory authorities. If they have a strict procedure to follow, it will not matter how you do your calculation. Aug 16, 2015 at 0:15

The method of modelling the chicken as a sphere, as mentioned on your other forum, might work something like this. Model the chicken as a sphere and use the heat equation, treating the surface as your boundary. The way to do this is discussed here. To obtain an approximate thermal diffusivity for a chicken you could use the equations you discovered above, which look like they are dependent on temperature only. This translates into the thermal diffusivity via:

$$\alpha = \frac{k}{\rho c_p}$$

where $k$ is the thermal conductivity, $\rho$ is the density of the chicken, and $c_p$ is the specific heat capacity, which can be found for a number of different meats here.

Even if this method doesn't sound quite like what you're looking for, you may still find using the heat equation in some way useful. It's often used for measuring heat and temperature changes in objects.

If the issue is the temperature of food (and the resultant rate of growth of bacteria), modeling a chicken as a sphere seems to be a bad idea because a chicken has plenty of parts sticking out. And unless the only bacteria on the chicken are in the chicken's center, tracking/predicting only the core temperature also seems a bad idea. The skin and the parts that stick out will warm up first, and will grow bacteria if they get warm enough even if the core of the chicken is frozen solid.

As pointed out in earlier comments and answers, every substance will have its own thermal conductivity. Moreover, the transition from frozen to unfrozen involves heat of fusion (e.g., an ice/water can stay at precisely 0 degrees C as heat flows out of it and the ratio of water to ice changes); and the effective heat of fusion of different foods will be different.

It will be very difficult (probably impossible) to come up with a single formula that covers all kinds of foods. An alternative approach might be to select samples of specific types of foods, insert small thermosensors into the samples at various depths & locations, and run the ambient temperature through its paces while you monitor the temperatures reported by the thermosensors.