# Calculation of temperature change in a refrigerated box

Let’s say I have a refrigerated box held at temperature T, and I also have a door that opens on the front face for S seconds. How does one go about calculating how long it takes for the temperature of the box to raise D degrees? Or can someone suggest keywords to search on so I can learn how to do this? All of the thermodynamics searches I have performed discuss temperature changes in plates, and I do not know how to extend this information to apply to an enclosed volume.

• Some more information is required. When you say that the door opens, do you imply that you're removing some form of ideal insulation instead? Are you asking if the area of the area not insulated is relevant? Also, the ambient temperature is relevant. And we usually end up with some form of temperature gradient, rather than all contents of the box changing their temperatures simultaneously. – user191954 Aug 28 '18 at 12:59
• I believe that a crude but good approximation would be to construct a transient 1D heat conduction model as simply treat the air volume as if it was a "plate". But as Chair suggests, you will end up with a temperature gradient. – user3408085 Aug 28 '18 at 13:20
• This may not be a "refrigeration" problem. Does the door open from the side, from the top, or from the bottom? How fast does the door open? For a door opening from the side, cold air will fall out of the refrigeration compartment, starting at the bottom, and warm air will take its place, starting from the top, due to the differences in density. The flow of these two air masses will control how the temperature changes in the refrigeration compartment. Thus, this is probably a flow problem rather than a refrigeration problem. – David White Aug 28 '18 at 15:08

## 1 Answer

A good place to start is the lumped-component model, which assumes that the entire system (here, the inside of the box) can be characterized by a single temperature $T$. That is, we're ignoring any temperature gradients within the material.

The typical approach is to perform an energy balance. We have a refrigeration mechanism that removes thermal energy at a constant rate, say, (call this $-Q$) and (with the door open) natural convection to the outside, which is at room temperature $T_\infty$. We could add up the rates of energy change and write something like

$$-Q-hA(T-T_\infty)=C\frac{dT}{dt}$$

where $h$ is a natural (or free) convection coefficient that characterizes the heat exchange with the room, $A$ is the exposed surface area, $C$ is the heat capacity of the cool area of the box and its refrigerated contents (a higher heat capacity means that the material is slower to change temperature), and $t$ is time. In other words, the speed at which the box changes temperature depends on the strength of cooling, the degree of convection, the exposed area, the temperature difference, and the heat capacity.

If we left the box open for a long time (so that steady state is achieved and $dT/dt$ approaches zero), we'd calculate that $T=T_\infty-Q/hA$, i.e., that the box is a little cooler than the room because of the refrigeration mechanism. So that makes sense.

Let's define the temperature difference $\Delta T=T-T_\infty$ to streamline the equation:

$$-Q-hA\Delta T=C\frac{d\,\Delta T}{dt}$$

The solution is

$$\Delta T=-\frac{Q}{hA}+\left(T_0-T_\infty+\frac{Q}{hA}\right)\exp(-hAt/C)$$

where $T_0$ is the initial temperature inside the box. This model therefore tells us that the temperature difference asymptotically approaches the steady-state value in an exponential manner when the door is opened. The time constant, as discussed here, is $C/hA$; this is the time (in seconds) that it takes to erase a fair amount of the original temperature difference. After a few time constants, we're nearly at the steady-state value. So estimating the time constant is the key to answering your question regarding quantitative temperature changes.

$A$ is the total exposed cooled area, and $C$ is the heat capacity of all of the cooled material. Estimating these values involves a whole additional set of assumptions, of course. Intuitively, areas within crevices are not going to participate as much in the heat transfer process. In addition, it may be challenging to estimate the amount of cooled material because in reality, temperature gradients will exist between the refrigerated part of the box and the outer surfaces and cooling mechanism. Estimates here may vary within an order of magnitude.

We have a similar degree of uncertainty regarding the natural convection coefficient $h$, which is notoriously challenging to estimate from first principles, as it depends on the gas density, viscosity, temperature, and flow rate and the system geometry. We often end up applying complex empirical relations that involve nondimensional relations such as the Rayleigh, Prandtl, and Reynolds numbers. As a first pass, I'd plug in $1-10\, \mathrm{Wm^{-2}K^{-1}}$ as an order-of-magnitude estimate for natural convection in air. However, $h$ could change substantially depending on whether the opening is facing upwards, to the side, or downwards. It's likely going to be necessary to calibrate these predictions with an actual system. However, this approach at least yields the insight that the characteristic time should scale up with $C$ and inversely with $hA$, which should match our intuition that a stocked fridge maintains its temperature better, for example.