A Mini-Fridge Investigation

Friends, on 7th June I will be giving a talk at the Cheltenham Science Festival with the illustrious Andrea Sella on the topic of Heat Pumps. And I am growing increasingly anxious.

Click on the image for a larger version. Sadly the publicity for our talk at Cheltenham has used the wrong title! But which title have Andrea and I chosen?

I’ve been working on the talk with Andrea for weeks, but there are still parts that we haven’t finalised, including exactly which demonstrations to include. In my search for demonstrations, I thought I would take a look at a mini-fridge that would be easy to position on a table-top. Forty pounds later, a mini-fridge arrived.

Click on Image for a larger version

I managed to get some nice thermal images, and while I was playing around, I thought I would see if I could work out the actual cooling power of such a tiny refrigerator.

Click on image for a larger version

Measurements

My idea was to measure the rate at which a glass of water cooled inside the fridge. From the definition of the heat capacity of water, I knew that to cool one gramme of water by 1 °C required the removal of 4.18 joules of energy. If I measured the time taken to cool a glass of water, I could work out the rate at which heat energy was being removed in joules per second i.e. watts.

So I poured 300.7 grams of water into a glass weighing 371.9. grams. I then placed the glass inside the refrigerator. I deployed four thermocouples to monitor the progress of the cooling.

• I put one thermocouple in the water.
• I put one thermocouple on the cold plate at the back of the inside of the fridge.
• I put one thermocouple on the hot plate at the back of the outside of the fridge
• I used one thermocouple to monitor the temperature of the ambient air.

I then used a datalogger to record the temperatures every 30 seconds. The results from the first 8 hours are shown below.

Click on image for a larger version. Graph showing the temperature versus time of the four thermocouples positioned around the fridge. See text for details of their location.

The data looked pretty much as I expected, with a couple of anomalies. I do not know the cause of the ‘spike’ at 5.2 hours, but the change in cooling rate just before six elapsed hours was caused by leaving open the door of the kitchen laboratory.

The graph below is the same as the previous graph, but has arrows to show the heat flows.

Click on image for a larger version. Heat flows naturally from the water to the cold sink. The heat pump then moves that heat from the cold sink to the hot sink at the rear of the refrigerator. Heat then flows naturally from the hot sink to the air in the room.

Cooling Power

To work out the cooling power, I started with the data of temperature versus time for the water.

Click on image for a larger version. Graph showing the temperature (°C) of the water versus time.

I then worked out the slope of graph above. The cooling rate is only a few thousandths of a degree per second, so in order to measure this with thermocouples with a resolution of only 0.1 °C, I averaged the cooling data over ±10 minutes.

Click on image for a larger version. Graph showing the cooling rate (°C/s) of the water versus time.

Just after the fridge is switched on, the cooling rate peaks at around 1.5 thousandths of a degree per second, or 5.4 °C/hour. The oscillations in the cooling rate are probably due to convection of the water in the glass.

To convert this cooling rate into an estimate of cooling power, one needs to multiply the data above by the heat capacity of the water and the glass, in this case, 1575 J/°C.

Click on image for a larger version. Graph showing the cooling power of the refrigerator.

The data suggests that the cooling power peaks at around 2.5 watts and then falls to just a fraction of a watt. The electrical power drawn was 40 W so overall the efficiency was around 6%.

Lowest Temperature

The cold plate in the back of the fridge gets impressively cold, cooling to below -5 °C. So why doesn’t the water cool to this temperature?

To investigate this, I plotted the rate of heat flow out of the water (in watts) not versus time, but versus the temperature difference between the water and the cold plate.

Click on image for a larger version. Graph showing the cooling power of the refrigerator as measured by the cooling rate of a glass of water, versus the difference in temperature (Delta T) between the water and the cold back plate.

The data show that the cooling rate is roughly proportional to the temperature difference between the water and the back plate. But the cooling rate falls to zero when there is around 9 °C of temperature difference. Why doesn’t the water keep on cooling?

The reason is that heat is flowing into the water through the walls of the refrigerator. The cooling power of the heat pump probably remains at around 2 or 3 W, but as the internal temperature of the fridge falls, heat ‘leaks’ through the insulating the walls.

Click on image for a larger version. A more elaborate version of the graph above. showing additionally heat flowing into the cold sink from the hot sink, and the room. The fridge uses insulation to try to minimise these flows, but they are not zero.

An analogy would be using a water pump to bail out a leaking vessel. When the rate of bailing is equal to rate at which water is leaking in, then the water level doesn’t change. Similarly, the colder the inside of the refrigerator becomes, the more significant heat leaks become. The lowest temperature occurs when the rate at which the heat pump removes heat is equal to the rate at which heat leaks from the environment.

Reflections

Friends, the mini fridge was a little more powerful than I expected, and I was pleased that I managed to remember how to operate the data logger. But none of this is relevant to the talk I am trying to finish with Andrea – somehow these simple measurement exercises seem very attractive when there is proper work to be done.