## 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.

### 26 Responses to “A Mini-Fridge Investigation”

1. David Regan Says:

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Hi Michael,

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• protonsforbreakfast Says:

David

I think something went wrong with the comment! Do try again

M

• David Regan Says:

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Hi Michael,

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• David R Says:

Hi Michael,

As always I really like and enjoy your work. Another angle you may wish to incorporate into your calculations on the ‘sustainability’ of the heat pump system being selected would be the choice of refrigerant used by the appliance manufacturer.

It is worth researching the global warming potential (GWP) for the refrigerant in each system and then overlay the potential impact that pending PFAS regulations may have on the the current ‘low’ GWP refrigerant availability later this decade.

• protonsforbreakfast Says:

David,

The GWP of the refrigerant is something I considered when purchasing my own heat pump. The working fluid in the Vaillant is propane with a GWP of 3 compared with the Ecodan heat pump which uses R32 (I think) with a GWP of 766. Although leaks from heat pumps are not very significant, there is a significant chance that working fluid will be released at the end of the heat pump life.

John Cantor discuss the issue here

https://heatpumps.co.uk/environmental/global-warming-potential-of-refrigerants/

He concludes (and I agree) that over the lifetime of a heat pump, even a total release of refrigerant will still result in reduced emissions, but even so, using CO2 or propane is obviously to be preferred,

Best wishes

M

2. Paul Rudman Says:

I had considered getting a mini-fridge to store solder paste, but was put off by the number of Amazon reviews saying that theirs had melted / smoked / crackled / other-worrying behaviours.
Good luck with the talk.

• protonsforbreakfast Says:

Paul, Good afternoon.

I have only run this for 20 hours but I was impressed. It was very quiet, drew just 40 W, and would actually keep things chilled.

M

3. Bob Pugh Says:

It would be interesting to know the efficiency of the fridge. A quick google suggests they are poor in terms of energy efficiency <5%. I don't suppose you logged the electrical input?

• protonsforbreakfast Says:

Bob, Good afternoon.

By good fortune I did log the electrical power. On switching on the power draw was 46 W, but within 15 minutes this reduced to 40.0 ±0.1 W for the duration of the experiment.

All the best

Michael

4. Dan Grey Says:

It has an unfavorable surface area/volume ratio, too. Would the ratio of heat leakage to cooling power be more favourable with a larger volume, everything else being equal?

• protonsforbreakfast Says:

Dan, I’ll make a thermal model. I suspect that the heat leak through the Peltier element itself is likely what limits the base temperature. But hopefully a spreadsheet will clarify what’s going on. M

• Dan Grey Says:

Ahhh hadn’t realised it was a Peltier device – like a “car cooler” device?

• Nick Cook Says:

Problem with peltier coolers they bear no real relationship to the heat pumps that are used for air con. The attend not to be terribly efficient unless you have a very small temperature differential between the hot and the cold sides.

Dan Grey
When you say a ‘car cooler’, I assume you mean the cool boxes you plug in to the cigarette lighter socket. The air con uses conventional refrigerant heat pump technology. The main problem though is because they are driven mechanically off the engine they are not completely sealed units, like fridges and freezers, which means the refrigerant eventually leaks out through the seals, which is not terribly good for the environment☹️.

• protonsforbreakfast Says:

Nick, Yes, they are not very efficient, and although I understand the Peltier effect in principle, I would need to do some serious thinking to honestly say that I understood how they worked.

And thank you for explaining why car AC units leak. As you say, not great for the environment.

M

5. PeterG Says:

Presumably you can work out the COP of the mini-fridge acting as a heat pump and explain why real heat pumps are better than mini-fridges.

• protonsforbreakfast Says:

Peter, I didn’t specifically measure the heat output of the device, but if it was using 40 W to operate and removing just 2.5 W from the environment, then it would have a COP of (40+2.5)/40 or around 1.06.

One of the big problems with Peltier devices is that the hot sink is just millimetres away from the cold sink, and so heat leaks back through teh Peltier element itself.

In a fridge with recirculating working fluid, the condenser (where the heat is released) is physically separated from the evaporator (where it is captured).

I don’t know for sure but I think that is probably the most important thing.

M

• Nick Cook Says:

Peltier coolers are basically thermal couples in reverse, usually multiple ones strong together in some series parallel combination.
It is important to choose the right combination of materials, commercial ones usually consist of antinomy and bismuth, and the correct operating conditions for them to work effectively.

• protonsforbreakfast Says:

Indeed. On peculiar aspect of them is that the required properties are summarised in a so-called figure of merit. This summarises the combination of properties that make a good Peltier junction.

Two of the requirements are (a) to have a low electrical resistance and (b) a high thermal resistance.

The need for a low electrical resistance is to minimise heating in the junction, and the need for a high thermal resistance is to stop heat flowing back from one side of the junction to another.

Normally these properties of materials are linked directly to each other via the so-called Weiddemann Franz Ratio (IIRC) which is roughly constant for a very wide range of materials.

There are better materials, but not a cost anyone is prepared to pay.

M

• Nick Cook Says:

I was aware of the figure of merit thing although I’m not familiar with what the better materials that might be, although it wouldn’t surprise me if the best are either radioactive or highly toxic. Bit like double glazing, the best gas you could put in the gap, from a thermal point of view, is actually radon!

6. Giovanni Says:

Hi,

I think that the mass of the glass should have been factored in the cooling load calculations. Being made of glass, the thermal mass of the water container certainly had an impact on the work needed by the fridge to bring the water T down. The air around the glass will also give a contribution to the overall cooling load.

I think that the mass of the glass will contribute to the cooling load as well and that would take more energy being denser than the water?

• protonsforbreakfast Says:

Giovanni, Good Morning.

I did include the effect of the glass, but it is not much as one might think.

The specific heat capacity of glass is 0.84 J/°C/kg and that of water is 4.2 J/°C/kg
So the 300.7 g of water had a heat capacity of 1263 J/°C while the 372 g of glass had a heat capacity of 312 J/°C

So the heat capacity of the glass was only 25% of the heat capacity of the water. But I took account of it anyway. While writing the article I just said “Water” when I should more properly have said “Glass + Water”.

All the best

Michael

7. Nick Cook Says:

Might I suggest that you have a selection of some nice cool beers, et.al., in it for after the event socialising 😋, see you there.

• protonsforbreakfast Says:

Nick: Thank you. I just met with Andrea Sella this afternoon and ‘Beers in the Fridge’ was one of Andreas suggestions too – great minds….

• Nick Cook Says:

Doesn’t suprise me, I’ve seen Andrea several times at the CSF with the ‘Over Ambitious Demo Challenge’, always an entertaining event, with some science thrown in for good measure.

8. Martin Grindrod Says:

Sold out! Yay for you – boo hoo for me! Thanks for all your blogs by the way, as a retired professional electronics engineer I am glad I have finally found some proper research, measurement and analysis – I was particularly interested in the effects of triple glazing and external wall insulation as I am actively looking at those as a prelude to switching to a Vaillant heat pump, I will be installing new radiators throughout so planning on sizing them to run at 40 degrees. Keep up the good work!

• protonsforbreakfast Says:

Martin,

Good Afternoon. Yes, I have just seen that the talk at Cheltenham is sold out! It’s a best Sella!

Well done on taking a measured approach to the challenge – and it is a challenge – of retrofitting your home. In case I haven’t said it yet, try reading your meters once a week and recording teh results in a spreadsheet – it’s incredibly powerful.

In summer you will see your baseline usage for hot water and cooking, and then as autumn and winter arrive you will see an astonishing correlation between gas usage and the outside temperature.

Do contact me if you think I might be able to help with the analysis.

M