$$r$$ Hypothetical rate of return for the NPV calculation
$$L_\text{fusion}$$ Latent heat of fusion of water
$$C_v$$ Heat capacity at constant volume
$$C_p$$ Heat capacity at constant pressure
$$\eta$$ Coefficient of refrigerator performance $$\frac{|\dot{Q_c}|}{\dot{W}}$$
$$W$$ Work done on the fluid
$$Q_c$$ Energy entering the refrigerant on the cool side

# A lot of thoughts about refrigerators

Photo credit to Obi Onyeador

This article is a collection of thoughts about refrigerators, one of the largest drivers of home energy bills outside of HVAC systems. It is a buyers guide and a users guide, discussing how to evaluate the economics of energy efficient appliances and whether it is cheaper to freeze your own water or buy it at the grocery store.

## The Consortium for Energy Efficiency

As I declared in this article, the controllable electricity use of any house is mostly a function of the appliances that add or remove heat. We should not be concerned with the efficiency of the stove, oven, or dryer because it is extremely easy to convert electrical energy (or almost any other kind of energy) to heat. (It is not that these appliance use negilible amounts of power, it is that the heaters are almost 100% efficient so there is little to be gained by chasing efficiency.) When something is malfunctioning and running inefficiently, the wasted energy is becoming heat. Heat is where all energy is trying to go. Because of the thermodynamics of the process, refrigerators and air conditioners are difficult to design so there is more diversity of efficiency on the market and these appliances are improving with technological innovation much more than a kitchen stove ever will.

Data on refrigerators in this article comes from the Consortium for Energy Efficiency, a group which promotes energy efficiency and is responsible for awarding those Energy Star labels to appliances. They test appliances under standardize conditions to provide metrics for direct comparison and approximate costs of normal use.

What does the electricity of a refrigerator cost? Obviously depends on how much electricity in your area costs but a modern full size refrigerator (25 cu ft) is going to cost 300-700 kWh per year, or $50-120 per year ($0.17 per kWh, the average I paid since 2013 when I rented my first apartment after college).

Ice makers are an additional large draw on the power consumption of the refrigerator. When regulatory agencies do their testing, a refrigerator's icemaker unit is not included in the electricity use measurement and they add approximately 50-100 kWhs per year on top of what is shown in the graph below. Ice makers are generally found on larger refrigerators so eliminating the ice-maker models from a plot disproportionately removes larger refrigerators.

[Caption] Data from 1193 refrigerators, CEE. Tier I, II, and II are energy efficiency standards, tier III is the energy star rating. [Top left] Histogram of refrigerator volume. [Top right] Histogram of refrigerator power use. [Bottom left] Scatter plot of refigerator power use vs volume. [Bottom right] Scatter plot of power use vs volume not including refrigerators with ice makers.

As refrigerators grow in size, they require more power. When shopping for a refrigerator of a given size, the choice will be to spend more upfront on a more efficient model or spend more later on electricity. Whenever we need to compare cash flows at different times, refer to the article on net present values. The plot below shows how much a savings of X kWh per year is worth in net present value terms over 15 years. It goes all the way to 1500 kWh per year because although modern refrigerators do not require that much energy, refrigerators from the 1990s and earlier often do. And some of those are still in service today. If you own an old refrigerator, consider buying a power monitoring device and measuring its power consumption for a week.

[Caption] [Top and bottom] Same plot, different range of x values - [0,1500] kWh per year on the top, [0,200] kWh per year on the bottom. [Left axis] NPV of refrigerator electricity use over 15 years at $0.17 per kWh (the average I paid the last 6 years). [Right axis] NPV of refrigerator electricity use over 15 years in today's kWhs. Multiply this figure by whatever the average cost of power is in your area to find the NPV in your area's prices. If you have a refrigerator over 15 years old, the odds are good it is inefficient enough you should replace it today, for the sake of the environment and your wallet. It is likely you can reduce your power usage by 500 to 1000 kWh per year by upgrading your old refrigerator. If your electricity costs about what mine does, that easily justifies a$750-1000 investment.

However if you are shopping for a refrigerator, the difference between the most efficient and the most inefficient refrigerator at a given size, is about 200 kWh per year which only justifies a price delta of $300. The efficiency difference between otherwise comparable models rarely justifies more than a$100 price difference.

### Rule of thumb

If the appliance lasts 15 years and electricity costs about $0.20 per kWh, a change of 1 kWh per year in efficiency is worth no more than a$1 increase in the price of the appliance.

## Extension to other electrical devices

The NPV graph above is also applicable to any other trade off between upfront cost and electricity savings. For example, LED vs conventional lighting. Each LED bulb is saving 50 W per hour on compared to conventional. If we assume a house has 25 light bulbs, on for 8 hours a day, that is an power difference of 3500 kWh per year. It is worth thousands of dollars to replace those conventional bulbs with LEDs but they cost less than \$2 ea on Amazon.

## The cost of buying ice

A college friend majoring in physics once told me "chemical engineers are only good for laying pipe and converting units" so I pointed out that physicists are not even good enough for that. Computing the cost of freezing water compared to buying ice is a unit conversion problem.

\begin{align} \rho_\text{cost density at grocery store} >& \bigg(C_p \Delta T + L_\text{fusion} \bigg) \frac{1}{\eta} \\ \mathrm{} 0.44\frac{1}{\mathrm{kg \, ice}} >& \left[ \left( 4200 \frac{\mathrm{J}}{\mathrm{kg}}\right) \left(23 \mathrm{\, C}\right) + 33400 \frac{\text{J}}{\text{kg}}\right]\left(\frac{1}{2} \frac{ \mathrm{J\, electricity}}{\mathrm{J \, removed \, from \, water}} \right) \left( 2.7\times10^{-7} \frac{\mathrm{kWh}}{1\mathrm{\, J}}\right) \left(\mathrm{}0.20 \mathrm{\, kWh^{-1}}\right) \\ \mathrm{} 0.44\frac{1}{\mathrm{kg \, ice}} >& \mathrm{\, } 0.012 \frac{1}{\mathrm{kg \, ice}} \end{align}

Ice is about 40x more expense at the grocery store than it is homemade.