formula for calculating coefficient of performance for energy efficiency of HVAC devices

Coefficient Of Performance (COP) For Heat Pumps and Air Conditioners

COP or Coefficient Of Performance is the most basic energy-efficiency metric of any heat engine. It’s very useful when comparing heat pumps, fridges, and air conditioners.

By definition, COP is the ratio of how much useful heat (or cold) an HVAC device will produce if we give it certain energy input. Basically, it tells us how much heat we can generate with every watt of energy we pay.

Example: We have a 1000W heat pump with a COP of 3.5. That means that we power it with 1000W and the heat pump gives us back 3500W worth of heat.  That’s a pretty energy-efficient heat pump. It will boil almost 10 gallons of water per hour.

For comparison: A 1000W heat pump with a COP of 2 will boil less than 6 gallons of water per hour.

Mind you that our electricity bill is the same in both cases. Running a 1000W heater for an hour costs about 13 cents. The COP 3.5 heater will boil a gallon of water for less than 1 cent while the COP 2 heater will boil a gallon of water for more than 1 cent.

It’s pretty obvious that it’s much better to have a COP 3.5 water heater than a COP 2 water heater, right?

Let’s look at how COP is calculated, what is the highest possible COP of a heat pump and how the electricity bill is affected by HVAC units with different COP values.

How To Calculate COP (Equation)?

Here is the formula of a COP:


where Q is heat the heater generates if we give it a certain amount of work (W).

For cooling, Q represents the heat that we take away from a cold reservoir. Air conditioner, for example, takes heat away from a room (cold reservoir).

Note: It’s entirely possible that COP for heating and COP for cooling is different. Mini split AC units, for example, are capable of cooling down space as well as heating it up. A good mini split system will usually have a cooling COP ratio of 2 or more, and a heating COP ratio of 3 or more.

If we apply the 1st law of thermodynamic and do a bit of derivation, we can see calculate the COP values for a theoretically 100% ideal heat pump and ideal air conditioner (we also call that a Carnot machine). Let’s do the heat pump first:

COP Of A Heat Pump

Here’s how you can calculate the theoretically maximum COP of a heat pump:

COPheat pump = Thot/(Thot-Tcold)

Thot is that cozy hot temperature we want to have during cold winters (let’s say 95F; that’s 298 in Kelvins).

Tcold is the cold temperature at which the heat pump starts to operate (let’s say 57F or 287K).

Theoretical maximum COP is calculated like this:

COPheat pump = 298K/(298K-287K) = 27,09

So, in theory, a heat pump can have a COP even above 20. Nonetheless, the real COP of a heat pump in practice is a lot lower.

Golden standard: The standard test to measure the COP of a heat pump is conducted with Thot = 95F (308K) and Tcold = 32F (273K). That means that in 100% ideal case the maximum COP is 8.8. But in practice, it’s lower.

In fact, the highest COP a heat pump can achieve is about 4.5. Any heat pump with a COP of above 3 has a very high energy-efficiency.

Here is a graph of how much more efficient high COP heat pumps are. We put the COP 2 as zero and calculated by how many percents more efficient the higher COP heat pumps are.

coefficient of efficiency for heat pump graph

You can see, for example, that a 3.2 COP pump is 60% more energy-efficient than 2 COP pumps.

Coefficient Of Efficiency Of Air Conditioners

OK, let’s calculate the maximal theoretical efficiency of a cooling device. Namely, an air conditioner or a refrigerator. By applying the 1st law of thermodynamics, we can deduce the ‘Carnot COP’ for a cooling device is calculated as such:

COPcooling= Tcold/(Thot-Tcold)

Tcold is the chill temperature you want to have in your room during the summer. Thot is the high heat wave temperature.

Let’s look at what would a COP of an air conditioner be in a standardized Thot = 95F (308K) and Tcold = 32F (273K) temperature interval. Plugging the temperature in the COP cooling equation above we get 7.8.

If you remember, the maximum coefficient of performance of a heat pump was 8.8. In you are in a market for an air conditioner, make sure to get one with a COP value of above 2. That’s a very high COP for a practical HVAC cooling device.

The problem usually is that you won’t find a COP value anywhere, not even on a specification sheet. Energy-efficiency is usually represented with such metrics as EER and SEER; all these as based on the COP of a HVAC divide. For example, we have compared the best portable air conditioners by comparing their EER ratings.

Seasonal COP Or SCOP

In 2013, SCOP or Seasonal Coefficient Of Performance was introduced. We know that COP is a measure of energy-efficiency for a heating or cooling device. What measuring SCOP is trying to achieve is to objectively measure energy-efficiency over the winter season (for heating) and summer season (for cooling).

Basically, the relationship between SCOP and COP is the same as with SEER and EER.

SCOP would give a much more realistic idea about how energy efficient an HVAC device is in practice; ie. in a real summer season.

Nonetheless, as of now, the SCOP has still considered a very new methodology of measuring seasonal cooling and heating efficiency. As such, you will rarely find the SCOP ratio on older devices. In fact, even the new devices rarely include SCOP in their specification sheets primarily because they have yet to measure it.

1 thought on “Coefficient Of Performance (COP) For Heat Pumps and Air Conditioners”

  1. I liked that you pointed out that you will want to consider what the heat pumps you have max output. It does seem like a good thing to know if you feel like your heat pump isn’t keeping your house warm enough during the winter. I know that I hate being cold so I would want to get a good pump in my house.


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