# Abrazolica

Efficiently Heating a House II

Out of all the conventional ways to heat a house using a heat pump is by far the best. The reason is that it takes less energy to move a quantity of heat than it does to generate that heat in the first place. A heat pump can take heat from the cold outside air, yes there is still heat in that air, and move or pump it into the warmer house. Heating a house with a heat pump is like having an inside out refrigerator where you're heating the inside and cooling the outside.

The basic thermodynamic operation of a heat pump is shown in the following figure. We have two heat reservoirs, the outside of the house at temperature $T_0$ and the inside at temperature $T_1\gt T_0$. With an input of energy $W$ (this is the part you have to pay for) the pump removes a quantity of heat $Q_0$ from the outside air and deposits a quantity of heat $Q_1=Q_0+W$ into the inside air. The efficiency of this process for a given heat pump is called its coefficient of performance $\mathbf{COP}$. It is defined as follows

$\mathbf{COP}=\frac{Q_1}{W}$

In other words, it's the ratio of what you get to what you pay for. If $\mathbf{COP}=2$ for example, then one unit of input energy will provide two units of heat energy to the inside of the house.

There is no single value for $\mathbf{COP}$ since it is generally a function of the temperatures $T_0$ and $T_1$. Ideally there should be a plot of $\mathbf{COP}$ for different values of $T_0$ and $T_1$ but we have not seen a single heat pump manufacturer that does this. Instead they use (for the case of heating performance) something called a heating seasonal performance factor $\mathbf{HSPF}$. I should mention here that there are different types of heat pumps. Some use the air as a heat source while others use ground or water as a heat source. The HSPF mainly applies to air source heat pumps.

$\mathbf{HSPF}$ is expressed as the heat output over an entire heating season measured in BTUs (British thermal units) divided by the total electricity used measured in watt-hours. This gives it the somewhat bizarre units of BTU/watt-hour. You can convert $\mathbf{HSPF}$ to the average $\mathbf{COP}$ over a heating season, which is unitless, by multiplying $\mathbf{HSPF}$ by $0.2931$, the number of watt-hours in a BTU.

High efficiency heat pumps have an $\mathbf{HSPF}$ of at least 8 or an average $\mathbf{COP}$ of $8\cdot 0.2931=2.3448$. By contrast, an electrical resistance heater, which can do no more than turn all of its input energy into heat, would have $\mathbf{COP}=1$ or $\mathbf{HSPF}=1/0.2931=3.4118$.

It is interesting to compare a real heat pump with an ideal heat pump which produces no entropy so that

$\frac{Q_1}{T_1}=\frac{Q_2}{T_2}$

with the temperatures measured in Kelvin. The $\mathbf{COP}$ can then be expressed in terms of the temperatures as

$\mathbf{COP}=\frac{T_1}{T_1-T_0}$

No heat pump operating between the temperatures $T_1\gt T_0$ can have a greater $\mathbf{COP}$ than this. As an example, if the outside temperature is 0°C and the inside temperature is 20°C then $T_0=273.15 K$, $T_0=293.15 K$ and the maximum $\mathbf{COP}$ that any heat pump can have is 14.6575.

Currently, in the United States, the minimum HSPF set by the government is 8.2, but will rise to 8.8 in 2023. You will pay more for higher HSPF. For example, a Senville 12,000 BTU mini split heat pump with HSPF of 8.5 costs 799.99 USD, while a Fujitsu 12,000 BTU mini split heat pump with HSPF of 12.5 costs 1539.00 USD. For this price difference of 739.01 USD the $\mathbf{COP}$ rises from 2.49 to 3.66.

Of course, reliability is another factor to consider when buying a heat pump. Consumer Reports has a reliability rating for numerous brands.

A blog post we found worthwhile reading on the topic of installing a heat pump yourself is "Our DIY Heat Pump Install – Free Heating and Cooling for Life?".

This post as a pdf

© 2010-2021 Stefan Hollos and Richard Hollos 