Air conditioners

Reference: Daniel V. Schroeder, An Introduction to Thermal Physics, (Addison-Wesley, 2000) – Problem 4.13.

An air conditioner is an example of a refrigerator in which the cold reservoir is the room to be cooled and the hot reservoir is the outside atmosphere. On a hot day, the rate at which heat leaks into an air conditioned room from the outside is roughly proportional to the temperature difference {T_{h}-T_{c}} between the outside and inside. In that case, the work done to remove an amount {Q_{c}} of heat in time {\Delta t} is

\displaystyle  W=Q_{h}-Q_{c}=Q_{h}-K\left(T_{h}-T_{c}\right) \ \ \ \ \ (1)

where {Q_{h}} is the heat expelled to the outside and {K} is a constant.

In an ideal refrigerator (e.g. one working on a reversed Carnot cycle) the entropy gained in absorbing {Q_{c}} is equal to the entropy lost in expelling {Q_{h}}, so

\displaystyle   \frac{Q_{c}}{T_{c}} \displaystyle  = \displaystyle  \frac{Q_{h}}{T_{h}}\ \ \ \ \ (2)
\displaystyle  Q_{h} \displaystyle  = \displaystyle  \frac{T_{h}}{T_{c}}Q_{c}\ \ \ \ \ (3)
\displaystyle  \displaystyle  = \displaystyle  K\frac{T_{h}}{T_{c}}\left(T_{h}-T_{c}\right) \ \ \ \ \ (4)

The work required to maintain a temperature of {T_{c}} is therefore

\displaystyle   W \displaystyle  = \displaystyle  K\frac{T_{h}}{T_{c}}\left(T_{h}-T_{c}\right)-K\left(T_{h}-T_{c}\right)\ \ \ \ \ (5)
\displaystyle  \displaystyle  = \displaystyle  \frac{K}{T_{c}}\left(T_{h}-T_{c}\right)^{2} \ \ \ \ \ (6)

Thus lowering the inside temperature by a small amount can have a large effect on the work required to maintain this temperature, and thus on the cost of running the air conditioner. For example, suppose the outside temperature is {30^{\circ}\mbox{ C}=303\mbox{ K}} and the inside temperature is {22^{\circ}\mbox{ C}=295\mbox{ K}}. If we wish to lower the inside temperature by only one degree, the extra work required is

\displaystyle  \frac{W_{294}}{W_{295}}=\frac{295}{294}\frac{9^{2}}{8^{2}}=1.27 \ \ \ \ \ (7)

We need to use 27% more power to achieve a single degree more cooling. This is one reason why it is much more economical to bear with a slightly higher indoor temperature on a hot day.

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