Modern Engineering Thermodynamics

7.6 Heat Engines Running Backward 217
IS THE EFFICIENCY OF HEAT PUMPS, AIR CONDITIONERS,
AND REFRIGERATORS GREATER THAN 100%?
If you look closely at the thermal efficiency equations for heat pumps, air conditioners, and refrigerators, you see that their
efficiency is going to be more than 100%, because under normal operating conditions, the numerator in their efficiency equation
is usually greater than the denominator. Consequently, their energy conversion efficiency is usually greater than 100%.
How can that be—nothing should have an energy conversion efficiency greater then 100%. But, it is correct. This is simply
due to the way in which the thermal efficiency formula (Eq. (4.70)) is structured:
Energy conversion efficiency = η E =
Desired energy result
Required energy input
(4.70)
This makes a heat pump much more attractive for domestic heating than, say, a purely resistive electrical heater. Electrical heaters
convert all their input electrical energy directly into thermal energy and therefore have energy conversion efficiencies of
100%, whereas most heat pumps have energy conversion efficiencies far in excess of 100% for the same electrical energy input.
Since this could be a problem in public advertising, the industry uses the phrase coefficient of performance (COP) instead of
efficiency. The COP is simply the pure efficiency number before it is converted into a percentage. For example, the COP of
a heat pump with an energy conversion efficiency of 450% is 4.5.
COP heat
= η heat
= jQ Hj
=
jW
pump pump in j
jQ H j
jQ H j − Q L
=
j _Q H j
=
j _W in j
j _Q H j
j _Q H j − _Q L
(7.17)
The desired energy result in the operation of a heat pump is heat addition to an already warm environment.
Therefore, its energy conversion efficiency can be determined from Eq. (4.70) and an energy balance on the
device (see Figure 7.9b) as
η heat
=
pump
Desired energy result
Required energy input = jQ Hj
jW in j
=
jQ H j
jQ H j − Q L
=
j _Q H j
j _W in j
=
j _Q H j
j _Q H j − _Q L
where, as in the previous section, we use the absolute values of certain terms to avoid improper or confusing
algebraic signs.
If the heat pump is modeled as a backward running Carnot heat engine, then Eqs. (7.15) and (7.17) can be
combined to yield the COP for a “reversible” (i.e., frictionless, etc.), or Carnot, heat pump as
COP Carnot
=
heat pump
T H
T H − T L
(7.18)
If the removal of heat Q L from a space is the desired result of a backward running heat engine, then the engine
is called a refrigerator when food is stored in the cooled space and an air conditioner when people occupy the
cooled space.
The energy conversion efficiency of a refrigerator or air conditioner can also be obtained from Eq. (4.70). As in
the case of a heat pump, these efficiencies are also normally greater than 100% and they too are commonly
represented with the pure number coefficient of performance label
COP refrig: or
= η refrig: or
=
air cond: air cond:
= Q L
jW in j
=
Desired energy result
Required energy input
Q L
jQ H j − Q L
=
_Q L
j _W in j
=
(7.19)
_Q L
j _Q H j − _Q L
For a backward running Carnot (i.e., reversible) heat engine, Eqs. (7.15) and (7.19) can be combined to give the
COP for a reversible refrigerator or air conditioner as
T
COP Carnot = L
(7.20)
refrig: or
T H − T L
air cond: