# Efficiency of parallel and sequential heat pumps

Consider two identical heat pumps, for example, split-system air conditioners. There're two ways to make them work - in parallel or sequential. Parallel means that "hot" radiators of the machines are put outside the room, while the "cold" radiators are put inside the room. Sequential means that the cold radiator of the first machine is inside the room, while the hot radiator is put inside, say, a large isolated box, where it is cooled by the cold radiator of the second machine, and the hot radiator of the second machine is outside the room.

In which case - parallel or sequential - the efficiency of the cooling would be better?

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Are you talking about heat engines or heat pumps? – Pygmalion Apr 26 '12 at 9:26
The air conditioner (cooler) is actually a heat pump, so, yes, I'm talking about heat pumps. – mbaitoff Apr 26 '12 at 10:03

Efficiency of one heat pump in cooling can be defined by expression

$$\eta = \frac{Q_C}{W},$$

that is heat that is taken from cooler reservoir divided by the work put into the pump.

If you have two pumps in parallel efficiency shall be the same, as you will have heat twice as large and work twice as large

$$\eta = \frac{2Q_C}{2W}.$$

If you however have two pumps in sequence equation reads as

$$\eta = \frac{Q_C}{2W}.$$

So in order to obtain at least same efficiency you should extract heat twice as large by each pump. So the efficiency of each pump should be twice as large for about half of the temperature change. So far so easy.

In the next step we must take into consideration exact cyclical thermodynamical processes. There are plenty of them that you can use and no can be exactly theoretically calculated. In such cases it is useful to observe the most efficient thermodynamical process, that is Carnot cycle and extract conclusions from it.

Efficiency of the heat engine based on Carnot cycle can be shown to be

$$\eta = \frac{T_C}{T_H-T_C}.$$

In case of two pumps in the first iteration intermediate temperature $T' = \frac{T_C+T_H}{2}$ in exactly in the middle. Efficiency of two pumps shall be

$$\eta_1 = \frac{T_C}{T'-T_C}, \eta_2 = \frac{T'}{T_H-T'}$$

Obviously $\eta_1 = 2 \eta$ and $\eta_2 > 2 \eta$, therefore two sequential pumps in cooling should be more efficient.

It is interesting that two sequential pumps in warming should be less efficient using the same arguments! I cannot find the error in the reasoning, so before accepting the answer, please wait some time that others check and give their comments.

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In case of sequential pumping you say that the amount of heat taken is the same as in case of parallel, but the heat exchanging is proportional to the temperature difference, and in sequential case the temperature differences are higher, so the amount of heat is not the same $Q_c$ for those cases. – mbaitoff Apr 26 '12 at 11:24
@mbaitoff I didn't exaclty understand you, but I will try to guess what is the problem you are talking about. If you have a heat pump and large temperature difference, the heat pump will extract only small amount of heat. If you have small temperature difference, the heat pump shall extract larger abount of heat in the same period of time. It is similar to rising 1kg to 10 m, and 10 kg to 1 m. You need equal work to do both. – Pygmalion Apr 26 '12 at 12:19
Heat pumping (in air conditioner) is achieved by subtracting the heat from the volume being cooled and by dissipating this heat into outer volume. Dissipating is only possible when coolant fluid temperature is lower than the outer volume temperature. When outdoor temperature rises, air conditioning becomes less effective and stops at all when those temperatures become equal, right? So, at the point of equilibrium $Q_c$ becomes zero, so $Q_c$ cannot be constant and depends on temperature difference. Am I right or am I missing something? – mbaitoff Apr 26 '12 at 17:50
Dissipating is only possible when coolant fluid temperature is higher than the outer volume temperature. – Pygmalion Apr 26 '12 at 17:54
If temperature difference is higher, then $Q_C$ effectively becomes smaller. Of course, for real systems there is certain temperature difference for which $Q_C = 0$. Theoretically this is possible only for infinite difference in temperatures – Pygmalion Apr 26 '12 at 17:56