Sunday, January 22, 2012

Beyond-Carnot heat pump (2)

In my post on 8 December 2011, I introduced a heat pump with a superior Coefficient of Performance to a Carnot heat pump.  The 2nd Law of Thermodynamics is not violated because my heat pump does not operate on a closed-loop cycle (as is required for Carnot heat pumps).  Today’s post explores how this out-performance depends on the inlet temperature to my heat pump.

Please read on …

By way of introducing the topic, the next part of today’s post is identical to my post of 8 December 2011. 

In 2004 I invented a new heat pump which I grandiosely called the Barton Drying Engine (BDE).  This has the following features:

> Mechanically expand a parcel of moist air; this lowers the temperature of the air as well as the partial pressures and densities of its component gases.

> Condensation forms immediately when dew point is reached, and any further expansion takes place along a moist adiabat.

> After the expansion is complete, collect the condensed droplets and remove them from the parcel of air.

> Re-compress the parcel of air back to ambient pressure; this involves an increase in temperature and partial pressures and densities.  Re-compression takes place along a dry adiabat.  Overall, work must be supplied to complete the cycle.

> The exhaust stream is hotter and drier than the inlet air stream.

I published a theoretical analysis of the BDE cycle in 2008 [1], with the main assumptions that air and water vapour are ideal gases with constant specific heat capacities.  The principal results are shown in Figure 1.
Figure 1: Results from [1] for the loss-free condensation heat pump.  The inlet air stream is saturated and at the temperature shown.  The results also depend on the expansion ratio.  Clockwise from top left: specific mass change in water vapour during cycle [kg/kg dry air], specific work output per cycle [J/kg dry air], temperature increase from inlet to outlet [°C], COP as a heat pump [dimensionless].

Here the Coefficient of Performance (bottom left diagram in Figure 1) was defined by

COP = ma CaP (T4-T1) / W

in which CaP is the specific heat capacity of dry air at constant pressure, ma is the mass of dry air in the parcel, T4 is the outlet temperature, T1 the inlet temperature and W is the work required to complete the cycle.  In words, the COP is the desired output (heat increase in the parcel of air) divided by the necessary input (work to complete the cycle).

We now come to the new material in today’s post.

We are going to compare the COP for the BDE to the COP for a Carnot heat pump operating over the same temperature range.  The emphasis will be on how the ratio COPBDE : COPCarnot varies with the inlet temperature.

Figure 2 shows the outlet temperature T4 for the BDE as a function of the expansion ratio r for six different inlet temperatures T1.  Note that the inlet air stream is saturated.
In the subsequent discussion, COP values are calculated as follows:
COPBDE = { ma CaP + [mv – δmv] CvP } (T4 – T1) / W,
COPCarnot = T4 / (T4 – T1).      [temperatures in K]
Here ma and mv are the original masses of air and vapour in the parcel of air that passes through the device, δmv is the amount of vapour condensed, T4 is the outlet temperature and T1 is the inlet temperature.  CaP and CvP are the heat capacities of air and water vapour at constant pressure, and W is the work required to complete the thermodynamic cycle.  COPBDE takes account of the heating of the water vapour component of the parcel of air.  This correction is minor.

As seen in Figure 1 (bottom-left), COPBDE does not vary significantly with inlet temperature, but does have a pronounced variation with the expansion ratio r.  (This is true whether or not heating of water vapour is considered; that is a minor effect compared to the heating of the dry air component.)

Figure 3 shows the ratio COPBDE : COPCarnot for the six cases in Figure 2.

For T1 = 5°C, the BDE in fact has sub-Carnot performance.

The ratio COPBDE : COPCarnot improves as the inlet temperature T1 increases, rising to a peak value of about 1.5 for T1 = 30°C at an expansion ratio of approximately 1.2.  It is conjectured that the ratio would continue to improve for higher inlet temperatures.  For all inlet temperatures there is a modest variation in the ratio COPBDE : COPCarnot as the expansion ratio r is allowed to vary.

The performance of the BDE depends on how much water can be condensed.  But the saturation water content of air increases faster than linearly with temperature, suggesting that the beyond-Carnot performance will become more pronounced as the inlet temperature is increased.  That has been confirmed by the results.

These results are affected strongly by inevitable losses in expansion and re-compression.  I have analysed these losses for the continuous-flow case, but the outcomes will have to remain a commercial secret for now.

Finally, I should point out that this beyond-Carnot performance might also be achievable – in the loss-free case – with other condensing heat pump dryers.


Thanks to Anthony Kitchener for his conjecture about the importance of the inlet temperature.  That has been confirmed by this study.


[1]  N G Barton, “An evaporation heat engine and condensation heat pump”, ANZIAM Journal, 49 (2008), 503-524.