According to Perry’s Chemical Engineers’ Handbook, seventh edition, page 10-37:
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In any continuous compression process the relation of absolute pressure p to volume V is expressed by the formula pVn = C = constant. The plot of pressure versus volume for each value of exponent n is known as the polytropic curve. Since the work W performed in proceeding from p1 to p2 along any polytropic curve is:
it follows that the amount of work required is dependent upon the polytropic curve involved and increases with increasing values of n. The path requiring the least amount of input work is n = 1, which is equivalent to isothermal compression. For adiabatic compression, (i.e., no heat added or taken away during the process), n = k = ratio of specific heat at constant pressure to that at constant volume.
Since most compressors operate along a polytropic path approaching the adiabatic, compressor calculations are generally based on the adiabatic curve.
The calculations in the July issue, however, ignored water vapor, humidity and condensate because water is a minor constituent of air. The presence of trace amounts of water vapor at the inlet of the compressor has minimal influence on the heat of compression. On the other hand, it is only the atmospheric humidity at the inlet of the compressor that accounts for all of the condensation that occurs in every compressor system. Water vapor is, therefore, an important constituent for purposes other than heat duty.
One more input needed
In addition to the thermometer and barometer the July article recommended for every compressor room, this article recommends a hygrometer be added to the instrument set so that the relative humidity can be incorporated into a more comprehensive set of calculations. Knowing the humidity allows the spreadsheet to produce additional outputs, specifically:
- Inlet moisture content.
- Mass flow rate of water entering the system.
- The inlet saturated vapor pressure.
- The actual inlet vapor pressure.
- The saturation humidity of the compressed air.
- The condensate rate.
- The heat duty attributable to water vapor.
The key operating equations
Compressing air reduces its ability to hold water in the vapor state. That is the underlying reason we need to handle compressor condensate. The formula that relates the humidity at saturation to the total system pressure is:
Hs = [ps/(P-ps)](Mv/Mg)
where Hs = saturation humidity in lb. vapor/lb. dry gas;
ps = saturation pressure of the vapor at a given temperature;
P = total system pressure;
Mv = molecular weight of the vapor;
and Mg = molecular weight of the gas.
In this case, the vapor is water having a molecular weight of 18.016 and the gas is air having an approximate molecular weight of 29. The saturation vapor pressure is given by the relationship:
Pvp-s= 0.08858 x 10(7.5Tc/237.7+Tc) where Pvp-s = saturation vapor pressure in psi and Tc = actual temperature in degrees Celsius.
The actual vapor pressure is given by another relationship: Pvp-a = (RH x Pvp-s)/100 where Pvp-a = actual vapor pressure in psi and RH is the relative humidity in percent.
Finally, the relationship connecting the moisture content, actual vapor pressure and pressure is:
M = (4354 x Pvp-a)/P
where M = moisture in grains of moisture per lb. of dry air.
Algorithmic overview
The spreadsheet calculates the moisture in the air and the mass flow of water into the compressor based on the inlet flow rate of air. Then, it calculates the saturation vapor pressure of the compressed air, which relates to the total amount of the inlet moisture that remains in the vapor state. The difference represents the amount of water that condenses and the spreadsheet converts the value to a flow rate in gpm.