Steam Engine Efficiency Calculations
For steam, gamma=Cp/Cv=1.32
V0= steam volume before expansion.
P0= steam pressure before expansion.
t = pressure difference between the hot and the cold sides.
r = volume expansion ratio within the cylinder. This should be controlled by an electronic valve.
T_out = theoretical temperature of the steam exiting the cylinder after expansion, assuming perfect adiabatic expansion and no condensation.
Within one engine cycle, there are two periods where work is done by the steam: the injection period when the hot side of the piston has pressure equal to the maximal one, and the expansion period when the steam injection is cut off, and the steam remaining within the cylinder expands.
The expansion is assumed to be adiabatic. In reality it is not, hence efficiency will be lower than the theoretic one.
During the injection period, the work is done by constant pressure against the low pressure at the other side of the piston, so the work equals
During the adiabatic expansion period, the pressure is proportional to 1/V^gamma where V is the volume, so the total work is the integration
V0*P0*(1/x^gamma-1/t)dx, from 1 to r.
Practical losses: friction 5%, generator 15%, parasitics 5%, fluid loss 10%. Overall 69%.
Water pump efficiency: 80% (mechanical drive).
Possible improvement: heat up the cylinder to high temperature. This can dry up the partial condensation, as well as can improve expansion efficiency, because the expansion will somewhat approach the isothermal process. But this may create difficulties for the lubricating oil.
If the cylinder heating is not done or is not efficient, the whole efficiency computation may suffer from a loss of 20% or more.
Cylinder heating should be efficient for multi-stage engines, by heating stages except the last one. It is not clear whether heating single-stage engines will be efficient. The situation is quite intricate, and can only be determined via practical tests.
We assume a maximal expansion ratio of r=5. In practice, this probably requires a high precision standard for the cylinder manufacturing. The piston compression ratio should not be lower than 15.
All examples are given for CHP cases, with condensed cooling side towards 60C.
Note. The calculations are approximative, especially for the high end. Steam nearing boiling point and at high temperature is not ideal gas, and data are hard to find for non-saturated steam.
enthalpy difference=2795-251=2544kJ/kg. For r=4: output=0.384*500*(0.96+1)=376kJ/kg, pump consumption = 1.25*0.5=0.6kJ/kg, net work 376kJ/kg, raw efficiency=14.8% (10.2% net electricity)
Comment: this case is of very low requirement, with a simple construction and requiring low solar concentration ratio, but the efficiency is too optimistic for a single stage engine, due to condensation and heat exchange in the cylinder. 8% is more realistic. For prototypes and starting products.
V0=0.238m^3/kg, t=10/0.2=50, enthalpy difference=2930-251=2679kJ/kg. For r=5: output=0.238*1000*(0.98+1.18)=514kJ/kg, pump consumption = 1.25*1=1kJ/kg, net work 513kJ/kg, raw efficiency=19.1%. (13.2% net electricity)
Comment: this is close to optimum for a single expansion home engine. 250C should be more or less the limit of what legislation will allow for a home, 10bar is reasonable, and increasing pressure for single stage does not give much more efficiency. But again the efficiency is probably too optimistic.
V0=0.158m^3/kg, t=15/0.2=75, enthalpy difference=2910-251=2659kJ/kg. First stage: r=5, t=10, with steam reheating to 230C. T_out=523K/10^0.24=28C. Average heat capacity=2kJ/kg.K, reheating energy input 404kJ/kg, total energy input 3063kJ/kg. Second stage: r=4, t=7.5, T_out=503K/7.5^0.24=37C. First stage output: 0.158*1500*(0.9+0.86)=417kJ/kg. Second stage output: 1.52*150*(0.87+0.72)=363kJ/kg. Pump consumption = 1.25*1.5=2kJ/kg, net output 778kJ/kg, raw efficiency=25.4%. (17.5% net electricity)
Comment: this should be the standard product. Double expansion with steam reheat, that's more serious. But the cylinders are easier to construct with a smaller r which can be further reduced without much penalty on the efficiency. The efficiency is competitive with commercial organic rankine systems. The last stage should use a slightly lower compression ratio to get a T_out closer to the cooler temperature. Cylinder heating of the first stage is a must.
60C-250C. P0=25 bar
V0=0.0952m^3/kg, t=25/0.2=125, enthalpy difference=2879-251=2628kJ/kg. Equal triple expansion stages: r=3.3 each, t=5 each. First stage T_out=523K/1.47=83C, reheated to 230C. Average heat capacity=2kJ/kg, reheating energy 294kJ/kg. Second stage T_out 513K/1.47=76C, reheated to 230C. Average heat capacity=2kJ/kg, reheating energy 308kJ/kg. Third stage T_out=493K/1.47=62C. Total energy input 3230kJ/kg. First stage output: 0.0952*2500*(0.8+0.53)=317kJ/kg. Second stage output: 0.457*500*(0.8+0.53)=304kJ/kg. Third stage output: 2.28*100*(0.8+0.53)=303kJ/kg. Pump consumption = 1.25*2.5=3kJ/kg, net output 924kJ/kg, raw efficiency=28.6%. (19.7% net electricity)
Comment: this is the high end home product. With triple expansion and double reheating, the cost is going to be higher, and it will require significant development efforts. Net efficiency might hopefully exceed 20%. Note that the Carnot efficiency is 35.6%; the raw efficiency is quite close.
V0=0.026m^3/kg, t=100/0.07=1400, enthalpy difference=2900-163=2737kJ/kg. Reheated triple stage. r=6 each, t=11.2 each. T_out=623K/1.8=73C. Average heat capacity=2.1kJ/kg.K, reheating energy input 2*582kJ/kg, total=3901kJ/kg. Output=3*0.026*10000*(0.91+0.92)=1427kJ/kg, pump consumption = 1.25*10=13kJ/kg, net work 1415kJ/kg, raw efficiency=36.6%.
Comment: this is to compare with first generation trough solar power plants who use 100 bar turbines. The efficiency calculation is in conformity with what is obtained by the turbines, but the computations here contain errors due to some guessed data values.