A three-stage, single acting compressor is required to compress 135m3 of free air per hour from 1bar to 64bar. Prove that with complete intercooling the stage pressure ratio should be equal to the cube root of the overall pressure ratio for the required power to a minimum. Calculate the power needed in each case, assuming n = 1.3.
If the mean piston speed is to be 140 m/min, calculate the piston areas, neglecting clearance. State any assumptions made.
Solution;
Neglecting clearance;
Assume "V_\\alpha" ="V_1"
For minimum work n=1
Process is isothermal;
Therefore;
"\\frac{P_2}{P_1}=\\frac{P_3}{P_2}=\\frac{P_4}{P_3}=r_p=(\\frac{P_4}{P_1})^\\frac13"
"r_p=(\\frac{64}{1})^\\frac13" =4
Power needed;
ip="\\frac{3n}{n-1}p_1V_1[(\\frac{p_4}{p_1})^\\frac{n-1}{3n}-1]"
ip="\\frac{3\u00d71.3}{0.3}\u00d71\u00d7100\u00d7\\frac{135}{3600}[(64)^\\frac{0.3}{3\u00d71.3}-1]"=18.379kW
Piston areas;
2NL=140m/min
"N=\\frac{140}{2L}=\\frac{70}{L}"
V1=V×N="V\u00d7\\frac{70}{L}"
Therefore;
"\\frac{135}{60}=\\frac{70V}{L}"
"\\frac VL=\\frac{135}{60\u00d770}"
But V=AL
A=0.0321m2
Area of LP cylinder=321cm2
Area of IP cylinder="\\frac{321}{r_p}=\\frac{321}{4}=80.25cm^2"
Area of HP cylinder="\\frac{321}{(r_p)^2}=\\frac{321}{4^2}=20.0625cm^2"
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