The power of a multistage compressor

[vishnu123]

TL;DR

how to calculate the power of a multistage compressor

is there any equation to find the work done, power and efficiency of a multistage compressor with a summation of calculating the power as so, instead of calculating individual power and summing at the end. any citiations is also good to know.
thanking you,
with regards,
vishnuvardhan

 

[Chestermiller]

Have you tried to derive such a relationship, at least for the reversible case?

 

[vishnu123]

I haven't tried i was looking for it. i found a relation to find the overall work of the compressor, but i need to find the individual pressure for each stages. i am looking for a relation where we can find the over all power with just few details.

 

[Chestermiller]

I mean, have you tried deriving it yourself? Do you know how to do a single compressor?

 

[vishnu123]

no i have'nt i wam looking for how to model the compressor. I don't have any details or design specifications of the compressor. all I know is jus the input pressurem of 1bar and massflow rate of the gas and the output pressure of 80bars. With the basic calculations formulas I can calculate the power required for a single stage. I need few details about, what's the optimal pressure ratio for the compressor and any relation to find the final output pressure at the last stage of the compressor.
the equations i know are:
Pressure ratio = (P_out/P_in)^(1/n)
total work = (n*k*P_in*V_in)/(k-1)*((P_(2*n)/P_in)^((k-1)/2n)-1)

where, n = number of stages
k = heat capacity ratio
V_in = inlet volume of gas
and i also need to find the efficiency of the compressor.

 

[Chestermiller]

no i have'nt i wam looking for how to model the compressor. I don't have any details or design specifications of the compressor. all I know is jus the input pressurem of 1bar and massflow rate of the gas and the output pressure of 80bars. With the basic calculations formulas I can calculate the power required for a single stage. I need few details about, what's the optimal pressure ratio for the compressor and any relation to find the final output pressure at the last stage of the compressor.
the equations i know are:
Pressure ratio = (P_out/P_in)^(1/n)
total work = (n*k*P_in*V_in)/(k-1)*((P_(2*n)/P_in)^((k-1)/2n)-1)

where, n = number of stages
k = heat capacity ratio
V_in = inlet volume of gas
and i also need to find the efficiency of the compressor.

This is a series of continuous flow compressors, right?

 

[vishnu123]

yes it is a series of continuous flow compressors, and i have designed the general model of the compressor using the basic equations. The final temperature out is very high, as i haven't used any intercooling in between the stages. I have used 3 stage compressor, where in third stage the pressure goes from 18.93 to 80 bar and temperaature rise of about 1100kelvin. Are there any equations to carryout intercooling of the air, apart from heat exchanger equations?