a common problem in the design of chemical processes is the steady state compres

Question

a common problem in the design of chemical processes is the steady state compression of gases from a low pressure p1 to a much higher pressure p2. we can gain some insight about optimal design of this process by considering adiabatic reversible compression of ideal gases with stage wise intercooling. if the compression is to be done in 2 stages, first compressing the gas from p1 to p* then cooling th egas at constant pressure down to the compressor inlet temperature t1 and then compressing the gas to p2 what should the cvalue of the intermediate pressure be to accomplish the compression with minimum work.

Explanation / Answer

FYI: the middle device ISN'T a "condenser" if it is handling gasses that don't condense. It is better called a heat exchanger for a more general term.
Work input of an adiabatic compression process, first compressor:
Win1 = m_dot*cp*(Tstar - T1)

Work input of an adiabatic compression process, first compressor:
Win2 = m_dot*cp*(T2 - T1).

Net work input:
Winnet = m_dot*cp*((T2 - T1) + (Tstar - T1))

Adiabatic Relation between T1 and Tstar:
Tstar = T1*(Pstar/P1)^((k-1)/k)

Adiabatic Relation between T2 and T1:
T2 = T1*(P2/Pstar)^((k-1)/k)

k is the adiabatic index of the gas flavor
KEEP ALL TEMPERATURES IN KELVIN

To minimize typing, let's define E, the exponent:
E = ((k-1)/k)

Update work equation:
Winnet = m_dot*cp*T1*(((P2/Pstar)^E - 1) + ((Pstar/P1)^E - 1))

Simplify:
Winnet = m_dot*cp*T1*((P2/Pstar)^E + (Pstar/P1)^E - 2)

Express as:
Winnet = m_dot*cp*T1*(P2^E*Pstar^(-E) + Pstar^E/P1^E - 2)

Take derivative of dWinnet/dPstar, as per optimization strategy:
dWinnet/dPstar = m_dot*cp*T1*(-E*P2^E*Pstar^(-E-1) + E*Pstar^(E-1)/P1^E)

Equate to zero, as per optimization strategy:
0 = m_dot*cp*T1*(-E*P2^E*Pstar^(-E-1) + E*Pstar^(E-1)/P1^E)

To hell with the leading factors:
0 = -E*P2^E*Pstar^(-E-1) + E*Pstar^(E-1)/P1^E

Solve for Pstar:
E*P2^E*Pstar^(-E-1) = E*Pstar^(E-1)/P1^E
P2^E*Pstar^(-E-1) = Pstar^(E-1)/P1^E
P2^E/Pstar^(E+1) = Pstar^(E-1)/P1^E

P2^E * P1^E = Pstar^(E-1) * Pstar^(E+1)

Add exponents:
P2^E * P1^E = Pstar^(2*E)

Take the 2Eth root of both sides:
Pstar = (P2^E * P1^E)^(1/(2*E))

Power raised to power means multiply exponents:
Pstar = P2^(E/(2*E)) * P1^(E/(2*E))

Cancel E:
Pstar = P2^(1/2) * P1^(1/2)

Or in otherwords:
Pstar = sqrt(P1*P2)

This is what mathematicians call the "geometric mean".

It turns out, that the optimal midway pressure is the geometric mean of the initial and final pressure.

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