Carbon dioxide (as an ideal gas) at 1 bar, 300 K enters a compressor operating at steady state and is compressed adiabatically to an exit state of 10 bar, 520 K. Ignoring kinetic and potential energy effects for the compressor, determine:
(a) the work input in kJ/Kg of CO2 flowing, (b) the rate of entropy production in kJ/kg K, and (c) the isentropic compressor efficiency. Do not assume constant specific heats.

a) -207.793 KJ/Kg

b) 0.08 [KJ/kg.K]

c) 0.808

Explanation:

Given

Temperature at state 1 T_1 = 300 K

Temperature at state 2 T_2 = 520 K

Pressure at state 1 P_1 = 1 bar

Pressure at state 2 P_2 = 10 bar

Required

The specific work input [KJ/kg]

The rate of entropy production [KJ/kg.K]

The isentropic compressor efficiency.

Assumption

Carbon dioxide in a compressor assembly is compressed adiabatically . The steam is a closed system.

Applying the ideal gas model.

Kinetic energy effects can be neglected.

Potential energy effects can be neglected.  

Solution  

Specific entropy at state 1 from Table A-3 at T_1 = 300 K and P_1 = 1 bar  

s_1 = 5.3754 [KJ/kg.K]

Specific entropy at state 2 from Table A-3 at T2 = 520 K and P_2 = 10 bar  

s_2 = 4.860 [KJ/kg.K]

The rate of entropy production could be defined as following.  

σ/m = s°(T_2) - s°(T_1) - R*ln(P_2/P_1)

(5.3754-4.860)-(8.314/44.01)*ln(10/1)

= 0.08 [KJ/kg.K]

Energy equation at compressor for actual process could be defined as following.  

Q-W = m [(h_2-h_1)+(V_2^2- V_1^2)/2 + g(Z_2-Z_1)]

As stray heat transfer is neglected at compressor.  

-W = m [(h_2-h_1)+(V_2^2- V_1^2)/2 + g(Z_2-Z_1)]

Potential energy play can be neglected.  

-W = m [(h_2-h_1)+(V_2^2- V_1^2)/2 ]

Kinetic energy play can be neglected.

-W = m [(h_2-h_1)]

The work of compressor for actual process could be defined as following.

W/m(turbine) = -(h_2-h_1)

214.2922-422.08588 = -207.793 KJ/Kg

Energy equation at compressor for ideal process could be defined as following.  

Q-W = m [(h_2s-h_1)+(V_2s^2- V_1^2)/2 + g(Z_2s-Z_1)]

As stray heat transfer is neglected at compressor.  

-W = m [(h_2s-h_1)+(V_2s^2- V_1^2)/2 + g(Z_2s-Z_1)]

Potential energy play can be neglected.  

-W = m [(h_2s-h_1)+(V_2s^2- V_1^2)/2]

Kinetic energy play can be neglected.

-W = m [(h_2s-h_1)

The specific work of compressor for ideal process could be defined as following.  

W/m(turbine ideal) = -(h_2s-h_1)

For the isentropic process.  

P_2/P_1 = P_r*(T_2)/P_r*(T_1)

The relative pressure for air at state 1 from table A-22 at T_1 = 290 K

P_r*(T_1)=2.2311

The relative pressure for air at state 2 could be defined as following.

P_r*(T_2) = P_r*(T_1)*P_2/P_1

               = 4.52

The specific enthalpy at state 2s could be defined as following by Interpolation using Table A-22 at Pr(T_2) = 4.062  

h_2s = 382.1402 KJ/Kg

The isentropic compressor efficiency could be defined as following.  

η_(turbine) = W_(ideal)/W_(act)

                  = (h_1-h_2s)/ (h_1-h_2)

                  = 0.808

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