Explanation Write the energy rate balance equation for one inlet and one outlet system. [ Q ˙ 1 + W ˙ 1 + m ˙ ( h 1 + V 1 2 2 + g z 1 ) ] − [ Q ˙ 2 + W ˙ 2 + m ˙ ( h 2 + V 2 2 2 + g z 2 ) ] = Δ E ˙ system (I) Here, the rate of heat transfer is Q ˙ , the rate of work transfer is W ˙ , the enthalpy is h and the velocity is V , the gravitational acceleration is g , the elevation from the datum is z and the rate of change in net energy of the system is Δ E ˙ system ; the suffixes 1 and 2 indicates the inlet and outlet of the system. The carbon dioxide flows at steady state through the compressor. Hence, the rate of change in net energy of the system becomes zero. Δ E ˙ system = 0 Since, the compressor is adiabatic, hence neglect the heat transfer rate. And also neglect the kinetic and potential energy changes. The work done is on the system (compressor) and the work done by the system is zero i.e. W ˙ 2 = 0 . The Equations (II) reduced as follows to obtain the work input. [ 0 + W ˙ 1 + m ˙ ( h 1 + 0 + 0 ) ] − [ 0 + 0 + m ˙ ( h 2 + 0 + 0 ) ] = 0 W ˙ 1 + m ˙ h 1 − m ˙ h 2 = 0 W ˙ 1 = m ˙ h 2 − m ˙ h 1 W ˙ 1 = m ˙ ( h 2 − h 1 ) (II) Here, the W ˙ 1 is the power input of the compressor. Write formula for enthalpy departure factor ( Z h ) . Z h = ( h ideal − h ) T , P R T cr (III) Here, the enthalpy at ideal gas state is h ideal , the enthalpy and normal state is h , the gas constant of methane is R , and the critical temperature is T cr ; The subscripts T , P indicates the correspondence of given temperature and pressure. Rearrange the Equation (III) to obtain h . h = h ideal − Z h R T cr (IV) Refer Equation (IV) express as two states of enthalpy difference (initial and final). h 2 − h 1 = ( h 2 − h 1 ) ideal − ( Z h 2 − Z h 1 ) R T cr (V) The change in enthalpy at ideal state is expressed as follow. ( h 2 − h 1 ) ideal = c p ( T 2 − T 1 ) Here, the specific heat is c p , the exit temperature is T 2 and the inlet temperature is T 1 . Substitute c p ( T 2 − T 1 ) for h 2 − h 1 in Equation (V). h 2 − h 1 = c p ( T 2 − T 1 ) − ( Z h 2 − Z h 1 ) R T cr (VI) Refer Table A-1, “Molar mass, gas constant, and critical-point properties”. The critical temperature and pressure of methane gas is as follows. T cr = 191.1 K P cr = 4.64 MPa The reduced pressure ( P R 1 ) and temperature ( T R 1 ) at initial state is expressed as follows

CONNECT FOR THERMODYNAMICS: AN ENGINEERI

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Chapter 12.6, Problem 74P

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