Problem 4.20: A Carnot cycle operates in a closed system using steam: saturated liquid at 20 bar (state A) is heated isothermally until it becomes saturated vapor (state B), expanded by reversible adiabatic process to 10 bar (state C), partially condensed to state D, and finally compressed by reversible adiabatic process to initial state A.

 

 

 

a)  Obtain the properties (T, P, V, U, H, and S) at the four states, A, B, C, and D, and summarize the results in a table.

 

 

 

b)  Calculate the amount of heat and work involved in each of the four legs of the process.

 

 

 

c)  Calculate the net amount of work that is produced.

 

 

 

d)  Calculate the ratio of the network over the amount of heat absorbed at the high temperature of the cycle. How does this value compare to the theoretical efficiency of the Carnot cycle that is calculated based on the two operating temperatures?

 

 

 

 

 

 

 

 

 

 

 

 

 

From Perry’s Handbook (p. 2–237 in Perry’s Chemical Engineers’ Handbook, 7th ed.) or other bibliographic source we find that the state is compressed liquid. Therefore, if the compressibility equation has three positive roots, the smallest will be selected. The calculation of the various parameters is summarized below:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

The cubic equation for Z is

 

Z3  − Z2  + 0.28682 Z − 0.0202872 = 0.

 

and has three positive roots:

 

Z1   = 0.105351, Z2   = 0.360538, Z3   = 0.534111.

 

We select the smallest root because the phase is liquid. With Z = 0.105351 we find

 

 

 

 

 

 

 

 

 

Comments Since residual properties represent corrections, their sign may be positive or negative. A negative value means that the actual property is less than what would be calculated by the ideal-gas equations.

 

Example 5.11: ΔH, ΔS Using the SRK

 

Use the Soave-Redlich-Kwong equation to calculate ΔH and ΔS for ethylene between T1   = 250 K, P1   = 30 bar, and saturated vapor at T2   = 170 K, P2   = 1.0526 bar.

 

Solution The calculation is done using eqs. (5.33) and (5.34):

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

The calculation requires the ideal-gas heat capacity and the residual properties at the two states. The ideal-gas heat capacity is

 

 

 

 

 

with T in kelvin. The residual properties at the initial state were calculated in Example 5.10 and the calculation in the final state is done in the same manner. The results are summarized below: