13-33
13-59 The mass fractions of components of a gas mixture are given. This mixture is compressed in a reversible, isothermal,
steady-flow compressor. The work and heat transfer for this compression per unit mass of the mixture are to be determined.
Assumptions All gases will be modeled as ideal gases with constant specific heats.
Properties The molar masses of CH4, C3H8, and C4H10 are 16.0, 44.0, and 58.0 kg/kmol, respectively (Table A-1).
Analysis The mole numbers of each component are
N CH4 
1 MPa
mCH4
60 kg

 3.75 kmol
M CH4 16 kg/kmol
N C3H8 
mC3H8
25 kg

 0.5682 kmol
M C3H8 44 kg/kmol
N C4H10 
mC4H10
15 kg

 0.2586 kmol
M C4H10 58 kg/kmol
qout
60% CH4
25% C3H8
15% C4H10
(by mass)
The mole number of the mixture is
N m  N CH4  N C3H8  N C4H10
100 kPa
20C
 3.75  0.5682  0.2586  4.5768 kmol
The apparent molecular weight of the mixture is
Mm 
mm
100 kg

 21.85 kg/kmol
N m 4.5768 kmol
The apparent gas constant of the mixture is
R
Ru
8.314 kJ/kmol  K

 0.3805 kJ/kg  K
21.85 kg/kmol
Mm
For a reversible, isothermal process, the work input is
P
win  RT ln 2
 P1

 1000 kPa 
  (0.3805 kJ/kg  K )(293 K)ln
  257 kJ/kg
 100 kPa 

An energy balance on the control volume gives
E  E
inout


Rate of net energy transfer
by heat, work, and mass
E system 0 (steady)


0
Rate of change in internal, kinetic,
potential, etc. energies
E in  E out
m h1  W in  m h2  Q out
W in  Q out  m (h2  h1 )
win  q out  c p (T2  T1 )  0 since T2  T1
win  q out
That is,
q out  win  257 kJ/kg
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