Page 229 - Chemical process engineering design and economics

Compressors, Pumps, and Turbines 211

To obtain an equation for calculating the work of compression, first apply
Bernoulli's equation, Equation 5.1, across the compressor. The first term, the ki-
netic energy term, is small compared to the other terms in the balance. The second
term is the change in potential energy, and it is also small. The last two terms are
the work done by the system and the friction loss. First, we consider frictionless
flow. Thus, the compressor work,
2
f d p
W = -| —— (5.8)
Ji P

Isentropic Compression

For isentropic compression of an ideal gas, the dependence of pressure on tem-
perature is given by

T 2 f P 2 1
—— =1 —— I (5.9)
T, I PI )

where k is the ratio of the heat capacity at constant pressure to the heat capacity at
constant volume. Equation 5.9 is derived in several thermodynamic texts and by
Bird et al. [9]. Table 5.5 contains values of k for several gases.
By integrating Equation 5.8 over an isentropic path using Equation (5.9),
it can be shown that the work of compression for an ideal gas,

k i)/k
rfp 2 v - i

W s = ————— I —— -1 I (5.10)
APi ) \

where k is assumed constant.
For a real gas, we define the compressibility factor, z, by

PV = n R T (5.11)
z
If the gas is ideal, z = 1. In Figure 5.14, the compressibility factor is plotted
as a function of reduce pressure and temperature. The compressibility factor in
Equation 5.11 will vary as the temperature and pressure changes from the com-
pressor inlet to the compressor outlet.

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