Page 69 - Separation process principles 2

34 Chapter 2 Thermodynamics of Separation Operations

At low pressure, the solid fugacity can be approximated equations apply only at near-ambient pressure, up to about
by vapor pressure to give, for the component in the solid 50 psia (345 kPa), for mixtures of isomers or components of
phase, similar molecular structure.
For the vapor, the molar volume and mass density are
( Pis) solid
Xi = computed from (I), the ideal-gas law in Table 2.4, which
Yi L ( Pis)liquid involves the molecular weight, M, of the mixture and the
universal gas constant, R. For a mixture, the ideal-gas law
assumes that both Dalton's law of additive partial pressures
EXAMPLE 2.2 and Amagat's law of additive pure-species volumes apply.
The molar vapor enthalpy is computed from (2) by inte-
Estimate the K-values of a vapor-liquid mixture of water and
grating, for each species, an equation in temperature for
methane at 2 atm total pressure for temperatures of 20 and 80°C.
the zero-pressure heat capacity at constant pressure, C$",
starting from a reference (datum) temperature, To, to the
SOLUTION
temperature of interest, and then summing the resulting
At the conditions of temperature and pressure, water will exist species vapor enthalpies on a mole-fraction basis. Typically,
mainly in the liquid phase and will follow Raoult's law, as given in To is taken as 0 K or 25°C. Although the reference pressure is
Table 2.3. Because methane has a critical temperature of -82SoC,
zero, pressure has no effect on the enthalpy of an ideal gas.
well below the temperatures of interest, it will exist mainly in the A common empirical representation of the effect of
vapor phase and follow Henry's law, in the form given in Table 2.3.
From Perry5 Chemical Engineers' Handbook, 6th ed., pp. 3-237
and 3-103, the following vapor pressure data for water and Henry's
law coefficients for CH4 are obtained: Table 2.4 Thermodynamic Properties for Ideal Mixtures

T, "C P for HzO, atm H for CH4, atm Ideal gas and ideal-gas solution:
20 0.02307 3.76 x lo4
80 0.4673 6.82 x lo4

K-values for water and methane are estimated from (3) and (6),
respectively, in Table 2.3, using P = 2 atm, with the following results:

The above K-values confirm the assumptions of the phase distribu- where the iirst term is s$
tion of the two species. The K-values for H20 are low, but increase Ideal-liquid solution:
rapidly with increasing temperature. The K-values for methane are
extremely high and do not change rapidly with temperature for this
example.

2.3 IDEAL-GAS, IDEAL-LIQUID-SOLUTION
MODEL

Design procedures for separation equipment require numer-
ical values for phase enthalpies, entropies, densities, and
phase-equilibrium ratios. Classical thermodynamics pro-
vides a means for obtaining these quantities in a consistent
manner from P-V-T relationships, which are usually Vapor-liquid equilibria:
referred to as equation-of-state models. The simplest model
applies when both liquid and vapor phases are ideal solu-
tions (all activity coefficients equal 1 .O) and the vapor is an
ideal gas. Then the thermodynamic properties can be
Reference conditions (datum): h, ideal gas at To and zero pressure; s, ideal
computed from unary constants for each of the species in gas at To and 1 atm pressure.
the mixture in a relatively straightforward manner using Refer to elements if chemical reactions occur; otherwise refer to
the equations given in Table 2.4. In general, these ideal components.