Multiphase systems: phase equilibrium, Raoult's law and humidity

The grain dryers at Kingfisher Bioenergy discharge a hot, humid exhaust. Before it reaches the stack it passes through the thermal oxidizer, and the plant is studying a heat recovery unit that would cool the exhaust to recover energy. The question Ingrid gives Theo is whether water will condense in the recovery unit, and if so at what temperature; condensate in the wrong place corrodes ducts and fouls heat exchangers. The same week, the tank farm asks at what temperature the natural gasoline used to denature the fuel ethanol starts to boil. Both questions are about equilibrium between phases, the subject of this lesson.

Phases and phase diagrams

A phase is a region of uniform composition and properties: a gas, a liquid, or a solid. Many process steps exist precisely to move material between phases: distillation, evaporation, condensation, absorption, drying and crystallization. Balances on these steps need one more piece of information beyond the conservation law, namely how the components distribute between phases at equilibrium.

For a pure substance, a phase diagram of pressure against temperature shows where it is solid, liquid or vapor. The line between liquid and vapor is the vapor pressure curve, which runs from the triple point to the critical point. At any temperature on that line, liquid and vapor coexist, and the pressure is the vapor pressure \(p^*\). A liquid boils when its vapor pressure equals the surrounding pressure; the normal boiling point is the temperature at which \(p^* = 1\ \text{atm}\). Above the critical temperature, no amount of pressure produces a separate liquid phase.

Vapor pressure correlations

Vapor pressure rises steeply with temperature. The Clausius-Clapeyron equation, derived from thermodynamics with a few approximations, predicts that \(\ln p^*\) is nearly a straight line against \(1/T\), with a slope set by the heat of vaporization. The empirical Antoine equation adds a third constant and fits data closely over a stated range. Antoine constants are tied to particular units, often mm Hg and degrees Celsius, so always check the units given with them.

The Gibbs phase rule

How many variables must be fixed to define a system at equilibrium? The Gibbs phase rule answers it for intensive variables, those that do not depend on the size of the system, such as temperature, pressure and composition. With \(c\) components and \(\pi\) phases and no reactions, the number of degrees of freedom is \(F = 2 + c - \pi\). Pure water with liquid and vapor present has \(F = 1\): fixing the temperature fixes the pressure. A two-component liquid in equilibrium with its vapor has \(F = 2\): fixing temperature and pressure fixes both phase compositions. This is a different count from the degree-of-freedom analysis of Lesson 3, which counts extensive variables such as flows as well.

Gas and liquid with one condensable component

Many process gases contain one component that can condense, typically water, mixed with others that cannot under the conditions of interest, such as air, nitrogen or carbon dioxide. If the gas is in contact with the liquid long enough to reach equilibrium, the gas is saturated and the partial pressure of the condensable component equals its vapor pressure at the system temperature. That is Raoult's law in its simplest form, for a liquid phase that is pure.

If a gas has a partial pressure of the condensable below its vapor pressure, it is superheated, and it can be cooled at constant pressure until the partial pressure equals the vapor pressure. That temperature is the dew point, where the first drop of liquid forms. The degrees of superheat are the difference between the actual temperature and the dew point. Several related measures describe how close a gas is to saturation.

Condensation and humidification balances

When a saturated gas is cooled or compressed, some vapor condenses. The balance is straightforward if the dry gas is used as a tie component: the dry gas flow is unchanged, the vapor leaving equals the dry gas times the molal saturation at the new conditions, and the condensate is the difference. Compression raises the partial pressure of the vapor, so compressing a saturated gas at constant temperature always condenses some liquid. This is why the carbon dioxide compressors have knockout drums after each intercooler.

Multicomponent vapor-liquid equilibrium: Raoult's and Henry's laws

When the liquid is a mixture, each component's partial pressure in the vapor depends on how much of it is in the liquid. For ideal solutions of similar molecules, such as hexane and heptane or benzene and toluene, Raoult's law states that the partial pressure equals the liquid mole fraction times the pure-component vapor pressure. For a gas dissolved in a liquid at low concentration, such as carbon dioxide or oxygen in water, Henry's law is used instead, with a Henry's constant from tables.

Raoult's law gives two standard calculations. At the bubble point, a liquid of known composition starts to boil: the sum of \(x_i p_i^*\) equals the total pressure, and the first bubble has \(y_i = x_i p_i^*/P\). At the dew point, a vapor of known composition starts to condense: the sum of \(y_i P/p_i^*\) equals one. Both need a trial-and-error search for the temperature, or for the pressure if the temperature is fixed. Between the bubble and dew points, a mixture is partly vapor and partly liquid, and a flash calculation finds how it splits.

# Vapor pressure, humidity and bubble/dew points with the Antoine equation
# Antoine constants (approximate): log10(p* / mmHg) = A - B / (T + C), with T in degrees C
ANT = {'water': (8.07131, 1730.63, 233.426),
       'n-hexane': (6.88555, 1175.817, 224.867),
       'n-heptane': (6.90253, 1267.828, 216.823)}
MMHG = 0.133322                      # kPa per mmHg

def psat(sp, T):                     # T in C, result in kPa
    A, B, C = ANT[sp]
    return 10 ** (A - B / (T + C)) * MMHG

def tsat(sp, p):                     # inverse: temperature (C) where psat = p (kPa)
    A, B, C = ANT[sp]
    from math import log10
    return B / (A - log10(p / MMHG)) - C

print(f"Water vapor pressure at 25 C {psat('water', 25):.2f} kPa, at 100 C {psat('water', 100):.1f} kPa")

# Dryer exhaust: 1.0 atm, 95 C, contains 18 mol% water vapor
P, T, y = 101.325, 95.0, 0.18
pw = y * P
print(f"\nDryer exhaust: p_water = {pw:.1f} kPa, saturation at {T:.0f} C = {psat('water', T):.1f} kPa")
print(f"Relative saturation {100 * pw / psat('water', T):.1f} %, dew point {tsat('water', pw):.1f} C")
print(f"Molal humidity {y / (1 - y):.3f} mol water per mol dry gas")

# Raoult's law bubble and dew points for a hexane/heptane liquid at 1 atm
def bubble_T(x, P):
    lo, hi = 0.0, 150.0
    for _ in range(80):
        T = (lo + hi) / 2
        s = sum(xi * psat(sp, T) for sp, xi in x.items()) / P
        lo, hi = (T, hi) if s < 1 else (lo, T)
    return T

def dew_T(y, P):
    lo, hi = 0.0, 150.0
    for _ in range(80):
        T = (lo + hi) / 2
        s = sum(yi * P / psat(sp, T) for sp, yi in y.items())
        lo, hi = (lo, T) if s < 1 else (T, hi)
    return T

z = {'n-hexane': 0.40, 'n-heptane': 0.60}
Tb, Td = bubble_T(z, P), dew_T(z, P)
yb = {sp: z[sp] * psat(sp, Tb) / P for sp in z}
print(f"\nBubble point of 40/60 hexane/heptane at 1 atm: {Tb:.1f} C, first vapor {yb['n-hexane']:.3f} hexane")
print(f"Dew point of the same mixture as a vapor: {Td:.1f} C")
for sp in z:
    print(f"Normal boiling point of {sp}: {tsat(sp, P):.1f} C")

Water vapor pressure at 25 C 3.16 kPa, at 100 C 101.3 kPa

Dryer exhaust: p_water = 18.2 kPa, saturation at 95 C = 84.5 kPa
Relative saturation 21.6 %, dew point 58.2 C
Molal humidity 0.220 mol water per mol dry gas

Bubble point of 40/60 hexane/heptane at 1 atm: 83.5 C, first vapor 0.622 hexane
Dew point of the same mixture as a vapor: 89.6 C
Normal boiling point of n-hexane: 68.7 C
Normal boiling point of n-heptane: 98.4 C

When Raoult's law fails

Ethanol and water are a famous exception. Their molecules attract each other differently from how each attracts its own kind, so the mixture's partial pressures depart strongly from Raoult's law, and the mixture forms an azeotrope near 95.6 percent ethanol by mass. Such systems need activity coefficients, which multiply the mole fraction in Raoult's law and are calculated from models such as Wilson, NRTL or UNIQUAC. Those models are the subject of thermodynamics courses; in this course, use Raoult's law only for mixtures of chemically similar molecules.