Gas-Vapor Phase Equilibrium Calculations

## 1.  Introduction

A Gas-Vapor (GV) System is an idealized model of a class of fluid mixture for which a liquid phase may condense from a gaseous phase, and this liquid phase consists entirely of only one of the mixture substances. This substance is called the condensable substance, and is denoted in the following as substance A. A real fluid mixture can be accurately modeled as a GV system when the following condition holds:

• The solubilities of the other substances in liquid A are very small.

This typically is the case in practise when:

• The system temperature, T, is below the critical temperature of the condensable substance, TcA, and the critical temperatures of the other substances are much lower than T.

The term gas-vapor refers to the fact that, since T < TcA, the gas phase of the pure condensable substance (A) is a vapor at the system T. Since the remaining substances are above their critical temperatures, in their pure state they can only exist in the gas phase.

A familiar example of a system that can be modeled as a GV system is a mixture of dry air1 and water vapor, referred to as moist air. For the primary constituents of moist air, Tc(H2O) = 647K, Tc(N2) = 126K, and Tc(O2) = 155K. Conditions of interest for moist air systems typically range from about 273K up to the critical temperature of water, at moderate (near-atmospheric) pressures. Under such conditions, moist air can be accurately modeled as a GV system. The most-air system is so important in practice that the special term psychrometry (or psychrometrics) is used to refer to the measurement and analysis of moist atmospheric air. Although moist air is important, in part due to air-conditioning applications, there are many other GV systems to which the same fundamental thermodynamic considerations apply, examples of which we give in Section 9.

For GV systems, a principal interest lies in determining the relationships among the variables temperature (T), pressure (P), and gaseous composition (mole fraction, y, of the condensable substance) under which a liquid phase can exist in equilibrium with the gas phase. (Other thermodynamic properties of the system are also of interest, but these are not considered in this tutorial.) For given values of P and y, the T at which a liquid phase may form is called the dew-point temperature of the mixture, Tdp. Similarly, for given values of T and y, the P at which a liquid phase may form is called the dew-point pressure of the mixture, Pdp.

In this tutorial, we present the general characteristics of GV systems, and describe the calculation of Tdp and Pdp. We give example calculations involving a moist-air system.

Finally, we emphasize that the particular forms of the relationships given herein do not directly extrapolate to more general types of vapor-liquid equilibria. These are the subject of a future tutorial.

## 2.   Review of Vapor-Liquid Equilibrium in Pure-Component Systems

The conditions in GV systems under which condensation occurs are related to the conditions for condensation for the pure condensable substance, A. The following facts are relevant (you might like to review them in a thermodynamics textbook of your choice):

• there is a unique curve involving P and T for pure A that describes the conditions under which a gas and a liquid phase can coexist. This curve is called the vapor pressure curve, p*(T ) of A.
• Values of P and T corresponding to 2-phase coexistence are called saturation values of the respective variables.
• p*(T ) is defined from a lower temperature Tt called the triple-point temperature, to an upper temperature Tc called the critical temperature. Tc is the highest temperature at which a liquid may exist.
• For a given T, if the total pressure is P, then
• if P > p*(T ), then the substance is in the liquid state.
• if P < p*(T ), then the substance is in the gaseous state.
• if P = p*(T ), then gaseous and liquid phases of the substance coexist (the masses of each phase depend on the total mass of the system and the volume of the container)

Although the thermodynamic analysis given in what follows provides the governing equations, in order to perform numerical calculations, knowledge of p*(T ) for the pure condensable substance is prerequisite information (and may be referred to as a constitutive relation for the problem). This may be available by means of tables or in the form of an analytical equation for the particular substance of interest.

## 3.  The GV Saturation Conditions: Simplest Approximations

In addition to the vapor-pressure curve of the condensable substance, an additional constitutive relation generally required is the equation of state (EOS) of the gaseous and liquid phases. The assumptions that:

1. the gas mixture obeys the ideal-gas equation of state (EOS)
2. the liquid phase properties are independent of P

considerably simplify the calculations for GV systems. In Section 8, we discuss how the calculations are modified when these ideality assumptions are relaxed.

The (saturation) condition for the simultaneous existence of the liquid and gas phases is, in general, a consequence of the equality of the chemical potentials of the condensable substance in each phase (or equivalently in this case, the equality of their fugacities). Under the approximations here, the condition that governs the condensation of liquid A is the same as that for pure A (Section 2), but with the total pressure P replaced by its partial pressure, pA, defined by

pA = y P 

where y is the mole fraction of substance A in the gas phase. Thus, the condition under which the liquid phase is present (the saturation condition) is

pA = y P = pA*(T ) 

Equation  is the key equation for SSC systems, relating the 3 variables (y, P, T) when both phases are present. This condition is a natural consequence of the ideal-gas EOS assumption, since gaseous A behaves as if the other substances are not present, but at a pressure pA, rather than the total pressure P.

Since the gas phase is always present for a GV system, the conditions determining the phase behavior are thus Equation  for the 2-phase case, and

y P < pA*(T ) 

in the single(gas)-phase case. A measure of the undersaturation of the gas can be defined as the relative saturation

RS = pA / pA*(T ) 

RS varies between 0 and 1 and is often expressed as a percentage. When RS < 1, the gas is undersaturated, and when RS = 1, it is saturated, and a liquid phase is also present in the system.

At saturation conditions (RS = 1), combining Equations  and  yields the condition

pA*(Tdp ) = RS  pA*(T ) 

Equation  relates the 3 variables Tdp, RS, and T. Note that Equation  has no explicit dependence on P. This is a consequence of the assumption of the ideal-gas EOS (in Section 8, we show how the P dependence arises when this assumption is improved).

## 4.  Special Case of Moist-Air System

Although Tdp, Pdp, and RS are defined for all GV systems, the following definitions are used only in the context of moist-air systems:

• The temperature, T, of the mixture is called the dry-bulb temperature.
• the term relative humidity, RH, is used rather than relative saturation, RS, defined in Equation .
• Another measure of the moisture content of air is the humidity ratio, HR, defined by
• HR = mv/ma 
• where mv is the mass of water vapor in a given volume of the mixture, and ma is the mass of the dry air in the same volume. HR and RH are related by the PvT behavior of the gas. When the gas phase is treated as ideal, then
• HR = 0.622  RH pA*/(P - RH  pA*)  
• Neither RH nor HR of a moist air mixture can be easily measured directly. Under certain assumptions, they can be determined from the temperature of a thermometer which has a wetted wick covering its bulb, over which the moist air is passed. This temperature is called the wet-bulb temperature, Twb. (For a discussion of the determination of RH and HR from Twb, see, for example, reference 2 below.

## 5.  Calculations of Pdp and Tdp for GV Systems

Pdp:

Equation  gives

Pdp = pA*(T)/ y  

This may be expressed in terms of RS using equation , to give

Pdp = P/RS  

Tdp:

For given pA, Equation  is a nonlinear equation for Tdp. For given RH and T, Equation  is a nonlinear equation for Tdp. (Note that the constitutive relation p*(T ) is required in all cases except for the determination of Pdp via Equation ).

For a given value of RS, the value of Tdp obtained from Equation  can be plotted against the value of T. For moist-air systems, this type of plot is called a psychrometric chart and such charts appear in many textbooks. The plots relate the 2 values of T with RH as a parameter, and also contain other thermodynamic information concerning the moist-air system.

Although charts are useful, they are a carry-over from the pre-computer era, and Tdp can be directly calculated from Equation . Knowledge of the basis for implementing such a procedure also allows calculations to be performed in the absence of charts (which may not be available for other GV systems). Although many constitutive relations for p*(T ) of water are available, of varying degrees of accuracy, for illustrative purposes we will use the following correlation3:

ln p* = A + B/T + C lnT + D T2 

where p* is in Pa, T is in K, and A=73.649, B=-7258.2, C=-7.3037, D=4.1653E-06.

Equation  can be solved using any of the popular computer algebra systems4 (Maple, Mathematica, Mathcad). For illustation, the following simple Maple commands calculate Tdp in a moist-air system for a dry-bulb temperature T=30°and a relative humidity of 50%.
C1:=73.649;
C2:=-7258.2;
C3:=-7.3037;
C4:=4.1653*10^(-6);
Pvap:=T->exp(C1 + C2/T + C3*ln(T) + C4*T^2);
T:= 30.;
RH:=.5;
fsolve(Pvap(Tdp+273.15)=RH*Pvap(T+273.15),Tdp,T-40..T+40);
Performing the above calculation at a range of RH values, the following results are obtained:

 RH(%) Tsat 10 -4.871421711 20 4.634686333 30 10.56084970 40 14.94381883 50 18.45158343 60 21.39096711 70 23.92957609 80 26.16943320 90 28.17745387

## 7.  GV Calculations Using EQS4WIN Lite

The Lite version of EQS4WIN can be used to perform the calculation of Tdp and Pdp. For example, Tdp cab be calculated as follows (using the case RH = 0.5, T = 30°for illustration):

First, find the vapor pressure, p*(T):

1. From the opening screen, click on the Database Problem Formulation button.
2. Select the elements H and O, gas and Pure phases, and the species H2O(gas) and H2O(liquid).
3. On the Data Input Screen, enter P = .041 atm, T = 30&#176C, 1 mole of H2O(gas), and 0 moles of H2O(liquid).
4. Click on Parameter Variation, check the box beside "Pressure", and enter P variation values of Step Size = 0.0001 and Steps = 10.
5. Click on Done and then New Calculation.
6. Reading the tabular output shows that p* is approximately 0.0419 atm (the phase change occurs at this value of P).

Second, find Tdp:

1. After performing the first step above, Tdb, go to the Data Input screen and enter T = 10&#176C.
2. Double-click on the grid cell under "Constraint" for H2(liquid), and the indication "Saturation" should appear in the grid cell.
3. Click on Inerts and enter 0.5 moles in the first cell.
4. Click on Parameter Variation, and enter a T variation of 20 steps and a step size of .5.
5. Click on Done and then New Calculation.
6. Read the tabular output and find the T value for which the mole fraction of H2O(gas) (equal to its partial pressure, since P = 1 atm.) is nearest to 0.02095 atm (=.5 * 0.0419). This is determined to be 18.5&#176C.

The value of Tdp can be refined by calculating more precise values of p*(30), and then by refining the calculation of Tdp in the final step.

## 8.  Improving the Simplest Approximations for GV Systems

If the solubilities of the other gases in the condensed liquid are significant, then more general phase equilibrium approaches must be used. This occurs, for example, if the critical temperatures of the remaining substances are below the system T, but not substantially so. We consider here only the relaxation of the approximations of Section 3.

In general, at saturation, we must equate the chemical potentials of the condensable component in each phase. This is equivalent to equating their fugacities. Thus we have, at saturation (2-phase) conditions,

fg(T,P,y)= f  liq(T,P) 

where T refers to Tdp, f is the fugacity, g denotes the gas phase, liq denotes the liquid phase and y is the mole fraction of the condensable component, A, in the gas. Using the fugacity coefficient &Oslash, we may re-write this as

pA = p*(T) [f liq(T,P)/f liq(T,P*)] [&Oslash; (T,P*,1)/&Oslash;(T,P,y)] 

The bracketed terms can be calculated from an EOS for pure condensed A and for the gas mixture, respectively, and Equation  may be re-written as

pA = p*(T) PC/&Oslash*  

where PC is the Poynting correction for the fugacity of pure liquid A and &Oslash* is the second bracketed term. PC is given by integrating

d (PC)/d P = vm/RT  

from p* (where PC = 1) to P at the mixture T, where vm is the molar volume of condensed A and R is the universal gas constant. Calculation of &Oslash* requires an EOS for the mixture. For example, if we assume the mixture obeys the virial equation of state up to and including the second virial coefficient6

ln &Oslash* = [(P - p*B2)/RT] + (P (1-y)2 delta/RT) 

where B2 is the second virial coefficient of pure A, and

delta = 2 B12 - B1 - B2  

where 1 refers to the second component of the mixture, B1 is its second virial coefficient, and B12 is the mixture second virial coefficient cross term.

## 9.  Other GV Systems

A binary mixture of n-hexane and nitrogen is another example of a GV system. The Tc values are respectively 507.43 K and 126.10 K. A situation that is analogous to a "dehumidification of moist air via cooling" is the following:

A mixture of n-hexane and nitrogen can be separated by passing it through a "cooler-condenser", in which the entering gas stream is cooled to condense the n-hexane. Assuming that the exit gas stream is in equilibrium with the condensed (n-hexane) liquid stream, the n-hexane partial pressure in the exit stream is the saturation pressure corresponding to the exit temperature.

Other GV systems are mixtures of water with each of the gases nitrogen, oxygen, methane, hydrogen, helium, neon, argon, krypton, xenon, carbon dioxide, and ethane. Properties of these mixtures are given in reference 5 given below. However, users of this reference should also have reference 6 available, which shows how errors in Reference 5 must be corrected.

## References

1. The term dry air refers to the usual mixture of gases that constitute the atmosphere at sea level, exclusive of water vapor. Although dry air consists of many species (N2 and O2 being the principal ones, in a 79:21 ratio), many other species are also present in varying small amounts. Furthermore, if no reactions occur, the composition of dry air is constant, and it may be considered to be a single substance for many purposes. Correspondingly, GV systems can usually be considered to be binary mixtures.
2. Thermodynamics, SI Version, 3rd edition, W. Z. Black and J. G. Hartley, HaperCollins Publishing Inc., New York, 1996, pp. 609-611.
3. 1989 AIChE DIPPIR compilation.