Gas-vapor Phase Equilibrium Calculations

Gas-Vapor Phase Equilibrium

Gas-Vapor Phase Equilibrium

Calculations

©1997, W. R. Smith. All rights reserved.

Last modified Sept. 18/97

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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

cA[/i], 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 *(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:

the gas mixture obeys the ideal-gas equation of state (EOS)

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 [1]

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 )

[2]

Equation [2] 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 [2]

for the 2-phase case, and

y P A*(T ) [3]

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 ) [4]

RS varies between 0 and 1 and is often expressed as a percentage.

When RS

At saturation conditions (RS = 1), combining Equations [2]

and [4] yields the condition

pA*(Tdp )

= RS pA*(T ) [5]

Equation [5] relates the 3 variables Tdp,

RS, and T. Note that Equation [5] 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 [4].

Another measure of the moisture content of air is the humidity

ratio, HR, defined by

HR = mv/ma [6]

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*) [7]

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 [i]Pdp

and Tdp for GV Systems

Pdp:

Equation [2] gives

Pdp = pA*(T)/ y

[8]

This may be expressed in terms of RS using equation [4], to

give

Pdp = P/RS [9]

Tdp:

For given pA, Equation [2] is a nonlinear

equation for Tdp. For given RH and T,

Equation [5] 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 [9]).

For a given value of RS, the value of Tdp

obtained from Equation [5] 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 [5]. 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 [10]

where p* is in Pa, T is in K, and

A=73.649, B=-7258.2, C=-7.3037, D=4.1653E-06.

Equation [5] 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):

From the opening screen, click on the Database Problem Formulation

button.

Select the elements H and O, gas and Pure phases, and the

species H2O(gas) and H2O(liquid).

On the Data Input Screen, enter P = .041 atm, T

= 30&#176C, 1 mole of H2O(gas), and 0 moles of

H2O(liquid).

Click on Parameter Variation, check the box beside "Pressure",

and enter P variation values of Step Size = 0.0001 and

Steps = 10.

Click on Done and then New Calculation.

Reading the tabular output shows that p*

is approximately 0.0419 atm (the phase change occurs at this

value of P).

Second, find Tdp:

After performing the first step above, Tdb,

go to the Data Input screen and enter T = 10&#176C.

Double-click on the grid cell under "Constraint"

for H2(liquid), and the indication "Saturation"

should appear in the grid cell.

Click on Inerts and enter 0.5 moles in the first cell.

Click on Parameter Variation, and enter a T variation

of 20 steps and a step size of .5.

Click on Done and then New Calculation.

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) [11]

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*)]

[Ø (T,P*,1)/Ø(T,P,y)] [12]

The bracketed terms can be calculated from an EOS for pure

condensed A and for the gas mixture, respectively, and Equation

[12] may be re-written as

pA = p*(T) PC/&Oslash*

[13]

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

[14]

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) [15]

where B2 is the second virial coefficient

of pure A, and

delta = 2 B12 - B1

- B2 [15]

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

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.

Thermodynamics, SI Version, 3rd edition, W. Z. Black

and J. G. Hartley, HaperCollins Publishing Inc., New York, 1996,

pp. 609-611.

1989 AIChE DIPPIR compilation.

Maple, Mathematica, and MathCad are trademarks of their respective

companies.

Moist Gases: Thermodynamic Properties, V. A. Rabinovich

and V. G. Beketov, Begell House, Inc., New York, 1995.

P. T. Eubank, Book review of Reference 5, J. Chem.

Eng. Data, 42, 412-413 (1997).

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