©1997, W. R. Smith. All rights reserved. 1. IntroductionA GasVapor (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:
This typically is the case in practise when:
The term gasvapor refers to the fact that, since T < T_{cA}, 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 air^{1} and water vapor, referred to as moist air. For the primary constituents of moist air, T_{c}(H_{2}O) = 647K, T_{c}(N_{2}) = 126K, and T_{c}(O_{2}) = 155K. Conditions of interest for moist air systems typically range from about 273K up to the critical temperature of water, at moderate (nearatmospheric) pressures. Under such conditions, moist air can be accurately modeled as a GV system. The mostair 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 airconditioning 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 dewpoint temperature of the mixture, T_{dp}. Similarly, for given values of T and y, the P at which a liquid phase may form is called the dewpoint pressure of the mixture, P_{dp}. In this tutorial, we present the general characteristics of
GV systems, and describe the calculation of Finally, we emphasize that the particular forms of the relationships given herein do not directly extrapolate to more general types of vaporliquid equilibria. These are the subject of a future tutorial.
2. Review of VaporLiquid Equilibrium in PureComponent SystemsThe 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):
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 ApproximationsIn addition to the vaporpressure 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:
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, p_{A}, defined
by 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 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 idealgas EOS assumption, since gaseous A behaves as if the other substances are not present, but at a pressure p_{A}, 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 2phase case, and in the single(gas)phase case. A measure of the undersaturation
of the gas can be defined as the relative saturation 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 [2]
and [4] yields the condition Equation [5] relates the 3 variables T_{dp},
RS, and T. Note that Equation [5] has no explicit dependence
on P. This is a consequence of the assumption of the idealgas
EOS (in Section 8, we show how the P dependence arises
when this assumption is improved). 4. Special Case of MoistAir SystemAlthough T_{dp}, P_{dp}, and RS are defined for all GV systems, the following definitions are used only in the context of moistair systems:
5. Calculations of P_{dp} and T_{dp} for GV SystemsP_{dp}: Equation [2] gives This may be expressed in terms of RS using equation [4], to
give T_{dp}: For given p_{A}, Equation [2] is a nonlinear equation for T_{dp}. For given RH and T, Equation [5] is a nonlinear equation for T_{dp}. (Note that the constitutive relation p^{*}(T ) is required in all cases except for the determination of P_{dp} via Equation [9]). For a given value of RS, the value of T_{dp} obtained from Equation [5] can be plotted against the value of T. For moistair 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 moistair system. Although charts are useful, they are a carryover from the
precomputer era, and T_{dp} 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 correlation^{3}: where p^{*} is in Pa, T is in K, and A=73.649, B=7258.2, C=7.3037, D=4.1653E06. Equation [5] can be solved using any of the popular computer
algebra systems^{4} (Maple, Mathematica, Mathcad). For
illustation, the following simple Maple commands calculate T_{dp}
in a moistair system for a drybulb temperature T=30°and
a relative humidity of 50%.
7. GV Calculations Using EQS4WIN LiteThe Lite version of EQS4WIN can be used to perform the calculation of T_{dp} and P_{dp}. For example, T_{dp} cab be calculated as follows (using the case RH = 0.5, T = 30°for illustration): First, find the vapor pressure, p^{*}(T):
Second, find T_{dp}:
The value of T_{dp} can be refined by calculating
more precise values of p^{*}(30), and then by refining
the calculation of T_{dp} in the final step. 8. Improving the Simplest Approximations for GV SystemsIf 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 (2phase)
conditions, where T refers to T_{dp}, 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 Ø,
we may rewrite this as The bracketed terms can be calculated from an EOS for pure
condensed A and for the gas mixture, respectively, and Equation
[12] may be rewritten as where PC is the Poynting correction for the fugacity of pure
liquid A and Ø^{*} is the second bracketed
term. PC is given by integrating from p^{*} (where PC = 1) to P at the
mixture T, where v_{m} is the molar volume
of condensed A and R is the universal gas constant. Calculation
of Ø^{*} 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 coefficient^{6} where B_{2} is the second virial coefficient
of pure A, and where 1 refers to the second component of the mixture, B_{1}
is its second virial coefficient, and B_{12} is
the mixture second virial coefficient cross term. 9. Other GV SystemsA binary mixture of nhexane and nitrogen is another example
of a GV system. The T_{c} 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 nhexane and nitrogen can be separated by passing it through a "coolercondenser", in which the entering gas stream is cooled to condense the nhexane. Assuming that the exit gas stream is in equilibrium with the condensed (nhexane) liquid stream, the nhexane 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
