Introduction | Contents | Cycle Thermodynamic Model

**Property Modeling**

- Introduction

Background

- Property Models

Experimental Mixture Data

- The Ideal Gas Equation of State

Cubic Equations of State

The Patel Teja Cubic Equation of State

Fugacity and the Enthalpy and Entropy Departures for a Pure Substance

- Enthalpy and Entropy Departures

- Mixing and Combining Rules

Enthalpy and Entropy Departures for a mixture

Property Calculation at a State for a Mixture

Fugacity for a component, i, in a mixture

Equilibrium States in Mixtures

- Single Phase, Non-Saturated Equilibrium States

Vapor-Liquid Equilibrium and Single Phase, Saturated Equilibrium States

Liquid-Liquid Equilibrium States

Vapor-Liquid-Liquid Equilibrium States

The mixtures encountered in this study were also modeled using the Patel-Teja equation of state. Since many of the mixtures use water or ammonia, both of which are polar fluids, the Patel-Teja equation of state can be readily extended to polar components (Georgeton, 1985). When modeling a mixture with a cubic equation of state, pure component parameters must be combined using "mixing rules" in order to obtain mixture parameters. Numerous mixing rules have been developed for particular mixtures. In this study, the mixing rules of Panagiotopoulos and Reid are used since they provide a "significant improvement" in the representation of binary phase equilibrium data for highly polar systems (Panagiotopoulos and Reid, 1985).

The first mixtures studied here are ammonia-water and ammonia-butane. Ammonia-water data are readily available and have already been correlated by S. Rizvi (Rizvi, 1985).

While ammonia-butane data are rare, three papers were found which provided sufficient data. The earliest vapor-liquid equilibrium measurements for the ammonia-butane system were performed by Webster Kay and Herbert Fisch who measured the system between 300 and 1500 psi (Kay, 1958). However, at pressures below 300 psi, the system exhibits two liquid phases as well as a vapor phase. Much later, E. Brunner performed measurments on 18 n-alkane and ammonia mixtures specifically to determine phase separation phenomenon (Brunner, 1988). Most recently, W. Vincent Wilding, Neil Giles, and Loren Wilson of Wiltec Research Company performed vapor-liquid-liquid equilibrium measurements on the ammonia-butane system at 0 C to precisely determine the miscibility limits and the compositions of the vapor phase in equilibrium with the two liquid phases (Wilding, 1996).

In 1965, Alfred Francis performed measurements on the ternary ammonia-butane-water system at 25 C (Francis, 1965).

Mixtures of ammonia with propane, pentane, and hexane were also needed for this study. Solubility data for ammonia with hexane were published by Kiyoharu Ishida in 1958 (Ishida, 1958). More recent liquid-liquid solubility data were published by Yokoyama, Hosaka, Kaminishi, and Takahashi in 1990 and 1991 for mixtures of ammonia with n-pentane, n-hexane, n-octane, n-heptane, n-decane, n-undecane, and n-dodecane (Yokoyama, 1990, 1991).

Mixture data for hydrogen chloride with butane and water were also used in this study. Vapor-liquid equilibrium data for the water-hydrogen chloride system were published by James Kao in 1970 (Kao, 1970). Vapor pressures of the water-hydrogen chloride system were measured by Sako, Hakuta, and Yoshitome (Sako, 1985). Cubic equation of state interaction parameters fitting the water-hydrogen chloride system to Kao's experimental data were reported by Stryjek and Vera (Stryjek, 1986).

Vapor-liquid equilibrium compositions measurements for the hydrogen chloride-n-butane system were published by Ottenweller, Holloway, and Weinrich (Ottenweller, 1943). Phase equilibrium measurements for the hydrogen chloride-ethane system were performed by Ashley and Brown (Ashley, 1972).

The compressibility, Z, is defined as:

For the ideal gas equation of state there is only one compressibility (Z=1) and hence only one, ideal gas, phase for any given state. This characteristic makes the ideal gas equation of state useful only for real gases whose compressibility is close to one.

The parameter a of equation 2-3 represents a rough measure of the attraction forces between the molecules, and the parameter b is related to the size of the molecule. Since equation 2-3 is cubic in volume, three volumes exist for any given temperature and pressure (see Figure 2-1). For a saturation pressure and temperature, the smallest of these is the saturated liquid's volume while the largest represents the saturated vapor's volume. Here, the middle root is of no significance (Smith, 87).

As seen in Figure 2-1 the critical point (P_{c}, T_{c},
and V_{c}) represents an inflection point in equation
2-3 which mathematically means that:

Usually, P_{c}, T_{c}, and V_{c} are known
and equations 2-4 and 2-5 may be simultaneously solved to provide
solutions for the parameters a and b:

When 2-6 and 2-7 are used in equation 2-4 at the critical point,
the van der Waals equation of state predicts the critical compressibility,
Z_{c}, to be 0.375. This is higher than the experimental
value of less than 0.30 for common substances. This fact, in
conjunction with its relative simplicity, causes the van der Waals
equation to yield poor agreement with experimental data, particularly
at the critical point and below (Patel, 1980). While the van
der Waals equation is easy to manipulate and predicts properties
of coexisting phases, its accuracy is limited.

Since 1873, numerous cubic equations of state have been developed
which far exceed the accuracy of the van der Waals equation while
still maintaining a relatively simple form. The most famous were
those proposed by Redlich and Kwong (1949) and Peng and Robinson
(1976). These equations included some form of temperature dependence
for the attraction parameter, *a*, thus vastly improving
the equation's accuracy. While the cubic equation of Peng and
Robinson is the most widely used equation for phase equilibria
calculations, it is not considered accurate for systems containing
large molecules or polar fluids. The Einstein cycle relies on
the two highly polar fluids, ammonia and water.

This added flexibility improves saturation property predictions for polar fluids while maintaining a somewhat simple form that doesn't require a large amount of pure component or mixture experimental data for accuracy. For these reasons plus good predictions for enthalpy and entropy departures, the Patel Teja equation of state is used for all the property modeling in this study.

As with the van der Waals equation of state, the parameters *a*
and *b* in the Patel-Teja equation are evaluated using the
conditions at the critical point given by equations 2-4 and
2-5. The condition for the third parameter, *c*, is:

Instead of having a fixed value of the critical compressibility fixed by a and b, the value is the arbitrary, substance specific, empirical parameter, . is determined by minimizing the errors between experimental saturated liquid volume and those calculated by the equation of state. Patel and Teja calculated for 38 substances covering most of the substances studied here (see Table 2-1). They also correlated to the acentric factor, , for non-polar substances yielding the following equation (Patel and Teja, 1982).

Satisfying the conditions of equations 2-4, 2-5, and 2-9 yields the following equations for the parameters a, b, and c.

where,

and is the smallest positive root of the following cubic equation:

Note that the value of for a given substance need only be solved for once. The term is given by:

where,

In equation 2-17a, F is a substance specific, empirical parameter also determined by minimizing errors between experimental saturated vapor pressures and those calculated by the equation of state. Patel and Teja calculated F for the same 38 substances and also correlated it to the acentric factor for non-polar substances (Patel and Teja, 1982):

See Table 2-1 for the values of and F used in this study.

When equation 2-2 for the compressibility is substituted into
the Patel-Teja cubic equation of state, the following expression
for the compressibility, Z, is obtained (Smith, 1995).

where,

Equation 2-19 will yield one or three positive real roots depending upon the number of phases in the system. If a vapor and liquid phase are present, the smallest root represents the liquid's compressibility while the largest root is that of the vapor's. Appendix A provides an analytical solution for a general cubic equation.

Thus, equations 2-8 thru 2-22 along with equation 2-2, provide P-v-T relations for all of the substances found in this study. However, this study also requires calculation of the equilibrium state of a vapor and a liquid (vapor-liquid equilibrium or VLE) as well as the properties of enthalpy and entropy.

it is possible to display the fugacity for a single phase of a substance in terms of the parameters of the Patel-Teja equation of state as (Patel, 1980):

where,

Recall now the phase rule of Gibbs:

where F is the degrees of freedom of the system (or the number of independent variables that must be arbitrarily fixed to establish the intensive state of the system), is the number of phases, and N is the number of chemical species (Smith, 1987). For the simplest case in this study, =1, and, N=1 so F =2. Hence, the state of a pure substance existing as a superheated vapor can be established by fixing two independent properties such as temperature and pressure. Using the set of equations 2-8 through 2-22 for a pure substance, specifying two independent intensive properties such as temperature and pressure does indeed provide sufficient information to calculate the equilibrium single phase state point of a pure substance (i.e. superheated vapor or subcooled liquid).

When the pure substance is saturated, two phases may occur in
equilibrium and Gibb's phase rule requires only one independent
intensive property to define the equilibrium state. However,
the set of equations 2-8 through 2-22 require more information
to determine the dependent properties. For example, if the saturation
pressure, P_{sat}, in Figure 2-1, is the independent property
defining the saturation state of pure butane, the equation of
state needs some other condition to determine which temperature,
or which volumes, also exist at this state since there are a range
of these dependent variables which satisfy equations 2-8 through
2-22. This "other condition", the equilibrium condition,
is supplied by the fugacity property.

The equilibrium condition for a vapor phase in equilibrium with
a liquid phase of a pure substance is:

To calculate the liquid fugacity in equation 2-29 the liquid's
compressibility is input to equation 2-24, and to calculate the
vapor fugacity, the vapor's compressibility is used. So, to determine
which saturation temperature should apply to a given saturation
pressure, where a vapor and liquid phase are known to exist, the
temperature is guessed and then appropriately adjusted until the
equality shown in equation 2-29 is satisfied. Similarly, a saturation
pressure may be calculated at a given temperature. Satisfying
the condition of equation 2-29 for a range of temperatures and
pressures below the critical point produces the *saturation
dome* in Figure 2-1.

These departures are calculated via departure functions which are first derived from Maxwell's relations:

where H^{o} and S^{o} are the respective enthalpy
and entropy at the hypothetical ideal gas reference state (Patel,
1980). Note, the fugacity equation (2-23) is actually a combination
of the ideal gas reference state (ideal gas behavior is approached
in the limit of P0) and fugacity departure
from this state, but the definition of fugacity is such that f/P1
as P0 making the ideal gas reference state
fugacity the same for all substances. However, the ideal gas
reference state enthalpy and entropy require the ideal gas specific
heat which is substance specific.

With the P-v-T relation provided by equation 2-8, the enthalpy and entropy departures can be expressed in terms of the Patel-Teja equation of state parameters as (Patel, 1980).:

See Appendix B for an expression for the partial derivative of parameter a with respect to T. For the vapor phase, the departure from the ideal gas reference state will be small relative to the liquid phase's departure since the departure for the liquid phase accounts for the property change due to vaporization.

All enthalpy and entropy calculations begin at the reference state,
state 1. At state 1, the substance is defined to be a saturated
liquid at P_{ref} and T_{ref} = T_{sat}(P_{ref}).
Here, the enthalpy and entropy are defined to be zero. Now the
saturated liquid undergoes the hypothetical process 1 to reach
state 2, an ideal gas state at the reference state's temperature,
T_{ref}. The enthalpy and entropy at state 2 are the
negative of the enthalpy and entropy departure values calculated
from equations 2-32 and 2-33 using P_{ref} and T_{ref}.

Next, the ideal gas at state 2 undergoes process 2 which takes
the substance from its reference temperature to the temperature
of the system's state, T_{sytem}. To determine the enthalpy
and entropy at state 3, the ideal gas specific heat for the substance,
c_{p,i.g.} is needed. In this study, the following temperature
dependent ideal gas specific heat equation is used:

and the substance specific constants C_{P,VAP,A}, C_{P,VAP,B},
C_{P,VAP,C}, and C_{P,VAP,D}, are obtained from
the property data bank in *The Properties of Gases and Liquids*
(Reid, 1987).

The final calculation departs the enthalpy and entropy from the
system ideal gas state (3) to the real saturated liquid and vapor
states. These departures are calculated using the temperature
and pressure of the system (T_{system} and P_{system})
in equations 2-32 and 2-33. Thus the calculation of the real
fluid properties is given by the following equations:

- sub-cooled liquids
- saturated liquids
- saturated liquids and vapors in equilibrium (VLE)
- two saturated liquids in equilibrium with a vapor (VLLE)
- saturated vapor

- super-heated vapor.

Extending the Patel-Teja cubic equation of state to mixtures is logical but complicated, therefore, the derivations of many of the following equations may be found in Appendix B.

The variable z_{i} represents the molar composition (concentration)
of component i in the phase of interest (in this study, z_{i}
= x_{i} for the liquid phase and z_{i }= y_{i}
for the vapor phase).

Equations 2-39 thru 2-41 represent what is known as the van der Waals 1-fluid "mixing rules". While this is the most common mixing rule, there are many other mixing rules, the choice of which is arbitrary. However, the use of the van der Waals 1-fluid mixing rules in this study is justified from prior successes.

Note, in equation 2-39, there is a new term when ij which has
yet to be defined, a_{ij}. This is known as the cross-interaction
term and can be defined with the following "combining rule"
(Patel and Teja, 1981):

The term k_{ij} is termed the binary interaction parameter
and is introduced so that the equation of state can be fit to
experimental data. In equation 2-42, it is assumed that k_{ij}=k_{ji}.
By relaxing this condition and adding a concentration dependent
second interaction parameter, Panagiotopoulos and Reid proposed
the following empirical modification to equation 2-42 in order
to better correlate cubic equations to complex phase behavior
in both non-polar and highly-polar systems (Panagiotopoulos and
Reid, 1985).

Note that if k_{ij }= k_{ji}, equation, 2-43 reduces
to equation 2-42. While the combining rule of Reid and Panagiotopoulos
has no physical significance, its improvement of the Patel-Teja
equation's ability to represent the highly polar mixtures found
in this study justify its use here.

As with property calculations for a pure substance, each component
undergoes a hypothetical process after which it behaves as an
ideal gas at its respective reference temperature and pressure
(State 2_{A}, State 2_{B}). This calculation
is performed by subtracting the liquid enthalpy and entropy departures
(calculated at the reference temperature and pressure) from the
reference enthalpy and entropy respectively for each component.

Next each component is taken from its respective ideal gas state
(2_{A}, 2_{B}) to an ideal gas state at the temperature
of the system (State 3_{A}, State 3_{B}). This
calculation requires the ideal gas specific heat for each component
as discussed earlier. Now components A and B are ideal gases
at the same temperature and pressure (T_{system}, P_{ref})
and can be ideally mixed. This mixing process requires information
on how much of each component will be present. If the final state
of the system of interest is that of a liquid, then the liquid
compositions must be used. Likewise, if the final state of the
system of interest is that of a vapor, then the vapor compositions
must be used.

The ideal mixing process (Process 3_{L} and 3_{V})
results in an ideal gas mixture of components A and B at T_{system},
P_{ref}. For ideal mixing, the mixture enthalpy and volume
are the composition weighted sum of each pure components enthalpy
or volume. However, the entropy of the mixture will be greater
than the composition weighted sum of each pure component entropy
by the ideal gas entropy increase of mixing.

In equation 2-44, z_{i} represents the mole fraction of
component i in the mixture and phase of interest.

Hypothetical processes 3_{L} and 3_{V} create
an ideal gas mixture at the system temperature, the reference
pressure, and the composition of either the liquid or vapor respectively
(State 4_{L} and State 4_{V}). Finally, the real
liquid at the system state is obtained by departing from this
ideal gas state, State 4_{L}, by process 4_{L}
to the final real liquid state, State 5_{L}, via the enthalpy
and entropy departures which are calculated at the temperature,
pressure, and liquid composition of the system, using the liquid
compressibility for enthalpy and entropy departure calculations.
The final real vapor state is similarly obtained by departing
from State 4_{V}, by Process 4_{V} to the final
real vapor state, State 5_{V}, via the enthalpy and entropy
departures. These departures are calculated at the temperature,
pressure, and vapor composition of the system, using the vapor
compressibility for enthalpy and entropy departure calculations.
The complete state point enthalpy and entropy calculations are
given by the following equations:

Equations 2-45 through 2-48 provide for all of the hypothetical processes needed to get to a state of interest for a two component mixture from the defined reference state of each pure substance.

Before integrating equation 2-45, the partial derivative of P
with respect to n_{i} must be determined using a modified
form of equation 2-8. The whole process is rather laborious and
the resulting equation for fugacity of the i^{th} component
is quite lengthy and complex. Therefore, the result is presented
in Appendix B. The result gives the fugacity of the i^{th}
component of a mixture in terms of equation of state parameters
and allows the fugacity of the i^{th} component to be
calculated given only temperature, pressure, and composition.

This equation allows the calculation of temperature composition diagrams and pressure composition diagrams for two component mixtures. Figure 2-4 displays a temperature composition diagram constructed using the Patel-Teja equation of state and equation 2-45 for the ammonia-butane system at 20.7 bar (300 psi). The lower line represents the bubble point temperature line.

For example, when a subcooled 50/50 liquid mixture of ammonia and butane is heated from 300K (point 1 on Figure 2-4), the first bubble of vapor will form at just above 316 K (point 2) and the mixture will be in vapor-liquid equilibrium. Upon further heating, the overall composition of both the vapor and the mixture will remain 50/50, but the compositions of the liquid and vapor phases will vary (see point 3). Finally, as the last drop of liquid is evaporated, the 50/50 mixture is now a saturated vapor (point 4). Further heating will superheat the vapor (point 5). Point 4 lies on the dew point line so named because this is where the first drop of condensation would form if it was approached via cooling from point 5. Note at a temperature around 316 K, the equilibrium vapor and liquid phases are at the same composition (~0.82). At this point, known as an azeotrope, the azeotropic mixture boils at a constant temperature with constant vapor and liquid phase compositions (similar to a pure substance).

It is important to note that there may be two compositions required when making vapor liquid equilibrium (VLE) calculations. At a saturated temperature and pressure in a mixture there are two cubic equations for the compressibility; one for the liquid and one for the vapor. The liquid compressibility cubic equation is obtained by evaluating equations 2-39 through 2-41 using liquid compositions. The vapor compressibility cubic equation is obtained by evaluating equations 2-39 through 2-41 using vapor compositions. The liquid's compressibility is still the smallest positive real root but of the liquid compressibility equation. The vapor's compressibility is the largest positive real root of the vapor compressibility equation.

In other words, at a given temperature and pressure, the compositions of the two liquid phases are actually dependent properties. For example, in the ammonia-butane system at 5.29 bar and 273.15 K, a two liquid phase equilibrium state exists in which the composition of butane is 0.8526 mole fraction in one of the liquid phases and 0.106 mole fraction in the other (Wilding, 1996). The equilibrium condition for liquid-liquid equilibrium is:

As in VLE calculations, LLE calculations also produce two different cubic equations for the compressibility due to the different compositions of each liquids phase. However, in LLE calculations, all the compressibilities are the smallest positive real roots.

As expected, VLLE calculations produce three different cubic equations for the compressibility since there are three sets of compositions. The compressibility of each liquid is represented by the smallest positive real root of each liquid's compressibility cubic equation while the compressibility of the vapor is the largest positive real root of the vapor's cubic compressibility equation.

Fortunately, the Patel-Teja equation of state predicts vapor-liquid-liquid equilibrium behavior. However, if it is not known a priori that such a phenomenon occurs in a mixture of interest, the results may be quite confusing.

The ammonia-butane mixture provides a good case in point. Early in this study, the only experimental data thought to exist was that of Kay and Fisch who only made VLE measurements down to 300 psi (Kay, 1958). However, ammonia-butane VLE mixtures in the Einstein cycle were required at 58 psi. Figure 2-5 shows a plot of the Patel-Teja equation of state fit to the 300 psi experimental ammonia-butane mixture data of Kay and Fisch (Kay and Fisch, 1958).

After fitting the experimental data at 300 psi, the pressure was lowered to 58 psi. The results predicted at these lower pressures were at first thought to be a computational error while in the end they were only a nightmare for the inexperienced. As seen in Figure 2-6, the equation of state fits the experimental data nicely at 20.7 bar (300 psi), but as soon as the pressure is decreased to 10 bar, a strange loop starts to occur in the dew point line while the bubble point line is also behaving rather badly. Actually, this is not the result of a programming error and is, mathematically, completely correct.

In fact, these loops are similar, in principle, to the van der Waals loops in a single component system. The equation of state is continuous through the two (and three) phase region which is why it goes through loops to get to the other phase(s).

It is **not** physically correct though. While Kay and Fisch
did not mention it in their paper, the ammonia-butane system experiences
vapor-liquid-liquid equilibrium below temperatures of 315 K.
By adjusting the program to allow for calculation of two liquid
phases as well as a vapor phase (using equation 2-47 instead of
2-45), the physically and mathematically correct temperature composition
diagram was constructed as shown in Figure 2-7.

For the ammonia-n-butane mixture, the equation of state was fitted
to the experimental data of Wilding at the P = 5.29 bar three
phase flash point (Wilding, 1996). While the parameters were
adjusted to match only the experimental data (T, P, x_{1},
x_{2}, y) at this point, agreement with the rest of the
data is quite good (Figure 2-8). Since no three-phase data could
be found for the other ammonia-nalkane mixtures, the kij, kji
values for the ammonia-nbutane mixtures were used.

For the ammonia-water mixture, the Patel-Teja equation of state was fitted to the equation of state developed by O.M. Ibrahim and S.A. Klein specifically for ammonia-water. The kij, kji parameters were adjusted so as to minimize the error in bubble and dew point temperatures at fifty points at a pressure of 4 bar (Figure 2-9).

The water-hydrogen chloride mixture was fitted to ten experimental data points taken by James Kao by minimizing the sum of the errors between the data and the equation of state (Kao, 1970). Figure 2-10 shows the agreement of the equation of state and the experimental data.

The nbutane-hydrogen chloride mixture was fitted to six experimental data points taken by Ottenweller, Holloway, and Weinrich in 1943. Figure 2-11 shows the agreement.

Experimental data on the ethane-hydrogen chloride system were taken from Ashley (Ashley, 1972). Figure 2-12 shows the result of using the nbutane-hydrogen chloride kij and kji for the ethane-hydrogen chloride data of Ashley.

While the data of Ashley and Brown were taken more recently and for HCL-ethane, using the mixture parameters fitted to the earlier data for HCl-n-butane works quite well.

With an ammonia-alkane mixture in the evaporator and condenser/absorber, the cycle utilizes the ammonia-water mixture as well. This is shown in Figure 2-14.

Hydrogen chloride may be substituted for ammonia, therefore, the characteristics of the n-butane-hydrogen chloride mixture and the water-hydrogen chloride mixture are shown in Figures 2-15 and 2-16.

Introduction | Contents | Cycle Thermodynamic Model