We now turn to the subject of non-reaction mixtures in more than one phase. We limit our discussion to two phases, but the work can be extended to more than two. We start by finding the conditions of equilibrium between two phases of a mixture of simple compressible substances.

Sections:

12.1 Conditions of Equilibrium

12.2 Electrochemical Potential and Gibbs Function

12.3 Properties of Two-Phase Mixtures

12.4 Distillation

12.5 Absorption Refrigeration

12.6 Chapter References

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12.1 Conditions of Equilibrium

Consider a closed, adiabatic system with constant volume containing two substances in two phases as shown. The two substances will be designated substance A and substance B. The liquid phase will be indicated using a single quotation sign and the vapor phase a double quotation sign. We assume that the intensive properties are uniform over the extent of each of the phases. The question is what are the conditions of equilibrium between to two phases.

The mass of each constituent is constant and is extensive, so

(12-1)

(12-2)

(12-3)

The volume, internal energy, and entropy are extensive,

(12-4)

(12-5)

(12-6)

Because the volume is held constant, we have

(12-7)

By the Conservation of Mass Principle for a closed system containing a non-reacting mixture, the masses of each constituent is constant,

(12-8)

(12-9)

The system is closed and adiabatic. Neglecting any kinetic energy changes, the internal energy of the system is also constant by the application of the Conservation of Energy Principle,

(12-10)

The Entropy Production Principle gives,

(12-11)

We developed an extensive form of the Gibbs' Equation in the last chapter. Applying the result given in Eqn. 11-10 to two constituents in each phase, we have,

(12-12)

and,

(12-13)

Solve these two equations for the differentials of entropy and combine with Eqn. 12-11,

(12-14)

or

(12-15)

The variations in energy, volume, and the masses are each independent of the other. The only way that the sum of these changes can be greater or equal to zero for the general case is if each are,

(12-16)

(12-17)

(12-18)

(12-19)

Equilibrium is reached when the processes each progress irreversibly to a final state. The first of these expressions shows that thermal equilibrium is reached when the temperatures of the two phases are equal. If they are unequal, heat will be transferred from one phase to the other to change the internal energies of both. For example, if the vapor is hotter than the liquid, Eqn. 12-16 indicates that the internal energy of the liquid will increase and the internal energy of the vapor will decrease until the two temperatures are equal.

When we have thermal equilibrium, the second result indicates that mechanical equilibrium is reached when the pressures of the two phases are equal. Please note that it has been assumed that the size of the system is large enough that the interface between the two phases is a plane surface. If we have a curved surface, as we would in a capillary tube, this would not be the appropriate result. In that case, we would have to consider the mass, energy, and entropy stored in the interface as well as in the two phases. (See Gibbs [1] and Kirkwood and Oppenheim [2])

The last two results give the conditions of chemical equilibrium, namely that the electrochemical potentials of each of the constituents must be equal across the interface between the liquid and the vapor. If they are not, there will be mass transfer across the interface until the potentials do become equal. An example is evaporation of liquid water into the surrounding air.

Please note that the relative amounts of the substances in the two phases is not a condition for equilibrium. In the general case, the mass fractions will be much different in the two phases. This allows us to separate mixtures by the process of distillation. Before we consider the distillation process, we must first look at the details of electrochemical potential and two-phase property relations.

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12.2 Electrochemical Potential and Gibbs Function

For the types of substances we are interested in, the electrochemical potential is the same thing as the Gibbs function. That this is the case is the subject of this section. We found that for a pure substance that Gibbs function is a function of temperature and pressure (see Eqn. 10-18). For mixtures, the extensive Gibbs function is,

(12-20)

We find that if we increase the amount of the masses by the same amount the Gibbs function is increased by the same amount,

(12-21)

Using Euler's theorem, we have,

(12-22)

where

(12-23)

This intensive property is the partial Gibbs function.

Using the Chain Rule of Calculus with Eqn. 12-20, we have

(12-24)

Recall Gibbs' Equation (Eqn. 11-10),

(12-25)

The extensive enthalpy and Gibbs functions are defined as,

(12-26)

Taking the differential,

(12-27)

Combine Eqns. 12-25 and 12-27,

(12-28)

Comparing Eqn. 12-24 with Eqn. 12-28 shows that

(12-29)

(12-30)

and

(12-31)

Therefore, one of the conditions of equilibrium may now be stated is that the partial Gibbs function is the same in both phases of a two-phase mixture for each constituent.

Differentiate Eqn. 12-22,

(12-32)

Subtract this from Eqn. 12-28 and use the result of Eqn. 12-31,

(12-33)

This is known as the Gibbs-Duhem Equation.

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12.3 Properties of Two-Phase Mixtures

The process of determining the properties of a two-phase mixture of two or more simple compressible substances is covered in this section. In general, the properties of the saturated liquid and the saturated vapor are desired as a function of temperature and the mass or mole fractions of the mixture in the liquid phase. The sum of mass or mole fractions is unity, so the number of independent mass or mole fractions is one less than the number of constituents making up the mixture. We will focus on a mixture of two constituents, so the properties will be functions of temperature (or pressure) and the mass or mole fraction of one of the substances. We usually pick the mass or mole fraction of the more volatile substance in the liquid phase as an independent parameter.

In addition to empirically developed property relations for each of the pure substances in both of the two phases, we must have a minimum of experimental data for the two-phase mixture. For example, we may experimentally find saturation pressures for different mixture temperatures and composition. As an example of the process of finding empirical expressions for a two-phase, two-component mixture, we will look at the case of a mixture of water and ammonia. We shall use the mathematical models given by Zieglar and Trepp [3] and the experimental data of Gillespie, et al. [4].

Zieglar and Trepp use the same p-v-T and specific heat correlations for water and ammonia,

(12-34)

(12-35)

The superscript, L, is for the liquid phase and the superscript, V, is for the vapor phase. From these expressions

The enthalpy and the entropy as functions of temperature and pressure may be found from these relations by the method outlined in Section 10.6. It is useful to combine these properties into the Gibbs function as a function of temperature and pressure. Further, the basis is shifted from mass to the mole, as indicated by the caret over the symbol for the property. For the liquid phase, this becomes,

(12-36)

and for the vapor phase,

(12-37)

These two equations, and many that follow, have been made dimensionless by using reduced thermodynamic properties defined as,

where

Zieglar and Trepp [3] give values for all of the constants used in Eqns 12-36 and 12-37.

These relations for the pure substance have the desirable feature of being separate for each phase and not dependent on the critical temperature or pressure of the substance. Thus, they may be evaluated for a state that could not be realized for the pure substance but can be realized in the mixture. The more volatile substance, in this case ammonia, can be in the liquid phase when mixed at a certain pressure whereas it would be vapor as a pure substance at that same pressure. Trying to use property correlations such as given by Reynolds [5] will not work for this development because they will not extrapolate into states that do no exist for the pure substance.

Zieglar and Trepp [3] propose a correlation for the liquid or vapor mixture of ammonia and water by adding an "excess" Gibbs function parameter to the ideal solution mixture relation,

(12-38)

where

The vapor phase is taken to be an ideal solution, so the "excess" Gibbs function term is zero. The following expression is used to fit the data for liquid mixtures of ammonia and water,

(12-39)

where

Values of the sixteen coefficients, E1 through E16, are evaluated by a least-squares procedure using the experimental two-phase data of Gillespie, et al. [4] and the conditions that the temperature, pressure, and chemical potentials must match in the two phases. The results of this is given in the program, AQUAM.EXE, which is available in both a DOS and a Windows version. Using this program, the user has the options of selecting inputs of,

1. Temperature, pressure, and ammonia mass fraction for liquid phase

2. Temperature, pressure, and water mass fraction for vapor phase

3. Temperature and liquid ammonia mass fraction for mixed phase

4. Pressure and liquid ammonia mass fraction for mixed phase

5. Temperature and water vapor mass fraction for mixed phase

6. Pressure and water vapor mass fraction for mixed phase

7. Temperature and pressure for mixed phase

When using the program for subcooled liquid or superheated vapor, the user must check that the state is possible by checking the pressure with saturated pressure for the same temperature and composition.

Finding the value of the exergy presents a new problem. Up to this point, we have discussed the substances N2 , O2 , Ar, CO2 , and H2O, which are the stable compounds of the elements N, O, Ar, C, and H, respectively, that exist in air, which is our natural environment. The chemical equilibrium term for exergy of these substances has been defined to be zero in the computation of their exergies. Now we are talking about a new substance, ammonia, NH3 . The chemical reaction that could produce ammonia is,

(Details about chemical reactions is the topic of the next chapter.) In Chapter 8, we found that the exergy of any substance based on a dead state that is in thermal and mechanical equilibrium with the environment is given by (cf. Eqn. 8-43),

(12-40)

where

(12-41)

and

(12-42)

This last function is the difference between the Gibbs function of a substance which is in thermal and mechanical equilibrium with the environment and a substance which is in the most stable configuration state in equilibrium with the environment. This has been referred to as the chemical equilibrium term of exergy.

In many cases, the environmental state is approximated as being at the standard temperature and pressure (STP). This state has a temperature of 25 C (77 F) and a pressure of one standard atmosphere. We shall use a superscript, , for the change at the STP state. Thus,

(12-43)

The details regarding this approximation are covered in the next chapter. The value of the standard Gibbs function comes from the chemical reaction given above,

(12-44)

The standard Gibbs function for each of the substances except ammonia is assigned a value of zero for exergy calculations. The enthalpies at the reference state are the enthalpies of formation. The entropies at the reference state are absolute entropies. Details of this are given in the next chapter. The Gibbs function of ammonia for the chemical reference state is,

In theory, this is to be added to the exergy based on the process going to thermal and mechanical equilibrium with the environment. However, it is found that the exergies for ammonia all are large values for any practical analysis when the chemical exergy is added. Thus, the ammonia and aqua-ammonia programs do not add this chemical exergy term. If the user of these programs wishes to have an exergy that includes the chemical contribution the above value may be added to the ammonia portion of the exergy.

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12.4 Distillation

We can conceptually think of a distillery as a device similar to the figure at the right. A source of thermal energy, frequently steam coils, is provided at the bottom of the still and a cooling, frequently by cold water, is provided at the top. A temperature gradient is established up and down the still, but the pressure is almost constant. The distillery is constructed in such a way that there is a large interface surface between the two phases all the way from the bottom to the top. This allows the two phases to be nearly in equilibrium at each temperature.

The generator produces vapor that contains a high percentage of the more volatile substance, which in our example of using aqua-ammonia is ammonia. The cooling coils condense the less volatile substance, which in our example is water. The composition of the liquid flowing down the tower and the vapor flowing up the tower changes at each elevation in the distillery. The vapor is becoming more rich of the more volatile substance as it rises and the liquid less rich as it flows down. The feed is introduced into the still at the level that best matches the composition of the feed. The top product is, in the limit, pure substance of the more volatile substance. The `bottoms,' as the bottom product stream is often called, is, in the limit, pure substance of the less volatile substance.

As an example, we will consider a feed (Station 3') of liquid aqua-ammonia at a temperature of 180 F and a pressure of 100 psia, which is the pressure of the whole column. The mass fraction of the ammonia in the feed is 25 per cent. Pure ammonia leaves the column as the top product (Station 1") and pure water leaves as the bottom product (Station 2'). Station 4 is at any arbitrary location between the feed and the top product streams. Liquid crosses this station flowing downward and vapor crosses flowing upward. They will be designated stations 4' and 4", respectively. The mass fraction of the liquid at this station is 0.65.

The steam heating will be done by the condensation of steam at a pressure of 200 psia. Saturated vapor at this pressures enters at Station 5 and leaves as saturated liquid at Station 6. The cooling water enters at Station 7 at 40 F and 30 psia and leaves at Station 8 at 50 F and 30 psia. The environment is at a temperature of 50 F and a pressure of 14.7 psia.

A state table for the process is as follows:

Station	T,F	p,psia	x	mf	h, Btu/lbm	w
1"	56.1	100.	sat vap	1.00	625.9	1.
2'	327.7	100.	sat liq	0.00	298.6	3.
3'	180.0	100.	sub	0.25	98.9	4.
4'	86.8	100.	sat liq	0.65	20.2	0.00287
4"	86.0	100.	sat vap	0.99900	645.8	1.00287
5"	382.0	200.	sat vap		1199.1	1.3607
6'	382.0	200.	sat liq		355.6	1.3607
7'	40.0	30.	sub		7.9	2.1917
8'	50.0	30.	sub		17.8	2.1917

The last column are values of mass flow rates normalized by the top product flow rate. The values for Stations 2-4 are found from the application of Conservation of Mass. Considering the whole column, we have,

(12-45)

Divide by the flow rate of Station 1 and define the symbol, w, to be the ratio of a flow rate at a given station and the flow rate of Station 1. We then have,

(12-46)

This yields the values of the dimensionless flow rate for Stations 2 and 3. The two stations at 4 are needed to be able to apply Conservation of Energy to the two portions of the column. A mass analysis of the upper region yields,

(12-47)

Again, form the dimensionless flow rate. The results are,

(12-48)

The dimensionless flow rate for the steam and the cooling water come from applying the Conservation of Energy Principle to the lower region and the upper region, respectively. These give,

(12-49)

(12-50)

Take as the mass flow rate of the ammonia produced to be 10 lbm/sec. Continuing the state table, we have,

Station	s, Btu/lbm-R	ex, Btu/lbm	, lbm/sec	, MW
1"	1.23341	109.17	10.0	1151.9
2'	0.47509	57.05	30.0	1806.4.
3'	0.29269	-21.76	40.0	-918.49
4'	0.18252	0.38	0.02865	0.01
4"	1.27180	109.72	10.02865	1161.0
5	1.54620	411.28	13.6072	5904.7
6	0.54390	78.64	13.6072	1129.1
7	0.01578	0.14	21.9173	3.31
8	0.03534	0.05	21.9173	1.05

The irreversibility of the process is found to be 901 kW.

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12.5 Absorption Refrigeration

Shown is a schematic of a typical absorption refrigeration system. The left hand side of this schematic shows a system that could be associated with a vapor compression refrigeration system. That is, we have the basic elements of a condenser, throttle valve, and evaporator. The subcooler has been added to increase the performance of the machine. The right hand side of the schematic is completely different, however. Instead of the consumption of power to drive a compressor, the major energy input to an absorption refrigeration machine is heat transfer from a thermal source at a temperature level of about 200 C (400 F).

The major three elements that are new are the absorber, the pump, and the distillation column. The heat exchanger has also been added to increase the performance. The refrigerant, ammonia in our example, is condensed and absorbed by the absorbent, water in our example, in the absorber. The requirement is that the refrigerant have a strong affinity with the absorber. Cooling is required for the absorbing process. The strong solution is then pumped up to a high pressure, essentially the operational pressure of the column and the condensation pressure of the condenser. The pump requires much less power for the same pressure rise compared with a vapor compressor. The refrigerant is purified in the distillation column, resulting is pure refrigerant at high pressure passing to the condenser.

An absorption refrigeration system is useful in those situations where a steam distribution system exists for process heating and comfort heating in the Winter. The steam may then be used for Summer cooling in absorption refrigeration machines at various locations in the system.

As an example, consider an absorption machine with known properties shown underlined in the following state table. Stations 14 and 15 are in equilibrium with each other. The brine has a density of 1280 kg/cu.m. and a specific heat of 2.847 kJ/kg-K. The enthalpy and entropy of the brine have been defined to be zero at Station 17. The chemical exergy at environmental temperature and pressure of the brine is estimated to be 17.11 kJ/kg. The environmental temperature is 22 C and the environmental pressure is 100 kPa. The load from the cold box is 250 kW and the temperature of the cold box is -2C. The efficiency of the strong aqua-ammonia pump is 75 per cent; the efficiency of the brine pump is 71 per cent.

Station	T, C	P, kPa	x	mf	h, kJ/kg	s, kJ/kg-K	ex, kJ/kg	, kg/sec	, kW
1	38.7	1500	Sat Vap	1	1470.90	4.8830	367.55	0.2190	80.49
2	38.7	1500	Sat Liq	1	365.12	1.3374	308.63	0.2190	67.59
3	38.7	1500	Sat Liq	1	365.12	1.3374	308.63	0.2190	67.59
4	16.6	1500	Sub	1	259.12	0.9849	306.64	0.2190	67.15
5	-23.8	160	0.1398	1	259.12	1.0485	287.89	0.2190	63.05
6	-23.8	160	Sat Vap	1	1412.60	5.6749	75.88	0.2190	16.62
7	23.0	160	Super	1	1517.70	6.0608	66.74	0.2190	14.62
8	38.0	160	Sat Liq	0.30527	21.03	0.6544	-65.93	1.4304	-94.30
9	38.1	1500	Sub	0.30527	26.77	0.6590	-61.55	1.4304	-88.04
10	111.2	1500	Sub	0.30527	351.14	1.5952	-13.36	1.4304	-19.11
11	145.0	1500	Sat Liq	0.17967	532.09	1.9264	27.79	1.2114	33.67
12	55.0	1500	Sub	0.17967	149.07	0.8975	-51.53	1.2114	-62.43
13	56.7	160	Sub	0.17967	149.07	0.9086	-54.81	1.2114	-66.39
14	91.0	1500	Sat Vap	0.97228	1644.80	5.3986	380.10	0.2302	87.50
15	91.0	1500	Sat Liq	0.43	265.59	1.4092	-11.55	0.0112	-0.13
16	-7.0	400	Sub		31.57	0.1205	53.15	8.0028	425.34
17	-18.0	200	Sub		0.00	0.0000	57.00	8.0028	456.15
18	-7.0	100	Sub		31.24	0.1202	52.93	8.0028	423.55
19	20.0	200	Sub		83.16	0.2930	0.13	15.4047	1.98
20	23.7	200	Sub		98.87	0.3475	0.34	15.4047	5.24
21	24.6	200	Sub		102.36	0.3542	0.38	15.4047	5.85
22	32.0	200	Sub		133.72	0.4620	0.80	15.4047	12.32
23	185.0	1123	Sat Vap		2782.00	6.5457	853.47	0.2594	221.38
24	185.0	1123	Sat Liq		785.19	2.1873	143.05	0.2594	37.11

The resulting rate table is as follows:

	, kW		, kW
Component	IN	OUT	IN	OUT	kW
Condenser					9.75
Subcooler					2.43
Throttle Valve					4.11
Evaporator					15.63
Cold Box HX			250.00		10.47
Cold Box					22.13
Brine Pump	2.64				0.85
Absorber					36.11
Aqua Pump	8.20				1.94
Aqua HX					27.16
Generator/Rect.					43.88
Reflux Cooler					6.42
Aqua Valve					3.97
Sum/Net	10.84		250.00		184.85

The availability transfer from the cold box heat exchanger to the cold box is 22.13 kW. Define the second law efficiency as,

(12-51)

That is, the ratio of what is wanted to what you have to purchase. This efficiency then works out to be 11.3 per cent. There is no rational first law coefficient of performance because you need to buy both low-grade and high-grade energy.

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12.6 Chapter References

1. Gibbs, J. W., The Scientific Papers, Volume One: Thermodynamics, Dover Publications, Inc., New York, 1961, pp. 219ff.

2. Kirkwood, J. G. and I. Oppenheim, Chemical Thermodynamics, McGraw-Hill Book Co., Inc., New York, 1961, pp 148ff.

3. Ziegler, B. and C. Trepp, "Equation of State for Ammonia-Water Mixtures," Revue Internationale du Froid, 7, March 1984, pp. 101-106.

4. Gillespie, P. C., W. V. Wilding, and G. M. Wilson, "Vapor-Liquid Equilibrium Measurements on the Ammonia-Water System from 313 to 589 K," AICHE Symposium Series 83, 1987, pp. 97-127.

5. Reynolds, W. C., Thermodynamic Properties in SI, Department of Mechanical Engineering, Stanford University, 1979.

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