1. Chapter 2
Multi-Phase Systems

2. Multiphase Systems
Virtually all commercial chemical processes involve operations in which material is transferred from one phase (gas, liquid, or solid) into another. These multiphase operations include all phase-change operations on a single species, such as freezing, melting, evaporation, and condensation, and most separation and purification processes, which are designed to separate components of mixtures from one another.
Most separations are accomplished by feeding a mixture of species A and B into a two- phase system under conditions such that most of the A remains in its original phase and most of the B transfers into a second phase. The two phases then either separate themselves under the influence of gravity—as when gases and liquids or two immiscible liquids separate—or are separated with the aid of a device such as a filter.
Pressure, volume, temperature (PVT) relations are required in simulating reservoirs, evaluating reserves, forecasting production, designing production facilities and designing gathering and transportation systems.

3. Multiphase Systems SINGLE-COMPONENT PHASE EQUILIBRIUM
At most temperatures and pressures, a single pure substance at equilibrium exists entirely as a solid, liquid, or gas; but at certain temperatures and pressures, two and even all three phases may coexist. Pure water is a gas at 130 ºC and 100 mmHg, for example, and a solid at – 40 ºC and 10 atm, but at 100 ºC and 1 atm it may be a gas, a liquid, or a mixture of both, and at approximately ºC and 4.58 mmHg it may be a solid, a liquid, a gas, or any combination of the three.
A phase diagrams of a pure substance is a plot of one system variable against another that shows the conditions at which the substance exists as a solid, a liquid, and a gas. The most common of these diagrams plots pressure on the vertical axis versus temperature on the horizontal axis. The boundaries between the single-phase regions represent the pressures and temperatures at which two phases may coexist. The phase diagrams of water and carbon dioxide are shown in Figure

4. A picture worth a thousand words applies Phase Diagram
SINGLE-COMPONENT PHASE EQUILIBRIUM
Phase diagrams
One of the primary challenges that we face as engineers is how to communicate large quantities of complex information to our superiors
The familiar saying
A picture worth a thousand words applies
Phase Diagram
A phase diagram is a concise graphical method of representing phase behaviour of fluids. It provides an effective tool for communicating a large amount of information about how fluid behave at different conditions
Two classes of fluids
Pure –component systems
Mixtures
Although most non-engineers believe that the only language an engineer speaks is mathematics, in actuality, the most effective means that engineers use to communicate information involves the use of pictures or diagrams.

5. T = 100 oC but more accurate response is T = 100 oC at 1 atm
SINGLE-COMPONENT PHASE EQUILIBRIUM
Pure-component systems
Water — the most plentiful substance on earth. What is the behavior of water under different conditions of pressure and temperature? Or even more specific, is there a single answer to the question: what is the boiling point of water? 
T = 100 oC but more accurate response is T = 100 oC at 1 atm
Vapor pressure of a liquid increases as temperature
increases (more liquid molecules escapes into
vapour phase)
The curve is called vapor pressure curve or boiling point
curve.
Dew Point (DP) & Bubble Point (BP) curve
Pressure, P
Temperature, T

6. SINGLE-COMPONENT PHASE EQUILIBRIUM
Pure-component systems
Vapor Pressure:
The pressure that the vapor phase of a fluid exerts over its own liquid at equilibrium at a given temperature.
Dew Point:
The pressure and temperature condition at which an infinitesimal quantity of liquid (a droplet) exists in equilibrium with vapor. It represents the condition of incipient liquid formation in an initially gaseous system.
Bubble Point: The pressure and temperature condition at which the system is all liquid, and in equilibrium with an infinitesimal quantity (a bubble) of gas. This situation is, in essence, the opposite of that of the dew point.

7. SINGLE-COMPONENT PHASE EQUILIBRIUM
Pure-component systems
For single-component systems, one single curve represents all three of these conditions (vapor pressure, dew point and bubble point conditions) simply because Vapor Pressure = Dew Point = Bubble Point for unary systems.
Sublimation
curve
Solidification
Boiling Point
Pressure, P
Temperature, T
Critical Point
Pressure, P
Temperature, T
Liquid
Gas
Triple Point

8. Its main purpose is to cause an increase in temperature of the system.
SINGLE-COMPONENT PHASE EQUILIBRIUM
Pure-component systems
Sensible Heat:
Its main purpose is to cause an increase in temperature of the system.
Latent Heat:
It serves only one purpose: to convert the liquid into vapor. It does not cause a temperature increase.
Pressure, P
Temperature, T
Critical Point
Triple Point
Liquid
Gas
Pressure, P
Temperature, T
Critical Point
Triple Point
Liquid
Gas A C B

9. Path AD: Isothermal compression Path DE: Isobaric heating
SINGLE-COMPONENT PHASE EQUILIBRIUM
Pure-component systems
Sequence of Paths
Path AD: Isothermal compression
Path DE: Isobaric heating
Path EB: Isothermal expansion
Pressure, P
Temperature, T A D E B

10. SINGLE-COMPONENT PHASE EQUILIBRIUM
PV Diagram for Pure-component systems
Pressure, P
Temperature, T E H H

11. SINGLE-COMPONENT PHASE EQUILIBRIUM
Phase diagrams
Suppose the system is initially at 20 ºC, and the force is set at a value such that the system pressure is 3 mmHg. As the phase diagram shows, water can only exist as a vapor at these conditions, so any liquid that may initially have been in the chamber evaporates, until finally the chamber contains only water vapor at 20 ºC and 3 mm Hg (point A on Figure ).
Now suppose the force on the piston is slowly increased with the system temperature held constant at 20 ºC until the pressure in the cylinder reaches 760 mm Hg, and thereafter heat is added to the system with the pressure remaining constant until the temperature reaches 130 ºC. The state of the water throughout this process can be determined by following path

13. Notice that the phase transitions—condensation at point B and evaporation at point D take place at boundaries on the phase diagram; the system cannot move off these boundaries until the transitions are complete.

14. Several familiar terms may be defined with reference to the phase diagram.
If T and P correspond to a point on the vapor–liquid equilibrium curve for a substance, P is the vapor pressure of the substance at temperature T, and T is the (more precisely, the boiling point temperature) of the substance at pressure .
The boiling point of a substance at P =1 atm is the normal boiling point of that substance.
If (T, P ) falls on the solid–liquid equilibrium curve, then T is the melting point or freezing point at pressure P.
If (T, P ) falls on the solid–vapor equilibrium curve, then P is the vapor pressure of the solid at temperature T, and T is the sublimation point at pressure P.
The point (T, P) at which solid, liquid, and vapor phases can all coexist is called the triple point of the substance.
The vapor–liquid equilibrium curve terminates at the critical temperature and critical pressure (Tc and Pc ). Above and to the right of the critical point, two separate phases never coexist.

15. Estimation of Vapor Pressures
The volatility of a species is the degree to which the species tends to transfer from the liquid (or solid) state to the vapor state. At a given temperature and pressure, a highly volatile substance is much more likely to be found as a vapor than is a substance with low volatility, which is more likely to be found in a condensed phase (liquid or solid).

16. Estimation of Vapor Pressures

17. EXAMPLE: Vapor Pressure Estimation Using the Clausius–Clapeyron Equation
The vapor pressure of benzene is measured at two temperatures, with the following results:
Calculate the latent heat of vaporization and the parameter B in the Clausius–Clapeyron equation
and then estimate p* at 42.2 ºC using this equation.
SOLUTION

18. The Clausius–Clapeyron equation is therefore:
The intercept B is obtained from Equation as
= ln 40 + (4213/280.8) = 18.69
The Clausius–Clapeyron equation is therefore:

20. GAS–LIQUID SYSTEMS: ONE CONDENSABLE COMPONENT
A law that describes the behavior of gas–liquid systems over a wide range of conditions provides the desired relationship. If a gas at temperature T and pressure P contains a saturated vapor whose mole fraction is yi (mol vapor/mol total gas), and if this vapor is the only species that would condense if the temperature were slightly lowered, then the partial pressure of the vapor in the gas equals the pure-component vapor pressure pi* (T) at the system temperature.

21. EXAMPLE: Composition of a Saturated Gas–Vapor System
Air and liquid water are contained at equilibrium in a closed chamber at 75 ºC and 760 mm Hg. Calculate the molar composition of the gas phase.
SOLUTION
Since the gas and liquid are in equilibrium, the air must be saturated with water vapor (if it were not, more water would evaporate), so that Raoult’s law may be applied:

22. Several important points concerning the behavior of gas–liquid systems and several terms used to describe the state of such systems are summarized here.
The difference between the temperature and the dew point of a gas is called the degrees of superheat of the gas.

23. EXAMPLE
A stream of air at 100 ºC and 5260 mmHg contains 10% water by volume. Calculate the dew point and degrees of superheat of the air.
SOLUTION

24. Humidity
If you are given any of the following quantities for a gas at a given temperature and pressure, you can solve the defining equation to calculate the partial pressure or mole fraction of the vapor in the gas; thereafter, you can use the formulas given previously to calculate the dew point and degrees of superheat.

25. EXAMPLE
Humid air at 75 ºC, 1.1 bar, and 30% relative humidity is fed into a process unit at a rate of 1000 m3 /h. Determine:
the molar flow rates of water, dry air, and oxygen entering the process unit,
the molal humidity, absolute humidity, and percentage humidity of the air, and
the dew point.
SOLUTION

26. Consequently,
The same result could have been obtained from the results of part 1 as (3.99 kmol H2O/h)/
(34.0 kmol BDA/h).

27. Raoult’s Law and Henry’s Law
Suppose A is a substance contained in a gas–liquid system in equilibrium at temperature and pressure. Two simple expressions—Raoult’s law and Henry’s law —provide relationships between pA, the partial pressure of A in the gas phase, and xA, the mole fraction of A in the liquid phase.
Raoult’s Law:
where 𝑝 𝐴 ∗ is the vapor pressure of pure liquid A at temperature T and yA is the mole fraction of A in the gas phase.
Raoult’s law is an approximation that is generally valid when xA is close to 1—that is, when the liquid is almost pure A
Henry’s Law:
where HA (T) is the Henry’s law constant for A in a specific solvent.
Henry’s law is generally valid for solutions in which xA is close to 0.

28. EXAMPLES: Raoult’s Law and Henry’s Law
Use either Raoult’s law or Henry’s law to solve the following problems.
A gas containing 1.0 mole% ethane is in contact with water at 20 ºC and 20 atm. Estimate the mole fraction of dissolved ethane.
An equimolar liquid mixture of benzene (B) and toluene (T) is in equilibrium with its vapor at 30 ºC. What is the system pressure and the composition of the vapor?
SOLUTION

30. Vapor–Liquid Equilibrium Calculations for Ideal Solutions
Suppose an ideal liquid solution follows Raoult’s law and contains species A, B, C, with known mole fractions xA, xB, xC. If the mixture is heated at a constant pressure to its bubble-point temperature TBP, the further addition of a slight amount of heat will lead to the formation of a vapor phase.
Since the vapor is in equilibrium with the liquid, and we now assume that the vapor is ideal (follows the ideal gas equation of state), the partial pressures of the components are given by Raoult’s law, Equation
Where is the vapor pressure of component at the bubble-point temperature. Moreover, since we have assumed that only A, B, C, … are present in the system, the sum of the partial pressures must be the total system pressure, P; hence,

31. The pressure at which the first vapor forms when a liquid is decompressed at a constant temperature is the bubble-point pressure of the liquid at the given temperature. Equation can be used to determine such a pressure for an ideal liquid solution at a specific temperature, and the mole fractions in the vapor in equilibrium with the liquid can then be determined as:
Let yi be the mole fraction of component in the gas. If the gas mixture is cooled slowly to its dew point, Tdb, it will be in equilibrium with the first liquid that forms. Assuming that Raoult’s law applies, the liquid-phase mole fractions may be calculated as:

32. At the dew point of the gas mixture, the mole fractions of the liquid components (those that are condensable) must sum to 1:
The dew-point pressure, which relates to condensation brought about by increasing system pressure at constant temperature, can be determined by solving Equation for :
Liquid mole fractions may then be calculated from Equation with Tdb replaced by the system temperature, T.

33. EXAMPLES: Bubble- and Dew-Point Calculations
1. Calculate the temperature and composition of a vapor in equilibrium with a liquid that is 40.0 mole% benzene–60.0 mole% toluene at 1 atm. Is the calculated temperature a bubble-point or dew-point temperature?
SOLUTION
Let A = benzene; B = toluene
Since the composition of the liquid was given, this was a bubble-point calculation
Equation may be written in the form
The solution procedure is to choose a temperature, evaluate and for that temperature from the Antoine equation using constants from Table B.4, evaluate f (Tdb) from the above equation, and repeat the calculations until a temperature is found for which f (Tdb) is sufficiently close to 0.

34. The solution is taken to be Tdb = 95. 1 ºC
The solution is taken to be Tdb = 95.1 ºC . At this temperature, Equation yields:
Furthermore, from Equation 6.4-5,

35. 2. Calculate the temperature and composition of a liquid in equilibrium with a gas mixture containing 10 mole% benzene, 10 mole% toluene, and the balance nitrogen (which may be considered non-condensable) at 1 atm. Is the calculated temperature a bubble-point or dew-point temperature?
2. Equation may be written as:

36. 3. A gas mixture consisting of 15 mole% benzene, 10 mole% toluene, and 75 mole% nitrogen is compressed isothermally at 80 °C until condensation occurs. At what pressure will condensation begin? What will be the composition of the initial condensate?

37. EXAMPLE: Bubble and Dew Point Calculations using Txy Diagrams
1. Using the Txy diagram, estimate the bubble-point temperature and the equilibrium vapor composition associated with a 40 mole% benzene–60 mole% toluene liquid mixture at 1 atm. If the mixture is steadily vaporized until the remaining liquid contains 25% benzene, what is the final temperature?.
2. Using the Txy diagram, estimate the dew-point temperature and the equilibrium liquid composition associated with a vapor mixture of benzene and toluene containing 40 mole% benzene at 1 atm. If condensation proceeds until the remaining vapor contains 60% benzene, what is the final temperature?
SOLUTION

38. Graphical Representations of Vapor–Liquid Equilibrium

39. EXAMPLE: Boiling Point of a Mixture
A mixture that is 70 mole% benzene and 30 mole% toluene is to be distilled in a batch distillation column. The column startup procedure calls for charging the reboiler at the base of the column and slowly adding heat until boiling begins. Estimate the temperature at which boiling begins and the initial composition of the vapor generated, assuming the system pressure is 760 mm Hg.
From the Txy diagram,
the mixture will boil at approximately 87 ºC . The initial vapor composition is approximately 88 mole% benzene and 12 mole% toluene .