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ERT 206/4 THERMODYNAMICS SEM 1 (2012/2013) Dr. Hayder Kh. Q. Ali hayderali@unimap.edu.my 1

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The Nature of Equilibrium The Phase rule. VLE Quantitative Behavior Raoult’s Law VLE by Modified Raoult’s Law 2

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3 In this chapter we first discuss the nature of equilibrium, and then consider two rules that give the number of independent variables required to determine equilibrium states. Then a qualitative discussion of vapor/liquid phase behavior. Introduce the simplest formulations that allow calculation of temperatures, pressures, and phase compositions for systems in vapor/liquid equilibrium. The first, known as Raoult's law, is valid only for systems at low to moderate pressures and in general only for systems comprised of chemically similar species. A modification of Raoult's law that removes the restriction to chemically similar species is treated. Finally the calculations based on equilibrium ratios or K-values are considered.

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Phase equilibrium Application – Distillation, absorption, and extraction bring phases of different composition into contact. Both the extent of change and the rate of transfer depend on the departure of the system from equilibrium. Quantitative treatment of mass transfer the equilibrium T, P, and phase compositions must be known. 4

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The nature of equilibrium A static condition in which no changes occur in the macroscopic properties of a system with time. At the microscopic level, conditions are not static. – The average rate of passage of molecules is the same in both directions, and no net interphase transfer of material occurs. An isolated system consisting of liquid and vapor phases in intimate contact eventually reaches a final state wherein no tendency exists for change to occur within the system. – Fixed temperature, pressure, and phase composition 5

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Phase rule vs. Duhem’s theorem (The number of variables that is independently fixed in a system at equilibrium) = (the number of variables that characterize the intensive state of the system) - (the number of independent equations connecting the variable): Phase rule: Duhem’s rule: – for any closed system formed initially from given masses of prescribed chemical species, the equilibrium state is completely determined when any two independent variables are fixed. – Two ? When phase rule F = 1, at least one of the two variables must be extensive, and when F = 0, both must be extensive. 6

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7 VLE: QUALITATIVE BEHAVIOR Vapor/Liquid equilibrium (VLE) is the state of coexistence of liquid and vapor phases. In this qualitative discussion, we limit consideration to systems comprised of two chemical species, because systems of greater complexity cannot be adequately represented graphically. When N = 2, the phase rule becomes F = 4 - π. Since there must be at least one phase (π = 1), the maximum number of phase-rule variables which must be specified to fix the intensive state of the system is three: namely, P, T, and one mole (or mass) fraction. All equilibrium states of the system can therefore be represented in three dimensional P-T- composition space. Within this space, the states of pairs of phases coexisting at equilibrium (F = 4 - 2 = 2) define surfaces. A schematic three-dimensional diagram illustrating these surfaces for VLE is shown in Fig. below:

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9 This figure shows schematically the P-T-composition surfaces which contain the equilibrium states of saturated vapor and saturated liquid for a binary system: The under surface contains the saturated-vapor states; it is the P-T-yl surface. The upper surface contains the saturated-liquid states; it is the P-T-xl surface. These surfaces intersect along the lines UBHC2 and RKAC1, which represent the vapor pressure-vs.-T curves for pure species 1 and 2.

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10 Because of the complexity of Fig. 10.1, the detailed characteristics of binary VLE are usually depicted by two-dimensional graphs. The three principal planes, each perpendicular to one of the coordinate axes. A vertical plane perpendicular to the temperature axis is outlined as ALBDEA. The lines on this plane form a P-x 1 -y 1 phase diagram at constant T. A horizontal plane perpendicular to the P axis is identified by HIJKLH. Viewed from the top, the lines on this plane represent a T-x 1 -y 1 diagram at constant P.

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12 The third plane, vertical and perpendicular to the composition axis, is indicated by MNQRSLM, this is the P-T diagram. At points A and B in Fig. 10.3 saturated-liquid and saturated-vapor lines intersect. At such points a saturated liquid of one composition and a saturated vapor of another composition have the same T and P, and the two phases are therefore in equilibrium.

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Within this space, the states of pairs of phases coexisting at equilibrium define surfaces. – The subcooled-liquid region lies above the upper surface; the superheated-vapor region lies below the under surface. – UBHC 1 and KAC 2 represent the vapor pressure-vs.-T curves for pure species 1 and 2. – C 1 and C 2 are the critical points of pure species 1 and 2. – L is a bubble point and the upper surface is the bubblepoint surface. – Line VL is an example of a tie line, which connects points representing phases in equilibrium. – W is a dewpoint and the lower surface is the dewpoint surface. Pxy diagram at constant T Txy diagram at constant P PT diagram at constant composition 15

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Fig 10.8 16

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Fig 10.9 17

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18 Simple models for VLE When thermodynamics is applied to vapor liquid equilibrium, the goal is to find by calculation the temperatures, pressures, and compositions of phases in equilibrium.

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Raoult’s law: – the vapor phase is an ideal gas (apply for low to moderate pressure) – the liquid phase is an ideal solution (apply when the species that are chemically similar) – where xi is a liquid-phase mole fraction, yi is a vapor-phase mole fraction, and Pi sat is the vapor pressure of pure species i at the temperature of the system. The product yi P on the left side of Eq. is known as the partial pressure of species i 19

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20 Ideal-solution behavior is often approximated by liquid phases wherein the molecular species are not too different in size and are of the same chemical nature although it provides a realistic description of actual behavior for a small class of systems, it is valid for any species present at a mole fraction approaching unity, provided that the vapor phase is an ideal gas.

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21 Dewpoint and Bubblepoint Calculations with Raoult's Law Engineering interest centers on dewpoint and bubblepoint calculations; there are four classes: In each case the name suggests the quantities to be calculated: either a BUBL (vapor) or a DEW (liquid) composition and either P or T. Thus, one must specify either the liquid-phase or the vapor-phase composition and either T or P.

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Binary system For bubblepoint calculation For dewpoint calculation 22

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Binary system acetonitrile (1)/nitromethane(2) conforms closely to Raoult’s law. Vapor pressures for the pure species are given by the following Antoine equations: (a) Prepare a graph showing P vs. x 1 and P vs. y 1 for a temperature of 75°C. (b) Prepare a graph showing t vs. x 1 and t vs. y 1 for a pressure of 70 kPa. (a) BUBL P At 75°C e.g. x 1 = 0.6 At 75°C, a liquid mixture of 60 mol-% (1) and 40 mol-% (2) is in equilibrium with a vapor containing 74.83 mol-% (1) at pressure of 66.72 kPa. Fig. 10.11 24

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Fig. 10.11 25

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(b) BUBL T, having P = 70 kPa Select tt vs. x 1 t vs. y 1 Fig. 10.12 26

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VLE modified Raoult’s law Account is taken of deviation from solution ideality in the liquid phase by a factor inserted into Raoult’s law: The activity coefficient, f (T, x i ) 27

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For the system methanol (1)/methyl acetate (2), the following equations provide a reasonable correlation for the activity coefficients: The Antoine equations provide vapor pressures: Calculate (a): P and {y i } for T = 318.15 K and x 1 = 0.25 (b): P and {x i } for T = 318.15 K and y 1 = 0.60 (c): T and {y i } for P = 101.33 kPa and x 1 = 0.85 (d): T and {x i } for P = 101.33 kPa and y 1 = 0.40 (e): the azeotropic pressure and the azeotropic composition for T = 318.15 K (a) for T = 318.15, and x 1 = 0.25 29

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(b): for T = 318.15 K and y 1 = 0.60 An iterative process is applied, with Converges at: (c): for P = 101.33 kPa and x 1 = 0.85 A iterative process is applied, with Converges at: 30

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(d): for P = 101.33 kPa and y 1 = 0.40 A iterative process is applied, with Converges at: 31

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(e): the azeotropic pressure and the azeotropic composition for T = 318.15 K Define the relative volatility: Azeotrope Since α 12 is a continuous function of x 1 : from 2.052 to 0.224, α 12 = 1 at some point There exists the azeotrope! 32

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VLE from K-value correlations A convenient measure, the K-value: – the “lightness” of a constituent species, i.e., of its tendency to favor the vapor phase. – When Ki is greater than unity, species i exhibits a higher concentration in the vapor phase; when less, a higher concentration in the liquid phase, and is considered a "heavy" constituent. 33

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– The Raoult’s law: – The modified Raoult’s law: 34

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Fig 10.13 35

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Fig 10.14 36

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For a mixture of 10 mol-% methane, 20 mol-% ethane, and 70 mol-% propane at 50°F, determine: (a) the dewpoint pressure, (b) the bubblepoint pressure. The K-values are given by Fig. 10.13. (a) at its dewpoint, only an insignificant amount of liquid is present: (b) at bubblepoint, the system is almost completely condensed: 37

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Flash calculations A liquid at a pressure equal to or greater than its bubblepoint pressure “flashes” or partially evaporates when the pressure is reduced, producing a two-phase system of vapor and liquid in equilibrium. Consider a system containing one mole of nonreacting chemical species: The moles of liquid The moles of vapor The liquid mole fraction The vapor mole fraction 38

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The system acetone (1)/acetonitrile (2)/nitromethane(3) at 80°C and 110 kPa has the overall composition, z 1 = 0.45, z 2 = 0.35, z 3 = 0.20, Assuming that Raoult’s law is appropriate to this system, determine L, V, {x i }, and {y i }. The vapor pressures of the pure species are given. Do a BUBL P calculation, with {z i } = {x i } : Do a DEW P calculation, with {z i } = {y i } : Since P dew < P = 110 kPa < P bubl, the system is in the two-phase region, 39

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