Report

Chapter 7 Equilibrium rela 1 1. Phase Rule C–Φ+2 F = number of degrees of freedom, or variance C= number of components Φ= number of phases 2 —— only temperature and pressure may affect th e.g.: In systems of two components, C=2；Φ=2； therefore, F = 2 - 2 + 2 = 2 •F = 2 •For binary distillation, F=2, there are four variables of interests: pressure, temperature, and the mole fractions of component A in liquid and vapor phases. If the pressure is fixed,only one variable, e.g., liquid phase mole fraction, can be changed independently, and temperature and vapor-phase mole fraction follow. 3 2. Equilibrium of Gas and Liquid and equilibrium stage Vapor Heater 加热 Liquid 4 P=pA+pB yA+yB=1 Vapor xA+xB=1 加热 Heater Liquid 5 Equilibrium State: No further changes in composition, temperature, o The chemical potential of the vapor and liquid pha The apparatus is performing as if it were an “equil 6 3. Thermodynamic relationships 1)Equilibrium ratio ( or equilibrium constant or K yA KA xA Where KA=EquilibriumKratio A yA=mole fractionof A in vapor ABcomponent KB xA= mole fraction of component A in liquid 7 • • The more volatile components in a mixture will have the higher values of KA, whereas less volatile components will have lower values of KA. 2)Relative volatility（相对挥发度）----key separation factor in distillation. （易挥发） AB KA KB (b) Where AB relative volatility ( A relative to B) 8 • 3)Ideal system • Ideal systems of vapor and liquid mixtures obey Dalton’s and Raoult’s law. Dalton’s law relates the concentration of a component present in an ideal gas or vapor mixture to its partial pressure. • p A PyA (a) 2 Where P=total pressure,force/length pA pA=partial pressure P of component A,force/length2 yA=mol fraction of component A,dimensionless(无因次) yA 9 • Raoult’s law p A p A0 x A p B p xB p (1 x A ) 0 B (b) 0 B Where pA=partial pressure of component A, force/length2 xA=molar fraction of component A, dimensionless(无因次) 10 Combining Equation(a) and (b) yields Therefore, Py P x A PyAA PP xx Py PPP KK A KAA P PP PPPA AB AA AB P AB B PP A A AA AA A AA BB BBB i ln P A i ln PP AAii C iTi ln Ci TT C i ii 11 P Vapor pressure of pure PA component i as a function ofAB temperature is correlated byAthe i P B Antoine equation: Bi Ai PyA PA x A Bi Ci Ai ln Pi Ai Ai Bi C T T B BP i Ci A A i i i KA Where ACiiPBi C i are Antoine constants R T T isCabsolute temperature, or K BTi P R i A K Ain Ktemperature CRless T sensitive • AB is to changes R i AB P A or B than is K K K R KB TK K A B AB (当温度变化不大时，可认为是常数，或取平均值) K B AB relative vol K R i Where i i 12 Where AB relative volatility ( K i B ln P A C T P p A pB since pA P xA A pB P x P (1 x A ) B B B 13 Bubble point equation: P p P f 2 (t ) xA 0 pA p f1 (t ) f 2 (t ) 0 B 0 B Dew point equation: p p P p f1 (t ) P f 2 (t ) yA xA 0 P P pA p P f1 (t ) f 2 (t ) 0 A 0 A 0 B 0 B 14 Combining Dalton’s law and Raoult’s law Py P A x A yields A Py P x p A PyAA PAA xAA P A P K Therefore, A P A KKA PA A PP P A P A P AB AB PA AB PPBB B Though not a strong function of temperature, B i B ln P A i B generally decreaseslnwith increasing temperature i i i P A i i ln P A C T i i i and increases with decreasing temperature CCi TT i 15 For a binary system, the relative volatility of the two components (A and B) is given by: y A / xA y B / xB y B 1 y A , xB 1 x A The above equations can be rearranged give (重新整理) 16 to AB x A yA 1 ( AB 1) x A Let xA=x；yA=y: Phase equilibrium equation: x y 1 ( 1) x The above equation is used to express the concentration of component A in the vapor as a function of its concentration in the liquid and relative volatility.表示在总压一定时,气液平衡时 的气液两相组成关系 17 4. Phase equilibrium diagrams for ideal systems •1) t-x-y diagram (boiling-point diagram) (Total pressure is constant) t •2 lines ; 3 regions •2 lines: Bubble point line (t-x curve, plotted from Bubble po Bubble point line 泡点线 x 18 Dew point line t point equation) (t-y curve,plotted（图示） from Dew 露点线 Dew point line y 19 t-x-y diagram (Total pressure is constant) The diagram gives the relation of composition 露点线 wit t 泡点线 x(y) 20 Superheated vapor region Two phase vapor/liquid region Dew 露点线 Point t Bubble 泡点线 Point Liquid region x(y) 21 Hint： (1)At equilibrium, temperature of liquid and vapor phases is the same. Dew point t Bubble point x t superheated tdew Dew point y (2)When x=y: tdew > tbubble tbubble bubble point (3) In two phase region: tcold x(y) y>x 22 2) x-y diagram The most used equilibrium curve in distillation is x-y diagram. It can be plotted from the t-x-y diagram. Question: What are the temperature relationships of points 1, 2 and 3？ 23 Hint: (1)The higher the x or y is, the lower the temperature is; (2) When α>1, y>x, equilibrium line is above diagonal; （3）When α=1, y=x equilibrium line is just diagonal; y α=4 α=2 α=1 α<1 x （4）α<1, equilibrium line is below diagonal. 24 5. Henry’s law • For low concentrations, the relationship between the concentration of a component in a liquid mixture and its partial pressure in the vapor (or gas) phase can be expressed as a linear relationship. The modified form of Raoult’s law, is known as Henry’s law. p A ExA (c) E law constant for component Where E=Henry’s A, force/lengthy2,A xAmolar m xA fraction of liquid. 25 E p A ExA • If both sides of eq.(c) are divided by the total pressure, thenE y A m xA E E m m P P Where m=modified Henry’s law constant(中文又 称相平衡常数), yA molar fraction of gas, xA molar fraction of liquid 26 6. Non-ideal systems Most liquid mixture are nonideal. Eq.(b) must be modified to include a correction factor called the liquid-phase activity coefficient. pA p xA A 0 A (d) Where γA=liquid phase activity coefficient, dimensionless 27 •If γ>1, positive deviations from ideality in the liquid occur; •if γ<1, there are negative deviations from ideality. P KA P A PA P AB K A A A P B PB A PA dependent KA and AB are on P B B composition. A A 28 Azeotropes （恒沸物） •When azeotropes are encountered, vapor and liquid compositions are equal, and components cannot be separated by conventional distillation. Minimum Boiling Azeotrope （最低恒沸点的溶液） Maximum Boiling Azeotrope （最高恒沸点的溶液） 29 7. Effect of total pressure on vapor/liquid equilibrium (1) Total Pressure↑→Bubble point↑ → α↓ → Separation difficulty↑ (2) Total Pressure changed→Azeotropic compositions changed •Total Pressure Effect on Water-Ethanol（乙醇） system P（kPa） 101.3 52.6 25.3 12.7 Azeotrope(mol fraction) 0.894 0.915 0.994 0.997 Assignments：21.1(a), (b)[p.706] 30