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1、1,Chapter 7 Equilibrium relationships and equilibrium stage operation,2,1. Phase Rule,F = C + 2 F = number of degrees of freedom, or variance C= number of components = number of phases 2 only temperature and pressure may affect the equilibrium state. e.g.: In systems of two components, C=2；=2； there
2、fore, F = 2 - 2 + 2 = 2,3,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 temp
3、erature and vapor-phase mole fraction follow.,2. Equilibrium of Gas and Liquid and equilibrium stage,4,5,P=pA+pB yA+yB=1,xA+xB=1,6,Equilibrium State: No further changes in composition, temperature, or pressure occur. The chemical potential of the vapor and liquid phases are equal. The apparatus is p
4、erforming as if it were an “equilibrium,” or“theoretical,” or “ideal” stage and the vapor and liquid compositions are the equilibrium compositions.,7,7,3. Thermodynamic relationships,1)Equilibrium ratio ( or equilibrium constant or K value).,Where KA=Equilibrium ratio yA=mole fraction of component A
5、 in vapor xA= mole fraction of component A in liquid,8,The more volatile（易挥发） 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.,(b),9,3)Ideal system Ideal syste
6、ms of vapor and liquid mixtures obey Daltons and Raoults law. Daltons law relates the concentration of a component present in an ideal gas or vapor mixture to its partial pressure.,10,Raoults law,Where pA=partial pressure of component A, force/length2 xA=molar fraction of component A, dimensionless(
7、无因次),(b),11,Therefore,Combining Equation(a) and (b) yields,12,Vapor pressure of pure component i as a function of temperature is correlated by the Antoine equation:,Where are Antoine constants is absolute temperature, or,is less sensitive to changes in temperature than is KA or KB,(当温度变化不大时，可认为是常数，或
8、取平均值),13,since,14,Bubble point equation:,Dew point equation:,15,Though not a strong function of temperature, generally decreases with increasing temperature and increases with decreasing temperature,16,For a binary system, the relative volatility of the two components (A and B) is given by:,The abov
9、e equations can be rearranged(重新整理) to give,17,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,18,4. Phase equilibrium diagrams for ideal systems,1) t-x-y diagram (boi
10、ling-point diagram) (Total pressure is constant) 2 lines ; 3 regions 2 lines: Bubble point line (t-x curve, plotted from Bubble point equation.) and Dew point line,19,Dew point line (t-y curve,plotted（图示） from Dew point equation),20,t-x-y diagram (Total pressure is constant) The diagram gives the re
11、lation of composition with temperature,21,Liquid region,Two phase vapor/liquid region,Superheated vapor region,22,Hint： (1)At equilibrium, temperature of liquid and vapor phases is the same.,(2)When x=y: tdew tbubble,(3) In two phase region: yx,23,2) x-y diagram The most used equilibrium curve in di
12、stillation 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？,24,Hint: (1)The higher the x or y is, the lower the temperature is; (2) When 1, yx, equilibrium line is above diagonal; （3）When =1, y=x equilibrium line is just
13、 diagonal; （4）1, equilibrium line is below diagonal.,25,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 Raoults law, is known as
14、Henrys law.,5. Henrys law,26,If both sides of eq.(c) are divided by the total pressure, then,Where m=modified Henrys law constant(中文又称相平衡常数), yA molar fraction of gas, xA molar fraction of liquid,27,6. Non-ideal systems Most liquid mixture are nonideal. Eq.(b) must be modified to include a correctio
15、n factor called the liquid-phase activity coefficient.,28,If 1, positive deviations from ideality in the liquid occur; if 1, there are negative deviations from ideality.,KA and are dependent on composition.,29,Azeotropes （恒沸物） When azeotropes are encountered, vapor and liquid compositions are equal,
16、 and components cannot be separated by conventional distillation.,30,7. Effect of total pressure on vapor/liquid equilibrium,(1) Total PressureBubble point Separation difficulty (2) Total Pressure changedAzeotropic 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,