To compute the composition of the top product D and the bottom product B which may be expected by use of a given distillation column operated at a given set of conditions, it is necessary to obtain a solution to equations of the following types:
1. Equilibrium relationships
2. Component-material balances
3. Total-material balances
4. Energy balances
Consider first the subject of equilibrium relationships.
A two-phase mixture is said to be in physical equilibrium if the following conditions are satisfied (Ref. 3).
1. The temperature Tv of the vapor phase is equal to the temperature TL of the liquid phase.
2. The total pressure Pl throughout the vapor phase is equal to the total pressure PL throughout the liquid phase. (1-1)
3. The tendency of each component to escape from the liquid phase to the vapor phase is exactly equal to its tendency to escape from the vapor phase to the liquid phase.
In the following analysis it is supposed that a state of equilibrium exists, Tv = Tl= T, Pv = PL= P, and the escaping tendencies are equal.
Now consider the special case where the third condition may be represented by Raoult's law
Pyt = PiXi
where x¡ and y¡ are the mole fractions of component i in the liquid and vapor phases, respectively, and P, is the vapor pressure of pure component / at the temperature T of the system.
The separation of a binary mixture by distillation may be represented in two-dimensional space while /i-dimensional space is required to represent the separation of a multicomponent mixture (i > 2). The graphical method proposed by McCabe and Thiele9 for the solution of problems involving binary mixtures is presented in a subsequent section. The McCabe-Thiele method makes use of an equilibrium curve which may be obtained from the "boiling-point diagram."
Construction and Interpretation of the Boiling-Point Diagram for Binary Mixtures
When a state of equilibrium exists between a vapor and a liquid phase composed of two components A and B, the system is described by the following set of independent equations where it is understood that Raoult's law is obeyed. Since the vapor pressures P A and PB depend upon T alone, Eq. (1-3) consists of four equations in six unknowns. Thus, to obtain a solution to this set of equations, two variables must be fixed. [Observe that this result is in agreement with the Gibbs phase rule: # + i= c + 2. For the above case, the number of phases = 2, the number of components c = 2, and thus the number of degrees of freedom V = 2, that is, the number of variables which must be fixed is equal to 2.] In the construction of the boiling-point diagram for a binary mixture, the total pressure P is fixed and a solution is obtained for each of several temperatures lying between the boiling-point temperature TA of pure A and the boiling-point temperature TB of pure B at the total pressure P. That is, when T = TA, PA = P and when T = TB, PB = P.
The solution of the set of equations [Eq. (1-3)] for xA in terms of PA, PB, and P is effected as follows. Addition of the first two equations followed by the elimination of the sum of the y's by use of the third expression yields
Equilibrium I PyB — PBxB
relationships \ _ 1
Elimination of xB by use of the fourth equation of the set given by Eq. (1-3) followed by rearrangement of the result so obtained yields
Mole fraction of A
Figure 1-6 The boiling-point diagram.
Mole fraction of A
From the definition of a mole fraction (0 < xA < 1), Eq. (1-5) has a meaningful solution at a given P for every T lying between the boiling-point temperatures Ta and Tb of pure A and pure B, respectively. After xA has been computed by use of Eq. (1-5) at the specified P and T, the corresponding value of yA which is in equilibrium with the value of xA so obtained is computed by use of the first expression of Eq. (1-3), namely,
By plotting T versus xA and T versus yA, the lower and upper curves, respectively, of Fig. 1-6 are typical of those obtained when component A is more volatile than B. Component A is said to be more volatile than component B, if for all T in the closed interval TA < T < TB, the vapor pressure of A is greater than the vapor pressure of B, that is, PA> PB. The horizontal lines such as CE that Join equilibrium, pairs (x, y), computed at a given T and P by use of Eqs. (1-5) and (1-6), are commonly called tie lines.
Example 1-1 (Taken from Ref. 6 by courtesy Instrument Society of America).
By use of the following vapor pressures for benzene and toluene [taken from The Chemical Engineer's Handbook, 2d ed., J. H. Perry (ed.) McGraw-
Hill, New York, 1941], compute the three equilibrium pairs (x, y) on a boiling-point diagram which correspond to the temperatures T = 80.02°C, T = 100°C, and T = 110.4oC. The total pressure is fixed at P = 1 atm. Given:
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