# Separation Of Multicomponent Mixtures By Use Of One Equilibrium Stage

Each of the separation processes considered in Sees. 1-2 and 1-3 consist of special cases of the general separation problem in which a multicomponent mixture is to be separated into two or more parts through the use of any number of plates.

The boiling-point diagram (Fig. 1-6) is useful for the visualization of the necessary conditions required for a flash to occur. Suppose that feed to be flashed has the composition X{ = xu (xlf A and #), and further suppose that this liquid mixture at the temperature T0 and the pressure P = 1 atm is to be flashed by raising the temperature to the specified flash temperature Tr = T2 at the specified flash pressure P = 1 atm. First observe that the bubble-point temperature of the feed TBP at P = 1 atm is The dew-point temperature, TDP, of the feed at the pressure P = 1 atm is seen to be T3. Then it is obvious from Fig. 1-6 that a necessary condition for a flash to occur at the specified pressure is that

In practice, the flash process is generally carried out by reducing the pressure on the feed stream rather than by heating the feed at constant pressure as described above.

To determine whether the feed will flash at a given Th and P, the above inequality may be used by determining the bubble-point and dew-point temperatures of the feed at the specified pressure P. In the determination of the bubble-point temperature of the feed at the specified P of the flash, the {x,} in Eq. (1-14) are replaced by the {X,} of the feed, and in the determination of the dew-point temperature at the specified pressure, the {#} in Eq. (1-17) are replaced by the {Xj}. Alternatively, the inequality given by Eq. (1-23) is satisfied if at the specified Tf and P

vFyFi Figure 1-7 The flash process.

The symbol represents the K value of component i evaluated at TF and P.

The two types of flash calculations which are commonly made are generally referred to as isothermal and adiabatic flashes.

### Isothermal Flash Process

The name " isothermal flash " is commonly given to the single-stage separation process shown in Fig. 1-7 for which the flash temperature Th and pressure P are specified as well as the total flow rate F and composition {X,} of the feed. The name " isothermal flash " originated, no doubt, from the fact that the temperature of the contents of the flash drum as well as the vapor and liquid streams formed by the flash is fixed at TF. The flash temperature TF is not necessarily equal to the feed temperature prior to its flashing.

For the set of specifications stated above, the problem is to find the total flow rates VF and Lt and the respective compositions {yFi} and {xFi} of the vapor and liquid streams formed by the flash process.

In addition to the c + 2 equations required to describe the state of equilibrium between the vapor and liquid phases [see Eq. (1-12)], c additional component-material balances which enclose the flash chamber are required to describe the isothermal flash process. Thus, the independent equations required to describe this flash process are as follows

Equilibrium relationships yFi= KFixFi (« = 1, 2 c)

Material balances FX, = -f- L,.<xf/ (/ = 1, 2, ..c)

Equation (1-26) is seen to represent 2c + 2 equations in 2c + 2 unknowns [K,,, Lf> I?«}. {*«}]•

This system of nonlinear equations is readily reduced to one equation in one unknown (say Vh) in the following manner. First observe that the total material balance expression (a dependent equation) may be obtained by summing each member of the third expression of Eq. (1-26) over all components to give

The relationships given by Eq. (1-26) may be reduced to one equation in one unknown in a variety of ways, and a variety of forms of the flash function may be obtained. One form of the flash function is developed below and a different form is developed in Chap. 4 in the formulation of multiple-stage problems. Elimination of the yF?s from the last expression given by Eq. (1-26) by use of the first expression, followed by rearrangement, yields

" Lf/F + VFKfJF Elimination of LF from Eq. (1-28) by use of Eq. (1-27) yields where