**7. Different equilibrium diagrams and strategy of using difference points**

Depending on different separation processes and different contact forms of phases, different equilibrium diagrams seem to be useful. The formulations that are carried out on each process as the mass and heat balance equations, often force the process calculations to be done using different equilibrium diagrams. Part of calculations usually related to converting different types of equilibrium data to each other.

In distillation operation, the Hxy diagram (phase enthalpies versus transferable mole fraction) sometimes is used as equilibrium diagram. The liquid-liquid extraction and leaching process calculations can be carried out in triangular diagrams or rectangular diagrams e.g. Nxy diagrams in leaching (N is in mass of solid matrix, marc per mass of liquid mixture; x and y are in mass of solute per mass of liquid mixture in heavy and light phases, respectively). The NXY diagram is used in liquid-liquid extraction when it assumes that a solvent is added to the feed solution and the transferable component is miscible in it (N is in mass of solvent added per mass of mixture with no solvent; X and Y are in mass of solute per mass of solvent free mixture in heavy and light phases, respectively). Fig. 17 shows different equilibrium diagrams which their tie lines are generated from usual equilibrium

Fig. 17. Useful equilibrium diagrams; (a1, a2) Hxy diagram useful in distillation operation when using the Ponchon-Savarit method for binary systems; (b1, b2) NXY diagram used in liquid-liquid extraction; (c1, c2) Nxy diagram used in leaching operation

Mass Transfer - The Skeleton of Purification Processes 801

Strategy of using difference points is used in separation processes with countercurrent form of contact to analyze the operations graphically. This plan is produced from this reality that the difference between total flow rates or component flow rates of two phases passed next to each other is constant. It means that as it is shown in Fig. 18a in distillation process for the loop shown between stage m and the bottom product of the column, total mass balance and

m m m1m1 W

where H, L and W are the flow rates of heavy phase, light phase and bottom product, respectively. The notations x and y are the mole fractions and m shows the stage number. As it is seen these equations can be written for all stages between feed tray and bottom of the tower and it means that the differences are constant. A similar equation can be added for heat balance and in that case the difference between energies is constant as well. The important subject is that in liquid-liquid extraction and leaching process when the contact form is countercurrent such equations can be developed and using a difference point to carry out the process calculations graphically seems to be necessary. Such points are brought in Fig. 18 for different processes of distillation, leaching and liquid-liquid extraction. The method that used to find number of stages graphically was described completely in ref. (Treybal, 1955).

Differential form of contact is a usual method to contact phases in different separation processes. High surface area for contacting phases, less pressure drop, simplicity and low cost are the factors that cause this type of mass exchanger to be more useful. Height and number of transfer units are of important parameters that need to be calculated in packed bed contactors. During contact between phases along the height of the tower, concentration of separating component in two phases changes continuously and this type of mass transfer changes the rate of two streams as well. Sometimes transport of mass is occurred in the case of transfer into stagnant components e. g. absorption of ammonia from air with water, and

Fig. 19. A schematic of packed bed tower (left) and an element of tower in z-direction (right)

sometimes it occurred as equimolar countertransfer e. g. in distillation process.

+ +

− = (27)

H x L y Wx +

m m1

HL W

− =

component mass balance equations can be written as

**8. Packed bed height** 

curves. As it is shown each point on xy or XY diagrams (e.g. M or N) can be converted to a tie line (KL or OP) on equilibrium curves laid on the top.

Fig. 18. Using difference point to carry out the processing calculations graphically (tie lines are shown as bold and other lines show the operating lines) (a) Distillation tower with 7 stages; (b) Leaching process with 3 stages; (c) Liquid-liquid extraction in 4 stage process (NXY equilibrium diagram); (d) Liquid-liquid extraction in 3 stage process (rectangular coordinates, C is the transferable component and B is the solvent used to separate C from its mixture with A)

curves. As it is shown each point on xy or XY diagrams (e.g. M or N) can be converted to a

(a) (b)

 (c) (d)

mixture with A)

Fig. 18. Using difference point to carry out the processing calculations graphically (tie lines are shown as bold and other lines show the operating lines) (a) Distillation tower with 7 stages; (b) Leaching process with 3 stages; (c) Liquid-liquid extraction in 4 stage process (NXY equilibrium diagram); (d) Liquid-liquid extraction in 3 stage process (rectangular coordinates, C is the transferable component and B is the solvent used to separate C from its

tie line (KL or OP) on equilibrium curves laid on the top.

Strategy of using difference points is used in separation processes with countercurrent form of contact to analyze the operations graphically. This plan is produced from this reality that the difference between total flow rates or component flow rates of two phases passed next to each other is constant. It means that as it is shown in Fig. 18a in distillation process for the loop shown between stage m and the bottom product of the column, total mass balance and component mass balance equations can be written as

$$\begin{aligned} \mathbf{H\_m} - \mathbf{L\_{m+1}} &= \mathbf{W} \\ \mathbf{H\_m x\_m} - \mathbf{L\_{m+1} y\_{m+1}} &= \mathbf{Wx\_W} \end{aligned} \tag{27}$$

where H, L and W are the flow rates of heavy phase, light phase and bottom product, respectively. The notations x and y are the mole fractions and m shows the stage number. As it is seen these equations can be written for all stages between feed tray and bottom of the tower and it means that the differences are constant. A similar equation can be added for heat balance and in that case the difference between energies is constant as well. The important subject is that in liquid-liquid extraction and leaching process when the contact form is countercurrent such equations can be developed and using a difference point to carry out the process calculations graphically seems to be necessary. Such points are brought in Fig. 18 for different processes of distillation, leaching and liquid-liquid extraction. The method that used to find number of stages graphically was described completely in ref. (Treybal, 1955).
