Material balance lever rules

Himmelbau (1995) or any of the general texts on material and energy balances listed at the end of Chapter 2. The Ponchon-Savarit graphical method used in the design of distillation columns, described in Volume 2, Chapter 11, is a further example of the application of the lever rule, and the use of enthalpy-concentration diagrams. 

All the methods presented above are based on the same principle i.e., the material balance using the lever rule. Therefore these are not limited to equilateral triangles, but equally valid for scalene triangles which frequently appear when dealing with subsystems. 

Since the point M lies in the two-phase region of the triangular diagram, the term mixture applies only on a scale larger than the size of the droplets formed. The droplet dispersion formed by agitation has sufficient interfacial area (see Section I.C) for equilibrium to be reached quickly, so that point M represents the mean of the extract composition (point E) and the raffinate composition (point R) which are connected by the appropriate tie-line. A further application of the inverse lever rule permits calculation of the relative amounts of extract and raffinate. In this example, the material balance based on 1 kg of feed is summarized as follows ... 

Graphical solutions to material balance problems involving equilibrium relationships offer the advantages of speed and convenience. Fundamental to all graphical methods is the so-called inverse lever rule, which is derived in Example 3.3 and applied in Example 3.4. 

This allows point M2 to be located by the lever rule. From Lw, a line through M2 locates V 2 with the terminal composition yn2 = 0-14. We can solve for Lw and V 2 by the lever rule or by material balances. 

If we stop the expansion momentarily at 10 bar (point C), we can use the diagram to obtain the compositions and relative amounts in the two phases. Construct the horizontal through C, then and are given by the intersections of the horizontal (a tie line) with the two-phase curves. These intersections are marked with triangles we find Xi = 0.449 and y = 0.937. On an isothermal Pxy diagram, tie lines are horizontal because at equilibrium both phases have the same pressure. The fraction of the system in the vapor phase can be determined by material balance (a lever rule) ... 

If til kmol of the ternary mixture at point B is added to /I2 kmol of the ternary mixture at point C, the composition of the new mixture lies on the line BC. This is shown by means of a material balance. The location of the mixing point M follows the lever, or mixture, rule such that... 

These definitions present convenient and simplifying properties ( ) the dimensions of the system are reduced, simplifying the depiction of equilibrium (figure 3.3) (Frey and Stichlmair, 19996 Barbosa and Doherty, 1987a) (ii) they have the same numerical values before and after reaction (m) they sum up to unity (iv) they clearly indicate the presence of reactive azeotropes when Xi =Yf, (v) the nonreactive hmits are well defined (vi) the number of linear independent transformed composition variables coincides with the number of independent variables that describe the chemical equihbrium problem and vii) the lever rule is valid as the chemical reaction no longer impacts the material balance. 

Assuming that the composition of the top vapor stream reaches the ternary azeotrope (with the minimum temperature of the whole system), the compositions of both the AO and the OR can be estimated. With the known fresh-feed flowrate and the lever rule, the flowrate of the OR and the makeup flowrate can be estimated by the inner and outer material balance envelopes, respectively. Second, from the tie-line information and the lever rule, the flowrate of the AO stream can also be estimated. Thirdly, by using the outer material balance envelop and the lever mle, the bottom flowrate can further be estimated. Finally by using the inner material balance envelop and the lever mle, the top vapor flowrate can be estimated. 

From Figure 8.14 one would immediately see that, because the fresh feed is more diluted, the material balance line drawn from the point of FF + D2 to the point of OR is very close to the point of ternary azeotrope. From the lever rule, the top vapor flowrate can be estimated to be much greater than the flowrate of the bottom IPA product, thus requiring a large recycle flowrate in the overall system (large OR and large AO as feed to the recovery column). 

In addition, we can determine the proportion of total moles in the vapor to the total moles in the liquid from the phase diagram. To find the relative amounts, we use the lever rule, which says the ratio of the amount of vapor in the system to the amount of liquid is given by the ratio of the line segments from the feed composition to the opposite curve, as shown in both phase diagrams the figure. The lever rule is a graphical result of applying overall and species material balances, as shown in Example 8.4. 

Apply the appropriate material balance equations to verify that the lever rule gives the relative amount of species in each phase along a tie line as depicted in Figure 8.2. 

The mass balances that lead to Equations (E8.4D) and (E8.4E) are general and not limited to the vapor and liquid phases thus, the lever rule can be applied to find the relative amounts of any two phases in equilibrium. The fraction of material present in one phase can be computed by taking the length of the tie line from the overall composition to the composition of the other phase and then dividing by the total length of the line. 

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