Equilibrium curve graphical determination

The equilibrium curve also can be transformed into these coordinates. These transformations are useful for graphical determinations of numbers of theoretical trays rather than for determination of numbers of transfer units. Example 13.13 employs both sets of units. 

The procedure for finding minimum raflux in each section of a complex fractionator is similar. First, the point of intersection of the component balance line with the 45° diagonal is found by using Eq. (2.38). Second, the point of intersection of the q-line for the feed or side product with the equilibrium curve is graphically determined. A straight line is then drawn between the two points. This line is the minimum reflux component balance line. The minimum reflux is found from the intercept of this line on they axis, using Eq. (2.40). 

The NTU values that characterise the concentration profile can be graphically determined if the operation line is parallel to the equilibrium curve in the loading diagram. In this case NTUp, = NTUg and reflects the theoretical stage number n,. A deviation of the ideal plug flow in the continuous and dispersed phase occurs for the following reasons ... 

As shown in Figure 27, plots of the equilibrium data, the y — x line, and the operating line allow a procedure to calculate the unknown variables for a system, typically, the outlet liquid and vapor compositions. The y = x line simplifies the graphical solution method and intersects the operating line at the feed composition, z. Thus, at this point, y = x = z. The unknown compositions, in the vapor and liquid product streams, are determined by the intersection of the operating line and the equilibrium curve. 

The McCabe-Thiele graphical method is applied using the Y-X diagram. The equilibrium curve is first plotted from the given data (Figure 6.3). A temperature coordinate is also included in the diagram (nonlinear) to determine the condenser, reboiler, and tray temperatures. 

In general, when the operating line and equilibrium curve are not straight, the NTU must be determined by numerical or graphical integration of Equation 15.3 or 15.5 or other equivalent form of these equations. 

Figure 13.11 shows a y-x diagram with the isothermal equilibrium lines at 25 and 31°C. Also shown is the approximate actual equilibrium curve location, taking into account heat effects in the column. For the linear equilibrium line, stages are determined by Equation (13.9) or by stepping graphically. For the curved (approximate actual) line, stages are stepped off as shown in Figure 13.11 ...

The operating lines, die q line, and die equilibrium curve [see Eq. (5.2-J 3)] are all y-x functions where, by convention, die mole fractions refer to the lighter, or mote volatile, component of the binary pair. It was on this haras that McCabe and Thiele1 developed their graphical approach to binary distillation stage determination and their approach is summarized in Fig. 5.3-6. 

Operating Line and "Equilibrium" Curve. Both terms are of importance for the graphical solution of a separation problem, i.e., for the graphical determination of the number of stages of a cascade. This method has been developed for the design of distillation columns by MacCabe and Thiele and should be well known. For all cases, the operating line represents the mass and material balances. In distillation, the equilibrium curve represents the thermodynamical va-por/liquid equilibrium. For an ideal binary system, the equilibrium curve can be calculated from Raoult s law and the saturation-pressure curves of the pure components of the mixture. In all other cases, however, for example, for all membrane processes, the equilibrium curve does not represent a thermodynamical equilibrium at all but will represent the separation characteristics of the module or that of the stage. 

For most cases, because of either a curved operating line or equilibrium curve, the relation between number of stages, compositions, and flow ratio must be determined graphically, as shown. For the special case where both are straight, however, with the equilibrium curve continuing straight to the origin (Yi - mX(), an analytical... 

Determine the number of theoretical stages by as the number of segments of a stepwise line that can be traced between the operating lines and the equilibrium curve starting from the top or from the bottom of the column. The graphical construction allows also the identification of the feed plate. 

For binary mixtures, percent vaporization or condensation is conveniently determined from graphical construction by methods similar to those of Section 3.5. Figure 7.2 shows the equilibrium curve and the y=x (45° line) curve for n-hexane in a mixture with n-octane at 101 kPa. Operating lines representing various percentages of feed vaporized are obtained by combining (7-4) and (7-5) to eliminate L to give for n-hexane (the more volatile component)... 

Mention should also be made of the graphical determination of equilibrium curves, which, however, is far more cumbersome than the arithmetic method. DetaUs may be found in Thormann [72] and Rosengart [84]. 

Fig. 1-57. Schematic of a countercurrent flow column with mass transfer from phase II to phase I (a) and an operating diagram to determine graphically the number of theoretical separation stages (b). EC Equilibrium curve...

If the balance line and equilibrium curve are parallel, NTUg, and NTU g are identical. NTUg, and NTUgg are then equal to the number of theoretical stages determined using a graphical method. 

Introduction and assumptions. A mathematical-graphical method for determining the number of theoretical trays or stages needed for a given separation of a binary mixture of A and B has been developed by McCabe and Thiele. The method uses material balances around certain parts of the tower, which give operating lines somewhat similar to Eq. (10.3-13), and the xy equilibrium curve for the system. 

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