The Equilibrium Curve

As mentioned above, the equilibrium curve in the case of membrane cascades is not at all a thermodynamic equilibrium but will represent the separation characteristics of the module or of the stage. This separation characteristic is in general a function of  

With respect to flow pattern, five different patterns are possible  

only the first case will be discussed. A detailed discussion of the other cases can be found elsewhere.10 11 

As Equation 30 indicates, the equilibrium curve in this case is influenced by the ideal separation factor and the pressure ratio r . 

For large pressure ratios, the equilibrium curve simplifies to  

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When the equilibrium curve is not straight, there is no strictly logical basis for the use of an overall transfer coefficient, since the value of m will be a function of position in the apparatus, as can be seen from Fig. 5-27. In such cases the rate of transfer must be calculated by solving for the interfacial compositions as described above. 

These equations are strictly valid only when m, the slope of the equilibrium curve, is constant, as noted previously. 

For the liquid-phase mass-transfer coefficient /cl, the effects of total system pressure can be ignored for all practical purposes. Thus, when using Kq and /cl for the design of gas absorbers or strippers, the primary pressure effects to consider will be those which affect the equilibrium curves and the values of m. If the pressure changes affect the hydrodynamics, then Icq, and a can all change significantly. 

It is sometimes permissible to assume constant relative volatility in order to approximate the equilibrium curve quickly. Then by applying Eq. (13-2) to components 1 and 2,... 

FIG. 13-34 Illustration of how equdihrium stages can he located on the x-q diagram through the alternating use of the equilibrium curve and the operating line. 

Once the design recovery for an absorber has been established, the operating curve can be constructed by first locating the point X9, y2 ou the diagram. The intersection of the horizontal hue corresponding to the inlet gas composition yi with the equilibrium curve y° = F x)... 

For stripping one begins by using the design specification to locate the point Xi, y. Then the iutersecfiou of the vertical huex = X9 with the equilibrium curve y° = F x) defines the theoretical minimum gas-to-liquid ratio. The actual value of is chosen to be about 20 to 50... 

Figure 14-4 shows the relationship between the operating curve and the equilibrium curve = F(x,) for a typical example involving solvent recovery, where yj and Xj are the interfacial compositions (assumed to be in equilibrium). Once y is known as a function of x along the operating curve, y can be found at corresponding points on the equilibrium curve by... 

For dilute systems in countercurrent absorption towers in which the equilibrium curve is a straight line (i.e., yj = mXi) the differential relation of Eq. (14-60) is formulated as... 

The end points of the operating line on an XY plot (Fig. 15-13) are X., Y, andXy, Y., and the number of theoretical stages can be stepped off graphically. The equilibrium curve is taken from the Hand type of correlation shown earlier (Fig. 15-9). When the equilibrium line is straight, its intercept is zero, and the operating line is straight, the number of theoretical stages can be calculated with one of the Kremser equations [Eqs. (l5-14) and (15-15)]. When the intercept of the eqnihbrinm line is not zero, the value of YJK, should be used... 

Detect pinched regions where the operating lines approach the equilibrium curve. 

Identify mislocated feed points where the feed, for binary mixtures, is not where the q-line intersects the equilibrium curve. This is not necessarily the case for multicomponent feed, however. 

Identify excessive reflux and reboil when there is too wide a gap between both operating lines and the equilibrium curve. 

Identify cases where feed or intermediate heat exchangers would be helpful, when only one of the operating lines is too far from the equilibrium curve. A large gap for the bottom section could indicate potential for feed preheat or an interreboiler. A large gap for the top section could portend precooler or intercondenser need. 

An important system in distillation is an azeotropic mixture. An azeotrope is a liquid mixture which when vaporized, produces the same composition as the liquid. The VLE plots illustrated in Figure 11 show two different azeotropic systems one with a minimum boiling point and one with a maximum boiling point. In both plots, the equilibrium curves cross the diagonal lines. 

These are azeotropic points where the azeotropes occur. In other words, azeotropic systems give rise to VLE plots where the equilibrium curves crosses the diagonals. Both plots are however, obtained from homogenous azeotropic systems. An azeotrope that contains one liquid phase in contact with vapor is called a homogenous azeotrope. A homogenous azeotrope carmot be separated by conventional distillation. However, vacuum distillation may be used as the lower pressures can shift the azeotropic point. Alternatively, an additional substance may added to shift the azeotropic point to a more favorable position. When this additional component appears in appreciable amounts at the top of the column, the operation is referred to as an azeotropic distillation. When the additional component appears mostly at the bottom of the column, the operation is called extractive distillation. 

The design of a distillation column is based on information derived from the VLE diagram describing the mixtures to be separated. The vapor-liquid equilibrium characteristics are indicated by the characteristic shapes of the equilibrium curves. This is what determines the number of stages, and hence the number of trays needed for a separation. Although column designs are often proprietary, the classical method of McCabe-Thiele for binary columns is instructive on the principles of design. 

Lj and are the pure liquid and inert gas loading rates, respectively, in units of Ib-moles/hr-ft. The second expression is the operating line on an equilibrium diagram. In all scrubbing application, where the transfer of solute is from the gas to the liquid, the operating line will lie above the equilibrium curve. When the mass transfer is from the liquid to the gas phase, the operating line will lie below the equilibrium curve. The latter case is known as stripping . 

Note that this equation holds for any component in a multi-component mixture. The integral on the right-hand side can only be evaluated if the vapor mole fraction y is known as a function of the mole fraction Xr in the still. Assuming phase equilibrium between liquid and vapor in the still, the vapor mole fraction y x ) is defined by the equilibrium curve. Agitation of the liquid in tire still and low boilup rates tend to improve the validity of this assumption. 

This takes the place of drawing the equilibrium curve and solving graphically, and is only necessary since the q is not 1.0 or zero. 

For a binary mixture the values of x and y can be obtained from the equilibrium curve. Select values of x and read the corresponding value of y from the equilibrium curve. Tabulate values of l/(y - x), and plot versus X], resulting in a graphical integration of the function dx (y - x) [130] between x and xj. This system would have no column internals and no reflux. 

The area under the curve between Xgo and is the value of the integrtil. Plot the equilibrium curve for the more volatile component on x - y diagram as shown in Figure 8-33. Then, select values of xd from the operating line hav ing the constant slope, L/V, from equation... 

Note that yo = xp on the diagonal of the equilibrium plot, and 7i and xj are points of intersection with the equilibrium curve. For an abnormal equilibrium curve (as compared to regular or normal shape) see Figure 8-34. 

To calculate the equilibrium curve at 90°F (constant temperature) for the system aqua ammonia-ammonia-inert vapors follow the steps listed ... 

To calculate the equilibrium curve taking the heat of solution into accoimt, i.e., operate adiabatically with liquid temperature variable, follow the steps ... 

Bisecting the vertical distance (line C) of Figure 9-77 between the 8% aqua operating line and the equilibrium curve B. 

Point (e) is the intersection of the equilibrium curve and the operating line and represents the equilibrium condition for tower oudet liquor, and the maximum liquor concentration. 

Note that the 10% (wt) aqua product operating line is shown on the diagram, but such a concentration cannot be reached when operating at 150 psig. A greater pressure is required in order to lower the equilibrium curve. 

If over a single stage, the equilibrium curve and the operating lines are assumed straight, then [127] ... 

Consider the equilibrium curve for AgCl shown below. Which of the following statements about a solution at point A on the curve are true ... 

Example. — Ice and water-vapour are in equilibrium at + 0 0077° C., under a pressure of 4 57 mm., and liquid water is in equilibrium with water-vapour at the same temperature and pressure ice, liquid water, and water-vapour are therefore in equilibrium under these conditions, and the equilibrium curves representing pressures as functions of temperature meet at a triple point (0 0077° C., 4 57 mm.). These curves are (Fig. 47) ... 

Figure 8.18 (Liquid + liquid) equilibria at p = 0.1 MPa for (.vin-QHu + Y2CH3OH). The intersections of the tie-lines with the equilibrium curve give the compositions of the phases in equilibrium,...

Figure 8.23. The solid and monotonically declining line to the right represents the equilibrium curve. The four curves represent lines with constant rates in the same plot. Since we want to operate at points where of maximum ammonia concentration, the optimal operation line is defined as the line running parallel to the equilibrium curve, passing through all the maxima. [Adapted from I. Dybkjaer, in Ammonia Catalysis and Manufacture (1995) Ed. A. Nielsen. Springer-Verlag, Berlin/Heidelberg, p. 199.]...

Figure 8.24. Left schematic diagram of an adiabatical three-bed, indirectly cooled reactor with two heat exchangers. Right a diagram showing the equilibrium curve to the upper right, the optimal operating line and the operation line for the reactor are to the left. [Adapted from C.J.H. Jacobsen, S. Dahl, A. Boisen, B.S. Clausen,...

The equilibrium curve and the optimal operation line are again plotted in an ammonia concentration versus temperature plot for each of the two sets of conditions in Fig. 8.26, but now together with the optimal catalyst curves for a few selected nitrogen bonding energies. The right-hand panel also shows the operating line, and it is now possible to estimate which catalyst should be where in the reactor. 

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