Operating Line and Equilibrium Curve

A large majority of the systems have operating lines and equilibrium curves which can be assumed as straight over the range covered by the design problem. For the conditions of a straight line equilibrium curve, y = mx, Colburn [10, 11] has integrated the relation above to obtain ... 

In some of the homework problems, the distance between the operating line and equilibrium curve becomes very small near the point of their intersection. When this happens, a very large number of steps might be required to make any progress on separation. 

Rule an infinite number of steps is required to get to an intersection of the operating line and equilibrium curve. 

Over a narrow range, any continuous curve becomes virtually linear. When both the operating line and equilibrium curve are straight lines, an analytic expression exists for calculating the number of steps ... 

Comment 2 If the operating line and equilibrium curve should cross (which is physically impossible), when evaluating (70) you will find yourself trying evaluate the log of a negative number. 

Comment 3 The situation in which the operating line and equilibrium curve are parallel (i.e. A = 1) is a special case (70) becomes indeterminant. However, since the driving force y -y is constant, the number of steps is just the total change in concentration divided by the driving force ... 

Once the operating lines are plotted, the equilibrium stages are stepped off between the operating line and equilibrium curve as described in Section 5.1.3. The optimum feed tray location is at the step that crosses from one operating line to the other over their intersection point. If the number of stages is fixed, the operating lines are determined by trial and error to match the specified number of stages. 

Nonparallel, Straight Operating Line, and Equilibrium Curve... 

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 15.3 Two-film mass transfer model, (a) operating line and equilibrium curve, (b)... 

Equations 15.30 and 15.31 are derived based on the assumption of constant operating line and equilibrium curve slopes (Vital et al., 1984). 

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. 

When the Murphree efficiency is constant for all trays, and under conditions such that the operating line and equilibrium curve are straight, the overall tray efficiency can be computed and the number of real trays can be determined analytically from... 

Calculate Rmin for intersection of top operating fne and vertical feed fne. This assumes there is no pinch point or intersection between the top operating line and equilibrium curve. 

The operating line is located by trial and error until five stages just fit between operating line and equilibrium curve. From the figure, yi = 0.079 lb. acetaldehyde/lb. water (corresponding to 7.33% acetaldehyde by weight). The slope of the operating line ... 

Operating lines and equilibrium curves are plotted on Figs. 7.22 and 7.23, stages are stepped oft, and the concentration x plotted against stage number on Fig. 7.24. Match-... 

The operating line and equilibrium curve are so close that cr and cr are almost identical. 

The minimum solvent-to-feed ratio is determined by the intercept of the operating line and equilibrium curve (dot-dashed line, Xextract.out = 0.53). Here we obtain a flow rate of 0.34 kmol solvent h and a minimum solvent-to-feed ratio of0.0034. 

FIGURE 15.1 Simple absorber straight, parallel, operating line, and equilibrium curve. 

Eor straight and parallel operating line and equilibrium curve. 

Figure 15.2 is a 7- Zdiagram for a simple absorber column section, similar to the process described above, but with nonparallel, straight operating line and equilibrium curve. From the geometry of the diagram, the following relationships may be written ... 

In general, the equilibrium curves calculated from the two curves do not match exactly, and each is used for its respective portion of the column. Plates are stepped off between the operating lines and equilibrium curves in the usual manner. 

We consider here the case where both the operating line and equilibrium curves are linear, tire latter represented by Henry s law ... 

In an attempt to obtain more consistency with mass transfer theory, the transfer unit concept has been proposed. The number of gas-phase transfer units (Nqg) be calculated by the Souders and Brown procedure using the absorption factor [11]. This equation assumes linear, although not parallel, operating line and equilibrium curve. 

Equation 4-9 assumes a straight operating line and equilibrium curve. This is valid for systems where the solute mass transferred is small compared with the liquid and gas mass flows, and the value of H or K is independent of solute concentration in the liquid phase. 

---