McCabe-Thiele method operating line

The McCabe-Thiele method employs the simplifying assumption that the molal overflows in the stripping and the rectification sections are constant. This assumption reduces the rectifying and stripping operating line equations to ... 

Operating Lines The McCabe-Thiele method is based upon representation of the material-balance equations as operating lines on the y-x diagram. The lines are made straight (and the need for the energy balance obviated) by the assumption of constant molar overflow. The liqmd-phase flow rate is assumed to be constant from tray to tray in each sec tiou of the column between addition (feed) and withdrawal (produc t) points. If the liquid rate is constant, the vapor rate must also be constant. 

The McCabe-Thiele method can be used for the design of columns with side streams and multiple feeds. The liquid and vapour flows in the sections between the feed and take-off points are calculated and operating lines drawn for each section. 

The plot at right shows points taken from the operating lines in the rigorous ChemSep separation and compares them with the ROL and SOL calculated using the McCabe-Thiele method. These calculated operating lines are the straight, solid lines on the plot. 

The McCabe-Thiele method will be used. The side draw is more concentrated in acetone than either feed and should therefore be located above the feeds. Calculate the internal liquid and vapor flow rates and the slope of the operating line between the condenser and the side draw ... 

The McCabe-Thiele method uses a graphical approach, where the equilibrium data, the operating lines, and the stepping of stages are done graphically. In this example, where the relative volatility is assumed constant, a numerical approach may be implemented. 

The McCabe-Thiele method hinges upon the fact that, as an approximation, the operating lines on the xy diagram can be considered straight for each section of a fractionator between points of addition or withdrawal of streams. This is true only if the total molar flow rates of liquid and vapor do not vary from stage to stage in each section of the column. This is the case if ... 

Total reflux corresponds to a situation where there is no feed, distillate, or bottoms and where the minimum number of stages is required to achieve a desired separation. It will be recalled that in the McCabe-Thiele method this situation corresponded to having the operating line coincide with the y = x line. 

This linear equation relates the composition of the vapour to the composition of the liquid at any level in the column (on the assumption of adiabatic operation) and rejjresents the operating line a — c (Fig. 59). If the molar heats of vaporization differ markedly this has to be taken into account when using the McCabe-Thiele method. 

Figure 2.3 McCabe—Thiele method draw operating lines.

Equilibrium data must be converted to ratio units, Y vs. X. These values can be greater than 1.0, since Y = y/(l - y) and X = x/(l - x). The Y = X line has no significance in absorption. As usual the stages are counted at the equilibrium curve. A minimum S/G ratio can be defined as shown in Figure 12-9. If the system is not isothermal, the operating line will not be affected, but the equilibrium line will be. Then the McCabe-Thiele method must be modified to include changing equilibrium curves. 

Minimum reflux, by definition, is that condition whereby a zone of an infinite number of stages exists immediately on each side of the feed location. In the McCabe-Thiele method of graphical representation and calculation, both operating lines intersect with the locus of the feed partitioning, all at a point on the fC-value plot (or equilibrium curve). It is therefore impossible to calculate away from this point of intersection, in either direction. 

This simple algorithm shows that in stagewise calculations alternate use is made of equilibrium data and mass balance calculations, which in this case are represented by the operating line equations. Although distillation is a thermal process, the McCabe-Thiele method sidesteps enthalpy balances. The key assumption is the one of equal molal latent heats of vaporisation. If this assumption is invalid it is necessary to introduce enthalpy balances in order to calculate the liquid and vapour flow rates. 

And we define a function to optimize including the McCabe-Thiele method for the number of trays assuming constant relative volatility. We compute the operating lines as a function of the compositions of the feed, the distillate, and the residue. Next, we use a while statement to compute the number of theoretical trays. Finally, we compute the cost of the vessel and that of the energy required at the boiler. 

The McCabe-Thiele method assumes that the liquid on each tray and the vapor above it are in equilibrium, and that between the trays the molar composition of the down-flowing liquid and the up-flowing vapor are related by the operation lines, Eq. (3.3.16) for the rectification section and Eq. (3.3.18) for the stripping section (Figure 3.3.13). 

To design such a process, the McCabe-Thiele method may be used to determine the number of theoretical separation stages, as examined in Sections 3.3.2-3.3.4 for distillation, absorption (gas scrubbing), and liquid-liquid-extraction. Thus, we obtain the number of theoretical extraction stages of a countercurrent extraction column based on the equilibrium curve (solubility of extract in the solvent for a given content in the solid) and the operating line. The latter depends on the extract content of the solid feed and residue, and on the in- and outlet extract concentration in the solvent The extract content of the feed is fixed, and the value of the residue is specified by the required degree of extraction. The inlet content of the extract in the solvent is also fixed, as either pure solvent is used or the value is specified by separation of the extract from the used solvent after the extraction. Therefore, the only parameter that is left is the outlet concentration of the extract in the solvent, which depends on the ratio of the solvent flow to the feed rate of the solid feedstock (mass balance). 

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