Minimum number of stages

The calculation can also be performed using the Kremser equation  

Rounding up, it is again determined that four equilibrium stages are needed. 

Another important concept is the calculation of the minimum number of equilibrium stages to perform a given separation. This is a theoretical concept since it occurs when one or both flowrates in a countercurrent cascade are infinite. It is useful to calculate since the actual number of stages will be larger (i.e., this is a limit). An analytical equation can be 

From the definition of a separation factor (Equation (2.9)), one can write 

The case of minimum number of equilibrium stages corresponds to infinite flow from and back into the cascade, such that no product is removed. In this case 

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The minimum reflux ratio (L/D)min been determined to be 1.017. Using the Brown and Martin graph [9], evaluate the theoretical number of trays at an operating reflux of 1.5 times the minimum. The minimum number of stages was determined to be 22.1 including the reboiler. See Figure 8-49. 

Nm = minimum number of stages at total reflux, including the reboiler, a, = average relative volatility of the component i with respect to the reference component. 

Minimum number of stages Fenske equation, equation 11.58 ... 

Having obtained the minimum number of stages from the Fenske Equation and minimum reflux ratio from the Underwood Equations, the empirical relationship of Gilliland10 can be used to determine the number of stages. The original correlation was presented in graphical form10. Two parameters (X and Y) were used to correlate the data ... 

A practical isotope separation plant can operate at neither minimum reflux (where the separation is zero, but the rate of production is high), nor at minimum number of stages (where the rate of production is zero, but the separation is high). A compromise is required. Since optimum reflux varies with stage number it is customary to employ tapered cascades for isotope separation. This results in marked savings in material hold-up, and in plant size and investment. 

Figure 14.7. Locations of operating points P and Q for feasible, total, and minimum extract reflux on triangular diagrams, and stage requirements determined on rectangular distribution diagrams, (a) Stages required with feasible extract reflux, (b) Operation at total reflux and minimum number of stages, (c) Operation at minimum reflux and infinite stages.

At minimum flows, pinch zones can be identified where the composition changes very little on successive stages. The profiles just intersect each other at the feed position. In Figure 3.23 (right-hand) a saddle pinch can be identified in the rectifying profile. This profile corresponds to the minimum reflux in a direct sequence, since the distillate is practically the pure component A. The minimum reflux is practically independent of the bottoms composition. Similarly, a minimum reboil ratio may be identified by simulation. The determination of the minimum number of stages is subtler, since the composition of the top distillate and bottoms are not independent, but the simulation may produce a reasonable estimation. 

The condition where the reflux and stripping ratios approach infinity is termed total reflux. No feed enters the column and no product leaves. Both component balance lines coincide with the 45° diagonal line and are therefore furthest away from the equilibrium curve. Total reflux sets the minimum number of stages required for the separation. For Example 2,1, Fig. 3,lid shows that the minimum number of stages required for the separation is 6. 

Determine the minimum reflux and minimum number of stages. 

The minimum number of stages is given by Fenske s equation (25)... 

Example 3.1 Evaluate the minimum number of stages for the depropanizer in Example 2.4 using each of the methods described above. Use the following data ... 

Assuming 20 theoretical stages (Example 3.4), and 12.4 to be the minimum number of Stages (Example 3.1),... 

Nm = minimum number of stages at total reflux, including the reboiler,... 

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