Stage calculations

The solution to a multi-component, multi-phase, multi-stage separation problem is found in the simultaneous or iterative solution of the material balances, the energy balance and the phase equilibrium equations (see Chapter 1). This implies that a sufficient number of design variables are specified so that the number of remaining unknown variables exactly equals the number of independent equations. When this is done, a separation process is said to be specified. 

The total input of material to stage n is Ln+ + V - units (mass or moles) per time and the total output is Ln + Vn units per time. Since, under steady flow, there is neither accumulation nor depletion, the input and the output are equal and the material balance can be written as  

This equation is the total material balance for the stage. Another balance can be written by equating input to output for component i giving the component i balance  

Obviously, if all the component balances for a system were added up, the sum would be equal to the total material balance. 

In most equilibrium stage processes, the general energy balance can be simplified by neglecting potential energy and kinetic energy. If, in addition, the process is workless and adiabatic, a simple enthalpy balance applies  

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Stage compositions in the TG method are obtained by stage-to-stage calculations from both ends toward the feed stage. With reference to Fig. 13-1, the calculations work with the ratios v /d, Jd, v /h, and instead of v or f directly. 

Stage-to-stage calculations shown in Tables 13-10 and 13-11 provide f/d and /h values for each stage. These are used in the following equations to calculate normahzed liquid concentrations for each component at each stage ... 

For the first time through a liqmd-liquid extrac tion problem, the right-triangular graphical method may be preferred because it is completely rigorous for a ternary system and reasonably easy to understand. However, the shortcut methods with the Bancroft coordinates and the Kremser equations become valuable time-savers for repetitive calculations and for data reduction from experimental runs. The calculation of pseudo inlet compositions and the use of the McCabe-Thiele type of stage calculations lend themselves readily to programmable calculator or computer routines with a simple correlation of equilibrium data. 

With the shaft speed and the tip speed calculated in Step 9 for the Section 2 stages, calculate an impeller diameter using Equation 5.15. 

Stage Calculated/chosen Equation used Remarks... 

Approximate number of stages = Hp calculated in (5) divide by H/stage calculated previously = No. stages... 

The problems relating to mass transfer may be elucidated out by two clear-cut yet different methods one using the concept of equilibrium stages, and the other built on diffusional rate processes. The selection of a method depends on the type of device in which the operation is performed. Distillation (and sometimes also liquid extraction) are carried out in equipment such as mixer settler trains, diffusion batteries, or plate towers which contain a series of discrete processing units, and problems in these spheres are usually solved by equilibrium-stage calculation. Gas absorption and other operations which are performed in packed towers and similar devices are usually dealt with utilizing the concept of a diffusional process. All mass transfer calculations, however, involve a knowledge of the equilibrium relationships between phases. 

This example illustrates the complexity and trial and error nature of stage-by-stage calculation. 

For a specification of not more than 1 mol of the light key in the bottom product and not more than 1 mol of the heavy key in the top product, and a reflux ratio of 2.5, make a stage-by-stage calculation to determine the product composition and number of stages required. 

It is necessary to estimate the composition of the non-keys so that they can be included in the stage calculations. As a first trial the following values will be assumed ... 

In each stage calculation it will necessary to estimate the stage temperatures to determine the K values and liquid and vapour enthalpies. The temperature range from top to bottom of the column will be approximately 120 — 60 = 60°C. An approximate calculation (Example 11.7) has shown that around fourteen ideal stages will be needed so the temperature change from stage to stage can be expected to be around 4 to 5°C. 

Fig. 6.1. A cartoon depiction of the phase space T and important phase space regions and their relationship. The important phase space regions of systems 0 (To ) and 1 (T) ) are abstractly represented by shaded and open oval shapes, respectively. These regions can be related in four ways (a) subset, (b) coincidence, (c) partial overlap, and (d) no overlap. Also sketched is the important phase space region of the intermediate M in a two-stage calculation (see section on Multiple-Stage Design ). The appropriate staging strategy differs according to the different overlap relationships between F0 and 77...

Note that for the cases Fig. 6. lc, d, depending on the difference between the 0 and 1 systems, it may be better to go with higher-order stage calculations. For example, for case (d) one can construct an intermediate MB bridging r0 and r, so that the relationship of Tg and r B r are both described by case (c), and then perform overlap sampling calculations between 0 and MB as well as 1 and MB. 

The same analysis can be repeated for precision. From the analysis in Sect. 6.4.2, we have (for a single-stage calculation)... 

Equilibrium segregation coefficients, for silicon impurities, 22 493t Equilibrium stages, calculation of, 10 756-759... 

A small subsection of the whole system (consisting of units 1 to 6) to show stage-by-stage calculations... 

Of course, the dose of a new drug is often not well defined early in the development process, so at this stage calculating... 

As the name implies, stage-by-stage calculations are useful for calculating from a composition and set of flows in one stage to the composition and flows in an adjoining stage. By successive repetition of the procedure it is possible to calculate from one end of a column of stages to the other end. 

The general method of stage-by-stage calculation for multicomponent systems was first shown by Lewis and Matheson (LI) and by Underwood (Ul) in 1932. The method of Lewis and Matheson was further improved by Robinson and Gilliland (Rl), but substantially unchanged. In its most basic form, the concept of the method is simple. Consider the example cited above. If the amounts of each component in both of the products could be exactly calculated, it would only be necessary to start at one end and calculate until a stage was reached at which the composition matched that of the other product. 

A wet gas reservoir produces 170,516 scf/STB of 0.646 specific gravity separator gas and 64.7°API stock-tank liquid. The separator operates at 555 psia and 89°F. The separator system is two-stage. Calculate the specific gravity of the reservoir gas. 

If in doubt regarding which tank should be used as the first stage, calculate the conversions for both configurations and compare. Note that accurate calculations are required in order to distinguish between the two.)... 

Figure 12. Effect of mean residence time and monomer concentration on steady-state particle number when a plug flow-type prereactor is used for the first stage (calculation conditions SF = 6.25 g/L H20 lF = 1.25 g/L H20 SCmc = 0.50...

Actual number of theoretical stages calculated for input actual reflux ratio of 2.0 = 9.60 (add two more stages for condenser and reboiler, or 12 theoretical total)... 

Examples of stage calculations for the most common chemical engineering separation methods are given later in this chapter. 

The equilibrium stage calculations discussed earlier can easily be applied to distillation in order to find the required operating and design parameters. Consider the following problem specification. 

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