Distillation material balance relations

The use of a y-x curve in the McCabe-Thiele method for binary distillation calculations brings up the matter of a flash-vaporization representation, in case the feedstream mixture is at saturation. An inspection of the y-x curve relative to a given feed composition shows that the equilibrium mixture varies along the curve over a range from the bubble point (where the liquid phase composition x is equal to that of the feed mixture x ) to the dew point (where the vapor composition y is equal to that of the feed mixture Xj.). Between the two is the region of flash vaporization, where the equilibrium compositions (y, x) respectively of phases V and L must satisfy the flash material balance relation F = V + L, where... 

Parameter Estimation Relational and physical models require adjustable parameters to match the predicted output (e.g., distillate composition, tower profiles, and reactor conversions) to the operating specifications (e.g., distillation material and energy balance) and the unit input, feed compositions, conditions, and flows. The physical-model adjustable parameters bear a loose tie to theory with the limitations discussed in previous sections. The relational models have no tie to theory or the internal equipment processes. The purpose of this interpretation procedure is to develop estimates for these parameters. It is these parameters hnked with the model that provide a mathematical representation of the unit that can be used in fault detection, control, and design. 

We are going to discuss a strategy of analysis of material balance problems that will enable you to understand, first, how similar these problems are, and second, how to solve them in the most expeditious manner. For some types of problems the method of approach is relatively simple and for others it is more complicated, but the important point is to regard problems in distillation, crystallization, evaporation, combustion, mixing, gas absorption, or drying not as being different from each other but as being related from the viewpoint of how to proceed to solve them. 

To accomplish this analysis, an overall material balance is written for the condenser as V = L + D, which relates the vapor (V) leaving the top stage, the liquid reflux returning (L) to the column from the condenser (reflux), and the distillate (D) collected. A material balance for component A is written as... 

The material balance equations, or operating lines (Equations 5.13 and 5.14) together with the feed stage heat balances (Equations 5.16 and 5.17), and the equilibrium relation, (Equation 5.12) form the basis for the McCabe-Thiele (McCabe and Thiele, 1925) graphical solution ofbinary distillation on a Y-X diagram. 

Equation 12.9 relates the bottoms composition, X to the distillate composition, E, at total reflux. The derivation of this equation on the basis of material balances and equilibrium relationships is equivalent to the graphical solution for binary mixtures described in Chapter 5. However, in the binary graphical solution, the material balance is represented by the operating curves, not necessarily at total reflux, while in the present derivation the material balance equation is obtained on the basis of total reflux. 

For the countercurrent contact with multiple stages in Fig. 10.3-2, the material-balance or operating-line equation (10.3-13) was derived which relates the concentrations of the vapor and liquid streams passing each other in each stage. In a distillation column the stages (referred to as sieve plates or trays) in a distillation tower are arranged vertically, as shown schematically in Fig. 11.4-1. 

Rigorous models of staged distillation processes are formulated by setting up material balance equations, equilibrium relations, summation equations, and enthalpy balance equations (MESH equations). In these models, the extent of nonlinearity may be very severe, particularly for azeotropic and reactive distillation systems. MESH system based mathematical models can thus yield multiple solutions (multiple steady states), a fact which has been observed by many researchers. 

Continuous Simple Distillation. Distillations that approximate this type are usually carried out on a continuous basis such that the liquid feed is added continuously to a well-mixed still in which a definite fraction is vaporized and removed and the excess unvaporized liquid is withdrawn from the still. An alternate arrangement is to preheat the feed and add it to a flash or disengaging section where vapor and liquid are separated and removed without additional heat requirements. In either case, assuming that the vapor and liquid leaving are in equilibrium with each other, the two fractions are related to each other by equilibrium constants and material balances. Thus for each component in a mixture the following material balance can be written ... 

Rayleigh equation An equation that relates the quantity to the concentration of the more volatile component in batch distillation. The equation is formulated from an unsteady-state material balance in which vapour leaving the still is in equilibrium wrtth the liquid. As the vapour is richer in the more volatile component, the composition of the Rquid and vapour are not constant ... 

The calculational base consists of equilibrium relations and material and energy balances. Equilibrium data for many binary systems are available as tabulations of x vs. y at constant temperature or pressure or in graphical form as on Figure 13.4. Often they can be extended to other pressures or temperatures or expressed in mathematical form as explained in Section 13.1. Sources of equilibrium data are listed in the references. Graphical calculation of distillation problems often is the most convenient... 

In the usual distillation problem, the operating pressure, the feed composition and thermal condition, and the desired product compositions are specified. Then the relations between the reflux rates and the number of trays above and below the feed can be found by solution of the material and energy balance equations together with a vapor-liquid equilibrium relation, which may be written in the general form... 

---