Cascade material balances

The total flow in the cascade is then given by the sum of equations 32 and 33, which can be simplified with the use of the cascade material balances ... 

We are now in a position to incorporate material balance into the synthesis procedure with the objective of allocating the pinch point as well as evaluating excess capacity of process MS As and load to be removed by external MSAs. These aspects ate assessed through the mass-exchange cascade diagram. 

Consider a cascade of n tanks in series. The material balance for A can be written as follows. 

The arguments presented above indicate that the large recycle and coolant flow rates wr and uq are the only manipulated inputs available in the fast time scale, and should be used to control the process temperatures. Likewise, the dynamics of the material-balance variables in the slow time scale are affected only by the small feed and effluent flow rates uq and up, which are thus the manipulated inputs that must be used to tackle control objectives involving the material balance. 6sp, the setpoints of the temperature controllers in the fast time scale, are also available as manipulated inputs in the slow time scale, a choice that leads to cascaded control configurations between the energy- and material-balance controllers. 

The models may be used to compare performance between several units or unit types. For example, a week of production data from the above-mentioned modified cascade unit was compared with simulated Stratco performance. The agreement between Amoco s Stratco simulation model predictions and material balance and performance predictions provided by the Stratford Engineering Corporation had been confirmed earlier. Model adjustments allowed a comparison at constant operating conditions to be made. Table I compares the performance of the modified cascade unit with the Stratco simulation model predictions. Acid consumption is 36% lower in the modified cascade unit. The Stratco unit does show a slight research octane advantage over the cascade. However, overall economics favor the modified cascade. in this case. 

In any separating element, stage, or cascade two material balance equations apply... 

The overall separation depends on the single stage separation factor, the number of separating elements, and design and operating characteristics of the cascade. Equations 5-8 are insufficient to determine all the variables. It is instructive to consider three types of cascades the minimum stage cascade, the minimum reflux cascade, and the ideal cascade. The material balance equations from the i + 1 th stage to the product of the cascade lead to... 

The performance of cascaded stages may also be described analytically on the basis of material balances and phase equilibrium relations. Consider, for simplicity, a binary mixture of components 1 and 2 and again assume component 1 to be the lower boiling, or more volatile, component. The equilibrium equations at stage j are... 

Liquid-liquid systems that obey Nemst s law can be particularly simple to analyze, since the equilibrium relationship is linear in composition. If the two liquid phases are nearly immiscible, and if the solute concentrations are low in either phase, then the operating or material balance line is also nearly linear. Writing a material balance around the j h stage and the raffinate end of the cascade in Figure 10.4 gives the operating line. 

For more efficient fractional extraction of two or more extractable components, the extracting-scrubbing cascade of Fig. 4.4 is employed. Nomenclature for flow rates, concentration, and stage number is shown in Fig. 4.14. With the same assumptions and approach as in Sec. 6.1, a material balance for any one of the components in the portion of the cascade below stage n in the extracting section is... 

However, an overall material balance for the extractable component written for the entire separation cascade of Fig. 4.14 is... 

In a recycle cascade such as Fig. 12.13, feed, product, and tails quantities and compositions (the external variables) must satisfy the material-balance relations... 

Two material-balance relations among internal variables may also be written for each stage. Consider the portion of the cascade from the product end down to, but not including, stage i ... 

The separative capacity is analogous to the heat duty of an evaporator or other process equipment. The separation potential is analogous to the enthalpy per mole of the streams entering or leaving an evaporator. Calculations of material balances and separative capacity in an isotope separation plant are made in similar fashion to conventional material and heat balances. A form for such calculations is illustrated in Table 12.8, which illustrates the calculation of the separative capacity of an isotope separation cascade producing 1 mol/day of at 0.80 mole... 

A simple relation between tp, t, and the cascade inventory may be derived as follows. The inventory of both components in the plant is assumed to remain constant at / during the start-up period. The average fraction of desired component in the plant changes from zp, the value throughout the plant at time zero, to x when the steady state is reached. The material-balance equation for desired component during the transient period is... 

MATERIAL BALANCES. Consider the portion of the cascade that includes stages 1 through , as shown by the section enclosed by the dashed line in Fig, 17.4. The total input of material to this section is L H- V +i moles per hour and the total output is A, + 1 moles per hour. Since, under steady flow, there is neither accumulation nor depletion, the input and the output are equal and... 

OPERATING-LINE DIAGRAM. For a binary system, the compositions of the two phases in a cascade can be shown on an arithmetic graph where x is the abscissa and y the ordinate. As shown by Eq. (17.2), the material balance at an intermediate point in the column involves the concentration of the L phase leaving stage , and the concentration of the V phase entering that stage. Equation (17.2) can be written to show the relationship more clearly ... 

OPERATING LINE. The equation for the operating line is obtained by writing material balances for that portion of the cascade consisting of the first n units, as shown by the control surface indicated by the dashed lines in Fig. 20.2. These balances are... 

Equation (17.24) cannot be used for the entire cascade if the solution entering with the unextracted solids, differs from L, the underflows within the system. Equations have been derived for this situation, but it is easy to calculate, by material balances, the performance of the first stage separately and then to apply Eq. (17.24) to the remaining stages. 

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. 

Constant Reflux. In general, the operating line of a cascade follows from a mass/molar balance and a material balance between the top and a certain stage n of the cascade. For constant reflux R = const resp. R/Pi = v = const,... 

The graphical approach to solution is developed in Figure 7.11 on right-triangular coordinates. A total material balance around the entire cascade is... 

Material balances may also be written for the whole cascade. A balance for (A + C) is of the form... 

We begin the development of graphical design procedures by deriving generalized material balance equations for countercurrent cascades. These are first applied to simple, single-section distillation, extraction, absorption, and... 

The internal flow rate ls- can be eliminated by successive substitution using material balances around successively smaller sections of the top of the cascade. For stages 1 through N — 2,... 

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