Constant-density system isothermal operation

The following example illustrates both an analytical and a graphical solution to determine the outlet conversion from a three-stage CSTR. 

A three-stage CSTR is used for the reaction A - products. The reaction occurs in aqueous solution, and is second-order with respect to A, with kA = 0.040 L mol-1 min-1. The inlet concentration of A and the inlet volumetric flow rate are 1.5 mol L-1 and 2.5 L min-1, respectively. Determine the fractional conversion (/A) obtained at the outlet, if Vj = 10 L, V2 = 20 L, and V3 = 50 L, (a) analytically, and (b) graphically. 

Equating the two right sides and rearranging, we obtain the quadratic equation 

Similarly, for stages 2 and 3, we obtain /A2 = 0.362, and /A3 = 0.577, which is the outlet fractional conversion from the three-stage CSTR. 

The system of three equations based on equation 14.4-1 for /A1, and /A3 may also be solved simultaneously using the E-Z Solve software (file exl4-9.msp). The same values are obtained. 

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The following example illustrates a simple case of optimal operation of a multistage CSTR to minimize the total volume. We continue to assume a constant-density system with isothermal operation. 

Isothermal Operation For a constant-density system, since... 

The volume of a recycle PFR (V) for steady-state, isothermal operation involving a constant-density system is given by equation 15.3-4, and is a function of the recycle ratio R for given operating conditions (cAo, cAU q0). Show that, as R - V becomes equal to the volume... 

A performance comparison between a BR and a CSTR may be made in terms of the size of vessel required in each case to achieve the same rate of production for the same fractional conversion, with the BR operating isothermally at the same temperature as that in the CSTR. Since both batch reactors and CSTRs are most commonly used for constant-density systems, we restrict attention to this case, and to a reaction represented by... 

The feed is charged all at once to a batch reactor, and the products are removed together, with the mass in the system being held constant during the reaction step. Such reactors usually operate at nearly constant volume. The reason for this is that most batch reactors are liquid-phase reactors, and liquid densities tend to be insensitive to composition. The ideal batch reactor considered so far is perfectly mixed, isothermal, and operates at constant density. We now relax the assumption of constant density but retain the other simplifying assumptions of perfect mixing and isothermal operation. 

To simplify the treatment for an LFR in this chapter, we consider only isothermal, steady-state operation for cylindrical geometry, and for a simple system (A - products) at constant density. After considering uses of an LFR, we develop the material-balance (or continuity) equation for any kinetics, and then apply it to particular cases of power-law kinetics. Finally, we examine the results in relation to the segregated-flow model (SFM) developed in Chapter 13. 

In this section, we apply the axial dispersion flow model (or DPF model) of Section 19.4.2 to design or assess the performance of a reactor with nonideal flow. We consider, for example, the effect of axial dispersion on the concentration profile of a species, or its fractional conversion at the reactor outlet. For simplicity, we assume steady-state, isothermal operation for a simple system of constant density reacting according to A - products. 

The axial dispersion plug flow model is used to determine the performance of a reactor with non-ideal flow. Consider a steady state reacting species A, under isothermal operation for a system at constant density Equation 8-121 reduces to a second order differential equation ... 

Introduction All of the examples up to this point have approached construction of the AR for various systems under different conditions. In the initial chapters of the book, we investigated lower dimensional systems under constant density, isothermal operation in concentration space. We have slowly relaxed many of these assumptions throughout the course of the book. In this final example, we wish to show how the constmction of the AR for a more realistic system might be addressed. 

In a classic paper Lapidus and Amundson (1952) studied liquid chromatography for isothermal operation with linear, independent isotherms when mass transfer is very rapid, but axial dispersion is inportant. Although the two-porosity model can be used (Wankat, 1990), the solution was originally obtained for the single-porosity model. Starting with Eq. [18-551. we substitute in the equilibrium expression Eq. [18-6al to remove the variable q (solid and fluid are assumed to be in local equilibrium). Since the fluid density is essentially constant in liquid systems, the interstitial fluid velocity Vj ter can be assumed to be constant. The resulting equation for each solute is... 

Jager etal. (1992) used a dilution unit in conjunction with laser diffraction measurement equipment. The combination could only determine, however, CSD by volume while the controller required absolute values of population density. For this purpose the CSD measurements were used along with mass flow meter. They were found to be very accurate when used to calculate higher moments of CSD. For the zeroth moment, however, the calculations resulted in standard deviations of up to 20 per cent. This was anticipated because small particles amounted for less then 1 per cent of volume distribution. Physical models for process dynamics were simplified by assuming isothermal operation and class II crystallizer behaviour. The latter implies a fast growing system in which solute concentration remains constant with time and approaches saturation concentration. An isothermal operation constraint enabled the simplification of mass and energy balances into a single constraint on product flowrate. 

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