Maximum mixedness model

Even for the maximum mixedness model, the calculation of an extreme concentration value is possible. This extreme value, in comparison with the previous one for the segregation model, represents the opposite end of the scale. The maximum mixedness model is relevant to microfluids and yields, even in this case, results that do not differ considerably from those that can be obtained by means of direct utilization of separate flow models. The model is, however, of importance in cases in which experimentally determined residence time functions are available for a reactor system. The maximum mixedness model is more difficult to visualize than the segregation model. However, its underlying philosophy can be described as follows  

It is worth mentioning that the first claim is also valid for the segregation model, whereas the second claim suggests that a volume element residing inside the reactor can attract new species. Additionally, this happens immediately at the time of entry into the system. Zwietering [2] described the maximum mixedness model mathematically. The derivation was based on the time that reveals the life expectation of a species in a reactor, which will be denoted here as t. Since the variable in question yields the time (fx) that an element with the age fa is going to remain in the reactor, the sum of these times must be equal to the residence time of the species, f  

Zwietering [2] concluded that the maximum mixedness model can be expressed as follows in terms of the generation rate  

In Equation 4.63, the expression X(fx) denotes the intensity function of t. Equation 4.63 can be solved by taking into accoimt that dc,/dfx = 0 at fx = oo. The c,-value at fx = 0 has to be determined. It should be noted that fx = 0 at the time the species leaves the reactor. Since a detailed description of Zwietering s philosophy would take too long, here it will suffice to state that he proved that X(fx), in Equation 4.63, can be replaced by X(f). If, further, the ratio q/co is denoted byy, and fx/f as 0x the left-hand side in Equation 4.63, dc,/dfx, becomes equal to Cq dy/t d0x. Since we are still able to show that the result of the calculation will be the same, although 0x would be replaced by 0 (= f/f), we will finally obtain f, an expression of practical use 

Equations 4.64 and 4.65 yield the final value of y, at 0 = 0, as the solution is obtained backwards, by starting with large values of 0 (in addition to the X values valid at these points) and ending up with 0 = 0. Similar to the segregation model, a precondition for the maximum mixedness model is that the residence time functions—or more precisely the intensity function—are available, either from theory or from experiments. 

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Maximum mixedness model The fluid in a flow reactor that behaves as a micro fluid. Mixing of molecules of different ages occurs as early as possible. 

A real system must lie somewhere along a vertieal line in Figure 9-5. Performanee is within the upper and lower points on this line namely maximum mixedness and eomplete segregation. Equation 9-10 gives the eomplete segregation limit. The eomplete segregation model with side exits and the maximum mixedness model are diseussed next. 

Micromixing between these two extremes (partial segregation) is possible, but not considered here. A model for (1) is the segregated-flow model (SFM) and for (2) is the maximum-mixedness model (MMM) (Zwietering, 1959). We use these in reactor models in Chapter 20. 

In addition to these two macromixing reactor models, in this chapter, we also consider two micromixing reactor models for evaluating the performance of a reactor the segregated flow model (SFM), introduced in Chapters 13 to 16, and the maximum-mixedness model (MMM). These latter two models also require knowledge of the kinetics and of the global or macromixing behavior, as reflected in the RTD. 

The segregated-flow reactor model (SFM) represents the micromixing condition of complete segregation (no mixing) of fluid elements. As noted in Section 19.2, this is one extreme model of micromixing, the maximum-mixedness model being the other. 

The maximum-mixedness model (MMM) for a reactor represents the micromixing condition of complete dispersion, where fluid elements mix completely at the molecular level. The model is represented as a PFR with fluid (feed) entering continuously incrementally along the length of the reactor, as illustrated in Figure 20.1 (after Zwieter-ing, 1959). The introduction of feed incrementally in a PFR implies complete mixing... 

For n > 1, the segregated flow model provides the upper bound on conversion, and the maximum-mixedness model defines the lower bound. 

For n < 1, the maximum-mixedness model sets the upper bound, while the lower bound is determined by the segregated-flow model. 

Equation (xi) must be numerically integrated, using either Ex(t) or E2 t), and the appropriate expressions for cA(t) and cD(t) (see E-Z Solve file ex20-5.msp). Table 20.1 gives the outlet concentration, conversion, yield, and selectivity obtained for each of the two cases, (d) Maximum-mixedness model For the maximum-mixedness model, the rate laws for A and D are substituted into Equation 20.4-6, and the two resulting ordinary differential equations (in dcA/dt and dcD/dt) must be numerically integrated. The respective equations are ... 

To obtain the results in Table 20.2, equations (xii) and (xiii) are solved numerically using E-Z Solve (file ex20-5.msp) and die initial conditions above for cA and cD. For N = 1, equation 13.4-2 is used for Ex(t) and is integrated to obtain W,(t), and similarly for N = 2 and equation 17.2-4. Note that most software for numerical integration cannot directly handle the negative step sizes required to solve the maximum-mixedness model... 

Identical performance is obtained from a CSTR and the maximum mixedness model, with E(t) based upon BMF. 

Comparison of the segregated-flow and maximum-mixedness models, with identical RTD functions, shows that the former gives better performance. This is consistent with the observations of Zwietering (1959), who showed that for power-law kinetics of order n > 1, the segregated-flow model produces the highest conversion. 

Show that, for a first-order reaction [(-rA) = AcA], the outlet concentration predicted by the maximum-mixedness model is... 

Figure 1.6. Four micromixing models that have appeared in the literature. From top to bottom maximum-mixedness model minimum-mixedness model coalescence-redispersion model three-environment model.

The tank-in-series (TIS) and the dispersion plug flow (DPF) models can be adopted as reactor models once their parameters (e.g., N, Del and NPe) are known. However, these are macromixing models, which are unable to account for non-ideal mixing behavior at the microscopic level. This chapter reviews two micromixing models for evaluating the performance of a reactor— the segregrated flow model and the maximum mixedness model—and considers the effect of micromixing on conversion. 

ASME CFSTR CFD CFM DIERS exp IR HA/AN HAZOP MM MMM American Society of Mechanical Engineers Continuous flow stirred tank reactor Computational fluid dynamics Computational fluid mixing Design Institute for Emergency Relief Systems exponential Infrared (spectroscopy) Hazard analysis Hazard and operability studies Michaelis-Menten Maximum-mixedness model... 

After studying this chapter the reader will be able to describe the cumulative F(t), external age E(t), and internal age I(t) residence-time distribution functions and to recognize these functions for PFR, CSTR, and laminar flow reactions. The reader will also be able to apply these functions to calculate the conversion and concentrations exiting a reactor using the segregation model and the maximum mixedness model for both single and multiple reactions. 

To answer this question we will model a real reactor in a number of ways. We shall classify each model according to the number of adjustable parameters that are extracted from the RTD data (see Table 13-1). In this chapter we discuss only the segregation and maximum mixedness models. Other models are discussed in Chapter 14. 

The segregation model and maximum mixedness model are further compared on the CD-ROM for reaction orders between zero and one. 

Cautions The segregation model and the maximum mixedness model will not give the proper boimds on the conversion for certain rate laws and for nonisothermal operation. These situations arise, for example, if the rate of reaction goes through both a maximum and a minimum when plotted as a function of conversion and the initial rate is higher than the maximum (Figure 13-18). 

If tracer tests are carried out isothermally and then used to predict nonisothermal conditions, one must couple the segregation and maximum mixedness models with the energy balance to account for variations in the specific reaction rate. For adiabatic operation and ACf> = 0,... 

Assuming that E(t) is unaffected by temperature variations in the reactor, one simply solves the segregation and maximum mixedness models, accounting for the variation of k with temperature [i.e., conversion see Problem P13-2(h)]. 

Example 13-8 Using Software to Make Maximum Mixedness Model Calculations... 

Use an ODE solver to determine the conversion predieted by the maximum mixedness model for the E(t) eurve given in Example El3-7. 

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