Heterogeneous Lumped Systems

For a heterogeneous system we write equation (6.67) out for each phase and account for the heat transfer Q between the two phases. For a nonadiabatic system the heat Qextemai 

To turn these heat-balance equations into nonisothermal heat balance design equations, we define the rate of reaction per unit volume (or per unit mass of catalyst, depending on the system) and the heat transfer per unit volume of the process unit (or per unit length), whichever is more convenient. 

The rate of heat transfer per unit volume of the process unit is given by 

Obviously for multiple reactions in each phase, we must change the equations accordingly. For N reactions (counted in j) in phase I and N reactions (counted in j) in phase II, the equation for phase I is 

Now our picture is almost complete and we can determine the heat-balance design equation for our distributed two phase system. 

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The Design Equations (Steady-State Models) for Isothermal Heterogeneous Lumped Systems... 

In this section we develop the heat-balance design equations for heterogeneous systems. Based on the previous sections it is clear how to use the heat-balance and heat-balance design equations that were developed earlier for homogeneous systems, as well as the principles that were used to develop the mass-balance and mass-balance design equations for heterogeneous systems for our purpose. We will start with lumped systems. 

Let us first review the most general material and heat balance equations and all of the special cases which can be easily obtained from these equations. This will be followed by the basic idea of how to transform these material and energy balance equations into design equations, first for lumped systems and then followed by the same for distributed systems. We will use homogeneous chemical reactors with multiple inputs, multiple outputs, and multiple reactions. It will be shown in Chapter 6 how to apply the same principles to heterogeneous system and how other rates (e.g., rates of mass transfer) can systematically replace (or is added to) the rates of reactions. 

This is a case with negligible mass transfer resistances, as described by the pseudohomogeneous model. For a full heterogeneous system, see Chapter 6. This situation is a bit more complicated compared to the lumped system. We will consider a two-phase system with no mass transfer resistance between the phases and the voidage is equal to s (see Fig. 4.9). The mass balance design equation over the element A/ is ... 

An important problem in catalysis is to predict diffusion and reaction rates in porous catalysts when the reaction rate can depend on concentration in a non-linear way.6 The heterogeneous system is modeled as a solid material with pores through which the reactants and products diffuse. We assume for diffusion that all the microscopic details of the porous medium are lumped together into the effective diffusion coefficient De for reactant. 

The analysis of stationary and nonstationary flow distributions in multiloop hydraulic systems with lumped, regulated, and distributed parameters and in heterogeneous systems was given in (Gorban et al., 2001, 2006 Kaganovich et al., 1997). In the concluding section of Section 5 the abundant capabilities of the flow MEIS are illustrated by the simplest example of stationary isothermal flow distribution of incompressible fluid in the three-loop circuit. It is shown how the degrees of order (laminar or turbulent modes) on the branches of this circuit are determined from calculation of the final equilibrium. 

VII.42. Heterogeneous mixtures of fissile material should assume an optimum spacing between fissile lumps such that maximum reactivity is achieved unless adequate structure is provided to ensure a known spacing or spacing range (e.g. reactor fuel pins in an assembly). It is important to realize that, with complex systems. 

On the other hand, the value of the second integral is very sensitive to the manner in which the fuel is disposed. If the fuel is in the form of discrete lumps fission neutrons will be much more likely to encounter and cause fission than if the fuel is uniformly divided. Thus the fast effect—i.e., essentially the second term in (13)—is considerably greater in lumped (heterogeneous) systems than in dispersed (homogeneous) systems. 

Fig. 1. Effects of homogeneous versus heterogeneous UOj-HjO system moderation bn the minimum criticality size of the system. Points for heterogeneous system correspond to the moderator/fuel volume ratio and fuel rod diameter for maximum material buckling at various fuel lump densities.

The dynamic terms for heterogeneous systems will be exactly the same as for the homogeneous systems (refer to Chapters 2 and 3), but repeated for both phases. The reader should take this as an exercise by just repeating the same principles of formulating the dynamic terms for both mass and heat and for both lumped and distributed systems. For illustration, see the dynamic examples later in this chapter. 

The increase in the resonance-escape probability has been attributed to two effects, namely, the self-shielding of the fuel lump and the physical separation of the two materials (refer to Secs. 10.1c and d). The purpose of the present section is to study these effects in detail. The treatment begins with an elementary analysis of the latter effect, which arises from purely geometric considerations. This study is followed by an outline of an analytical procedure (based on nuclear-resonance parameters) for computing the integrals which appear in the expression for the resonance-escape probability [see, for example, (10.25) and (10.37)]. Application is made first to homogeneous systems, and the results are later extended to heterogeneous systems. 

The heterogeneous system formulas (10.159) and (10.166) for the contribution to based on the NR and NRIA approximations may be used to deduce the dependence of the effective resonance integral upon the surface-to-mass ratio (Ay/My) of the fuel lump. In the cases wherein the NRIA formula applies, it frequently occurs that (< or / 1. Then the form (10.166) is well approximated by... 

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