Steady-state continuity equations for

The quasi steady state continuity equation for the reactant A involved in the single reversible reaction A = B can be written, for the case of a pseudo hoinogeneouB reactor models... 

The steady state continuity equations for solute in the mobile phase and in the stationary phase are ... 

The steady-state continuity equation for A in the pore, accounting for diffusion in the pore axial direction and effective diffusion inside the solid at the pore wall, is... 

The approach to be followed in the determination of rates or detailed kinetics of the reaction in a liquid phase between a component of a gas and a component of the liquid is, in principle, the same as that outlined in Chapter 2 for gas-phase reactions on a solid catalyst. In general the experiments are carried out in flow reactors of the integral type. The data may be analyzed by the integral or the differential method of kinetic analysis. The continuity equations for the components, which contain the rate equations, of course depend on the type of reactor used in the experimental study. These continuity equations will be discussed in detail in the appropriate chapters, in particular Chapter 14 on multiphase flow reactors. Consider for the time being, by way of example, a tubular type of reactor with the gas and liquid in a perfectly ordered flow, called plug flow. The steady-state continuity equation for the component A of the gas, written in terms of partial pressure over a volume element dV and neglecting any variation in the total molar flow rate of the gas is as follows ... 

The steady-state continuity equation for the key component may be written... 

Steady-state continuity equation for A in the liquid phase ... 

When reaction occurs on the pore walls simultaneously with diffusion, the process is not a strictly consecutive one and both phenomena must be considered together. For a first-order reaction, equimolar counterdiffusion and isothermal conditions the steady-state continuity equation for A can be written when the / coordinate is oriented from the center line to the surface of the slab ... 

Fig. 6 Trapped carrier density profiles calculated from the steady-state continuity equation for open and short circuit conditions. A1l,o = 10 cm- ...

Example 3-7 Consider the steady-state continuity equation for an incompressible fluid in cylindrical coordinates. Using the turbulence representations for the velocities, determine the time-average contributions to this representation. 

The steady-state continuity equations which describe mass balance over a fluid volume element for the species in the stagnant film which are subject to uniaxial diffusion and reaction in the z direction are... 

In other words, when the magnitude of the left-hand term is negligible, only the incompressible steady-state continuity equation V V = 0 remains. For isothermal, nonreacting flow, it is only when velocity variations are responsible for density variations where compressibility effects are important. 

The main reaction i.e, benzene hydrogenation occuring inside porous Ni-catalyst pellets is accompanied by poisoning reaction in which the thiophene presented in the feed stream reacts irtevcrsibly with the catalytic active sites. An analysis was made assuming isothermal behaviour [7], the same effective diffusivity for reactant and poison and that the steady-state continuity equation represents a good approximation at all times [8,9]. Under these conditions the mass balances for benzene, thiophene and catalyst activity are... 

Plug flow is a simplified and idealized picture of the motion of a fluid, whereby all the fluid elements move with a uniform velocity along parallel streamlines. This perfectly ordered flow is the only transport mechanism accounted for in the plug flow reactor model. Because of the uniformity of o>nditions in a cross section the steady-state continuity equation is a very simple ordinary differential equation. Indeed, the mass balance over a differential volume element for a reactant A involved in a single reaction may be written ... 

The partial oxidation of propene (PP) to propylene oxide over a Ag-Re catalyst immobilized in a porous ceramic tube membrane reactor was studied by a continuous forced flow system under a differential reactor condition. The steady state rate equations for the production of propylene oxide (PO) and carbon dioxide were separately proposed by two different reaction pathways as follows. 

The system of Eqs. [18], [54], and [55] is usually solved iteratively, with each iteration defined by the successive solution of the three equations. An initial guess is first supplied for the force field P in Eq. [54], which is then solved on a discrete grid to provide the components of the cnrrent density /. The divergence of J is then computed with the steady-state continuity equation (Eq. [55]) to obtain the charge distribution that, in turn, is used in the forcing function of Poisson s equation. From the gradient of the computed potential, one derives a new (better) approximation to the force P that is used to start... 

The Entrance Velocity Field For an isothermal, steady state, incompressible flow of a Newtonian fluid being symmetrical in the azimuthal direction, the governing equations are the Navier-Stokes equations and the steady state continuity equation. In dimensionless form the equations are ... 

The time derivatives are dropped for steady-state, continuous flow, although the method of false transients may still be convenient for solving Equations (11.11) and (11.12) (or, for variable Kh, Equations (11.9) and (11.10) together with the appropriate auxiliary equations). The general case is somewhat less complicated than for two-phase batch reactions since system parameters such as V, Vg, Vh and At will have steady-state values. Still, a realistic solution can be quite complicated. 

Equation 9.1-17 is the continuity equation for unsteady-state diffusion of A through the ash layer it is unsteady-state because cA = cA(r, a To simplify its treatment further, we assume that the (changing) concentration gradient for A through the ash layer is established rapidly relative to movement of the reaction surface (of the core). This means that for an instantaneous snapshot, as depicted in Figure 9.3, we may treat the diffusion as steady-state diffusion for a fixed value of rc i.e., cA = cA(r). The partial differential emiatm. 

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. 

The derivation of the material-balance or continuity equation for reactant A is similar to that of equations 19.4-48 and -49 for nonreacting tracer A, except that steady state replaces unsteady state (cA at a point is not a function of t), and a reaction term must be added. Thus, using the control volume in Figure 19.15, we obtain the equivalent of equation 19.4-48 as ... 

There is of course an equation for each species, subject to conservation of total mass given by the continuily equation, and we implicitly used conservation of total mass as stoichiometric constraints, Njo — Nj)lvj are equal for all 7 in a batch (closed) system or (Fjo — Fj)/vj are equal for aU j in a steady-state continuous reactor. 

Inhomogeneous systems. If Eq. 21-46 is an inhomogeneous system, that is, if at least one Ja is different from zero, then usually all eigenvalues are different from zero and negative, at least if the equations are built from mass balance considerations. Again, the eigenvalue with the smallest absolute size determines time to steady-state for the overall system, but some of the variables may reach steady-state earlier. In Illustrative Example 21.6 we continue the discussion on the behavior of tetrachloroethene (PCE) in a stratified lake (see also Illustrative Example 21.5). Problem 21.8 deals with a three-box model for which time to steady-state is different for each box. 

Beginning with a mass-conservation law, the Reynolds transport theorem, and a differential control volume (Fig. 4.30), derive a steady-state mass-continuity equation for the mean circumferential velocity W in the annular shroud. Remember that the pressure p 6) (and hence the density p(6) and velocity V(6)) are functions of 6 in the annulus. 

Thus the species continuity equation for the steady, constant cross-sectional plug flow is stated as... 

In this section we have presented modeling results for industrial type IV FCC units that produce high octane number gasoline from gas oil. Such units consist of two connected bubbling fluidized beds with continuous circulation of the catalyst between the two vessels, the reactor and the regenerator. The steady-state design equations are nonlinear transcendental equations which can be solved using the techniques described in the earlier chapters of the book. 

For the solution of this problem, the momentum and continuity equations for the steady-state flow of an incompressible viscoelastic fluid are given by... 

Considering both the mass balance corresponding to a CSTR and the kinetic equation, a dynamic model was established and validated for steady-state continuous operation of the EMR (Fig. 10.6). Moreover, the model was analyzed to foresee the deviations of the steady-state that occur when short-term changes in operational conditions take place (Fig. 10.7). Higher Orange II loading rates resulting from a more concentrated influent or a variation of HRT caused an increase in dye concentration in the effluent, which was predicted accurately by the dynamic model. The validation was also observed when modifications in the peroxide addition rate occurred. 

This is the continuity equation for turbulent flow when the mean motion is two-dimensional. It will be noted that this equation has exactly the same form as the continuity equation for two-dimensional steady laminar flow with the mean values of the velocity components substituted in place of the steady values that apply in laminar flow. This result can, in fact, be deduced by intuitive reasoning and simply states that if an elemental control volume through which the fluid flows is considered, then over a sufficiently long period of time, the fluctuating components contribute nothing to the mass transfer through this control volume. 

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