Component balance axial dispersion

Axial Dispersion. Enthusiastic modelers sometimes add axial dispersion terms to their two-phase, piston flow models. The component balances are... 

The UASB tractor was modeled by the dispensed plug flow model, considering decomposition reactions for VFA componaits, axial dispersion of liquid and hydrodynamics. The difierential mass balance equations based on the dispersed plug flow model are described for multiple VFA substrate components considaed... 

From the axial dispersion flow model the component balance equation is 9Ca 9Ca 3 Ca, p... 

At any level in the transition region, there will be a balance between the mixing effects attributable to (a) axial dispersion and to (b) the segregating effect which will depend on the difference between the interstitial velocity of the liquid and that interstitial velocity which would be required to produce a bed of the same voidage for particles of that size on their own. On this basis a model may be set up to give the vertical concentration profile of each component in terms of the axial mixing coefficients for the large and the small particles. 

The boundary conditions normally associated with Equation (9.14) are known as the Danckwerts or closed boundary conditions. They are obtained from mass balances across the inlet and outlet of the reactor. We suppose that the piping to and from the reactor is small and has a high Re. Thus, if we were to apply the axial dispersion model to the inlet and outlet streams, we would find Din = Dout = 0, which is the definition of a closed system. See Figure 9.8. The flux in the inlet pipe is due solely to convection and has magnitude Qi ain. The flux just inside the reactor at location z = 0+ has two components. One component, Qina(0+), is due to convection. The other component, —DAc[da/dz 0+, is due to diffusion (albeit eddy diffusion) from the relatively high concentrations at the inlet toward the lower concentrations within the reactor. The inflow to the plane at z = 0 must be matched by material leaving the plane at z = 0+ since no reaction occurs in a region that has no volume. Thus,... 

This approach is based on the similarity between a TCC and a SMB unit, such that the flow rates in a TCC can be converted easily to the equivalent ones in a SMB unit. In the frame of equilibrium theory - that is, a model assuming one-dimensional flow - adsorption equilibrium between solid and liquid phase and neglecting axial dispersion, the following mass balances are obtained for each component i in every section j of a TCC unit ... 

The equation of the ADF model flow can be obtained by making a particular species mass balance, as in the case of a plug flow model. In this case, for the beginning of species balance we must consider the axial dispersion perturbations superposed over the plug flow as shown in Fig. 3.31. In the description given below, the transport vector has been divided into its convective and dispersion components. 

The Flow Equation. Consider a differential cross-sectional slice, dx, at distance x from the feed end of the devolatilizer. A volatile component material balance across this slice will include net inputs due to mean axial flow and axial dispersion (the latter arising from the nip mixing action), and depletion through the regenerated surface films. In addition to the three assumptions made above, it is assumed that uniform conditions prevail throughout the length—i.e., constant Uy p, S, Wy D y etc.-and that the effect of axial dispersion may be characterized by a constant axial eddy diffusivity, E. The steady-state material balance for a volatile component across dx reduces to ... 

The stationary phase concentration is given by the isotherm equation (Eq. 2.4). The mobile phase concentration is denoted simply by Q. Since the axial dispersion coefficient is nil, the mass balance equation for component i (Eq. 2.2) simplifies to... 

In the ideal model, we assume that the column efficiency is infinite, hence the rate of the mass transfer kinetics is infinite and the axial dispersion coefficient in the mass balance equation (Eq. 2.2) is zero. The differential mass balances for the two components are written ... 

The classical formulation of the GRM includes a mass balance equation in each of the two fractions of the mobile phase, axial dispersion, intraparticle diffusion, and the kinetics of adsorption-desorption. In dimensionless form, the following set of equations is written for each component of the system, as follows. 

As explained in Sec. 4.4.4, the movement of components through a chromatography column can be modelled by a two-phase rate model, which is able to handle multicomponents with nonlinear equilibria. In Fig. 1 the column with segment n is shown, and in Fig. 2 the structure of the model is depicted. This involves the writing of separate liquid and solid phase component balance equations, for each segment n of the column. The movement of the solute components through the column occurs by both convective flow and axial dispersion within the liquid phase and by solute mass transfer from the liquid phase to the solid. 

Notice that the molar density of key-limiting reactant A on the external surface of the catalytic pellet is always used as the characteristic quantity to make the molar density of component i dimensionless in all the component mass balances. This chapter focuses on explicit numerical calculations for the effective diffusion coefficient of species i within the internal pores of a catalytic pellet. This information is required before one can evaluate the intrapellet Damkohler number and calculate a numerical value for the effectiveness factor. Hence, 50, effective is called the effective intrapellet diffusion coefficient for species i. When 50, effective appears in the denominator of Ajj, the dimensionless scaling factor is called the intrapellet Damkohler number for species i in reaction j. When the reactor design focuses on the entire packed catalytic tubular reactor in Chapter 22, it will be necessary to calcnlate interpellet axial dispersion coefficients and interpellet Damkohler nnmbers. When there is only one chemical reaction that is characterized by nth-order irreversible kinetics and subscript j is not required, the rate constant in the nnmerator of equation (21-2) is written as instead of kj, which signifies that k has nnits of (volume/mole)"" per time for pseudo-volumetric kinetics. Recall from equation (19-6) on page 493 that second-order kinetic rate constants for a volnmetric rate law based on molar densities in the gas phase adjacent to the internal catalytic surface can be written as... 

To take into account a hmited mixing behavior in the hquid phase, the axial dispersion model with a dispersion confident can be used (see Figure 2.4). The back-mixing of the hquid superposes the net hquid flow (superficial hquid velocity Vj ). The mass balance for the individual component i in the volume element Avol gives... 

The developed dynamic reactor model for the simulation studies of the unsteady-state-operated trickle-flow reactor is based on an extended axial dispersion model to predict the overall reactor performance incorporating partial wetting. This heterogeneous model consists of unsteady-state mass and enthalpy balances of the reaction components within the gas, liquid and catalyst phase. The individual mass-transfer steps at a partially wetted catalyst particle are shown in Fig. 4.5. 

As in Section 8.1 we consider an element of the bed through which a stream containing concentration c,(z,r) of adsorbabie species / is flowing. Assuming that the flow pattern can be described by the axially dispersed plug flow model, the differential fluid phase mass balance equation for each component is... 

The final goal of the axial dispersion model is its utilization in the modeling of chemical reactors. Below we will consider steady-state models only, which imply that the mass balance of a reacting component i can be written as... 

Mathematical models for different kinds of gas-liquid reactors are based on the mass balances of components in the gas and liquid phases. The flow pattern in a tank reactor is usually close to complete backmixing. In the case of packed and plate columns, it is often a good approximation to assume the existence of a plug flow. In bubble columns, the gas phase flows in a plug flow, whereas the axial dispersion model is the most realistic one for the liquid phase. For a bubble column, the ideal flow patterns set the limit for the reactor capacity for typical reaction kinetics. 

This simple analysis for an isothermal and equilibrium controlled process can be extended to concentrated systems in which u must remain within the differential of the second term in equation (6.19). The analysis can also be extended to systems which include more than a single adsorbable component. Consider the case of a feed stream which contains only two adsorbable components, i.e. a system which does not include a non-adsorbing carrier fluid. In this case both components can be ejqiected to be concentrated in the fluid and hence the variation in fluid velocity over the MTZ must be taken into account. Two differential fluid phase mass balance equations must be written, one for each component. Equation (6.31) is shown for component 1. The axial dispersion term is retained to create a general equation. 

Consider the case when the flow pattern can be assumed to be axially dispersed plug flow. By not making the assumption that the system is dilute, the differential fluid phase material balance for each component is given by ... 

For each component, a mass balance equation can be written using an infinitesimal element of the bed as a control volume this is equation (6.37). The terms in this equation account for convection and axial dispersion (if applicable) into and out of the control volume, together with the rate of adsorption and the accumulation in the fluid phase. 

It is noted that the accumulation term and the axial dispersion term may often be neglected. It may be helpful to illustrate the nature of the last two terms on the r.h.s. of Eq. (7.4.24). If we were to establish a balance on water vapor in the gas phase, then drying of a porous solid would correspond to the physical transfer of the component in question from the solid to the gas. In contrast, the gas phase oxidation of hydrogen to water vapor would have to be represented by a chemical reaction term. 

The tube reactor containing the catalyst bed was described with a dynamic axial dispersion model. The flows, dispersion effects, and interfacial fluxes are illustrated in Fig. 9.11. The balance equation for a component in the gas-phase volume element can be written as follows... 

Where t is time, z are the axial position in the column, qt is the concentration of solute i in the stationary phase in equilibrium with Cu the mobile phase concentration of solute /, u is the mobile phase velocity, Da is the apparent dispersion coefficient, and F is the phase ratio (Vs/Vm). The equation describes that the difference between the amounts of component / that enters a slice of the column and the amount of the same component that leaves it is equal to the amount accumulated in the slice. The fist two terms on the left-hand side of Eq. 10 are the accumulation terms in the mobile and stationary phase, respectively [109], The third term is the convective term and the term on the right-hand side of Eq. 10 is the diffusion term. For a multi component system there are as many mass balance equation, as there are active components in the system [13],... 

All of this translates into the pseudo-homogeneous model that has radial dispersion and axial convection as the driving components. Given the extensive treatment in Chapter 5, we will not rederive these equations. For a reaction component A, the mass balance is... 

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