Dispersion coefficients mass balance

Pollutants emitted by various sources entered an air parcel moving with the wind in the model proposed by Eschenroeder and Martinez. Finite-difference solutions to the species-mass-balance equations described the pollutant chemical kinetics and the upward spread through a series of vertical cells. The initial chemical mechanism consisted of 7 species participating in 13 reactions based on sm< -chamber observations. Atmospheric dispersion data from the literature were introduced to provide vertical-diffusion coefficients. Initial validity tests were conducted for a static air mass over central Los Angeles on October 23, 1968, and during an episode late in 1%8 while a special mobile laboratory was set up by Scott Research Laboratories. Curves were plotted to illustrate sensitivity to rate and emission values, and the feasibility of this prediction technique was demonstrated. Some problems of the future were ultimately identified by this work, and the method developed has been applied to several environmental impact studies (see, for example, Wayne et al. ). 

The design engineer can use the dispersion coefficients determined in this way for the calculation of the real course of concentrations, c, of any component in the dispersed d) and continuous (c) phases along the countercurrent column. If the mass transfer between the two phases, the actual task of an extractor, is included in the balance, the balance equations for an element of height dh of the extractor for stationary conditions is ... 

In the above derivation of the mass balance equation, we assume that the column is radially homogeneous, the compressibility of the mobile phase is negligible, the axial dispersion coefficient is constant, and the temperature is unchanged. Furthermore, no diffusion in the stationary phase is assumed. 

The band profile obtained as a numerical solution of Equation 10.10 gives the concentration distribution as a function of the reduced time at the column end, i.e., at location x= 1, regardless of the column length. The band profile depends only on the column efficiency, the boundary conditions, the phase ratio, and the sample size (which is part of the boundary conditions). The mobile phase velocity has been eliminated from the mass balance equation and the apparent axial dispersion coefficient has been replaced by the plate number. 

Commercial reactors are non isothermal and often adiabatic. In a noniso-thermal gas-liquid reactor, along with the mass dispersions in each phase, the corresponding heat dispersions are also required. Normally, the gas and liquid at any given axial position are assumed to be at the same temperature. Thus, in contrast to the case of mass, only a single heat-balance equation (and corresponding heat-dispersion coefficient) is needed. Under turbulent flow conditions (such as in the bubble-column reactor) the Peclet number for the heat dispersion is often assumed to be approximately equal to the Peclet number for the mass dispersion in a slow-moving liquid phase. 

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],... 

In general, the overall balance for the mass transport streams (Eqs. 6.23 and 6.24) at the column inlet and outlet has to be fulfilled. In Eq. 6.92 the closed boundary condition is obtained by setting the dispersion coefficient outside the column equal to zero. In open systems, the column stretches to infinity and in these limits concentration changes are zero. 

Because of the analogy between simulated and true counter-current flow, TMB models are also used to design SMB processes. As an example, the transport dispersive model for batch columns can be extended to a TM B model by adding an adsorbent volume flow Vad (Fig. 6.38), which results in a convection term in the mass balance with the velocity uads. Dispersion in the adsorbent phase is neglected because the goal here is to describe a fictitious process and transfer the results to SMB operation. For the same reason, the mass transfer coefficient feeff as well as the fluid dispersion Dax are set equal to values that are valid for fixed beds. 

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... 

Haarhoff and Van der Linde [68] have given a more direct mathematical demonstration of this result in the case of a moderately overloaded column, with a parabohc isotherm. It leads to the fundamental equation of the equilibriiun-dispersive model, in which the diffusion coefficient in the diffusive term of the mass balance equation (Eq. 2.2) is replaced by the apparent dispersion coefficient (Eq. 2.38). 

In contrast to the equilibrium-dispersive model, which is based upon the assumptions that constant thermod3mamic equilibrium is achieved between stationary and mobile phases and that the influence of axial dispersion and of the various contributions to band broadening of kinetic origin can be accounted for by using an apparent dispersion coefficient of appropriate magnitude, the lumped kinetic model of chromatography is based upon the use of a kinetic equation, so the diffusion coefficient in Eq. 6.22 accounts merely for axial dispersion (i.e., axial and eddy diffusions). The mass balance equation is then written... 

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 ... 

In the equilibrium-dispersive model of chromatography, however, we assume that Eq. 10.4 remains valid. Thus, we use Eq. 10.10 as the mass balance equation of the component, and we assume that the apparent dispersion coefficient Da in Eq. 10.10 is given by Eq. 10.11. We further assume that the HETP is independent of the solute concentration and that it remains the same in overloaded elution as the one meastued in linear chromatography. As shown by the previous discussion this assxunption is an approximation. However, as we have shown recently [6], Eq. 10.4 is an excellent approximation as long as the column efficiency is greater than a few hundred theoretical plates. Thus, the equilibriiun-dispersive model should and does account well for band profiles under almost all the experimental conditions used in preparative chromatography. In the cases in which the model breaks down because the mass transfer kinetics is too slow, and the column efficiency is too low, a kinetic model or, better, the general rate model (Chapter 14) should be used. 

In particular cases simplified reactor models can be obtained neglecting the insignificant terms in the governing microscopic equations (without averaging in space) [9]. For axisymmetrical tubular reactors, the species mass and heat balances are written in cylindrical coordinates. Himelblau and Bischoff [9] give a list of simplified models that might be used to describe tubular reactors with steady-state turbulent flow. A representative model, with radially variable velocity profile, and axial- and radial dispersion coefficients, is given below ... 

It is very interesting to note that the virtual-mass force mainly acts through the effective volume coefficient y, and tends to reduce the momenffim-exchange terms. The final step is to use the definitions of the mass-average moments introduced in Chapter 4 to derive the force terms in the macroscale momenffim balance. For the disperse-phase momentum balance (see Eq. (4.85) on page 123), this procedure leads to... 

Thermal axial dispersion must be treated with care. Even if axial dispersion of mass is negligible, the same may not be true for heat transport. The dispersion coefficient that appears in the thermal Peclet number is very different from the dispersion coefficient of the mass Peclet number. The combination of a plug-flow model for the mass balance and a dispersion... 

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... 

Results from the previous section in this chapter illustrate how and when interpellet axial dispersion plays an important role in the design of packed catalytic tubular reactors. When diffusion is important, more sophisticated numerical techniques are required to solve second-order ODEs with split boundary conditions to predict non-ideal reactor performance. Tubular reactor performance is nonideal when the mass transfer Peclet number is small enough such that interpellet axial dispersion cannot be neglected. The objectives of this section are to understand the correlations for effective axial dispersion coefficients in packed beds and porous media and calculate the mass transfer Peclet number based on axial dispersion. Before one can make predictions about the ideal vs. non-ideal performance of tubular reactors, steady-state mass balances with and without axial dispersion must be solved and the reactant concentration profiles from both solutions must be compared. If the difference between these profiles with and without interpellet axial dispersion is indistinguishable, then the reactor operates ideally. 

The linear driving force (LDF) model can be classified in the group of equilibrium transport dispersive models (Fig. 9.5). For this model it is no longer assumed that the mobile and the stationary phases are permanently in equilibrium state, so that an additional mass-balance equation for the stationary phase is required. Assuming a linear concentration gradient an effective mass-transfer coefficient keff is implemented, where all mass-transfer resistances and the diffusion into the pores of the particle are lumped together. In this model a constant local equilibrium between the solid and the liquid in the pores is assumed. 

The two equations for the mass and heat balance, Eqs. (4.10.125) and (4.10.126) or the dimensionless forms represented by Eqs. (4.10.127), (4.10.128) and (4.10.130), consider that the flow in a packed bed deviates from the ideal pattern because of radial variations in velocity and mixing effects due to the presence of the packing. To avoid the difficulties involved in a rigorous and complicated hydrodynamic treatment, these mixing effects as well as the (in most cases negligible contributions of) molecular diffusion and heat conduction in the solid and fluid phase are combined by effective dispersion coefficients for mass and heat transport in the radial and axial direction (D x, Drad. rad. and X x)- Thus, the fluxes are expressed by formulas analogous to Pick s law for mass transfer by diffusion and Fourier s law for heat transfer by conduction, and Eqs. (4.10.125) and (4.10.126) superimpose these fluxes upon those resulting from convection. These different dispersion processes can be described as follows (see also the Sections 4.10.6.4 and 4.10.7.3) ... 

When gas phase adsorption takes place in a large column, heat generated due to adsorption cannot be removed from the bed wall and accumulated in the bed because of poor beat transfer characteristics in packed beds of particles. A typical model of this situations is an adiabatic adsorption. The fundamental relations for this case are Eqs. (8-22), (8-38), (8-39) and (8-40), which are essentially similar to those employed by Pan and Basmadjian (1970). Thermal equilibrium between particle and fluid is assumed and oidy axial dispersion of heat is taken into account while mass transfer resistance between fluid phase and particle as well as axial dispersion is considered. This situation is identical with the model employed in the previous section. For further simpliHcation, axial dispersion effect may be involved in the overall mass transfer coefficient of the linear driving force model as discussed in Chapter S. In this case, after further justifiable simplifications such as negligible heat capacity and accumulation of adsorbate in void spaces, a set of basic equations to describe heat and mass balances can be ven as follows. 

Assuming plug flow in the channel (a trivial change would be to add an axial dispersion coefficient), the mass balances in the channel taking into account also the devolatilization from the pool (mass transfer coefficients k y,) thus becomes those given in Eqs. (38). 

In chemical reaction engineering single phase reactors are often modeled by a set of simplified ID heat and species mass balances. In these cases the axial velocity profile can be prescribed or calculate from the continuity equation. The reactor pressure is frequently assumed constant or calculated from simple relations deduced from the area averaged momentum equation. For gases the density is normally calculated from the ideal gas law. Moreover, in situations where the velocity profile is neither flat nor ideal the effects of radial convective mixing have been lumped into the dispersion coefficient. With these model simplifications the semi-empirical correlations for the dispersion coefficients will be system- and scale specific and far from general. 

Wakao and Funazkri (1978) revealed that when mass transfer coefficients were measured by experiments involving adsorption or evaporation, the mass balance for the bed (see Chapter 6) should include a term accounting for axial dispersion. Previous correlations of experimental data were based upon a mass balance equation for the packed bed ignoring axial dispersion. It was shown that the mass transfer coefficient could be expressed in terms of the dimensionless Sherwood number (Sh) by the relation... 

In Section 1.3, the concept of mass balance was applied to finite control volumes with well-defined boundaries, such as lakes. Mass balance, however, also can be expressed in an infinitesimal control volume, mathematically considered to be a point. Conservation of mass is expressed in such a volume by the advection-dispersion-reaction equation, which states that the rate of change of chemical storage at any point in space, dC/dt, equals the sum of both the rates of chemical input and output by physical means and the rate of net internal production (i.e., sources minus sinks). The inputs and outputs that occur by physical means (advection and Fiddan transport) are expressed in terms of the fluid velocity (V), the diffusion/ dispersion coefficient (D), and the chemical concentration gradient in the fluid (dC/dx). The net contribution by internal sources and sinks of the chemical is represented by r. In one dimension, the advection-dispersion-reaction equation for a fixed point is... 

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