Mass balances with convection

Hence, molar flow rates are linear functions of conversion because dF/H is the same for all components, based on stoichiometry and the mass balance with convection and one chemical reaction (see equation 3-15) ... 

If the reactor is well stirred, then the molar densities of reactant A and product B in the kinetic rate law are expressed in terms of conversion / via stoichiometry and the steady-state mass balance with convection and chemical reaction ... 

It is possible to avoid some of the instabilities associated with guessing the dimensionless axial concentration gradient at the inlet to a non-ideal tubular reactor by solving the one-dimensional mass transfer equation backwards from outlet to inlet. In practice, it is necessary to introduce a new dimensionless independent spatial variable which increases as one travels backward through the packed catalytic tubular reactor. This stipulation is required by conventional ODE solvers. Hence, if = 1 — f, then the two coupled ODEs which represent the mass balance with convection, interpellet axial dispersion, and nth-order irreversible chemical reaction are rewritten as follows ... 

Step 5. Calculate the mass transfer Peclet number which appears in the mass balance with convection and axial dispersion. 

Steacfy-State Binaiy Fickian Diffusion and Mass Balances with Convection... 

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. 

These models are designed to define the complex entrance effects and convection phenomena that occur in a reactor and solve the complete equations of heat, mass balance, and momentum. They can be used to optimize the design parameters of a CVD reactor such as susceptor geometry, tilt angle, flow rates, and others. To obtain a complete and thorough analysis, these models should be complemented with experimental observations, such as the flow patterns mentioned above and in situ diagnostic, such as laser Raman spectroscopy. 

This chapter provides analytical solutions to mass transfer problems in situations commonly encountered in the pharmaceutical sciences. It deals with diffusion, convection, and generalized mass balance equations that are presented in typical coordinate systems to permit a wide range of problems to be formulated and solved. Typical pharmaceutical problems such as membrane diffusion, drug particle dissolution, and intrinsic dissolution evaluation by rotating disks are used as examples to illustrate the uses of mass transfer equations. 

With this equation, we can now discuss a generalized mass balance equation. We still use Figure 1 to show the derivation. Based on Eq. (5), the net contribution by diffusion and convection now becomes... 

The TDE solute module is formulated with one equation describing pollutant mass balance of the species in a representative soil volume dV = dxdydz. The solute module is frequently known as the dispersive, convective differential mass transport equation, in porous media, because of the wide employment of this equation, that may also contain an adsorptive, a decay and a source or sink term. The one dimensional formulation of the module is ... 

As noted earlier, air-velocity profiles during inhalation and exhalation are approximately uniform and partially developed or fully developed, depending on the airway generation, tidal volume, and respiration rate. Similarly, the concentration profiles of the pollutant in the airway lumen may be approximated by uniform partially developed or fully developed concentration profiles in rigid cylindrical tubes. In each airway, the simultaneous action of convection, axial diffusion, and radial diffusion determines a differential mass-balance equation. The gas-concentration profiles are obtained from this equation with appropriate boundary conditions. The flux or transfer rate of the gas to the mucus boundary and axially down the airway can be calculated from these concentration gradients. In a simpler approach, fixed velocity and concentration profiles are assumed, and separate mass balances can be written directly for convection, axial diffusion, and radial diffusion. The latter technique was applied by McJilton et al. 

The algebraic equations for the orthogonal collocation model consist of the axial boundary conditions along with the continuity equation solved at the interior collocation points and at the end of the bed. This latter equation is algebraic since the time derivative for the gas temperature can be replaced with the algebraic expression obtained from the energy balance for the gas. Of these, the boundary conditions for the mass balances and for the energy equation for the thermal well can be solved explicitly for the concentrations and thermal well temperatures at the axial boundary points as linear expressions of the conditions at the interior collocation points. The set of four boundary conditions for the gas and catalyst temperatures are coupled and are nonlinear due to the convective term in the inlet boundary condition for the gas phase. After a Taylor series expansion of this term around the steady-state inlet gas temperature, gas velocity, and inlet concentrations, the system of four equations is solved for the gas and catalyst temperatures at the boundary points. 

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,... 

The mass balance in the reactor is derived under the following assumptions (i) unsteady state operation, (ii) convective laminar Newtonian flow in the axial direction z (the Re)molds number is below the transition regime), (iii) diffusion in the z direction is negligible with respect to convection, (iv) symmetry in the y direction (the lamp length is much larger than the reactor width), and (v) constant physical properties. The local mass balance for a species i in the reactor and the corresponding initial and boundary conditions are... 

In addition to the Navier-Stokes equations, the convective diffusion or mass balance equations need to be considered. Filtration is included in the simulation by preventing convection or diffusion of the retained species. The porosity of the membrane is assumed to decrease exponentially with time as a result of fouling. Wai and Fumeaux [1990] modeled the filtration of a 0.2 pm membrane with a central transverse filtrate outlet across the membrane support. They performed transient calculations to predict the flux reduction as a function of time due to fouling. Different membrane or membrane reactor designs can be evaluated by CFD with an ever decreasing amount of computational time. 

Well-Mixed Cell Model. A conceptually simple approach is based on the representation of the airshed by a three-dimensional array of well-mixed vessels (34, 35, 36). As before, we assume that the airshed has been divided into an array of L cells. Instead of using the array simply as a tool in the finite-difference solution of the continuity equations, let us now assume that each of these cells is actually a well-mixed reactor with inflows and outflows between adjacent cells. If we neglect diffusive transport across the boundaries of the cells and consider only convective transport among cells, a mass balance on species i in cell k is given by... 

Figure 9 Examples of models proposed for the chemical structure of the terrestrial mantle, (a) Whole mantle convection with depletion of the entire mantle. Some subducted slabs pass through the transition zone to the coremantle boundary. Plumes arise from both the core-mantle boundary and the transition zone. This model is not in agreement with isotopic and chemical mass balances, (b) Two-layer mantle convection, with the depleted mantle above the 660 km transition zone and the lower mantle retaining a primitive composition, (c) Blob model mantle where regions of more primitive mantle are preserved within a variously depleted and enriched lower mantle, (d) Chemically layered mantle with lower third above the core comprising a heterogeneous mixture of enriched (mafic slabs) and more primitive mantle components, and the upper two-thirds of the mantle is depleted in incompatible elements (see text) (after Albarede and van der Hilst, 1999).

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],... 

A different numerical strategy to simulate multiphase mixing was introduced by Mann and Mann and Hackett. The idea of the method, called the network-of-zone, is to subdivide the flow domain in a set of small cells assumed to be mixed perfectly. The cells are allowed to exchange momentum and mass with their neighboring cells by convective and diffusive fluxes. Brucato and Rizzuti and Brucato et al. applied this idea to the modeling of solid-liquid mixing. An unsteady mass balance for the particles was derived to estimate the solid distribution in the vessel, namely ... 

Under plug flow conditions the convective transport is completely dominant over the diffusive mass transport term. The fluid moves like a plug and the diffusive term can be neglected. The conditions for plug flow are closely satisfied for narrow and long tubular reactors when the viscosity is low. However, this approximation is clearly best for fully developed turbulent flow, for which the velocity profiles are relatively fiat. For dynamic conditions, the species mass balance is a PDF with z and t as the independent variables. The Eulerian species mass balance (1.301) reduces further to ... 

Step 1 It is easy to add the convective diffusion equation by starting with the fluid flow model of the T-sensor. Choose Multiphysics/Model Navigator. A window appears with a hst of possible equations. Scroll down and select Chemical Engineering Module/Mass Balance/Convection and Diffusion/Steady-state Analysis. Click Add. Now FEMLAB will solve both equations. 

A section of a fixed-bed catalytic reactor is shown in Fig. 13-4. Consider a small volume element of radius r, width Ar, and height Ar, through which reaction mixture flows isothermally. Suppose that radial and longitudinal diffusion can be expressed by Pick s law, with and Dj as effective diffusivities, based on the total (void and nonvoid) area perpendicular to the direction of diffusion. We want to write a mass balance for a reactant over the volume element. With radial and longitudinal diffusion and longitudinal convection taken into account, the input term is... 

---