Boundary conditions dispersion model

First principle mathematical models These models solve the basic conservation equations for mass and momentum in their form as partial differential equations (PDEs) along with some method of turbulence closure and appropriate initial and boundary conditions. Such models have become more common with the steady increase in computing power and sophistication of numerical algorithms. However, there are many potential problems that must be addressed. In the verification process, the PDEs being solved must adequately represent the physics of the dispersion process especially for processes such as ground-to-cloud heat transfer, phase changes for condensed phases, and chemical reactions. Also, turbulence closure methods (and associated boundary and initial conditions) must be appropriate for the dis-... 

Ghoreishy, M. H. R. and Nassehi, V., 1997. Modelling the transient flow of rubber compounds in the dispersive section of an internal mixer with slip-stick boundary conditions. Adv. Poly. Tech. 16, 45-68. 

Comparison of solutions of the axially dispersed plug flow model for different boundary conditions... 

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 = 0, which is the definition of a closed system. See... 

These boundary conditions are really quite marvelous. Equation (9.16) predicts a discontinuity in concentration at the inlet to the reactor so that ain a Q+) if D >0. This may seem counterintuitive until the behavior of a CSTR is recalled. At the inlet to a CSTR, the concentration goes immediately from to The axial dispersion model behaves as a CSTR in the limit as T) — 00. It behaves as a piston flow reactor, which has no inlet discontinuity, when D = 0. For intermediate values of D, an inlet discontinuity in concentrations exists but is intermediate in size. The concentration n(O-l-) results from backmixing between entering material and material downstream in the reactor. For a reactant, a(O-l-) [Pg.332]

An Eulerian-Eulerian (EE) approach was adopted to simulate the dispersed gas-liquid flow. The EE approach treats both the primary liquid phase and the dispersed gas phase as interpenetrating continua, and solves a set of Navier-Stokes equations for each phase. Velocity inlet and outlet boundary conditions were employed in the liquid phase, whilst the gas phase conditions consisted of a velocity inlet and pressure outlet. Turbulence within the system was account for with the Standard k-e model, implemented on a per-phase basis, similar to the recent work of Bertola et. al.[4]. A more detailed description of the computational setup of the EE method can be found in Pareek et. al.[5]. 

Bunimovich et al. (1995) lumped the melt and solid phases of the catalyst but still distinguished between this lumped solid phase and the gas. Accumulation of mass and heat in the gas were neglected as were dispersion and conduction in the catalyst bed. This results in the model given in Table V with the radial heat transfer, conduction, and gas phase heat accumulation terms removed. The boundary conditions are different and become identical to those given in Table IX, expanded to provide for inversion of the melt concentrations when the flow direction switches. A dimensionless form of the model is given in Table XI. Parameters used in the model will be found in Bunimovich s paper. 

In the formulation of the boundary conditions, it is presumed that there is no dispersion in the feed line and that the entering fluid is uniform in temperature and composition. In addition to the above boundary conditions, it is also necessary to formulate appropriate equations to express the energy transfer constraints imposed on the system (e.g., adiabatic, isothermal, or nonisothermal-nonadiabatic operation). For the one-dimensional models, boundary conditions 12.7.34 and 12.7.35 hold for all R, and not just at R = 0. 

The Smoluchowski-Levich approach discounts the effect of the hydrodynamic interactions and the London-van der Waals forces. This was done under the pretense that the increase in hydrodynamic drag when a particle approaches a surface, is exactly balanced by the attractive dispersion forces. Smoluchowski also assumed that particles are irreversibly captured when they approach the collector sufficiently close (the primary minimum distance 5m). This assumption leads to the perfect sink boundary condition at the collector surface i.e. cp 0 at h Sm. In the perfect sink model, the surface immobilizing reaction is assumed infinitely fast, and the primary minimum potential well is infinitely deep. 

Breakthrough Behavior for Axial Dispersion Breakthrough behavior for adsorption with axial dispersion in a deep bed is not adequately described by the constant pattern profile for this mechanism. Equation (16-128), the partial differential equation of the second order Fickian model, requires two boundary conditions for its solution. The constant pattern pertains to a bed of infinite depth—in obtaining the solution we apply the downstream boundary condition cf — 0 as NPeC, —> < >. Breakthrough behavior presumes the existence of a bed outlet, and a boundary condition must be applied there. 

Equation 5-9 together with appropriate boundary and initial conditions forms the fundamental basis for dispersion modeling. This equation will be solved for a variety of cases. 

The TIS model is relatively simple mathematically and hence easy to use. The DPF or axial dispersion model is mathematically more complex and yields significantly different results for different choices of boundary conditions, if the extent of backmixing is large (small Pe,). On this basis, the TIS model may be favored. 

Figure El2.2a shows the boundary conditions X0 and Yx. Given values for m, Nox, and the length of the column, a solution for Y0 in terms of vx and vY can be obtained Xx is related to Y0 and F via a material balance Xx = 1 - (Yq/F). Hartland and Meck-lenburgh (1975) list the solutions for the plug flow model (and also the axial dispersion model) for a linear equilibrium relationship, in terms of F ...

This equation too is solved with the same boundary conditions as Eq. (148). A series of equations results when different combinations of fluids are used. There is no change for the first stage. All the terms of equation of motion remain the same except the force terms arising out of dispersed-phase and continuous-phase viscosities. The main information required for formulating the equations is the drag during the non-Newtonian flow around a sphere, which is available for a number of non-Newtonian models (A3, C6, FI, SI 3, SI 4, T2, W2). Drop formation in fluids of most of the non-Newtonian models still remains to be studied, so that whether the types of equations mentioned above can be applied to all the situations cannot now be determined. 

Figure 13.7 Various boundary conditions used with the dispersion model.

The solution of Eq. (173) poses a rather formidable task in general. Thus the dispersed plug-flow model has not been as extensively studied as the axial-dispersed plug-flow model. Actually, if there are no initial radial gradients in C, the radial terms will be identically zero, and Eq. (173) will reduce to the simpler Eq. (167). Thus for a simple isothermal reactor, the dispersed plug flow model is not useful. Its greatest use is for either nonisothermal reactions with radial temperature gradients or tube wall catalysed reactions. Of course, if the reactants were not introduced uniformly across a plane the model could be used, but this would not be a common practice. Paneth and Herzfeld (P2) have used this model for a first order wall catalysed reaction. The boundary conditions used were the same as those discussed for tracer measurements for radial dispersion coefficients in Section II,C,3,b, except that at the wall. 

A number of authors [46 to 48] employ the single sphere model in which the packed bed is considered as a set of equal spheres that are under the same state of extraction, and the fluid flowing around them is solute-free. That is, equation (3.4-90) would be valid, but without the generation term [46], The transport at the solid-fluid interface obeys the boundary condition (Eqn. 3.4-94) with C = 0 (fluid-flows at a large velocity). Under these assumptions, there is an analytical solution to the above problem (without axial dispersion) in terms of the Biot number (Bi = k, R/De), included in the following equation ... 

This is the standard dispersion model for P, and the aim of investigating more complex situations has often been to reduce them to this form with D = De, an effective dispersion coefficient that wraps up the complexities of the underlying situation in a single quantity. Whether this is wise is another matter. For example, in a packed bed the flow is obviously very complex, but both theory [4] and experiment can be invoked to justify an effective Peclet number, URIDe, of about 2. The question that hangs over the use of Eq. (33) is that it is a parabolic equation, with infinite signal speed and controversial boundary conditions. 

We have seen that the basic P model has the form of a first-order partial differential Eq. (22) describing each narrow slice as a little batch reactor being transported through the reactor at constant speed. This equation was so elementary that it could be solved at sight in Eq. (30). When we added a longitudinal dispersion term governed by Fick s law and took the steady state, Eq. (40), we had a second-order o.d.e. with controversial boundary conditions. This is the model with ( ) = c(z)lcm and Pe = vLID, Da = kL/v,... 

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