Dispersion boundary conditions

Surface waves at an interface between two innniscible fluids involve effects due to gravity (g) and surface tension (a) forces. (In this section, o denotes surface tension and a denotes the stress tensor. The two should not be coiifiised with one another.) In a hydrodynamic approach, the interface is treated as a sharp boundary and the two bulk phases as incompressible. The Navier-Stokes equations for the two bulk phases (balance of macroscopic forces is the mgredient) along with the boundary condition at the interface (surface tension o enters here) are solved for possible hamionic oscillations of the interface of the fomi, exp [-(iu + s)t + i V-.r], where m is the frequency, is the damping coefficient, s tlie 2-d wavevector of the periodic oscillation and. ra 2-d vector parallel to the surface. For a liquid-vapour interface which we consider, away from the critical point, the vapour density is negligible compared to the liquid density and one obtains the hydrodynamic dispersion relation for surface waves + s>tf. The temi gq in the dispersion relation arises from... 

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. 

For axial dispersion in a semi-infinite bed with a linear isotherm, the complete solution has been obtained for a constant flux inlet boundary condition [Lapidiis and Amundson,y. Phy.s. Chem., 56, 984 (1952) Brenner, Chem. Eng. Set., 17, 229 (1962) Coates and Smith, Soc. Petrol. Engrs. J., 4, 73 (1964)]. For large N, the leading term is... 

Boundary Conditions In normal operation with closed ends, reactant is brought in by bulk flow and carried away by both bulk and dispersion flow. At the inlet where L = 0 or r = 0,... 

With these two-point boundary conditions the dispersion equation, Eq. (23-50), may be integrated by the shooting method. Numerical solutions for first- and second-order reaciions are plotted in Fig. 23-15. 

In any case, like frequency analysis, examining the uncertainties and sensitivities of the results to changes in boundary conditions and assumptions provides greater perspective. The level of effort required for a consequence analysis will be a function of the number of different accident scenarios being analyzed the number of effects the accident sequence produces and the detail with which the release, dispersion, and effects on the targets of interest is estimated. The cost of the consequence analysis can typically be 25% to 50% of the total cost of a large QRA. 

Two types of boundary conditions are considered, the closed vessel and the open vessel. The closed vessel (Figure 8-36) is one in which the inlet and outlet streams are completely mixed and dispersion occurs between the terminals. Piston flow prevails in both inlet and outlet piping. For this type of system, the analytic expression for the E-curve is not available. However, van der Laan [22] determined its mean and variance as... 

The criterion for the validity of Equation 8-141 is Npg 1.0. A rough rule-of-thumb is Npg > 10. If this condition is not satisfied, the correct equation depends on the boundary conditions at the inlet and outlet. A procedure for determining dispersion coefficient Dg [ is as follows ... 

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

The main purpose with flow visualization is to make the airflow field or the emission and transport of air contaminants visible and thereby possible to study. In technical terms, flow visualization gives possibilities to study airflow field and contaminant dispersion and changes in it depending on general changes in geometry, boundary conditions, inlet and exhaust airflow, etc. It is... 

Turbulence may arise by two mechanisms. First, it may result either from a violent release of fuel from under high pressure in a jet or from explosive dispersion from a ruptured vessel. The maximum overpressures observed experimentally in jet combustion and explosively dispersed clouds have been relatively low (lower than 1(X) mbar). Second, turbulence can be generated by the gas flow caused by the combustion process itself an interacting with the boundary conditions. 

In the application of the multienergy concept, a particular vapor cloud explosion hazard is not determined primarily by the fuel-air mixture itself but rather by the environment into which it disperses. The environment constitutes the boundary conditions for the combustion process. If a release of fuel is anticipated somewhere, the explosion hazard assessment can be limited to an investigation of the environment s potential for generating blast. 

The overall set of partial differential equations that can be considered as a mathematical characterization of the processing system of gas-liquid dispersions should include such environmental parameters as composition, temperature, and velocity, in addition to the equations of bubble-size and residence-time distributions that describe the dependence of bubble nucleation and growth on the bubble environmental factors. A simultaneous solution of this set of differential equations with the appropriate initial and boundary conditions is needed to evaluate the behavior of the system. Subject to the Curie principle, this set of equations should include the possibilities of coupling effects among the various fluxes involved. In dispersions, the possibilities of couplings between fluxes that differ from each other by an odd tensorial rank exist. (An example is the coupling effect between diffusion of surfactants and the hydrodynamics of bubble velocity as treated in Section III.) As yet no analytical solution of the complete set of equations has been found because of the mathematical difficulties involved. To simplify matters, the pertinent transfer equation is usually solved independently, with some simplifying assumptions. 

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. 

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. 

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