Fluid model equations boundary conditions

Koelman and Hoogerbrugge (1993) have developed a particle-based method that combines features from molecular dynamics (MD) and lattice-gas automata (LGA) to simulate the dynamics of hard sphere suspensions. A similar approach has been followed by Ge and Li (1996) who used a pseudo-particle approach to study the hydrodynamics of gas-solid two-phase flow. In both studies, instead of the Navier-Stokes equations, fictitious gas particles were used to represent and model the flow behavior of the interstial fluid while collisional particle-particle interactions were also accounted for. The power of these approaches is given by the fact that both particle-particle interactions (i.e., collisions) and hydrodynamic interactions in the particle assembly are taken into account. Moreover, these modeling approaches do not require the specification of closure laws for the interphase momentum transfer between the particles and the interstitial fluid. Although these types of models cannot yet be applied to macroscopic systems of interest to the chemical engineer they can provide detailed information which can subsequently be used in (continuum) models which are suited for simulation of macroscopic systems. In this context improved rheological models and boundary condition descriptions can be mentioned as examples. 

Smoluchowski theory [29, 30] and its modifications fonu the basis of most approaches used to interpret bimolecular rate constants obtained from chemical kinetics experiments in tenus of difhision effects [31]. The Smoluchowski model is based on Brownian motion theory underlying the phenomenological difhision equation in the absence of external forces. In the standard picture, one considers a dilute fluid solution of reactants A and B with [A] [B] and asks for the time evolution of [B] in the vicinity of A, i.e. of the density distribution p(r,t) = [B](rl)/[B] 2i ] r(t))l ] Q ([B] is assumed not to change appreciably during the reaction). The initial distribution and the outer and inner boundary conditions are chosen, respectively, as... 

In the SMB operation, the countercurrent motion of fluid and solid is simulated with a discrete jump of injection and collection points in the same direction of the fluid phase. The SMB system is then a set of identical fixed-bed columns, connected in series. The transient SMB model equations are summarized below, with initial and boundary conditions, and the necessary mass balances at the nodes between each column. 

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. 

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. 

Lu et al. [7] extended the mass-spring model of the interface to include a dashpot, modeling the interface as viscoelastic, as shown in Fig. 3. The continuous boundary conditions for displacement and shear stress were replaced by the equations of motion of contacting molecules. The interaction forces between the contacting molecules are modeled as a viscoelastic fluid, which results in a complex shear modulus for the interface, G = G + mG", where G is the storage modulus and G" is the loss modulus. G is a continuum molecular interaction between liquid and surface particles, representing the force between particles for a unit shear displacement. The authors also determined a relationship for the slip parameter Eq. (18) in terms of bulk and molecular parameters [7, 43] ... 

Prediction of the breakthrough performance of molecular sieve adsorption columns requires solution of the appropriate mass-transfer rate equation with boundary conditions imposed by the differential fluid phase mass balance. For systems which obey a Langmuir isotherm and for which the controlling resistance to mass transfer is macropore or zeolitic diffusion, the set of nonlinear equations must be solved numerically. Solutions have been obtained for saturation and regeneration of molecular sieve adsorption columns. Predicted breakthrough curves are compared with experimental data for sorption of ethane and ethylene on type A zeolite, and the model satisfactorily describes column performance. Under comparable conditions, column regeneration is slower than saturation. This is a consequence of non-linearities of the system and does not imply any difference in intrinsic rate constants. 

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 ... 

Summarizing, the model of the screw channel flow is governed by eqns. (8.99), (8.105) and (8.106) with boundary conditions eqns. (8.100), (8.101) and (8.104). The constitutive equation that was used by Griffith is a temperature dependent shear thinning fluid described by... 

Under quite general conditions on the geometry of the flow domain and the data we show that the model has a solution that satisfies the equations and boundary conditions in an integrated or weak sense. Clearly, the fluid velocity q, as well as the electrical charge c are solved independent of the chemistry. This part of the model ((81-3) and (9i 2)) is standard and its solution is straightforward. The challenging non-standard issue is the description of the chemistry ((84) and (93-5)), in particular the multi-valued dissolution rate in (95). Existence is demonstrated by regularization, where (94,5) are replaced by... 

The above analytical solution was expanded to three dimensions. In such a way, the reactor geometry or the channel can be designed. An appropriate simplified model, given in [38], can be derived from the diffusion equation. Appropriate boundary conditions at the channel walls account for the heterogeneous wall reaction. The concentration of a species A which reacts at the channel wall irreversibly to a species B was given as a function of the lateral channel dimensions y and z and the axial channel dimension xv For an inert gas and for y and z equal to zero (coordinate center indicated in Figure 3.94), Eq. (3.13) reduces to the solution of a non-reactive fluid given above ... 

For fixed or chosen values of the parameters, the model equations (eqs. 1-4) along with the initial and boundary conditions (eqs. 5) are solved iteratively by a centered-in-space, forward-in-time, finite difference scheme to obtain (i) the hexene and hexene oligomer concentration profiles in the pore fluid phase, and (ii) the coke (extractable + consolidated) accumulation profde. The effectiveness factor (rj) is estimated from the hexene concentration profile as follows ... 

This equation is derived by integrating Eq.( 11-29) with boundary condition)/ = 0, T = To at r = 0. Although the model has some elastic character the viscous response dominates at all but short times. For this reason, the element is known as a Maxwell fluid. 

Solving the previous set of equations, especially with realistic boundary conditions, is a formidable task and a lot of issues are still unanswered. This is not surprising because of the complexity of the equations, and because of their recent derivation, around 1950 for the first nonlinear models, the Oldroyd models. On the other hand, the mathematical theory for the Euler and the Navier-Stokes equations for incompressible Newtonian fluids is still not complete though these equations were derived in 1755 md 1821 respectively ... 

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