Continuous fluid phase balance equations

Single-phase forced convection through microchaimels can be studied by determining the velocity distribution and the temperature field in the fluid region with the aid of the mass conservation principle continuity equation), the fluid momentum balance equations, and the energy balance equation [1] ... 

The balance equations for the continuous fluid phase follow from the conservation of mass and momentum and are given by... 

As will be shown later the equation above is identical to the mass balance equation for a continuous stirred-tank reactor. The recycle can be provided either by an external pump as shown in Fig. 5.4-18 or by an impeller installed within the reaction chamber. The latter design was proposed by Weychert and Trela (1968). A commercial and advantageously modified version of such a reactor has been developed by Berty (1974, 1979), see Fig. 5.4-19. In these reactors, the relative velocity between the catalyst particles and the fluid phases is incretised without increasing the overall feed and outlet flow rates. 

The above six equations for continuity and force balance do not, however, afford a complete description of a heterogeneous particle-fluid system in which a dense phase and a dilute phase coexist. An additional constraint needs to be identified to account for the stability of the system. 

The composition boundary values entering into Eqs. (All) represent external values for Eqs. (A10). With some further assumptions concerning the diffusion and reaction terms, this allows an analytical solution of the boundary-value problem [Eqs. (A10) and (All)] in a closed matrix form (see Refs. 58 and 135). On the other hand, the boundary values need to be determined from the total system of equations describing the process. The bulk values in both phases are found from the balance relations, Eqs. (Al) and (A2). The interfacial liquid-phase concentrations xj are related to the relevant concentrations of the second fluid phase, y , by the thermodynamic equilibrium relationships and by the continuity condition for the molar fluxes at the interface (57,135). 

The creeping flow of a single fluid phase through a stationary permeable media is the most basic physical situation of interest. In this case, the volume-averaged overall mass balance yields the equation of continuity 3... 

The steady-state fluid mechanics problem is solved using the Fluent Euler-Euler multiphase model in the fluid domains. Mass, momentum and energy balances, the general forms of which are given by eqn. (4), (5), and (6), are solved for both the liquid and the gas phases. In solid zones the energy equation reduces to the simple heat conduction problem with heat source. By convention, / =1 designates the H2S04 continuous liquid phase whereas H2 bubbles constitute the dispersed phase 0 =2). 

Tomiyama [148] and Tomiyama and Shimada [150] adopted a N + 1)-fluid model for the prediction of 3D unsteady turbulent bubbly flows with non-uniform bubble sizes. Among the N + l)-fluids, one fluid corresponds to the liquid phase and the N fluids to gas bubbles. To demonstrate the potential of the proposed method, unsteady bubble plumes in a water filled vessel were simulated using both (3 + l)-fluid and two-fluid models. The gas bubbles were classified and fixed in three groups only, thus a (3 + 1)- or four-fluid model was used. The dispersions investigated were very dilute thus the bubble coalescence and breakage phenomena were neglected, whereas the inertia terms were retained in the 3 bubble phase momentum equations. No population balance model was then needed, and the phase continuity equations were solved for all phases. It was confirmed that the (3 + l)-fluid model gave better predictions than the two-fluid model for bubble plumes with non-uniform bubble... 

Several extensions of the two-fluid model have been developed and reported in the literature. Generally, the two-fluid model solve the continuity and momentum equations for the continuous liquid phase and one single dispersed gas phase. In order to describe the local size distribution of the bubbles, the population balance equations for the different size groups are solved. The coalescence and breakage processes are frequently modeled in accordance with the work of Luo and Svendsen [74] and Prince and Blanch [92]. 

Bove [16] proposed a different approach to solve the multi-fluid model equations in the in-house code FLOTRACS. To solve the unsteady multifluid model together with a population balance equation for the dispersed phases size distribution, a time splitting strategy was adopted for the population balance equation. The transport operator (convection) of the equation was solved separately from the source terms in the inner iteration loop. In this way the convection operator which coincides with the continuity equation can be employed constructing the pressure-correction equation. The population balance source terms were solved In a separate step as part of the outer iteration loop. The complete population balance equation solution provides the... 

The PBE is a simple continuity statement written in terms of the NDE. It can be derived as a balance for particles in some fixed subregion of phase and physical space (Ramkrishna, 2000). Let us consider a finite control volume in physical space O and in phase space with boundaries defined as dO. and dO., respectively. In the PBE, the advection velocity V is assumed to be known (e.g. equal to the local fluid velocity in the continuous phase or directly derivable from this variable). The particle-number-balance equation can be written as... 

The forces that give rise to the phenomena spoken of appear because of the alteration in stresses at the interface between two immiscible fluid phases. For a curved interface there is a difference in pressure between the two fluids given by the Young-Laplace equation. This pressure difference is termed the capillary pressure, and since the normal stress component at the interface must be continuous, then that pressure added to the hydrostatic pressure must balance. A balance can always be achieved under static conditions. In addition, the tangential stress must also be continuous at the interface. However, if there... 

Cross-flow drying in a plug-flow, continuous fluid bed is a case when axial dispersion of flow is often considered. Let us briefly present a method of solving this case. First, the governing balance equations for the solid phase will have the following form derived from Eqnations 3.10 and 3.11... 

Also, the depletion of oxygen from the gas phase is rather low and usually compensated by the desorption of carbon dioxide. The methodology is attractive because it permits a separation of fluid dynamics (momentum balances, continuity equations, and turbulence model) from material balance equations for the state variables of interest. Figure 8 illustrates how results from the fluid dynamic simulations (mean velocities turbulent dispersion coefficient... 

When adsorption takes place with suspended adsorbent particles in a vessel, adsorbate is transported from the bulk fluid phase to the adsorption sites in the adsorbent particle In this type of situauon, changes in the amount adsorbed or concentration in the fluid phase can be predicted by solving the set of differential equations describing the mass balances in the particle, at the outer surface and between the paruclc and the fluid phase In this chapter the adsorption uptake relations are shown for several typical situations These are applicable to batch adsorption in a liquid stirred tank, batch measurement of gas adsorption by gravity method or by pressure method as shown in Fig 5 1 Also adsorption in a shallow bed is a typical example of application of the treatment for batch adsorption with continuous flow (Fig 5 2)... 

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