Pressure correction equation

Here, cr is the condensation rate constant, Aw is the partial pressure of water vapor, A at is the saturation pressure, Mh2o is the molecular weight of the water vapor. A corresponding source term has to be added to water vapor conservation equation and pressure correction equation (mass source). The liquid water velocity is assumed to be same as the gas velocity inside the gas channel. However, inside the porous region, the convection term is replaced by capillary diffusion term and the equation becomes... 

The corrected velocities are assumed to satisfy continuity equations. If the corrected velocity expressions (Eq. (6.38)) are substituted in the discretized continuity equation, pressure correction equations can be derived. However, velocity corrections as given by Eq. (6.39) involve velocity corrections at neighboring nodes and unless some approximations are made, it is not possible to obtain the desired pressure correction equations. In SIMPLE algorithm, the first term comprising velocity corrections at the neighboring nodes is neglected to yield a simplified expression for velocity corrections ... 

Substitution of this velocity correction into the discretized form of the continuity equation then leads to a pressure correction equation of the following form ... 

The overall algorithm is exactly the same as that of SIMPLE, the only exception being that Eq. (6.43) is used instead of Eq. (6.40) to derive the discretized pressure correction equation. The more consistent approximation proposed in SIMPLEC reduces the need for under-relaxing the pressure. 

The pressure correction equation is solved and the velocity field is corrected using the derived pressure correction field. For PISO, a second pressure correction equation is solved to correct the pressure and velocities again. 

For single-phase flows, pressure is shared by three momentum equations and requires special algorithms to compute the pressure field. Most of these algorithms (discussed in the previous chapter) use one continuity equation and three momentum equations to derive pressure and/or pressure correction equations. However, for multiphase flows, there is more than one continuity equation. Answers to questions such as which continuity equation should be used to derive pressure equations are not obvious. As discussed in the previous chapter, it is customary to employ iterative techniques to solve single-phase flow equations. Such iterative techniques can, in principle, be extended to simulate multiphase flows. In practice, however, the process... 

Several alternatives may be used to derive suitable pressure or pressure correction equations. In this section, we will discuss a specific option based on the work of Spalding (1980) and Carver (1984). This option has been used to simulate gas-liquid flows in stirred vessels (Ranade and van den Akker, 1994) and bubble columns (Ranade, 1992 1997) and was found to be quite robust. The method is illustrated here for two-fluid models. It can be extended to more than two phases following the same general principles. The overall method is an extended version of the SIMPLER... 

If the flows are unsteady, the terms containing apo can be added on both sides of Eq. (7.10) (refer to Section 6.4). It must be noted that for multiphase flows, the inflow and outflow terms require considerations of interpolations of phase volume fractions in addition to the usual interpolations of velocity and the coefficient of diffusive transport. The source term linearization practices discussed in the previous chapter are also applicable to multiphase flows. It is useful to recognize that special sources for multiphase flows, for example, an interphase mass transfer, is often constituted of terms having similar significance to the a and b terms. Such discretized equations can be formulated for each variable at each computational cell. The issues related to the phase continuity equation, momentum equations and the pressure correction equation are discussed below. 

Usual interpolation rules and definitions of velocity and pressure corrections, similar to single-phase flows (Eq. (6.29)), can be used to derive a pressure correction equation from the discretized form of the overall continuity (normalized) equation. The momentum equation for multiphase flows (Eq. (7.16)) can also be written in the form of Eq. (6.28) for single-phase flows. Again, following the approximation of SIMPLE, one can write an equation for velocity correction in terms of pressure correction,/ ... 

Solve the momentum equations for all phases based on the guessed pressure field (or that obtained from solution of the pressure correction equation during... 

The pressure correction equation is solved based on the normalized overall continuity equation. 

Application of the discretized continuity equation (12.161) and velocity corrections on the form (12.166) produces the following pressure-correction equation ... 

Another way of treating the last term in the pressure-correction equation (12.167) is to approximate it rather than neglecting it. One could approximate the velocity correction term e neighbor nodes by use of the... 

Yet another similar method of this kind called SIMPLER (SIMPLE Revised) was proposed by Patankar [I4f] to improve the convergence rate, as compared to the SIMPLE procedure. In this method, the pressure-correction equation (12.167) is solved first with the last term neglected as in SIMPLE. The pressure correction so obtained is used only to correct the velocity field v to obtain so that it satisfies continuity by use of (12.166) in which the first term on the RHS is neglected. The new pressure field is calculated from pressure equation (12.164) using J2nb nbvl h instead of 6-... 

The convergence of the SIMPLE algorithm can be improved by underrelaxation so that only a portion of p is added to after the pressure-correction equation is solved ... 

Assemble and solve the pressure-correction equation to obtain p. ... 

For the PISO algorithm, solve the second pressure-correction equation and correct both velocities and pressure again. 

To explain the modifications required to the pressure-correction equation previously presented for incompressible flow to deal with compressible flows. 

The second term in the mass flux correction relation (12.175), is due to the compressibility effects. It involves the correction to the density at the grid cell volume face. If the SIMPLE method is to be extended to compressible flows, we must thus express the density correction in terms of the pressure correction to enable an efficient update of the density corrections in the pressure correction equation. 

It is noted that the given pressure-correction method can be applied for arbitrary Mach number flows. At low Mach numbers (almost incompressible flow), the Laplacian term dominates and we recover the Poisson equation. On the other hand, at high Mach number (highly compressible flow), the convective term dominates, reflecting the hyperbolic nature of the flow. Solving the pressure-correction equation is then equivalent to solving the continuity equation for density. Thus, the pressure correction method automatically adjusts... 

The velocity- and pressure correction equations in IPSA are frequently derived using the SIMPLEC method (i.e., the SIMPLE- Consistent approximation) by van Doormal and Raithby [191]. 

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 iterative procedure is considered to give a converged solution if the absolute normalized residuals for all the variables as well as the mass source b of the pressure correction equation are less than a prescribed small value, denoting the convergence criterion. The absolute normalized residual is defined as ... 

The pressure correction equation is derived from the liquid continuity equation and the liquid velocity correction equation formulas. The SIMPLE Consistent (SIMPLEC) -approximation proposed by van Doormal and Raithby [23] is used to derive the velocity correction formulas. 

The second class of methods, which seem to be more popular among the commercial CFD codes, relies on the pressure-correction approach. Here, the velocity components are solved in an uncoupled manner without using the continuity equation as a constraint. An equation for pressure or a change in pressure (hence the term pressure correction ) is derived that will alter the velocity field so as to better satisfy the continuity equation. The precise formulation and the iterative procedures may differ as long as iterative application of the momentum and the pressure-correction equations produces a solution that satisfies both the momentum and the continuity equations. Algorithms that differ in this fashion are employed... 

So-called false-time-step relaxation is used to achieve stationarity. The semi-implicit method, which considers the pressure-Hnk of the pressure correction equation and the Reynolds equations, is the SIMPLEST algorithm. The sets of algebraic equations for each variable are solved iteratively by means of the ADI technique. An example of the simulated flow field is illustrated in Fig. 3. Good agreement can then be achieved between measured flow details and the simulation results for vessels and impellers of different geometry [1]. 

Substitute a velocity correction formula, expressed in terms of pressure corrections at the main grid points in the discretized continuity equation (discretized over the main control volume, not the staggered one), to get the pressure correction equation in the following form (with T and B denoting the top and bottom grid points relative to the point P in a three-dimensional space) ... 

No attempt is made for the direct solution of momentum equations. In the velocity correction expression, i.e., Ug = Onb nb + b + (pp — PeMb- the first term in the right-hand side is essentially dropped, which enables one to cast the pressure correction equation in a general conservative form. 

The approximation introduced in the derivation of the pressure correction equation by... 

The solution strategy is somewhat varied by the last step since the approach used to linearize and solve the discretized equations varies with the solver type. The two commonly employed solvers in the FVM 2se pressure-based and density-based solvers [ 12,16]. In both methods the velocity field is obtained from the momentum equations. In the density-based approach, the continuity equation is used to obtain the density field, while the pressure field is determined from an equation of state. On the other hand, the pressure-based solver extracts the pressure field by solving the pressure or pressure correction equation, which is obtained by manipulation of the momentum and continuity equations [16]. Implementation of the pressure-based solver via the so-called Semi-Implicit Method for Pressure-Linked Equations (SIMPLE) algorithm [12] is explained later. Details of the density-based solver are extensively covered elsewhere [16] and will not be discussed here. 

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