Convection equations

Reviews of concentration polarization have been reported (14,38,39). Because solute wall concentration may not be experimentally measurable, models relating solute and solvent fluxes to hydrodynamic parameters are needed for system design. The Navier-Stokes diffusion—convection equation has been numerically solved to calculate wall concentration, and thus the water flux and permeate quaUty (40). 

A simplified model usiag a stagnant boundary layer assumption and the one-dimension diffusion—convection equation has been used to calculate wall concentration ia an RO module. The iategrated form of this equation, the widely appHed film theory (41), is given ia equation 8. 

The general forms of the convection equations are given below in a simple form. More accurate equations can be found from the latest research results presented in technical journals. 

In some convection equations, such as for turbulent pipe flow, a special correction factor is used. This factor relates to the heat transfer conditions at the flow inlet, where the flow has not reached its final velocity distribution and the boundary layer is not fully developed. In this region the heat transfer rate is better than at the region of fully developed flow. 

The solution of a pure convection equation for a scalar field O ... 

Figure 2.38 Finite-difTerence grid for the solution of the one-dimensional convection equation. The F, j. 1 2 denote the fluxes from node i to the neighboring nodes.

Among interface capturing methods, one of the most popular and most successful schemes is the volume-of-fluid (VOF) method dating back to the work of Hirt and Nichols [174]. The VOF method is based on a volume-fraction field c, assuming values between 0 and 1. A value of c = 1 indicates cells that are filled with phase 1, and phase 2 corresponds to c= 0. Intermediate values of c indicate the position of the interface between the phases however, the goal is to maintain a sharp interface in order to identify the different fluid phases uniquely. Volumes assigned to the different phases are moving with the local flow velocity W , and therefore the evolution of c is determined by a convection equation ... 

In practice, the Peclet number can always be ignored in the diffusion-convection equation. It can also be ignored in the root boundary condition unless C > X/Pc or A, < Pe. Inspection of the table of standard parameter values (Table 2) shows that this is never the case for realistic soil and root conditions. Inspection of Table 2 also reveals that the term relating to nutrient efflux, e, can also be ignored because e < Pe [Pg.343]

Solve the convection equation of high order (3rd order) essentially non-oscillatory (ENO) upwind scheme (Sussman et al., 1994) is used to calculate the convective term V V

velocity field P". The time advancement is accomplished using the second-order total variation diminishing (TVD) Runge-Kutta method (Chen and Fan, 2004). 

Let us try to preserve convection. If we ignore Il6 but maintain the more correct turbulent boundary layer convection (Equation (12.24)), then... 

Convection refers to bulk directional (instead of random) motion of a fluid (see Chapter 3). In the presence of convection, a one-dimensional mass transport (including both diffusion and convection) equation can be obtained by adding a convective term to the diffusion equation ... 

We have been able to get a feel for the Fokker-Planck equation by turning it in to a Diffusion-Convection equation. A mathematician on Greg Stephanopoulos viva confessed that he would have had to translate the D-C equation into an F-P equation Tot homines, quot aditus. 

P 49] CFD simulations were made for monitoring the flow patterns within a droplet which is generated at a concentric separation-layer micro mixer [39], Diffusion-convection equations of two user scalars have to be solved in addition to the corresponding equation for the volume fraction of the fluids within a multiphase CFD simulation. 

The convection equations include conservation of mass, momentum and energy. The dimensionless form of the equations are listed below. 

These convection equations may be written in vector form as... 

The equation is linear if u is a known function. This may be considered to describe a time-dependent one-dimensional heat convection equation for a problem with a known flow field. For a fluid with constant properties in the temperature range considered, the momentum equation is decoupled from the energy equation. In the following example, the... 

The boundary layer equations were derived in a previous chapter, or may be deduced from the general convection equations in the early part of this chapter. For two-dimensional, steady flow over a flat plate of an incompressible, constant-property fluid, the continuity, x-momentum and the energy equations are as follows ... 

Chow, L.C. and Tien, C.L. An Examination of Four Differencing Schemes for Some Elliptic-Type Convection Equations , Numerical Heat Transfer, Vol. 1, 1978, pp. 87-100. 

Consider the natural-convection equations available. Heat-transfer coefficients for natural convection may be calculated using the equations presented below. These equations are also valid for horizontal plates or discs. For horizontal plates facing upward which are heated or for plates facing downward which are cooled, the equations are applicable directly. For heated plates facing downward or cooled plates facing upward, the heat-transfer coefficients obtained should be multiplied by 0.5. 

Interfaclal region, J is the volume flux through the vesicle, <3 the "reflection" coefficient, represents some specific interaction of the species with vesicle. Since at the Interface of charged vesicles both diffusion and migration fluxes are much larger than the flux due to convection, equation 16 can be written ... 

Pohlhausen (P4) in 1921 presented the direct solutions of the convection equations for the laminar boundary layer on the upstream portion of a flat... 

The parameter j is a measure of the lag to achieve a uniform heating rate and is associated with the position of the cold spot or slowest heating point, the can size, and the IT (Ball and Olson, 1957) basically, these three factors determine the time to achieve a uniform heating rate. Although is usually defined as the time required to traverse one logarithmic cycle on the temperature scale, the physical meaning of j is more complex and is associated with the mode of heat transfer. Ball and Olson (1957) derived analytical solutions for h in both ideal thermal convection (Equation 8.62) and conduction (Equation 8.63), respectively ... 

Solutions of Convection Equations for a Flat Plate 376 The Energy Equation 378... 

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