Convection terms

The convection term in the equation of motion is kept for completeness of the derivations. In the majority of low Reynolds number polymer flow models this term can be neglected. 

The integrals in Equation (3.32) are found using a quadrature over the element domain The viscoelastic constitutive equations used in the described model are hyperbolic equations and to obtain numerically stable solutions the convection terms in Equation (3.32) are weighted using streamline upwinding as (inconsistent upwinding)... 

Note that in a Lagrangian system the convection terms in Equation (5.11) vanish. 

In the Couette flow inside a cone-and-plate viscometer the circumferential velocity at any given radial position is approximately a linear function of the vertical coordinate. Therefore the shear rate corresponding to this component is almost constant. The heat generation term in Equation (5.25) is hence nearly constant. Furthermore, in uniform Couette regime the convection term is also zero and all of the heat transfer is due to conduction. For very large conductivity coefficients the heat conduction will be very fast and the temperature profile will... 

These three terms represent contributions to the flux from migration, diffusion, and convection, respectively. The bulk fluid velocity is determined from the equations of motion. Equation 25, with the convection term neglected, is frequently referred to as the Nemst-Planck equation. In systems containing charged species, ions experience a force from the electric field. This effect is called migration. The charge number of the ion is Eis Faraday s constant, is the ionic mobiUty, and O is the electric potential. The ionic mobiUty and the diffusion coefficient are related ... 

Equations for several methods are given here, as taken from the book by Finlayson (Ref. 107). If the convective term is treated with a centered difference expression, the solution exhibits oscillations from node to node, and these only go away if a very fine grid is used. The... 

Lagranglan codes are characterized by moving the mesh with the material motion, u = y, in (9.1)-(9.4), [24]. The convection terms drop out of (9.1)-(9.4) simplifying all the equations. The convection terms are the first terms on the right-hand side of the conservation equations that give rise to fluxes between the elements. Equations (9.1)-(9.2) are satisfied automatically, since the computational mesh moves with the material and, hence, no volume or mass flux occurs across element boundaries. Momentum and energy still flow through the mesh and, therefore, (9.3)-(9.4) must be solved. 

As the (n)th plate of the column acts as the detecting cell, there can be no heat exchanger between the (n-l)th plate and the (n)th plate of the column. As a consequence, there will be a further convective term in the differential equation that must account for the heat brought into the (n)th plate from the (n-l)th plate by the flow of mobile phase (dv). Thus, heat convected from the (n-l)th plate to plate (n) by mobile phase volume (dv) will be... 

Substituting for f(v) from equation (53) in equation (47) and inserting the extra convection term from (54),... 

It should also be remembered that the discretization scheme influences the accuracy of the results. In most CFD codes, different discretization schemes can be chosen for the convective terms. Usually, one can choose between first-order schemes (e.g., the first-order upwind scheme or the hybrid scheme) or second-order schemes (e.g., a second-order upwind scheme or some modified QUICK scheme). Second-order schemes are, as the name implies, more accurate than first-order schemes. However, it should also be remembered that the second-order schemes are numerically more unstable than the first-order schemes. Usually, it is a good idea to start the computations using a first-order scheme. Then, when a converged solution has been obtained, the user can continue the calculations with a second-order scheme. 

The convection term is given by convective heat transfer coefficient inside surface temperature of wall element g 6, room air temperature) ... 

Equation (15) is derived under the assumption that the amount of adsorbed component transferred by flow or diffusion of the solid phase may be neglected. This assumption is clearly justified in cases of fixed-bed operation, and it is believed to be permissible in many cases of slurries or fluidized beds, since the absolute amount of adsorbed component will probably be quite low due to its low diffusivity in the interior of the catalyst pellet. The assumption can, however, be waived by including in Eq. (15) the appropriate diffusive and convective terms. 

Delete the radial convection term but otherwise run the full simulation. This gives avgC = 0.5197. Now add the radial term to get 0.5347. The change is in the correct direction since velocity profile elongation hurts conversion. 

In order to increase the accuracy of the approximation to the convective term, not only the nearest-neighbor nodes, but also more distant nodes can be included in the sum appearing in Eq. (37). An example of such a higher order differencing scheme is the QUICK scheme, which was introduced by Leonard [82]. Within the QUICK scheme, an interpolation parabola is fitted through two downstream and one upstream nodes in order to determine O on the control volume face. The un-... 

When transient problems are considered, the time derivative appearing in Eq. (32) also has to be approximated numerically. Thus, besides a spatial discretization, which has been discussed in the previous paragraphs, transient problems require a temporal discretization. Similar to the discretization of the convective terms, the temporal discretization has a major influence on the accuracy of the numerical results and numerical stability. When Eq. (32) is integrated over the control volumes and source terms are neglected, an equation of the following form results ... 

The diffusive and convective terms in Eq. (20-10) are the same as in nonelectrolytic mass transfer. The ionic mobility Uj, (g mol cm )/(J-s), can be related to the ionic-diffusion coefficient D, cmVs, and the ionic conductance of the ith species X, cmV(f2-g equivalent) ... 

In these equations the designation for dimensionless concentration c, has been dropped. Note that in the above equation, the finite differencing of the convection term has been done over two neighbouring segments. Again special relationships apply to the end segments, owing to the absence of axial dispersion, exterior to the cake. 

The model equations follow from Sec. 4.5.2.2. Here, except for the end sections, the average temperatures of the sections are used to calculate the convection terms. 

Strictly speaking, in this formulation the effective diffusion coefficient, is replaced by an empirical dispersion coefficient, D, to account for the effect of water flow on diffusion. However, in practice, the rate of transpirational water flow is sufficiently slow that dispersion effects are minimal and Eq. (8) can be used without error. This is because the Peclet number (see Sect. F.2) is small. For the same reason, in almost all cases diffusion is the most important process in moving nutrients to the root and the convection term can be omitted entirely. 

Hence by substitution of the convection term and from Fick s second law of diffusion (eqn. 3.2), we obtain... 

Under common experimental conditions the convective term does not contribute significantly to the temperature profile, and thus Eq. (27) can be simplified to... 

An example of a vapor pressure profile is shown in Figure 11, where it is assumed that the relative humidity within the chamber is 80%, the critical relative humidity of the solid is 40%, and the thickness of the diffusion layer (8) is 1 cm. From the figure, note that the relative humidity profile is linear and we could have made the simplifying assumption that the convective term is negligible. By ignoring the convective term, Eq. (42) simplifies to... 

All the methods have a limit to the time step that is set by the convection term. Essentially, the time step should not be so big as to take the material farther than it can go at its velocity. This is usually expressed as a Courant number limitation. 

The correlation was tested against nine experimental sets of data with a mean-square error of 22%. Visser and Valk (1993) subsequently modified the particle convection term of the model of Borodulya et al. for low gas velocities. Their results indicated improved agreement for low velocity ranges. 

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). 

In this Eq. (Js)n is the Jacobi matrix for the solid phase, which contains the derivatives of the mass residuals for the particulate phase to the solid volume fraction. Explicit expressions for the elements of the Jacobi matrix can be obtained from the continuity for the solid phase and the momentum equations. For example for the central element, the following expression is obtained from the solid phase continuity equation, in which the convective terms are evaluated with central finite difference expressions ... 

Thus in these equations for the steady-state, the convective terms are balanced by the transfer terms. Here the KLa value is based on the total volume Vx and is assumed to have the same value for each component. The linear velocities, expressed in terms of the volumetric flow rates and the empty tube cross-section, are for the gas and liquid phases... 

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