Transport equations closed forms

Assuming local thermal equilibrium, i.e. the equality of the averaged fluid and solid temperature, a transport equation for the average temperature results which still contains and integral over the fluctuating component. In order to close the equation, a relationship between the fluctuating component and the spatial derivatives of the average temperature of the form... 

These two transport equations for k and e form an inherent part of any k i model of RANS-simulations. As the result of closing the turbulence modeling such that no further unknown variables and equations are introduced, the e-equation does contain some terms that are still the result of modeling, albeit at the very small scales (e.g., Rodi, 1984). 

The application of DQMOM to the closed composition PDF transport equation is described in detail by Fox (2003). If the IEM model is used to describe micromixing and a gradient-diffusivity model is used to describe the turbulent fluxes, the CFD model will have the form... 

Equation 20.24 can be solved analytically to give closed-formed answers to simple problems. Many mass transport problems, however, including all but the most straightforward in reactive transport, require the equation to be evaluated numerically (e.g., Phillips, 1991). There are a variety of methods for doing so, including... 

Because die outlet concentrations will not depend on it, micromixing between duid particles can be neglected. The reader can verify this statement by showing that die micromixing term in the poorly micromixed CSTR and the poorly micromixed PFR falls out when die mean outlet concentration is computed for a first-order chemical reaction. More generally, one can show that die chemical source term appears in closed form in die transport equation for die scalar means. 

This assumption leads to relationships for a and b of the form of (5.144) and (5.145), respectively, but with the mixture-fraction mean and variance replaced by the mean and variance of Y. The joint PDF can thus be closed by solving four transport equations for... 

Note that due to the assumption that f and Y are independent, a covariance transport equation is not required to close the chemical source term. However, it would be of the form of (3.136) and have a non-zero chemical source term (f S (T, f). Thus, since the covariance should be null due to independence, the covariance equation could in theory be solved to check the validity of the independence assumption. 

Transported PDF methods combine an exact treatment of chemical reactions with a closure for the turbulence field. (Transported PDF methods can also be combined with LES.) They do so by solving a balance equation for the joint one-point, velocity, composition PDF wherein the chemical-reaction terms are in closed form. In this respect, transported PDF methods are similar to micromixing models. 

We shall see that a conditional acceleration model in the form of (6.48) is equivalent to a stochastic Lagrangian model for the velocity fluctuations whose characteristic correlation time is proportional to e/k. As discussed below, this implies that the scalar flux (u,

joint velocity, composition PDF level, and thus that a consistent scalar-flux transport equation can be derived from the PDF transport equation. 

To understand the behavior of the movement of the contaminant in ground-water, people solve Eq. (1) forward in time. In solving this equation forward in time, one assumes that the plume is originated from somewhere and will travel through the porous media due to advection and dispersion. The conventional procedure to solve Eq. (1) is to use finite difference or finite element methods. For simple cases, closed-form solutions exist. Quantitative descriptions of the processes forward in time are well understood. Multidimensional models of these processes have been used successfully in practice [50]. Numerical solute transport models were first developed about 25 years ago. When properly applied, these models can provide useful information about transport processes and can assist in the design of remedial programs. 

While it may be elegant to obtain analytic closed-form model solutions, such as Equations (8.52) and (8.53) (introduced by Sangren and Sheppard as solutions to their model governing equations [178]), modeling of transport in biological systems... 

Optimum temperatures are obtained from the analytical expressions describing the optimum conditions, along with the derivative of the centerline reactant fraction with respect to the deposition modulus. Using an analytical solution to provide the derivative of the centerline reactant fraction, a closed-form implicit expression for the optimum temperature is obtained for the special case of no normalized reaction yield. For the more general case, this derivative is computed from numerical solutions to the equations governing transport and deposition. Optimum temperatures are presented graphically for a very wide range of the normalized preform thickness and normalized reaction yield. 

In order to close the set of modeled transport equations, it is necessary to estimate turbulent viscosity or if the k-e model is used, the turbulent kinetic energy, k and turbulent energy dissipation rate, s. The modeled forms of the liquid phase k and s transport equations can be written in the following general format (subscript 1 denotes... 

Work directly with the unclosed terms in the moment-transport equations to find a functional form to close them (Struchtrup, 2005). For example, a spatial flux involving moment might be closed using a gradient-diffusion model involving... 

A popular method for closing a system of moment-transport equations is to assume a functional form for the NDF in terms of the mesoscale variables. Preferably, the parameters of the functional form can be written in closed form in terms of a few lower-order moments. It is then possible to solve only the transport equations for the lower-order moments which are needed in order to determine the parameters in the presumed NDF. The functional form of the NDF is then known, and can be used to evaluate the integrals appearing in the moment-transport equations. As an example, consider a case in which the velocity NDF is assumed to be Gaussian ... 

The integral terms in parentheses are known for the family of orthogonal polynomials. With finite N and known moments, this system of linear equations has the form M = AC and can be solved to find fhe expansion coefficients Ca(t, x). Thus, the presumed NDF n (t, X, Vp) is a unique function of a finite set of moments, and the latter are found by solving the moment-transport equations using n to close the unclosed terms. The fact that... 

A increasingly popular method for closing the moment-transport equations is to assume a discrete form for the phase-space variables. Taking the velocity NDF as an example, the velocity phase space can be discretized on a uniform, symmetric lattice centered at Vp = 0. For illustration purposes, let us assume that A = 16 lattice points are used and denote the corresponding velocities as Ua. The formal definition of the discrete NDF is... 

The moment-transport equations discussed above become more and more complicated as the order increases. Moreover, these equations are not closed. In quadrature-based moment methods, the velocity-distribution function is reconstructed from a finite set of moments, thereby providing a closure. In this section, we illustrate how the closure hypothesis is applied to solve the moment-transport equations with hard-sphere collisions. For clarity, we will consider the monodisperse case governed by Eq. (6.131). Formally, we can re-express this equation in conservative form ... 

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