Diffusion second-order moment

Second-Order Moment. The linearity of y U /2L vs. l/t/B2 is shown in Figure 4. From the slope of the straight line, the axial dispersion coefficient D can be calculated. With the assumption that kR = Z)Ab, Da and Di can be calculated from the second and third terms in the bracket of the right-hand side of Equation 6 by varying the particle size. The results are given in Table II. As expected, both inter- and intracrystalline diffusion coefficients increase with temperature. The values obtained for Di in Na mordenite are somewhat smaller than those obtained by Satterfield and Frabetti (7) and Satterfield and Margetts (8) which were obtained at a lower temperature. However, Frabetti reported that diffusion co-... 

Hong et al. [45] used numerical solutions of the FOR model, under linear conditions, to determine the internal diffusion coefficient of rubrene in the particles of Symmetry-Ci8, with methanol/water solutions (90 to 100% methanol) as the mobile phase. The results derived from the analytical solution of the model in the Laplace domain and from the first and second order moments were in excellent agreement. 

A large number of explicit numerical advection algorithms were described and evaluated for the use in atmospheric transport and chemistry models by Rood [162], and Dabdub and Seinfeld [32]. A requirement in air pollution simulations is to calculate the transport of pollutants in a strictly conservative manner. For this purpose, the flux integral method has been a popular procedure for constructing an explicit single step forward in time conservative control volume update of the unsteady multidimensional convection-diffusion equation. The second order moments (SOM) [164, 148], Bott [14, 15], and UTOPIA (Uniformly Third-Order Polynomial Interpolation Algorithm) [112] schemes are all derived based on the flux integral concept. 

For simplicity, let mo(t,x) = 1, m t,x) so that the variance at x = 0 is initially zero. The exact solution for the second-order moment is miit x) = 2t + x. The corresponding weights and abscissas are Wi(t,x) = W2(t,x) = 1/2 and i(f,x) = -fiit x) = V2t + x for 0 < t. Thus, atr = 0, the abscissas form an X that separates into two parts for 0 < t. Obviously, the same thing could occur for higher-order moments, and the DQMOM must be modified to treat such cases. In practice, this diffusion-induced change in the number of degenerate moments occurs for problems in which one of the internal coordinates is generated by a source term, for example, for products of a chemical reaction. 

In the case of weak collisions, the moment changes in small steps AJ (1 — y)J < J, and the process is considered as diffusion in J-space. Formally, this means that the function /(z) of width [(1 — y2)d]i is narrow relative to P(J,J, x). At t To the latter may be expanded at the point J up to terms of second-order with respect to (/ — /). Then at the limit y -> 1, to — 0 with tj finite, the Feller equations turn into a Fokker-Planck equation... 

As discussed in Chapter 5, the complexity of the chemical source term restricts the applicability of closures based on second- and higher-order moments of the scalars. Nevertheless, it is instructive to derive the scalar covariance equation for two scalars molecular-diffusion coefficients ra and I, respectively. Starting from (1.28), p. 16, the transport equation for ((,) can be found following the same steps that were used for the Reynolds stresses. This process yields34... 

The authors have studied the properties of this density in detail [24, 33] and have observed some similarities between the PF and the AMC as well as some differences. Both these methods are utilized in the context of a single-point closure. Therefore, in both cases the magnitudes of moments up to second order must be provided externally. Also, neither method accounts for migration of the scalar s bounds as mixing proceeds. This is portrayed by the evolution of the conditional statistics of the scalar namely, the conditional expected dissipation and the conditional expected diffusion [41, 42]. 

As one might expect, the Frost model gives rather poor bond quadrupole moments owing to the diffuse nature of spherical gaussian charge distributions. Amos et al, have additionally extended the calculations to second-order bond properties.64... 

Equations (8.10)—(8.12), tensorial ranks and boundary conditions (8.14)-(8.15) notwithstanding, embody a structure similar in format and symbolism to their counterparts for the transport of passive scalars, e.g., the material transport of the scalar probability density P (Brenner, 1980b Brenner and Adler, 1982), at least in the absence of convective transport. As such, by analogy to the case of nonconvective material transport, the effective kinematic viscosity viJkl of the suspension may be obtained by matching the total spatial moments of the probability density Pu to those of an equivalent coarse-grained dyadic probability density P j, valid on the suspension scale, using a scheme (Brenner and Adler, 1982) identical in conception to that used to determine the effective diffusivity for material transport at the Darcy scale from the analogous scalar material probability density P. In particular, the second-order total moment M(2) (sM, ) of the probability density P, defined as... 

In fact, this expression is the same as the second-order expression found when working directly with the diffusion term written using the transported moment set. In general, the second-order spatial reconstruction on unstructured grids will also be realizable. Perhaps for this reason, the diffusion term is not usually identified as a potential source of unrealizable moment sets. 

Models Considering Membrane and Liquid Film Diffusion. Models considering membrane and liquid film diffusion are quite complex as they are of second order in nature, and the solution to these models require numerical analysis or a method of moments due to their complexity (Sobotka et al., 1982). Linek et al. (1985), Ruchti et al. (1981), and Dang et al. (1977) suggested that while these models are more complex and involved, their solutions are much superior to any first-order model. However, due to their complexity, they are typically not used and the reader is referred to the literature for more information concerning these models. 

Errors in advection may completely overshadow diffusion. The amplification of random errors with each succeeding step causes numerical instability (or distortion). Higher-order differencing techniques are used to avoid this instability, but they may result in sharp gradients, which may cause negative concentrations to appear in the computations. Many of the numerical instability (distortion) problems can be overcome with a second-moment scheme (9) which advects the moments of the distributions instead of the pollutants alone. Six numerical techniques were investigated (10), including the second-moment scheme three were found that limited numerical distortion the second-moment, the cubic spline, and the chapeau function. 

The drift velocity and diffusivity for a Stratonovich SDE may be obtained by using Eq. (2.239) to calculate the first and second moments of AX to an accuracy of At). To calculate the drift velocity, we evaluate the average of the RHS of Eq. (2.244) for AX . To obtain the required accuracy of At), we must Taylor expand the midpoint value of that appears in Eq. (2.244) to first order in AX about its value at the initial position X , giving the approximation... 

In fact, if the control parameter K is large enough, the variable E, is an uncorrelated fluctuation, and, in the long-time limit, it can be thought of as a noisy function of the continuous time t. The solution of Eq. (270) yields results agreeing with ordinary statistical mechanics, namely, a diffusion process making the second moment of x increase as a linear function of time. However, in the quantum case this linear increase has an upper bound in time. At times of the order of... 

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