Chemical source term higher-order

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... 

In Chapter 5, we will review models referred to as moment methods, which attempt to close the chemical source term by expressing the unclosed higher-order moments in terms of lower-order moments. However, in general, such models are of limited applicability. On the other hand, transported PDF methods (discussed in Chapter 6) treat the chemical source term exactly. 

Thus, the ASM scalar flux in a first-order reacting flow will decrease with increasing reaction rate. For higher-order reactions, the chemical source term in (3.102) will be unclosed, and its net effect on the scalar flux will be complex. For this reason, transported PDF methods offer a distinct advantage terms involving the chemical source term are closed so that its effect on the scalar flux is treated exactly. We look at these methods in Chapter 6. 

For higher-order reactions, a model must be provided to close the covariance source terms. One possible approach to develop such a model is to extend the FP model to account for scalar fluctuations in each wavenumber band (instead of only accounting for fluctuations in In any case, correctly accounting for the spectral distribution of the scalar covariance chemical source term is a key requirement for extending the LSR model to reacting scalars. 

The problems associated with coupling packed columns to a mass spectrometer are s re severe than those encountered with capillary columns. Conventional pacdced columns are operated at much higher flow rates, 20 to 60 al/ain, and although this diminishes the influence of dead volumes in the interface on sample resolution, it poses a problem due to the pressure and volume flow rate restrictions of the mass spectrometer. The interface must provide a pressure drop between column and mass spectrometer source on the order of 10 to 10, it must reduce the volumetric flow of gas into the mass spectrometer without diminishing the mass flow of sample by the same amount, and it must retain the integrity of the sample eluting from the column in terms of the separation obtained and its chemical constitution [3,25,26]. To meet the above requirements the interface must function as a molecular separator. 

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. 

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