Transport equation inert

Like the turbulent energy spectrum discussed in Section 2.1, a transport equation can be derived for the scalar energy spectrum lipjn. t) starting from (1.27) and (1.28) for an inert scalar (see McComb (1990) or Lesieur (1997) for details). The resulting equation is21... 

The scalar spectral transport equation can be easily extended to the cospectrum of two inert scalars Eap(K, t). The resulting equation is... 

For turbulent mixing of an inert scalar mean scalar transport equation reduces to... 

Like the Reynolds stresses, the scalar flux obeys a transport equation that can be derived from the Navier-Stokes and scalar transport equations. We will first derive the transport equation for the scalar flux of an inert scalar from (2.99), p. 48, and the governing equation for inert-scalar fluctuations. The latter is found by subtracting (3.89) from (1.28) (p. 16), and is given by... 

Reynolds averaging of (3.91) then yields the transport equation for the inert-scalar flux 27... 

The transport equation for the variance of an inert scalar (

and Reynolds averaging the resultant expression. This process leads to an unclosed term of the form 2[Pg.103]

The inert-scalar-variance transport equation can then be written as... 

The transport equation for the scalar dissipation rate of an inert scalar can be derived starting from (3.90). We begin by defining the fluctuating scalar gradient as... 

Table 3.2. The scalar statistics and unclosed quantities appearing in the transport equations for inhomogeneous turbulent mixing of an inert scalar. 

The scalar statistics used in engineering calculations of high-Reynolds-number turbulent mixing of an inert scalar are summarized in Table 3.2 along with the unclosed terms that appear in their transport equations. In Chapter 4, we will discuss methods for modeling the unclosed terms in the RANS transport equations. 

The first factor occurs even in homogeneous flows with two inert scalars, and is discussed in Section 3.4. The second factor is present in nearly all turbulent reacting flows with moderately fast chemistry. As discussed in Chapter 4, modeling the joint scalar dissipation rate is challenging due to the need to include all important physical processes. One starting point is its transport equation, which we derive below. 

The filtered transport equation for an inert, passive scalar has the form... 

The transport equation for the mean of an inert scalar was derived in Section 3.3 ... 

The failure of first-order moment closures for the treatment of mixing-sensitive reactions has led to the exploration of higher-order moment closures (Dutta and Tarbell 1989 Heeb and Brodkey 1990 Shenoy and Toor 1990). The simplest closures in this category attempt to relate the covariances of reactive scalars to the variance of the mixture fraction (I 2). The latter can be found by solving the inert-scalar-variance transport equation ((3.105), p. 85) along with the transport equation for (f). For example, for the one-step reaction in (5.54) the unknown scalar covariance can be approximated by... 

The three-dimensional transport equation for inert pollutant dispersion results from timesmoothing the equation of continuity of the emitted substance. In Cartesian coordinates the distribution of a pullutant is given by the partial differential equation of second order for the concentration C(x, y, z, t) 111 ... 

Eqs. (l)-(5) are still the basic sorption and transport equations used today for "ideal systems, penetrant-polymer systems in which both (Jo and Do are pressure and concentration independent. This "ideal" behavior is observed in sorption and transport of permanent and inert gases in polymers well above their Tg. 

In those limited cases where the diffusion equation for the membrane/support composite is solved in conjunction with the governing mass transport equations for the tube and shell sides [Sun and Khang, 1988 and 1990 Agarwalla and Lund, 1992], Equation (10-5) applies with = 0 for catalytically inert membranes. In addition, either Equations (10-36) for plug flows or Equation (10-54) for perfect mixing needs to be solved for the... 

Dispersion Models Based on Inert Pollutants. Atmospheric spreading of inert gaseous contaminant that is not absorbed at the ground has been described by the various Gaussian plume formulas. Many of the equations for concentration estimates originated with the work of Sutton (3). Subsequent applications of the formulas for point and line sources state the Gaussian plume as an assumption, but it has been rigorously shown to be an approximate solution to the transport equation with a constant diffusion coefficient and with certain boundary conditions (4). These restrictive conditions occur only for certain special situations in the atmosphere thus, these approximate solutions must be applied carefully. 

In a series of papers Lathouwers and Bellan [43, 44, 45, 46] presented a kinetic theory model for multicomponent reactive granular flows. The model considers polydisersed particle suspensions to take into account that the physical properties (e.g., diameter, density) and thermo-chemistry (reactive versus inert) of the particles may differ in their case. Separate transport equations are constructed for each of the particle types, based on similar principles as used formulating the population balance equations [61]. 

In scalar mixing studies and for infinite-rate reacting flows controlled by mixing, the variance of inert scalars is of interest since it is a measure of the local instantaneous departure of concentration from its local instantaneous mean value. For non-reactive flows the variance can be interpreted as a departure from locally perfect mixing. In this case the dissipation of concentration variance can be interpreted as mixing on the molecular scale. The scalar variance equation (1.462) can be derived from the scalar transport equation... 

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