Derivation of transport equation

In contrast to moment closures, the models used to close the conditional fluxes typically involve random processes. The choice of the models will directly affect the evolution of the shape of the PDF, and thus indirectly affect the moments of the PDF. For example, once closures have been selected, all one-point statistics involving U and 0 can be computed by deriving moment transport equations starting from the transported PDF equation. Thus, in Section 6.4, we will look at the relationship between (6.19) and RANS transport equations. However, we will first consider the composition PDF transport equation. 

We have seen that the joint velocity, composition PDF treats both the velocity and the compositions as random variables. However, as noted in Section 6.1, it is possible to carry out transported PDF simulations using only the composition PDF. By definition, / (0 x, t) can be found from /u, (V, 0 x, t) using (6.3). The same definition can be used with the transported PDF equation derived in Section 6.2 to find a transport equation for /0(t/ x, t). 

The transport equation for the joint composition PDF can be found by integrating the velocity, composition PDF transport equation, (6.19), over velocity phase space  

Taking each contribution individually yields the following simplifications. Accumulation  

Collecting all terms, the composition PDF transport equation reduces to 

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Problem 2-13. Derivation of Transport Equations. Consider the arbitrary fluid element depicted in the figure. If we have a flow containing several species that are undergoing reaction (a source/sink per unit volume) and diffusion (a flux of each species in addition to convection), derive the equation that governs the conservation of each species. The source of species i that is due to reaction is denoted as Rt (units of mass of i per unit time per unit volume) and the total mass flux of species i (diffusion and convection) is given by (p u + ji), in which p, is the mass of species i per unit volume, u is the total mass average velocity of the fluid and j, is the diffusive flux of species i. Note that both u and j, are vectors. We are not using index notation in this problem ... 

Problem 2-15. Derivation of Transport Equation for a Sedimenting Suspension. There are many parallels among momentum, mass, and energy transport because all three are derived from similar conservation laws. In this problem we derive a microscopic balance describing the concentration distribution (x, t) of a very dilute suspension of small particles suspended in an incompressible fluid undergoing unsteady flow. [Note cj>(. t) is the local volume fraction of particles in the fluid (i.e. volume of particles/volume of fluid) and hence is dimensionless. ... 

Grahman, T, Philosophical Transactions of the Royal Society of London A 140, 1, 1850. Gray, WG, A Derivation of the Equations for Multiphase Transport, Chemical Engineering Science 30, 229, 1975. 

Basic elements of transport equations are the laws expressing conservation of mass and conservation of momentum. The former is self-evident the latter derives from Newton s second law (stating that the sum of forces acting on a system equals the rate of production of momentum in that system). For details on the basic premises and features, refer to the specialised literature [24—29]. 

The model turbulent energy spectrum given in (2.53) was introduced to describe fully developed turbulence, i.e., the case where / , (/<. t) does not depend explicitly on t. The case where the turbulent energy spectrum depends explicitly on time can be handled by deriving a transport equation for the velocity spectrum tensor 4> (k, t) starting from the Navier-Stokes equation for homogeneous velocity fields with zero or constant mean velocity (McComb 1990 Lesieur 1997). The resultant expression can be simplified for isotropic turbulence to a transport equation for / ,(/<. t) of the form14... 

As discussed in Section 2.1, in high-Reynolds-number turbulent flows the scalar dissipation rate is equal to the rate of energy transfer through the inertial range of the turbulence energy spectrum. The usual modeling approach is thus to use a transport equation for the transfer rate instead of the detailed balance equation for the dissipation rate derived from (1.27). Nevertheless, in order to understand better the small-scale physical phenomena that determine e, we will derive its transport equation starting from (2.99). 

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... 

The derivation of the transport equation for g xg,p is analogous to that used to derive the transport equation for the scalar covariance. The resultant expression is... 

Other two-equation models have been developed using various combinations of k and e to derive alternative transport equations to replace (4.47). The most popular is, perhaps, the k-co model (see Wilcox (1993) for a detailed discussion of its advantages), where the turbulence frequency co = e/k is used in place of s. The standard transport equation for... 

A joint statistic of particular interest is the scalar flux uitransport equation, and to compare the result to (3.102) on p. 84. 

Clearly, the actual pressure head in each phase depends on the fluid configuration within the pores. Hux equations for each of the three phases can be combined with mass conservation equations to derive governing transport equations. 

The particle turbulent kinetic energy is governed by its own transport equation. Similarly to the derivation of the -equation, the p-equation is given by... 

Derivations of conservation equations from the viewpoint of kinetic theory usually do not exhibit explicitly the diffusion terms, such as diffusion stresses, that appear on the right-hand sides of equations (49), (50), and (51), since it is unnecessary to introduce quantities such as afj specifically in these derivations. Kinetic-theory developments work directly with the left-hand sides of equations (49), (50), and (51). Transport coefficients (Appendix E) are defined only in terms of these kinetic-theory quantities because prescriptions for calculating the individual continua transports, afj and qf, are unduly complex. Moreover, measurement of diffusion stresses is feasible only by direct measurement of diffusion velocities, followed by use of equation (24). Therefore, it has not been fruitful to study the diffusion terms which, in a sense, may be viewed as artifacts of the continuum approach. 

In the most systematic application of this approach, Harlow and co-workers at Los Alamos have derived a transport equation for the full Reynolds stress tensor pu u j. They have coupled this equation with a scalar dissipation transport equation and have utilized with various semi-empirical approximations to evaluate the numerous unknown velocity, velocity-pressure, and velocity-temperature correlations which appear in the formulation. While this treatment is fairly vigorous, extensive compu-... 

The special features of the different routes of administration are dealt with in separate sections of this chapter, after a brief summary of the general properties of biological membranes and drug transport, a knowledge of which is important in understanding all absorption processes. It is impossible to be comprehensive in this one chapter, but we will concentrate on factors unique to the routes discussed, such as the properties of the vehicle in topical therapy, and the aerodynamic properties of aerosols in inhalation therapy, to give a flavour of the different problems that face formulators. Where attempts have been made to quantify absorption, equations are presented, but the derivations of most equations have been omitted. 

We may easily carry out a linear response theory derivation of transport properties based on the quantum-classical Liouville equation that parallels the... 

Darcy s law describes fluid flux in porous media, and must be combined with the continuity equation to develop flow equations. From the flow equations, the spatial and temporal pressure and velocity distributions can be estimated that are needed for the transport equations. The derivation of flow equations starts with the continuity equation, which states that the change in mass or volume within a control volume equals the net flux across the control volume boundary, plus sources and sinks within the control volume. For water within porous media, the continuity equation on a mass basis is ... 

A this point a few mathematical prerequisites are required before the derivation of the equations of change can be discussed. First, the three kinds of time derivatives involved in this task are defined. Secondly, the transport theorem is introduced. 

This is the total energy equation, for which the potential energy term is expressed in terms of the external force F. By use of the momentum equation we can derive a transport equation for the mean kinetic energy, and thereafter extract the mean kinetic energy part from the equation (i.e, the same procedure was used manipulating the continuum model counterpart in chap. 1, sect. 1.2.4). The result is ... 

Gray WG (1975) A Derivation of the Equations for Multi-Phase Transport. Chem Eng Sci 30 229-233... 

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