One-velocity transport equation

L. S. Bohl and F. D. Judge, Variational treatment of the one-velocity transport equation for an arbitrary slab lattice. Transactions of the American Nuclear Society, vol. 1, no. 2, June, 1958. 

P. B. Daitch et al., CEPTR An IBM-704 Code to Solve the P, Approximation to the One-Velocity Transport Equation in Cylindrical Geometry, Combustion Engineering report MPC-20 (Sept. 1959). 

The one-velocity calculations require as Input a collision cross section ( t). end a neutron production per collision. Both were determine from soluUons of the one- velocity transport equation for two delayed critical experiments a 7-in.-diam (17.78 cm) by 4.971-in.-hi (12 63 cm) cylinder of -the- U(93) metal and a IS-in. (38.1)cm) diam by 3.03-ln.-h di (T.Bg cm) cylinder. The solutUms, Which were obtaii from Su transport calculations with the DDK cote , using a 16 X 16. grid of space points, yielded values of vZt/7 as a function oi Zt. The pair Of values common to both experiments was then used as input to the Monte Carlo calculations for all the geometries.. ... 

In the presentation which follows, we derive the formal solution to the one-velocity transport equation (7.15) for several media of practical importance. The general treatment follows that of Case, de Hoffmann, and Placzek. The reader is referred to the work of these authors for a much more complete development of the general methods and results. 

The solution to the general time-dependent system is easily developed from the above results. In the notation of the present method, the non-stationary one-velocity transport equation is... 

For the functions 0c(r) and ji(r) which appear in (8.95) we use first-order approximations of the solutions to the transport equation. The basis for this choice is derived from the following argument. It was shown previously in Sec. 7.4d that the elementary solution to the one-velocity transport equation for the source-free infinite multiplying... 

The Eulerian gas velocity field required in both the mass balance and the above transport equation for nh is found by an approximate method first, the complete field of liquid velocities obtained with FLUENT is adapted downward because the power draw is smaller under gassed conditions next, in a very simple way of one-way coupling, the bubble velocity calculated from the above force balance is just added to this adapted liquid velocity field. This procedure makes a momentum balance for the bubble phase redundant this saves a lot of computational effort. 

As another application of this method of asymptotic integration, we shall consider the problem of the Fourier coefficients pfU(P t) i 1 the limit of long times. As mentioned above, we do not wish to give here a detailed proof of the transport equation for pk] p] t) (see, for instance, Ref. 31). The main result of this analysis is, however, very simple in the limit of long times (t —> oo), the correlations are entirely determined by the velocity distribution function p< p t). One has ... 

Thus, in Chapter 6, the transport equations for /++ x, t) and the one-point joint velocity, composition PDF /u+V, + x. / ) are derived and discussed in detail. Nevertheless, the computational effort required to solve the PDF transport equations is often considered to be too large for practical applications. Therefore, in Chapter 5, we will look at alternative closures that attempt to replace /++ x, t) in (3.24) by a simplified expression that can be evaluated based on one-point scalar statistics that are easier to compute. 

In one-point models for turbulent mixing, extensive use of conditional statistics is made when developing simplified models. For example, in the PDF transport equation for /++ x, r), the expected value of the velocity fluctuations conditioned on the scalars appears and is defined by... 

For high-Reynolds-number flows, the two terms on the right-hand side involving T will be negligible.126 The inhomogeneous transport equation will thus have only one additional term due to the conditional velocity (U f). 

The procedure followed above can be used to develop a multi-environment conditional LES model starting from (5.396). In this case, all terms in (5.399) will be conditioned on the filtered velocity and filtered compositions,166 in addition to the residual mixture-fraction vector = - . In the case of a one-component mixture fraction, the latter can be modeled by a presumed beta PDF with mean f and variance (f,2>. LES transport equations must then be added to solve for the mixture-fraction mean and variance. Despite this added complication, all model terms carry over from the original model. The only remaining difficulty is to extend (5.399) to cover inhomogeneous flows.167 As with the conditional-moment closure discussed in Section 5.8 (see (5.316) on p. 215), this extension will be non-trivial, and thus is not attempted here. 

The screw rotation analysis leads to the model equation for the extruder discharge rate. There are now two screw-rotation-driven velocities, and and a pressure-driven velocity, Pp that affect the rate. and transport the polymer fluid at right angles to one another. In order to calculate the net flow from screw rotation It Is necessary to resolve the two screw-rotation-driven velocities into one velocity, Vpi, that can be used to calculate the screw rotation-driven flow down the screw parallel to the screw axis (or centerline) as discussed in Chapter 1 and as depicted in Fig. 7.14. The resolved velocity will then be integrated over the screw channel area normal to the axis of the screw. 

The purpose of this chapter is to provide a comprehensive discussion of some simple approaches that can be employed to obtain information on the rate of heat and mass transfer for both laminar and turbulent motion. One approach is based on dimensional scaling and hence ignores the transport equations. Another, while based on the transport equations, does not solve them in the conventional way. Instead, it replaces them by some algebraic expressions, which are obtained by what could be called physical scaling. The constants involved in these expressions are determined by comparison with exact asymptotic solutions. Finally, the turbulent motion is represented as a succession of simple laminar motions. The characteristic length and velocity scales of these laminar motions are determined by dimensional scaling. It is instructive to begin the presentation with an outline of the basic ideas. 

In this expression one term vanishes because V V = 0 for an incompressible flow and Vw = 0 because the divergence of the curl of a vector vanishes (vorticity is the curl of the velocity vector). For the same reason the last term on the right-hand side of the vorticity equation also vanishes. As a result the vorticity-transport equation is further reduced to... 

A numerical analysis using FlumeCAD was made, solving the incompressible Navier-Stokes equation for the velocity and pressure fields [70], The steady-state velocity field was then used in the coupled solution of three species transport equations (two reagents and one product). Further details are given in [70],... 

One-dimensional reactive flow. As a preliminary trial problem in our computations of reactive flow, we have studied a simple situation in which a fluid obeying first-order chemical kinetics moves at constant velocity and temperature in the positive x direction. Here only the mass-transport equation is operative, and it takes the simple form ... 

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