Constant-density flow

This is the same result as for residence time t in constant-density flow, equation 2.4-8. However, in this case, density is not constant through the PFR, and the result for T is different from that for t obtained in (a). 

For a first-order, single-phase reaction (A - products, (—rA) = a a) taking place in the tanks, calculate the fiactional conversion of A(/a) leaving the second tank based on the SFM, if q = 0.5 m3 min-1 (constant-density flow), V = 10 m3, and jfcA = 0.1 min-1. 

For the constant-density flows considered in this work,27 the fundamental governing equations are the Navier-Stokes equation for the fluid velocity U (Bird et al. 2002) ... 

Although this choice excludes combustion, most of the modeling approaches can be directly extended to non-constant-density flows with minor modifications. 

This expression is found from (1.27) using the continuity equation for a constant-density flow ... 

For a constant-density flow, the continuity equation is linear, and reduces to V U = 0. Reynolds averaging then yields... 

For constant-density flows, the probabilities are equal to the volume fractions. 

We again assume constant-density flow so that U is solenoidal. The first equality requires this assumption. 

For constant-density flows, the scalar fields have no direct effect on the velocity field so that the conditional acceleration is independent of composition ... 

The primary task in Lagrangian PDF modeling is thus to find appropriate functional forms for the drift (a, a >) and diffusion (Bj/u, Bj/, B j/, B ) coefficients. Fortunately, in many applications only a small sub-set of the coefficients will be non-zero. For example, for constant-density flows, au and Bf/f/ are independent of ( ), and both Bjand B u are... 

For constant-density flow, the fundamental relationship (Pope 2000) between the fluid-particle PDF and the Eulerian PDF of the flow is... 

From the continuity equation, it then follows for constant-density flow that the PDF of X (t) is uniform 128... 

Thus, in summary, the two necessary conditions for correspondence between the notional-particle system and the fluid-particle system in constant-density flows are... 

In order for condition (2) to hold, it will be necessary to choose the coefficients properly in (6.153) and (6.154). We will return to this question below. First, however, note that for constant-density flow, the first line of (6.167) leads to a relationship between the one-point Eulerian moments and the conditional Lagrangian moments. For example, let Q(V, ip) be an arbitrary function. It follows that ... 

At the end of the chemical-reaction step, all particle properties (w n>, X(n), fl(n>) have been advanced in time to t + At. Particle-field estimates of desired outputs can now be constructed, and the MC simulation is ready to perform the next time step. For a constant-density flow, the particle-field estimates are not used in the FV code. Thus, for stationary flow, the particle properties can be advanced without returning to the FV code. For unsteady or variable-density flow, the FV code will be called first to advance the turbulence fields before calling the PDF code (see Fig. 7.3). 

For simplicity, we shall assume that (U)(x, t) and governing equations solved by the FV code are given by... 

In constant-density flows,48 the particle fields must be used to find estimates for the Reynolds stresses uu X and the composition means 0 X4. The latter can be found using any of the methods described earlier (e.g., LCME or LLME). Estimates for the Reynolds-stress tensor are needed at the grid nodes in the FV code. As in (7.46), bi-linear basis functions can be employed to estimate each component at grid node xa ... 

For steady, constant density flow systems, the internal and external NPDs, in a parallel fashion to RTD, can be shown to be related as follows ... 

The energy equation for constant density flow becomes... 

Note the space-time dependence for the perturbation field is without any restrictions, at this stage. In the absence of body force for constant density flows, the governing equations for the disturbance held in small perturbation analysis is obtained from the linearized perturbation equations given by. 

When deriving these expressions, it was assumed that velocity at all the cell faces is positive. In other cases, suitable modifications to include appropriate upstream nodes (in place of 0ww and 0ss) should be made. It can be seen that the continuity equation indicates that the last term inside the bracket of Eq. (6.19) will always be zero for constant density flows. The behavior of numerical methods depends on the source term linearization employed and interpolation practices. Before these practices are discussed, a brief discussion of the desired characteristics of discretization methods will be useful. The most important properties of the discretization method are ... 

For calculating one-dimensional flow behavior in a packed bed, the Ergun equation is generally sufficient. For single-phase, constant-density flow in two or three dimensions, Darcy s law can be substituted into the continuity equation (V v = 0) and simplified to obtain the Laplace equation ... 

An important group of fluid mechanics problems involves ID, incompressible, steady flow with friction and shaft work. Included in this category are constant density flows through pumps and agitators. 

The relative importance of the specific physical phenomena mentioned strongly depends on the type of flow under consideration. In this section, the discussion is limited to single-phase, constant density flows under isothermal conditions with constant viscosity and equal diffusivities. The emphasis is placed on the modeling of turbulent mixing and on the interactions between turbulent mixing and chemical reactions in non-premixed turbulent reacting flows. 

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