Variable density flow

Solution It is easy to begin the solution. In piston flow, molecules that enter together leave together and have the same residence time in the reactor, t. When the kinetics are first order, the probabiUty that a molecule reacts depends only on its residence time. The probability that a particular molecule will leave the system without reacting is exp(— F). For the entire collection of molecules, the probability converts into a deterministic fraction. The fraction unreacted for a variable density flow system is... 

For variable-density flows, the transport equation for the density-weighted PDF is used as the starting point. The resulting PDF codes use the particle mass as an intrinsic random variable. The particle density and specific volume can be computed based on the particle properties. 

The random selection in step (iii) is carried out by generating uniform random numbers U e [0, 1], For example, the index of a random particle selected from a set of N particles will be n = intup(//N) where intuP() rounds the argument up to the nearest integer. Note that for constant-density, statistically stationary flow, the effective flow rates will be constant. In this case, steps (i) and (ii) must be completed only once, and the MC simulation is advanced in time by repeating step (iii) and intra-cell processes. For variable-density flow, the mean density field ((p)) must be estimated from the notional particles and passed back to the FV code. In the FV code, the non-uniform density field is held constant when solving for the mean velocity field.15... 

In variable-density flows, this relation is used to couple the PDF code to the FV code by replacing the mean density predicted by the FV code with p X. Because the convergence behavior of the FV code may be sensitive to errors in the estimated mean density, particle-number control is especially critical in variable-density Lagrangian PDF codes. 

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 variable-density flows, the LCME is appropriate for estimating cell-centered particle fields that are passed back to the FV code. 

For variable-density flow, two additional mean equations are added to solve for the mean density and the energy (Jenny et al. 2001 Muradoglu et al. 2001). 

See Jenny el al. (2001) for a discussion of particle-field estimation for variable-density flows. 

For variable-density flow, Muradoglu etal. (2001) identify a third independent consistency condition involving the mean energy equation. 

Gupta, N. Bair, E. S. 1997. Variable-density flow in the midcontinent Basin and Arches Region of the United States. Water Resources Research, 33, 1785-1802. 

It is useful to think of two different kinds of forces, one that acts over the volume of a fluid element and the other that acts on the element s surface. The most common body force is exerted by the effect of gravity. If an element of fluid is less dense than its surroundings (e.g., because it is warmer), then a volumetric force tends to accelerate it upward—hot air rises. Other fields (e.g., electric and magnetic) can exert volumetric body forces on certain fluids (e.g., ionized gases) that are susceptible to such fields. Here we are concerned mostly with the effect of gravity on variable-density flows,... 

Chemical reactions may also affect turbulence by releasing energy and modifying the fluid properties locally. The influence can be quite significant in variable density flows (e.g. combustion). Nevertheless, in many computational models of constant density reactive flow processes, it is implicitly assumed that chemical reactions do not affect scalar mixing rates. 

The third system constitutes a low Mach number internal gas flow of a reactive mixture, i.e., a variable density flow [77]. 

In this book variable density flows are considered compressible in accordance with the view of [119], i.e., even though the density variations are not necessarily induced by pressure or compressibility effects, the terms variable density flow and compressible flow are then equivalent phrases. This view may not agree with all the textbooks of classical gas d3mamics. 

The transient term vanishes for incompressible flows, as the fluid properties are constant. The transient term is retained here to emphasize that this method can be applied for compressible flows as well. For compressible flow both velocity and density appear as dependent variables in the continuity equation. Nevertheless, the discrete form of the mass balance can still be interpreted as a constraint equation for pressure. Compressible and reactive variable density flows are considered shortly in the present section. 

Jakobsen et al [81] and Lindborg et al [118] did apply similar algorithms for variable density flows to simulate the performance of chemical processes in fixed bed reactors. The fractional steps defining the elements of these algorithms are sketched in the following ... 

For reactive variable density flows, the pressure is interpreted as a projection operator which projects an arbitrary vector field into a vector field which fulfill the continuity equation. 

In variable-density flow, additional estimated terms are needed for the mean energy equation (Jenny etal. 2001). 

Numerical studies have been carried out in three-dimensional (x, y, z) geometry. The system of equations of the mathematical model includes the continuity equation, a generalization of Darcy law for the case of variable density flow, equation of thermal energy conservation, and closing relationships for the calculation of the pore solution density and viscosity. 

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