Unsteady Flow Systems

The residence time distribution is normally considered a steady-state property of a flow system, but material leaving a reactor at some time 8 wiU have a distribution of residence times regardless of whether the reactor is at steady 

The washout function for an unsteady reactor is dehned as 

t) = Fraction of material leaving the reactor at time 6 that remained in the system for a duration greater than t i.e., that entered before time 9 — t. 

A simple equation applies to a variable-volume CSTR  

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Matching the flow between the impeller and the diffuser is complex because the flow path changes from a rotating system into a stationary one. This complex, unsteady flow is strongly affected by the jet-wake of the flow leaving the impeller, as seen in Figure 6-29. The three-dimensional boundary layers, the secondary flows in the vaneless region, and the flow separation at the blades also affects the overall flow in the diffuser. 

Iim88] Lim, H.A., Lattice gas automata of fluid dynamics for unsteady flow, Complex Systems 2 (1988) 45-58. 

Ozawa M, Akagawea K, Sakaguchi T (1989) Flow instabilities in paraUel-channel flow systems of gas-liquid two-phase mixtures. Int 1 Multiphase Flow 15 639-657 Peles YP (1999) VLSI chip cooling by boiling-two-phase flow in micro-channels. Dissertation, Faculty of Mechanical Engineering, Technion-Israel Institute of Technology, Haifa Peles YP, Yarin LP, Hetsroni G (2001) Steady and unsteady flow in heated capUlary. Int J Multiphase Flow 22 577-598... 

In this section we present the system of quasi-one-dimensional equations, describing the unsteady flow in the heated capillary tube. They are valid for flows with weakly curved meniscus when the ratio of its depth to curvature radius is sufficiently small. The detailed description of a quasi-one-dimensional model of capillary flow with distinct meniscus, as well as the estimation conditions of its application for calculation of thermohydrodynamic characteristics of two-phase flow in a heated capillary are presented in the works by Peles et al. (2000,2001) and Yarin et al. (2002). In this model the set of equations including the mass, momentum and energy balances is ... 

Equations (1.1) to (1.3) are diflerent ways of expressing the overall mass balance for a flow system with variable inventory. In steady-state flow, the derivatives vanish, the total mass in the system is constant, and the overall mass balance simply states that input equals output. In batch systems, the flow terms are zero, the time derivative is zero, and the total mass in the system remains constant. We will return to the general form of Equation (1.3) when unsteady reactors are treated in Chapter 14. Until then, the overall mass balance merely serves as a consistency check on more detailed component balances that apply to individual substances. 

It is also shown that the. one-dimensional, unsteady flow eqs 2.2.1 and 2.2.2 form a hyperbolic system with two characteristic directions, while the steady plane flow eqs 2.2.4 2.2.5 have the toots for... 

In conclusion to this section, research in the RTD area is always active and the initial concepts of Danckwerts are gradually being completed and extended. The population balance approach provides a theoretical framework for this generalization. However, in spite of the efforts of several authors, simple procedures, easy to use by practitioners, would still be welcome in the field of unsteady state systems (variable volumes and flow rates), multiple inlet/outlet reactors, variable density mixtures, systems in which the mass-flowrate is not conserved, etc... On the other hand, the promising "generalized reaction time distribution" approach could be developed if suitable experimental methods were available for its determination. 

Steady fiow will occur with fine powders if the target flow rate (feed rate through the system) is below the critical flow rate that occurs when the solids stress is balanced by the air pressure at the outlet. The target flow rate is often controlled by a feeder, such as at the inlet to a compression machine (press feed frame). The critical flow rate, and the flow properties tests used to determine it, is described in more later in this chapter. At target flow rates exceeding the critical flow rate, unsteady flow can occur by two different modes described below. 

Zijl, W., 1989. Three-dimensional flow systems analysis II, Steady and unsteady nested flow systems and their relation to field-scale convective-dispersive transport of chemical constituents. TNO Institute of Applied Geoscience, Delft, The Netherlands, Report no. OS 90-14... 

Gas-solid reactions are carried out on a commercial basis using fixed-bed, moving-bed, and fluidized-bed reactors. The fixed-bed reactor is an unsteady-state system as reactive gas is fed on a continuous basis through the reactor that is packed with a finite quantity of solid reactant. The solid is depleted and breakthrough of the gas reactant occurs after a certain reaction time. In the moving-bed reactor, both solid and gas are fed on a continuous basis and overall operation is steady state. The fluidized-bed reactor, where small solid particles are fluidized by upward flow of gas, also operates in a steady-state manner. Diffusional reaction resistances are reduced because of the small solid particles while solid backmixing reduces solid concentration gradients and promotes isothermal operation. 

Most of the studies mentioned above adopted a kind of A — e model to describe the liquid turbulence in the system, whereas there is less consensus regarding whether the dispersed phase should be considered turbulent or laminar, or even whether any deviating stress terms at all should remain in the dispersed phase equations. However, even the k — e model predictions are questioned by Deen et al [30] and Bove et al [15]. This group showed that only low frequency unsteady flow is obtained using the k — e model due to overestimation of the turbulent viscosity. These model predictions were found not to... 

When simulating unsteady flows and time accuracy is required, the iterations must be continued within each time step until the entire system of non-linear equations is satisfied in accordance with an appropriate convergence criterion. For steady flows, it is common either to take an infinite time step and iterate until the steady non-linear equations are satisfied, or march in time without requiring full satisfaction of the non-linear equations at each time step. However, both of these approaches may become unstable if the initial guesses are not sufficiently close to the exact solution, hence in some complex cases the time step must be restricted to ensure that the simulation does not diverge/explode. 

For unsteady flows the system of non-linear equations are linearized in the iteration process within each time step, since all the solvers are limited to linear systems. The iterative process is thus performed on two different levels. The solver iterations are performed on provisional linear systems with fixed coefficients and source terms until convergence. Then, the system coefficients and sources are updated based on the last provisional solution and a new linearized system is solved. This process is continued until the non-linear system is converged, meaning that two subsequent linear systems give the same solution within the accuracy of a prescribed criterion. A standard notation used for the different iterations within one time step is that the coefficient and source matrices are updated in the outer iterations, whereas the inner iterations are performed on provisionally linear systems with fixed coefficients. On each outer iteration, the equations solved are on the form ... 

Chaos was discovered and studied since almost a century ago, and has been mostly thought of in the context of turbulence. The concept of chaotic advection in laminar flows was introduced in the early 1980s by Aref [7]. Since then, a substantial number of investigators have demonstrated that chaotic advection occurs in a wide variety of laminar flows ranging from creeping flow to potential flow, and in different flow systems including unsteady two-dimensional flow, and both steady and time-dependent three-dimensional flows [8-12]. 

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