Quasi-steady-state-flow

Example 15.3 The Flow Analysis Network Method Clearly Eq. E15.2-22 is identical to Eq. E15.2-21. This is the basis for the flow analysis network (FAN) method developed by Tadmor et al. (30) to solve two-dimensional steady or quasi-steady state flow problems in injection molds and extrusion dies. In two-dimensional flows the pressure distribution is obtained by dividing the flow region into an equal-sized mesh of square elements... 

Similarly, denoting by the quasi-steady-state flow-rate of particles between the secondary and the primary minimum, one can write... 

Mechanical degradation of polymers has been studied for more than 70 years in several flow fields encompassing strong elongation components. In certain flow fields the streamlines are symmetric with a stagnation point. In the vicinity of the stagnation point, the dwell time of the fluid element is longer than the timescale for coU extension. Such flow is referred to as quasi-steady-state-flow (QSSF). hi most other cases the dwell time is shorter than the coil extension time and the flow is referred to as fast-transient-flow (FTF). 

Assume a quasi-steady state flow for time period < < t. The equilibrium scour depth y, is calcu-... 

The theoretical scheme we applied here are totally as same as our previous papers [3, 7]. The non-Newtonian and non-isothermal 3-D flow analysis is conducted and the dynamic motion of the screw rotation is solved under the hypothesis of quasi steady state flow field. 

In a short time period, the dynamic model shown in Equation (3.13.1.1) at quasi-steady-state condition, OTR to microbial cells would be equal to oxygen molar flow transfer to the liquid phase.4... 

Typical for the spectroscopic character of the measurement is the rapid development of a quasi-steady state stress. In the actual experiment, the sample is at rest (equilibrated) until, at t = 0, oscillatory shear flow is started. The shear stress response may be calculated with the general equation of linear viscoelasticity [10] (introducing Eqs. 4-3 and 4-9 into Eq. 3-2)... 

It may be assumed that the accumulation of hydrogen within the pellet is negligible and that it may be treated as being in a quasi-steady-state condition. The finite difference form of Fick s first law may be used to determine the flow rate of hydrogen through the pellet. The diffusion constant appearing in this equation may be considered as an effective Knudsen diffusion coefficient. 

U-shaped curve, we have mixtures that can be ignited for a sufficiently high spark energy. From Equation (4.25) and the dependence of the kinetics on both temperatures and reactant concentrations, it is possible to see why the experimental curve may have this shape. The lowest spark energy occurs near the stoichiometric mixture of XCUi =9.5%. In principle, it should be possible to use Equation (4.25) and data from Table 4.1 to compute these ignitability limits, but the complexities of temperature gradients and induced flows due to buoyancy tend to make such analysis only qualitative. From the theory described, it is possible to illustrate the process as a quasi-steady state (dT/dt = 0). From Equation (4.21) the energy release term represented as... 

In a transported PDF simulation, the chemical source term, (6.249), is integrated over and over again with each new set of initial conditions. For fixed inlet flow conditions, it is often the case that, for most of the time, the initial conditions that occur in a particular simulation occupy only a small sub-volume of composition space. This is especially true with fast chemical kinetics, where many of the reactions attain a quasi-steady state within the small time step At. Since solving the stiff ODE system is computationally expensive, this observation suggests that it would be more efficient first to solve the chemical source term for a set of representative initial conditions in composition space,156 and then to store the results in a pre-computed chemical lookup table. This operation can be described mathematically by a non-linear reaction map ... 

The governing equation is therefore identical with that for the irrotational flow of an ideal fluid through a circular aperture in a plane wall. The stream lines and equipotential surfaces in this rotationally symmetric flow turn out to be given by oblate spheroidal coordinates. Since, from Eq. (157), the rate of deposition of filter cake depends upon the pressure gradient at the surface, the governing equation and boundary conditions are of precisely the same form as in the quasi-steady-state approximation... 

The aforementioned numerical experiments, namely quasi-static drainage and steady-state flow simulations, are specifically designed to study the influence of microstructure and wetting characteristics on the underlying two-phase behavior and flooding dynamics in the PEFC CL and GDL. 

Figure 2.21 Temperature and species concentration profiles in a reverse-flow reactor (a) flow direction is left to right (b) flow is reversed to be from right to left (c) periodic quasi-steady state [47] (by courtesy of ACS).

Example 6.14 Squeezing Flow between Two Parallel Disks This flow characterizes compression molding it is used in certain hydrodynamic lubricating systems and in rheological testing of asphalt, rubber, and other very viscous liquids.14 We solve the flow problem for a Power Law model fluid as suggested by Scott (48) and presented by Leider and Bird (49). We assume a quasi-steady-state slow flow15 and invoke the lubrication approximation. We use a cylindrical coordinate system placed at the center and midway between the plates as shown in Fig. E6.14a. 

Fig. 15.6(c)]. At the center of each element there is a node. The nodes of adjacent elements are interconnected hy links. Thus, the total flow field is represented by a network of nodes and links. The fluid flows out of each node through the links and into the adjacent nodes of the network. The local gap separation determines the resistance to flow between nodes. Making the quasi-steady state approximation, a mass (or volume) flow rate balance can be made about each node (as done earlier for one-dimensional flow), to give the following set of algebraic equations... 

The model of the slow dynamics of the system consists therefore of a set of coupled DAEs of nontrivial index, since the variables z (that physically correspond to the net material flows of the system in the slow time scale) are implicitly fixed by the quasi-steady-state constraints, rather than explicitly specified in the dynamic model. Also, note that the DAE model (4.27) has a well-defined index only if the flow rates u1 which appear in the algebraic constraints are specified as functions of the state variables x. This is typically accomplished via a control law u (x). 

Equation (5.12) effectively corresponds to the dynamics of the individual process units that are part of the recycle loop. The description of the fast dynamics (5.12) involves only the large flow rates u1 of the recycle-loop streams, and does not involve the small feed/product flow rates us or the purge flow rate up. As shown in Chapter 3, it is easy to verify that the large flow rates u1 of the internal streams do not affect the total holdup of any of the components 1,..., C — 1 (which is influenced only by the small flow rates us), or the total holdup of I (which is influenced exclusively by the inflow Fjo, the transfer rate Af in the separator, and the purge stream up). By way of consequence, the differential equations in (5.12) are not independent. Equivalently, the quasi-steady-state condition 0 = G (x)u corresponding to the dynamical system (5.12) does not specify a set of isolated equilibrium points, but, rather, a low-dimensional equilibrium manifold. 

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