Quasi-steady flow

Four aspects of unsteady fluid flow will be considered in this chapter quasi-steady flow as in the filling or emptying of vessels, incremental calculations, start-up of shearing flow, and pressure surge in pipelines. 

In equation 10.2, hf is the head loss due to friction and is given by the equation 

In equation 10.3, XL, is the equivalent length of the oudet pipe (including the contraction at its inlet) and d, is the pipe s inside diameter u is the mean velocity in the outlet pipe. It has been assumed in writing equations 10.1 and 10.2 that the fluid s velocity in the tank is so low that it can be neglected. 

In an infinitesimal time interval St the liquid level in the tank changes by an amount Sz and the volume of liquid in the tank by SV. Thus, equating the rate of loss of liquid from the tank to the flow rate through the exit pipe 

Note that, for this case of emptying a tank, 8z and 5F are negative. 

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When one gas diffuses into another, as A into B, even without the quasi-steady-flow component imposed by the burning, the mass transport of a species, say A, is made up of two components—the normal diffusion component and the component related to the bulk movement established by the diffusion process. This mass transport flow has a velocity Aa and the mass of A transported per unit area is pAAa. The bulk velocity established by the diffusive flow is given by Eq. (6.58). The fraction of that flow is Eq. (6.58) multiplied by the mass fraction of A, pA/p. Thus,... 

The use of the compressibility term can be described as follows. The greater the stiffness a system model has, the more quickly the flow reacts to a change in pressure, and vice versa. For instance, if all fluids in the system are incompressible, and quasi-steady assumptions are used, then a step change to a valve should result in an instantaneous equilibrium of flows and pressures throughout the entire system. This makes for a stiff numerical solution, and is thus computationally intense. This pressure-flow solution technique allows for some compressibility to relax the problem. The equilibrium time of a quasi-steady model can be modified by changing this parameter, for instance this term could be set such that equilibrium occurs after 2 to 3 seconds for the entire model. However, quantitative results less than this timescale would then potentially not be captured accurately. As a final note, this technique can also incorporate flow elements that use the momentum equation (non-quasi-steady), but its strength is more suited by quasi-steady flow assumptions. 

As an example, if only quasi-steady flow elements are used with volume pressure elements, a model s smallest volume size (for equal flows) will define the timescale of interest. Thus, if the modeler inserts a volume pressure element that has a timescale of one second, the modeler is implying that events which happen on this timescale are important. A set of differential equations and their solution are considered stiff or rigid when the final approach to the steady-state solution is rapid, compared to the entire transient period. In part, numerical aspects of the model will determine this, but also the size of the perturbation will have a significant impact on the stiffness of the problem. It is well known that implicit numerical methods are better suited towards solving a stiff problem. (Note, however, that The Mathwork s software for real-time hardware applications, Real-Time Workshop , requires an explicit method presumably in order to better guarantee consistent solution times.)... 

Denoting by Na the quasi-steady flow-rate of particles between the bulk of the fluid and the secondary minimum, one obtains... 

Timewise steady flow (or quasi-steady flow)... 

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... 

The quasi-one-dimensional model described in the previous chapter is applied to the study of steady and unsteady flow regimes in heated micro-channels, as well as the boundary of steady flow domains. The effect of a number of dimensionless parameters on the velocity, temperature and pressure distributions within the domains of liquid vapor has been studied. The experimental investigation of the flow in a heated micro-channel is carried out. 

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... 

At 100 s, the burning rate has not significantly changed, and the compartment has reached a quasi-steady condition with countercurrent flow at the doorway. At this new time, compute the following ... 

In practice the resistance of the exit pipe of the tank shown in Figure 10.1 will be sufficiendy large that the flow rate will be relatively low and consequendy conditions in the tank, in particular the fluid head causing the flow, will change only slowly. In these circumstances the emptying operation can be treated as quasi-steady. In view of this, Bernoulli s equation, which is valid only for steady flow, may be used. 

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 ... 

Turbulent mass transfer near a wall can be represented by various physical models. In one such model the turbulent flow is assumed to be composed of a succession of short, steady, laminar motions along a plate. The length scale of the laminar path is denoted by x0 and the velocity of the liquid element just arrived at the wall by u0. Along each path of length x0, the motion is approximated by the quasi-steady laminar flow of a semiinfinite fluid along a plate. This implies that the hydrodynamic and diffusion boundary layers which develop in each of the paths are assumed to be smaller than the thickness of the fluid elements brought to the wall by turbulent fluctuations. Since the diffusion coefficient is small in liquids, the depth of penetration by diffusion in the liquid element is also small. Therefore one can use the first terms in the Taylor expansion of the Blasius expressions for the velocity components. The rate of mass transfer in the laminar microstructure can be obtained by solving the equation... 

The quasi-steady laminar model is now employed to describe the heat transfer near the wall. Note that while the shear stress at the wall can be related easily to the pressure drop for the flow in a tube, it is more difficult to establish a relation between these two quantities for a packed or fluidized bed. However, while for the flow in a tube the dissipated energy is not uniform over the section... 

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