Flow Geometries

It may seem odd that such different geometries, some with curved streamlines and some 

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Fig. 17. Comparison of the predictions of k-Q model with experimental data for a turbulent jet inside a 5° conical duct, (a) Flow geometry and inlet conditions, where geometry =1.6 cm, = 16 cm, L = 64 cm, 0 = 5°-, flow conditions, p = 0.998 g/mc, p = 0.01 g/cm-s, Uj = 40 cm/s,...

Fig. 18. Jet trajectory of a round jet in bounded cross flow where J = Pj V j p (a) flow geometry, ratio of height of tunnel to diameter of injection tube HID) = 12 and (b) flow streamlines where the data points are experimental deterrninations and the lines correspond to calculated predictions for (—)...

Characterization and influence of electrohydro dynamic secondary flows on convective flows of polar gases is lacking for most simple as well as complex flow geometries. Such investigations should lead to an understanding of flow control, manipulation of separating, and accurate computation of local heat-transfer coefficients in confined, complex geometries. The typical Reynolds number of the bulk flow does not exceed 5000. 

Laminar and Turbulent Flow, Reynolds Number These terms refer to two distinct types of flow. In laminar flow, there are smooth streamlines and the fuiid velocity components vary smoothly with position, and with time if the flow is unsteady. The flow described in reference to Fig. 6-1 is laminar. In turbulent flow, there are no smooth streamlines, and the velocity shows chaotic fluctuations in time and space. Velocities in turbulent flow may be reported as the sum of a time-averaged velocity and a velocity fluctuation from the average. For any given flow geometry, a dimensionless Reynolds number may be defined for a Newtonian fluid as Re = LU p/ I where L is a characteristic length. Below a critical value of Re the flow is laminar, while above the critical value a transition to turbulent flow occurs. The geometry-dependent critical Reynolds number is determined experimentally. 

FIG. 6-14 Loss coefficients for flow in hends and curved pipes (a) flow geometry, (h) loss coefficient for a smooth-waUed hend at Re = 10, (c) Re correction factor, (d) outlet pipe correction factor (From D S. Millet] Internal Flow Systems, 2d. ed., BHRA, Cranfield, V.K., 1990.)... 

Turbulent flow occurs when the Reynolds number exceeds a critical value above which laminar flow is unstable the critical Reynolds number depends on the flow geometry. There is generally a transition regime between the critical Reynolds number and the Reynolds number at which the flow may be considered fully turbulent. The transition regime is very wide for some geometries. In turbulent flow, variables such as velocity and pressure fluctuate chaotically statistical methods are used to quantify turbulence. 

To determine the appropriate injection rate, a field test should first be performed at one of the industry-sponsored full-scale loop test facilities. The optimum mixture, its injection rate, and location of injcciioii points will be a function of flow geometry, fluid properties, pressure leinpcrature relationships, etc., that will be encountered in the actual field application. The appropriate injection rate and location of injection jii iiiis can be determined from this test by observing pressure increases, which indicate that hydrate plugs are forming. 

The flow behavior of the polymer blends is quite complex, influenced by the equilibrium thermodynamic, dynamics of phase separation, morphology, and flow geometry [2]. The flow properties of a two phase blend of incompatible polymers are determined by the properties of the component, that is the continuous phase while adding a low-viscosity component to a high-viscosity component melt. As long as the latter forms a continuous phase, the viscosity of the blend remains high. As soon as the phase inversion [2] occurs, the viscosity of the blend falls sharply, even with a relatively low content of low-viscosity component. Therefore, the S-shaped concentration dependence of the viscosity of blend of incompatible polymers is an indication of phase inversion. The temperature dependence of the viscosity of blends is determined by the viscous flow of the dispersion medium, which is affected by the presence of a second component. 

Chemical reducing conditions and flow geometries are optimized to minimize EC. Reducing conditions especially must not be too severe (as may happen with excess hydrazine) to prevent autocat-alytic EC mechanisms proliferating. 

Using different elongational flow geometries, the CS transition of a flexible polymer chain as well as the phenomenon of hysteresis for the reversed process of SC transition has been confirmed experimentally by the Bristol group [8, 9], the Cal-Tech group [10, 53] and the Paris group [11, 54]. 

The above description refers to a Lagrangian frame of reference in which the movement of the particle is followed along its trajectory. Instead of having a steady flow, it is possible to modulate the flow, for example sinusoidally as a function of time. At sufficiently high frequency, the molecular coil deformation will be dephased from the strain rate and the flow becomes transient even with a stagnant flow geometry. Oscillatory flow birefringence has been measured in simple shear and corresponds to some kind of frequency analysis of the flow... 

As opposed to stagnant elongational flow which necessitates highly specific flow geometries, transient elongational flow is readily obtained with some simple arrangements which will be described below. 

The flow geometry is time-dependent with the constant advance of the piston during the experiment. The flow field, however, was not significantly perturbed by this displacement except when the plunger reached a position a few millimeters... 

Fig. 37. Strain rate distribution along the centerline in a 2-dimensional hyperbolic flow (the flow geometry is shown as an insert). The solid curve, redrawn according to ref. 131, corresponds to a viscoelastic fluid (the spike at x = 2 is a calculation artefact) the dotted curve is calculated with POLYFLOW for a Newtonian liquid...

According to some recent results (Sect. 5.5), the dependence of K on e and qs is more involved than suggested by Eq. (94). The dependence of K on ris is much weaker than a direct proportionality and the correct flow parameter to be used in Eq. (94) should be the local fluid kinetic energy ( v2) rather than the strain rate (e). For a constant flow geometry, however, the two variables v and e are interchangeable. At the present stage and in order not to complicate unduly the kinetic scheme, it will be assumed that the rate constant K varies with the MW and strain rate as ... 

As mentioned in Sect. 4.2, transient elongational flow can be found in a variety of experimental situations. Generally, the flow geometries are not well-defined and the flow field can only be estimated qualitatively. 

Figure 8.5 shows another flow geometry for which rectangular coordinates are useful. The bottom plate is stationary but the top plate moves at velocity 2m. 

Figure 3.34 Slug-flow geometry. (From Taitel and Barnea, 1990. Copyright 1990 by Academic Press, Orlando, FL. Reprinted with permission.)...

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