Flow approximations

The sohds are also assumed to be in plug flow. As part of the plug flow approximation, the gas and soHds are assumed isothermal in the radial direction at a given axial location. Detailed models for kiln heat transfer are available (20,21). 

From Fig. 5.3 it may be seen that this exceeds the shear rate for non-laminar flow (approximately 30 s ) so that the entry to this region would need to be streamlined. Fig. 5.3 also shows that the extensional strain rate, k, in the tapered entry region should not exceed about 15 s if turbulence is to be avoided. 

In some microfluidic applications liquid is transported with a comparatively low velocity. In such cases, a liquid volume co-moving with the flow experiences inertial forces which are small compared with the viscous forces acting on it. The terms appearing on the left-hand side of Eq. (16) can then be neglected and the creeping flow approximation is valid... 

An indicator of the validity of the creeping flow approximation is the dimensionless Reynolds number ... 

In open rod bundles, transverse flow between subchannels is detectable by variations in hydraulic conditions, such as the difference in equivalent diameter in rod and shroud areas (Green et al., 1962 Chelemer et al., 1972 Rouhani, 1973). The quality of the crossflow may be somewhat higher than that of the main stream (Madden, 1968). However, in view of the small size of the crossflow under most circumstances, such variation generally will not lead to major error in enthalpy calculations. The homogeneous flow approximation almost universally used in subchannel calculations appears to be reasonable (Weisman, 1973). The flow redistribution has a negligible effect on the axial pressure drop. 

The reactor design is similar to conventional flow tubes(J7 ). The main difference is use of higher pressures and faster flow velocities. Both these quantities shift the flow conditions from laminar towards turbulent. To date all workers have interpreted their data using a plug flow approximation which is likely to be... 

Often times concentrated polymeric solutions cannot be treated as Newtonian fluids, however, and this tends to offset the simplifications which result from the creeping flow approximation and the fact that the boundaries are well defined. The complex rheological behavior of polymeric solutions and melts requires that nonlinear constitutive equations, such as Eqs. (l)-(5), be used (White and Metzner, 1963) ... 

Note that setting one of the terms on the left side of the equation equal to zero yields either the batch reactor equation or the steady-state PFTR equation. However, in general we must solve the partial differential equation because the concentration is a function of both position and time in the reactor. We will consider transients in tubular reactors in more detail in Chapter 8 in connection with the effects of axial dispersion in altering the perfect plug-flow approximation. 

We will now find the RDT for several models of tubular reactors. We noted previously that the perfect PFTR cannot in fact exist because, if flow in a tube is sufficiently fast for turbulence (Rco > 2100), then turbulent eddies cause considerable axial dispersion, while if flow is slow enough for laminar flow, then the parabolic flow profile causes considerable deviation from plug flow. We stated previously that we would ignore this contradiction, but now we will see how these effects alter the conversion from the plug-flow approximation. 

In fact, the plug-flow approximation is even better than this calculation indicates because of radial mixing, which wiU occur in a laminar-flow reactor. A fluid molecule near the wall will flow with nearly zero velocity and have an infinite residence time, while a molecule near the center will flow with velocity 2u. However, the molecule near the wall will diffuse toward the center of the tube, and the molecule near the center will diffuse toward the wall, as shown in Figure 8-7. Thus the tail on the RTD will be smaller, and the spike at t/2 will be broadened. We will consider diffusion effects in the axial direction in the next section. 

Whereas inviscid flow is a useful reference point for high Reynolds number flows, a different simplification known as the creeping flow approximation applies at very low Re. From Eq. (1-3), the terms on the right-hand side dominate as Re 0, so that the convective derivative may be neglected. In dimensional... 

The creeping flow approximation has found wide application in problems such as lubrication, injection molding, and flow through porous media. Its application to rigid and fluid particles is discussed in Chapters 3 and 4. However, a fundamental difficulty, first recognized by Oseen, arises in applying Eq. (1-33) or (1-36) to particles in unbounded media. This difficulty, and Oseen s attempt to overcome it, are discussed in Chapter 3. 

The system considered in this chapter is a rigid or fluid spherical particle of radius a moving relative to a fluid of infinite extent with a steady velocity U. The Reynolds number is sufficiently low that there is no wake at the rear of the particle. Since the flow is axisymmetric, it is convenient to work in terms of the Stokes stream function ij/ (see Chapter 1). The starting point for the discussion is the creeping flow approximation, which leads to Eq. (1-36). It was noted in Chapter 1 that Eq. (1-36) implies that the flow field is reversible, so that the flow field around a particle with fore-and-aft symmetry is also symmetric. Extensions to the creeping flow solutions which lack fore-and-aft symmetry are considered in Sections II, E and F. 

Section II shows that the dimensionless external velocity field uJU, UqIU) is a function of dimensionless position r/a, 0) and k for creeping flow. The dimensionless concentration defined in Eq. (1-45) is a function of these quantities and of the Peclet number, Pe = 2aU/. Hence the Sherwood number, Sh = is a function of k and Pe (with additional dependence on Re unless the creeping flow approximation is valid). The exact solution of Eqs. (3-39) to (3-42) with the Hadamard-Rybczynski velocity field is not available for all values of Pe and K, but several special cases have been treated. 

Taking detectable departure from fore-and-aft symmetry as the upper limit of Stokes flow, Coutanceau found that increasing I increased the range of validity of the creeping flow approximation." The upper limit of Stokes flow was proposed as ... 

The term includes the effect of departures from the creeping flow approximations. In creeping flow, Kp = Kjj = = K, but at higher Re the relationships are more complex. In general, any of these ratios is a function of X and... 

Sutterby (S7) gave a useful tabulation of the viseosity ratio, defined in Eq. (9-8), for relatively low Re and a. These values, intended primarily to correct for departures from Stokes law in falling sphere viscometry, are shown in Fig. 9.6. Reynolds number is defined using the measured Uj and defined in Eq. (9-9). The curve for a = 0 accounts for departures from the creeping flow approximations in an unbounded fluid, and the relative displacement of the other curves indicates the wall effect. 

The average plasma clearance is 600 to 980 ml/min with a blood clearance of 1.0 to 1.6 L/min, which is close to hepatic blood flow. This indicates that the rate of metabolism of THC is dependent on hepatic blood flow. Approximately 70% of a dose of THC is excreted in the urine (30%) and... 

The three-dimensional, fully parabolic flow approximation for momentum and heat- and mass-transfer equations has been used to demonstrate the occurrence of these longitudinal roll cells and their effect on growth rate uniformity in Si CVD from SiH4 (87) and GaAs MOCVD from Ga(CH3)3 and AsH3 (189). However, gas expansion in the entrance zone combined with flow obstructions, such as a steeply sloped susceptor, can also produce... 

It features the three chemical processes flowing approximately one to the other. The joint administration and workshop facilities are located in the south-western corner of the plant. There is a central control room for all operations, and the rail link is extended into the despatch area. 

The hydrodynamics is then described by the system of Navier-Stokes equations in the film-flow approximation (Shilkin et al., 2006) ... 

Before initiating treatment it is important that a sample of urine be sent for culture and sensitivities. A mid-stream urine sample (MSU) and clean catch urine are the most commonly collected specimens and are recommended for routine use. The first part of the voided urine is discarded and without interrupting the flow approximately 10 mL is collected in a sterile container. 

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