Transport Reynolds number

Eurther research on convective transport under low Reynolds number, quasicontinuum conditions is needed before the optimal design of such a micro heat exchanger is possible. The cooling heat exchanger is usually thermally linked to a relatively massive substrate. The effects of this linkage need to be explored and accurate methods of predicting the heat-transfer and pressure-drop performance need to be developed. 

Using this simplified model, CP simulations can be performed easily as a function of solution and such operating variables as pressure, temperature, and flow rate, usiag software packages such as Mathcad. Solution of the CP equation (eq. 8) along with the solution—diffusion transport equations (eqs. 5 and 6) allow the prediction of CP, rejection, and permeate flux as a function of the Reynolds number, Ke. To faciUtate these calculations, the foUowiag data and correlations can be used (/) for mass-transfer correlation, the Sherwood number, Sb, is defined as Sh = 0.04 S c , where Sc is the Schmidt... 

Turbulent Diffusion FDmes. Laminar diffusion flames become turbulent with increasing Reynolds number (1,2). Some of the parameters that are affected by turbulence include flame speed, minimum ignition energy, flame stabilization, and rates of pollutant formation. Changes in flame stmcture are beHeved to be controlled entirely by fluid mechanics and physical transport processes (1,2,9). 

The problems that arise when experiments are carried out in a greatly reduced scale can be overcome if the Reynolds number is high and the flow pattern is governed mainly by fully developed turbulence. It is possible to ignore the Reynolds number, the Schmidt number, and the Prandtl number because the structure of the turbulence and the flow pattern at a sufficiently high level of velocity will be similar at different supply velocities and therefore independent of the Reynolds number. The transport of thermal energy and mass by turbulent eddies will likewise dominate the molecular diffusion and will therefore also be independent of the Prandtl number and the Schmidt number. 

Optimal hydraulics is the proper balance of hydraulic parameters (flowrate and equivalent nozzle size) that satisfy chosen criteria of optimization. Hydraulic quantities used to characterize jet bit performance include hydraulic horsepower, jet impact force, jet velocity, Reynolds number at the nozzle, generalized drilling rate or cost per foot drilled. While designing the hydraulic program the limitations due to cuttings transport in the annulus and pump performance characteristics must be included. 

Because concentrated flocculated suspensions generally have high apparent viscosities at the shear rates existing in pipelines, they are frequently transported under laminar flow conditions. Pressure drops are then readily calculated from their rheology, as described in Chapter 3. When the flow is turbulent, the pressure drop is difficult to predict accurately and will generally be somewhat less than that calculated assuming Newtonian behaviour. As the Reynolds number becomes greater, the effects of non-Newtonian behaviour become... 

Water is pumped at 1.4 m3/s from a tank at a treatment plant to a tank at a local works through two parallel pipes, 0.3 m and 0.6 m diameter respectively. What is the velocity in each pipe and, if a single pipe is used, what diameter will be needed if this flow of water is to be transported, the pressure drop being the same Assume turbulent flow, with the friction factor inversely proportional to the one quarter power of the Reynolds number. 

These observations are consistent with the proposed mechanism of the reaction being diffusion controlled in the laminar flow regime. The mass transport is aided by the velocity gradient and thus the reaction rate increases as the Reynolds number is increased. 

LES is suitable for the simulation of turbulence at moderate Reynolds numbers. Turbulence generation, transport, and dissipation are described very accurately... 

Dimensionless numbers (Reynolds number = udip/jj., Nusselt number = hd/K, Schmidt number = c, oA, etc.) are the measures of similarity. Many correlations between them (known also as scale-up correlations) have been established. The correlations are used for calculations of effective (mass- and heat-) transport coefficients, interfacial areas, power consumption, etc. 

The Peclet number Pe = v /Dc, where Dc is the diffusion coefficient of a solute particle in the fluid, measures the ratio of convective transport to diffusive transport. The diffusion time Tp = 2/D< is the time it takes a particle with characteristic length to diffuse a distance comparable to its size. We may then write the Peclet number as Pe = xD/xs, where xv is again the Stokes time. For Pe > 1 the particle will move convectively over distances greater than its size. The Peclet number can also be written Pe = Re(v/Dc), so in MPC simulations the extent to which this number can be tuned depends on the Reynolds number and the ratio of the kinematic viscosity and the particle diffusion coefficient. 

The side-by-side diffusion cell has also been calibrated for drug delivery mass transport studies using polymeric membranes [12], The mass transport coefficient, D/h, was evaluated with diffusion data for benzoic acid in aqueous solutions of polyethylene glycol 400 at 37°C. By varying the polyethylene glycol 400 content incrementally from 0 to 40%, the kinematic viscosity of the diffusion medium, saturation solubility for benzoic acid, and diffusivity of benzoic acid could be varied. The resulting mass transport coefficients, D/h, were correlated with the Sherwood number (Sh), Reynolds number (Re), and Schmidt number (Sc) according to the relationships... 

To summarize, a comprehensive understanding of turbulent transport is not yet achieved, and information will be needed from optical as well as from further mass-transfer measurements. The latter will have to be made at high Reynolds numbers (> 50,000 in channel flow) and at very high Schmidt numbers (> 10,000) to yield critical information about the transfer process. 

In Chapter 11, we indicated that deviations from plug flow behavior could be quantified in terms of a dispersion parameter that lumped together the effects of molecular diffusion and eddy dif-fusivity. A similar dispersion parameter is usefl to characterize transport in the radial direction, and these two parameters can be used to describe radial and axial transport of matter in packed bed reactors. In packed beds, the dispersion results not only from ordinary molecular diffusion and the turbulence that exists in the absence of packing, but also from lateral deflections and mixing arising from the presence of the catalyst pellets. These effects are the dominant contributors to radial transport at the Reynolds numbers normally employed in commercial reactors. 

Most of the available data have been recorded under conditions such that only the terms for eddy transport and conduction through the solid are significant. Equation 12.7.19 requires that /c increase with particle diameter, mass velocity, and the conductivity of the solid. It is consistent with data for low conductivity solids, but some discrepancies arise for very high conductivity solids (108). At Reynolds numbers greater than 40, the contribution of the molecular conduction term is negligible. 

The rate of agitation, stirring, or flow of solvent, if the dissolution is transport-controlled, but not when the dissolution is reaction-con-trolled. Increasing the agitation rate corresponds to an increased hydrodynamic flow rate and to an increased Reynolds number [104, 117] and results in a reduction in the thickness of the diffusion layer in Eqs. (43), (45), (46), (49), and (50) for transport control. Therefore, an increased agitation rate will increase the dissolution rate, if the dissolution is transport-controlled (Eqs. (41 16,49,51,52), but will have no effect if the dissolution is reaction-controlled. Turbulent flow (which occurs at Reynolds numbers exceeding 1000 to 2000 and which is a chaotic phenomenon) may cause irreproducible and/or unpredictable dissolution rates [104,117] and should therefore be avoided. 

The viscosity (dynamic, 17, or kinematic, v) and density, p (Eq. 47), influence the dissolution rate if the dissolution is transport-controlled, but not if the dissolution is reaction-controlled. In transport-controlled dissolution, increasing 17 or v will decrease D (Eq. 53), will increase h (Eqs. 46 and 49) and will reduce J (Eqs. 51 and 52). These effects are complex. For example, if an additional solute (such as a macromolecule) is added to the dissolution medium to increase 17, it may also change p and D. The ratio of 17/p = v (Eq. 47) and D directly influence h and J in the rotating disc technique, while v directly influences the Reynolds number (and hence J) for transport-controlled dissolution in general [104]. 

In chemical reacting systems, the Reynolds number of the flow is not the only source of computational challenges. Indeed, even for laminar reacting flows the chemical source term can be extremely stiff and tightly coupled to the diffusive transport terms. Averaging, as done above to treat turbulent flows, does not... 

The left-hand sides of Eqs. (25)-(29) have the same form as Eq. (5) and represent accumulation and convection. The terms on the right-hand side can be divided into spatial transport due to diffusion and source terms. The diffusion terms have a molecular component (i.e., /i and D), and turbulent components. We should note here that the turbulence models used in Eqs. (26) and (27) do not contain corrections for low Reynolds numbers and, hence, the molecular-diffusion components will be negligible when the model is applied to high-Reynolds-number flows. The turbulent viscosity is defined using a closure such as... 

When the two liquid phases are in relative motion, the mass transfer coefficients in either phase must be related to the dynamical properties of the liquids. The boundary layer thicknesses are related to the Reynolds number, and the diffusive transfer to the Schmidt number. Another complication is that such a boundary cannot in many circumstances be regarded as a simple planar interface, but eddies of material are transported to the interface from the bulk of each liquid which change the concentration profile normal to the interface. In the simple isothermal model there is no need to take account of this fact, but in most industrial circumstances the two liquids are not in an isothermal system, but in one in which there is a temperature gradient. The simple stationary mass transfer model must therefore be replaced by an eddy mass transfer which takes account of this surface replenishment. 

Chapter 2 reviews the statistical theory of turbulent flows. The emphasis, however, is on collecting in one place all of the necessary concepts and formulae needed in subsequent chapters. The discussion of these concepts is necessarily brief, and the reader is referred to Pope (2000) for further details. It is, nonetheless, essential that the reader become familiar with the basic scaling arguments and length/time scales needed to describe high-Reynolds-number turbulent flows. Likewise, the transport equations for important one-point statistics in inhomogeneous turbulent flows are derived in Chapter 2 for future reference. 

The two terms on the right-hand side of this expression appear in closed form. However, the molecular transport term vV2 (Ut) is of order Re 1, and thus will be negligible at high Reynolds numbers. 

The molecular transport term vV2(m m ) is closed, but negligible (order ReL 1) in high-Reynolds-number turbulent flows. The production term... 

As discussed in Section 2.1, in high-Reynolds-number turbulent flows the scalar dissipation rate is equal to the rate of energy transfer through the inertial range of the turbulence energy spectrum. The usual modeling approach is thus to use a transport equation for the transfer rate instead of the detailed balance equation for the dissipation rate derived from (1.27). Nevertheless, in order to understand better the small-scale physical phenomena that determine e, we will derive its transport equation starting from (2.99). 

Table 2.4. The turbulence statistics and unclosed quantities appearing in the transport equations for high-Reynolds-number inhomogeneous turbulent flows. 

For high-Reynolds-number homogeneous turbulent flows,22 the right-hand side of the dissipation-rate transport equation thus reduces to the difference between two large terms 23... 

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