Diffusion Reynolds analogy

At present analytical solutions of the equations describing the microscopic aspects of material transport in turbulent flow are not available. Nearly all the equations representing component balances are nonlinear in character even after many simplifications as to the form of the equation of state and the effect of the momentum transport upon the eddy diffusivity are made. For this reason it is not to be expected that, except by assumption of the Reynolds analogy or some simple consequence of this relationship, it will be possible to obtain analytical expressions to describe the spatial variation in concentration of a component under conditions of nonuniform material transport. 

Little detailed experimental information is available on the value of eddy transport properties under conditions of simultaneous thermal and material transport. If it is assumed that the Reynolds analogy is applicable, it follows that the eddy diffusivity and eddy conductivity are equal and independent of cross linking. Such an assumption is probably not true since it is to be expected that a substantial part of the eddy transport is associated with molecular transport particularly as the eddies become small in accordance with Kolmogoroff s (K10) principle. For this reason it is to be expected that temperature gradients in turbulent streams will influence to some extent the material transport in the same... 

Obtain the Taylor-Prandtl modification of the Reynolds analogy between momentum and heat transfer and give the corresponding analogy for mass transfer. For a particular system a mass transfer coefficient of 8.71 x 10-6 m/s and a heat transfer coefficient of 2730 W/m2K were measured for similar flow conditions. Calculate the ratio of the velocity in the fluid where the laminar sub-layer terminates, to the stream velocity. Molecular diffusivity = 1.5 x 10 9 m2/s. Viscosity = 1 mN s/m2. Density = 1000 kg/m3. Thermal conductivity = 0.48 W/m K. Specific heat capacity = 4.0 kJ/kg K. 

Obtain the Taylor-Prandtl modification of the Reynolds analogy between momentum transfer and mass transfer (equimolecular counterdiffusion) for the turbulent flow of a fluid over a surface. Write down the corresponding analogy for heat transfer. State clearly the assumptions which are made. For turbulent flow over a surface, the film heat transfer coefficient for the fluid is found to be 4 kW/m2 K. What would the corresponding value of the mass transfer coefficient be, given the following physical properties Diffusivity D = 5 x 10 9 m2/s. Thermal conductivity, k = 0.6 W/m K. Specific heat capacity Cp = 4 kJ/kg K. Density, p = 1000 kg/m3. Viscosity, p = 1 mNs/m2. 

Qfis may ask the reason for the functional form of Eq. (6-4). Physical reasoning, based on the experience gained with the analyses of Chap. 5, would certainly indicate a dependence of the heat-transfer process on the flow field, and hence on the Reynolds number. The relative rates of diffusion of heat and momentum are related by the Prandtl number, so that the Prandtl nunfber is expected to be a significant parameter in the final solution. We can be rather confident of the dependence of the heat transfer on the Reynolds and Prandtl numbers. But the question arises as to the correct functional form of the relation i.e., would one necessarily expect a product of two exponential functions of the Reynolds and Prandtl numbers The answer is that one might expect this functional form since it appears in the flat-plate analytical solutions of Chap. 5, as well as the Reynolds analogy for turbulent flow. In addition, this type of functional relation is convenient to use in correlating experimental data, as described below. 

Toor, H. L., Turbulent Diffusion and the Multicomponent Reynolds Analogy, AIChE J, 6, 525-527 (1960). 

Reynolds analogy allows estimates to be made of SO2 deposition velocity (V ) based on heat transfer or skin friction tests (or theory), of which the literature abounds. In so doing, one must realize that such a calculation deals only with the delivery of pollutant to the surface, through diffusion. If we assume that the concentration is zero at the surface (perfect absorption), we have tacitly assumed that the physical chemistry is not limiting, which will only be the case with reactive materials such as zinc or calcareous stones. For less reactive materials, the surface concentration in the pollutant profile may not be zero, leading to an interaction between physical and chemical processes. Such a situation may occur if the pH in the liquid film drops too low to permit additional SO2 dissolution, as given by Henry s law. Buffering of the film with corrosion products can prevent this from... 

Other analogies. The Reynolds analogy assumes that the turbulent diffusivitiess, a, and are all equal and that the molecular difTusivities p/p, a, and are negligible compared to the turbulent diffusivities. When the Prandtl number (p/p)/a is 1.0, then pip = a also, for = 1.0,/i/p = D g. Then,(p/p + e,) = (a + a,) = (D g + e, ) and the Reynolds analogy can be obtained with the molecular terms present. However, the analogy breaks down when the viscous sublayer becomes important since the eddy difTusivities diminish to zero and the molecular diffusivities become important. 

Prandtl modified the Reynolds analogy by writing the regular molecular diffusion equation for the viscous sublayer and a Reynolds-analogy equation for the turbulent core region. Then since these processes are in series, these equations were combined to produce an overall equation (Gl). The results also are poor for fluids where the Prandtl and Schmidt numbers differ from 1.0. 

Von Karman further modified the Prandtl analogy by considering the buffer region in addition to the viscous sublayer and the turbulent core. These three regions are shown in the universal velocity profile in Fig. 3.10-4. Again, an equation is written for molecular diffusion in the viscous sublayer using only the molecular diffusivity and a Reynolds analogy equation for the turbulent core. Both the molecular and eddy diffusivity are used in an equation for the buffer layer, where the velocity in this layer is used to obtain an... 

The fluid temperature profile for fully turbulent flow in a pipe can be obtained by Reynolds analogy for momentum and heat transfer. Reynolds assumed that in turbulent flow molecular diffusivity can by neglected, that is, gm v, hS> a. 

Turbulent diffusivity based closure models for the scalar fluxes describing turbulent transport of species relate the scalar flux to the mean species concentration gradient according to Reynolds analogy between turbulent momentum and mass transport. The standard gradient-diffusion model can be written ... 

For the gas phase, the turbulent diffusivity D was deduced from the computed from the kc — c model adopting a Reynolds analogy ... 

The Reynolds analogy is found by experiment to be accurate for gases, but not for liquids. We can rationalize this on the basis of the transport coefficients involved. We expect turbulent mixing to take place at two levels a macroscopic level, where eddies are dominant, and a microscopic level, where diffusion, conduction, and viscosity are important. For gases, these microscopic processes are about the same because... 

The technique developed in Section 22.2 (Reynolds splitting) to describe transport by turbulent diffusion can also be applied to dispersion. By analogy to Eq. 22-28, the... 

The eddy diffusitives for momentum and heat, and Ejj, respectively, are not properties of the fluid but depend on the conditions of flow, especially on all factors that affect turbulence. For simple analogies, it is sometimes assumed that and jf are both constants and equal, but when determined by actual velocity and temperature measurements, both are found to be functions of the Reynolds number, the Prandtl number, and position in the tube cross section. Precise measurement of the eddy diffusivities is diflScult, and not all reported measurements agree. Results are given in standard treatises. The ratio Sh/sm also varies but is more nearly constant than the individual quantities. The ratio is denoted by i/f. For ordinary liquids, where Np > 0.6, is close to 1 at the tube wall and in boundary layers generally and approaches 2 in turbulent wakes. For liquid metals is low near the wall, passes through a maximum of about unity at j/r X 0.2, and decreases toward the center of the pipe. ... 

An alternative interpretation may be ascribed to the Reynolds number, consistent with our earlier analogy of the similarity of momentum, heat, and mass transport. We may then interpret the dimensionless parameters appearing in the energy and diffusion equations in an analogous manner that is. 

In Section 15.2.1 we noted that Pick derived his model for mass transfer pardy by analogy to Fourier s law of heat transfer and that one reason Pick s model was rapidly accepted was this close analogy to Fourier s law. Shordy after Pick s developments, Osborne Reynolds (yes, the Reynolds number is named after him) stated that heat or mass transfer in a moving fluid should be the result of both normal diffusion processes and eddies caused by the fluid motion. At the time, he had not yet discovered the difference between laminar motion (only normal diffusion operates) and turbulent motion (both molecular and eddy diffusion occur). We now know that Reynolds was correct only for turbulent flow. Since eddies depend on fluid velocity, the easiest functional form is to assume that eddy diffusion is linearly dependent on velocity. Then the equation for mass transfer becomes... 

Viscosity can apparently be viewed as conduction of impulse, analogous to thermal conduction or material diffusion (as will be discussed in the sequel). The flow pattern inside a body or along a bocty with diameter d (for example a tube) depends on the flow velocity v and can be characterized by the Reynolds number ... 

Abstract In this chapter, the two CMT models, i.e., c — Eci model and Reynolds mass flux model (in standard, hybrid, and algebraic forms) are used for simulating the chemical absorption of CO2 in packed column by using MEA, AMP, and NaOH separately and their simulated results are closely checked with the experimental data. It is noted that the radial distribution of Di is similar to a, but quite different from fit. It means that the conventional assumption on the analogy between the momentum transfer and the mass transfer in turbulent fluids is unjustifled, and thus, the use of CMT method for simulation is necessary. In the analysis of the simulation results, some transport phenomena are interpreted in terms of the co-action or counteraction of the turbulent mass flux diffusion. 

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