Reynolds analogy assumptions

This shows that the mean velocity and temperature profiles are similar when the Reynolds analogy assumptions are adopted. 

The original Reynolds analogy involves a number of simplifying assumptions which are justifiable only in a limited range of conditions. Thus it was assumed that fluid was transferred from outside the boundary layer to the surface without mixing with the intervening fluid, that it was brought to rest at the surface, and that thermal equilibrium was established. Various modifications have been made to this simple theory to take account of the existence of the laminar sub-layer and the buffer layer close to the surface. 

Derive an expression relating the pressure drop for the turbulent flow of a fluid in a pipe to the heat transfer coefficient at the walls on the basis of the simple Reynolds analogy. Indicate the assumptions which are made and the conditions under which you would expect it to apply closely. Air at 320 K and atmospheric pressure is flowing through a smooth pipe of 50 mm internal diameter, and the pressure drop over a 4 m length is found to be 150 mm water gauge. By how much would you expect the air temperature to fall over the first metre if the. wall temperature there is 290 K ... 

Obtain an expression for the simple Reynolds analogy between heat transfer and friction. Indicate the assumptions which are made in the derivation and the conditions under which you would expect the relation to be applicable. 

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 ... 

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... 

Similar assumptions to those used in the Reynolds analogy are now introduced i.e., it is assumed in the outer layer that ... 

In the discussion of the use of the Reynolds analogy for the prediction of the heat transfer rate from a flat plate it was assumed that when there was transition on the plate, the x-coordinate in the turbulent portion of the flow could be measured from the leading edge. Develop an alternative expression based on the assumption that the momentum thickness before and after transition is the same. This assumption allows an effective origin for the x-coordinate in die turbulent portion of the flow to be obtained. 

Equation (5-114) is called the Reynolds analogy for tube flow. It relates the heat-transfer rate to the frictional loss in tube flow and is in fair agreement with experiments when used with gases whose Prandtl numbers are close to unity. (Recall that Pr = 1 was one of the assumptions in the analysis.)... 

The numerical results of the various Reynolds analogy factors are compared in Fig. 6.36 for laminar Prandtl numbers ranging from those characteristic of gases to those of oils and for Re = 107. Results for very low laminar Prandtl numbers, characteristic of liquid metals, are not shown because the assumptions for the velocity distributions in the various analyses are... 

Equation (9 16) is known as the steady-state heat conduction equation and is completely analogous to the creeping-motion equation of Chaps. 7 and 8. It can be seen that convection plays no role in the heat transfer process described by (9 16) and (9 17). Thus the form of the velocity field is not relevant, and in spite of the initial assumption (9 15), there is no dependence of 0o on the Reynolds number of the flow. The solution of (9 16) and (9-17) depends on only the geometry of the body surface, represented in (9 17) by S. 

So far the closure problem for the system of Reynolds equations has not been theoretically solved in a conclusive way. In engineering calculations, various assumptions that the Reynolds stresses depend on the average turbulent flow parameters are often adopted as closure conditions. These conditions are usually formulated on the basis of experimental data, dimensional considerations, analogies with molecular rheological models, etc. 

THE REYNOL NAIjOGY. The simplest and oldest analogy equation is that of Reynolds, whic]i derived for flow at high Reynolds numbers in straight round tubes. It is b sbd, liowever, on several questionable assumptions ... 

The solid line in the figure represents the general trend of the experimental observations of Refs. 35 and 85, confirming the fact that the heat transfer reduction always exceeds the friction factor reduction. This contradicts the common assumption of the validity of the Reynolds or Colburn analogy made in a number of heat transfer studies of viscoelastic fluids [101-106],... 

Mass balances obtained so far were formulated and deduced in an inertial frame fixed with distant stars. But their form is the same in any frame (even a noninertial one), i.e., formulae (3.58)-(3.68) are independent of the frame. This may be seen from the assumption of objectivity of scalar mass density p ((3.30) is valid). Using the last formulae from Rem. 12 with objective density p as the scalar we find general validity of mass balance (3.59) in any frame. In some new frame, Reynolds theorem (3.24) may be quite analogously deduced and used and then, by localization, all remaining formulae (3.60)-(3.68) are valid in any frame. Indeed, e.g., (3.63) is valid in any frame because of the objectivity of the material derivative of the objective scalar p and, see (3.16) divv = trD, because the trace of objective tensor (3.54) is objective. Similarly, so is (3.65) with the same po (reference is unique for all actual configurations) and by (3.50). This is also (3.68) for q> from such a new frame. 

The vast majority of studies of turbulent transport have dealt widi momentum transport, so that an analogy between heat or mass transfer and momentum transport is highly desirable. Rather more stringent assumptions are required to demonstrate that such an anak is to be expected. Based on experimental observations such an analog appears to hold under high Reynolds number conditions, but it is not as successful as those relating heat transfer and mass transfer. 

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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