Analogy Solutions Reynolds

The simplest analogy solution is that due to Reynolds. This analogy has many limitations but does have some practical usefulness and serves as the basis for more refined analogy solutions. 

Eqs. (7.122) to (7.125) can be simultaneously solved to give the variationsanalogy solutions can then be used to obtain the heat transfer rate, the Reynolds analogy, for example, giving ... 

Some simple methods of determining heat transfer rates to turbulent flows in a duct have been considered in this chapter. Fully developed flow in a pipe was first considered. Analogy solutions for this situation were discussed. In such solutions, the heat transfer rate is predicted from a knowledge of the wall shear stress. In fully developed pipe flow, the wall shear stress is conventionally expressed in terms of the friction factor and methods of finding the friction factor were discussed. The Reynolds analogy was first discussed. This solution really only applies to fluids with a Prandtl number of 1. A three-layer analogy solution which applies for all Prandtl numbers was then discussed. 

Consider the flow of water through a 65-mm diameter smooth pipe at a mean velocity of 4 m/s. The walls of the pipe are kept at a uniform temperature of 40°C and at a certain section of the pipe the mean water temperature is 30°C. Find the heat transfer coefficient for this situation using both the Reynolds analogy and the three-layer analogy solution. 

In Madejski s full model,l401 solidification of melt droplets is formulated using the solution of analogous Stefan problem. Assuming a disk shape for both liquid and solid layers, the flattening ratio is derived from the numerical results of the solidification model for large Reynolds and Weber numbers ... 

Equation (31) is similar to Eq. (7) except that it takes into account the effect of kinematic viscosity on the eddy properties near the wall. An iterative solution is required for the solution of Eqs. (30) and (31). Throughout Deissler s recent analysis (D3) the Reynolds analogy has been assumed, and throughout the turbulent core Deissler shows that... 

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. 

J5 Consider the wall-injection problem in an axisymmetric setting, where a uniform injection velocity flows through the wall of a cylindrical tube. There is a mean velocity U that enters through one end of the tube. Following a procedure analogous to the flow-between-plates problem (Section 5.6), develop a solution for the velocity profiles and the wall shear stress as characterized by the product of a Reynolds number and a friction factor. 

It may be noted mat, in those problems, an arithmetic mean temperature difference is used rather man a logarithmic value for ease of solution. This is probably justified in view of me small temperature changes involved and also me approximate nature of me Reynolds analogy.)... 

Webb, R.L., A Critical Evaluation of Analytical Solutions and Reynolds Analogy Equations for Turbulent Heat and Mass Transfer in Smooth Tubes , Warme- und Stoffuber-... 

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. 

The empirical correlations presented above, with the exception of Eq. (6-7), apply to smooth tubes. Correlations are, in general, rather sparse where rough tubes are concerned, and it is sometimes appropriate that the Reynolds analogy between fluid friction and heat transfer be used to effect a solution under these circumstances. Expressed in terms of the Stanton number,... 

We start this chapter with a general physical description of the convection mechanism. We then discuss (he velocity and thermal botmdary layers, and laminar and turbitlent flows. Wc continue with the discussion of the dimensionless Reynolds, Prandtl, and Nusselt nuinbers, and their physical significance. Next we derive the convection equations on the basis of mass, momentiim, and energy conservation, and obtain solutions for flow over a flat plate. We then nondimeiisionalizc Ihc convection equations, and obtain functional foiinis of friction and convection coefficients. Finally, we present analogies between momentum and heat transfer. 

We then return briefly to consider the creeping-flow approximation of the previous two chapters. We do this at this point because we recognize that the creeping-flow solution is exactly analogous to the pure conduction heat transfer solution of the preceding section and thus should also not be a uniformly valid first approximation to flow at low Reynolds number. We thus explain the sense in which the creeping-flow solution can be accepted as a first approximation (i.e., why does it play the important role in the analysis of viscous flows that it does ), and in principle how it might be corrected to account for convection of momentum (or vorticity) for the realistic case of flows in which Re is small but nonzero. 

When the dimensionless form of the thermal energy equation is compared with the dimensionless Navier-Stokes equation, it is clear that the Peclet number plays a role for heat transfer that is analogous to the Reynolds number for fluid motion. Thus it is natural to seek approximate solutions for asymptotically small values of the Peclet number, analogous to the low-Reynolds-number approximation of Chaps. 7 and 8. 

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. 

Hence, the ordinate in Fig. 6.22 can also be used in conjunction with Eq. 6.107 or 6.109 to calculate the cross flow skin friction coefficient for cases of very small yaw angles (ts =1). Note that Iaw is equal to unity because the solution of Eq. 6.102 with Pr = 1 and an insulated surface is / = 1. Although the trends exhibited in Figs. 6.21 and 6.22 are generally similar, it must be cautioned that such large variations in the Reynolds analogy factor occur that the latter is no longer a useful concept. The heat transfer parameter for a cooled surface shows a rather small variation with Pp for Pp > Vi, a fact first utilized in Ref. 44 to obtain relatively simple expressions for the local heat flux to blunt bodies in hypersonic flow. 

This relation may be regarded as the counterpart of Stokes equations, Eqs. (7)-(8). Roughly speaking, Eq. (301) bears the same relationship to Eq. (299) at very small Peclet (and Reynolds) numbers as do Stokes equations to the complete Navier-Stokes equations at very small Reynolds numbers. Hence, many of the results of Section II pertaining to the solutions of Stokes equations have analogs in the theory of heat- (and mass-) transfer at asymptotically small Peclet numbers. It will suffice, therefore, to illustrate these analogs by a few salient examples. 

TTie analytical solution for a fully developed, steady viscous flow in a curved tube of circular cross section was developed by Dean in 1927, who expressed the ratio of centrifugal inertial forces to the viscous forces (analogous to the definition of Reynolds number Re) by the dimensionless Dean number. 

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