Momentum-Transfer Analogies

There are many more data available for pressure-drop friction and heat transfer than for mass transfer. The similarity of Eqs. (3.46) to (3.48) for the transfer processes of momentum, mass, and heat, and the identical solutions, Eq. (3.52), lead to the possibility of deducing the mass-transfer characteristics for other situations from knowledge of the other two. 

In the case of turbulent flow, the differential equations will contain time-averaged velocities and in addition the eddy diffusivities of momentum, mass, and heat transfer. The resulting equations cannot be solved for lack of information about the eddy diffusivities, but one might expect results of the form 

Successful completion of an analogy requires, therefore, knowledge of how the ratios E /v, Ep/E., and Ef /E. vary with distance from the fluid interface. It is customary arbitrarily to set Ep/E = Eh/E, = 1, despite experimental evidence to the contrary and to make some arbitrary decision about /v. With these assumptions, however, experimental velocity profiles permit prediction of the profiles and coefficients for mass and heat transfer. Further, assuming only Ep = Ef (which is much more reasonable) permits information on heat transfer to be converted directly for use in mass-transfer calculations and vice versa. 

Let us sum up and further generalize what can be done with the analogies  

Friction factors and velocity profiles can be expected to correlate with the corresponding heat- and mass-transfer quantities only if E E j = Ej in turbulent flow and in extension to viscous sublayers only if d/ au- For either laminar or turbulent flow, the friction factors must indicate skin friction and not form drag as well. In general, it is safest to avoid the friction analogy to mass and heat transfer. 

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Dimensionless heat transfer coefficient Forced convection (heat, mass, and momentum transfer analogy)... 

The procedure of Mason and Evans has the electrical analog shown in Figure 2.2, where voltages correspond to pressure gradients and currents to fluxes. As the argument stands there is no real justification for this procedure indeed, it seems improbable that the two mechanisms for diffusive momentum transfer will combine additively, without any interactive modification of their separate values. It is equally difficult to see why the effect of viscous velocity gradients can be accounted for simply by adding... 

Despite the fact Chat there are no analogs of void fraction or pore size in the model, by varying the proportion of dust particles dispersed among the gas molecules it is possible to move from a situation where most momentum transfer occurs in collisions between pairs of gas molecules, Co one where the principal momentum transfer is between gas molecules and the dust. Thus one might hope to obtain at least a physically reasonable form for the flux relations, over the whole range from bulk diffusion to Knudsen streaming. 

Analogy between Momentum and Heat Transfer The interrelationship of momentum transfer and heat transfer is obvious from examining the equations of motion and energy. For constant flmd properties, the equations of motion must be solved before the energy equation is solved. If flmd properties are not constant, the equations are coupled, and their solutions must proceed simultaneously. Con-... 

The analogy has been reasonably successful for simple geometries and for fluids of very low Prandtl number (liquid metals). For high-Prandtl-number fluids the empirical analogy of Colburn [Trans. Am. Tn.st. Chem. Ting., 29, 174 (1933)] has been veiy successful. A J factor for momentum transfer is defined asJ =//2, where/is the friction fac tor for the flow. The J factor for heat transfer is assumed to be equal to the J factor for momentum transfer... 

Circular Tubes Numerous relationships have been proposed for predicting turbulent flow in tubes. For high-Prandtl-number fluids, relationships derived from the equations of motion and energy through the momentum-heat-transfer analogy are more complicated and no more accurate than many of the empirical relationships that have been developed. 

The analogy between heat and mass transfer holds over wider ranges than the analogy between mass and momentum transfer. Good heat transfer data (without radiation) can often be used to predict mass-transfer coefficients. 

There are strict limitations to the application of the analogy between momentum transfer on the one hand, and heat and mass transfer on the other. Firstly, it must be borne in mind that momentum is a vector quantity, whereas heat and mass are scalar quantities. Secondly, the quantitative relations apply only to that part of the momentum transfer which arises from skin friction. If form drag is increased there is little corresponding increase in the rates at which heat transfer and mass transfer will take place. 

Derive the Taylor-Prandtl modification of the Reynolds analogy between heat and momentum transfer and express it in a form in which it is applicable to pipe flow. 

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 ... 

An analogy exists between mass transfer (which depends on the diffusion coefficient) and momentum transfer between the sliding hquid layers (which depends on the kinematic viscosity). Calculations show that the ratio of thicknesses of the diffnsion and boundary layer can be written as... 

No and Kazimi (1982) derived the wall heat transfer coefficient for the forced-convective two-phase flow of sodium by using the momentum-heat transfer analogy and a logarithmic velocity distribution in the liquid film. The final form of their correlation is expressed in terms of the Nusselt number based on the bulk liquid temperature, Nuft ... 

There are hence two fundamentally analogous ways of probing a crystal lattice by varying the momentum transfer. In the first (classical) case the wave vector, k, is held constant and the angle of scatter, 0, is varied (angular-dispersive). Alternatively, it is possible to hold 9 constant and to allow k to vary (energy-dispersive). 

As discussed in the previous chapter, most early efforts at trying to theoretically predict heat transfer rates in turbulent flow concentrated on trying to relate the wall heat transfer rate to the wall shear stress [1],[2],[3],[41. The reason for this is that a considerable body of experimental and semi-theoretical knowledge concerning the shear stress in various flow situations is available and that the mechanism of heat transfer in turbulent flow is obviously similar to the mechanism of momentum transfer. In the present section an attempt will be made to outline some of the simpler such analogy solutions for boundary layer flows, attention mainly being restricted to flow over a flat plate. 

Turbulent heat transfer is analogous to turbulent momentum transfer. The turbulent momentum flux postulated by Eq. (5-59) carries with it a turbulent... 

The log-mean-temperature-difference and effectiveness approaches are presented in heat-exchanger analysis since both are in wide use and each offers its own advantages to the designer. A brief introduction to diffusion and mass transfer is presented in order to acquaint the reader with these processes and to establish more firmly the important analogies between heat, mass, and momentum transfer. 

Liquid metals constitute a class of heat-transfer media having Prandtl numbers generally below 0.01. Heat-transfer coefficients for liquid metals cannot be predicted by the usual design equations applicable to gases, water, and more viscous fluids with Prandtl numbers greater than 0.6. Relationships for predicting heat-transfer coefficients for liquid metals have been derived from solution of Eqs. (5-38a) and (5-38b). By the momentum-transfer-heat-transfer analogy, the eddy conductivity of heat is = k for small IVp,. Thus in the solu-... 

The analogy among heat, mass, and momentum transfer was studied and a more generalized presentation of the data on heat- and mass-transfer coefficients was made (El, Gl, G5, S3, W3). 

Discussion This example shows the great utility of momentum-heat transfer analogies in that the convection heat transfer coefficient can be obtained from a knowledge of friction coefficient, which Is easier to determine. 

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