Analogies between momentum, heat, and mass

The term Csm/Cr (the ratio of the logarithmic mean concentration of the insoluble component to the total concentration) is introduced because hD(CBm/Cr) is less dependent than hD on the concentrations of the components. This reflects the fact that the analogy between momentum, heat and mass transfer relates only to that part of the mass transfer which is not associated with the bulk flow mechanism this is a fraction Cum/Cr of the total mass transfer. For equimolecular counterdiffusion, as in binary distillation when the molar latent heats of the components are equal, the term Cem/Cj- is omitted as there is no bulk flow contributing to the mass transfer. 

Simple form of analogy between momentum, heat and mass transfer... 

The radial dispersion coefficient for this case is, of course, the average eddy diffusivity as discussed in works on turbulence (H9). If the various analogies between momentum, heat, and mass transport are used. 

The gas film coefficient is dependent on turbulence in the boundary layer over the water body. Table 4.1 provides Schmidt and Prandtl numbers for air and water. In water, Schmidt and Prandtl numbers on the order of 1,000 and 10, respectively, results in the entire concentration boundary layer being inside of the laminar sublayer of the momentum boundary layer. In air, both the Schmidt and Prandtl numbers are on the order of 1. This means that the analogy between momentum, heat, and mass transport is more precise for air than for water, and the techniques apphed to determine momentum transport away from an interface may be more applicable to heat and mass transport in air than they are to the liquid side of the interface. 

Heat and mass transfer are taking place simultaneously to a surface under conditions where the Reynolds analogy between momentum, heat and mass transfer may be applied. The mass transfer is of a single component at a high concentration in a binary mixture, the other component of which undergoes no net transfer. Using the Reynolds analogy, obtain a relation between the coefficients for heat transfer and for mass transfer. 

Equation 8.10 through Equation 8.13 are the convection tranter equations. When appropriate boundary conditions are included, these equations can be solved to determine spatial variations of u, v, T, and Ca in the boundary layers. In this section, these reduced equations will be scaled to establish important analogies between momentum, heat, and mass transfer, while identifying key design parameters. 

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 ... 

We can summarize this brief discussion of the similarity between momentum, heat, and mass transfer as follows. An elementary consideration of the three processes leads to the conclusion that in certain simplified situations there is a direct analogy between them. In general, however, when three- rather than one-dimensional transfer is considered, the momentum-transfer process is of a sufficiently different nature for the analogy to break down. Modification of the simple analogy is also necessary when, for example, mass and momentum transfer occur simultaneously. Thus, if there were a net mass transfer toward the surface of Fig. 2.6, the momentum transfer of Eq. (2.48) would have to include the effect of the net diffusion. Similarly, mass transfer must inevitably have an influence on the velocity profile. Nevertheless, even the limited analogies which exist are put to important practical use. 

There is a great deal of theoretical and experimental information from micrometeorological research on the transfer of momentum, heat, and mass at solid and liquid surfaces and across their associated air boundary layers (hence the term boundary layer models for relationships arising from this approach). Based on the analogy between transfer of momentum and mass, it has been shown that k is proportional to the friction velocity in air (u ) and that k is also proportional to Sc. Apart from an assumption that the surface was smooth and rigid, it was also necessary to assume continuity of stress across the interface in order to convert the velocity profile in air to the equivalent profile in the water (Deacon, 1977). The relationship developed by Deacon is as follows ... 

An elementary consideration of the processes of momentum, heat, and mass transfer leads to the conclusion that in certain simplified situations there is a direct analogy between them. In general, however, modification of the simple analogy is necessary when mass and momentum transfer occur simultaneously. Nevertheless, even the limited analogies which exist are put to important practical use, as will be seen in Chapter 2. 

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. 

Obtain the Taylor-Prandtl modification of the Reynolds analogy between momentum and heat transfer and write down the corresponding analogy for mass transfer. For a particular system, a mass transfer coefficient of 8,71 x 10 8 m/s and a heat transfer coefficient of 2730 W/m2 K 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. 

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. 

Tn Section 15.5.4 we consider the analogy between heat and mass transfer. The dimensionless group for heat transfer analogous to Sc is the Prandti number Pr, which is the ratio of momentum diffusivity to thermal diffusivity. 

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