Momentum-transfer coefficients

Here V represents the local volume of a computational cell and Va the volume of particle a. The 5-function ensures that the drag force acts as a point force at the (central) position of this particle. In Eq. (42), [ > is the momentum transfer coefficient, which will be discussed in more detail in Section III.D. The gas phase density p is calculated from the ideal gas law ... 

Vanoni was considered an authority on the mechanics of sediment transport by streams and rivers. He performed research on the hydrodynamics of sediment transport, coastal engineering, and hydraulic structures. He has written seminal papers particularly on sediment transport, including the bedload equations, distribution of suspended sediment, stream degradation, bed forms, and total sediment discharge relations. His 1946 work deals with the sediment distribution demonstrating that the then assumed values result in different than the actual momentum transfer coefficients. He also was able to show that suspended sediment tends to reduee the turbulent momentum transfer and henee the flow resistance. Suspended sediment may also cause secondary flow in a river. He was awarded for this paper the 1950 Karl Emil Hilgard Prize by ASCE. 

The interphase momentum transfer coefficient P is frequently modeled by a combination of the Ergun equation and the Wen and Yu correlation, but in this model, the improved drag relation by Beetstra et al. (2007), based... 

Knowing 0, the solid phase pressure and the solid phase bulk and shear viscosities can be calculated from formulae derived by Tun et al. [1984]. The granular temperature conductivity, k, has also been formulated by Tun et al. [1984]. y has been modeled in terms of 0 by Jenkins and Savage [1983]. For dense regimes, the interphase momentum transfer coefficient, / , can be calculated from the Ergun equation already encountered in Chapter 11 on fixed bed reactors [Gidaspow, 1994]. For dilute regimes, a correlation has been proposed by Wen and Yu [1966]. 

Both the theoretical and empirical results of the analogy theories relate one transport coefficient to another. In this case, they are k, the MTC, and Cf, the coefficient of skin friction or the momentum transfer coefficient. This is a fortuitous outcome, because as will be seen in the next few paragraphs much information is available on the coefficients of friction, Cf, for fluid moving over solid surfaces in natural systems. 

In the drag force, the influence of the gas on the particle is described with interphase momentum transfer coefficient P p which is calculated depending on the porosity of the soHd phase e. For the dense regimes, when porosity is smaller than 0.8, the coefficient is calculated according to the Ergun correlation (Ergun, 1952) ... 

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. 

Another property of gases which appears in the Reynolds and the Schmidt numbers is the viscosity, which results from momentum transfer across the volume of the gas when drere is relative bulk motion between successive layers of gas, and the coefficient, y], is given according to the kinetic theoty by the equation... 

Because the mechanisms governing mass transfer are similar to those involved in both heat transfer by conduction and convection and in momentum transfer (fluid flow), quantitative relations exist between the three processes, and these are discussed in Chapter 12. There is generally more published information available on heat transfer than on mass transfer, and these relationships often therefore provide a useful means of estimating mass transfer coefficients. 

In addition to momentum, both heat and mass can be transferred either by molecular diffusion alone or by molecular diffusion combined with eddy diffusion. Because the effects of eddy diffusion are generally far greater than those of the molecular diffusion, the main resistance to transfer will lie in the regions where only molecular diffusion is occurring. Thus the main resistance to the flow of heat or mass to a surface lies within the laminar sub-layer. It is shown in Chapter 11 that the thickness of the laminar sub-layer is almost inversely proportional to the Reynolds number for fully developed turbulent flow in a pipe. Thus the heat and mass transfer coefficients are much higher at high Reynolds numbers. 

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. 

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 ... 

When two or more phases are present, it is rarely possible to design a reactor on a strictly first-principles basis. Rather than starting with the mass, energy, and momentum transport equations, as was done for the laminar flow systems in Chapter 8, we tend to use simplified flow models with empirical correlations for mass transfer coefficients and interfacial areas. The approach is conceptually similar to that used for friction factors and heat transfer coefficients in turbulent flow systems. It usually provides an adequate basis for design and scaleup, although extra care must be taken that the correlations are appropriate. 

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 ... 

The coefficient, p, of the viscosity resisting dislocation motion is the shear stress at the glide plane, x divided by the frequency of momentum transfer, v. The maximum value that x can have is about Coct/47i, and as mentioned above v = 1013/sec for the Al atoms,so p = Coct./47rv = 4x 10 3 Poise.This is comparable to the dislocation viscosity coefficients in other metallic systems. Another view of the viscosity is Andrade s theory in which ... 

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