Momentum transport diffusion

Chapter 4 Mass, Heat, and Momentum Transport Analogies. The transport of mass, heat, and momentum is modeled with analogous transport equations, except for the source and sink terms. Another difference between these equations is the magnitude of the diffusive transport coefficients. The similarities and differences between the transport of mass, heat, and momentum and the solution of the transport equations will be investigated in this chapter. 

It is apparent that the source and sink terms can be different between mass, heat, and momentum transport. There is another significant difference, however, related to the magnitude of the diffusion coefficient for mass, heat, and momentum. 

It is seen that we are comparing kinematic viscosity, thermal diffusivity, and diffu-sivity of the medium for both air and water. In air, these numbers are all of the same order of magnitude, meaning that air provides a similar resistance to the transport of momentum, heat, and mass. In fact, there are two dimensionless numbers that will tell us these ratios the Prandtl number (Pr = pCpv/kj = v/a) and the Schmidt number (Sc = v/D). The Prandtl number for air at 20°C is 0.7. The Schmidt number for air is between 0.2 and 2 for helium and hexane, respectively. The magnitude of both of these numbers are on the order of 1, meaning that whether it is momentum transport, heat transport, or mass transport that we are concerned with, the results will be on the same order once the boundary conditions have been made dimensionless. 

Prandtl s mixing length hypothesis (Prandtl, 1925) was developed for momentum transport, instead of mass transport. The end result was a turbulent viscosity, instead of a turbulent diffusivity. However, because both turbulent viscosity and turbulent diffusion coefficient are properties of the flow field, they are related. Turbulent viscosity describes the transport of momentum by turbulence, and turbulent diffusivity describes the transport of mass by the same turbulence. Thus, turbulent viscosity is often related to turbulent diffusivity as... 

Over the last four decades or so, transport phenomena research has benefited from the substantial efforts made to replace empiricism by fundamental knowledge based on computer simulations and theoretical modeling of transport phenomena. These efforts were spurred on by the publication in 1960 by Bird et al. (6) of the first edition of their quintessential monograph on the interrelationships among the three fundamental types of transport phenomena mass transport, energy transport, and momentum transport. All transport phenomena follow the same pattern in accordance with the generalized diffusion equation (GDE). The unidimensional flux, or overall transport rate per unit area in one direction, is expressed as a system property multiplied by a gradient (5)... 

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. 

As stated, this equation finds limited practical application. It requires knowledge of both velocity profiles and its solution requires vorticity boundary conditions that also depend on the velocity profiles. The principal reason to write the equation is to make the point that vorticity is transported within the boundary layer by convection and diffusion in a manner analogous to momentum transport. 

In this particular problem, in which the diffusion effects drive the momentum transport, by increasing the order in the interpolation, which finally represents an increased order in the derivative, the approximation given in eqn. (7.17) is improved. This is related with the idea of the form in which the information is traveling (explained later in Chapter 8). Both interpolations were chosen in a way that the function in the imaginary nodes, i 1/2, was expressed in terms of nodes which are at both sides of the imaginary nodes. In other words, the information for the function at this imaginary... 

Transport phenomena modeling. This type of modeling is applicable when the process is well understood and quantification is possible using physical laws such as the heat, momentum, or diffusion transport equations or others. These cases can be analyzed with principles of transport phenomena and the laws governing the physicochemical changes of matter. Transport phenomena models apply to many cases of heat conduction or mass diffusion or to the flow of fluids under laminar flow conditions. Equivalent principles can be used for other problems, such as the mathematical theory of elasticity for the analysis of mechanical, thermal, or pressure stress and strain in beams, plates, or solids. 

The usefulness of the flow terms as common characteristics for transport processes allows them to illustrate such seemingly diverse processes as convection, momentum transport (viscosity), diffusion and heat conductance. To simplify the written expression, the flux components of the four processes are expressed in Eq. (7-3) in the direction of one axis of the coordinate system whereby, instead of the partial derivative for the function, a variable and useful form of the derivative expression is used ... 

When the transport is considered without turbulence we have, in general, Dj- u is the cinematic viscosity for the momentum transport a = A,/(pCp) is the thermal diffusivity and D is the diffusion coefficient of species A. Whereas with turbulence we have, in general, Dj-, w, is the cinematic turbulence viscosity for the momentum transport a, =, /(pCp) is the thermal turbulence diffusivity and D t is the coefficient of turbulent diffusion of species A frequently = a = D t due to the hydrodynamic origin of the turbulence. 

The thermal diffusion factor a is proportional to the mass difference, (mi — mo)/(mi + m2). The thermal diffusion process depends on the transport of momentum in collisions between unlike molecules. The momentum transport vanishes for Maxwellian molecules, particles which repel one another with a force which falls off as the inverse fifth power of the distance between them. If the repulsive force between the molecules falls off more rapidly than the fifth power of the distance, then the light molecule will concentrate in the high temperature region of the space, while the heavy molecule concentrates in the cold temperature region. When the force law falls off less rapidly than the fifth power of the distance, then the thermal diffusion separation occurs in the opposite sense. The theory of the thermal diffusion factor a is as yet incomplete even for classical molecules. A summary of the theory has been given by Jones and Furry 15) and by Hirschfelder, Curtiss, and Bird 14), Since the thermal diffusion factor a for isotope mixtures is small, of the order of 10", it remained for Clusius and Dickel (8) to develop an elegant countercurrent system which could multiply the elementary effect. 

Figure 8 Enhancement of intrabed mass transfer by axial flow through the bed. A Parallel flow driven by the pressure difference between inlet and outlet B momentum transport by diffusing molecules C convective momentum transport by penetrating eddies.

At higher gas velocities in the channels, the effective intrabed diffusivity increases substantially, reaching values as high as or even higher than the diffusivity in the bulk gas. This enhancement of intrabed diffusivity cannot be ascribed solely to parallel flow as calculated by the Eigun equation, since a 2.5-9-times-higher axial gas velocity is needed to account for the experimentally found enhancement of the intrabed diffusivity. The substantial enhancement must be attributed to momentum transport through the screen [6]. 

S = ntul/f V can also be interpreted as a dimensionless relaxation time r, where tn/f is a characteristic time for particle motion and v/u] h a characteristic time for the turbulent fluctuations. Hence S" " = r". The viscous sublayer is the region near a smooth wall where momentum transport is dominated by the viscous forces, which are large compared with eddy diffusion of momentum. Fol lowing the usual practice and taking the sublayer thickness to extend to y = 5, particles with a slop distance < 5 would not reach the wall if the sublayer were truly stagnant. 

We shall see later that it is often advantageous to consider the temporal or spatial development of a velocity field in terms of diffusion and convection processes. In this framework, the basic concept of diffusion over a distance -Jvi in a time increment t plays a critical role in helping to determine the extent of boundary influence on the flow. It should be recognized however, that the class of unidirectional flows is a special case in that the direction of diffusion is always at right angles to the direction of motion. In most flows this will not be true, and the influence of the boundary geometry will be propagated by means of momentum transport by both diffusion and convection. 

Both UL/a and UL/D are termed the Peclet number and usually given the same symbol. When there is no reason for confusion, we shall do the same otherwise we distinguish between the two by reference to the thermal Peclet number or the diffusion Peclet number. The Peclet number plays a similar role in heat and mass transport as the Reynolds number in momentum transport. The thermal and diffusion Peclet numbers may be written somewhat differently to bring out their relation to the Reynolds number in particular. 

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