Turbulent flow momentum transfer

In turbulent flow, momentum transfer is often characterized by an eddy viscosity Ey defined by analogy with Newton s Law of Viscosity. The time averaged momentum flux (shear stress) in the y direction owing to the gradient of the time averaged velocity in Ak z direction is given by... 

In the less turbulent flow through the straight channel of a monolith, momentum transfer from the fluid to the wall is less effective and in the case of two-phase, countercurrent annular flow, momentum transfer between gas and liquid will also be less than in the interstitial channels of a packed bed. The lower rates of momentum transfer, which is the reason for the higher permeability of monoliths, should in principle improve the possibility for achieving countercurrent flow of gas and liquid at realistic velocities. 

In turbulent flow, momentum is constantly fed into the layer adjacent to the wall because of the momentum transfer between layers at different velocities. The kinetic energy of the fluid elements close to the wall does not decrease as rapidly as in laminar flow. This means that turbulent boundary layers do not become detached as quickly as laminar boundary layers. Heat and mass transfer close to the wall is not only promoted by turbulence, the fluid also flows over a larger surface area without detachment. At the same time the pressure resistance is lower because the fluid flow does not separate from the surface for a longer flow path. 

Hughmark employed this u to derive a correlation for Son and Hanratty (1967) and Hughmark (1971,1974) correlated wall to fluid heat transfer in pipe flow based on the relatively simple and well-established boundary layer theory. In the case of pipe flow, momentum transfer is solely by skin friction because of the geometry involved. Nonetheless, this approach was extended to particle-fluid mass transfer in turbulent flow. The correlation proposed was of the following form ... 

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. 

Inertial forces are developed when the velocity of a fluid changes direction or magnitude. In turbulent flow, inertia forces are larger than viscous forces. Fluid in motion tends to continue in motion until it meets a sohd surface or other fluid moving in a different direction. Forces are developed during the momentum transfer that takes place. The forces ac ting on the impeller blades fluctuate in a random manner related to the scale and intensity of turbulence at the impeller. 

Computational fluid dynamics (CFD) is the numerical analysis of systems involving transport processes and solution by computer simulation. An early application of CFD (FLUENT) to predict flow within cooling crystallizers was made by Brown and Boysan (1987). Elementary equations that describe the conservation of mass, momentum and energy for fluid flow or heat transfer are solved for a number of sub regions of the flow field (Versteeg and Malalase-kera, 1995). Various commercial concerns provide ready-to-use CFD codes to perform this task and usually offer a choice of solution methods, model equations (for example turbulence models of turbulent flow) and visualization tools, as reviewed by Zauner (1999) below. 

In turbulent motion, the presence of circulating or eddy currents brings about a much-increased exchange of momentum in all three directions of the stream flow, and these eddies are responsible for the random fluctuations in velocity The high rate of transfer in turbulent flow is accompanied by a much higher shear stress for a given velocity gradient. 

In streamline flow, E is very small and approaches zero, so that xj p determines the shear stress. In turbulent flow, E is negligible at the wall and increases very rapidly with distance from the wall. LAUFER(7), using very small hot-wire anemometers, measured the velocity fluctuations and gave a valuable account of the structure of turbulent flow. In the operations of mass, heat, and momentum transfer, the transfer has to be effected through the laminar layer near the wall, and it is here that the greatest resistance to transfer lies. 

Equation 11.12 does not fit velocity profiles measured in a turbulent boundary layer and an alternative approach must be used. In the simplified treatment of the flow conditions within the turbulent boundary layer the existence of the buffer layer, shown in Figure 11.1, is neglected and it is assumed that the boundary layer consists of a laminar sub-layer, in which momentum transfer is by molecular motion alone, outside which there is a turbulent region in which transfer is effected entirely by eddy motion (Figure 11.7). The approach is based on the assumption that the shear stress at a plane surface can be calculated from the simple power law developed by Blasius, already referred to in Chapter 3. 

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 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. 

Just as the velocity fluctuations give turbulent flow extra kinetic energy, so they generate extra momentum transfer. Consider the transfer of x-component momentum across a plane of area 8y8z perpendicular to the x-coordinate direction. The momentum flow rate is the product of the mass flow rate (pvx8y8z) across the plane and the velocity component vx ... 

It is the large scale eddies that are responsible for the very rapid transport of momentum, energy and mass across the whole flow field in turbulent flow, while the smallest eddies and their destruction by viscosity are responsible for the uniformity of properties on the fine scale. Although it is the fluctuations in the flow that promote these high transfer rates, it is... 

If the fluid in the pipe is in turbulent flow, the effects of molecular diffusion will be supplemented by the action of the turbulent eddies, and a much higher rate of transfer of material will occur within the fluid. Because the turbulent eddies also give rise to momentum transfer, the velocity profile is much flatter and the dispersion due to the effects of the different velocities of the fluid elements will be correspondingly less. 

Authors efforts in this part of the work have been concentrated on developing turbulence closures for the statistical description of two-phase turbulent flows. The primary emphasis is on development of models which are more rigorous, but can be more easily employed. The main subjects of the modeling are the Reynolds stresses (in both phases), the cross-correlation between the velocities of the two phases, and the turbulent fluxes of the void fraction. Transport of an incompressible fluid (the carrier gas) laden with monosize particles (the dispersed phase) is considered. The Stokes drag relation is used for phase interactions and there is no mass transfer between the two phases. The particle-particle interactions are neglected the dispersed phase viscosity and pressure do not appear in the particle momentum equation. 

The transfer of heat and/or mass in turbulent flow occurs mainly by eddy activity, namely the motion of gross fluid elements that carry heat and/or mass. Transfer by heat conduction and/or molecular diffusion is much smaller compared to that by eddy activity. In contrast, heat and/or mass transfer across the laminar sublayer near a wall, in which no velocity component normal to the wall exists, occurs solely by conduction and/or molecular diffusion. A similar statement holds for momentum transfer. Figure 2.5 shows the temperature profile for the case of heat transfer from a metal wall to a fluid flowing along the wall in turbulent flow. The temperature gradient in the laminar sublayer is linear and steep, because heat transfer across the laminar sublayer is solely by conduction and the thermal conductivities of fluids are much smaller those of metals. The temperature gradient in the turbulent core is much smaller, as heat transfer occurs mainly by convection - that is, by... 

In process operations, simultaneous transfer of momentum, heat, and mass occur within the walls of the equipment vessels and exchangers. Transfer processes usually take place with turbulent flow, under forced convection, with or without radiation heat transfer. One of the purposes of engineering science is to provide measurements, interpretations and theories which are useful in the design of equipment and processes, in terms of the residence time required in a given process apparatus. This is why we are concerned here with the coefficients of the governing rate laws that permit such design calculations. 

Comparing these equations with the x- and y- Navier-Stokes equations for two-dimensional laminar flow shows that in turbulent flow extra terms arise due to the presence of the fluctuating velocity components. These extra terms, which arise because the Navier-Stokes equations contain nonlinear terms, are the result of the momentum transfer caused by the velocity fluctuating components and are often termed the turbulent or Reynolds stress terms because of their similarity to the viscous stress terms which arise due to momentum transfer on a molecular scale. This similarity can be clearly seen by noting that the x-wise momentum equation, for example, for laminar flow can be written as ... 

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. 

This chapter has been concerned with flows in wb ch the buoyancy forces that arise due to the temperature difference have an influence on the flow and heat transfer values despite the presence of a forced velocity. In extemai flows it was shown that the deviation of the heat transfer rate from that which would exist in purely forced convection was dependent on the ratio of the Grashof number to the square of the Reynolds number. It was also shown that in such flows the Nusselt number can often be expressed in terms of the Nusselt numbers that would exist under the same conditions in purely forced and purely free convective flows. It was also shown that in turbulent flows, the buoyancy forces can affect the turbulence structure as well as the momentum balance and that in turbulent flows the heat transfer rate can be decreased by the buoyancy forces in assisting flows whereas in laminar flows the buoyancy forces essentially always increase the heat transfer rate in assisting flow. Some consideration was also given to the effect of buoyancy forces on internal flows. 

If we can expect that the eddy momentum and energy transport will both be increased in the same proportion compared with their molecular values, we might anticipate that heat-transfer coefficients can be calculated by Eq. (5-56) with the ordinary molecular Prandtl number used in the computation. It turns out that the assumption that Pr, = Pr is a good one because heat-transfer calculations based on the fluid-friction analogy match experimental data very well. For this calculation we need experimental values of C/ for turbulent flow. 

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. 

---