Turbulent flow deviating velocities

In turbulent flow the velocity is fluctuating in all directions. In Fig. 3.10-3 a typical plot of the variation of the instantaneous velocity in the x direction at a given point in turbulent flow is shown. The velocity v is the deviation of the velocity from the mean velocity in the x-direction of flow of the stream. Similar relations also hold for the y and z directions. 

There will be velocity gradients in the radial direction so all fluid elements will not have the same residence time in the reactor. Under turbulent flow conditions in reactors with large length to diameter ratios, any disparities between observed values and model predictions arising from this factor should be small. For short reactors and/or laminar flow conditions the disparities can be appreciable. Some of the techniques used in the analysis of isothermal tubular reactors that deviate from plug flow are treated in Chapter 11. 

In Sect. 3.2, the development of the design equation for the tubular reactor with plug flow was based on the assumption that velocity and concentration gradients do not exist in the direction perpendiculeir to fluid flow. In industrial tubular reactors, turbulent flow is usually desirable since it is accompanied by effective heat and mass transfer and when turbulent flow takes place, the deviation from true plug flow is not great. However, especially in dealing with liquids of high viscosity, it may not be possible to achieve turbulent flow with a reasonable pressure drop and laminar flow must then be tolerated. 

Deviation from the ideal plug flow can be described by the dispersion model, which uses the axial eddy diffusivity (m s ) as an indicator of the degree of mixing in the flow direction. If the flow in a tube is plug flow, the axial dispersion is zero. On the other hand, if the fluid in a tube is perfectly mixed, the axial dispersion is infinity. For turbulent flow in a tube, the dimensionless Peclet number (Pe) deflned by the tube diameter (v dlE-Q is correlated as a function of the Reynolds number, as shown in Figure 10.3 [3] dz is the axial eddy diffusivity, d is the tube diameter, and v is the velocity of liquid averaged over the cross section of the flow channel. 

In practice, there is always some degree of departure from the ideal plug flow condition of uniform velocity, temperature, and composition profiles. If the reactor is not packed and the flow is turbulent, the velocity profile is reasonably flat in the region of the turbulent core (Volume 1, Chapter 3), but in laminar flow, the velocity profile is parabolic. More serious however than departures from a uniform velocity profile are departures from a uniform temperature profile. If there are variations in temperature across the reactor, there will be local variations in reaction rate and therefore in the composition of the reaction mixture. These transverse variations in temperature may be particularly serious in the case of strongly exothermic catalytic reactions which are cooled at the wall (Chapter 3, Section 3.6.1). An excellent discussion on how deviations from plug flow arise is given by DENBIGH and TURNER 5 . 

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. 

In all cases the pressure differences must be in such limits that stream-line flo y is operative, since the equations do not apply to turbulent flow. This conffition may be tested by using different driving pressures P and showing that the times for equal volumes are iuversely proportional to P. According to Gruneisen3. the velocity u cm./sec. for which the deviation from Poiseuille s law reaches 0-1 per cent is given by ... 

For calculation of the heat and mass transfer it is sufficient to consider turbulent flow to be hydrodynamically fully developed after an entry-length of xe/d 10. Then the small deviations from the end value of the velocity profile hardly affect the heat and mass transfer coefficients. 

It is the fluctuating element of the velocity in a turbulent flow that drives the dispersion process. The foundation for determining the rate of dispersion was set out in papers by G. 1. Taylor, who first noted the ability of eddy motion in the atmosphere to diffuse matter in a manner analogous to molecular diffusion (though over much larger length scales) (Taylor 1915), and later identified the existence of a direct relation between the standard deviation in the displacement of a parcel of fluid (and thus any affinely transported particles) and the standard deviation of the velocity (which represents the root-mean-square value of the velocity fluctuations) (Taylor 1923). Roberts (1924) used the molecular diffusion analogy to derive concentration profiles... 

The averaging process creates a new set of variables, the so-called Reynolds stresses, which are dependent on the averages of products of the velocity fluctuations UjUj (which for i = j simply represent the standard deviations of the velocity components). This creates a closure problem, which is one of the fundamental issues that has to be addressed in the modeling of turbulent flows. Importantly, Equation 3.2.12 also indicates that the Reynolds stress terms, which in line with Taylor s fundamental result should be related to the dispersion parameters, are coupled to the gradients of the mean flow velocity. 

To assess the physical deviation between the average of products and the product of averages a momentum velocity correction factor can be defined by Cm = vz) / v1)a- By use of the Hagen-Poiseuille law (1.353) and the power law velocity profile (1.354) it follows that at steady state Cm has a value of about 0.95 for turbulent flow and 0.75 for laminar flow [55]. In practice a value of 1 is used in turbulent flow so that v1)a is simply replaced by the averaged bulk velocity vz) - On the other hand, for laminar flows a correction factor is needed. For more precise calculations a simplified (not averaged ) 2D model is often considered for ideal axisymmetric pipe flows [52, 69]. 

However, in other cases the model predictions deviate much more from each other and were in poor agreement the experimental data considering the measurable quantities like phase velocities, gas volume fractions and bubble size distributions. An obvious reason for this discrepancy is that the breakage and coalescence kernels rely on ad-hoc empiricism determining the particle-particle and particle-turbulence interaction phenomena. The existing param-eterizations developed for turbulent flows are high order functions of the local... 

The a factors in the kinetic energy (or velocity head) term represent a correction factor to account for the deviation from ping flow throngh the conduit. For a Newtonian flnid in laminar flow in a circular tube, the profile is parabolic and the valne of a is 2. For a highly tnrbnlent flow, the profile is much flatter and a 1.06 (depending on the Reynolds number), although for practical purposes it is usually assumed that a = 1 for turbulent flow. 

DEVIATING VELOCITIES IN TURBULENT FLOW. A typical picture of the variations in the instantaneous velocity at a given point in a turbulent flow field is shown in Fig. 3.3. This velocity is really a single component of the actual velocity vector, all three components of which vary rapidly in magnitude and direction. Also, the instantaneous pressure at the same point fluctuates rapidly and simultaneously with the fluctuations of velocity. Oscillographs showing these fluctuations provide the basic experimental data on which modern theories of turbulence are based. 

REYNOLDS STRESSES. It has long been known that shear forces much larger than those occurring in laminar flow exist in turbulent flow wherever there is a velocity gradient across a shear plane. The mechanism of turbulent shear depends upon the deviating velocities in anisotropic turbulence. Turbulent shear stresses are called Reynolds stresses. They are measured by the correlation coefficients of the type defined in Eq. (3.15). 

Although Ey and are analogous to fj. and v, respectively, in that all these quantities are coefficients relating shear stress and velocity gradient, there is a basic difference between the two kinds of quantities. The viscosities n and v are true properties of the fluid and are the macroscopic result of averaging motions and momenta of myriads of molecules. The eddy viscosity and the eddy diffusivity are not just properties of the fluid but depend on the fluid velocity and the geometry of the system. They are functions of all factors that influence the detailed patterns of turbulence and the deviating velocities, and they are especially sensitive to location in the turbulent field and the local values of the scale and intensity of the turbulence. Viscosities can be measured on isolated samples of fluid and presented in tables or charts of physical properties, as in Appendixes 8 and 9. Eddy viscosities and diffusivities are determined (with difficulty, and only by means of special instruments) by experiments on the flow itself. 

In Chapter 2, the design of the so-called ideal reactors was discussed. The reactor ideahty was based on defined hydrodynamic behavior. We had assumedtwo flow patterns plug flow (piston type) where axial dispersion is excluded and completely mixed flow achieved in ideal stirred tank reactors. These flow patterns are often used for reactor design because the mass and heat balances are relatively simple to treat. But real equipment often deviates from that of the ideal flow pattern. In tubular reactors radial velocity and concentration profiles may develop in laminar flow. In turbulent flow, velocity fluctuations can lead to an axial dispersion. In catalytic packed bed reactors, irregular flow with the formation of channels may occur while stagnant fluid zones (dead zones) may develop in other parts of the reactor. Incompletely mixed zones and thus inhomogeneity can also be observed in CSTR, especially in the cases of viscous media. 

In Section 2-2, we used Figure 2-7 to illustrate the instantaneous velocity versus time signal and the mean velocity. A third velocity used widely for turbulent flows is the root-mean-square (RMS) velocity, or the standard deviation of the instantaneous velocity signal. Because the average fluctuation is zero by definition, the RMS velocity gives us an important measure of the amount or intensity of turbulence, but many different signals can return the same mean velocity and RMS fluctuating velocity, so more information is needed to characterize the turbulence. 

These methods hardly take spatial distributions of velocity field and chemical species or transient phenomena into account, although most chemical reactors are operated in the turbulent regime and/or a multiphase flow mode. As a result, yield and selectivity of commercial chemical reactors often deviate from the values at their laboratory or pilot-scale prototypes. Scale-up of many chemical reactors, in particular the multiphase types, is still surrounded by a fame of mystery indeed. Another problem relates to the occurrence of thermal runaways due to hot spots as a result of poor local mixing effects. 

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