Developing turbulent

In general, V For laminar Newtonian flow the radial velocity profile is paraboHc and /5 = 3/4. For fully developed turbulent flow the radial... 

For laminar flow a = 0.5. For fully developed turbulent flowO.88 < a < 0.98, but can be taken to be unity with Httle error. 

The minimum velocity requited to maintain fully developed turbulent flow, assumed to occur at Reynolds number (R ) of 8000, is inside a 16-mm inner diameter tube. The physical property contribution to the heat-transfer coefficient inside and outside the tubes are based on the following correlations (39) ... 

Model Experiments in the Case of Fully Developed Turbulent Flow 1183... 

There is also a standardized method based on the estimation of the flow rate on one measurement point only, In this method the velocity probe is placed in the duct so that the measured local velocity is equal to the mean axial velocity. In fully developed turbulent duct flow, this distance from the wall... 

The problems that arise when experiments are carried out in a greatly reduced scale can be overcome if the Reynolds number is high and the flow pattern is governed mainly by fully developed turbulence. It is possible to ignore the Reynolds number, the Schmidt number, and the Prandtl number because the structure of the turbulence and the flow pattern at a sufficiently high level of velocity will be similar at different supply velocities and therefore independent of the Reynolds number. The transport of thermal energy and mass by turbulent eddies will likewise dominate the molecular diffusion and will therefore also be independent of the Prandtl number and the Schmidt number. 

A reduced scale of the model requires an increased velocity level in the experiments to obtain the correct Reynolds number if Re < Re for the prob lem considered, but the experiment can be carried out at any velocity if Re > RCj.. The influence of the turbulence level is shown in Fig. 12.40. A velocity u is measured at a location in front of the opening and divided by the exhaust flow rate in order to obtain a normalized velocity. The figure show s that the normalized velocity is constant for Reynolds numbers larger than 10 000, which means that the flow around the measuring point has a fully developed turbulent structure at that velocity level. The flow may be described as a potential flow with a normalized velocity independent of the exhaust flow rate at large distances from the exhaust opening— and far away from surfaces. 

Fully developed turbulent flow, Nr over 10,000, in a tank containing four equally spaced baffles having a width of 10% of the tank diameter ... 

This may be compared with fully developed turbulent flow along a flat sheet or tube when ... 

In Regime I, for example, there are many possible domain sizes and, consequently, Q D) 0 for many different ) s moreover, the maximum domain size increases with the total lattice size [kaneko89]. Regimes II and III, on the other hand, allow only a relatively few domain sizes so that Q D) 0 for only a few domain sizes D. In addition, there appears to be a cutoff size Dc (which does not depend on lattice size) such that Q(D > Dc) = 0. In the case cf fully developed turbulence in Regime IV, Q D) oc expf—constant. D), which is indicative of the random generation of domains. 

Fully Developed Turbulence burst regions spread throughout the entire lattice so that it becomes very difficult to discern any laminar regions. 

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. 

For fully developed turbulent flow in a pipe, the whole of the flow may be regarded as lying within the boundary layer. The cross-section can then conveniently be divided into three regions ... 

As is well known, fluid dynamics is the study of motion and transport in liquids and gases. It is primarily concerned with macroscopic phenomena in nonequilibrium fluids and covers such behavior as diffusion in quiescent fluids, convection, laminar flows, and fully developed turbulence. 

Many experimental results have been published, which deal with shear stress in biological systems. Most of them use laminar flow systems such as viscosimeters, flow channels or flasks and very small agitated vessels which are not relevant to technical reactor systems with fully developed turbulent flow. On the other hand the geometric and technical parameters are often not sufficiently described. Therefore it is not possible to explain the complex mechanism of force in bioreactors only on the basis of existing results from biological systems. 

The validity of Eqs. (3-5) are bond on the condition of fully developed turbulent flow which only exists if the macro turbulence is not influenced by the viscosity. This is the case if the macro turbulence is clearly separated from the dissipation range by the inertial range. This is given if the macro scale A is large in comparison to Kolmogorov s micro scale qp Liepe [1] and Mockel [24] found out by measurement of turbulence spectra s the following condition ... 

Fig. 1. Dimensionless stress in fully developed turbulent flow given by the theory of isotropic turbulence...

In order to use Eqs. (3) and (4) or the data given in Fig. 1, for the calculation of maximum turbulent fluctuation velocity the maximum energy dissipation e , must be known. With fully developed turbulence and defined reactor geometry, this is a fixed value and directly proportional to the mean mass-related power input = P/pV, so that the ratio ,/ can be described as an exclusive function of reactor geometry. In the following, therefore details will be provided on the calculation of power P and where available the geometric function ,/ . 

To avoid gas-liquid mass transfer Hmitation, which would have a negative influence on productivity, in correctly operated bioreactors there are turbulent flow conditions with more or less pronounced turbulence, for which the Reynolds stress formula (Eq. (2)) can be used. Whereas, as a rule there is fully developed turbulent flow in technical apparatuses (see condition (6) and explanations in Sect. 8), this is frequently not the case in laboratory fermenters. Equations (3) and (4) are then only valid to a limited extent. 

In shake flasks there is neither undisturbed laminar flow nor fully developed turbulent flow. However, stress can be estimated approximately using Eqs. (2-4). 

It could be shown (see Sect. 6) that in stirred vessels with baffles and under the condition of fully developed turbulence, particle stress can be described by Eqs. (2) and (4) alone. The turbulent eddys in the dissipation range are decisive for the model particle systems used here and many biological particle systems (see Fig. 2), so that the following equation applies to effective stress ... 

This equation should generally valid for all particle systems and working conditions with (9p-9)/9 1, dp/qL< 6 and A/qL> 125... 250. The last condition of fully developed turbulent flow is very important. To small values A/qp which mostly corresponds to Reynolds numbers Re <10 (small reactors, higher viscosity s of media and small power input) leads to an distinct reduction of stress. That was observed by the investigations in 166] which were carried out with the... 

A constant value of the friction factor f = 0.009 is assumed, for fully developed turbulent flow and a relative pipe roughness e = 0.01. The assumed constancy of f, however, depends upon the magnitude of the discharge Reynolds number which is checked during the program. The program also uses the data values given by Szekely and Themelis (1971), but converted to SI. 

In the flow region between laminar and fully developed turbulent flow heat-transfer coefficients cannot be predicted with certainty, as the flow in this region is unstable, and the transition region should be avoided in exchanger design. If this is not practicable the coefficient should be evaluated using both equations 12.11 and 12.13 and the least value taken. 

As developed by Dukler, Deissler s expression for e was used for the region near the wall, and, von Karman s relationship was used for highly developed turbulent flow. 

Expressions for Fully Developed Turbulent Mass Transfer... 

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