Fully Developed Turbulent Flow

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

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... 

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 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 ... 

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...

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). 

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. 

Fig. 12. Snapshot from a two-phase DNS of colliding particles in an originally fully developed turbulent flow of liquid in a periodic 3-D box with spectral forcing of the turbulence. The particles (in blue) have been plotted at their position and are intersected by the plane of view. The arrows denote the instantaneous flow field, the colors relate to the logarithmic value of the nondimensional rate of energy dissipation.

For fully developed turbulent flow in rough pipes,/is independent of the Reynolds number, as shown by the nearly constant friction factors at high Reynolds number in Figure 4-7. For this case Equation 4-33 is simplified to... 

Guess a value for the velocity at point 2. If fully developed turbulent flow is expected, then this is not required. 

Thus the velocity of the liquid discharging from the pipe is 3.66 m/s. The table also shows that the friction factor/changes little with the Reynolds number. Thus we can approximate it using Equation 4-34 for fully developed turbulent flow in rough pipes. Equation 4-34 produces a friction factor value of 0.0041. Then... 

Determine the Fanning friction factor / from Equation 4-34. This assumes fully developed turbulent flow at high Reynolds numbers. This assumption can be checked later but is normally valid. 

The excess head loss terms 2 Kt are found using the 2-K method presented earlier in section 4-4. For most accidental discharges of gases the flow is fully developed turbulent flow. This means that for pipes the friction factor is independent of the Reynolds number and that for fittings Kf = and the solution is direct. 

Assume fully developed turbulent flow to determine the friction factor for the pipe and the excess head loss terms for the fittings and pipe entrances and exits. The Reynolds number can be calculated at the completion of the calculation to check this assumption. Sum the individual excess head loss terms to get 2 Kf. 

A stringent requirement for PF, nearly in accordance with fluid mechanics, is that it be fully developed turbulent flow. For this, there is a minimum value of Re that depends on D and on e, surface roughness ... 

Consider a fully developed turbulent flow through a pipe of circular cross section. A turbulent boundary layer will exist with a thin viscous sublayer immediately adjacent to the wall, beyond which is the buffer or generation layer and finally the fully turbulent outer part of the boundary layer. 

Equations 2.58, 2.70 and 2.71 enable the velocity distribution to be calculated for steady fully developed turbulent flow. These equations are only approximate and lead to a discontinuity of the gradient at y+ = 30, which is where equations 2.70 and 2.71 intersect. The actual profile is, of course, smooth and the transition from the buffer zone to the fully turbulent outer zone is particularly gradual. As a result it is somewhat arbitrary where the limit of the buffer zone is taken often the value y+ = 70 rather than j + = 30 is used. The ability to represent the velocity profile in most turbulent boundary layers by the same v+ - y+ relationships (equations 2.58, 2.70 and 2.71) is the reason for calling this the universal velocity profile. The use of in defining v+ and y+ demonstrates the fundamental importance of the wall shear stress. 

Wu, Ruff and Faeth12491 studied the breakup of liquid jets with holography. Their measurements showed that the liquid volume fraction on the spray centerline starts to decrease from unit atZ/<70=150 for non-turbulent flows, whereas the decrease starts at aboutZ/<70=10 for fully developed turbulent flows. Their measurements of the primary breakup also showed that the classical linear wave growth theories were not effective, plausibly due to the non-linear nature of liquid breakup processes. 

As discussed in Chapter 2, a fully developed turbulent flow field contains flow structures with length scales much smaller than the grid cells used in most CFD codes (Daly and Harlow 1970).29 Thus, CFD models based on moment methods do not contain the information needed to predict x, t). Indeed, only the direct numerical simulation (DNS) of (1.27)-(1.29) uses a fine enough grid to resolve completely all flow structures, and thereby avoids the need to predict x, t). In the CFD literature, the small-scale structures that control the chemical source term are called sub-grid-scale (SGS) fields, as illustrated in Fig. 1.7. 

Table 2.1. The principal length and time scales, and Reynolds numbers characterizing a fully developed turbulent flow defined in terms of the turbulent kinetic energy k, turbulent dissipation rate e, and the kinematic viscosity v. 

From this definition, it can be observed that T,(k. t) is the net rate at which turbulent kinetic energy is transferred from wavenumbers less than k to wavenumbers greater than k. In fully developed turbulent flow, the net flux of turbulent kinetic energy is from large to small scales. Thus, the stationary spectral energy transfer rate Tu(k) will be positive at spectral equilibrium. Moreover, by definition of the inertial range, the net rate of transfer through wavenumbers /cei and kdi will be identical in a fully developed turbulent flow, and thus... 

In a fully developed turbulent flow, the rate at which the size of a scalar eddy of length l,P decreases depends on its size relative to the turbulence integral scale L and the Kolmogorov scale ij. For scalar eddies in the inertial sub-range (ij < Ip, < Lu), the scalar mixing rate can be approximated by the inverse of the spectral transfer time scale defined in (2.68), p. 42 8... 

In general, the scalar Taylor microscale will be a function of the Schmidt number. However, for fully developed turbulent flows,18 l.,p L and /, Sc 1/2Xg. Thus, a model for non-equilibrium scalar mixing could be formulated in terms of a dynamic model for Xassociated with working in terms of the scalar spatial correlation function, a simpler approach is to work with the scalar energy spectrum defined next. 

In a fully developed turbulent flow,22 the scalar spectral transfer rate in the inertial-convective sub-range is equal to the scalar dissipation rate, i.e., T k) = for /cei < < Kn. Likewise, when Sc 1, so that a viscous-convective sub-range exists, the scalar trans-... 

Taylor (T4, T6), in two other articles, used the dispersed plug-flow model for turbulent flow, and Aris s treatment also included this case. Taylor and Aris both conclude that an effective axial-dispersion coefficient Dzf can again be used and that this coefficient is now a function of the well known Fanning friction factor. Tichacek et al. (T8) also considered turbulent flow, and found that Dl was quite sensitive to variations in the velocity profile. Aris further used the method for dispersion in a two-phase system with transfer between phases (All), for dispersion in flow through a tube with stagnant pockets (AlO), and for flow with a pulsating velocity (A12). Hawthorn (H7) considered the temperature effect of viscosity on dispersion coefficients he found that they can be altered by a factor of two in laminar flow, but that there is little effect for fully developed turbulent flow. Elder (E4) has considered open-channel flow and diffusion of discrete particles. Bischoff and Levenspiel (B14) extended Aris s theory to include a linear rate process, and used the results to construct comprehensive correlations of dispersion coefficients. 

It was also found that for fully developed turbulent flow the above values were essentially independent of the Reynolds number. 

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