Flow range Re

The power input in stirred tanks can be calculated using the equation P = Ne pnM if the Newton number Ne, which at present still has to be determined by empirical means, is known. For stirred vessels with full reinforcement (bafQes, coils, see e.g. [20]), the only bioreactors of interest, this is a constant in the turbulent flow range Re = nd /v> 5000-10000, and in the non-aerated condition depends only on geometry (see e.g. [20]). In the aerated condition the Newton number is also influenced by the Froude number Fr = n d/g and the gas throughput number Q = q/nd (see e.g. [21-23]). 

Flow velocities and flow range (Re) for EKI mixing under EOF conditions... 

The drawn-in curve is valid for the laminar flow range (Re < 2300) and corresponds to the analytical expression ( process equation )... 

Fig. 33 Work-sheet for determining optimal operating conditions for heat removal in a vessel with an anchor stirrer with two different wall clearances (D/d <= 1.00 - without wiper blades -and D/d = 1.10) in the laminar flow range (Re < 100) from [59]...

Fig. 47 clearly shows that the contact angle, , is satisfactorily taken into account by the function Ku x sin =/(Bd). Only now it is also obvious that the capillarity-buoyancy number CB exerts no influence on the hanging film. The liquid viscosity, 11, proves to be irrelevant. This is not surprising because of the fact that the respective measurement were executed in the turbulent flow range, Re = 4.15 x 103-1.42 x 10s. 

In the turbulent flow range (Re > 10 ) the following process characteristics apply for individual stirrer types in baffled vessels with H/D = 1 ... 

There can be Httle doubt that for thin Uquids and large stirred tanks, i.e. in the flow range Re > 10 , the above discussed 3. range inevitable appears, because the macro-mixing (coarse mixing) is strongly scale-dependent. 

Fig. 4.32 Gas throughput characteristic of the "three-edged hollow stirrer in turbulent flow range (Re > W). For installation conditions see Fig. 4,31.

In the flow range Re = 10 -10 the heat transfer characteristic can be represented to a good approximation in the form... 

In the laminar flow range Re < 10 the heat conduction dominates (which is periodically disturbed by the stirring device rotating past in the neighborhood of the wall) and the effect of density and viscosity disappears entirely. Then the following relationship applies for the heat transfer characteristic ... 

For this flow range (Re >400), they determined the following heat transfer-characteristic... 

In the case of homogenization of equal fractions of acid and alkali (( = 1), added axially and isokineticaly, the following relationship was found for the turbulent flow range Re = (3-17) x 10 ... 

The Colebrook foriTuila (Colebrook, y. Inst. Civ. Eng. [London], II, 133-156 [1938-39]) gives a good approximation for the/-Re-( /D) data for rough pipes over the entire turbulent flow range ... 

Glass and silicon tubes with diameters of 79.9-166.3 iim, and 100.25-205.3 am, respectively, were employed by Li et al. (2003) to study the characteristics of friction factors for de-ionized water flow in micro-tubes in the Re range of 350 to 2,300. Figure 3.1 shows that for fully developed water flow in smooth glass and silicon micro-tubes, the Poiseuille number remained approximately 64, which is consistent with the results in macro-tubes. The Reynolds number corresponding to the transition from laminar to turbulent flow was Re = 1,700—2,000. 

Maynes and Webb (2002) presented pressure drop, velocity and rms profile data for water flowing in a tube 0.705 mm in diameter, in the range of Re = 500-5,000. The velocity distribution in the cross-section of the tube was obtained using the molecular tagging velocimetry technique. The profiles for Re = 550,700,1,240, and 1,600 showed excellent agreement with laminar flow theory, as presented in Fig. 3.2. The profiles showed transitional behavior at Re > 2,100. In the range Re = 550-2,100 the Poiseuille number was Po = 64. 

Celata et al. (2006) studied experimentally the drag in glass/fused silica microtubes with inner diameter ranging from 31 to 259 jam for water flow with Re > 300. The drag measurements show that the friction factor for all diameters agrees well with predictions of conventional theory A = 64/Re (for the smallest diameter 31 pm, the deviations of experimental points from the line A = 64/Re do not exceed... 

Experimental and numerical analyses were performed on the heat transfer characteristics of water flowing through triangular silicon micro-channels with hydraulic diameter of 160 pm in the range of Reynolds number Re = 3.2—84 (Tiselj et al. 2004). It was shown that dissipation effects can be neglected and the heat transfer may be described by conventional Navier-Stokes and energy equations as a common basis. Experiments carried out by Hetsroni et al. (2004) in a pipe of inner diameter of 1.07 mm also did not show effect of the Brinkman number on the Nusselt number in the range Re = 10—100. 

Early studies of the transition to turbulence relied on flow visualization techniques for liquid flow through arrays of spheres. Jolls and Hanratty (1966) found a transition from steady to unsteady flow in the range 110<7 e< 150 for flow in a dumped bed of spheres at N — 12, and they observed a vigorous eddying motion that they took to indicate turbulence at Re — 300. In regular beds of spheres, Wegner et al. (1971) found completely steady flow with nine regions of reverse flow on the surface of the sphere for Re — 82, and similar flow elements but with different sizes in an unsteady flow at Re — 200. Dybbs and Edwards (1984) used laser anemometry and flow visualization to study flow... 

If the particle Re is well above the creeping flow range, mean drag may be increased or decreased by freestream turbulence. The most significant effect is on the critical Reynolds number. As noted in Chapter 5, the sharp drop in Cd at high Re results from transition to turbulence in the boundary layer and consequent rearward shift in the final separation point. Turbulence reduces Re, presumably by precipitating this transition." ... 

The only rigid particle for which accelerated motion beyond the creeping flow range has been considered in detail is the sphere. Odar and Hamilton (06) suggested that Eq. (11-11) be extended to higher Re as ... 

In the range Re < 20, the proportionality NecxRe is found, thus resulting in the expression Ne Re = R/= const. Density is irrelevant here because we are dealing with the creeping flow region. 

In the range Re > 50 (vessel with baffles) or Re > 5 x 10" (unbaffled vessel), because the Newton number Ne = P/(pn d ) remains constant. In this case, viscosity is irrelevant we are dealing with a turbulent flow region. 

In the turbulent flow range, which appears in industrially rough (= smooth) pipes at Re > 106, the following applies ... 

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