Newton number

FIG. 20-38 Newton number as a Function of Reynolds number for a horizontal stirred bead mill, with fluid alone and with various filling fractious of 1-mm glass beads [Weit and Schwedes, Chemical Engineering and Technology, 10(6), 398 04 (1987)]. (N = power input, W d = stirrer disk diameter, m n = stirring speed, 1/s i = liquid viscosity, Pa-s Qj = feed rate, mVs.)... 

Ne = Newton number, depends on unit design. Can range from 0.8 to 1.9, for example. 

Function of fluid properties, such as, Ll, k and cp Shaft speed of rotation, revolutions per second Newton number, depends on design Force number, consistent units, dimensionless = F/(PN2D )... 

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]). 

Using Eqs. (11) and (2) the stirrer performance and the resultant Newton number Ne for the laminar flow range can be derived with the formula P = 2ti... 

The results presented here were found by investigations with a special cyUn-der system [45,48]. This system was constructed for an existing Searle viscosimeter (rotation of inner cylinder), such that the gap widths were large in relation to the reference floe diameter of the floccular system used, so that the formation of the floes and their disintegration in the cylinder system are not impaired. For this system, with r2 = 22 mm, rj = 20.04 mm, and Li = 60 mm (r2/ri > 1.098), the following Newton number relationships were determined from the experimental values collected by Reiter [38] for the Taylor number range of 400 < Ta < 3000 used here ... 

In the case of stirred vessels the values A/riL can be calculated by the following equation using the geometry parameter d/D, H/D, the Newton number Ne, the Reynolds number Re = nd /v, the energy dissipation ratio e/e and the related macro scale A/d. For standard turbines e.g. Mockel [24] found the value A/d = 0.08 close to the impeller. Corresponding to this the maximum of the dissipation ratio ,/ has to be used which can be estimated by Eq. (20). 

If the Newton number, Ne, in relation to the Reynolds number, Re, of the mixer is known from experiments and the rotor speed, n, the mixer diameter, dji and the mass density pc of the continuous liquid are given. 

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. 

Table 2.6 Newton number and stirrer geometric characteristics of some common stirrer types.

The other friction factors in common use are the Newton number and Fanning s friction factor. The relationship between the three is ... 

From this, after the well known procedure, the following four pi-numbers result rir = F/(p l2 v2) = Ne (Newton number) n2 = v /I v = Re-1 (Reynolds number) n3 = g 1/v2 = Fr-1 (Froude number)... 

Dividing this functional dependence (6.12) by p l2 v2, in order to transform Ff into the Newton number of form drag, leads to the following proportionality... 

The (appropriately assembled) dimensional matrix undergoes only one linear transformation to produce the two pi-numbers (Ne - Newton number Re - Reynolds number) ... 

For liquid-liquid mixtures, the calculations of mixing time and power (or Newton) number outlined above are valid for unbaffled vessels only as long as the vortex created by the stirrer does not reach the stirrer head. Otherwise, gas entrainment occurs and the physical properties of the system change. The depth of the liquid-gas interface at the vessel axis with respect to static liquid surface level, HL, can be related to the Froude and Galileo numbers. Some of the reported relationships are summarized in Table XIV. The value of H, at which the vortex reaches the upper impeller blades level can be expressed as... 

For a given Reynolds number, the Newton number for a reactor with h/D = 2 is somewhat higher than the one for h/D = 1. The data for both h/D values compare reasonably well with those reported by Zlokarnik (1967) for a conventional mixing vessel with a propeller stirrer. For high Reynolds numbers Ne - 0.3, which is considerably smaller than Ne - 5 to 6 under similar conditions for conventional reactors with turbine stirrers. 

Archimedes number Bingham number Bingham Reynolds number Blake number Bond number Capillary number Cauchy number Cavitation number Dean number Deborah number Drag coefficient Elasticity number Euler number Fanning friction factor Froude number Densometric Froude number Hedstrom number Hodgson number Mach number Newton number Ohnesorge number Peclet number Pipeline parameter... 

Assuming that the groups II2, II3, II4 are essentially constant, the 11-space can be reduced to a simple relationship Lfo = /(Lfi), that is, the value of Newton number Np at any point in the process is a function of the specific amount of granulating liquid. 

When corrections for gross vortexing, geometric dissimilarities, and powder bed height variation were made, data from all mixers (Fielder PMA 25, 100 and 600 L) correlated to the extent that allows predictions of the optimum end-point conditions. The linear regression of Newton number (power) on the product of adjusted Reynolds number, Froude number, and the Geometric number (in log/log domain) yields (Fig. 12) an equation of the form ... 

The fluctuating hydraulic forces, which operate on stirrer heads [273], are expressed as Newton numbers Ncf = F/ pn d ) with the forces F operating on the stirrer sur-... 

Weber [572] has recently drawn attention to this problem by reminding us of the fact that fully developed turbulence cannot be realized in unbaffied laboratory vessels. Only at Re > 10 does the friction factor Cf in stirrer flow (counterpart of the Newton number) become constant. This can only be attained by stirring water in tanks with D > 1 m ... 

Ne is the so-called Newton number (often termed the power number) and Re is the Reynolds number. These numbers are defined as follows ... 

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