Scaling-up, criterion for

This is the scale-up criterion for the roasting time of meat of the same kind (a, p = idem). It states that in doubling the mass of meat, the cooking time will increase by 22/3 = 1.58. 

This relationship is reproduced in Fig. 3.12a with the test results from Hoogen-dom and den Hartog [220]. It shows that FIs = const applied in the laminar range for several axially conveying stirrer types (helical ribbon stirrers propeller stirrers in a draught tube), i.e. the stirrer power per unit volume, which is required for a particular homogenization time, is here proportional to the viscosity n of the liquid and independent of the tank diameter. FIj and therefore P/D P/V in this range is actually a scale-up criterion for these stirrers. 

In the range 112 IO is no longer constant for these stirrer types, but decreases rapidly only to start strongly increasing again from 112 = 10 see also Fig. 3.12b, from [611]. In general P/V is by no means suitable as a scale-up criterion for homogenization operations. 

Kipke cited, as a possible reason for this discrepancy the fact, that the iVss, which applies for liquids at rest, is tacitly used in the calculations of [112 and 114]. When, however, the turbulence field is determining for the sinking velocity and this changes in the macro-range by /z, then it can be imagined that for particles with particular dimensions relative to the micro- and macro-scales no generally valid scale-up criterion for all boundary conditions is to be expected [277]. 

FIGURE 9.27 Power per unit volume as the scale-up criterion for solids distribution. (From Mak etal., 1990.)... 

Okufi et al. [88] studied the scale-up effect on the DSD of n-heptane in water, and reported that the rule of equal impeller tip speed provided the best scale-up criterion for equal interfacial areas per unit volume of dispersion, while the criterion of equal input power per unit volume resulted in the production of smaller droplets. Scully [89] used the well-known correlation between the Sauter mean diameter... 

A.7.1.2 Scale-Up Criterion for Solid Suspension This subject has received far more attention in the literature than the corresponding condition of gas dispersion. Different criteria have been employed by various investigators for predicting the scale-up rule for just solid suspension. The two main criteria employed are as follows ... 

Figure 9.20 The data show that the inlet jet Reynolds number (Re ) is a good scale up criterion for geometrically similar scaled CIJ mixers CIJ-d2 data are also shown to evidence that a variation of the inlet tube/chamber diameter ratio may influence the final particle size. Both PEGylated copolymer nanospheres and nanocapsules are shown, evidencing different dependence on Re. quenched samples (quench ratio = 1) in the upper graph, non-quenched in the lower graph, measured after solvent (acetone) evaporation. Nanocapsules (containing Miglyol , MR=1.26) , scale down , reference (CIJ-dl) , scale up A, CIJ-d2. Polymer nanospheres , scale down O, reference (ClJ-dl) O, scale up , CIJ-d2.

Obviously the latter criterion is much more conservative than the former one. This would result in even larger reactors on the larger scale. Moreover a situation could occur where the parameter (p becomes > 1, which would have an adverse effect on the meso-mixing (see section 4,2,2,4), Bourne pointed out however, that a considerable gain in possible reactor capacity is obtained by using multiple feed tubes. When the feed tube diameters are increased proportionally to the scale of the reactor, the scale-up criterion for reactors with feed tubes is... 

The criterion of maintaining equal power per unit volume has been commonly used for dupHcating dispersion qualities on the two scales of mixing. However, this criterion would be conservative if only dispersion homogeneity is desired. The scale-up criterion based on laminar shear mechanism (9) consists of constant > typical for suspension polymerization. The turbulence model gives constant tip speed %ND for scale-up. 

The scale-up criterion that is probably most widely used for mixing-limited unit operations is based on constant power input per unit volume according to (Harnby etal., 1992). 

We must draw the following conclusion A particular scale-up criterion that is valid in a given type of apparatus for a particular process is not necessarily applicable to other processes occurring in the same device. 

The basic principles employed in the preparation of parenteral products do not vary from those widely used in other sterile and non-sterile liquid preparations. However, it is imperative that all calculations are made in an accurate and most precise manner. Therefore, an issue of a parenteral solution scale-up essentially becomes a liquid scale-up task, which requires a high degree of accuracy. A practical yet scientifically sound means of performing this scale-up analysis of liquid parenteral systems is presented below. The approach is based on the scale of agitation method. For singlephase liquid systems, the primary scale-up criterion is equal liquid motion when comparing pilot-size batches to a larger production-size batches. 

For mixing operations, the scale-up criterion is customarily given as an overall equation (Montane et al., 2003) ... 

Satisfactory results were obtained with the pilot-scale fermentor at a rotational impeller speed N of 1.5 s and air rate (30 °C) of 0.5 in min T The density and viscosity of the broth are 1050 kg m and 0.002 Pa s, respectively. Data from the turbine impellers [3] showed that kf a can be correlated by Equation 12.13, with values m = n = 2/3. Using k a as the scale-up criterion, estimate the impeller speed and the air rate for the production-scale fermentor that will give results comparable with the pilot-scale data. 

For homogeneous chemical reactions, the power per volume can be used as a scale-up criterion. As a rule of thumb, the intensity of agitation can be classified based on the power input per 1,000 gallon as shown in Table 9.4... 

For the scale-up of the gas-liquid contactor, the volumetric mass-transfer coefficient kLa can be used as a scale-up criterion. In general, the volumetric mass-transfer coefficient is approximately correlated to the power per volume. Therefore, constant power per volume can mean a constant kLa. 

The wall effect on the performance of fluidized beds can be reflected by the differences of the overall hydrodynamic behavior in beds with different diameters. Practically, in order to scale up a pilot fluidized bed to a full-size commercial unit, the size of the pilot unit should be large enough to exclude the wall effect. A critical scale-up diameter (Dcs) for the pilot fluidized bed is thus defined as the minimum bed diameter beyond which the bed behavior is almost independent of the bed size. Calculate the critical scale-up diameter for fluidized beds without internals, using the following criterion ... 

Scale-Up at Constant Re/. For a given fluid, Re oc NO2 = constant, i.e. N cc D 2. Scale-up at constant Rei implies constant Po, yielding, for a given fluid, P oc N2 If. Therefore, scale-up at constant Re gives P oc N2D5 cc (D 2)2 x D5 oc D l. Power requirement decreases as scale increases This is not a suitable scale-up criterion. 

Convective bulk transport is also an extremely important factor in the suspension of solids in a stirred tank (this is also responsible for the flow pattern at the tank bottom). P/V cannot be used as a scale-up criterion in this process either. Measurements have shown that the minimum rotational speed, ncrit, of the stirrer which is necessary for the suspension (whirling-up) of particles in the turbulent regime is given by the appropriate Froude number ... 

Because the Froude number is the scale-up criterion here, we formulate it for a given material system by P/V as follows ... 

It is difficult to predict from fundamental considerations that constant power per unit volume should be a generally significant scale-up criterion. In fact, as Rushton (R8) shows, use of equal power per volume for scaling can result in serious error in many cases. Thus, the successful application of this concept to certain operations must be regarded as a somewhat fortuitous result of the specific interactions present in those particular cases. 

Another more general scale-up criterion is that of impeller Reynolds number, DzNp/fi. This approach, strongly recommended by Rushton (R8), is based on the observation that in many mixing operations the performance of the system can be successfully described by relations like Eq. (44) or (45), involving the familiar Nusselt or Sherwood numbers and the impeller Reynolds number. For geometrically-similar systems and constant material properties, these relations reduce to the forms... 

Emulsion life expectancy for a formulation may be conservatively scaled up from 2-in. pipe-loop tests at the same velocity by demonstrating that the emulsion will survive transport for the desired actual distance in the pilot plant. Pilot-plant transport is a more severe test of emulsion life than transport in larger lines. The conservative nature of this scale-up criterion tends to dictate specification of some excess surfactant for a large-scale application beyond the minimum quantity required. 

Fig. 3.12a, b Investigation of the possible scale-up criterion P/V = const for different stirrer types and different flow conditions a from [220], b from [611]. 

Final Discussion from the Viewpoint of the Dimensional Analysis No industrial stirring operation in Germany in connection with the scale-up criterion has caused more stir and none was more controversially discussed than the suspension of solid particles. It is noteworthy that the authors without exception ascribed to dimensional analysis the blame for the inappropriate or deficient, in the sense of scale-up criteria, at any rate inadequate pi space. 

If in the scale-up criterion the expression is replaced by the equivalent expression for the turbulent flow range, P/pV, the following expression is obtained... 

For an animal cell culture, satisfactory results were obtained with a pilot fermentor, 0.3 m in diameter, with a liquid height of 0.3 m (clear liquid), at a rotational impeller speed N of 1.0s 1 (impeller diameter 0.1m) and an air rate (30 °C) of 0.02 m3 min-1. The density and viscosity of the broth are 1020 kg m-3 and 0.002 Pas, respectively. The kLa value can be correlated by Equation 7.36. When kLa is used as the scale-up criterion, and the allowable impeller tip speed is 0.5 m s-1, estimate the maximum diameter of a geometrically similar stirred tank. 

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