The target quantity

During dispersion a steady-state situation is established, in which the droplets formed coalesce to ever larger droplets and are broken up anew by the stirrer. The droplet size distribution of the dispersion produced by stirring is usually characterized by an average droplet size 32, which is defined by the expression  

The average droplet diameter 32 is obtained from the sum of the volumes of all 

The following relationship exists between 32 and the interfacial area a per unit liquid volume  

An extensive literature exists over coalescence in L/L systems, because this is of crucial importance for the dimensioning of extraction columns. On the other hand this process has been much less investigated during dispersion in stirred tanks. This is because it is not the only process involved, it being constantly disturbed by the dispersion action of the stirrer, and thus producing a steady-state droplet size distribution. 

the capillary number = p /r/n, which describes the strength of the flow field, and 

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Frequently the signal intensity of the analyte A is the target quantity, the influences on which are described by Eq. (3.16a). Handling all the influences (interferences and other factors) in the same way and holding xA at any constant value so that a0 = yAo + SAAxA, Eq. (3.16a) can be written... 

In analytical chemistry the target quantity y which has to be optimized is frequently the signal intensity, absolute or relative (signal-to-noise ratio), but occasionally other parameters like yields of extractions or chemical reactions, too. The classical way to optimize influences, e.g., in an optimization space as shown in Fig. 5.3a is to study the factors independently one after the other. In Fig. 5.3b,c it can be seen that an individual optimum will be found in this way. 

We facilitate the procedure by systematically listing the target quantity and all the parameters influencing it ... 

The relevance list of this task consists of the target quantity (mixing power P) and the following parameters stirrer diameter d, density p, kinematic viscosity v of the liquid, and stirrer speed ... 

Mixing time 6 is the time necessary to completely homogenize an admixture with the liquid contents of the vessel. It can easily be determined visually by a decolorization reaction (neutralization, redox reaction in the presence of a color indicator). The relevance list of this task consists of the target quantity (mixing time 6) and of the same parameters as in the case of mixing power— on condition that (contrary to Example 3) both liquids have similar physical properties ... 

The flne grinding of solids in mills of different shape and mode of operation is used to produce the finest particles with a narrow particle size distribution. Therefore—as in the previous example—the target quantity is the median value dso of the particle size distribution. 

Once preformulation screening has identified which excipients are able to stabilize the API, a series of prototype formulations can be developed, which will be more reflective of the targeted quantities of excipients and drugs present in the final... 

Example 2 The Determination of the Pi Set for the Stirrer Power in the Contact Between Gas and Liquid. We examine the power consumption of a turbine stirrer (so-called Rushton turbine see inset in Fig. 1) installed in a baffled vessel and supplied by gas from below (see Sketch 2). We facilitate the procedure by systematically listing the target quantity and all the parameters influencing it ... 

Quantities of the square core matrix may eventually appear in all of the dimensionless numbers as fillers, whereas each element of the residual matrix will appear in only one dimensionless number. For this reason the residual matrix should be loaded with essential variables like the target quantity and the most important physical properties and pro-cess-related parameters. 

With the basic dimensions of mass, length, and time denoted as [M], [L], and [T], respectively, the relevance list for the target quantity H [ML T ] (mechanical tensile strength of the tablet) included ... 

Multiple regression of the target quantity (H k) on a combined data set (including data for two materials, two presses, two speed levels for each press, and a... 

Equation 23.6 gives the approximate relationship between the charging current and the peak faradaic current, ip (the target quantity), for a test compound of concentration C° (mM) assuming normal double-layer capacitances [1],... 

The simplest method to produce droplets is to drip a liquid slowly out of a capillary under the influence of gravity. Due to the low shear rate, the viscosity of the liquid will, as a rule, have no influence. In this case the target quantity, particle diameter (here droplet diameter) dp, will depend only on the wetted capillary diameter d, the surface tension o and the weight gp of the dripping liquid ... 

Roasting is an endothermic process. The meat is ready when a certain temperature distribution (T) is reached within it. The target quantity is the time duration, 0, necessary to achieve this temperature field. 

The nature of the steps which have to be carried out now makes this dimensional matrix less than ideal because it is necessary to know that each of the individual elements of the residual matrix will appear in only one of the dimensionless numbers, while the elements of the core matrix may appear as fillers in the denominators of all of them. The residual matrix should therefore be loaded with essential variables such as the target quantity and the most important physical properties and process-related parameters. Variables with an, as yet, uncertain influence on the process must also be included in this group. If, later, these variables are found to be irrelevant, only the dimensionless number concerned will have to be deleted while leaving the others unaltered. 

This example, taken from [12], belongs to the field of heat transfer by convection. Here the heat transfer coefficient, h, represents the target quantity. This quantity can be determined only via the general heat transfer equation... 

Flotation is a depletion process which obeys a time law of the 1st order and is described by the flotation rate constant, kF. The target quantity, solids discharge A , corresponds to the chemical conversion, X, in 1st order reactions ... 

This aim fully describes the target quantity of a foam centrifuge, s. sketch in Fig. 4. One has to determine the minimum rotational speed, nmin, required to set the foam flowing. (According to [17], a resulting foam density of ca. p = 0.50 kg/1 should suffice.)... 

Occasionally, one could have read about the failures of the Theory of Similarity or of its limits. However, this criticism has arisen when, due to some physical reasons, a complete similarity could not be achieved (see e.g. remarks of Damkohler [113] on p. 183) or the scale-up criterion could not have been worked out with certainty because the measuring conditions did not allow it (false model scale, wanted sensitivity of the target quantity, non-availability of the model material system, ignorance about relevant physical properties, such as in foams and sludges, etc.). 

In surface aeration, the absorption rate is also measured with 02 electrodes in the liquid volume. By this method, the liquid-side overall mass transfer coefficient, kLa, is determined (a - volume-related mass transfer area = surface of all gas bubbles in the liquid volume). Due to the fact that the mass transfer in surface aeration occurs almost solely in the liquid surface, A, and by no means in the liquid volume, V, the measured kLa has to be multiplied by V to obtain the target quantity kLA = kLa V. 

Due to the approximate uniformity of the intensively mixed gas/liquid system and, therefore, the intensity character of the target quantity kLa, the influencing process quantities (stirrer power P, air throughput q) have to be formulated in an intensive manner as well. Now, the question arises whether, in addition to the volume-related stirrer power, P/V, the gas throughput also has be formulated as a volume-related one (q/V), or if its inclusion as the so-called superficial velocity vG = q/S (as accurate... 

For a given geometry of the set-up, the relevance list for this problem contains the power consumption, P, as the target quantity, the stirrer diameter, d, as the characteristic length and a number of physical properties of the liquid and the gas (the latter are marked with an apostrophe) Densities, p and p, kinematic viscosities, v and v, surface tension, a, and an unknown number of still unknown physical properties, S, which describe the coalescence behaviour of finely dispersed gas bubbles and by this, indirectly, their hold-up in the liquid. The process parameters are the stirrer speed, n, and the gas throughput, q, which can be adjusted independently, as... 

This pi-equation, taken from [68], represents a physical correlation. It is absolutely useless for scaling up purposes To predict the target quantity dp max, knowledge of another target quantity, namely the film thickness, 8, is also necessary ... 

Russian researchers [70] have found that in glass tubes the same Ku value is obtained at Bd > 6. This discrepancy caused Eichhom [71] to carry out a detailed dimensional-analytical examination. He first discovered that the lower critical vG corresponded to a critical film thickness and to a critical shear rate in the phase boundary G/L. Therefore, there are three parameters independent of each other, which could be regarded as target quantities. However, vG can be measured more accurately and more easily than the others, therefore, it is accepted as the target quantity. 

All of them utilise a high energy input to produce very fine droplets of the disperse (mostly oil) phase. The aim of this operation is, likewise in Example 27, the narrowest possible droplet size distribution. It is normally characterised by the Sau-ter mean diameter d32 [67] or by the median d50 of the size distribution. d32 and dso, respectively, therefore represent the target quantity of this operation. 

The target quantity in this process is the temperature, T, of the liquid. We will include the cylinder diameter, d, as characteristic geometric parameter in the relevance list. The physical properties are the density, p, the viscosity, p, the heat capacity, Cp, the thermal conductivity, k, and the temperature coefficients of viscosity, y0, and of density, p0- The process parameters are the experimental time, t, gravitational acceleration, g (because of the density differences on account of the temperature field) and the two characteristic temperatures ... 

The selection of the volume-related and, therefore, intensively formulated variable kLa, this being the target quantity of the mass transfer process, implies the following consequences ... 

Since the target quantity kLa is an intensity variable, the process parameters must also be formulated intensively. 

The target quantity E will depend on the parameters of mass transfer kL, DA, DB and reaction kinetics k2, cA , cB, whereby k2 stands for the rate constant of a 2nd order reaction ... 

And, inversely, once the regression line is established, you can calculate Newton power number Np (which is the target quantity for scale-up) and net power AP (which can be observed in real time as a true indicator of the target end-point) for any point on the line. 

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