Volume density distribution

Figure 2. volume sum distribution Q3 (x) and volume density distribution q3 (x)... 

Droplet size distributions are, as are all particle size distributions, either represented as volume density distributions 3(dp) or as cumulative volume distributions Qi (dp) ... 

Fig. 6.1 Volume density distribution <73 (dp) as a function of dp for the 6-blade turbine stirrer at four different stirrer speeds in the material system trichloroethene/water (y> = 0.2) from [166]... 

Change of the volume density distribution with stirring after [166]... 

Figure 9.30 Size characterization of an azo pigment, yielded in the pilot plant microreactor, compared to the batch standard. Left volume density distribution right TEM magnification (by courtesy of Wiley-VCH Verlag GmbH from [66]).

Gas hold-ups were determined from the EIT reconstructions by using the resistivity distribution of continuous phase as a reference. Gas-liquid resistivity distributions were compared to the reference and three-dimensional images were formed.If resistivity is assumed to depend linearly on the gas hold-up, relative differences in gas hold-up are obtained from EIT results between the various locations. Actually, the relation between the conductivity and the gas hold-up is slightly non-linear (Mwambela and Johansen 2001) and therefore calibration experiments are needed to determine gas volume density distributions. Due to the fluctuating nature of gas-liquid flow some abnormal resistivity distributions were obtained with EIT and the averaging of several experiments is necessary to get accurate resistivity fields from the mixed tank. Abnormal resistivity distributions were also found at the boundaries like at the liquid surface and the bottom of the tank. 

Figure 13.9 gives an example of droplet volume density distributions of emulsions obtained by pressing an emulsion premix through a membrane at transmembrane pressure differences varying from 3 bar to 11 bar. These pressure differences are 7.5- to 27.5-fold the minimum pressure difference required (capillary pressure). A hydrophilic polyamide membrane with a mean pore size of 0.8 pm was used. The emulsion premix consisted of 20% dispersed phase (vegetable oil). As continuous phase water containing emulsifier Tween 80 at a concentration of 2% was used. The Sauter diameter of the emulsion premix was X3,2 = 25 pm. 

Influence of Repeated Processing Repeated processing results in smaller droplets and narrower droplet size distributions of the fine emulsions. Figure 13.10 depicts an example of the volume density distributions of an emulsion with a dispersed phase of 30% after the first, second and third pass at 9 bar through a membrane with a mean pore size of 0.8 pm. In this case, at least two passes through the membrane are required in order to obtain a monomodal droplet size distribution. 

Influence of Emulsifier Concentration Emulsions of small droplet sizes and narrow droplet size distributions can be obtained at high emulsifier concentration. Figure 13.12 shows the volume density distributions of emulsions of a disperse phase fraction of cp = 72% and two different emulsifier (Tween 80) concentrations, 2.4% and 4.6%. The production parameters were trans-membrane pressure difference of 12 bar, three passes and membrane mean pore size of 0.8 pm. In both cases the emulsifier concentration is above the critical micelle concentration (CMC). 

Figure 13.12 Volume density distributions for emulsions with different emulsifier concentrations. Disperse phase fraction

For convenient comparison of the different measuring methods upper figures on the following pages contain the statistical evaluation of the volume density distribution function of the split samples. 

Figure 5 Laser diffraction measurements of the same 10 split samples, mean volume density distribution function q(x), standard deviation s and variation coefficient v, ...

The volume density distribution of the ten split samples measured by wet sieving is evaluated in figure 7. Relative errors below 15% can be achieved in the range between 35 and 200 /urn. 

Exemplarily, for the first classification step leading to sample 1, the volume density distribution of the initial sample qm is shown in Fig. 16a together with the mass weighted coarse and fines fraction. The classification is analyzed by distinguished parameters which are commonly used in the field of particle technology [61,66] ... 

Fig. 2.3.3. Number and volume density distributions for a sample powder...

If one of the distributions is known, the others can be calculated at least this is the case if one assumes the particles to be spherical (Allen, 1990). For instance, we can calculate the volume density distribution from the number density distribution. Since the number fraction of particles in the diameter interval x — 1/2 dx and x + 1/2 dx is /jv(x)dx, then ... 

We have to choose the proportionality constant so that the area under our volume density distribution becomes unity ... 

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