Interface height

Figure 10. Plot of interface height and solids concentration versus time.

This expression is based on our earlier assumption that the time required to thicken the sludge is independent of the interface height of the compression zone. An approximate solution to this expression can be obtained if we assume mj/m to be constant, i.e., an average mass ratio in the thickening zone from top to bottom. Then,... 

For each UVP file created, one gas-liquid interface height time series hn(N) is obtained n — 1,2,3 (one time series for each transducer) and N is the velocity or ultrasound echo intensity profile number. The gas-liquid... 

It is desirable to consider the interface heights of the three transducers with transducers 1 and 3 being to the left and to the right of transducer 2, respectively, as shown in Figure 9b. When only one transducer is used, the gas-liquid interface is assumed to be flat as in Figure 10a (line p-p ) then the area occupied by the liquid phase near the wall of the pipe and above the gas-liquid interface is ignored. With three transducers, however, a more accurate gas-liquid free surface shape may be obtained... 

A correlation analysis is performed to align the positions of the gas-liquid interface detected by the UVP transducers as if they are located on the same x-y plane. As mentioned earlier, the gas-liquid interface height series from the UVP transducers 1, 2, and 3 are h (N), h2(N), and h3(N), respectively. Each series consists of 4,096 elements (Nmax), i.e., the number of velocity/ultrasound echo intensity profiles captured, N = 0,1,. ..,4,095. 

As mentioned in Section 3.2, owing to the position of the UVP transducers along the pipe, the gas-liquid interface height is determined first by transducer 2, then transducer 1 and finally transducer 3. The positions of the gas-liquid interface heights determined by the transducers are similar but differ slightly. The value of i in Equation (3) that maximizes the correlation coefficient between series 2 and 1 is denoted z 2—i and the value that maximizes the correlation coefficient between... 

Not only do the series have the same number of elements but also the gas-liquid interface heights of the series appear as if the three transducers are located in the x-y plane where transducer 2 is originally located. The next step determines the curvature of the interface from these three points. 

We consider a square area as shown in Figure 11. This area is divided into 40,000 square elements. When summing the areas of the square elements in the circle in Figure 11a, the value obtained differs from the circular area by 0.1%. The locations of transducers 1, 2, and 3 and the locations of the gas-liquid interface heights are X, x 2, and x3 and Si( i,i/i), s2(x2,y2), and s3( 3,1/3), respectively. In the range x, free surface is calculated by a cubic polynomial function (spline interpolation), as shown by Kreyszig (1999). In the ranges 0 < x < Xj and x3 liquid interface is calculated by a linear extrapolation. The slope of... 

Figure 11 (a) Gas-liquid interface height position and (b) channel distribution. 

Width of the dead zone of the interface height step Converse piezoelectric constant Dielectric displacement... 

At this point, we now follow Lipowsky (1987) to discuss the effect of short range versus long range forces. In the spirit of fig. 6, we describe the problem in terms of an effective potential for the interface height h above the surface. For the short range case we have (f = 1 — T/TL)... 

Wolf and Villain (WV) [8], Villain, Lai, and Das Sanna (VLD) [9] and others [10]. The rate equations for the film growth resulting from these models are modified versions of the Langevin stochastic equation [10] for the variation with time of the interface height. 

According to the scaling theory, the discreteness of the depositing material is the main factor in turning the growing surface into a self-afiBne fractal surface. The standard deviation of the interface height can be expressed as... 

Figure 4. Interface height vs. time for gravity settling of unconditioned effluent...

Residence time for settUr too short] interface height of the continuous phase de-creases/[inaccurate sensing of interface] /turbulence in the continuous phase/ flowrate in continuous phase > expected for example > 3 L/s m /sludge setties and reduces effective height of continuous phase/[phase inversion] /inlet conditions faulty. 

It is usual to operate either type of pulse column with the interface near the top, as with a static column, i.e. with the aqueous phase continuous. It is sometimes advantageous, however, to run with the solvent phase continuous, e.g. when it is particularly important to avoid the carry-over of solids from the interface into the solvent phase. The interface height is achieved in the normal manner, i.e. by balancing the column against a limb of aqueous phase which is leaving it and is vented to atmosphere. Adjustments to the height of the balancing limb would allow the interface to be in any desired position. 

Figure 14.3-4. Batch sedimentation restdts (a) original uniform suspension, b) zones of settling after a given time, (c) compression of zone D after zones B and C disappear, (d) clear liquid interface height z versus lime of settling.

By measuring the interface height and solids concentrations in the dilute and concentrated suspensions, a graphic representation of the sedimentation... 

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