A New Particle Size Distribution Apparatus based on Unbalance by Centrifugal Sedimentation

A new apparatus for the measurement of particle size distribution based on the unbalance caused by centrifugal sedimentation was made on an experimental bases. The size distributions obtained by this method with a constant revolution of the rotor were compared with those measured by other methods using the same sample. Furthermore, in order to shorten the measuring time and expand the measuring size range, some additional attempts were conducted with a linear acceleration followed by a constant revolution of the rotor. They gave similar results of size distribution to the measurements at a constant revolution.


Introduction
The principle of sedimentation of particulate materials in a liquid phase is used most widely and reliably among various principles for the measurement of particle size distribution. One of the most popular methods based on this principle adapts the opacity with the particles settling by a centrifugal force. This method has two disadvantages. One of them is the difficulties caused by the fact that the size of particles in the submicron range becomes the same order to the wave length of light. The other is that the optical extinction coefficient varies with the particle size and the material.
A new method based on the displacement of the center of gravity caused by particle settling was proposed to overcome the disadvantages of light extinction methods!).
In this report, we first introduce a test apparatus that was made to measure size distribution applying this method and combining a personal computer for immediate analysis. Then we show some examples of the measurement using this instrument. In order to expand the range of measuring size it was examined to increase rotor speed of the centrifuge with the measurement time.

Measuring principle and procedures
Pouring homogeneous suspension including a sample powder into a cell and setting it in a rotor, the balance of the rotor is arranged before measuring. The rotor being set in motion, the particles settle in the centrifugal field and the center of gravity of the cell is shifted from its initial position.
A sedimentation curve in terms of the unbalance can be obtained by the instrument which detects the deviation of the center of gravity making use of a balancing machine.
The equation of motion for centrifugal sedimentation is written with an angular velocity w and a particle radius r as follows.  (2) Substituting Eq. (2) for (1 ), and applying the boundary condition x =xi at t=O, we obtain or X ln--=Kr 2 w 2 t xi (3) (4) For the displacement of the center of gravity in the cell by the particle settling, we assume two processes as illustrated in a) The displacement of the center of gravity caused by the particles which are located between xi and Xm at t=O and settle out completely by t=t. b) The displacement of the center of gravity resulted from the sedimentation of the particles which are located between x 0 and xi at t =0 and do not reach the cell bottom by t=t. The displacements defined in above a) and b) are shown with E a and E b, respectively.
Mass of particles rna positioned between xi and xm is given: where rn is the mass of particles in the cell.
Letting the initial eccentric distance be e 0 and the one after the displacement of the center of gravity be e 1 , the balancing moment takes the following forms in each case, where M is the total mass of the rotating system. Combining Eq. (6) and Eq. (7) _

_ma (Xm-Xi)(Pp-PL)
Ea-e 1 -e 0 --- For particles settling from xi to Xm with timet (9) Reforming the above, Substituting Eq. (10) for Eq. (5), On the other hand, the balance moment for the mass of particles mb positioned between x 0 and xi at t=O gives: where e 0 ', e/ are the initial eccentric distance and the one after the displacement of the center of gravity by sedimentation respectively and x 0 ' is the interface position between the particle suspension and the supernatant liquid containing no particles. Similar to Eq. (4), we obtain the following form for x 0 ' And from this relation where mb is given by the proportional allotment of m, Substituting Eqs. (9) and (17) for Eq. (16), simplification gives Total eccentric displacement e for the rotor is the sum of ea and eb and given from Eq. (12) and Eq. (18),

2(xm -Xo)
In the balancing test, centrifugal force P located on the rotor is expressed in general (20) where e is eccentric distance. The value of Me is determined by the centrifugal force P with a known w and thus we defined an unbalance U as follows,

U=Me
Now assuming that the unbalanced state of the actual rotor can be realized being replaced by a completely balanced rotor with attaching a Fig. 2 Equivalent unbalance on a rotor weight having mass of m at a radius R as illustrated in Fig. 2.
The unbalance of the measuring rotor is obtained from Eq. (19) with setting e = e, When the rotor revolution varies as shown in Fig. 3, angular speed w is given by the following form, w = w 0 +at (24) With this relation introduced, Eqs. (3) and (4) are rewritten as follows, where Tz is defined in the following equation, The unbalance of the period before reaching constant rotation is derived from Eq. (23)' replacing T a by T z, The unbalance U is given The first time on the right hand side U(Te) is derived from Eq.

(32)
The second term U(te) is calculated in the same way, where Te (te) is given as The unbalance U(ti) is therefore calculated as a sum of products of the mass mi and the coefficient A (ri, ti) of the particles having radius ri at t = ti. of particles mi whose diameter is Dpi (=2rd and the particle size distribution.

Measurement apparatus
The system developed to determine particle size distributions is shown in Fig. 4. We use a dynamic balancing machine type FH-214G prepared by Akashi Seisakusho, Ltd. reformed for the attachment of the rotor and the cell shown in Fig. 5.
The block diagram of the unbalance detecting unit is shown in Fig. 6. The balancing machine like this "hard" type has condenser pickups which detects the amplitude of vibration of the bearing supported by springs having constant k. The vibration is produced by the rotor unbalance caused by the centrifugal sedimen tation of particles. The tachometer detects the rotor speed and the signal of the rotation is transferred to the measuring circuit, which makes analog processing of the unbalance with the vibration amplitude X, spring constant k and angular velocity w and convert it in the The personal computer receives the real-time signal converted to BCD coded data by the digital voltmeter through the isolated parallel interface. The computer saves the data inputs of the unbalance and corresponding time to a flexible disk unit.
After the measurement, the computer processes the data saved in the disk with the excution of the analizing program and transmit the size distribution to an X-Y plotter or a printer. The computer we used is NEC PC-980 1 personal computer with 16 bit CPU and neumerical co-processing LSI. On the measurement, the computer is taking the data and simultaneously sends the control signal through the D/ A converter to regulate the rotor revolution. The frequency converter is Toshiba TOSVERT-130 and controls the rotation according to the computer instruction. The resolution of the balancing machine used for this work is 0.5 ~m and 8 mg converted in terms of weight.
Concentration of the suspension was set at 3 wt% as usually applied for gravitational sedimentation method. The suspension of about 30 ml was weighed accurately and used for the measurement. After adjustment of the initial balance of the rotor is performed, the computer conducts the measurement automatically.

Results
The measurements were performed on the following three conditions.  Fig. 3. The last condition in the above-mentioned was planned to shorten the measuring time and to expand the measuring size range. Figure 7 shows an example of sedimentation curve of the unbalance with a constant rotation. The horizontal axis presents the measuring time and the vertical axis indicates the relative unbalance from the base in the figure. The unbalance has a unit [g-mm] as seen from Eq. (21). Figure 8 indicates the results of four measurements for the same sample on the condition of constant rotation to examine the reappearance. The measurements give small fluctuations of the results in the finer size range. The results are shown in Fig. 9, compared with other methods, such as sedimentation balance, electric resistance and photo extinction. The figure indicates that sedimentation balance gave the finest, having good agreement with this method, while photo extinction   Fig. 11 Comparison of the measuring conditions measure in the coarser particle size range, Kanto loam was employed as substitute material to compare with. The rotation speed was set at 1300 r.p.m. in this case, because the Kanto loam was relatively coarse and would have settled out immediately at 2500 r.p.m. Decreasing the rotation produces smaller vibration amplitude and so reduces the resolution of pickup as seen from Fig. 6. To avoid decreasing the resolution and rapid settling of particles, the measuring rotation was set at 1300 r.p.m. There was a little difference in the results between the constant rotation and the linear acceleration. It may come from the error in the determination of a correcting values. The signal band detected by the pickup varied with rotor revolution and therefore would require blank tests and correction of the directly obtained values. We will examine the correction more exactly.

Conclusion
A new method for particle size distribution measurement making use of a dynamic balancing machine was introduced and a test apparatus was made based on this method.
It has advantages of the direct relation with the displacement of the center of gravity caused by the particle settling and no connection with properties depending on the materials such as the extinction coefficient in the case of photo extinction. The other advantage of the measurement is comparatively short time required to determine the size distribution. In this instrument, as the particle settling in any