Electronic Journal of Polish Agricultural Universities (EJPAU) founded by all Polish Agriculture Universities presents original papers and review articles relevant to all aspects of agricultural sciences. It is target for persons working both in science and industry,regulatory agencies or teaching in agricultural sector. Covered by IFIS Publishing (Food Science and Technology Abstracts), ELSEVIER Science - Food Science and Technology Program, CAS USA (Chemical Abstracts), CABI Publishing UK and ALPSP (Association of Learned and Professional Society Publisher - full membership). Presented in the Master List of Thomson ISI.
2007
Volume 10
Issue 1
Topic:
Civil Engineering
ELECTRONIC
JOURNAL OF
POLISH
AGRICULTURAL
UNIVERSITIES
Siwiec T. 2007. THE SPHERICITY OF GRAINS OF FILTRATION BEDS APPLIED FOR WATER TREATMENT ON EXAMPLES OF SELECTED MINERALS, EJPAU 10(1), #30.
Available Online: http://www.ejpau.media.pl/volume10/issue1/art-30.html

THE SPHERICITY OF GRAINS OF FILTRATION BEDS APPLIED FOR WATER TREATMENT ON EXAMPLES OF SELECTED MINERALS

Tadeusz Siwiec
Department of Civil Engineering and Geodesy, Water Supply and Sewage Systems Section, Warsaw Agricultural University, Poland

 

ABSTRACT

The paper describes several methods of determining the sphericity of bed grains and their verification in respect to selected minerals, such as: anthracite, barite, chalcedonite, diatomite, clinoptylolite, nevtraco, sand and pyrolusite. In order to make the results more reliable the tests were performed for 10 fractions of each bed. Since the above mentioned minerals, exploited in technical processes, are not uniform, it was necessary to apply intermediary methods, which enable to determine the averaged sphericity. The testing consisted in measuring other parameters which, together with sphericity, are components of suitable calculation formulas. Formulas of Ergun, Wen and Yu, Narshimhan, Dharmajah and Cleasby, as well as that by Cleasby and Fan, were applied.

It has been shown that reliable results may be obtained only by performing a quite long series of tests, because even small errors in measuring led to noticeable deviations from average values. Most reliable results were provided by formulas of Wen and Yu, as well as Narshimhan. The method of Dharmajah and Cleasby raises doubts, because sphericity obtained by this method gave results that increased with the growing velocity of rinsing, which should not happen, as sphericity is a constant value for a material.

Key words: sphericity, bakwashing rapid filters, filtration beds.

INTRODUCTION

A filtration bed is not built of any mineral. As defined by Kawamura [10], an ideal filtration mineral should have such size (granulation) and density that it stops as much suspended matter as possible, enables to obtain the best filtrate and is easy to purify by using the lowest possible amount of water. Minerals which meet these criteria to a lesser or higher degree include: quartz sand, garnet, basalt, anthracite and active carbons, active beds, as pyrolusite, defeman, dolomite. The beds differ from one another in one significant feature of major consequences for filtration and rinsing processes. That significant feature is the method of their obtaining. Quartz sand may be exploited from under soil layer, as sand deposits are quite common at certain depths under ground, or as river sand. The other group are minerals obtained as a result of rock crushing. The difference between them consists in differentiation in their shapes. The pit sand, and river sand in particular, has rounded shape, somewhat like spherical shape, whereas crushed minerals exhibit irregular shapes with sharp edges, because during crushing they break along grain edges or edges of cleavage. It appears that the phenomena occurring during filtration and rinsing are strongly affected by the grain shape and by derivative properties resulting from grain shape.

Prior to being filled in the filter, the bed material is sieved through suitable sieves of predetermined mesh. Obviously, it is not possible to obtain grains of single diameter, because if a bed is sieved through sieve with holes of, for example, 1.0 mm, and then through 0.8-mm sieve, the grains retained on the latter sieve are those within the range of 0.8 mm to 1.0 mm; grains of intermediate granulations are occurring with the same probability. If a bed would be built of a perfectly spherical mineral, then a substitute diameter equal to arithmetical average of two extreme sieve hole sizes could be used for calculations. In actual situations however, one has to deal with minerals other than perfectly spherical, whereas crushed minerals may have shapes that are elongated, cylindrical, lamellar, pyramidal and even completely irregular. In such case, sieves may retain grains of dimensions other than the sieve hole size, as larger elongated particles may be among them as a result of vertical positioning of smallest dimensions or jamming of grains with largest dimensions, oriented horizontally. For that reason, it was necessary to develop certain equivalent values to describe averaged parameters of beds. They include equivalent (substitute) diameter, sphericity (shape factor) and porosity.

The substitute diameter is calculated on the basis of specific surface from the formula (1); [13]

       (1)

where: ni – number of grains of specific fraction, SAi and SVi – shape factors of surface A and the volume V of fraction “i”, d – grain diameter of fraction “i”.

As the total volume of solid particles forming a fixed bed is smaller than the volume of the bed, the ratio of free spaces between these particles to bed volume is called porosity, which may be determined by different methods [32]. Most frequently, it is calculated as [28]:

       (2)

where: ρs – specific density of the skeleton [12, 19], ρp – apparent density calculated as the ratio of grain mass to grain volume, as if they had no pores [34].

THE METHODS OF SPHERICITY CALCULATION

Sphericity is defined as the ratio of surface of a sphere having the same volume as the particle to the surface of that particle [2]. For particles with regular shapes the sphericity may be determined on the base of that definition. For particles with irregular shapes the sphericity is determined with approximate methods. One of them is the measurement of three characteristic dimensions, which may be obtained with optical method, or density and volumetric, or gas-expansion methods [32]. According to Dharamajah and Cleasby [6], as well as Zenz and Othmer [36], with these dimensions being known, the sphericity may be calculated from the following formula

       (3)

where: r is the smallest dimension of the particle, whereas s and t are the remaining dimensions.

Intuitively, one can imagine that a relationship exists between sphericity and porosity, as well as that porosity varies with/during bed compacting. Razumow [30], as well as Cooke and Rowe [5], indicated that, for uniform spherical particles, porosity depends only on the number of points of contact of individual spheres and not on grain diameter. Thus, theoretically, the porosity of such bed may range from 0.259 – for strongly compacted bed to 0.476 – for a loose bed in which each sphere has only 6 contact points with the neighboring spheres [30]. As in practice a bed is never built of uniform and perfect spheres, the porosity may deviate significantly from these values. The relationship between sphericity and porosity and the degree of compacting (densification) is presented in Fig. 1 [3,13,34].

Fig. 1. The relationship between sphericity, porosity and bed compacting [3]

As may be seen, at specific sphericity, especially with shapes close to spherical, bed compacting affects the porosity quite significantly. This relationship is of significant importance for the practice, because the lower porosity the better water purification and the higher hydraulic losses during water flow through the bed.

In case of water purification filters, the use of a direct method of sphericity determination is not justified, because the filter always contains a bed built of a mixture of particles of different sphericities. Therefore, it is better to determine sphericity with indirect methods, for instance with the use of Ergun’s formula, which is correct in the initial phase of the rinsing process when bed rests still on the gravel layer [1, 16, 20, 21]. In this way the sphericity was determined by many researchers for various minerals [4, 6].

       (4)

k1 i k2 are determined as follow:

The first (Stokes’) term in that formula, which accounts for viscosity, dominates during laminar flow, whereas the second (Newton’s) term dominates during turbulent flow. In the intermediate flow range both terms assume negligible values. If a measurement is carried out of all terms in the formula (4), except for ψ, then sphericity calculation is no longer a problem.

When the rinsing velocity is gradually increasing, water stream loosens the bed slowly and causes its lifting to a so-called fluid or fluidized state when the bed behaves like a liquid. The rinsing velocity at which the bed begins to loosen and to rise is called a minimum flow velocity of fluidization and is designated as Vmf, and the bed porosity in that state – εmf. Porosity εmf became for many researchers a good point of reference when defining the relationship between εmf and sphericity ψ. Narshimhan’s formula is one example [34]

       for 0.35 < εmf < 0.55        (5)

Other, also simple formulas, are those obtained by Wen and Yu [35] through juxtaposition of sufficiently great number of results of measurements by various researchers, and then approximation with suitable functions. These formulas bind sphericity and porosity parameters in the “mf” (minimum fluidization) state and have the following form:

          (6)

          (7)

Cleasby and Fan [4] proposed a relationship which, they claim, correlates well with the results of tests and a determination coefficient R2=0.833. The formula is:

       (8)

This formula is derived from Richardson-Zaki law [31],

            (9)

which binds rinsing velocity V and the velocity of free sedimentation of bed grains in water with bed porosity ε and the exponent n. This formula is correct within the fluid state of bed. If it is brought to logarithmic form, one may obtain an expression:

            (10)

which, on the logV = log(ε) plot, is a straight line with inclination n. When ε = 1 is substituted, i.e. a case when the entire bed has flown out of the filter the velocity V will be equal to the Vs velocity and it will be the point intersection with the vertical axis on the plot. This velocity is designated as Vi and it goes into the formula (10). Experimental test with natural minerals show that Vs ≠ Vi. Thus, if Vs and Vi are determined experimentally, the sphericity, ψ, may be calculated.

In their in-depth tests, Dharmajah and Cleasby [6] analyzed various minerals and attempted to find relationships between sphericity and densities of water and grains, granulation, porosity, etc. In addition to their own measurements, they utilized those by Gunasingham at al. [9] and those by Cleasby and Fan [4]. They reached the following relationship:

            (11)

where: ReB – the Reynold’s number, as modified by Blake, which is:

            (12)

where: dz – substitute (equivalent) grain diameter, ψ – sphericity of grains, V – water rinsing velocity, ν – kinematical coefficient of viscosity, ε – bed porosity.

The shape coefficient expresses the inverse of sphericity [11, 18, 22, 34] although more often the shape coefficient is expressed with the formula [14]

            (13)

The example of relationship between sphericity, shape factor and porosity is presented in Table 1.

Table 1. Sphericity, shape factor and porosity for granulated minerals [7,17]

Description

Sphericity, ψ

Shape factor, S

Porosity, ε

Spherical

1.0

6.0

0.38

Raunded

0.98

6.1

0.38

With grooves

0.94

6.4

0.39

Pointed

0.81

7.4

0.40

Angular

0.78

7.7

0.43

Crushed

0.70

8.5

0.48

Cleasby and Fan [4] introduced a concept of dynamic shape factor (DSF), which they defined in two ways:

            (14)

or

            (15)

where: Vs – grain sedimentation velocity, Vn – sedimentation velocity of spherical grains of equivalent volume calculated with the use of equations, u and z are constants bound with each other as follows:

u + z = 1            (16)

and

            (17)

where

            (18)

It should be stressed that ψ ≠ DSF, though there is a certain relationship between them: if one is bigger the other is bigger as well.

Cleasby and Fan [4] used the DSF concept and proposed a formula similar in form to the formula (8)

            (19)

Gunasingham at al. [9] introduced the concept of hydraulic equivalent grain diameter dh and, after that, the hydraulic shape factor, designated ψh or Ω.

            (20)

where: dh – diameter of sphere having the same density and sedimentation velocity as the particle in liquid of the same density and viscosity (hydraulic diameter for short), dz – diameter of sphere having the same volume as the particle. In some papers ψh is confused or mixed up with ψ, despite of the fact that their measurement show different values. Differences between ψh and DSF were identified by Dharmajah and Cleasby [6], as well as by Cleasby and Fan [4], and they are presented in Fig. 2.

Fig. 2. Dependence between hydraulic shape factor ψh and dynamic shape factor DSF and the logarithm of Reynolds number of the particle sedimentation velocity [6]

The literature shows no consistency in methods used for hydraulic diameter determination. For spherical particles, Churchill [3] defined and recommended to use the formula:

            (21)

where: Vchan – volume of channels in bed volume, Fchan – surface of channels in bed volume whereas Allen [15]:

            (22)

where designations are as in previous formulas.

EXPERIMENTS

The following eight beds, which have already found practical applications or are under intensive study, were selected for tests:

The products were obtained in commercial quantities with mixed granulation. Therefore, the first step prior to actual testing was to prepare beds in suitable grain fractions. The beds were screened so as to enable grading of 10 following fractions: 3.15 – 5.0 mm, 2.0 – 3.15 mm, 1.5 – 2.0 mm, 1.25 – 1.5 mm, 1.0 – 1.25 mm, 0.8 – 1.0 mm, 0.63 – 0.8 mm, 0.5 – 0.63 mm, 0.4 – 0.5 mm and 0.315 – 0.4 mm.

The granulation ranges resulted from sieve set, used by the laboratory.

A substitute diameter was assigned to each fraction range and for each bed. The measurement consisted in determining the mass of single bed grain, and then, knowing its density, the diameter of a ball of the same volume as the grain was calculated using the following formula

            (23)

where: m – mass of grains, ρs – their density and k – the grain number.

As individual grain fractions contained grains of different shapes and sizes, limited only by the size of sieve, differing numbers of grains were added to the weighing vessel. In case of courser fractions (3.15 – 5.0 mm) 30 grains were put each time, 50 grains each - for smaller fractions and so on, ending with over 3000 grains each being counted for the smallest fraction. This growing number of grains for smaller granulations was used in order to reduce the effect of weighing error.

Density of minerals was determined by standard method [25] using pycnometer and denaturized alcohol.

Volumetric density may have different values, depending on the degree of grain compaction. Since filtration bed is quite loose during rinsing process, once it is ended, the bed is in the state of bulk-density conditions. That is why the porosity calculation utilized volumetric density determined by standard method [27, 29] with steel cylinders and funnels for continuous grain feeding.

If density of mineral and bulk density values are known, porosity can be calculated according to the standard method [28]

            (24)

where: εl – porosity, ρs – density of bed grains, ρp – bulk density.

The microscope-inspections, as well as initial density and hydraulic testing, found out that some minerals exhibited significant internal porosity with consequent long degassing of grains when immersed in liquid. This phenomenon applied primarily to diatomite and clinoptylolite, and less to chalcedonite, nevtraco and barite. Since the intergranular porosity rather than internal/grain porosity plays a significant role in the flow through both settled and fluidized beds, the effect of the latter should be eliminated in calculations.

Therefore, additional experiment was carried out by immersing grains of all beds in kerosene of density precisely determined (by pycnometric method). Once the pores were filled with kerosene (air bubbling has stopped), grains were removed onto a screen and left for kerosene to drip off. Seen under microscope, the internal pores in grains remained filled with kerosene with convex meniscus, while flat surfaces were covered with kerosene film. In order to remove this excess of kerosene, the grains were gently rolled on paper of average absorptivity. The final stage was the weighing of grains with their pores filled with kerosene. Using the mass of dry grains, that of grains saturated with kerosene, and knowing the densities of kerosene and that of the minerals, the volume of pores per 1 g of bed (ml/g) was calculated. This information permitted correcting of porosity calculated from the formula (25) by internal porosity

            (25)

where: ρp – bulk density [kg/m3], vw – specific volume of pores [m3/kg], ρs – density of the mineral [kg/m3].

The porosity value, as determined for each bed and each fraction from the formula (25), was used in further tests and calculations.

The sedimentation velocity was testing with use of a 2m-long column of transparent material (PLEXI) of internal diameter 52 mm. Appropriate levels were marked on column wall and the time of grain sinking was measured on sections of 0.5 m and 1.0 m. The lengths of sections and sinking times became a basis for calculating of sedimentation velocity Vs. In order to obtain reliable results, the sedimentation velocity was measured for 100 grains of each mineral and each fraction.

All measurements of sinking velocity, as well as bed fluidization testing described in the following part, were performed in water at the temperature of 13°C. The constancy of that temperature was observed, because water was taken from a well and, after its conditioning, was stored in tank. This tank of 1000 m3 capacity is buried under ground under a quite thick layer of soil, thus ensuring constant temperature conditions inside.

The testing of bed expansion and head losses during water flow through the bed in both steady state and during fluidization was conducted with the use stand shown in Fig. 3.

Fig. 3. Diagram of stand for bed testing

The testing stand was built of a 2m-long transparent tube of PLEXI plastic material with internal diameter of 52 mm. A strainer, made of plastic filtration nozzle, with additional small holes drilled in its upper surface to enable rather uniform water flow, was installed in the lower part of the tube. A gravel layer of quartz sand in granulation from 3.15 do 5.0 mm was poured first onto the strainer. The height of the gravel layer ranged from 13.0 to 14.0 cm. Then a layer of tested bed was poured onto that gravel layer. Its height was 50 cm for courser beds and 40 cm – for beds of finer granulation. These heights resulted from the distance of the top surface of bed from column outlet, so as to allow for expected expansion and to avoid grain escape to drain. These strata thicknesses were determined during earlier, preliminary tests.

Each series of tests began by filling the column with water to approximately half of the height and with gravel layer filling. Then water was pumped with high intensity through column inlet in order to fluidize the layer. This operation permitted removing of any impurities which could get in among gravel grains, and enabled obtaining flat horizontal top surface of that layer.

The second part of measurements consisted in filling the tested bed on thus prepared gravel layer. Then valves were opened to enable water to flow through the column from the bottom up for the bed ‘settle down’ and to flush any dust fraction out, as well as to remove air. Once the flowing water was clean and contained no air bubbles, the measurement started from the lowest intensity to higher and higher intensities, with water flow velocity being measured with flow meter Promag A Endress+Hauser firm, as well as the upper surface level.

Head losses were measured as a differences of water mirror levels in a piezometer connected to the column under the strainer, as well as the water mirror level in column resulting from installed overflow, through which water is drained after bed rinsing. Obviously, the level of water mirror over the overflow was not constant and depended on its flow velocity/intensity. For this reason, two water mirror levels, in piezometer and column, were read during each measurement. The difference in readings showed the amount of head loss.

As the measurement of pressure loss in filtration bed layer was the primary objective, all measurements were preceded by measurements of head loss in the strainer. They were carried out identically as described earlier, with the only difference that they were carried out with column without gravel layer and bed. After several series of strainer head loss measurements the results were plotted on the diagram and a curve of the function hstrainer = 0,2196·Q2 + 0,1948·Q with coefficient R2 = 0,9996 was obtained, which was then utilized in all tests for suitable subtraction of head losses in the strainer from total head losses.

The second element that have to be accounted for were head losses in the flow through the gravel layer. The difference in approach here resulted from the fact that, while the strainer remained the same all the time, i.e. its head losses could be determined once and the curve used for all measurements, the gravel layer was not always so. That is why each measurement of rinsing parameters consisted of two parts.

The first part was executed after gravel layer ‘settling’ (as clarified above) and consisted in pressure loss measurements in water flow through the strainer and that layer. Measurements were carried out identically as for water flow through the strainer. Their results were specified in tabular form of functional dependence of head loss differences versus flow velocity/intensity. The head loss was therefore a sum of head losses in water flow through the strainer and through the gravel layer.

The second part of measurements consisted in pouring the bed to be tested on a previously tested gravel layer. Then, once it has settled and degasified the measurement began from the lowest flow velocity/intensity to higher and higher flow velocities, while measuring for each flow its value and the water mirror levels in the column and piezometer. Measurements were carried out so that the first group of readings referred to bed resting still on the gravel layer, and the second group – to bed fluidized state. In this second state the top level of bed layer was additionally measured, so that expansion could be calculated as well. Such measurement pictured the sum of head losses in the strainer, in gravel layer and in the bed. As all 3 components ‘cooperate with each other in series’, thus head losses in bed were determined by deducting from latest indications of head loss in the strainer and those in the gravel layer.

TEST RESULTS AND DISCUSSION PROPERTIES OF MATERIALS

Since the tests were carried out on natural materials, which are in use in water treatment filters, the results of measurements of density and porosity differed slightly between individual fractions. It was because of natural admixtures occurring in them. Therefore, such parameters as density and porosity were determined separately for each fraction and these individual results for specific fraction, rather than the averaged result for the mineral, were taken for further testing. The results of measurements of density ρs, substitute (equivalent) diameter, dz, and intergranular porosity in loose state εloose, are specified in Table 2.

Table 2. The properties of tested bed materials

Parameters

Granulations [mm]

3.15÷5.0

2.0÷3.15

1.5÷2.0

1.25÷1.5

1.0÷1.25

0.8÷1.0

0.63÷0.8

0.5÷0.63

0.4÷0.5

0.315÷0.4

Anthracite

ρs [kg/m3]

1793.2

1750.7

1776.5

1743.2

1742.9

1773.2

1745.2

1763.5

1794.1

1.8009

stand. dev.

26.2

14.9

28.3

4.1

23.7

9.1

21.4

20.1

29.9

16.3

dz [mm]

3.877

2.544

1.883

1.368

1.083

0.905

0.824

0.607

0.441

0.351

εloose [-]

0.491

0.486

0.491

0.539

0.580

0.582

0.568

0.544

0.553

0.562

stand. dev.

0.011

0.006

0.011

0.008

0.006

0.007

0.008

0.006

0.007

0.005

Barite

ρs [kg/m3]

4235.4

4047.8

4071.3

4237.7

4181.0

4165.6

4096.0

4038.9

3989.0

3906.6

stand. dev.

260.1

232.5

319.0

165.8

122.0

65.5

9.5

154.2

30.9

123.2

dz [mm]

4.251

2.882

2.059

1.490

1.217

1.009

0.764

0.590

0.507

0.403

εloose [-]

0.472

0.453

0.459

0.490

0.483

0.465

0.457

0.459

0.452

0.463

stand. dev.

0.029

0.026

0.035

0.016

0.013

0.018

0.003

0.017

0.006

0.014

Chalcedonite

ρs [kg/m3]

2488.0

2560.0

2563.6

2541.5

25.39.5

2653.5

2627.0

2701.6

2657.7

2681.7

stand. dev.

8.1

49.4

48.1

39.9

48.0

99.0

78.4

94.4

80.7

104.1

dz [mm]

3.370

2.522

1.763

1.359

1.124

0.788

0.640

0.524

0.404

0.321

εloose [-]

0.503

0.474

0.512

0.518

0.519

0.511

0.498

0.491

0.483

0.477

stand. dev.

0.003

0.009

0.007

0.005

0.007

0.013

0.011

0.12

0.011

0.014

Diatomite

ρs [kg/m3]

2428.7

2323.1

2354.9

2360.4

2391.5

2380.8

2421.1

2432.1

2377.6

2358.3

stand. dev.

97.6

38.8

15.9

6.1

43.5

25.3

10.1

40.0

42.3

29.5

dz [mm]

3.730

2.320

1.644

1.345

1.047

0.829

0.635

0.514

0.414

0.329

εloose [-]

0.513

0.466

0.490

0.496

0.492

0.493

0.501

0.506

0.514

0.483

stand. dev.

0.013

0.008

0.005

0.004

0.007

0.005

0.004

0.007

0.006

0.005

Clinoptylolite

ρs [kg/m3]

2391.3

2407.1

2393.7

2371.5

2380.5

2375.2

2378.2

2376.0

2339.6

2349.0

stand. dev.

23.9

8.4

9.6

14.7

6.3

19.3

15.3

12.1

19.5

15.4

dz [mm]

3.449

2.247

1.698

1.295

1.015

0.792

0.647

0.516

0.394

0.313

εloose [-]

0.517

0.528

0.547

0.502

0.532

0.522

0.515

0.528

0.517

0.534

stand. dev.

0.006

0.005

0.006

0.003

0.002

0.004

0.005

0.003

0.003

0.04

Nevtraco

ρs [kg/m3]

2620.8

2630.8

2635.5

2627.2

2666.8

2647.3

2639.3

2676.1

2673.3

2656.0

stand. dev.

26.8

38.2

5.6

30.8

24.8

19.7

39.4

18.9

18.5

21.1

dz [mm]

3.337

2.335

1.824

1.500

1.211

0.911

0.698

0.639

0.405

0.321

εloose [-]

0.512

0.510

0.513

0.513

0.521

0.528

0.525

0.532

0.546

0.551

stand. dev.

0.007

0.008

0.004

0.005

0.005

0.003

0.006

0.004

0.003

0.004

Sand

ρs [kg/m3]

2667.1

2664.0

2659.5

2655.5

2656.6

2649.6

2665.0

2657.3

2657.6

2657.1

stand. dev.

7.0

16.1

11.9

23.5

24.7

11.9

17.1

25.5

22.3

7.6

dz [mm]

4.427

2.804

1.899

1.324

1.172

0.947

0.749

0.583

0.431

0.329

εloose [-]

0.430

0.427

0.423

0.421

0.423

0.418

0.417

0.412

0.410

0.418

stand. dev.

0.003

0.004

0.003

0.005

0.005

0.003

0.004

0.006

0.005

0.002

Pyrolusite

ρs [kg/m3]

4006.2

4188.2

4231.1

4109.9

4134.7

4184.4

3998.8

3811.9

3808.3

3838.1

stand. dev.

27.7

20.6

150.4

106.9

114.2

51.7

35.6

70.7

69.8

127.4

dz [mm]

3.729

2.443

1.771

1.374

1.141

0.880

0.678

0.486

0.423

0.336

εloose [-]

0.517

0.508

0.492

0.482

0.495

0.461

0.489

0.520

0.515

0.506

stand. dev.

0.008

0.006

0.014

0.010

0.010

0.006

0.004

0.007

0.007

0.013

stand. dev. – standard deviation

As may be seen from the table the density results of some minerals (especially barite and pyrolusite) exhibited considerable spread, because of the presence of lighter admixtures e.g. sand in pyrolusite and schist in barite.

The greatest differences in results, confirmed under microscope, occurred for barite, which contained schist impurities so soft that they crumbled during screening but were hard to remove. From all fractions of barite they were removed hydraulically through intensive flow of water from bottom upwards, but this method has not succeeded to eliminate them entirely. That is why, in case of courser fractions, they were additionally removed manually, but it could not be done in respect to the finest fractions.

SPHERICITY

In order to determine sphericity and shape factors with the help of formulas (4), (5), (6), (7), (8), (19) and (20), it was necessary to determine additional parameters, such as head losses and porosity at minimum fluid velocity of fluidization. Using diagrams such as those, shown as examples in Fig. 4, the above values were determined.

Fig. 4. The variability of specific pressure losses (A) and porosity (B) versus rinsing velocity for barite granulation 1.0-1.25 mm

Using such diagrams as in Fig. 4 for all beds and fractions, the flow velocities Vmf and εmf, i.e. conditions from which the fluid state commences, were determined. The flow velocity Vi, clarified through formula (10), was determined using diagrams, such as that shown as example in Fig. 5.

Fig. 5. The relationship describing the function of logarithmic form of the Richardson-Zaki formula (10) for selected beds.

With the use of diagrams, such as that in Fig. 5, and approximating inclined points with linear function, it was possible to determine logVi for ε = 1, i.e. log ε = 0, and thus the Vi values.

Sphericities of individual beds were determined with the use of formulas, referenced by numbers in the first column of Table 3.

Table 3. The values of sphericity ψ, Ω and DSF, determined from respective formulas with the use of experimental data

Parametry

Granulacje [mm]

3.15÷5.0

2.0÷3.15

1.5÷2.0

1.25÷1.5

1.0÷1.25

0.8÷1.0

0.63÷0.8

0.5÷0.63

0.4÷0.5

0.315÷0.4

Anthracite

ψ (4)

0.648

0.751

0.707

0.662

0.680

0.658

0.679

0.630

0.549

-

ψ (5)

0.665

0.677

0.683

0.524

0.446

0.434

0.456

0.516

0.487

-

ψ (6)

0.631

0.644

0.603

0.500

0.442

0.433

0.448

0.493

0.471

-

ψ (7)

0.611

0.631

0.571

0.434

0.365

0.355

0.373

0.426

0.399

-

ψ(8)

0.492

0.372

0.432

0.658

0.392

0.342

0.413

0.410

0.077

-

ψh(21) i (19)

0.445

0.613

0.723

0.879

0.854

0.827

0.842

0.807

0.687

-

DSF(18)

0.323

0.211

0.265

0.505

0.228

0.185

0.248

0.245

0.019

-

Barite

ψ (4)

0.654

0.768

0.725

0.695

0.680

0.689

0.757

0.823

0.795

0.771

ψ (5)

0.693

0.744

0.725

0.661

0.665

0.672

0.703

0.715

0.734

0.728

ψ (6)

0.661

0.723

0.699

0.627

0.631

0.639

0.674

0.688

0.711

0.703

ψ (7)

0.658

0.755

0.717

0.605

0.611

0.623

0.677

0.699

0.735

0.722

ψ(8)

0.284

0.244

0.389

0.277

0.390

0.494

0.318

0.347

0.248

0.578

ψh(21) i (19)

0.343

0.454

0.554

0.647

0.670

0.733

0.837

0.930

0.990

1.048

DSF(18)

0.140

0.111

0.226

0.134

0.227

0.325

0.165

0.190

0.113

0.414

Chalcedonite

ψ (4)

0.690

0.980

0.847

0.773

0.879

0.871

0.914

0.996

1.125

1.194

ψ (5)

0.626

0.717

0.613

0.572

0.579

0.581

0.611

0.638

0.641

0.647

ψ (6)

0.591

0.690

0.578

0.541

0.547

0.549

0.577

0.603

0.606

0.612

ψ (7)

0.555

0.701

0.536

0.486

0.494

0.497

0.534

0.571

0.575

0.584

ψ(8)

0.346

0.341

0.551

0.566

0.458

0.559

0.565

0.528

0.498

0.541

ψh(21) i (19)

0.372

0.437

0.551

0.675

0.728

0.818

0.901

0.962

1.069

1.135

DSF(18)

0.188

0.184

0.384

0.401

0.289

0.394

0.453

0.360

0.329

0.373

Diatomite

ψ (4)

0.763

0.913

0.883

0.883

0.900

0.819

0.829

0.833

0.842

0.967

ψ (5)

0.639

0.710

0.656

0.657

0.663

0.640

0.612

0.597

0.595

0.651

ψ (6)

0.604

0.681

0.621

0.622

0.629

0.605

0.577

0.563

0.562

0.616

ψ (7)

0.573

0.688

0.597

0.599

0.608

0.575

0.535

0.516

0.514

0.590

ψ(8)

0.395

0.351

0.502

0.368

0.382

0.362

0.389

0.497

0.732

0.793

ψh(21) i (19)

0.405

0.475

0.650

0.660

0.770

0.849

0.944

1.016

1.093

1.134

DSF(18)

0.231

0.193

0.333

0.207

0.219

0.202

0.225

0.328

0.594

0.672

Clinoptylolite

ψ (4)

0.647

0.661

0.644

0.822

0.697

0.770

0.832

0.847

0.987

1.130

ψ (5)

0.589

0.568

0.511

0.630

0.560

0.573

0.605

0.552

0.568

0.608

ψ (6)

0.556

0.537

0.490

0.595

0.530

0.542

0.571

0.523

0.537

0.573

ψ (7)

0.507

0.482

0.422

0.560

0.472

0.487

0.526

0.463

0.482

0.530

ψ(8)

0.281

0.303

0.402

0.742

0.440

0.345

0.457

0.320

0.243

0.228

ψh(21) i (19)

0.394

0.427

0.602

0.765

0.772

0.843

0.884

0.979

1.040

1.032

DSF(18)

0.137

0.154

0.237

0.607

0.273

0.188

0.288

0.167

0.109

0.099

Nevtraco

ψ (4)

0.637

0.654

0.650

0.633

0.600

0.653

0.694

0.582

0.729

0.749

ψ (5)

0.610

0.603

0.598

0.581

0.575

0.555

0.553

0.530

0.492

0.505

ψ (6)

0.575

0.569

0.564

0.549

0.543

0.526

0.524

0.504

0.475

0.484

ψ (7)

0.532

0.524

0.517

0.497

0.489

0.467

0.465

0.440

0.404

0.415

ψ(8)

0.680

0.659

0.548

0.537

0.501

0.589

0.517

0.683

0.742

0.908

ψh(21) i (19)

0.460

0.573

0.674

0.744

0.821

0.936

1.027

1.041

1.276

1.390

DSF(18)

0.531

0.506

0.381

0.370

0.332

0.426

0.349

0.534

0.607

0.826

Sand

ψ (4)

0.745

0.815

0.898

0.993

0.894

0.899

0.916

0.949

1.059

1.046

ψ (5)

0.791

0.812

0.823

0.816

0.803

0.816

0.815

0.824

0.840

0.817

ψ (6)

0.787

0.818

0.834

0.824

0.805

0.824

0.823

0.836

0.861

0.826

ψ (7)

0.863

0.918

0.947

0.928

0.894

0.928

0.926

0.951

0.997

0.932

ψ(8)

0.705

0.841

0.974

0.906

0.861

0.726

0.863

0.698

0.955

1.597

ψh(21) i (19)

0.452

0.612

0.774

0.919

0.995

1.089

1.301

1.305

1.476

1.669

DSF(18)

0.561

0.735

0.921

0.823

0.763

0.587

0.765

0.553

0.893

1.964

Pirolusite

ψ (4)

0.707

0.739

0.807

0.849

0.800

0.802

0.804

0.655

0.761

0.815

ψ (5)

0.615

0.612

0.638

0.638

0.632

0.692

0.633

0.576

0.553

0.587

ψ (6)

0.580

0.577

0.603

0.603

0.597

0.661

0.598

0.544

0.524

0.554

ψ (7)

0.539

0.535

0.571

0.571

0.563

0.657

0.564

0.490

0.464

0.504

ψ(8)

0.728

0.587

0.559

0.625

0.584

0.625

0.407

0.278

0.336

0.622

ψh(21) i (19)

0.381

0.489

0.588

0.687

0.747

0.827

0.924

0.913

0.998

1.106

DSF(18)

0.589

0.424

0.393

0.467

0.420

0.466

0.242

0.135

0.180

0.464

No results for the finest fraction of anthracite is caused by the fact that this fraction did not expand even at very low flow velocities (below the limit of measurability), so the minimum fluidization conditions could not be measured.

As may be seen from Table 3, the results describing sphericity of individual minerals and fractions, obtained with the use of individual methods, differed from one another. When analyzing the results summarized in Table 3 the values specified in two bottom rows for each mineral should be treated separately, because, in the light of descriptions for individual formulas and their ranges of application there is no simple relationship between sphericity ψ and ψh and dynamic shape factor (DSF).

The analysis of individual values for sphericity shows that Ergun formula ψ (4) is not very suited for defining this parameter. For chalcedonite, clinoptylolite sand and nevtraco, it shows a rising tendency with decreasing granulation of grains and, for grains below 0.5 mm it offers results that are hardly acceptable. Values of ψ for fine granulations are close to or even exceed 1. It should be stressed that, in accordance with the definition, ψ = 1.0 for a sphere. The calculated value of sphericity for diatomite and pyrolusite and, to a lesser degree for anthracite, exhibited surprisingly high values.

Formulas of Wen and Yu ψ (6), ψ (7), as well as that Narshimhan ψ (5), correlate well with each other, giving similar results.

Since the results obtained with the use of different formulas deviated from one another, additional stereoscopic microscopic examination of appearance of random grains from individual beds were conducted within the framework of this effort. It provided the opportunity to compare various grains with each other, and to determine the degree of their elongation, shape, edge sharpness, etc. Also, it became possible to find out, which of the beds had more regular, sphere-like shapes, so that their grain sphericity should have higher values, closer to 1. Sand, and then barite exhibited the highest sphericity, while chalcedonite, diatomite, clinoptylolite and nevtraco as well as pyrolusite followed the leaders quite closely. Anthracite had the lowest sphericity and microscopic examination showed that large number of its grains had elongated, stick-like shapes with sharp ends. These additional studies enhanced the impression that results obtained from formulas (5), (6) and (7) are the closest to reality.

Formula (8) provides variable results. Those for anthracite, barite, chalcedonite, diatomite and clinoptylolite appear too low, clearly lower than for the remaining minerals. It gave overvalued results for course and fine fractions of nevtraco, and for fine fractions only for sand. For pyrolusite, the results exhibited no steady tendency.

Results from Allen’s formula ψh(22) tend to have higher and higher values with decreasing granulation and are higher than 1 for sand of granulation 0.8-1.0 mm. Similarly results that deviate from all others are exhibited for DSF.

To summarize, values of sphericity calculated from formulas of Wen and Yu ψ (6), as well as Narshimhan ψ (5) may be treated as trustworthy results. Results obtained from these formulas adequately determine sphericity of tested minerals.

In recent years some authors adduce the method of Cleasby and Fan [4]. Its model is presented with formulas (11) and (12). While using the results of experiments, sphericity calculations with these formulas were carried out for studied minerals. An example of sphericity results for anthracite of selected granulations, as well as those for a predetermined granulation of 1.0-1.25 mm for individual minerals were specified in Fig. 6.

Fig. 6. Sphericity calculation with the Cleasby and Fan method [4], formulas (11) and (12) for anthracite of selected granulations (A), as well as those for various minerals of similar granulation of 1.0-1.25 mm (B)

As may be seen from Fig. 6, the method proposed by the authors has not proven itself for the analyzed minerals. The analysis of random results does not show it, but a methodic study indicates that the model is very sensitive to rinsing velocity variation. Sphericity value rises with growing rinsing velocity, which is not correct as it is a constant value for a material.

SUMMARY AND CONCLUSIONS

The presented results of this study, referred to 8 different minerals over the range of 10 different granulations, have shown that formulas published up to now and claiming to enable determination of sphericity of minerals, not always prove to be correct.

Utilization of Ergun’s model (formula 4) for fine granulations of sand, clinoptylolite, diatomite and chalcedonite exhibited a rising tendency and, for very fine granulations of 0.315-0.4 mm and 0.5-0.4 mm even exceeded the maximum of one.

Simple-to-apply formulas of Narshimhan (5) and Wen and Yu (6) and (7) passed their verification quite well. The results do not deviate much from each other and, compared to visual examination of the grains, are within acceptable quantities.

Similarly to Ergun’s formula, results obtained from formula (8) showed poor correlation with the above ones. The results appeared to have considerable spread between individual granulations of the same mineral, e.g. for sand of finest granulation the sphericity result from this model was approx. 1.6.

It was shown that ψh, i.e. sphericity calculated from hydraulic diameter, as well as DSF (dynamic shape factor), cannot be compared with sphericity. Results ψ obtained for analyzed minerals are widely different for ψh and DSF.

The method proposed by Cleasby and Fan (formulas 11 and 12) cannot be recommended, because of rising values of sphericity in function of water flow velocity.

In summary, to determine sphericity, as a significant practical factor for designing rapid filters the formulas proposed by Narshimhan and by Wen and Yu, which gave acceptable results and are simple, are believed worthy of recommendation. The only unknown is the porosity at the beginning of fluidization. However, as the latter parameter is not too difficult to determine at low cost. Such study may be carried out for any bed. The knowledge of that parameter and the subsequent calculations of sphericity permit correct designing of filter rinsing system, as well as effective operating.

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Accepted for print: 23.02.2007


Tadeusz Siwiec
Department of Civil Engineering and Geodesy,
Water Supply and Sewage Systems Section,
Warsaw Agricultural University, Poland
Nowoursynowska St. 159, 02-776 Warsaw, Poland
Phone: +48 22 59 35 161
email: Tadeusz_Siwiec@sggw.pl

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