THE EXPERIMENTAL VERIFICATION OF RICHARDSON-ZAKI LAW ON EXAMPLE OF SELECTED BEDS USED IN WATER TREATMENT

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

ABSTRACT

The paper presents and discusses the characteristics of various methods of determining the exponent for Richardson-Zaki [34] formula, which binds the flow velocity of rinsing water, sedimentation of bed grains in water, as well as bed porosity at various states of expansion. The characteristic of methods served for verification of their correlation with the results received from experimental tests, where the exponent was the degree of inclination in the relationship log V = n · log ε + log V_s, which is a logarithmic form of Richardson-Zaki formula. The tests were carried out for 8 minerals such as anthracite, barite, chalcedonite, diatomite, clinoptylolite, nevtraco, sand and pyrolusite. In order to make results more trustworthy, they were carried out for 10 fractions of each bed. It was shown that the formulas, from which the value of exponent n is commonly calculated, give results not differing much from one another, but considerably different from experimental ones, where the exponent was the degree of inclination of the above mentioned function. The results close to experimental ones were given by the formula containing sphericity of particles developed by Dharmajah and Cleasby [5]. Experimental results of testing enabled to develop the exponential functions describing the relationship of exponent n on Reynolds number in the state of free sedimentation.

Key words: backwashing filters, bed expansion, porosity, fluidization.

INTRODUCTION

The process of rinsing rapid filters consists in pumping rinsing water through bed in direction from the bottom upwards. It causes bed loosening and carrying away of impurities left there in filtration process. It should be stressed that different phenomena occur during the flow of water through steady bed than through the one in loosened state.

The course of these phenomena is presented in Fig. 1.

As may be seen in Fig. 1, slow increasing of the speed of rinsing has, at first, no effect on bed behavior, porosity remains constant and, under laminar flow, the losses are linearly variable, as it results from the Kozena-Carman formula [19]. The sum of hydraulic losses and grain buoyancy is lower than the forces of gravity. As it flows between grains, the stream of rinsing water is characterized by very differentiated local speeds, both in respect of value and direction. In places where water stream is of high speed, individual grains may become displaced or slightly lifted. Increasing in flow velocity of rinsing stream raises the speeds of individual, local streams, which, in their flow among grains, affect higher lying grains. Gradual increasing of flow velocity raises the hydrodynamic force and when the sum of that force and displacement force becomes equal to the force of gravity, the bed becomes weightless. From that moment any increase in flow velocity will translate into raising the bed height, i.e. bed expansion. From such state the bed is called fluidal or fluidized bed, as is it behaves like a liquid. The upper limit for rinsing speed increase is such bed expansion that the lightest grains, those in the upper part of the fluidized bed, reach the edge of the filter outlet, and begin to flow over, out of the filter.

A visualization of fluidization phase provides evidence of complex processes taking place at that time, which makes bed behavior during rinsing difficult to describe [13]. In such conditions, impurities that were stopped during filtration, through filtering, sedimentation or other forces on bed grain surfaces, are carried away beyond the bed. Important role in the removing of impurities from grain surfaces is played by shearing forces and by collisions of grains. Weaver [38] claims that collision energy significantly affects the results of grain cleaning and that it depends on granulation of bed grains.

Bed expansion significantly influences the effectiveness of its cleaning for two reasons. One is the fact that in loosened state it is easier to carry off impurities from intergranular spaces beyond the filter, and the other is the existence of shearing forces during water flow around grains; they help to tear particles of impurities off the grains to which they stick.

A consequence of expansion is the gradual rise in intergranular porosity [30]. This porosity is related to bed porosity in settled state and to bed height in that state, as well as in fluidized state. The relationship between porosities and bed heights in any state is reflected by the formula

(1)

where: L₀ and ε₀, L_mf and ε_mf, L_e and ε_e, are height and porosity in bed state: settled, at fluidization onset and at any fluidized state, respectively.

MODELING OF BED EXPANSION DURING RINSING

The basic formula utilized by the majority of researchers for correlation of rinsing speed and bed porosity is Richardson-Zaki formula [34]

(2)

where: V – water flow velocity, V_s – sedimentation velocity of bed grains in water, ε – bed porosity, n – exponent.

If both speeds, V and V_s, are clearly determined, the exponent is not a constant value and depends on several factors. Fair et al. [8] assumed n = 5, whereas Richardson and Zaki [34] recommended the following formulas:

The above formulas were then recommended and utilized by Leva [15], Amirtharajah and Cleasby [1], Razumow [31], as well as by Cleasby and Kuo-Shuh [4]

In his later publication Richardson [33] corrected previously proposed formulas and offered the following summary:

In the above formulas, Re_s is the Reynolds number calculated for free sedimentation velocity of bed grains, V_s, d – is the substitute diameter of grains, D – internal diameter of filtration column. The above formulas are correct for spherical grains. Therefore, for grains of other shapes, Richardson and Zaki [33] proposed a correction of exponent n, as well as a formula, which, as they claim, is correct for Re > 500

n = 2.7 · P ^0.16          (12)

where

(13)

where, in turn, d_z – diameter of sphere having the same surface as the grains, d_s – diameter of circle with surface area of the particle projected, when it lies in most stable position.

Formulas expressing the exponent n may be divided into two groups. One is formulas different for various ranges of the Re_s number, and the other – formulas, which the authors consider universal and the same for a large range of the Re_s number.

In addition to the above discussed, the following formulas or recommended values may be counted among the first group formulas utilized in successive years by other researchers [3, 10, 32]

For Re_s > 500 Yang i Renken [40] recommended a lower value of n = 2.39.

The second group of formulas does not depend on Reynolds number ranges, and their author believed they are universal for beds and rinsing speeds applied for regeneration of typical filters. These formulas have forms combined with Archimedes number, Ar or Reynolds number, Re_s and are presented below

(27)

Garside i Al-Dibouni [2, 41]

(28)

By modifying that formula with a relationship that binds Reynolds number with Archimedes number, they obtained [2, 41]:

(29)

Limtrakul at al. [16]

(30)

Rowe [17]

(31)

Khan i Richardson [7, 17]

(32)

Rowe [37]

(33)

Moldavsky at al. [18]

(34)

The formula (34) published in the paper [18], contains the editorial mistake consisting in incorrect multiplier: 0.0043. It should read: 0.043.

The research presented below consisted in finding exponent n from the above formulas for experimental data and comparing them with the results obtained from experimental research referring to the formula.

EXPERIMENTS

The following 8 beds, which have already found practical application or are under intensive research were selected for testing:

• Anthracite – imported from Ukraine by “Tlen” cooperative in Czempin,

• Barite – from Przedsiębiorstwo Produkcyjno Handlowo Usługowe “R & S” Ltd. Co. in Boguszów Gorce near Wałbrzych [26],

• Chalcedonite – from ‘InowłódŸ’ mine, ‘Teofilów’ deposit, exploited by MIKROSIL Ltd. Co. from Radom [36],

• Diatomite – obtained from Diatomite Product Dept. in Jawornik Ruski near Bircza, of the Specjalistyczne Przedsiębiorstwo Górnicze GÓRTECH Ltd. Co. in Krakow [11],

• Clinoptylolite – from ZEOCEM a.s. near Bystre, Slovakia [23, 24],

• Roasted dolomite called nevtraco – offered by the Eurowater company,

• Sand – from Tomaszowskie Kopalnie Wyrobów Mineralnych “Biała Góra” mines in Tomaszow Mazowiecki,

• Pyrolusite – from Gabon, purchased through Centrala Zaopatrzenia Hutnictwa in Katowice.

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 (35)

(35)

where: m – mass of grains, ρ_s – their density and n – 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 counted in 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 using pycnometer [25] and denaturized alcohol.

Volumetric density of each mineral may have different values, depending on the degree of grain compaction. Since 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 and bulk density values are known, porosity can be calculated according to the standard method [28]

(36)

ε_l – porosity, ρ_s – density of bed grains, ρ_p – bulk density.

Inspection under microscope, 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 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 (37) by internal porosity

(37)

where: ρ_p – bulk density [kg/m³], v_w – specific volume of pores [m³/kg], ρ_s – density of the mineral [kg/m³].

The porosity value, as determined for each bed and each fraction from the formula (37) was used in further tests and calculations.

The sedimentation velocity was tested with use of 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 V_s. 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, were performed in water at the temperature of 13°C. The constancy of that temperature was observed, because water was taken from a drilled well and, after its conditioning, was stored in reserve and surge tank. This tank of 1000 m³ capacity is buried under ground under a quite thick layer of soil, thus ensuring constant temperature conditions in it.

The testing of bed expansion and head losses during water flow through the bed in both steady state and during fluidization was conducted with use of the stand shown in Fig. 2.

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 bed 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 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. The latter was particularly important for bed with significant internal porosity, especially clinoptylolite and diatomite. 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.

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, substitute (equivalent) diameter, d_z, and intergranular porosity in both loose and compacted state are specified in Table 1.

The densities specified in Table 1 exhibit a relative spread, which is unavoidable because of the presence of lighter admixtures, as well as the inherent measurement error.
The measurement error applied in equal measure to all minerals and fractions, whereas any admixtures influenced all materials to different degrees. The greatest differences in results, confirmed under microscope, are noticeable for barite, which was contaminated with 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.

Expansion
The variability of bed expansion during rinsing is defined as a ratio of the bed height in expanded and settled state to the height of bed in settled state, expressed with the formula [14]

(38)

As bed expansion variations are caused by the rinsing stream, a natural relationship expresses it as a function of rinsing speed. Within the framework of this paper, several curves of such relationship(s) were plotted for all beds and fractions and, for example, a curve for one granulation, 1.0-1.25 mm is presented in Fig. 3. and Fig. 4 on the other hand shows examples of expansion variations for selected 5 fractions of chalcedonite bed.

The diagrams (Fig. 3 and Fig. 4) indicate that expansion variations versus rinsing speed are clearly linear functions. The assessment of test results for all beds and all fractions permits a conclusion that this linearity was common for all analyzed cases.

Fig. 3 shows that the curves tend to form groups. The highest inclination is exhibited by anthracite, then diatomite and clinoptylolite, followed by sand, nevtraco and chalcedonite, the lowest – pyrolusite and barite. The curves show clearly a significant effect of density on function inclination. Fig. 4 in turn shows that, within a single mineral, granulation of grains has effect on function inclination. Both these parameters are bound with each other in/by Archimedes number (formula 27) [9]. In logarithmic formula (2), the formula becomes a linear relationship

(39)

Once a diagram is plotted for experimental data, logV = f(logε) is a straight line with inclination of the exponent n. Examples of diagrams for selected fractions of diatomite, clinoptylolite, sand and pyrolusite are presented in Figs 5 and Fig. 6.

The diagrams permit obtaining the additional information. If porosity equal to one is substituted in the formula, thus expressing a phenomenon of bed being carried out of filtration column, then the formula shows that V = V_s, meaning that, with the use of data from Fig. 5 and 6, one may obtain values corresponding to sedimentation velocity of individual bed grains.

The results of experimentally determined values of exponent n as the inclination of functions, presented as examples in Fig. 5 and 6, are specified in Tables 2, 3, 4, 5, 6, 7, 8 and 9 and designated n exp. For comparison, Tables 2, 3, 4, 5, 6, 7, 8 and 9 specify also the values of exponent n as calculated for all minerals and all fractions using the formulas from other models. These are designated as form. with the number of the respective formula. Letters Re_s designate the Reynolds number referred to experimentally determined sedimentation velocity of individual grains.