## Abstract

This article presents the results of experimental studies for information support of the hydraulic compartment of the mathematical model of physicochemical iron removal from groundwater. It is experimentally established that the bound water in the deposit significantly exceeds the mineral component (chemical iron) for any waters with the different levels of iron oxidation. The deposit dehydration takes place at its boundary with the liquid phase. As a result, ferrous iron is gradually oxidized and ferric iron transforms from hydroxide forms to oxide forms, which leads to a decrease in the deposit volume and a increase in its density and strength. Further, the coefficient of the deposit hydrodynamic stability is proposed to use to take into account changes in the properties of the deposit that accumulates in the granular medium. It is experimentally determined that the iron amount remaining in the medium after backwashing increases simultaneously with the deposit's age. It will lead to a gradual reduction of the operational dirt capacity of the granular medium and a need to have measures of the filtration properties restoration or to replace the filter medium entirely.

## HIGHLIGHTS

The properties of the deposit change during the iron removal from water at the rapid filter.

The coefficient of the deposit hydrodynamic stability.

The bound water is dominated in the deposit at the rapid filters.

The residual contaminants in the granular medium increase.

The undetachable kinetics of the physicochemical iron removal of groundwater.

### Graphical Abstract

## INTRODUCTION

The global population growth, industrial and agriculture development led to a steady increase in the fresh water usage (Monastyrov *et al.* 2019). The rational use of the water resources is important for the creation of the material and technical basis for the furthering mankind's existence (Polyakov *et al.* 2019; Haile & Gabbiye 2022).

The purification of water from natural sources for drinking purposes is conducted to bring the indicators of these waters in accordance with sanitary norms and rules (Dushkin & Shevchenko 2020). The groundwater that corresponds to the hygienic safety plays an important role in solving the problem of providing the population with fresh drinking water (Tugay *et al.* 2004; Maharjan *et al.* 2022). At the same time, iron removal must be performed frequently for water from underground sources (Orlov 2008; Ahmad *et al.* 2019) due to the high concentration of iron compounds in the Earth's crust (Mansoor 2012).

The iron in natural waters can be in various forms (Peng *et al.* 2020). The iron concentration in groundwater ranges from fractions of mg·dm^{−3} to tens of mg·dm^{−3} (Orlov 2008). The iron is usually in the form of ferrous ions, which are stable in aqueous solutions in the presence of the free carbon dioxide and in the absence of oxidants in waters of the deep horizons (Romanovski *et al.* 2021). The iron does not remain stable when water rises to the surface but constantly changes its state under the influence of oxygen, carbon dioxide, water, organic substances and microorganisms. Usually the amount of the oxygen dissolved in drinking water determines the state (form) of iron (Chaturvedi & Dave 2012). The ferrous iron compounds dissolve quite easily and form the iron bicarbonate Fe(НCO_{3})_{2} in the underground horizons of geological layers of the iron spar FeCO_{3} under the influence of the carbon dioxide and artesian water (Tugay *et al.* 2004).

The oxidation (oxygen, chlorine, potassium permanganate, etc.) and the filtration are used for iron removal from water in a natural or artificial granular medium (Orlov *et al.* 2016; Galangashi *et al.* 2021), the biosorption (Marsidi *et al.* 2018), the ion exchange (Tugay *et al.* 2004), the absorption by the activated carbon (Mekhelf *et al.* 2020), the barometric technologies based on nano- (nanofiltration (NF)) or ultrafiltration (Du *et al.* 2017).

The technological scheme of the water treatment in the technological process is usually determined on the basis of the water quality indicators of the water supply source, the consumer requirements for the treated water quality, the size of the construction sites, the automation degree, etc. (Twort *et al.* 2006; Orlov 2008). The experience of the plants operation in similar conditions and the pilot studies directly at the source of the water supply and also the use of the mathematical modeling can be taken into account when choosing a water treatment scheme (Stiriba *et al.* 2017). However, the last-mentioned method allows the researcher to select the most optimal technological scheme and its design and operational parameters (Epoyan *et al.* 2018).

The use of the method of the contact iron removal in the granular filters is economically attractive in some cases (Tugay *et al.* 2004; Orlov 2008), including the floating foam polystyrene medium (Orlov *et al.* 2016).

## MATERIALS AND METHODS

The nonlinear mathematical model which consists of three interconnected compartments is developed to describe the processes of the physicochemical iron removal from water at the granular filters (Poliakov & Martynov 2021). The first two compartments describe the behaviour of two forms of iron (ferrous and ferric) (Martynov & Poliakov 2021), and the third compartment describes the hydraulic aspects of the filter medium.

This article focuses on the hydraulic compartment of the mathematical model. The hydraulic compartment includes:

- – the boundary condition at the bed outlet:where [g·m
^{−3}] is the deposit concentration; [-] is the function that characterizes the ratio of the deposit specific volume and the hydroxide iron in it; [g·m^{−3}] is the concentration of deposited hydroxide particles; [m·s^{−1}] is the hydraulic conductivity; [m·s^{−1}] is the hydraulic conductivity in the clean bed; [m·s^{−1}] is the filtration rate; [m] is the piezometric head; [m] is the coordinate on the vertical axis; [m] is the height of the filter medium; [m] is the head at the bed outlet.

The experimental studies under the laboratory and the industrial conditions were conducted to provide the information support for the mathematical model of the physicochemical iron removal from water. The results of these studies as related to the first two compartments of the mathematical model are presented in the article (Poliakov & Martynov 2021). The following questions were discussed for the information support of the hydraulic compartment of the mathematical model:

the establishment of the deposit composition (the ratio between mineral component and bound water);

measuring the residual amount of the deposit in the medium after backwashing;

determining the deposit's hydrodynamic stability according to its age (polymerization and dehydration);

reduction of the medium throughput ability due to its contamination;

the substantiation of the possibility of using the undetachable filtering in the mathematical model.

The experimental studies of the filtration characteristics of the granular filters during iron removal from water were conducted at the Department of Water Supply, Sewerage and Drilling of the National University of Water and Environmental Engineering. The plants described in Martynov & Poliakov (2021) were used for this purpose. The granular medium of different types and different granules sizes were studied (foam polystyrene, quartz sand, zeolite, granite crushed stone). The ‘floating’ foam polystyrene medium was kept in a submerged state by a copper grid when it was used and the water supply was carried out in the lower part of a filter column. The ‘heavy’ medium (zeolite) was placed on the supporting layers during the studies. The water consumption was measured by the volumetric method and the head losses in the filtering columns were controlled using a piezometer shield.

Some studies of the head losses increase in the foam polystyrene and zeolite medium were conducted at the production filters at the operating plants of iron removal from water.

### Methodology of the bound water determination in the grain medium

The use of the polystyrene foam is impossible because the determination of the amount of the bound water implies that the samples are heated to a temperature higher than 100°С. The hydrochloric acid is used for the ferric compounds removal from the medium surface. Therefore, the use of the zeolite medium is also undesirable. The quartz sand is the most suitable material for experiments. It is resistant to high temperatures and the effect of hydrochloric acid. The quartz sand fraction by diameter of 1.2–1.5 mm was selected. After that, it was thoroughly washed with water. Further, it was dried out to a constant mass at the temperature of 105–110 °C and cooled in the desiccator.

^{−3}and the filtration rate of 5 m·h

^{−1}per day with monitoring head losses growth. The free water was drained off after filtration from the medium. The medium sample was weighed. Separately, a control sample was taken after calcination, which was kept in water for one day to evaluate the possible reduction of the sand mass. Then the same operations were carried out with it with the other samples. The control sample was also used for the bound water evaluation without the iron compounds accumulation in it. The porcelain cup was being dried out to a constant weight at the temperature of 105–110 °C. Then the medium sample was placed in it and evaporated to dry on a water bath. The cup with the obtained dry medium was first wiped with filter paper, which was moistened with dilute hydrochloric acid (to remove possible scum or dirt from the sand) and then with dry paper. The cup was being covered with a watch glass and after that was dried in a thermostat during 2–3 hours. The cup was weighed after cooling in the desiccator. Then the cup was placed into the thermostat again for 1 hour. Then the cup was weighed again. The medium desiccation was considered complete if the weight of the cup has not changed, otherwise the drying operation was repeated. The content of the physicomechanically bound water, g·m

^{−3}, was calculated by the formula:where [mg] is the mass of the cups with dry medium; [mg] is the cups weight with medium before drying; [ml] is the volume of the sample being tested.

The medium samples continued to be heated in a thermostat at the temperature of 200–250 °C to a constant mass to remove physicochemical bound water. The physicochemically bound water content was determined the same way as the physicomechanically bound water. The bound water from the loading sample was calcined in a muffle oven at a temperature of 500–700 °C to a constant temperature to remove the bound water chemically. The content of chemically bound water was determined in the same way as for physicomechanically bound water. The iron should be in the anhydrous iron oxide form, (hematite) at the end of the experiment according to (9).

^{−3}], [dm

^{3}] are the total iron average concentration in solution, washed out from the medium and its volume.

### Research methodology of medium deposit hydrodynamic stability of the water iron removal filter

The deposit accumulated in the medium pores changes its properties during the filtration process. The dehydration of the deposit with the ferric iron gradual oxidation and the ferric iron transition from hydroxide forms into oxide forms occurs at the surface of deposit contact with a liquid phase. This process causes a decrease in both the deposit volume and its density, and also increase in strength. The so-called deposit ‘aging’ takes place. The deposit stops to contact with the liquid phase when new portions of the deposit stick to the surface of the ‘old’ deposit. In this case dehydration should not occur. Only a change in the geometric structure of the deposit and final oxidation of the ferric iron is possible. Based on the above, it can be predicted that the ‘older’ deposit will not be washed out as easily from the medium and the consumptions of the pure water for filter backwashing will increase significantly. Therefore, it is important to study the characteristics of the strengths durability the ferrous deposit depending on its age (duration of filtration). The methodology of determining the deposit strength is not available at present due to the complexity of the experiments. So, an indirect evaluation methodology of this effect is developed.

The experimental plant for the study of the deposit hydrodynamic stability was similar to the experimental plant for the study of the bound water in a granular medium. Foam polystyrene with the grain diameters of 1.5–2.0 mm (sifted on sieves), 90 mm high was being poured into a column that simulates the elementary layer of a real filter medium. The filtration column was additionally equipped with a stainless steel net to keep the foam polystyrene medium in the flooded state. The filtration rate regulator in the form of a glass tube was arranged in front of the filter column because upward filtration was planned. Water expenditure was determined using the volumetric method, the pressure in front of the filter column was measured using a ruler, the total iron concentration was determined by the Rodanid method.

The studies were conducted in two modes:

Mode I is the filtration (deposit accumulation in a medium). A model iron solution was directed to the filtration column at different points of times (6 h, 15 h, 24 h, 48, 72 and 96 h). The total iron concentration in the model solution was about 1.5 mg·dm^{−3}. The constant filtration rate 5 m·h^{−1} was supported.

Mode II is the deposit washout. The deposit was washed out from the medium by an upward flow performed with different durations at a given (fixed) washout rate until the total iron concentration in the leftover water practically has not changed. The samples were taken at certain time intervals to determine the iron concentration in the washed water. To determine the amount of the deposit that remained in the filter medium, a part of the medium was being unloaded in a separate container and then was thoroughly washed. The total iron concentration and the water volume in this container were determined.

The washout duration was taken at intervals of 15 minutes. The graphs of the deposit washout kinetics end with a horizontal line in all of the studies at different filtration duration.

The studies using the above experimental plant included three stages. Their goals were:

- 1.
To determine the dependence of the coefficient of the deposit hydrodynamic stability from the deposit age.

- 2.
To establish the dependence of the coefficient of the deposit hydrodynamic stability from the washout rate.

- 3.
To verify the hypothesis about the absence of the deposit detachment at the filtration mode with a significant increase in the filtration rate.

## RESULTS AND DISCUSSION

### Hydraulic characteristics of a clean granular medium

*et al.*2004). The hydraulic conductivity in the majority of these formulae depends on the medium porosity, the surface granule characteristics and the water viscosity, i.e.:

The Kozeny–Karman formula for the ideal soil model is widely used in water treatment practice for the determination of the head losses in the filter medium with laminar fluid flow. Dependences for two components are expedient to use in the general case in which one component takes into account viscosity forces (the hydraulic conductivity is present in a linear dependence on the filtration rate); another component takes into consideration inertial forces (the power on the filtration rate is higher than one).

_{10}= 2.05 mm, d

_{80}= 4.85 mm, d

_{ek}= 2.8 mm, k

_{un}= 2.4. The studies were conducted for different heights of foam polystyrene filter medium: 0.78 m, 0.9 m, 1.0 m. The overall results are given in Table 1 and Figure 2.

No . | Name . | Symbol, calculation formula . | Unit . | Value . | ||
---|---|---|---|---|---|---|

1 | Water temperature | °С | 12 | |||

2 | coefficient of kinematic water viscosity | m^{2}·s^{−1} | 1.25·10^{−6} | |||

3 | Equivalent grain diameter | m | 0.0028 | |||

4 | Medium porosity | fractions of a unit | 0.4 | |||

5 | Grain form coefficient | – | 1.05 | |||

6 | Density of water | t·m^{−3} | 1 | |||

7 | Medium height | m | 0.78 | 0.9 | 1.0 | |

8. Kozeny–Karman formula: , cm | ||||||

8.1 | Empirical Kozeny coefficient | – | 11.3 | |||

8.2 | Average relative error of the study (at V ≤ 10 m/h) | % | 8.3 | 14.2 | 7.2 | |

8.3 | Average relative error of the study | % | 9.9 | |||

9. Ergun's formula: , cm | ||||||

9.1 | Empirical coefficient | – | 400 | |||

9.2 | Empirical coefficient | – | 3.5 | |||

9.3 | Average relative error | % | 6.9 | 11.2 | 7.1 | |

9.4 | Average relative error of the study | % | 8.4 |

No . | Name . | Symbol, calculation formula . | Unit . | Value . | ||
---|---|---|---|---|---|---|

1 | Water temperature | °С | 12 | |||

2 | coefficient of kinematic water viscosity | m^{2}·s^{−1} | 1.25·10^{−6} | |||

3 | Equivalent grain diameter | m | 0.0028 | |||

4 | Medium porosity | fractions of a unit | 0.4 | |||

5 | Grain form coefficient | – | 1.05 | |||

6 | Density of water | t·m^{−3} | 1 | |||

7 | Medium height | m | 0.78 | 0.9 | 1.0 | |

8. Kozeny–Karman formula: , cm | ||||||

8.1 | Empirical Kozeny coefficient | – | 11.3 | |||

8.2 | Average relative error of the study (at V ≤ 10 m/h) | % | 8.3 | 14.2 | 7.2 | |

8.3 | Average relative error of the study | % | 9.9 | |||

9. Ergun's formula: , cm | ||||||

9.1 | Empirical coefficient | – | 400 | |||

9.2 | Empirical coefficient | – | 3.5 | |||

9.3 | Average relative error | % | 6.9 | 11.2 | 7.1 | |

9.4 | Average relative error of the study | % | 8.4 |

Consequently, the best conformity of the experimental and calculation data by the Kozeny–Karman's formula was observed at the Kozeny coefficient . The average measurement error for the foamed polystyrene medium was 9.9% at different heights. The greatest overlap of the experimental and calculation data obtained using the Ergun formula was observed at the coefficients and . The average measurement error for the foam polystyrene medium at different height was 8.4%. The Ergun formula allows to determine the initial head losses more exactly in comparison with the simpler Kozeny-Karman formula, but this correction is insignificant.

### Deposit composition in a partially clogged medium

The amount of the bound water in granular filter mediums depends on the nature and dispersion of the medium as well as the nature and composition of impurities. It should change in the bed when the form of the iron, its concentration and the concentration of other impurities in the deposit vary. The concentrations of the chemical iron mass in various forms (ferrous and ferric iron), in a free and immobilized states can be used in the clarification compartment of the developed mathematical model of physicochemical iron removal from water. It is quite convenient, because the total and ferric iron mass concentrations are determined by the photocolorimetric method. First of all, we need to know the change in the medium porosity to determine the head losses dynamics in granular medium during the water filtration with excessive iron concentration. It was proposed to establish dynamics in the third compartment (hydraulic) of the mathematical model using the ratio of the deposit mass concentration in the medium (in its solid phase) to the solid particles mass concentration per unit of the deposit volume. However, the value of the last concentration cannot be obtained experimentally and we need to solve the inverse mathematical task. It is more convenient to use the volumetric concentrations to determine the head losses. There is a need to know not only the ratio of chemical iron to the bound water, but also the ratio of the mass to the volumetric concentration to recalculate the corresponding concentrations from the clarification compartment to the hydraulic compartment of the mathematical model. To a first approximation, the ratio of the mass concentration to the volumetric concentration can be determined by the known total moisture capacity of the sand: mass is 40%, volume is 25%. Based on that, the ratio of the volumetric and the weight content of sand saturated with moisture is 40/25 = 1.6.

So, the bound water content is 10.17–10.34% of the medium mass including the bound water and the ferruginous deposit. The main mass of the bound water is removed at 105 °C, which is 94.4–96.1% of the total amount of the bound water. The bound water content exceeds the iron content () in 21.7–23.3 times. Slightly higher values were recorded when water with only one form of iron (ferrous or ferric) was being filtered.

### The dependence of the hydraulic conductivity on the deposit concentration

The value was obtained from the experimental studies. The value of the coefficient *a* noted above is . The graph of the volumetric concentration dependence on the deposit mass concentration is given in (Ojha & Graham 1992). Based on that, the ratio of the two quantities is in the range of 8–21. The coefficient takes large value for iron removal from water in comparison with the filtration of the surface water.

The calculations were performed using dimensionless experimental data (Table 2). The change in the hydraulic conductivity is slower if the coefficient decreases and consequently the head losses in the medium will increase more slowly. The value of the coefficient was found using the approximation criterion in MS Excel as an optimization function applying the ‘Solution Search’.

0 | 0.074 | 0.208 | 0.34 | 0.524 | 0.638 | 1 | |

0 | 2.72 | 7.65 | 12.51 | 19.28 | 23.48 | 36.8 | |

Experimental, , m·h^{−1} | 75 | 35 | 12.5 | 5.6 | 3.2 | 2.5 | 2.4 |

Experimental | 1.0 | 0.47 | 0.17 | 0.07 | 0.04 | 0.03 | 0.02 |

, an approximation criterion | |||||||

Calculated | 1.0 | 0.76 | 0.47 | 0.29 | 0.15 | 0.10 | 0.03 |

, an approximation criterion | |||||||

Calculated | 1.0 | 0.50 | 0.14 | 0.05 | 0.02 | 0.021 | 0.01 |

, an approximation criterion | |||||||

Calculated | 1.0 | 0.33 | 0.05 | 0.01 | <0.01 | <0.01 | <0.01 |

0 | 0.074 | 0.208 | 0.34 | 0.524 | 0.638 | 1 | |

0 | 2.72 | 7.65 | 12.51 | 19.28 | 23.48 | 36.8 | |

Experimental, , m·h^{−1} | 75 | 35 | 12.5 | 5.6 | 3.2 | 2.5 | 2.4 |

Experimental | 1.0 | 0.47 | 0.17 | 0.07 | 0.04 | 0.03 | 0.02 |

, an approximation criterion | |||||||

Calculated | 1.0 | 0.76 | 0.47 | 0.29 | 0.15 | 0.10 | 0.03 |

, an approximation criterion | |||||||

Calculated | 1.0 | 0.50 | 0.14 | 0.05 | 0.02 | 0.021 | 0.01 |

, an approximation criterion | |||||||

Calculated | 1.0 | 0.33 | 0.05 | 0.01 | <0.01 | <0.01 | <0.01 |

*et al.*2004):where , are the empirical coefficients.

The calculated values of the hydraulic conductivity differ significantly from the experimental values when the coefficient deviates from 1 by 0.2 to either side. Therefore, the value was assumed for further analysis and the coefficient was selected (Table 3).

0 | 0.074 | 0.208 | 0.34 | 0.524 | 0.638 | 1 | |

0 | 2.72 | 7.65 | 12.51 | 19.28 | 23.48 | 36.8 | |

Experimental, , m·h^{−1} | 75 | 35 | 12.5 | 5.6 | 3.2 | 2.5 | 2.4 |

Experimental | 1.0 | 0.47 | 0.17 | 0.07 | 0.04 | 0.03 | 0.02 |

, , , an approximation criterion | |||||||

Calculated | 1.0 | 0.81 | 0.53 | 0.32 | 0.14 | 0.07 | <0.01 |

, , , an approximation criterion | |||||||

Calculated | 1.0 | 0.70 | 0.35 | 0.15 | 0.04 | 0.01 | <0.01 |

, , , an approximation criterion | |||||||

Calculated | 1.0 | 0.53 | 0.15 | 0.03 | 0.01 | <0.01 | <0.01 |

0 | 0.074 | 0.208 | 0.34 | 0.524 | 0.638 | 1 | |

0 | 2.72 | 7.65 | 12.51 | 19.28 | 23.48 | 36.8 | |

Experimental, , m·h^{−1} | 75 | 35 | 12.5 | 5.6 | 3.2 | 2.5 | 2.4 |

Experimental | 1.0 | 0.47 | 0.17 | 0.07 | 0.04 | 0.03 | 0.02 |

, , , an approximation criterion | |||||||

Calculated | 1.0 | 0.81 | 0.53 | 0.32 | 0.14 | 0.07 | <0.01 |

, , , an approximation criterion | |||||||

Calculated | 1.0 | 0.70 | 0.35 | 0.15 | 0.04 | 0.01 | <0.01 |

, , , an approximation criterion | |||||||

Calculated | 1.0 | 0.53 | 0.15 | 0.03 | 0.01 | <0.01 | <0.01 |

The best conformity of the experimental and theoretical values *k* is observed at , . This dependence causes significant errors when the amount of the deposit in the medium is high (close to the ultimate values). It is associated with the term , which requires the correct definition of the deposit limit saturation for the medium pore space . Therefore, it is suggested to use the exponential dependence (20) for practical calculations.

The use of the proposed formulae for increasing the head losses in the granular medium with the corresponding empirical coefficients obtained under the laboratory conditions was tested using experimental data. This information was obtained at the operational plants of the physicochemical iron removal from groundwater. The kinetic coefficients were determined taking into account the features of the autocatalytic reaction, conditions of the iron delivery and fixation in the solid phase, the sorption (adsorption) coefficients from the main parameters were determined by the formulae given in the previous article of this series (Martynov & Poliakov 2021).

### Coefficient of deposit hydrodynamic stability

In the first series of the experiments the characteristics of the deposit were determined based on its age. The parameters of the experimental plant are shown in Table 4.

No. . | The name of the indicator . | Unit . | Value . |
---|---|---|---|

1 | Filtration duration | hour | 0.25; 0.6; 24; 48; 72; 96 |

2 | The iron concentration in the model solution | mg·dm^{−3} | 1.5 |

3 | Filtration rate | m·h^{−1} | 5 |

4 | Washing rate | m·h^{−1} | 41.38 |

5 | Washing intensity | L·s^{−1}·m^{−2} | 11.49 |

No. . | The name of the indicator . | Unit . | Value . |
---|---|---|---|

1 | Filtration duration | hour | 0.25; 0.6; 24; 48; 72; 96 |

2 | The iron concentration in the model solution | mg·dm^{−3} | 1.5 |

3 | Filtration rate | m·h^{−1} | 5 |

4 | Washing rate | m·h^{−1} | 41.38 |

5 | Washing intensity | L·s^{−1}·m^{−2} | 11.49 |

The results that characterize the deposit washout are of particular interest. They are expressed as a percentage of the total iron which accumulated within the medium in the filtration mode. The least amount of iron was washed out during the longest filtration, which was explained by the higher deposit strength. At the same time the graphs for the filtration durations of 6, 15 and 24 hours practically coincide which points to the similarity of the deposit strength properties. The efficiency of the iron washout from the medium decreases when the deposit age increases and for 6 h is 67.1%; for 15 h is 68.0%; for 24 h is 66.8%; for 48 h is 57.2%; for 72 h is 49.9% and for 96 h is 36.9%.

*G*nearly doubles at the end of the 4-day filtration comparing to the 2-day filtration.

The behaviour of the same ‘age’ deposit as depends on the washout rate was determined during the second series of the experiments. A set of experimental parameters is shown in Table 5.

No. . | The name of the indicator . | Unit . | Value . |
---|---|---|---|

1 | Filter run | h | 24 |

2 | Iron concentration in the model solution | mg·dm^{−3} | 1.5 |

3 | Filtration rate | m·h^{−1} | 5 |

4 | Washout rate | m·h^{−1} | 5; 20.7; 41.4; 62.1; 75.9 |

No. . | The name of the indicator . | Unit . | Value . |
---|---|---|---|

1 | Filter run | h | 24 |

2 | Iron concentration in the model solution | mg·dm^{−3} | 1.5 |

3 | Filtration rate | m·h^{−1} | 5 |

4 | Washout rate | m·h^{−1} | 5; 20.7; 41.4; 62.1; 75.9 |

^{−1}to 41.4 m·h

^{−1}, the peak value of the iron concentration increases at first and then decreases (from 41.4 m·h

^{−1}to 75.9 m·h

^{−1}), which is explained by a much larger water volume during the first 20 s of washout and respectively by the sample dilution. The amount of the deposit that is being washout from the medium grows as the washout rate increases. Therefore, the 82.7% of the deposit was washed out at a washout rate of 75.9 m·h

^{−1}. The coefficient of the deposit hydrodynamic stability decreases as the washout rate increases according to a nonlinear law that can be described with sufficient accuracy (R

^{2}= 0.995) by the logarithmic dependence (Figure 7).

^{rd}experiments series). At first, water with the iron concentration of 1.5 mg·dm

^{−3}was being filtered at the filtration rate of 5 m·h

^{−1}the entire day. Next, the water free of iron was supplied in the same filtration direction (ascending) with the same filtration rate (5 m·h

^{−1}) during 6.5 h. Thereby, the hypothesis about a decrease in the value of the deposit strength characteristic was tested (possible deposit detachment) by supplying water free of iron for a prolonged period of time. The study results are shown in Figure 8.

The iron concentration in water used for backwashing was 0.253 mg·dm^{−3} at the beginning of the washout. It was 0.06 mg·dm^{−3} on the 20th s of the washout. Further, the iron concentration in the washed water was 0.06–0.01 mg·dm^{−3} for the next 6.5 h of the washout. It is noticeable that the iron concentration is higher than normal at the beginning of the washout that is related to the iron washout from the liquid phase. Therefore, the deposit washout from the medium does not occur at the washout rates similar to the filtration rates. Therefore, it is appropriate to use the undetachable kinetics of iron removal from water.

## CONCLUSIONS

The legitimacy of using undetachable filtration at the contact iron removal from groundwater was proven experimentally.

It was established that the accelerated increase in mechanical energy expenditure to support the required productivity of a rapid filter serves as one of the main limitations for the technological process of iron removal. The main reason is the intensive accumulation of the deposit that mainly consists of the bound water. It was found experimentally that for water with varying degrees of iron oxidation the amount (volume) of the bound water exceeds the volume of the iron at 21.7–23.3 times.

A sharp decrease in the medium throughput ability is a direct consequence of a progressive reduction of the pore space. The current hydraulic conductivity is justified to use as a measure of such nonlinear effect. The exponential dependence is recommended for determining hydraulic conductivity of a partially clogged porous medium with the bound water under consideration.

The deposit accumulated in the granular medium changes its properties during the filtration and becomes ‘old’, that leads to a decrease in the deposit volume and an increase in its density and strength. The coefficient of deposit hydrodynamic stability is proposed to use for the estimation of the deposit strength characteristics, which grows when the age of the deposit increases too. This effect will be taken into account when determining the filtration duration in a series of consecutive filter runs.

We are planning to use the mathematical model to develop an engineering calculation method of filtration characteristics, technological parameters for physicochemical iron removal taking into account the gradual decrease in the medium clarifying potential in the series of filter runs. This will allow us to appropriately choose the rational duration of the filter run and the medium service life before it exhausts its operational resource. These questions will be discussed in detail in the last article of this research series.

## DATA AVAILABILITY STATEMENT

All relevant data are included in the paper or its Supplementary Information.

## CONFLICT OF INTEREST

The authors declare there is no conflict.