### Preparation of metal solutions

For preparation of synthetic water solutions, analytical grades of Iron (III) chloride (with a chemical formula FeCl_{3}.6H_{2}O) and Manganese (II) sulfate (with a chemical formula MnSO_{4}.H_{2}O) were dissolved in tap water to give a concentration of 1000 mg/l and diluting when necessary.

### Preparation of nano-biosorbants and nanosilica

The raw tea leaves (TL) and rice straw (RS) were used as biosorbents. Samples were collected from tea plants and rice paddies in the North of Iran. At first, samples were several times washed with tap water for removal of such all traces as oil and dirt. The material was dried at room temperature for 3 days; for more drying, raw adsorbents were placed into oven with a temperature of 70 °C until its weight became constant. The Planetary ball mill was used to reduce adsorbent particles sizes. The adsorbents were placed in a ball mill at 600 rpm for 20 min. then; they were placed in a furnace at 400 °C for 3 h. After 24 h passed, the biosorbents char were taken out of the furnace and stored in closed containers. Biosorbents char include tea leaves-derived char (TLC) and rice straw-derived char (RSC). Silicon oxide nano powder (nano-silica) with chemical formula SiO_{2}, amorphous structure, and hydrophilic type were purchased from Fine-Nano Company (Made By US-NANO) with 99.5% purity.

### Fixed bed column studies

The adsorption process was carried out using Fe and Mn solution with a concentration of 5 mg/l as an artificial sample as well as 0.1 g of nano-biosorbents. A feed water reservoir (with 20 L capacity) was chosen with a heat source inside, its temperature controlled by a thermo regulator. Moreover, the inlet stream was placed over the reservoir; the feed flow was pumped by a variable flow pump (WS123) at a flow rate of 10 ml/min. For the next step, the stream was conducted by a three-way pipe. One stream returned to the feed water reservoir and the other entered the flow meter. Subsequently, the stream entered the top of the column. The process was conducted in a continuous adsorption process with a fixed bed column (diameter of 1 cm and a height of 50 cm). The absorption column had an outer protective body, covered by a heating element. At the column outlet, collection of the samples was done at different times for each test. The time intervals for the process ranged from 1, 2, 5, 10, 20 to 900 (min) minutes, respectively. The process temperature was 30 ℃. In addition, the pH value was 7.9. The flow was stopped once the column was fully marked by C_{t}=C_{0}. For each test, samples were collected at a volume of 15 ml for analysis using atomic adsorption (Agilent 240–280 Series AA) as effluent Fe and Mn ions concentration (C_{t}). The column was washed with distilled water after each test. The schematic diagram of the fixed-bed column is presented in Fig. 1.

The breakthrough curve is usually described as the ratio of the ions concentration at the outlet to the column inlet ratio (C_{t}/C_{0}) in the function of time for the fixed-bed column. The amount of adsorbed Fe and Mn was calculated by Eq. (1):

$${q}_{mathrm{total}}=frac{Q*A}{1000}=frac{Q}{1000}{int }_{0}^{{mathrm{t}}_{mathrm{total}}}{mathrm{C}}_{mathrm{ad}}mathrm{dt}=frac{Q}{1000}{int }_{0}^{{mathrm{t}}_{mathrm{total}}}({mathrm{C}}_{0}-{mathrm{C}}_{mathrm{t}})mathrm{dt}$$

(1)

where _{qtotal} indicates the maximum removal capacity of column in mg, *Q* is the volumetric flow rate circulating through the column in ml/min, *A* is the area under the breakthrough curve in m^{2}, ({t}_{mathrm{total}}) is the total flow time in min, ({mathrm{C}}_{mathrm{ad}}) is the absorbed removal concentration in mg/l. C_{0} and C_{t} are the Fe and Mn ions concentrations (mg/l) in inlet and outlet flows, respectively. The equilibrium capacity of the column (mg/g) was obtained by Eq. (2), where *m* is the dry absorbent mass (*g*):

$${q}_{mathrm{eq}}=frac{{q}_{mathrm{total}}}{m}$$

(2)

({m}_{mathrm{total}}) is the total mass absorbed into the column in mg, which is obtained using Eq. (3):

$${m}_{mathrm{total}}=frac{{mathrm{C}}_{0}Q{t}_{mathrm{total}}}{1000}$$

(3)

%*R* is the total removal percentages of iron and manganese, which is obtained using Eq. (4):

$$mathrm{%}R=frac{{q}_{mathrm{total}}}{{m}_{total}}times 100$$

(4)

In other words, (%*R*) is the ratio of the total removed ions in the column to the total amount of ions delivered to the column.

### Mathematical description

Various simple mathematical models have been developed to describe and possibly predict the dynamic behavior of the bed in column performance. The most famous and widely applied models used in column performance theory are Thomas and Adams–Bohart [36].

#### Thomas model

Thomas model is one of the most common and widely used methods in expressing the theory of absorption column performance. It can predict the theoretical Breakthrough curve based on laboratory data (time and Output to input concentration ratio). The linearized form of the Thomas model can be expressed as follows:

Equation (5)

$$mathbf{l}mathbf{n}left(frac{{mathbf{C}}_{0}-{mathbf{C}}_{{varvec{t}}}}{{mathbf{C}}_{{varvec{t}}}}right)=mathbf{l}mathbf{n}left[mathbf{e}mathbf{x}mathbf{p}left(frac{{{varvec{K}}}_{mathbf{T}mathbf{h}}{{varvec{q}}}_{0}{varvec{M}}}{{varvec{Q}}}right)-1right]-{{varvec{K}}}_{mathbf{T}mathbf{h}}{mathbf{C}}_{0}{varvec{t}}$$

(5)

Here, *K*_{Th} (ml/mg min) is the Thomas rate constant; *q*_{0} (mg/g) is the equilibrium uptake of metal ions. *M* as the amount of adsorbent packed inside the column is denoted by (mg). C_{0} (mg/l) is the influent concentration, C_{t} (mg/l) is the outlet concentration at each time; and *Q* (ml/min) is the flow rate. Because the amount (expleft(frac{{K}_{Th}{q}_{0}M}{Q}right)) is much larger than one, the equation is written as follows:

$$mathbf{l}mathbf{n}left(frac{{mathbf{C}}_{0}-{mathbf{C}}_{mathbf{t}}}{{mathbf{C}}_{mathbf{t}}}right)=left(frac{{{varvec{K}}}_{mathbf{T}mathbf{h}}{{varvec{q}}}_{0}{varvec{M}}}{{varvec{Q}}}right)-{{varvec{K}}}_{mathbf{T}mathbf{h}}{mathbf{C}}_{0}{varvec{t}}$$

(6)

where *q*_{0} and *K*_{Th} amounts could be obtained through the slope and width of the origin of the linear graphs, respectively, obtained by (mathrm{ln}left(frac{{mathrm{C}}_{0}-{mathrm{C}}_{mathrm{t}}}{{mathrm{C}}_{mathrm{t}}}right)) in terms of *t* [37].

#### Adams–Bohart model

Adams–Bohart provides a simple basic model for describing the initial region of absorption, often adapting to failure curves with very high accuracy. It has aided researchers to determine important and key parameters, such as adsorption rate constants and adsorption capacity. The mathematical equation of the model can be written as Eq. (7):

$$mathbf{l}mathbf{n}left(frac{{mathbf{C}}_{0}}{{mathbf{C}}_{{varvec{t}}}}-1right)=mathbf{l}mathbf{n}left[mathbf{e}mathbf{x}mathbf{p}left(frac{{{varvec{K}}}_{{varvec{B}}{varvec{A}}}{{varvec{N}}}_{0}{varvec{H}}}{{{varvec{U}}}_{0}}right)-1right]-{{varvec{K}}}_{{varvec{B}}{varvec{A}}}{mathbf{C}}_{0}{varvec{t}}$$

(7)

As the amount (mathrm{exp}left(frac{{K}_{BA}{N}_{0}H}{{U}_{0}}right)) is much larger than one, the equation is written as follows:

$$mathbf{l}mathbf{n}left(frac{{mathbf{C}}_{0}}{{mathbf{C}}_{mathbf{t}}}-1right)=frac{{{varvec{K}}}_{{varvec{B}}{varvec{A}}}{{varvec{N}}}_{0}{varvec{H}}}{{{varvec{U}}}_{0}}-{{varvec{K}}}_{{varvec{B}}{varvec{A}}}{mathbf{C}}_{0}{varvec{t}}$$

(8)

Here, C_{0} (mg/l) is the inlet concentrations. C_{t} (mg/l) is the outlet concentration at each time. *H* (cm) is the bed height and *U*_{0} (cm/min) is the superficial velocity, which is obtained through dividing the flow rate by the cross section of the column. *K*_{BA} (l/mg min) is the Kinetic coefficient or constant Adams–Bohart absorption rate; and *N*_{0} is volumetric absorption capacity (mg/l). *N*_{0} and *K*_{BA} amounts could be obtained by the slope and width of the origin of the linear graphs, respectively; they are obtained practically by ({varvec{l}}{varvec{n}}left(frac{{mathbf{C}}_{0}}{{mathbf{C}}_{{varvec{t}}}}-1right)) in terms of t [38].

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