Partial Desalination of Saline Irrigation Water Using [Fe_xO_y(OH)_z(H₂O)_m)^n+/−]

Abstract

Arable crop yields decrease with increased irrigation water salinity. The low wholesale value ($/m³) of most crops coupled with a relatively high irrigation water demand (m³/ha) ensures that desalination, or partially desalination, of saline irrigation water is not economically viable in most locations unless the partially desalinated water can be delivered to the field for an incremental processing cost of less than about $0.2/m³. Decreases in irrigation water salinity by 20–50% can (depending on crop variety, location, and initial water salinity) have the potential to substantially increase crop yields (e.g., by 20% to more than 500%). Air stable, metal complexes ([Fe_xO_y(OH)_z(H₂O)_m)^n+/−]) form an inexpensive, reusable, desalination catalyst which can allow batches of irrigation water to be partially desalinated at source for an incremental cost of less than $0.1/m³. The small footprint of the reactor units and low capital cost indicates that this technology could provide an economically viable solution for the provision of 1 to 400 m³/d of partially desalinated irrigation water.

Tabulated Data and Analyses

Desalination Model

The current ASMC desalination model assumes that the ASMC acts as a Dubinin-Astakhov adsorption desorption catalyst [11, 19]. If the ASMC catalyst is given the notation [Fe⁺-X⁻]_ads, then Cl⁻ (or ClO⁻, Cl_xO_yⁿ⁻) ions are adsorbed as [8, 11,12,13]:

$$ {\left[{\mathrm{Fe}}^{+}-{X}^{-}\right]}_{\mathrm{ads}}+2{\mathrm{Cl}}^{-}={\left[{\mathrm{Fe}}^{3+}-{\left({\mathrm{Cl}}^{-}\right)}_2-{X}^{-}\right]}_{\mathrm{ads}}+2{\mathrm{e}}^{-}\left(\mathrm{anodic}\right) $$

(6)

X = anions, e.g., [O_y(OH)_z(H₂O)_m)ⁿ⁻]. Sodium ions are adsorbed as:

$$ {\left[{\mathrm{Fe}}^{+}-{X}^{-}\right]}_{\mathrm{ads}}+2{\mathrm{Na}}^{+}+2{\mathrm{OH}}^{-}={\left[{\mathrm{Fe}}^{3+}-{\left({\mathrm{Na}\mathrm{O}}^{-}\right)}_2-{X}^{-}\right]}_{\mathrm{ads}}+2{\mathrm{H}}^{+}+2{\mathrm{e}}^{-}\, \left(\mathrm{anodic}\right) $$

(7)

$$ {\left[{\mathrm{Fe}}^{3+}-{\left({\mathrm{NaO}}^{-}\right)}_2-{X}^{-}\right]}_{\mathrm{ads}}+2{\mathrm{H}}^{+}+2{\mathrm{e}}^{-}\, ={\left[\left[{\mathrm{Fe}}^{+}-{X}^{-}\right]{\left[\mathrm{NaOH}\right]}_2\right]}_{\mathrm{ads}}\, \left(\mathrm{cathodic}\right) $$

(8)

The desalination desorption environment can contain a flora of aerobic iron bacteria, e.g., Leptothrix discophora [12]. This implies that desorption takes the form [12]:

$$ {\left[\left[{\mathrm{Fe}}^{3+}-{\left({\mathrm{Cl}}^{-}\right)}_2-{X}^{-}\right]\right]}_{\mathrm{ads}}+{\mathrm{Cl}}^{-}+2{\mathrm{e}}^{-}={\left[{\mathrm{Fe}}^{+}-{X}^{-}\right]}_{\mathrm{ads}}+3{\mathrm{Cl}}^{-}\left(\mathrm{cathodic}\right) $$

(9)

$$ {\left[\left[{\mathrm{Fe}}^{+}-{X}^{-}\right]\ {\left[{\mathrm{Na}}^{+}-{\mathrm{OH}}^{-}\right]}_2\right]}_{\mathrm{ads}}={\left[{\mathrm{Fe}}^{+}-{X}^{-}\right]}_{\mathrm{ads}}+{2\mathrm{Na}}^{+}+0.{5\mathrm{O}}_2+{\mathrm{H}}_2\mathrm{O}+{2\mathrm{e}}^{-}\ \left(\mathrm{anodic}\right) $$

(10)

Desorption into dead-end porosity results in:

$$ {{\mathrm{Na}}^{+}}_{\mathrm{aq}}+{{\mathrm{Cl}}^{-}}_{\mathrm{aq}}={\left[{\mathrm{Na}}^{+}{\mathrm{Cl}}^{-}\right]}_{\mathrm{aq},\mathrm{s}} $$

(11)

The associated water may contain concentrations of NaOH, NaHClO, HClO (e.;g., [6, 8, 11]) and related species, e.g.,

$$ {{\mathrm{Na}}^{+}}_{\mathrm{aq}}+{\mathrm{OH}}^{-}={\left[{\mathrm{Na}}^{+}{\mathrm{OH}}^{-}\right]}_{\mathrm{aq},\mathrm{s}} $$

(12)

$$ {{\mathrm{Na}}^{+}}_{\mathrm{aq}}+{\mathrm{OH}}^{-}+{\mathrm{Cl}}^{-}={\left[{\mathrm{Na}}^{+}{\mathrm{Cl}}^{-}{\mathrm{OH}}^{-}\right]}_{\mathrm{aq}} $$

(13)

$$ {\mathrm{OH}}^{-}+{\mathrm{Cl}}^{-}={\left[{\mathrm{Cl}}^{-}{\mathrm{OH}}^{-}\right]}_{\mathrm{aq}} $$

(14)

Examples of halite precipitation in an ASMC pellet following desalination are provided in Fig. 10. The halite can contain holdfasts associated with the aerobic iron bacterium Leptothrix discophora (Fig. 10).

Fig. 10

Type A (ST [6]) catalyst examples following desalination. Reflected light. (a) polished section showing halite infilled porosity; Field of view = 0.91 mm; (b) cleaved section showing halite growing from the ASMC into the pores; Field of view = 0.91 mm; (c) detail of halite in the cleaved section showing embedded “doughnut shaped” holdfasts of the filamentous bacteria Leptothrix discophora; Field of view = 0.15 mm

Desalination Rate Constant

The observed desalination reaction rate constant k_observed [22, 51] is:

$$ k={k}_{observed}={k}_{normalized}\ {P}_w\ {a}_s=\ln \left({\mathrm{C}}_{t= 0}/{\mathrm{C}}_{t=n}\right)/t=\left[\mathrm{A}\right]\ \exp \left(-{\mathrm{E}}_{\mathrm{a}}/\mathrm{RT}\right) $$

(15)

k_normalized = reaction rate constant (moles/unit volume/unit time) normalized to 1 g Fe/L water and a surface area of 1 m²/g Fe; P_w = Fe concentration, g Fe/L; a_s = is a measure of the degree of contact between a reactive surface and the environment (e.g., m²/g Fe). t = time spent in the reaction environment, seconds; C_t = 0 = ion concentration entering the reaction environment at time t = 0; C_t = n = ion concentration leaving the reaction environment at time t = n; k = observed rate constant; E_a = activation energy of the reaction; R = universal gas constant; [A] = the pre-exponential factor; T = temperature, K; as a first approximation the desalination reaction can be considered to be first order. Desalination associated with Type B catalysts (Fig. 1) either ceases or switches to a lower rate constant, after a short period of operation [6, 8, 11]. The impact on k of catalyst aging, catalyst poisoning, catalyst reactivation, catalyst aging, and catalyst deactivation varies with catalyst formulation, water chemistry, and reactor type. k can remain constant or increase with increasing feed water salinity [6, 8, 11, 13].

Expected Desalination When the Amount of Desalination Is Equilibrium Limited

ASMC desalination is equilibrium limited (e.g., Fig. 1) for both Type A and Type B catalysts. The equilibrium salinity varies with catalyst (e.g., [6, 8, 11]).

Prediction of reactor performance under a range of operating conditions (batch size, temperature, pressure, batch duration, and P_w) can be undertaken for equilibrium limited reactions [49]:

$$ {\mathrm{C}}_{t=n}={\mathrm{C}}_{t= 0}\ 1/\exp \left({P}_w\ {\mathrm{k}}_{\mathrm{a}}\ {k}_{actual}\ t\right) $$

(16)

C_e = Equilibrium NaCl concentration in the product water, g/L, C_t = n ≥ C_e. k_a = The catalyst activity relative to the design activity; t = Batch duration, seconds. Mean C_e (regression equation) for catalyst K (Fig. 2b) approximates to:

$$ {\mathrm{C}}_{\mathrm{e}}=-0.0591\ {\left({\mathrm{C}}_{t= 0}\right)}^2+0.743\ \left({\mathrm{C}}_{t= 0}\right)\, -\mathrm{Mean} $$

(17)

$$ {\mathrm{C}}_{\mathrm{e}}=-0.0798\ {\left({\mathrm{C}}_{t= 0}\right)}^2+0.9025\ \left({\mathrm{C}}_{t= 0}\right)\, -\mathrm{Upper}\ \mathrm{Limit} $$

(18)

$$ {\mathrm{C}}_{\mathrm{e}}=-0.0581\ {\left({\mathrm{C}}_{t= 0}\right)}^2+0.6287\ \left({\mathrm{C}}_{t= 0}\right)\, -\mathrm{Lower}\ \mathrm{Limit} $$

(19)

R² (mean) = 86.83%; n = 19; C_i = 1.3–5.9 g/L; Temperature = −5 °C to 15 °C; 800 L/batch; P_w = 0.072 g Fe/L.

Statistical Analyses

All agricultural landowners will want to be able to predict the expected salinity of the partially desalinated irrigation water prior to making a commercial investment decision to install an ASMC desalination reactor + catalyst.

This decision will be based on the desalination distributions associated with commercial scale trials. The initial data required is the desalination associated with each of nt batches of water from a single reactor train (Fig. 3).

The probability distribution is calculated using the nonparametric rank order probability assessment eq. [2]:

$$ \mathrm{Probability}=\mathrm{Observation} \operatorname {rank}\ \mathrm{number}/\left(\mathrm{Total}\ \mathrm{Number}\ \mathrm{of}\ \mathrm{Observations}+1\right) $$

(20)

Each observation refers to the salinity associated with a specific batch of water exiting a single reactor train. The relationship between desalination and probability can be described (for a specific catalyst) using a regression equation, e.g.,:

$$ \mathrm{Desalination}=\mathrm{a}\left[P\right]+\mathrm{b} $$

(21)

$$ \mathrm{Desalination}=\mathrm{a}{\left[P\right]}^5+\mathrm{b}{\left[P\right]}^4+\mathrm{c}{\left[P\right]}^3+\mathrm{d}{\left[P\right]}^2+\mathrm{e}\left[P\right]+\mathrm{f} $$

(22)

Where a to f are regression constants, P = probability (lower and upper limits are 0 and 1). P is replaced by a random number (R_n) in a Monte Carlo analysis. Eq. 21 assumes a linear correlation and Eq. 22 assumes a polynomial correlation. The accuracy of this regression analysis increases as nt increases.

Monte Carlo Analysis: Salinity of Water Leaving the Desalination Reactor

Eq. 21, 22 allow a Monte Carlo analysis to be undertaken (e.g., [2]) to determine the expected desalination associated with placing n reactor trains in parallel within a reactor, where the desalination associated with each iteration is calculated as follows:

$$ \mathrm{Iteration}\ \mathrm{Desalination},{\mathrm{I}}_{\mathrm{d}}\, =\, \sum \left(\mathrm{a}\left[{\mathrm{R}}_{\mathrm{n}}\right]+\mathrm{b}\right)/n $$

(23)

$$ \mathrm{Iteration}\ \mathrm{Desalination},{\mathrm{I}}_{\mathrm{d}}\, =\sum \left(\mathrm{a}{\left[{\mathrm{R}}_{\mathrm{n}}\right]}^5+\mathrm{b}{\left[{\mathrm{R}}_{\mathrm{n}}\right]}^4+\mathrm{c}{\left[{\mathrm{R}}_{\mathrm{n}}\right]}^3+\mathrm{d}{\left[{\mathrm{R}}_{\mathrm{n}}\right]}^2+\mathrm{e}\left[{\mathrm{R}}_{\mathrm{n}}\right]+\mathrm{f}\right)/n $$

(24)

The summation is for reactor trains 1 to n where each train has the same capacity (weighting). a to f = constants. A rank order analysis (Eq. 20) of the I_d values will provide the anticipated desalination distribution associated with a reactor unit containing parallel reactor trains. The number of iterations required for a Monte Carlo analysis is between 100 and 10,000.

Monte Carlo Analysis: Salinity of Water in the Product Water Tank

Each batch iteration associated with a single reactor train or a multiple reactor train reactor will discharge water into a product water storage tank. The salinity of the product water tank is a function of its size and can be calculated as:

$$ \mathrm{Tank}\ \mathrm{Salinity},{\mathrm{I}}_{\mathrm{t}}\, ={\mathrm{C}}_{\mathrm{t}=0}\ \left(1-\left(\sum \left(\mathrm{a}\left[{\mathrm{R}}_{\mathrm{n}}\right]+\mathrm{b}\right)/m\right)\right){-} \mathrm{Single}\ \mathrm{train} $$

(25)

$$ \mathrm{TankSalinity},{\mathrm{I}}_{\mathrm{t}}\, ={\mathrm{C}}_{\mathrm{t}=0}\left(1-\left(\sum \left(\mathrm{a}{\left[{\mathrm{R}}_{\mathrm{n}}\right]}^5+\mathrm{b}{\left[{\mathrm{R}}_{\mathrm{n}}\right]}^4+\mathrm{c}{\left[{\mathrm{R}}_{\mathrm{n}}\right]}^3+\mathrm{d}{\left[{\mathrm{R}}_{\mathrm{n}}\right]}^2+\mathrm{e}\left[{\mathrm{R}}_{\mathrm{n}}\right]+\mathrm{f}\right)/m\right)\right) $$

(26)

$$ \mathrm{Tank}\ \mathrm{Salinity},{\mathrm{I}}_{\mathrm{t}}\, ={\mathrm{C}}_{\mathrm{t}=0}\ \left(1-\left(\sum \left(\sum \left(\mathrm{a}\left[{\mathrm{R}}_{\mathrm{n}}\right]+\mathrm{b}\right)/n\right)/m\right)\right){-} \mathrm{Multiple}\ \mathrm{trains} $$

(27)

$$ {\displaystyle \begin{array}{ll}& \mathrm{Tank}\ \mathrm{Salinity},{\mathrm{I}}_{\mathrm{t}}\, \\ {}& ={\mathrm{C}}_{\mathrm{t}=0}\ \left(1-\left(\sum \left(\sum \left(\mathrm{a}{\left[{\mathrm{R}}_{\mathrm{n}}\right]}^5+\mathrm{b}{\left[{\mathrm{R}}_{\mathrm{n}}\right]}^4+\mathrm{c}{\left[{\mathrm{R}}_{\mathrm{n}}\right]}^3+\mathrm{d}{\left[{\mathrm{R}}_{\mathrm{n}}\right]}^2+\mathrm{e}\left[{\mathrm{R}}_{\mathrm{n}}\right]+\mathrm{f}\right)/n\right)/m\right)\right)\end{array}} $$

(28)

Where the summation is for feed water batches 1 to m, and m batches are required to fill the tank. The desalination represented by the tank after m batches is (∑ (a[R_n] + b)/m) or (∑ (∑ (a[R_n] + b)/n)/m). Since each batch of water contains a volume of water, V_n, it follows that the volume of water in the tank after m batches is mV_n.

99% Confidence Limits on the Mean Desalination

The mean and standard deviation associated with m batches of product water defines the upper and lower 99% confidence limits for a specific value of m, where (BS2846 [16]):

$$ 99\%\mathrm{confidence}\ \mathrm{limit}=\mathrm{mean}+/-\left(\mathrm{Standard}\ \mathrm{deviation}\ \left({\mathrm{t}}_{0.995}/{(m)}^{0.5}\right)\right) $$

(29)

t_0.995 = two sided 99% confidence value for the Students t test associated with m samples. Tabulated in BS2846 [16]). If m is very large (e.g., >200), then t_0.995 approximates to z = 2.576.