Equilibrium Data and its Analysis with the Freundlich Model in the Adsorption of Arsenic(V) on Granular Ferric Hydroxide

Journal Pre-proofs Equilibrium Data and its Analysis with the Freundlich Model in the Adsorption of Arsenic(V) on Granular Ferric Hydroxide Kenneth Walsh, Sebastian Mayer, Dirk Rehmann, Thomas Hofmann, Karl Glas PII: DOI: Reference:

S1383-5866(19)33775-X https://doi.org/10.1016/j.seppur.2020.116704 SEPPUR 116704

To appear in:

Separation and Purification Technology

Received Date: Revised Date: Accepted Date:

20 September 2019 11 February 2020 11 February 2020

Please cite this article as: K. Walsh, S. Mayer, D. Rehmann, T. Hofmann, K. Glas, Equilibrium Data and its Analysis with the Freundlich Model in the Adsorption of Arsenic(V) on Granular Ferric Hydroxide, Separation and Purification Technology (2020), doi: https://doi.org/10.1016/j.seppur.2020.116704

This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

© 2020 Published by Elsevier B.V.

Equilibrium Data and its Analysis with the Freundlich Model in the Adsorption of Arsenic(V) on Granular Ferric Hydroxide Kenneth Walsha,b,*, Sebastian Mayera, Dirk Rehmannb, Thomas Hofmanna, Karl Glasa a Technical

University of Munich, Chair of Food Chemistry and Molecular Sensory Science, Lise-Meitner-Strasse 34, 85354 Freising, Germany

b Weihenstephan-Triesdorf

University of Applied Sciences, Institute of Food Technology,

Am Hofgarten 4, 85354 Freising, Germany *Corresponding author: Kenneth Walsh (email: [email protected])

Abstract An accurate understanding of equilibrium conditions is an essential starting point for dimensioning and modelling efforts in adsorption systems. However, key assumptions and methodologies in the generation and analysis of adsorption data sets might be contributing to the error in the description of the equilibrium. This study investigates the potential impacts of the equilibration time, test adsorbent particle size selection, and the regression technique applied, on the fitted coefficients of the Freundlich equilibrium model in the adsorption of arsenate on Granular Ferric Hydroxide. The choice of regression algorithm was found to impact primarily the exponent describing the energetic heterogeneity of the adsorption surface and resulting non-linearity of the isotherm, nf. Non-linear regression with a hybrid error function on the observed solute concentration rather than loading was found to be a suitable approach. Insufficient equilibration time was found to impact predominantly the strength of adsorption, given by Kf, when applying a small adsorbent size fraction, and both coefficients when applying a coarser technical size fraction. Application of the small size fractions in lieu of the larger in accelerated trials appeared to over-estimate both coefficients, but is a reasonable alternative to long contact times. Keywords: Adsorption, Equilibrium, Linear Regression, Non-Linear Regression, Statistical Analysis, Arsenic, Granular Ferric Hydroxide

Manuscript re-submitted to the Journal of Separation & Purification Technology February 2020

Definitions and Nomenclature 𝛼

Proportional error in the observed solute concentration

𝛽

Absolute error in the observed solute concentration [mg/L]

𝑐𝑎𝑑𝑠

Adsorbent concentration in grams dry mass per litre liquid phase [g/L]

𝑐0

Initial liquid phase solute concentration [mg/L] 𝑐𝑒𝑞

Equilibrium liquid phase solute concentration [mg/L]

𝑐𝑚𝑜𝑑

Model (predicted) equilibrium liquid phase solute concentration [mg/L]

𝑐𝑜𝑏𝑠

Observed (measured) liquid phase solute concentration [mg/L]

𝑐𝑤𝑙

Solute concentration at the intersection of the model and work-lines [mg/L]

𝐶𝑥

Dimensionless solute concentration, i.e. (𝑐𝑥 𝑐0)

𝑞𝑒𝑞

Loading, i.e. solid phase solute concentration, at equilibrium, i.e. [mg/g]

𝑞𝑚𝑎𝑥

Maximum loading theoretically possible, i.e. in equilibrium with c0 [mg/g]

𝑞𝑚𝑜𝑑

Model (predicted) equilibrium loading [mg/L]

𝑞𝑜𝑏𝑠

Observed loading derived from the mass balance with the observed solute concentration [mg/g]

𝑞𝑤𝑙

Loading at the intersection of the model and work-lines [mg/g]

𝑞𝑒𝑞,𝑝𝑟𝑒𝑑

PSOM-predicted equilibrium loading [mg/g]

𝑄𝑥

Dimensionless loading, i.e. (𝑞𝑥 𝑞𝑚𝑎𝑥)

DB

Batch adsorption distribution parameter

mads

Dry mass of adsorbent [g]

Vl

Fluid volume [L]

𝑖

Subscript for the observation number in the data set

𝑛

Number of observations in the data set

𝑟

Regression residual

reps

Number of repetitions, i.e. number of artificial data sets

𝑥𝑜𝑏𝑠

Observed predictor

𝑦𝑜𝑏𝑠

Observed response

𝑦𝑚𝑜𝑑

Model (predicted) response, i.e. f(xobs)

GFH

Granular ferric hydroxide

BET

Brunauer-Emmett-Teller (isotherm)

XRD

X-Ray-Diffraction

𝐿𝐹

Linear fit

𝑁𝐿𝐹

Non-linear fit

𝑆𝑆𝑅

Sum of squared residuals (error function in regression) 𝐻𝑌𝐵 Hybrid error function (error function in regression)

MPSD

Marquardt’s percent standard deviation (error function in regression)

𝑅𝑀𝑆𝐸

Residual mean squared error

PSOM

Pseudo Second-Order Model

“Work-line

Distance along a bottle-point’s work-line from the data point to the isotherm

2

distance”

1.

model curve

Introduction The removal of arsenic from ground water through application of technical oxide-

based adsorbents is a well-established treatment in the production of potable water. Granular ferric hydroxide (GFH) is a leading example of this technology and is capable of achieving high adsorption capacities with a simple process set-up [1]. Both in simple dimensioning calculations for a new application of the material and more detailed dynamic modelling of the system, an understanding of the adsorption equilibrium in the given conditions is essential. The adsorption equilibrium of arsenic on ferric hydroxide is highly sensitive to pH and competing ions such as phosphate, silicate and vanadium in the water matrix [2–8]. Predictive methods for the determination of adsorption equilibria based on surface complexation models have been shown to have some merit [4,9]. However, the most common approach, and typically the most accurate, remains the experimental determination of the equilibrium [10]. Adsorption equilibrium studies are therefore an essential starting point for further work with a given water matrix. They typically involve batch trials with the technique referred to as the bottle-point method [10]. This is described later in further detail. The resulting adsorption isotherm data set is then fit to an equilibrium model. For the adsorption of arsenic on porous iron hydroxides, the Freundlich model has been found to represent the relationship well [5,11–14]. The purpose of this study is to revisit some of the assumptions and methods in the determination of the adsorption equilibrium of arsenic on GFH in aqueous systems with the bottle-point method. As the description of the equilibrium is the basis of many dimensioning and modelling efforts, it is important to understand the nature and extent of recurring sources of error in the resulting estimated coefficients.

3

Firstly, a common practice for large porous inorganic adsorbents like GFH is to apply smaller particle size fractions in order to shorten the time required to reach equilibrium [1,4,9,11,13,15,16]. Such material can be gained via grinding and/or sieving of the coarse media and is typically assumed to otherwise have the same fundamental properties. Similarly, the adsorption equilibrium derived with the fine fraction is then assumed to be representative of the coarse adsorbent. With the same chemical composition, crystal structure, and physical properties excepting particle size, this would seem to be a reasonable assumption in ideal conditions. Worch [10] raises the possibility that a smaller size fraction is not necessarily representative of the original adsorbent, and that this possible source of error should be weighed against that of the long equilibrium times required for the latter. Excessively long equilibration times are impractical and can result in errors due to side-reactions such as adsorbent dissolution [10,17]. However, to shorten the equilibration time when applying the coarse granular media risks vastly underestimating the equilibrium. It is therefore one goal of this study to assess the nature of the resulting error of applying a representative powder and the original coarse granular media in the determination of the equilibrium in a synthetic water matrix. Secondly, while application of small particle sizes allows a much quicker equilibration time, it is understood that the data generated over short periods of time is that of a pseudo-equilibrium [2,10,18]. That the overall capacity might be underestimated with pseudo-equilibrium data is clear, but how might the equilibrium model coefficients be impacted? A more detailed understanding of the influence of equilibration time in combination with the size fraction applied is therefore sought. Finally, numerous studies have assessed the impact that the applied regression method can have on equilibrium coefficients in isotherm data sets [19–27]. For one, 4

regression of a linearly-transformed data set of solid phase concentration, or loading, over solute concentration has previously been described as being statistically flawed [28–30]. Also, the non-linear regression of loading over solute concentration in its native form violates assumptions of a controlled-errors-in-variables model [20,21]. A solution to this regression problem has been identified, namely to fit the observed solute concentration remaining in solution against the adsorbent concentration, the original experiment predictor and response variables. In this study, the validity of selected linear and non-linear regression techniques is first assessed theoretically from a practitioner’s viewpoint. Their application to several artificial and real data scenarios then explores the impact of the regression technique on the coefficients of the Freundlich model. Following this, adsorption data with a synthetic water matrix containing arsenic and two different particle size fractions of GFH, the coarse media (granulate) and a smaller representative size fraction (powder), is presented. The progression in time of the isotherms is used to assess the error resulting from particle size fraction selection and equilibration time. 2.

Theoretical analysis of the regression of isotherm data a. Derivation and nature of the data set An adsorption isotherm describes the loading of a solute on a solid phase, such as

a technical adsorbent or soil sample, as a function of its concentration in solution. This data is typically gained with the bottle-point method [10]. For each data point in the isotherm, a set volume of fluid, Vl , with a set or measured initial solute concentration, c0,obs , is mixed with a set dry mass of adsorbent, mads. After a given equilibration time, typically with mixing of the suspension via shaking, the solute concentration remaining in solution, cobs, is quantified. The loading, qobs, is not typically quantified directly, but rather calculated via mass balance on the basis of the observed change in

5

solute concentration, as per Equation 1. The dimensionless version is presented in Equation 5. 𝑞𝑜𝑏𝑠 = (𝑐0,𝑜𝑏𝑠 ― 𝑐𝑜𝑏𝑠).𝑉𝑙 𝑚𝑎𝑑𝑠

Eq. 1

It is assumed here that the quantity of solute dissolved in the liquid within the pores of the adsorbent mass is negligible in comparison to that in the bulk liquid and adsorbed on the adsorbent surface. Therefore, the pore liquid may be disregarded in the mass balance. With much higher liquid volume than adsorbent mass, this is a reasonable assumption. Loss of mass to other sources, e.g. via adsorption on to the bottle’s surface, is also assumed to be negligible – an assumption easily tested with control samples without adsorbent. The adsorbent mass and liquid volumes are assumed to contribute negligibly to error. The error in the initial solute concentration is dependent on the experimental set-up. With existing natural water, for example, the initial concentration must be quantified and can therefore have a larger degree of error associated with it. In this case, the adsorbent concentration (mads/Vl) is by default the varied parameter. If the water is synthetic, the initial solute concentration is well known and shouldn’t be subject to as great an error, and either the initial solute concentration or the adsorbent concentration can be varied. In either case, the experimental predictor-response pair is not equivalent to the variables of the isotherm model, which expresses the solute concentration as the predictor and loading as the response. There are several sources of error in the observation of the equilibrium solute concentration. The uncertainty in the quantification of arsenic in water by inductively coupled plasma-mass spectrometry/-optical emission spectrometry is generally described in a proportional sense [31–33]. This, along with error from dilution, would give a similar proportional variance across the measurement range, resulting in a heteroscedastic data set. Other experimental errors might impact all observations 6

equally independent of the observations’ magnitudes. Sample contamination could be an example of this. Finally, inherent in this regression problem is the assumption that the observed concentrations represent the true equilibrium. In reality, this might not be the case with insufficient equilibration time, for example. This error, while systematic, would not impact each data point equally. The higher the equilibrium loading, the more time is required to reach it and therefore, the further away the system is from equilibrium with insufficient time. This would therefore contribute to heteroscedasticity in the data set. The error in the observed solute concentration is propagated into the derived loading when it is calculated with the mass balance in Equation 1. The same applies to any errors present in the observed initial concentration and in the adsorbent concentration. While the error in an observed initial concentration would be conserved between data points, the resulting shift in the observed loading is proportional to the inverse of the adsorbent concentration. So, in the regression of derived loading over observed solute concentration, both the predictor and the response contain errors. Furthermore, these errors are fundamentally interdependent and correlated. b. The Freundlich Model The Freundlich model is shown in Equation 2. The equilibrium loading, qeq, is given as a function of the strength of adsorption, Kf, the equilibrium solute concentration in solution, ceq, and the energetic heterogeneity of the adsorption surface, nf. A higher value of nf means a more convex isotherm, which is advantageous for typical adsorption processes. The model assumes that multilayer adsorption is possible, such that there is no saturation loading at high concentrations.

()

𝑞𝑒𝑞 = 𝐾𝑓.𝑐𝑒𝑞

7

1 𝑛𝑓

Eq. 2

c. Regression There are several important considerations with this regression problem, including the variables to be regressed, linear versus non-linear regression, and the error function. Particularly the question of linear or non-linear regression and the error function applied has been considered numerous times [23,24,26,27,34,35]. i.

Linear regression following transformation of ceq and qeq

One method of deriving coefficients from adsorption equilibrium data is the linear regression on transformed data sets [18,36,37], which before the ready availability of modern computing power was a practical approach. For the Freundlich isotherm model in Equation 2, taking the logarithm results in a linear function, as shown in Equation 3. 1

𝑙𝑜𝑔10(𝑞𝑜𝑏𝑠) = 𝑙𝑜𝑔10(𝐾𝑓) + 𝑛𝑓.𝑙𝑜𝑔10(𝑐𝑜𝑏𝑠)

Eq. 3

Linear transformation of the data set is simple, rapid and would seem to result in a decent estimation of coefficient values [10]. It is therefore likely sufficient for many commercial and industrial applications, where generous safety margins are included in dimensioning efforts. It is also possible to ascertain with this approach whether or not the model is suited to the data set, judging simply by its linearity after transformation. In applications where the coefficients are to be applied in modelling tools or compared and used in further analyses, a more detailed consideration of the method’s validity and of potentially better alternatives is appropriate. Linear transformation alters the native error structure of the data set [19,28–30]. This can be useful when out of a heteroscedastic data set a homoscedastic one is derived. Homoscedasticity is an assumption of linear regression, after all. As will be shown later, however, the log-transformation does not result in homoscedasticity in representative isotherm data sets and is unlikely to do so in real ones. Therefore, the

8

transformation in this case does not serve a purpose other than simplification of the regression. In addition, linear transformation inherently weights the regression in favour of low observation values. That is, two equal absolute residuals in the logarithm of loading at different regions in the plot have the same proportional residual in the loading. For example, a 0.5 mg/g error at 5 mg/g would have the same impact on the regression as a 5 mg/g error at 50 mg/g in the log-transformed data set. This is also not necessarily a negative outcome, as observation weighting is a useful tool in regression. It should be applied rationally, however. The regression on the calculated loading is already subject to an observation weighting, as the propagation of the error in the observed solute concentration is dependent on the magnitude of the adsorbent concentration. Adding further obscurity to the weighting via linear transformation purely for simplification of the regression is therefore difficult to justify. In comparison, non-linear regression is not subject to the assumption of homoscedasticity in the data set. ii.

Selection of variables and error function for regression

Regarding both linear and non-linear regression of loading against solute concentration, an assumption for the application of controlled-errors-in-variables models (model type I, e.g. the commonly applied sum of least squares) is also violated. Namely, it is assumed that the predictor is not subject to significant error [38,39]. In this regression problem however, the “predictor” is in fact a measured variable, cobs, while the “response”, qobs, is calculated. So, there is non-negligible error in both axes of the plot. This error duality could be taken into account with an errors-in-variables approach (model type II) like total least squares, or orthogonal distance regression, as previously investigated by other authors [22,40]. However, this approach has been

9

found to deal poorly with correlated errors, like those that characterise this regression problem [41]. Two possibilities have been considered to deal with the presence of errors in both axes and their interdependency. The first would be to calculate the residual in a manner that reflects the correlation between the two variables. It is known exactly how loading varies in relation to the observed concentration, namely the mass balance used to calculate loading in the first place, given in Equation 1. This is presented in the isotherm plot as a work-line, as shown in Figure 1-A. Assuming that the initial solute concentration and adsorbent concentration are subject to negligible error, the position of the data point can vary only along the work-line. Therefore, the distance from the data point {cobs,qobs} to the model curve along this work-line {cwl,qwl} would be a sensible error for minimisation in the non-linear regression. This error can be described as per Equation 4. Due to the difference in magnitude of the solute concentration and the loading, dimensionless quantities are recommended, i.e. Cobs = cobs/c0 and Qobs = qobs/qmax, where qmax is the theoretical (model) loading in equilibrium with c0. Alternatively, the largest solute concentration in the isotherm data set to be fitted would also be a suitable normalising variable for the dimensionless quantities. 𝑊𝑜𝑟𝑘–𝐿𝑖𝑛𝑒 𝐸𝑟𝑟𝑜𝑟 = [(𝐶𝑜𝑏𝑠 ― 𝐶𝑤𝑙)2 + (𝑄𝑜𝑏𝑠 ― 𝑄𝑤𝑙)2]

and

12

Eq. 4

where 𝐶𝑜𝑏𝑠 + 𝐷B.𝑄𝑜𝑏𝑠 = 𝐶𝑤𝑙 + 𝐷B.𝑄𝑤𝑙 = 1

Eq. 5

𝐷B = 𝑚𝑎𝑑𝑠.𝑞𝑚𝑎𝑥 (𝑉𝐿.𝑐0)

Eq. 6

This differs from orthogonal distance regression, which estimates the normal distance to the model curve. The different distances are shown conceptually in Figure 1-B. With a vertical, horizontal or orthogonal distance regression, the corresponding points on the model curve from the data point couldn’t exist, as they don’t lie on its work-line and thus violate the mass balance. 10

(A)

(B)

Figure 1: Conceptualisation of the data point variance in the non-linear data set of loading against concentration assuming negligible error in initial concentration and adsorbent concentration (A), and different errors for minimisation in the non-linear regression (B).

In the steep region of the model curve at low observed solute concentrations, a data point with a small horizontal residual in concentration can have a large vertical residual in loading. However, in reality the error in loading is only that to the intersection of the model curve and the work-line and is therefore only small. Taking the standard vertical distance would then exaggerate this point’s error. Conversely, in the flatter region of the curve at high concentrations, a small vertical residual in loading is derived from a large one in concentration, which is then trivialised.

11

The distance along the work-line between the data point and model curve presented in Equation 4 could be described with the solute concentration alone by substituting in the mass balance in Equation 5. This leads to the second possibility, namely to avoid the problem of varying axes of error entirely and plot solute concentration or loading against the true predictor, adsorbent concentration (or initial solute concentration, if this is the varied parameter in the experimental design). This manner of regression has been applied previously, where it was argued to be more statistically valid than existing methods and found to result in different fitted coefficients to existing methods in some conditions [42,43]. It is shown conceptually in Figure 2. As the error in loading and by extension the error in the work-line distance can be described as a function of the error in solute concentration, regression of the solute concentration alone is arguably more justified. It is also a valid application of a controlled-errors-in-variables model, given the assumption of negligible error in the predictor, adsorbent concentration. Taking the vertical error in cobs against cads does not violate the mass balance and is the same as taking the horizontal component of the work-line error in qobs against cobs. In addition, the determination of residuals for confidence interval estimation of the fitted coefficients is not complicated by the inclusion of multiple variables, as orthogonal distance regression or regression of the work-line distance would be. In either approach, the intersection of the model curve and the work-line needs to be determined. This can be achieved by setting the model loading to equal the observed loading and thereby combining both equations to derive an implicit expression for the solute concentration at this point, cwl. This is shown in Equation 7.

(𝑐0 ― 𝑐𝑤𝑙) 𝑐𝑎𝑑𝑠 = 𝐾𝑓.𝑐( 𝑛𝑓) 𝑤𝑙 1

12

Eq. 7

Figure 2: Conceptual depiction of the regression with the experimental predictor-response variables of the isotherm data set

An additional possibility would be to plot the derived loading, in this case qwl, rather than the solute concentration as the response to adsorbent concentration. However, this would mean that the predictor is present in both axes. This should be avoided, as the independence of the errors is no longer maintained [44]. The remaining consideration is which error function is to be minimised. The considered application of a weighting factor on the residuals is reasonable at this point. Numerous authors have made a topic of the selection of the error function previously, for example [23–25,35]. In this study, three representative error functions are applied: the sum of squared residuals (SSR), Marquardt’s percent standard deviation (MPSD) and the Hybrid (HYB) function. These are functions based on squared absolute residuals, squared proportional residuals and a mixture thereof, respectively, as shown in Equations 8-10. Static scalars or operations that can be found in these functions but have no bearing on the regression result, e.g. the division by the degrees of freedom or taking the square root of the whole expression, are not included. 𝑆𝑆𝑅 = ∑

𝑛

[(𝑦𝑜𝑏𝑠,𝑖 ― 𝑦𝑚𝑜𝑑,𝑖)2]

𝑖=1

13

Eq. 8

𝑀𝑃𝑆𝐷 = ∑ 𝐻𝑌𝐵 = ∑

𝑛

𝑛

[(𝑦𝑜𝑏𝑠,𝑖 ― 𝑦𝑚𝑜𝑑,𝑖)2 𝑦𝑜𝑏𝑠,𝑖2]

Eq. 9

𝑖=1

[(𝑦𝑜𝑏𝑠,𝑖 ― 𝑦𝑚𝑜𝑑,𝑖)2 𝑦𝑜𝑏𝑠,𝑖]

𝑖=1

Eq. 10

where y = c or q, depending on the regression problem, and n = number of data points.

3.

Materials and Methods All theoretical work and data analysis was conducted with MATLAB® 2018b. a. Generation of artificial data sets As a basis for the theoretical analysis of the selected regression approaches,

artificial data sets with known “true” Freundlich coefficients were generated. Equilibrium solute concentrations between 0.0005 and 1 mg/L were defined, from which the equilibrium loading was determined with the isotherm model and a Freundlich model coefficient pair similar to later experimental results of Kf = 50 mg. 1 ―1 nf) nf) ( ( ( ― 1) L .g .mg

and nf = 5. The required adsorbent concentrations were then

derived as per the mass balance with an initial solute concentration of 5 mg/L. Pseudorandom noise was applied to derive the “observed” solute concentrations, cobs, with Equation 11. 𝑐𝑜𝑏𝑠 = (1 + 𝛼).(𝑐𝑒𝑞 + 𝛽)

Eq. 11

The set equilibrium concentrations were subjected to a proportional (α) and an absolute (β) error, which were applied independently and based on a normal distribution of the errors across the data set. They were derived with MATLAB®’s normally distributed random number generator, randn. The proportional error was adjusted to a span of ± 0.2 around a mean of null, the absolute error to a one-sided distribution from 0 to 0.0025 mg/L. In an additional scenario, error with a span of α =

14

± 0.1 was applied to the initial starting concentration to derive a c0,obs as well. The observed loadings were then determined with the observed solute concentrations. This process was repeated for the given number of artificial data sets, reps = 100; a number chosen to ensure stability in the results while not costing excessive computing time. The random number generator was reset before running the code, to ensure reproducible noise. b.

Experimental isotherm determination Experimental materials and methods, particle analyses, and the resulting

adsorption data for this study are described in full in the published data set [45]. The key conditions are summarized below. i.

Adsorbent

Table 1 summarises the properties of the fine (powder) and coarse (granulate) adsorbent size fractions used in this study. With the exception of the particle size, they are similar to each other and to the specification for GFH [46]. This is based on a comparison of the specific surface area, pore size distribution, chemical composition and crystal structure. The pore size distribution was found to be unimodal, with diameters between roughly 2.5 and 9.5 nm. This finding is in agreement with previously reported particle analyses that have shown that the specific surface area of GFH is independent of particle size [8,48]. A small difference in pore size distribution between particle size fractions has been reported for GFH, from a BET-analysis which also found that some of the internal surface is present in micropores under 2 µm in a bimodal distribution [11]. Table 1: Summary of adsorbent properties – described in full in [45] Property

Analysis

Powder

Granulate

D10 [µm]

Laser-diffraction & automated image analysis

29.1 ± 0.9

416 ± 39

D50 [µm]

“

44.6 ± 1.8

692 ± 76

15

a

D90 [µm]

“

70.3 ± 1.6

1119 ± 88

D4,3 [µm] (volume mean)

“

48.4 ± 1.3

732 ± 67

Specific surface area [m²/g(dry)]

N2-BET-analysis

302

300

Isotherm form per [47]

N2-BET-analysis

Type IV a/b

Type IV a/b

Description per [47]

N2-BET-analysis

Mesoporous, ink-bottle pores

Mesoporous, ink-bottle pores

Approx. pore size [nm] (min. / meana / max.)

N2-BET, DFT-model with silica as reference

2.5 / 4 - 5 / 9.5

2.3 / 4 - 5 / 9.5

Composition

XRD, Raman spec. & specification [46]

Ferrihydrite, Akaganéite

Ferrihydrite, Akaganéite

mean of the pore surface area distribution

ii.

Synthetic water matrix

The synthetic water matrix was composed of distilled water (≤ 3 µS/cm), arsenic acid standard for an initial concentration of 4.00 mg/L As, BES buffer at 0.480 g/L and NaNO3 at a total concentration of 1.2 g/L (NaNO3 was added both directly and via neutralisation with NaOH of the HNO3 present in the arsenate standard). The pH of the solution was set to 7.25 ± 0.05. The ionic strength was calculated to be 21 mM, while an electrical conductivity of 1.57 mS/cm was measured. iii.

Bottle-point adsorption trials

After addition of the starting solution and adsorbent into HDPE bottles, the bottles were shaken at 90 rpm for 4 weeks. Following this, further equilibration up to 26 weeks in total occurred without shaking. The temperature was constant over the entire 26 weeks at 20 ± 2 °C. The pH value was tested and readjusted to 7.25 ± 0.05 as required on multiple occasions with 2 % HNO3 and NaOH. Sampling occurred at 1, 4, 8 and 26 weeks, for which a set constant volume of 6 mL was removed after at least 15 mins settling time. The concentration of arsenic was determined externally via inductively coupled plasma mass spectrometry (ICP-MS). Loading was then calculated with the mass balance in Equation 1.

16

c.

Regression algorithms Table 2 outlines the regression algorithms applied to both artificial and real

isotherm data sets. LF-SSR-LQ is the standard linear regression of transformed data sets, while NLF-SSR-Qa is the standard non-linear regression of loading against observed solute concentration. The remaining algorithms apply the selected error functions to observed concentrations (NLF-SSR-C, NLF-HYB-C and NLF-MPSD-C) and observed loadings (NLF-SSR-Qb) relative to the intersection of the work line and model curve. Table 2: Applied regression algorithms LFSSR-LQ

NLFSSR-Qa

NLFSSR-C

NLFHYB-C

NLFMPSD-C

NLFSSR-Qb

Predictor variable

log10(cobs)

cobs

cads

cads

cads

cads

Response variable

log10(qobs)

qobs

cobs

cobs

cobs

qobs

Predicted response

log10(qmod)

qmod

cwl

cwl

cwl

qwl

Algorithm

derived via… MATLAB®

model at cobs

| model & work-line intersection at cads

function applied in regression

fitlm

nlinfit

nlinfit

nlinfit

nlinfit

nlinfit

Error function

SSR

SSR

SSR

HYB

MPSD

SSR

via weighting factor...

1

1

1

1/cobs,i

1/cobs,i 2

1

MATLAB® function applied for confidence intervals of coefficients

coefCI

nlparci

nlparci

nlparci

nlparci

nlparci

Linear regression was conducted with MATLAB®’s fitlm function with all default options, to which coefCI was applied to determine coefficient confidence intervals. Non-linear regression was undertaken with MATLAB®’s nlinfit function also with default conditions. The selected error functions were realised via observation weighting factors. The covariance-based confidence interval determination was conducted with the nlparci function on the non-weighted residuals with a 95 % confidence interval. nlpredi was also applied to determine prediction intervals of the fit at a 95 % confidence interval in some cases. Without applied noise in the artificial

17

data sets, all regression algorithms returned the set coefficients with no error within six significant figures. d.

Prediction of the true equilibrium with the pseudo-second-order model The time progression of the isotherm data was further analysed to estimate the

true equilibrium conditions. The pseudo-second-order model [49], or PSOM, has been found on numerous occasions to describe the adsorption kinetics of arsenic on porous ferric hydroxides well [18,50]. As the PSOM is a reaction-based kinetic model, it does not have a relevant theoretical background for diffusion-limited adsorption kinetics and should therefore be considered as an empirical model in this system [10]. However, the model has been applied successfully in the past to predict the equilibrium and equilibrium time of adsorption systems based on preceding measurements [51,52]. The accuracy of the prediction is likely to decrease the further away from true equilibrium the data set is, as has been found previously [53]. The equilibrium loading, here specified as qeq,pred, is one of the fitted parameters of the PSOM, as shown in Equation 12, so it is not necessary to reapply the model to make a prediction. Non-linear SSR regression with nlinfit on qobs against time was applied to derive the equilibria predictions for each data point. Freundlich model coefficients were then determined using nlinfit, with a hybrid error function and qeq,pred as the response variable, rather than solute concentration. 𝑑𝑞𝑡 𝑑𝑡 = 𝑘2(𝑞𝑒𝑞,𝑝𝑟𝑒𝑑 ― 𝑞𝑡)2

Eq. 12

where qt = loading at time t, k2 = PSOM rate coefficient, and qeq,pred = predicted loading at equilibrium. e.

Assessment of regression algorithms The performance of the regression approaches was assessed with two different

purposes in mind: the ability of the regression to find the correct coefficients in

18

artificial data sets to then make accurate predictions of the loading (Equations 13 to 18), and the quality of the fit to the observed data in real data sets (Equation 19). –

The average value of Kf and nf across all reps data sets 1

𝑟𝑒𝑝𝑠

𝐾𝑓,𝑚𝑒𝑎𝑛 = 𝑟𝑒𝑝𝑠 . ∑𝑖 = 1(𝐾𝑓,𝑖) 1

Eq. 13

𝑟𝑒𝑝𝑠

𝑛𝑓,𝑚𝑒𝑎𝑛 = 𝑟𝑒𝑝𝑠 . ∑𝑖 = 1(𝑛𝑓,𝑖) –

–

The standard deviation in the values of Kf and nf across all reps data sets 𝑟𝑒𝑝𝑠

1

∑𝑖 = 1(𝐾𝑓,𝑖 ― 𝐾𝑓,𝑚𝑒𝑎𝑛)2

1

∑𝑖 = 1(𝑛𝑓,𝑖 ― 𝑛𝑓,𝑚𝑒𝑎𝑛)2

𝑆𝑡.𝐷𝑒𝑣(𝐾𝑓) =

𝑟𝑒𝑝𝑠.

𝑆𝑡.𝐷𝑒𝑣(𝑛𝑓) =

𝑟𝑒𝑝𝑠.

Eq. 15

𝑟𝑒𝑝𝑠

Eq. 16

The average sum of errors in the coefficients across all reps data sets: 𝑟𝑒𝑝𝑠

1

𝑒𝑟𝑟𝑜𝑟𝑐𝑜𝑒𝑓𝑓,𝑚𝑒𝑎𝑛 = 𝑟𝑒𝑝𝑠 .

–

Eq. 14

∑ (| 𝑖=1

|+ |

|)

𝐾𝑓,𝑖 ― 𝐾𝑓,𝑡𝑟𝑢𝑒

𝑛𝑓,𝑖 ― 𝑛𝑓,𝑡𝑟𝑢𝑒

𝐾𝑓,𝑡𝑟𝑢𝑒

𝑛𝑓,𝑡𝑟𝑢𝑒

Eq. 17

The mean root mean squared error in the predicted loadings relative to the set loadings: 1

𝑟𝑒𝑝𝑠

𝑅𝑀𝑆𝐸𝑞,𝑚𝑜𝑑,𝑚𝑒𝑎𝑛 = 𝑟𝑒𝑝𝑠 .

∑ [ (∑ 1

𝑛.

𝑖=1

–

𝑛

)]

[𝑞𝑖,𝑗,𝑒𝑞 ― 𝑞𝑖,𝑗,𝑚𝑜𝑑]2

𝑗=1

0.5

Eq. 18

The root mean squared error in the fitted concentration relative to the observed concentration as an indicator of the ability of the regression to fit the observed data points: 1

𝑅𝑀𝑆𝐸𝑐,𝑜𝑏𝑠,𝑚𝑒𝑎𝑛 = 𝑟𝑒𝑝𝑠 .

𝑟𝑒𝑝𝑠

∑ [ (∑ 1

𝑛.

𝑖=1

𝑛

)]

[𝑐𝑖,𝑗,𝑜𝑏𝑠 ― 𝑐𝑖,𝑗,𝑚𝑜𝑑]2

𝑗=1

0.5

Eq. 19

errorcoeff,mean is useful to assess how well the regression approach found the set coefficients over all artificial data sets, while RMSEq,mod,mean indicates how much error would result from applying the fitted coefficients to predict the loading at the set concentrations. As an indicator of the regression validity, the MATLAB® ttest was applied to the regression residuals to test for a mean of zero. Residuals were assessed visually in 19

selected data sets and found to resemble a normal distribution sufficiently enough to make the assumption of a normal distribution on which this test is based. The t-test statistics for all reps data sets were added together, as per Equation 20. ttestres,sum therefore describes how many regressions were statistically fallible. 𝑟𝑒𝑝𝑠

𝑡𝑡𝑒𝑠𝑡𝑟𝑒𝑠,𝑠𝑢𝑚 = ∑𝑖 = 1(𝑡–𝑡𝑒𝑠𝑡 𝑜𝑛 𝑑𝑎𝑡𝑎 𝑠𝑒𝑡 𝑖′𝑠 𝑟𝑒𝑔𝑟𝑒𝑠𝑠𝑖𝑜𝑛 𝑟𝑒𝑠𝑖𝑑𝑢𝑎𝑙𝑠)

Eq. 20

where test statistic “1” is a rejection of the hypothesis that the residuals of data set i have a mean of 0 at a confidence level of 95 %. f.

Summary of artificial isotherm data regression analysis The procedure for the theoretical comparison of the regression algorithms with

artificial data sets is summarized as follows: a) Set Kf and nf and derive an equidistantly spaced plot of qeq over ceq b) Derive the required values of cads for a set c0 with the mass balance c) Apply normally distributed pseudo-random noise for “observed” solute concentrations, cobs, as per Equation 11, with: i.

Scenario 1 – proportional error in ceq

ii.

Scenario 2 – proportional and absolute error in ceq

iii.

Scenario 3 – proportional and absolute error in ceq and in c0

d) Calculate the observed loading qobs with the new cobs and the mass balance e) Apply regression algorithms to determine Freundlich model coefficients and their confidence intervals f) Repeat d) – f) for reps unique data sets g) Apply the performance indicators in Equations 13-20 to the generated sets of equilibrium coefficients and residuals.

4.

Results and Discussion a. Analysis of artificial data sets

20

The three scenarios of noise in the artificial data sets are presented in Figure 3. Firstly, the variance in the data points along the work-lines derived with pseudorandom noise is visualised in Figure 3-A & -C. In the first scenario shown in Figure 3-A & -B, where only proportional errors were applied, the log-transformation achieved homoscedasticity in the observed solute concentrations. The loading isn’t improved to the same degree, with the variance in the y-axis still increasing with observation size. With additional absolute errors as in Figure 3-C & -D, homoscedasticity is not achieved in either variable with the transformation. While the small absolute errors are hardly noticeable in the native data set compared to the previous example, the impact they have on small observations in the linearly transformed data set is clear. Finally, additional error in the initial solute concentration shifts each data set vertically which, with all data points plotted together, greatly increases the scatter. This is shown in Figure 3-E & -F. b. Impact of regression approach on fitted coefficients for artificial data sets The results of the regression of the artificial data sets in all scenarios is summarised in Figure 4. In the first scenario given in Figure 4–A and –B (proportional errors in the equilibrium concentration only), all regression algorithms returned mean coefficients to within 1.1 % of the set values. The highest accuracy and precision, as well as the lowest mean error in the prediction of loading, was gained with the linear regression. The reduced heteroscedasticity in data sets with this type of noise is advantageous, even if not removed completely from the loading.

21

Linearly-Transformed Data Set

Native Data Set 1.8

(A)

(B)

1.7

1

1.6

0.8

Log10(Loading)

Calculated Dimensionless Loading

1.2

0.6

0.4

0

0.2

0.4

0.6

1.4 1.3

cobs =(1+α).ceq

1.2

Set true isotherm Fictitious data points

1.1

0.2

0

1.5

0.8

1

Set true isotherm Fictitious data points

1 -3.5

1.2

-3

1.8

(C)

-2

-1.5

-1

-0.5

0

0.5

(D)

1.7

1

1.6

0.8

Log10(Loading)

Calculated Dimensionless Loading

1.2

-2.5

Log10(Solute Concentration)

"Observed" Dimensionless Solute Concentration

0.6

0.4

0

0.2

0.4

1.4 1.3

cobs =(1+α).(ceq + β)

1.2

Set true isotherm Fictitious data points

1.1

0.2

0

1.5

0.6

0.8

1

Set true isotherm Fictitious data points

1 -3.5

1.2

-3

1.2

-2.5

-2

-1.5

-1

-0.5

0

0.5

Log10(Solute Concentration)

"Observed" Dimensionless Solute Concentration 1.8

(E)

(F)

1 1.6

0.8

Log10(Loading)

Calculated Dimensionless Loading

1.7

0.6

c0,obs =(1+α).c0

0.4

cobs =(1+α).(ceq + β)

1.5 1.4 1.3 1.2 1.1

0.2 1

Set true isotherm Fictitious data points 0

0

0.2

0.4

0.6

0.8

1

1.2

0.9 -3.5

Set true isotherm Fictitious data points -3

-2.5

-2

-1.5

-1

-0.5

0

0.5

Log10(Solute Concentration)

"Observed" Dimensionless Solute Concentration

Figure 3: All 100 randomly generated original (A, C & E) and linearly transformed (B, D & F) artificial

(1 nf).g( ― 1).

isotherm data sets (x), based on a “true” isotherm (o) of n = 20 points with Kf,true = 50 mg.L mg

( ―1 nf)

and nf,true = 5. Data points in A & B contain proportional errors in equilibrium solute

concentration, in C & D proportional and absolute errors in the equilibrium solute concentration, and in E & F proportional and absolute errors in the initial and equilibrium solute concentrations.

22

Across all algorithms, the fitted value of nf was subject to greater variation with standard deviations at up to 13.3 % than that of Kf with 2.3 %. This was true in particular for the two SSR functions with errors along the work-line, NLF-SSR-C and NLF-SSR-Qb. The latter resulted in the highest mean errors, likely as it considers absolute and unweighted residuals and thereby favours larger observation values. It was therefore not beneficial to determine the error along the work-line in this case, compared the vertical error as per NLF-SSR-Qa. A purely normally distributed proportional error is unlikely to be representative of the true nature of the error in real data sets, but provides a basis for comparison. With application of an additional absolute error to the observed equilibrium solute concentrations, Figure 4–C and –D, the regression returned more diverse results. As previously discussed, this can result in large proportional errors at small concentrations. The algorithms which weigh these errors more heavily, the linear regression (LF-SSR-LQ) and the MPSD error function on solute concentration (NLFMPSD-C), therefore yielded the greatest errors in the fitted coefficients. The remaining regression algorithms did not change as noticeably from the previous scenario. Despite again showing high variation in the fitted values of nf, the Hybrid function on observed concentration (NLF-HYB-C) gave the lowest mean errors in the coefficients and in the prediction of loading. NLF-SSR-C and NLF-SSR-Qb showed the greatest accuracy but the lowest precision in the mean fitted coefficient values, particularly regarding nf. The region of the data set more responsible for the shape of the isotherm, at low observation values, is less strongly weighted with these algorithms. This could lead to a stiffness in the error function to variation of nf and therefore, greater variability in its fitted value.

23

(A)

(B)

(C)

(D)

(E)

(F)

Figure 4: Comparison of the results of the regression algorithms across all reps = 100 artificial data

(1 nf).g( ― 1).mg( ―1 nf)

sets, each with n = 20 points with Kf,true = 50 mg.L

and nf,true = 5, with (A & B)

normally distributed proportional errors in equilibrium solute concentrations, (C & D) normally distributed proportional and absolute errors in equilibrium solute concentrations, and (E & F) normally distributed proportional and absolute errors in initial and equilibrium solute concentrations. (A, C & E) show the coefficients’ mean fitted values relative to the set one, with the standard deviation given as error bars. (B, D & F) show the total mean error as per Equation 17 and the mean RMSE in the prediction of loading as per Equation 18.

24

The inclusion of error in the initial solute concentration resulted in a mean observed value of 4.97 mg/L and standard deviation of 0.24 mg/L over 100 repetitions. Figure 4–E and –F show the impact this had on the regression. In this scenario, the means of Kf remained roughly the same, as would be expected with a mean c0,obs close to the true value. However, the standard deviations increased by roughly 200 % relative to the previous scenario. This resulted in all algorithms suffering an increase in the mean RMSE in the prediction of loading. The largest error was gained with the linear regression, the smallest once again with the hybrid function on observed concentration, NLF-HYB-C. The range of set adsorbent concentrations was such that the impact of error in the observed initial concentration was similar across all data points. This meant that the shape of the curve, and therefore the fitted values of nf, was not greatly impacted. Concerning the validity of the regressions, the MPSD function on work-line concentration (NLF-MPSD-C) struggled to return a viable fit. A quarter of the regressions with this error function failed the t-test in the presence of applied absolute errors, where large proportional errors can exist at low observation values. It is difficult to gain a well-balanced distribution in the residuals in this case, as these particular errors are one-sided. Therefore, this finding could be at least partially an artefact of the function used to apply noise to the data sets. In an additional analysis, the set value of nf was varied between 1 (linear) and 10 (highly convex) in order to test if the magnitude of the coefficient has an impact on the variance in fitted coefficients and on the regression algorithms’ performance. The entire process of artificial data set generation and regression was repeated for each value of nf, using the second artificial data scenario with both proportional and absolute errors in the observed equilibrium concentration. The value of Kf was kept

25

constant. Figure 5 shows how both the error in predicted loadings and the variation in Kf generally decreased across all algorithms with an increasing value of nf. (A)

(B)

(C)

(D)

Figure 5: Dependency of regression error and precision on the set value of nf, across reps = 100

(1 nf).g( ― 1).mg( ―1 nf)

artificial data sets, each with n = 20 data points based on Kf,true = 50 mg.L

and a normally distributed proportional and absolute error in observed equilibrium concentrations (i.e. scenario 2). A & B show the mean error in the coefficients and in the prediction of loading relative to set values, while C & D show the variation in the fitted coefficients themselves.

With decreasing linearity and thereby a flatter region of large observation values, the fitted values of Kf become more accurate. The linear regression and the non-linear regression on relative errors in concentration, NLF-MPSD-C, showed the greatest dependency on the value of nf, likely due their heightened sensitivity to the 26

shape determining region of the isotherm. They generally yielded the worst results across the tested range of nf. On the other hand, the Hybrid function on observed concentration, NLF-HYB-C, had the lowest mean error and the some of the lowest variance in fitted coefficients at all values of nf. An interesting observation was the presence of a minimum in the mean errors and in the variance of the fitted values of nf at set values of 2 – 4 for some of the algorithms. This was true particularly for the non-linear regression on vertical errors, NLF-SSR-Qa. With increasing steepness in the initial region of the curve, the overrepresented vertical residuals in loading from small deviations in concentration become more pronounced and grow in their impact on the regression. This rapidly outweighs the benefit of greater accuracy in the fitted value of Kf. Figure 6 shows the results of a similar analysis with variation of Kf.

Figure 6: Dependency of the accuracy of the regression on Kf, across reps = 100 artificial data sets, each with n = 20 data points based on nf,true = 5 and a normally distributed proportional and absolute error in observed solute concentrations. Standard deviations in the fitted coefficients (i.e. as shown in C & D in Figure 5) also did not change with variation in Kf and are not shown here.

27

(1 nf).g( ― 1).

Variation of the set value of this coefficient between 10 and 90 mg.L ―1 nf) ( mg

was found to have no impact on the variability and error in the fitted

coefficients. Only the mean RMSE in predicted loading increased, which is to be expected with increasing magnitude of the loadings. The extent of this impact was not equal for all algorithms, however. NLF-HYB-C again showed the best performance across the range of set values. Overall, nf would appear to be the more decisive coefficient when considering the regression algorithm. This refers not only to the choice of algorithm having a larger effect on the fitted value of nf than on Kf, but also to the larger impact that the magnitude of nf itself has on the performance of the regression. c. Impact of regression approach on fitted coefficients with real data sets Table 3, Table 4 and Figure 7 show the results of the regression algorithms on real isotherm data sets with the powder and granulate adsorbent after an equilibration time of 26 weeks. In this case, there is the possibility of errors in the observations, but also of the model being an imperfect representation of the data set. Note that the provided confidence intervals are those of the fitted coefficients relative to the data set. Therefore, they do not provide the same information regarding the precision of the result as the standard deviation in the theoretical analysis. The results of this analysis mirror those of the artificial data sets. Firstly, the greatest variation between regression approaches was seen with the fitted values of nf, with a range of 8 – 45% around the mean compared to 7 – 26 % for Kf in the granulate data set. The impact of regression choice was less pronounced in the powder data set, with variation from the mean of 8 – 20 % in nf and 1 – 7 % in Kf. The linear and MPSD algorithms again resulted in substantially lower fitted values 28

of nf and accordingly, higher values of Kf. Considering the fits in Figure 7, their insensitivity to the largest observation values is clear and they would appear to be a poor fit of the data as a whole. This supports the conclusion that these algorithms are overly controlled by residuals at low observation values. The greater variability in fitted coefficient values seen in the granulate data set is likely an artefact of the much larger number of data points at low observation values. This would have accentuated the impact of the over-sensitivity of the linear and MPSD algorithms. Table 3: Comparison of the results of different regression techniques on powder isotherm data at 26 weeks ± 3 hours Algorithm Kf a intervala

± 95 % confidence nf [-]

± 95 % confidence interval RMSE in vertical q

[mg/g]b

RMSE in work-line q

[mg/g]b

LFSSR-LQ

NLFSSR-Qa

NLFSSR-C

NLFHYB-C

NLFNLFMPSD-C SSR-Qb

60.09

59.16

58.27

58.73

63.69

58.28

3.56

2.30

0.86

1.51

5.54

0.75 d

8.13

8.47

10.77

10.03

7.552

10.71

1.11

1.11

2.26

1.64

1.233

2.59

2.06

1.97

3.35

2.84

3.42

3.30

0.85

0.61

0.44

0.48

2.01

0.44

0

0

0

0

1

0

T-Test on Residualsc

Table 4: Comparison of the results of different regression techniques on granulate isotherm data at 26 weeks ± 3 hours Algorithm

LFSSR-LQ

NLFSSR-Qa

NLFSSR-C

NLFHYB-C

NLFNLFMPSD-C SSR-Qb

Kf a

54.08

46.03

44.41

45.44

62.54

44.32

10.37

3.25

0.68

1.68

13.41

0.48 d

3.67

4.49

7.19

5.86

3.55

7.91

0.50

0.54

1.08

0.79

0.53

1.31

RMSE in vertical q [mg/g] b

4.61

2.83

6.85

4.67

7.71

8.02

RMSE in work-line q [mg/g] b

2.21

0.77

0.29

0.51

3.38

0.28

0

0

0

0

0

0

± 95 % confidence intervala nf [-] ± 95 % confidence interval

T-Test on Residuals

c

( nf).g( ― 1).mg( 1

―1

)

nf

a

with units [mg.L

b

]

both based on Equation 18, but with different derivation of the model loading, qmod or qwl

c

Result of t-test on residuals of regression, 0 = null hypothesis that {mean or residuals = 0} cannot be rejected at a 95 % confidence level.

d

Bold = smallest confidence interval (relative to fitted to value) or lowest error

29

70

60

60

Calculated Loading [mg/g]

Calculated Loading [mg/g]

70

50 40 30

Data of Powder at 26 Weeks LF-SSR-LQ NLF-SSR-Qa NLF-SSR-C NLF-HYB-C NLF-MPSD-C NLF-SSR-Qb

20 10 0

0

0.5

1

1.5

50 40 30

Data of Granulate at 26 Weeks LF-SSR-LQ NLF-SSR-Qa NLF-SSR-C NLF-HYB-C NLF-MPSD-C NLF-SSR-Qb

20 10

2

0

0

0.5

1

1.5

Observed Solute Concentration [mg/L]

Observed Solute Concentration [mg/L]

Figure 7: Comparison of the fitted isotherms with different regression techniques on the powder (left) and granulate (right) isotherm data at 26 weeks ± 3 hours

The smallest uncertainty in Kf and the largest uncertainty in nf in both data sets were gained with the SSR functions on work-line residuals, NLF-SSR-C and NLF-SSRQb. Both are less sensitive to the shape-determining low observation values. The difference between taking the vertical residual or the work-line residual in loading, NLF-SSR-Qa and NLF-SSR-Qb respectively, is also highlighted. The regression using vertical errors weighs the steep region of the curve more strongly, due to its overrepresentation of errors in this region. This results in a lower fitted value of nf, seemingly as the fit is pulled away from the y-axis. The Hybrid function on observed concentration appears again to be a good compromise in the weighting of the observations. That the findings of this analysis mirror those of the artificial data sets allows two conclusions to be drawn. Firstly, the nature of the error inherent in the real isotherm data sets was not entirely misrepresented in the artificial data sets. Secondly, the real isotherm data set is sufficiently accurately represented by the Freundlich model that the choice of regression algorithm controls the goodness-ofthe-fit, not the suitability of the model. 30

To summarise, the choice of regression approach and error function clearly have an impact in the determination of isotherm model coefficients, largely due to their particular error sensitivities. This impact was predominately seen in the fitted value of nf. The linear regression was found to give a good result in close-to-ideal data sets, but struggled otherwise. This approach and the non-linear MPSD function on solute concentration showed increased sensitivity to errors at low observation values, such that these controlled the fit and resulted in lower values of nf. That the MPSD function resulted in worse fits than the linear regression implies that the choice of error function, or observation weighting, is just as important as which errors are minimised or the theoretical validity of the regression technique. SSR functions on work-line residuals returned some of the smallest mean errors in the fitted coefficients and predicted loadings in artificial data sets. However, they were found to be insensitive to errors at low concentrations and experienced larger uncertainty in the value of nf. When low observation values are subject to a higher relative error, this is advantageous. However, if there is a legitimate mechanism behind the deviation of small observations from the model, i.e. model error rather than observational error, these models will not detect it. In comparison, the SSR on vertical error in loading was strongly impacted by errors in the steep region curve, resulting in lower values of nf. The Hybrid function on solute concentration appears to be a good compromise in the regression of the isotherm data considered in this work, while also being statistically valid. It does not weigh either region of the isotherm too heavily. It is thereby capable of picking up legitimate mechanisms not foreseen by the model without being impacted excessively by errors at low observation values.

31

d. Influence of equilibration time on fitted coefficients The progression of the isotherm over 26 weeks with the powder size fraction of the adsorbent is shown in Table 5 and Figure 8. As described later, the data set at 26 weeks was found to be an accurate representation of the equilibrium and is therefore taken as the basis for comparison. Table 5: Regression with the hybrid error function on solute concentration (NLF-HYB-C) in the powder data set over time and on the PSOM predictions of the true equilibria 1 Week

4 Weeks

50.13

54.12

55.78

58.73

1.94

1.44

2.05

1.51

% of equilibrium Kf at 26 weeks

85 %

92 %

95 %

100 %

% of granulate equilibrium Kf

104 %

113 %

116 %

122 %

10.89

10.39

10.33

10.03

3.20

1.90

2.59

1.64

RMSE in vertical q [mg/g]

2.97

2.64

3.87

2.84

RMSE in work-line q [mg/g]

1.24

0.76

0.75

0.48

0

0

0

0

Kf

a

± 95 % confidence interval a b

nf [-] ± 95 % confidence interval

T-Test on

Residuals c

( nf).g( ― 1).mg( 1

―1

8 Weeks

26 Weeks

)

nf

a

with units [mg.L

]

b

as predicted by extrapolation of the granulate’s data set with the PSOM (described later)

c

result of t-test on residuals of regression, 0 = null hypothesis that {mean or residuals = 0} cannot be rejected at a 95 % confidence level, such that regression’s validity is not rejected.

The value of Kf after 1 week was found to be 15 % lower than at equilibrium. It follows then, that 1 week would be an insufficient time to derive accurate estimates for this particular system. At 8 weeks the capacity is shown to reach 95 %, which given the uncertainty could be considered equivalent. On the other hand, the shape of the curve as given by nf did not appear to be greatly impacted by insufficient equilibration time. Therefore, if uncertainty existed as to whether the data gained in an adsorption trial was representative of the adsorbents’ true equilibrium, it could be assumed at least that the value of nf would not develop further.

32

70

Data 1 Week Data 4 Weeks Data 8 Weeks Data 26 Weeks Fit 26 Weeks Prediction Interval of Fit at 26 Weeks

Calculated Loading [mg/g]

60 50 40 30 20 10 0

0

0.5

1

1.5

2

Observed Solute Concentration [mg/L] Figure 8: Progression of the powder’s isotherm over time, the non-linear fit with the hybrid function on observed solute concentration (NLF-HYB-C) at 26 weeks, and the prediction interval at a 95 % confidence level for the fit

The slow kinetics of these equilibrium adsorption trials are unlikely to be improved through mixing conditions, as intraparticle diffusion should rapidly become limiting [10]. One approach to reducing the required equilibration time would be to further decrease the tested particle size. Otherwise, in the absence of additional data proving that the system is at equilibrium after short reaction times, the sensitivity of any applied dimensioning and simulation tools to variation in Kf should be tested. In a test of the ability of the PSOM to predict the equilibrium from preceding measurements, the model was applied to the data of the powder up to 8 weeks. The predicted loadings qeq.pred and the resulting equilibrium coefficients are compared to those of the data at 26 weeks in Table 6. The PSOM was a good fit of the data, with a RMSE relative to observed loadings, averaged over all bottle points, of 0.19 mg/g. Table 6: Freundlich model coefficients from regression of the equilibria predicted by the PSOM applied to partial and complete data sets

33

Prediction 1

Prediction 2 – all data

Using data up to week number..

8

26

Kf a fitted to PSOM predictions

55.75

57.09

± 95 % confidence interval

0.85

0.77

100%

102%

95%

97%

12.24

11.34

3.80

2.90

122%

113%

Mean RMSE in PSOM [mg/g] b

0.19

0.68

Mean (qobs – qeq,pred) [mg/g] c

0.74

0.34

0.93

0.43

a

% of Kf of data at 8 weeks % of Kf of data at 26 weeks (equilibrium) nf [-] ± 95 % confidence interval % of nf of data at 26 weeks (equilibrium)

St.Dev. (qobs – qeq,pred) [mg/g]

( nf).g( ― 1).mg( 1

―1

c

)

nf

a

with units [mg.L

b

]

average error in the non-linear regressions of qobs against time with the PSOM (quality of PSOM regression)

c

in the comparison of data points at 26 weeks and the PSOM predicted equilibria (quality of prediction)

The analysis resulted in an underestimation of the equilibrium value of Kf by 5 %. In fact, the prediction at this point in time implied that the adsorption was already complete. Late stage adsorption was therefore misrepresented. The discrepancy might be due to the changing adsorbent concentration over time with the removal of a fluid sample at each point in time. It increased to 110 % of the starting value in the final adsorption phase, with an average of 105 % over the whole 26 weeks. So, a lower change in loading occurred than would have been the case with constant adsorbent concentration, giving a smaller final equilibrium loading. It could also simply be caused by compensation for the different fitted values of nf, which were overestimated by the prediction. Considering the existing adsorption kinetic up to 26 weeks and that both PSOM predictions were lower than the final observation, the data set at 26 weeks is assumed to be representative of the final equilibrium. The progression over 26 weeks of the isotherm of the coarse adsorbent size fraction is shown in Table 7 and Figure 9.

34

Table 7: Regression with the hybrid error function on solute concentration (NLF-HYB-C) in the granulate data set over time and on the PSOM predictions of the true equilibria 1 Week

4 Weeks

9.13

21.24

31.62

45.44

48.07

0.67

1.21

1.21

1.68

0.54

% equilibrium prediction Kf

19 %

44 %

66 %

95 %

100 %

% powder equilibrium Kf b

16 %

36 %

54 %

77 %

82 %

nf [-]

3.72

3.45

4.00

5.86

8.18

0.87

0.50

0.43

0.79

1.38

RMSE in vertical q [mg/g]

2.34

1.59

1.52

4.67

-

RMSE in work-line q [mg/g]

2.09

0.85

0.59

0.51

-

0

0

0

0

-

Kf 1 ± 95 % confidence

interval a

± 95 % confidence interval

T-Test on Residuals c

( nf).g( ― 1).mg( 1

―1

8 Weeks 26 Weeks Prediction

)

nf

a

with units [mg.L

b

]

equilibrium of powder adsorbent at 26 weeks

c

result of t-test on residuals of regression, 0 = null hypothesis that {mean or residuals = 0} cannot be

rejected at a 95 % confidence level, such that regression’s validity is not rejected.

70

Granulate Data 1 Week Granulate Data 4 Weeks

60

Granulate Data 8 Weeks Granulate Data 26 Weeks

Loading [mg/g]

50

PSOM Predictions of Granulate Equilibria

40

Confidence Intervals of PSOM Predictions

30

Fit of PSOM-Predicted Equilibria Fines Data 26 Weeks Fit of Fines Data

20

Prediction Intervals of Isotherm Fits

10

0

0

0.5

1

1.5

2

2.5

Solute Concentration [mg/L]

Figure 9: Progression of the granulate’s isotherm with time, PSOM predictions of the bottlepoint equilibria and the resulting isotherm estimate, compared to the powder equilibrium and its prediction interval

The system did not appear to achieve equilibrium within the tested time period. Lacking this data, the PSOM was applied over all time points to make a prediction of 35

the true equilibria, from which an isotherm was derived. The results are also shown in Figure 9. The limited number of data points for the PSOM regression resulted in large confidence intervals in the predicted equilibrium loadings, shown as bands in the same planes as the work lines. These large uncertainties are not considered in the thin prediction interval of the Freundlich model fit. The prediction implies that the final data set at 26 weeks was roughly 5 % removed from the true equilibrium in regards to Kf, just outside this data point’s confidence interval. After 1 week, the granulate had achieved less than 20 % of the final equilibrium, and at 8 weeks only 66 %. Furthermore, unlike the powder isotherm, the value of nf is seen to change throughout the trial. While the increased distance from equilibrium likely has strong bearing on this, even its value at 26 weeks is removed from that of the predicted final equilibrium. Thus, without a very long equilibration time and repeated measurements to ensure that the system is at equilibrium, it should be assumed that both coefficients are inaccurate, not just Kf. e. Application of small size fractions in accelerated adsorption trials Comparing the two size fractions of adsorbent at 26 weeks, the value of Kf of the granulate 22 % smaller than that of the powder. The shape of the isotherm was also dissimilar, with a 40 % lower value of nf. While the choice in regression algorithm has a substantial impact on this coefficient, none of them resulted in a convergence of value of nf for the two size fractions, as was shown earlier in Table 3 and Table 4. This result could simply be an artefact of the large number of data points at very low concentrations in the granulate’s data set, rather than any difference in the energetic heterogeneity of the adsorption surface. Taking the PSOM-predicted final equilibrium of the granulate instead, the difference in Kf is reduced slightly to 18 %, while that in nf is reduced considerably

36

to 22 %. It is possible that the equilibrium is underestimated with this prediction methodology, as was seen previously with the powder. Particularly if the late phase adsorption is misrepresented, this could have had a greater impact on the larger particle size fraction with its overall slower kinetic. However, as the granulate was apparently as close to its equilibrium at 26 weeks as the powder was at 8 weeks, it is unlikely that the PSOM-prediction is solely at fault for the disparity between the size fractions. Other mechanisms might also have contributed. These include experimental artefacts such as the impacts of long equilibration times being stronger with slower adsorption speed, through to genuine mechanisms which would, even with a perfect experimental design, result in reduced equilibrium adsorption capacity with increasing particle radius. Yet it was not disproven that the two size fractions have the same adsorption equilibrium. The final equilibrium of the coarse adsorbent wasn’t reached within 26 weeks, and most of the confidence intervals of the PSOM-predicted equilibria overlapped with the isotherm of the powder. However, it is clear that application of either size fraction to determine the equilibrium of the coarse media would have its own particular challenges. In process dimensioning calculations based on the equilibrium alone, coefficients that reflect any capacity-reducing mechanisms of the applied particle size would be appropriate. So, application of the coarse size fraction should result in the best estimate of its own equilibrium. However, the 26 week minimum and roughly 40 week optimum (based on extrapolation of the PSOM) in the tested conditions is impractical and likely prone to error. It is also not possible to then differentiate this error from the legitimate mechanisms. Furthermore, without existing knowledge of the system, it would not be sure that applying such large equilibration times are sufficient. Therefore, repeated measurements over even

37

longer periods would be necessary. Application of this size fraction otherwise risks vastly underestimating the adsorption equilibrium, as seen here in both Kf and nf with the data prior to 26 weeks. The question then arises, why the practitioner shouldn’t simply test the full size adsorbent in a dynamic application trial from the start, rather than conduct a static isotherm experiment over the same long period of time. Application of a smaller particle size fraction allows a drastically reduced experimental time while delivering a decent approximation of the equilibrium. An over-estimation of both coefficients could result, in this case by roughly 23 % relative to the prediction of the coarse size fraction’s equilibrium. In fact, the closest estimation of the equilibrium of the coarse adsorbent was gained after 1 week when applying the powder. In this case, only the value of nf was misrepresented, in a range of values that has little impact on the curvature of the isotherm. That is, the difference in the curvature between nf = 2 and 4 is much larger than between 8 and 10. This good comparability of the powder data at 1 week won’t necessarily be conserved in other experimental conditions. For process simulation, equilibrium coefficients have an additional purpose. In most dynamic models for this system, the adsorption is described on the basis of equilibrium and diffusion coefficients. Taking for example film diffusion, the driving force is the difference between the bulk solute concentration and the concentration at the outer surface of the particle, which is considered to be in equilibrium with the loading at the outer surface [10]. This equilibrium would not be subject to any capacity-reducing mechanisms within the particle. Assuming that the fine powder is a better representation of the pure material surface, with fewer such mechanisms at play, its equilibrium would be a more appropriate basis for modelling the mass transport. However, in the simulation of a fixed-bed reactor with the powder’s

38

equilibrium coefficients, the loading achieved in larger adsorbent particles behind the mass transport zone would be overestimated, leading to earlier breakthrough than predicted. 5.

Conclusion The choice of regression algorithm, insufficient equilibration time, and particle

size selection were all found to impact Freundlich model coefficients in the determination of the adsorption equilibrium of arsenic on GFH. The choice of regression approach, namely which errors are minimised and with which error function, was found to have its greatest impact on the value of nf, rather than Kf. By avoiding over-weighting of any one region of the isotherm and inherently considering the mass balance of the trial points, the hybrid error function applied to the observed solute concentration was found to be ideal for the tested data sets. Insufficient equilibration time was found to impact only the value of Kf when testing small size fractions of the adsorbent. This coefficient reached 85 % of its final equilibrium value after 1 week, 92 % after 4 weeks and 95 % after 8 weeks. The value of nf remained stable. On the other hand, both coefficients were inaccurately represented when applying the coarse granulate with insufficient time, likely as it was further away from equilibrium than the fine fraction was. Impractically long equilibration times were required to gain an accurate estimate when applying this large particle size fraction. The results suggest that applying a small particle size fraction in an accelerated adsorption trial to predict the equilibrium of the coarse media would result in overestimation of the coefficients by up to 23 %. Taking instead the pseudo-equilibrium of the powder after 1 week contact time was found to relatively accurately predict

39

the equilibrium of the granular media in the tested conditions. Given that the application of both fine and coarse size fractions to determine the adsorption equilibrium of GFH is subject to error, the greatly shortened equilibration time and likely more accurate representation of mass transport makes the smaller size fraction a preferable option. 6.

Acknowledgements This work was conducted within the cooperative project MikroAd,

KF2444907SA4, funded through the Zentrale Innovationsprogramm Mittelstand (ZIM) of the Federal Ministry for Economic Affairs and Energy, with the support of Aqua Technologie Nörpel, Hydroisotop GmbH, the Water Systems Engineering research group at the Technical University of Munich and the Institute of Food Technology at the Weihenstephan-Triesdorf University of Applied Science. The authors would also like to thank the work group of Dr. Uwe Blum at the Soil and Climate department of the Bavarian State Research Centre for Forest and Forestry (LWF) for their analytical support. Finally, the constructive input of the reviewers was also greatly appreciated. 7.

Competing Interests

The authors declare no competing interests for this work. 8.

References

[1]

W. Driehaus, M. Jekel, U. Hildebrandt, Granular ferric hydroxide—a new adsorbent for the removal of arsenic from natural water, Journal of Water Supply: Research and Technology—AQUA 47 (1998) 30–35. https://doi.org/10.2166/aqua.1998.0005.

[2]

G. Amy, H.-w. Chen, A. Drizo, U.v. Gunten, P. Brandhuber, R. Hund, Z. Chowdhury, S. Kommeni, S. Sinha, M. Jekel, K. Banerjee, Adsorbent treatment technologies for arsenic removal, Denver, CO, 2005. ISBN 1-58321-399-6.

[3]

M. Kanematsu, T.M. Young, K. Fukushi, P.G. Green, J.L. Darby, Individual and combined effects of water quality and empty bed contact time on As(V)

40

removal by a fixed-bed iron oxide adsorber: implication for silicate precoating, Water research 46 (2012) 5061–5070. https://doi.org/10.1016/j.watres.2012.06.047. [4]

M. Kanematsu, T.M. Young, K. Fukushi, P.G. Green, J.L. Darby, Arsenic(III, V) adsorption on a goethite-based adsorbent in the presence of major co-existing ions: Modeling competitive adsorption consistent with spectroscopic and molecular evidence, Geochimica et Cosmochimica Acta 106 (2013) 404–428. https://doi.org/10.1016/j.gca.2012.09.055.

[5]

M. Kanematsu, T.M. Young, K. Fukushi, D.A. Sverjensky, P.G. Green, J.L. Darby, Quantification of the effects of organic and carbonate buffers on arsenate and phosphate adsorption on a goethite-based granular porous adsorbent, Environmental science & technology 45 (2011) 561–568. https://doi.org/10.1021/es1026745.

[6]

M. Stachowicz, T. Hiemstra, W.H. van Riemsdijk, Multi-competitive interaction of As(III) and As(V) oxyanions with Ca(2+), Mg(2+), PO(3-)(4), and CO(2-)(3) ions on goethite, Journal of colloid and interface science 320 (2008) 400–414. https://doi.org/10.1016/j.jcis.2008.01.007.

[7]

L. Wang, D.E. Giammar, Effects of pH, dissolved oxygen, and aqueous ferrous iron on the adsorption of arsenic to lepidocrocite, Journal of colloid and interface science 448 (2015) 331–338. https://doi.org/10.1016/j.jcis.2015.02.047.

[8]

A. Sperlich, Phosphate adsorption onto granular ferric hydroxide (GFH) for wastewater reuse, Doctoral Thesis, 2010. ISBN 978-3-89720-560-4.

[9]

M. Kersten, S. Karabacheva, N. Vlasova, R. Branscheid, K. Schurk, H. Stanjek, Surface complexation modeling of arsenate adsorption by akagenéite (βFeOOH)-dominant granular ferric hydroxide, Colloids and Surfaces A: Physicochemical and Engineering Aspects 448 (2014) 73–80. https://doi.org/10.1016/j.colsurfa.2014.02.008.

[10] E. Worch, Adsorption Technology in Water Treatment: Fundamentals, Processes, and Modeling, Walter de Gruyter GmbH & Co. KG, Berlin/Boston, 2012. ISBN 978-3-11-024022-1. [11] M. Badruzzaman, P. Westerhoff, D.R.U. Knappe, Intraparticle diffusion and adsorption of arsenate onto granular ferric hydroxide (GFH), Water research 38 (2004) 4002–4012. https://doi.org/10.1016/j.watres.2004.07.007. [12] M. Usman, I. Katsoyiannis, M. Mitrakas, A. Zouboulis, M. Ernst, Performance Evaluation of Small Sized Powdered Ferric Hydroxide as Arsenic Adsorbent, Water 10 (2018) 957. https://doi.org/10.3390/w10070957. [13] A. Sperlich, S. Schimmelpfennig, B. Baumgarten, A. Genz, G. Amy, E. Worch, M. Jekel, Predicting anion breakthrough in granular ferric hydroxide (GFH) adsorption filters, Water research 42 (2008) 2073–2082. https://doi.org/10.1016/j.watres.2007.12.019. 41

[14] O.S. Thirunavukkarasu, T. Viraraghavan, K.S. Subramanian, Arsenic removal from drinking water using granular ferric hydroxide, WSA 29 (2003). https://doi.org/10.4314/wsa.v29i2.4851. [15] L. Boels, K.J. Keesman, G.-J. Witkamp, Adsorption of phosphonate antiscalant from reverse osmosis membrane concentrate onto granular ferric hydroxide, Environmental science & technology 46 (2012) 9638–9645. https://doi.org/10.1021/es302186k. [16] P. Suresh Kumar, L. Korving, K.J. Keesman, M.C.M. van Loosdrecht, G.-J. Witkamp, Effect of pore size distribution and particle size of porous metal oxides on phosphate adsorption capacity and kinetics, Chemical Engineering Journal 358 (2019) 160–169. https://doi.org/10.1016/j.cej.2018.09.202. [17] M. Kosmulski, Chemical properties of material surfaces, Surfactant science series 102, Marcel Dekker, New York, 2001. ISBN 0-8247-0560-2. [18] B. Xie, M. Fan, K. Banerjee, J.H. van Leeuwen, Modeling of arsenic(V) adsorption onto granular ferric hydroxide, Journal - American Water Works Association 99 (2007) 92–102. https://doi.org/10.1002/j.15518833.2007.tb08083.x. [19] C.H. Bolster, G.M. Hornberger, On the Use of Linearized Langmuir Equations, Soil Science Society of America Journal 72 (2008) 1848. https://doi.org/10.2136/sssaj2006.0304. [20] C.P. Schulthess, D.K. Dey, Estimation of langmuir constants using linear and nonlinear least squares regression analyses, Soil Science Society of America Journal 60 (1996) 433. https://doi.org/10.2136/sssaj1996.03615995006000020014x. [21] M.K. Bothwell, L.P. Walker, Evaluation of parameter estimation methods for estimating cellulase binding constants, Bioresource Technology 53 (1995) 21–29. https://doi.org/10.1016/0960-8524(95)00046-H. [22] M.I. El-Khaiary, Least-squares regression of adsorption equilibrium data: comparing the options, Journal of hazardous materials 158 (2008) 73–87. https://doi.org/10.1016/j.jhazmat.2008.01.052. [23] G.-N. Moroi, E. Avram, L. Bulgariu, Adsorption of Heavy Metal Ions onto Surface-Functionalised Polymer Beads. I. Modelling of Equilibrium Isotherms by Using Non-Linear and Linear Regression Analysis, Water Air Soil Pollut 227 (2016) 737. https://doi.org/10.1007/s11270-016-2953-5. [24] D.D. Marković, B.M. Lekić, V.N. Rajaković-Ognjanović, A.E. Onjia, L.V. Rajaković, A new approach in regression analysis for modeling adsorption isotherms, TheScientificWorldJournal 2014 (2014) 930879. https://doi.org/10.1155/2014/930879. [25] J.F. Porter, G. McKay, K.H. Choy, The prediction of sorption from a binary mixture of acidic dyes using single- and mixed-isotherm variants of the ideal

42

adsorbed solute theory, Chemical Engineering Science 54 (1999) 5863–5885. https://doi.org/10.1016/S0009-2509(99)00178-5. [26] E.B. Simsek, U. Beker, Equilibrium arsenic adsorption onto metallic oxides Isotherm models, error analysis and removal mechanism, Korean J. Chem. Eng. 31 (2014) 2057–2069. https://doi.org/10.1007/s11814-014-0176-2. [27] A. Terdputtakun, O.-a. Arqueropanyo, P. Sooksamiti, S. Janhom, W. Naksata, Adsorption isotherm models and error analysis for single and binary adsorption of Cd(II) and Zn(II) using leonardite as adsorbent, Environ Earth Sci 76 (2017) 83. https://doi.org/10.1007/s12665-017-7110-y. [28] M.I. El-Khaiary, G.F. Malash, Common data analysis errors in batch adsorption studies, Hydrometallurgy 105 (2011) 314–320. https://doi.org/10.1016/j.hydromet.2010.11.005. [29] F. Harrison, S.K. Katti, Hazards of linearization of Langmuir's model, Chemometrics and Intelligent Laboratory Systems 9 (1990) 249–255. https://doi.org/10.1016/0169-7439(90)80075-H. [30] T.A. Osmari, R. Gallon, M. Schwaab, E. Barbosa-Coutinho, J.B. Severo, J.C. Pinto, Statistical Analysis of Linear and Non-Linear Regression for the Estimation of Adsorption Isotherm Parameters, Adsorption Science & Technology 31 (2013) 433–458. https://doi.org/10.1260/0263-6174.31.5.433. [31] DIN EN ISO 17294-2:2017-01, Wasserbeschaffenheit_- Anwendung der induktiv gekoppelten Plasma-Massenspektrometrie (ICP-MS)_- Teil_2: Bestimmung von ausgewählten Elementen einschließlich Uran-Isotope (ISO_17294-2:2016); Deutsche Fassung EN_ISO_17294-2:2016, Beuth Verlag GmbH, Berlin. https://doi.org/10.31030/2353617. [32] U.S. Environmental Protection Agency, Determination of Trace Elements in Waters and Wastes by Inductively Coupled Plasma – Mass Spectrometry, 5.4th ed., 1994. [33] I.G. Tanase, D.E. Popa, G.E. Udriştioiu, A.A. Bunaciu, H.Y. Aboul-Enein, Estimation of the uncertainty of the measurement results of some trace levels elements in document paper samples using ICP-MS, RSC Adv. 5 (2015) 11445– 11457. https://doi.org/10.1039/C4RA12645A. [34] N. Ayawei, A.N. Ebelegi, D. Wankasi, Modelling and Interpretation of Adsorption Isotherms, Journal of Chemistry 2017 (2017) 1–11. https://doi.org/10.1155/2017/3039817. [35] K.Y. Foo, B.H. Hameed, Insights into the modeling of adsorption isotherm systems, Chemical Engineering Journal 156 (2010) 2–10. https://doi.org/10.1016/j.cej.2009.09.013. [36] J. Ćurko, M. Matošić, V. Crnek, V. Stulić, I. Mijatović, Adsorption Characteristics of Different Adsorbents and Iron(III) Salt for Removing As(V) from Water,

43

Food technology and biotechnology https://doi.org/10.17113/ftb.54.02.16.4064.

54

(2016)

250–255.

[37] S. Sinha, G. Amy, Y.-M. Yoon, N.-G. Her, Arsenic Removal from Water Using Various Adsorbents: Magnetic Ion Exchange Resins, Hydrous Ion Oxide Particles, Granular Ferric Hydroxide, Activated Alumina, Sulfur Modified Iron, and Iron Oxide-Coated Microsand, Environmental Engineering Research 16 (2011) 165–173. https://doi.org/10.4491/eer.2011.16.3.165. [38] N.R. Draper, H. Smith, Applied regression analysis, Third edition ed., Wiley series in probability and statistics Texts and references section, Wiley, New York, Chichester, Weinheim, Brisbane, Singapore, Toronto, 1998. ISBN 9780471170822. [39] G.A.F. Seber, C.J. Wild, Nonlinear regression, Wiley-Interscience paperback series, Wiley, New York, 2004. ISBN 0-471-47135-6. [40] F. Yan, Y. Chu, K. Zhang, F. Zhang, N. Bhandari, G. Ruan, Z. Dai, Y. Liu, Z. Zhang, A.T. Kan, M.B. Tomson, Determination of adsorption isotherm parameters with correlated errors by measurement error models, Chemical Engineering Journal 281 (2015) 921–930. https://doi.org/10.1016/j.cej.2015.07.021. [41] G. Valsami, A. Iliadis, P. Macheras, Non-linear regression analysis with errors in both variables: estimation of co-operative binding parameters, Biopharmaceutics & drug disposition 21 (2000) 7–14. https://doi.org/10.1002/1099-081X(200001)21:1<7:AIDBDD210>3.0.CO;2-F. [42] C.H. Bolster, Revisiting a statistical shortcoming when fitting the Langmuir model to sorption data, Journal of environmental quality 37 (2008) 1986– 1992. https://doi.org/10.2134/jeq2007.0461. [43] N. Barrow, The Description of Phosphate Adsorption Curves, Journal of Soil Science 29 (1978) 447–462. https://doi.org/10.1111/j.13652389.1978.tb00794.x. [44] A. Sayago, M. Boccio, A. Asuero, Fitting Straight Lines with Replicated Observations by Linear Regression: The Least Squares Postulates, Critical Reviews in Analytical Chemistry 34 (2004) 39–50. https://doi.org/10.1080/10408340490273744. [45] K. Walsh, S. Mayer, Adsorption Equilibrium Study of Arsenic(V) on Granular Ferric Hydroxide: Experimental Methods, Particle Characterisation & Equilibrium Data, Technical University of Munich, 2019. https://doi.org/10.14459/2017mp1515279. [46] GEH Wasserchemie GmbH & Co. KG, Product specification: GEH 102 - Granular Ferric Hydroxide for Drinking Water Treatment, 2018, https://www.gehwasserchemie.de/en/applications/drinking-water/.

44

[47] M. Thommes, K. Kaneko, A. v. Neimark, J.P. Olivier, F. Rodriguez-Reinoso, J. Rouquerol, K.S.W. Sing, Physisorption of gases, with special reference to the evaluation of surface area and pore size distribution (IUPAC Technical Report), Pure and Applied Chemistry 87 (2015) 160. https://doi.org/10.1515/pac-2014-1117. [48] I. Hilbrandt, A. Ruhl, M. Jekel, Conditioning Fixed-Bed Filters with Fine Fractions of Granulated Iron Hydroxide (µGFH), Water 10 (2018) 1324. https://doi.org/10.3390/w10101324. [49] Y.S. Ho, G. McKay, Pseudo-second order model for sorption processes, Process Biochemistry 34 (1999) 451–465. https://doi.org/10.1016/S00329592(98)00112-5. [50] T. Mahmood, S.U. Din, A. Naeem, S. Tasleem, A. Alum, S. Mustafa, Kinetics, equilibrium and thermodynamics studies of arsenate adsorption from aqueous solutions onto iron hydroxide, Journal of Industrial and Engineering Chemistry 20 (2014) 3234–3242. https://doi.org/10.1016/j.jiec.2013.12.004. [51] T. Liu, M. Yang, T. Wang, Q. Yuan, Prediction Strategy of Adsorption Equilibrium Time Based on Equilibrium and Kinetic Results To Isolate Taxifolin, Ind. Eng. Chem. Res. 51 (2012) 454–463. https://doi.org/10.1021/ie201197r. [52] Y.-S. Ho, Pseudo-Isotherms Using a Second Order Kinetic Expression Constant, Adsorption 10 (2004) 151–158. https://doi.org/10.1023/B:ADSO.0000039870.28835.09. [53] W. Plazinski, J. Dziuba, W. Rudzinski, Modeling of sorption kinetics: the pseudo-second order equation and the sorbate intraparticle diffusivity, Adsorption 19 (2013) 1055–1064. https://doi.org/10.1007/s10450-0139529-0.

45

Highlights for the manuscript titled “Equilibrium Data and its Analysis with the Freundlich Model in the Adsorption of Arsenic(V) on Granular Ferric Hydroxide” The following highlights are submitted for this manuscript: 

Common existing regression approaches are statistically unsound



The regression can be conducted using residuals which uphold the mass balance



Regression approach impacts mainly the apparent energetic heterogeneity, nf



Equilibration time and applied particle size fraction impact both coefficients



Fine particle size fractions give a reasonable estimate of the equilibrium

Manuscript: “Equilibrium Data and its Analysis with the Freundlich Model in the Adsorption of Arsenic(V) on Granular Ferric Hydroxide” Authors: Kenneth Walsh, Sebastian Mayer, Dirk Rehmann, Thomas Hofmann, Karl Glas Affiliations: Technical University of Munich, Weihenstephan-Triesdorf University of Applied Sciences

Author Statement Authors contributed with the following actions:

Kenneth Walsh: Conceptualisation, methodology, software, validation, formal analysis, investigation, data curation, visualization, project administration, writing – original draft, writing – review & editing.

Sebastian Mayer: Conceptualization, methodology, formal analysis, investigation, project administration, validation, visualization, writing – review & editing.

Dirk Rehmann: Conceptualization, funding acquisition, project administration, resources, supervision, writing – review & editing.

Thomas Hofmann: Funding acquisition, resources, supervision.

Karl Glas: Conceptualization, methodology, funding acquisition, project administration, resources, supervision, visualization, writing – review & editing.

Term Conceptualization

Definition Ideas; formulation or evolution of overarching research goals and aims Methodology Development or design of methodology; creation of models Software Programming, software development; designing computer programs; implementation of the computer code and supporting algorithms; testing of existing code components Validation Verification, whether as a part of the activity or separate, of the overall replication/ reproducibility of results/experiments and other research outputs Formal analysis Application of statistical, mathematical, computational, or other formal techniques to analyze or synthesize study data Investigation Conducting a research and investigation process, specifically performing the experiments, or data/evidence collection Resources Provision of study materials, reagents, materials, patients, laboratory samples, animals, instrumentation, computing resources, or other analysis tools Data Curation Management activities to annotate (produce metadata), scrub data and maintain research data (including software code, where it is necessary for interpreting the data itself) for initial use and later reuse Writing - Original Preparation, creation and/or presentation of the published Draft work, specifically writing the initial draft (including substantive translation) Writing - Review & Preparation, creation and/or presentation of the published Editing work by those from the original research group, specifically critical review, commentary or revision – including pre- or post-publication stages Visualization Preparation, creation and/or presentation of the published work, specifically visualization/ data presentation Supervision Oversight and leadership responsibility for the research activity planning and execution, including mentorship external to the core team Project Management and coordination responsibility for the research administration activity planning and execution Funding acquisition Acquisition of the financial support for the project leading to this publication

Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: