# Multilevel Factorial Optimisation of Acid-to-Ore Ratio and Solids Concentration in Hydrometallurgical Recovery of Nickel from Laterite Ore

Okechukwu Vincent Dickson, Thomas Deleau, Fabienne Espitalier, Christophe Coquelet, Julien Lombart

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Okechukwu Vincent Dickson, Thomas Deleau, Fabienne Espitalier, Christophe Coquelet, Julien Lombart. Multilevel Factorial Optimisation of Acid-to-Ore Ratio and Solids Concentration in Hydrometallurgical Recovery of Nickel from Laterite Ore. Results in engineering, 2025, 27, pp.106744. 10.1016/j.rineng.2025.106744. hal-05215119

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Research paper

# Multilevel factorial optimisation of Acid-to-Ore ratio and solids concentration in hydrometallurgical recovery of nickel from laterite ore

Okechukwu Vincent Dickson<sup>a,b,*</sup> <sup>□</sup>, Thomas Deleau<sup>b</sup>, Fabienne Espitalier<sup>b,□</sup>, Christophe Coquelet<sup>b,c</sup>, Julien Lombart<sup>a</sup>

## ARTICLE INFO

### Keywords:

- Nickel extraction

- Hydrometallurgy

- Design of experiment

- Multilevel factorial design

- Statistical approach

- Process optimisation

## ABSTRACT

This study applies a multilevel factorial design to optimise nickel (Ni) extraction in a High-Pressure Acid Leaching (HPAL) autoclave, examining the influence of slurry solid content and acid-to-ore (A/O) ratio on Ni yield, selectivity (Ni/(Al+Fe)), and scale formation. A regression model was established to predict these responses across varying A/O ratios and solid contents. Using Minitab® statistical software, response surface methodology (RSM) was employed to identify process conditions that balance maximised Ni yield, enhanced selectivity and reduced scale formation. To predict outputs beyond the experimental range, the complex models generated in Minitab® were simplified into ‘linear + quadratic + interaction terms’ using the least squares method in Matlab®. Results indicate that higher Ni yield (> 95 wt.%) is achieved at a lower slurry solid content (< 27 wt.%) and higher A/O ratios (> 0.35), whereas higher solution purity (selectivity > 3) is favoured at lower A/O ratios with preferably lower solid content (~22 wt.%). Scale formation increases significantly with higher solid content and A/O ratios exceeding 0.35. Optimisation revealed that a compromise between Ni yield, selectivity, and scale minimisation is necessary, with the highest composite desirability (0.68) obtained at 22 wt. % solid content and an A/O ratio of 0.35. The findings provide valuable insights for industrial leaching process optimisation, highlighting the trade-offs between maximising Ni recovery, limiting impurity dissolution and minimising scale formation.

## 1. Introduction

Electric vehicle (EV) adoption is accelerating worldwide, playing a pivotal role in efforts to lower greenhouse gas emissions and mitigate climate change [1]. With growing emphasis on a circular economy and the rapid expansion of battery-related sectors, research into nickel recovery from laterite ores is both timely and indispensable. The extraction of critical metals such as Ni and Co from primary resources, especially laterite ores, remains essential to meet growing industrial demand [2]. There are various methods for recovering Ni from laterite ores, including pyrometallurgical processes (e.g., smelting and ferro-nickel production) and hydrometallurgical techniques (e.g., Atmospheric Acid Leaching and High-Pressure Acid Leaching) [3].

The atmospheric acid leaching (AAL) process is more advantageous than high-pressure acid leaching (HPAL) because of its lower construction materials costs for conventional equipment and maintenance, easier

technology and process control, and lower energy consumption [4]. However, the AAL process has not been well tested commercially due to the inconsistency of nickel recovery and high acid consumption [3]. In addition, the main minerals of laterite ore in the form of goethite (α-FeOOH) are difficult to break at atmospheric pressure [5], resulting in a longer time and a low level of Ni selectivity. Although companies like Eramet have explored atmospheric leaching methods for extracting nickel (Ni) and cobalt (Co) from laterite ores [6,7], recent developments indicate a strategic shift towards HPAL processes driven by the increasing demand for battery-grade materials, particularly for electric vehicles.

In the industry, HPAL is broadly applied for iron-rich, limonitic ores due to its broader feed flexibility and high recoveries [3,5]. HPAL technology allows efficient processing of laterite ores, achieving high metal recovery up to 95 % in shorter leaching times, and production of high-purity nickel and cobalt essential for lithium-ion batteries [4],

<footnote><sup>a</sup> Prony Resources New Caledonia SAS, Usine du Grand Sud, Route de Kwa Neie-Prony 98834 Yaté, NC, France</footnote>

<footnote><sup>b</sup> IMT Mines Albi, UMR CNRS 5302, RAPSODEE Centre, 81000 Albi, France</footnote>

<footnote><sup>c</sup> Mines Paris, PSL University, Centre for Energy Environment Processes (CEEP), 77300 Fontainebleau, France</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: Okechukwu_vincent.dickson@mines-albi.fr (O.V. Dickson).</footnote>
thereby supporting the energy transition. However, HPAL brings sig-
nificant capital and operational expenditures, technical complexity, and
frequent ramp-up issues in commercial plants due to energy and acid
usage [8,9] and substantial greenhouse gas emissions. This makes
project feasibility and profitability challenging [3,10]. The complex
chemistry and variable mineralogy of laterite ores also cause differences
in how the ore behaves during processing, leading to difficulties in
maintaining consistent and stable plant operation at a large scale [10].

To address these challenges, research is increasingly focused on
developing sustainable hydrometallurgical alternatives that incorporate
greener reagents, eutectic solvents and processes to reduce the overall
carbon footprint of nickel and cobalt production [11]. Despite these
advances, alternative hydrometallurgical methods are still limited to
laboratory and pilot scale. Until large-scale eco-friendly solutions
become industrially feasible, HPAL remains the leading technology due
to its proven reliability and strong performance, especially for producing
battery-grade nickel and cobalt [12].

At Prony Resources, New Caledonia, a persistent issue in the HPAL autoclaves' operation is scale formation, which causes operational downtime and hinders production efficiency [13–15]. Thus, achieving an optimal balance between nickel recovery, product purity, and scaling control is crucial for improving HPAL performance and ensuring sufficient nickel supply for the rapidly expanding battery industry, since obtaining high nickel recovery and purity simultaneously with low scale formation under the same operational conditions and processing time is generally not feasible.

Previous studies have identified temperature, agitation speed, acid-
to-ore ratio, and slurry solid content as the key factors influencing
nickel recovery, selectivity, and scale formation in HPAL [16].

The optimal leaching temperature for nickel extraction from laterite
ores lies in the range 250–275 °C, a condition widely recognised in HPAL
operations as crucial for efficient dissolution of Ni-bearing goethitic or
silicate-bearing substrates [5]. Achieving slurry suspension homogene-
ity without splashing or settling is essential to ensure even acid distri-
bution during injection. Various authors have demonstrated that the
effectiveness of agitation depends primarily on reactor geometry and
slurry density, rather than a universal stirring speed; hence, no specific
rpm qualifies as “one-size-fits-all good practice” for HPAL [17–21].
Although sufficient agitation enhances Ni recovery, it does not fully
prevent scaling due to mineral precipitation from supersaturated solu-
tions, particularly Al and Fe hydroxides, which remain problematic even
at high mixing rates [22].

No single value can be universally prescribed considering the slurry
solid content and acid-to-ore ratio. Studies indicate that lower solid
content and an acid-to-ore ratio optimised for ore mineralogy deliver
higher Ni yields [5,23]. However, this does not guarantee elimination of
scaling; careful tuning of these two parameters at optimal temperature
and agitation is essential to achieve both moderate-to-high selectivity
and high Ni yield while minimising scaling in HPAL, though scaling
cannot be completely avoided.

During HPAL of laterite ore, nickel is preferentially leached from
silicate (saprolite) and lateritic phases, while iron and aluminium,
initially associated with goethitic and aluminous compounds, tend to
precipitate under high acid-to-ore (A/O) ratios and temperature condi-
tions [3]. Thermodynamically, at elevated A/O ratios and high sulphuric
acid concentrations, Fe³⁺ and Al³⁺ species reach supersaturation more
readily. These ions preferentially form stable precipitates such as he-
matite, jarosite, and alunite [22]. The increasing activity of sulphate,
coupled with elevated proton concentration, depresses the solubility of
Fe and Al salts in the leachate, leading to higher impurity dissolution
and subsequent precipitation. By contrast, lower A/O ratios reduce
proton and sulphate activities, raising the Fe and Al solubility limits and
suppressing precipitation, which enhances Ni selectivity (i.e. purity) in
the pregnant leach solution, but this is disadvantageous to Ni yield [5].

In the field of hydrometallurgy (HPAL), optimising process param-
eters is crucial for maximising metal recovery and ensuring the

economic feasibility of extraction operations [3,24]. Design of Experi-
ments (DOE) methodologies provide a structured framework for exam-
ining the influence of multiple variables on desired outcomes, enhancing
the quality, efficiency, and performance of leaching systems [25,26].
DOE is a robust statistical methodology employed across various disci-
plines to systematically plan, conduct, and analyse experiments. Its
primary purpose is to identify the relationships between factors affecting
a process and the resulting outputs, thereby facilitating the optimisation
of systems, processes, and products [27]. By methodically varying fac-
tors and analysing their effects, DOE, particularly Box-Behnken, Central
composite design and multilevel factorial design, enables the identifi-
cation of optimal conditions [28,29], leading to informed
decision-making and process improvements.

In a factorial design, each factor is examined at specific levels to
assess its impact on a process or system. Two-level factorial designs,
where each factor is tested at two levels (e.g., high and low), are
commonly employed for screening purposes [25]. These designs help
identify the primary factors influencing the process by evaluating main
effects and interactions efficiently; however, they may not detect
non-linear relationships or curvature effects. To address this, three-level
factorial designs are utilised, introducing an intermediate level for each
factor [25,30]. This approach allows for the modelling of quadratic ef-
fects and provides a more comprehensive understanding of factor in-
fluences, facilitating process optimisation [28,31,32]. A full factorial
design encompasses all possible combinations of factor levels. For k
factors, each at r levels, the total number of experimental runs required
is r<sup>k</sup>. For instance, a two-level factorial design with k factors necessitates
2<sup>k</sup> runs, while a three-level design requires 3<sup>k</sup> runs [25,33]. It is
important to note that as the number of factors increases, the total
number of runs grows rapidly, which can make full factorial designs
resource-intensive. In such cases, fractional factorial designs may be
considered to reduce the number of runs while still providing valuable
insights [27]. By employing statistical models in experimental design,
researchers can develop predictive equations that describe the rela-
tionship between process variables and outcomes, facilitating process
optimisation.

Multilevel factorial designs are increasingly employed in engineering research to investigate the simultaneous influence of multiple process variables and their interactions. For instance, Pimsamarn et al. [34] applied a multilevel factorial design combined with response surface methodology to optimise energy yield during the torrefaction of durian peel under different atmospheric conditions, demonstrating the robustness of this approach in process optimisation across diverse applications.

This new study focuses on the use of a full multilevel factorial design involving nine experiments with two factors, each at three levels, to optimise the leaching process for nickel (Ni) extraction and find a trade-off between the 3 crucial responses (outcomes) in the industry. The responses evaluated are Ni yield (wt.%), selectivity (Ni/Al+Fe), and scaling rate (g/run). By systematically varying the levels of the selected factors, the study aims to generate a regression model that explains the effects of these factors on the responses [30], intending to maximise Ni yield and selectivity while minimising scaling in the HPAL at 265 °C. Such a model can serve as a predictive tool for optimising leaching conditions, thereby enhancing Ni recovery and purity. This approach not only aids in identifying optimal conditions but also contributes to a deeper understanding of the underlying mechanisms influencing leaching efficiency [30,35]. The outcomes of this research are anticipated to offer practical guidelines for the hydrometallurgical industry in optimising process efficiency.
## 2. Methodology

### 2.1. Leaching experimental setup and procedures

The leaching experiments followed a standard HPAL procedure widely used in industry and research, where the ore sample is placed inside a sealed autoclave, heated to the target temperature (typically 240–270 °C), and then acid is injected to initiate leaching under controlled high-pressure conditions, enabling efficient dissolution of nickel [5].

In this study, experiments were performed in a custom high-pressure autoclave (Fig. 1a) at 265 °C, an optimal temperature selected based on industrial HPAL operations at Prony Resources, New Caledonia. The laterite ore used contained both saprolite and limonite, typical of New Caledonian deposits. Saprolite zones, rich in Mg and garnierite silicates, usually contain 1.5–2.5 % Ni (dry basis), while limonitic zones, dominated by Fe oxyhydroxide-rich goethite, contain 1.2–1.7 % Ni with high iron content (~45 %) [36].

The experiment employed a factorial design combining three levels of slurry solid content (22, 26, 30 wt. %) and three acid-to-ore (A/O) mass ratios, 0.25 (acid underdosage), 0.35 (industrial theoretical dosage), and 0.45 (acid overdosage), reflecting typical industrial parameters for throughput optimisation. Solids content chosen in this range mimics operational standards seen in commercial HPAL plants, where these values balance throughput and leaching kinetics without excessive viscosity increase.

The minimum agitation speed required to maintain the slurry mixture in suspension was estimated at 654 ± 65 rpm using the correlation in Eq. (1) [37], which accounts for the slurry density and the geometry of the autoclave tank (Fig. 1b) used in the leaching experiments.

$$N_{js} = A \left[ \left( g \left( \frac{\rho_s - \rho_l}{\rho_l} \right) \right)^{0.5} \frac{d_p^{0.17} \varnothing_{\text{mass}}^{0.23}}{N_p^{0.33} \mathbf{D}^{0.67}} \left( \frac{\mathbf{D}}{\mathbf{d}} \right) \right] \quad (1)$$

Where $N_{js}$ is the minimum impeller speed to achieve 'just-suspension' (rpm/60), and A is a constant dependent on the tank, impeller geometry and particle properties (ore size, diameter and viscosity of liquid) [37]. g is the acceleration due to gravity (9.81 m/s²), $N_p$ is the power number of the Mixell-TT impeller (1.35) for two propellers, d is the impeller diameter (0.07 m), $\rho_s$ and

$\rho_l$ are the density of solid ore and liquid (water + acid) (kg/m³), $d_p$ is the particle diameter (m) and $\varnothing_{\text{mass}}$, the mass ratio in weight percentage is expressed in Eq. (2) [38,39].

$$\varnothing_{\text{mass}} = \frac{m_s}{m_l} \times 100 \quad (2)$$

$m_s$ and $m_l$ are the respective masses (kg) of solid and liquid used for the leaching experiment. The range of parameters used to obtain the minimum speed using Eq. (1) is shown in Table 1.

The obtained minimum speed was determined based on the maximum solid concentration of 30 wt.%, an acid-to-ore (A/O) ratio of 0.45, and a particle size of 300 µm, under a temperature of 25 °C, using a Mixell-TT axial flow impeller. For all nine experiments, an agitation speed of 800 rpm (power input ≈ 7.44 W estimated from Eq. (3), varying with slurry density) was used to ensure effective mixing of the slurry and acid in the autoclave.

$$\mathbf{P} = \rho_{sl} \mathbf{N}_{js}^3 \mathbf{N}_p \mathbf{d}^5 \quad (3)$$

where P is dissipated power (W), and the slurry density ($\rho_{sl}$) (kg/m³).

For each batch experiment, the laterite ore was weighed according to the desired slurry solid content and loaded into the autoclave (Fig. 1a). A total mass of 2149 g comprising ore, water, and acid was maintained across all experimental runs to ensure consistency, with the proportions of water, acid, and ore adjusted according to the specified acid-to-ore ratio used. The slurry was heated to 265 °C (a pressure of 50 bar) while being stirred. Once the target temperature was reached, sulphuric acid (98 wt.% purity) was injected to commence leaching. To promote efficient initial dispersion of the acid in the slurry, the agitation speed was temporarily increased to 1000 rpm during the first 5 min of acid injection, after which, the speed was maintained at 800 rpm throughout the experimental duration until 24 hrs. Due to the fast kinetics of HPAL, the industrial continuous process at Prony operates with a short residence time of about 25 min. This duration was used as the main timeframe in the DOE to assess nickel yield and selectivity. At 25 min, leachate samples were taken, filtered, and analysed via ICP-AES (Inductively Coupled Plasma-Atomic Emission Spectroscopy) following the method described by Minh et al. [40], to determine concentrations of nickel and major impurities (aluminium and iron) in HPAL. To study scale formation, the batch experiments were extended to 24 h to promote scale deposition on the autoclave walls, simulating the

<figure><img src="image_2.png" /><figcaption>Fig. 1a. High-Pressure Acid Leaching Batch Autoclave Setup Used for the Leaching Experiments in This Study.</figcaption></figure>
<figure><img src="image_3.png" /><figcaption>Fig. 1b. Pictorial representation of the geometry of the autoclave tank used for the leaching experiment (designed by Top Industrie®).</figcaption></figure>

<caption>Table 1</caption>

Range of parameters defined by the sample, the tank and impeller geometry.

<table><thead><tr><th>Parameter</th><th>Studied range covered</th></tr></thead><tbody><tr><td>ρ<sub>s</sub> - ρ<sub>i</sub> (kg/m<sup>3</sup>)</td><td>2400</td></tr><tr><td>d<sub>p</sub> (μm)</td><td>d<sub>p</sub> ≤ 160</td></tr><tr><td>X (%)</td><td>28 - 43</td></tr><tr><td>D (m)</td><td>0.135</td></tr><tr><td>d/D</td><td>0.519</td></tr><tr><td>Y/D</td><td>0.333</td></tr><tr><td>Vessel base shape</td><td>Flat</td></tr><tr><td>Dynamic viscosity (mPa.s)</td><td>1 - 3</td></tr></tbody></table>

scaling behaviour observed in the first autoclave compartment of continuous HPAL operations at Prony Resources, New Caledonia.

Ni yield and selectivity were determined using Eqs. (4) and (5). After each leaching run, the scale deposits adhering to the internal walls of the autoclave were carefully recovered, dried at 60 °C and weighed using a precision analytical balance. This measurement captures the total mass of precipitated residue generated during the leaching process.

$$ \mathbf{X}_i = \left[ \frac{\mathbf{C}_{\text{filtrate}} \times \mathbf{V}_{\text{filtrate}}}{(\mathbf{C}_{\text{filtrate}} \times \mathbf{V}_{\text{filtrate}}) + (\mathbf{m}_{\text{residue}} \times \mathbf{w}_{\text{residue}})} \right] \times 100 \quad (4) $$

$$ \text{Selectivity} = \left[ \frac{\mathbf{m}_{\text{Ni, filtrate}}}{\mathbf{m}_{\text{Al, filtrate}} + \mathbf{m}_{\text{Fe, filtrate}}} \right] \quad (5) $$

Where: Xᵢ is the elemental yield of i (wt.%) at time t, C<sub>filtrate</sub> is the concentration (mg/L) of the element in the leached filtrate at time t, measured by ICP-AES. V<sub>filtrate</sub> is the volume (L) of the leached filtrate collected at time t is computed from density and mass measurement. w<sub>residue</sub> is the total mass of leached solid residue collected at time t. m<sub>residue</sub> is the mass fraction of the element in the leached solid residue at t. m<sub>Ni, filtrate</sub>, m<sub>Al, filtrate</sub> and m<sub>Fe, filtrate</sub> are the mass concentrations of the measured nickel, aluminium and iron obtained in the leaching experiments.

## 2.2. Statistical optimisation of leaching parameters using DOE

Striking out the effect of temperature and hydrodynamic (agitation) after selecting an optimal conditions based on prior tests, industrial operation and our batch autoclave tank geometry, a slurry solid content

d = Mixell-TT Impeller (diameter 70 mm)

T = tank height (201 mm)

D = tank diameter (135 mm)

Y = off bottom clearance (45 mm)

B = baffle (width 13.5 mm)

C = Teflon control bar

of 22, 26 and 30 wt.% and acid-to-ore mass ratios of 0.25, 0.35 and 0.45 as described in the experimental part were used for the Design of Experiment, resulting in an experimental order of 2 factors and 3 levels (3²) factorial design to develop a nonlinear mathematical model describing the relationships between the independent variables (factors) and the dependent variables (responses).

**Key components of DOE and their meaning in this paper:**

1. **Factors:** Variables that are controlled during the experiment. They can be categorical (qualitative, e.g. gender, colours) or continuous (quantitative, e.g. volume, mass). Continuous variables (slurry solid content and acid-to-ore ratio) are the 2 factors used for this study.

2. **Levels:** Different settings of the factors. 0.22, 0.26, 0.30 solid wt. fraction and 0.25, 0.35, 0.45 acid-to-ore ratio.

3. **Response:** The output of the experiments. The outputs used for this study are nickel yield (wt.%), the purity of Ni (Selectivity) and scale quantity (g) obtained in a given experiment.

The resulting Ni yield, selectivity, and scaling rate analysed from the leaching experiments were optimised using Minitab® statistical software. The factors, levels, and responses, summarised in Table 2 (Section 3), were incorporated into Minitab's general full factorial design module, with a significance level (α) of 0.05, corresponding to a 95 % confidence interval. To assess the individual and combined effects of the factors on the responses, their levels were systematically varied, as depicted in the cube plot (Fig. 2). Minitab® generated a model corresponding to the inputted responses for the multilevel factorial design. This process involved fitting the collected data using regression analysis and estimating the effects of each factor and their interactions on the response variable. Model assumptions, including the homogeneity of variance, were verified through residual plots.

To enhance model interpretability, a stepwise regression technique, specifically backward elimination, was employed to remove less significant predictors, retaining only the most impactful ones. Minitab® utilises t-tests (Eq. (6)) to assess the significance of each coefficient in the regression analysis [41]. For each coefficient, a t-value is calculated as the ratio of the coefficient estimate to its standard error. The associated P-value indicates whether the coefficient significantly differs from zero. A P-value below the predetermined significance level (commonly 0.05) suggests that the corresponding predictor variable has a statistically significant effect on the response variable.
<caption>**Table 2**</caption>

3² Full factorial experimental design on isothermal experiment at 265 °C and the experimental results of Ni yield, Selectivity and Scaling rate obtained in this paper.

<table><thead><tr><th rowspan="2">Experimental runs (3<sup>2</sup> factorial design)</th><th colspan="2">Factors and levels</th><th colspan="3">Responses</th></tr><tr><th>Initial solid content (wt.%)</th><th>Initial acid: ore wt. ratio</th><th>Ni yield (wt.%) at 25 min</th><th>Selectivity at 25 min</th><th>Scaling rate (g/run) ± 0.1 mg observed after 24 h of experiment</th></tr></thead><tbody><tr><td>1</td><td>-1</td><td>-1</td><td>72.81 ± 0.04</td><td>4.67 ± 0.08</td><td>0.114</td></tr><tr><td>2</td><td>-1</td><td>0</td><td>94.86 ± 0.01</td><td>2.07 ± 0.04</td><td><sup>*</sup>No observation</td></tr><tr><td>3</td><td>-1</td><td>+1</td><td>97.08 ± 0.01</td><td>0.84 ± 0.02</td><td>0.746</td></tr><tr><td>4</td><td>0</td><td>-1</td><td>67.50 ± 0.05</td><td>2.50 ± 0.03</td><td><sup>*</sup>No observation</td></tr><tr><td>5</td><td>0</td><td>0</td><td>93.11 ± 0.01</td><td>1.98 ± 0.02</td><td><sup>*</sup>No observation</td></tr><tr><td>6</td><td>0</td><td>+1</td><td>94.09 ± 0.01</td><td>0.69 ± 0.01</td><td>1.006</td></tr><tr><td>7</td><td>+1</td><td>-1</td><td>73.19 ± 0.03</td><td>3.33 ± 0.22</td><td>4.062</td></tr><tr><td>8</td><td>+1</td><td>0</td><td>91.17 ± 0.01</td><td>2.37 ± 0.07</td><td>1.959</td></tr><tr><td>9</td><td>+1</td><td>+1</td><td>91.25 ± 0.01</td><td>0.78 ± 0.01</td><td>4.861</td></tr></tbody></table>

<figure><img src="image_4.png" /><figcaption>**Fig. 2.** Cube plot of the multilevel 3² factorial design used to identify the optimal level of the two studied factors.</figcaption></figure>

$$t = \frac{\hat{\beta}}{\text{SE}(\hat{\beta})} \qquad (6)$$

where, $\hat{\beta}$ and $SE(\hat{\beta})$ are the estimated coefficient and the standard error of the coefficient

In the generated model, the effects of the factors on responses are considered significant when the standardised effect value exceeds the t-value threshold line. This threshold is determined by the confidence interval and degrees of freedom (DF) in the experimental data. The threshold value is derived from the t-distribution, considering the DF (Eq. (7)) and the significance level after fitting the regression model.

$$\mathbf{DF} = \mathbf{N} - \mathbf{p} \qquad (7)$$

Where N is the total number of runs across all factor combinations and replicates, and p is the number of regression coefficients estimated, including the intercept, the main effects (solid content and A/O ratio),

and their interaction term in our study, following standard ANOVA and regression theory for DOE.

## 2.3. Statistical model definition

The three-level full factorial design is among the most widely used optimisation approaches due to its ability to model complex response surfaces. However, it is typically applied when studying only two or three input factors, as the required number of experiments increases significantly with additional factors [28]. A key advantage of this design lies in its capability to model second-order response surfaces, as each factor is assessed at three to five levels. This enables the evaluation of response surface curvature [31,32]. The model incorporates both linear and quadratic terms, along with interactions, as represented in Eq. (8) [25].

$$\hat{\mathbf{y}}_{i} = \beta_{0} + \sum_{i}^{k} \beta_{i} \mathbf{x}_{i} + \sum_{j}^{k} \beta_{j} \mathbf{x}_{j}^{2} + \sum_{ij}^{k} \beta_{ij} \mathbf{x}_{i} \mathbf{x}_{j} + \epsilon \qquad (8)$$

Where $\hat{y}_i$: is the predicted response variable.

$\beta_0$: Constant term, which represents the baseline value of the response ($\hat{y}_i$).

$\beta_{ij}$: Coefficients for each independent variable $x_i$ and $x_j$, representing the change in $\hat{y}_i$ for a unit change in that variable.

$x_i$, $x_j$: Independent variables (factors) such as acid-to-ore ratio and solid content.

$\beta_{ij}x_i x_j$: Interaction term between $x_i$ and $x_j$ and their constant $\beta_{ij}$, indicating how one variable effect depends on the level of the other.

The error term ($\epsilon$) is expressed in Eq. (9).

$$\epsilon = \mathbf{y}_i - \hat{\mathbf{y}}_i \qquad (9)$$

The difference between the actual observation $y_i$ and the corresponding fitted value $\hat{y}_i$ is the residual (error). The estimated sum of squares (SS$_E$), a key metric in ANOVA regression analysis used to represent the unexplained variability in the response variables that are not accounted for by the regression model is given in Eq. (10) [25,30].

$$\mathbf{SS}_E = \sum_{i=1}^{n} (\mathbf{y}_i - \hat{\mathbf{y}}_i)^2 = \mathbf{y}\mathbf{y} - \hat{\mathbf{y}}'\mathbf{y} \qquad (10)$$

$\hat{\mathbf{y}}'$ and $\mathbf{X}'$ is the transpose of the estimated regression coefficients and design matrix of independent variables, respectively. $\mathbf{y}_i$ is the observed dependent variable value, and $\hat{\mathbf{y}}_i$ is the predicted value.

The measure of the amount of reduction in the variability of y obtained by using the regressor variables $R^2$ is given in Eq. (11).

$$R^2 = 1 - \frac{\mathbf{SS}_E}{\mathbf{SS}_T} \qquad (11)$$

The total sum of squares ($SS_T$) is expressed as the summation of the estimated ($SS_E$, due to residuals) and regression ($SS_R$, due to the model) sum of squares in Eq. (12). The sum of squares due to the model (regression) is given in Eq. (13) [25], where n is the number of observations.

$$SS_T = SS_E + SS_R \qquad (12)$$

$$SS_R = \hat{\beta}' \mathbf{X}' \mathbf{y} - \left( \sum_{i=1}^{n} \mathbf{y}_i \right)^2 / n \qquad (13)$$

The Root Mean Squared Error for the least squares method used for simplifying the obtained regression model, considering the linear + quadratic + interaction terms, is obtained from Eq. (14).

$$\text{RMSE} = \sqrt{\frac{1}{n} \sum_{i=1}^{n} (\mathbf{y}_i - \hat{\mathbf{y}}_i)^2} \qquad (14)$$

<footnote>* For the DOE on Minitab®, the value of scales was set to 0.01 g in experiments where no scale formation was observed on the walls of the reactor; Levels: Low (-1) = 22 wt.% solids, A/O = 0.25; Medium (0) = 26 wt.% solids, A/O = 0.35; High (+1) = 30 wt.% solids, A/O = 0.45. Scaling rate (g/run) and scale mass (g) are used interchangeably in this paper.</footnote>
## 3. Results and discussion

Table 2 presents the results of the 3² full factorial design conducted under isothermal HPAL conditions at 265 °C. The table summarises the factors and levels used (initial solid content and acid-to-ore ratio) along with the corresponding responses, including nickel yield and selectivity at 25 minutes, and the scale formation rate observed after 24 hours of each batch experiment.

### 3.1. Predictive regression models for responses (Ni yield, selectivity and scaling)

Experimental Ni yield, selectivity, and scaling rate data were modelled based on varying solid fraction and acid-to-ore ratios. An initial model generated by Minitab® from Analysis of Variance (ANOVA) was used to estimate the responses based on pre-determined coefficients. Due to the complexity of the Minitab®-generated model, a simplified quadratic regression model was then developed using least squares, incorporating linear, squared, and interaction terms. Model performance was assessed using R² and RMSE to ensure reliability and predictive accuracy.

Following the expression in Eq. (8), the developed regression equation for the prediction of Ni yield (%), selectivity and scaling (g) is shown in Eqs. (15), (16) and (17), respectively (see Sections 3.2 – 3.4). Table 3 shows the comparison of the experimental value with the predicted values and their error term (ε).

### 3.2. Effect of slurry solid content and acid-to-ore ratio on Ni yield

The regression equation in Eq. (15) quantifies the effects of solid content and acid-to-ore ratio on Ni yield during leaching of laterite ore in the HPAL at 265 °C. The negative intercept (–17.94) in the equation suggests that the model predicts a theoretical Ni yield below zero when both parameters are absent, though this has no physical meaning and primarily serves as a baseline. The linear term associated with the solid content (–566.24 × S) indicates that an increase in solid content generally reduces Ni yield, likely due to decreased acid availability and/or mass transfer limitations in the slurry. Conversely, the positive coefficient of the acid-to-ore ratio (956.93 × A) suggests that increasing acid concentration enhances Ni extraction, which is expected given that a higher acid dosage promotes the dissolution of Ni-bearing minerals.

The quadratic terms reveal nonlinear effects on Ni yield. The positive coefficient for solid content squared (1305.00 × S²) suggests that beyond a certain threshold, increasing the solid content may counteract the initial negative impact, potentially due to changes in reaction kinetics effects. In contrast, the negative coefficient for acid-to-ore ratio squared (–1042.80 × A²) indicates a reduction in efficiency at higher

acid concentrations, which could be attributed to secondary reactions, impurity precipitation, or acid supersaturation effects. The interaction term (–420.88 × S × A) further highlights the complex interplay between these factors, showing that when both solid content and acid-to-ore ratio increase simultaneously, Ni yield decreases more than would be expected from their individual effects. This suggests an antagonistic interaction, possibly due to excessive acid consumption leading to impurity dissolution or restricted acid diffusion at higher slurry densities.

$$ \text{Ni yield (\%)} = -17.94 + (-566.24 \times \mathbf{S}) + (956.93 \times \mathbf{A}) \\ + (1305.00 \times \mathbf{S}^2) + (-1042.80 \times \mathbf{A}^2) \\ + (-420.88 \times \mathbf{S} \times \mathbf{A}) + \epsilon \quad (15) $$

Where ‘S’ denotes the solid content of the slurry (expressed as a ratio, e.g. 0.22 for 22 wt.%), and ‘A’ is the acid-to-ore ratio used for the leaching process. The same notation is used in Eqs. (16) and (17).

The ANOVA results of the statistical modelling in Table 3 indicate that the model explains nearly all response variability, with an R² value of 98.6 %. These high R² values indicate a well-fitting statistical model, consistent with best practices in experimental design [25,30]. Table 4 presents the ANOVA results for assessing factor effects and interactions. The regression coefficient for solid content (e.g. coef 2.094) suggests that a one-unit increase in slurry solid content raises Ni yield by 2.094 units, assuming other variables remain constant. Conversely, the acid dosage coefficient (e.g., 15.230) implies that each one-unit increase in acid dosage reduces Ni yield by 15.230 units. The constant term (β₀ = 86.373) represents the theoretical baseline of Ni yield (in the experimental range) in the absence of major effects from the solid content and acid dosage contributions.

For each coefficient, Minitab calculates a t-value, which is the ratio of the coefficient estimate to its standard error. The associated P-value indicates the significance of the variable. A t-value > 2, a t-value > –2 and a P-value ≤ 0.05 are considered statistically significant and generally indicate a large effect size, suggesting a strong relationship between the variables (Table 4).

Fig. 3 presents the effects and interactions of solid content and acid-to-ore ratio (A/O) on Ni yield based on the multilevel experimental design. The A/O ratio (B) is a key factor in Ni extraction, with Ni yield increasing with a higher A/O ratio. The interaction plot reveals a sharp increase in Ni yield from A/O 0.25 to 0.35, followed by a minimal gain at A/O 0.45. This trend aligns with previous studies on nickel leaching, which emphasise the importance of acid concentration in optimising metal recovery [42].

Slurry solid content and its interaction with A/O (A and AB) have a negligible impact, as indicated by their proximity to the significance threshold. This suggests that optimising the A/O ratio is more influential than adjusting solid content. Increasing solid content from 22 to 26 wt.%

<table><caption>**Table 3**
Experimental and Predicted values for Ni yield, Selectivity and Scale mass (g).</caption><thead><tr><th colspan="2">levels (3<sup>2</sup>)</th><th colspan="4">Ni yield (wt.%)</th><th colspan="4">Selectivity</th><th colspan="4">Scale mass (g)</th></tr><tr><th>Solid ratio</th><th>A/O ratio</th><th>Experimental value</th><th>Predicted Eq. (15)</th><th>ε</th><th>Experimental value</th><th>Predicted Eq. (16)</th><th>ε</th><th>Experimental value</th><th>Predicted Eq. (17)</th><th>ε</th><th>Experimental value</th><th>Predicted Eq. (17)</th><th>ε</th></tr></thead><tbody><tr><td>0.22</td><td>0.25</td><td>72.81±0.04</td><td>71.55</td><td>1.26</td><td>4.67±0.08</td><td>4.16</td><td>0.50</td><td>0.114</td><td>0.30</td><td>-0.19</td><td></td><td></td><td></td></tr><tr><td>0.22</td><td>0.35</td><td>94.86±0.01</td><td>95.42</td><td>-0.56</td><td>2.07±0.04</td><td>2.52</td><td>-0.44</td><td>0.010<sup>*</sup></td><td>-0.47<sup>*</sup></td><td>0.48<sup>*</sup></td><td></td><td></td><td></td></tr><tr><td>0.22</td><td>0.45</td><td>97.08±0.01</td><td>98.43</td><td>-1.35</td><td>0.84±0.02</td><td>0.84</td><td>0.00</td><td>0.746</td><td>1.03</td><td>-0.28</td><td></td><td></td><td></td></tr><tr><td>0.26</td><td>0.25</td><td>67.50±0.05</td><td>69.75</td><td>-2.25</td><td>2.50±0.03</td><td>3.06</td><td>-0.56</td><td>0.010<sup>*</sup></td><td>0.32<sup>*</sup></td><td>-0.31</td><td></td><td></td><td></td></tr><tr><td>0.26</td><td>0.35</td><td>93.11±0.01</td><td>91.93</td><td>1.18</td><td>1.98±0.02</td><td>1.72</td><td>0.26</td><td>0.010<sup>*</sup></td><td>-0.41<sup>*</sup></td><td>0.42<sup>*</sup></td><td></td><td></td><td></td></tr><tr><td>0.26</td><td>0.45</td><td>94.09±0.01</td><td>93.26</td><td>0.83</td><td>0.69±0.01</td><td>0.35</td><td>0.34</td><td>1.006</td><td>1.12</td><td>-0.11</td><td></td><td></td><td></td></tr><tr><td>0.30</td><td>0.25</td><td>73.19±0.03</td><td>72.13</td><td>1.07</td><td>3.33±0.22</td><td>3.19</td><td>0.14</td><td>4.062</td><td>3.53</td><td>0.53</td><td></td><td></td><td></td></tr><tr><td>0.30</td><td>0.35</td><td>91.17±0.01</td><td>92.62</td><td>-1.45</td><td>2.37±0.07</td><td>2.16</td><td>0.21</td><td>1.959</td><td>2.84</td><td>-0.88</td><td></td><td></td><td></td></tr><tr><td>0.30</td><td>0.45</td><td>91.25±0.01</td><td>92.27</td><td>-1.02</td><td>0.78±0.01</td><td>1.10</td><td>-0.32</td><td>4.861</td><td>4.41</td><td>0.45</td><td></td><td></td><td></td></tr><tr><td colspan="2"></td><td>R<sup>2</sup> = 0.986</td><td colspan="2"></td><td colspan="2">RMSE = 0.35 (16.46 %)</td><td colspan="2"></td><td colspan="2">RMSE = 0.46 g (32.24 %)</td><td colspan="2"></td><td></td></tr><tr><td colspan="2"></td><td colspan="2"></td><td colspan="2"></td><td colspan="2"></td><td colspan="2"></td><td colspan="2"></td><td colspan="2"></td><td></td></tr></tbody></table>

<footnote>* No scale observed, experimental value set to 0.001 to allow for statistical analysis. Error on the scaling (experimental value) = error on the precision weighing balance, 0.0001 g. RMSE: Root Mean-Square Error on predicted model for each response.</footnote>
<table><caption>**Table 4**
General Factorial Regression (ANOVA): Ni yield (wt.%) versus Solid content and Acid dosage.</caption><thead><tr><th>Source</th><th>DF</th><th>Adj SS</th><th>Adj MS (Adj SS / DF)</th><th>F-Value</th><th>P-value</th></tr></thead><tbody><tr><td>Model (SS<sub>R</sub>)</td><td>9</td><td>2184.97</td><td>242.77</td><td>1242.07</td><td>1.31 × 10<sup>-11</sup></td></tr><tr><td>Linear</td><td>4</td><td>2133.80</td><td>533.45</td><td>2729.20</td><td>1.44 × 10<sup>-12</sup></td></tr><tr><td>Solid content</td><td>2</td><td>40.89</td><td>20.44</td><td>104.59</td><td>1.84 × 10<sup>-6</sup></td></tr><tr><td>Acid dosage</td><td>2</td><td>2092.91</td><td>1046.46</td><td>5353.82</td><td>3.11 × 10<sup>-13</sup></td></tr><tr><td>Solid*Acid</td><td>4</td><td>49.99</td><td>12.50</td><td>63.93</td><td>4.13 × 10<sup>-6</sup></td></tr><tr><td>Error (SS<sub>E</sub>)</td><td>8</td><td>1.56</td><td>0.20</td><td></td><td></td></tr><tr><td>Total (SS<sub>T</sub>)</td><td>17</td><td>2186.54</td><td></td><td></td><td></td></tr></tbody></table>

Adj: Adjusted, SS: Sum of Squares, MS: Mean Square

<table><thead><tr><th colspan="6">Coefficients (Coef)</th></tr><tr><th rowspan="2">Term</th><th rowspan="2">Coef</th><th rowspan="2">SE Coef</th><th rowspan="2">t-value</th><th rowspan="2">P-value</th><th colspan="2">Means</th></tr><tr><th>Fitted Mean</th><th>SE Mean</th></tr></thead><tbody><tr><td>Constant</td><td>86.373</td><td>0.104</td><td>828.87</td><td>1.3E-11</td><td>-</td><td>-</td></tr><tr><td colspan="7"><strong>Solid content (fraction)</strong></td></tr><tr><td>0.22</td><td><strong>2.094</strong></td><td>0.147</td><td><strong>14.21</strong></td><td>&lt;0.0001</td><td>88.467</td><td>0.180</td></tr><tr><td>0.26</td><td>-1.392</td><td>0.147*</td><td>-9.44</td><td>&lt;0.0001</td><td>84.981</td><td>0.180</td></tr><tr><td>0.30</td><td>-0.702</td><td>0.147*</td><td>-4.78</td><td>&lt;0.0001</td><td>85.671</td><td>0.180</td></tr><tr><td colspan="7"><strong>Acid dosage</strong></td></tr><tr><td>0.25</td><td>-15.230</td><td>0.147</td><td>-103.35</td><td>&lt;0.0001</td><td>71.143</td><td>0.180</td></tr><tr><td>0.35</td><td>6.952</td><td>0.147</td><td>47.17</td><td>&lt;0.0001</td><td>93.325</td><td>0.180</td></tr><tr><td>0.45</td><td>8.278</td><td>0.147</td><td>56.34</td><td>&lt;0.0001</td><td>94.651</td><td>0.180</td></tr><tr><td colspan="7"><strong>Solid content*Acid dosage</strong></td></tr><tr><td>0.22×0.25</td><td>-0.652</td><td>0.208*</td><td>-3.13</td><td>0.0018</td><td>72.585</td><td>0.313</td></tr><tr><td>0.22×0.35</td><td>-0.215</td><td>0.208*</td><td>-1.03</td><td>0.0302</td><td>95.203</td><td>0.313</td></tr><tr><td>0.22×0.45</td><td>0.867</td><td>0.208*</td><td>-4.16</td><td>&lt;0.0001</td><td>97.612</td><td>0.313</td></tr><tr><td>0.26×0.25</td><td>-2.427</td><td>0.208</td><td>-11.64</td><td>&lt;0.0001</td><td>67.324</td><td>0.313</td></tr><tr><td>0.26×0.35</td><td>1.158</td><td>0.208*</td><td>5.56</td><td>&lt;0.0001</td><td>93.091</td><td>0.313</td></tr><tr><td>0.26×0.45</td><td>1.269</td><td>0.208*</td><td>6.11</td><td>&lt;0.0001</td><td>94.528</td><td>0.313</td></tr><tr><td>0.30×0.25</td><td>3.079</td><td>0.208</td><td>14.81</td><td>&lt;0.0001</td><td>73.519</td><td>0.313</td></tr><tr><td>0.30×0.35</td><td>-0.943</td><td>0.208*</td><td>-4.53</td><td>&lt;0.0001</td><td>91.680</td><td>0.313</td></tr><tr><td>0.30×0.45</td><td>-2.136</td><td>0.208</td><td>-10.27</td><td>&lt;0.0001</td><td>91.813</td><td>0.313</td></tr></tbody></table>

The fitted mean average of each experimental campaign gave the constant coefficient value. Smaller SE (Standard Error) Coef leads to larger t-value and lower P-value, making it easier to identify significant factors. (*) has a high SE coefficient value > 10% of the coefficient, indicating a large amount of variability or uncertainty in the estimate of that particular effect.

<figure><img src="image_5.png" /><figcaption>**Fig. 3.** Interaction (left) and Pareto chart of the standardised effect (right) on Ni yield (wt.%).</figcaption></figure>

reduced Ni yield by three units, while a further increase to 30 wt.% resulted in a slight one-unit gain at a constant A/O ratio. Similar observations have been reported in hydrometallurgical studies, where excessive solid content can reduce mass transfer efficiency that promotes goethite dissolution in Ni-extraction [19].

### 3.3. Effect of slurry solid content and acid-to-ore ratio on selectivity

In the regression equation for selectivity (Eq. (16)) derived from Table 5, the positive intercept (40.47) represents the baseline selectivity when both factors are at their coded zero levels. However, in a

practical sense, setting both $S = 0$ and $A = 0$ is unrealistic, as it would imply a system with no solids and no acids. Because of this, the intercept does not have a direct physical interpretation but rather serves as a mathematical reference point in the regression equation. It adjusts the model to fit the data properly within the experimental range, ensuring that the predicted values remain accurate when realistic values of $S$ and $A$ are used. The negative linear coefficient for solid content ($-231.69 \times S$) indicates that increasing the solid content reduces selectivity, likely due to decreased acid availability and increased impurity dissolution. Similarly, the negative coefficient for acid-to-ore ratio ($-32.45 \times A$) suggests that higher acid concentrations also lower selectivity, possibly
<table><caption>**Table 5**

General Factorial Regression (ANOVA): Selectivity versus Solid content and Acid dosage.</caption><thead><tr><th>Source</th><th>DF</th><th>Adj SS</th><th>Adj MS<br>(Adj SS / DF)</th><th>F-Value</th><th>P-value</th></tr></thead><tbody><tr><td>Model</td><td>9</td><td>26.84</td><td>2.98</td><td>7930.61</td><td>7.93 ×<br>10<sup>-15</sup></td></tr><tr><td>Linear</td><td>4</td><td>23.89</td><td>5.97</td><td>15,880.84</td><td>1.26 ×<br>10<sup>-15</sup></td></tr><tr><td>Solid<br>content</td><td>2</td><td>1.90</td><td>0.95</td><td>2529.97</td><td>6.21 ×<br>10<sup>-12</sup></td></tr><tr><td>Acid dosage</td><td>2</td><td>21.98</td><td>10.99</td><td>29,231.71</td><td>3.50 ×<br>10<sup>-16</sup></td></tr><tr><td>Solid*Acid</td><td>4</td><td>2.95</td><td>0.74</td><td>1960.44</td><td>5.39 ×<br>10<sup>-12</sup></td></tr><tr><td>Error</td><td>8</td><td>3E-03</td><td>4E-04</td><td></td><td></td></tr><tr><td>Total</td><td>17</td><td>26.85</td><td></td><td></td><td></td></tr></tbody></table>

due to excessive acid dissolution of unwanted elements alongside nickel.

The quadratic terms show nonlinear effects on selectivity. The positive coefficient for solid content squared (385.00 × S²) suggests that at higher solid contents, selectivity may improve again, indicating an optimal range where impurity dissolution is minimised. In contrast, the small negative coefficient for acid-to-ore ratio squared (−1.65 × A²) implies a slight decrease in selectivity at very high acid concentrations, which could be due to undesired secondary reactions. The interaction term (77.19 × S × A) is positive, indicating that under certain conditions, increasing both solid content and acid-to-ore ratio together may enhance selectivity, potentially due to shifts in leaching dynamics that favour nickel extraction over impurities [19,22].

$$ \text{Selectivity} = 40.47 + (-231.69 \times \mathbf{S}) + (-32.45 \times \mathbf{A}) \\ + (385.00 \times \mathbf{S}^2) + (-1.65 \times \mathbf{A}^2) + (77.19 \times \mathbf{S} \times \mathbf{A}) \\ + \epsilon \qquad (16) $$

As shown in Fig. 4, the A/O ratio and its interaction with solid content (S < 30 wt.% and A/O < 0.45) significantly impact the purity of the leached Ni solution. Although an increase in A/O increases the yield of Ni, the purity of the leached solution decreases due to the rapid extraction of other impurities along with Ni [5,20]. In Fig. 4(right), B (0.45) and B(0.25) represent high and low A/O ratios, respectively. A high A/O ratio (B(0.45)) reduces the selectivity of Ni, as it promotes the co-extraction of impurities such as Al and Fe. Conversely, a low A/O ratio (B(0.25)) enhances Ni selectivity due to the limited availability of acid, which restricts the dissolution of competing impurities relative to Ni.

<figure><img src="image_6.png" /><figcaption>**Fig. 4.** Interaction (left) and Pareto chart of the standardised effect (right) on Selectivity.</figcaption></figure>

At a constant A/O, solid content has less effect on selectivity, as indicated by the P-value of 6.21 × 10⁻¹² in Table 5 compared to a P-value of 3.50 × 10⁻¹⁶ for the A/O ratio that shows a greater effect. At higher solid content, acid availability per unit mass of ore is limited, affecting nickel dissolution relative to impurities, an effect comparable to lowering the A/O ratio at a given solid content. Conversely, lower solid content at a constant acid dosage enhances Ni dissolution but also increases impurity dissolution, reducing purity, similar to raising the A/O ratio at a given solid content.

Although leaching at a lower A/O ratio (0.25) improves selectivity by reducing impurity dissolution, it also lowers Ni yield due to insufficient acid availability. Analysis of variance, the regression model, and associated statistical parameters were used to evaluate the effects and interactions of these factors on Ni selectivity, yielding an R² of 91.9 % (Table 3).

### 3.4. Effect of slurry solid content and acid-to-ore ratio on scaling

The negative linear coefficients for solid content (−482.25 × S) and A/O ratio (−77.68 × A) in Eq. (17) suggest that increasing either parameter initially reduces scaling. However, the strong positive quadratic term for solids (1000.63 × S²) indicates that at higher solid content, scaling increases significantly, likely due to supersaturation effects. Supersaturation occurs when the concentration of dissolved metal salts (e.g., Fe-, Al-sulphates) in the autoclave liquor exceeds their solubility limit under prevailing temperature and pH conditions [5]. At high solid contents, more mineral material is available for dissolution, which rapidly increases dissolved species concentrations in the solution. As the leaching reaction progresses and acid is consumed, As leaching progresses and acid is consumed, localised pH rises promote precipitation of minerals such as hematite and alunite from the supersaturated solution. These precipitates nucleate and grow on available surfaces, accelerating scale formation once deposition begins [5,22,43,44].

Similarly, the positive quadratic term for acid (113.25 × A²) suggests a similar trend, though to a lesser extent. The small interaction term (9.25 × S × A) implies minimal combined influence. Overall, scaling is minimised at moderate conditions but increases at high solid content and acid levels.

$$ \text{Scaling (g)} = 69.80 + (-482.25 \times \mathbf{S}) + (-77.68 \times \mathbf{A}) \\ + (1000.63 \times \mathbf{S}^2) + (113.25 \times \mathbf{A}^2) + (9.25 \times \mathbf{S} \times \mathbf{A}) \\ + \epsilon \qquad (17) $$

Fig. 5 shows the significance of factors affecting the leaching process with respect to scale formation. Both the solid content, A/O and their interaction have effects on the mass of scale formed, with the effect of

<figure><img src="image_7.png" /></figure>
<figure><img src="image_8.png" /><figcaption>Fig. 5. Interaction (left) and Pareto chart of the standardised effect (right) on scaling rate (g/run).</figcaption></figure>

solid being more significant as seen in Table 6 with a P-value of 1.11 × 10⁻¹⁶. Keeping the A/O constant, at higher solid content, the increased slurry density leads to greater supersaturation and precipitation, promoting more scale formation [45]. This aligns with the statistical significance of A30 (30 wt.% solid content) shown by its deviation from the threshold line in Fig. 5(right). Conversely, at lower solid content, lower slurry density results in lower ion concentration, limiting scale-forming minerals precipitation. The effect is less pronounced, as indicated by its proximity to the threshold line. Similarly, higher A/O ratios (B0.45) accelerate the dissolution of metal ions and contribute to supersaturation, leading to more scale mass formation [46]. Lower A/O ratios (B0.25 and B0.35) provide less excess acid (Fig. 5), reducing the availability of dissolved species for precipitation, which explains their smaller effect magnitudes.

The regression analysis in Table 6 supports these observations, showing that factors with higher absolute effect magnitudes contribute significantly to the variation in scale formation. The presence of interaction effects (AB), though smaller, suggests that the impact of solid content on scaling is dependent on the A/O ratio, reinforcing the need to optimise both parameters simultaneously.

For the determination of the scaling rate, the linear + quadratic + interaction term with RMSE of 32.24 % (Table 3) did not perform well compared to Ni yield and selectivity. This is because the encrustation process is inherently nonlinear, driven by complex precipitation and growth dynamics of secondary minerals (e.g. hematite, kieserite and hydronium alunite) [22]. Crystallisation processes are highly sensitive to local thermodynamic and hydrodynamic conditions such as supersaturation levels, pH, temperature gradients, and flow patterns, which

can fluctuate within the autoclave during leaching [47]. Under such conditions, nucleation and growth rates do not follow a fixed or uniform pattern, even when the bulk operating parameters remain constant, leading to variability in scaling behaviour [22].

In HPAL specifically, studies have shown that the formation of scale minerals such as alunite, jarosite, and hematite depends on subtle shifts in acid concentration, temperature, and slurry composition, all of which influence precipitation kinetics and crystal morphology [5,48]. Furthermore, the complex interplay between fluid dynamics and mineral supersaturation within autoclaves creates localised zones where scaling can be highly variable and difficult to predict using simple regression models [22,45,49]. Such factors necessitate nonlinear or mechanistic models that can capture the coupled effects of thermodynamics, kinetics, and transport phenomena on scale formation in HPAL systems.

Due to this complexity, the regression model for the scaling was simplified by considering a 2-level factorial design utilising the low and high points of the factors (solid content and acid-to-ore ratio) with the introduction of a centre point to account for curvature, as shown in Table 7. The generated model follows the order of linear + interaction term in Eq. (18), capturing the majority of the experiments with scale formation in this study performed better in comparison to Eq. (17).

$$ \text{Response Y} = \beta_0 + \beta_1 S_p + \beta_2 A + \beta_3 (S_p \times A) + \epsilon \quad (18) $$

$$ \begin{array}{l} \text{Scale (g)} = -10.948 + 0.4669 \times \text{Solid}(\%) + 1.125 \times \text{Acid} \\ \quad : \text{Ore} + 0.0923 \times \text{Solid}(\%) \times \text{Acid} : \text{Ore} - 2.4143 \times \text{CP} \end{array} \quad (19) $$

Where $β_0, β_1, β_{i,j}, x_i, x_j$ and $ε$ are as defined in the 3-level DOE (Equations (8)). The solid content 'S_p' in Eq. (19) is expressed in percentage (wt.%). E.g. 22 wt.%. 'A' is the acid-to-ore ratio expressed as

<caption>**Table 6**</caption>

General Factorial Regression: Scale mass (g) versus Solid content and Acid dosage.

<table><caption>**Table 7**</caption><thead><tr><th>Source</th><th>DF</th><th>Adj SS</th><th>Adj MS<br>(Adj SS / DF)</th><th>F-Value</th><th>P-value</th></tr></thead><tbody><tr><td>Model</td><td>8</td><td>53.785</td><td>5.976</td><td>9348.23</td><td>1.11 ×<br>10<sup>-16</sup></td></tr><tr><td>Linear</td><td>4</td><td>50.071</td><td>12.518</td><td>19,581.28</td><td>1.11 ×<br>10<sup>-16</sup></td></tr><tr><td>Solid<br>content</td><td>2</td><td>43.024</td><td>21.512</td><td>33,650.73</td><td>1.11 ×<br>10<sup>-16</sup></td></tr><tr><td>Acid dosage</td><td>2</td><td>7.047</td><td>3.524</td><td>5511.83</td><td>1.27 ×<br>10<sup>-14</sup></td></tr><tr><td>Solid*Acid</td><td>4</td><td>3.711</td><td>0.928</td><td>1451.23</td><td>1.24 ×<br>10<sup>-12</sup></td></tr><tr><td>Error</td><td>9</td><td>8E-03</td><td>9E-04</td><td></td><td></td></tr><tr><td>Total</td><td>17</td><td>53.790</td><td></td><td></td><td></td></tr></tbody></table>

Experimental and Predicted values of Scale mass (g) Using 2² Factorial Design in Eq. (19).

<table><thead><tr><th colspan="2">levels (2<sup>2</sup>)</th><th colspan="3">Scale mass (g)</th></tr><tr><th>Solid ratio</th><th>A/O ratio</th><th>Exp. value</th><th>Predicted</th><th>ε</th></tr></thead><tbody><tr><td>0.22</td><td>0.25</td><td>0.114</td><td>0.113</td><td>0.001</td></tr><tr><td>0.22</td><td>0.45</td><td>0.746</td><td>0.744</td><td>0.002</td></tr><tr><td>0.30</td><td>0.25</td><td>4.062</td><td>4.033</td><td>0.029</td></tr><tr><td>0.30</td><td>0.45</td><td>4.861</td><td>4.811</td><td>0.050</td></tr><tr><td>0.26</td><td>0.35</td><td>*0.010</td><td>0.011</td><td>-0.001</td></tr><tr><td colspan="5">R<sup>2</sup> = 0.997</td></tr><tr><td colspan="5">RMSE = 0.0037 g (1.89 %)</td></tr></tbody></table>

<footnote>* No scale formation observed.</footnote>
0.25, 0.35 or 0.45. CP is the centre point.

Similar to the ‘linear + quadratic terms” in Eqs. (15) – (17), the parameters in Eq. (19) show that scale formation follows yet another pattern. Both solid content and A/O have positive effects, with coefficients of 0.4669 and 1.125, respectively, indicating that increasing either factor leads to more scaling. The interaction term of 0.0923 is also positive, though small, implying a slight reinforcing effect when both parameters are high. However, the most striking observation in this equation is the strongly negative coefficient of -2.4143 for the centre point, which suggests a nonlinear relationship in scaling behaviour. This could mean that at intermediate values of S<sub>p</sub> and A, the scaling tendency does not follow a simple linear trend and may be influenced by additional factors.

The centre point (CP) is set to 1 only when conditions are precisely at the centre point (26 % solid and 0.35 A/O). Otherwise, CP is set to 0. Since the scaling response is non-linear, this approach allows the model to distinguish the effects specifically associated with the centre point from those influenced by other factors.

The results obtained when Eqs. (17) and (19) are used to predict the extent of scaling in comparison with the experimental data are shown in Fig. 6. The orange-circled points in Fig. 6(A and B) denote experimental conditions set at 0.01 g, under which no scale formation was observed.

## 3.5. Multilevel response optimisation

Fig. 7 summarises the effects of solid content and acid-to-ore ratio on Ni yield, selectivity and scale mass formed during laterite ore leaching at 265 °C in the autoclave. Generally, higher A/O increased nickel extraction, with Ni yield > 95 % achieved at a moderate solid content (< 28 wt.%). The best selectivity (≥ 4) is obtained at low solid content (< 24 wt.%) and low A/O ratios < 0.30. However, this region does not correspond to the highest Ni yield. Minimal scale formation (<1 g) is achieved at lower solid contents (< 28 wt.%) with an adequate amount of acid. Increasing the A/O ratio and solid content leads to more scaling. The results of the 3 responses show varying patterns, either by increasing the A/O ratio and vice versa to achieve a favourable Ni yield and selectivity or reducing the quantity of scale formed in the process. A compromise must be made between these parameters to balance high Ni yield, high selectivity, and low scale formation. While increasing A/O improves yield, it also increases impurity dissolution and scale formation. Lowering A/O improves selectivity and reduces scaling but compromises Ni yield. Thus, an optimal operating window must be identified where Ni yield remains high while minimising scale and impurity dissolution.

Table 8 presents the multilevel response optimisation results for balancing Ni yield, Ni selectivity, and scale formation. The goal was to find compromise solutions that maximise Ni yield and selectivity while minimising scaling. Five optimal solutions were identified, each prioritising one of these outputs.

To minimise scale formation while maintaining moderate selectivity, a lower slurry solid content and medium acid dosage are required, resulting in a high Ni yield between 93 and 95 %. Among the identified solutions, Solution 1 (22 wt.% solid content and an A/O ratio of 0.35) demonstrated the highest composite desirability (0.68), as seen in Table 8 and gave the best trade-off between these three outputs.

Table 9 presents the optimum settings derived from the DOE analysis in Minitab. The associated 95 % confidence intervals represent the range within which the true mean of each response is expected to lie under the same experimental conditions. For Ni yield, the predicted value is 95.20 %, with a narrow 95 % confidence and prediction interval (94.48–95.92 %) and a standard error of 0.313, indicating high model precision and reliability. Selectivity is optimised at 2.06, with a low standard error of 0.014 and a tight confidence interval of 2.03–2.09, demonstrating strong predictive accuracy and consistency. Scale formation is minimised at 0.01 g/run, with a standard error of 0.021 and prediction bounds ranging from -0.04 to 0.06. Although the lower bound includes a slightly negative value, likely due to model curvature arising from the uncertainty in precipitation reaction and behaviour in the autoclave, the overall low predicted scale indicates favourable operating conditions.

## 4. Conclusion

This study demonstrates the critical role of slurry solid content and acid-to-ore ratio in optimising Ni yield, selectivity, and scale formation. Based on the experimental studies and design of experiments conducted at the optimised temperature of 265 °C and an agitation speed sufficient to maintain a homogenous mixture without solid settling or slurry splashing, a regression model was successfully developed to predict Ni yield, selectivity, and scale formation under different A/O ratios and solid contents, providing a reliable tool for process optimisation.

The experimental design and statistical modelling provided a strong framework for identifying optimal process conditions. The acid-to-ore ratio emerged as a critical factor influencing Ni extraction yield, selectivity (leachate purity), and scale formation. Increasing A/O ratios (0.45) generally enhanced Ni yield, but this came at the cost of reduced selectivity and increased scale formation. Conversely, lowering A/O (0.25) improved selectivity and minimised scaling but resulted in lower Ni recovery.

The slurry's solid content does not significantly impact Ni extraction rate or purity when an equivalent acid dosage is applied. However, it plays a notable role in accelerating scale formation at higher solid content (30 wt.%).

Optimal leaching conditions were achieved using theoretical acid dosages (A/O = 0.35) and a lower solid content ≤ 26 wt.% (preferably

<figure><img src="image_9.png" /><figcaption>Fig. 6. Experimental versus predicted data for the scale formation regression model. (A) Eq. (17), (B) Eq. (19).</figcaption></figure>
<figure><img src="image_10.png" /><figcaption>Fig. 7. Optimisation Trade-offs: Effect of Solid Content and Acid/Ore Ratio on Ni Yield, Selectivity, and Scale mass.</figcaption></figure>

<table><caption>**Table 8**
Multilevel Response Optimisation on Minitab® – Ni yield (wt.%), Selectivity and Scale (g).</caption><thead><tr><th colspan="5">Optimisation condition</th><th rowspan="2">Solution</th><th rowspan="2">Solid (%)</th><th rowspan="2">Acid: Ore</th><th colspan="3">Fits</th><th rowspan="2">Composite Desirability</th></tr><tr><th>Response</th><th>Goal</th><th>Lower</th><th>Target</th><th>Upper</th><th>Ni yield (wt.%)</th><th>Selectivity</th><th>Scale (g)</th></tr></thead><tbody><tr><td>Ni yield</td><td>Maximise</td><td>67.15</td><td>98.15</td><td>-</td><td>1</td><td>22.0</td><td>0.35</td><td>95.20</td><td>2.06</td><td>0.01</td><td>0.68</td></tr><tr><td>Selectivity</td><td>Maximise</td><td>0.68</td><td>4.67</td><td>-</td><td>2</td><td>26.0</td><td>0.35</td><td>93.09</td><td>1.97</td><td>0.01</td><td>0.65</td></tr><tr><td>Scaling</td><td>Minimise</td><td>-</td><td>0.01</td><td>4.86</td><td>3</td><td>30.0</td><td>0.35</td><td>91.68</td><td>2.36</td><td>1.94</td><td>0.59</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td>4</td><td>22.0</td><td>0.25</td><td>72.59</td><td>4.62</td><td>0.11</td><td>0.56</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td>5</td><td>22.0</td><td>0.45</td><td>97.52</td><td>0.84</td><td>0.74</td><td>0.32</td></tr></tbody></table>

The set optimisation condition cannot exceed or fall below the values obtained from the experimental campaign.

<table><caption>**Table 9**
Confidence Intervals of the Optimum Multiple Response Prediction.</caption><thead><tr><th>Variable</th><th>Setting</th><th>Response</th><th>Fit</th><th>SE Fit</th><th>95 % Lower Confidence Bound</th><th>95 % Lower Prediction Bound</th></tr></thead><tbody><tr><td>Solid (%)</td><td>22</td><td>Ni yield (wt.%)</td><td>95.20</td><td>0.313</td><td>94.48</td><td>95.92</td></tr><tr><td rowspan="2">Acid:Ore</td><td rowspan="2">0.35</td><td>Selectivity</td><td>2.06</td><td>0.014</td><td>2.03</td><td>2.09</td></tr><tr><td>Scale rate (g/run)</td><td>0.01</td><td>0.021</td><td>-0.04</td><td>0.06</td></tr></tbody></table>

22 wt.% specific to the range conducted in this study), yielding improved solution purity and reducing scale formation risk, although with a slightly lower Ni yield compared to a higher acid-to-ore ratio (A/O = 0.45). Controlling the acid dosage during leaching is crucial. This balanced approach allows for efficient extraction with minimal scale buildup, promoting a more stable and manageable continuous leaching process in the industry. These findings emphasise the necessity of a process compromise to achieve optimal efficiency in hydrometallurgical Ni extraction.

### Disclaimer

Portions of information contained in this publication are printed with permission of Minitab®, LLC. All such material remains the exclusive property and copyright of Minitab, LLC. All rights reserved. www.minitab.com.

### Funding

This work was funded by the Association Nationale Recherche Technologie (ANRT-CIFRE), France, under grant [number 2021/1702].

### CRediT authorship contribution statement

**Okechukwu Vincent Dickson:** Writing – review & editing, Writing – original draft, Visualization, Methodology, Investigation, Formal analysis, Data curation, Conceptualization. **Thomas Deleau:** Writing – review & editing, Supervision, Formal analysis. **Fabienne Espitalier:** Writing – review & editing, Visualization, Validation, Supervision, Software, Investigation, Data curation. **Christophe Coquelet:** Writing – review & editing, Supervision, Formal analysis, Data curation. **Julien Lombart:** Validation, Supervision, Resources, Project administration,

Funding acquisition.

### Declaration of competing interest

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.

### Acknowledgements

The authors wish to thank Prony Resources, New Caledonia, Chaire MINAUMET - Mines Paris - PSL, and the Association Nationale Recherche Technologie (ANRT-CIFRE), France, for supporting this study.

### Data availability

Data will be made available on request.

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