Sedimentation Analysis of Galena Concentrate and Predictability of Effective Particle Depth Based on Settling Time and Particle Diameter

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Figure 2. Pulverized Pb-Zn ore

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Figure 3. Galena Concentrate used for sedimentation analysis

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3. Results and Discussion

3.1. Model Formulation

Experimental data obtained from research workwere used for this work. Computational analysis of the experimental data shown in Table 1, gave rise to Table 2 which indicate that;

Table 1. Results of Sedimentation Analysis

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(3)

Introducing the values of N, K, S, and S_e into equation (3)

(4)

(5)

Where

(β) = Effective depth of particle (m)

(γ) = Particle diameter (μm)

(α)= Settling time (mins.)

N, K, S and S_eare equalizing constants with values 1.5 x 10^-3, 2 x 10^-⁵, 5 x 10^-⁴and 0.2176 respectively (determined using C- NIKBRAN ^[14])

Table 2. Variation of β - 1.5 x10^-3with 2 x 10^-5 γ²-5 x 10^-4γ + 0.2176

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4. Boundary and Initial Condition

concentrate for the sedimentation analysis was prepared and associated apparatus set according to the conventional procedure (BS 1377:1975, Test 6(B)). Before the start of the leaching process, the hydrometer was assumed to be initially free of attached unwanted bacteria and other micro organism. The effect of oxygen on the process was assumed to be atmospheric since it is an open system process. In all cases, weight of galena used was 50g. The range of the settling time considered: 4-300 seconds. A constant leaching temperature of 25^oC and average ore grain size of 150µm were also used. Details of the experimental technique and other process conditions are as presented in the experiment.

A non-zero gradient was assumed for the liquid scalar at the bottom of the particles and for the gas phase at the top of the particles. The sides of the particles were assumed to be symmetries.

4.1. Model validation

The validity of the model is strongly rooted on equation (4) where both sides of the equation are correspondingly approximately equal to 0.2. Table 2 also agrees with equation (4) following the values of β - 1.5 x 10^-3and 2 x 10^-5 γ²- 5 x 10^-4γ + 0.2176 evaluated from the experimental results in Table 1. Furthermore, the derived model was validated by comparing the effective depth of particles predicted by the model and that obtained from the experiment. This was done using various analytical techniques.

4.2. Computational Analysis

Computational analysis of the experimental and model-predicted settling rate was carried out to ascertain the degree of validity of the derived model. This was done by comparing the settling rates obtained by calculations involving experimental results with the model-predicted results.

Settling rate S_R (m/mins.) was calculated from the equation;

(6)

Equation (6) is detailed as

(7)

Where

ΔS_D = Change in the settling depths S_D₂, S_D₁ within a range of settling time: t₁- t₂.

Δt = Change in the leaching timest₂, t₁_.

Therefore, a plot of effective depth of particles against settling time as in Figure 1 using experimental results in Table 1, and substitution of points (0.1333, 0.2111) and (5, 0.2234) for (t_1,S_D₁ ) and (t_2,S_D2) respectively into the mathematical expression in equation (7) gives 0.0025 m/mins as the settling rate of the galena particles during the actual sedimentation process.

Similarly, a plot of effective depth of particles against settling time as in Figure 2 using model-predicted results, and substitution of points (0.1333, 0.2147) and (5, 0.2243) for (t_1,S_D₁ ) and (t_2,S_D2) respectively into the mathematical expression in equation (7) gives 0.002 m/mins as the model-predicted settling rate of the galena particles. These settling rates evaluated from experiment and derived model indicate proximate agreement and hence validity of the model.

Figure 4. Coefficient of determination between effective depth of particle and settling time as obtained from the experiment

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Figure 5. Coefficient of determinationbetween effective depth of particle and settling time as predicted by model

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4.3. Statistical Analysis

The standard errors (STEYX) incurred in predicting the effective particle depth for each value of settling time & particle diameter considered as evaluated from experiment and derived model are 0.0018 and 0.0027 & 0.0005 and 0.0024 respectively. The standard errors were evaluated using Microsoft Excel 2003 ^[7].

Figure 6. Coefficient of determination between effective depth of particle and particle diameter as obtained from the experiment

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The correlations between effective depth of particle and settling time as well as effective depth of particle and particle diameter as obtained from derived model and experimental results were calculated. This was done by considering the coefficients of determination R2 from Figure 1-Figure 4, using the equation;

(8)

The evaluated correlations are shown in Table 3 and Table 4 for experimental and model-predicted results which are in proximate agreement.

Figure 7. Coefficient of determination between effective depth of particle and particle diameter as predicted by model

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Table 3. Comparison of the correlations between effective depth of particle and settling time as evaluated from experimental and derived model

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Table 4. Comparison of the correlations between extraction rate and final pH of leaching solution as evaluated from experimental and derived model

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4.4. Graphical Analysis

Comparative graphical analysis of Figure 8 and Figure 9 shows very close alignment of the curves from model-predicted effective particle depth (MoD) and that of the experiment (ExD). The degree of alignment of these curves is indicative of the proximate agreement between both experimental and model-predicted effective particle depth.

Figure 8. Comparison of effective depths of particles (relative to settling time) as obtained from experiment and derived model

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Figure 9. Comparison of effective depths of particles (relative to particle diameter) as obtained from experiment and derived model

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Graphical analysis of Figure 8-Figure 10 show very close alignment of the curves from derived model (D-MoD) and experimental (ExD) results of effective particle depth. It is strongly believed that the degree of alignment of these curves is indicative of the proximate agreement between ExD and MoD predicted results. Figure 10 shows a two-factorial dependency of effective particle depth; on settling time and particle diameter. This indicates that for any pre-determined values of settling time and particle diameter within the boundary conditions, the effective depth reached by the particles during descend could be predicted.

Figure 10. Comparison of effective depths of particles (jointly relative to settling time and particle diameter) as obtained from experiment and derived mode

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4.5. Deviational Analysis

Comparative analysis of effective particle depth from experiment and derived model revealed deviations on the part of the model-predicted values relative to values obtained from the experiment. This is attributed to the fact that the surface properties of the galena particles and the physiochemical interactions between the particles and the dispersing agent (sodium hexametaphosphate) which played vital roles during the sedimentation processwere not considered during the model formulation. This necessitated the introduction of correction factor, to bring the model-predicted effective particle depth to those of the corresponding experimental values.

Deviation (Dn) of model-predicted effective particle depth from that of the experimentis given by

(9)

Correction factor (Cr ) is the negative of the deviation i.e

(10)

Therefore

(11)

Where

P^D = Model-predicted effective particle depth (m)

E^D = effective particle depth from experiment (m)

Cr = Correction factor (%)

Dn = Deviation (%).

Figure 11. Comparison of effective depths of particles (relative to deviation of model-predicted results from experimental values)

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Introduction of the corresponding values of Cr from equation (11) into the model gives exactly the corresponding experimental effective particle depth:

(12)

Figure 11 shows that the maximum deviation of the model-predicted effective particle depth from the corresponding experimental values is less than 3% and quite within the acceptable deviation limit of experimental results.

These figures show that the least and highest magnitudes of deviation of the model-predicted effective particle depth (from the corresponding experimental values) are +0.22 and + 2.29% which corresponds to effective particle depths: 0.2227 and 0.2147 m and settling times: 4 and 0.0667 mins. and particle diameters: 1.97 and 14.83 μm respectively.

Figure 12. Comparison of effective depths of particles (relative to correction factor to model-predicted results)

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Figure 13. Variation of model-predicted effective particle depth with associated deviation and correction factor

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