Process stream inference and response surface methodology execution

A detailed account of the experimental procedure is provided in Section 11.2 (of Appendix C). The experimental design used was a 4-factor Box-Behnken design to determine concentrate and tailings stream responses as feed grade (𝑥1), feed SG (𝑥2), feed flow rate (𝑥3) and primary splitter settings (𝑥4) change. Measured responses include concentrate and tailings flow rates and densities obtained during sampling. The remainder of this section discusses the inference used to determine grade(s) and recovery(s) – to be used in regression modelling.

Solids densities and solids composition must be determined from total mass flowrates, solids mass flow rates, volumetric flowrates and density measurements that were collected, from feed and product

47 streams, after experimentation. Equation 4.3 can be used to determine the solids density of feed or product streams (incompressibility of liquid and solids phases is assumed).

1 𝑝_{𝑠𝑙𝑢𝑟𝑟𝑦} = 𝑋₂ 𝑝_{𝑠𝑜𝑙𝑖𝑑𝑠}+ 1−𝑋₂ 𝑝_𝑤 (Eq. 4.3) Where: • Slurry density is 𝑝_{𝑠𝑙𝑢𝑟𝑟𝑦}.

• Water density 𝑝_𝑤 is equal to 0.99 g/cm3_.

• 𝑋₂ represents solids mass fraction. • Solids density is 𝑝_{𝑠𝑜𝑙𝑖𝑑𝑠}.

A rough estimation of solids composition can be obtained by setting up a calibration curve between solids density and the concentration of a MOI. The chromite separation problem is simplified by considering the feed slurry as a combination of HM and light mineral (LM) species which is separated in the spiral product streams. Using this simplification, the MOI of this problem becomes the HM and the spiral concentration problem becomes concerned with HM recovery and grade in the concentrate stream (stream 7 in Figure 11.8). The shape of the calibration curve is given by Equation 4.4.

𝑝𝑠𝑜𝑙𝑖𝑑𝑠= 𝑏1+ 𝑏2𝑋3 (Eq. 4.4) Where:

• 𝑏₁ and 𝑏2 are regression parameters. • 𝑋₃ is the HM grade (as mass fraction).

The regression parameters of Equation 4.4 can be solved by using the solids density and composition analysis results from the representative samples obtained from homogenised ore to be used as spiral feed during the experimental tests. HM serves as a good indicator of 𝐶𝑟2𝑂3 and 𝐹𝑒2𝑂3 content and LM is an indicator of 𝑆𝑖𝑂2 and 𝐴𝑙2𝑂3 content.

If XRF analysis of all experimental samples are available then 𝐶𝑟2𝑂3 grade and recovery can be directly computed, however, obtaining true 𝑝𝑠𝑜𝑙𝑖𝑑𝑠 values can still be challenging. XRF provides major component composition analysis of solids but does not provide composition of different mineral phases (which in turn can be used to determine solids density precisely). Equation 4.3 is still required when all samples can be analysed via XRF which also implies that Equation 4.4 can be prepared to validate XRF results (for future composition analysis).

Because all stream measurements are repeated 3 times during experimentation it is possible to prepare rough estimates of all measurement expected values and variances. Once these values are available it is possible to perform reconciliation of the measurements using the spiral equipment mass balance and assuming all measurement distributions follow the Gaussian distribution. It is assumed that gross

48 errors are not present. It is also assumed that steady-state is achieved. Because every stream around the spiral concentrator, was observed, reconciliation implies that improved variable expected values and variances can be obtained.

The connectivity matrix 𝑀 for the mass flow rates of the spiral in Figure 11.8 is 𝑀 = [1, −1, −1]. Total mass flow rate, solids fractions and HM grades can be reconciled to give more precise expected values and variances for the measured variables. The reconciliation problem can be stated, using Equation 3.11 from Section 3.5, as Equation 4.5.

min 𝑋 𝐽 = (𝑌 − 𝑋) 𝑇_𝑉−1_{(𝑌 − 𝑋) (Eq. 4.5)} s.t. [ 1 −1 −1 0 0 0 0 0 0 0 0 0 1 −1 −1 0 0 0 0 0 0 0 0 0 1 −1 −1 ] [ 𝑋1 𝑋1⨀𝑋2 𝑋1⨀𝑋2⨀𝑋3 ] = 𝑧(𝑋) = 0 Where:

• 𝑌 contains measured mass flow rate, solids fraction and HM grade averages of feed and product streams.

• 𝑉 is the variance values flow rate, solids fraction and HM grade variables. • 𝑋 = [𝑋𝑋21

𝑋3 ].

Equation 4.5 can be solved using successive linearization (see Section 11.3.1). The linearized constraints (with linearization performed around 𝑌) are given by Equation 4.6. During successive linearization, constraints are linearized around the current iteration’s 𝑋̂ (after the initial iteration). Solutions of Equation 4.5 will have the same form as Equation 3.13. 5 Successive linearization steps were used to perform the data reconciliation of experimental data. Variance updates can be performed as show by Equation 3.14. 𝑧(𝑌) + [ 𝑀 0 0 𝑀⨀𝑋2𝑇 𝑀⨀𝑋1𝑇 0 𝑀⨀(𝑋1⨀𝑋2)𝑇 𝑀⨀(𝑋1⨀𝑋2)𝑇 𝑀⨀(𝑋1⨀𝑋2)𝑇 ] (𝑋 − 𝑌) = 0 (Eq. 4.6)

After spiral concentrator feed flow rates, solids fractions and grades (of the feed and product streams) have been reconciled, new estimates of spiral feed settings can be attained (giving precise estimates of the feed conditions obtained during experimentation – see Section 9.3.1). New feed HM grade estimates are directly obtained after reconciliation which, in turn, can be used to find feed 𝑝𝑠𝑜𝑙𝑖𝑑𝑠 according to Equation 4.4. Finally, feed slurry density can be calculated with Equation 4.3 which can be used to find the feed volumetric flow rate. The spiral output responses considered for modelling are tailings and concentrate HM grade, HM recovery and concentrate-tailings flow ratio (simply the mass

49 flow rate of concentrate divided by the mass flow rate of tailings). A flow ratio model gives a method with which to determine the overall mass balance around a spiral. Grade and recovery models allow calculation of solids and HM balances around a spiral. The same feature set is used to formulate all response models. Lastly, interface responses are also modelled to determine detection variance for sensor placement (see Sections 4.5 and 11.3).

Regression modelling of spiral responses were performed using MATLAB’s fitlm() (for linear or full quadratic models) and stepwiselm() (for linear or full quadratic model parameter selection) functions from MATLAB’s Statistics and Machine learning Toolbox. The function stepwiselm() reduces full quadratic models (see Equation 9.14 in Section 9.2) by keeping the significant parameters that minimize some model criterion. Default settings for stepwiselm() were used during spiral response model fitting which means that parameters are removed from the full quadratic model in a way that minimizes the regression model squared sum of errors. This will also tend to reduce F-test p-values resulting in models with improved statistical significance. Parameters are removed one by one based on their coefficient t-test p-values (Montgomery, 2001: 412). The parameter with the highest p-value is removed and then regression is repeated to find the new coefficient with the highest p-value. This process is repeated until none of the remaining coefficients have p-values higher than some threshold. Linear terms with p-values higher than the default threshold can still be included in the model if interaction or higher order terms of those linear terms are significant. The default coefficient p-value threshold, of stepwiselm() as implemented in MATLAB, is 0.1.