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Rare earth elements are strategic commodities in many countries, and an important resource for the growing modern technology industry. As such, there is an increasing interest for development of rare earth element processing, and this work is a part of further development of chromatography as a rare earth element separation process method. Process optimization is pivotal for process development, and it is common that several competing objectives must be regarded. Chromatographic separation processes often consider competing objectives, such as productivity, yield, pool concentration and modifier consumption, which leads to Pareto optimal solutions. Adding robustness to a process is of great importance to account for process disturbances and uncertainties but generally comes with reduced performance of the other process objectives as a trade off. In this study, a model-based robust multi-objective optimization was carried out for batch-wise chromatographic separation of the rare earth elements samarium, europium and gadolinium, which was considered highly un-robust due to the neighbouring peaks proximity to the product pooling horizon. The results from the robust optimization were used to chart the required operation point changes for keeping the amount of failed batches at an acceptable level when a certain level of process disturbance was introduced. The loss of process performance due to the gained robustness was found to be in the range of 10% - 20% reduced productivity when comparing the robust and un-robust Pareto solutions at Pareto points with identical yield. The methodology presented shows how to increase robustness to a highly un-robust system while still keeping multiple objectives at their optima.

Rare earth elements (REE) are extensively used in modern technological industries [

Since mathematical modelling offers a cost efficient and powerful approach for assessing preparative chromatography [

Approaches for introducing robustness to process optimizations are readily available in literature, and robust optimization for chromatography in particular are, amongst others, presented in [

The main focus of this study was to perform a robust multi-objective optimization of chromatographic REE separation. In this context, an experimentally validated process model from a previous study [

The chromatography model in this work is based on an experimental study of batch chromatography separation of Sm, Eu and Gd [

The chromatography model in this study has been presented in a previous publication [

∂ c α ∂ t = − ∂ ∂ z ( c α v int − D app, α ∂ c α ∂ z ) − ( 1 − ε c ) ε c + ( 1 − ε c ) ε p ∂ q α ∂ t , (1)

∂ q α ∂ t = k kin , α ( c α K e q , α q max , α [ 1 − ∑ γ ∈ { Sm,Eu,Gd } q γ q max , γ ] − q α c S ν α ) , (2)

where c α and q α are the mobile and solid phase concentration of component α ∈ { Sm , Eu , Gd , S } , v int is the quotient of superficial velocity over total porosity, D app,α the apparent dispersion coefficient, and ε c and ε p the column and particle void fractions. Here, c S denotes the concentration of the modifier (i.e. nitric acid), k kin , α a parameter describing the kinetics, K e q , α the equilibirum constant regarding adsorption and desorption, ν α a parameter describing the ion-exchange characteristics, and q max , α the maximum concentration of adsorbed components. The model does not consider modifier ions on the solid phase, therefore Equation (2) and its associated part in Equation (1) are omitted (i.e. ∂ q α / ∂ t ≡ 0 ) when α = S . Equation (1) is complemented with Danckwert boundary conditions [

c α ( t , z 0 ) v int − D app,α ∂ c α ∂ z ( t , z 0 ) = ( c load , α v int Π ( t , t 0 , Δ t load ) if α ∈ { Sm,Eu,Gd } , c mix , S v int if α = S , (3)

∂ c α ∂ z ( t , z f ) = 0, ∀ α ∈ { Sm , Eu , Gd , S } (4)

where c load , α is the injected load concentration, and Π ( t , t 0 , Δ t load ) ∈ { 0,1 } a rectangular function in the temporal horizon [ t 0 , Δ t load ] . The dynamics of the modifier concentration in the upstream mixing tank, c mix , S , are given by:

d c mix , S d t = 1 τ mix ( u ( t ) − c mix , S ) , (5)

u ( t ) = ( u 0 , if t ≤ Δ t load + Δ t wash , u 0 + t − ( Δ t load + Δ t wash ) , if t > Δ t load + Δ t wash , (6)

where τ mix is the residence time, u is the elution gradient described by the initial value, u 0 , and the slope of the linear elution gradient, Δ u , expressed as

Δ u = u f − u 0 t f − ( Δ t load + Δ t wash ) .

The first-order spatial derivative in Equation (1) was approximated using a method-of-lines and finite volume method with 100 grid points where z k = k Δ z is the discretized spatial coordinate and k ∈ [ 1 , 100 ] . The first order derivative was approximated as a two-point backward difference, and the second-order derivative was approximated as a three-point central difference. The model parameter values from [

The multi-objecitve optimization problem formulation in this work resembles that of a previous study [

cut-times [ t c , t f ] are defined as:

δ load , α d Y α d t = c α ( t , z f ) v int A c Π ( t , t c , t f ) , (7)

d P α d t = 1 V c 1 t f + t r δ load , α d Y α d t , (8)

where δ load , α = c load , α v int A c Δ t load is the total amount of injected sample, A c and V_{c} the column cross-sectional area and volume, and t r = 2 V c Q ˙ − 1 the regeneration and re-equilibration time following the final cut-time. Thus, the objective becomes to determine an optimal elution gradient, u , batch load, δ load , α , and pooling cut-times, [ t c , t f ] , that maximizes Y α ( t f ) and P α ( t f ) , while fulfilling the target component purity constraint given by:

X α ( t f ) = δ load , α Y α ( t f ) ∑ β ∈ { Sm , Eu , Gd } δ load , β Y β ( t f ) , (9)

where the numerator is the captured amount of the target component in [ t c , t f ] and the denominator represents the total amount of captured components.

The weighted sum scalarization method was used to combine the objectives in Equations (7)-(8) to a single objective with the weight for productivity defined as ω ∈ [ 0,1 ] , and the weight for yield defined as 1 − ω . The decision variables are the free operating parameters, i.e. Δ t load , t c , t f , u 0 and u f , which in turn determine the trajectories x = ( c α ( t , z k ) , c S ( t , z k ) , c mix , S ( t ) , q α ( t , z k ) , P Eu ( t ) , Y Eu ( t ) , X Eu ( t ) ) . The resulting optimization problem can then be set in the framework for min-min optimal control:

min . − ( ω ∫ t 0 t f d P Eu d t d t + ( 1 − ω ) ∫ t 0 t f d Y Eu d t d t ) , (10a)

w .r .t . p = ( Δ t load , u 0 , u f ) ∈ ℝ 3 ,

s .t . p L ≤ p ≤ p U , (10b)

( x , t c , t f ) = argmin . − ( ω ∫ t 0 t f d P Eu d t d t + ( 1 − ω ) ∫ t 0 t f d Y Eu d t d t ) , (10c)

w .r .t . ( t c , t f ) ∈ ℝ 2 ,

s .t . x ˙ = F ( t , x ( t ) , t c , t f , p ) , x ( t 0 ) = x 0 , (10d)

X Eu , L − X Eu ( t f ) ≤ 0, (10e)

t c , L ≤ t c ≤ t c , U , t f , L ≤ t f ≤ t f , U , (10f)

∀ t ∈ [ t 0 , t f ] , ∀ z ∈ [ z 0 , z f ] .

The solution of Equation (10) will result in a Pareto front situated on the boundary of the feasible region. This implies that a process disturbance, however slight, can cause a batch failure in terms of not meeting the purity constraint [

In order to formulate a robust counterpart of Equation (10), we consider a set of bounded distributed disturbances, p ˜ , on the free operating parameters, p (i.e Δ t load , u 0 and u f ), and define X ˜ Eu as the cumulative purity distribution of the model responses that are produced from p ˜ . A purity constraint back-off term, X BF , is introduced in order to make the purity constraint robust with respect to the disturbances. The back-off term can essentially be seen as a safety margin that amplifies the purity inequality constraint in Equation (10e) so that the purity requirement, X Eu, L , still can be met for the considered set of bounded disturbances. The success rate is defined as the fraction of batches in the disturbance set that fulfil the purity requirement, X Eu, L , and Φ X Eu signifies the desired success rate. The following robust counterpart of Equation (10) is then given by:

min . ∫ − ∞ X Eu , L + X BF X ˜ Eu d X Eu − Φ X Eu , (11a)

w .r .t . X BF ,

s .t . x ˜ ˙ = F ˜ ( t , x ( t ) , t c , t f , p ˜ ) , (11b)

p ˜ ~ N ( p , σ p 2 ) , (11c)

p = argmin . − ( ω ∫ t 0 t f d P Eu d t d t + ( 1 − ω ) ∫ t 0 t f d Y Eu d t d t ) , (11d)

w .r .t . p = ( Δ t load , u 0 , u f ) ∈ ℝ 3 ,

s .t . p L ≤ p ≤ p U , (11e)

( x , t c , t f ) = argmin . − ( ω ∫ t 0 t f d P Eu d t d t + ( 1 − ω ) ∫ t 0 t f d Y Eu d t d t ) , (11f)

w .r .t . ( t c , t f ) ∈ ℝ 2 ,

s .t . x ˙ = F ( t , x ( t ) , t c , t f , p ) , x ( t 0 ) = x 0 , (11g)

( X Eu , L + X BF ) − X Eu ( t f ) ≤ 0, (11h)

t c , L ≤ t c ≤ t c , U , t f , L ≤ t f ≤ t f , U , (11i)

∀ t ∈ [ t 0 , t f ] , ∀ z ∈ [ z 0 , z f ] .

A decomposition strategy was adopted to transform the robust MOP into three levels: 1) the upper-level optimization problem given by Equations (11a)-(11c) with respect to X BF , 2) the mid-level optimization problem given by Equations (11d)-(11e) with respect to p, and 3) the lower-level optimization problem given by Equations (11f)-(11i) and constrained by the ODE system, F , governed by Equations (1), (2), (5), (7), (8), (9). Essentially, Equation (11) was solved by using the simulated system response, x ˜ , for an uncertainty set of the free operating parameters, p ˜ , to evaluate the cumulative distribution function of X ˜ Eu . The back-off term, X BF , in the purity inequality constraint, Equation (11h), was then incrementally increased to gain more successful batches in X ˜ Eu , and thereby achieving a more robust process. This procedure was repeated iteratively until Equation (11a) was fulfilled, at which point Equation (11) was considered to be solved.

In this study, the desired success rate, Φ X Eu , was set to 0.95, and the decision variable boundaries are presented in

The robust optimization method in this study is similar to the methods presented in [

methodology as described in [

As a first step, the nominal and non-robust Pareto front was obtained by solving the MOP as defined in Equation (10). This was carried out through MATLAB’s fmincon function with a sequential quadratic programming algorithm, the BFGS formula for updating the approximation of the Hessian matrix, and central differences to estimate the gradient of the objective function and constraint functions. Then an uncertainty set, p ˜ , with a normal distribution, assuming no covariance between the free operating parameters p , a standard deviation σ , and sampling size of 10.000 was obtained via MATLAB’s lhsnorm function. The uncertainty set was applied to the investigated operating points on the nominal Pareto front, and the model responses were used to evaluate the cumulative purity distribution, X ˜ Eu , of the uncertainty set.

Then, an initial investigation of the back-off terms impact on X ˜ Eu was conducted by creating new Pareto fronts with an incrementally increased back-off and observing how X ˜ Eu changes when p ˜ is applied to the investigated points on the new Pareto fronts. At this stage, it is of particular interest to investigate how the fraction of batches that fulfil the purity requirement in the perturbed set, changes with an increased back-off. This provides an estimate of the required back-off to meet a certain success rate for a given purity constraint.

The required back-off for a given point on the nominal Pareto front was obtained by applying MATLAB’s fminbnd function on the upper level of the robust counterpart problem in Equation (11), with suitable boundaries obtained from the previous back-off investigation. The mid- and lower-level optimization problems in Equation (11) were solved by MATLAB’s fmincon function with a sequential quadratic programming algorithm, the BFGS formula for updating the approximation of the Hessian matrix, and central differences to estimate the gradient of the objective function and constraints. The procedure comprises an evaluation of the cumulative distribution function of X ˜ Eu based on x ˜ and p ˜ , as obtained from the mid- and lower-level optimization problem for a given initial X BF . X BF is then varied for the upper level optimization problem through MATLAB's fminbnd function, resulting in new x ˜ , p ˜ and cumulative distribution functions of X ˜ Eu to be evaluated. This continues until a X BF that produces a cumulative distribution function of X ˜ Eu corresponding to the desired success rate Φ X Eu is obtained.

Optimization Method BenchmarkingThe proposed optimization method was compared with a previous optimization method [

The robust multiobjective optimizations of the studied system were carried out with a product purity requirement, X Eu , L , of 0.95 and 0.99 respectively, and the target success rate, Φ X Eu , was set to 0.95. The perturbed process parameters were the injected load concentration, c load , α , and the modifier concentration in the upstream mixing tank, c mix , S . Several values for the uncertainty set standard deviation, σ , were investigated. However, a standard deviation exceeding 0.01 did not result in achieving the target success rate even when the back-off was set to the maximum limit, i.e. X Eu , L + X BF = 1 . Therefore, only results for σ = 0.01 are presented in this study. This should be interpreted as that the system is very un-robust, and sensitive to even the slightest process disturbances. However, this was expected since the studied elements are extremely similar in both chemical and physical properties, resulting in a minute separation selectivity which in turn makes the separation very difficult and unforgiving towards process perturbations.

The nominal un-robust Pareto fronts are presented by the outermost fronts in

The results from the initial investigation on how the success rate, Φ X Eu , for the investigated points on the nominal Pareto front increases with an increased back-off are shown in

It should be noted that the Pareto point corresponding to yield as a single objective, i.e. ω = 0 , has been omitted since the point for maximum yield is considered as an undesirable operating point due to the drastic decrease in productivity.

It is somewhat counter intuitive that the success rate should decrease with an increased objective weight for yield, since a higher yield typically is associated with an increased peak separation which in turn should result in an increased robustness. The decrease of robustness can be explained by observing how the decision variables change with an increased back-off for the 0.95 purity requirement case in

This has the important implication of that a high objective weight on productivity will result in pooling cut times occurring closer to the Eu elution peak centre and farther away from the neighbouring peaks. When a higher yield is desired, the pooling horizon will increase in order to capture more of the target molecules, and this will move the pooling close to, and even into, the neighbouring elution peaks as long as the purity requirement is met. For this reason, a higher weight on yield will demand a higher back-off on purity in order to meet the desired success rate. This is due to that when a perturbation is introduced, the neighbouring peaks may move closer to, and even intrude, the pooling horizon, and a higher purity requirement will move the pooling cut times farther away from the neighbouring peaks. The farther away the cut times are from the neighbouring peaks in the nominal case, the higher disturbance can be tolerated since there is more room available for the neighbouring peaks to move before they impact the purity of the target peak. This is illustrated in

The results from the robust optimizations can be seen in

handle the given process disturbances satisfactorily. However, the proposed method should be favoured since it produces a robust Pareto front with higher objective values compared to the benchmark method, which implies that the benchmark method should be considered more restrictive. Further, the benchmark method generates operating points that cannot be considered Pareto optimal, which is the case for the points with ω = 1 on front (c) in

Applying robustness to a point on the nominal Pareto front with a given ω will result in a change of both productivity and yield, as can be seen in

The required back-off, that meets the robustness requisite according to the proposed method, is presented in

This study has shown that the proposed optimization method can be used for robust multi-objective optimization of chromatographic rare earth element separation, and provided with expected performance changes for the robustification of the studied process. It has been highlighted that the studied system is highly un-robust, and that the system’s lack of robustness is largely due to the neighbouring peaks’ proximity to the product pooling horizon. The system can only cope with slight process disturbances, which in turn demands use of process equipment with high reliability. We show how the optimal solution of a chromatographic separation is affected by introducing robustness in a brute force manner. For future studies, it would be of interest to employ the proposed optimization method on additional chromatography schemes, as well as other chromatography separation applications than rare earth elements.

This study has been performed within ProOpt and Process Industry Centre at Lund University, and financed by VINNOVA.

Knutson, H.-K., Holmqvist, A., Andersson, N. and Nilsson, B. (2017) Robust Multi-Objective Optimization of Chromatographic Rare Earth Element Separation. Advances in Chemical Engineering and Science, 7, 477-493. https://doi.org/10.4236/aces.2017.74034