Team:Calgary/Metal Separation Modelling
PACLan Modelling
Design
To produce an efficient and productive lanthanide recovery system, the produced LanM is immobilized on cellulose beads (CBs) using a cellulose-binding module (CBM). The immobilized LanM-CB complex can then be packed in a fixed-bed adsorber column. In order to design an absorber unit that can be optimized for industrial processes, we must first use modelling and experimentation techniques to gain a better understanding of the operation of an adsorber.
Our motivation for developing a fixed-bed adsorber column was based on the ability to re-use and recycle the LanM-CBs. In addition the packed column design allows for greater mixing of the process fluid within the column, thus leading to a more efficient separation process. The steps for the operation of a fixed-bed adsorber column to recover REEs is as follows
Adsorption: Pumping E-waste leachate containing REEs as the incoming stream at a given input velocity
Washing: pumping acidic (pH >2.5) washing solution to eliminate non-REE metals from the column. The solution containing the mixture of non-REE metal ions can be fed into existing industry standards for metal recovery.
Elution: pumping acidic (pH < 2.5) elution fluid to denature LanM [1] to release REEs and elute out a high-yield and pure REE liquid product
Figure 1.Proposed three-step metal separation process
While the use of a fixed-bed adsorber column for bioseparation processing is common , the system we are proposing has not been studied or implemented at an industrial scale d. This posed challenges in the initial stages of our design process as we needed to determine whether or not the proposed system, the Packed Adsorber Column for Lanthanides (PACLan), was a viable design . Thus, we needed to first employ modelling simulations to examine the effectiveness and viability of the proposed system, and how to optimize process parameters such that the process would meet the requirement for industry standards. To evaluate the model simulations, a prototype can be built and tested at the laboratory scale. An overview of laboratory scale experiments of lanthanide adsorption onto LanM and cellulose can be found here.
Overview
We implemented a simple yet robust set of models to answer the following questions:
Does the model demonstrate that PACLan is a viable solution to meet our operational criteria?
How do we change the controllable parameters to optimize the system?
How does the viability of the PACLan design change if we take into account non-idealities reflective of real-world processes?
Mass transfer phenomena in PACLan
For the Nd3+ to be separated from the fluid phase and adsorbed onto the lanM-CBs, Nd3+ ions must:
Travel to the functionalized bead through the process of forced convection
Diffuse through the film of fluid surrounding the particle
Find its way onto a LanM-CBM complex
Bind to LanM
These steps do not occur instantaneously, and the rates at which these happen are driven by the physics of the operation, such as the effects of concentration and fluid velocity [2].
When the PACLan recovery process is implemented on a larger scale, REE-rich leachate is passed through a column packed with the LanM-CBs. In the operation of this unit, we first assume that the column is packed with clean beads i.e. LanM-CBs with no Nd3+ on them. The mass transfer phenomena from the feed stream into the functionalized beads starts as soon as the feed stream becomes in contact with the bed; as a result, the concentration of Nd3+ in the passing fluid decreases along the length of the column until it reaches zero. Assuming that the concentration of the feed stream entering the column remains constant, the absorbing LanM-CBs will become saturated over time. When the fluid reaches LanM-CBs that have not yet been saturated, mass-transfer phenomena continue and the Nd3+ concentration continues to decrease along the length of the column. The region of the adsorber column in which the concentration of Nd3+ is changing is called the mass transfer zone (MTZ) [2].
As the bed of LanM-CBs becomes saturated with the Nd3+, the feed stream must travel further up the column for the recovery of the Nd3+ from the fluid phase to occur. Hence, the MTZ is said to be “moving” along the bed as the amount of saturated LanM-CBs increases over time. When the MTZ reaches the end of the column and the entire bed is saturated, no more recovery from the fluid phase can occur. Beyond this point, Nd3+ ions will not be recovered by the LanM-CBs which would result in a loss of potential profit by the operator of the process. The point at which this occurs is called the breakthrough point, and it is defined by [2]:
Threshold relative outlet concentration: The relative concentration (C/C0) of the outlet stream at which the bed of functionalized beads is considered saturated. This is calculated by dividing the actual outlet stream concentration by the concentration of the feed stream going to the inlet. Upon saturation of LanM-CBs, the outlet stream concentration will eventually approach that of the inlet feed stream and result in a relative concentration of 1.
Time at breakthrough: The time at which the threshold relative outlet concentration is reached.
Figure 2.Adsorption process and mass transfer zone movement
Evaluating the Performance of PACLan
We evaluated the performance of PACLan through various design parameters. These relate to how well PACLan operates relative to the identified needs of stakeholders.
Operational Criteria
The selection of appropriate operational criteria, which is made up of the treatment objective and process goal, allows for a target process and result to be defined. The treatment objective identifies the quality of the final product, while the process goal outlines the rate at which the unit should operate to ensure a time-efficient system for industrial integration of our process.
Treatment Objective: Despite the relatively high concentration of REEs in electronic waste feedstock, leachates derived from solubilization of this feedstock are still considered dilute solutions of REEs. A process design that minimizes the waste of REEs is needed. As such, we defined the breakthrough concentration to be at C/C0 = 0.01, so that REEs are not wasted before the column is considered saturated.
Process Goal: Upon liaising with electronic waste processing facilities in Alberta, eCycle Solutions revealed that a standard facility is capable of processing 500-600 metric tonnes of e-waste monthly. Using the material fraction of steel in e-waste [3] (47.9%) and the composition of a steel stream (0.06% REE)[4], the mass rate of lanthanides processed by the facility is estimated to be 5.748 kg/day. In order to develop working values for a prototype that can be experimented in conjunction with the protein produced by our wet lab, we envisioned the prototype to have a capacity of a chromatography column corresponding to the scale of possible LanM production in our lab. Hence, the process goal on which the prototype is based will be 1% of the estimated industrial processed lanthanide mass rate, which is 0.5748g/day .
Modeling affinity bioseparation in PACLan
Modeling the concentration of the outlet stream was needed as it indicates the degree of bed saturation and, by extension, the movement of the mass transfer zone along the column over time. Coupled partial differential equations (PDEs) describing the movement of Nd3+ in the fluid phase and the transfer of Nd3+ onto the LanM-CBMs were utilized to quantify the adsorption behavior in PACLan [5,6].
Governing Equations
Description
Equation 1 describes the mass balance of the Nd3+ in the fluid as it passes through the column. Term 1 accounts for the convective transfer of Nd3+ i.e. movement of Nd3+ due to the flow of the fluid, term 2 accounts for the accumulation of Nd3+ over time, and term 3 accounts for the net adsorption of Nd3+ onto the lanmodulin-functionalized cellulose beads.
Equation 2 describes the mass balance of the Nd3+ onto the LanM-CBMs. Term 1 accounts for the adsorption of the Nd3+ onto the cellulose bead, while term 2 represents the desorption of Nd3+ from the bead due to lanmodulin-Nd3+ unbinding.
Initial and Boundary Conditions
Explanation
For the first boundary condition, c(0,t) = cin since the concentration of the fluid at the bottom of the column is equal to the concentration of the feed stream into the inlet. For the first boundary condition, q(z,0) = 0 because we assume that the bed of functionalized beads does not have any Nd3+ on it. For the second boundary condition, the time derivative of concentration at z = 0 is 0 because the concentration of the feed stream is assumed to stay constant at the inlet. For the third boundary condition, the spatial derivative at z = L is 0 because the concentration at the end of the column does not change as the adsorption only occurs in the bed of functionalized beads.
Model Assumptions
The system is under isothermal (constant temperature) conditions [5].
No chemical reaction occurs in the column other than the LanM-Nd3+ binding interaction [5].
Radial dispersion: spread of Nd3+ across the cross-section of the column is negligible (plug-flow conditions) [5].
Packing material is made up of spherical beads that are porous and have size uniformity [5].
The LanM-Nd3+ affinity interaction is 2nd order in the forward direction (binding) and 1st order in the reverse direction (unbinding) [6].
The average liquid phase concentration is equal to the concentration inside the pores and the surface of the beads.
Axial dispersion is negligible: the spread of Nd3+ longitudinally in the column is negligible.
Variables and Parameters
Table 1. Constant process parameters utilized in modeling PDEs
Table 2. Dependent and independent variables examined in the given model
qmax**: Since qmax is the maximum amount of adsorbed metal per volume of bead, we can calculate it by assuming that LanM binds to 3 moles of Nd3+ and using the protein loading i.e. amount of protein immobilized onto the bed along with LanMs molecular weight and cellulose bead density.
Column Design Parameters
The mass transfer phenomena in the adsorber column is a product of the intertwined chemical phenomena of adsorption and the physical phenomena of fluid flow. So, we first needed to determine the design parameters for the column i.e. the length and the diameter of the actual column to be filled with LanM-CBs. To do this, we ensured that:
The maximum flow rate was achieved, which is related to the diameter of the column
The pressure drop generated is minimized, which is related to the length of the column
S1 is the used column capacity representing how much of the column has been used before tb
S2 is the feed wasted representing how much went out of the column before tb
S3 is the unused column capacity representing how much of the column has been used after the tb (or while the MTZ is moving out of the outlet)
The highest concentration of feed possible, since higher feed concentrations lead to higher processing capacities
The fastest flow velocity possible, since faster velocities lead to higher processing capacities
The least protein loading possible, since (a) they do not affect the processing capacity as much as other controllable parameters and (b) protein production is foreseen as a bottleneck to implementing the system.
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The flow rate through the column is given by
Where Q is the volumetric flow rate, and D is the diameter
The pressure drop generated across the column, which is physically a bed of packed spheres, is described by the Ergun equation [9]:
Table 3.Process parameters constants used in Ergun equation
Figure 3.Flow velocity as a function of flow rate for various column diameters
At a constant flow velocity, it can be observed that as the diameter increases, the corresponding volumetric flow rate for that flow velocity also increases. This is because the increase in the cross-sectional area of the pipe allows for more volume of fluid to pass through at a single instance. We would ideally want the maximum possible flow rate so we can process more volume of feed per unit time. Therefore, the ideal column diameter is 80 mm.
Figure 4.Pressure as a function of flow rate for various column lengths and various flow rates at a specific column diameter and flow velocity
At constant flow rate, the pressure drop generated is directly proportional to the length of the column. At a vertical column configuration, this direct proportionality is due to the increase in weight per unit area as more fluid and more LanM-CBs are present at longer column lengths. Given that we determined an 80-mm diameter would be ideal, it can be observed that the ideal column length is 600 mm as it generates the least pressure drop.
Adsorption Efficiencies and Capacity
The relative outlet concentration curve provides a basis for evaluating the technical performance PACLan through different quantities calculated from it. The time at the breakthrough (tb) indicates the point at which the column should be eluted to regenerate the LanM-CBs. Then, the different areas of the regions divided by the curve and the tbrepresent different quantities [14]:
Figure 5.Anatomy of a breakthrough curve
From these areas, various efficiencies and quantities can be calculated:
Solute Recovery Efficiency (SRE) [14] represents the extent to which Nd3+ is recovered from the feed stream
Bed Utilization Efficiency (BUE)[14] represents the extent to which the bed of LanM-CBs is utilized prior to tb
Processing Capacityrepresents how much Nd3+ can be recovered per unit time. This is directly compared to the process goal to determine the viability of the solution.
Baseline Viability of PACLan
Figure 5.Relative concentration of the outlet stream as a function of time using initial design parameters
The breakthrough point occurs at around t = 106.5 s. It very quickly rises to a relative outlet concentration of C/C0 = 1 (as shown by the steepness of the curve), thus indicating that the MTZ is not long i.e. the column was almost immediately saturated after some of the Nd3+ breaks through. This suggests that the bed is being utilized efficiently.
Table 4.Evaluation parameters for the viability of PACLan using initial parameters
From Table 4, it can be observed that the SRE of Nd3+ ions from the stream is very high. This, along with the large time needed to reach the breakthrough point, demonstrates that PACLan is excellent at removing Nd3+ from the stream. The very high BUE indicates that the bed is being used very efficiently since not a lot of Nd3+ leaks out of the column from the point of breakthrough until the column saturation point. The amount of Nd3+ recovered along with the time until breakthrough is reached provides a processing capacity of 0.865 g/day, which is greater than the minimum required scale-down processing capacity of 0.5748 g/day, thus signifying that PACLan is a viable system at the prototype level.
Sweeping through Parameters to Optimize Performance
Despite PACLan’s apparent viability as a solution at the prototype level, we explored how we can better optimize the system by changing controllable operation variables. This will guide future iterations of the prototype and potential scale-up endeavors as to how PACLan’s performance can be made better.
Changing feed concentration
The initial feed concentration used in the initial designs was the highest expected concentration of REEs present in electronic waste. So, we tested out if PAClan can perform at lower feed concentrations since it may reflect the leachate we receive from the bioleaching process more realistically.
Figure 6.Relative concentration of the outlet stream as a function of time at different concentrations of the feed stream
We observed that as the concentration of the feed decreased, the breakthrough time increased. This is because a low-concentration stream has fewer Nd3+ ions to saturate the LanM-CBs, so the entire column bed will take a longer time to be saturated as a whole.
Changing flow velocity
The initial flow velocity used before was an estimate that we determined from our prototype design simulations. So, we tested out if PACLan can perform well at higher flow velocities since this could allow for higher volumes of REE leachate to be processed per day.
Figure. Relative concentration of the outlet stream as a function of time at different flow velocities of the feed stream
Figure 7.Relative concentration of the outlet stream as a function of time at different flow velocities of the feed stream
We observed that as the flow velocity increased, the breakthrough time decreased. This can be attributed to the fact that more Nd3+ ions are transported onto the LanM-CBs over time if the flow of fluid is faster, thereby saturating the column bed quicker and reaching the breakthrough point in a shorter amount of time.
Changing protein loading
The initial protein loading onto the functionalized bead was determined from the literature based on the binding capacity of CBMs onto cellulose. This was considered the ideal case as we might not be able to load as much protein as we would like. Furthermore, protein production is a resource-intensive process. So, we explored how well PACLan recovers Nd3+ even at low protein loadings.
Figure 8.Relative concentration of the outlet stream as a function of time at different protein loadings of the bead
We observed that as the protein loading decreased, the breakthrough time decreased. This can be due to the fact that lower protein loadings mean fewer sites for Nd3+ to bind. This means that it is easier to saturate the functionalized bead, which translates to a faster saturation of the column bed and thus shorter breakthrough times.
In terms of adsorption efficiencies, the SRE was consistently high across various parameter values. This might be attributed to the fact that the threshold relative outlet concentration was set to C/C0 = 0.01 and that the system is extremely effective in binding Nd3+ from the solution. In terms of BUE, we observed that the BUE was slightly affected by changing parameters, with protein loading having the largest effect and feed concentration having the least. This can be due to the fact that bed utilization is directly affected by the amount of protein onto the bead since if there is not enough protein for binding, the Nd3+ ions will leak out faster than they can be bound by the LanM. Lastly, we noted that the flow velocity has the largest impact on the processing capacity and protein loading has the least. Larger flow velocities translate to more Nd3+ ions exposed to the functionalized beads and bound onto lanM per unit time, thus increasing the processing capacity. Therefore, to optimize the system, it is best to have:
Taking into Account Non-Ideality
The model described above is for ideal plug flow conditions i.e. the Nd3+ does not diffuse longitudinally or axially as it travels along the column length. However, real packed-bed adsorber columns have conditions that make it conducive to axial dispersion, or the longitudinal spread of the solute along the column as it flows. To account for this spread, we can add a term to the fluid mass balance [5].
The axial dispersion term accounts for the spread of the Nd3+ along the column as it flows through it. The axial dispersion coefficient is determined through empirical correlations using dimensionless numbers relevant in fluid mechanics [15, 16].
Figure 9.Axial dispersion through time
The Peclet (Pe) number is a dimensionless group representing the ratio of the convective strength to diffusive strength, while the Reynolds number (Re) is a dimensionless group representing the ratio of inertial force to the viscous force within the fluid body [17]. These were used to determine DL, the axial dispersion coefficient.
Baseline Viability of PACLan accounting for axial dispersion
Figure 10.Relative concentration of the outlet stream as a function of time using initial design parameters and accounting for axial dispersion
The breakthrough point occurs at around t = 67.5 s. It slowly rises to the relative outlet concentration of C/C0 = 1 (as shown by the gradual increase of the curve), thus indicating that the MTZ is long i.e. the column takes time to be saturated after some of the Nd3+ initially breaks through. This suggests that the bed is not being utilized efficiently.
Table 5.Evaluation parameters for the viability of PACLan using initial parameters and accounting for axial dispersion
From Table 4, it can be observed that the SRE of Nd3+ ions from the stream is very high. This means that accounting for axial dispersion does not drastically affect the SRE. The low BUE, however, indicates that the bed is not being used efficiently because Nd3+ leaks out of the column from the breakthrough point up to the column saturation point. The amount of Nd3+ recovered along with the time until breakthrough is reached provides a processing capacity of 0.862 g/day, which is greater than the minimum required scale-down processing capacity of 0.5748 g/day. This signifies that PACLan is still viable even after accounting for non-ideality like axial dispersion.
Sweeping through Parameters to Optimize Performance accounting for Axial Dispersion
Despite PACLan’s apparent viability as a solution at the prototype level, we explored how we can better optimize the system by changing controllable variables. This will guide future iterations of the prototype and potential scale-up endeavors as to how PACLan’s performance can be made better.
We determined that PACLan is still viable even after accounting for non-ideality. Despite this, we performed another iteration of our parameter sweep to optimize the performance of our solution. The final parameter values we derive from this sweep are more applicable for prototype development since they do not assume ideality.
Changing feed concentration
Figure 11.Relative concentration of the outlet stream as a function of time at different concentrations of the feed stream accounting for axial dispersion
Changing flow velocity
Figure 12.Relative concentration of the outlet stream as a function of time at different flow velocities accounting for axial dispersion
Changing protein loading
Figure 13.Relative concentration of the outlet stream as a function of time at different protein loadings of the bead accounting for axial dispersion
The observed trends between the parameters and the breakthrough time in the ideal case also held in the model where non-ideality was taken into account, with the apparent difference of the shape of the relative outlet concentration curve, suggesting that axial dispersion decreases the BUE in all cases.
In terms of the SRE, we noted that it was consistently high across various parameter values. However, they are a bit lower than in the ideal case indicating that a bit more of the feed stream is wasted when considering account axial dispersion. Increasing the flow velocity and decreasing the protein loading minimally reduced the SRE. For the former, this can be attributed to the increase in the volume of the feed processed at a single point in time as the flow velocity increases thus leading to an increase in the amount of feed wasted. For the latter, reducing the amount of protein on the bead leads to a reduction of recovered Nd3+ ions and thus a lower SRE.
In terms of BUE, we observed that axial dispersion led to an overall decrease in BUE. Furthermore, the trend observed in the ideal case was also applied in the non-ideal one as protein loading had the greatest impact on BUE and the feed concentration had the least. It should be noted that the differences in BUE by changing protein loading were more pronounced in the non-ideal case. This can be attributed to additive effects of protein loading and axial dispersion onto the length of the mass-transfer zone and thus the BUE.
Lastly, the observed trends between the changing parameters and the processing capacity are consistent in the non-ideal case. We also noted that despite taking into account axial dispersion, the processing capacities were in the same order of magnitude as in the ideal case. This suggests that despite axial dispersion affecting the SRE minimally and BUE profoundly, its effects on processing capacity are negligible, further implying that PACLan fares well in realistic conditions.
Results of Optimization
Given the parametric sweeps, we propose the following final parameters for the system
Table 6.Final optimized parameters for PACLan
Figure 14.Relative concentration of the outlet stream as a function of time of the optimized system
Table 7.Evaluation parameters for optimized PACLan performance
We observed that optimizing the parameters led to a significant decrease in BUE, but the high SRE and processing capacity allow PACLan to perform in accordance with our process goal, thus demonstrating the viability of PACLan in a scaled-down operating environment determined through stakeholder liaison.
Conclusions
We found that PACLan as a solution for lanthanide recovery is viable both in ideal and non-ideal conditions using literature-derived parameters. After optimization via parametric sweeps, we further improved PACLan performance to significantly exceed the process requirements, thus making it conducive for prototyping and implementation. The effect of changing parameters on the adsorption efficiencies and processing capacity were not equal; protein loading had minimal effects while flow velocity had the greatest impact. This suggests that protein production will not be a major barrier to implementation as the system can perform well despite low protein yields.
Future Directions
The current model has numerous assumptions whose validity may not hold in real-world conditions. More rigorous modeling that does not necessitate making such assumptions is needed to guarantee that PACLan is a viable solution that can perform well after implementation.
Having said this, the trends found in the changes in adsorption efficiency and processing capacities after changing the controllable parameters engender several recommendations in the implementation of the system. First, the concentration of the feed stream must be as high as possible. Therefore, there might be a need to implement a concentrating stage prior to using PACLan. Second, the flow velocity must be as high as possible. This might necessitate the use of greater energy through a pumping mechanism. Furthermore, higher velocities will cause fluidization in the bed, a phenomenon that might render our model predictions invalid. So, great caution must be taken when changing the flow velocity. Lastly, the protein loading must be as low as possible. As of the moment, this recommendation is justifiable as protein production, especially of novel proteins such as LanM , is costly and complex. However, as protein production becomes more mainstream and research and development operations push lanmodulin production further, this recommendation may no longer be applicable. Validation of the model is necessary. Hence, the construction of the adsorber prototype should be undertaken to provide empirical results that will allow the model to be refined.
References
