Optimization of the Passive Recovery of Uranium ... - ACS Publications

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Optimization of the Passive Recovery of Uranium from Seawater M. Flicker Byers* and E. Schneider Nuclear and Radiation Engineering Program, The University of Texas at Austin, University Station C2200, Austin, Texas 78712, United States S Supporting Information *

ABSTRACT: The aim of this study is to optimize the design and deployment conditions utilized by a technology for passively collecting uranium from seawater in order to achieve the minimum uranium production cost. Given the complicated empirically driven nature of the cost calculation, which uses discounted cash flow techniques to follow the life cycle costs of a unit mass of adsorbent, the cost calculation tool is treated as a black box model, and thus the minimization employs a derivative free optimization method, chosen to be the Nelder−Mead simplex method. By use of an illustrative base case, the uranium production cost is minimized, resulting in a 20% savings. From there, sensitivity analyses are considered in order to provide insight as to how the optimal deployment conditions are determined and to shed light on significant cost drivers. It is determined that optimizing the deployment conditions and improving the uranium binding kinetics can significantly decrease production cost. The results presented in this paper can inform the direction of future research. Furthermore, as the technology continues to evolve, the methodology developed for this optimization will remain relevant and the optimization can continue to be used to guide design and R&D decisions.

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INTRODUCTION The aim of this paper is to optimize the deployment conditions utilized by a passive uranium from seawater collection system that is currently under development by a consortium of national laboratories along with university partners. Although much debate surrounds predictions regarding availability of future supplies of conventionally mined uranium, it is undeniable that nuclear power as an energy source would benefit from increased supply security. Due to its relatively high solubility in water, uranium is present in the ocean at a concentration of 3.3 ppb.1 While this low concentration hinders the economic feasibility of its recovery, the magnitude of uranium contained in the oceans, some 4 billion tonnes, would have a transformative effect on issues of supply security. At their current state of maturity, technologies for uranium recovery from seawater chiefly serve to establish a production cost ceiling for uranium. That being said, there is great interest in reducing this cost along with its uncertainty. This analysis considers an adsorbent that consists of a high density polyethylene (HDPE) backbone co-grafted with an amidoxime ligand, to afford uranium affinity, and a comonomer, to increase hydrophilicity. The buoyant adsorbent is moored to the ocean floor in a kelp field like structure. After a predetermined period of time it is winched up so the uranium may be eluted off the braided adsorbent. Functional groups on the braids then need to be regenerated before they can be redeployed. This process is repeated until it is no longer economically advantageous to continue to reuse the adsorbent due to the degradation it suffers with each reuse. The unit cost of producing uranium is a function of the performance of the technology for adsorbing uranium from the ocean, which is in turn affected by many specific cost drivers. Given all of the feedbacks between adsorbent production options, deployment conditions, and adsorbent performance, © XXXX American Chemical Society

the determination of the optimal deployment scenario is nontrivial. As this technology continues to evolve and progress to more detailed stages of the design process, the number of cost driving design parameters will increase. For that reason, manual optimization of system parameters to minimize uranium production cost will become increasingly impractical. Given the complicated empirically driven nature of the cost calculation, the cost calculation tool will be treated as a black box model; thus, the minimization employs a derivative free optimization method. A literature review was conducted to explore applicable existing algorithms, and case studies of other engineering cost minimizations were considered and compared to the optimization problem at hand.2−11 Ultimately an algorithm was selected, the Nelder−Mead simplex method. This is used to find the optimal deployment conditions for a predefined base case. A previous economic analysis determined that adsorbent uptake and number of adsorbent uses are the most significant cost drivers.12 Therefore, the deployment conditions to be optimized include the number of uses, along with length of campaign and ocean temperature, as they have significant impacts on uptake. Verification of the method is achieved by comparing the solutions to a brute force calculation using discrete values for the decision variables. Finally sensitivities to adsorbent performance, input parameters, and process costs are explored. Special Issue: Uranium in Seawater Received: September 2, 2015 Revised: November 2, 2015 Accepted: November 19, 2015

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DOI: 10.1021/acs.iecr.5b03242 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research

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METHODOLOGY Cost Estimation Methodology. The economic analysis provides an estimate of the cost to recover uranium from seawater on an industrial scale via the passive collection process currently under development by a consortium of national laboratories led by Oak Ridge National Lab (ORNL). The proposed system consists of three major steps whose costs are considered individually and then summed. First the fibrous adsorbent must be produced via radical polymerization. Once braided, the adsorbent is then sent out to sea to be moored to the bottom of the ocean. After a predetermine collection period the loaded adsorbent is retrieved and the uranium is eluted off in an acidic chemical bath, regenerated in a base solution and the process repeated until it is no longer economically advantageous to continue. It is worth noting that the reconditioning of the adsorbent only happens when the adsorbent is to go back into the ocean and thus takes place N − 1 times, where N is the number of adsorbent deployments. The methodology used to calculate the net present value of the costs associated with the recovery of uranium from seawater relies primarily on the Economic Modeling Working Group cost estimation guidelines as a reference.13 The uranium production cost is estimated by considering the sum of the individual costs incurred by one unit mass of adsorbent over its lifetime. This cost has several components, the first of which is associated with the initial production of the adsorbent. In calculating the adsorbent production cost, it is assumed that all chemicals can be utilized with 100% efficiency and those that are recyclable can be recovered at a rate of 90%. Next incurred is the cost of mooring a unit mass of adsorbent to the ocean floor, winching it back up, and transporting it to a mothership for elution. Finally, there are costs associated with the removal and purification of the uranium and regeneration of the adsorbent before its next deployment, with the same assumptions regarding chemical use as in adsorbent production. The capital, operating, and decommissioning costs associated with each of the three major steps are evaluated, and a timeline of when the costs are incurred is developed. These are then summed using a discounted cash flow technique, wherein the time value of money is taken into account.14 The methodology used in the current cost analysis builds upon previously published cost estimates for this same process.15−20 Updates to the previously published methodology and a more streamlined description of the cost calculation can be found in the appendix in the Supporting Information. For the highest level of detail, the reader is still encouraged to refer to ref 15. Parameter Space. In the case of optimizing the recovery of uranium from seawater the objective function, uranium production cost, is a function of the various cost inputs along with adsorbent performance. The uranium uptake of the adsorbent is governed by a number of system and design parameters including length of soaking time in the ocean, number of adsorbent uses before ultimate disposal, and ocean temperatures. While there are certainly other factors that can have an effect on adsorbent performance, these three will be the focus of this optimization, as they have the most clearly defined empirical models and in practice could be easily manipulated without significant design overhauls. The details of these governing relationships will be described in the following section, as each of these factors has its own unique effects on

both adsorbent performance and cost. Additionally the constraints unique to each parameter will be defined. Length of Campaign. The relationship governing adsorbent uptake as a function of immersion time, ti, is described as follows. The uranium complexation with amidoxime is presumed to follow the one site ligand saturation model21,22 shown in eq 1. This model states that the maximum uptake of the adsorbent, C, is a function of the theoretical saturation capacity, βmax, and time required to reach half the saturation point, KD.

C=

Βmax ti KD + ti

(1)

The values for saturation capacity and half saturation time are unique to the adsorbent and must be determined experimentally. Time series data for uranium uptake received from Pacific Northwest National Lab (PNNL) are fitted to the one site ligand saturation model to yield values for KD and βmax of 23 days and 5.4 g U/kg ads, respectively,23 giving rise to a relationship between length of campaign and adsorbent uptake as depicted in Figure 1.

Figure 1. Adsorbent uptake as a function of immersion time.

Realistically, it can be expected that the length of campaign should not exceed the time it takes to reach 95% of the saturation capacity because beyond this point little gain will be realized by extended soaking times. If C is set to 0.95βmax, then the maximum campaign length for the case of the reference adsorbent is 399 days. Temperature. The relationship between uranium uptake of the ORNL adsorbent and ocean temperature was recently quantified and is here shown to have a significant effect on the final production cost. PNNL measured time series data for adsorbent uptake at three temperatures and observed a linear relationship between ocean temperature and uranium uptake for a fixed 56 day campaign length.24 In order to carry out economic analyses, it is necessary to relate the measured uptakes to both the temperature and immersion time, requiring us to generalize the one-site ligand saturation model to predict adsorbent uptake also as a function of temperature. Hence the time series measurements are used to create temperature dependent models for the kinetic parameters discussed earlier, βmax and KD. Given the observed linear relationship, a linear regression was performed on the kinetic parameters for all of the adsorbent types analyzed in the PNNL marine experiments. The relationships predicting βmax and KD as a function of temperature for the reference adsorbent type used in this analysis can be seen in Figure 2. These models for βmax and KD are then placed back into the one site ligand saturation model discussed previously, eq 1, to B


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Figure 3. Total uranium uptake. Figure 2. Temperature dependence of kinetic parameters.

with each reuse, it can be assumed that no more than 50 uses would be economically feasible. Biofouling. Although marine experiments have long resulted in discoloration of deployed adsorbent, the effect of biofouling on uptake has only recently been studied and is therefore largely uncertain. Due to the preliminary nature of existing data, significant extrapolation is used to create different hypothetical models in order to examine the potential impact biofouling could have on uranium cost. Literature research on marine biofouling suggests that the degree of oceanic biofouling is proportional to at least two factors already otherwise incorporated to the cost analysis, length of campaign,26 and temperature.27 Biofouling will likely also depend on numerous other factors including ocean pH, opacity affecting the extent of the eutrophic zones, and oxygen content, although these have not been the focus of marine test carried out to date. Since there is not yet sufficient data to definitively quantify the relationships between campaign length and temperaturedependent effects of biofouling on uranium uptake, two candidate models will be used. Initially, the effects of biofouling will first be assumed to be strictly a function of ocean temperature. Then, biofouling is made a function of both ocean temperature and length of campaign. The data set regarding the effects of biofouling on uptake comes from experiments run at PNNL. Adsorbent fibers were exposed to filtered seawater and light for 56 days with their uranium uptakes periodically checked. The uptake of these fouled fibers is compared to the negative control, nonfouled fibers, to determine the effects on adsorbent uptake.28 These time series data are plotted in Figure 4 to show that the

predict adsorbent uptake as a function of both immersion time and ocean temperature, T [°C], as seen in eq 2. C=

(0.312T − 0.813)ti (0.58T + 11.1) + ti

(2)

In most cases, adsorbents were analyzed at three temperatures, but for the reference fiber type, most of the data from the experiment conducted at 32 °C were not usable, leaving only the two lower temperature experiments. Nonetheless, the use of the linear model has been justified by the high R2 values obtained by the same linear regression performed on the other fiber types tested, which did include three temperatures. Further verification of this model arises from the empirical data in two ways. The time series uptake at the lower temperature values as predicted by the regression model is consistently within 5% of the laboratory measured uptake at each time point. Furthermore, even in the case of the missing data set, one time point was recoverable and agreed with the model prediction to within 5%. Further explanation of the linear temperature-uptake model can be found in ref 24. Number of Uses. The number of times the adsorbent is used, N, also has an effect on the cost. Although reusing the adsorbent circumvents the original production cost, the acidic chemical baths administered as part of the elution process degrade the uranium binding sites. In the base case the adsorbent suffers degradation, d [% loss in uptake/reuse], at a rate of 5% of the uptake achieved on the previous use, as this loss rate was seen in previous experiments.25 This loss in uptake compounds with each use has a nontrivial effect on the total uranium recovered by a unit mass of adsorbent over its lifetime, Ctot [g U/kg adsorbent lifetime]. Equation 3 sums the recovery from each use, where C is the uptake realized on the first use determined earlier in eq 2. i=1

C tot =

⎛

∑ C⎜⎝1 − N

d ⎞ ⎟ 100 ⎠

i−1

(3)

After multiple uses the accumulated degradation eventually becomes high enough such that the marginal cost of another mooring, elution, and regeneration cycle outweighs the marginal benefit. The cumulative lifetime uranium uptake for a unit mass of adsorbent as a function of number of uses is depicted in Figure 3 using a degradation rate of 5% loss per reuse. The number of uses must be a non-negative integer. The treatment of this integer constraint will be discussed later in the methodology. Since the adsorbent suffers 5% loss in capacity

Figure 4. Initial biofouling data from PNNL.28 C


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Industrial & Engineering Chemistry Research saturation capacity of the fouled fibers suffered a roughly 20% decrease, as did the measured uptake at all time points, while the half saturation time was nearly unchanged, indicating no rate interference. The PNNL investigators that measured the biofouling concluded that the film simply blocks physical accessibility to uranium binding sites and beyond that does not appear to interfere with the uptake mechanism. Therefore, it seems unlikely that the rate of uptake by the remaining available sites should change, meaning the one-site ligand saturation model will continue to be used but with a nominally lower saturation capacity. Temperature Dependent, Time Independent Biofouling. It may be expected that the negative effects of marine organism activity on capacity would increase with time. Initial experimental data, however, have indicated that biofouling has little effect on rate of uptake but simply decrease the uptake and capacity almost as soon as the adsorbent is placed in the water. It is hypothesized that this is a result of a difference in many orders of magnitude between the time scale of campaign length and marine organism reproduction cycle. The literature shows that in the case of oceanic fouling a biofilm typically forms within the first few hours of immersion.29 Since the first data point is not taken until 7 days into soaking, it is possible that by this point the fibers have already reached their carrying capacity of microorganisms. If the net population is not changing with time, this would lead to the observed effects of biofouling decreasing the maximum capacity essentially instantaneously and then having no further impacts that are dependent on soaking time. While the uranium complexation with amidoxime favors warmer waters, it is likely that a competing feedback of increased biofouling also exists at these elevated temperatures. Beyond visual inspection, though, no empirical correlation quantitatively relating loss in uptake and presence of microorganisms has yet been obtained. Therefore, in order to define a candidate model for this relationship as a function of temperature, we first turn to the literature to examine marine bacterial growth as a function of water temperature. This relationlship is then used to scale the loss in uptake deomstrated in the single temperature experiment displayed in Figure 4. A general formula relating heterotrophic bacterial specific growth rate, G in units of inverse time, and ocean temperature, T [°C], found in the literature27 and seen in eq 4, is used to correlate the effects of biofouling to water temperature. log(G) = − 1.54 + 0.052T

Figure 5. Uptake for a fixed 60 day campaign as a function of temperature for the case of temperature dependent and time independent biofouling.

reference 60 day campaign. The hypothesized model shown here suggests that, for a fixed campaign, beyond some optimal temperatures the uptake begins to decrease due to competing temperature dependent effects. Eventually microorganisms may become so abundant that they colonize a large enough surface of the fibers such that the increase in uptake caused by higher temperatures has been negated by temperature induced biofouling. It is in this range where the optimization becomes most telling, as it can determine the point at which these two competing effects offset each other. uptake(T ) = uptake(Tno fouling) ⎛ ⎞ 10−1.54 + .052T × ⎜1 − −1.54 + (0.052)(20) × 20%⎟ ⎝ ⎠ 10

(5)

Temperature and Time Dependent Biofouling. Since no data currently exist to suggest a relationship between length of campaign, ti, and loss due to biofouling, a linear model is drawn from the existing data to explore the effects of a potential scenario. For the purposes of this hypothesis, it is assumed that the effects of biofouling begin upon the adsorbents’ immediate contact with water and increase linearly, passing through the point (56 days, 20% loss in uptake), which comes from the data displayed previously in Figure 4. This decreased uptake relative to the unfouled fibers is calculated in eq 6, for any given temperature. ⎛ 0.353ti ⎞ ⎟ uptake = uptake(no fouling) × ⎜1 − ⎝ 100 ⎠

(4)

(6)

To relate the fouling back to temperature, the specific growth rate as a function of temperature is again used. The loss predicted at a given length of campaign by eq 6 is multiplied by the temperature dependent loss calculated in eq 5. These combined effects of biofouling on uptake can be seen in eq 7 and Figure 6.

Since previous experiments have shown that temperature appears to affect the adsorbent capacity but not the kinetics, this temperature dependent biofouling is applied to the uptake ultimately achieved and does not affect the rate. No suitable correlation between growth and temperature which might be directly applicable to fouling of the adsorbent surface, thus loss in uptake, has been determined experimentally. Therefore, the single known datum point, a 20% decrease at 20 °C, is used to normalize all other temperature dependent growth rates to a loss in adsorbent capacity. The effect of biofouling on uptake is computed using a ratio of the growth rate at any given temperature, T, to the growth rate at 20 °C, scaled by the reference 20% loss. This temperature dependent loss is shown in eq 5. Figure 5 provides a graphical example of this effect by showing the uptake realized by a unit mass of adsorbent as a function of temperature for the

uptake(T ,ti) = uptake no fouling (T ,ti) ⎛ ⎛ 10−1.54 + 0.052T ⎞⎞ × ⎜⎜1 − 0.353ti⎜ −1.54 + (0.052)(20) ⎟⎟⎟ ⎠⎠ ⎝ 10 ⎝

(7)

It is worth noting that for some water temperature and immersion time combinations the uptake predicted by the model reaches a maximum and begins to decrease. This is likely nonphysical, as uranium already taken up would not be lost by continued immersion unless the bioactivity caused the D


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process ends when the objective functions at all vertices are approaching the same value. This is achieved when the standard error of the objective function values, ε, at the given simplex falls below a predetermine ε value. At each iteration this convergence criterion is checked using eq 8. i=n+1

ε=

∑ i=1

(yi − y ̅ )2 n

(8)

Although this direct search method is designed for unconstrained optimization, the constraints are imposed through logical conditions embedded in the code. This is implemented by forcing any nonfeasible decision variables back to their boundary values and continuing the algorithm unobstructed. This is sufficient because it is unlikely that the optimal values of decision variables will be at or near the boundaries of the feasible region. In fact if this does turn out to be the case, then a different parameter should be considered for optimization, as there is little insight to be gained if the optimal objective function is achieved by pushing a given parameter to the edge of its allowed domain. A brute force sweep, seen in Figure 7, will be used to verify the implementation of this method and the appropriate convergence criterion.

Figure 6. Uptake as a function of temperature for the case of temperature and time dependent biofouling.

adsorbent to lose its physical integrity, which has not been seen experimentally thus far. Therefore, beyond the time at which maximum uptake is reached the uptake is assumed to have reached saturation and remains constant. Cost Minimization Methodology. In the case of uranium from seawater, the objective function is made up of a combination of linear and nonlinear functions. The tool used to compute the uranium production cost is an Excel spreadsheet with many built-in dependencies and conditional operations. Given the complicated empirically driven nature of the cost calculation and its existing platform, the cost calculation tool is treated as a black box model, meaning only the inputs and outputs of the calculation are considered without regard to its internal workings. While the system and cost calculation models are available in previous publications15,20 and summarized in the appendix in the Supporting Information, the complexity of the system makes the analytical determination of derivatives impractical. Likewise, it is not possible to prove the existence and continuity of numerically evaluated derivatives across the entire input space of interest. For that reason, a derivative-free optimization method was chosen. A literature review along with case studies of other engineering economic optimizations was conducted to select an applicable method.2−11 Due to its ease of coupling with black boxes and quick convergence with this objective function, the Nelder−Mead simplex method will be used for this optimization. A quasi-Newton method was also implemented but was found to require a significantly larger number of executions of the black box model per optimization run. Since these objective function calculations are time-consuming, the Nelder−Mead simplex method performed strongly, as it only requires one objective function value per iteration Implementation of the Nelder−Mead Simplex Method. This method follows the directed movement of a simplex, a shape of n + 1 vertices in n dimensions, through the parameter space until it arrives at the minimum. The number of dimensions corresponds to the number of decision variables being optimized. Each vertex represents a set of decision variables and their associated objective function value. With each iteration the simplex moves through the feasible region by replacing at least one vertex with a new set of conditions and their calculated objective function value. The algorithm used in this analysis mirrors that originally published by the creators of the method, which assumes that the objective function is to be minimized.7 The convergence criterion used to terminate the method is based on the objective function values, y, since no derivatives are available for the determination of a stationary point. The

Figure 7. Region of minimum cost for 15 uses as determined by brute force calculation.

Given that the current problem formulation deals with three variables (days of campaign, number of uses, and temperature), the simplex utilized is a tetrahedron. With each iteration the simplex moves in such a way that the vertex corresponding to the highest uranium production cost is replaced with a new set of deployment conditions yielding a lower cost. The initial simplex is generated by randomly perturbing the user specified starting point to produce three additional points. This unique stochastic addition to an otherwise deterministic algorithm is implemented in order to remove starting point dependence and increase the chances of converging on a global as opposed to local minima. The standard simplex algorithm is then used until the tetrahedron collapses to roughly a single point.7 The appropriate convergence criterion is selected by iteratively running the algorithm with smaller ε values, where ε is calculated as the standard error of the uranium production costs, ucu, of each vertex of the current simplex as seen in eq 9. E


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Industrial & Engineering Chemistry Research Table 1. Resulting Optima Used in Convergence Criteria Determination ε

number of iterations

brute force 1 × 10−2 1 × 10−4 1 × 10−6 1 × 10−8 1 × 10−10 1 × 10−15 1 × 10−20 n+1

ε=

∑ i=1

3 624 5 236 6 280 7 152 9 620 11 204 12 164

time (s)

minimum cost ($/kg U)

optimal days

optimal number of uses

optimal temp (°C)

29 534 30 907 35 984 57 571 66 236 64 955 77 990

599 604.65 597.69 597.69 597.53 597.52 597.52 597.47

30 38.6 32.4 32.4 30.9 30.9 30.9 30.1

15 13 15 15 14 14 14 15

26 24.5 25.5 25.5 25.7 25.7 25.7 25.9

accurately reflects those most likely to be adopted upon industrial scale-up. However, in order to clearly distinguish economic impacts of important cost drivers and provide a starting point for optimization, it is necessary to consider a single scenario yielding an illustrative cost. For the purposes of this analysis such a base case yielding a uranium production cost of $749/kg is defined according to Table 2. The values in

(uc u, i − uc u)2 n

(9)

The results of these trials for an illustrative base case are shown in Table 1. Also in the table is the approximate minimum cost found by the brute-force parameter sweep. The algorithm quickly finds and improves upon this minimum. Increasing the convergence criteria beyond 1 × 10−10 offers little to no decrease in the minimum cost returned by the algorithm but does increase the required time. Below this value the computed minima show slight fluctuations, since the minimum is very shallow. The fluctuations however are on the order of tens of cents and are therefore negligible. These fluctuations may arise due to discrete value requirements used in the cost calculation, such as number of work boats required to service the adsorbent field or number of chains used to moor adsorbent to the ocean floor. The existence of discrete values gives rise to numerous local minima that differ from the global minimum by a very small, insignificant amount. Since sensitivity analyses will be conducted requiring the algorithm to be run many times, this seemingly small increase in time will have a non-negligible effect in the total run time. Therefore, an ε value of 1 × 10−10 is considered sufficient. In addition to the convergence criteria discussed above derived from the standard method, a stationary iteration, i.e., having all the same vertices as the previous iteration, also ceases the algorithm. Once either convergence criterion is reached, this predicted minimum is recorded and the process initialized again by generating three random points around this new minimum. To avoid convergence on a local minimum, the algorithm undergoes 10 local restarts, using the predicted minimum of the previous run as the new starting point, as recommended by the literature.30 Also unique to this analysis is the integer constraint on the number of uses of a single unit mass of adsorbent. The algorithm generally returns noninteger values for decision variables, but the physicality of this integer constraint must be imposed. Therefore, this minimization deals with this issue by rounding the continuous value outputted by the standard algorithm both up and down to end up with two possibilities for number or uses. Both of these use values are input to the black box cost model with all other decision variable values remaining constant. The number of uses resulting in the lower of the two costs is the one selected to continue the algorithm.

Table 2. Base Case Description parameter

illustrative value

illustrative uranium production cost degree of grafting number of uses alkaline solution degradation rate length of campaign ocean temperature loss in uptake due to biofouling

749 250 6 NaOH 5 60 20 27

unit $/kg U % deployments % loss per reuse days °C %

this case come from ORNL laboratory data,31,32 deployments by the JAEA that provide the degradation rate datum,33 and previous cost estimates.15,18−21,33 This case is used for illustrative purposes only and is not meant to be interpreted as the most likely combination of parameters. The uranium production cost for this base case is displayed in Figure 8, illustrating the contribution from capital and operating costs for the three main process steps. For the sake of visual clarity the disposal and decommissioning cost associated with each step are included with the capital cost. The adsorbent production operations, which make the largest contribution to the final cost, are dominated by chemical and material expenses that make up approximately 60% of this cost, with the remainder resulting from contingency and other miscellaneous operating costs, utilities, and labor. The first case to be optimized is the use of the illustrative reference case as the initial guess while allowing the length of campaign, number of uses, and ocean temperature to vary. Running the optimization returns a cost of $597/kg U, a 20% savings compared to the original $749. The implementation of the minimization algorithm can be verified by comparing the result it returns with a brute force pseudo-minimization of the same case. Table 3 shows that the Nelder−Mead algorithm returns a uranium production cost value that is marginally lower than that predicted by the brute force calculation. This is unsurprising, as the brute force calculation can only calculate the cost at selected discrete values of each cost-driving variable. In Table 3 the results of the coarse-interval brute force calculation run earlier are included along with a more finely meshed sweep where temperature and days of campaign are intervals of one and two integer units,

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RESULTS Base Case Optima. The chemical and engineering processes associated with the braided adsorbent system are evolving with time. Ongoing changes to the technology make it impossible to define a reference set of parameters that F


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Figure 8. Base case uranium production cost as a sum of capital and operating cost for each process step.

sensitivity analysis is intended to illustrate the degree to which biofouling is a significant cost driver but should not be used to draw conclusions regarding the likely bounding values of the effect of biofouling. The resulting optimal production cost as a function of these parameters is plotted in Figure 9. Each (deployment cost,

Table 3. Optimized Reference Case: Comparison of Brutue Force and Nelder−Mead Algorithm Minimization Results parameter

brute force: coarse interval

brute force: fine interval

Nelder−Mead algorithm

minimized cost length of campaign number of uses temperature loss in uptake due to biofouling

$599/kg U 30 days 15 uses 26 °C 22%

$598/kg U 32 days 15 uses 26 °C 23%

$597/kg U 30.1 days 15 uses 25.9 °C 21%

respectively. The optimal values for the length of campaign, number of uses, and temperature broadly agree across all three optimization approaches. Sensitivity Analyses. The technology used in the recovery of uranium from seawater is evolving. Additionally, each cost input and adsorbent characteristic has its own uncertainty. Therefore, many elements affecting the final uranium production cost are subject to considerable uncertainty. In anticipation of potential changes to recovery technology and the performance models and to provide intuition for determining optimal conditions, sensitivity analyses to various decision variables and cost input sensitivities are conducted. Input Cost and Biofouling Performance Variation. The first sensitivity explores the effects of changing selected input costs along with the relationship between length of campaign and effects of biofouling. Both the deployment and mooring cost for a unit mass of adsorbent and the rate of loss in uptake due to biofouling are varied to reflect their uncertainties. To create the upcoming sensitivity plots, the optimization is run many times, with each run multiplying the base case deployment and mooring cost and rate of time dependent biofouling by a factor ranging between 0.5 and 1.5. This range of biofouling rates is arbitrarily chosen, since the degree of time dependence of biofouling remains largely uncertain. Hence the

Figure 9. Minimized uranium production cost as a function of deployment cost and biofouling slope multipliers.

biofouling rate) point in Figure 9 is associated with a unique optimal deployment strategy. Figures 10−12 plot the optimal campaign length, number of uses of the adsorbent, and ocean temperature that gives rise to the minimized cost surface of Figure 9. In order to clearly visualize the trends in optima seen in these figures, the data were smoothed using a penalized leastsquares function found in the literature.34 To further take advantage of the smoothing algorithm, some elements of each G


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Figure 12. Optimal temperature as a function of deployment cost and biofouling slope multipliers. Figure 10. Optimal days of campaign as a function of deployment cost and biofouling slope multipliers.

little effect on the operations component of the deployment cost, which is dominated by the sailors’ labor expense. The increased frequency of service does increase this cost due to a greater number of ships required. This is however overshadowed by the significant decrease in capital cost associated with a smaller field requiring fewer expensive mooring chains used to anchor the adsorbent to the seabed. The more dominant effect on optimal campaign length is, as expected, the rate of biofouling. Not surprisingly, as the rate of time dependent biofouling increases, the ideal length of campaign decreases, since the marginal uptake in uranium is offset by the heightened loss due to microorganism growth. This highlights the importance of improving kinetics in order to recover sufficient uranium before the effects of biofouling become substantial. As expected, lower deployment costs favor more uses of the same mass of adsorbent. Degradation of the adsorbent causes lower uranium uptake with each use, and at increasing deployment costs this decreased mass of uranium is no longer of sufficient value to justify the cost of retrieval. The rate of biofouling has negligible effects on the optimal number of uses because each deployment is treated identically with respect to biofouling. The acidic chemicals used to elute uranium off of the fibers are assumed to remove all microorganisms such that each time the adsorbent is redeployed, it begins with no loss due to biofouling. Although the fouling multiplier only directly affects the degree of biofouling as a function of time, an increase in this slope favors lower ocean temperatures. As the rate of biofouling speeds up, lower ocean temperatures are favored to combat the loss in uptake. This again highlights the need for improved speed of adsorbent recovery. If the severity of biofouling as a function of time does turn out to be more significant than observed thus far, then faster kinetics will not only ameliorate time dependent fouling but also allow the adsorbent to take advantage of higher ocean temperatures to further increase uptake. The ideal ocean temperature is largely unaffected by deployment costs. Currently there is no dependence of deployment cost on temperature; however, this may change as the cost analysis continues to develop. Factors affecting deployment cost may realistically include the correlation between location of deployment, water temperature, and bioactivity, and the dependence of these factors and mooring

matrix containing the optima are left blank allowing for a smoother interpolation that is less sensitive to small fluctuations. There are two reasons for the irregularities in some of the contour plots of Figures 9−11 where cost-

Figure 11. Optimal number of uses as a function of deployment cost and biofouling slope multipliers.

minimizing values of individual system parameters are being identified. The minima are often quite shallow so that even under stringent convergence criteria some fluctuation in the derived cost-minimizing parameter values is still observed. In addition, several of the system parameters (e.g., number of uses) are integer-valued. When these are varied as part of each cost minimization calculation, they give rise to discrete, sometimes substantial cost responses. The cost of adsorbent deployment is a complicated function of the length of immersion. Shorter campaigns require a smaller adsorbent field with a higher servicing rate, meaning that less uranium is recovered per deployment event but more uranium is recovered per unit area of the field as the campaign is shortened. Since the unit deployment and mooring cost include the capital, operating, and decommissioning cost of deployment activities, shorter immersion times become slightly favorable as the unit deployment cost increases. Length of campaign has H


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there are no underlying principles that make a third order polynomial physically significant. As expected, the cost of uranium recovery decreases as the number of uses increases and approaches the optimal value in the reference case conditions. The increase in total uranium uptake over the adsorbent lifetime effectively recoups more of the costly production expense. Consequently, fewer uses of the adsorbent favor longer campaign lengths in order to recover as much uranium as possible throughout the shortened adsorbent lifetime. This effect is most dramatic in the case of a single adsorbent use. Multiple reuses of the adsorbent favor shorter campaign lengths with a reduction in overall field size and to take advantage of the rapid increase in rate of uptake at short campaign lengths discussed earlier (Figure 1). When reuse is not an option, however, it is most advantageous to allow the adsorbent to approach its capacity, even when this capacity has been lowered due to biofouling. This trend does not continue as the ocean temperature approaches the maximum considered, where the biofouling becomes so severe that even a single use scenario favors short campaign lengths due to the significant colonization of fiber surface by microorganism growth.

cost on deployment depth. At present, deployments are assumed to take place at 100 m depth. Fixed Number of Uses. The next sensitivity analysis constrains one part of the parameter space to investigate how other decision variables are affected. The number of uses is removed from the optimization and is analyzed at three constant values, 1, 6, and 15. A single use scenario is considered as a hedge against the possibility that degradation turns out to be significantly higher than expected. Six uses of a unit mass of adsorbent are considered as a historically relevant case. Fifteen uses are also considered, as this is the optimal number of uses identified earlier for the reference case. The minimum uranium production cost and its corresponding optimal length of campaign and ocean temperature for these three scenarios are displayed in Figures 13 and 14. The optimal days of campaign

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CONCLUSION This analysis presented the optimization of deployment conditions to minimize the uranium production cost for a passive recovery system. First the system under consideration, as developed under ORNL and partners, was described. Given the complexity and platform of the cost calculation, it was determined that the problem be treated as a black box optimization. Therefore, derivative free optimization algorithms were considered and the Nelder−Mead simplex method was ultimately selected. The parameter space of the problem was described in detail. The base case was defined and optimized with respect to three decision variables each with a unique influence on adsorbent performance: days of campaign the adsorbent spends in the ocean, number of uses of each adsorbent unit mass before disposal, and ocean temperature. The length of time the adsorbent soaks increases the uranium uptake but also the degree of oceanic biofouling. While each reuse of adsorbent results in additional uranium recovery without the expense of producing fresh adsorbent, prior to each reuse the adsorbent must undergo a costly regeneration process that causes it to suffer a loss in uptake. Warmer waters are known to increase the capacity of the adsorbent to take up uranium but also presumably enhance the effects of biofouling. Optimization of the base case was conducted to yield a minimum uranium production cost 20% lower than a historical reference case cost. A brute force calculation verified this result. Next, sensitivity analyses were conducted to provide intuition for determining optimal deployment conditions. First the cost associated with deploying and mooring a unit mass of adsorbent and the rate of biofouling for the base case were multiplied by a factor ranging from 0.5 to 1.5 to account for the uncertainties embedded in the cost calculation and adsorbent performance models. These changes had significant impacts on the optimal decision variable values, indicating that the optimal scenario will continue to change with the technology and as better characterization of the fibers becomes available with continued experimentation and development. Increasing both the deployment cost and rate of biofouling generally favored a shorter length of campaign. Since deployment capital cost is more sensitive to changes in field

Figure 13. Minimized uranium production cost as a function of temperature for a fixed number of uses.

Figure 14. Optimal days of campaign as a function of temperature for a fixed number of uses.

data seen in Figure 14 are fit to a third order polynomial, and the resulting equation is plotted to reduce noise and more easily visualize trends. This was purely for aesthetic reasons; I


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(8) Leps, M.; Sejnoha, M. New Approach to Optimization of Reinforce Concrete Beams. Comput. Struct. 2003, 81 (18−19), 1957− 1966. (9) Jayabalan, V.; Chaudhuri, D. Cost Optimization of Maintenance Scheduling for a System with Assured Reliability. IEEE Trans. Reliab. 1992, 41 (1), 21−25. (10) Santarelli, M.; Pellegrino, D. Mathematical Optimization of a RES-H2 Plant Using a Black Box Algorithm. Renewable Energy 2005, 30, 493−510. (11) Abendroth, R.; Salmon, C. Sensitivity Study of Optimum RC Restrained End T-Sections. Journal of Structural Engineering. 1986, 112, 1928−1943. (12) Kim, J.; Tsouris, C.; Mayes, R.; Oyola, Y.; Saito, t.; Janke, C.; Dai, S.; Schneider, E.; Sachde, D. Recovery of Uranium from Seawater: A Review of Current Status and Future Research Needs. Sep. Sci. Technol. 2013, 48 (3), 367−387. (13) Economic Modeling Working Group of the Generation IV International Forum. Cost Estimating Guidelines for Generation IV Nuclear Energy Systems, revision 4.2; OECD Nuclear Energy Agency, 2007. (14) Park, C. Contemporary Engineering Economics, 5th ed.; Pearson Education Inc.: Upper Saddle River, NJ, 2011. (15) Schneider, E.; Sachde, D. The Cost of Recovering Uranium from Seawater by a Braided Polymer Adsorbent System. Science and Global Security 2013, 21 (2), 134−163. (16) Kim, J.; Tsouris, C.; Oyola, Y.; Janke, C.; Mayes, R.; Dai, S.; Gill, G.; Kuo, L.; Wood, J.; Choe, K.; Schneider, E.; Lindner, H. Uptake of Uranium from Seawater by Amidoxime-Based Polymeric Adsorbent: Field Experiments, Modeling, and Updated Economic Assessment. Ind. Eng. Chem. Res. 2014, 53 (14), 6076−6083. (17) Lindner, H.; Schneider, E. Review of Cost Estimates for Uranium Recovery from Seawater. Energy Economics. 2015, 49, 9−22. (18) Schneider, E.; Lindner, H. Updates to the Estimated Cost of Uranium Recovery from Seawater. Proc. Pac. Basic Nucl. Conf. 2014, 19. (19) Flicker Byers, M.; Schneider, E.; Chen, J. Sensitivity of Seawater Uranium Cost to System and Design Parameters. Proc. Global Nucl. Fuel Cycle Conf. 2015, 21, 969−977. (20) Byers, M. Optimization of the Passive Recovery of Uranium from Seawater. Master’s Thesis, The University of Texas at Austin, 2015; unpublished. (21) Schneider, E.; Gill, G. Characterization and Deployment Studies and Cost Analysis of Seawater Uranium Recovered by a Polymeric Adsorbent System. Presented at the International Symposium on Uranium Material for the Nuclear Fuel Cycle: Exploration, Mining, Production, Supply and Demand, Economics and Environmental Issues, Vienna, Austria, Jun 23−27, 2014. (22) Gill, G.; Kuo, L.; Janke, C.; Park, J.; Jeters, R.; Bonheyo, G.; Pan, H.; Wai, C.; Khangaonkar, T.; Bianuccik, L.; Wood, J.; Warner, M.; Peterson, S.; Abrecht, D.; Mayes, R.; Tsouris, C.; Oyola, Y.; Strivens, J.; Schlafer, N.; Addlemand, R.; Chouyyok, W.; Das, S.; Buesseler, K.; Breier, C.; D’alessandro, E. The Uranium from Seawater Program at PNNL: Overview of Marine Testing, Adsorbent Characterization, Adsorbent Durability, Adsorbent Toxicity, and Deployment Studies. Ind. Eng. Chem. Res. 2015, submitted. (23) Kuo, L.; Janke, C.; Wood, J.; Strivens, E.; Gill, G.; et al. Characterization and Testing of Amidoxime-Based Adsorbent Materials to Extract Uranium from Natural Seawater. Ind. Eng. Chem. Res. 2015, DOI: 10.1021/acs.iecr.5b03267. (24) Gill, G.; Kuo, L.-J.; Wood, J.; Janke, C. Complete Laboratory Evaluation and Issue a Report on the Impact of Temperature on Uranium Adsorption. Pacific Northwest National Laboratory: Sequim, WA, Sep 20, 2014. (25) Sugo, T.; Tamada, M.; Seguchi, T.; Shimizu, T.; Uotani, M.; Kashima, R. Recovery System for Uranium from Seawater with Fibrous Adsorbent and Its Preliminary Cost Estimation. Journal of the Atomic Energy Society of Japan 2001, 43, 1010−1016.

size than the operating cost, the more frequent servicing of smaller fields necessary for shorter immersion times becomes advantageous. Increasing the deployment cost also decreased the optimal number of adsorbent uses because the degradation suffered with each use has a cumulative effect. This highlights the need for a more durable adsorbent, especially should deployment be more expensive than anticipated. Another important conclusion of the sensitivity analysis was that if biofouling is in fact dependent on length of campaign, then the improvement of uptake kinetics will become an important objective for reducing uranium production costs. Although many cost estimates exist for uranium recovery from seawater, a systematic optimization framework has never been developed. The novelty of this research is the application of an existing optimization algorithm in order to illuminate areas of R&D focus. Cost minimization through system and design parameter alterations indicates what type of return would be achieved by improving various components of the system. Furthermore, even as the technology continues to evolve, the methodology developed for this optimization will remain relevant. Updates to the adsorbent performance parameters can be made with little to no change to the optimization framework. Additionally, as the number of factors known to affect adsorbent performance increases, the optimization can easily be expanded to include any number of decisions variables with minimal modification to the methodology.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.iecr.5b03242. Additional details of cost estimation methodology (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This research was supported by the U.S. DOE Office of Nuclear Energy, under Contract DE-AC05-00OR22725 with ORNL, managed by UT-Battelle, LLC.

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REFERENCES

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