Time-Dependent Implementation of Argonne's Model for Universal

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Time-Dependent Implementation of Argonne’s Model for Universal Solvent Extraction Kurt Frey, John F. Krebs, and Candido Pereira* Chemical Sciences and Engineering, Argonne National Laboratory, 9700 South Cass Avenue, Argonne, Illinois 60439, United States S Supporting Information *

ABSTRACT: Argonne’s Model for Universal Solvent Extraction (AMUSE) simulates multistage counter-current solvent extraction processes for species of interest to spent nuclear fuel reprocessing; it is a model of a liquid−liquid extraction unit operation. This work extends the model from its original steady-state implementation to include a time-dependent description of all species of interest. Major components of this extension include a differential mass transfer term, a description of interstage flow rates, and a reaction network for plutonium reduction. The mass transfer term has been formulated using a lumped efficiency term in place of a mass transfer coefficient; the plutonium reduction reactions have been formulated to ensure consistency at all operating conditions. Several nonequilibrium behaviors during simulations have been identified, which allows for improved safety monitoring during process start up and disturbance response.

1. INTRODUCTION AND BACKGROUND Argonne National Laboratory maintains Argonne’s Model for Universal Solvent Extraction (AMUSE): a simulation tool used to describe multistage, counter-current solvent extraction processes of interest to spent nuclear fuel reprocessing.1 It is a model of a liquid−liquid extraction unit operation. This tool is an expansion of the Generic TRUEX model2 (GTM) to additional systems of interest (e.g., PUREX, UREX, etc.). AMUSE simulations were originally designed to provide steadystate process descriptions. This work expands that description to provide a dynamic representation of the solvent extraction process. A dynamic representation allows for examination of start-up conditions as well as possible transients due to disturbances. The solvent extraction process simulated in AMUSE is constructed using a series of well-mixed stages. These stages may represent physical equipment (i.e., centrifugal contactors or mixer settlers) or be theoretical divisions (i.e., axial length in a pulsed column). 1.1. Context. Computer modeling and simulation of nuclear fuel recycle has a long history, primarily due to interest in plutonium and uranium extraction (PUREX) as a method for obtaining material for nuclear weapons but also for radioactive waste treatment and regeneration of used nuclear fuel. A complete review of all relevant work is beyond the scope of this paper. Dedicated review articles of this field have been conducted since 19653 and continued to the present.4 Development of numerical tools in this area remains an active field of research.5 Only models and applications similar to the current work will be addressed here. Spent nuclear fuel reprocessing involves several unit operations: disassembly, dissolution, separation, concentration, conversion, and solidification. The AMUSE utility is relevant to separations, particularly counter-current aqueous solvent extraction (i.e., liquid−liquid extraction) operations. This model is not appropriate for nonaqueous separation processes (e.g., electrochemical or molten salt separations) or other unit operations required for reprocessing of used nuclear fuel. © 2012 American Chemical Society

Primary motivation for this work stemmed from the need for the AMUSE model to incorporate dynamic phenomena. Although solvent extraction operations are continuous processes, from a systems perspective, they are operated in a batch or semibatch manner to accommodate inventory restrictions particular to radioactive materials.6−8 This manner of operation implies that start-up, shut-down, and other transient behaviors of the solvent extraction operation are of significant interest (primarily for operationally safety but also for material accountancy and related issues). Several other dynamic simulation utilities have been developed and are available for use on systems relevant to the dynamic AMUSE model. In particular, SEPHIS, PUBG, CUSEP, and SX Process9,10,6,5 all are directly applicable to systems simulated using the AMUSE utility. Several characteristics of AMUSE make it unique from the other models relevant to aqueous solvent extraction in the nuclear fuel cycle indicated here. Foremost is the library of component data that is available to the AMUSE model. In addition to uranium, plutonium, and nitric acid, AMUSE supports up to 50 other species common to dissolved fuel assemblies, including thermodynamic data relevant to these species in several different extraction processes. Additionally, implementation of the extraction kinetics in the dynamic AMUSE model is unique; molar fluxes due to extraction are described explicitly. Both SEPHIS and CUSEP assume rapid equilibrium within each stage of extraction. The PUBG model supports nonequilibrium extraction kinetics in a manner similar to AMUSE but requires data for mass transfer coefficients and estimated interfacial area. Dynamic implementation of AMUSE automatically estimates these values given a lumped efficiency parameter. Received: Revised: Accepted: Published: 13219

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Figure 1. Representation of the stage model used in AMUSE.

quantity flows (given as the product of volumetric flow rates Qi and species concentrations Ci in those flows) in addition to species generation from reaction Rreact and species flux from mass transfer Rext. The summation in eq 1 involves the nQ different flows relevant to a particular stage and phase of interest. These flows are described in the Interstage Flow Rates section. Water, the principal component of the aqueous phase, and organic solvent and diluent, the principal components of the organic phase, are considered immiscible (i.e., the molar quantity of water in the organic phase is zero; the molar quantities of organic solvent and diluent in the aqueous phase is zero). Two additional state variables per stage are required to describe these phases. The total aqueous-phase volume and total organic-phase volume (Vaq and Vorg) were selected as state variables instead of the molar quantities of the principal components. These variables correspond to overall mass balances on the aqueous and organic phases, which have been rearranged to be explicit in the rate of change of phase volume.

2. METHODOLOGY Dynamic implementation of AMUSE has been formulated as a set of ordinary differential equations (ODEs) that describe the evolution of a selected set of state variables (y) with time. These equations have been formulated as an initial value problem on a specified time interval given initial values y(t0) = y0. For simulations where steady-state conditions are of interest (y′ = 0), the simulation is allowed to continue until the relative rate of change for all state variables is less than 0.05% per unit time (an adjustable threshold). Steady-state results of the dynamic implementation of AMUSE are of particular interest because they can be compared with the results from the previous (original) version of AMUSE. For this work, unit time was selected as the minute and state variables varying less than 0.05%/min were considered to have achieved their steady-state value. Each of the state variables in the dynamic model corresponds to a component mass balance. Within each stage of the simulation, a nonprincipal component (i.e., not water, organic solvent, or diluent) may partition between the aqueous and the organic phase. Therefore, two state variables (aqueous phase molar quantity and organic phase molar quantity; Naq and Norg) are required per stage for each component of interest; these variables correspond to species mass balances in each phase and for each stage. dN = dt

dV = dt

Q

c

c

⎛ ∂ρ dNj ⎞ ∂ρ dT ⎜ · ⎟ − V ∂T dt ⎝ ∂Nj dt ⎠

∂ρ

ρ + V ∂V

(2)

Equation 2 pertains to both aqueous and organic phase volumes and describes the rate of change dV/dt in the volume V for a particular phase (i.e., aqueous or organic) in a given stage. This expression includes a summation involving nQ total mass flows (given as the products of volumetric flow rates Qi and densities of those flows ρi) plus the change in mass due to species extraction (given as summation involving the nc

∑ (Q i·Ci) + R react + R ext nQ

∑n (Q i·ρi ) + ∑n (MWj·R ext, j) − V ∑n

(1)

Equation 1 pertains to both phases and describes the rate of change dN/dt of the molar quantity N for a particular phase in a given stage. This expression involves the sum of the molar 13220

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components present, their mass transfer terms Rext,j, and their respective molecular weights MWj). For consistency between eqs 1 and 2, Rext,j has units of moles per time and the species molecular weights MWj are used in eq 2 to convert from moles to masses. The additional terms in eq 2 are needed to express the equation explicitly as dV/dt; a more detailed derivation is provided as Supporting Information. Extractant concentrations are assumed to be uniform throughout the organic phase and constant throughout the system. No balances on extractant species are required. 2.1. Numerical Techniques. MATLAB was used to implement the dynamic version AMUSE because of its capabilities in solving systems of ODEs. In particular, the ODE15s function in MATLAB was found to be the most useful in calculating trajectories for the state variables.11 ODE15s itself is an implementation of numeric differentiation formulas for solving ODEs. Options specific to the ODE15s solver include increasing absolute tolerances on state variable accuracy from a default value of 1e−6 to a value of 1e−12. Additionally, these absolute tolerances are enforced on state variable values by the ODE15s algorithm after a preliminary scaling has been separately applied by the AMUSE implementation. State variable values within dynamic AMUSE simulations span several orders of magnitude. This span is a consequence of dramatically different component concentrations within feeds to the process. A typical feed is comprised of spent nuclear fuel that has been dissolved in strong acid; concentrations often vary from molar to micromolar.12 Molar quantities are scaled by AMUSE relative to the total molar quantities present in the feed streams prior to using the ODE15s solver. This scaling ensures that tolerances on a trace component are roughly equivalent to tolerances on more concentrated components. Species synthesized during the solvent extraction process that are not present in the feed remain unscaled; volumes also remain unscaled. 2.2. Stage Description. Both current (dynamic) AMUSE simulations and previous (original) AMUSE simulations consist of a variable number of user-specified of stages, Figure 1. Each stage within the simulation consists of two phases that are denoted aqueous and organic. These phases are assumed to be well mixed but immiscible. The total system volume is entirely comprised of these well-mixed stage volumes, and no interstage volumes are considered. Concentrations in outlet flows from a stage are assumed to be equal to the concentrations within their respective stages. The summations in eqs 1 and 2 over the nQ different flows to each stage are performed over the flows described here, where nQ is a maximum of six. Each stage receives up to four different entering flows: a pure aqueous feed, a pure organic feed, and two inlet interstage flows from adjacent stages. Correspondingly, each stage also has two interstage outlet flows. Simulations are assumed to be counter-current, so one of the interstage outlet flows is directed from the current stage to next stage, while the other is directed from the current stage to the prior stage. Any of these flows may be zero (e.g., there is no feed onto a particular stage or the stage may be an end stage with no prior or next stage present). Interstage outlet flows do not necessarily connect directly to the inlet flows on adjacent stages. Each of the two outlet flows from a stage is characterized by a parameter describing the fraction of the flow that exits the system. When the exiting fraction of the flow is zero, all of the flow continues to the inlet of the adjacent stage. These exiting fractions are used to

represent sample streams or system effluents. Additionally, interstage outlet flows are not necessarily pure phase flows. Each of the two outlet flows from a stage is characterized by an additional parameter describing the carryover fraction (i.e., the fraction of the flow that is the nonprimary phase for that flow). When the carryover fraction of the flow is zero, there is no entrainment and the flow corresponds to a pure phase. These carryover fractions are used to represent imperfect separation of the aqueous and organic phases. Note that the carryover fraction has an implicit maximum of 0.50; a carryover fraction of 0.50 implies that a flow is an equal mixture (by volume) of the aqueous and organic phases. These entrainment parameters can be used to approximate the mutual solubility of the primary components as needed. By convention, stage outlets connected to the inlets of the next stage (i.e., stage with a high index number) are predominantly organic, and stage outlets connected to the inlets of the prior stage (i.e., stage with a lower index number) are predominantly aqueous phase. During simulation, the direction of the flows is not density dependent; flows directed from a lower stage index to a higher stage index are always assumed to be organic dominant, with the carryover parameter describing the fraction of the flow that is aqueous. Typically the organic phase is the less-dense phase, but for some solvent extraction processes (e.g., CCD-PEG extraction) the organic phase will be the denser of the two phases. Each stage is also characterized by a tunable parameter for the organic phase and a tunable parameter for the aqueous phase. These parameters are denoted the maximum organic phase volume Vmax,org and maximum aqueous phase volume Vmax,aq, although they do not always correspond to the actual maximum phase volume. During simulation, the total aqueous and organic phase volumes may exceed the Vmax,aq and Vmax,org parameter values. These tunable parameters are most easily estimated as the phase holdups at steady state recorded from process data. Approximate expressions for interstage flow rates require these tunable parameters as described in the following section. Carryover fractions and exiting fractions must be specified for each of the stage outlets. It should be noted that the carryover fraction in the dynamic AMUSE model applies to both the exiting and the nonexiting fractions of the outlet flow, while the carryover fraction in the original AMUSE model applies to only the nonexiting fraction.13 2.3. Interstage Flow Rates. An expression for the outlet flow rates from each stage must be provided for the system of equations in the dynamic AMUSE model to be fully specified, but these expressions are strongly dependent on the selection of physical equipment used in solvent extraction. Approximate outlet flow rates for each stage are determined by assuming phase volumes, densities, and temperatures are constant (i.e., dV/dt = 0, dρ/dt = 0, dT/dt = 0) and the extraction rate is assumed to be zero (i.e., Rext = 0). 0=

∑ (Q i†·ρi ) nQ

(3)

Equation 3 appends a dagger superscript to the flow rate variables Qi† to indicate that these flow rates are calculated assuming steady-state conditions and zero interphase mass transfer; steady-state flow rates are only used as an intermediate value when approximating actual process flow rates. Additional 13221

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back-extraction). The expression in eq 6 relates the extraction rate for an organic phase component to a mass transfer coefficient korg→aq, the interfacial mass transfer area A, the steady-state distribution ratio of the solute of interest D, and the current aqueous and organic concentrations of that solute Caq and Corg, respectively. Current organic concentrations are determined from a ratio of state variables (Corg = Norg/Vorg). Interphase molar fluxes for the aqueous phase are equal and opposite in sign with respect to the expression in eq 6.

details on this calculation have been included as Supporting Information. Phase densities in eq 3 are allowed to vary from stage to stage (feed streams may also have different densities). The instantaneous outlet flow rates for each stage in the dynamic model are then calculated according eq 4.

Q out =

V Q† Vmax out

(4)

Equation 4 applies to both the outlet flow rate directed to the next stage (organic dominant) and the outlet flow rate directed to the prior stage (aqueous dominant). Organic-dominant outlet flow rates are proportional to the ratio of the current instantaneous organic phase volume to the maximum organic phase volume parameter Vmax,org; aqueous-dominant outlet flow rates are proportional to the ratio of the instantaneous aqueous phase volume to the maximum aqueous phase volume parameter Vmax,aq. As previously indicated, the maximum phase volume parameters are tunable values used to approximate outlet flow rates. Transient stage volumes may exceed the value of this parameter, and a significant mass transfer rate between phases may result in phase volumes that exceed the value of the maximum stage volume parameters even at steady state. Equation 4 is a reasonable approximation for a generic stage in a solvent extraction operation where holdups are known approximately but is also the expression that has the most potential for refinement given more specific system knowledge or a concrete description of the physical equipment used. As a consequence of using eq 4, the ratio of phase volumes within a stage is approximately equal to the ratio of the maximum phase volume parameters. 2.4. Phase Densities. In the dynamic AMUSE model, instantaneous densities need to be calculated for both phases in order to describe the evolution of the phase volume state variables with time. Equation 5 has been used to calculate both aqueous phase and organic phase densities. ρ = ρ0 +

R ext,org = −R ext,aq

The mass transfer coefficient and interfacial area are not readily available quantities and both strongly dependent on process specifications. An approximation of the product of these two values is provided in eq 8. korg → aqA =

nc

(5)

In eq 5, the phase density ρ has been expressed as the sum of the primary component density in that phase ρ0 (water or organic solvent and diluent) with additional contributions from each of the other components. Concentrations are determined from the state variables (i.e., C = N/V for each of the nc components); molecular weights MWj and partial molar volumes at infinite dilution Vj∞ must be known for each component. The derivation and use of eq 5 has been detailed in the Supporting Information. 2.5. Mass Transfer. Deriving an expression for the rate of mass transfer between aqueous and organic volumes was required for dynamic implementation of AMUSE. The rate of mass transfer from the organic phase to the aqueous phase can be represented generically using a mass transfer coefficient as in eq 6. Equation 6 describes the net transfer from the aqueous phase to the organic phase. R ext,org = korg → aqA(DCaq − Corg)

eff 1 − eff

⎛ 1 ⎜ + ⎝ ∑ Q org,enter

⎞ D ⎟ ∑ Q aq,enter ⎠

(8)

Equation 8 relates the product of the mass transfer coefficient and interfacial area for a given stage to the component distribution ratio D in that stage, the stage efficiency term eff, and summations of the entering aqueous and organic flow rates for that stage (these flow rates include feed flows to the process occurring on the specified stage as well as all inlet flows, including carryover, from other stages). The distribution ratio is the ratio of organic concentration to aqueous concentration of solute at steady state (i.e., Corg,SS/ Caq,SS) and is equivalent to the ratio of mass transfer coefficients (i.e., kaq→org/korg→aq). Distribution ratios must be known for each component. The efficiency term is a user-specified parameter. In this context, the efficiency describes the fractional approach of the mass transfer driving force to zero (i.e., steady state). This is approximately equivalent to the Murphree efficiency for each phase. Also note that a stage efficiency fraction of one (i.e., 1.0 or 100% efficiency) is undefined. A detailed derivation of eq 8 has been included in the Supporting Information. 2.6. Thermophysical Property Data. Equilibrium distribution ratios (D) used to calculate the mass transfer rates, as well as molecular weights (MW), partial molar volumes at infinite dilution (Vj∞), and primary component densities (ρ0), are available from Argonne’s Speciation and Partitioning Equilibria Calculator (SPEC). SPEC is a FORTRAN implementation of Argonne’s Spreadsheet Algorithm for Speciation and Partitioning Equilibria (SASPE)1 as a shared library. This library has been used to provide property data to simulations in both the MATLAB and the Aspen Custom Modeler environments. No modifications were made to any of the SASPE data in the SPEC implementation. Thermophysical property data from SPEC and SASPE are only relevant to nitric acid processes. Both aqueous and organic phase charge neutrality is enforced using the nitrate ion (NO3−), with separate charge balances performed on each phase. Nitrate concentrations cannot be specified independently. Additionally, SPEC and SASPE assume a minimum aqueous concentration for hydrogen ion (H+). This ion must be present as a component in all simulations that use these thermophysical property data. Data are available for the uranium extraction (UREX) process, plutonium−uranium extraction (PUREX) process,

∑ (MWj·Cj) − ρ0 ∑ (V̅ j∞·Cj) nc

(7)

(6)

Note that eq 6 applies only to the extraction rate with respect to the organic phase. In this work, positive mass transfer rates denote flow of material from the aqueous phase to the organic phase (i.e., extraction); negative mass transfer rates denote flow of material from the organic phase to the aqueous phase (i.e., 13222

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Figure 2. Schematic of the flowsheet representing the three-step solvent extraction process: extraction, solvent wash, and solvent regeneration.

four variants of the transuranic extraction (TRUEX) process, and CCD-PEG-based extraction processes. 2.7. Reactions. Chemical transformations within a phase are included through the Rreact term in eq 1. This term is not required for the system of equations in the dynamic AMUSE model to be fully specified and is zero for all species not participating in reactions. All phase volumes are assumed to be uniform, so reactions included in the AMUSE model must be either homogeneous or able to be expressed intensively (e.g., using a specific surface area). Only one reaction, the reduction of plutonium, has been implemented in the current work. Reduction of the component Pu4+ to the component Pu3+ using hydroxyl ammonium nitrate ((NH3OH+)(NO3−); HAN) was selected for implementation because it is of particular interest to nuclear fuel reprocessing. The two oxidation states of plutonium behave differently during the solvent extraction process, and HAN is the preferred reductant because its reaction products (N2, N2O, and H2O) do not contribute to production of solid waste.14,15 This reaction rate has been studied by many authors;16−19 the reaction rates described by Barney have been used in this work.16 In this work, eq 9 is used to represent the overall reaction for reduction of Pu4+ to Pu3+. Pu 4 + + NH3OH+ → Pu 3 + + 2H+

R react = kA

[Pu 4 +][NH3OH+] ⎛ 1 ⎜ + 2 − ⎝ [H ] (Kd + [NO3 ]) 2X + X +

⎞ ⎟ 1⎠

(10)

Quantities appearing in square brackets describe concentrations. Equation 10 involves a dimensionless reaction coordinate X that is calculated according to eq 11. X=

kA[Pu 3 +]2 [H+]2 (Kd + [NO−3 ]) 2k O[Pu 4 +][NH3OH+]

(11)

In eqs 10 and 11, the equilibrium constant Kd = 0.33 mol/L. The two reaction rate constants kA and kO are calculated using eqs 12 and 13, respectively. Units on these constants are mol2/ L2/min and mol5/L5/min, respectively. Temperature (T) has units of Kelvin. ⎛ −11 574 ⎞ ⎟ kA = 1.8518 × 1019 exp⎜ ⎝ T ⎠

(12)

⎛ −15 600 ⎞ ⎟ k O = 3.8796 × 1022 exp⎜ ⎝ T ⎠

(13)

A detailed motivation for eq 11 is provided as Supporting Information. The reaction described here is only appropriate in the presence of excess HAN. When the concentration of HAN is roughly equal to or less than the concentration of Pu4+, then a competing reaction that generates nitrous oxide instead of nitrogen gas is relevant.16 Additionally, the presence of nitrous acid contributes toward a third, autocatalytic pathway that consumes HAN and generates nitrous oxide.20 Neither of the nitrous oxide generating reactions has been included in this work, although generation of N2O does represent a potential safety concern. Operating conditions are typically selected to maximize the nitrogen gas forming reaction described in this section. 2.8. Stage Temperature Profiles. Only phase volumes and molar quantities are currently included as state variables;

(9)

Only Pu4+, NH3OH+, Pu3+, and H+ are species tracked during simulation. Mass is not conserved in the reaction described by eq 9. Gaseous products such as molecular nitrogen (1/2N2) are assumed to exit the system. Products such as water (H2O) that are not tracked explicitly are accounted for through density variations caused by changes in the concentrations of explicitly accounted species. The rate of reaction used to describe eq 9 is given in eq 10. This rate has a coefficient of −1 for Pu4+ and NH3OH+, a coefficient of 1 for Pu3+, and a coefficient of 2 for H+. 13223

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These initial conditions reflect typical a start-up procedure, where the two primary components are fed to the system and allowed to achieve near-equilibrium stage contents prior to introducing any extractable components of interest (e.g., radionuclides). For extraction operations where the density of the feed is sufficiently different from the density of the primary components absent the extractable components, an additional step may be included in the start-up procedure where an intermediate feed of appropriate density or composition is introduced prior to the feed containing components of interest. The goal of this stepped start-up procedure is to ensure that the interstage flow rates are nearly constant while extracting components of interest. 3.2. Components and Physical Properties. Hydrogen ion concentrations (i.e., nitric acid concentrations) are the most important factor in determining the distribution values for all components of interest. Accurately simulating acid concentrations during solvent extraction is essential to accuracy in the concentrations for all other components. For this example simulation, the only extractable component included in the simulation is nitric acid. Concentrations of nitric acid for the four feed streams in this example are listed in Table 1; nitric acid concentrations are

stage temperatures are included in the dynamic AMUSE model, but insufficient physical property data are available to completely specify a set of energy balances. However, the temperature of each stage is accessible as a specified parameter and used when calculating the phase densities and component distribution ratios. During the derivation of eq 2, it was assumed that the temperatures vary. If a dynamic stage temperature profile is available, the current implementation of the model supports its use.

3. RESULTS AND DISCUSSION An example flowsheet was constructed to test the dynamic AMUSE model. This flowsheet was intended to describe a combined extraction, wash, and regeneration process. The first group of eight stages extracts components of interest from the aqueous phase into the organic phase, the second group of eight stages serves to scrub undesired components extracted into the organic phase back into the aqueous phase, and the final group of four stages strips all components introduced into the organic phase back into an aqueous phase. Typically, the regenerated organic phase effluent from the strip section is then reintroduced into the extraction section. Recycling and rerouting of exiting streams is not supported in the current implementation of the model, but these behaviors are not relevant to the current analysis. A schematic of the example flowsheet is presented in Figure 2. Interstage flows have been omitted except where these flows involve nonzero exiting fractions. Feed flows have been omitted except where their flow rates are nonzero. Flow rates were selected as 10, 20, 30, and 40 L/min for the three aqueous feeds and one organic feed. Carryover fractions on all outlet flows have been specified at 0.5% (i.e., the carryover fraction is 0.005 for all interstage flows). Holdup values (i.e., Vmax parameter values) for each phase in each stage have been set at 20 L. Selection of parameter values results in an organic to aqueous phase ratio that is roughly 1:1 for all stages during the simulation. Flow rates and parameter values were selected arbitrarily to provide plausible values for residence times and phase volume ratios for stages representing annular centrifugal contactors. Rotor diameters can vary widely (diameters from 2 to 50 cm are common), resulting in a wide range of potential flow rates and holdups. Each individual stage depicted in Figure 2 uses the stage representation from Figure 1. All feed and stage process temperatures are held constant at 25 °C. A UREX-type process was simulated (i.e., the aqueous phase consists of water; the organic phase consists of tributyl phosphate (TBP) as the extractant and n-dodecane as the organic solvent), with a uniform concentration of 1.1 M TBP throughout the organic phase (30% TBP v/v). 3.1. Consistent Initialization. Initialization of the set of ODEs in the dynamic AMUSE model requires values for the aqueous and organic phase volumes in each stage as well as initial molar quantities for each phase in all stages. Initial volumes were selected to be equal to their respective Vmax parameter values: 20 L for both aqueous and organic phases. These volumes are the steady-state volumes in the absence of any other components and tend to be in the neighborhood of the steady-state volumes regardless of stage composition. Hydrogen ion quantities were initialized to 2.0e−6 mol in all aqueous phases, which corresponds to initial concentrations of 1.0e−7 M (i.e., pH = 7). All other molar quantities in aqueous and organic phases were initialized to zero.

Table 1. Feed Concentrations Used during Simulation of the Example Flowsheet feed stream aqueous, stage 8 aqueous, stage 16 aqueous, stage 20 organic, stage 1

contents [H+] [H+] [H+] [H+]

= = = =

1.50 0.30 0.06 0.00

M M M M

typically zero in organic feed streams of fresh solvent, and aqueous nitric acid concentrations were selected arbitrarily to span the range of relevant distribution ratios in a 30% v/v TBP solvent extraction process (see Figure 3). Explicit thermophysical properties have been used for this example simulation in lieu of values from the SPEC library. Nitric acid distribution ratios have been reported in the literature for water−kerosene with 30% v/v TBP system at ambient temperature (23−25 °C).21,22

Figure 3. Mass transfer coefficient−interfacial area products for all 20 stages as a function of simulation time. Experimental data taken from Burns and Hanson21 and Davis.22 13224

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these values using eq 8 is a reasonable alternative when data are unavailable. Actual mass transfer coefficients for nitric acid at the conditions relevant to the present simulation (i.e., 80% stage efficiency) should be on the order of 1e−4 m/s for kaq→org,23 which implies a specific surface area on the order of 1000 m−1. A specific surface area of 1000 m−1 corresponds to a mean droplet size on the order of a few millimeters. Droplet sizes in the range of hundreds of micrometers, which are more characteristic of centrifugal contactors,24 correspond to a specific surface area of roughly 10 000 m−1. For nitric acid, this implies that the efficiency for each stage with respect to nitric acid is better than 95%. Tracking the evolution of concentrations with time was one of the primary motivations for reformulating the AMUSE model as a dynamic tool. Trajectories for the aqueous nitric acid concentrations in all 20 stages are presented in Figure 4.

These experimental data for the nitric acid distribution ratios have been described using a single-parameter correlation.21 This parameter can be interpreted physically as the concentration-based equilibrium constant for the nitric acid− TBP complex in the organic phase. An explicit relationship for this constant is given in eq 14. K=

[HNO3(TBP)]org +

[H ]aq [NO−3 ]aq [TBP]org

(14)

A value of 0.22 was determined by least-squares regression of the data. Actual TBP−HNO3 complexes are not always 1:1 and may incorporate water molecules, but eq 14 is a useful approximation for this example simulation. For the 30% v/v TBP organic phase (approximately equivalent to 1.1 M TBP in the organic phase), organic phase nitric acid saturation occurs at an aqueous phase nitric acid concentration of 1.0−1.5M. Explicit densities have also been used to simplify this example simulation. At 25 °C and low nitric acid concentrations (