End-Point Prediction Modeling for Semibatch Hydroxide Precipitation

Ind. Eng. Chem. Res. 2003, 42, 5429-5436

5429

End-Point Prediction Modeling for Semibatch Hydroxide Precipitation Randy D. Weinstein,* Kenneth R. Muske, and John P. Dawson Department of Chemical Engineering, Villanova University, Villanova, Pennsylvania 19085

An equilibrium solubility model for the accurate prediction of the end point of semibatch hydroxide precipitation reactions in high ionic strength aqueous solutions is presented. We are interested in end-point and solution-pH predictions to determine the amount of base addition necessary to exactly achieve the base hydroxide solubility limit for the removal of heavy metals without excess base addition. Experimental verification of the model predictions is carried out using a chloride, iron, and magnesium system that is a model of the industrial system of interest. As shown by the experimental results, very good end-point predictions are possible even with the incorporation of a number of simplifying assumptions in the activity coefficient models. Introduction Hydroxide precipitation is a common process for the removal of heavy metals from acidic, aqueous waste streams.1 In a hydroxide semibatch precipitation system, alkali hydroxide is added to a batch reaction vessel containing the aqueous waste until the desired end point is achieved. This end point is typically specified as either a target pH that results in the heavy-metal hydroxides precipitated out of solution or the solubility limit of the alkali hydroxide added to the system. Accurate endpoint control of these systems ensures that the heavy metals present are completely precipitated from solution without the excessive addition of alkali hydroxide. We present a practical end-point prediction modeling approach for semibatch precipitation reaction systems. The motivation for this work is improved control of a series of semibatch precipitation reactors used to remove actinides from acidic waste streams generated in the nuclear materials processing facility at Los Alamos National Laboratory.2,3 Each reactor is initially charged with effluent to which base hydroxide is added until the neutralization reaction is complete. Recovery of the precipitated actinide hydroxides is then accomplished by filtration of the reactor contents. The acidic waste streams consist of hydrochloric acid solutions, typically between 1 and 2 N in concentration, with actinide loadings ranging from 1 to 5 g/L. These semibatch reactors were initially operated using aqueous potassium hydroxide. The aqueous hydroxide, however, resulted in a large fraction of submicron particles in the precipitate that are difficult to remove from solution by filtration.4 For this reason, and the possibility of undesired calcium hydroxide coprecipitation, solid magnesium hydroxide powder addition was implemented. Precipitation occurs as the powder is dissolved by the acid, yielding a larger mean precipitate particle size and a significant reduction in the amount of submicron particles produced.2 The result was a reduced settling time of the precipitated heavymetal hydroxides with faster and more efficient filtering of the neutralized solution. In addition, any trace amounts of sodium, potassium, or calcium present will * To whom correspondence should be addressed. Tel.: (610) 519-4954. E-mail: [email protected].

not coprecipitate with the heavy metals because magnesium hydroxide has a lower solubility limit, and hence the minimal possible solid radioactive waste is produced. The objective of end-point control in this process is to add magnesium hydroxide until its solubility limit is achieved. Because of the solubility difference between magnesium hydroxide and the actinide hydroxides, all of the actinides will be precipitated at this point while the magnesium will remain in solution.5 If the end point is not achieved, however, unprecipitated actinides may be left in solution and the batch will have to be reprocessed. Because reprocessing can significantly reduce capacity, the typical operating practice was to add excess magnesium hydroxide to the reactor to ensure complete actinide precipitation. This practice produced undissolved magnesium hydroxide in the precipitate that resulted in an increase in the high-level solid radioactive waste produced by the process. The magnesium nuclei in the precipitate also generated neutron radiation from (R, n) reactions with R particles arising from the R decay of the actinides. The neutron radiation formed by these interactions is a processing concern that must be minimized while ensuring complete actinide recovery from the effluent. Consistently achieving the end point with each batch minimizes the high-level solid radioactive waste and (R, n) neutron radiation produced by the process while ensuring complete actinide precipitation. However, precise control at the solubility limit is difficult to achieve with conventional feedback control based on pH measurements because of the nonlinear increase in pH at the end point coupled with the inability to reliably add small amounts of magnesium hydroxide powder with the pneumatic transport system installed to isolate the reactors. Therefore, a discrete model-based endpoint control approach was developed.3 The equilibrium solubility model initially used in the model-based endpoint controller neglected the effect of composition on the activity coefficients of the ionic components in solution. The result was increased uncertainty in the model updates based on pH measurements and consistent underprediction of the end point. In this work, the effect of composition on the activity coefficients is considered using excess Gibbs free energy and infinitedilution models for these coefficients. Verification of this model is provided by a comparison of the model-

10.1021/ie030036a CCC: $25.00 © 2003 American Chemical Society Published on Web 09/23/2003

5430 Ind. Eng. Chem. Res., Vol. 42, No. 22, 2003

predicted hydrogen ion potential to experimental data collected over the range of typical reactor operating conditions. End-Point Prediction Model An equilibrium solubility model is used to describe the magnesium hydroxide precipitation reaction. This model is based on the algebraic charge balance and equilibrium solubility of the ions in solution. The result is a steady-state model that can be used to predict the magnesium hydroxide solubility limit and solution pH for end-point control. Because of the relatively fast rate of dissolution of magnesium hydroxide powder in the acid solution and the requirement that each batch be certified for actinide removal before being allowed to leave the facility, a steady-state model is sufficient in the target application. Equilibrium Solubility Model. A charge balance for the ionic species in solution is shown in eq 1 where

3[M3+] + 2[Mg2+] + [A+] + [H+] ) [OH-] + [Cl-] (1) [M3+] represents the heavy-metal-ion concentration, [Mg2+] represents the magnesium ion concentration, [A+] represents the trace alkali-metal-ion concentration that may be present, and [Cl-] represents the chloride ion concentration, where the brackets indicate the molar concentration of the species. A standard potassium hydroxide titration is performed on each batch prior to the neutralization reaction. This titration determines the initial moles of acid and heavy metals in the charged effluent. Because hydrochloric acid is the only acid present, all of the metals are dissolved as chlorides, and the alkali metals do not precipitate, it is reasonable to assume that this quantity is also the difference between the moles of chloride and the moles of alkali-metal ions present in solution. This difference remains constant throughout the reaction because magnesium hydroxide addition cannot precipitate chloride or alkali-metal ions. Therefore, the charge balance in eq 1 can also be expressed as

3[M3+] + 2[Mg2+] + [H+] ) [OH-] + [C-]

(2)

in which [C-] represents the difference between the chloride ion and alkali-metal-ion concentrations in the effluent determined by the titration. The equilibrium constant for the autoprotolysis of water and the equilibrium solubility products of the magnesium and heavymetal hydroxides are represented as the ratio of the activities raised to the stoichiometric coefficients as shown in eqs 3-5, in which γi is the activity coefficient

KW )

γH[H+]γOH[OH-] γH2O[H2O]

(3)

KMg ) γMg[Mg2+](γOH[OH-])2

(4)

KM ) γM[M3+](γOH[OH-])3

(5)

of the species i and the activity of the pure solids is unity. The hydroxide solubility products are valid only when solid hydroxide is present in the system. When

no solid is present, the product of the equilibrium concentrations will be less than the solubility product. This model assumes that the heavy metals present can be represented by a single species with the solubility product of the most soluble metal hydroxide. With such a large difference between the solubility of the heavymetal and magnesium hydroxides and a relatively low concentration of heavy metals in this process, this assumption is justified. Simulation studies with varying solubility products and multiple heavy-metal species result in essentially no change in the predicted end points with this assumption.3 This model also assumes that any alkali-metal ions that may be present in solution can be represented by a single species. Because these ions do not precipitate, this assumption is also justified. Activity Coefficient Models. The activity coefficients for H+, Mg2+, M3+, OH-, and H2O are required as a function of composition at the reaction conditions, which are typically 25 °C and 1 bar. The excess Gibbs free-energy model of Clegg et al. for ternary ion solutions of unsymmetrical electrolytes6 is used to calculate the hydronium, magnesium, chloride, and solvent (water) activity coefficients. We choose the excess Gibbs free-energy model described in ref 6 because it was developed for mixtures including unsymmetrical electrolytes and mixing parameters for the major components of interest are available in the literature. We also note that Pitzer’s method significantly outperformed other multicomponent solution activity coefficient prediction methods in the determination of chloride salt solubilities in aqueous electrolyte solutions.7 The mixing parameters required for this model are typically obtained experimentally using solubility, osmotic pressure, vapor pressure, freezing point depression, or electrochemical potential data from the corresponding ionic solutions.8 In this work, the model is applied to solutions in which the heavy-metal species consist of a mixture of actinides with the possibility of trace amounts of iron, chromium, and nickel due to the dissolution of stainless steel by hydrochloric acid. Because mixture parameter data for actinide systems are not present in the literature and are difficult to obtain experimentally and the complexity of the excess Gibbs free-energy models increases substantially with the number of components considered, the following simplifying assumptions are made that allow the use of readily available mixing parameter data within a computationally tractable ternary activity coefficient model. We note that the concentration of heavy-metal ions in solution is relatively low and is essentially equal to zero at the end point, where we have the most interest in modeling accurately. Because the concentration of heavy-metal ions is much lower than the concentration of the other positive ions in solution, their effect on the mixing parameters and solution ionic strength used in the activity coefficient model can reasonably be neglected. In addition, the hydroxide ion concentration is also very low over the pH range of interest and hence can also be neglected in the mixing parameters. With these simplifications, only the effect of H+, Mg2+, and Cl- is considered for obtaining the mixing parameters for the excess Gibbs free-energy model. For this ternary system, the pure-component and ternary mixing parameters are presented in ref 6 and used to compute the activity coefficients of these three ions as a function

Ind. Eng. Chem. Res., Vol. 42, No. 22, 2003 5431

Figure 1. Activity coefficients for a 1 N solution.

of the solution composition. The activity coefficient of water computed by this model varies by less than 2% over the range of solution compositions considered in this work. Therefore, it is assumed to be a constant and incorporated into the value for KW. We refer the reader to the original work of Pitzer and co-workers6,9-11 and the summary presented in ref 7 for the rather lengthy equations and do not replicate them here. Because magnesium hydroxide is used to neutralize the solution, the pH does not increase above 8.7 during the precipitation reaction. Therefore, an infinite-dilution model can reasonably be employed to predict the hydroxide ion activity coefficient. The infinite-dilution model developed by Pitzer as presented in Tester and Modell12 is used to calculate the hydroxide ion activity coefficient where Ix is the ionic strength based on mole

[

xIx(1 - 2xIx) 2 ln(1 + F*xIx) + ln(γOH) ) -A F* 1 + (1 + F* I )

A)

[

x

](

6 1 (2 × 10 )πNAFs 3 Ms

F* ) r

0.5

e2 4π0DkT

(2 × 106)e2NAFs Ms0DkT

)

x

]

(6)

1.5

(7)

(8)

fraction, NA is Avogadro’s number, Fs is the solvent density, Ms is the solvent molecular weight, e is the elementary electron charge, 0 is the vacuum permittivity, D is the static dielectric constant, k is the Boltzmann constant, T is the absolute temperature, and r is the hard-core hydroxide ionic radius. This model is a basic extension of Debye-Hu¨ckel theory that accounts for long-range ion-ion interactions. We note that more sophisticated models are available, such as that presented in ref 13, but cannot be justified for this system

based on the uncertainty in the parameters required for the system of interest. Because of the use of a ternary ion activity coefficient model, the heavy-metal-ion activity coefficient is not computed directly. In this model, we assume that the heavy-metal-ion activity coefficients behave in the same manner as the magnesium ion activity coefficient. An approximation to the relative magnitude of the difference between these coefficients can be determined from potentiometric titration of a known heavy-metal solution. We note that this assumption does not have a significant effect on the end-point prediction and hence is reasonable to use in this situation. End-Point Model Calculations The end point in this work is the solubility limit of the magnesium hydroxide. The amount of magnesium hydroxide necessary to reach the end point can be determined from the solution of the set of nonlinear algebraic equations shown in eqs 2-5 for the unknowns [H+], [OH-], [Mg2+], and [M3+]. The concentration [C-] is assumed known from the initial titration. We note, however, that there are measurement errors associated with the initial titration, volume charged to the reactor, and magnesium hydroxide addition. These errors, along with the modeling error introduced by the simplifying assumptions, can lead to errors in the calculated magnesium hydroxide end-point addition. For this reason, the end point is approached by a series of magnesium hydroxide additions in which the pH measurement is used to update the model online.3 Therefore, the model must also be able to predict the hydrogen ion concentration prior to the end point. To be of practical use for end-point control, the model predictions are based on quantities that either are known or can be estimated from the available measurements on the system. Noting that the heavy-metal-ion, and any trace alkali-metal-ion, concentrations are typically quite small allows for the approximation [H+]0 )


Figure 2. Activity coefficients for a 2 N solution.

[Cl-]0 ) [C-], where the subscript 0 indicates the initial concentration and C- is determined by the initial titration. The magnesium ion concentration is determined from the known magnesium hydroxide addition

mh [Mg2+] ) MhV

(9)

where mh is the mass of magnesium hydroxide added, V is the reactor volume, Mh is the magnesium hydroxide molecular weight, and is the fractional purity of the powder. To simplify the calculations required for the model, the activity coefficients γH, γOH, and γMg are expressed as quadratic functions of the magnesium ion concentration for a series of initial acid concentrations representing the normal range found in the effluent stream. The resulting activity coefficient expressions for component i are of the form

γi ) ai[Mg2+]2 + bi[Mg2+] + ci

(10)

where ai, bi, and ci are functions of the initial acid concentration [H+]0 estimated from the initial titration. The quadratic functions are fit to the activity coefficient values that are computed offline using the excess Gibbs free-energy model for γH and γMg and the infinite dilution model for γOH. The quadratic function form results in the sum of square residuals on the order of 10-6 for initial hydrochloric acid concentrations ranging from 1 to 2 N. This error is considered to be well within the necessary accuracy for the model. Figures 1 and 2 present the activity coefficients as a function of hydrogen ion concentration for initial acid concentrations of 1 and 2 N, respectively. Because the heavy-metal-ion concentration is relatively low, the appropriate set of quadratic model parameters used in practice can be determined from the initial acid strength only, which is obtained by titration. The determination of the component concentrations is carried out by solving the appropriate system of

nonlinear algebraic equations presented in eqs 2-5. Prior to the heavy-metal precipitation, the solubility model need only consider the autoprotolysis of water. Because the heavy-metal-ion concentration is much less than the other positive ions in solution, they are ignored in the charge balance equation, resulting in the nonlinear system

2[Mg2+] + [H+] - [OH-] - [C-] ) 0 γH[H+]γOH[OH-] - KW ) 0 γH - aH[Mg2+]2 - bH[Mg2+] - cH ) 0 γOH - aOH[Mg2+]2 - bOH[Mg2+] - cOH ) 0 (11) for the four unknowns [H+], [OH-], γH, and γOH based on the magnesium ion concentration computed using eq 9 and the activity coefficient representation in eq 10. When the heavy-metal solubility product exceeds the solubility constant, heavy-metal hydroxide precipitate forms, resulting in the nonlinear system

3[M3+] + 2[Mg2+] + [H+] - [OH-] - [C-] ) 0 γH[H+]γOH[OH-] - KW ) 0 γM[M3+](γOH[OH-])3 - KM ) 0 γH - aH[Mg2+]2 - bH[Mg2+] - cH ) 0 γM - aM[Mg2+]2 - bM[Mg2+] - cM ) 0 γOH - aOH[Mg2+]2 - bOH[Mg2+] - cOH ) 0 (12) that describes the semibatch precipitation system prior to the end point for the six unknowns [H+], [OH-], [M3+], γH, γOH, and γM. The end point is the solubility limit of the magnesium hydroxide. The magnesium hydroxide concentration at the end point can be determined from the solution of


Figure 3. 1 N HCl solution with 1 g/L Fe loading.

the full set of nonlinear algebraic equations.

3[M3+] + 2[Mg2+] + [H+] - [OH-] - [C-] ) 0

Experimental Section

γH[H+]γOH[OH-] - KW ) 0 2+

- 2

γMg[Mg ](γOH[OH ]) - KM ) 0 γM[M3+](γOH[OH-])3 - KMg ) 0 γH - aH[Mg2+]2 - bH[Mg2+] - cH ) 0 γM - aM[Mg2+]2 - bM[Mg2+] - cM ) 0 γMg - aMg[Mg2+]2 - bMg[Mg2+] - cMg ) 0 γOH - aOH[Mg2+]2 - bOH[Mg2+] - cOH ) 0 (13) Solution of the system of nonlinear algebraic equations outlined in eqs 11-13 is performed using a Newton-Raphson iteration as described by Dennis and Schnable14 and Kelley.15 The resulting linear system is

Jδx ) -r

(14)

in which J is the Jacobian of the nonlinear algebraic system, r is the residual vector of the nonlinear system of equations, and δx is the correction to the vector x of unknowns. An analytical Jacobian is used and updated at every iteration. Solution of the linear system in eq 14 is performed using LU decomposition. The vector of unknowns is updated at each iteration k as follows:

xk+1 ) xk + Rkδx

nent values is essential in order to obtain a reasonable condition number for the Jacobian matrix in eq 14.

(15)

where Rk ∈ (0, 1] is selected at each iteration based on a line search to retain convergence. Because the component concentrations in the system can differ by over 10 orders of magnitude, adaptive scaling of the compo-

Because of the toxicity and radiation problems associated with actinide solutions, we verify the solubility model in this work using nonradioactive effluent solutions containing iron as the heavy metal. This procedure has been a reliable indication of the performance on the actual process in the past.2 We consider a low metal loading, 1 g/L, a moderate metal loading, 2 g/L, and a high metal loading, 5 g/L, at acid concentrations of 1 and 2 N and conduct a series of semibatch precipitation reactions at each of these conditions to compare with the model. Each condition was tested using three to six neutralization experiments. Procedure. A 2 N HCl acid (Fisher Scientific certified 1.995-2.005 N) was used to create the 2 N acid solutions. The 1 N acid solutions were created by dilution with DIUF water (Fisher Scientific). The HCl solution was added to a volumetric flask containing iron(III) chloride (Fisher Scientific ACS anhydrous) to obtain 100 mL of solution with an iron concentration of 1, 2, or 5 g/L. Neutralization was carried out using magnesium hydroxide powder (Aldrich Chemical certified 95% pure). Potentiometric titration of the magnesium hydroxide indicated an apparent purity of 97%. This difference was most likely due to the presence of calcium impurities, which would behave similarly to magnesium in the neutralization reaction. No attempt was made to correct for the presence of calcium in the activity coefficient models because it was considered to be insignificant in comparison to the other positive ions present. Hydrogen ion potentials were measured with a Fisher Accumet 925 pH meter and an Orion 9165 combination pH probe. A Teflon-sheathed type T thermocouple was used to measure the solution temperature, which remained at 23 ( 2 °C at equilibrium throughout the experimental study. A pH 7 buffer solution (Fisher Scientific certified (0.01 pH) was used




to verify the accuracy of the pH probe prior to each experiment. The initial hydrogen ion potential was measured for each solution before any base addition. The first addition of magnesium hydroxide was approximately half that required to bring the solution to the end point. After this addition, approximately 0.05 g of powder was added batchwise until the end point was reached. The hydrogen ion potential was measured after steady state was achieved for each base addition. Complete neutralization was indicated by no further increase in the hydrogen ion potential. Results and Conclusions. The solutions in this study contain five predominant ions, H+, Mg2+, Fe3+, Cl-, and OH-. For this model system, the ferric ion activity coefficient is assumed to be the same as the magnesium ion coefficient. This assumption is sup-

ported by the fact that the pure solution Pitzer9 and Pitzer and Mayorga10,11 model parameters for ferric chloride and magnesium chloride are almost identical. The measured hydrogen ion potential was compared to that predicted by the Nernst equation

E ) Ex +

RT log(γH[H+]) nF

(16)

where γH[H+] was predicted by the equilibrium solubility model presented in this work for the experimental conditions considered. Figures 3-5 present the model-predicted and experimental hydrogen ion potential as a function of magnesium hydroxide addition for an initial acid concentration of 1 N. As shown in these figures, the character of the




titration curve is captured by the model, but the hydrogen ion potential is consistently underpredicted by the model prior to the start of iron hydroxide precipitation. After this point and up to the end point of the reaction, the model prediction improves. We note a similar behavior of the model prediction with an initial acid concentration of 2 N, as shown in Figures 6 and 7; however, the underprediction is larger. A possible explanation for these results is that the simplifying assumptions made in the activity coefficient models are not valid because the magnesium ion concentration increases prior to the precipitation of iron hydroxide. As the iron hydroxide precipitates from solution, these assumptions appear to become a more representative description of the system. Because we are more interested in the behavior of the system as it

approaches the end point, the performance of the model is considered acceptable for end-point control. The significant additional complexity to the model that would be required to improve the prediction prior to the iron hydroxide precipitation would contribute little to improving the end-point control. As shown in this work, a reasonably simple implementation of ion activity coefficient models for high ionic strength solutions can provide very adequate predictive capabilities that can be used for end-point control. Literature Cited (1) Patterson, J. W. Industrial Wastewater Treatment Technology, 2nd ed.; Butterworth: Boston, 1985. (2) Palmer, M. J.; Fife, K. W. Magnesium Hydroxide as the Neutralizing Agent for Radioactive Hydrochloric Acid Solutions;

5436 Ind. Eng. Chem. Res., Vol. 42, No. 22, 2003 Technical Report LA-12975-MS; Los Alamos National Laboratory: Los Alamos, NM, 1995. (3) Muske, K. R.; Palmer, M. J. End Point Control of an Actinide Precipitation Reactor. Proceedings of the 1997 American Control Conference; American Automatic Control Council: San Diego, CA, 1997; p 2439. (4) Walton, A. G. The Formation and Properties of Precipitates; Krieger: Melbourne, FL, 1979. (5) Cleveland, J. M. The Chemistry of Plutonium, 2nd ed.; American Nuclear Society: LaGrange Park, IL, 1979. (6) Clegg, S. L.; Pitzer, K. S.; Brimblecombe, P. Thermodynamics of Multicomponent, Miscible, Ionic Solutions: 2. Mixtures Including Unsymmetrical Electrolytes. J. Phys. Chem. 1992, 96, 9470. (7) Zemaitis, J. F.; Clark, D. M.; Rafal, M.; Scrivner, N. C. Handbook of Aqueous Electrolyte Thermodynamics; Design Institute for Physical Property Data, American Institute of Chemical Engineers: New York, 1986. (8) Pitzer, K. S. Thermodynamics, 3rd ed.; McGraw-Hill: New York, 1995. (9) Pitzer, K. S. Thermodynamics of Electrolytes: 1. Theoretical Basis and General Equations. J. Phys. Chem. 1973, 77, 268.

(10) Pitzer, K. S.; Mayorga, G. Thermodynamics of Electrolytes: 2. Activity and Osmotic Coefficients for Strong Electrolytes with One or Both Ions Univalent. J. Phys Chem. 1973, 77, 2300. (11) Pitzer, K. S.; Mayorga, G. Thermodynamics of Electrolytes: 3. Activity and Osmotic Coefficients for 2-2 Electrolytes. J. Solution Chem. 1974, 3, 539. (12) Tester, J. W.; Modell, M. Thermodynamics and Its Applications, 3rd ed.; Prentice-Hall: Upper Saddle River, NJ, 1997. (13) Lu, X.; Maurer, G. Model for Describing Activity Coefficients in Mixed Electrolyte Aqueous Solutions. AIChE J. 1993, 39, 1527. (14) Dennis, J. E.; Schnable, R. B. Numerical Methods for Unconstrained Optimization and Nonlinear Equations; PrenticeHall: Englewood Cliffs, NJ, 1983. (15) Kelley, C. T. Iterative Methods for Linear and Nonlinear Equations; SIAM: Philadelphia, PA, 1995.

Received for review January 14, 2003 Revised manuscript received June 19, 2003 Accepted July 24, 2003 IE030036A

End-Point Prediction Modeling for Semibatch Hydroxide Precipitation

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