- Email: [email protected]
A thermodynamic modeling approach for solubility product from struvite-k
Computational Materials Science 157 (2019) 51–59
Contents lists available at ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
A thermodynamic modeling approach for solubility product from struvite-k a
a,b
Yuanquan Yang , Jun Liu a b
a,⁎
b
a,⁎
, Baomin Wang , Runqing Liu , Tingting Zhang
T
Faculty of Infrastructure and Engineering, Dalian University of Technology, Dalian 116024, China School of Materials and Engineering, Shenyang Ligong University, Shenyang 110159, China
ARTICLE INFO
ABSTRACT
Keywords: Potassium and phosphorus recovery Solubility product Struvite-k Thermodynamic model
Crystallization of struvite-k could be an economical and sustainable alternative for potassium and phosphorus recovery from wastewater and ashes of biomass combustion. Knowledge with regards to thermodynamics that are involved in the formation of struvite-k in reactor is key in determining the optimal conditions for an efficient process. However, less research work has been done on struvite-k solubility product. A thermodynamic model for struvite-k precipitation was in this study thus proposed to determine the struvite-k solubility product, as well as conversion rate for struvite-k and its accompanying crystal cattiite over a pH value range of 10.5–12.5. The model was based on numerical equilibrium prediction of involved system Mg-K-PO4-H2O. The model was presented by a set of nonlinear equations that were solved by an optimization strategy with a three-step method. The method consists of a preliminary search by Lagarias Simplex search method for initialization, Successive Quadratic Programming (SQP) for resolution of system, and extrapolating calculated data to zero ionic strength for determination of struvite-k solubility product. A pKstruvite-k value of 10.872 for struvite-k at 25 °C was obtained in the model. This novel model will provide a theoretical reference for recovery of potassium and phosphate, as well as its possible application in magnesium potassium phosphate cement (MKPC).
1. Introduction Water pollution has been a worldwide problem from which phosphorous plays a major role, due to its potential for inducing-eutrophication. One proposed solution to this problem is phosphorous recovery from the wastewater by precipitation method [1]. Normally, struvite (MgNH4PO4·6H2O) precipitation method is seen as an effective way for phosphorous removal, and has gained increasing interest. However, it won’t work well when it comes to the disposal of wastewater that is rich in potassium and phosphate. One alternative solution is to use struvite-k (MgKPO4·6H2O) precipitation method. Struvite-k is similar to struvite but contains potassium instead of ammonium, and it can be produced from very pure form or even impurities present in solution reactor, and sometimes from hydration of magnesium potassium phosphate cement (MKPC) [2–6]. Solubility product is equilibrium constant at constant temperature that describes a solid being dissolved into its constituent ions [2,7], which is however a key parameter in determining the optimal conditions for an efficient process in the formation of struvite-k. To improve the product quality and thus minimizing the associated production costs, it is an important point to predict the struvite-k precipitation potential, yield and purity for potassium and phosphate recovery.
⁎
Several thermodynamic models have been developed to allow reasonable prediction of equilibrium constant Ksp of struvite [8–10]. These models are based on liquid-solid physicochemical equilibrium, which at least considers the involved ions, dissolved and solid species, such as NH4+, Mg2+, PO43−, H3PO4 and MgNH4PO4·6H2O. However, ionic species e.g. HPO42−, H2PO4−, MgPO4−, MgOH+ and MgH2PO4+, and dissolved species e.g. MgHPO4 also take part in the precipitation process. Crutchik, D. et al. [11] considered the struvite crystallization as a reversible reaction, and the included species were Mg2+, NH4+, PO43−, H2PO4−, HPO42−, and MgOH+, MgPO4−. Note that, in this model the authors considered struvite as the only solid phase. Hanhoun et al. [1] developed a model to predict the equilibrium constant (Ksp) for struvite at variable temperatures. They computed the Ksp values using a robust numerical method, which combined a genetic algorithm for initialization purpose and a standard Newton-Raphson method implemented within MATLAB environment. Many researchers, as reviewed above, reported the constant Ksp of struvite. Thus far, little work has however been done on the determination of solubility product values for struvite-k [7,12,13]. Bennett et al. [7] obtained the Ksp for struvite-k which provides the best match of the theoretical concentration values with the experimental data, and the theoretical concentration values were generated using PHREEQC software. Taylor et al. [12] achieved the Ksp for
Corresponding authors. E-mail addresses: [email protected] (B. Wang), [email protected] (T. Zhang).
https://doi.org/10.1016/j.commatsci.2018.10.037 Received 9 September 2018; Received in revised form 26 October 2018; Accepted 28 October 2018 0927-0256/ © 2018 Elsevier B.V. All rights reserved.
Computational Materials Science 157 (2019) 51–59
Y. Yang et al.
struvite-k by an iterative procedure. Luff and Reed [13] determined the Ksp for struvite-k by theoretically calculated. However, the results differ much great from each other (pKsp by Taylor et al. [12], Bennett [7] and Luff and Reed [13] are of 10.6, 11 and 11.7, respectively). It is believed that it was due to the different thermodynamic models, objective function and solution methods for this system that leads to the discrepancies. It is therefore urgent and supplementary to establish a robust and effective model for the struvite-k system. Even though there is similarity between the crystal structure of struvite and struvite-k, their crystallization process and associated precipitation are quite different. Research into a thermodynamic modeling approach for the solubility product from the struvite-k is thus necessary. To solve this problem, the final crystallized product from the K-Mg-PO4-H2O system needs to be first confirmed. Several studies have focused on the processes involved in the crystallization of K-Mg-PO4H2O system. For Mg/P ratios comprised of 4 to 12, Chau et al. [14] reported that MgHPO4·7H2O precipitated first, then Mg2KH (PO4)2·15H2O and was finally converted into struvite-k. The precipitation sequence depended on pH, and MgHPO4·7H2O tended to precipitate at 4–6 pH, while struvite-k was formed at pH above 7. Mg2KH (PO4)2·15H2O was an intermediate phase between MgHPO4·7H2O and the struvite-k. Wang et al. [3] showed that the finally hydrated product was struvite-k, with Mg/P ratios being ranged from 5:1 to 45:1. However, Lahalle [15] investigated the precipitated hydrate from the K-MgPO4-H2O system in diluted suspension, and they found that crystallized process for K-Mg-PO4-H2O system was a multi-step process, where MgHPO4·7H2O was precipitated first, and was then destabilized to form Mg2K(PO4)2·15H2O, which was finally converted into struvite-k and cattiite (Mg3(PO4)2·22H2O). The experimental findings about the final hydration products, as reviewed above, seem to be inconsistent. It is suggested that the final hydration products are associated with the w/c ratio (water to cementitious materials ratio by weight, here the cementitious materials refer to the sum of the solid component). The final hydration products struvite-k by [14] and [3] were obtained with w/c ratios lower than 10, while the struvite-k and cattiite by [15] were precipitated in diluted suspension with a much greater w/c of 100. To obtain a reliable Kstruvite-k value, the extended Debye-Huckel method proposed by Davies [16] was chosen in this work to calculated the activity coefficient of each ion and complex. It was however required ionic strength inferior to 0.1 M, each ion and complex for this system should be therefore diluted for enough times. Considering a w/c of approximate 400 herein was adopted, the struvite-k and cattiite were assumed as the two final hydration products. It should be highlighted that increase in the number of dissolved and solid species leads to a more complicated model for the K-Mg-PO4H2O system. Thus, analytical solutions are no longer viable and hence numerical solution is required [1]. The objective of this paper was therefore to develop a thermodynamic modeling approach for the determination of Ksp for struvite-k and find an appropriate resolution to solve this model. In this model, the involved ions and considered complexes were K+, Mg2+, PO3− 4 , − − + HPO2− 4 , H2PO4 , H3PO4(aq), MgPO4 , MgHPO4(aq), MgH2PO4 , + + – + − MgOH , H , OH , Na and Cl (since chloride and sodium exists in the form of MgCl2·5H2O and NaOH, respectively), while the assumed precipitated solid were struvite-k and cattiite. A thermodynamic model for struvite-k system was proposed and established here, moreover, a robust and effective three-step method, which was coupled with Gibbs free energy and scalar function, were proposed. The study is organized into three parts as follows: First, a thermodynamic modeling was established for the K-Mg-PO4H2O system at various pH values using extended Debye-Huckel activity coefficient modeling. Second, this study presents the resolution for the thermodynamic model, a three-step method that combines a fminsearch function to approximate the bounds and initial values for the following calculation, and a fmincon function for the resolution (implemented within MATLAB environment), and then a final determination of
Table 1 Mix proportions, initial pH and final pH for the batches. Batch
Mix proportion
P1 P2 P3 P4 P5
KCl
KH2PO4
MgCl2·6H2O
0.016 mol/L 0.016 mol/L 0.016 mol/L 0.016 mol/L 0.016 mol/L
0.004 mol/L 0.004 mol/L 0.004 mol/L 0.004 mol/L 0.004 mol/L
0.004 mol/L 0.004 mol/L 0.004 mol/L 0.004 mol/L 0.004 mol/L
Initial pH
Final pH
10.5 11 11.5 12 12.5
8.44 8.56 8.86 11.07 12.13
struvite-k solubility product. A two-step method that previously combined initial solution with further resolution was successfully applied in a related problem for the resolution of calcium phosphate precipitate model [17] and struvite precipitate model [1]. Third, model validation for this system was performed. 2. Materials and methods Experiments were performed in a continuous stirred bath reactor (1L) at a rotational speed of around 90 rpm. The reactor was tightly enclosed to avoid incorporation of CO2. Deionised water was taken to prepare the synthetic solution. KCl, KH2PO4 and MgCl2·6H2O solutions were used as the sources of K+, PO43− and Mg2+, respectively. The initial phosphorous concentration was 4 mmol/L, with a K/PO4/Mg ratio of 5/1/1 (Table 1). In our previous test, it was found that struvitek could not be detected by XRD and SEM-EDS method at pH values lower than 10.5. Therefore, NaOH (1 M) was used in this work to adjust the solution of KH2PO4 to a specified initial pH (10.5, 11, 11.5, 12 and 12.5 respectively) before adding MgCl2·6H2O. This procedure was performed to obtain a specified pH and to avoid the precipitation of struvite-k before the beginning of each experiment. The pH and conductivity for each batch were recorded by pH meter (Crison pH 29) and conductivity meter (Crison CM 35), respectively. Moreover, these experiments were conducted at a fixed temperature of 25 °C. Samples were immediately filtered through a 0.45 μm membrane filter, and then washed using distilled water followed by ethanol. The samples were then dried at ambient temperature for a week. Residual concentrations of magnesium and potassium were analyzed by ICP-OES (PerkinElmer, Avio 200) after filtration and the concentration of PO43− was analyzed by atomic absorption (AA). Each batch was repeated 3 times and measurements were conducted 3 times for each trial, giving 9 data points per sample. X-ray diffraction analysis (XRD, Rigaku D/ MAX-IIIC, Cu-kα radiation) was applied to identify the composition of the resulting crystallized product. The XRD was performed over the range between 10° and 90° 2θ, with a scanning rate of 5° 2θ per minute at a stepping size of 0.02°. The chemical composition and appearance of the resulting crystallized product were analyzed by SEM (Hitachi S-3400N) test coupled with energy dispersive spectroscopy. 3. Thermodynamic modeling 3.1. Calculation for struvite-k solubility product Struvite-k precipitates when its product from the ionic activities of potassium, magnesium and phosphate exceed the struvite-k solubility product (when the solution is supersaturated). The precipitation reaction is thus represented by an equilibrium constant Kstruvite-k, which can be computed according to Eq. (1).
Kstruvite
k
=
K+· Mg 2 +· PO43
=
+ 2+ K +·[K ]· Mg 2 +·[Mg ]· PO43 +
·[PO43 ]
Here, [] indicates concentration and Kc = [K ][Mg therefore, 52
2+
][PO4
(1) 3−
],
Computational Materials Science 157 (2019) 51–59
Y. Yang et al. n i
= Kstruvite k / K c
cattiite
(2)
i=1
lg( i ) = ADH Zi2 ((
(9)
I 1+
I
)
0.3I )
Xstruvite
where A (= ADH
n i
A(
k
Zi2 )
I 1+
I
0.3I )
[K+] = Ktotal
Kstruvite
( [PO43 ]
PO43
Mg3 (PO4 )2 ·22H2 O
Ptotal· Xstruvite
(13)
k
– Function one: Gibbs free energy for the system In our modeling, the objective function one to be minimized was the Gibbs free energy G (Eqs. (23a)–(23c)) for the system. It presents the thermodynamic potential that is minimized when a system reaches chemical equilibrium, which is a linear combination of the chemical potential from each component. According to the second law of thermodynamics, there is a general natural tendency to achieve a minimum of the Gibbs free energy for the system that tends to reach chemical equilibrium. The Gibbs free energy function could be established given by chemical potential values and concentrations for the involved ions and complexes, without considering the mass and electroneutrality conservation balances. It is therefore an effective and direct method to obtain the initial concentration of the involved ions and complexes. This method has previously been successfully applied in determining the equilibrium constant for calcium phosphate precipitation [17]. Table 2 Equilibrium constants for the system Mg-K-PO4-H2O.
) (7)
The cattiite precipitation equation can be expressed as Eq. (8):
3Mg2 + + 2PO34 + 22H2 O
(12)
Xcattiite )
(14)
(6)
k
k
Furthermore, the concentrations of ions or complexes were determined from eight equilibrium reactions (Eqs. (15)–(22)) proposed in Table 2, as well as their respective equilibrium constants at 25 °C. In this work, two methods were adopted to construct the objective functions:
The struvite-k supersaturation ratio was defined by the βstruvite-k parameter: K +)
Xstruvite
= 3[PO43 ] + 2[HPO42 ] + [H2 PO4 ] + [MgPO4 ] + [OH ] + [Cl ]
To model the evolution of struvite-k precipitation, the phosphate conversion rates were calculated as a function of pH, with respect to struvite-k and cattiite crystallization, by considering the mass and electroneutrality conservation balances, as well as the supersaturation ratio for each species. During the reaction, the aqueous species considered here were Mg and P-containing ions or complexes, as well as K+, H+, OH–, Na+ and Cl−, as described in the above section. The struvite-k precipitation equation can be written as Eq. (6):
) ([K+]
(11)
[K+] + 2[Mg 2+] + [MgH2 PO4+] + [MgOH+] + [H+] + [Na+]
3.2. Thermodynamic modeling for struvite-k precipitation
Mg 2 +
Psolution Ptotal
– the electroneutrality requirement gives:
(1 + I 0.3I ) is linear. The intersection of this line with the ordinate axis (I = 0) gives the value of -lgKstruvite-k from which Kstruvite-k was calculated.
( [Mg 2 +]
+ Xcattiite = 1
– a mass balance for potassium:
(5)
MgKPO4 ·6H2 O
k
+ [MgH2 PO4+] = Ptotal (1
is a positive constant. A plot of –lgKc against
Mg2 + + K+ + PO34 + 6H2 O
(10)
+ X cattiite)
[PO43 ] + [HPO42 ]+ [H2 PO4 ]+[H3 PO4 ](aq) + [MgPO4 ] + [MgHPO4 ](aq)
(4)
lg Kstruvite
k
– a mass balance for phosphate:
(3)
Ci·Zi2
Ptotal (X struvite
where Xstruvite-k and Xcattiite were the conversion ratios for phosphate relative to struvite-k and cattiite forms, respectively, and defined as Eq. (11).
In this equation, Ci is the concentration of ion or complexes of I [19]. It is of great importance to consider all the ionic species in the solution when determining the ionic strength. In this work, the ionic strength was calculated from the ions and complexes involved in struvite-k precipitation. The ions and complexes included K+, Mg2+, PO3− 4 , − − + HPO2− 4 , H2PO4 , H3PO4(aq), MgPO4 , MgHPO4(aq), MgH2PO4 , MgOH+, H+, OH–, Na+ and Cl−(since chloride and sodium exist in the form of MgCl2·5H2O and NaOH, respectively). Combining Eqs. (2) and (3),
struvite k
)2
K cattiite
= Mgtotal
where ADH is the Deby-Huckel parameter that takes a value of 0.509 at 25 °C [1]. Zi is valence of ion or complexes of I, and I is the ionic strength, which is determined by Eq. (4).
=
PO43
[Mg2 +] + [MgPO4 ] + [MgHPO4 ](aq) + [MgH2 PO+4 ] + [MgOH+]
The activity coefficients approach unity at low ionic strength [18]. To obtain a reliable Kstruvite-k value, the thermodynamic solubility product from the struvite-k at successively lower ionic strengths was determined by extrapolating calculated data to zero ionic strength. In this work, the extended Debye-Huckel equation (Eq. (3)) proposed by Davies was chosen due to its simplicity and accuracy at ionic strength inferior to 0.1 M.
I
)3 ( [PO43 ]
Mg 2 +
– a mass balance for magnesium:
n = the number of ions and complexes in the solution Kstruvite-k = solubility product of struvite-k in terms of activity Kc = solubility product of struvite-k in terms of concentration γi = activity coefficient of the ion or complexes i
- lg K c =
( [Mg 2 +]
The different mass balances in the liquid phase include:
where
I = 0.5·
=
(8)
The cattiite supersaturation ratio was defined by the βcattiite parameter: 53
Eq. No.
Equilibrium
Equilibrium constant ki at 25 °C
References
(15) (16) (17) (18) (19) (20) (21) (22)
H3PO4 ↔ H+ + H2PO4− H2PO4− ↔ H+ + HPO42− HPO42− ↔ H+ + PO43− MgOH+ ↔ Mg2+ + OH– MgPO4− ↔ Mg2+ + PO43− MgHPO4 ↔ Mg2+ + HPO42− MgH2PO4+ ↔ Mg2+ + H2PO4− H2O ↔ H+ + OH–
k1 = 7.42 × 10−3 k2 = 6.37 × 10−8 k3 = 3.80 × 10−13 k4 = 2.58 × 10−3 k5 = 3.74 × 10−7 k6 = 8.21 × 10−4 k7 = 4.15 × 10−2 kw = 1.00 × 10−14
[20] [21] [22] [1] [1] [1] [1] [16]
Computational Materials Science 157 (2019) 51–59
Y. Yang et al.
energy. Note that the chemical potential values available online are not yet sufficient to calculate the Gibbs free energy during this reaction. For this reason, partial chemical potential values were computed from Eq. (6), Eq. (8), Eqs. (18)–(21), and Eqs. (24)–(26) [1].
Table 3 Chemical potential values for involved ions and complexes in the system at 25 °C. Species
μi (kJ/mol)
References
H3PO4 H2PO4− HPO42− PO43− K+ Mg2+ OH– H+ MgOH+ MgPO4− MgHPO4 MgH2PO4+ Cl− Na+ MgKPO4·6H2O Mg3(PO4)2·22H2O
−1142.65 −1130.40 −1089.26 −1018.80 −283.27 −454.80 −157.29 0 −597.3 −1436.9 −1526.4 −1577.3 −240.17 −261.905 −1696.30 −3461.45
[17] [17] [17] [17] [16] [16] [17] [17] Calculated Calculated Calculated Calculated [16] [16] Calculated Calculated
nik µik
µiL = µiL (T ) + RT ln([x i ]
(26)
– Function two: A scalar function, composed of sum of squares of several equations. The function two to be minimized was the sum of squares of Eqs. (10), (12)–(14). For example, f1 = x2 + y2 + 3z2 and 2 2 f2 = 2x + 3y + z, then the scalar function was minimized as G = f12 + f22, by ensuring that f1 = 0 and f2 = 0 when the function G
where μiL and μiS are the chemical potentials of ions and complexes of i in the liquid and solid, respectively. Table 3 gives the chemical potential values which were considered for computation of Gibbs free
Influent [Mgtotal], [Ktotal], [Ptotal] and [Natotal]
Thermodynamic Model for this system Simplex search method
Initialization Step1 Xstruvite-k
Xcattiite
[K]liquid
[Mg]liquid
[P]liquid
…
Resolution of the Model Refinement
SQP
Step2
Step3
(25)
where ΔGR(T) represented the free energy of reaction (Eq. (25)) involved in the equilibrium. The reaction equation is simply as follows: A + B ↔ C. The free energy was thus given by Eq. (25). In thermodynamics, the Gibbs free energy for a system that is held at constant temperature and pressure can be expressed in a simple fashion as Eq. (26). It was noted that in this calculation, the values for Kcattiite and Kstruvite-k at 25 °C (1.04 × 10−24 and 2.4 × 10−11, respectively) were assumed to be known.
(23c)
µiS = µiS (T )
µi dNi i=1
(23b)
xi )
[ GfA (T ) + GfB (T )]
n
dG =
(23a)
i=1 k=1
(24)
GR (T ) = GfC (T )
N
G=
GR (T ) RT
ln(Ki ) =
Xstruvite-k
Xcattiite
Kc
…
I
Determination of Kstruvite-k Fig. 1. Schematic program for each computation. 54
Initialized values and bounds
Computational Materials Science 157 (2019) 51–59
Y. Yang et al.
Fig. 2. XRD patterns of crystallized products from the synthetic solutions at different initial pH values (25 °C).
– 16 bounded variables (14 aqueous ion or complexes and 2 solid species) – 12 equality constraints (Eq. (10), Eqs. (12)–(14) and Eqs. (15)–(22)) – Two inequality constraints (precipitation of struvite-k and cattiite) – Two functions to minimize (Gibbs free energy and a scalar function for several variables)
12.00 pH Electrical conductivity
3.6
pH
11.50
3.5
11.25
11.00
3.4
10.75
Electrical conductivity(mS/cm)
11.75
3.7
4. Determination of Kstruvite-k The objective was to calculate the equilibrium state from the initial concentrations of Mg2+, K+ and PO43− ions. Then, it was possible to calculate the ionic strength I and solubility product of struvite-k in terms of concentration (Kc), and thus the solubility product of struvite-k in terms of activity was determined. The initialization step still constituted difficulty: where Genetic algorithms are commonly used for complex problems [1,17], and are usually applied to initialize the concentrations of ions in the solution at equilibrium state. However, when a set of nonlinear constraints were embedded, it sometimes failed. Simplex search method by Lagarias et al. [23] is a direct search method that does not use numerical or analytic gradients and is generally referred to as unconstrained nonlinear optimization. Thus, the fminsearch function within MATLAB environment, which uses the Lagarias simplex search method was applied for determining the accurate bounds and initial values for the following calculation. For each computation, a three-step was implemented (Fig. 1): Step 1: Initialization for Xstruvite-k, Xcattiite, I ion concentrations, and bounds for the following calculation. Approximate values for 2− − − [K+], [Mg2+], [PO3− 4 ], [HPO4 ], [H2PO4 ], [H3PO4](aq), [MgPO4 ], + + – [MgHPO4](aq), [MgH2PO+ ], [MgOH ], [H ], [OH ], X and struvite-k 4 Xcattiite were determined using the Simplex search method by Lagarias (fminsearch function within MATLAB environment). A coefficient i (ranging from 1 to 10) was applied thereafter to extend the obtained values. A matrix was then denoted as lower bounds = {0}, and matrix denoted upper bounds = {i*values} were obtained and applied for accurate bounds in the following calculation. It should be emphasized that the coefficient was adopted to reduce the fluctuation of results
3.3 10.50
0
2
4
6
8
10
12
Time/h Fig. 3. Evolution of pH and electrical conductivity over time for the solution at initial pH of 12.
was minimized to zero. Even though the Gibbs free energy function could solve this model in reasonable computational times and is less time-consuming, it is unable to get accurate solution for this system, with regards to the fact that partial of the chemical values online are not yet sufficient and some of them are from calculated. An accurate objective function was thus here required, and a scalar function composed of sum of squares of the mass and electroneutrality conservation balances function was suitable. It could avoid the adverse effect of the uncertain chemical values on its final results. Most importantly, the initial values calculated by the Gibbs free energy function method could help to solve the scalar function more precisely. Thus, we coupled the two functions to calculate the equilibrium state. Details for the two approaches are available as supplementary materials (Appendix A). To solve this system, the precipitation of struvite-k and cattiite were taken into account, leading to optimization problem with two inequalities. The thermodynamic model contained:
55
Computational Materials Science 157 (2019) 51–59
Y. Yang et al.
Mg
P
1000
1000
800
800
O
600 400
Mg O
K
600 400
200 0
P
1200
counts
counts
1200
200
K 0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
0
5.0
0.0
0.5
1.0
1.5
Energy/keV
1000
Mg
P
3.0
3.5
4.0
4.5
5.0
Mg
1000
O
600
400
O
K
800 600 400
200
0
2.5
P
1200
counts
counts
800
2.0
Energy/keV
200
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
Energy/keV
0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
Energy/keV
Fig. 4. SEM morphologies of resulting crystallized products coupled with EDS results of interest area: (a) & (b) P4; (c) & (d) P2.
56
Computational Materials Science 157 (2019) 51–59
Y. Yang et al.
indicated by the location of intensity peaks, corresponding to the PDF #35-0812 and PDF #44-0775, respectively. The results indicated that the main precipitated crystal were struvite-k or cattiite or both of them for all the experimental runs. In addition, the quantity of cattiite increased as pH increased over the range of 10.5–11.5. Struvite-k tended to be favored after that (at pH value above 11.5), while the cattiite was hardly detected at pH of 12.5. However, both (over the pH range of 10.5–12.5) were confirmed by SEM-EDS method (It will be discussed at the end of this section). During the experimental runs, pH evolution and electrical conductivity for the solutions were measured with time (Fig. 3). Results showed that both pH and electrical conductivity almost did not evolve over 10 h. To guarantee that a quasi-equilibrium state was reached, the time periods for each run were extended to 7 days. SEM-EDS was conducted to further validate that the resulting crystallized products were struvite-k and cattiite. For the sake of illustration, only results of P2 (pH = 11) and P4 (pH = 12) are presented (Fig. 4). The SEM microphotographs showed that the resulting crystalized products were composed of prim-like crystal (Fig. 4b) and multilayered sheet-like crystal (Fig. 4d). SEM-EDS microanalysis procedure is a qualitative elemental analysis [11], which provides information about the involved elements that are present in the resulting crystallized product, and the EDS analysis was conducted on the P2 and P4. The SEM-EDS microanalysis results indicated that area B (Fig. 4b) and area D (Fig. 4d) shared the common elements of Mg, K, P, and O, and the ratios of Mg, K and P were approximately 1:1.13:0.95 and 1:1.06:1.02, respectively (Table 4). This implied that the prism-like crystal in area B and area D was struvite-k, which is close to the theoretical ratio of 1:1:1 for struvite-k. In terms of area A and area C, the main shared elements were Mg, P and O, and the ratios of Mg and P were 0.94:1 and 1.60:1, respectively. It could be postulated that the multi-layered sheet-like crystal was cattiite, taking into account that the theoretical Mg/P ratio was 1.5:1. Moreover, it was noted that, potassium was also identified in area A, however, the quantity was too small. The accompanying potassium could have resulted from the deposited struvite-k on the cattiite, considering that the tiny particles of struvite-k than the cattiite with sheet-like crystal structure (from Fig. 3b it could also found that partial struvite-k particles lean close to the sheet-like crystal).
Table 4 Elements in the specified areas for P3 and P5 scanned by energy dispersive spectroscopy. Element
area A
area B
area C
area D
53.23 15.22 17.14 14.41
56.56 26.71 16.73 –
53.23 15.12 16.14 15.51
Atomic/% O Mg P K
48.42 22.90 24.44 4.24
Table 5 Predicted and experimentally determined concentration of potassium and magnesium. K/mol⋅L−1 Predicted pH10.5 pH11 pH11.5 pH12 pH12.5
0.0198 0.0197 0.0195 0.0173 0.0163
Mg/mol⋅L−1 Measured 0.0198 0.0196 0.0195 0.0177 0.0163
Predicted
Measured −4
1.6620 × 10 1.5930 × 10−4 9.1520 × 10−5 7.0890 × 10−5 4.5000 × 10−5
1.6520 × 10−4 1.5920 × 10−4 9.0520 × 10−5 7.0290 × 10−5 4.3200 × 10−5
Table 6 Predicted conversion rates, pKstruvite-k and mean pKstruvite-k. pH
Xstruvite-k
Xcattiite
pKstruvite-k
Mean pKstruvite-k
pH10.5 pH11 pH11.5 pH12 pH12.5
0.0370 0.0686 0.1125 0.6510 0.9068
0.6142 0.5943 0.5763 0.2208 0.0545
10.9460 10.8790 10.8690 10.8210 10.8440
10.8720
from the system. In this step, Eq. (23a) (Gibbs free energy for the system) was used as the objective function to be minimized. Step 2: Resolution of system. Resolution of system was implemented by Successive Quadratic Programming (fmincon function within MATLAB environment), with initialized values and bounds given by step 1. In this step, the squared sum of Eq. (10), Eqs. (12)–(14) was the objective function to be minimized, and Eqs. (7) and (9) were treated as constraint equations. Then, the mean values for the involved parameters were computed (refer to Appendix A). Step 3: Determination of Kstruvite-k. The Kstruvite-k was determined by extrapolating calculated data to zero ionic strength, given by the ionic strength I and solubility product of struvite-k in terms of concentration (Kc) by step 2. The required data input for the three-step methods were as follows:
5.2. Model validation with Kstruvite-k values at various initial pH values To validate the feasibility of the established model in this work, the concentrations of residual potassium and magnesium, determined by the model and measured values, (Table 5) were compared. The results indicated that the predicted values were well in agreement with experimentally measured ones. The predicted conversion rates are also presented in Table 6, where they indicate that lower initial pH (< 12) favored the formation of cattiite but suppressed the precipitation of struvite-k. The conversion rate for struvite-k gave 3.7%, 6.8% and 11.25%, respectively with the increased pH values from 10.5 to 11.5, which was within low content. These results were also consistent with the XRD results that the struvite-k was hardly detected due to its very low content. The results were also consistent with SEM results, that few struvite-k crystals were observed. Moreover, when the initial pH was above the critical value of 12, the conversion rate for struvite-k increased abruptly, and the rates reached 65.1% and 90.68%, respectively, at 12 and 12.5 initial values. These results were also confirmed by the XRD results (Fig. 2), as well as SEM-EDS results, in which the main crystal was struvite-k, (Fig. 4a and b) where large quantity of the struvite-k crystals with prim-like structure were observed. The Kstruvite-k varied with temperature according to Van’t Hoff equation [24], and it theoretically depends only on temperature and not pH concentration or particle size [7]. Thus, we averaged the five determined pKstruvite-k values (pKstruvite-k is the negative log of the Kstruvitek), and obtained a mean pKstruvite-k value of 10.872. Fig. 5 shows
Mg_total: Magnesium concentration in the solution (mol/L) K_total: Potassium concentration in the solution (contains the initial potassium in the form of KH2PO4 and potassium in the form of KCl, mol/L) P_total: Concentration of P in the solution (mol/L) Na_total: Sodium concentration in the solution (mol/L) (mol/L) 5. Results and discussions 5.1. Identification of resulting crystallized products Solid phase characterization was conducted to confirm that the resulting crystallized products were k-struvite and cattiite. Fig. 2 shows results from the XRD analysis performed on the crystallized products at different initial pH values. The presence of k-struvite and cattiite were 57
Computational Materials Science 157 (2019) 51–59
Y. Yang et al.
12.0
11.5
pKstruvite-k
k in terms of concentration (Kc) by step 2. In the model, a pKstruvite-k value of 10.872 for struvite-k at 25 °C was obtained. This novel model will therefore provide theoretical reference for the recovery of potassium and phosphate from wastewater and other fields, such as its application in magnesium potassium phosphate cement (MKPC).
This work Taylor (1963) A.M Bennett (2010) Luff and Reed (1980)
CRediT authorship contribution statement 11.0
Yuanquan Yang: Writing - original draft, Data curation, Investigation, Writing - review & editing. Jun Liu: Funding acquisition, Supervision. Baomin Wang: Funding acquisition, Supervision, Writing - review & editing. Runqing Liu: Methodology, Project administration, Resources. Tingting Zhang: Writing - review & editing.
10.5
10.0
8
9
10
11
12
13
Acknowledgement
14
The authors would like to acknowledge the financial support from the National Natural Science Foundation of China (NSFC, No. 51578108 and No. 51878116) and the Fundamental Research Funds for the Central Universities (No. DUT18ZD211). The financial support from the University Innovation Team Project of Liaoning Province (No. LT2015024) and the Research and Development (No.17-48-2-00) are also gratefully acknowledged.
pH Fig. 5. Kstruvite-k at different pH values for this work and other literature results. (Taylor [12], Bennett [7] and Luff and Reed [13]).
comparison between Kstruvite-k at different pH values for this work and other literature results. The predicted pKstruvite-k decreased slightly as the pH was increased, which is approximately the same as the value determined by Taylor [12], and higher than the value by Bennett [7]. But this value was considerably lower than the value by Luff and Reed [13]. It should be pointed out that the determined Kstruvite-k value by Luff and Reed was theoretically calculated and the accompanying pH was not pointed out, while the values by Taylor and Bennett were obtained at constant pH. In our current work, only the initial pH was controlled and it evolved within the reaction process. Moreover, the selected different chemical species and equilibrium solution were not the same, which may be the reasons that led to the gap between our work and the reported literature. However, it is still an open question to identify the effect of pH, selected chemical species and equilibrium solution on the predicted Kstruvite-k.
Appendix A. Supplementary material Supplementary data to this article can be found online at https:// doi.org/10.1016/j.commatsci.2018.10.037. References [1] M. Hanhoun, L. Montastruc, C. Azzaro-Pantel, B. Biscans, M. Frèche, L. Pibouleau, Temperature impact assessment on struvite solubility product: A thermodynamic modeling approach, Chem. Eng. J. 167 (1) (2011) 50–58. [2] M.L. Rouzic, T. Chaussadent, G. Platret, L. Stefan, Mechanisms of k-struvite formation in magnesium phosphate cements, Cem. Concr. Res. 91 (2017) 117–122. [3] A.J. Wang, N. Song, X.J. Fan, J.M. Li, L.J. Bai, Y. Song, R. He, Characterization of magnesium phosphate cement fabricated using pre-reacted magnesium oxide, J. Alloy. Compd. 696 (2016). [4] B. Xu, B. Lothenbach, A. Leemann, F. Winnefeld, Reaction mechanism of magnesium potassium phosphate cement with high magnesium-to-phosphate ratio, Cem. Concr. Res. 108 (2018) 140–151. [5] H. Lahalle, C. Cau Dit Coumes, C. Mercier, D. Lambertin, C. Cannes, S. Delpech, S. Gauffinet, Influence of the w/c ratio on the hydration process of a magnesium phosphate cement and on its retardation by boric acid, Cem. Concr. Res. 109 (2018) 159–174. [6] B. Xu, H. Ma, Z. Li, Influence of magnesia-to-phosphate molar ratio on microstructures, mechanical properties and thermal conductivity of magnesium potassium phosphate cement paste with large water-to-solid ratio, Cem. Concr. Res. 68 (2015) 1–9. [7] A.M. Bennett, Potential for potassium recovery as K-struvite, 2015. [8] R.E. Loewenthal, U.R.C. Kornmuller, Modelling struvite precipitation in anaerobic treatment systems, Water Sci. Technol. (1994). [9] E.V. Musvoto, M.C. Wentzel, G.A. Ekama, Integrated chemical–physical processes modelling—II. Simulating aeration treatment of anaerobic digester supernatants, Water. Res. 34 (6) (2000) 1868–1880. [10] J. Wang, Y. Song, P. Yuan, J. Peng, M. Fan, Modeling the crystallization of magnesium ammonium phosphate for phosphorus recovery, Chemosphere 65 (7) (2006) 1182–1187. [11] D. Crutchik, J.M. Garrido, Kinetics of the reversible reaction of struvite crystallisation, Chemosphere 154 (2016) 567–572. [12] A.W. Taylor, A.W. Frazier, E.L. Gurney, Solubility products of magnesium ammonium and magnesium potassium phosphate, Trans. Faraday Soc. 59 (487) (1963) 1580–1584. [13] B.B. Luff, R.B. Reed, Thermodynamic properties of magnesium potassium orthophosphate hexahydrate, J. Chem. Eng. Data 25 (4) (1980) 310–312. [14] C.K. Chau, F. Qiao, Z. Li, Potentiometric study of the formation of magnesium potassium phosphate hexahydrate, J. Materl. Civil Eng. 24 (24) (2012) 586–591. [15] H. Lahalle, C.C.D. Coumes, A. Mesbah, D. Lambertin, C. Cannes, S. Delpech, S. Gauffinet, Investigation of magnesium phosphate cement hydration in diluted suspension and its retardation by boric acid, Cem. Concr. Res. 87 (2016) 77–86. [16] H.E. Barner, R.V. Scheuerm, Handbook of Thermochemical Data for Compounds and Aqueous Species, Wiley, 1979. [17] L. Montastruc, C. Azzaro-Pantel, L. Pibouleau, S. Domenech, Use of genetic algorithms and gradient based optimization techniques for calcium phosphate
6. Conclusions This research work was aimed at developing a thermodynamic modeling approach for determination of Ksp for struvite-k and find an appropriate resolution to solve this model. It is necessary to obtain a robust model, to make it feasible to represent thermodynamic equilibrium for various initial concentrations and pH values, and thus obtaining the Kstruvite-k as well as the conversion rates for truvite-k and cattiite. The model is based on identified involved reactions and equilibrium constants at 25 °C as provided by literatures, as well as by Gibbs free energy of each component provided or by calculation. The model consists of three steps. A first step involves initialization of Xstruvite-k, Xcattiite, I, ion concentrations as well as bounds for the following calculation. Simplex search method by Lagarias (fminsearch function within MATLAB environment) was applied for determining the accurate bounds and initial values for the following calculation. Gibbs free energy for the system was used as the objective function to be minimized. A second step involved resolution of the system, which was implemented by Successive Quadratic Programming (fmincon function within MATLAB environment), with initialized values and bounds given by step 1. A scalar function was thus used as the objective function to be minimized, to guarantee the computation robustness. Kc as well as the conversion rates for struvite-k and cattiite were obtained by providing the initial concentration of each involved component. Then, a third step was final determination of the Kstruvite-k, in which Kstruvite-k was determined by extrapolating calculated data to zero ionic strength, given by the ionic strength I and solubility product of struvite58
Computational Materials Science 157 (2019) 51–59
Y. Yang et al. precipitation, Chem. Eng. Process. Process Intensif. 43 (10) (2004) 1289–1298. [18] M.I. Bhuiyan, D.S. Mavinic, R.D. Beckie, A solubility and thermodynamic study of struvite, Environ. Technol. 28 (9) (2007) 1015–1026. [19] G.N. Lewis, M. Randall, The activity coefficient of strong electrolytes, J. Am. Chem. Soc. (5) (1921) 1112–1154. [20] D.D. Wagman, W.H. Evans, V.B. Parker, I. Hallow, S.M. Baily, R.H. Schumm, Selected values of chemical properties, Natl. Bur. Stand. Tech. Note (270) (1968). [21] W.H.E.D.D. Wagman, V.B. Parker, I. Halow, S.M. Bailey, R.H. Schumm, Selected
values of chemical properties, Natl. Bur. Stand. Tech. Note (270) (1969). [22] W.H.E.D.D. Wagman, V.B. Parker, I. Halow, S.M. Bailey, R.H. Schumm, K.L. Churney, Selected values of chemical thermodynamics properties, Natl. Bur. Stand. Tech. Note (170) (1970). [23] J.C. Lagarias, M.H. Wright, P.E. Wright, J.A. Reeds, Convergence properties of the Nelder-Mead simplex method in low dimensions, Siam J. Optim. A Publ. Soc. Ind. Appl. Math. 9 (1) (2014) 112–147. [24] V.L. Snoeyink, D. Jenkins, D. Jenkins, Water Chemistry, Wiley, New York, 1980.
59
Contents lists available at ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
A thermodynamic modeling approach for solubility product from struvite-k a
a,b
Yuanquan Yang , Jun Liu a b
a,⁎
b
a,⁎
, Baomin Wang , Runqing Liu , Tingting Zhang
T
Faculty of Infrastructure and Engineering, Dalian University of Technology, Dalian 116024, China School of Materials and Engineering, Shenyang Ligong University, Shenyang 110159, China
ARTICLE INFO
ABSTRACT
Keywords: Potassium and phosphorus recovery Solubility product Struvite-k Thermodynamic model
Crystallization of struvite-k could be an economical and sustainable alternative for potassium and phosphorus recovery from wastewater and ashes of biomass combustion. Knowledge with regards to thermodynamics that are involved in the formation of struvite-k in reactor is key in determining the optimal conditions for an efficient process. However, less research work has been done on struvite-k solubility product. A thermodynamic model for struvite-k precipitation was in this study thus proposed to determine the struvite-k solubility product, as well as conversion rate for struvite-k and its accompanying crystal cattiite over a pH value range of 10.5–12.5. The model was based on numerical equilibrium prediction of involved system Mg-K-PO4-H2O. The model was presented by a set of nonlinear equations that were solved by an optimization strategy with a three-step method. The method consists of a preliminary search by Lagarias Simplex search method for initialization, Successive Quadratic Programming (SQP) for resolution of system, and extrapolating calculated data to zero ionic strength for determination of struvite-k solubility product. A pKstruvite-k value of 10.872 for struvite-k at 25 °C was obtained in the model. This novel model will provide a theoretical reference for recovery of potassium and phosphate, as well as its possible application in magnesium potassium phosphate cement (MKPC).
1. Introduction Water pollution has been a worldwide problem from which phosphorous plays a major role, due to its potential for inducing-eutrophication. One proposed solution to this problem is phosphorous recovery from the wastewater by precipitation method [1]. Normally, struvite (MgNH4PO4·6H2O) precipitation method is seen as an effective way for phosphorous removal, and has gained increasing interest. However, it won’t work well when it comes to the disposal of wastewater that is rich in potassium and phosphate. One alternative solution is to use struvite-k (MgKPO4·6H2O) precipitation method. Struvite-k is similar to struvite but contains potassium instead of ammonium, and it can be produced from very pure form or even impurities present in solution reactor, and sometimes from hydration of magnesium potassium phosphate cement (MKPC) [2–6]. Solubility product is equilibrium constant at constant temperature that describes a solid being dissolved into its constituent ions [2,7], which is however a key parameter in determining the optimal conditions for an efficient process in the formation of struvite-k. To improve the product quality and thus minimizing the associated production costs, it is an important point to predict the struvite-k precipitation potential, yield and purity for potassium and phosphate recovery.
⁎
Several thermodynamic models have been developed to allow reasonable prediction of equilibrium constant Ksp of struvite [8–10]. These models are based on liquid-solid physicochemical equilibrium, which at least considers the involved ions, dissolved and solid species, such as NH4+, Mg2+, PO43−, H3PO4 and MgNH4PO4·6H2O. However, ionic species e.g. HPO42−, H2PO4−, MgPO4−, MgOH+ and MgH2PO4+, and dissolved species e.g. MgHPO4 also take part in the precipitation process. Crutchik, D. et al. [11] considered the struvite crystallization as a reversible reaction, and the included species were Mg2+, NH4+, PO43−, H2PO4−, HPO42−, and MgOH+, MgPO4−. Note that, in this model the authors considered struvite as the only solid phase. Hanhoun et al. [1] developed a model to predict the equilibrium constant (Ksp) for struvite at variable temperatures. They computed the Ksp values using a robust numerical method, which combined a genetic algorithm for initialization purpose and a standard Newton-Raphson method implemented within MATLAB environment. Many researchers, as reviewed above, reported the constant Ksp of struvite. Thus far, little work has however been done on the determination of solubility product values for struvite-k [7,12,13]. Bennett et al. [7] obtained the Ksp for struvite-k which provides the best match of the theoretical concentration values with the experimental data, and the theoretical concentration values were generated using PHREEQC software. Taylor et al. [12] achieved the Ksp for
Corresponding authors. E-mail addresses: [email protected] (B. Wang), [email protected] (T. Zhang).
https://doi.org/10.1016/j.commatsci.2018.10.037 Received 9 September 2018; Received in revised form 26 October 2018; Accepted 28 October 2018 0927-0256/ © 2018 Elsevier B.V. All rights reserved.
Computational Materials Science 157 (2019) 51–59
Y. Yang et al.
struvite-k by an iterative procedure. Luff and Reed [13] determined the Ksp for struvite-k by theoretically calculated. However, the results differ much great from each other (pKsp by Taylor et al. [12], Bennett [7] and Luff and Reed [13] are of 10.6, 11 and 11.7, respectively). It is believed that it was due to the different thermodynamic models, objective function and solution methods for this system that leads to the discrepancies. It is therefore urgent and supplementary to establish a robust and effective model for the struvite-k system. Even though there is similarity between the crystal structure of struvite and struvite-k, their crystallization process and associated precipitation are quite different. Research into a thermodynamic modeling approach for the solubility product from the struvite-k is thus necessary. To solve this problem, the final crystallized product from the K-Mg-PO4-H2O system needs to be first confirmed. Several studies have focused on the processes involved in the crystallization of K-Mg-PO4H2O system. For Mg/P ratios comprised of 4 to 12, Chau et al. [14] reported that MgHPO4·7H2O precipitated first, then Mg2KH (PO4)2·15H2O and was finally converted into struvite-k. The precipitation sequence depended on pH, and MgHPO4·7H2O tended to precipitate at 4–6 pH, while struvite-k was formed at pH above 7. Mg2KH (PO4)2·15H2O was an intermediate phase between MgHPO4·7H2O and the struvite-k. Wang et al. [3] showed that the finally hydrated product was struvite-k, with Mg/P ratios being ranged from 5:1 to 45:1. However, Lahalle [15] investigated the precipitated hydrate from the K-MgPO4-H2O system in diluted suspension, and they found that crystallized process for K-Mg-PO4-H2O system was a multi-step process, where MgHPO4·7H2O was precipitated first, and was then destabilized to form Mg2K(PO4)2·15H2O, which was finally converted into struvite-k and cattiite (Mg3(PO4)2·22H2O). The experimental findings about the final hydration products, as reviewed above, seem to be inconsistent. It is suggested that the final hydration products are associated with the w/c ratio (water to cementitious materials ratio by weight, here the cementitious materials refer to the sum of the solid component). The final hydration products struvite-k by [14] and [3] were obtained with w/c ratios lower than 10, while the struvite-k and cattiite by [15] were precipitated in diluted suspension with a much greater w/c of 100. To obtain a reliable Kstruvite-k value, the extended Debye-Huckel method proposed by Davies [16] was chosen in this work to calculated the activity coefficient of each ion and complex. It was however required ionic strength inferior to 0.1 M, each ion and complex for this system should be therefore diluted for enough times. Considering a w/c of approximate 400 herein was adopted, the struvite-k and cattiite were assumed as the two final hydration products. It should be highlighted that increase in the number of dissolved and solid species leads to a more complicated model for the K-Mg-PO4H2O system. Thus, analytical solutions are no longer viable and hence numerical solution is required [1]. The objective of this paper was therefore to develop a thermodynamic modeling approach for the determination of Ksp for struvite-k and find an appropriate resolution to solve this model. In this model, the involved ions and considered complexes were K+, Mg2+, PO3− 4 , − − + HPO2− 4 , H2PO4 , H3PO4(aq), MgPO4 , MgHPO4(aq), MgH2PO4 , + + – + − MgOH , H , OH , Na and Cl (since chloride and sodium exists in the form of MgCl2·5H2O and NaOH, respectively), while the assumed precipitated solid were struvite-k and cattiite. A thermodynamic model for struvite-k system was proposed and established here, moreover, a robust and effective three-step method, which was coupled with Gibbs free energy and scalar function, were proposed. The study is organized into three parts as follows: First, a thermodynamic modeling was established for the K-Mg-PO4H2O system at various pH values using extended Debye-Huckel activity coefficient modeling. Second, this study presents the resolution for the thermodynamic model, a three-step method that combines a fminsearch function to approximate the bounds and initial values for the following calculation, and a fmincon function for the resolution (implemented within MATLAB environment), and then a final determination of
Table 1 Mix proportions, initial pH and final pH for the batches. Batch
Mix proportion
P1 P2 P3 P4 P5
KCl
KH2PO4
MgCl2·6H2O
0.016 mol/L 0.016 mol/L 0.016 mol/L 0.016 mol/L 0.016 mol/L
0.004 mol/L 0.004 mol/L 0.004 mol/L 0.004 mol/L 0.004 mol/L
0.004 mol/L 0.004 mol/L 0.004 mol/L 0.004 mol/L 0.004 mol/L
Initial pH
Final pH
10.5 11 11.5 12 12.5
8.44 8.56 8.86 11.07 12.13
struvite-k solubility product. A two-step method that previously combined initial solution with further resolution was successfully applied in a related problem for the resolution of calcium phosphate precipitate model [17] and struvite precipitate model [1]. Third, model validation for this system was performed. 2. Materials and methods Experiments were performed in a continuous stirred bath reactor (1L) at a rotational speed of around 90 rpm. The reactor was tightly enclosed to avoid incorporation of CO2. Deionised water was taken to prepare the synthetic solution. KCl, KH2PO4 and MgCl2·6H2O solutions were used as the sources of K+, PO43− and Mg2+, respectively. The initial phosphorous concentration was 4 mmol/L, with a K/PO4/Mg ratio of 5/1/1 (Table 1). In our previous test, it was found that struvitek could not be detected by XRD and SEM-EDS method at pH values lower than 10.5. Therefore, NaOH (1 M) was used in this work to adjust the solution of KH2PO4 to a specified initial pH (10.5, 11, 11.5, 12 and 12.5 respectively) before adding MgCl2·6H2O. This procedure was performed to obtain a specified pH and to avoid the precipitation of struvite-k before the beginning of each experiment. The pH and conductivity for each batch were recorded by pH meter (Crison pH 29) and conductivity meter (Crison CM 35), respectively. Moreover, these experiments were conducted at a fixed temperature of 25 °C. Samples were immediately filtered through a 0.45 μm membrane filter, and then washed using distilled water followed by ethanol. The samples were then dried at ambient temperature for a week. Residual concentrations of magnesium and potassium were analyzed by ICP-OES (PerkinElmer, Avio 200) after filtration and the concentration of PO43− was analyzed by atomic absorption (AA). Each batch was repeated 3 times and measurements were conducted 3 times for each trial, giving 9 data points per sample. X-ray diffraction analysis (XRD, Rigaku D/ MAX-IIIC, Cu-kα radiation) was applied to identify the composition of the resulting crystallized product. The XRD was performed over the range between 10° and 90° 2θ, with a scanning rate of 5° 2θ per minute at a stepping size of 0.02°. The chemical composition and appearance of the resulting crystallized product were analyzed by SEM (Hitachi S-3400N) test coupled with energy dispersive spectroscopy. 3. Thermodynamic modeling 3.1. Calculation for struvite-k solubility product Struvite-k precipitates when its product from the ionic activities of potassium, magnesium and phosphate exceed the struvite-k solubility product (when the solution is supersaturated). The precipitation reaction is thus represented by an equilibrium constant Kstruvite-k, which can be computed according to Eq. (1).
Kstruvite
k
=
K+· Mg 2 +· PO43
=
+ 2+ K +·[K ]· Mg 2 +·[Mg ]· PO43 +
·[PO43 ]
Here, [] indicates concentration and Kc = [K ][Mg therefore, 52
2+
][PO4
(1) 3−
],
Computational Materials Science 157 (2019) 51–59
Y. Yang et al. n i
= Kstruvite k / K c
cattiite
(2)
i=1
lg( i ) = ADH Zi2 ((
(9)
I 1+
I
)
0.3I )
Xstruvite
where A (= ADH
n i
A(
k
Zi2 )
I 1+
I
0.3I )
[K+] = Ktotal
Kstruvite
( [PO43 ]
PO43
Mg3 (PO4 )2 ·22H2 O
Ptotal· Xstruvite
(13)
k
– Function one: Gibbs free energy for the system In our modeling, the objective function one to be minimized was the Gibbs free energy G (Eqs. (23a)–(23c)) for the system. It presents the thermodynamic potential that is minimized when a system reaches chemical equilibrium, which is a linear combination of the chemical potential from each component. According to the second law of thermodynamics, there is a general natural tendency to achieve a minimum of the Gibbs free energy for the system that tends to reach chemical equilibrium. The Gibbs free energy function could be established given by chemical potential values and concentrations for the involved ions and complexes, without considering the mass and electroneutrality conservation balances. It is therefore an effective and direct method to obtain the initial concentration of the involved ions and complexes. This method has previously been successfully applied in determining the equilibrium constant for calcium phosphate precipitation [17]. Table 2 Equilibrium constants for the system Mg-K-PO4-H2O.
) (7)
The cattiite precipitation equation can be expressed as Eq. (8):
3Mg2 + + 2PO34 + 22H2 O
(12)
Xcattiite )
(14)
(6)
k
k
Furthermore, the concentrations of ions or complexes were determined from eight equilibrium reactions (Eqs. (15)–(22)) proposed in Table 2, as well as their respective equilibrium constants at 25 °C. In this work, two methods were adopted to construct the objective functions:
The struvite-k supersaturation ratio was defined by the βstruvite-k parameter: K +)
Xstruvite
= 3[PO43 ] + 2[HPO42 ] + [H2 PO4 ] + [MgPO4 ] + [OH ] + [Cl ]
To model the evolution of struvite-k precipitation, the phosphate conversion rates were calculated as a function of pH, with respect to struvite-k and cattiite crystallization, by considering the mass and electroneutrality conservation balances, as well as the supersaturation ratio for each species. During the reaction, the aqueous species considered here were Mg and P-containing ions or complexes, as well as K+, H+, OH–, Na+ and Cl−, as described in the above section. The struvite-k precipitation equation can be written as Eq. (6):
) ([K+]
(11)
[K+] + 2[Mg 2+] + [MgH2 PO4+] + [MgOH+] + [H+] + [Na+]
3.2. Thermodynamic modeling for struvite-k precipitation
Mg 2 +
Psolution Ptotal
– the electroneutrality requirement gives:
(1 + I 0.3I ) is linear. The intersection of this line with the ordinate axis (I = 0) gives the value of -lgKstruvite-k from which Kstruvite-k was calculated.
( [Mg 2 +]
+ Xcattiite = 1
– a mass balance for potassium:
(5)
MgKPO4 ·6H2 O
k
+ [MgH2 PO4+] = Ptotal (1
is a positive constant. A plot of –lgKc against
Mg2 + + K+ + PO34 + 6H2 O
(10)
+ X cattiite)
[PO43 ] + [HPO42 ]+ [H2 PO4 ]+[H3 PO4 ](aq) + [MgPO4 ] + [MgHPO4 ](aq)
(4)
lg Kstruvite
k
– a mass balance for phosphate:
(3)
Ci·Zi2
Ptotal (X struvite
where Xstruvite-k and Xcattiite were the conversion ratios for phosphate relative to struvite-k and cattiite forms, respectively, and defined as Eq. (11).
In this equation, Ci is the concentration of ion or complexes of I [19]. It is of great importance to consider all the ionic species in the solution when determining the ionic strength. In this work, the ionic strength was calculated from the ions and complexes involved in struvite-k precipitation. The ions and complexes included K+, Mg2+, PO3− 4 , − − + HPO2− 4 , H2PO4 , H3PO4(aq), MgPO4 , MgHPO4(aq), MgH2PO4 , MgOH+, H+, OH–, Na+ and Cl−(since chloride and sodium exist in the form of MgCl2·5H2O and NaOH, respectively). Combining Eqs. (2) and (3),
struvite k
)2
K cattiite
= Mgtotal
where ADH is the Deby-Huckel parameter that takes a value of 0.509 at 25 °C [1]. Zi is valence of ion or complexes of I, and I is the ionic strength, which is determined by Eq. (4).
=
PO43
[Mg2 +] + [MgPO4 ] + [MgHPO4 ](aq) + [MgH2 PO+4 ] + [MgOH+]
The activity coefficients approach unity at low ionic strength [18]. To obtain a reliable Kstruvite-k value, the thermodynamic solubility product from the struvite-k at successively lower ionic strengths was determined by extrapolating calculated data to zero ionic strength. In this work, the extended Debye-Huckel equation (Eq. (3)) proposed by Davies was chosen due to its simplicity and accuracy at ionic strength inferior to 0.1 M.
I
)3 ( [PO43 ]
Mg 2 +
– a mass balance for magnesium:
n = the number of ions and complexes in the solution Kstruvite-k = solubility product of struvite-k in terms of activity Kc = solubility product of struvite-k in terms of concentration γi = activity coefficient of the ion or complexes i
- lg K c =
( [Mg 2 +]
The different mass balances in the liquid phase include:
where
I = 0.5·
=
(8)
The cattiite supersaturation ratio was defined by the βcattiite parameter: 53
Eq. No.
Equilibrium
Equilibrium constant ki at 25 °C
References
(15) (16) (17) (18) (19) (20) (21) (22)
H3PO4 ↔ H+ + H2PO4− H2PO4− ↔ H+ + HPO42− HPO42− ↔ H+ + PO43− MgOH+ ↔ Mg2+ + OH– MgPO4− ↔ Mg2+ + PO43− MgHPO4 ↔ Mg2+ + HPO42− MgH2PO4+ ↔ Mg2+ + H2PO4− H2O ↔ H+ + OH–
k1 = 7.42 × 10−3 k2 = 6.37 × 10−8 k3 = 3.80 × 10−13 k4 = 2.58 × 10−3 k5 = 3.74 × 10−7 k6 = 8.21 × 10−4 k7 = 4.15 × 10−2 kw = 1.00 × 10−14
[20] [21] [22] [1] [1] [1] [1] [16]
Computational Materials Science 157 (2019) 51–59
Y. Yang et al.
energy. Note that the chemical potential values available online are not yet sufficient to calculate the Gibbs free energy during this reaction. For this reason, partial chemical potential values were computed from Eq. (6), Eq. (8), Eqs. (18)–(21), and Eqs. (24)–(26) [1].
Table 3 Chemical potential values for involved ions and complexes in the system at 25 °C. Species
μi (kJ/mol)
References
H3PO4 H2PO4− HPO42− PO43− K+ Mg2+ OH– H+ MgOH+ MgPO4− MgHPO4 MgH2PO4+ Cl− Na+ MgKPO4·6H2O Mg3(PO4)2·22H2O
−1142.65 −1130.40 −1089.26 −1018.80 −283.27 −454.80 −157.29 0 −597.3 −1436.9 −1526.4 −1577.3 −240.17 −261.905 −1696.30 −3461.45
[17] [17] [17] [17] [16] [16] [17] [17] Calculated Calculated Calculated Calculated [16] [16] Calculated Calculated
nik µik
µiL = µiL (T ) + RT ln([x i ]
(26)
– Function two: A scalar function, composed of sum of squares of several equations. The function two to be minimized was the sum of squares of Eqs. (10), (12)–(14). For example, f1 = x2 + y2 + 3z2 and 2 2 f2 = 2x + 3y + z, then the scalar function was minimized as G = f12 + f22, by ensuring that f1 = 0 and f2 = 0 when the function G
where μiL and μiS are the chemical potentials of ions and complexes of i in the liquid and solid, respectively. Table 3 gives the chemical potential values which were considered for computation of Gibbs free
Influent [Mgtotal], [Ktotal], [Ptotal] and [Natotal]
Thermodynamic Model for this system Simplex search method
Initialization Step1 Xstruvite-k
Xcattiite
[K]liquid
[Mg]liquid
[P]liquid
…
Resolution of the Model Refinement
SQP
Step2
Step3
(25)
where ΔGR(T) represented the free energy of reaction (Eq. (25)) involved in the equilibrium. The reaction equation is simply as follows: A + B ↔ C. The free energy was thus given by Eq. (25). In thermodynamics, the Gibbs free energy for a system that is held at constant temperature and pressure can be expressed in a simple fashion as Eq. (26). It was noted that in this calculation, the values for Kcattiite and Kstruvite-k at 25 °C (1.04 × 10−24 and 2.4 × 10−11, respectively) were assumed to be known.
(23c)
µiS = µiS (T )
µi dNi i=1
(23b)
xi )
[ GfA (T ) + GfB (T )]
n
dG =
(23a)
i=1 k=1
(24)
GR (T ) = GfC (T )
N
G=
GR (T ) RT
ln(Ki ) =
Xstruvite-k
Xcattiite
Kc
…
I
Determination of Kstruvite-k Fig. 1. Schematic program for each computation. 54
Initialized values and bounds
Computational Materials Science 157 (2019) 51–59
Y. Yang et al.
Fig. 2. XRD patterns of crystallized products from the synthetic solutions at different initial pH values (25 °C).
– 16 bounded variables (14 aqueous ion or complexes and 2 solid species) – 12 equality constraints (Eq. (10), Eqs. (12)–(14) and Eqs. (15)–(22)) – Two inequality constraints (precipitation of struvite-k and cattiite) – Two functions to minimize (Gibbs free energy and a scalar function for several variables)
12.00 pH Electrical conductivity
3.6
pH
11.50
3.5
11.25
11.00
3.4
10.75
Electrical conductivity(mS/cm)
11.75
3.7
4. Determination of Kstruvite-k The objective was to calculate the equilibrium state from the initial concentrations of Mg2+, K+ and PO43− ions. Then, it was possible to calculate the ionic strength I and solubility product of struvite-k in terms of concentration (Kc), and thus the solubility product of struvite-k in terms of activity was determined. The initialization step still constituted difficulty: where Genetic algorithms are commonly used for complex problems [1,17], and are usually applied to initialize the concentrations of ions in the solution at equilibrium state. However, when a set of nonlinear constraints were embedded, it sometimes failed. Simplex search method by Lagarias et al. [23] is a direct search method that does not use numerical or analytic gradients and is generally referred to as unconstrained nonlinear optimization. Thus, the fminsearch function within MATLAB environment, which uses the Lagarias simplex search method was applied for determining the accurate bounds and initial values for the following calculation. For each computation, a three-step was implemented (Fig. 1): Step 1: Initialization for Xstruvite-k, Xcattiite, I ion concentrations, and bounds for the following calculation. Approximate values for 2− − − [K+], [Mg2+], [PO3− 4 ], [HPO4 ], [H2PO4 ], [H3PO4](aq), [MgPO4 ], + + – [MgHPO4](aq), [MgH2PO+ ], [MgOH ], [H ], [OH ], X and struvite-k 4 Xcattiite were determined using the Simplex search method by Lagarias (fminsearch function within MATLAB environment). A coefficient i (ranging from 1 to 10) was applied thereafter to extend the obtained values. A matrix was then denoted as lower bounds = {0}, and matrix denoted upper bounds = {i*values} were obtained and applied for accurate bounds in the following calculation. It should be emphasized that the coefficient was adopted to reduce the fluctuation of results
3.3 10.50
0
2
4
6
8
10
12
Time/h Fig. 3. Evolution of pH and electrical conductivity over time for the solution at initial pH of 12.
was minimized to zero. Even though the Gibbs free energy function could solve this model in reasonable computational times and is less time-consuming, it is unable to get accurate solution for this system, with regards to the fact that partial of the chemical values online are not yet sufficient and some of them are from calculated. An accurate objective function was thus here required, and a scalar function composed of sum of squares of the mass and electroneutrality conservation balances function was suitable. It could avoid the adverse effect of the uncertain chemical values on its final results. Most importantly, the initial values calculated by the Gibbs free energy function method could help to solve the scalar function more precisely. Thus, we coupled the two functions to calculate the equilibrium state. Details for the two approaches are available as supplementary materials (Appendix A). To solve this system, the precipitation of struvite-k and cattiite were taken into account, leading to optimization problem with two inequalities. The thermodynamic model contained:
55
Computational Materials Science 157 (2019) 51–59
Y. Yang et al.
Mg
P
1000
1000
800
800
O
600 400
Mg O
K
600 400
200 0
P
1200
counts
counts
1200
200
K 0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
0
5.0
0.0
0.5
1.0
1.5
Energy/keV
1000
Mg
P
3.0
3.5
4.0
4.5
5.0
Mg
1000
O
600
400
O
K
800 600 400
200
0
2.5
P
1200
counts
counts
800
2.0
Energy/keV
200
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
Energy/keV
0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
Energy/keV
Fig. 4. SEM morphologies of resulting crystallized products coupled with EDS results of interest area: (a) & (b) P4; (c) & (d) P2.
56
Computational Materials Science 157 (2019) 51–59
Y. Yang et al.
indicated by the location of intensity peaks, corresponding to the PDF #35-0812 and PDF #44-0775, respectively. The results indicated that the main precipitated crystal were struvite-k or cattiite or both of them for all the experimental runs. In addition, the quantity of cattiite increased as pH increased over the range of 10.5–11.5. Struvite-k tended to be favored after that (at pH value above 11.5), while the cattiite was hardly detected at pH of 12.5. However, both (over the pH range of 10.5–12.5) were confirmed by SEM-EDS method (It will be discussed at the end of this section). During the experimental runs, pH evolution and electrical conductivity for the solutions were measured with time (Fig. 3). Results showed that both pH and electrical conductivity almost did not evolve over 10 h. To guarantee that a quasi-equilibrium state was reached, the time periods for each run were extended to 7 days. SEM-EDS was conducted to further validate that the resulting crystallized products were struvite-k and cattiite. For the sake of illustration, only results of P2 (pH = 11) and P4 (pH = 12) are presented (Fig. 4). The SEM microphotographs showed that the resulting crystalized products were composed of prim-like crystal (Fig. 4b) and multilayered sheet-like crystal (Fig. 4d). SEM-EDS microanalysis procedure is a qualitative elemental analysis [11], which provides information about the involved elements that are present in the resulting crystallized product, and the EDS analysis was conducted on the P2 and P4. The SEM-EDS microanalysis results indicated that area B (Fig. 4b) and area D (Fig. 4d) shared the common elements of Mg, K, P, and O, and the ratios of Mg, K and P were approximately 1:1.13:0.95 and 1:1.06:1.02, respectively (Table 4). This implied that the prism-like crystal in area B and area D was struvite-k, which is close to the theoretical ratio of 1:1:1 for struvite-k. In terms of area A and area C, the main shared elements were Mg, P and O, and the ratios of Mg and P were 0.94:1 and 1.60:1, respectively. It could be postulated that the multi-layered sheet-like crystal was cattiite, taking into account that the theoretical Mg/P ratio was 1.5:1. Moreover, it was noted that, potassium was also identified in area A, however, the quantity was too small. The accompanying potassium could have resulted from the deposited struvite-k on the cattiite, considering that the tiny particles of struvite-k than the cattiite with sheet-like crystal structure (from Fig. 3b it could also found that partial struvite-k particles lean close to the sheet-like crystal).
Table 4 Elements in the specified areas for P3 and P5 scanned by energy dispersive spectroscopy. Element
area A
area B
area C
area D
53.23 15.22 17.14 14.41
56.56 26.71 16.73 –
53.23 15.12 16.14 15.51
Atomic/% O Mg P K
48.42 22.90 24.44 4.24
Table 5 Predicted and experimentally determined concentration of potassium and magnesium. K/mol⋅L−1 Predicted pH10.5 pH11 pH11.5 pH12 pH12.5
0.0198 0.0197 0.0195 0.0173 0.0163
Mg/mol⋅L−1 Measured 0.0198 0.0196 0.0195 0.0177 0.0163
Predicted
Measured −4
1.6620 × 10 1.5930 × 10−4 9.1520 × 10−5 7.0890 × 10−5 4.5000 × 10−5
1.6520 × 10−4 1.5920 × 10−4 9.0520 × 10−5 7.0290 × 10−5 4.3200 × 10−5
Table 6 Predicted conversion rates, pKstruvite-k and mean pKstruvite-k. pH
Xstruvite-k
Xcattiite
pKstruvite-k
Mean pKstruvite-k
pH10.5 pH11 pH11.5 pH12 pH12.5
0.0370 0.0686 0.1125 0.6510 0.9068
0.6142 0.5943 0.5763 0.2208 0.0545
10.9460 10.8790 10.8690 10.8210 10.8440
10.8720
from the system. In this step, Eq. (23a) (Gibbs free energy for the system) was used as the objective function to be minimized. Step 2: Resolution of system. Resolution of system was implemented by Successive Quadratic Programming (fmincon function within MATLAB environment), with initialized values and bounds given by step 1. In this step, the squared sum of Eq. (10), Eqs. (12)–(14) was the objective function to be minimized, and Eqs. (7) and (9) were treated as constraint equations. Then, the mean values for the involved parameters were computed (refer to Appendix A). Step 3: Determination of Kstruvite-k. The Kstruvite-k was determined by extrapolating calculated data to zero ionic strength, given by the ionic strength I and solubility product of struvite-k in terms of concentration (Kc) by step 2. The required data input for the three-step methods were as follows:
5.2. Model validation with Kstruvite-k values at various initial pH values To validate the feasibility of the established model in this work, the concentrations of residual potassium and magnesium, determined by the model and measured values, (Table 5) were compared. The results indicated that the predicted values were well in agreement with experimentally measured ones. The predicted conversion rates are also presented in Table 6, where they indicate that lower initial pH (< 12) favored the formation of cattiite but suppressed the precipitation of struvite-k. The conversion rate for struvite-k gave 3.7%, 6.8% and 11.25%, respectively with the increased pH values from 10.5 to 11.5, which was within low content. These results were also consistent with the XRD results that the struvite-k was hardly detected due to its very low content. The results were also consistent with SEM results, that few struvite-k crystals were observed. Moreover, when the initial pH was above the critical value of 12, the conversion rate for struvite-k increased abruptly, and the rates reached 65.1% and 90.68%, respectively, at 12 and 12.5 initial values. These results were also confirmed by the XRD results (Fig. 2), as well as SEM-EDS results, in which the main crystal was struvite-k, (Fig. 4a and b) where large quantity of the struvite-k crystals with prim-like structure were observed. The Kstruvite-k varied with temperature according to Van’t Hoff equation [24], and it theoretically depends only on temperature and not pH concentration or particle size [7]. Thus, we averaged the five determined pKstruvite-k values (pKstruvite-k is the negative log of the Kstruvitek), and obtained a mean pKstruvite-k value of 10.872. Fig. 5 shows
Mg_total: Magnesium concentration in the solution (mol/L) K_total: Potassium concentration in the solution (contains the initial potassium in the form of KH2PO4 and potassium in the form of KCl, mol/L) P_total: Concentration of P in the solution (mol/L) Na_total: Sodium concentration in the solution (mol/L) (mol/L) 5. Results and discussions 5.1. Identification of resulting crystallized products Solid phase characterization was conducted to confirm that the resulting crystallized products were k-struvite and cattiite. Fig. 2 shows results from the XRD analysis performed on the crystallized products at different initial pH values. The presence of k-struvite and cattiite were 57
Computational Materials Science 157 (2019) 51–59
Y. Yang et al.
12.0
11.5
pKstruvite-k
k in terms of concentration (Kc) by step 2. In the model, a pKstruvite-k value of 10.872 for struvite-k at 25 °C was obtained. This novel model will therefore provide theoretical reference for the recovery of potassium and phosphate from wastewater and other fields, such as its application in magnesium potassium phosphate cement (MKPC).
This work Taylor (1963) A.M Bennett (2010) Luff and Reed (1980)
CRediT authorship contribution statement 11.0
Yuanquan Yang: Writing - original draft, Data curation, Investigation, Writing - review & editing. Jun Liu: Funding acquisition, Supervision. Baomin Wang: Funding acquisition, Supervision, Writing - review & editing. Runqing Liu: Methodology, Project administration, Resources. Tingting Zhang: Writing - review & editing.
10.5
10.0
8
9
10
11
12
13
Acknowledgement
14
The authors would like to acknowledge the financial support from the National Natural Science Foundation of China (NSFC, No. 51578108 and No. 51878116) and the Fundamental Research Funds for the Central Universities (No. DUT18ZD211). The financial support from the University Innovation Team Project of Liaoning Province (No. LT2015024) and the Research and Development (No.17-48-2-00) are also gratefully acknowledged.
pH Fig. 5. Kstruvite-k at different pH values for this work and other literature results. (Taylor [12], Bennett [7] and Luff and Reed [13]).
comparison between Kstruvite-k at different pH values for this work and other literature results. The predicted pKstruvite-k decreased slightly as the pH was increased, which is approximately the same as the value determined by Taylor [12], and higher than the value by Bennett [7]. But this value was considerably lower than the value by Luff and Reed [13]. It should be pointed out that the determined Kstruvite-k value by Luff and Reed was theoretically calculated and the accompanying pH was not pointed out, while the values by Taylor and Bennett were obtained at constant pH. In our current work, only the initial pH was controlled and it evolved within the reaction process. Moreover, the selected different chemical species and equilibrium solution were not the same, which may be the reasons that led to the gap between our work and the reported literature. However, it is still an open question to identify the effect of pH, selected chemical species and equilibrium solution on the predicted Kstruvite-k.
Appendix A. Supplementary material Supplementary data to this article can be found online at https:// doi.org/10.1016/j.commatsci.2018.10.037. References [1] M. Hanhoun, L. Montastruc, C. Azzaro-Pantel, B. Biscans, M. Frèche, L. Pibouleau, Temperature impact assessment on struvite solubility product: A thermodynamic modeling approach, Chem. Eng. J. 167 (1) (2011) 50–58. [2] M.L. Rouzic, T. Chaussadent, G. Platret, L. Stefan, Mechanisms of k-struvite formation in magnesium phosphate cements, Cem. Concr. Res. 91 (2017) 117–122. [3] A.J. Wang, N. Song, X.J. Fan, J.M. Li, L.J. Bai, Y. Song, R. He, Characterization of magnesium phosphate cement fabricated using pre-reacted magnesium oxide, J. Alloy. Compd. 696 (2016). [4] B. Xu, B. Lothenbach, A. Leemann, F. Winnefeld, Reaction mechanism of magnesium potassium phosphate cement with high magnesium-to-phosphate ratio, Cem. Concr. Res. 108 (2018) 140–151. [5] H. Lahalle, C. Cau Dit Coumes, C. Mercier, D. Lambertin, C. Cannes, S. Delpech, S. Gauffinet, Influence of the w/c ratio on the hydration process of a magnesium phosphate cement and on its retardation by boric acid, Cem. Concr. Res. 109 (2018) 159–174. [6] B. Xu, H. Ma, Z. Li, Influence of magnesia-to-phosphate molar ratio on microstructures, mechanical properties and thermal conductivity of magnesium potassium phosphate cement paste with large water-to-solid ratio, Cem. Concr. Res. 68 (2015) 1–9. [7] A.M. Bennett, Potential for potassium recovery as K-struvite, 2015. [8] R.E. Loewenthal, U.R.C. Kornmuller, Modelling struvite precipitation in anaerobic treatment systems, Water Sci. Technol. (1994). [9] E.V. Musvoto, M.C. Wentzel, G.A. Ekama, Integrated chemical–physical processes modelling—II. Simulating aeration treatment of anaerobic digester supernatants, Water. Res. 34 (6) (2000) 1868–1880. [10] J. Wang, Y. Song, P. Yuan, J. Peng, M. Fan, Modeling the crystallization of magnesium ammonium phosphate for phosphorus recovery, Chemosphere 65 (7) (2006) 1182–1187. [11] D. Crutchik, J.M. Garrido, Kinetics of the reversible reaction of struvite crystallisation, Chemosphere 154 (2016) 567–572. [12] A.W. Taylor, A.W. Frazier, E.L. Gurney, Solubility products of magnesium ammonium and magnesium potassium phosphate, Trans. Faraday Soc. 59 (487) (1963) 1580–1584. [13] B.B. Luff, R.B. Reed, Thermodynamic properties of magnesium potassium orthophosphate hexahydrate, J. Chem. Eng. Data 25 (4) (1980) 310–312. [14] C.K. Chau, F. Qiao, Z. Li, Potentiometric study of the formation of magnesium potassium phosphate hexahydrate, J. Materl. Civil Eng. 24 (24) (2012) 586–591. [15] H. Lahalle, C.C.D. Coumes, A. Mesbah, D. Lambertin, C. Cannes, S. Delpech, S. Gauffinet, Investigation of magnesium phosphate cement hydration in diluted suspension and its retardation by boric acid, Cem. Concr. Res. 87 (2016) 77–86. [16] H.E. Barner, R.V. Scheuerm, Handbook of Thermochemical Data for Compounds and Aqueous Species, Wiley, 1979. [17] L. Montastruc, C. Azzaro-Pantel, L. Pibouleau, S. Domenech, Use of genetic algorithms and gradient based optimization techniques for calcium phosphate
6. Conclusions This research work was aimed at developing a thermodynamic modeling approach for determination of Ksp for struvite-k and find an appropriate resolution to solve this model. It is necessary to obtain a robust model, to make it feasible to represent thermodynamic equilibrium for various initial concentrations and pH values, and thus obtaining the Kstruvite-k as well as the conversion rates for truvite-k and cattiite. The model is based on identified involved reactions and equilibrium constants at 25 °C as provided by literatures, as well as by Gibbs free energy of each component provided or by calculation. The model consists of three steps. A first step involves initialization of Xstruvite-k, Xcattiite, I, ion concentrations as well as bounds for the following calculation. Simplex search method by Lagarias (fminsearch function within MATLAB environment) was applied for determining the accurate bounds and initial values for the following calculation. Gibbs free energy for the system was used as the objective function to be minimized. A second step involved resolution of the system, which was implemented by Successive Quadratic Programming (fmincon function within MATLAB environment), with initialized values and bounds given by step 1. A scalar function was thus used as the objective function to be minimized, to guarantee the computation robustness. Kc as well as the conversion rates for struvite-k and cattiite were obtained by providing the initial concentration of each involved component. Then, a third step was final determination of the Kstruvite-k, in which Kstruvite-k was determined by extrapolating calculated data to zero ionic strength, given by the ionic strength I and solubility product of struvite58
Computational Materials Science 157 (2019) 51–59
Y. Yang et al. precipitation, Chem. Eng. Process. Process Intensif. 43 (10) (2004) 1289–1298. [18] M.I. Bhuiyan, D.S. Mavinic, R.D. Beckie, A solubility and thermodynamic study of struvite, Environ. Technol. 28 (9) (2007) 1015–1026. [19] G.N. Lewis, M. Randall, The activity coefficient of strong electrolytes, J. Am. Chem. Soc. (5) (1921) 1112–1154. [20] D.D. Wagman, W.H. Evans, V.B. Parker, I. Hallow, S.M. Baily, R.H. Schumm, Selected values of chemical properties, Natl. Bur. Stand. Tech. Note (270) (1968). [21] W.H.E.D.D. Wagman, V.B. Parker, I. Halow, S.M. Bailey, R.H. Schumm, Selected
values of chemical properties, Natl. Bur. Stand. Tech. Note (270) (1969). [22] W.H.E.D.D. Wagman, V.B. Parker, I. Halow, S.M. Bailey, R.H. Schumm, K.L. Churney, Selected values of chemical thermodynamics properties, Natl. Bur. Stand. Tech. Note (170) (1970). [23] J.C. Lagarias, M.H. Wright, P.E. Wright, J.A. Reeds, Convergence properties of the Nelder-Mead simplex method in low dimensions, Siam J. Optim. A Publ. Soc. Ind. Appl. Math. 9 (1) (2014) 112–147. [24] V.L. Snoeyink, D. Jenkins, D. Jenkins, Water Chemistry, Wiley, New York, 1980.
59