Size-dependent growth kinetics of struvite crystals in wastewater with calcium ions

Abstract Kinetic parameters describing continuous reaction crystallization of struvite from aqueous solutions containing also calcium ions (from 100 to 2000 mg Ca2+/kg) were estimated. Test results were compared with kinetic data of struvite manufactured from real cattle liquid manure. Kinetic model for ideal MSMPR (Mixed Suspension Mixed Product Removal) crystallizer was used assuming dependence of crystal growth rate G on its size L (size-dependent growth, SDG MSMPR model). Based on nonlinear regression and statistical analysis, one from within five considered G(L) models was selected – Rojkowski exponential (RE) model – rendering the experimental population density distributions the best. It was concluded, that calcium ions influenced all components of struvite manufacturing process disadvantageously. A rise of Ca2+ concentration in a feed from 100 to 2000 mg/kg increased nucleation rate ca. 160-time, whereas growth rate of nuclei up to macroscopic size G0 decreased more than 10-time. Linear (larger) crystal growth rate G¥ was nearly two-times smaller: 1.71·10–8 m/s (100 mg Ca2+/kg) – 9.10·10–9 m/s (2000 mg Ca2+/kg). Resulting in a product with deteriorated quality. Mean size of the crystals decreased nearly two-times (to 18.4 μm), non-homogeneity within product population enlarged and calcium fraction in the product increased. The product, beside struvite MgNH4PO4·6H2O, also contained hydrated amorphous calcium phosphate(V) Ca3(PO4)2·nH2O (ACP). It was observed, that 5-times smaller concentration of phosphate(V) ions in a feed and magnesium ions excess in relation to phosphate(V) and ammonium ions (1.2 : 1 : 1) influenced all kinetic parameters of continuous struvite reaction crystallization advantageously.


Introduction
Modern technologies designed for efficient phosphate(V) ions recovery from various wastewaters [1] or from liquid manure [2] are commonly based on fixing of these ions into sparingly soluble magnesium and ammonium salt MgNH 4 PO 4 •6H 2 O -struvite (phosphorus recycling) [3]. The process should be operated under rational technological control, resulting in struvite of the possibly best quality [4,5]. The crystalline product should not contain excessive amount of impurities, which are present in wastewater, especially heavy and toxic metals [3,6]. If wastewater contains additionally calcium ions, calcium phosphates also co-precipitate during struvite reaction crystallization. These form different crystalline or amorphous phases, like CaHPO 4 •2H 2 O (brushite, DCPD), Ca 4 H(PO 4 ) 3 •2.5H 2 O (OCP), Ca 3 (PO 4 ) 2 •nH 2 O (ACP, n may be also 0), Ca 5 (PO 4 ) 3 OH (hydroxyapatite, HAP) and other compounds depending on adjustable process control parameters (mainly pH and process / residence time) and specific chemical composition of wastewater [7]. The effect of the calcium ions presence on struvite reaction crystallization run and product properties was investigated by Le Corre [8] (batch variant of reaction crystallization) and Hutnik [9,10] (continuous approach). Both chemical and phase composition of such product change significantly [3]. Calcium fraction itself can reach even up to 30 mass % [11].
Liquid manure is wastewater representing high calcium load [2,6]. Crystal product manufactured from liquid manure should be thus regarded as a blend of struvite and calcium phosphates(V) [12,13], what essentially influences its application as mineral fertilizer [14]. Also hydroxides and salts of some impurities co-present originally in liquid manure were detected in a product [15,16]. Final phase and chemical composition of the product, as well as its crystal size distribution (CSD) depend mainly on liquid manure composition and on assumed parameters of continuous reaction crystallization [15,17]. From cattle liquid manure (CLM), of composition shown in Table 1, solid products representing from 58 to 62 mass % of struvite, from 33 to 36 mass % of amorphous hydrated calcium phosphate(V) (ACP), 3.2 -3.6 mass % of halite and 1.2 -1.6 mass % of sylvine [16] were obtained.
The results of experimental research and corresponding kinetic modeling data concerning continuous reaction crystallization process of sparingly soluble phosphates from aqueous solutions containing originally phosphate(V) and calcium ions are presented. Process tests ran in a continuous laboratory DT MSMPR (Draft Tube, Mixed Suspension Mixed Product Removal) type crystallizer [18] with internal circulation of suspension driven by propeller mixer. From the product CSDs nucleation rates and linear crystal growth rates of struvite (for nuclei and for larger crystals) were determined. Kinetic model of continuous mass crystallization in MSMPR crystallizer, SDG (Size-Dependent Growth) MSMPR model was used in calculations [19][20][21][22][23]. Generally, the SDG model is based on assumption of nonlinear G(L) function, where G -crystal linear growth rate, L -linear crystal size. Struvite crystals, based on experimental lnn(L) data observation, satisfy assumptions of this phenomenon. The results were also compared with kinetic parameters [11] estimated with the most simplified SIG (Size Independent Growth) MSMPR kinetic model [18]. Kinetic data from both models were also related to struvite nucleation and crystal growth observed in real cattle liquid manure (CLM) [24]. Knowledge of kinetic relations in struvite continuous reaction crystallization process can be practically utilized e.g. for elaboration of phosphates recovery technology based on aqueous solutions with high calcium content, including liquid manure.  [9,10] and kinetic parameters values [11] of continuous struvite reaction crystallization process. Nucleation rate B and crystal linear growth rate G calculated with SIG MSMPR model. Process temperature: 298 K, pH 9, mean residence time of suspension in a crystallizer τ 900 s.

Setup and procedure
The research was carried out in a continuous laboratory experimental plant with automatic data processing (IKA Labortechnik). Operation, monitoring and recording of the experimental data were done with PC computer (IKA Labworldsoft and BioScadaLab software). The continuous process ran in DT MSMPR type crystallizer (constructional details presented elsewhere [18]), of working volume V w 0.6 dm 3 . Mixing intensity, temperature, inflow streams of feed and alkalizing solution, as well as outflow rate of product crystals suspension from the crystallizer unit were strictly centrally controlled and adjusted by PC computer. The crystallizer was continuously provided with aqueous solution of predetermined concentrations of phosphate(V), magnesium and ammonium ions, as well as calcium ions. Moreover, alkalizing solution (20 mass % aqueous solution of NaOH) was injected when necessary for stabilization of the predetermined pH level. Experimental stand and details of testing method were presented elsewhere [10], whereas calculation procedures are presented in [17].
The effects of three process parameters on struvite continuous reaction crystallization process: molar ratio of substrates in a feed (PO 4 3-: Mg 2+ : NH 4 + as 1 : 1 : 1 and 1 : 1.2 : 1), inlet concentration of phosphate(V) ions (0.20 and 1.0 mass % of PO 4 3-), as well as concentration of calcium ions (from 100 to 2000 mg Ca 2+ /kg of solution) were investigated. Process temperature was set as 298 K, pH as 9 and mean residence time of suspension in a crystallizer τ as 900 s. For comparison, struvite was also manufactured from real liquid manure derived from animal-breeding farm [15], containing 0.34 mass % of phosphate(V) ions and calcium ions (1200 mg Ca 2+ /kg solution), together with other accompanying impurities (Cu, Fe, K, Mn, Mo, Na, Zn and others) (CLM in Table 1). For each product CSD was determined using solid particle analyzer Coulter LS 230 and Beckman Coulter LS 13 320. On this basis distribution parameters were determined and then used for estimation of nucleation rate B and linear growth rate of struvite G(L) (for nuclei and larger crystals). Identification of product crystals shape, as well as calculation of crystal volume shape factor k v were done based on images provided by scanning electron microscope JEOL JSM 5800LV. Chemical compositions of mother liquor and solid product were determined with atomic absorption spectrometer iCE 3000 (metals), spectrophotometer UV-VIS Evolution 300 (phosphates(V)) and by titration method (ammonium, formalin method). Additionally, the X-ray fluorescence spectrometer PANalytical Magi'X PW2424 was used for solid phase analysis (phosphorus and metals). Identification of product phases and quantification of their fractions were done with X-ray diffractometer PANalytical XPERT PRO MPD and Rietveld method ( Figure 1).

Crystal product characteristic
Solid particles were manufactured from the aqueous solutions containing phosphate(V) and calcium ions not only of diverse crystal sizes, but also representing different chemical composition (Table 1). Mean size of struvite crystals varied from 12.4 to 48.6 mm. Particles of amorphous calcium phosphate were small, with size below 2 mm, the most often forming agglomerates on struvite crystal surface ( Figure 1). Depending on crystallizer feed composition and its work parameters, struvite crystals with shape coefficient k v 0.042 -0.049 were obtained. Crystals length to width ratio varied from 4.0 (k v 0.049) to 4.3 (k v 0.042), thus within relatively narrow range. In scaning electron microscope images ( Figure 1) one can distinguish dominating tubular crystals, some with longitudinal fractures (in respect to axis, trough), all with deformed, shreded ends. Only a few crystals demonstrated the prismatic shape, which is specific for struvite. Presence of not only crystalline struvite MgNH 4 PO 4 •6H 2 O, but also hydrated amorphous calcium phosphate(V) Ca 3 (PO 4 ) 2 •nH 2 O was confirmed in the products. Calcium content in solid phase varied from 0.3 up to even 29.6 mass % (Table 1). Product from cattle liquid manure (CLM) represented 58.6 mass % of struvite, 36.2 mass % of amorphous phase, 3.6 mass % of halite, as well as 1.6 mass % of sylvite [16].
From experimental CSDs of products the particle number distributions were calculated (population density distributions, n(L)). From these distribution courses (presented in [11]) one can observe, that for struvite crystals of size L > 40 mm these relationships can be modeled with linear function, representing the most simplified SIG MSMPR kinetic model [18,25]. Linear growth rates G of struvite crystals and nucleation rates B calculated [11,17] with this model are presented in Table 1. These results should be thus regarded as the estimated values only.  Table 1) [11], b) calcium ions in a real cattle liquid manure (CLM in Table 1) [16], c) calcium ions and inorganic impurities in synthetic wastewater [11], d) XRD pattern of crystal product -identification of struvite.

Size Dependent Growth (SDG) kinetic model
For struvite crystals of size L < 40 µm population density distributions n(L) in lnn(L) -L coordinates show significant nonlinearity ( Figure 2). This characteristic, curved course may be interpreted as manifestation of some more complex kinetic mechanisms of crystal phase growth than it could result from the most simplified SIG MSMPR approach. In particular, this shape of population density distribution can result from dispersion of crystal growth rates within investigated population (Growth Rate Dispersion -GRD, where crystals of the same size in the same environment grow with different rates), complex secondary nucleation phenomena, size-dependent solubility in small size range or size-dependent relative slip velocities in crystal-solution system modifying by hydrodynamic action the convection/diffusion conditions of mass transfer from liquid phase into solid structure. Individual contributions of all these phenomena are, however, undistinguishable based on analysis of crystal population density distribution shape only. Nevertheless, one can assume, that for the process design all these partial phenomena may be considered together because all these produce qualitatively identical nonlinear effect in lnn(L) -L coordinates [18,25]. This way their net effect may be described using size-dependent growth (SDG) kinetic models -as convenient, practical mathematical approach for determination of "conventional" kinetic parameters of the process.
For continuous mass crystallization, assuming steady state in a crystallizer with perfectly mixed suspension and removal of not classified, representative product (ideal MSMPR crystallizer), crystal population balance can be written in a form of differential equation (1) [25]: (1) Considering in crystal population balance (Eq. (1)) kinetic mechanism of size-dependent crystal growth rate (G(L) ≠ const, SDG MSMPR model) requires preliminary assumption of some form of G(L) dependence, following by solving of corresponding equation (2): (2) where: n -population density, n 0 -nuclei population density. Some solutions of Eq. (2), obtained as n(L) analytical expressions corresponding to theoretical G(L) models assumed, are presented in Table 2 [26]. Knowing nuclei population density, n 0 , and their linear growth rate, G 0 , nucleation rate B can be calculated from the following relation (13): Two available SDG MSMPR models - Table 2, Canning and Randolph (CR) model (Eq. (3)) and ASL model (Eq. (5)) -predict monotonic rise of crystal linear growth rate with crystal size increment. Such prediction is, however, inconsistent with the results of experimental research. These demonstrated, that beyond some limiting crystal size, corresponding G practically stabilizes. This way the best approach is provided by three Rojkowski SDG models (Eqs. (7), (9) and (11) in Table 2), predicting both finite, minimal growth rate for nuclei (of size L = 0) and finite, maximal growth rate for large crystals. Ethical approval: The conducted research is not related to either human or animal use.

Results and discussion
Experimental CSDs (Table 1) were recalculated into corresponding population density distributions, n(L).  Table 1 and 4).
These distributions, presented graphically as lnn(L), demonstrated concave upward for the smallest particles, of size L < 40 µm. For modeling of n(L) data courses five SDG MSMPR models were preliminary tested. Their G(L) functions and corresponding analytical equations of population density distributions n(L) are presented in Table 2.
Eqs. (4), (6), (8), (10) and (12) were then fitted (using nonlinear regression analysis) to experimentally determined population density distributions of struvite crystals (7 experimental n(L) distributions). For each experimental data set and for each model ( Table 2) the parameters of n(L) distributions were calculated, making determination of variance (Eq. (14) possible: (4) where RMSD2 -variance, n calc is population density value calculated by the model, n exp is experimental population density value, p is number of experimental data.
Comparison of sum of variances Σ(RMSD) 2 for all test runs made selection of n(L) form modeling of experimental data the best possible (assumed criterionminimal Σ(RMSD) 2 ). This way selection of corresponding theoretical G(L) model ( Table 2) was possible. These verifying calculation results are presented in Table 3. From the table it results, that two SDG MSMPR models described experimental data the best. These were: Rojkowski exponential RE (Eqs. (7) and (8), Table 2) and Rojkowski hyperbolic RH (Eqs. (9) and (10), Table  2) models; (Σ(RMSD) 2 = 0.1363 and 0.2225, respectively). Quality of other models was significantly worse, Σ(RMSD) 2 varied from ca. 1 to 3.6. For further calculations, covering determination of kinetic parameters of the continuous process, the best model was selected. It was Rojkowski exponential -RE SDG MSMPR model (Eqs. (7)-(8)), because of the smallest sum of variances Σ(RMSD) 2 and relatively simple G(L) dependency form. Parameters of both RE SDG MSMPR model equations - (7) and (8), namely: nuclei population density n 0 , minimal linear growth rate (of nuclei, zero-size crystals) G 0 , maximal linear growth rate (for the largest crystals, of theoretical infinite size) G ∞ , nucleation rate B (= n 0 G 0 ), as well as model constant a -are all presented in Table 4.
Exemplary graphical comparison of experimental population density distributions n exp (L) of solids produced from aqueous solutions of phosphate(V) ions (1 mass %) with 100, 1500 and 2000 mg Ca 2+ /kg (inlet concentration) and ncalc(L) courses predicted by Rojkowski exponential RE SDG MSMPR kinetic model (Eq. (8), tests No. 1, 3 and 4 in Table 1 and 4) is demonstrated in Figure 2. From the comparison it results, that RE SDG MSMPR model precisely renders curvature of the n(L) course in lnn(L)-L coordinates. The calculated kinetic parameter values (see Table 4) may be thus regarded as reliable numerical data for nominal description of kinetics of continuous reaction crystallization process of struvite in presence of calcium ions.
From the analysis of kinetic data presented in Table 4 it results, that calcium ions affected all kinetic components of struvite reaction crystallization process disadvantageously. Table 2: Size-dependent growth G(L) kinetic models and corresponding analytical n(L) forms.

G(L) kinetic model
Corresponding analytical n(L) form [26] Canning and Randolph (CR) model [19]   Abegg, Stevens, Larson (ASL) model [20] Rojkowski (hyperbolic model RH) [22] aL L aG G aL Increase in concentration of these ions from 100 to 2000 mg Ca 2+ /kg caused rise of nucleation rate B of struvite from 1.00·10 14 up to 1.61·10 16 1/(sm 3 ), what produced significant enlargement of the nuclei number in a crystallizer -nuclei population density n 0 raised nearly 1800-time. Linear crystal growth rate G ∞ of larger crystals decreased by nearly 2-times as much, from 1.71•10 -8 to 9.10•10 -9 m/s. Also growth rate of nuclei G 0 decreased, ca. 10-times, from 2.30·10 -11 to 2.09·10 -12 m/s. In result, number of small crystals in agitated suspension enlarged. In effect mean size of solids removed from the crystallizer decreased from 34.2 to 18.4 mm (see Table 1), thus by 46%. Calcium content in a product increased, however, from 0.3 to 8.4 mass %, what additionally raised the non-homogeneity within product population (coefficient of variation CV 83.0 → 85.5%, Table 1). Especially concentration of small amorphous particles of calcium phosphate, ACT, increased [11]. Lower concentration of phosphate(V) ions in a feed (0.20 mass %) together with 20% excess of magnesium ions in a process system influenced the continuous struvite reaction crystallization process advantageously. In such conditions lower nucleation rates B and higher crystal growth rates G ∞ were observed. From this comparison it results, that stronger effect on raise of struvite crystals growth resulted from concentration of phosphate(V) ions. The 5-time lower concentration caused 17 -19% increase in growth rate G ∞ (tests 3 and 5, 6 and 7 in Table 4). Under these conditions final mean size of struvite crystals enlarged by 62 -65% (crystal mean size L m reached ca. 47 -49 µm). However, in presence of 20-proc. magnesium ions excess linear struvite crystals growth rate G ∞ raised by 1 -2% only (tests 3 and 6, as well as 5 and 7 in Table 4). Product quality, however, slightly improved (L m in Table 1). Compared data corresponded to presence of 1500 mg Ca 2+ /kg in a feed, where pH of mother liquor was 9 and mean residence time of suspension -900 s, while process temperature -298 K.
Kinetic calculations results of struvite continuous reaction crystallization in a real cattle liquid manure (CLM) [16] containing 0.34 mass % of phosphate(V) ions, 1200 mg/kg of calcium ions and specific impurities, which concentrations were provided in detail in Table  1, are also presented in Table 4 for comparison. Despite many impurities in CLM, kinetic parameters values of struvite reaction crystallization process did not deviate significantly from the ones corresponding to simple systems with calcium ions (tests No. 1 -7 in Table 4). Nuclei population density n 0 of struvite crystals from CLM was ca. 3·10 26 1/(m m 3 ), their growth rate G 0 exceeded 6·10 -12 m/s, whereas nucleation rate B was below 2·10 15 1/(sm 3 ). These represent advantageous values, for which one should expect manufacturing of properly shaped struvite crystals. However, experimental data are different. Struvite of small sizes, non-homogeneous, of deformed shapes and with clear trend to caking and agglomeration (see Figure  1b) were removed from the crystallizer. Mean size of these crystals reached 12.4 µm only, and CV exceeded 100% (CLM, Table 1). The reason explaining the observed poor   (7) and (8)): nuclei population density n 0 , nucleation rate B, minimal G 0 and maximal G ∞ crystal linear growth rate, as well as RE SDG MSMPR model parameter a -estimated for continuous struvite reaction crystallization process in DT MSMPR crystallizer. Corresponding process conditions -see Table 1. * according to [24] quality of the product manufactured from CLM may be, among others, the smallest value of linear crystal growth rate G ∞ : only 8.48·10 -9 m/s (CLM, Table 4). In solution, where inlet concentration of calcium ions was higher by 300 mg Ca 2+ /kg, while concentration of phosphate(V) ions was 1.0 mass % (0.34 mass % in liquid manure, CLM), linear crystal growth rate G ∞ was 1.47·10 -8 m/s (test No. 3 in Table 4). From this solution product of significantly higher quality: L m 28.9 µm, CV 85.4% was manufactured (Table  1). In general, one can conclude, that impurities present in real liquid manure (CLM) influenced the continuous reaction crystallization of struvite in diverse way. These affected nuclei population density n 0 of struvite and growth rate of its nuclei G 0 advantageously. In consequence nucleation rate B (= n 0 G 0 ) of struvite was smaller than the values calculated for the solutions with calcium ions of similar concentration (compare CLM and tests No. 3, 5-7 in Table 4). In liquid manure linear crystal growth rate G ∞ of struvite decreased by 2-times on average: to 8.48·10 -9 m/s. Composition of impurities observed in liquid manure influenced also struvite crystal shape disadvantageously (compare Figure 1a and 1b). In general, quality of the product manufactured from liquid manure CLM should be regarded as unsatisfactory in respect to market demands [27,28]. Improvement of the product quality is, however, possible, among others, through elongation of residence time of struvite suspension in a crystallizer [15,16,24]. Comparison of G(L) courses predicted by RE SDG MSMPR model (Eq. (7) Tables 1 and 4) is presented in Figure 3, where G(L) determined for struvite produced from cattle liquid manure CLM is also added for comparison. With the size enlargement of struvite crystals, linear growth rate increases from some finite growth rate of nuclei (crystals "formally" of L = 0) G 0 , in m/s: 2.30·10 -11 (100 mg Ca 2+ / kg), 4.09·10 -12 (1500 mg Ca 2+ /kg) and 2.09·10 -12 (2000 mg Ca 2+ /kg) up to asymptotic, maximal G ∞ reaching: 1.71·10 -8 , 1.47·10 -8 and 9.10·10 -9 m/s, appropriately. The limiting, constant G ∞ value, which beyond some initial L range is practically stable and size-independent, is visibly influenced by calcium ions concentration in a feed. The higher this concentration is, the more limiting value of G ∞ is shifted towards smaller crystals. For 100 mg Ca 2+ /kg in a feed linear growth rate G ∞ stabilizes for the crystals of size larger than 140 µm, for 1500 mg Ca 2+ /kg -for crystals of L > 120 µm, whereas for 2000 mg Ca 2+ /kg -for the crystals of L > 80 µm. In liquid manure linear growth rate raises from G 0 6.46·10 -12 up to 8.48·10 -9 m/s. For crystals of size L > 100 µm it reaches practically constant value. From all presented data it results, that in struvite reaction crystallization process small crystals grow slower, larger ones -faster, up to maximal, stable value, practically dependent on concentration of calcium ions in a process system. All these affect product's CSD and homogeneity of its population. It is of great importance in practice, in elaboration of technology of phosphate(V) ions recovery from liquid manure, as well as in developing of rational strategy of this process steering in a continuous crystallizer.

No. Kinetic parameters -RE SDG MSMPR model
Other divalent cation present in struvite reaction crystallization environment, Mg 2+ , was also subject of the authors' more thorough research (e.g. [17]). However, Mg 2+ ion is substrate in struvite synthesis reaction, thus should be analyzed together with other process parameters. The 20% excess of magnesium ions in respect to PO 4 3- (7) in Table 2 with model parameters values presented in Table 4). From the comparison it results, that presence of calcium ions enhances changes in process kinetics for the nearly identical process conditions (T, pH, t).

Conclusions
Size-Dependent Growth (SDG) kinetic model for ideal MSMPR crystallizer was assumed for the modeling of experimental data derived from continuous reaction crystallization process of phosphate(V) ions recovery from aqueous wastewater solutions containing also calcium ions. The 5 different SDG MSMPR n(L) models derived from 5 different theoretical G(L) relations were considered. Based on results of nonlinear regression analysis the n(L) equation the best modeling of population density distribution of solid product was selected. It was Rojkowski exponential (RE SDG MSMPR) model. Kinetic parameters of struvite continuous reaction crystallization in presence of calcium ions (from 100 to 2000 mg Ca 2+ /kg of feed) were calculated based on this model.
Considering the experimental data and kinetic calculations it was generally concluded, that increase in inlet concentration of calcium ions produced increase in nucleation rate of struvite from 1.00·10 14 1/(sm 3 ) (100 mg Ca 2+ /kg) up to 1.61·10 16 (1/(sm 3 ) (2000 mg Ca 2+ /kg), simultaneously observing decrease of growth rate of these nuclei G 0 : from 2.30·10 -11 to 2.09·10 -12 m/s. Linear crystal growth rate G ∞ of struvite also decreased, from 1.71·10 -8 to 9.10·10 -9 m/s. These are significant changes in kinetic parameters of struvite continuous reaction crystallization. Nucleation rate higher by more than two orders of magnitude, at 2-time lower crystal growth rate G ∞ caused, that mean struvite product size L m decreased from 34.2 to 18.4 µm, while calcium fraction in a product increased from 0.3 up to 8.4 mass %. Amorphous calcium phosphate ACP appeared in solid phase. The final product represents thus a mix of both these phosphates, what is not, however, essential drawback in practical application of such product in agriculture as a valuable mineral fertilizer slowly dosing its nutrient content [27,28].
It was also concluded, that the decrease of concentration of phosphate(V) ions (from 1.0 to 0.2 mass % of PO 4 3-) in a feed, as well as magnesium ions excess in respect to phosphate(V) and ammonium ions inlet concentrations (as 1.2 : 1 : 1) constrained nucleation rate of struvite and influenced its crystal growth rate advantageously (G ∞ (1.72 -1.76)·10 -8 m/s). Mean size L m of solid product reached even 48.6 µm. However, in these conditions calcium content in a product was even 29.6 mass %.
Calculation results according to RE SDG MSMPR model can be regarded as moderately reliable for kinetic description of continuous, complex struvite reaction crystallization process from wastewaters containing calcium ions. -calcium content in a product, mass % CV -coefficient of (crystal size) variation, % G -crystal linear growth rate, m/s G 0 -linear growth rate of nuclei, m/s G ¥ -linear crystal growth rate (of larger particles), m/s k v -crystal volume shape factor (coefficient), L -crystal size, m L 84 , L 16