Article pubs.acs.org/crystal
Growth Kinetics of Synthetic Hydromagnesite at 90 °C Subrahmaniam Hariharan and Marco Mazzotti* ETH Zurich, Institute of Process Engineering, Sonneggstrasse 3, CH-8092 Zurich, Switzerland S Supporting Information *
ABSTRACT: The seeded growth kinetics of synthetic hydromagnesite was investigated at 90 °C and under low fugacities of CO2 ( 0 ⎩0
Fin − Fout − FMT = 0
where Fout and FMT are the molar flow rates of the gas leaving the reactor and dissolving in the aqueous solution, respectively. The mass balances for CO2 in the gas phase can be written as ⎛ Vg ⎞ dyCO 2 ⎟⎟ FinyCO ,in − FoutyCO ,out − FMT = ⎜⎜ 2 2 ⎝ Vg,m ⎠ dt
yCO (t0) = yCO ,0
where mw is the mass of water in the reactor and Qin is the solvent flow rate into the reactor corresponding to the addition of MgCl2 solution to create supersaturation. Mass balances for the different elements in solution, namely, Na, Cl, Mg, and C, can be written as
d(m w cCl) = Q in(t )cCl,in dt
(5)
d(m w c Mg) dt
2
d(m w cC) = −4R ppt + FMT dt
yCO + yN + yH O = 1 2
(13)
2
3.3. Solid Phase. The growth of hydromagnesite seeds in a well-stirred reactor can be described by the following population balance equation (PBE):
(6)
∂f ∂f +G =0 ∂t ∂L
(7)
with initial conditions:
(14)
where L is the characteristic length of the particle, f is the particle size distribution; i.e., f dL represents the total number of particles with characteristic length between L and L + dL, and G is the size independent growth rate. The initial and the boundary conditions, when assuming no nucleation, can be written as
m w (t0) = m w,0 , c Na(t0) = c Na,0 , cC(t0) = cC,0 , c Mg(t0) = 0, and cCl(t0) = 0
2
The value of yCO2,out is different from yCO2 as the gas stream leaving the reactor is completely dry. Vg is the volume of the gas phase, Vg,m is the molar volume of the gas phase, and yl is the mole fraction of the component l in the gas phase of the reactor such that 2
= Q in(t )c Mg,in − 5R ppt
(11)
2
The value of yCO2,0 for each experiment is provided in Table 1. yCO2,in and yCO2,out are the mole fractions of the CO2 in the gas stream that enters and leaves the reactor, respectively, with yCO 2 yCO ,out = 2 yCO + yN (12)
(3)
(4)
(10)
with: 2
d(m w c Na) =0 dt
(9)
(8)
The values of mw, cNa0, and cC0 for each experiment are reported in Table 1. Rppt is the overall precipitation rate of (MgCO3)4· Mg(OH)2·4H2O, and FMT is the molar flow rate of the gaseous CO2 dissolving in the aqueous solution; their values are derived later in eq 20 and eq 21, respectively. The total concentration of each element in the above equations is the sum of the concentrations of the speciated forms shown in Figure 4. All concentrations are expressed in the molality scale, and the concentrations of the speciated forms are calculated by solving a set of nonlinear equations consisting of the equilibrium relations from reactions 2−8 in Table S5.1 of the Supporting Information, the material balances for the different speciated forms, and the overall charge balance simultaneously. The activity coefficients for the different speciated ionic forms are calculated using the B-dot equation, according to the standard procedures used in the geochemical equilibrium software
f (L , 0) = f0 (L)
(15)
f (0, t ) = 0
(16)
For seeded growth experiments, the size-independent growth rate at a constant T can be expressed by two different functional forms: (1) according to surface nucleation mechanism ⎛ B⎞ G = kσ 5/6 exp⎜ − ⎟ ⎝ σ⎠
(17)
or (2) through an empirical power-law expression G = kσ p
(18)
where σ is the relative supersaturation defined as 321
DOI: 10.1021/acs.cgd.6b01546 Cryst. Growth Des. 2017, 17, 317−327
Crystal Growth & Design
Article
Figure 5. Quality of fitting of the CO2 sorption experiments. The experimental (circles) and the simulated (lines) conductivity profiles for experiments MT03−MT14 are shown. (19)
σ = ln S
Vg,m, were calculated from the Redlich−Kwong equation of state as described elsewhere.16 The values of yH2O and Vg,m (used in eqs 13 and 10, respectively) are constants at the experimental conditions and equal to 0.73 and 31.6 L mol−1. The flow of gas in the downstream piping was modeled according to procedures described in Section S3 of the Supporting Information. 4.1. CO2 Sorption Kinetics. Equations 3−13 and 21 were solved in order to describe CO2 sorption in water. Speciation calculations were performed at each time step, and the conductivity of the solution, κ, was then calculated using eq S2.1 (see the Supporting Information) and fitted to the experimentally measured conductivity profiles of experiments MT03−MT14. The volumetric mass transfer coefficient kMT in eq 21 was estimated from the fit to be 4.49 × 10−5 m3 min−1. Figure 5 demonstrates the high quality of fit that we obtain from fitting one parameter to the 12 experiments. Full simulation results that include the profiles of the downstream gas compositions and flow rates are provided in Section S7 of the Supporting Information. 4.2. Solubility of Hydromagnesite. The solubility of hydromagnesite, expressed as the total Mg concentration, cMg, that is in equilibrium with the solid particles, at different f CO2 is shown in Figure 6. The experimental values for cMg were calculated from the experimentally measured conductivity values at equilibrium. The solubility product for hydromagnesite expressed as log Ksp fitted quite accurately to the four experimental data points equals −40.59. This is in very good agreement with the value of −40.47 that is calculated at 90 °C by assuming a constant reaction enthalpy from the 25 °C data reported by Drever;17 this is the same assumption made by Wang and Li in their study.8 Figure 6 also shows the solubility curve for crystalline natural hydromagnesite at 90 °C (log Ksp = −44.0515). Xiong7 showed that synthetic hydromagnesite precipitated in laboratory environments are more amorphous and more soluble than crystalline natural hydromagnesite. Our
On the basis of eqs 17−19 and eq 2, the parameter m is lumped in with the growth rate parameters in the growth rate expressions. Therefore, the choice of m either equal to 5 or to 15 would not affect the quality of fitting of the experimental profiles. In this study, we assume that m = 15. The overall precipitation rate, Rppt, can then be defined as R ppt =
3k vρc G M
∫0
∞
fL2 dL
(20) −1
where kv = π/6 is the shape factor and M = 467.64 g mol is the molar mass of hydromagnesite. 3.4. Liquid-Film. On the basis of the film theory for mass transfer,14 the kinetics of CO2 sorption can be written as ⎛ f K1 ⎞ CO2 FMT = kMTρs ⎜⎜ − cCO2(aq)⎟⎟ ⎝ γCO2(aq) ⎠ * (aq) − cCO (aq)) = kMTρs (cCO 2 2
(21)
where kMT is the overall volumetric mass transfer coefficient, ρs is the density of the solution, f CO2 is the bulk gas phase fugacity of CO2, K1 is the equilibrium constant of the reaction CO2(g) ⇌ CO2(aq), γCO2(aq) is the activity coefficient of CO2(aq) computed from the equations provided in Section S6 of the Supporting Information, and cCO2(aq) is the concentration of CO2(aq) in the bulk liquid.
4. RESULTS AND DISCUSSION All experiments in this entire study were performed at T = 90 °C under atmospheric pressure conditions (P = 0.95 bar). The volume of the reactor excluding the probes, stirrer, etc. was measured to be 135.3 mL. For all calculations, the density of the solution assumed to be a constant and equal to that of water at 90 °C is ρs = 965.4 kg m−3. The fugacity of CO2, f CO2, the mole fraction of water, yH2O, and molar volume of the gas, 322
DOI: 10.1021/acs.cgd.6b01546 Cryst. Growth Des. 2017, 17, 317−327
Crystal Growth & Design
Article
and obtained values of p between = 0.85 and 1.1. The supersaturations investigated in their study are similar to those in our study. The difference in the estimated value of p between the two studies could be because they measured the growth rates of crystalline hydromagnesite, while the precipitate that we have formed over our shorter experimental times is partly amorphous. Berninger et al.11 assumed the transport limited growth rate model to fit the desupersaturation profiles for the growth of natural hydromagnesite at low supersaturations. However, a visual inspection of their fitted profiles suggests that the quality of fitting could have been significantly improved if they had not constrained their model to use p = 1. Nevertheless, it is also likely that the difference between the two studies could once again be the result of the nature of hydromagnesite precipitated from the solution. Figure 7 shows also the simulated and experimental downstream gas composition profiles. The simulations show a small enrichment of the downstream gas in CO2 during the precipitation process. It should be noted that the downstream gas composition profiles were not fitted and that the small mismatch between the experimental and the simulated profiles is within the uncertainty of the gas A composition (±2% relative error, see section 2.1) and the calibration error for the MS. The simulations are also able to successfully describe the final PSD of the grown particles. Among all the experiments performed in this study, only the final PSD from experiment HY05 shows the presence of a significant amount of fines (
