# Modeling of Limestone Dissolution for Flue Gas Desulfurization with Novel Implications

^{1}

^{2}

^{*}

## Abstract

**:**

_{2}released by fuel combustion. Limestone dissolution plays a major role in the process. Nevertheless, there is a need for improvements in the optimization of the WFGD process for scale-up purposes. The mathematical model has been tested by comparison with experimental data from several mild acidic dissolution assays of sedimentary and metamorphic limestone. We have found that R

^{2}⊂ 0.92 ± 0.06 from dozens of experiments. This fact verifies the model qualifications in capturing the main drivers of the system.

## 1. Introduction

_{2}are released by the process. For instance, some super-high sulfur contents can be found in particular samples from China [10]. SO

_{2}and any forms of SO

_{x}should be removed carefully from the produced flue gas, since SO

_{2}can react with the atmospheric moisture and produce sulfuric acid, which is precipitated then under the form of acid rains. This damages all living species [17].

_{2}in the form of gypsum [18] (Table 1). This technology is not new, and very little changes have been implemented for this process in recent years.

## 2. Materials and Methods

#### 2.1. Experimental Procedure

#### 2.2. Mathematical Modeling

^{+}consumption rate; thus, the equation above can be re-written as:

## 3. Results

#### 3.1. Limestone Analysis

#### 3.2. Reactivity Estimation of Limestone

_{3}content. The rotating electrode experiments of Lund et al. (1975) [5] on calcite dissolution in hydrochloric acid solutions suggest that at low pH, the reaction can be considered irreversible, and the absorption of hydron over the solid calcite surface and subsequent reaction of the adsorbed hydron with the solid calcite matrix is the main reaction mechanism during the acidic dissolution of calcium carbonate. These facts are later confirmed by Shiraki et al. (2000) using Atomic Force Microscopy [40]. Therefore, from the model expressed in Equation (25), which is derived from mass balance considerations only, the parameter k’ will necessarily depend upon the activity of the adsorbed hydrogen ion on the calcite particle surface [41] ${\theta}_{H+}{C}_{{H}^{+}}^{0}$ as follows:

_{A}is the impeller diameter (0.03 m), N is the agitation rate (2300 rpm), and µ is the dynamic viscosity, which can be considered tend to have the same order of magnitude as water, since the solid loading is low and the shear rate is high [44]. The high value of the Reynolds number indicates a context of a fully developed turbulent flow, which points to the convective mass transfer as the dominant mechanism of calcite dissolution. This fact is confirmed by the value of $D{a}_{I}$, which in this context is several orders of magnitude lower than $D{a}_{II}$. When the solid motion is rapid enough, as in this case, $D{a}_{I}$ << $D{a}_{II}$, and it turns out that ${\theta}_{H+}$ is proportional to the ${C}_{{H}^{+}}$ of the bulk. Therefore, by averaging the characteristic times of each involved process, k

_{p}stands as [2,45]:

**Table 3.**Relevant physical parameters for the description of calcium carbonate particles dissolution in turbulent media.

Quantity | Symbol | Units | Min Value | Median Value | Max Value | Reference |
---|---|---|---|---|---|---|

Reynolds number (agitated tank) | Re | none | 42,000 | This Work | ||

Convective mass transfer coefficient | ${k}_{L,A}$ | 10^{−3} dm/s | 0.01 | 2 | 12 | [2,43] |

Limestone particle size | d_{p} | 10^{−6} m | 150 | 320 | 500 | This Work |

Solid calcite concentration | ${C}_{S}$ | g/L | 2 | 3 | 5 | This Work |

Calcite absolute contact area | A | dm^{2} | 3 | 4 | 5 | This Work |

Conductivity (local) | σ | mS/cm^{2} | 0.4 | 100 | 10^{5} | [46] |

Hydron concentration | ${C}_{H+}$ | mol/L | 10^{−6} | 6 × 10^{−6} | 2 × 10^{−3} | This Work |

Hydron surface activity coefficient | ${\theta}_{H+}$ | none | 0 | 0.2 | 1 | [2,47,48] |

Hydron ion diffusivity in water | ${D}_{H+}$ | 10^{−5} cm^{2} s^{−1} | 9.3 | [2,46] | ||

Hydroxide anion diffusivity in water | ${D}_{OH-}$ | 10^{−5} cm^{2} s^{−1} | 5.273 | [46] | ||

Hydrogen Carbonate ion diffusivity in water | ${D}_{HC{O}_{3}-}$ | 10^{−5} cm^{2} s^{−1} | 1.185 | [46] | ||

Carbonate ion diffusivity in water | ${D}_{\frac{1}{2}C{O}_{3}2-}$ | 10^{−5} cm^{2} s^{−1} | 0.923 | [46] | ||

Calcium ion diffusivity in water | ${D}_{\frac{1}{2}Ca2+}$ | 10^{−5} cm^{2} s^{−1} | 0.79 | 0.792 | 0.84 | [2,46] |

Magnesium ion diffusivity in water | ${D}_{\frac{1}{2}Mg2+}$ | 10^{−5} cm^{2} s^{−1} | 0.706 | [46] | ||

Boundary layer thickness (mass transfer) | δ | 10^{−5} m | 1 | 2.4 | 5 | [5,43] |

Diffusive mass transfer coefficient | ${k}_{L,D}$ | 10^{−3} dm/s | 0.14 | 2.8 | 9.3 | This Work [2] |

Effective reactivity parameter | ${k}_{r}{}_{,eff}$ | M^{−2} dm^{−2} s^{−1} | 10^{5} | 10^{6} | 10^{7} | This work [2,41] |

First Damköhler number | $D{a}_{I}$ | none | 10^{4} | 10^{6} | 10^{9} | This work |

Second Damköhler number | $D{a}_{II}$ | none | 30 | 300 | 3000 | This work |

_{r,eff}.

_{r,eff}of each run varies even within the same experimental conditions of agitation, which can be explained by the variability in the local conductivity of the dissolving media. The surface activity of the hydrogen ion ${\theta}_{H+}$ is closely related to the ionic mobility of the nearby media [34,47]. Carletti et al. (2016) [9] report the non-uniformity of conductivity for the very same system, which is measured by Electrical Resistance Tomography. Then, the fluctuations of local conductivity at the element of fluid level play a major role explaining the errors of the linear fitting observed in Figure 7 and Figure 8. Recent studies over calcite dissolution rates indicate that its active surface is not homogeneous but rather intricately distributed over the exposed layer [49]. A stochastic pattern of superficial detachment is expected during the process that is complex enough to ascertain that the reactivity of the particles is not constant; nevertheless, in practice, the observed reactivity is a robust statistical estimator due to numerical compensation, as shown in Table 4. The actual reactive portion of the outer layer rounds is reported as 20% of the grain surface [48], which depends upon the composition of the bulk (especially the conductivity of the medium), the instantaneous size distribution of the particles, the interfacial tension, and the agitation conditions [2,33,37].

## 4. Discussion

^{2}, which in none of the dozens of experiments we carried out is observed lower than 0.85. Actually, for these samples, the average determination coefficient is 0.92 +/− 0.06. As a matter of fact, the fitting of this model with experimental data for the case of limestone dissolution is rather complicated, considering the complexity of the interfacial mass transfer in turbulent media, and it is a great step forward toward understanding the limestone dissolution in detail, which is the key for the optimization, scaling up, and de-risking of flue gas desulfurization.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Latin Symbols | Significance | Units |

${a}^{\prime}$ | Polynomial multiplicative factor | dimensionless |

A | Absolute contact area | dm^{2} |

${A}_{p}$ | Particles’ superficial area | dm^{2} |

${b}^{\prime}$ | Polynomial multiplicative factor | dimensionless |

$c$ | Substitute variable $c={C}_{{H}^{+}}^{0}$ | dimensionless |

${c}^{\prime}$ | Polynomial multiplicative factor | dimensionless |

${C}_{{H}^{+}}$ | Hydrogen ion concentration | M |

${C}_{{H}^{+}}^{0}$ | Initial Hydrogen ion concentration | M |

${C}_{s}$ | Undissolved limestone concentration | g/L (or M) |

$d$ | Shape factor | dimensionless |

d_{p} | Limestone particle size | µm |

D | Average ionic diffusivity | cm^{2} s^{−1} |

${D}_{H+}$ | Hydrogen ion diffusivity in water | cm^{2} s^{−1} |

${D}_{OH-}$ | Hydroxide anion diffusivity in water | cm^{2} s^{−1} |

${D}_{HC{O}_{3}-}$ | Hydrogen carbonate ion diffusivity in water | cm^{2} s^{−1} |

${D}_{\frac{1}{2}C{O}_{3}2-}$ | Carbonate ion diffusivity in water | cm^{2} s^{−1} |

${D}_{\frac{1}{2}Ca2+}$ | Calcium ion diffusivity in water | cm^{2} s^{−1} |

${D}_{\frac{1}{2}Mg2+}$ | Magnesium ion diffusivity in water | cm^{2} s^{−1} |

$D{a}_{I}$ | First Damköhler number | dimensionless |

$D{a}_{II}$ | Second Damköhler number | dimensionless |

$k$ | Reaction rate constant | M^{−2} dm^{−2} s^{−1} |

${k}^{\prime}$ | Model rate parameter | M^{−1} dm^{−2} s^{−1} |

${k}_{L}$ | Interfacial mass transfer coefficient | dm/s |

${k}_{L,A}$ | Convective mass transfer coefficient | dm/s |

${k}_{L,D}$ | Diffusive mass transfer coefficient | dm/s |

${k}_{p}$ | Lumped rate parameter | M^{−1} dm^{−2} s^{−1} |

${k}_{r}$ | Reactivity parameter | M^{−2} dm^{−2} s^{−1} |

${k}_{r}{}_{,eff}$ | Effective reactivity parameter | M^{−2} dm^{−2} s^{−1} |

$N$ | Agitation rate | rpm |

$P$ | Substitute variable $\gamma ={P}^{3}$ | dimensionless |

${R}_{o}$ | Characteristic length of the particle | dm |

Re | Reynolds number (agitated tank) | dimensionless |

${V}_{p}$ | Particle volume | dm^{3} |

$x$ | Substitute variable $x={C}_{{H}^{+}}$ | dimensionless |

$X$ | Conversion | dimensionless |

$z$ | Substitute parameter $z=\left(1-\frac{1}{d}\right)$ | dimensionless |

Greek symbols | Significance | Units |

${\varnothing}_{\mathrm{A}}$ | Agitator impeller diameter | m |

$\gamma $ | Substitute variable $\gamma =1-\frac{x}{c}$ | dimensionless |

δ | Mass transfer boundary layer thickness | m |

${\theta}_{H+}$ | Hydrogen surface activity coefficient | dimensionless |

$\mu $ | Dynamic viscosity | Pa.s |

$\rho $ | Mass density | Kg/m^{3} |

σ | Local conductivity | mS/cm^{2} |

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**Figure 1.**Experimental set-up for limestone dissolutions assays. In the figure, the Malvern Mastersizer 3000 is shown on the left side with its Hydro Unit. Temperature and pH are measured by a pH meter, and the sensor is connected to a pH-controlling unit (center). For experiments where the pH is required to be constant, a peristaltic pump with a scale (right side) can be used.

**Figure 2.**SEM images for sedimentary limestone (

**a**) and metamorphic limestone (

**b**). Åbo Akademi University.

**Figure 5.**Hydrogen ion concentration (

**a**) and absolute contact area (

**b**) time series for the metamorphic limestone sample registered for each step.

**Figure 6.**Hydrogen ion concentration (

**a**) and absolute contact area (

**b**) time series for the sedimentary limestone sample registered for each step. The sampling time period of pH (therefore ${C}_{{H}^{+}}$) was 5 s and the area measurement frequency was every 30 s.

**Figure 7.**Correlation between the left-hand side of Equation (25) and t·${C}_{{H}^{+}}$ for the metamorphic (

**a**) and sedimentary (

**b**) limestone samples. The absolute value of the regression slope is reported in the figure; it can be considered as the mean area multiplied by an estimator of the effective reactivity.

**Figure 8.**Correlation between the left-hand side of Equation (25) and t·$({C}_{{H}^{+}})$ A for the metamorphic (

**a**) and sedimentary (

**b**) limestone samples. The absolute value of the regression slope is reported in figure. This represents an estimator of the effective reactivity.

**Table 1.**Main steps and reactions in Wet Flue Gas Desulfurization (WFGD) [11].

Rate Determining Steps | Reactions |
---|---|

Absorption of gaseous SO_{2} in liquid water | ${\mathrm{SO}}_{2}+{\mathrm{H}}_{2}\mathrm{O}\rightleftarrows {\mathrm{H}}^{+}+{\mathrm{HSO}}_{3}^{-}$ ${\mathrm{HSO}}_{3}^{-}\rightleftarrows {\text{}\mathrm{H}}^{+}+{\mathrm{SO}}_{3}^{2-}$ |

Oxidation of ${\mathrm{HSO}}_{3}^{-}$ (liquid phase) | ${\mathrm{HSO}}_{3}^{-}+\frac{1}{2}{\mathrm{O}}_{2}\rightleftarrows {\mathrm{H}}^{+}+{\mathrm{SO}}_{4}^{2-}$ |

Solid limestone is dissolving in acidic environment (pH 5.5, industrial process) | ${\mathrm{HSO}}_{4}^{2-}\rightleftarrows {\mathrm{SO}}_{4}^{2-}+{\mathrm{H}}^{+}$ ${\mathrm{CaCO}}_{3}\rightleftarrows {\mathrm{Ca}}^{2+}+{\mathrm{CO}}_{3}^{2-}$ ${\mathrm{CO}}_{2}+{\mathrm{H}}_{2}\mathrm{O}\rightleftarrows {\mathrm{HCO}}_{3}^{-}+{\mathrm{H}}^{+}$ ${\mathrm{HCO}}_{3}^{-}\rightleftarrows {\mathrm{CO}}_{3}^{2-}+{\mathrm{H}}^{+}$ ${\mathrm{H}}_{2}\mathrm{O}\rightleftarrows {\mathrm{H}}^{+}+{\mathrm{OH}}^{-}$ |

Crystallization of gypsum | ${\mathrm{Ca}}^{2+}+{\mathrm{SO}}_{4}^{2-}+2{\mathrm{H}}_{2}\mathrm{O}\rightleftarrows {\mathrm{CaSO}}_{4}\xb72{\mathrm{H}}_{2}\mathrm{O}$ |

**Table 2.**Sample type, density, CaCO

_{3}content, and composition (wt %) given by X-Ray Fluorescence (XRF).

Sample | ρ (kg/m^{3}) | CaCO_{3} wt % | CaO wt % | Al_{2}O_{3} wt % | SiO_{2} wt % | MgO wt % |
---|---|---|---|---|---|---|

Metamorphic Limestone | 2720 | 98.5 | 54.5 | 0.13 | 0.5 | 0.59 |

Sedimentary Limestone | 2703 | 99.1 | 55.2 | 0.01 | 0.05 | 0.32 |

Experiment | k_{r,eff} A(10 ^{7} M^{−2} s^{−1}) | Goodness of Fit (r^{2}) | Averaged Contact Area (dm^{2}) | k_{r,eff}(10 ^{7} M^{−2} dm^{−2} s^{−1}) | Goodness of Fit (r^{2}) |
---|---|---|---|---|---|

Metamorphic_step1 | 0.159 ± 0.008 | 0.9329 | 3.76 ± 0.33 | 0.038 ± 0.001 | 0.9634 |

Metamorphic_step2 | 0.129 ± 0.009 | 0.8835 | 3.78 ± 0.31 | 0.031 ± 0.002 | 0.9216 |

Metamorphic_step3 | 0.112 ± 0.008 | 0.8683 | 3.72 ± 0.38 | 0.027 ± 0.002 | 0.9126 |

Metamorphic_step4 | 0.132 ± 0.010 | 0.8719 | 3.80 ± 0.28 | 0.032 ± 0.002 | 0.9085 |

Metamorphic_step5 | 0.109 ± 0.008 | 0.8512 | 3.74 ± 0.36 | 0.026 ± 0.002 | 0.8935 |

Sedimentary_step1 | 0.636 ± 0.034 | 0.9020 | 3.45 ± 0.70 | 0.137 ± 0.005 | 0.9589 |

Sedimentary_step2 | 1.018 ± 0.057 | 0.9124 | 3.74 ± 0.35 | 0.223 ± 0.008 | 0.9619 |

Sedimentary_step3 | 0.409 ± 0.032 | 0.8527 | 3.78 ± 0.30 | 0.097 ± 0.006 | 0.9109 |

Sedimentary_step4 | 0.327 ± 0.030 | 0.8035 | 3.75 ± 0.34 | 0.079 ± 0.006 | 0.8730 |

Sedimentary_step5 | 0.428 ± 0.035 | 0.8329 | 3.74 ± 0.36 | 0.101 ± 0.006 | 0.9062 |

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**MDPI and ACS Style**

De Blasio, C.; Salierno, G.; Sinatra, D.; Cassanello, M.
Modeling of Limestone Dissolution for Flue Gas Desulfurization with Novel Implications. *Energies* **2020**, *13*, 6164.
https://doi.org/10.3390/en13236164

**AMA Style**

De Blasio C, Salierno G, Sinatra D, Cassanello M.
Modeling of Limestone Dissolution for Flue Gas Desulfurization with Novel Implications. *Energies*. 2020; 13(23):6164.
https://doi.org/10.3390/en13236164

**Chicago/Turabian Style**

De Blasio, Cataldo, Gabriel Salierno, Donatella Sinatra, and Miryan Cassanello.
2020. "Modeling of Limestone Dissolution for Flue Gas Desulfurization with Novel Implications" *Energies* 13, no. 23: 6164.
https://doi.org/10.3390/en13236164