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Estimation of nucleation and growth rate of sodium bicarbonate crystals in a steady-state bubble column reactor

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Abstract

In the sodium bicarbonate bubble column, carbon dioxide gas CO2, as a gaseous mixture of CO2 and air, is continuously injected into sodium carbonate Na2CO3 and bicarbonate NaHCO3 brine. In this work, a molar balance has been produced for flows and components, on the basis of Danckwerts theory for mass transfer between gas and liquid phases, and a population balance has been derived to obtain the nucleation and growth formula. For validation the model results were compared with experimental data. The effects of several conditions on the sodium bicarbonate crystal nucleation and growth rate were investigated. Nucleation and growth rate change under different operating conditions, and, especially, a formula derived at one temperature cannot be used at a different temperature. Production of sodium bicarbonate crystals is reduced by increasing the temperature of the liquid, reducing the gas pressure, or reducing the mole fraction of CO2 in the gas phase.

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Abbreviations

a :

Growth rate coefficient

b :

Supersaturation order in the growth formula

B 0 :

Nucleation (#/s kg solution)

\( C_{{{\text{CO}}_{2} e}} \) :

Carbon dioxide concentration in the liquid bulk (mol/m3)

c :

Nucleation rate coefficient

\( C_{{{\text{CO}}_{2} {\text{i}}}} \) :

Carbon dioxide concentration at the gas–liquid interface (mol/m3)

d :

Magma density order in the nucleation formula

d B :

Average diameter of bubbles (m) \( d_{\text{B}} = 26 \times d_{\text{R}} \cdot \left( {\frac{{g \cdot d_{\text{R}}^{2} \cdot \rho_{\text{l}} }}{\delta }} \right)^{ - 0.5} \cdot \left( {\frac{{g \cdot D_{\text{l}}^{3} }}{{\upsilon_{\text{l}}^{2} }}} \right)^{ - 0.12} \cdot \left( {\frac{{U_{\text{g}} }}{{\sqrt {g \cdot d_{\text{R}} } }}} \right)^{ - 0.12} \)

\( D_{{{\text{CO}}_{2} }} \) :

Carbon dioxide molecular diffusion coefficient (m2/s) \( D_{{{\text{CO}}_{2} }} = 5.35 \times 10^{ - 0.12} \times \frac{T}{{\left( {{\raise0.7ex\hbox{${\mu_{\text{l}} }$} \!\mathord{\left/ {\vphantom {{\mu_{\text{l}} } {10^{ - 3} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${10^{ - 3} }$}}} \right)^{1.035} }} \)

D g :

Gas phase radial dispersion coefficient (m2/s) \( D_{\text{g}} = 5 \times d_{\text{R}} \times \frac{{U_{\text{g}} }}{{\varepsilon_{\text{g}} }} \)

D l :

Liquid phase radial dispersion coefficient (m2/s) \( D_{\text{l}} = 2.7 \times d_{\text{R}}^{1.4} \cdot (0.219 \times U_{\text{g}}^{0.77} + 0.122 \times U_{\text{g}}^{0.1} ) \)

d R :

Column diameter (m)

E :

Enhancement factor \( E = \sqrt {1 + \frac{{D_{{{\text{CO}}_{2} }} \cdot k}}{{K_{\text{l}}^{2} }}} \)

e :

Supersaturation order in the nucleation formula

g :

Gravity (m/s2)

G :

Molar velocity of gas (mol/m2 s)

G 0 :

Growth rate (μm/s)

G FR :

Gas flow rate (m3/s)

h :

Column height (m)

H :

Henry constant (kmol/atm m3) \( \log \left( {{\raise0.7ex\hbox{$H$} \!\mathord{\left/ {\vphantom {H {H_{\text{w}} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${H_{\text{w}} }$}}} \right) = - K_{\text{s}} \cdot I;\,\log \left( {H_{\text{w}} } \right) = \frac{1140}{T} - 5.3,\;K_{\text{s}} = 0.06,\;I = 6.2 \)

I 0 :

Height of column at which sodium bicarbonate reaches saturation concentration

k :

Constant rate of first-order reaction (1/s) \( k = \left[ {2.2 \times 10^{7} \times \exp \left( { - \frac{71500}{R \cdot T}} \right)} \right] - \left[ {3.2 \times 10^{ - 6} \times \rho_{{{\text{H}}_{2} {\text{O}}}} \cdot \exp \left( { - \frac{13800}{R \cdot T}} \right)} \right] \)

K l :

Liquid phase mass-transfer coefficient (m/s) \( K_{\text{l}} = 0.6 \times \frac{{D_{{{\text{CO}}_{2} }} }}{{\alpha_{\text{g}} d_{\text{R}} }} \cdot \left( {\frac{{\mu_{\text{l}} }}{{D_{\text{l}} .\rho_{\text{l}} }}} \right)^{0.5} \cdot \left( {\frac{{g.\rho_{\text{l}} }}{\delta }} \right)^{0.62} \cdot \left( {\frac{g}{{\left({\mu_{\text{l}} }/{\rho_{\text{l}} } \right)^{2} }}} \right)^{0.31} \cdot \varepsilon_{\text{g}}^{1.1} \)

L :

Molar velocity of liquid (mol/m2 s)

L FR :

Liquid mass flow rate (kg/s)

L nu :

Size of crystals (μm)

M T :

Magma density (g crystal/kg solution)

N :

Flux of mass transfer (mol/m2 s)

n :

Population density (no./μm kg solution)

n Dis :

Flux of mass transfer by dispersion (mol/m2 s)

nu:

Number of nucleons created in dz \( {\text{nu}} = \frac{{B_{0} \cdot \rho_{\text{l}} \cdot \pi \cdot d_{\text{R}}^{2} \cdot ({\text{d}}z)}}{4} \)

P 0 :

Gas pressure at bottom of column (atm)

\( P_{{{\text{CO}}_{2} }} \) :

Carbon dioxide partial pressure in the gas phase (atm) \( P_{{{\text{CO}}_{2} }} (z) = P_{0} - \rho_{\text{l}} \cdot \left( {1 - \varepsilon_{g} } \right) \cdot z \)

pH:

Acidity of the liquid phase \( {\text{pH}} = 10.33 + \log \left( {\frac{{x_{{{\text{Na}}_{2} {\text{CO}}_{3} }} }}{{x_{{{\text{NaHCO}}_{3} }} }}} \right) \)

Q :

Liquid flow rate (m3/s)

r :

Rise of size of nucleons from dz to the bottom of the column (m3) \( r = \frac{\pi }{6}\left[ {\frac{{G_{0} \times 10^{ - 6} }}{{U_{\text{l}} }} \cdot (h - z)} \right]^{3} \)

R :

Gas constant (j/g mol K)

S :

Molar velocity of solid (mol/m2 s)

T :

Liquid temperature (K)

t :

Time

U g :

Gas phase velocity (m/s) \( U_{\text{g}} = \frac{{4 \times G_{\text{FR}} }}{{\pi \cdot d_{\text{R}}^{2} \cdot \varepsilon_{\text{g}} }} \)

U l :

Liquid phase velocity (m/s) \( U_{\text{l}} = \frac{{4 \times L_{\text{FR}} }}{{\pi \cdot d_{\text{R}}^{2} \cdot \varepsilon_{\text{l}} \cdot \rho_{\text{l}} }} \)

V :

Volume of element (m3)

w :

Weight fraction of component in liquid phase

Δw :

Supersaturation (g NaHCO3/kg solution)

x :

Mole fraction of components in liquid phase

\( x_{{{\text{NaHCO}}_{3} }}^{*} \) :

Mole fraction of sodium bicarbonate at supersaturation \( \log \left( {x^{*} } \right) = 6.71535 - \frac{843.0681}{T} - 2.24336 \times \log (T) \)

y :

Mole fraction of components in gas phase

z :

Height of column (m)

α g :

Gas–liquid interface (m2/m3) \( \alpha_{\text{g}} = \frac{{6 \times \varepsilon_{\text{g}} }}{{d_{\text{B}} }} \)

α s :

Solid–liquid interface (m2/m3)

δ :

Liquid surface tension (N/m)

\( \varepsilon_{\text{g}} \) :

Gas holdup \( \frac{{\varepsilon_{\text{g}} }}{{(1 - \varepsilon_{\text{g}} )^{4} }} = 0.2 \times \left( {\frac{{9.8 \times d_{\text{R}} \cdot \rho_{\text{l}} }}{\delta }} \right)^{\frac{1}{8}} \cdot \left( {\frac{{g \cdot d_{\text{R}}^{2} \cdot \rho_{\text{l}}^{2} }}{{\mu_{\text{l}}^{2} }}} \right)^{\frac{1}{12}} \cdot \left( {\frac{{U_{\text{g}} }}{{\sqrt {d_{\text{R}} \cdot g} }}} \right) \)

\( \varepsilon_{\text{l}} \) :

Liquid holdup

μ j :

jth moment of population density \( \mu_{j} = \int_{0}^{\infty } {L_{\text{nu}}^{j} \cdot n \cdot {\text{d}}L_{\text{nu}} } \)

μ l :

Liquid dynamic viscosity (N s/m2)

\( \rho_{{{\text{H}}_{2} {\text{O}}}} \) :

Water density (kg/m3)

ρ l :

Liquid density (kg/m3)

\( \upsilon_{\text{l}} \) :

Liquid kinematic viscosity (m2/s)

i:

At inlet point position of element

I0 :

At I 0 point position

in:

At inlet point position of column

O:

At outlet point position of element

References

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Authors and Affiliations

  1. Department of Environment, College of Natural Resources, Islamic Azad University, Bandar Abbas, Iran

    Rouhollah Soltani Goharrizi

  2. Department of Chemical Engineering, Faculty of Engineering, Shahid Bahonar University of Kerman, Jomhoori Blvd., Post Code 76175, Kerman, Iran

    Bahador Abolpour

Authors
  1. Rouhollah Soltani Goharrizi
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  2. Bahador Abolpour
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Correspondence to Bahador Abolpour.

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Goharrizi, R.S., Abolpour, B. Estimation of nucleation and growth rate of sodium bicarbonate crystals in a steady-state bubble column reactor. Res Chem Intermed 41, 1459–1471 (2015). https://doi.org/10.1007/s11164-013-1285-y

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  • DOI: https://doi.org/10.1007/s11164-013-1285-y

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