https://doi.org/10.1016/j.cej.2022.138999
“The strategy adopted to simulate the experimental results relies on the classical homogeneous model for unstirred bubble column tank reactors operated continuously for the gas phase and in batch mode for the liquid one. Isothermal and isobaric conditions are assumed, consistently with the experimental procedure. Due to the concentration gradient (driving force), carbon dioxide molecules diffuse from the gas bubbles to the liquid phase. In order to model this phenomenon, the two films theory is adopted. Rigorously, the reactive absorption process should be conceived by considering the film and the bulk sections for both phases along with the corresponding material balance equations for each of them. Therefore, the dynamic variation of the species concentrations results in a system of partial differential equations that takes into account the interface flow rate and the reactive phenomena taking place in the liquid phase. This model should be then coupled with the relevant boundary and initial conditions, related to bulk concentrations, the interface equilibrium and the molar transfer rate taking place at the gas/liquid interface. However, the unavailability of experimental data related to the time evolution of species concentration along the film, would make such a rigorous model uselessly complicated. For this reason, in this work, the CO2 molar flow rate (NCO2), through the gas/liquid interface, is modelled by considering the global mass transfer coefficient, KLCO2 and the enhancement factor, E, that describes the influence of a chemical reaction on the mass transfer rate:(1)NCO2=SEKLCO2CCO2∗-CCO2.
where (CCO2) is the carbon dioxide concentration into the liquid bulk and (CCO2∗) represents its equilibrium concentration while the surface where mass transfer occurs is indicated by S.
To accurately describe the capture process, ammonia losses should also be considered and modelled via eq. (2):(2)NNH3=SKLNH3CNH3∗-CNH3.
3.1.1. Henry constants
By relying on the Henry’s law [25], the equilibrium concentration (Ci∗) of the ith species can be evaluated through eq. (3) [26].(3)Ci∗=PyiHij,
where P is the total pressure of the capture system, yi is the molar fraction of the ith species within the gas phase and Hij is the Henry constant for the ith species in the jth solvent.
The Henry constant of a specific gas is strictly dependent on solvent properties and system temperature. Eqs. (4), (5) show the correlations adopted to evaluate the Henry constant of CO2 (HCO2H2O) and NH3(HNH3H2O) in water, respectively as a function of temperature [27]:(4)HCO2H2O=13.4104exp-23501T-1298.15(5)HNH3H2O=10.59exp-42001T-1298.15.
The “N2O analogy”, proposed by [28] and expressed by eq. (6), is used to estimate the CO2 solubility in water/ammonia solution, HCO2w/a [29]:(6)HCO2w/a=HN2Ow/aHCO2H2OHN2OH2O,
Where the N2O Henry constant in water solution (HN2OH2O) can be expressed by eq. (7) [12]:(7)HN2OH2O=8.55·106exp-2284T,
and eq. (8) can be used to correlate HN2Ow/a with ammonia concentration [30]:(8)HN2Ow/a=0.155+8.17·10-3NH3·106exp-1.14·103T.
3.1.2. Enhancement factor
The enhancement factor (E) is defined as the ratio between the absorption rate of a gas component into a liquid phase in the presence of one or more reactions involving the absorbed gas species and the absorption rate when reactions are not taking place. However, it is well known [31] that no general analytical expressions to calculate this value are available in the literature since it strictly depends on the reaction kinetics, the physiochemical properties of the species and the reaction regime. In this work, the approach proposed in the literature [31] is taken into account where the Hatta number, Ha (eq. (9)), and the enhancement factor infinite, Einf(eq. (10)), are obtained to firstly determine the reactive regime occurring:(9)Ha=kappDCO2KLCO2(10)Einf=1+DNH3NH3νNH3DCO2PyCO2HCO2w/a,
where kapp is the apparent kinetic constant of the reactive phenomena that determine the CO2 consumption into the liquid phase [12], [32], [33], Di is the diffusion coefficient of the ith species and νNH3 is the ammonia stoichiometric coefficient in the reaction involving CO2 consumption.
In this work, based on the experimental conditions, two different regimes are possible, i.e. the pseudo first order one that occurs when ammonia is in excess compared to CO2 and the fast intermediate regime that takes place when CO2 accumulates in the liquid phase [31].
If the pseudo first order condition is satisfied (3<Ha≪Einf) [12], [32], [33], the enhancement factor can be computed through eq. (11):(11)E=Hatanh(Ha)
As the process evolves, the absorption process falls into the fast intermediate regime, during which the enhancement factor can be approximate through the eq. (12) [34], [35], proposed by [36] for the first time:(12)E=-Ha22(Einf-1)+Ha44(Einf-1)2+EinfHa2(Einf-1)+1.
3.1.3. Mass transfer coefficient
The global mass transfer coefficient, KL, as expressed by eq. (13), depends on both local transfer coefficients, i.e. kl and kg, that contribute to the global mass transfer resistance [37]:(13)1EKL=1Ekl+1Hijkg.
Nevertheless, in the mass transfer model of bubble columns, the gas phase resistance is commonly assumed to be negligible (Ekl<< kgHij) [38], [39], [40], since the material transfer resistance is assumed to be concentrated in the homogeneous liquid-phase only (KL → kl).
The local transfer coefficients expression is strictly dependent on the system configuration and geometry. By relying on experimental observations, some correlations for mass transfer coefficients have been developed for standard cases (e.g., fluid flow through a packed bed of particles, gas bubbles rising in a tank, falling films, flow over surfaces and within tubes). In this work, based on the configuration of the experimental setup, the correlation referring to rising gas bubbles reported in eq. (14) is used [26], [41], [42]:(14)kl=2DDb+0.31Nsc-23Δϱgμlϱl213
where D is the gas diffusion coefficient, Nsc is the Schmidt number, ϱl and μl represent the liquid phase density and dynamic viscosity, respectively, the symbol Δϱ accounts for the density difference between liquid and gas phase, g is the gravity acceleration and Db is the bubble diameter.
Within the solution, the bubble size distribution is not uniform, due to coalescence and break-up phenomena that are strongly influenced by the physical properties of the solution and the system fluid dynamics. As a consequence, the Sauter mean diameter (d32) is considered to represent the average size of the bubbles [43]. By referring to the specific experimental condition such as type of sparger and column geometry, eq. (15), proposed by [44] to evaluate d32, was selected:(15)d32=d0ϱlϱg0.07271σl3ϱlgμl4-0.04322HCDC0.16528UGgDC0.27752d0DC-0.71397
where d0 is the pore diameter of the sparger, σl is the interfacial tension of the liquid, ϱg is the gas density, HC and DC are the height and the diameter of the column whereas UG is the superficial gas velocity, estimated by knowing the gas volumetric flow rate and the column section. The superficial gas velocity (UG) could be then used to assess the theoretical hold-up (ε), through eq. (16), in order to compare the experimental value with the theoretical one:(16)ε=gDC2ϱlσl-1.167gϱl2DC3μl20.3317UGgDC0.3196ϱlϱg1.048HSDC1.4948d0DC-0.862.
The relations used to quantify the physical properties of the system are reported in the supplementary materials. Once the Sauter diameter is assessed, the interfacial surface, S, can be evaluated by multiplying the total bubble volume in the solution obtained from hold-up measurements and the bubble surface to volume ratio.
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