Modeling of Column Apparatus Processes

The fundamental problem of the one-phase processes modeling in the column apparatuses comes from the complicated hydrodynamic behavior of the flow, and as a result, the velocity distribution in the column is unknown.

The concentrations of the transferred substance in the phases are presented as kg mol of the transferred substance in 1 m³ of the phase volume.

The modeling of three-phase (gas–liquid–solid) interphase mass transfer processes in column apparatuses [1‐4] is used in the case of absorption and adsorption in two-component (\(i_{0} = 2\)), three-phase (\(j = 1,2,3\)) systems.

The theoretical procedure (II.5–II.15) presented in the Part II will be used for creation of average-concentration models of simple and complex chemical processes in one-phase column apparatuses. On this basis, the effect of the velocity radial non-uniformity will be analyzed and methods for model parameter identification (Boyadjiev in Int J Heat Mass Transf 49:796–799 [1], Boyadjiev in Trans Acad 3:7–22 [2], Boyadjiev in Theoretical chemical engineering. Modeling and simulation. Springer, Berlin [3]) proposed.

The theoretical procedure (II.5–II.15) presented in Part II will be used for the creation of average-concentration models of absorption, adsorption, and catalytic processes in two-phase systems.

A new approach for the column apparatuses modeling uses convection–diffusion-type models and average-concentration models. All these new types of models (Boyadjiev in Theoretical chemical engineering. Modeling and simulation. Springer, Berlin [1], Doichinova and Boyadjiev in Int J Heat Mass Transf 55:6705–6715 [2], Boyadjiev in J Pure Appl Math: Adv Appl 10:131–150 [3]) are characterized by the presence of small parameters at the highest derivatives. As a result, the model equations have no exact solutions and approximate (asymptotic) solutions have to be obtained (Мищенко and Розов in Дифференциальные уравнения с малым параметром и релаксационные колебания. Изд. “Наука”, Москва [4], O’Malley in Introduction to singular perturbations. Academic Press, New York [5], Boyadjiev et al. in J Eng Thermophys 24:371–380 [6]). In these cases, the use of the conventional software (MATLAB) for solving the model differential equations is difficult and this difficulty may be eliminated by an appropriate combination with the perturbation method.

In the cases of physical absorption [1–4] in a high countercurrent gas–liquid column, the mass transfer process model has to be presented in two-coordinate systems (see 3.​1.​8):

In the cases of a non-stationary chemical adsorption in gas–solid systems, the presence of mobile (gas) and immobile (solid) phases in lengthy processes leads to a non-stationary process in the immobile phase and stationary process in the mobile phase, practically. As a result different coordinate systems have to be used in the gas and the solid phase model.

The new approach for the modeling of the processes in column apparatuses (Boyadjiev in Theoretical chemical engineering. Modeling and simulation. Springer, Berlin, Heidelberg, 2010, [1]; Doichinova, Boyadjiev in Int J Heat Mass Transf 55:6705–6715, 2012, [2]; Boyadjiev in Pure Appl Math Adv Appl 10(2):131–150, 2013 [3]) presents the convection–diffusion and average-concentration models of the column chemical reactors (in Chaps. 2 and 5), where the radial velocity component is equal to zero in the cases of a constant axial velocity radial non-uniformity along the column height:

The new approach for the modeling of the processes in column apparatuses (Boyadjiev in Theoretical chemical engineering. Modeling and simulation. Springer, Berlin, Heidelberg, 2010, [1]; Doichinova, Boyadjiev in Int J Heat Mass Transf 55:6705–6715, 2012, [2]; Boyadjiev in Pure Appl Math Adv Appl 10(2):131–150, 2013 [3]) presents the convection–diffusion and average-concentration models of the column chemical reactors (Boyadjiev and Boyadjiev in Bulgaria Chem Commun 49(3):711–719, 2017 [4], in the cases of an axial modification of the axial velocity radial non-uniformity along the column height (see Chap. 10). This problem will be solved in the cases of the absorption processes in a co-current column (Boyadjiev and Boyadjiev Bulgaria Chem Commun 49(3):711–719, 2017 [5].

In Chap. 11 were presented the convection–diffusion and average-concentration models (Boyadjiev in Theoretical chemical engineering. Modeling and simulation. Springer, Berlin, 2010) [1], (Doichinova and Boyadjiev in Int J Heat Mass Transfer 55:6705–6715, 2012) [2] and (Boyadjiev in J Pure Appl Math: Adv Appl 10(2):131–150, 2013) [3] of the gas absorption processes in the co-current columns, where the radial velocity component is not equal to zero, in the cases of an axial modification of the axial velocity radial non-uniformity along the column height (Boyadjiev in Bulg Chem Commun 49(3):711–719, 2017) [4]. This possibility will be used for modeling of the gas absorption processes in the counter-current columns, where the problem is complicated (Boyadjiev in J Eng Thermophys 24(3):247–258, 2015) [5], (Boyadjiev in Bulg Chem Commun 49(3):720–728, 2017) [6] because the mass transfer process models have to be presented in two-coordinate systems (in a one-coordinate system, one of the equations has no solution due to the negative Laplacian value).

In Chaps. 10–12 were presented convection–diffusion and average-concentration models of chemical (Boyadjiev and Boyadjiev in Bul Chem Commun 49(3):706–710 [1]), co-current absorption (Boyadjiev and Boyadjiev in Bul Chem Commun 49(3):711–719 [2]) and countercurrent absorption (Boyadjiev and Boyadjiev in Bul Chem Commun 49(3):720–728 [3]) processes in industrial column apparatuses, where an axial modification of the axial velocity radial non-uniformity along the column height exists and the radial velocity component is not equal to zero. This problem is solved in the cases of the physical and chemical adsorption processes in the industrial column apparatuses (Boyadjiev and Boyadjiev in J Eng Thermophys 27(1) [4]).

In Chaps. 10–13 were presented convection-type and average-concentration models of chemical (Boyadjiev and Boyadjiev in Bulg Chem Commun 49(3):706–710 [1]), co-current absorption (Boyadjiev and Boyadjiev in Bulg Chem Commun 49(3):711–719 [2]), countercurrent absorption (Boyadjiev and Boyadjiev in Bulg Chem Commun 49(3):720–728 [3]), and non-stationary adsorption (Boyadjiev and Boyadjiev in J Eng Thermophys 27(1):82–97 [4]) processes in industrial column apparatuses, where the radial velocity component is not equal to zero in the cases of an axial modification of the axial velocity radial non-uniformity along the column height. This problem will be solved in the cases of the catalytic reactions in gas–solid systems (physical and chemical adsorption mechanisms) in the industrial column apparatuses (Boyadjiev and Boyadjiev in J Eng Thermophys [5]).

The chemical absorption of average soluble gases (ASG) in the case of slow chemical reaction (e.g., absorption of CO₂ with aqueous solutions of NaOH, where Henry’s number in the system CO₂/H₂O is \(\chi^{{20\;{^\circ{\text{C}}}}} = 1.16\)) is possible to be used for waste gas purification. The absorption process intensification has to be realized through intensification of the convective mass transfer in the gas phase (in gas–liquid drops system) and in the liquid phase (in liquid–gas bubbles system). This theoretical result is applied in a new method and bizonal apparatus for gas absorption [1]. In the upper equipment zone, a physical absorption (as a result of the short reaction time, i.e., short existence of the absorbent drops) is realized in a gas–liquid drops system and the big convective transfer in the gas phase leads to decrease of the mass transfer resistances in this phase. In the lower zone, a chemical absorption in a liquid–gas bubbles system takes place and the big convective transfer in the liquid phase lowers the mass transfer resistances in this phase.

Different companies (Babcock & Wilcox Power Generation Group, Inc., Alstom Power Italy, Idreco-Insigma-Consortium) propose methods and apparatuses for waste gas purification from SO₂ using two-phase absorbent (CaCO₃ suspension). The adsorption (absorption) of SO₂ on materials derived from natural carbonates (c) [3] has the drawback of waste accumulation. The basic problem of the carbonate absorbents is that its chemical reaction with SO₂ lead to CO₂ emission (every molecule of SO₂ absorbed from the air is equivalent to a molecule of CO₂ emitted in the air), because the ecological problems (greenhouse effects) of SO₂ and CO₂ are similar. The large quantity of by-products is a problem, too. Another drawback of these methods is the impossibility for regeneration of the absorbents.

Countercurrent absorbers are used for purification of the large amounts of waste gases emitted from the combustion plants. The gas velocity (and as a result the absorbers diameter, too) is limited by the rate of the absorbent drops fall in an immobile gas medium, i.e., the gas velocity must be less than the drops velocity (~4 (m s⁻¹) practically).

The column apparatuses are the main devices for separation and chemical processes realization in chemical, power, biotechnological, and other industries. They are different types as plate columns, packed bed columns, bubble columns, trickle columns, catalyst bed columns, etc.