# A Review on Gas-Liquid Mass Transfer Coefficients in Packed-Bed Columns

^{*}

## Abstract

**:**

_{y}[kmol·m

^{−2}·s

^{−1}], liquid-side coefficient per unit surface k

_{x}[kmol·m

^{−2}·s

^{−1}], interfacial packing area a

_{e}[m

^{2}·m

^{−3}], which constitute the ingredients to assess the mass transfer rate of packed-bed columns. The models have been reported in the original form provided by the authors together with the geometric and model fitting parameters published in several papers to allow their adaptation to packings different from those covered in the original papers. Although the work is focused on a collection of carefully described and ready-to-use equations, we have tried to underline the criticalities behind these models, which mostly rely on the assessment of fluid-dynamics parameters such as liquid film thickness, liquid hold-up and interfacial area, or the real liquid paths or any mal-distributions flow. To this end, the paper reviewed novel experimental and simulation approaches aimed to better describe the gas-liquid multiphase flow dynamics in packed-bed column, e.g., by using optical technologies (tomography) or CFD simulations. While the results of these studies may not be easily extended to full-scale columns, the improved estimation of the main fluid-dynamic parameters will provide a more accurate modelling correlation of liquid-gas mass transfer phenomena in packed columns.

## 1. Introduction

_{y}and k

_{x}equations. The authors provided a number of calibration parameters based on extensive experimental tests conducted under specific operating conditions (different gas and liquid loads, pressure and temperature and several chemical-physical properties of gas and liquid) and specific packing both random and structured.

_{y}) and the liquid-side (k

_{x}) mass-transfer coefficients and of the effective interfacial area (a

_{e}) reported in a rigorous and complete form together with the fitting parameters of the models necessary for their use. This study also provides the ranges of models’ validity and applicability together with their main pros and cons, to help the reader in selecting the most suitable one for specific packing/application. Finally, the last part of the paper describes the new possibilities to refine the proposed correlations offered by new experimental findings and modeling approaches available in the last 20 years literature.

## 2. Mass Transfer Models for Packed-Bed Columns

_{x}and k

_{y}) per surface unit [kmol·m

^{−2}·s

^{−1}] and wet effective surface area to the mass transfer (a

_{e}, [m

^{2}·m

^{−3}]).

_{G}and liquid Re

_{L}numbers, Schmidt gas Sc

_{G}and liquid Sc

_{L}numbers, Froude liquid number Fr

_{L}, Weber liquid number We

_{L}, Kapitza liquid number Ka

_{L}, Graetz liquid number Gr

_{L}), characteristic dimensions of packing (e.g., equivalent or hydraulic diameters, specific packing sizes and surface area, corrugation or inclination angle of packing sheet, void and holes fractions of the packing) and physical and fluid-dynamics properties of gas and liquid streams (e.g., molecular weight, density, viscosity, surface tension and diffusivity). The equations also included a series of model fitting parameters to adapt their correlations to specific types and models of packings.

#### 2.1. Onda et al., 1968 (The OTO Model)

_{e}) is identical with the gas-liquid interface. However, they derived a new empirical equation for wet surface area, taking into account the liquid surface tension (σ

_{L}, [N·m

^{−1}]) and the critical surface tension of packing material (σ

_{c}, [N·m

^{−1}]) as a model parameter:

_{n}is the nominal surface area [m

^{2}·m

^{−3}], Re

_{L}is the Reynolds liquid number, We

_{L}is the Weber liquid number, and Fr

_{L}is the Froude liquid number. In Table 1 are listed the critical surface tension (σ

_{c}) values for different packing materials [46].

_{Ls}[m·s

^{−1}] is the superficial liquid velocity, ρ

_{L}[kg·m

^{−3}] is the mass liquid density, μ

_{L}[kg·m

^{−1}·s

^{−1}] is the mass liquid viscosity, and g is the acceleration of gravity (9.81 m·s

^{−2}).

_{e}equation can be applicable within ±20% error to the column filled with Raschig rings, Berl saddles, Spheres and Rods made of ceramic, glass, polyvinylchloride and also coated with paraffine film.

_{x}) data are referred to gas absorption into water, desorption from water, gas absorption of CO

_{2}into water adding a non-foaming surfactant (Newpol PE-61) and gas absorption of pure CO

_{2}into methanol and carbon tetrachloride. The authors obtained the following correlation:

_{L}

^{OTO}is a proportionality model factor set to 0.0051, Re

_{L}is given by Equation (2) replacing the nominal surface area (a

_{n}) by the wet surface area (a

_{e}), d

_{p}[m] is the diameter of a sphere possessing the same surface area as a piece of packing (i.e., the packing sizes), ρ

_{x}[kmol·m

^{−3}] is the molar liquid density, and Sc

_{L}is the Schmidt liquid number defined by

_{L}[m

^{2}·s

^{−1}] is the gas diffusivity in the liquid phase. The exponent of Re

_{L}in Equation (5) coincides with that derived on the wet surface area basis by Vankrevelen and Hoftijzer [47] and Fujita and Hayakawa [48] and also is nearly equal to 0.61 of that derived by Norman and Sammak [49].

_{x}) for gas absorption and desorption in packed columns have been correlated within an error of ±20% for organic solvents as well as water.

_{y}) data are referred to gas absorption processes reported in the literature [45,48,50,51,52]. The empirical equation that best represents experimental data is reported below.

_{G}

^{OTO}is a proportionality model factor set to 5.23 for packing sizes above 15 mm and to 2.0 for sizes below 15 mm, D

_{G}[m

^{2}·s

^{−1}] is the gas diffusivity in the gas phase, and ρ

_{y}[kmol·m

^{−3}] is the molar gas density; Re

_{G}and Sc

_{G}are, respectively, the Reynolds and the Schmidt gas numbers, which are expressed as

_{Gs}[m·s

^{−1}] is the superficial gas velocity, ρ

_{G}[kg·m

^{−3}] is the mass gas density, and μ

_{G}[kg·m

^{−1}s

^{−1}] is the mass gas viscosity. The k

_{y}equation for gas absorption is also applicable to the vaporization process with ±30% error. Onda et al. [6] also found that the difference between the mass transfer data for absorption and that for vaporization is quite small and could be neglected.

#### 2.2. Bravo et al., 1985 (The BRF Model)

_{y}) was estimated using the relationship proposed by Johnstone and Pigford [54] for counter-current evaporation in a wetted-wall column as

_{G}

^{BRF}is a proportionality factor set to 0.0338, d

_{eq}(equivalent diameter, [m]) is the characteristic packing dimension, Sc

_{G}is the Schmidt gas number defined as in Equation (9), Re

_{G}is the Reynolds gas number for the BRF model which is defined in Equation (11):

_{Ge}and u

_{Le}are the gas and liquid effective velocity through the channel. The equivalent diameter (d

_{eq}) can be calculated through the knowledge of the packing dimensions that are, in particular, its base width (B

_{p}), slant height (S

_{p}) and height (H

_{p}) of packing corrugation as

_{p}, S

_{p}and H

_{p}) of structured packings are shown in Table 2.

_{Ge}) is given by the following expression:

_{p}[m

^{3}·m

^{−3}] is the void volumetric fraction of the packing, and θ

_{c}[°] is the inclination or corrugation angle. The liquid effective velocity (u

_{Le}), instead, is assessed through the classical falling film equation [55] as

^{−1}·s

^{−1}], expressed as

_{s}, [m·m

^{−2}]) in Equation (15) is given by the following expression:

_{x}), which is expressed as

_{e}[s] is the exposure time, that is, the time it takes for a fluid element to flow between corrugation channels, and C

_{L}

^{BRF}is a model parameter, and its value is set to 2. According to this theory, the liquid-side mass transfer resistance could be neglected in comparison to the gas-side one. The exposure time (t

_{e}) is defined as the ratio between the slant height of the corrugation (S

_{p}) and the effective liquid velocity through the channel (u

_{Le}).

_{e}), the authors assumed the wet surface equal to the nominal surface (a

_{n}), due to the corrugation and the capillarity of the packing which led to assume a unitary wettability efficiency.

_{e}) based on Shi and Mersmann equation [9]:

_{p}) with the hydraulic equivalent diameter (d

_{eq}) in Equation (20) to reduce average deviation [57].

#### 2.3. Bravo et al., 1992 (The SRP Model)

_{e}). Following the study by Shi and Mersmann [9] that proposed an expression base on fluid hydraulics over an inclined plane, Bravo and Fair [58] developed an equation for randomized packing under distillation and absorption/stripping conditions. For structured packing, Fair and Bravo [59] and Bravo et al. [10] observed a relatively influence by gas flow rate and a much more dependence by liquid rate.

_{e}) in SRP model was based on the extensive study by Shi and Mersmann [9] about hydraulic consideration for sheet-metal packing.

_{SE}) which accounts for variations of surface packing (lancing, fluting, etc.), and the second is a correction factor for total liquid hold-up due to effective wetted area (F

_{t}).

_{SE}represents a corrective parameter based on observation of liquid flow on packing surfaces relative to distillation experiments by McGlamery [60]. Some values for F

_{SE}are reported in the studies of Rocha et al. [33,61] for several packing types. In Table 3 are shown the nominal surface area (a

_{n}), void volumetric fraction (ε

_{p}) and surface enhancement factor (F

_{SE}) for some common packings.

_{t}:

_{c}[°] is the contact angle which accounts for surface material wettability, Re

_{L}is the Reynolds liquid number, We

_{L}is the Weber liquid number, and Fr

_{L}is the Froude liquid number for SRP model.

_{c}in Equation (22) could be calculated for two different conditions:

_{G}

^{SRP}is a proportionality Sherwood gas number factor, and its value is 0.054 by Rocha et al. [33] and Re

_{G}is the Reynolds gas number for SRP model. In this formulation, the side dimension of a corrugation cross section (S

_{p}) was used as characteristic dimension of packing in place of the equivalent diameter (d

_{eq}) adopted in BRF model [8]. The calculation of effective gas velocity (u

_{Ge}) takes into account of the corrugation packing angle (θ

_{c}), the space occupied by liquid using the void volumetric fraction of packing (ε

_{p}) and the liquid hold-up (h

_{L}).

_{L}, [m

^{3}·m

^{−3}] is the volumetric fraction of the liquid hold-up which was expressed in terms of the liquid film thickness (δ

_{f}, [m]), the side dimension of a corrugation cross section (S

_{p}) and a correction factor for total hold-up (F

_{t}).

_{eff}[m·s

^{−2}] is the effective gravity related to a force balance on the liquid film which flows on the packing surface. The effective gravity can be expressed as a function of the liquid-gas densities, the pressure drops (ΔP/Z, [Pa·m

^{−1}]) and K

_{1}constant (which depends only on the packing type [61]).

_{eff}= 0, it means that the flooding condition occurs.

_{1}parameter is constant for a particular shape of packing regardless of size or surface characteristic.

_{flood}, [Pa·m

^{−1}] is a pressure drops per meter of packing at flooding condition, and it is a specific packing parameter like to K

_{1}constant. Fair and Bravo [64] observed that the flooding appears in the 900–1200 Pa·m

^{−1}range for the packing investigated. For the sake of simplicity, the value of ΔP/Z

_{flood}was fixed at 1025 Pa·m

^{−1}in order to calculate K

_{1}.

_{e}) adopted by BRF model:

_{E}

^{SRP}is the surface renewal factor for SRP model. Murrieta [65] estimated, with experiments on oxygen-air-water system, that the value was slightly less than unity (C

_{E}

^{SRP}~0.9) for structured packings.

_{Le}) is defined as

_{E}

^{SRP}, so-called surface renewal factor) for exposure time evaluation:

_{L}

^{SRP}is the same model parameter in BRF model (C

_{L}

^{BRF}= C

_{L}

^{SRP}).

#### 2.4. Billet and Schultes, 1993 (The BS Model)

_{L}[s] is the time necessary for the renewal of interface area.

_{h}is the characteristic dimension of the packing (hydraulic diameter, [m]) and h

_{L}is the liquid hold-up (below the loading point [66]) for this model:

_{L}

^{BS}is a specific constant parameter for the liquid phase.

_{G}, [s]) required for renewal of the contact area between phases can be defined as

_{G}) is comparatively short in the packed-beds and for the gas mass transfer can be assumed the Higbie analogy [56]:

_{G}

^{BS}is a specific constant parameter for the liquid phase. In this model, Re

_{G}is the Reynolds gas number expressed as

_{L}

^{BS}and C

_{G}

^{BS}depend on the characteristic constructions and material of the packing [11,15] and dedicated experimental tests are needed for their determination. Billet and Schultes estimated the values of the C

_{L}

^{BS}and C

_{G}

^{BS}for several random and structured packings [11,15]. Table 4 reports some values of the BS model fitting parameters.

_{OG}) was also estimated at around 14.1%.

#### 2.5. Brunazzi and Paglianti, 1997 (The BP Model)

_{2}from water into air and in column working co-currently and the absorption of chlorinated compounds with two commercial high boiling liquids (Genosorb 300 and Genosorb 1843). The tests were performed in a column working counter-currently, using structured packings, such as Mellapak 250.Y and BX.

_{y}) is equal to that of the SRP model (C

_{G}

^{BP}= C

_{G}

^{SRP}= 0.054).

_{G}) is defined as in Equation (11), and the characteristic dimension of the channels (d

_{h}) is the same of BS model in Equation (40).

_{Ge}) set out in the Reynolds gas number (Re

_{G}) is defined hereafter:

_{x}) depended on the packing height (Z, [m]) and the mixing factor; for this reason, they introduced two dimensionless numbers to describe the liquid phase: the Graetz (Gr

_{L}) and the Kapitza (Ka

_{L}) liquid numbers, which were correlated to k

_{x}through the following equation.

_{L}), b is the functional dependence on Graetz liquid number, and c is the functional dependence on Kapitza liquid number.

_{L}[°] is the slope of the steepest descent line with respect to the horizontal axis. This geometric parameter can be computed using Spekulijak and Billet correlation [67], once the corrugation angle (θ

_{c}) and the packing dimension parameters (B

_{p}and H

_{p}) are known:

_{L}) was expressed as in Equation (9), while the Reynolds liquid number (Re

_{L}) was defined as below:

_{Le}) is a function of the superficial liquid velocity (u

_{Ls}), of the dynamic liquid hold-up (h

_{L}) and the slope of the steepest descent line with respect to the horizontal axis of the packing (θ

_{L}).

_{f}):

_{f}) is a function of liquid properties, geometric characteristics of the packing, liquid velocity (u

_{Ls}) and the dynamic liquid hold-up (h

_{L}). If the liquid-phase is Newtonian liquid and flows in laminar conditions, the liquid film thickness can be calculated as

_{e}), the authors used a model based on experimental measurements of the liquid hold-up [68,69]:

_{Lo}is the dynamic viscosity of water at 20 °C [kg·m

^{−1}·s

^{−1}]; k

_{hL}is a proportionality factor, which is set to 0.0169 for u

_{Ls}< 0.0111 m·s

^{−1}, otherwise to 0.0075; x is a model parameter, which is set to 0.37 for u

_{Ls}< 0.0111 m·s

^{−1}and to 0.59 otherwise. Instead, for BX packings Torelli [71] proposed the following empirical equation to evaluate the liquid hold-up.

_{x}a

_{e}and k

_{y}a

_{e}within an error of ±15% and ±19%, respectively, for desorption of CO

_{2}from water in air and for absorption of chlorinated solvents with Genosorb 300 and Genosorb 1843 using Mellapak 250.Y.

#### 2.6. Olujić et al., 2004 (The Delft Model)

_{y,lam}, [kmol·m

^{−2}·s

^{−1}]) and turbulent (k

_{y,turb}, [kmol·m

^{−2}·s

^{−1}]) regime conditions:

_{G,lam}is the Sherwood gas number for laminar flow; Sh

_{G,turb}is the Sherwood gas number for turbulent flow, and d

_{hG}[m] is the hydraulic diameter for the gas phase. The gas-side mass transfer coefficients referred to laminar and turbulent flow are defined below:

_{G}

^{Delft}is a proportionally coefficient for laminar flow case, and its value is 0.664; l

_{G,pe}[m] is the length of gas flow channel in a packing element, h

_{pe}[m] is the height of structured packing element unit, φ is the fraction of the triangular flow channel occupied by liquid, Re

_{Grv}is the Reynolds gas number based on relative effective velocities between gas and liquid, and ξ

_{GL}is the gas-liquid friction factor by Colebrook and White [75]. The expressions provided for these parameters are:

_{Ge}is the effective gas velocity (expressed as BP model in Equation (52)), and u

_{Le}is the effective liquid velocity.

_{L}, shown in Equation (57).

_{f}). Olujić et al. [73] proposed, based on experimental evidence, that the liquid film thickness (δ

_{f}) is not significantly affected by the gas flow rate in the pre-loading region. Therefore, δ

_{f}can be determined through correlations developed for liquid films in stagnant gas. If the liquid film flows in laminar regime on the packing surface, it can be estimated as

_{L}) as a product of the nominal surface packing area (a

_{n}) and the liquid film thickness (δ

_{f}):

_{L}

^{Delft}is a proportionality Sherwood liquid number factor set to 2, while C

_{E}

^{Delft}(the surface renewal factor) is fixed at 0.9 as suggested by Murrieta [65].

_{hG}was used in place of the S

_{p}[10] and d

_{eq}[8]. The expression provided by the authors for this parameter is

_{e}) cannot exceed the nominal surface area (a

_{n}) and that in the case of uniform distribution the liquid in the bed, poor distribution occurs only at low liquid flows. Therefore, the authors obtained an empirical formulation in which the percentage of wettability of the packing is a function of the liquid flow:

_{p}[m

^{3}·m

^{−3}] is the volumetric fraction of packing surface area occupied by holes, A and B are the constants depending by type and size of packing.

_{e}for structured packings:

_{c}can be evaluated from Table 2, for different construction materials.

_{e}of the Delft model. This formulation has been developed both for unperforated and perforated packings and takes into account of the fraction of packing surface area occupied by holes (Ω

_{p}). For unperforated packings, (Montz B1 series) Ω

_{p}= 0, while for the perforated packings with common size of the holes around 4 mm (e.g., Montz BSH, Koch-Glitsch Flexipac and Sulzer Mellapak packings), the perforated fraction is generally in the range of 10–15% (Ω

_{p}= 0.1–0.15).

_{op}[bar] is the operating pressure; the specific surface area of 250 m

^{2}·m

^{−3}, the corrugation angle of 45° and the atmospheric pressure are taken as reference parameters. The correction term on the right hand-side of Equation (81), with the exponent n described by the above expression, reduces the size of effective area predicted by Equation (80) to the extent corresponding to the observed effects in the experiments by Olujić et al. [13].

#### 2.7. Hanley and Chen, 2012 (The HC Model)

_{2}, p/o-Xylene, Chlorobenzene/Ethylbenzene, i-Cyclobutane/n-Butane, Cyclohexane/n-Heptane and cis-Decalin/trans-Decalin) using Flexipac and Mellapak type packings. In particular, they focused on several packed-column mass-transfer/interfacial area correlations found in commercially available simulation software like Aspen Technology’s Aspen Rate Based Distillation component [77]. The results demonstrated that predicted HETPs are more than 20% above or below the experimental results [14]. Consequently, they used a new data fitting procedure related to distillation and acid gas absorption with amines operations in order to develop a set of dependable and dimensionally consistent correlations for the mass transfer related quantities k

_{x}, k

_{y}and a

_{e}available for metal Pall rings, metal IMPT, sheet metal structured packings of Mellapak type and metal gauze structured packings in the X configuration.

_{y}), the authors derived a correlation from the classical hydraulic analogy proposed by Chilton and Colburn [78] which is reasonably accurate for flows in which no form drag is present:

_{G}

^{HC}and β are model fitting parameters (listed in Table 7), Sc

_{G}is the Schmidt gas number given by Equation (9), d

_{h}[m] is the hydraulic diameter given by Equation (40), Re

_{G}is the Reynolds gas number, whose expression is reported below:

_{Gs}[m·s

^{−1}] is the superficial gas velocity. It is important to underline that in this model, the Reynolds number for the gas phase (Re

_{G}) is linearly dependent on the gas-phase superficial velocity (u

_{Gs}), and the gas-phase effective velocity is not required. In particular, in Equation (83) β is the functional parameter for the Reynolds gas number, Re

_{G}. According to the abovementioned analogy, the random packings are considered to have open structures; therefore, form drag should be small and the friction factor is found to be weakly dependent on the Reynolds number. Consequently, the gas-side mass transfer coefficient (k

_{y}) is a linear function of the Reynolds number (β = 1).

_{x}), the authors considered that the induced shear at the liquid interface due to the turbulent counter-current flow of the gas interferes with the path of the liquid film in most packed columns. There is much less general consensus for the liquid-film mass transfer coefficient correlation under this case. Two different models are often chosen to describe mass transfer phenomena in a turbulent liquid film, which are the penetration/surface renewal model and the film model [56,79]. According to these theories, the Sherwood liquid number (Sh

_{L}) is a function of two dimensionless numbers, which are the liquid Reynolds (Re

_{L}) and Schmidt (Sc

_{L}) numbers; the former is showed below, whereas the latter is given by Equation (6).

_{L}) can generally approximated by a 1/3 power function [78]. Instead, for the Reynolds liquid number (Re

_{L}), Potnis and Lenz [80] studied liquid desiccant systems for gas drying using random as well as structured packings reporting that the exponent of Re

_{L}ranged from 0.9 to 1.2: for this reason, it was set by authors to 1. The liquid-side mass transfer coefficient (k

_{x}) is given by the following expression:

_{L}

^{HC}is a model fitting parameters (listed in Table 7), and d

_{h}[m] is the hydraulic diameter, which is given by Equation (40).

_{y}) and liquid-side (k

_{x}) mass transfer coefficients. For the packings abovementioned, the authors considered adjusting factors F

_{θ,G}and F

_{θ,L}, respectively, which are dependent on the corrugation angle (θ

_{c}, [°]):

_{θ,G}) and the liquid-side (F

_{θ,L}) mass transfer dependence on crimp inclination angle (θ

_{c}, [°]), which are set to −3.072 and 4.078 for Mellapak type packings, respectively.

_{e}), the authors proposed a dependence upon Reynolds gas and liquid numbers (Re

_{G}and Re

_{L}), Froude liquid number (Fr

_{L}) and Weber liquid number (We

_{L}). The last two dimensionless parameters Fr

_{L}and We

_{L}are given by the following equations, respectively.

_{e}) is determined using Equation (91), where η, κ, λ, ν, χ, ω and ψ are model fitting parameters listed in Table 8 for metal Pall rings, IMTP random packings, sheet metal and gauze metal packings.

_{e}); κ, λ, ν and χ represent the wet surface area functional parameters for Reynolds gas and liquid numbers, Weber liquid number and Froude liquid number, respectively; ω represents the wet surface area dependence parameter on the gas to liquid density ratio; ψ represents the wet surface area dependence parameter on the gas to liquid viscosity ratio; C

_{m}

^{HC}is a correction factor related to construction material. The authors specified that this factor (C

_{m}

^{HC}) is equal to 1 for metal packings and 0.75 for plastic ones. Further, no adjustment is made to account for the expected mass transfer improvement of Y type over X type ones.

## 3. Final Considerations and Models Refinement

#### 3.1. Model Comparison and Field of Application

_{x}a

_{e}/k

_{y}a

_{e}or HETPs.

_{y}and k

_{x}). It should be noted that when used for structured packings, only the void fraction and nominal surface area of the packing data are needed, unlike the other correlations which require other characteristic dimensions of the packing. On the contrary, the SRP model requires the use of four fitting parameters, one of which for the interfacial area. Generally, the model equations proposed for a

_{e}calculation are not calibrated through the use of fitting parameters because the interfacial area is difficult to measure experimentally, and in fact, a calibration procedure on the coefficients k

_{x}a

_{e}and k

_{y}a

_{e}is preferred, introducing fitting parameters in the equations for the gas-side (k

_{y}) and liquid-side (k

_{x}). However, despite the authors’ efforts to revise the previous version, this model provides an error of about ±24% considering the experimentally measured HETP values. Furthermore, among the other correlations examined, the SRP model couples a predictive model for pressure drops in the mass transfer model to calculate the variation in liquid hold-up and film thickness with an iterative algorithm. This complication makes this correlation more complex in use but on the other hand allows to estimate pressure drops and mass transfer coefficients simultaneously.

_{x}is based on an iterative algorithm on the height of the packing column which significantly increases the computational efforts required. Despite this, the errors found compared to the experiments are ±15% for the liquid-side and ±19% for the gas-side coefficient.

#### 3.2. New Insights on the Characterization of Liquid Distribution in Packed Columns

_{2}scrubbing with concentrated aqueous NaOH in a Mellapak 250.Y column, varying the gas flow rate. The authors showed that the dependence of the gas-side mass transfer coefficients on the gas flow rate agrees very well with the experimental data, with errors in the range of 3%. These predictions are better than the results obtained with different semi-empirical correlations, such as Bravo et al. [10] and Olujic et al. [13].

_{2}from air into aqueous solutions containing sodium hydroxide under three different operating conditions. The simulation study highlighted the influence of random packing geometry on separation efficiency. It is found that a small hydraulic diameter, an increased frequency of liquid film mixing and an increased liquid hold-up lead to higher absorption rates. In contrast, the frequency of gas or jet/droplet mixing has a negligible influence on the absorption rate. These performed ‘‘virtual experiments” bring about useful recommendations for the development of random packings. The method can be also extended to support the design and optimization of structured packings [1,91] to understand the impact that the structured packing geometry, physical parameters of fluids and the shape of flow have on physical and reactive mass transfer phenomena.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## List of Symbols

a | Proportionality coefficient for the Sherwood liquid number in the BP model, [-] |

A | Constant depending by type and size packing in Delft model, [-] |

a_{e} | Wet effective surface area of packing, [m^{2}·m^{−3}] |

a_{n} | Nominal surface area of packing, [m^{2}·m^{−3}] |

b | Functional parameter for Graetz liquid number in the BP model, [-] |

B | Functional parameter for superficial liquid velocity in Delft model, [-] |

B_{p} | Base width of a packing corrugation, [m] |

c | Functional parameter for Kapitza liquid number in the BP model, [-] |

C_{E}^{SRP} | Surface renewal factor of the packing in the BRF model, [-] |

C_{E}^{Delft} | Surface renewal factor of the packing in the Delft model, [-] |

C_{G}^{BP} | Gas proportionality factor in the BP model, [-] |

C_{G}^{BRF} | Gas proportionality factor in the BRF model, [-] |

C_{G}^{BS} | Gas-side specific constant in the BS model, [-] |

C_{G}^{Delft} | Gas-side proportionality coefficient for laminar flow case in the Delft model, [-] |

C_{G}^{HC} | Gas proportionality factor in the HC model, [-] |

C_{G}^{SRP} | Gas proportionality factor in the SRP model, [-] |

C_{G}^{OTO} | Gas proportionality model factor in the OTO model, [-] |

C_{L}^{BP} | Liquid-side proportionality model factor in the BP model, [-] |

C_{L}^{BRF} | Liquid-side proportionality model factor in the BRF model, [-] |

C_{L}^{BS} | Liquid-side proportionality model factor in the BS model, [-] |

C_{L}^{HC} | Liquid-side proportionality model factor in the HC model, [-] |

C_{L}^{SRP} | Liquid-side proportionality model factor in the SRP model, [-] |

C_{L}^{OTO} | Liquid-side proportionality model factor in the OTO model, [-] |

C_{m}^{HC} | Correction factor related to construction material, [-] |

d | Characteristic dimension of the liquid film, [m] |

d_{eq} | Equivalent diameter, [m] |

D | Column diameter, [m] |

D_{G} | Gas diffusivity in the gas phase, [m^{2}·s^{−1}] |

d_{h} | Hydraulic diameter, [m] |

d_{hG} | Hydraulic diameter of triangular gas flow channel, [m] |

D_{L} | Gas diffusivity in the liquid phase, [m^{2}·s^{−1}] |

d_{p} | Diameter of a sphere possessing the same surface are as a piece of packing, [m] |

Fr_{L} | Froude liquid number, [-] |

F_{SE} | Surface enhancement factor in the SRP model, [-] |

F_{t} | Correction factor for total hold-up due to effective wetted area in the SRP model, [-] |

F_{θ,G} | Gas-side mass transfer coefficient dependence on crimp inclination angle, [-] |

F_{θ,L} | Liquid-side mass transfer coefficient dependence on crimp inclination angle, [-] |

g | Acceleration of gravity, [m·s^{−2}] |

g_{eff} | Effective acceleration of gravity, [m·s^{−2}] |

Gr_{L} | Graetz liquid number, [-] |

H | Flow distance, [m] |

HETP | Height equivalent to a theoretical plate, [m] |

h_{L} | Volumetric liquid hold-up, [m^{−3}·m^{−3}] |

H_{p} | Peak height of a packing corrugation, [m] |

h_{pe} | Height of structured packing element unit, [m] |

Ka_{L} | Kapitza liquid number, [-] |

K_{1} | Parameter for packing type dependence, [-] |

k_{hL} | Proportionality factor for liquid hold-up in BP model, [-] |

k_{x} | Liquid-side mass transfer coefficient per surface unit, [kmol·m^{−2}·s^{−1}] |

k_{y} | Gas-side mass transfer coefficient per surface unit, [kmol·m^{−2}·s^{−1}] |

k_{y,lam} | Gas-side mass transfer coefficient for laminar regime, [kmol·m^{−2}·s^{−1}] |

k_{y,turb} | Gas-side mass transfer coefficient for turbulent regime, [kmol·m^{−2}·s^{−1}] |

l_{G,pe} | Length of the triangular gas flow channel in a packing element, [m] |

n | Correction exponent for the effective area in Delft model, [-] |

P_{op} | Operating pressure, [bar] |

P_{s} | Perimeter per unit cross-sectional area, [m] |

Re_{G} | Reynolds gas number, [-] |

Re_{Grv} | Reynolds gas number based on relative effective velocity between gas and liquid, [-] |

Re_{L} | Reynolds liquid number, [-] |

Sc_{G} | Schmidt gas number, [-] |

Sc_{L} | Schmidt liquid number, [-] |

Sh_{L} | Sherwood liquid number, [-] |

Sh_{G,lam} | Sherwood gas number for laminar flow, [-] |

Sh_{G,lam} | Sherwood gas number for turbulent flow, [-] |

S_{p} | Slant height of a packing corrugation, [m] |

t_{e} | Exposure time, [s] |

t_{G} | Gas contact time, [s] |

t_{L} | Time necessary for renewal of interface area, [s] |

u_{Ge} | Gas effective velocity through the packing channel, [m·s^{−1}] |

u_{Le} | Liquid effective velocity through the packing channel, [m·s^{−1}] |

u_{Gs} | Superficial gas velocity, [m·s^{−1}] |

u_{Ls} | Superficial liquid velocity, [m·s^{−1}] |

We_{L} | Weber liquid number, [-] |

Z | Packing height, [m] |

## Greek Symbols

α | Liquid-side mass transfer coefficient dependence on crimp inclination angle in the HC model, [-] |

β | Functional parameter for Reynolds gas number in the HC model, [-] |

γ | Gas-side mass transfer dependence on crimp inclination angle in the HC model, [-] |

Γ | Liquid flow per unit length of perimeter, [kg·m^{−1}·s^{−1}] |

γ_{c} | contact angle accounts for surface material wettability, [°] |

δ_{f} | Liquid film thickness, [m] |

ΔP/Z | Total pressure drops per meter of packing, [Pa·m^{−1}] |

ΔP/Z_{floood} | Pressure drops per meter of packing at flooding condition, [Pa·m^{−1}] |

ε_{p} | Void volumetric fraction of the packing, [m^{−3}·m^{−3}] |

ζ_{GL} | Interaction coefficient for gas-liquid friction losses in the Delft model, [-] |

η | Proportionality coefficient for the wet surface area in the HC model, [-] |

θ_{c} | Inclination or corrugation angle, [°] |

θ_{L} | Slope of the steepest descent line with respect to the horizontal axis, [°] |

κ | Functional parameter for Reynolds gas number in the HC model, [-] |

λ | Functional parameter for Reynolds liquid number in the HC model, [-] |

μ_{G} | Mass gas viscosity, [kg·m^{−1}·s^{−1}] |

μ_{L} | Mass liquid viscosity, [kg·m^{−1}·s^{−1}] |

μ_{Lo} | Dynamic viscosity of water at 20 °C, [kg·m^{−1}·s^{−1}] |

ν | Functional parameter for Weber liquid number in the HC model, [-] |

ρ_{G} | Mass gas density, [kg·m^{3}] |

ρ_{y} | Molar gas density, [kmol·m^{−3}] |

ρ_{x} | Molar liquid density, [kmol·m^{−3}] |

ρ_{L} | Mass liquid density, [kg·m^{−3}] |

σ_{c} | Critical surface tension of packing material, [N·m^{−1}] |

σ_{L} | Liquid surface tension, [N·m^{−1}] |

φ | Fraction of the triangular flow channel occupied by liquid, [-] |

χ | Functional parameter for Froude liquid number in the HC model, [-] |

ψ | Wet surface area dependence parameter on the gas to liquid viscosity ratio in the HC model, [-] |

ω | Wet surface area dependence parameter on the gas to liquid density ratio in the HC model, [-] |

Ω_{p} | Fraction of packing surface area occupied by holes, [m^{−3}·m^{−3}] |

## Abbreviations

BP | Referred to the work of Brunazzi and Paglianti (1997) |

BRF | Referred to the work of Bravo et al. (1985) |

BS | Referred to the work of Billet and Schultes (1993) |

Delft | Referred to the work of Olujić et al. (2004) |

HC | Referred to the work of Hanley and Chen (2012) |

OTO | Referred to the work of Onda et al. (1968) |

SRP | Referred to the work of Bravo et al. (1992) |

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**Table 1.**Critical surface tension for different packing materials [46].

Packing Material | Ceramic | Steel | Plastic | Carbon |
---|---|---|---|---|

σ_{c} [N·m^{−1}] | 0.061 | 0.075 | 0.033 | 0.056 |

Structured Packing | B_{p}mm | S_{p}mm | H_{p}mm |
---|---|---|---|

Flexipac 1.Y, Metal/Plastic | 12.7 | 9 | 6.4 |

Mellapak 2.Y, Metal/Plastic | 33 | 21.5 | 13.8 |

Mellapak 125.Y, Metal/Plastic | 55 | 37 | 24.8 |

Mellapak 250.Y, Metal/Plastic | 24.1 | 17 | 11.9 |

Mellapak 350.Y, Metal/Plastic | 15.3 | 11.9 | 8.9 |

Mellapak 500.Y, Metal/Plastic | 9.6 | 8.1 | 6.53 |

Mellapak Plus 252.Y, Metal/Plastic | 24.1 | 17 | 11.9 |

Mellapak 250.YS, Metal/Plastic | 24.1 | 17 | 11.9 |

Mellapak 250.X, Metal/Plastic * | 24.1 | 17 | 11.9 |

Sulzer BX, Metal/Plastic | 24.1 | 17 | 11.9 |

Structured Packing | a_{n}m ^{−2}·m^{−3} | ε_{p}m ^{−3}·m^{−3} | S_{p}mm | F_{SE}- |
---|---|---|---|---|

Flexipac 2.Y, Metal | 233 | 0.95 | 18 | 0.35 |

Gempak 2A, Metal | 233 | 0.95 | 18 | 0.34 |

Gempak 2AT, Metal | 233 | 0.95 | 18 | 0.312 |

Intalox 2T, Metal | 213 | 0.95 | 22.1 | 0.415 |

Maxpak, Metal | 229 | 0.95 | 17.5 | 0.364 |

Mellapak 250.Y, Metal | 250 | 0.95 | 17 | 0.35 |

Mellapak 350.Y, Metal | 350 | 0.93 | 11.9 | 0.35 |

Mellapak 500.Y, Metal | 500 | 0.91 | 8.1 | 0.35 |

Sulzer BX, Metal | 492 | 0.9 | 17 | 0.35 |

**Table 4.**Characteristic data of several packings and specific constants C

_{L}

^{BS}and C

_{G}

^{BS}[15].

Random/Structured Packing | Material | Size mm | a_{n}m ^{−2}·m^{−3} | ε_{p}m ^{−3}·m^{−3} | C_{L}^{BS}- | C_{G}^{BS}- |
---|---|---|---|---|---|---|

Raschig | Metal | 0.3 | 315 | 0.96 | 1.5 | 0.45 |

Super-Rings | Metal | 0.5 | 250 | 0.975 | 1.45 | 0.43 |

Metal | 1 | 160 | 0.98 | 1.29 | 0.44 | |

Metal | 2 | 97.6 | 0.985 | 1.323 | 0.4 | |

Metal | 3 | 80 | 0.982 | 0.85 | 0.3 | |

Plastic | 2 | 100 | 0.96 | 0.377 | 0.337 | |

Raschig rings | Ceramic | 50 | 95 | 0.83 | 1.416 | 0.21 |

Ceramic | 38 | 118 | 0.68 | 1.536 | 0.23 | |

Ceramic | 25 | 190 | 0.68 | 1.361 | 0.412 | |

Ceramic | 15 | 312 | 0.69 | 1.276 | 0.401 | |

Ceramic | 13 | 370 | 0.64 | 1.367 | 0.265 | |

Ceramic | 10 | 440 | 0.65 | 1.303 | 0.272 | |

Carbon | 25 | 202.2 | 0.72 | 1.379 | 0.471 | |

Ralu Flow | Plastic | 1 | 165 | 0.94 | 1.486 | 0.36 |

Plastic | 2 | 100 | 0.945 | 1.27 | 0.32 | |

Pall Rings | Metal | 50 | 112.6 | 0.951 | 1.192 | 0.41 |

Metal | 38 | 139.4 | 0.965 | 1.012 | 0.341 | |

Metal | 25 | 223.5 | 0.954 | 1.44 | 0.336 | |

Plastic | 50 | 111.1 | 0.919 | 1.139 | 0.368 | |

Plastic | 38 | 151.1 | 0.906 | 0.856 | 0.38 | |

Plastic | 25 | 225 | 0.887 | 0.905 | 0.446 | |

Ceramic | 50 | 155.2 | 0.754 | 1.278 | 0.333 | |

Ralu rings | Metal | 50 | 105 | 0.975 | 1.192 | 0.345 |

Metal | 38 | 135 | 0.965 | 1.277 | 0.341 | |

Metal | 25 | 215 | 0.96 | 1.44 | 0.336 | |

Plastic | 50 | 95.2 | 0.983 | 1.52 | 0.303 | |

Plastic | 38 | 150 | 0.93 | 1.32 | 0.333 | |

Plastic | 25 | 190 | 0.94 | 1.32 | 0.333 | |

Hiflow rings | Metal | 50 | 92.3 | 0.977 | 1.168 | 0.408 |

Metal | 25 | 202.9 | 0.962 | 1.641 | 0.402 | |

Metal | 50 | 117.1 | 0.925 | 1.478 | 0.345 | |

Plastic | 50 hydr. | 118.4 | 0.925 | 1.553 | 0.369 | |

Plastic | 50 S | 82 | 0.942 | 1.219 | 0.342 | |

Plastic | 25 | 194.5 | 0.918 | 1.577 | 0.39 | |

Ceramic | 50 | 89.7 | 0.809 | 1.377 | 0.379 | |

Ceramic | 38 | 111.8 | 0.788 | 1.659 | 0.464 | |

Ceramic | 20 | 286.2 | 0.758 | 1.744 | 0.465 | |

NOR PAC rings | Plastic | 50 | 86.6 | 0.947 | 1.08 | 0.322 |

Plastic | 35 | 141.8 | 0.944 | 0.754 | 0.425 | |

Plastic | 25 type B | 202 | 0.953 | 0.883 | 0.366 | |

Plastic | 25 | 197.9 | 0.92 | 0.976 | 0.41 | |

Glitsch rings | Metal | 30 PMK | 180.5 | 0.975 | 1.92 | 0.45 |

Metal | 30 P | 164 | 0.959 | 1.577 | 0.398 | |

VSP rings | Metal | 50 | 104.6 | 0.98 | 1.416 | 0.21 |

Metal | 25 | 199.6 | 0.975 | 1.361 | 0.412 | |

Envi Pac rings | Plastic | 80 | 60 | 0.955 | 1.603 | 0.257 |

Plastic | 60 | 98.4 | 0.961 | 1.522 | 0.296 | |

Plastic | 32 | 138.9 | 0.931 | 1.517 | 0.459 | |

Bialecki rings | Metal | 50 | 121 | 0.966 | 1.721 | 0.301 |

Metal | 35 | 155 | 0.967 | 1.412 | 0.39 | |

Metal | 35 | 176.6 | 0.945 | 1.405 | 0.377 | |

Metal | 25 | 210 | 0.956 | 1.462 | 0.331 | |

Raflux rings | Plastic | 15 | 307.9 | 0.894 | 1.913 | 0.37 |

TOP-Pac rings | Aluminum | 50 | 105.5 | 0.956 | 1.326 | 0.389 |

Berl saddles | Ceramic | 38 | 164 | 0.7 | 1.568 | 0.244 |

Ceramic | 25 | 260 | 0.68 | 1.246 | 0.387 | |

Ceramic | 13 | 545 | 0.65 | 1.364 | 0.232 | |

Intalox saddles | Ceramic | 13 | 625 | 0.78 | 1.677 | 0.488 |

DIN-PAK | Plastic | 70 | 110.7 | 0.938 | 1.527 | 0.326 |

Plastic | 47 | 131.2 | 0.923 | 1.69 | 0.354 | |

Ralu Pak | Metal | YC-250 | 250 | 0.945 | 1.334 | 0.385 |

Metal | 250 | 250 | 0.975 | 0.983 | 0.27 | |

Impulse packing | Metal | 250 | 250 | 0.975 | 0.983 | 0.27 |

Plastic | 100 | 91.4 | 0.838 | 1.317 | 0.327 | |

Montz packing | Metal | B1-200 | 200 | 0.979 | 0.971 | 0.39 |

Metal | B1-300 | 300 | 0.93 | 1.165 | 0.422 | |

Plastic | C1-200 | 200 | 0.954 | 1.006 | 0.412 | |

Euroform | Plastic | PN-110 | 110 | 0.936 | 0.973 | 0.167 |

**Table 5.**Characteristic parameters in Equation (53) for different packings [12].

Type of Packing | Material | a - | b - | c - |
---|---|---|---|---|

Mellapak Y | Metal/Plastic | 16.43 | 0.915 | 0.09 |

Sulzer BX | Plastic | 63.10 | 0.676 | 0.09 |

Structured Packing | a_{n}m ^{−2}·m^{−3} | θ_{c}° | ε_{p}m ^{−3}·m^{−3} | H_{p}mm | B_{p}mm | S_{p}mm | h_{pe}m |
---|---|---|---|---|---|---|---|

Montz B1-250, Metal | 244 | 45 | 0.98 | 12 | 22.5 | 16.5 | 0.197 |

Montz B1-250.60, Metal | 245 | 60 | 0.98 | 12 | 22.3 | 16.4 | 0.211 |

Montz B1-400, Metal | 394 | 45 | 0.96 | 7.4 | 14 | 10.3 | 0.197 |

Montz B1-400.60, Metal | 390 | 60 | 0.96 | 7.4 | 14.3 | 10.3 | 0.215 |

Montz BSH-400, Metal | 378 | 45 | 0.97 | 7.4 | 15.1 | 10.6 | 0.194 |

Montz BSH-400.60, Metal | 382 | 60 | 0.97 | 7.4 | 14.8 | 10.5 | 0.215 |

Mellapak 250.X, Metal/Plastic * | 250 | 60 | 0.98 | 17 | 24.1 | 11.9 | 0.223 |

**Table 7.**Liquid-side and gas-side characteristic parameters in the HC model for different metal packings [14].

Packing Type | C_{G}^{HC}- | β - | γ - | C_{L}^{HC}- | α - |
---|---|---|---|---|---|

Pall rings | 0.00104 | 1.0 | - | 1.0 | - |

IMTP | 0.00473 | 1.0 | - | 1.0 | - |

Mellapak | 0.0084 | 1.0 | −3.072 | 0.33 | 4.078 |

Sulzer X | 0.3516 | 0.5 | - | 12 | - |

**Table 8.**Effective wet surface area characteristic parameter in the HC model for different metal packings [14].

Packing Type | η - | κ - | λ - | ν - | χ - | ω - | ψ - |
---|---|---|---|---|---|---|---|

Pall rings | 0.25 | 0.134 | 0.205 | 0.075 | −0.164 | −0.154 | 0.195 |

IMTP | 0.332 | 0.132 | −0.102 | 0.194 | −0.2 | −0.154 | 0.195 |

Mellapak | 0.538 | 0.1455 | −0.1526 | 0.2 | −0.2 | −0.033 | 0.090 |

Sulzer X | 2.308 | −0.274 | 0.246 | 0.248 | −0.161 | −0.180 | 0.233 |

**Table 9.**Summary of the validity and applicability ranges of the correlations examined in this Section, based on packing/application, experimental conditions, number of fitting parameters and estimated error adopted by the authors for their formulations.

Models | Application | Column Size | Operating Conditions | Packing Type | Error | Fitting Param | ||||
---|---|---|---|---|---|---|---|---|---|---|

D, m | Z, m | P, atm | T, K | F_{G}, Pa^{0.5} | F_{L}, m/h | |||||

OTO | Absorption/Desorption | 0.06–0.1 | 0.1–0.3 | 1.0 | 293–298 | 0.75–2.95 | up to 295 | Raschig rings, Berl saddles, Spheres, Rods | ±30% ^{1} | 2 |

BRF | Distillation | 0.43 | 3.0 | 0.33–4.14 | 334–427 | 0.6–3.2 | 9.35 | Sulzer BX | 47% ^{2}8.0% ^{3} | 2 |

SRP | Distillation Absorption | 0.43 | 3.0 | 0.33–20.4 | 334–427 | 0.2–3.6 | 9.35 | Sulzer BX, Gempak 2A, Gempak 2AT, Intalox 2T, Flexipac 2Y, Maxpak, Mellapak: 250Y, 350Y, 500Y | ±24% ^{4} | 4 |

BS | Distillation Absorption/Desorption | 0.06–1.4 | 0.15–3.95 | 0.033–1.0 | 288–407 | 0.01–2.77 | up to 118.20 | See Table 4 | ±8.3% ^{5}±12.4% ^{6} | 2 |

BP | Absorption/Desorption | 0.05–1.0 | 0.42–1.89 | 1.00 | 298 | 0.5–3.1 | 1.2–79.2 | Sulzer BX, Mellapak: 125Y, 250Y, 500Y | ±15% ^{5}±19% ^{6} | 4 |

Delft | Distillation | 0.2–1.4 | 3.4–6.0 | 0.33–4.14 | 334–427 | 0.5–4.0 | 9.0–35 | Montz: B1-250,B1-400 B1-250.60 B1-400.60 BSH-400 BSH-400.60 | ±12% ^{4} | 3 |

HC | Distillation Absorption/Desorption | See BRF, SRP and BS model | ±10% ^{4} | 10 (random) 11/12 (structured) |

^{1}refers to the range errors of the two coefficients;

^{2}refers to the average error with respect to the experimental HETPs using original a

_{e}equation;

^{3}refers to the average error with respect to the experimental HETPs using revisited a

_{e}equation;

^{4}refers to the range error with respect to the experimental HETPs;

^{5}refers to the range error for k

_{x}coefficient;

^{6}refers to the range error for k

_{y}coefficient.

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**MDPI and ACS Style**

Flagiello, D.; Parisi, A.; Lancia, A.; Di Natale, F. A Review on Gas-Liquid Mass Transfer Coefficients in Packed-Bed Columns. *ChemEngineering* **2021**, *5*, 43.
https://doi.org/10.3390/chemengineering5030043

**AMA Style**

Flagiello D, Parisi A, Lancia A, Di Natale F. A Review on Gas-Liquid Mass Transfer Coefficients in Packed-Bed Columns. *ChemEngineering*. 2021; 5(3):43.
https://doi.org/10.3390/chemengineering5030043

**Chicago/Turabian Style**

Flagiello, Domenico, Arianna Parisi, Amedeo Lancia, and Francesco Di Natale. 2021. "A Review on Gas-Liquid Mass Transfer Coefficients in Packed-Bed Columns" *ChemEngineering* 5, no. 3: 43.
https://doi.org/10.3390/chemengineering5030043