# Experimental evaluation of novel gas liquid contactor packing based on triply periodic minimal surfaces (TPMS)

Gregory Homem Alves César Ribeiro, David Rouzineau, Michel Meyer

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Gregory Homem Alves César Ribeiro, David Rouzineau, Michel Meyer. Experimental evaluation of novel gas liquid contactor packing based on triply periodic minimal surfaces (TPMS). Chemical Engineering Science, 2025, 301, pp.120690. 10.1016/j.ces.2024.120690. hal-04932988

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# Experimental evaluation of novel gas liquid contactor packing based on triply periodic minimal surfaces (TPMS)

Gregory Homem Alves César Ribeiro *, David Rouzineau, Michel Meyer

Laboratoire de Génie Chimique, Université de Toulouse, CNRS, INPT, UPS, Toulouse, France

<figure><img src="image_3.png" /></figure>

## ARTICLE INFO

### Keywords:

Additive manufacturing

Innovative packing

Gas liquid contactor

TPMS

Complex structures

Liquid holdup

## ABSTRACT

Additive manufacturing has widened the possibilities of design for gas-liquid packing geometries in recent years, allowing for new unexplored shapes. A novel design approach based on triply periodic minimal surfaces (TPMS<fnref n="1" />), an interesting class of smoothened curved surfaces, led to the conception of new periodic structures with complex geometrical features. The present study aims to assess the applicability of these new TPMS-based geometries as packing for gas-liquid contactors and the effect of their topology on performance. Liquid holdup, pressure drop, flooding and mass transfer efficiency were evaluated for three TPMS-based prototypes, with distinct characteristic dimensions. A new experimental setup was developed for the estimation of liquid holdup based on the measure of the liquid weight during operation by a set of load cells. Results show good repeatability and precision, allowing to distinguish liquid holdup between the prototypes for various gas and liquid flow rates. Two of the studied prototypes presented higher holdup than the 5 ppi ceramic foam and the 15 mm Raschig rings for lower liquid rates. Prototypes also showed good results for pressure drop, with flooding reached at gas load factors above 3 Pa⁰.⁵. In terms of mass transfer efficiency, HETP results for the new prototypes remained between 0.35m and 0.45m, comparable to Mellapak 250Y.

## 1. Introduction

In recent years, the emerging use of additive manufacturing (AM<fnref n="2" />) in chemical engineering have brought new possibilities in terms of complex design systems. The shorter time span between conception, production and experimental validation of small-scale devices accelerates the design optimization procedure with lower costs. The conception of 3D-printed structures has surged as a key technique mainly first seen in reactor engineering, with periodic open cellular structures (POCS<fnref n="3" />). These spatially uniform structures showed improvements in gas-liquid performance compared to state-of-the-art geometries such as foams, for instance, better liquid distribution and lower pressure drop (Ferroni et al., 2022; Klumpp et al., 2014). Regarding gas-liquid separation technologies, performance benchmark relies on the compromise of lower pressure drop while ensuring high mass transfer efficiency, often compared to corrugated metal sheet such as MellapakPlus® (Sulzer®) or Montz-Pak B1® (Koch-Glitsch®). Several studies have replicated and

adapted industry-standard structured packings produced by 3D printing, comparing experimental results with corrugated metal sheet packing (Bolton et al., 2019; Riese et al., 2022; Sun et al., 2021).

Other studies however have focused on the conception of POCS-based packing for gas-liquid separation. Lämmermann et al. (2016) proposed a hybrid form of POCS unit cell based on the combination of the Diamond and Kelvin cell geometries (Lämmermann et al., 2016). The hybrid geometry, with higher surface area, enhanced liquid distribution compared to results from the original structures. Similarly, a methodology for the development of AM-produced packing by Neukäufer et al. (2019) highlighted the versatility of potential shapes for POCS unit cells, involving both CFD simulation and experimental evaluation to optimize and validate new geometries (Neukäufer et al., 2019). In the Chemical Engineering Laboratory of Toulouse (LGC<fnref n="4" />), the performance of wire-based packing was studied regarding hydrodynamics and mass transfer. Results pointed out the strong potential of these structures as gas-liquid contactors for their effect of increasing

<footnote>* Corresponding author at: Laboratoire de Génie Chimique, 4 Allée Emile Monso CAMPUS INP – ENSIACET, 31400 Toulouse, France
*E-mail address: gregory.homemalvescesarribeiro@toulouse-inp.fr* (G. Homem Alves César Ribeiro).</footnote>

<footnote n="1">¹ Triply periodic minimal surfaces.</footnote>

<footnote n="2">² Additive manufacturing.</footnote>

<footnote n="3">³ Periodic open cellular structures.</footnote>

<footnote n="4">⁴ Laboratoire de Génie Chimique de Toulouse / Chemical Engineering Laboratory of Toulouse.</footnote>
gas-liquid interfacial area due to the cylindrical curved shape as compared to flat surfaces (Dejean et al., 2020; Kawas et al., 2021). In these studies, the observation and classification of wetting regimes as function of packing dimensions and liquid flow rate put in evidence the impact of the packing morphology on the wetting rate. A good wetting would correspond to the condition in which a stable liquid film completely covers the packing surface, an ideality often not achieved in practice for larger scale contactors.

On the urge of exploiting novel geometries, a particular class of surfaces, known as triply periodic minimal surfaces (TPMS), have gained relevant interest in process engineering, aiming mostly for heat transfer purposes (Al-Ketan et al., 2021). The self-supported intertwined and smoothly curved shape of TPMS led to its interest also for gas-liquid systems, namely for CO₂ capture technologies. Periodic in all three directions, TPMS divide space in two separate domains and can become actual objects by placing a non-zero thickness wall on its surface or filling one side with solid. Gas-liquid pressure drop and liquid distribution of TPMS were first assessed by CFD simulation by Fu et al. (2019), suggesting the possibility of reaching high interface area, which were later endorsed by experimental results (Fu et al., 2019; Singh et al., 2022). In terms of mass transfer efficiency, TPMS canonical forms such as Gyroid and Diamond showed promising results with flooding conditions comparable to Mellapak 250Y. In light of the interesting features of the TPMS, a study of our group presented a novel design approach based on the modification of their characteristic surface equations. In a first stage, a new surface was derived by combining mathematical terms from canonical TPMS, forming an octahedral lattice with cylindrical-like struts as connections. Later on, the surface equation was adapted in order to control its characteristics dimensions such as strut thickness and orientation. Finally, the study evaluated three TPMS-based prototypes, each with distinct dimensions using CFD simulation for gas-solid pressure drop. Gas flow behavior was also assessed through gas velocity profile on the packing section as well as parameters associated with dispersion such as residence time and tortuosity. Considering the final use of these new prototypes for gas-liquid separation systems, the present study focuses on their experimental evaluation in terms of gas-liquid pressure drop, flooding, liquid holdup and mass transfer efficiency. An experimental pilot was conceived for the simultaneous measurement of pressure drop and liquid holdup for countercurrent gas-liquid systems and is discussed in Section 2. Results are further discussed in Section 3 considering the effect of each characteristic prototype dimension as well as a comparison of the prototypes with standard packing.

## 2. Materials and methods

### 2.1. Experimental setup

Among the evaluated parameters, the liquid holdup provides an indication of the quality of contact between the liquid and the solid and is of particular interest for reactive systems. For reactive distillation

systems, a potential use of the TPMS-based prototypes, ensuring a higher liquid holdup might lead to better conversion. The estimation of liquid holdup in packed towers may be carried out through various methods, which can be classified by their observation scale (Al-Dahhan and Highfill, 1999). Integral measurement methods provide information about the overall bed volume and are the most commonly used, including drainage, weighting and tracer methods. Semi-integral measurement methods are based on the measure of a section of the bed volume, namely through absorbance or electrical conductivity through a line or grid. Furthermore, local measurement methods can provide information at specific points of the volume through a probe or by computed tomography. Regarding integral measurement techniques, Table 1 presents a summary of the advantages and drawbacks of all three main methods, the required equipment and literature references demonstrating their application.

An experimental pilot was designed based on the weighting method, measuring the weight of the packing during gas-liquid operation to estimate liquid holdup. The arrangement allows for the simultaneous measurement of pressure drop and liquid holdup for consecutive tests at different flow rates without operation shutdown. Also, as the weight is measured constantly with a short timestep and high precision, a better observation of the system stabilization is provided with more complete statistical information on the measure. The setup is composed of a glass column of 150 mm diameter and length of 1 m containing the packing to be studied. The system scheme is given in Fig. 1A and an image of the weighted system is shown in Fig. 1B. The weighted system, including the packing and column glassware, is suspended over a symmetric set of three load cells measuring its total weight over time. Each load cell is composed of a stainless-steel flex sensor of 50 kg maximum capacity, with adjusted precision of 10 g and measure timestep of one second (ARPEGE MASTERK©). A custom metal rim was conceived to support the weighted column over the balanced triangular set of load cells. The chosen configuration in which the system is suspended by the load cells at its upper end and not entirely positioned over a scale improves the measurement stability as the system's own weight prevents any disturbances. The system of load cells is shown in Fig. 1C.

A dry air flow is inserted at the bottom of the column, passing through the packing and exiting at the top of the column. Air flow rate is read by an upstream flowmeter and a manometer. Pressure drop is measured between bottom and top extremities of the column through a Keller-Series-30 differential pressure transmitter. Liquid is pumped from a container into the top of the column and is then recycled back into the feed container in a closed loop. A tree-like liquid distributor developed at the LGC is used for the experiments and is described in a previous work (da Cunha et al., 2023). A hydraulic syphon system is positioned at the column bottom before the liquid outlet and is used to estimate the residual liquid holdup, which is accounted in the measured weight but does not represent effective packing holdup. A graduated column is used to measure the liquid level, which can be controlled by adjusting the height of the hydraulic syphon. The liquid mass retained at the syphon can then be subtracted from the total measured weight. Moreover, as the

<table><caption>**Table 1**

Integral measurement methods for liquid holdup.</caption><thead><tr><td>Method</td><td>Advantages</td><td>Drawbacks</td><td>Required equipment</td><td>Refs.</td></tr></thead><tbody><tr><td>Drainage</td><td>■ Minimal modification</td><td>■ Disturbed operation, low precision for static liquid holdup, error due to residual volumes (e.g. piping)</td><td>■ Adapted scale, instantaneous closing valves for both gas and liquid inputs</td><td>(Al-Dahhan and Highfill, 1999; Ellenberger and Krishna, 1999; Heymes et al., 2006; Riese et al., 2022; Xu et al., 1997)</td></tr><tr><td>Weighting</td><td>■ Undisturbed operation, multipoint measurement</td><td>■ Potential instability of weighted system</td><td>■ Custom weighting system (e.g. scale or load cells)</td><td>(Al-Dahhan and Highfill, 1999; Crine and Marchot, 1981; Levec et al., 1986; Nemec et al., 2001; Shulman et al., 1955; van Baten et al., 2001; Zakeri et al., 2012)</td></tr><tr><td>Tracer</td><td>■ Undisturbed operation, RTD curve for hydrodynamic studies</td><td>■ Risk of fouling, repeatability of tracer injection, representative measurement of output</td><td>■ Tracer injection system, signal measurement device (e.g. conductometer)</td><td>(Alicilar et al., 1994; Kolodziej et al., 2005; Muzen and Cassanello, 2005; Schubert et al., 1986)</td></tr></tbody></table>
<figure><img src="image_4.png" /><figcaption>Fig. 1. Experimental pilot for pressure drop and liquid holdup measurements: (A) Setup scheme; (B) Weighted system containing the packing; (C) Column top and the system of load cells.</figcaption></figure>

air inlet direction is perpendicular to the column axis, a force is added up to the measured weight as the air pushes the column. The added force is constant for each air flow rate and is remeasured after every change of packing as it depends on the mechanical stress of the connecting bolts between the supporting rim and the suspended column. This value is also subtracted from the measured weight.

Once the packing is installed and the system is still dry, the load cells are tared so that only the net liquid weight is measured. Load cells sensitivity is recalibrated near the smallest readable value to increase precision. Both gas-liquid pressure drop and liquid holdup values are transmitted and registered. Experiments consist on varying the air input velocity on a range of 0 to $3m.s^{-1}$ for each value of liquid flow rate from 8.5 to $21 m^3.h^{-1}.m^{-2}$. The applied liquid is distilled water. Before gas-liquid studies, a gas-solid test is carried out to estimate dry pressure drop and the effect of air entry on the weight measure, as previously explained. Liquid is then pumped into the column at high flow rate and the weight is monitored until its value becomes stable. For each pair of liquid and gas flow rate, the weight and pressure drop are registered during ten minutes to ensure stability. Air flow rate is progressively increased until flooding is reached and operation is shut down. For both measured parameters, an average value is considered as the measure is stable.

Another experimental setup was used to evaluate geometries in terms of HETP (height equivalent to theoretical plate). This parameter is an indicator of the mass transfer efficiency of a gas-liquid contactor packing and gives an approximative size of packing that corresponds to one theoretical separation plate. Lower values of HETP imply the structure is capable of a better gas-liquid separation efficiency with a smaller column size, which is desired for packing design. The experimental setup to measure HETP is a distillation column, operating at total reflux, composed by a 150 mm glass column containing one meter of packing height. The reboiler, containing a known mixture of cyclohexane and n-heptane, is heated by a feed vapor stream at 8 bar. The heat duty is controlled by a pressure control valve, regulated by a pressure drop set point. Pressure drop is measured between the upper and lower parts of the packing. Once reached steady state, with bottom and top temperatures stable, samples are collected from liquid distillate, liquid bottom and vapor bottom products. Samples are then analyzed using a refractometer (Abbe ATAGO NAR-3 T) for molar fraction of each component. The number of theoretical plates achieved by the structure is estimated using the Wilson thermodynamic model for liquid vapor equilibrium curve and the molar fractions observed at the bottom and

top of the column. The packing total height is then divided by the number of theoretical plates, resulting in the HETP, expressed in meters. The HETP pilot is described in more detail in a previous study of our group (Kawas et al., 2021).

## 2.2. Evaluated packing geometries

The novel design approach is based on the modification of the equations describing the TPMS. A novel structure was derived from the canonical I-WP surface and its unit cell is shown in Fig. 2. This structure presented interesting geometric features such as the corrugations over the struts and the hollow cubic body appearing at the center of the unit cell. The characteristic equations of the I-WP structure and the derived TPMS are given in Equations (1) and (2), respectively. The isosurface constant, which corresponds to the right-side term of the surface equation, equals 2.9 for the modified TPMS. The appearance of the central cavity in the structure allowed for higher values of specific surface area and porosity compared to canonical TPMS for the same unit cell size, highlighting the potential of this new complex shape.

$$F_{I-WP}(x, y, z) = 2(\cos x \cos y + \cos y \cos z + \cos x \cos z) - (\cos 2x + \cos 2y + \cos 2z) \quad (1)$$

$$ \begin{aligned} F_{I-WP}(x, y, z) &+ 0.0976(\cos 4x \cos 4y \cos 4z + \sin 4x \sin 4y \sin 4z) \\ &- 0.0732(\cos 2x + \cos 2y + \cos 2z) \\ & = 2.83 \end{aligned} \quad (2) $$

<figure><img src="image_5.png" /><figcaption>Fig. 2. Modified TPMS structure.</figcaption></figure>
The characteristic equation of the newly-found TPMS was subsequently adapted to control the strut diameter, strut orientation and the width of the central cavity. Also, a global axial deformation of the structure was applied to prevent free gas passages in the axial direction. An example of the control of characteristic dimensions is shown in Fig. 3. The strut diameter is increased in Fig. 3A, accordingly for the strut-vertical axis angle in Fig. 3B, for the cavity size in Fig. 3C and for the axial deformation of the structure in Fig. 3D. The equation-based design approach proved to be capable of fine tuning the structure dimensions and provided new possibilities of design. Two key techniques were used to adapt the equations: changing the scaling factor of each spatial coordinate x, y and z and changing the values of the isosurface constant (right-side term). Replacing these two terms with precisely chosen mathematical expressions led to a broader control of the unit cell characteristics. A detailed description of the mathematical tools applied in this design methodology is provided in another study by our group.

In regard of the new design possibilities and the previous experimental results on wire-based lattices (Dejean et al., 2020; Kawas et al., 2021), three variations of the TPMS-based structure were evaluated in this study. The modified parameters were the strut diameter *D* and the size of the cell central cavity *L*₁ as shown in Fig. 4A. The cell central cavity is of interest to increase the liquid holdup, aiming in particular applications in reactive distillation. From *Prototype 1* to *Prototype 2*, the strut diameter is reduced, with specific surface area kept constant. *Prototype 3* has the same strut diameter and specific surface area as *Prototype 2* but with larger cavity size. All three prototypes present the same strut orientation (angle between vertical axis and struts). The characteristic dimensions of the TPMS-based prototypes are given in Table 2. Specific surface area is kept constant by adjusting the size of the unit cell. The characteristic dimensions of the prototypes are the unit cell horizontal size *L*, unit cell vertical size *L*<sub>z</sub>, the strut diameter *D*, the angle between struts and the vertical axis θ and the outer central cavity size *L*<sub>1</sub>, indicated in Fig. 4A. Visualization of the TPMS-based prototypes and creation of the CAD files for 3D-printing was carried out using the software nTopology (version 4).

Cylindrical pieces of the studied geometries were printed by Materialise© using selective laser sintering (SLS) technology with Polyamide-12. Each printed packing element measures 145 mm in diameter and presents variable height around 150 mm, resulting in 6 pieces in total. Packing total height varied then from 800 mm to 900 mm as function of the studied geometry, to ensure an integer number of unit cell repetitions. A CAD model of the packing piece for *Prototype 1* and the 3D-printed packing piece are shown in Fig. 4B and 4C, respectively. X-ray

computed tomography of the printed structures revealed a good 3D printing quality with accurate representation of the CAD model and no defects despite the complex shape.

## 3. Results and discussion

### 3.1. Liquid holdup

#### 3.1.1. Validation of pilot accuracy

Visual inspection from preliminary tests for liquid holdup measurement pointed out the presence of liquid external to the packing volume, which must also be subtracted from the measured weight. This residual liquid holdup can be estimated by running the same range of gas and liquid load ranges with the empty column. Results prove that a non-negligible liquid holdup is cumulated at the peripheral regions of the measured system, namely the column bottom glass piece and the liquid distributor. The residual holdup is likewise dependent on the liquid flow rate. In order to establish a coherent value for each liquid load, an exponential fit was derived from several estimations, as shown in Fig. 5. Most measured values are within an error margin of 5 % of the exponential fit model. Different values obtained for the same liquid load can be explained by the hysteresis effect of the system wetting. Thus, the need of a well-established residual liquid holdup curve to be used for all further experiments.

Considering the novelty of the measuring system arrangement for liquid holdup, it is necessary to validate its accuracy by comparing its results with available data from literature for commonly studied packings. For this purpose, the evaluated packing consisted of glass Raschig rings of dimension 15 mm and ceramic foams of 5 ppi. Characteristics of evaluated benchmark packings are given in Table 3. The Kelvin cell geometry was also evaluated and results will be discussed later in Section 3.1.2. For Raschig rings, experimental results were compared with the established liquid holdup model from Otake et Okada (Otake and Okada, 1953). The model considers the total liquid holdup as the sum of static and dynamic liquid holdup. The former corresponds to the liquid volume retained by the packing with no gas or liquid flow, by effect of viscosity and surface tension. The latter is the liquid volume retained by the packing during gas-liquid operation. The contribution of static liquid holdup is derived from the general correlation using the Eötvos number given by Kramer et al. (Kramer, 1998). Dynamic holdup is estimated from a correlation based on various sizes of Raschig rings. Total liquid holdup before the loading point is then evaluated for a liquid load range of 8.5 to 11.3 $m^3 \cdot h^{-1} \cdot m^2$ and compared to the

<figure><img src="image_6.png" /><figcaption>**Fig. 3.** Example of modifications on characteristic dimensions by the equation-based approach.</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>Fig. 4. (A) The three evaluated TPMS-based prototypes and their characteristic dimensions; (B) The CAD model of the packing piece (Prototype 1); (C) 3D-printed packing piece in Polyamide-12 (Prototype 1).</figcaption></figure>

<table><caption>**Table 2**
Dimensions of the evaluated prototypes.</caption><thead><tr><th>Prototype</th><th>1</th><th>2</th><th>3</th></tr></thead><tbody><tr><td>Unit cell horizontal size L [mm]</td><td>12.45</td><td>11.5</td><td>12.15</td></tr><tr><td>Unit cell vertical size L<sub>z</sub> [mm]</td><td>24.9</td><td>23</td><td>24.3</td></tr><tr><td>Strut diameter D [mm]</td><td>2.25</td><td>1.6</td><td>1.6</td></tr><tr><td>Strut-vertical axis angle θ [°]</td><td>35</td><td>35</td><td>35</td></tr><tr><td>Outer cavity size L<sub>1</sub> [mm]</td><td>4.3</td><td>4.3</td><td>6.6</td></tr><tr><td>Void rate ε [%]</td><td>85</td><td>89</td><td>89</td></tr><tr><td>Specific surface area a [m<sup>-2</sup>]</td><td>234</td><td>234</td><td>234</td></tr></tbody></table>

<table><caption>**Table 3**
Characteristics of evaluated benchmark packings.</caption><thead><tr><th>Packing</th><th>Material</th><th>Dimension</th><th>Specific surface area</th><th>Void rate</th></tr></thead><tbody><tr><td></td><td></td><td></td><td>m<sup>2</sup>.m<sup>-3</sup></td><td>%</td></tr><tr><td>Raschig rings</td><td>Glass</td><td>15 mm</td><td>310</td><td>69.0 %</td></tr><tr><td>Foam</td><td>Ceramic</td><td>5 ppi</td><td>640</td><td>92.0 %</td></tr><tr><td>Kelvin cell</td><td>Aluminum</td><td>17.45 mm</td><td>268</td><td>86.4 %</td></tr></tbody></table>

literature correlation. All measures show good accordance with the predicted values, within a range of 5 % deviation, as shown in Fig. 6. Moreover, total liquid holdup of 5 ppi ceramic foam was measured

<figure><img src="image_8.png" /><figcaption>Fig. 5. Residual liquid holdup.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 6. Liquid holdup of 15 mm Raschig rings – Experimental and literature model.</figcaption></figure>
for liquid loads of 8.5 and 11.3m³·h⁻¹·m² and compared with two sets of literature data for lower (Lévêque et al., 2009) and higher (Stemmet et al., 2006) ranges of liquid load. Substantially higher liquid flow rates were used in the latter study as it aimed to evaluate the transition from trickle to pulse and bubble flow. In order to compare the liquid holdup tendency with respect to liquid flow rate, an exponential fit is derived from literature data. As shown in Fig. 7, the measured values follow the same trend as the literature values, within a range of 20 % deviation. Both studies, carried out with distinct packing geometries, demonstrate the good accuracy achieved by the weighting system as an estimator for liquid holdup.

### 3.1.1.1. Results for TPMS-based prototypes and other packings.
Results for total liquid holdup of the TPMS-based prototypes are provided in Fig. 8, at equivalent scale for comparison and shown as function of liquid flow rate and gas load factor. The error bars displayed consider the total propagated uncertainty due to load cell precision, standard deviation of the measure and user-dependent error on reading the liquid level of the hydraulic syphon. Measures are shown as function of gas load factor, which consists of a standard hydrodynamic indicator, used for commercial packing characterization. Eq. (3) provides the definition of gas load factor $F_g$, where $U_g$ is the superficial gas velocity and $\rho_g$ the gas density.

$$F_g = U_g \sqrt{\rho_g} \quad (3)$$

For all three prototypes, the liquid holdup remains mostly constant for lower gas loads and starts to increase from the loading point onwards. The loading point correspond to the condition at which the upward gas flow starts to interact with the falling liquid film and is reached at lower gas loads for higher liquid loads. For Prototype 1 and 3, the loading point is observed at a gas load of 1.4 $Pa^{0.5}$ whereas for Prototype 2 it is observed at 1.8 $Pa^{0.5}$. This behavior is also remarked with respect to flooding point. For 1 and 3, flooding is reached at 3 $Pa^{0.5}$ and for 2 flooding is not reached for the studied range of liquid load (up to 3 $Pa^{0.5}$). Compared to Prototype 1, Prototype 2 has the same central cavity size and specific surface area but with shorter and thinner struts. This last prototype has the highest values of liquid holdup among the three before the loading point. The observed behavior can be partly explained by its higher number of cavities per m³ of packing compared to 1 and 3, an increase of 26 % and 18 %, respectively. A larger volume of liquid is thus expected to be retained at the cavity in the unit cell compared to the liquid retained over the struts. The results also point out the effect of the enlargement of cavity size from Prototype 2 to Prototype 3 with a decrease of liquid holdup.

Fig. 9 represents liquid holdup values before the loading point as function of the number of cavities per packing volume for all three prototypes and at different liquid loads. Liquid holdup does not vary linearly with cavity density, which demonstrates that other geometrical features impacted this parameter. As Prototypes 2 and 3 have the same

strut diameter, it can be inferred from Fig. 9 that the larger cavity size of 3 contributed to increase liquid holdup as well. Furthermore, the slope of the liquid holdup curves with respect to gas load beyond the loading point is considerably higher for Prototypes 1 and 3. Such difference may be also related to the partition of liquid in the cavity and over the struts. The upward gas stream interacts with the falling liquid film over the struts by friction and leads to higher film thickness for higher gas loads. For Prototype 2, the liquid volume over the struts has a smaller contribution to total liquid holdup hence its weaker sensitivity to gas flow rate.

Experimental results for liquid holdup of TPMS-based prototypes (before loading point) are compared with the other evaluated benchmark packings in Fig. 10. Among all packings, the Kelvin cell presents the highest values of total liquid holdup reaching values above 10 %. Moreover, the impact of liquid load over holdup is not uniform for all packings. For Raschig rings and the ceramic foam, the liquid holdup increases faster with the increment of liquid flow rate, also reaching flooding at lower liquid loads compared to other packings. However, for operations requiring lower flow rate ranges, such as reactive distillation, the TPMS-based prototypes allow for higher values of liquid holdup compared to the ceramic foam, as for Prototype 2 reaching up to 8 %.

### 3.1.1.2. Static and dynamic liquid holdup.
The liquid holdup in packed columns is known to be the sum of two contributions, the static and the dynamic liquid holdup. The static liquid holdup consists of the liquid fraction that remains in the packing after operation is stopped. This portion of liquid is attached to the pores and narrow parts of the packing by effect of capillary forces. The dynamic holdup corresponds to the fraction of liquid actively flowing across the packing volume and is dependent on the liquid flow rate. It is known that static holdup has a considerably higher residence time than the dynamic holdup, thus is usually not sought to be predominant in the packing. The liquid holdup partition in static and dynamic was also evaluated in this study by the use of the presented experimental setup. After total liquid holdup was measured at different liquid loads and gas flow rates, as described in Section 2.1, the gas and liquid inlets were shutdown but the weight continued to be measured for at least two hours until it reached a stable value.

The tracking of the weight decay allows for a more precise choice of which value to consider for static holdup, as shown in Fig. 11 for Prototype 1. In this case, with no gas or liquid flow, only the liquid retained at the hydraulic syphon will be subtracted from the measured weight. For all three prototypes, the weight reached a stable value after one hour from shutdown. The experimental results for static holdup for the TPMS-based prototypes as well as the relative contribution of static in total holdup for all liquid loads before the loading point are shown in Table 4. The highest contribution of static liquid holdup is observed for Prototype 2, above 30 %, whilst for Prototype 3 it remains at approximately 10 %. This result indicates that a considerable portion of the observed liquid holdup for Prototype 2 corresponds to the liquid retained at narrow regions of the packing, particularly at the cavity. The hypothesis is

<figure><img src="image_10.png" /><figcaption>Fig. 7. Liquid holdup of ceramic foam (5 ppi) – Experimental and literature data.</figcaption></figure>
<figure><img src="image_11.png" /><figcaption>Fig. 8. Total liquid holdup of TPMS-based prototypes.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>Fig. 9. Total liquid holdup as function of number of cavities per packing volume.</figcaption></figure>

confirmed by the low static liquid holdup contribution for Prototype 3, which has a larger cavity size and thus does not retain as much liquid by capillary forces at the cavity. The liquid volume retained at the cavity is anticipated to present greater residence time and poorer mixing compared to other areas of the structure.

3.1.1.3. Modelling of total liquid holdup. Liquid holdup can be usually predicted for various types of packing based on empirical models and correlations available in literature, which require a large amount of data to build and validate. However, several models are based on a physical description of the falling liquid film and are further adjusted to consider phenomena such as gas–liquid interaction, wetting rate and geometrical features particular to each packing. One of the main models is presented by Billet et al. (Billet, 1995) for total liquid holdup $h_L$, estimated as in

Equation (4), where $u_L$ is the volumetric liquid flow rate per column section area and $a$ the packing specific surface area. The model brings a correction factor for the effective wetted area $a_h$ using a packing-specific coefficient $C_h$, as in Equations (5) and (6). The value of the coefficient $C_h$ is usually obtained empirically for each specific packing. The applied liquid Reynolds and Froude numbers are shown in Equations (7) and (8).

$$h_L = \left( \frac{12\mu_L u_L a^2}{g\rho_L} \right)^{1/3} \left( \frac{a_h}{a} \right)^{2/3} \quad (4)$$

$$\left(\frac{a_h}{a}\right)_{Re_L<5} = C_h Re_L^{0.15} Fr_L^{0.1} \quad (5)$$

$$\left(\frac{a_h}{a}\right)_{Re_L>5} = 0.85 C_h Re_L^{0.25} Fr_L^{0.1} \quad (6)$$
<figure><img src="image_13.png" /><figcaption>Fig. 10. Total liquid holdup of various packings (before loading point) as function of liquid flow rate.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>Fig. 11. Measured weight after shutdown for estimation of static liquid holdup for Prototype 1.</figcaption></figure>

<table><caption>**Table 4**

Results for static liquid holdup for TPMS-based prototypes.</caption><thead><tr><th rowspan="2">Prototype</th><th>Static liquid holdup</th><th colspan="4">Contribution of static in total holdup [%]as function of liquid flow rate [m<sup>3</sup>·h<sup>-1</sup>·m<sup>-2</sup>]</th></tr><tr><th>%vol</th><th>8.5</th><th>12.7</th><th>17.0</th><th>21.2</th></tr></thead><tbody><tr><td>1</td><td>1.1 %</td><td>17 %</td><td>16 %</td><td>16 %</td><td>15 %</td></tr><tr><td>2</td><td>3.0 %</td><td>38 %</td><td>36 %</td><td>34 %</td><td>32 %</td></tr><tr><td>3</td><td>0.8 %</td><td>11 %</td><td>10 %</td><td>10 %</td><td>9 %</td></tr></tbody></table>

$$Re_L = \frac{\rho_f u_L}{a \mu_L} \qquad (7)$$

$$Fr_L = \frac{u_L a^2}{g} \qquad (8)$$

Considering the wetted area $a_h$ as equivalent to specific surface area $a$, it is possible to estimate the maximum value of liquid holdup (total wetting) for the TPMS-based prototypes using Billet's model. As all three prototypes have the same specific area, the model would predict equivalent liquid holdup. Another approach is to predict the liquid holdup through the use of numerical CAD tools, by adding an artificial layer of liquid over the packing surface. The thickness of a falling liquid film around a cylindrical shape can be calculated for a given liquid flow rate according to the Nusselt approach. After adding the artificial static liquid layer over the packing surface, the void rate of the packing is numerically recalculated by the CAD tool, giving an estimation of the corresponding liquid holdup. However, the TPMS-based prototypes present the hollow central cavity that is filled with the same amount of liquid independently of liquid load. Thus, the liquid volume within the cavity has to be considered separately from the liquid retained over the struts. A visual representation of the cavity being artificially filled with liquid by the CAD tool is shown in Fig. 12. Liquid holdup is estimated by the CAD-based numerical approach for the studied range of liquid loads. Experimental results for *Prototype 1* are compared with the Billet model results and the geometrical model results in Fig. 13. As the CAD-based geometrical model considers an ideal complete wetting of the packing, a correction factor was multiplied to the calculated value and curves are shown in Fig. 13.

As can be observed in Fig. 13, the sensitivity of liquid holdup with respect to liquid load is better described by the geometrical model curves. Moreover, using a wetting correction factor of 75 %, the geometrical model fits the experimental values within their error margin for *Prototype 1*. Normally, the effective wetted area is dependent on liquid flow rate as shown by Eqs. (3) and (4), but a good accuracy is achieved with a constant correction value for wetting in this case. Another key result obtained from the geometrical model is the partition
<figure><img src="image_15.png" /><figcaption>**Fig. 12.** Representation of Prototype 1's unit cell: (A) Dry; (B) Cavity artificially filled with liquid by CAD tool.</figcaption></figure>

of liquid holdup in the cavity and over the struts, with the former accounting for 40–50 % of the total holdup. This value is in accordance with the effect of cavity density on liquid holdup, as previously discussed. The applicability of the geometrical model to accurately predict liquid holdup for a TPMS-based structure is yet to be improved as the correction factor may vary for each structure. However, its main function in this study is to endorse the hypothesis of the cavity contribution in the liquid holdup.

## 3.2. Pressure drop

For all three prototypes, pressure drop was measured for the dry system and at different values of liquid load. The pressure drop value for each condition is obtained after the filtered average value no further

varies based on a Student's t-test. The upper value of gas load for each liquid load corresponds to the imminent condition to flooding. Fig. 14 shows the experimental results for dry and gas-liquid pressure drop for the three TPMS-based prototypes. For all prototypes, the gas-liquid pressure drop curves present Ergun-like behavior, e.g. quadratic with respect to gas velocity, for gas loads below the loading point. As observed for the liquid holdup curves, the loading point of Prototype 2 is reached at a higher gas load compared to other prototypes.

Gas-liquid pressure drop curves show strong similarity for all three prototypes. An average increase of 30–40 % in pressure drop is observed from liquid load of 8.5 to 21.2 $m^3.h^{-1}.m^2$. The presence of liquid has stronger impact on pressure drop for Prototype 3, as an average increase of 230 % is observed between the dry system and the lowest liquid load value. For Prototypes 1 and 2, this increase is of 190 %. The stronger effect of liquid load over the pressure drop can be explained by the liquid holdup results. For the dry system, the pressure drop presents the lowest values due to the larger cavity size of Prototype 3 and higher void rate. As inferred from liquid holdup results, the larger cavity contributes for higher liquid volume retained at the cavity and thus gas stream passages are more disturbed compared to Prototype 2. For purpose of comparison, pressure drop results for Prototype 1 are compared with other evaluated benchmark packings in Fig. 15, for a liquid flow rate of 8.5 $m^3.h^{-1}.m^2$. Raschig rings are expected to produce higher pressure drop levels due to their low void rates. However, both Kelvin cell and ceramic foam present higher void rates compared to Prototype 1 and generate higher pressure drop. These results are in accordance with the observed liquid holdup performance for the various packings and point out the compromise between achieving high liquid holdups and pressure drop.

## 3.3. Mass transfer efficiency

As for mass transfer efficiency, HETP results for all three prototypes are shown in Fig. 16. Lower HETP values indicate higher mass transfer efficiency of the structure. All three prototypes present the same specific surface area, thus the observed differences in mass transfer efficiency should be related to distinct wetting behavior caused by the particular geometrical features of each prototype. It is to be noted that HETP

<figure><img src="image_16.png" /><figcaption>**Fig. 13.** Total liquid holdup of Prototype 1: Experimental, geometrical model and Billet model.</figcaption></figure>
<figure><img src="image_17.png" /><figcaption>Fig. 14. Pressure drop results for TPMS-based prototypes.</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>Fig. 15. Pressure drop results for Prototype 1 and other packings at liquid flow rate of..8.5m³·h⁻¹·m²</figcaption></figure>

measurements are carried out at total reflux for each gas load factor, implying that liquid flow rate increases accordingly, unlike for other measurements shown in this study. *Prototype 1* presents the lowest HETP of all three prototypes for the entire gas load range. The loss in mass transfer efficiency from *Prototype 1* to *2* can be explained by the change in partition of liquid holdup between the cavity and over the struts. Despite the observed higher liquid holdup values for *Prototype 2*, the liquid is mostly retained at the cavity and effective interfacial area is reduced. Contribution of static liquid holdup also corroborates with this hypothesis. Furthermore, thinner strut diameter may lead to lower interfacial area over the struts and stronger instability of the falling liquid film. Liquid detachment reduces interfacial area and hinder mass transfer. From *Prototype 2* to *3*, a slight improvement in HETP can be observed for gas loads below flooding point. Here, it is the cavity size enlargement that allowed for higher mass transfer, despite of the

decrease of strut length. It can be inferred that the wider cavity section prevented the cavity to be filled with liquid for lower gas loads. The remaining deviation between *Prototype 1* and *3* may be due to smaller strut diameter and length, undermining mass transfer over the struts. Aiming towards HETP improvement, a potential fourth prototype would have the same strut diameter and length as *Prototype 1* but with cavity size as *Prototype 3*.

Furthermore, a different HETP profile is observed for prototypes with smaller strut diameter, reaching a maximum at the middle of gas load range and then decreasing near flooding point. Usually, HETP remains independent from gas load factors for structured packing. A change of liquid film flow regime may take place as liquid load increases which would explain the non-uniformity of HETP for TPMS-based structures. Similar trend was observed in a previous study in wire-based packing structures (Kawas, 2021). However, results for *Prototype 2* show that
<figure><img src="image_19.png" /><figcaption>Fig. 16. HETP results for TPMS-based prototypes and Mellapak 250Y.</figcaption></figure>

liquid holdup does not present a strong sensitivity to gas load near the loading point, hence suggesting that mass transfer takes place primarily over the struts. Experimental data for HETP of TPMS-based prototypes are also compared with industry standard Mellapak 250Y in Fig. 16. Prototype 1 shows slightly lower HETP in comparison to this structured packing. However, wet pressure drop for Mellapak 250Y is notably lower and flooding point reached at higher gas loads. Regarding the other benchmark packings evaluated in this study, the TPMS-based structures do not reach the same level of mass transfer efficiency. For instance, at a gas load of $1Pa^{0.5}$, experiments with the 15 mm Raschig rings and the Kelvin cell showed HETP values of 0.25m whereas for the 5 ppi ceramic foam it reaches 0.19m (Al-Maqaleh, 2022; Lévèque et al., 2009). A systematic comparison between the studied structures is provided in Table 5. The results reveal that the TPMS-based prototypes present strong performance in terms of pressure drop (as well as flooding) and liquid holdup but with lower levels of mass transfer efficiency compared to benchmark structures. A compromise is thus observed for the three evaluated performance parameters: higher mass transfer and holdup usually require higher pressure drop and flooding limitations. An adequate choice of the geometrical parameters investigated in this study (unit cell size, strut diameter and cavity size) should be able to result in an optimized structure, with enhanced mass transfer while keeping the advantage of pressure drop and liquid holdup.

## 4. Conclusions and perspectives

A novel design approach based on triply periodic minimal surfaces

<caption>Table 5</caption>

Comparison of various packing structures in terms of pressure drop, liquid holdup and mass transfer.

<table><thead><tr><th>Packing</th><th>Pressure drop<sup>a</sup></th><th>Total liquid holdup<sup>a</sup></th><th>HETP<sup>b</sup></th></tr><tr><td></td><td>mbar/m</td><td>%vol</td><td>m</td></tr></thead><tbody><tr><td>Raschig rings (15 mm)</td><td>8</td><td>5.3 %</td><td>0.25<sup>1</sup></td></tr><tr><td>Ceramic foam (5 ppi)</td><td>2</td><td>6.5 %</td><td>0.19<sup>1</sup></td></tr><tr><td>Kelvin cell (17.45 mm)</td><td>4</td><td>9.8 %</td><td>0.25<sup>2</sup></td></tr><tr><td>Mellapak 250Y</td><td>0.3<sup>3</sup></td><td>3.0 %<sup>3</sup></td><td>0.4</td></tr><tr><td>Prototype 1</td><td>0.5</td><td>6.3 %</td><td>0.38</td></tr><tr><td>Prototype 2</td><td>0.6</td><td>8.0 %</td><td>0.51</td></tr><tr><td>Prototype 3</td><td>0.5</td><td>7.5 %</td><td>0.44</td></tr></tbody></table>

(TPMS) led to the conception of new gas liquid column packing prototypes with interesting geometrical features. The unit cell of the structures is composed of cylindrical-like struts connecting to a central hollow cavity. Among the characteristic dimensions, the diameter of the struts and the size of the central cavity were chosen to be analyzed. In order to assess the effect of these features on the packing's performance, the present study proposed to experimentally evaluate liquid holdup, pressure drop and mass transfer of three TPMS-based prototypes varying the chosen parameters and with equal specific surface area. With respect to liquid holdup, the results showed that as the number of cavities per volume increased, the total holdup increased accordingly. Moreover, results indicated that increasing the cavity size also contributed to drive liquid holdup. In regard to pressure drop, all three prototypes showed similar values for the studied range of gas and liquid loads. The loading and flooding points were reached at higher gas load factors for the prototype with higher liquid holdup. Geometric modeling of the liquid film with CAD tools and experimental evaluation of the static holdup of the prototypes suggested that a considerable portion of liquid is retained at the cavity. Results for HETP, on the other side, showed that the prototype with the highest holdup was the least efficient in mass transfer. This result confirms that the retained liquid at the cavity has a lower contribution to mass transfer compared to the liquid portion flowing over the struts. In comparison with benchmark column packings, also evaluated in this study, the TPMS-based prototypes showed better pressure drop, flooding and liquid holdup than the Raschig rings and the ceramic foam for lower liquid loads. However, in spite of a higher pressure drop, the Kelvin cell shows the highest range of liquid holdup and better mass transfer efficiency.

The design possibilities of the TPMS-based packings allow for its adaptation towards different gas-liquid applications, such as reactive distillation, concentric heat integrated distillation columns, CO<sub>2</sub> capture and catalytic supports. In future studies, the application of the TPMS-based prototypes as heterogeneous reactive distillation packings will be evaluated in a pilot scale setup, in order to demonstrate its synergistic effect on reaction yield and separation efficiency. Furthermore, the equation-based approach brings up the possibility of conceiving an anisotropic packing structure, by establishing a spatial profile for its characteristic dimensions, aiming to improve hydrodynamics, reaction rate, mass transfer and also heat transfer in the case of heat integrated columns. The methodology and required mathematical tools for the design of anisotropic TPMS-based structures will be discussed later in a future study.

### Nomenclature

<table><tr><td>a</td><td>Specific surface area (m<sup>2</sup>·m<sup>-3</sup>)</td></tr><tr><td>a<sub>h</sub></td><td>Effective specific surface area (m<sup>2</sup>·m<sup>-3</sup>)</td></tr><tr><td>C<sub>h</sub></td><td>Packing-specific coefficient for wetting rate correlation (–)</td></tr><tr><td>D</td><td>Strut diameter (m)</td></tr><tr><td>F<sub>g</sub></td><td>Gas load factor (Pa<sup>0.5</sup>)</td></tr><tr><td>Fr<sub>L</sub></td><td>Liquid Froude number</td></tr><tr><td>g</td><td>Gravitational acceleration (m·s<sup>-2</sup>)</td></tr><tr><td>h<sub>L</sub></td><td>Total liquid holdup (%vol/vol)</td></tr><tr><td>L</td><td>Unit cell horizontal size (m)</td></tr><tr><td>L<sub>z</sub></td><td>Unit cell vertical size (m)</td></tr><tr><td>L<sub>1</sub></td><td>Outer size of the central cavity (m)</td></tr><tr><td>Re<sub>L</sub></td><td>Liquid Reynolds number (–)</td></tr><tr><td>u<sub>L</sub></td><td>Superficial liquid velocity (m·s<sup>-1</sup>)</td></tr><tr><td>U<sub>g</sub></td><td>Superficial gas velocity (m·s<sup>-1</sup>)</td></tr><tr><td>ε</td><td>Void rate (%)</td></tr><tr><td>θ</td><td>Angle between struts and the vertical z axis (°)</td></tr><tr><td>μ<sub>L</sub></td><td>Liquid viscosity (Pa·s)</td></tr><tr><td>ρ<sub>g</sub></td><td>Gas density (kg·m<sup>-3</sup>)</td></tr><tr><td>ρ<sub>L</sub></td><td>Liquid density (kg·m<sup>-3</sup>)</td></tr><tr><td>AM</td><td>Additive manufacturing</td></tr><tr><td>CAD</td><td>Computer-aided design</td></tr><tr><td>HETP</td><td>Height equivalent to theoretical plate</td></tr><tr><td>LGC</td><td>Laboratoire de Génie Chimie de Toulouse</td></tr><tr><td>POCS</td><td>Periodic open cellular structure</td></tr><tr><td>TPMS</td><td>Triply periodic minimal surface</td></tr><tr><td>3D</td><td>Three-dimensional</td></tr></table>

<footnote><sup>a</sup> Gas load factor of 1 Pa<sup>0.5</sup>, liquid flow rate of 8.5 m<sup>3</sup>·h<sup>-1</sup>·m<sup>-2</sup>; <sup>b</sup> Gas load factor of 1 Pa<sup>0.5</sup>, liquid flow rate of 5 m<sup>3</sup>·h<sup>-1</sup>·m<sup>-2</sup>; <sup>1</sup> (Lévèque et al., 2009) <sup>2</sup> (Al-Maqaleh et al., 2022); <sup>3</sup> (Brunazzi et al., 2002); <sup>4</sup> (Suess and Spiegel, 1992).</footnote>
CRediT authorship contribution statement

**Gregory Homem Alves César Ribeiro:** Writing – review & editing, Writing – original draft, Validation, Methodology, Data curation, Conceptualization. **David Rouzineau:** Supervision, Methodology, Data curation, Conceptualization. **Michel Meyer:** Writing – review & editing, Supervision, Project administration, Methodology, Conceptualization.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability

Data will be made available on request.

Acknowledgment

The authors gratefully acknowledge the funding of this work by the French National Research Agency (ANR).

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