Determination of mass transfer resistances in trickle bed reactors

(cid:129) The Mass Transfer Resistances in TBR were calculated using a new methodology. (cid:129) The MTR were found close to those reported in the literature. (cid:129) The wetting e ﬃ ciency of the bed was approximated. (cid:129) The wetting e ﬃ ciency was predicted from available correlation in the literature. A methodology for determining the mass transfer resistances of three-phase reactions in trickle bed reactors is presented. The hydrogenation of styrene over Pd/C was used as a case study to demonstrate the methodology. The gas-liquid mass transfer resistance was experimentally approximated by changing the palladium content of the bed while the chemical reaction resistance was calculated by using the observed chemical reaction rate constant which has been experimentally approximated in two di ﬀ erent stirred tank reactors by hydrogenating the same compound under the same temperature and using the same solvent and active catalytic metal. The liquid-solid mass transfer resistance was calculated by subtracting the gas-liquid mass transfer resistance and the chemical reaction resistance from the overall resistance. The wetting e ﬃ ciency of the bed was estimated from the experimental data and it was compared to the literature by using the dimensionless numbers of Re, Fr, Mo, Ga, We and Stk. The speci ﬁ c gas-liquid and liquid-solid mass transfer coe ﬃ cients found in agreement with the literature.


Introduction
The trickle bed reactors are employed for gas-liquid-solid reactions and they consist of a stationery catalytic bed through which gas and liquid flow. The gas constitutes the continuous phase while the liquid trickles down on the particles of the stationery catalytic bed. The stationery bed consists of coarse particles (supporting material) which are coated with the active catalytic phase. Many times, catalytically nonactive inert coarse particles are added into the bed in order to increase the available area for gas-liquid mass transfer, improve mixing and reduce temperature gradients.
Regarding the gas-liquid-solid reaction, it is a complicated combination of physical and chemical processes. With respect to the first, a three-phase reaction involves mass transfer from gas to liquid phase, from liquid to solid phase and within solid phase (Fig. 4) [1][2][3]. Finally, the chemical reaction takes place on catalyst surface and involves interactions of the gas and liquid reactants with the active sites of catalyst. Each of the physical and chemical processes contributes to the overall reaction rate in different extent. An indication of the degree which each process affects the overall reaction rate is given by the mass transfer coefficients and the intrinsic reaction rate constant. Therefore, the determination of the mass transfer coefficients and the intrinsic reaction rate constant is necessary for designing a trickle bed reactor.
The determination of the gas-liquid mass transfer coefficient in packed beds relies on techniques which do not use the actual reacting system. Instead, according to them, the calculation of the gas-liquid mass transfer coefficient is based on the absorption/desorption rate of another reacting or non-reacting system in the same bed. The most common techniques available in the literature are the physical absorption/desorption of oxygen or carbon dioxide in water while they flow through the bed, some researchers who have used these techniques are V. Spechia et al. [4], T. Hirose et al. [5] and Goto and Smith [6] (other systems can be used as well). Other techniques, which involve fast chemical reactions, are the chemical absorption of carbon dioxide https://doi.org/10.1016/j.cej.2018. 08.194 and the sulphite method [7].
Regarding the liquid-solid mass transfer, many researchers have dealt with the experimental approximation of the liquid-solid mass transfer coefficient in packed beds. As in the case of gas-liquid mass transfer coefficient, most of the techniques do not use the actual reacting system. Namely, the dissolution method [6], the electrochemical method [8], the ion-exchange method [9] and the dynamic adsorption method [10] are available in the literature. Zheng et al. [11] developed the magnetic resonance imaging method for determining the liquidsolid mass transfer coefficient, although it provides a methodology which uses the actual system, the need of the nuclear magnetic resonance magnets may make this method difficult in use, especially for larger than bench scale apparatus.
The objective of this article is to propose a newly developed methodology for determining the mass transfer resistances of a threephase reaction which takes place in a trickle bed reactor using the actual reacting system. To demonstrate the proposed methodology, the hydrogenation of styrene over Pd/C was chosen as case study. According to the proposed methodology, (a) the gas-liquid mass transfer resistance is approximated as the intercept of the linear regression model between (i) the reciprocal of the palladium concentration of the reactor bed, V /W L P d , and (ii) the overall mass transfer resistance, Ω , H ,tot 2 (b) the chemical reaction resistance is calculated by using experimental data obtained from the hydrogenation of the same molecule over the same active phase of catalyst but conducted in semibatch stirred tank reactor and (c) the liquid-solid mass transfer  Abbreviation GB glass beads STR stirred tank reactor TBR trickle bed reactor resistance is calculated by subtracting the gas-liquid mass transfer resistance and the chemical reaction resistance from the overall mass transfer resistance.

Materials
Methanol 99.9% and styrene 99%, purchased from Sigma Aldrich, were used as solvent and substrate, respectively. Decane 99%, purchased from Sigma Aldrich, was used as internal standard for quantitative analysis of reaction mixture samples in gas chromatography. Compressed pure hydrogen (UN: 1049) was purchased from BOC ( Table 1).
The catalytic beds consist of mixtures of Ballotini solid soda glass beads (diameter 2.85-3.3 mm, Sigmund Lindner GMBH) and activated carbon extrudates (Johnson Matthey). Two types of extrudates where used: the base activated carbon support, which is catalytically non-active, and the same support with 1.25% palladium (Pd/C, eggshell). Both types of extrudates have the same physical dimensions; mean diameter 0.00198 m and mean length 0.00320 m. Fig. 1 shows the glass beads, the catalytically non-active carbon support and the 1.25% Pd/C catalytic extrudates while Table 2 outlines the physical characteristics of the extrudates and glass beads. The jacket contains a spiral baffle to ensure a good distribution of the recirculating heating media using a Huber Unistat heat exchanger to control temperature. Thermocouple ports traverse the jacket and are placed as shown in Fig. 3 with 5 K-type thermocouples connected to a PicoLog thermocouple data recorder and a Pt100 sensor (in the 4th port from the top) connected to the heat exchanger for bed temperature control.

Experimental setup
The reactor is pressurised by a continuous stream of nitrogen at the gas outlet of the system with the pressure being controlled at 5 barg by a manual back pressure regulator (BPR). Hydrogen flow is controlled and measured by a Bronkhorst mass flow meter and directly charged to the reactor headspace. Safety precautions include a flame arrestor upstream the mass flow meter, 2 safety relief valves in the nitrogen supply, and 2 more in the hydrogen supply. Temperatures, pressures and the hydrogen mass flow rate are logged on a computer.
The reactor is fed from the top with the liquid phase using an KNAUER HPLC pump 1800 Smartline (R-Pump 1). There is a three-way valve which switches between the pure solvent and the substrate solution. The liquid phase is collected in the vessel R-T3 while there are three drain points which can be used to by-pass blockages in the rig. The level of the trickle bed reactor is maintained by observing the level indicators (LI1 and LI2) and using the KNAUER HPLC pump 100 Smartline (R-Pump 2) which is attached downstream the reactor outlet. The back-pressure regulator R-BPR is attached at the outlet of the R-Pump 2 to ensure the system pressure does not push material through the pump. During the steady state operation, the bed of the reactor should not be submerged in the liquid phase. The reactor bed consists of 232 g glass beads mixed with 2 g of activated carbon extrudate. The ratio of the 1.25% Pd/C catalytic extrudates to the catalytically non-active carbon extrudates ranged between 0% and 12.7% w/w. To ensure a good distribution the glass beads and the extrudates where charged in 5 equal portions whilst manually shaking the reactor.

Bed voidage
To calculate the bed voidage, Φ b , the volume of 2 g activated carbon extrudates was considered negligible comparing to the volume of 232 g glass beads. Therefore, the volume of the bed, V b , was taken equal the volume of 232 g glass beads, V GB . The latter was approximated by charging 232 g glass beads into 0.3 L of methanol and calculating the volume displacement.

Liquid holdup and liquid residence time
From the various holdup measurement techniques [12] we selected the draining method for its ease of implementation. The reactor was filled with 232 g glass beads mixed with 2 g catalytically non-active carbon extrudates. To eliminate any dead time and experimental error to the estimation of the liquid hold-up, related to the pipe network, the apparatus downstream the valve R-V5 was not used. Pure methanol was fed to the reactor (atmospheric pressure, N2) at 5, 10, 20 mL/min. After 30 min the inlet and outlet valves (R-V2, R-V3 and R-V5) were simultaneously closed. The liquid in the column then partitions in two fractions (i) V fd the free draining liquid found below the bed, and (ii) V st the stagnant liquid held in the bed through surface tension. The volume of the free draining liquid was recorded by emptying the liquid content, the volume of stagnant liquid was resulted from the weight difference between the drained column, and the dry column.

Hydrogenation of styrene
Three different ratios of 1.25% Pd/C catalytic extrudates to catalytically non-active carbon extrudates were used; specifically, 3.9%, 6.7% and 12.7% w/w. The reactor was filled with the intended for the experiment mixture of glass beads and extrudates. Before the start of the reaction the rig was purged with nitrogen and flushed with methanol to ensure all air had been removed from the rig before flowing hydrogen and to avoid any contamination of residuals of prior experiments. The catalyst was activated by flowing hydrogen through the bed for 30 min. To initiate the reaction, the valve R-V.IN was switched to substrate solution (concentrations shown in Table 4 and 5 mL/min) while hydrogen was flowing through the bed (60 mL/min). The styrene hydrogenations took place under 6 bara and 32°C. The reactor was sampled, regularly, from the stream F12 and the samples were analysed using gas chromatography. The experimental conditions are summarised in Table 5.

Mass transfer in series model
Models used to describe multiphase reactions require both physical transport and chemical processes to be modelled. Transport by diffusion is slow compared to transport by (turbulent) convection. As the fluid flow near interfaces is slow (e.g. non-slip boundary conditions for fluids Chemical Engineering Journal xxx (xxxx) xxx-xxx on particles), diffusion becomes the dominant transport process near interfaces and this is well explained by Dankwert's surface renewal model of mass transfer which relates mass transfer from an interface into a fluid by: The mass transfer rate (MTR) has the units of

MTR·Ω ΔC
(2) Comparing Eqs. (1) and (2), the mass transfer resistance is the inverse of the mass transfer coefficient. As Fig. 4 depicts, for the trickle bed reactor there are two main interfaces; the gas-liquid and the liquidsolid.

Mass transfer at the gas-liquid interface
The interfacial surface area, available for mass transfer from gas to liquid phase, is developed by the trickling liquid onto the external surface of the solids in the bed, α bed , and it is given in units of surface per units of bed volume. The wetting efficiency, f , ranges between 0 and 1, for completely unwetted and completely wetted beds, respectively. Due to the use of pure hydrogen, its gas-liquid interfacial concentration is given by Henry's law (Eq. (4)) considering that it is in equilibrium with the hydrogen pressure in the gas phase.
The Henry constant, H E , is given by Eq. (5) where H E in Mpa, T in K and P H2 in Pa [12].
The interfacial surface area, available for mass transfer from liquid to solid phase, is developed by the trickling liquid onto the external surface of 1.25% Pd/C catalytic extrudates in the bed and it is given in units of surface per units of palladium weight, α act.pel 'Pd . Finally, the hydrogen reacts at the catalyst surface with the adsorbed molecules of styrene. It is common to express the reaction rate with a 1st-order observed rate constant in order to make easy the combination of the chemical reaction step with the external mass transfer steps [13]: The observed rate constant is a complex function between the surface concentration of reagents and temperature. In addition, the effect of mass transfer in the solid catalyst pores may be modelled using the Thiele modulus. In this work pore diffusion limitation is ignored as we used an eggshell catalyst with palladium deposited only near the surface of the catalyst pellets. Typically, the observed reaction rate constant is proportional to the quantity of catalyst, or more exact, the quantity of palladium nano particles, as we have no pore resistance.
The number of moles of hydrogen in the three phases is now modelled with a transient mass balance: Typically, the concentration of hydrogen is low, so the Bodenstein assumption applies: So, all mass transfer rates are equal; the rate at which gas is adsorbed from the gas phase is equal to the rate it reacts on the catalyst and the concentration of hydrogen in the fluid at various locations have

Table 2
Characteristics of the glass beads and pellets in the bed, (r = radius and L = length); external surface area of the pellets without considering the pores.  Table 3 Technical characteristics of the reactor bed for calculating the liquid hold-up.  reached a "steady state", though it may slowly change as the concentration of the reactant, here styrene reduces, as the reaction progresses. The slowest (limiting) step defines the mass transfer rate of the multiphase reaction and this is similar to the way current flows through resistances in series:  Table 6 summarises the definition and expression of the mass transfer resistances of a hydrogenation which takes place in the trickle bed reactor.

Surface reaction model
To describe mathematically the mechanism of the surface reaction between the styrene and hydrogen, we adopted the Langmuir-Hinshelwood model. We used palladium catalyst which is a transition metal where the hydrogen is dissociatively chemisorbed [14][15][16][17][18]. We assumed that • styrene and hydrogen compete for the same sites, • styrene is consecutively hydrogenated by two different hydrogen atoms which have been dissociatively chemisorbed onto the active sites of catalyst, • the first hydrogen addition is non-reversible.  Table 5 Experimental conditions for determining the gas-liquid mass transfer resistance.
The volume of the bed and the bed voidage were calculated at 0.095 L and 0.4, respectively. Table 3 outlines the technical characteristics of the reactor bed for calculating the liquid holdup.
The liquid hold-up and the residence time have been plotted against the liquid flow rate and the liquid in the reactor in Fig. 5. The upper x axis which corresponds to the volume of the liquid in the reactor has been scaled taking into account its dependence on the liquid flow rate. Therefore, one can read the corresponding volume of liquid in the reactor for a certain liquid flow rate.

Determination of gas-liquid mass transfer resistance
If one rewrites the expression of the overall mass transfer resistance of hydrogen as Eq. (19) indicates and observes the mass transfer rate of hydrogen, at different palladium content in the bed, W , Pd but under the same liquid flow rate, pressure, temperature and overall bed weight (i.e. sum of weight of glass beads, 1.25% Pd/C extrudates and catalytically non-active extrudates); and plots the Ω H ,tot To change the palladium content in the bed, W Pd , the ratio of the 1.25% Pd/C extrudates to catalytically non-active extrudates was varying while their total weight was keeping constant. The bed compositions which were used at the experiments for determining the gasliquid mass transfer resistance are presented in Table 7.
The mass transfer rate of hydrogen was calculated from styrene consumption rate which is given by Eq. (20). Styrene outlet concentration was given by analysing the reactor outlet samples in gas chromatography.
To evaluate the dependence of reaction rate on the catalyst loading, the consumption rate has been plotted in Fig. 6 against (i) the palladium content of the bed and (ii) the weight of 1.25% Pd/C extrudates in the bed. At the left y axis, the consumption rate is expressed in molar amount per minute, while, at the right axis of the same figure the consumption rate has been divided by the volume of liquid in the reactor. As it was expected, the reaction rate depends linearly on the catalyst loading.
The overall mass transfer resistance of hydrogen has been plotted against the reciprocal of the palladium concentration in the bed in Fig. 7. The linear regression on the data allowed the approximation of the gas-liquid mass transfer resistance which is given by the intercept of the linear regression model. Table 8 presents the results of the linear regression and the 95% confidence intervals of the model parameters.

Specific effective gas-liquid mass transfer coefficient calculation
The specific effective gas-liquid mass transfer coefficient, k ·f, L was calculated by using the reciprocal of the intercept and approximating the external surface area of the bed per unit volume of the bed, α bed . The proportion of 1.25% Pd/C extrudates and catalytically non-active extrudates to glass beads in the bed is about 4%. This means that methanol and hydrogen meet four extrudates, either with palladium or catalytically non-active, every hundred glass beads, therefore, it is likely the solvent to have been saturated with hydrogen before they come in contact on the extrudates. Consequently, the gas-liquid mass transfer was assumed that took place on the interfacial area developed by the glass beads and the external surface area created by the extrudates did not contribute to the interfacial area for gas-liquid mass transfer.
To calculate the external surface area of the bed available for gasliquid mass transfer, first, the external surface area of one glass bead was calculated and it was multiplied by the total number of glass beads in the bed. The number of the glass beads in the bed was approximated by dividing the total weight of the glass beads in the bed by the average weight of a single glass bead. Table 9 summarises the calculated values of the gas-liquid mass transfer coefficient and the external surface area per unit volume of the bed which is available for gas-liquid mass transfer.

Comparing to the literature
Gas-liquid mass transfer coefficients of trickle bed reactors calculated by different researchers using different fluids and beds were collected. Because different experimental conditions, fluids and beds were used in these works, the Reynolds numbers of liquid phase were calculated for each case. The Reynolds numbers varied between 0.46 and 23.89. Details of the experimental conditions of each work are summarised in Table 10. Then, all the available values of the gas-liquid mass transfer coefficient including the one of this work were plotted against the Reynolds number (Fig. 8). The calculated value of our work fits well to the others' data. The gas-liquid mass transfer coefficient depends linearly on the Re -0.5942 which is very close to the well-known correlation of Gupta and Thodos [20], given by Eq. (21), for the heat

Film thickness and wetting efficiency approximation
The specific gas-liquid mass transfer coefficient was theoretically calculated by adopting the concept of the stagnant film theory, from which it is defined as the ratio between the diffusion coefficient and the thickness of the stagnant film through which the mass transfer occurs (Eq. (23)).
In addition, the film thickness was calculated by dividing the overall liquid hold-up by the external surface area of the bed per unit volume of the bed, α bed (Eq. (24)) [21]. For non-completely wetted bed, the liquid is distributed in a smaller surface area resulting in thicker film.  Table 11 outlines the diffusion coefficient of hydrogen in methanol, the external surface area of the bed per unit volume of the bed, the liquid hold-up and the calculated values of the film thickness and the mass transfer coefficient.
The theoretically calculated gas-liquid mass transfer coefficient is higher than the one which was calculated by applying linear regression on the data set of the overall mass transfer resistance and the reciprocal of the palladium content in the bed. This is an indication of non-completely wetted bed during the reactor operation. The wetting efficiency, f, was estimated at 48.8% by dividing the effective value of gas-liquid mass transfer coefficient by the theoretical one. Therefore, the actual thickness of the film at the gas-liquid interface is 48.8% thicker and equal to 0.339·10 -3 m, since the liquid volume was distributed in a smaller surface area. The film thickness is about the 11% of the characteristic length of the glass beads.

Comparing to the literature
To compare the calculated value of the wetting efficiency to those available in literature, the results of the work of Julcour-Lebigue et al. [22] were used. They implemented a step injection of a coloured liquid at the inlet of a bed of adsorbing particles, in combination with image processing to calculate the wetting efficiency of systems with different characteristics and under several experimental conditions. Then, they calculated the dimensionless numbers of Reynolds, Weber, Stokes, Morton, Froude and Galileo for the different conditions and they fitted their experimental data to Eq. (25), where N is the dimensionless number. They found that using more than 3 dimensionless numbers in the correlation does not improve the optimization criteria which they used. The exponents, x i , for different combinations of dimensionless numbers and the predicted value of the wetting efficiency of our work     are presented in Table 12. The lowest relative difference between the experimental and predicted wetting efficiency is 8.6% (overestimation) and it is given when the Weber and Stokes numbers (in bold Table 12) are used in Eq. (25). All the combinations of dimensionless numbers overestimate the wetting efficiency, this may happen because the effect of gas velocity has not been taken into account.

Determination of chemical reaction resistance
The determination of the chemical reaction resistance needs the approximation of the observed chemical reaction rate constant, The development of the expression of the overall mass transfer resistance of hydrogen (Eq. (14)) has been based on assuming that the surface reaction between styrene and hydrogen is 1st-order with respect to hydrogen and zero-order regarding styrene. As it has been already mentioned, this was assumed in order to make easy the combination of the chemical reaction step with the external mass transfer steps. However, the surface reaction model which has been defined in Section 3.2 suggests a half-order reaction rate with respect to hydrogen when styrene is in excess. In this case, the reaction rate is given by Eq. (16). To encounter the assumption of 1st-order reaction rate law, we compare the Eqs. (7) and (16) Adopting the methodology which has been introduced by Stamatiou and Muller [3], the observed chemical reaction rate constant, ′ k obs Pd , was calculated from Eq. (28), for the hydrogenation of styrene over Pd/C in two different semi-batch stirred tank reactors using the slopes of Fig. 9.  [27] n-hexane Fig. 8. Gas-liquid mass transfer coefficient against liquid Reynolds number for different works.

Table 11
Summary of gas-liquid mass transfer coefficient for theoretical calculation.

Table 12
Exponential factors of dimensional numbers taken from Julcour-Lebigue, et al. [22] and predicted wetting efficiency.
The factor β is defined as the ratio of the chemical reaction resistance to the overall mass transfer resistance [3]. Therefore, it varies with pressure and it can be calculated from Eq. (29).
The values of the observed chemical reaction rate constant, ′ k obs Pd , are similar for both semi-batch reactor setups. Therefore, taking into account that the initial concentration of styrene was the same throughout all the experiments, it is considered that the term of ′ k · K /K 1 Pd H St is independent of reactor setup as long as the chemical reaction takes place over the same active phase of catalyst, under the same temperature and using the same solvent.
Using Eq. (17) and knowing the styrene concentration at the outer surface of the catalyst particle, C , St, S the term of ′ k · K /K 1 Pd H st was calculated from the mean of the observed chemical reaction rate constant, ′ k¯, obs Pd in the semi-batch reactors, and it is presented in Table 15.
Regarding the concentration of styrene at the outer surface of the catalyst particle, in the semi-batch reactors, it was taken equal to the mean of styrene concentration in the liquid phase as far as styrene is in excess.
Since the term of ′ k · K /K 1 Pd H St is independent of the reactor setup and because the hydrogenation of styrene in the trickle bed reactor took place under the same temperature, over the same active phase of catalyst and using the same solvent as in the semi-batch reactors, the chemical reaction resistance in the trickle bed reactor was calculated from Eq. (30) by using the reciprocal of ′ k · K /K 1 Pd H S t , which was calculated for the semi-batch reactors and it is outlined in Table 15.
Regarding the wetting efficiency, f, and the effectiveness factor, ε; the first has been approximated in Section 4.3 and it is outlined in Table 13, the latter was considered unity because an eggshell type of catalyst was used. Fig. 10 illustrates the conversion of styrene against its initial concentration in the liquid phase under constant mass transfer rate. The conversion for all the experiments is lower than 2%. Consequently, the concentration of styrene in the liquid phase is assumed constant along the reactor bed and equal to its inlet concentration. As it is shown in Fig. 11, the liquid-solid mass transfer resistance is at least 4 times higher than the other two resistances. This means that the liquid-solid mass transfer of hydrogen affects the overall reaction rate in greater extent, than the other two processes, making it independent of styrene concentration. This explains why the conversion of styrene, for constant active pellets amount and under constant mass transfer rate decreases as its inlet concentration increases. The factor β of the trickle bed reactor was calculated by Eq. (31).  Table 16 summarises the necessary variables for calculating the chemical reaction resistance from Eq. (30) and the chemical reaction resistance for the different values. The results of the chemical reaction resistance are visualised in the Fig. 11 where have been plotted in bar chart form for the different values of palladium concentration in the bed, styrene inlet concentration and external surface area of active pellets per unit volume of bed. The increase of palladium content needs the addition of 1.25% Pd/C extrudates in the bed which means that the palladium content was not feasible to be increased selectively and without increasing the external surface area of the active pellets in the same time.

Determination of liquid-solid mass transfer resistance
The liquid-solid mass transfer resistance was calculated from Eq. (32) by using the values of the overall mass transfer resistance, the gasliquid and the chemical reaction resistance. Table 17 outlines the results of the liquid-solid mass transfer resistance. The calculated values of the liquid-solid mass transfer resistance are illustrated in Fig. 11 where they have been plotted in bar chart form for the different values of palladium content in the bed, styrene inlet concentration and external surface area of active pellets per unit volume of bed. (32)

Specific liquid-solid mass transfer coefficient calculation
The external surface area of the active pellets per weight of palladium, ′ α Act.pel Pd , was approximated as it is necessary for calculating the specific liquid-solid mass transfer coefficient, k S , from the value of the liquid-solid mass transfer resistance. The external surface area of one active pellet was calculated and it was multiplied by the total number of active pellets in the bed. The number of the active pellets in the bed was approximated by dividing the total weight of the active pellets in the bed by the average weight of a single active pellet. The external surface available for liquid-solid mass transfer resistance was varying due to the need of changing the palladium content in the bed by changing the weight of 1.25% Pd/C extrudates. Table 18 introduces the external surface area and the mean specific liquid-solid mass transfer coefficient considering the wetting efficiency which has been estimated in Section 4.3.

Comparing to the literature
To compare the obtained value of the liquid-solid mass transfer coefficient, k s , to those available in literature, the dimensionless Sherwood, Schmidt and Reynolds numbers, Sh, Sc and Re respectively, were employed. For encountering the non-spherical shape of the pellets, the shape factor, γ, were used in the calculation of the Sherwood and Reynolds numbers. Taking into account the bed void, their expressions for a packed bed, are given by Eqs. (34)-(36) [23]. The dimensionless numbers, the shape factor and the superficial liquid velocity are summarised in Table 19.      The Sherwood number is an indicator of the relative contribution of the convective and diffusive mass transfer. In the case of the studied system, the Sherwood number is high enough to allow the omission of the diffusive mass transfer contribution. Consequently, the most common function found in the literature to correlate the liquid-solid mass transfer coefficient is according to Eq. (33).
To identify the factors B and m, several experimental values of liquid-solid mass transfer coefficients in a range of Reynolds number are necessary. Because in the present study, the liquid-solid mass transfer coefficient was calculated in a single Reynolds number, this is infeasible. Therefore, several correlations with different factors which are reported in the literature were tried. The one which predicts better the experimental liquid-solid mass transfer coefficient is given by Satterfield et al. [24] who studied the liquid-solid mass transfer in packed beds with downward concurrent gas-liquid flow and they reported factors B and m equal to 8.18 and 0.26, respectively. The abstract of Miyashita et al. [25], who studied the transport phenomena in low Reynolds numbers (< 550), and reported value of exponent of Reynolds number in the range between 0.11 and 0.33.

Conclusions
The gas-liquid mass transfer resistance of the three-phase styrene hydrogenation in a trickle bed reactor was determined by changing the palladium content of the bed. The observed chemical reaction rate constant was shown to be independent of the reactor setup by calculating it in two different stirred tank vessels operated in semi-batch. Taking advantage of this independence the observed chemical reaction rate constant was used to calculate the chemical reaction resistance in the trickle bed reactor. The liquid-solid mass transfer resistance was calculated by using the values of gas-liquid mass transfer and chemical reaction resistances. The overall wetting efficiency of the reactor bed was calculated as well. The values of the mass transfer resistances and the wetting efficiency were found to be close to those found in the literature.