Dynamic liquid-solid mass transfer in solid foam ... - ACS Publications

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Dynamic Liquid−Solid Mass Transfer in Solid Foam Packed Reactors at Trickle and Pulse Flow Johannes Zalucky,† Markus Schubert,*,† Rüdiger Lange,‡ and Uwe Hampel†,§ †

Institute of Fluid Dynamics, Helmholtz-Zentrum Dresden-Rossendorf, Bautzner Landstraße 400, 01328 Dresden, Germany Chair of Chemical Engineering and Process Plants and §AREVA Endowed Chair of Imaging Techniques in Energy and Process Engineering, Technische Universität Dresden, 01062 Dresden, Germany

‡

S Supporting Information *

ABSTRACT: The effective liquid−solid mass transfer (LSMT) has been investigated in solid foam packed reactors under cocurrent gas−liquid downflow. Based on a comprehensive literature analysis, the limiting current technique was adapted successfully to solid foam packings enabling measurements of dynamic mass transfer coefficients at different axial packing positions. The flexible reactor setup was used to analyze the appropriateness of various setup configurations. The LSMT coefficients are presented for two different solid foam pore densities for a wide range of gas and liquid velocities covering trickling and pulsing flow regime. To illustrate the difference in the LSMT dynamics in terms of electric current compared to the liquid holdup at pulse flow, a compartment model is developed. Excellent agreement is achieved between simulations and measurement data. Eventually, the effective mass transfer data are compared with conventional random and structured packings.

1. INTRODUCTION In gas−liquid−solid reactors, the individual mass transfer steps govern the overall reactor performance.1 In reactors with perfectly wetted catalysts, all reactants must cross the liquid− solid interface either from or to the solid surface. Thus, the liquid−solid mass transfer (LSMT) denotes a very crucial step in such systems and has been the subject of many investigations. The LSMT is basically accessible via reactive, dissolution, or electrochemical techniques.2−4 While the reactive techniques require complex experimental setups and profound knowledge on the reaction kinetics, the dissolution techniques depend strongly on the precipitation of reproducible coating layers. Furthermore, the complexity of both techniques increases dramatically, when applying them to complex solid structures due to limited access to the solid network. Therefore, the electrochemical technique, also known as the limiting current technique, has been applied often.3,5,6 At first glance, the principle is rather simple. A part of the installed solid structures (e.g., catalyst packing, etc.) is replaced by morphologically similar electrodes, and an aqueous electrolyte solution is used as the liquid phase. Subsequently, the LSMT coefficient can be measured directly via the occurring electric current at the electrodes. Beside the effective LSMT coefficients, the limiting current technique enables measuring continuously the shear stress related effects of fast dynamic processes such as the passage of liquid pulses. In continuation of our recent studies7,8 on the hydrodynamics in open-cell solid foams, the current work aims at © XXXX American Chemical Society

applying the electrochemical technique to solid foams operated at gas−liquid trickling and pulsing flow. Large specific surface area, low two-phase pressure drop, high porosity, and liquid holdups8 as well as new operation windows9 make solid foams a promising replacement for packed beds with random catalyst particles operated in the cocurrent downflow mode. While the hydrodynamic behavior of descending gas−liquid flows in solid foams has already been addressed widely,10−15 the knowledge about their mass transfer characteristics is still rather limited.3,14,16 Especially the LSMT in solid foams was only studied once, however, only in terms of effective time-averaged data.3 Thus, the main objective of the present work is the investigation of the effective LSMT in the trickling regime and, in particular, the mass transfer dynamics in the pulsing flow regime. The mass transfer behavior is studied at different axial positions and for two different solid foam pore densities. In addition, a compartment model based on first principles is proposed to assist the interpretation of the dynamic mass transfer effects at pulse flow. Prior to that, the limiting current technique is comprehensively revisited, and preliminary studies were performed to Special Issue: Tapio Salmi Festschrift Received: Revised: Accepted: Published: A

April 14, 2017 July 4, 2017 July 5, 2017 July 5, 2017 DOI: 10.1021/acs.iecr.7b01578 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research

In the last 50 years, more than 25 LSMT studies using the electrochemical technique were published. However, the proposed configurations varied strongly. Thus, to adapt the method for solid foams properly, a comprehensive literature review was carried out, which is summarized below. Originally, Eisenberg et al.17 introduced the limiting current technique for diffusion measurements in diluted solutions, which was the basis for the famous Wilke-Chang correlation for single component diffusion coefficients. The first LSMT measurements were reported by Trueb.6 Later, Mizusha18 highlighted the usage of this technique for local investigations of miscellaneous heat and mass transfer phenomena via the Chilton-Colburn analogy, shear stress studies, and even for velocity measurements in a comprehensive review.18 Selman and Tobias19 discussed important fundamental physical and electrochemical effects of the limiting current technique, such as diffusion double layer potentials, overpotential contributions, and time-constants. Meanwhile, the electrochemical technique has been widely used to study the LSMT at various geometries. A selection of studies for chemical process applications is given in Table 1.

adapt the electrochemical technique to solid foam packed reactors.

2. THE LIMITING CURRENT TECHNIQUE REVISITED 2.1. Basic Principle. The limiting current technique was for the first time reported by the group of Eisenberg, Wilke, and Chang17 and can be described as follows: By applying an electric potential between two electrodes in an electrolyte solution, the ion flux of the electrochemically converted species is directly proportional to the measured electric current, which is given by Faraday’s law. In an ideal environment without secondary reactions or mass transfer limitations, the ion flux increases linearly with the applied potential in accordance to Ohm’s law. In nonideal systems, however, mass transfer limitations and secondary reactions lead to the development of two additional characteristic regions depending on the applied potential (Figure 1).

Table 1. Selected Mass Transfer Studies for Chemical Application Using the Limiting Current Technique solid geometry/ packing type pipes spherical particles cylindrical particles pall rings corrugated sheets solid foams

Figure 1. Example of the limiting current curve with kinetic region (I), limiting current region (II), and secondary reaction region (III).

At low potential, the curve follows Ohm’s law and is only limited by the electrode kinetics. Thus, this part is often referred to as kinetic region. At higher potential, the LSMT becomes rate limited, and the curve reaches a plateau corresponding to the limiting current. Increasing the potential further, the current increases again indicating secondary reactions for example due to electrolysis. In the range of the limiting current plateau, the limiting ion species restricts the conversion at the corresponding working electrode. The limiting current Ilim = nlim ̇ ·z·F

(1)

(2)

Assuming a negligible small species concentration at the solid surface csurf, the total mass transfer rate can be determined directly by φkLSA = Ilim/(cbulk ·z ·F )

(3)

Accordingly, the specific mass transfer coefficients is available via φkLSa = φkLSA /VC

20

Trueb, Rode et al. Appel and Newman,21 Bartelmus,22 Burghardt et al.,2 Chou et al.,23 Jolls and Hanratty,24 Joubert and Nicol,5 Latifi et al.4 Sedahmet et al.,25 Rao and Drinkenburg,26 Zewail et al.,27 Highfill and Al-Dahhan28 Gostick et al.29,30 Battista and Böhm1 Mohammed et al.,3 Montillet et al.31

Although most of the investigations above used the limiting current technique for the LSMT measurements, others also studied the effective solid surface wetting,1,5 the flow regime transitions,32 and the local shear stress on a solid surface.20 Beside the various geometries (solid packings), the setups differed widely in terms of alignment and design of electrodes, fluid compositions, and countermeasures against degenerative effects, which influence crucially the results. 2.2. Electrode Configuration. Electrode Alignment. Basically, downstream and radial electrode alignments have been applied. In the first case, the rate-limiting working electrode is located at the desired reactor height and operated against the counter electrode installed downstream in the reactor. While the working electrode mimics the respective packing shape, the counter electrode may be of any design such as meshes,4 pipe walls,6,20 or foils2 but can also be shaped like the packings.1,3,5,29,30 In radial alignments, either the reactor wall25,32 or metal foils between wall and packing23,26,28 denote the counter electrode. Electrode Distance. In previous studies, the distance between working and counter electrodes covered a wide range from a few millimeters22,32,33 to almost one meter.5 In several publications, however, the electrode distance has not been specified,2,23,26 although the fluid system’s low conductivity denotes a major electrical resistance. The impact of the electrode distance was also studied using COMSOL

is proportional to the ion flux ṅlim, the number of transferred electrons z, and the Faraday constant F. The ion flux at limited effective mass transfer rate φkLSA is given by nlim ̇ = φkLSA ·(cbulk − csurf )

studies 6

(4)

where V is the volume of the electrode segment. B

DOI: 10.1021/acs.iecr.7b01578 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research Multiphysics,34 which revealed a nonlinear decrease of the kinetic slope with increasing axial distance. Thus, the electrode distance should be kept as small as possible. Electrode Size and Area. Various sizes of the working electrodes were reported. Microelectrodes embedded in inert packings,32 single particles,2,4,6,23−26,28,29,33,35 or particles with individual contacting18,20 were applied to study small-scale phenomena. However, such small electrodes suffer from the risk of low wetting and nonrepresentative shear stress.20 Thus, larger composite electrodes of several directly coupled particles5,21,22,27 were used to yield more integral mass transfer data. In addition, large electrodes such as metallic foam electrodes3,31 or corrugated sheets1 were installed to provide information at reactor scale. In a prestudy, the effects of lateral size and axial length of electrodes built of multiple particle layers were analyzed via single-phase simulation using COMSOL Multiphysics.34 It was found that the evolving electric field clearly determines the upper size limit of the electrodes. In the simulations, electrode particles in the fourth layer and beyond were shielded practically and did not contribute to the overall electric current anymore. The counter electrode area is commonly chosen equal to or larger than the working electrode to ensure mass transfer limitations solely at the working electrode. Furthermore, larger counter electrodes avoid biasing effects at low wetting conditions. In the literature, the area ratio between counter and working electrode ranges from unity3,31,35 to more than 1000.6,32 Electrode Material. To obtain stable limiting current plateaus, the selection of proper electrode material is essential to harmonize it with redox couple and support electrolyte as it affects the overpotentials of desired redox reactions and unwanted side reactions. Nickel has been the preferred electrode material for the studies summarized in Table 1, although platinum4,22,32 and stainless steel21 were also used. Taama et al. demonstrated that nickel and platinum electrodes provide comparable performance and stability, if applied with carbonate or hydroxide-based electrolytes.36 Beside pure metals, some authors also applied coated brass24 and nickel20 electrodes. Electrode Deactivation. Applying nickel electrodes, surface deactivation occurs due to the oxidative formation of Ni(OH)2 and NiO(OH), which can only be suppressed by frequent surface polishing and activating37 requiring a plain and accessible surface. To prevent such deactivation, the experiments were commonly performed in opaque setups at nitrogen atmosphere, while only a few studies were conducted with air as the gas phase.23,25,26,35 2.3. Electrolyte System. Working Electrolyte. A main advantage of the electrochemical technique is the fully reversible redox reaction at the electrode, which makes it applicable for setups with large volumes, complex electrode geometries, and setup assemblies even at larger throughputs. Thereby, the electrolyte system requires fast electrode kinetics, appropriate overpotential positions of the desired and undesired reactions, sufficiently high ion solubility, high chemical stability, and reasonable handling. The redox system of potassium iron complexes K3Fe(III)(CN)6/K4Fe(II)(CN)6 is the almost solely applied system fulfilling these requirements. Only Latifi et al. used Fe2+/Fe3+ salts in dimethyl sulfoxide; however, the system is not further specified.4 Support Electrolyte. To increase the electrolyte conductivity and to suppress transport processes driven by ion-migration, a

support electrolyte is required.35 Although hydroxide ions are known for their degenerative ligand exchange with ferrocyanide and ferricyanide complexes under certain conditions,19,38 sodium and potassium hydroxides are the most common support electrolytes due to their high solubility and ion mobility in water. Nevertheless, potassium and sodium salts of chloride,22,37 carbonate,36,37 and nitrate21,37 were also used. Szanto et al.37 and Taama et al.36 proofed the superiority of carbonate salts compared to others in terms of long-term stability observed in experiments with rotating disk electrodes. Electrolyte Concentrations. Another important system parameter is the concentration of the various electrolyte salts. Earlier, an equimolar composition of oxidized and reduced species was preferred,6,21,35 while later on up to 40 times excess concentrations of the non-limiting species were applied.20 The support electrolyte concentration, which has a significant effect on the physicochemical and electrochemical properties,34 was commonly between 0.5 and 3.0 M. The influence of the electrolyte concentration was also highlighted in the results of Saroha39 as significant bias, which showed that effects beyond the change in viscosity or diffusion coefficients may occur. Electrolyte Aging. Exposing the electrolyte to light and air3,5,37 results in aging of the solution, which is attributed to partial ligand elimination in ferrocyanide38 and ligand exchanges in both iron complexes,19 respectively. Furthermore, an autoreduction of ferrocyanide to ferricyanide has been described at the rate of 0.85% molar conversion per day.6 These effects are counterbalanced by using opaque setups and nitrogen atmosphere as mentioned above. This comprehensive review indicates that the experiment design itself has already a decisive impact on the investigated parameters. Therefore, it is important to note that a direct comparison of the LSMT behavior obtained from different setups, configurations, and operating conditions should be done carefully.

3. EXPERIMENTAL SECTION For the application of the limiting current concept to solid foam packed reactors operated with descending gas−liquid flow, the following conclusions were drawn: 1. Since the implementation of microelectrodes in solid foams is even more difficult than for particle packings, only integral measurements are carried out at different axial positions. Therefore, the concept of Mohammed et al.3 is adapted by using alternating anodes and cathodes as working and counter electrodes installed between the solid foam segments. 2. The axial distance between the electrodes is kept minimal, i.e. the height of one SiSiC foam element, which corresponds to 100 mm length. 3. Pure nickel foam electrodes of similar nominal morphological properties are applied. 4. Based on the findings in the preliminary studies on the maximal electrode sizes, only thin nickel foam electrodes are applied. 5. The area ratio between working and counter electrode is slightly higher than unity. The mass transfer limitation is assigned to the working electrode by using nonequimolar concentrations of the reactive species. 6. Three working electrodes are operated in parallel against four working electrodes. 7. Due to its superior properties concerning electrode corrosion and electrolyte degeneration, sodium carbonate is C


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Figure 2. Schematic representation of reactor setup and electric circuit.

used as the support electrolyte. As a compromise between altered liquid properties and electrolyte conductivity, the sodium carbonate concentration is set to 1 M. 8. All nickel electrodes are chemically and electrochemically pretreated to maximize their kinetic activity. The setup consists of a closed liquid flow loop with supply unit and solid foam packed bed and is depicted schematically in Figure 2. The supply unit consists of the closed and opaque liquid loop and the continuous gas supply branch. The liquid is circulated using the rotary piston pump (E-2, Waukesha Cherry-Burrell Universal 1 Series 060-UI) from the opaque liquid reservoir (E1, also serving as gas−liquid separator) via the Coriolis mass flow meter (E-3, Rheonik Coriflow RHM 20 GMT G1”) and the heat exchanger to the reactor. The heat exchanger (E-4, FUNKE SSCF 302-W-1) was coupled to the thermostat (Lauda Proline RP 845) controlling the liquid temperature (υ = 30 °C) via downstream thermocouple. A second thermocouple (E-5) is installed at the reactor head to verify the actual inlet temperature. The gas-side supply unit provided the nitrogen from 30 MPa gas cylinders (E-6). A side stream (dotted line) of about 5 to 10 mL/min was fed directly to the ring injector submerged in the liquid reservoir to ensure nitrogen atmosphere in the electrolyte storage tank. In the main branch (solid line), the nitrogen flow rate was adjusted using a thermal mass flow controller (E-6, Omega FMA-2613A). The reactor unit consists of PMMA tube segments of 100 mm inner diameter flanged between head and bottom plate. The head plate provides four large gas inlet holes and carries the multipoint liquid distributor. The liquid distributor consists of 12 liquid inlet holes with 4 mm diameter and 4 gas holes through which the gas bypasses from the reactor head space to the packing section. The liquid distributor was located 50 mm above the packing top. At the packing bottom, a coarse metal sieve served as a holding tray for the solid foam packings. All

installed packings consisted of eight SiSiC solid foam elements with nominal properties provided in Table 2. Table 2. Nominal Properties of Applied Ceramic SiSiC Foams and Metallic Nickel Foams specification SiSiC-20 SiSiC-30 RCM Ni-1116 RCM Ni-1723

pore density 20 30 20 30

ppi ppi ppi ppi

nominal surface area 1000 1600 1000 1600

m2/m3 m2/m3 m2/m3 m2/m3

hydraulic porosity 0.87 0.89 0.95 0.95

The SiSiC solid foam elements of 20 and 30 ppi nominal pore density were provided by IKTS Fraunhofer (Dresden, Germany). Each cylindrical foam segment had a diameter of 100 mm and a height of 100 mm. Nickel foam discs with approximately 98 mm diameter and 3 mm thickness were used as electrodes. The reticulated foam electrodes RCM Ni-1116 and Ni-1723 were provided by RECEMAT BV (Dodewaard, The Netherlands). Although not wholly identical in terms of nominal properties and foam morphology, pure nickel electrodes were preferred over electroplated foams, because nickel-plate alloys are known to disturb the electrode kinetics. Moreover, thin nickel-plated ceramic foams were not feasible technically. Prior to the mass transfer experiments, the electrodes were submerged in an ultrasonic bath of diluted oxalic acid (10% (m/m)) to remove the passivated oxidation layer. The electrodes were connected electrically via pure nickel wires (Goodfellow, 99.99% purity) welded to the electrode to prevent the formation of a local cell. The short, insulated nickel wires were attached to the electronics via ports installed at the reactor wall. Since the SiSiC solid foam elements feature semiconducting characteristics, the electrodes were encapsulated via 3 mm thick polyurethane foam slices of 20 ppi pore density (see Figure 2). The assembly of the packing as well as the electronic circuit are also highlighted in Figure 2. The complete reactor unit was wrapped in 10 mm ARMAFLEX D


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thickness cover the whole cross section of the reactor, the measured current is an integral value for the cross section at a certain axial position. However, the order of the electrodes in relation to the flow direction and the wiring in terms of single stacks or sandwich configurations according to Table 3 were studied.

insulation to prevent thermal losses and the intrusion of electrolyte-degrading light. The circuit consisted of three parallel electric current measurement branches, each containing a small precision resistor made from nickel wire with about 0.1 Ω, which was attached to the anode serving as working electrode WE (usually #2, #4, and #6) via copper wires. The other side of the circuit was attached to one or two cathodes serving as counter electrodes CE with the same connection scheme. The electric current is adjusted via a digital power supply (Elektro-Automatik PS 8032-20T) allowing the precise adjustment of potentials. The electric current flow over each precision resistance was determined by measuring the potential drop via fast amplifiers. The three amplified voltages were recorded alongside with the applied potential at 1 kHz sampling rate using an ADC acquisition device (Measurement Computing FS1208). The bias of the electric circuits was determined by measuring the currents in the short-circuit mode using PeakTech amperemeter (0−200 mA) and VoltCraft amperemeter (0−10 A). For the preparation of the electrolyte solution, approximately 70 L of deionized water were saturated with nitrogen for several minutes in the opaque liquid reservoir. Afterward, the electrolyte salts were added slowly under continuous nitrogen flushing and liquid stirring, while preventing any light exposure of the solution. The gas outlet of the closed reservoir was connected to the local exhaust funnel. Prior to the measurements, the liquid was heated up by circulating the liquid until reaching thermal equilibrium. For the final electrochemical activation of the electrodes, the reactor section was flooded prior to the mass transfer measurements, and the electrodes were operated as cathodes and anodes, respectively, at the potential range of electrolysis (Φ > 1.5 V). After the activation, the setup was operated in the fully developed pulse flow to ensure complete wetting of the packing (also known as KAN startup mode) before adjusting the desired flow conditions. The superficial liquid uLS and gas velocities uGS were varied between 0.01 and 0.04 m/s and between 0.2 and 1.0 m/s, respectively. The solution contained 1 M Na2 CO 3, 0.03 M K4Fe(II)(CN)6, and 0.001 M K3Fe(III)(CN)6 and was kept at 30 °C under nitrogen atmosphere without light exposure. For each set of measurements, a fresh electrolyte solution was prepared as described above. As the outlet of the gas−liquid separator was open to the atmosphere, the reactor pressure was nearly atmospheric. Basically, the electric currents were acquired in two different modes. In the “plateau mode”, the applied (electric) potential was slowly increased from 0 V to about 1.3 V to measure not only the limiting current plateau but also the slope of the “kinetic” ramp. The maximum sweep rate was 10 mV per second. In the “dynamic mode”, the applied voltage was set to 1.0 V corresponding to the typical plateau range. Then, the signal was recorded for at least 60 s. Commonly, the working electrodes #2, #4, and #6, located at the axial position Z = L/D = {2, 4, 6} respectively, were operated as cathodes, while the counter electrodes #1, #3, #5, and #7 at Z = {1, 3, 5, 7} were connected as anodes. The cathodes were operated in parallel, and their corresponding currents were recorded synchronously.

Table 3. Summary of Different Electrode Configurations Applied configuration type single stack, CE above single stack, CE below single sandwich multiple sandwiches

working electrode(s)

counter electrode(s)

position of counter electrode

#2

#1

above

#2

#3 to #7 (one at a time) #1, #3 (parallel) #1, #3, #5, #7 (parallel)

below

#2 #2, #4, #6

above and below above and below

For all electrode configurations, the flow conditions were kept constant at trickle flow (uLS = 0.02 m/s, uGS = 0.2 m/s), and the electric current at the working electrode WE#2 was measured with different counter electrodes connected. The results of the configuration study are presented in Figure 3. In Figure 3a, the limiting current curves measured for 20 ppi foams for the single sandwich configuration (dashed black line) and the single stack configurations (colored lines) are shown. At first glance, there is a remarkable difference in the slope of the kinetic range as well as a clear shift of the limiting current range toward higher potentials Φ. Moreover, a difference is observed between the configurations with the counter electrode below (solid blue line, ΔZ = 1) and above (dotted red line, ΔZ = −1). In Figure 3b the effect of the configuration on the limiting current is highlighted. Contrary to the 20 ppi foams (plain columns), the effect of the configuration is much more pronounced for 30 ppi foams (cross-hatched columns). In the latter case, the limiting current I decreases by 4% to 20% when increasing the axial distance ΔZ of the electrodes by a factor of 2 and 3, respectively. At axial distance ΔZ > 3, a complete collapse of the limiting current curve was encountered for 30 ppi (not shown), which was not the case for 20 ppi foams. Beyond that, a 10% difference in the limiting current value is measured for 30 ppi foams between the configurations with the counter electrode above (ΔZ = −1) and below (ΔZ = 1). This effect traces back to the stacked configurations with dense electrodes, where the “rear side” of the working electrode is shielded from the electric field and does not contribute to the same extent to the measured electric current.34 Thus, the shear stress at the upper side of the working electrode is dominant for the counter electrode above (ΔZ = −1), while the lower side is over-represented for the counter electrode below (ΔZ = 1). In the dense 30 ppi foams, the struts form a denser shield inhibiting a complete penetration by the electric field. In contrast, the 20 ppi electrodes still provide more openings for the electric field, i.e., are completely penetrated. Hence, no difference is encountered there between the above and below configuration. This might also explain the curve collapse at electrode distances ΔZ > 3, where three unconnected electrodes are located in between. Thus, it is debatable, to which extent mass transfer results for electrode distances of few millimeters only22,32,33 and up to one meter5 are really comparable, when considering such a pronounced impact of the electrode distances on the limiting current.

4. RESULTS 4.1. Effect of Electrode Configuration. Prior to the comprehensive mass transfer study concerning the effects of flow rate and pore density, the impact of the electrode configurations was addressed. Since the electrodes of 3 mm E


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Figure 3. Effect of the electrode configuration on the limiting current curve of the working electrode WE#2 at trickle flow conditions (uLS = 0.02 m/ s, uGS = 0.2 m/s): a) limiting current curves for 20 ppi foams, b) height of the limiting current for 20 and 30 ppi foams for different configurations, and c) development of the kinetic slope depending on the axial distance between working and counter electrode for both pore densities.

Figure 4. Typical limiting current curves at uLS = 0.02 m/s, uGS = 0.2 m/s in a 30 ppi foam packing showing a) trickle flow (WE#2), b) transition flow (WE#4), and c) pulse flow (WE#6).

relation between ΔZ and dI/dΦ is less obvious, which is also due to the sudden collapse of the limiting current curve at ΔZ > 3. Using the 30 ppi foams in a single sandwich configuration (diamond with cross), the slope is significantly larger than in any other configuration. Thereby, the slope is as high as for the configuration with the counter electrode installed above (crossed red triangle) and below (crossed blue triangle) the working electrode together. This also accounts for the corresponding 20 ppi configurations (empty symbols). Since the slope represents the relation between kinetic and electric potential (= driving force), such behavior is reasonable. Finally, the following conclusions can be drawn for the preferred configuration to study the LSMT at various flow conditions for 20 and 30 ppi foams reliably: (i) Using the sandwich configuration with short axial distance yields the widest limiting current plateaus and reduces the total electric resistance of the setup. This is even more important for the finer solid foams. The chosen distance of 100 mm (ΔZ = 1) denotes a reasonable compromise between electric resistance and undisturbed hydrodynamics.8 (ii) At least for 20 ppi foams, the synchronous wiring of multiple working and counter electrodes (“multiple sandwiches”) does not invoke a significant bias to the measure-

In addition to the single stack configurations with only one working electrode, the limiting current was also measured at WE#2 when operated with two other (independent) working electrodes in parallel. This multiple sandwich configuration is also shown in Figure 3b (white column). Since the WE#2 was sharing the lower counter electrode CE#3 with the working electrode WE#4, a 5% decrease in the limiting current is encountered for the finer foams compared to the single sandwich configuration. The difference is attributed to the altered electric field. In the coarser foam, only a very minor difference was observed. The effect of the electrode configuration is even more pronounced in the slope of the kinetic range as summarized in Figure 3c using the same symbols as in Figure 3b. The slope drops with increasing distance between the electrodes for both pore densities. In principal, the resistances for the electric flux consist of the contributions of the solid−liquid interfaces at cathode and anode as well as of the ohmic drop in between. The latter is a function of the specific conductivity of the liquid and the distance between the two electrodes. The evolution of the total electric resistance is indicated by the dashed line in Figure 3c which represents the linear regression through the reciprocal slope values of the 20 ppi foams (open symbols). For the finer foams (crossed symbols), however, the mathematical F


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Industrial & Engineering Chemistry Research ments. In 30 ppi foams, the synchronous operation is accompanied by a bias of 5% in the limiting current value. In the upcoming sections, the multiple sandwich configuration with three independent working electrodes and four shared counter electrodes was used to study the interaction between hydrodynamics and liquid−solid mass transfer. 4.2. Flow Regime Impact. To reveal the typical limiting current curves at different gas−liquid−solid interactions, the reactor packed with 30 ppi solid foam was operated at a flow condition (uLS = 0.02 m/s, uGS = 0.8 m/s), which is close to the low-to-high-interaction regime transition. It is known that the onset of pulse flow is initially observed at the reactor bottom, while stable trickle flow still remains at the reactor top.7,40,41 The electric current curves at the electrodes #2, #4, and #6 (left to right in Figure 4) were measured independently and synchronously. The topmost working electrode (WE#2, Figure 4a) features the typical plateau (horizontal line) of the limiting current curve for trickle flow with very narrow distribution of the current values. In contrast, the second working electrode (WE#4, Figure 4b) shows a wide distribution of values in both the “kinetic” range (Φ < 0.5 V) as well as the plateau range. The “gap” at approximately Φ = 0.75 V is attributed to the random nature of the dynamics at the transition from lowinteraction (trickle flow) to high-interaction (pulse flow) regime occurring occasionally during the measurements. While the fluctuations in the plateau range result from the variation of the shear stress induced by traveling liquid pulses, the minor fluctuations in the kinetic range are attributed to variations of the effectively used area of the electrode. In comparison, the kinetic slope in Figure 4a is lower than the maximum slope in Figure 4b but higher than the lowest possible slope in Figure 4b. This indicates also a partial dewetting of the electrode between the passage of two traveling pulses.5 Figure 4c shows the typical behavior of a fully developed pulse flow. The kinetic range is only subjected to moderate fluctuations, while the plateau range features significant current variations encapsulated between the plateau value (which is slightly lower than in Figure 4a) and the kinetically limiting slope (which is slightly lower than in Figure 4b). Due to the large mass transfer during the pulse passage, a clear upper limiting current cannot be distinguished below Φ < 1 V. Although the peak currents are significantly higher, the values are more frequently located near the lower plateau range (at Ilim = 0.18 A) than at the kinetic limit. It should be mentioned that, although the properties of the electrolyte were similar to deionized water (see Table 4), significant changes concerning the regime transition were encountered as shown in the flow maps in Figure 5. The shaded area thereby denotes the range of transition velocities, where neither stable trickle flow nor stable pulse flow was encountered. For the sake of comparability with our previous work,9 only the data from the lowest working electrode WE#6 were taken into account. At this axial position,

Figure 5. Flow regime map as a function of flow rates, liquid properties, and type of liquid distributor for a) 20 ppi and b) 30 ppi solid foams (GLY: 51% (m/m) aqueous glycerol solution, ELEC: electrolyte solution, H2O: deionized water, SN: spray nozzle, MPD: multipoint distributor).

the gas and liquid flow rates considered in Figure 4c belong to the pulse flow regime as indicated by the red circle in Figure 5b. For both foam pore densities, the trickle-to-pulse transition region is shifted significantly toward lower superficial liquid and gas velocities. Likely, the effect is mainly attributed to the change in surface tension, which is known to have a very pronounced effect in the slightly hydrophobic SiSiC foams.9 4.3. Effective Liquid−Solid Mass Transfer Coefficients. In Figure 6, the determined effective LSMT coefficients ϕkLSa are shown as a function of the superficial liquid velocity for two different gas superficial velocities. For the sake of clarity, only the data of the central working electrode at Z = 4 (WE#4) are shown. The dynamic fluctuations of the value over time are indicated by the shaded areas around the corresponding timeaveraged value. The width of the so-called violin plot corresponds to the distribution density function. At trickle flow (empty symbols), the mass transfer coefficient increases linearly with the liquid flow rate. At these conditions, the variation of the mass transfer with time is smaller than the size of the symbol, and, thus, it is not shown here. For trickling liquid (uLS = 0.01 m/s), the gas phase has no influence on the mass transfer. At pulse flow (solid symbols, plain violin plots), however, both the average values as well as the temporal

Table 4. Physicochemical Properties of the Three Solutions in Figure 5 liquid

H2O

ELEC

GLY

temperature ϑ in °C viscosity ηL in mPa s density ρL in kg/m3 surface tension σL in mN/m

20 1 1000 72.5

30 1.4 1100 65

20 6 1150 65 G


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Figure 6. Effective LSMT coefficients as a function of the superficial liquid velocity at Z = 4 for solid foams of a) 20 ppi and b) 30 ppi (shaded areas represent the distribution density function (histogram) in transition (hatched) and pulse regime (plain), empty and solid symbols correspond to trickle and pulse flow, respectively, half-empty symbols indicate the trickle-to-pulse transition).

Figure 7. Effective LSMT coefficients as a function of the superficial gas velocity at Z = 4 for solid foams of a) 20 ppi and b) 30 ppi (shaded areas represent the distribution density function (histogram) in transition (hatched) and pulse regime (plain), empty and solid symbols correspond to trickle and pulse flow, respectively, half-empty symbols indicate the trickle-to-pulse transition).

variation increase significantly. In the range of the flow regime transition (half-empty symbols, striped violin plots), the temporal variation is strongly asymmetric with the distribution maxima tending to the lower value range as already shown in Figure 4b. At fully developed pulse flow, the mass transfer values approach the Gaussian distribution, which stands in contrast to the asymmetric liquid holdup distribution highlighted in our previous study.9 Comparing 20 and 30 ppi solid foams, the LSMT coefficient increases with higher pore density due to the higher specific surface area. To account for the increase in the specific surface area, the ranges of the y axis of Figure 6a and b were adapted corresponding to the same range of φkLS. However, the mass transfer coefficients do not increase fully proportionally to the pore density (1000 m2/m3 vs 1600 m2/m3). The corresponding effect of the gas velocity is summarized in Figure 7 at two different liquid flow rates. The effect of the gas flow rate becomes large, when changing from trickle (empty symbols) to pulse flow (solid symbols) at the lower liquid velocity uLS = 0.02 m/s. At pulse flow conditions (solid symbols), however, the φkLSa values increase only moderately with further increase in the gas velocity.

4.4. Electric Current Dynamics. The dynamics of the electric current as a measure for the effective LSMT coefficient is highlighted in Figure 8 for different gas superficial velocities at top (#2, Z = 2), central (#4, Z = 4), and bottom (#6, Z = 6) regions of the packing. In addition, the corresponding violin plots are shown on the right-hand side. Pulse frequency f p and pulse velocity vp for the respective conditions were determined via the moving average method7 and cross-correlation of the electric current signals, respectively. At trickle flow conditions (uGS = 0.2 m/s) all electrodes exhibit nearly constant electric currents with only minor variations. With increasing liquid flow rate (uGS = 0.4 m/s), WE#6 (Z = 6, green line) experiences initial pulses. When the pulse front reaches the electrode, the electric current increases almost instantaneously up to a factor of 3. The decay period, however, shows a comparably slow decrease and reaches base values eventually below the corresponding trickle flow value. Due to inertia effects, the curve features the typical shark fin shape also known from other packings operated at pulse flow.42,43 Increasing the gas flow rate further (uGS = 0.6 m/s), the frequency of approaching local pulses f p reach about 3.7 Hz and the impact of subsequent pulses overlap due to their inertia. At uGS = 0.8 m/s, fully developed pulse flow is observed H


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pulse volume.7 Furthermore, a decrease in liquid holdup during the base period is encountered.7 Averaged over time (see blue curve in Figure 7b), the effective LSMT coefficient is the highest at the highest gas flow rate, although the variation range is smaller than for uGS = 0.8 m/s (see Figure 8, right side). The statistical analysis of the time-averaged mass transfer data reveal distinct relation between φkLSa and the product of pulse frequency and velocity. While recognized in the statistics of short signal segments including a few pulses, the relation is also well represented in the time-average data of f p × vp (average pulse frequency × average pulse velocity). As shown in Figure 9, this accounts for minimal (downward blue triangles), average (black circles), and maximal (upward red triangle) mass transfer rate taken from the respective histograms (see violin plots in Figure 6 and Figure 7). For 20 ppi foams, the effective mass transfer coefficient increases linearly with f p × vp and appears to be related to the liquid acceleration induced by passing pulses. Although less distinct, a similar trend is observed for 30 ppi foams. 4.5. Time Scale Analysis. In our previous study,7 the dynamic liquid distribution was examined with high spatiotemporal resolution using ultrafast X-ray computed tomography. This study revealed the hydrodynamic properties of pulsing in solid foam packed reactors. The discovered time scales of the liquid holdup pulses were significantly shorter compared to the current pulses studied in the present work, which is highlighted by two representative time series displayed in Figure 10. Figure 10a represents the temporal evolution of the liquid holdup average over the cross section for the 20 ppi foam, which was obtained using ultrafast X-ray computed tomography (CT) described previously.7,8 The lower graph in Figure 10c represents the time series of the electric current signal of one working electrode. Although the experimental conditions were different in terms of fluid properties (water vs electrolyte solution and air vs nitrogen) and liquid velocity (uLS = 0.04 m/s vs 0.02 m/s), the gas velocity uGS as well as pulse frequency f p and velocity vp agree very well. The gray-shaded pulses were discretized from the corresponding profile of liquid holdup and electric current, respectively, according to the thresholding method proposed by Zalucky et al.7 While the liquid holdup profile features a rapid increase and decrease during the pulse passage, the electric current pulses show a significantly slower

Figure 8. Temporal variation of the electric current at various heights with characteristic parameters in 30 ppi foams at uLS = 0.02 m/s (blue curves of Z = 4 correspond to blue data in Figure 7b).

at WE#4 and WE#6 (red and green time series). The frequency f p of the larger pulses, which merged from the frequent local pulses at higher axial position, decreases to 2.6 Hz. Due to the lower pulse frequency, the electric current drops to the trickle flow value in the base period between consecutive pulses. At the highest gas velocity (uGS = 1 m/s), the base period between two consecutive pulses shrinks significantly. Consequently, the characteristic pulse decay time is not sufficient to reach the base electric current. A pulse velocity vp of 1.2 m/s between electrodes #4 and #6 was estimated via cross-correlation. This number confirms previous findings, whereby the increase in gas velocity is accompanied by an increase in pulse velocity and

Figure 9. Relation between effective mass transfer coefficients and pulse properties for a) 20 ppi and b) 30 ppi (each set comprising min, avg, max corresponds to one particular operating condition in terms of superficial gas and liquid velocity as exemplarily shown). I


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current increase cannot be attributed solely to the increase in shear stress or liquid holdup. Basically, the pulse passage has two local effects. First, the drag of the pulse increases the local shear stress by the viscous forces. Second, the liquid approaching with a pulse exchanges the liquid film, which covers the packing surface. Thereby, both effects contribute to an increase in the transfer of the ion species from the liquid bulk phase to the electrode surface and, thus, to an increase in the measured current.

5. DYNAMIC MASS TRANSFER MODELING 5.1. Modeling Approach and Solution Strategy. Inspired by natural liquid waves two fundamental mechanism concepts for the impact of a liquid pulse can be imagined phenomenologically: the tsunami hypothesis and the surface wave hypothesis. In the latter, the liquid pulses resemble oceanic surface waves, which roll over a more or less stagnant liquid bulk reservoir (the static liquid holdup). Moving along the surface, a wave exerts some momentum to the bulk and exchanges liquid with a liquid circulation zone close to the surface. Unlike surface waves, the complete water column is set to movement during a tsunami and forms large scaled eddies reaching from the top to the bottom of the water column. Consequently, the liquid zones are exchanged completely. To mimic the effects of both hypotheses, a simplified model with seven compartments has been developed and implemented in MATLAB. Each compartment thereby represents one of the seven electrodes connected via entering and leaving streams of both reactive components K4Fe(II)(CN)6 and K3Fe(III)(CN)6. Each compartment with the volume VC (volume of one porous electrode) consists of three modeled zones, namely the static zone “s”, the trickle zone “tr”, and the pulse zone “p” as depicted for the topmost working electrode WE#2 in Figure 11.

Figure 10. Qualitative comparison of liquid holdup time series (a), absolute gradient (b, eq 5), and electric current time series (c) at comparable pulse frequencies and velocities.

modulation due to the inertia effects mentioned above. The average pulse duration in the current signal (τ ̅ ≈ 100 ms) is about three times the liquid holdup duration (τ ̅ ≈ 33 ms). A right-skewed liquid holdup distribution is a distinct feature of the fully developed pulse flow in solid foams,7 while the electric current distribution is fully symmetric (not shown). The time constants of the liquid holdup evolution are accessible via the absolute gradients of the tomographic images calculated from each pixel (x, y) according to

∑ |εL(x , y , t + 1) − εL(x , y , t )|

(5)

The absolute gradient profile is shown in Figure 10b at each time step t. The profile features an exponential decay of a PT1 element and − in contrast to the liquid holdup profile7 − a stable baseline. Moreover, it illustrates how long the local holdup is distorted due to the pulse passage. By fitting the exponential function ⎛ t ⎞ y(t ) = K ·exp⎜ − ⎟ ⎝ T1 ⎠

(6)

against the absolute gradients from the CT data, the time constant T1 was determined. While different K values were determined for different pulses at the same flow condition, T1 depends only on the pore density as shown in Table 5, regardless of the gas and liquid velocity or pulse parameters. Table 5. Time Constants of the Exponential Decay (PT1 Element) Determined from CT Holdup Dataa pore density

T1 in 10−3 s

R2

20 ppi 30 ppi 45 ppi

17.8 ± 0.14 16.1 ± 0.12 13.8 ± 0.14

0.9286 0.9164 0.9087

Figure 11. Scheme of one compartment with volume VC shown for the limiting component K3Fe(III)(CN)6 at WE#2.

a

Thereby, the static zone represents the solely reactive zone, where the educt is consumed in a first-order reaction with respect to educt concentration ceduct,s and applied potential Φ with the reaction rate

Although not studied in the present work, the data for 45 ppi foams are also given for the sake of completeness.

With increasing pore density (higher ppi number) the time constants decrease with increasing pore density due to increasing capillary forces acting in finer foams. Since a stable baseline in the limiting current did not exist, no such decay relation could be determined. Considering the fundamentally different evolution of the electric current with time (see Figure 10), the increase and decrease in the LSMT in terms of the

R i = kR ·Φ·ceduct,s

(7)

at electrode i = {2, 4, 6}, where kR is the intrinsic reaction rate. While the reaction rate is the driving sink at the working electrodes, the reaction rates of the counter electrodes i = {1, 3, 5, 7} J


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Figure 12. Scheme for compartment coupling for one of the two components.

1 1 R i = + ·R i − 1 + ·R i + 1 2 2

compartments, the holdup peaks are shifted by the arbitrary pulse run time

(8)

Δτp = Δz /vp

are driven by and coupled to the electric currents occurring at the working electrodes above (i+1) and below (i−1). Between all zones of one compartment, transient mass transfer is imposed with a transfer rate (φkLSA)j,z↔α from zone z to another zone α. As shown in Figure 12, the corresponding zones are furthermore connected to the neighbor compartments via convective inlet and outlet flow, respectively. For the pulse domain, the net change in the amount of each substance j dnj ,p dt

=

d(VC·cj ,p·εL,p) dt

i.e. the time which the pulse requires for passing the axial distance between the (electrode) compartments. The timevariant mass transfer rate, in turn, is coupled to the pulse holdup profile of the compartment via a damped linear oscillation ODE. The influence of the dynamic inertia accounts for the dynamic behavior of the limiting currents, which differ from the liquid holdup profiles as discusses above. Finally, the inlet concentrations of the compartments are coupled with the outlet concentrations of the corresponding compartment in the upstream direction. In case of the trickling flux, the concentration at the electrode i + 1 is defined as

α = tr ,s

=

∑

nj̇ ,p ↔ α +

̇ ∑ nconv

(9)

is given by the mass transfer fluxes to trickle “tr” and static “s” zones on the one hand. On the other hand, the net sum of the convective fluxes ṅconv is provided via the pulse zone interface AR · εL,p, where AR denotes the cross section of the reactor. In detail, the molar balances for the component j in the pulse zone “p” are formulated as VC·

d(cj ,p·εL,p) dt

cj ,tr,in(t )|i + 1 = cj ,tr(t − Δτtr)|i

∑

Δτtr = Δz /vtr

cj ,p,in(t )|i + 1 = cj ,p(t − Δτp)|i

(10)

dcj ,tr dt

α = p,s

=

∑

cj ,p,in =

(φkLSA)j ,tr ↔ α ·(cj ,tr − cj , α)

+ vtr ·AR ·εL,tr ·(cj ,tr,in − cj ,tr)

dcj ,s dt

α = tr ,p

=

∑

∑ cj ,p·εL,p/∑ εL,p

(17)

Eventually, the set of 42 ordinary differential equations of the molar balances (3 zones × 2 components × 7 electrodes) was implemented in MATLAB R2015a as an initial value problem using the factory ode45 solver with the NonNegative option. 5.2. Mechanistic Studies and Sensitivity Analysis. To carry out several generic studies, the limiting current measurement from Figure 10 was chosen. The relevant parameters of this reference case are given in Table S1 in the Supporting Information. To ensure quasi steady-state dynamics, a time of at least 10 s was simulated. Since the liquid properties of the electrolyte solution differ from the properties of deionized water as discussed above, the holdup parameters were approximated to comply with the liquid-side mass balance. Based on the experiences from previous studies,7,8 the parameters were specified to ensure realistic trickle velocities and physically sound relations between superficial liquid velocities and liquid holdup. The objective of this simulation approach is rather to assist the interpretation of the

(11)

where εL,tr is the constant trickling holdup, and vtr is the interstitial trickling velocity. Finally, the molar balance of the static zone “s” is given by εL,s ·VC·

(16)

with the pulse run time defined in eq 13. In contrast to the trickle concentration, however, complete back-mixing of the concentrations within one pulse is assumed (after leaving one compartment). Thus, the concentration within one pulse is calculated according to

where εL,p is the time-variant liquid holdup of the pulse, and cj,p,in and cj,p are the inlet flow and actual concentration. Accordingly, the molar balance of the trickling zone “tr” is given by εL,tr ·VC·

(15)

assuming plug flow with a trickling velocity vtr between two consecutive compartments. In a similar manner, the inlet concentration for the pulse zone is defined as

((φkLSA)j ,p ↔ α ·(cj ,p − cj , α))

+ vp·AR ·εL,p·(cj ,p,in − cj ,p)

(14)

with the trickle time-of-flight

α = tr ,s

=

(13)

(φkLSA)j , s ↔ α ·(cj , s − cj , α) − νj·R i (12)

Per compartment and component, three molar balances are solved for the time-variant concentrations cj,α in zone α with the reaction rate Ri. Furthermore, pulse holdup εL,p, effective mass transfer (φkLSA)j,α as well as the inlet concentrations cj,α,in are given as discrete values for each time step, compartment, and component, respectively. The pulse holdup is given as a continuous profile of fast sinusoidal jumps followed by the firstorder exponential decay described earlier. Between the K


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Industrial & Engineering Chemistry Research fundamental dynamic behavior of mass transfer in pulsing flow than a complete quantitative agreement. The qualitative character arises from the necessary parameter estimations, since the pulse decay constants and liquid holdup data were only determined for deionized water. While the decay constants were kept for the electrolyte solution, the holdup parameters were estimated in order to fulfill the liquid-side mass balance. With regard to the intention of the model, the assumptions are justified and inevitable for the formulation of the model. An example of the model’s dynamic prediction of the reference conditions according to Table S1 is shown in Figure 13.

Table 6. List of Different Mass Transfer Cases Studied for Figure 13a Case A1 A2 A3 A4 A5

φkLSa|p→s = = = = =

f(t) f(t) 0 0 0

φkLSa|p→tr = = = = =

f(t) f(t) f(t) f(t) 2 · f(t)

φkLSa|tr→s = = = = =

f(t) const const f(t) f(t)

f(t) denotes a time-variant function of φkLSa which varies between 0.042 1/s and 0.17 1/s, i.e. the measured range from Figure 10c.

a

In Case A1 (applied for the reference case shown in Figure 13) the individual mass transfer rates between all zones are identical and vary over time (compare Figure 13a, dashed line) in accordance to the ODE coupling between φkLSa and liquid holdup. In Case A2, on the other hand, a constant value is assumed for the mass transfer rate between pulse and static zone (φkLSa ≡ 0.042 1/s). By their results, Cases A1 and A2 with three mass transfer paths yield the highest average electric current as well as the highest range of variation as depicted by the violin plots in Figure 14, although Case A1 clearly outperforms the others in terms of average value and dynamic fluctuation range.

Figure 13. Time series of periodic liquid holdup during pulse flow and corresponding LSMT (a), predicted concentrations in the static, trickle, and pulse zone (b), and calculated electric currents (c) for the working electrode WE#2 at reference conditions (Table S1).

Figure 14. Electric current predicted for the configuration given in Table 6.

For the reference condition shown in Figure 10 and Table S1, the liquid holdup pulses were assumed evolving in a regular periodic manner (Figure 13a, solid line) traveling through the reactor with the specified delay corresponding to the time-offlight at a given pulse velocity. In addition, the corresponding φkLSa profile is given, which is slightly delayed due to the inertia. In Figure 13b, the concentration profiles of the limiting component K3Fe(III)(CN)6 for each zone of the working electrode are depicted. While the Fe3+ concentration of the pulse zone (dotted blue line) increases significantly due to the convective inflow of pulses, the Fe3+ concentration in the trickle zone (dashed magenta line) drops at the pulse arrival, because the mass transfer rate between trickle and static zone is increased during the pulse passage. While the Fe3+ concentrations in pulse and trickle zone fluctuate about 0.1 mol/m3 between minimum and maximum, the concentration in the static zone (solid line) varies only slightly (0.04 mol/m3) during one period. The calculated electric current in Figure 13c, on the other hand, varies significantly between 0.22 and 0.30 A. To evaluate the mechanism of the pulsing further, different test Cases A1 to A5 were simulated with various mass transfer paths and dynamics summarized in Table 6.

In the Cases A3, A4, and A5, the direct transfer from the pulse zone to the static zone is omitted (φkLSa|p→s = 0), which mimics the sequential transfer of the surface wave hypothesis in different variations. In combination with the constant transfer rate from trickle to static interface (Case A3), the current’s dynamics becomes negligible, and the electric current reaches only one-third compared to Case A1. Although the volume of the trickle zone is significantly smaller compared to the other two zones, the trickling film damps the mass transfer oscillation completely. In contrast, the dynamics of the trickle-to-static transfer φkLSa|tr→s are considered in Cases A4 and A5. As the overall transfer is strictly limited by the transfer from trickle to static zone (φkLSa|p→s = 0), the amplitude of the modulation at the pulse-to-trickle interface − which is the only difference between Cases A4 and A5 − has almost no influence. It can be concluded that the mass transfer of Case A1 is the most appropriate approach in terms of predicted electric current values and their dynamics. Using the mass transfer Case A1, a sensitivity analysis for the holdup parameters was carried L


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Figure 15. Results of the sensitivity analysis for variation of static (a), trickle (b), and pulse (c) holdup deviations from the reference case highlighted by the gray pattern.

out one at a time. The sensitivity of the model for the prediction of the electric current is shown in Figure 15 for a range of static, trickle, and pulse holdup values, respectively. The reference case is highlighted by the gray pattern. Thereby, the static liquid holdup revealed the largest impact on the dynamic range of the simulated electric currents Isim. The sensitivity toward the static holdup is related to the storage effect of the stationary film and the fluctuation of the trickle concentration resulting from it. The trickle holdup, in contrast, has only minor influence due to the trickle velocity, which affects the concentrations in the trickle zone. At large pulse holdup fluctuations, the average electric current decreases slightly, while the fluctuations are damped slightly. The latter is attributed to the increase in the interstitial trickle velocity vtr, which is calculated from the liquid-side mass balance and which increases with smaller pulse holdups. 5.3. Case Study for Different Scenarios. To mimic the more random dynamics of natural pulsing, the previously regular periodic pulse profile was converted into an irregular profile, where the pulse height and the time between two consecutive pulses were modulated up to 20% with a fixed pattern. Accordingly, the modulation also affects the mass transfer profile via the ODE coupling. Regardless of the modulation, all parameters’ time-average values were preserved at the same values as for the regular periodic simulation Case A1, which also accounts for time-variant transport between all zones. In Figure 16, a snapshot of the corresponding profiles is shown. In contrast to the regular periodic profiles, the modulated profiles feature faster sequences of pulse but also longer periods with no pulse passage. The simulated electric current profile in Figure 16c reflects the measured current profile (compare Figure 10c) very well. Nevertheless, the distribution of the current values in Figure 17 (Scenario A = irregular equivalent to Case A1) still deviates slightly from the measured data in terms of the dynamic range. Additionally, the simulations revealed clearly that even the most accurate measurement of the instantaneous mass transfer rate in solid foams is disguised by the magnitude of the static liquid holdup, the magnitude of the mass transfer rates, and the fast sequence of pulses, at least to some extent. For conventional packings with lower static holdup (see violin Figure 15a, to the left) the extent of disguise is significantly lower.

Figure 16. Simulation of the measurement data in Figure 10 with irregular a) holdup and mass transfer time series for the calculation of the concentration profiles b) and the electric current profile at WE#2 c).

In addition to the simulation of the experiment in Figure 10c (Scenario A), various scenarios were simulated to test for their effect on the electric current’s dynamic and average, which indicate the average conversion of the limiting species. For the adjustment of parameters tabulated in Figure 17, the profound knowledge from our previous studies was taken into account.7,8 In Scenario B, regime transition conditions were chosen, assuming slow but frequent pulses of small volume. In contrast, stable pulse flow conditions were assumed in Scenarios C and D, once with moderate liquid, gas, and pulse velocities and once with high throughputs, respectively. The comparison of the three scenarios reveals that Scenarios C and D are only slightly superior to Scenario B, though the expenses in terms of required gas throughputs (and the resulting pressure drop) are not balanced out. The impact for processes with gas-phase limitations, however, is not considered in the simulation. With Scenarios E and F, an attempt was made to mimic a liquid with higher viscosity or lower surface tension, M


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(PUXX) is included in Figure 18 alongside with data for packings of cylindrical (CylXX)26 and spherical particles

Figure 18. Comparison of the determined specific mass transfer rate ranges against literature data for different packings (CylXX: cylindrical particles; SphXX: spherical particles, StPck: stuctured packing; PUXX: polyurethane foams. XX refers to particle length, diameter, and pore density, respectively).

(SphXX)5,22,43 of different diameters as well as data for a structured packing (StPck)1 operated at cocurrent gas−liquid downflow. Since geometric surface area a and transfer coefficient kLSa are also strongly interdependent, the effective mass transfer rate φkLSa was chosen for comparison instead of the commonly used Sherwood number. It should be mentioned furthermore that the prevailing flow regime might be different since it depends strongly on the solid structure. Additionally, there is some discrepancy in the diffusion coefficients due to different electrolyte compositions and temperature. Thus, the Schmidt (Sc = ηL/(ρL · DL)) number of the different studies ranges from 1560 to 2300, which are annotated below the labels in Figure 18. Although featuring only a moderately high Schmidt number of about 1840, the highest specific LSMT rates were obtained for the solid foams applied in this study. However, the comparably higher liquid fluxes have to be considered, which contribute to higher shear stress. At lower liquid velocities, however, the data for 20 ppi foams also coincides with the mass transfer coefficients for cylindrical particles26 of 6 × 6 mm (Cyl6) and 3 × 3 mm (Cyl3). The 30 ppi foams, however, outperform the cylindrical particles already at moderately high liquid throughputs. The LSMT performance of spherical particles22 with 3.8 mm (Sph3.8) diameter can be located between both foam packings. In pulse flow, the pulses propel faster through the solid-foam packed reactor (vp ≤ 1.7 m/s) than in conventional packed beds (vp ≤ 1.2 m/s),7 which invokes large shear stress at the static-pulse interface. It can be concluded that the superior performance of solid foam packings is mostly a benefit of their low flow resistance, which allows operation at higher gas and liquid throughputs without the expense of higher pressure drop. In addition to the “delayed” transition to pulse flow in solid foams and their low affinity for industrially less relevant bubbly flow, this new operation window is quite promising. The short contact time of elevated liquid throughputs, on the other hand, requires the application of catalytic systems with high kinetic activity.

Figure 17. Comparison of electric currents from measurement (Scenario A) and different simulation data with applied scenarios (Scenarios B to F) listed in the table.

respectively. Due to the higher viscosity in Scenario E, the trickle holdup and pulse holdup were higher, and the pulse frequency was assumed to be lower than for less viscous liquids. Additionally, the pulse decay time T1 and the specific mass transfer rates were assumed to drop by a factor of 2 in comparison to Scenario B. Despite that, both the range and the average value of the electric current shrink to about 30% and 40% of Scenario B, respectively. The contrary effect is encountered for Scenario F, for which a longer pulse decay time T1 of 27 ms was specified to mimic lower surface tension. Although pulse and trickle holdups were adjusted slightly above and below Scenario C, respectively, the dynamic range of Scenario F exceeds its counterpart C by far due to the higher pulse frequency (6 Hz vs 4 Hz). In contrast, the same average current is achieved in the simulations.

6. COMPARISON WITH OTHER PACKINGS So far, only Mohammed et al.3 studied the LSMT in solid foams. Both studies agree well concerning the enhanced mass transfer rates at higher superficial liquid velocities. However, contradicting results were reported for the influence of the gas flow and of the pore density. Although the effective mass transfer coefficients φkLSa of the two pore densities in the present work were roughly in the same range of specific surface areas, the discrepancies can be mainly attributed to the exceptional properties of compressible polyurethane foams with thin triangular-concave struts and lower specific surface areas used by Mohammed et al.3 Compared to metal or ceramic foams, liquid holdup and pressure drop,8,12 liquid distribution patterns,8,11,15,44 and regime transition3,9,12,14 already showed significant deviations. Nevertheless, the polyurethane data N


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7. SUMMARY In this work, the established limiting current technique was analyzed comprehensively and adopted for the liquid−solid mass transfer measurements in solid foam packed reactors operated with descending gas and liquid flow. Based on the literature review and preliminary experiments, an optimized setup was applied to a reactor packed with solid foams made from SiSiC and embedded nickel foam electrodes of similar morphology. In particular, the effects of various electrode configurations were addressed for two different foam pore densities. The LSMT was studied simultaneously at three axial positions for a wide range of gas and liquid velocities. Besides providing comprehensive data on the effective LSMT coefficient in trickle and pulse flow, a detailed analysis of the dynamics during the passage of liquid pulses was carried out. Thereby, the time series of electric currents revealed dynamic behavior differing significantly from the liquid holdup timeseries. By introducing a compartment model for the interaction of pulse-induced holdup and mass transfer fluctuations, the dynamic behavior was simulated successfully. The model revealed that the higher inertia of the electric current is attributed to the comparatively large static holdups in solid foam packings. The explorative study furthermore indicated that the liquid properties affect the mass transfer dynamics significantly. In terms of measured specific LSMT rates, the solid foam clearly outperforms conventional packings of similar specific surface area. Even though, the elevated LSMT rates are also attributed to larger operation windows in terms of gas and liquid flow rates due to the low pressure drop of the solid foams.

■

MPD = multipoint liquid distributor LSMT = liquid−solid mass transfer ppi = pores per (linear) inch, classification unit for the foam pore density PUXX = polyurethane (foam) with XX ppi classification SiSiC = silicon-infiltrated silicon-carbide (foam) SN = spray nozzle SphXX = spherical particle packing with XX mm nominal particle diameter StPck = structured packing Latin symbols

A = interfacial area [m2] a = interfacial area per volume of reactor [1/m] D = column diameter [m] DL = liquid diffusion coefficient [m2/s] cFe3+ = concentration of limiting component K3Fe(III)(CN)6 [mol/m3] cj,α = concentration of component j in zone α [mol/m3] cj,in,α = inlet concentration of component j in zone α [mol/ m3] F = Faraday’s constant [C/s·mol] f p = pulse frequency [Hz] I = measured current [A] Ilim, Isim = measured and simulated limiting current [A] kLS = liquid−solid mass transfer coefficient [m/s] kR = intrinsic reaction rate [1/V·s] L = axial position from packing’s top [m] ṅlim = limiting ion flux at the interface [mol/m3s] uGS = superficial gas velocity [m/s] uLS = superficial liquid velocity [m/s] Ri = reaction rate at electrode i [mol/m3s] Sc = Schmidt number (ηL/(ρL · DL)) [-] T1 = pulse decay time [s] t = time [s] VC = compartment volume [m3] vp = linear pulse velocity [m/s] vtr = interstitial trickle velocity [m/s] Z = L/D = dimensionless axial position [-] z = ion coordination number [-]

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.iecr.7b01578. Complete set of simulation parameters (PDF)

■

Greek symbols

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Markus Schubert: 0000-0002-6218-0989 Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS The authors acknowledge the Helmholtz Association for support of the research within the frame of the Helmholtz Energy Alliance “Energy Efficient Chemical Multiphase Processes” (HEA-E0004).

■

■

α = modeled pulse, static or trickle zone (α = p, s, or tr) Δτp = pulse time-of-flight [s] Δτtr = trickle time-of-flight [s] εL,α = liquid holdup of zone α [-] ηL = dynamic liquid viscosity [Pa·s] ϑ = temperature [°C] vj = stochiometric coefficient [-] ρL = liquid density [kg/m3] σL = liquid surface tension [N/m] φ = wetting efficiency [-] Φ = applied potential [V]

REFERENCES

(1) Battista, J.; Böhm, U. Mass Transfer in Trickle-Bed Reactors with Structured Packing. Chem. Eng. Technol. 2003, 26, 1061. (2) Burghardt, A.; Bartelmus, G.; Jaroszynski, M.; Kolodziej, A. Hydrodynamics and Mass Transfer in a Three-Phase Fixed-Bed Reactor with Cocurrent Gas-Liquid Downflow. Chem. Eng. J. 1995, 58, 83. (3) Mohammed, I.; Bauer, T.; Schubert, M.; Lange, R. Liquid-Solid Mass Transfer in Tubular Reactors with Solid Foam Packings. Chem. Eng. Sci. 2014, 108, 223.

NOMENCLATURE

Abbreviations

CT = computed tomography CylXX = cylindrical particle packing with XX mm nominal particle diameter ELEC = electrolyte solution GLY = glycerol solution H2O = deionized water O


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