Experimental demonstration of the heat transfer — pressure drop trade-off in 3D printed baffled logpile structures

Shaping of catalytic material by 3D printing allows for greater design freedom, which can be used to optimize reactor operating windows. A promising concept in this regard is the use of structures with porous baffles, which induce a cross-flow regime that offers enhanced heat transfer at relatively low pressure drop. In this work, eighteen novel cylindrical 3D printed baffled logpile structures were designed and their heat transfer — pressure drop trade-off was quantified experimentally. It was found that the performance of these full-scale structures could be estimated from previous pseudo-2D computational fluid dynamics simulations for variations in configuration and baffle gap spacing. Moreover, the structures with 50μm baffle gap spacing demonstrated superior heat transfer performance over a packed bed of pellets, as hypothesized. The number of baffles per unit length was introduced as a novel design variable. Remarkably, a reduction of this parameter led to comparable heat transfer performance while achieving a significant decrease in pressure drop. Finally, positioning of consecutive baffles under an angle was demonstrated to have a favorable effect. The results were correlated to facilitate reactor design considerations. Overall, this work sheds light on the process intensification potential of 3D printed baffled logpile structures as novel structured catalysts, enabled by offering enhanced heat transfer characteristics at relatively low pressure drop, both of which can be tailored to meet specific process requirements.


Introduction
The design of chemical reactors involves balancing catalyst holdup, heat transfer, residence time distribution, internal mass transfer and pressure drop.The key design variables in this trade-off are the reactor dimensions and the catalyst geometry.Since conventionally used randomly packed beds of pellets often lack sufficient radial heat transfer, a reduction in reactor diameter is used as compensation.However, achieving sufficient production capacity then requires numerous of these small tubes, and such a multi-tubular configuration is relatively expensive [1,2].Therefore, it would be favorable to intensify the heat transfer within the chemical reactor by optimizing of the catalyst geometry.Significant progress has been made in this area through the use of cellular structures, such as honeycomb monoliths or foams, which feature a support with a high thermal conductivity [3][4][5].By avoiding point contacts and their associated heat transfer resistance, these structures present a fundamental improvement over the randomly packed beds [6,7].Washcoating is the default method of incorporating catalyst material in a cellular structure, but this limits the catalyst holdup [8].
For further process intensification, packed foams and packed Periodic Open Cellular Structures (POCS) have recently been proposed and demonstrated in a number of publications, mainly from researchers at the Politecnico di Milano [9][10][11][12][13][14][15].In this reactor configuration, small catalyst particles are packed into the foam (or POCS, a 3D printed foam with more controlled properties) to achieve higher catalyst holdup compared to washcoating.The heat transfer is significantly intensified in comparison to conventional packed beds, although there may be some limitations related to the particle size relative to the foam cell size, enforcing minor limitations on the reactor design.
Another promising concept for chemical process intensification is the direct 3D printing of catalyst material, eliminating the need for a backbone structure.This is typically achieved through Direct Ink Writing (DIW, also called micro-extrusion), producing so-called logpile structures [16][17][18][19].The 3D printing process offers greater design freedom, for example enabling independent control of particle size and porosity [20][21][22][23].In the scientific literature, two stacking configurations are commonly employed; aligned and staggered [24].In the aligned stacking configuration, where all stacked layers follow the same pattern, the structures resemble honeycomb monoliths.However, https://doi.org/10.1016/j.cej.2024.149092Received 20 October 2023; Received in revised form 31 December 2023; Accepted 23 January 2024 by nature of the stacking there are gaps in the transverse direction.The staggered stacking configuration is very similar, but each stacked layer is offset from the previous one.Previous experimental work conducted by our group studied the transverse dispersion coefficient (responsible for the fluid-phase contribution to the overall heat transfer performance) in different variations of these structures [25].This study demonstrated that the geometric freedom enables adjustable transverse dispersion behavior, but generally the extent of transverse dispersion was relatively limited.In addition to this, although point contacts are avoided by the 3D printing of structures, the low thermal conductivity of ceramic-like porous catalyst powders limits the solid-phase contribution [26][27][28].In light of the relatively limited benefits, it could be questioned whether these geometries justify the use of the relatively complex 3D printing technology.
In order to increase the process intensification potential of 3D printed logpile structures, the incorporation of baffles within these structures was recently proposed by our group [29].This concept was demonstrated through pseudo-2D Computational Fluid Dynamics (CFD) simulations.The proposed baffles are envisioned as densified zones within the logpile structure, rather than solid objects, thus guiding the fluid around the reactor in a cross-flow regime without complete blockage.Achieving such a flow regime in conventional packed bed reactors requires the use of non-catalytic internals to guide the flow.Hence, the proposed structures exploit the unique advantages of 3D printing, as it is relatively straightforward to incorporate densified zones in 3D printed logpile catalyst structures.In addition to this, the 3D printing of catalytic reactor modules as standalone reactor internals provides an advantage compared to reactors with non-catalytic internals and structured catalyst, since the complexity of the packing procedure is reduced.
For the proposed baffled logpile catalyst structures, the simulations showed that porous baffles can achieve a heat transfer performance only slightly lower than conventional solid baffles, while offering significantly reduced pressure drop.In addition to this, the heat transfer -pressure drop trade-off was much more favorable in these structures compared to the packed bed.Although this modeling study serves as a good starting point, experimental demonstration of the concept for full 3D modules is required, along with additional design parameter studies.
To this end, the performance of the proposed 3D printed baffled logpile catalyst structures will be experimentally investigated in this work.Specifically, the heat transfer -pressure drop trade-off will be mapped for various structure designs, using cylindrical polymeric structures.Previous modeling work has already established that the length of the baffle should be fixed at the highest possible value, while the spacing of the gaps in the densified baffle zones is the preferred tuning variable [29].The current investigations will examine the influence of this variable in the cylindrical configuration, where the orthogonal baffle placement is expected to yield different behavior.Additionally, the number of baffles per unit length of reactor and baffle rotation (i.e.positioning of consecutive baffles under an angle) will be evaluated.The simulations did not consider such geometries due to computational constraints, but these variables play a crucial role in reactor scale-up.The aim is to establish correlations between the results and the design parameters of the structure, facilitating wider application of the conclusions in reactor design considerations.
This paper is organized as follows.The Methods section first introduces the design variables of the structures under study and describes the experimental setup and methods.This is followed by an overview of the data treatment procedure, required to extract performance metrics from the measured pressure and temperature profiles.This methodology is then validated for the exemplary case of a packed bed of spheres.Following this, the main results are presented, including the pressure drop and heat transfer performance of eighteen structures of varying configuration, baffle gap spacing and number of baffles.Finally, a summary of the findings is presented.

Experimental
3D printed cylindrical modules were tested in a stainless steel tube with an inner diameter of 44.2 mm and an internal height of 205 mm.Nitrogen gas was introduced from the bottom of the tube through a porous distributor, regulated by a Bronkhorst F-201AV mass flow controller, with flow rates up to 26 L min −1 .The top of the tube was sealed with a lid, which included thermowells and a gas outlet, and was bolted and sealed with a rubber O-ring.To monitor the pressure drop over the structure, a Keller Series PD-23 differential pressure transmitter with a range of 0 to 200 mbar and a measurement accuracy of 0.2% was attached to the inlet and outlet.The side wall of the tube was equipped with evenly spaced electrical tracing, producing a maximum power output of 80 W. A thermocouple was placed at the axial center between the tracing and the tube to regulate the power output based on a target temperature.The tracing was insulated with 38 mm of glass wool and a 3 mm layer of Armaflex tape to minimize radial heat losses to the environment.Thermocouples with a diameter of 1.5 mm were placed at four different positions inside the removable lid to monitor the temperature within the structure.
For the evaluation of the pressure drop, the flow rate was increased to 26 L min −1 in fourteen steps.Each flow rate was maintained for six minutes while a data point was recorded every 30 s.During the transition between flow rates, a temporary non-steady state occurred.To ensure that measured pressure drop data corresponded to the steady state value, the first 60 and last 30 s of each flow rate measurement were excluded.During these measurements, a thermocouple positioned at the center of the structure monitored the temperature.
For the evaluation of heat transfer, the electrical tracing was set to a temperature of 353 K and steady state axial temperature profiles were recorded using sliding thermocouples at the center of the structure and near the reactor wall.This evaluation was conducted for five flow rates, corresponding to superficial velocities ranging from 0.03 to 0.37 m s −1 .The steady state was deemed reached when the center and nearwall temperatures were constant for ten minutes, which took between 20 min and 3 h depending on the flow rate and the initial temperature.The temperature profile was then recorded in 1 cm increments, with each measurement taken when the temperature remained stable for 30 s.

3D printed structure
Due to the brittleness of 3D printed catalyst structures, achieving proper sealing for the current experiments is challenging.Additionally, the determination of heat transfer and pressure drop does not require catalytic activity.In light of these considerations, polymeric structures were used, benefiting from a faster and more cost-effective printing process compared to the 3D printing of catalyst powders.Specifically, polyethylene terephthalate glycol (PETG) has been selected as the printing polymer thanks to its higher glass transition temperature, allowing for higher temperature operation compared to the more commonly used polylactic acid (PLA).Furthermore, PETG exhibits a slightly higher thermal conductivity than PLA (between 0.162 and 0.225 W m −1 K −1 for PETG and between 0.032 and 0.17 W m −1 K −1 for PLA) and this is comparable to the thermal conductivity of catalyst powders [26,27,[30][31][32].
All the structures were designed using the Blender software, sliced for printing in PrusaSlic3r and printed on a Prusa MK3S Fused Deposition Modelling (FDM) machine equipped with a 0.4 mm nozzle.Following previous experimental work from our group, the logpile structures were designed by stacking 1.5 mm cylinders with an axial stacking offset of 80% [25].An exemplary 3D printed logpile structure is visualized in Fig. 1.In this work, variations of configuration, baffle gap spacing and number of baffles per unit length are investigated, resulting in a total of eighteen structures.Porous baffle structures and split-recombine structures are the main configurations studied.The baffle gap spacing was varied for the porous baffle configuration, mirroring previous pseudo-2D simulations from our group [29].Values ranging from 0 to 250 μm are studied, along with an aligned logpile structure without baffles.For the variation of baffle gap spacing, the number of baffles per unit length was maintained at 208 m −1 , along with a baffle opening of 2 unit cell sizes, resulting in a slight variation in baffle fraction based on the spacing.The number of baffles per unit length ( baf f le , with unit m −1 ) was varied for both configurations, using a fixed baffle gap spacing of 50 μm.The visual representation shown in Fig. 2 provides a brief overview of these variations.In addition to the previous variations, two exemplary structures were designed with consecutive baffles under an angle to explore beyond the orthogonal 2D projection onto the logpile grid (where the baffles align with a single axis throughout the structure).The first structure is a porous baffle geometry with a baffle gap spacing of 50 μm and 208 baffles per meter, where the logpile stacking is rotated by 45°after every axially repeating unit (e.g. two baffles).The second structure is an alternative split-recombine design, alternating two 'splitting' layers with an offset of 90°with two 'recombining' layers with the same offset, with a baffle gap spacing of 50 μm and 119 baffles per meter.In this context, 'splitting' layers refer to layers where the baffle leads the gas past the wall and 'recombining' layers refer to layers where the baffle leads the gas through the center of the structure.
As the geometric variations result in varying porosity and relative baffle fraction for the different structures, an overview of these parameters is given in Table 1.The relative baffle fraction is defined as the area of the baffle divided by the cross-sectional area of the module.
The logpile structure of 149 mm high was integrated into a cylindrical module with a wall thickness of 2 mm, on a supporting grid of 1 mm high.Fig. 2 shows the position of the four thermowells with a wall thickness of 1 mm.It is worth noting that the wall thermowell is located on the inside of the structure, which does not fully represent the imposed wall temperature.Unfortunately, placing a thermowell on the outside of the structure, in contact with the stainless steel tube wall, could not be realized since this caused excessive channeling of the gas flow, hindering accurate measurements.While the internal structure had a height of 150 mm, the module was printed with a height of 205 mm to ensure its stability within the tube and to prevent displacement due to the gas flow.The overhead region featured two holes to facilitate the mounting and removal of the structure from the tube.To ensure adequate sealing of the structure, it was wrapped with a layer of PTFE thread seal tape.The scale of the structures currently under investigation is on the lower limit of relevance for industrial application, but it is relevant for the current state of the DIW technology.Scaling up with this 3D printing technology might pose challenging since post-processing becomes more time-consuming and complex as the size of the printed sample is increased.Alternatively, scaling of structures could also be achieved by stacking multiple units (both axially and radially).However, the feasibility of this approach was not yet studied in detail in the literature and further investigation is required before implementation can be considered.Therefore, the findings and correlations obtained from the current scale are relevant for DIW catalyst structures at the current state of the technology, but anticipating the performance of larger-scale structures may require additional experiments and simulations.

Heat transfer data evaluation
Experimental measurements provide axial temperature profiles, which serve as a solid basis for comparing the performance of different structures.However, the performance is more commonly expressed using heat transfer parameters.To calculate these parameters, the imposed wall temperature is useful.Although the power output of the  electric tracing was regulated to achieve a temperature of 353 K (at the axial center of the tube), its uniform heat flux, combined with heat losses from the bottom and top of the reactor, resulted in a temperature profile along the axial coordinate even without gas flow through the structure.It is anticipated that this temperature profile is constant along the radial coordinate at steady state without flow.Fig. 3 shows the profiles of four different structures at both the center and the nearwall thermowell.As hypothesized, minimal variation was observed between the different thermocouples and structures at the same axial position, validating this profile as the imposed wall temperature (at the outside of the structures' wall), independent of the structure itself.To facilitate subsequent heat transfer calculations, the profile was fitted to a second-degree polynomial, using the _  function from the  . toolbox in Python 3.10.This equation accurately describes the trend, as shown in Fig. 3 and testified by the low Mean Absolute Percentage Error (MAPE) of 0.17%.
For laminar convective flows with a uniform axial velocity, radial heat transfer is commonly described as conductive heat penetration as a function of the local residence time, with  = ∕ (where  is the superficial velocity).The general thermal energy balance for this situation yields Eq. ( 1), with the thermal diffusivity defined in Eq. ( 2).For the case of an imposed wall temperature profile, an analytical solution is provided by Carslaw and Jaeger [33], given in Eq. ( 3).In this equation,  () represents the wall temperature profile as a function of integration variable  and   is the n th root of  0 ( t   ).Considering a seconddegree polynomial wall temperature profile in the form of Eq. ( 4), integration yields Eq. (5).By fitting the experimentally obtained center temperature profile, using the outer radius of the center thermowell as , the effective radial thermal conductivity of the structure can be determined.This fitting process utilizes the  function from the  . toolbox in Python 3.10, with the MAPE as minimization target (Eq.( 6)).
To summarize, the assumptions made in this analysis are the following: • The local heat transfer along the azimuthal angle () is not explicitly considered, but is incorporated into the effective radial thermal conductivity.This assumption requires careful consideration on the measured near-wall temperature, since its value may be locally influenced by the cross-flowing gas in baffled structures.A detailed discussion on this influence is given in Section 4.1.• The axial thermal dispersion is negligible, which is generally valid for sufficiently high axial thermal Péclet numbers.This will be verified in Section 3.1.• The velocity profile is flat, which is generally valid in packed beds with a moderate to high  t ∕ p .In the current setup,  t ∕ p = 27.• The fluid temperature is constant at  = 0 by nature of Eq. ( 5), and the exact value is extrapolated from the polynomial description of the measured center temperature profile.However, due to the thermal contact between the inlet of the tube and the electrically traced wall in contact with the structure, this assumption is not strictly accurate.To account for the (lumped) heat transfer in this section, a fictive entrance length needs to be defined and fitted to the experimental data.• The heat transfer resistance is fully located inside of the structure.This is likely true, but given the relatively low thermal conductivity of PETG, the thermal resistance of the thermowell walls may be relatively high.In addition to this, the wall-to-bed heat transfer can be considered high in the 3D printed structure thanks to the direct connection between wall and structure, but this is not necessarily the case for the randomly packed bed [34].Therefore, this assumption will also be given detailed attention in Section 3.1.
The fitted effective radial thermal conductivities can in turn be correlated to the operating conditions and design parameters, which is commonly done in the form of Eq. (7).In this equation  represents the stagnant bed contribution and the term including  represents the convective contribution.The particle Reynolds number, defined in Eq. ( 8) with the superficial velocity and particle diameter as characteristic velocity and characteristic length, is used in this correlation with the density and viscosity calculated through Eqs. ( 9) and (10) (using  S = 1.62 × 10 −5 Pa s K −0.5 and  S = 171 K).The _  function from the  . toolbox in Python 3.10 is employed to determine the coefficients.To evaluate the quality of the fit, the MAPE is again used.

Pressure drop data evaluation
To accurately measure the pressure drop induced by the structure, it is necessary to account for the pressure drop across the entire setup, including upstream and downstream tubing.To this end, a baseline measurement was conducted using an empty module without the logpile structure.Additionally, the repeatability of the measurements was assessed by re-sealing the empty module with PTFE thread seal tape five times and calculating the Coefficient of Variation (COV).The COV of time-averaging was also determined.The results, shown in Fig. 4, demonstrate acceptably low COV values, generally below 10 −2 .At lower superficial velocities, the repeatability COV is slightly larger, but this is due to the low overall pressure drop at these conditions.All measurements were conducted within a relatively small temperature range, between 292 K and 296 K.
The baseline values were subtracted from the measurements with structures to isolate the structures' pressure drop.This pressure drop can then be correlated to the operating characteristics using Eq. ( 11), where  and  are fitted coefficients.The _  function from the  . toolbox in Python 3.10 was again employed to determine these coefficients, using the MAPE as a metric to evaluate the quality of the fit.

Packed bed validation
To validate the methodology used in this work, experiments were conducted with a randomly packed bed of monodisperse glass spheres of 1.5 mm diameter.Correlations are available in the literature to describe the pressure drop and heat transfer in this system.To enable this measurement and mirror the design of the logpile structures, a packed bed holder was employed.During the design phase, it was found that the thermowells could not be 3D printed as free standing structures, which is why a 1 mm supporting cross was incorporated over the entire height, as shown in Fig. 5.The holder was loaded with the glass spheres, and based on the mass of the particles, the average porosity was determined to be 42.2%.This relatively high value is likely attributed to the increased wall effects resulting from the presence of the supporting structure.
In addition to this, heat transfer measurements were conducted with a packed bed of cylinder-like PETG pellets (also shown in Fig. 5).The thermal conductivity of these pellets is the same as the 3D printed geometries and this allows for a more balanced comparison of the heat transfer performance.The pellets have an approximate height-overdiameter ratio of 1 to 2 and were sieved to a particle size between 1.7 and 2.5 mm.The porosity of this bed was calculated to be 40.0%.

Heat transfer -monodisperse glass spheres
The axial temperature profiles obtained in the packed bed are depicted as markers in Fig. 6.Triplicate experiments were conducted for the lower, middle and higher superficial velocities, and the small error bars indicate satisfactory repeatability of the measurements (an average COV of 0.0017 was achieved).It was observed that for  > 0.09 m, the center thermocouple recorded a higher temperature than the near-wall temperature, which is due to significant cooling through the tube lid.Since this may lead to complications in the data evaluation, only the data for  < 0.08 m was considered for all subsequent experiments.A comparison between the stagnant profile (without flow) and the nearwall profiles with flow reveals that the convective flow cools down the wall significantly, and thus the driving force should be determined based on the measured near-wall temperature rather than the imposed temperature.
The right-hand side of Fig. 6 shows the fitted curves based on Eq. (5).During the evaluation of this equation, it was observed that the solution oscillates along the radial coordinate for a small number of roots in the summation, resulting in inaccuracies.This is graphically demonstrated in Fig. 7 for the lowest and highest superficial velocities under study.For  > 1000, the results stabilize and the MAPE for  = 1000 compared to  = 10000 is just 0.24% for these experiments.Therefore,  = 1000 was adopted.
It was previously mentioned that a fictive entrance length is required in the fitting procedure to account for the heat transfer in the inlet region.By minimization of the MAPE of the fitted curves, the entrance length was established to be 11 mm, resulting in a MAPE of only 0.71%.It is worth noting that this length is longer than the thickness of the porous distributor (3.6 mm, considering negligible heat  transfer upstream of the distributor).However, this discrepancy can be attributed to the basis of the fictive entrance length, which represents the length required to transmit heat through a structure with the fitted effective radial thermal conductivity.In reality, the porous distributor possesses a higher effective thermal conductivity.
Before considering the validation, it is important to assess the validity of the model assumptions in the context of the packed bed configuration adopted.
Firstly, the assumption of negligible axial dispersion was evaluated by calculating the axial thermal Péclet number according to the methodology of Dixon and Cresswell [35] (the equations are shown in Appendix A).The obtained values ranged from 0.25 to 1.4 with increasing superficial velocity.These values indicate that convection is the main mechanism of axial heat transfer for the highest superficial velocities, but regardless of the superficial velocity, dispersion is not negligible since   e,a < ℎ∕ t (the height-over-radius ratio is approximately 4 in the section of the bed under study) [36].When assessing the validation results, an influence of axial dispersion is thus likely, which decreases with increasing superficial velocity.
Secondly, the assumptions of negligible resistance of the wall-to-bed heat transfer and conduction through the thermowell were examined.Again using the methodology of Dixon and Cresswell [35], the wall heat transfer coefficient was calculated.The associated resistance for steady state heat transfer was calculated through Eq. ( 12), which was compared to the resistance of the bed, calculated through Eq. ( 13) (using the tube radius and the outer radius of the center thermowell as  2 and  1 , respectively).The resistance of the bed ranged from approximately 19.7 K W −1 at low superficial velocity to 16.3 K W −1 at the highest superficial velocity, while the resistance of the wall-to-bed heat transfer was in the order of 0.52 K W −1 , significantly smaller.Although these are the values for steady state heat conduction, an approximation of the relative importance is obtained and it is indeed justified to neglect this resistance.Similarly, the resistance of the center thermowell was calculated through Eq. ( 13) as 6.43 K W −1 (using the outer and inner radii of the center thermowell as  2 and  1 , respectively), which is also smaller than the resistance of the bed.However, it is in the same order of magnitude and is therefore potentially not negligible, with a more pronounced influence at increasing superficial velocity.
[44] noted a general agreement on the form of the equations used in phenomenological modeling (e.g.without knowledge of the exact radial porosity and velocity profiles).The main disagreement between studies lies in the value of   f,r (∞).In light of this, instead of comparing models from different researchers, this validation utilizes the equations proposed by Dixon and Cresswell [35] with varying   f,r (∞), ranging from 8 to 12 as suggested by Dixon [44].The constituting equations can be found in Appendix A, and the results are presented in Table 2.
The values were calculated with a particle thermal conductivity of 1.4 W m −1 K −1 [45].From the validation results, it is observed that there is a relatively good agreement for the three highest velocities and   f,r (∞) = 8.This is consistent with the literature, since for  t ∕ p > 10,   f,r (∞) is expected to be below 10 [44].However, at lower superficial velocities, the calculated values are slightly higher than expected, and show an opposite trend.This is likely a result of the non-negligible axial heat dispersion at these conditions.The observed discrepancy may also partially be attributed to either the measurement error or the fact that the packed bed holder may not represent an ideal packed bed, potentially leading to channeling and thermal conduction via the supporting structure (3D printed out of PETG with a thermal conductivity of approximately 0.20 W m −1 K −1 , thus similar to the fitted value) [46].Despite this, the methodology is validated, although the results at lower superficial velocity should be interpreted with caution.These results can still provide estimates of the properties of the structures.In this validated methodology, there is no need for the temperature measurements at 0.5 t , so these data are discarded in subsequent sections.In Appendix B, it is shown that the presence of the additional two thermowells had no noticeable influence on the measured pressure drop.

Pressure drop -monodisperse glass spheres
The measured average pressure drop, obtained from triplicate experiments, is shown in Fig. 8, along with the predictions from three widely used correlations from the literature, which are presented in Appendix C [47].For the experimental data points, the baseline pressure drop was determined using the empty packed bed holder.The pressure drop of the holder was found to be similar to the empty module, but on average 2.7% higher than the values in Fig. 4 (at the same inlet flow rate).This increase is attributed to the additional wall effects arising from the presence of the packed bed support structure.
The COV values for repeatability and time-averaging of the triplicate measurements for both the empty holder and the packed bed are consistent with the values observed for the empty module; between 10 −3 and 10 −2 for superficial velocities higher than 0.10 m s −1 .The COV of repeatability was approximately 2.6 times higher than that of time-averaging, which can be attributed to slight variations in porosity (ranging from 41.8% to 42.5%) and temperature (between 292 K and 296 K) among the measurements.While the reported COV values are relatively low, they represent the total pressure drop, with the packed bed accounting for between 8% and 16% of this pressure drop (again, for superficial velocities higher than 0.10 m s −1 ).Hence, the error bars are relatively large.
It is observed that the measured values initially align on the high side of the correlations, but generally follow the trends.The closest agreement is found with the Montillet correlation, with a MAPE of 10.3%.This correlation distinguishes between  < 0.40 and  > 0.40, making it suitable for the current measurements with a relatively high porosity [47].While the MAPE value may seem relatively large, it is significantly increased by errors at lower pressure drops.For instance, the MAPE for  > 0.15 m s −1 is only 5.5%.Considering the measurement uncertainties, the validation is considered acceptable.

Heat transfer -PETG pellets
The effective radial thermal conductivity for the packed bed of PETG pellets is shown in Fig. 9.The temperature curves were fitted with a MAPE of only 0.7%, validating the reliability of the methodology.The literature lacks detailed correlations for heat transfer in particle systems with shape and size distributions.However, a comparison of the experimental results with the effective radial thermal conductivity obtained using the correlations by Dixon and Cresswell [35] (using  = 2.5 and   f,r (∞) = 5 for 2.1 mm cylinders with a solid-phase thermal conductivity of 0.2 W m −1 K −1 ) yields a MAPE of 4.4%, and adequately captures the trend [44].The calculated values are also shown in Fig. 9, alongside the calculated values for a bed of monodisperse 1.5 mm spheres, which are considerably lower.Using these values as a reference, the performance of the 3D printed geometries can be benchmarked.

Exemplary structures and influence of thermowell position
In this section, the performance of three exemplary structures is examined closely to provide a foundation for later evaluations.Specifically, the aligned geometry, and the porous baffle and split-recombine geometries with a baffle gap spacing of 50 μm and 208 baffles per meter are considered.The analysis includes triplicate measurements and assessment of the error of repeatability, which will not be repeated for subsequent measurements due to the high time requirements of such experiments.
Particular attention is dedicated to the influence of the thermowell position.As demonstrated in the previous section, the cooling of the wall imposes a near-wall temperature profile which depends on the flow rate.In the baffled logpile structures, the heat transfer is not uniform over the azimuthal angle (), and the degree of cooling varies depending on whether the near-wall measurement is taken in the baffle opening.To elucidate this influence, a study was conducted on the placement of the thermowell.In the porous baffle configuration, the original module features the near-wall thermowell in the baffle, but in the split-recombine configuration it is located in the opening of the 'recombining' layer.To mitigate the measurement bias caused by the (cool) cross-flowing gas in this opening, the thermowell was insulated with a 1 mm layer of stagnant air.To further study the severity of the cooling effect, the porous baffle was rotated by 90°to position the near-wall thermowell in the baffle opening.This leads to six different variations, as visualized in Fig. 10.

Heat transfer
The effective radial thermal conductivity for the three exemplary configurations and the six thermowell variations is shown in Fig. 11.A notable observation from these results is that, regardless of thermowell variation, the baffled geometries consistently outperform the aligned configuration.Comparing the results of this configuration to the randomly packed beds in Fig. 9, it is evident that the aligned geometry exhibits significantly lower performance compared to the bed of cylinder-like PETG pellets, while its performance is similar to that of a packed bed of 1.5 mm PETG spheres.Furthermore, for  > 0.10 m s −1 (or  p ≈ 20), it is observed that all variations of baffled structures exhibit an effective radial thermal conductivity exceeding the thermal conductivity of the solid material (0.2 W m −1 K −1 ).
To further analyze the influence of the thermowell configuration, the measured temperature profiles are provided in Fig. 12.It can be seen that the center temperatures of the different variations largely overlap, indicating that the near-wall thermowell does not significantly affect the heat transfer within the structure.However, notable differences are observed in the near-wall temperature profiles.As expected, insulating the near-wall thermowell leads to higher measured temperatures, and this significantly lowers the effective radial thermal conductivity due to the increased driving force.Similarly, rotation of the relative position of the thermowell in the porous baffle structure leads to lower measured temperatures, but the impact on the effective radial thermal conductivity is less substantial.Based on these findings, all measurements for the porous baffle structures will be conducted without rotation or insulation.For the split-recombine structures, the trends are less pronounced.However, the curve with the   insulation layer corresponds most closely to the porous baffle curve without insulation, and therefore this design will be adopted.This choice may slightly underestimate the performance, but it allows for a more balanced relative comparison.Fig. 13 shows the complete temperature profiles of all reference structures with the selected thermowell configurations.For the triplicate measurements, the error bars are barely visible in most instances, indicating excellent repeatability with an average COV of 0.0016.The use of Eq. ( 5) to describe the center temperature trend is also accurate, with an average MAPE of 1.25%.
The effective radial thermal conductivity was fitted as a function of the Reynolds and Prandtl numbers for these exemplary structures and the fitted coefficients are tabulated in Table 3.As expected, both the stagnant and the convective contribution are significantly higher in the baffled geometries.The increase in stagnant contribution may be attributed to the decrease in porosity.However, this alone does   5) (bottom).Error bars represent the standard deviation to illustrate the repeatability of the triplicate experiments for the lower, middle and higher superficial velocities.The superficial velocities are slightly higher for the split-recombine structure due to the insulated thermowell.
not explain the difference between the two baffled structures, which is likely a result of the difference in the number of cylinders directly connected to the center thermowell.Interestingly, the pre-factor for the convective contribution is very similar for both baffled structures, which contrasts previous 2D CFD simulations, where a larger difference was observed (the overall Nusselt number was approximately one-third lower for the split-recombine structure compared to the porous baffle structure) [29].In the 3D configuration adopted in this work, it is anticipated that the relationship between the relative baffle fraction, the degree of cross-flow and the contact with the wall will differ from the pseudo-2D geometries studied.Consequently, due to this complex interplay between different contributions, it is not expected that the fitted trends exhibit exactly the same behavior.
The effective radial thermal conductivity of the porous baffle structure is on average 46% higher than the packed bed of PETG pellets, and approximately 107% higher than the calculated values for a packed bed of 1.5 mm PETG spheres.As expected, the split-recombine structure performs slightly worse, but still exceeds both packed beds with values of 30% and 85%, respectively.

Pressure drop
The pressure drop data for the six thermowell variations is shown in Fig. 14.A slight decrease in pressure drop is observed for the insulated thermowell compared to the original configuration.This could be attributed to a slight increase in channeling, although the effect is minimal and does not overshadow the overall trends among the structures.Notably, high pressure drops were obtained for the porous baffle structure with the insulated and rotated thermowell.This is due to the added insulation layer, which obstructs the baffle opening significantly (as visualized in Fig. 10), leading to increased pressure drop.
Based on these observations, the novel rotated geometries will be designed without an insulating layer, as the baffle opening in these configurations aligns with the near-wall thermowell in some layers.In this case, the marginal benefit of a more balanced near-wall measurement is outweighed by the substantial increase in pressure drop.
In Fig. 14, the data for the exemplary structures with the selected thermowell configuration is based on triplicate measurements, with the COV of time-averaging consistently in the order of 10 −3 for all

Table 3
Fitted coefficients based on Eqs. ( 7) and ( 11) for the effective radial thermal conductivity and the pressure drop of the three exemplary structures.structures.The error bars, or the absence of them for the aligned structure, represent the COV of repeatability, which is observed to correlate with the magnitude of the pressure drop.This is reasonable since the tendency for the fluid to bypass the structure (and the importance of proper sealing) depends on the pressure drop.Despite these variations, the repeatability is considered adequate.The fitted coefficients of the pressure drop correlation are tabulated in Table 3, and the fitted curves are also shown in Fig. 14.The quality of fitting is good for the baffled structures, as evidenced by the MAPE values which align with the COV of repeatability.However, Eq. ( 11) seems to poorly describe the pressure drop in the aligned structure.This is likely due to two reasons.Firstly, as discussed in Section 3.2, lower pressure drop values have relatively high errors.Secondly, the description in the form of Eq. ( 11), based on the superficial velocity and the particle diameter, is less suitable for the aligned configuration since it consists of relatively undisturbed axial channels with a different characteristic length scale.

Structure
Qualitatively, the relative pressure drop values are in agreement with the expectations from previous pseudo-2D CFD studies [29].To quantitatively compare the fitted correlations of the 2D CFD studies and the experimental work, the pressure drop is plotted as a function of  ∕ to account for porosity differences between simulations and experiments in Fig. 15.As anticipated, the experimental pressure drops are consistently higher due to the additional resistance of the supporting cylinders and the roughness of the printed structures.Interestingly, the increase from 2D to 3D for the porous baffle structure is similar to that of the regular logpile structures, with pressure drops ranging from 2.2 to 1.4 times higher, compared to a factor two increase in pressure drop for the regular structures [29].The split-recombine structure presents a different trend, with pressure drops increased by a factor of 3.6 to 2.5.Hence, the postulated benefits of this geometry over the porous baffle structure are slightly less pronounced in cylindrical modules compared to pseudo-2D simulations.This may be a result of the difference in baffle fraction between the 'splitting' and the 'recombining' layers in the cylindrical configuration.

Baffle gap spacing
The impact of the baffle gap spacing on the effective radial thermal conductivity and the pressure drop is shown in Figs.16 and 17, respectively.For both parameters, the structures show a clear trend as a function of the baffle gap spacing.
As expected, the heat transfer -pressure drop trade-off is more favorable for the porous baffles compared to the 0 μm spacing, as the pressure drop decreases more significantly than the effective radial thermal conductivity.Table 4 quantifies this trade-off and also presents the corresponding values from the pseudo-2D CFD simulations for Ref. [29].This is achieved through the parameters  e,r ,  v and , representing the change in heat transfer and pressure drop parameters of porous baffle geometries relative to the 0 μm spacing.These yield negative values in all cases, as both the heat transfer and pressure drop parameters are lower in the geometries with porous baffles.It was already acknowledged that the translation from simulated 2D geometries to cylindrical configurations is not entirely exact, due to three reasons.Firstly, the simulations do not include solid-phase heat transfer.Secondly, the baffle fraction and baffle gap spacing could not be controlled independently in the structures in this work.Finally, for this comparison specifically, the structure with a baffle gap spacing of 0 μm represents connected cylinders in this work, but a 'plate-like' baffle in the simulations.Nevertheless, both experimental and simulation results consistently indicate a more favorable trade-off for porous baffles compared to non-porous baffles.For the remaining investigations, a baffle gap spacing of 50 μm will be used since the effective radial thermal conductivity of this structure remains significantly higher than the packed bed of PETG pellets.Both performance parameters were correlated to the operating conditions and the baffle gap spacing, with the fitted curves already shown in Figs.16 and 17.The correlations for the effective radial thermal conductivity and the pressure drop are shown in Eqs. ( 14) and (15), respectively.Eq. ( 14) features a baffle gap spacing dependency for both the stagnant and the convective contribution to the effective radial thermal conductivity, accounting for the difference in solid holdup and the varying degree of cross-flow, respectively.The correlation was established with a MAPE of 8.3%, contrasting the average MAPE of 2.4% for fitting of the individual curves.
Eq. ( 15) uses a constant Reynolds exponent based on the exemplary porous baffle structure's value (Table 3).Although the fit in Fig. 17 is qualitatively good, the MAPE for all data points is 76.9%, which is very high.This is strongly skewed by the low values of the 125 μm and 250 μm baffle gap spacings, with MAPE values above 100%.However, for  > 0.10 m s −1 , the MAPE decreases to 11.6%, with an average value of 2.6% for the three lower baffle gap spacing values, and MAPE values of 19.4% and 30.9% for 125 μm and 250 μm, respectively.
Despite the relatively high errors for the curves of both performance parameters, global correlations are preferred over individual correlations to enhance the practical applicability of the data.Moreover, it should be noted that the current description of the data does not require

Number of baffles per unit length
The influence of the number of baffles per unit length on the effective radial thermal conductivity and the pressure drop is shown in Figs.18 and 19, respectively.A clear trend is observed for the pressure drop, indicating that as the number of baffles per unit length decreases, the pressure drop decreases.Remarkably, the effective radial thermal conductivity does not exhibit a similar trend.While the porous   baffle structure with 208 baffles per meter appears to have the highest effective radial thermal conductivity, for the split-recombine configuration this is the structure with 139 baffles per meter.However, all points are relatively closely clustered together, making it challenging to deconvolute the underlying trend.The likely explanation is that lower values may result in less intense cross-flow but improved contact time with the wall.Moreover, the baffles locally obstruct the fluid from contacting the wall, thereby increasing the overall surface available for heating with decreasing number of baffles per unit length.Nevertheless, there are limits to this compensating effect, as illustrated by the lower values for  baf f le = 52 m −1 and 26 m −1 for the porous baffle configuration and  baf f le = 83 m −1 for the split-recombine configuration.Regarding the heat transfer -pressure drop trade-off, the other structures present interesting opportunities.For instance, using a splitrecombine structure with 104 baffles per meter rather than a porous baffle structure with 417 baffles per meter maintains, on average, 92.7% of the heat transfer at just 9.2% of the pressure drop.All of these structures demonstrate their potential by outperforming the packed bed of PETG pellets in terms of heat transfer.Moreover, the structures with the lowest number of baffles per unit length of both configurations even exhibit a pressure drop slightly below that of the aligned reference geometry, while achieving effective radial thermal conductivity values approximately 50% higher.
Due to the complex relationship between the relevant effects influencing the effective radial thermal conductivity, attempts to create global correlations were abandoned.However, the pressure drop was successfully correlated to the operating conditions and the number of baffles per unit length, as shown in Eqs. ( 16) and ( 17) for the porous baffle and the split-recombine configuration, respectively (already presented in Fig. 19).Similar to the baffle gap spacing correlations, these pressure drop correlations exhibit qualitatively acceptable accuracy.
However, the MAPE values for all data points fitted through Eqs.(16) and (17) 11) with the fitted coefficients from Table 5 are shown as dashed lines.

Baffle rotation
The effective radial thermal conductivity and the pressure drop for the two novel rotated baffled structures are shown in Figs.20 and 21, respectively.As was the case for the previous results, it is evident that the split-recombine structure outperforms the porous baffle structure, both by exhibiting higher heat transfer at high superficial velocities and lower pressure drop over the entire investigated range.However, it should be noted that the number of baffles per unit length differs between the two structures.For a more appropriate assessment, the rotated porous baffle structure was compared to the porous baffle structure with a baffle gap spacing of 50 μm and 208 baffles per meter.On average, the effective radial thermal conductivity is 8.5% higher in the rotated structure, while the pressure drop is 28.3% lower.For the split-recombine structure, a direct comparison with the same number of baffles per unit length is not possible, as a 90°rotation is only possible by maintaining an even amount of axial layers between baffles, which is not the case for structures without rotation.Nonetheless, for reference, the structure with a baffle gap spacing of 50 μm and 104 baffles per meter can be considered.In this case, the effective radial thermal conductivity is on average 24.4% higher, while the pressure drop is decreased by 17.0%.The correlations for these results were derived, and the fitted coefficients are tabulated in Table 5.The quality of fit is considered adequate.Of particular interest is the fact that the coefficient  attains its highest value achieved thus far (with the exception of the structure with a baffle gap spacing of 0 μm), suggesting that the heat transfer properties scale favorably with increasing superficial velocity.
To illustrate the favorable heat transfer -pressure drop tradeoff in the structures with rotated baffles, the effective radial thermal conductivity is plotted as a function of the pressure drop for these geometries and two reference geometries in Fig. 22.The curves were constructed by calculating the effective radial thermal conductivity and pressure drop using the established correlations over the range of superficial velocities studied in this work.It is seen that the splitrecombine structure with rotated baffles offers superior performance.Specifically, it offers approximately two to threefold higher effective radial thermal conductivity compared to the aligned structure and the reference packed bed at the higher superficial velocities.

Conclusions
This study has experimentally demonstrated the favorable heat transfer -pressure drop trade-off in 3D printed baffled logpile structures for the first time.This novel class of geometries was first proposed in previous work from our group based on 2D CFD simulations, and it was shown in this work that these results can be used to estimate the trade-offs for the full-scale cylindrical modules with variations in configuration and baffle gap spacing.While the use of porous baffles provides a more favorable heat transfer -pressure drop trade-off compared to non-porous baffles, the relative effect was slightly lower in the experimentally tested modules due to the additional resistance of the supporting cylinders.In addition to the configuration and baffle gap spacing, two new design variables were introduced; the number of baffles per unit length and rotation of the baffles.Remarkably, a reduction of the former led to a significant decrease in pressure drop, up to 80%, while maintaining comparable heat transfer performance.In addition to this, rotation of the baffles for the split-recombine structure

Fig. 1 .
Fig. 1.Internal logpile structure for a split-recombine geometry with  gap = 50 μm and  baf f le = 104 m −1 .The geometry as modeled in Blender is shown on the left, the 3D printed structure is shown on the right.

Fig. 2 .
Fig. 2. Structure cross-sections to visualize the different configurations (top left) and variations of number of baffles per unit length for a split-recombine geometry (center left).Also shown is the top view of the modules for a porous baffle geometry, including thermowells, to show variations of the baffle gap spacing (bottom left).Right-hand side shows consecutive baffles (from top to bottom) in the rotated configurations.

Fig. 3 .
Fig. 3. Center and near-wall temperatures as a function of the axial position for four different structures without gas flow.The second-degree polynomial fit based on all data points is shown as a dashed line.

Fig. 4 .
Fig. 4. Baseline pressure drop as a function of the superficial velocity (top) and the COV's associated with the measurement (bottom).Circles represent the repeatability COV, based on five measurements, and squares represent the average COV of time-averaging of the individual measurements.

Fig. 5 .
Fig. 5. Visualization of the empty packed bed holder (left) and the two packing materials (right).

Fig. 6 .
Fig. 6.Temperature profiles as a function of the superficial velocity and axial position for the packed bed of monodisperse glass spheres.Dashed lines represent the polynomial description of the near-wall profile (left) and the result of fitting of Eq. (5) (right).Error bars represent the standard deviation to illustrate the repeatability of the triplicate experiments for the lower, middle and higher superficial velocities.The legend is applicable to both plots.

Fig. 7 .
Fig. 7. Calculated effective radial thermal conductivity as a function of  (the number of roots included in the summation in Eq. (5)) for the packed bed of monodisperse glass spheres.

Fig. 8 .
Fig. 8. Pressure drop as a function of the superficial velocity for the packed bed of monodisperse glass spheres.Three different correlations from the literature are shown as lines.Error bars represent the standard deviation to illustrate the repeatability of the triplicate experiments.The shaded area represents the standard deviation of the empty packed bed holder measurements.

Fig. 9 .
Fig. 9. Effective radial thermal conductivity as a function of the superficial velocity for the packed bed of PETG pellets.Values calculated using the correlations of Dixon and Cresswell [35], both for 2.1 mm cylinders and 1.5 mm spheres, are shown as dashed lines.

Fig. 11 .
Fig. 11.Effective radial thermal conductivity as a function of the superficial velocity for the three exemplary structures with  gap = 50 μm and  baf f le = 208 m −1 and varying thermowell configuration.

Fig. 12 .
Fig. 12. Near-wall (orange) and center (blue) temperature profiles as a function of axial position for variations of thermowell configuration in porous baffle (left) and split-recombine (right) structures with  gap = 50 μm and  baf f le = 208 m −1 at the highest superficial velocity.

Fig. 13 .
Fig. 13.Temperature profiles as a function of the superficial velocity and axial position for the three exemplary structures with  gap = 50 μm and  baf f le = 208 m −1 .Dashed lines represent the polynomial description of the near-wall profile (top) and the result of fitting of Eq. (5) (bottom).Error bars represent the standard deviation to illustrate the repeatability of the triplicate experiments for the lower, middle and higher superficial velocities.The superficial velocities are slightly higher for the split-recombine structure due to the insulated thermowell.

Fig. 14 .
Fig. 14.Pressure drop as a function of the superficial velocity for the three exemplary structures with  gap = 50 μm and  baf f le = 208 m −1 and varying thermowell configuration.For the selected thermowell configurations, fitted curves based on Eq. (15) with the coefficients in Table 3 are shown as dashed lines and error bars represent the standard deviation to illustrate the repeatability of the triplicate experiments.

Fig. 15 .
Fig. 15.Comparison of the pressure drop predicted by 2D CFD simulations and the current measurements for the exemplary baffled structures with  gap = 50 μm and  baf f le = 208 m −1 .Source: 2D CFD results are taken from Rosseau et al. [29].

Fig. 16 .
Fig. 16.Effective radial thermal conductivity as a function of the superficial velocity and the baffle gap spacing for porous baffle structures with  baf f le = 208 m −1 .Fitted curves based on Eq. (14) are shown as dashed lines.

Fig. 17 .
Fig. 17.Pressure drop as a function of the superficial velocity and the baffle gap spacing for porous baffle structures with  baf f le = 208 m −1 .Fitted curves based on Eq. (15) are shown as dashed lines.corrections for the baffle fraction, which is likely specific to this data set. e,r = 0.1776 exp ( −1429 gap ) + 0.01691 exp ( −15239 gap )  p   (14)  ℎ =  f  2 2 p 20388 exp ( −15740 gap )  0.8866 p

Fig. 18 .
Fig. 18.Effective radial thermal conductivity as a function of the superficial velocity and the number of baffles per unit length for porous baffle (left) and split-recombine (right) structures with  gap = 50 μm.

Fig. 19 .
Fig. 19.Pressure drop as a function of the superficial velocity and the number of baffles per unit length for porous baffle (left) and split-recombine (right) structures with  gap = 50 μm.Fitted curves based on Eqs.(16) and (17) for porous baffle and split-recombine structures, respectively, are shown as dashed lines.y-Axes for the two plots do not represent the same range.

Fig. 20 .
Fig. 20.Effective radial thermal conductivity as a function of the superficial velocity for the two rotated baffled structures.Fitted curves based on Eq. (7) with the fitted coefficients from Table5are shown as dashed lines.

Fig. 21 .
Fig. 21.Pressure drop as a function of the superficial velocity for the two rotated baffled structures.Fitted curves based on Eq. (11) with the fitted coefficients from Table5are shown as dashed lines.

Table 1
Porosity and relative baffle fraction as a function of the configuration and design parameters.The split-recombine structures are considered with an insulated thermowell, which slightly influences the porosity.Two numbers are tabulated for the relative baffle fraction of the split-recombine geometries to indicate first the 'splitting' and then the 'recombining' layer.

Table 2
[35]ctive radial thermal conductivity for the packed bed of monodisperse glass spheres at different superficial velocities, as calculated from experimental temperature profiles and obtained with the equations of Dixon and Cresswell[35]with varying   f ,r (∞).

Table 4
Heat transfer -pressure drop trade-off for the porous baffle structures with  baf f le = 208 m −1 as a function of the baffle gap spacing, relative to the structure with 0 μm spacing.Source: 2D CFD results are taken from Rosseau et al. [29].
were 53.7% and 19.9%, respectively.The curve for the lowest number of baffles per unit length has a MAPE of 177% for the porous baffle configuration and 53.3% for the split-recombine configuration, thus heavily skewing the error.Despite this effect, the correlations are considered valuable for future reactor design considerations. (17))

Table 5
(11)ed coefficients based on Eqs.(7)and(11)for the effective radial thermal conductivity and pressure drop of the rotated baffled structures.