Assessment of transport phenomena in catalyst effectiveness for chemical polyolefin recycling

Since the dawn of agitated brewing in the Paleolithic era, effective mixing has enabled efficient reactions. Emerging catalytic chemical polyolefin recycling processes present unique challenges, considering that the polymer melt has a viscosity three orders of magnitude higher than that of honey. The lack of protocols to achieve effective mixing may have resulted in suboptimal catalyst effectiveness. In this study, we have tackled the hydrogenolysis of commercial-grade high-density polyethylene and polypropylene to show how different stirring strategies can create differences of up to 85% and 40% in catalyst effectiveness and selectivity, respectively. The reaction develops near the H2–melt interface, with the extension of the interface and access to catalyst particles the main performance drivers. Leveraging computational fluid dynamics simulations, we have identified a power number of 15,000–40,000 to maximize the catalyst effectiveness factor and optimize stirring parameters. This temperature- and pressure-independent model holds across a viscosity range of 1–1,000 Pa s. Temperature gradients may quickly become relevant for reactor scale-up.

In fluid mixing processes, such as those in stirred vessels, the shear rate is influenced by various factors, including velocity gradients, impeller design, flow characteristics, and fluid properties.Viscosity gradients arise due to the variation in fluid velocities within the mixing system.Near the impeller blades, these gradients can be significant, leading to higher shear rates.Herein, we considered the simple model of a rotating cylinder 1 with D and L as the diameter and height L, respectively, in a concentric cylindrical vessel of diameter D r .This model enabled an analytical relationship between applied torque (τ) and between shear stress (Equation S2) and shear rate and stirrer and vessel geometry (Equation S3), that were applied to the analysis of the three types of stirrers studied in this work, where x represents the radial distance from the vertical rotation axis.For the case of small clearance between the tip of the blade and the vessel wall, the simplification x = D/2 and rearranging of Equations S1-S3 leads to a simplified expression for the representative value of viscosity in terms of operating conditions and geometry (Equation S4), which allows estimation of viscosity when torque is measured during catalyst testing.This set of relationships was used in Fig. 1c to estimate stirring rates in our setup from shear rates obtained from rheological analysis, revealing that under typical stirring rates, viscosity can be considered independent from stirring rate and temperature.

Supplementary Note 2 | Experimentally available range of operating conditions.
The parallel reactor setup used in this study comprised three reactors that can be operated independently as shown in Supplementary Fig.
where the velocity of the stirrer tip has been taken as representative for the system and the following orders of magnitude have been considered: • Density of the polymer melt ∼10 3 kg m −3 • Diameter of the reactor ∼ 10 −1 m • Viscosity of the melt ∼ 10 2 Pa s Supplementary Note 4 | Estimation of penetration depth of polymers into large pores.
For the case of characteristic pore lengths in the order of micrometres, typically encountered in shaped catalysts, 3 a microscopic description of the capillary penetration of the molten polymer using the Washburn equation could be appropriate (Equation S6). 4 cos( ) Modified versions of the Washburn equation are needed to describe the behavior of molten polymers at the nanoscale, which, nonetheless, converge to Equation S6 in the case of micrometer or larger scales. 5,6According to it, the penetration length, λ of a liquid into a pore of radius r with a contact angle φ, and a surface tension γ has the functional dependency λ ∝ (t•r•μ −1 ) 0.5 , where t represents time.Taking a surface tension 7 of 35 mJ m −2 and assuming a contact angle of 45° between the melt and catalyst particles, 5 the Washburn equation predicts a characteristic time of ca.0.1 s for high M w polyolefins to penetrate a pore with r = 10 −6 m even when considering viscosities under unstirred conditions (i.e., zero shear rate, Fig. 1).This small characteristic time suggests that, even in the case of relatively large error in the estimation of the equation parameters, catalysts typically used in industry, comprising an active phase in the form of particles and a binder among them and showing intraparticle pore sizes commonly in the micrometre scale, 3 could be successfully used for processing of consumer grade polyolefins.However, this estimation seems at odds with recent visualization on the lack of accessibility of PP with high M w into commercial fluid catalytic cracking catalyst particles, 8 hinting at further phenomena at play requiring further studies.

Supplementary Note 5 | Estimation of characteristic dimension of polymer molecules.
The maximum length of a representative chain of HDPE 200 is ca. 3 μm, if modelled as C 20000 H 40000 (M w ∼ 200 kDa ) and considering a C-C bond distance of δ = 0.154 nm. 16wever, the numerous internal spatial degrees of freedom such as rotation around each C-C bond makes it is uncommon to observe it in a fully extended state.The typical dimension (Λ) for folded polymer chains in a uniform melt can be deduced from the Freely Jointed Chain model, 17 predicting Λ = Γ 0.5 •δ , were Γ represents the number of C-C bonds in the backbone.
This model provides a minimum boundary for Λ in our conditions, since polymer chains tend to unfold under shear stress. 18This model thus predicts a very mild dependency of the typical chain size with M w , resulting in Λ ∼ 22 nm for HDPE 200 . 19upplementary Note 7 | Estimation of typical scale lengths of the reaction front.
The penetration and reaction of hydrogen into the melt upon contact with the catalyst particles was modeled assuming the diffusional transport of hydrogen into the melt and its homogeneous reaction in a film according to the thin film theory used to describe fluid-fluid reactions. 9Pseudo first-order kinetics were assumed, where the volumetric reaction rate is given by r = k r c H2 .This enabled the prediction of the evolution of c H2 with z, being z the coordinate penetrating into the melt with zero value at the H 2 -melt interface.Under these conditions, the following mass balance holds under steady-state conditions (Equation S7): with boundary conditions: where equilibrium at the interface reflected by application of the Henry's law has been assumed.Solving this differential equation enables describing the decay of c H2 in the melt The estimation of c H2,int and k r contained in Equation S8 led different profiles depicted in We then used a slightly larger value of D H2 = 1•10 −8 m 2 s −1 compared to measurements and simulations for another available small molecule like methane under similar conditions (D CH4 = 4•10 −9 -9•10 −9 m 2 s −1 ). 11,12r the calculation of the rate constant k r , we estimated r based on the following assumptions: • We modelled HDPE 200 conservatively as C 10000 H 20000 , which gives a molecular weight of 105 kDa, to account for variability in chain lengths present in HDPE 200 and assure a lower bound for r.
• Assumed its full conversion into C 50 H 100 (included in the range of products denoted as residue in the manuscript), which also contributes to find a lower bound for the consumption of hydrogen, as all tests provided significant yields of C 1 -C 45 products.
• With this data we calculated the consumption of hydrogen per unit time, considering that the full transformation of 1 mol of C 10000 H 15000 into C 50 H 100 requires 200 moles of H 2 .
Considering the amount of plastic processed (0.5 g) and 4 h as reaction time, the hydrogen consumption resulted to be 6•10 −7 mol H2 s −1 .This value is in agreement with other reports. 13Considering the volume of the melt (4 Following this approach, the gray region in Fig. 2b indicating the regions for poorly and highly active catalysts could be found, evidencing that the reaction is mostly confined to the first millimeters of the melt.Using these parameters and a estimation for the mass transfer coefficient of hydrogen 10 of k H2 = 10 −6 m s −1 , a value of 5 was calculated for the Hatta number (ratio between the observed reaction rate and diffusion rate for hydrogen, Equation S9), reinforcing the assumption of the reaction be mostly confined to the vicinity of the H 2 -melt interface.A first factor to analyze the motion of the particles is the relative density between the melt (910 kg m −3 for PP 340 and 1000 kg m −3 for HDPE 200 ) 14 and catalyst particles, since values below that of the molten polymer (e.g., some zeolites) could help particles reach the top of the melt more easily.In most cases, like metal oxides or carbonaceous materials, catalyst particles show a larger density than the molten polymer (ρ p /ρ m ∼ 4 in this study).However, the estimated values for the Archimedes number (Ar), giving the ratio between gravitational and viscous forces (Equation S10), varies from 10 −8 (dense materials) to 10 −7 (less dense materials) for typical sieve fractions in the order of 10 −1 mm.This implies that the density of the catalyst is expected to play a negligible role in the motion of particles.Increase particle circulation increases conversion as exposes particles more often to the reactive environment located at the vicinity of the H 2 -melt interface (Supplementary Note 3).
The z-Reynolds number of particles, defined using the velocity along the z-axis (Re p,z ), is thus an appropriate non-dimensional descriptor for the vertical circulation of particles.Since particles predominantly follow the flow field of the polymer melt (Supplementary Fig. 11), characterized by vertical circulation patterns (Fig. 3b and Supplementary Video 4), high values of Re p,z indicate frequent exposure of particles to the reaction front.This parameter is not observable as varies for each particle in space and time.The maximum value reached by Re p,z (Re p,z,max , Equation S11) is not observable either, but can be phenomenologically related to the tip velocity of the stirrer via simulations using the herein defined shape factor (K s ).This parameter thus describes the ability of different stirrer geometries to transfer the rotational kinetic energy of the stirrer into vertical kinetic energy of particles.Extended Data Fig. 2 shows noticeable differences of up to two times among the three investigated stirrer Extended Data Fig. 3 shows the simulated distribution of particle vertical velocities for the three geometries (Supplementary Table 7 and Supplementary Fig. 13 for PP 340 ).The maximum vertical particle velocity in the reactor, v p,z,max achieved by propeller and impeller types was 50-60% larger than the turbine.The relation between particle circulation and performance could be quantitatively described after defining the maximum Reynolds number of catalyst particles along the z-axis (Re p,z,max , Equation S11 and Extended Data Table 2) as a dimensionless descriptor for vertical catalyst particle circulation.This approach manages to retain the different features among stirrers observed in Fig. 3, where the propeller generally favours xy-velocities since it keeps particles separated in two distinct regions, whereas the impeller shows particle circulation in a narrow range of z-values.
Re p,z,max can thus be estimated from the melt properties, stirring rate, and particle and stirrer geometry after considering an average viscosity μ (see Fig. 1) and through the herein defined shape factor, K s = v p,z,max /v tip (Extended Data Table 2, and Supplementary Note 9), which is dependent on the stirrer type and can be estimated from simulations and tabulated (Extended Data Fig. 4 and Supplementary Table 11).This made possible to correlate performance parameters and Re p,z,max (Extended Data Fig. 3).The most noticeable result is the opposite trend of the selectivity to liquid products with increasing Re p,z,max for both polymers.In the case of HDPE 200 , the use of turbines (Re p,z,max ∼ 0.1•10 −4 ) is recommended, whereas for PP 340 , impellers are more advantageous (Re p,z,max ∼ 2.6•10 −4 ) to reach this fraction.This effect could be linked to the lower yields to C 1 -C 45 products obtained for the more chemically resistant PP 340 (Supplementary Table 6).Stirrers promoting larger conversions by larger particle circulation (propeller, impeller) may facilitate progression of chain lengths towards the liquid range for this type of waste.A simple criterion through the estimation of Re p,z,max is thus available to predict conditions and stirrer geometries to tune selectivity for these two feedstocks.
Supplementary Note 11 | Characterization of the hydrogen-melt interface.
Since the reaction is predominantly constrained to the vicinity of the hydrogen-melt interface (Supplementary Note 7), effective catalyst testing requires increasing through stirring the magnitude of this interface (which equals the cross section of the vessel under no stirring).
This parameter is not experimentally attainable.We estimated the relative efficiency of different stirring configurations via CFD simulations after defining the hydrogen fraction (χ H2 ) in a volume close to the H 2 -polymer melt as a proxy for the hydrogen-melt interface magnitude (see Methods for more details).The magnitude of this interface can be related to the degree of turbulence, and therefore average Reynolds number, which physically reflects on the hills and valleys formed under operation (Fig. 4b,c).However, the average Reynolds number close to the interface cannot be easily determined.Nevertheless, it can be inversely correlated to the power number (N p , defined in Equation S12) regardless of the stirrer geometry for the case of Re < 10 (laminar flow), 1,15 a threshold that is two orders of magnitude larger than maximum values presented in Fig. 4d.N p is thus a more convenient descriptor for the magnitude of the hydrogen-melt interface.Given that the density of a polymer melt is ca.four orders of magnitude larger than that of hydrogen, the simplification described in Equation S12 can be applied and N p calculated from observable variables and χ H2 .
However, accurate torque (τ) monitoring or control is not currently a widely available feature in test benches.We employed the stirrer model of concentric cylinders described in the Equations S1-S3 to relate the torque with the geometry of the stirrer and viscosity, arriving at Equation S13, where the assumption of a homogenous value of viscosity independent of stirring rate and temperature (Fig. 1c) under typical conditions was made.
All variables in Equation S13 are observable or part of the design process, except χ H2 .CFD simulations enabled to correlate χ H2 and N p (Extended Data Fig. 5) and revealed its dependency with stirrer and plastic type, while displaying a general tendency to plateau in a wide range of stirring rates.
These results may allow the estimation of N p and χ H2 based on an iterative approach.
Nevertheless, χ H2 = 0.20-0.22 for a wide range of stirring rates, which allows the direct estimation of N p with a sufficient accuracy and thus calculation of the effectiveness factor from Fig. 4d and Fig. 5.However, it is relevant to notice that simulations did not account for the evolution of product distribution toward lighter alkanes, making the overall viscosity decrease over time in real operation, and thus expectedly increasing the degree of turbulence for a constant stirring rate as the reaction progresses.This reasoning may be behind the shift in optimal stirring rates observed between experimental results (Fig. 4a) and simulations (Extended Data Figure 5).With this in mind, we calculated N p values in Fig. 4d using N values extracted from Fig. 4a and corresponding χ H2 from nominal stirring rates in Extended Data Fig. 5 to offer a valuable criterion to catalysis practitioners.
The criterion presents a range of N p for which the effectiveness factor is maximized for both plastics.According to Equation S13, it is then possible to select different combinations of stirring rates, stirrer, and vessel geometry to achieve optimal N p ranges available in Fig. 4d and Fig. 5.
Supplementary Table 2

Fig. 2b .
Fig. 2b.The Henry's coefficient under operating conditions could be estimated 10 to be H H2 ≈ 0.4•10 5 mol H2 m −3 Pa −1 .From this and considering that p H2 = 20•10 5 Pa under standard testing conditions, the concentration of H 2 at the interface is c H2,int ≈ 50 mol m −3 .The diffusivity of H 2 in molten polyolefins is not available in the open literature for temperatures above 500 K.
10 −6 m 3 ), r = 0.15 mol H2 s −1 m −3 and therefore k r = 0.15/50 = 3•10 −3 s −1 .This value, according to Equation S8 and Fig. 2b, indicates that the concentration of hydrogen decays below 10% of the concentration at the interface within ca. 8 mm.If the target product is changed to C 20 H 40 as more representative according to observed product distributions (Supplementary Tables5,6), then k r ≈ 10 −2 s −1 , whereas the extreme case of full conversion into methane would have rendered k r ≈ 0.15 s −1 .

Supplementary Note 8 |
Influence of catalyst particle density in their motion.

Supplementary Note 9 |
Characterization of catalyst particle circulation in the vessel.
1.The range of operating conditions is representative of the current state of the art and is provided below for reference when interpreting CFD simulations.The maximum amount of processable plastic was about 1 g.Larger amounts led to poor conversion due to inefficient overall stirring originated from the Non-Newtonian nature of the polymer melt, showing much larger viscosities (and thus decreased Reynolds numbers) in regions separated from the blades, where the shear rate is lower.The maximum operating temperature is 623 K mainly due to limitations of the o-ring sealings of the vessel.The maximum feasible hydrogen pressure is 150 bar, which suggests the lack of operational limitations on this aspect.The stirring capabilities are limited to the maximum torque provided by the engine (70 N cm), that resulted in a maximum stirring rate of ca.1500 rpm under typical conditions for consumer grade plastics like HDPE 200 and PP 340 .Monitoring of the torque is possible in our setup, enabling indirect calculation of the average viscosity of the melt over the course of the experiment, as explained in the Supplementary Note 1.The flow regime in all cases was laminar, since a minimum stirring rate of 10 5 s −1 (∼10 6 rpm) is required to achieve an average Reynolds number of 10 4 giving access to turbulent flow 2 (Equation S5):

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Accumulated number of articles in the field of plastic hydrogenolysis or hydrocracking, counted with respect to the type of stirring and type of feedstock used (M w > 100 kDa is labelled as commercial grade).

Table 4 |
Viscosity obtained at different temperatures from rheological analysis of PP 340.