Main

Some 12,000 years ago, our ancestors already possessed a rudimentary understanding of the benefits of agitation during brewing, when process efficiency was not a pressing concern1. The landscape changed dramatically with the inauguration of the chemical industry after the construction of the first soda ash plant in Widnes in 18472. This marked the start of an era in which reducing costs and environmental pollution in large-scale processes became the imperative. Fast forward to the 20th century, pioneers such as G. Damköhler3, E. W. Thiele4 and J. J. Carberry5,6 conceptualized the notion of the ‘catalyst effectiveness’ and devised quantitative criteria to assess it in heterogeneously catalyzed reactions, integrating the concepts of catalyst design and reaction engineering. The catalytic chemical recycling of polyolefins, with the potential for processing more than 60% of global plastic waste7, is a prominent example of a catalyst design and reaction engineering challenge for contemporary chemical engineers8.

Focusing on the emerging hydrogenolysis strategy to tackle polyolefin waste, design efforts to find active catalytic surfaces offering control over the cleavage of polyolefins, which are more chemically resistant than functionalized polymers9,10, has steadily gained momentum over the past decade11,12,13,14,15,16,17,18. In contrast, reports on reaction engineering are scarce, despite distinctive features requiring careful attention, as extrapolation from already mature and related technologies, such as pyrolysis19,20,21,22, is not straightforward. The main reason is that the multiphasic operation, typically in batch mode, compounded by the presence of non-Newtonian polymer melts with viscosities exceeding that of honey by up to three orders of magnitude, can lead to highly ineffective mixing if technologies with high power per unit volume, such as mechanical stirring, are not used23,24. A recent report has raised awareness about transport phenomena in this field, exemplified by the application of classical diffusion theory to analyze external mass transport limitations in polyolefin depolymerization under equilibrium25. The inability of high-average-molecular-weight polyolefin chains to access the interior of porous catalyst particles has also been claimed to be a factor26,27,28, whereas for low-average-molecular-weight polyolefins, the accessibility of different types of plastic to the pores is mostly seen as a catalyst design strategy to tune selectivity29,30.

Increasing attention is directed toward low-molecular-weight plastics (<100 kDa, with limited commercial relevance) compared with consumer-grade plastics with molecular weights (Mw) > 100 kDa. Magnetic (or often unspecified) stirring is, however, becoming the most popular mixing strategy regardless of the Mw of the studied plastic (Fig. 1a and Supplementary Tables 1 and 2). The lack of quantitative criteria regarding stirring in current testing protocols31 raises the question of the impact on catalyst effectiveness, with a lack of standardization and limited catalyst benchmarking, which are among the most prominent obstacles to the scaling up of catalytic technologies. Plastic recycling urgently needs modern chemical engineering tools to fully exploit catalyst design efforts.

Fig. 1: Literature analysis and influence of melt viscosity.
figure 1

a, Aggregated number of scientific publications on the hydrogenolysis/hydrocracking of polyolefins, classified according to the stirring configuration and molecular weight. Low Mw is defined as <100 kDa and high Mw is defined as >100 kDa. See Supplementary Table 2 for numerical values. b, Dependency of the viscosity of PP340 and HDPE200 on shear rate at different temperatures (Supplementary Tables 3 and 4). The approximate stirring rates required to reach the equivalent shear rates in a typical reactor used for the catalytic tests in this study are included. More details are available in Supplementary Note 1. The viscosity of water at 298 K is shown for comparison. c, Maximum viscosity of fluids amenable to the magnetic or mechanical stirrers usually available in research laboratories. High-molecular-weight (and some low-molecular-weight) plastics require mechanical stirring, as shown in Supplementary Videos 1 and 2.

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Here we report quantitative guidelines for maximizing three-phase contact in this field of reaction engineering and demonstrate them for the hydrogenolysis of commercial-grade high-density polyethylene (HDPE) and polypropylene (PP). The guidelines are derived from a combination of experimental, theoretical and simulation studies, which led to a simple quantitative criterion based on the dimensionless power number to optimize catalyst effectiveness factors.

Results

Viscosity of molten polyolefins and stirring performance

The mixing of highly viscous substances (viscosity (μ) ≥ 10 Pa s)23 is mainly characterized by the difficulty in reaching turbulent flows due to high viscous energy dissipation leading to excessive power consumption and local hot spots in the reaction medium. Polymer melts are known to be non-Newtonian fluids with a viscosity dictated by shear rate (a measure of the rate at which parallel internal surfaces slide past one another; Extended Data Table 1) and temperature, potentially leading to local variations in the reactor that could affect the mixing regime during operation32,33. We selected HDPE (Mw = 200 kDa, denoted HDPE200) and PP (Mw = 340 kDa, denoted PP340) grades found in consumer goods such as plastic caps, cars or textiles, with the characteristic Mw used to label the polymers obtained from their melt flow index34.

We first conducted a rheological analysis, the results of which are shown in Fig. 1b (see Supplementary Tables 3 and 4 and Methods for details). As expected, increasing the shear rate led to a decrease in the viscosity at all of the tested temperatures, converging toward a distinctive value for each polymer (~500 Pa s for HDPE200 and ∼320 Pa s for PP340). Laminar flows with inherently low mixing capabilities are thus expected in polyolefin chemical recycling (Supplementary Note 1)35. Estimation of the stirring rates (N) required in a typical reactor vessel to reach the equivalent shear rates (secondary horizontal axis in Fig. 1b) revealed that temperature and local variations in viscosity in the melt under typical stirring rates can be disregarded (see Supplementary Note 1 for details).

The torque (τ) required to stir a non-Newtonian fluid is proportional to the product of its viscosity and a power law of the stirring rate, that is, τ ∝ μNα (ref. 36), where α is an experimentally determined parameter dependent on the fluid. Taking into account that molten HDPE200 and PP340 have viscosities approximately one million times greater than that of water at room temperature (\(\mu_{{\rm{H}}_2{\rm{O}},298{\rm{K}}}\) ≈ 0.001 Pa s), we verified that the magnetic stirrers commonly available in laboratories are incapable of stirring high-molecular-weight polyolefins and presumably do not allow good control over the stirring rate for low-molecular-weight ones, as illustrated in Fig. 1c and Supplementary Videos 1 and 2. Magnetic stirrers are suitable for viscosities lower than ~1.5 Pa s (ref. 37), whereas mechanical stirrers can be functional up to 105 Pa s (ref. 38).

Catalyst evaluation and computational fluid dynamics simulations

Experiments were developed in a four-parallel reactor set-up (see Methods, Supplementary Fig. 1 and also Supplementary Video 3 for the stirring configuration). Ruthenium nanoparticles supported on titania, a state-of-the-art catalyst for the conversion of HDPE200, was used throughout the study (see Supplementary Fig. 2 for the characterization of the Ru/TiO2 catalyst)39. The factors driving the performance (listed in Extended Data Fig. 1) within the experimental limitations described in Supplementary Note 2 were systematically varied and translated into computational fluid dynamics (CFD) simulations, including experimentally obtained viscosity dependencies, to describe the hydrogen–catalyst–melt contact over time (see Methods for a general description, Supplementary Note 3 for scope and Supplementary Fig. 3 for convergence plots).

Initial experiments revealed optimum hydrogen pressures for HDPE200 and PP340 in accordance with the literature, with a pressure of 20 bar selected for subsequent tests (Extended Data Fig. 2, Supplementary Fig. 4 and Supplementary Tables 5 and 6)40,41, likely due to competitive adsorption of the polyolefin and hydrogen on the metal surface (Supplementary Fig. 5). Simulations at different temperatures confirmed minimum differences in the distribution of viscosities and Reynolds numbers (Extended Data Fig. 1 and Supplementary Table 7), in line with the observations depicted in Fig. 1b showing viscosity values almost independent of temperature at shear rates equivalent to stirring rates larger than approximately 15 r.p.m. in laboratory reactors (Supplementary Note 1). A commonly reported reaction temperature (498 K) was thus chosen for further analyses after performance tests (Extended Data Fig. 1)42,43.

Internal and external mass transport limitations

We first evaluated the ability of polymer chains to penetrate micropores and mesopores (see Supplementary Note 4 for the case of micrometer-sized pores usually found in shaped catalysts). The Freely Jointed Chain model (Supplementary Note 5) predicts a typical dimension for the folded chain (Λ) of ~ 22 nm for HDPE20044,45,46. For chain lengths below Λ, polymer chains tend to gradually favor the linear conformation47. The relevant pore size that polyolefins, or their liquid products following reaction, may not be able to penetrate thus ranges from Λ ≈ 1 nm (C6) to Λ ≈ 100 nm (high-molecular-weight polyethylene, Mw ≈ 5,000 kDa). These scales are represented in Fig. 2a and suggest that internal mass transport limitations in the case of porous catalysts may increase in relevance even for polyolefins with very high Mw as the reaction progresses toward shorter chain products, and therefore further studies are required. Internal mass transport phenomena were disregarded for simplicity as the Ru/TiO2 used in our study showed an average pore size of 6 nm within a very low specific pore volume of 0.02 cm3 g−1. Experiments at different stirring rates using a particle diameter of 0.6 mm support this assumption (Supplementary Fig. 6).

Fig. 2: Characteristic lengths in catalytic polyolefin hydrogenolysis.
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a, Typical magnitudes of relevant length scales in the catalytic processing of consumer-grade plastics. Chain size refers to the typical dimensions of folded chains in low- and high-molecular-weight polymers. Reaction front refers to the characteristic penetration length of hydrogen in the melt before its concentration drops below 10% of its value at the H2-melt interface. b, Simulated decay of relative hydrogen concentration with distance from the H2–melt interface for different reaction rate constants (kr). \({\rm{C}}_{{{\rm{H}}_2},{\rm{int}}}\) refers to the concentration of H2 at the H2-melt interface. Pseudo-first-order kinetics (\(r=k_{\rm{r}}c_{{\rm{H}}_2}\)) were considered, as explained in more detail in Supplementary Note 5. The area shaded in gray indicates the typical range of kr values calculated in our experiments and reported in the literature. c, Schematic representation of the circulation of catalyst particles in the reaction vessel, with those exposed to hydrogen and polymer melt shown in green as active particles toward hydrogenolysis, as deduced from b, highlighting the fact that the reaction is mostly constrained to the vicinity of the H2–melt interface.

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Regarding external mass transport limitations, an earlier study determined negligible external hydrogen gradients to catalyst particles immersed in the melt if equilibrium bulk concentrations of H2 are reached25. However, the simulation of H2 diffusion into molten HDPE200 in the absence of reaction for a range of hydrogen pressures, times and viscosities (Supplementary Fig. 7 and Supplementary Notes 6 and 7) revealed a characteristic time for equilibration beyond typically reported reaction times. In view of this, we computed the decay of the H2 concentration at the H2–melt interface assuming the direct reaction of H2 with the melt after estimating that the observed reaction rate is around five times that of the diffusion rate of H2, as provided by the Hatta number (Extended Data Table 2 and Supplementary Note 7)48. Figure 2b shows the results for different reaction rate constants (kr), defined according to the expression for pseudo-first-order kinetics, \(r=k_{\rm{r}}c_{{\rm{H}}_2}\), where r is the rate of the reaction and \(c_{{\rm{H}}_2}\) is the concentration of H2. A suitable range of kr was estimated from the typical hydrogen consumption and reaction times observed in our study and reported in the literature (see Supplementary Note 7 for details)30. Poorly active catalysts not yielding any liquid products are characterized by kr values of ∼1.5 × 10−3 s−1, whereas highly active systems able to provide 100% conversion into methane are expected to present kr values of ∼0.1 s−1. These values translate into a range of concentration decays, highlighted in gray in Fig. 2b. Typically, we obtained values for kr of ∼0.01 s−1. As observed in Fig. 2b, the concentration of hydrogen drops below 10% of the interface value within a few millimeters in all cases, strongly suggesting that the reaction is mostly confined to the vicinity of the H2–melt interface with a typical length (λ) of ∼10−3 m, leaving most of the melt non-reactive (Fig. 2a). A representation in which only catalyst particles exposed to this region are active toward the reaction (Fig. 2c) is thus a suitable approximation to study the role of agitation in catalytic performance.

Impact of catalyst particle circulation on performance

The previous analysis shows the benefit of stirring configurations maximizing the presence of catalyst particles in the vicinity of the H2–melt interface. A first analysis based on the ratio of gravitational and viscous forces given by the Archimedes number (Ar; Extended Data Table 2) predicted Ar = 10−8–10−7 and therefore that the density of the catalyst is expected to play a negligible role in particle motion (Supplementary Note 8). Nevertheless, the average catalyst particle diameter (dp) is important as it determines the tendency of particles to follow melt streamlines according to the Stokes number (Stk; Extended Data Table 2)49. Considering λ as the characteristic length, Stk ≈ 10−2–100 for dp = 10−4–10−3 m. These values indicate that small catalyst particles will closely follow streamlines, whereas larger ones may deviate from them to an extent comparable to λ.

We evaluated the importance of dp by comparing the catalytic performance of three different catalyst sieve fractions (0–0.2, 0.2–0.4 and 0.4–0.6 mm) in the hydrogenolysis of HDPE200 (Fig. 3a and Supplementary Tables 5 and 6). Equivalent experiments with PP340 did not show substantial variation in the total yield due to its low reactivity effectively limiting the performance of the catalyst (Supplementary Fig. 8). However, the same trend was confirmed by using the shorter and thus more reactive PP12 (Supplementary Table 8 and Supplementary Fig. 9). We found the smallest sieve fraction to be beneficial, producing a 40% greater yield of the C1–C45 products compared with the largest sieve fraction under the tested conditions. CFD simulations for dp = 0.2, 0.4 and 0.6 mm, keeping the same stirrer geometry, predicted differences in the particle trajectories. Larger particles on average required longer times to leave the bottom of the reactor and tended toward a more irregular occupancy of the vessel volume (Fig. 3b and Supplementary Fig. 10 for the three modeled particle sizes). Having determined the benefits of smaller sieve fractions, dp = 0.2 mm was used for the rest of the simulations in this study.

Fig. 3: Influence of catalyst particle motion on performance.
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a, Variation in the product distribution for the hydrogenolysis of HDPE200 with catalyst sieve fraction using a propeller stirrer after 4 h. b, Corresponding three-phase CFD simulations using discrete phase modeling showing the trajectories of 0.2 and 0.6 mm catalyst particles under steady-state conditions: the total number of particles (np, with np = 196 for 0.2 mm and np = 58 for 0.6 mm; top) and representative initial trajectories of individual particles (bottom). c, Variation in the product distribution for the hydrogenolysis of HDPE200 with stirrer type after 4 h. d, Corresponding CFD simulations (top views) of catalyst particle trajectories for different stirrers, colored according to the H2 fraction in the vicinity. Simulations for other sieve fractions and parallel analyses for PP340 can be found in Supplementary Figs. 8–10 and Supplementary Tables 5–7. Simulated particle trajectories are presented in Supplementary Video 4. Reaction and simulation conditions: T = 498 K, \({p}_{{{\rm{H}}}_{2}}\) = 20 bar, catalyst/plastic ratio = 0.05 and stirring rate = 750 r.p.m.

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The stirrer imposes the flow pattern that catalyst particles follow (Supplementary Fig. 11). Figure 3c shows the product distribution for the catalytic hydrogenolysis of HDPE200 using three different stirrer geometries under the same conditions (Supplementary Tables 5 and 6 and Supplementary Fig. 12 for PP340). The yield of C1–C45 products was not greatly affected by the stirrer type, whereas the product distribution shifted from gas to liquid fractions, with the amount of gaseous product decreasing in the order impeller > propeller > turbine, highlighting that stirring strategies can tune selectivity, as a consequence of tuning the activity, given that the hydrogenolysis is a series of reactions, and must be reported to facilitate benchmarking. The total number of carbon–carbon bonds followed the same trend, as determined using the recently published procedure for calculating the number of backbone scission, isomerization and demethylation events15 (Supplementary Table 9). Figure 3d and Supplementary Video 4 show critical differences between the stirrers. Propellers tend to split the catalyst particles into two separate zones with either high or low H2 concentration. Impellers tend to keep catalyst particles circulating around the mid plane, where the H2 concentration is high due to the V shape adopted by the H2–melt interface. Impellers are thus better suited to optimizing catalyst use. The turbine is poorly efficient in transferring particles to H2-rich zones, leading to the modest generation of gaseous products (which require more molecules of H2 per molecule of polymer). These effects can be quantitatively understood by considering the maximum value of the vertical component of the particle Reynolds number (Rep,z,max; Extended Data Figs. 3 and 4, Supplementary Notes 9 and 10, Supplementary Tables 7,10 and 11, and Supplementary Fig. 13) as the first performance descriptor. Rep,z,max can be derived from the melt properties, stirring rate, and particle and stirrer geometries and can be linked to activity and therefore changes in selectivity (Extended Data Fig. 2), offering a first tool to predict performance trends.

Criterion to maximize the catalyst effectiveness factor

The yield of C1–C45 products did not monotonically increase with stirring rate, as shown in the catalytic tests for both polymers (Fig. 4a). The existence of an optimum rate, in accord with some reports in the literature on the conversion of low-molecular-weight plastics50,51, led us to study the influence of stirring rate on the extent of the H2–melt interface. CFD simulations developed for the three stirrer types (Fig. 4b) predicted the potential of propellers and impellers to increase the interface. The ability of simulations to reproduce the V shape of the H2–melt interface for highly viscous plastics is confirmed in Supplementary Video 3. CFD simulations of the impeller at different stirring rates strongly hint at a relationship between stirring rate and the H2–melt interfacial area (Fig. 4c). Small variations in the distance between the base of the stirrer and the bottom of the vessel also led to small changes in the H2–melt interface. However, an excessive distance (the top of the stirrer at the free melt surface) led to a decrease in the interface (Supplementary Fig. 14). In general, the shear-thinning character of molten plastics makes stirring only effective in the imaginary volume occupied by the stirrer under rotation or slightly beyond, making it advisable to minimize the distance between the stirrer and reactor walls.

Fig. 4: Criterion for maximizing the effectiveness factor.
figure 4

a, Variation in the product distribution with stirring rates for HDPE200 and PP340 with the impeller stirrer. b,c, Two-phase CFD simulations of the hydrogen fraction in the mid z–x plane for different stirrer types (b) and different stirring rates for the impeller stirrer (c). d, Correlation between the effectiveness factor, defined as the ratio between the yield of C1–C45 and maximum yield of C1–C45 in a, and the modified power number for HDPE200 and PP340, calculated using the stirring rates in a and the simulated fraction of H2 in Extended Data Fig. 5. Reaction and simulation conditions: T = 498 K, \({p}_{{{\rm{H}}}_{2}}\) = 20 bar and catalyst/plastic ratio = 0.05.

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Given the difficulty of calculating the extent of the H2–melt interface, we defined as a proxy the fraction of hydrogen (\(\chi_{{\rm{H}}_2}\); Extended Data Table 2) in a volume contained between the bottom of the stirrer and the H2–melt interface (Extended Data Fig. 1) when there is no stirring. The average Reynolds number in this region could serve as a descriptor for \(\chi_{{\rm{H}}_2}\) as more turbulence (larger Re values) may lead to more pronounced hills and valleys on the surface of the melt. However, Re is not observable. For the case of Re ≪ 1 as studied here, the power and Reynolds number are inversely linearly correlated. The power number (Np) expresses the relationship between resistance and inertia forces and can be written in terms of \(\chi_{{\rm{H}}_2}\) and observable variables such as the average density of the melt (\({\bar{\rho }}\)), the reactor diameter (D) and the average density of the melt (ρm) (equation (1), Extended Data Table 2 and Supplementary Note 11)24,52.

$$\begin{array}{l}{N}_{{\rm{p}}}=\\\displaystyle\frac{2\uppi N\tau }{60\bar{\rho }(N/60)^{3}{D}^{5}}=\displaystyle\frac{7200\uppi \tau }{[\,{\chi }_{{\rm{H}}_2}{\rho }_{{\rm{H}}_2}+(1-{\chi }_{{\rm{H}}_2}){\rho }_{{\rm{m}}}]{N}^{\,2}{D}^{5}}\approx \displaystyle\frac{7200\uppi \tau }{(1-{\chi }_{{\rm{H}}_2}){\rho }_{{\rm{m}}}{N}^{\,2}{D}^{5}}\end{array}$$
(1)

Extended Data Fig. 5 shows the relationship between \(\chi_{{\rm{H}}_2}\) and Np obtained from CFD simulations for HDPE200 and PP340 under the same conditions as used in Fig. 4a. Propeller and impeller stirrers yielded volcano behavior, with a maximum \(\chi_{{\rm{H}}_2}\) ≈ 0.20–0.30 for Np ≈ 104–105, shifted toward slightly lower values in the case for PP340. The difference in the optimal rates in Fig. 4a and Extended Data Fig. 5 can be ascribed to the lower average viscosity in reaction compared with the simulations (Supplementary Note 12), although in practice this has a small impact as \(\chi_{{\rm{H}}_2}\) displays values of around 0.2 for a broad range of stirring rates.

Plots of the effectiveness factor (η), defined as the ratio between the yield of C1–C45 products and the maximum yield of C1–C45 products over a series of experiments (equation (2) and Extended Data Table 1), versus the corresponding Np values based on the results presented in Fig. 4a show the optimal Np ranges for the two polymers (\(N_{{\rm{p}},{\rm{HDPE}}_{200}}\) ≈ 2 × 104 to 3 × 104 and \(N_{{\rm{p}},{\rm{PP}}_{340}}\) ≈ 1.5 × 104 to 2.5 × 104) to achieve high η values (Fig. 4d) and serve as a guide for the design of catalytic tests for performance optimization. From equation (1) and Extended Data Fig. 5, it is possible, for a given stirrer geometry (stirrer type and D), to select the stirring rate (N) and torque (τ) to be applied to deliver the desired Np value. Nevertheless, torque control is not a widely available feature of current reactor systems for catalyst evaluation, hindering the applicability of this criterion.

$$\eta =\frac{{{\rm{Yield}}\;{\rm{C}}}_{1}-{{\rm{C}}}_{45}}{{({\rm{Yield}}\;{\rm{C}}_{1}-{{\rm{C}}}_{45})}_{\max }}$$
(2)

Use of the concentric cylinders model to describe the stirrer geometry (Supplementary Note 1) gives access to analytical relationships between viscosity, shear rate and torque, leading to an alternative expression for Np (Fig. 5 and Supplementary Note 11) that now includes contributions from the melt properties, stirring rate, fluid dynamics (through \(\chi_{{\rm{H}}_2}\)), and reactor and stirrer geometry (through D, Dr and L; Extended Data Fig. 1). All of the variables are either directly observable or design parameters, except \(\chi_{{\rm{H}}_2}\), which is available from Extended Data Fig. 1 and Supplementary Table 12 and depends on the stirrer type and plastic under treatment. Practitioners of catalysis can thus select appropriate combinations of stirring rate and reactor geometry to achieve the optimal Np ranges for a certain plastic. We note that deviations from the optimal range led to differences of up to 85% in activity and 40% in selectivity (Fig. 4a).

Fig. 5: Application of the developed criteria for maximizing the effectiveness factor.
figure 5

Parameters that can be approximated under typical reaction conditions or are known a priori are indicated. Ranges of optimal stirring rates for a given reactor and stirrer geometry can thus be calculated. \({\bar{\mu }}\) refers to the average viscosity of the melt, Dr refers to the diameter of the stirrer and L to the height of the stirrer blades (Extended Data Table 1).

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In the most common case where the geometries of the stirrers and reactor are given, a first approximation to the optimal ranges of Np can be obtained from the values provided in Fig. 5. Melt densities and average viscosities at typical operation temperatures (Fig. 1b) and a reasonable value for \(\chi_{{\rm{H}}_2}\) of ∼0.2 (Extended Data Fig. 5) allow a straightforward calculation of stirring rate ranges. For example, in the case of D = 2 cm, L = 1 cm and Dr = 2.5 cm, the approximate ranges for high catalyst effectiveness factors would be N = 880–1,300 r.p.m. for HDPE200 and N = 760–1,100 r.p.m. for PP340 This criterion was shown to be valid in the range of most reported operation pressures (20–30 bar), with lower pressures (10 bar) showing behavior compatible with H2 depletion as the limiting factor (Supplementary Table 5 and Supplementary Fig. 15). These results, together with the small variation in viscosity at commonly applied temperatures (Fig. 1), make this criterion pressure- and temperature-independent under most reported conditions.

Model scope and future directions

As the average chain length of the hydrocarbons decreases due to cleavage, so does the viscosity, spanning six orders of magnitude until reaching values close to water (Fig. 1). Thus, the ability of the criterion to predict performance as the reaction progresses was next investigated.

We hypothesized that the transition from the initial non-Newtonian character to a Newtonian character, facilitating the creation of turbulence24, may change the structure of the H2–melt interface. The Freely Jointed Chain model predicts a transitioning chain length of around C200 (Supplementary Note 5). With this in mind, we simulated stirring patterns for HDPE100 (non-Newtonian, a proxy for low conversion stages), a hypothetical C200 under Newtonian and non-Newtonian regimes, and eicosane (C21, Newtonian, a proxy for high conversion stages). The results clearly reflect the transition from a single H2–melt interface to an abundance of H2 bubbles populating the melt (Extended Data Fig. 6 and Supplementary Fig. 16), as supported by direct observations when turbulence starts to dominate as viscosity decreases (Supplementary Video 3). We then performed catalytic tests on HDPE100 and eicosane (Supplementary Table 8 and Fig. 6a,b), calculated \(\chi_{{\rm{H}}_2}\) (Supplementary Table 13 and Supplementary Fig. 17) and applied the criterion (Fig. 6c). The non-Newtonian melts of HDPE200 and HDPE100 exhibited very similar trends, with identical optimal stirring rates (although different Np due to different viscosities), whereas eicosane displayed a C-shaped relationship between effectiveness and Np, clearly suggesting the need for a different modeling strategy for the later stages of the reaction (or for the case of catalytic hydrogenolysis of surrogate molecules or the often-used very-low-molecular-weight plastics). The transition seems to occur at Np = 102–103, corresponding to viscosities of around 3–30 Pa s at 1,000 r.p.m., therefore validating the proposed criterion until the later stages of the reaction.

Fig. 6: Model scope and influence of thermal gradients.
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a,b, Variation in the product distribution with stirring rate and two-phase CFD simulations of the hydrogen fraction in the mid z–x plane for HDPE100 (a) and eicosane (b). c, Correlation between the effectiveness factor, defined as the ratio between the yield of C1–C45 and the maximum yield of C1–C45 for HDPE and between the yield of methane and the maximum yield of methane for eicosane, and the modified power number, calculated using the simulated H2 fractions in Extended Data Fig. 5, using an impeller as stirrer. d, Temperature distribution in the mid x–y plane for different stirrer geometries when the thermocouple reaches the operation temperature (498 K, at the position indicated). e, Temporal evolution of the temperature distribution in the x–z plane for different reactor diameters (Dr). Reaction and simulated conditions: T = 498 K, \({p}_{{{\rm{H}}}_{2}}\) = 20 bar and catalyst/plastic ratio = 0.05.

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In addition to the analysis of mass transport limitations, we also investigated heat transport constraints. We simulated the largest possible temperature gradient within the reactor during operation with three different stirrer configurations when the temperature at the thermocouple reaches the set temperature (498 K in our case, equal to that imposed on the reactor walls). Figure 6d shows the temperature distribution in the reactor, which resembles that of the Reynolds number distribution (Extended Data Fig. 2), with gradients of approximately 100 K for the best impeller and propeller geometries. This led us to conduct temporal simulations to predict the time for the gradient to reduce to less than 10 K. Figure 6e (top) shows a time of 15 min for the worst-case scenario of walls at the set temperature and the interior at room temperature at t = 0 with a stagnant and non-reactive melt (see Methods for more details), representing only 6% of the operation time (4 h). From a forward-looking perspective, more refined models able to stepwise predict product distributions and thus recommend optimal operation times will be possible after incorporating kinetic descriptions for the catalyst under study. Alternatively, developing operando tools to track viscosity could also guide optimal reaction times. We also highlight the generality of the applied analysis that could be adapted to future reactor architecture operating in continuous mode. In this direction, processes such as continuous reactive extrusion53 are first steps, which would also enable the online analysis of products.

Discussion

The importance of stirring strategies for optimizing the potential of catalytic materials in hydrogenolysis and hydrocracking has been quantitatively established for virgin consumer-grade polyolefins. Our results show that mechanical stirring is highly recommended, even for low-molecular-weight plastics. The reaction can be assumed to develop in a millimeter-scale region next to the H2–melt interface for moderately and highly active catalysts. Stirring can thus be seen as a means to maximize the presence of particles in this region. Stirrer geometries largely determine the location of particles and thus performance, which decrease in the order impeller > propeller > turbine. A temperature- and pressure-independent criterion to maximize the catalyst effectiveness factor based on the observed correlation between power number and H2–melt interface has been developed based on the power number. For a given stirrer and reactor geometry, stirring rates for non-Newtonian melts (viscosities greater than 3–30 Pa s) can easily be calculated to operate within optimal power number values (2 × 104 to 3 × 104 for HDPE200 and 1.5 × 104 to 2.5 × 104 for PP340). Future criteria incorporating heat transport gradients will be key for the successful scale-up of this technology. This work provides readily implementable tools to maximize and tune the performance of catalysts, facilitating the standardization of catalyst evaluation and underscoring the key role of engineering considerations in catalyst development programs.

Methods

Rheological measurements

The viscosity of HDPE200 and PP340 (Supplementary Table 1) was measured using an Ares-G2 rheometer equipped with a separate motor and transducer (TA Instruments). A parallel plate geometry with a diameter of 25 mm was used. The temperature was regulated by convection, and flow curves were generated using a shear rate sweep from 50 s−1 to 0.01 s−1. The shear rate and viscosity were fitted assuming the plastic melts to be Carreau fluids.

Catalyst synthesis

Commercially available titanium oxide (anatase, Sigma–Aldrich) was used as the support and calcined in ceramic crucibles in static nitrogen (5 h, 573 K and 5 K min−1) before synthesis. Typically, 0.30 g TiO2 was dry-impregnated with 1.04 cm3 ruthenium nitrosyl nitrate (Ru(NO)(NO3)x(OH)y, x + y = 3) in dilute HNO3 (0.015 g cm−3; Sigma–Aldrich) to achieve 5 wt% while mixing with a glass-coated magnetic stir bar (VWR chemicals) until the solvent had evaporated. Residual solvent was removed under vacuum (12 h, 353 K and 80 mbar). Finally, the sample was heated in nitrogen at 573 K. The prepared catalyst was then pressed and sieved into different sieve fractions (0.0−0.2, 0.2−0.4 and 0.4−0.6 mm).

Catalyst characterization

High-angle annular dark-field scanning transmission electron microscopy and energy-dispersive X-ray spectroscopy were carried out on a probe-corrected Titan Themis microscope operated at 300 kV. Samples were pretreated to remove adventitious compounds in Ar/O2 plasma before being inserted into the microscope. X-ray photoelectron spectroscopy (XPS) was conducted using a Physical Electronics Quantera SXM spectrometer. Monochromatic Al Kα radiation (1,486.6 eV) generated by an electron beam (15 kV and 49.3 W) was used to irradiate the samples with a spot size of 200 μm. The finely ground samples were pressed into indium foil (99.9%, Alfa Aesar) and then mounted onto the sample holder. During measurement, electron and ion neutralizers were operated simultaneously to suppress undesired sample charging. High-resolution spectra were obtained using pass energies of 55 eV, while the Au f7/2 signal at 84 ± 0.1 eV was used for calibration. All calculations were performed with the CasaXPS (Casa software), and the relative sensitivity factors used for quantification were taken from the instrument. All spectra were deconvoluted into Gaussian–Lorentzian components after application of the Shirley background. X-ray diffraction was conducted using a Rigaku SmartLab diffractometer equipped with a D/teX Ultra 250 detector and Cu Kα radiation (λ = 0.1541 nm) operating in Bragg−Brentano geometry. Data were acquired in the 2θ range of 20–80° with an angular step size of 0.025° and a counting time of 1.5 s per step. Temperature-programmed desorption experiments were conducted using a Micromeritics AutoChem II 2920 instrument. Signals were acquired with a pre-equipped thermal conductivity detector (TCD) and an attached Pfeiffer Vacuum OmniStar GSD 320 O mass spectrometer (MS). Samples of ~0.1 g were dried at 473 K for 1 h, followed by saturation with either n-C7H16 or 2,4-dimethylpentane (contained in an attached vaporizer). In the subsequent desorption experiment, a heating ramp (313–873 K at 10 K min−1) was applied while using both the TCD and MS to monitor the evolution of the probe gas.

Catalyst evaluation

The catalysts were evaluated in a parallel batch reactor set-up (BüchiGlasUster) fitted with an electrical heating jacket and active cooling unit (chilled water system). Typically, 0.5 g virgin polyethylene or polypropylene and 0.025 g catalyst were placed inside a glass inset, which was then placed inside the reactor (Supplementary Fig. 1). Catalyst particles were always found at the bottom of the reactor after reaction regardless of the catalyst activity. The distance between the base of the stirrer and the bottom of the reactor was minimized, and it is highly recommended to use a volume of melt that closely matches the envelope of the stirrer to minimize melt zones between the stirrer blades and reactor walls, where stirring is highly ineffective due to the shear-thinning nature of molten plastics. The reactors were flushed first with nitrogen and then with hydrogen before pressurization to the desired pressure. All reactor systems were equipped with mechanical stirring, temperature/pressure control systems and gas sampling lines. A schematic of the reactor set-up, including the most relevant dimensions, is shown in Extended Data Fig. 1, and the experimental limitations are described in Supplementary Note 2. Instantaneous values of temperature, pressure and stirring torque were recorded using the SYSTAG Flexsys software54. The temperature in the reactor was measured using a thermocouple placed in the space (immersed in the polyolefin melt) between the stirrer and reactor wall. Before the reaction, the reactor was weighed along with all its content. The reaction mixture was heated and stirred (at 498 K and 750 r.p.m. unless otherwise specified) for a set reaction time. After the reaction, the vessels were cooled with circulating chilled water. Three different commercially available stirrer types (acquired from BüchiGlasUster) were used, namely, a turbine, propeller and impeller. They all had the same blade height (1 cm) and diameter (2.8 cm). The glass inset containing the reaction mixture had a diameter of 3 cm (Extended Data Fig. 1) and was inserted into a stainless-steel reactor placed in a heating jacket. The glass inset was fabricated so that its outer diameter closely fitted the inner diameter of the reactor (Supplementary Fig. 1). A detailed description of the experimentally available range of operating conditions is provided in Supplementary Note 2.

Product analysis

The gaseous products were collected from the headspace of the reactor using a sampling cylinder and analyzed using a gas chromatograph (HP Agilent 6890) equipped with a 25 m × 0.53 mm × 20 μm column (Agilent J&W PoraPlot Q column) and flame ionizing detector (FID). A temperature ramp of 308–573 K (5 K min−1) was applied, while the inlet and FID were held at 573 K and 473 K, respectively. The gas chromatograph (GC) columns were calibrated as per a procedure reported elsewhere for products C1–C45 (ref. 55); the calibration was performed for products C1–C5, detected by FID, using a standard refinery gas mixture (Agilent P/N 5080-8755). The reactor inset was then weighed to calculate the amount of gas formed. The products remaining inside the inset were dissolved in dichloromethane using sonication and filtered using a syringe for GC-FID and 1H NMR analysis. GC-FID analysis was performed on a GC (HP Agilent 6890) equipped with a 15 m × 0.25 mm × 0.10 μm column (HP DB-5 HT). A temperature ramp of 313–648 K (4 K min−1) was applied, while the FID detector was held at 613 K. The initial and final hold times were set at 2 and 10 min, respectively. Calibration was performed for C7–C40 alkanes using a certified reference mixture (C7–C40 in hexane, 1 mg cm−3, traceCERT, Sigma–Aldrich). For each 1H NMR experiment, conducted on a 300 MHz Bruker Ultrashield spectrometer, 0.45 cm3 of sample and 0.05 cm3 of [D2]dichloromethane were mixed and analyzed using a solvent suppression method reported elsewhere2. The signals in the aliphatic region of the 1H NMR spectra were numerically integrated to identify the ratio of primary to secondary carbon atoms. The areas corresponding to each carbon type were normalized by the number of hydrogen atoms bonded to the carbon.

CFD simulations

CFD simulations were performed with Ansys Academic Research Fluent (Release 2023 R1) using a double-precision steady-state solver. Supplementary Note 3 describes the scope and limitations of the simulations when applied to polyolefin hydrogenolysis. The geometry of the reactor was created in Ansys SpaceClaim on a 1:1 scale with the actual dimensions. The geometry was meshed using the watertight geometry workflow of Ansys Fluent Meshing, resulting in 0.5 million polyhedral cells. The stirring within the reactor was simulated using the moving reference frame method. Fluid flow was computed using the renormalized group k-ε model with swirl-dominated flow and Menter–Lechner near-wall treatment. The experimentally measured viscosity of the plastic melt was modeled using the Carreau–Yasuda model and implemented in Fluent as an interpreted function. The power number was calculated as a user-defined function within Fluent. The pressure–velocity coupled solver used the Rhie–Chow momentum-based flux type auto-selected by Fluent, which was run until residual values of 10−5 were reached for continuity, x, y and z velocities, turbulent kinetic energy (k) and the rate of dissipation of kinetic energy (e).

The two-phase system of hydrogen and molten polyolefin was simulated using the volume-of-fluid (VOF) method with sharp interfaces, continuum surface stress and the no-slip condition. The VOF simulations were achieved by solving the continuity equation (3), where ρq, αq and vq refer to the density, volume fraction and velocity of a given phase q, \({\dot{m}}_{sq}\) and \({\dot{m}}_{qs}\) refer to the mass transfer between phases q and s within the multiphase system, t refers to time and S indicates a mass transfer source (none in this case). The validity of this equation is subject to equation (4). A condition of no mass transfer between the phases was assumed, which results in equation (5). The volume fraction of the secondary phase (in this case molten plastics) was first computed, followed by the calculation of the primary phase (hydrogen) using the constraint that the sum of the volume fractions of all phases must be 1. The volume fraction equation was solved in this case using an implicit scheme for time discretization that uses equation (6), where V refers to the volume of the cell, Uf refers to the volume flux through the face, based on a normal velocity, n refers to the previous iteration step, while n + 1 is the current iteration step, αq,f is the face value of the qth volume fraction computed through either the first- or second-order upwind scheme and Δt is the infinitesimally small time step considered for iterative computation. The volume used to compute the H2 fraction was a cylinder delimited by the free surface of the melt, the reactor walls and reached 2 mm below the stirrer base after preliminary tests to determine the volume showing the best compromise to account for the H2–melt interface under all of the tested conditions.

$$\frac{1}{{\rho }_{q}}\left[\frac{\partial }{\partial t}({\alpha }_{q}{\rho }_{q})+\nabla ({\alpha }_{q}{\rho }_{q}{\bf{v}}_{q})=S{\alpha }_{q}+\mathop{\sum }\limits_{s=1}^{n}({\dot{m}}_{sq}-{\dot{m}}_{qs})\right]$$
(3)
$$\mathop{\sum }\limits_{q=1}^{n}{\alpha }_{q}=1$$
(4)
$$\frac{\partial }{\partial t}({\alpha }_{q}{\rho }_{q})+\nabla ({\alpha }_{q}{\rho }_{q}{\bf{v}}_{q})=0$$
(5)
$$\frac{{\alpha }_{q}^{n+1}{\rho }_{q}^{n+1}-{\alpha }_{q}^{n}{\rho }_{q}^{n}}{\Delta t}V+\sum _{f}{\rho }_{q}^{n+1}{U}_{\!f}^{\;n+1}{\alpha }_{q,\,f}^{n+1}=\left[S{\alpha }_{q}+\mathop{\sum }\limits_{s=1}^{n}({\dot{m}}_{sq}-{\dot{m}}_{qs})\right]V$$
(6)

Particles within the melt were simulated using the discrete phase modeling method of Ansys Fluent with the particles being allowed to interact with the continuous phase. Particles were introduced at the bottom of the vessel during simulations to reflect the fact that they were always found at the bottom of the reactor after reaction regardless of the catalyst activity. Virtual mass force and pressure gradient force models were employed along with two-way turbulence coupling. The trajectory of each particle was computed using the force balance on the particle provided by equation (7), where vp and vm are the velocities of the particles and fluid (multiphase mix of plastic melt and hydrogen), respectively, gx is the gravitational acceleration (9.81 m s−2), and ρp and ρm are the densities of the particle and fluid, respectively. The drag force (FD) was computed using equation (8), where CD is the coefficient of drag, Rep is the particle Reynolds number (computed with equation (9)) and dp is the particle diameter. Fx1 represents the virtual mass force (equation (10)) and Fx2 is the pressure gradient force (equation (11)).

$$\frac{{\rm{d}}{v}_{{\rm{p}}}}{{\rm{d}}t}={F}_{{\rm{D}}}({v}_{{\rm{m}}}-{v}_{{\rm{p}}})+\frac{{g}_{x}(\;{\rho }_{{\rm{p}}}-{\rho }_{{\rm{m}}})}{{\rho }_{{\rm{m}}}}+{F}_{\!x1}+{F}_{\!x2}$$
(7)
$${F}_{{\rm{D}}}=\frac{18\mu {C}_{{\rm{D}}}{\mathrm{Re}}_{{\rm{p}}}}{24{\rho }_{{\rm{p}}}{d}_{{\rm{p}}}^{\,2}}$$
(8)
$${{\rm{Re}}}_{{\bf{p}}}=\frac{{\rho }_{{\rm{m}}}{d}_{{\rm{p}}}({v}_{{\rm{p}}}-{v}_{{\rm{m}}})}{\mu }$$
(9)
$${F}_{x1}=\frac{1}{2}\frac{{\rho }_{{\rm{m}}}}{{\rho }_{{\rm{p}}}}\frac{{\rm{d}}}{{\rm{d}}t}({v}_{{\rm{m}}}-{v}_{{\rm{p}}})$$
(10)
$${F}_{x2}=\frac{1}{2}\left(\frac{{\rho }_{{\rm{m}}}}{{\rho }_{{\rm{p}}}}\right){v}_{{{\rm{p}}}_{i}}\frac{\partial v}{\partial {x}_{i}}$$
(11)

The sources of the discrete phase model were updated every flow iteration. The particles were injected into the two-phase system of molten polyolefin and hydrogen after the VOF model had been fully resolved. Particle injection was achieved using an injection file generated by a Python script to randomly generate particles at coordinates (x, y, z) near the base of the stirrer.

Hydrogen diffusion across a static polyolefin melt was simulated with the COMSOL Multiphysics software using a time-dependent dilute species diffusion model (in all three dimensions) constructed using Fick’s law of diffusion. The concentration of hydrogen was varied from 25 to 100 mol m−3, corresponding to different pressures of hydrogen. The diffusivity of hydrogen through the polyolefin was varied between 10−8 and 5 × 10−8 m2 s−1, corresponding to different polyolefin Mw, as explained in Supplementary Note 6. The time-dependent temperature profile in a static polyolefin melt (with the reactor wall temperature set at 498 K) was simulated with the COMSOL Multiphysics software56.

The steady-state temperature profiles in polyolefin melts stirred with various stirrer geometries were simulated by adding energy dissipation equations to the volume-of-fluid and discrete particle method (VOF-DPM) model in ANSYS Fluent (ref. 57). The heat transfer coefficients and heat capacities were taken from libraries embedded within ANSYS. As the stirring simulations were always conducted under steady-state conditions, the reported results correspond to the case where the temperature at the position of the thermocouple reached the set temperature.