This article presents an arithmetic, called superposition relaxation, for bracketing the graph of a multivariate factorable function on a compact domain between a pair of underestimating and overestimating functions that are both separable. Propagation rules are established for affine and nonlinear composition operations, with a focus on exploiting global monotonicity and convexity properties in the composition. The local convergence properties of this arithmetic are also analyzed in both the pointwise and Hausdorff sense, including conditions under which quadratic pointwise convergence propagates through composition. Parameterizations of the univariate summands in a superposition relaxation either as piecewise-constant or continuous piecewise-linear functions are discussed for a practical implementation. It is shown through numerical case studies that superposition relaxations can be consistently tighter than McCormick relaxations, including for the relaxation of artificial neural networks. But superposition relaxations also incur a higher computational cost than McCormick relaxations. Further investigations are thus warranted as applications in global optimization seek to balance a relaxation's tightness with its computational cost.

Spatially dispersed industrial sites contribute substantially to industrial CO2 emissions. Hydrogen is a key decarbonisation option; however, its high flammability and the logistics required to supply dispersed sites raise safety concerns that remain insufficiently addressed. This work investigates the inherent safety of hydrogen supply strategies for spatially dispersed industrial sites. Centralised configurations are examined in which hydrogen production and conditioning are shared to exploit economies of scale, while truck-based delivery connects dispersed end users. Three centralised options—compressed gaseous hydrogen, liquid hydrogen, and liquid organic hydrogen carrier (LOHC)—are benchmarked against decentralised electrolysis. A newly developed inherent safety assessment framework enables ex-ante evaluation through hydrogen supply potential and inherent hazard indices (HSPI and HSHI), integrating centralised operations, road transport, and on-site processing. Applied to a reference set of spatially dispersed industrial sites, the results show that conditioning-intensive hydrogen supply options can increase inherent hazard levels, with HSPI values up to one order of magnitude above decentralised electrolysis. When accident scenario credibility is incorporated, transport operations emerge as the dominant contributor to inherent hazard. Within the centralised configurations, LOHC-based supply exhibits the lowest HSPI and HSHI values and therefore the most favourable inherent safety performance.

Hydrothermal liquefaction (HTL) converts wet municipal solid waste (MSW) into energy-dense bio-oil, yet up to 50% of the feedstock carbon is retained in the resulting aqueous phase (HTL-AP), representing a significant resource loss. This study proposes a process integrating HTL with an existing anaerobic digestion (AD) facility to recover this carbon alongside the biodegradable waste fraction. In the integrated configuration, the non-digestible fraction is valorized via HTL, while the HTL-AP is partially recycled to the HTL reactor and partially co-digested with the biodegradable fraction. Experimental data from real MSW feedstock are integrated within a commercial process simulator to conduct a process-wide analysis. The integrated process recovers over 50% of the incoming carbon and achieves an energy efficiency of 49-52% on a higher-heating-value basis, substantially outperforming standalone AD, which recovers less than 10%. These results demonstrate a viable pathway for upgrading existing waste treatment infrastructure towards more circular, resource-efficient operations.

Activity coefficients are key thermodynamic quantities for describing phase equilibria, but their experimental determination entails laborious and costly phase-equilibrium measurements, making predictive approaches highly desirable. The potential of machine learning for such predictions has received growing attention as an alternative to physics-based models that require experimental data or expensive calculations for parameterization. We propose a physics-informed edge-enhanced graph attention network (PEGAT) to predict activity coefficients in multicomponent mixtures, where each molecule is encoded as a graph in which the nodes correspond to atoms and the edges to chemical bonds. The excess Gibbs free energy of the mixture is predicted using the proposed model, including a nonlinear transformation in the final layer to ensure that the excess Gibbs free energy vanishes for pure components. To further enforce thermodynamic consistency, the relevant activity coefficients are obtained via the Gibbs–Duhem relation. Unlike machine-learning models developed primarily for binary systems, the proposed framework is directly applicable to arbitrary multicomponent mixtures. The PEGAT model is evaluated using a mixed dataset comprising both binary and ternary mixture data and demonstrates high predictive accuracy. Further validation on representative mixtures shows close agreement between predicted and reference activity coefficients. The results confirm that improved thermodynamic consistency can be achieved by embedding hard physical constraints into the graph neural network architecture. However, they also highlight that unphysical behaviors may still be predicted despite these constraints.

Machine learning models are increasingly used to model chemical process systems, yet they often lack principled uncertainty quantification and mechanisms to enforce physical constraints. We propose a probabilistic neural network framework that guarantees satisfaction of linear equality constraints within a given tolerance, while capturing aleatoric uncertainty. Compared to state-of-the-art methods, our formulation demonstrates improved predictive accuracy, uncertainty calibration, and adherence to constraints on reduced data. It also demonstrates competitive performance, but with significantly faster training times when evaluated on large data regimes. We evaluated this on two batch reactor case studies, enforcing mass balances.

The model-based determination of maximally-informative campaigns involving multiple parallel experimental runs remains a challenging task. Effort-based methodologies are well suited to the design of such experiment campaigns through discretizing the experiment control domain into a finite sample of candidate experiments. However, this approach can lead to suboptimal results if the discretization fails to cover the experiment domain sufficiently well. We present a comprehensive computational framework that combines an effort-based optimization step with a gradient-based refinement as part of an iterative procedure. The convexity of classical design criteria in the effort space allows for a globally optimal effort selection over the discretization, which is exploited to warm-start the gradient-based search for a refined discretization. Our framework also considers parametric model uncertainty by formulating risk-inclined, risk-neutral and risk-averse design criteria, and it enables the solution of exact designs in the effort-based step. Through the case study of a fed-batch fermentation, we show that the integrated effort-based optimization with gradient-based refinement procedure consistently outperforms an effort-only optimization. The results demonstrate the benefits of robust design approaches compared to their local counterparts, and establish the computational tractability of the framework in computing robust experiment campaigns with up to a dozen dimensions.

The ability to guarantee a single homogeneous liquid phase is a key consideration in computer-aided mixture/blend design (CAMbD). In this article, we investigate the use of a classifier surrogate of the phase stability condition within a CAMbD optimisation model for designing solvent mixtures with guaranteed phase stability properties. We show how to develop such classifiers for describing multiple candidate mixtures over range of compositions and temperatures based on the generation of phase stability data using thermodynamic models such as UNIFAC. We test the approach on two solvent design case studies and illustrate its effectiveness in enabling the in silico design of stable mixtures, simultaneously providing a probability of phase stability as an interpretable metric.

The definition of strategies for operation of process networks is a key research focus in process systems engineering. This challenge is commonly formulated as a numerical constraint satisfaction problem, where most practical algorithms are limited to identifying inner approximations to the feasible operational envelope. Sampling-based approaches so far have only been developed for formulations that required coordinated operation of the units within the network. We propose a decomposition approach that enables decentralized operation for acyclic muti-unit processes by sampling. Our methodology leverages problem structure to decompose unit-wise and deploys surrogate models to couple the resultant subproblems. We demonstrate it on a serial, batch chemical reactor network. In future research, we will extend this framework to consider the presence of uncertain unit parameters robustly.

Inadequate waste disposal methods currently employed around the globe are leading to substantial damage to both local human populations and the surrounding environment. Simultaneously, the chemical sector is facing pressure to shift from an unsustainable linear economy model and towards a more circular one. Municipal solid waste (MSW) gasification is a technology which is garnering increased attention as it offers a unified solution to both these issues. In this work, the economic and environmental performance of MSW gasification coupled with three chemical manufacturing routes (methanol, olefins via methanol-to-olefins, and ethanol) were compared to landfilling and incineration as a waste disposal method. Detailed process simulation is coupled with the monetisation of endpoint environmental impacts to determine the enviro-economic cost (EEC) of waste processing. Despite the high capital expenditure for all of the gasification routes, the break-even gate fees are competitive with both landfilling and incineration if the CO2 produced is recovered and sent to storage (CCS). However, if the CO2 is recycled and utilised (CCU), the break-even gate fees are significantly greater due to a higher green H2 demand. Both methanol and MTO routes offer significant reductions in environmental impacts due to offsetting fossil-based chemical manufacturing. All gasification routes have predicted EECs lower than both landfilling and incineration, with methanol coupled with CCS resulting in the lowest overall EEC at-0.21 pound/kgMSW. Methanol production with CCU via MSW gasification is likely to be the most feasible option in the short term as it does not rely on the existence of CO2 pipeline infrastructure, but it would still need policy support via higher landfilling and carbon taxes. Overall, this work highlights the potential environmental benefit in coupling chemical manufacture with MSW gasification.

Applying model-based design of experiments to compute maximally-informative campaigns with multiple parallel runs is challenging. Herein, we develop a systematic framework for recasting an experiment design problem for model parameter precision as one of discrimination between multiple rival models with different uncertain parameter realizations. We use an algebraic upper bound on the Bayes Risk as information criterion and apply a search procedure that iterates between an effort-based optimization step followed by a gradient-based refinement step. Through the case study of a fed-batch reactor, we show that a Bayes Risk discrimination strategy can provide highly-informative experimental campaigns to improve parameter precision, while being computationally advantageous compared to conventional FIM-based design strategies and capable of handling structurally unidentifiable problems. Copyright (c) 2025 The Authors.

Fractionation of lignocellulosic biomass is a crucial step to provide cellulose, lignin, and hemicellulose for further processing. This paper is concerned with modelling biomass fractionation using the ionoSolv process, which employs low-cost ionic liquid water mixtures, with a special focus on describing the effect of acid:base ratio of the mixture on process performance. We build on an existing semi-mechanistic modelling framework describing the solvent extraction of three main biopolymers from woody biomass for varying fractionation temperature, time, and solids loading. Since the effect of acidity is poorly understood from a mechanistic standpoint, we use sparse regression with lasso regularisation to incorporate it in the semi-mechanistic model. We investigate both polynomial and exponential functional forms and find that the latter yields more physically-consistent results. This enabled us to recalibrate the parameters of the combined semi-mechanistic and sparse data-driven models simultaneously to accurately predict the effect of varying acid:base ratio. This hybrid modelling framework opens new opportunities for further analysis and optimisation of ionic liquid-based biomass fractionation processes. Copyright (c) 2025 The Authors. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/)

Reducing the carbon intensity of maritime transport is essential to achieve global emission reduction targets. Electro-fuels (e-fuels) represent a promising cleaner alternative to conventional marine fossil fuels, offering potential lifecycle greenhouse gas reductions when synthesised from renewable electricity and low-carbon feedstocks. While techno-economic and environmental assessments of e-fuels exist, their broader sustainability implications, spanning technological, economic, environmental and safety factors together, remain largely unexplored. This study introduces a quantitative framework to assess the sustainability of ship fuel systems that integrates key performance indicators (KPIs) across these four areas. A case study is conducted to compare the sustainability of carbon-based e-fuels (e-methanol and e-diesel) and carbon-free e-fuels (hydrogen and ammonia) against marine diesel oil (MDO) under multiple decision-making perspectives. The robustness of the overall sustainability-based ranking of fuel alternatives, as derived under each perspective, against uncertainties in the individual KPIs is confirmed via sensitivity analysis. Environmental and safety aspects are found to be critical in comparing the sustainability of alternative fuels. Both e-methanol and e-diesel achieve higher overall sustainability than MDO, irrespective of the decision-making perspective. Ammonia and hydrogen are hindered by safety concerns in the short term, although ammonia also shows long-term potential for sustainable shipping subject to appropriate risk management and the implementation of inherently safer design measures. Overall, the proposed framework enables a comprehensive assessment of alternative fuel systems for cleaner shipping, guiding future sustainability-driven policy and technology development.

Biofilm systems present a promising approach for microalgae production by reducing water and energy costs while improving productivity and operational efficiency. However, this technology is still in its infancy, particularly for high-value compounds production. To confirm its potential at large scale, mathematical models are required to better understand biofilm behavior under varying environmental conditions and to predict productivity. In this study, a dynamic model was developed to estimate astaxanthin production by Haematococcus lacustris biofilms on a rotating system. It incorporates well-established dynamics, accounting for nitrogen limitation and photoacclimation, while introducing a novel hypothesis correlating astaxanthin dynamics with those of chlorophyll. The model predicts key biofilm traits, including biomass density, intracellular nitrogen, and pigment quotas, demonstrating its ability to simulate changes in light and nitrogen conditions and assess their impact on biofilm physiology. Furthermore, the possibility of dynamically altering the life cycle of H. lacustris within a biofilm was demonstrated both experimentally and mathematically, enabling reversible transitions between green and red stages. This reversion facilitates continuous astaxanthin production through repeated harvest and regrowth cycles. This was assessed through the development of an optimization strategy that maximized astaxanthin productivity by adjusting light intensity over time and determining the optimal harvest frequency. The model provides a valuable framework for optimizing astaxanthin production in microalgal biofilms, enabling the development of continuous production systems and supporting the scale-up of biofilm technology.

The discovery of chlorophyll f-containing photosystems, with their long-wavelength photochemistry, represented a distinct, low-energy paradigm for oxygenic photosynthesis. Structural studies on chlorophyll f-containing photosystem I could identify some chlorophyll f sites, but none among the photochemically active pigments, and thus concluded that chlorophyll f plays no photochemical role. Here, we report two cryo-electron microscopy structures of far-red photosystem I from Chroococcidiopsis thermalis PCC 7203, allowing the assignment of eight chlorophyll f molecules, including the redox active A-1B. Simulations of absorption difference spectra induced by charge separation indicated that the experimental spectra can be reproduced only by considering the presence of a chlorophyll f at the A-1B site. The chlorophyll f locations, wavelength assignments, and conserved far-red-specific residues provide functional insights for efficient use of long-wavelength photons.

Certifying feasibility in decision-making, critical in many industries, can be framed as a constraint satisfaction problem. This paper focuses on characterising a subset of parameter values from an a priori set that satisfy constraints on a directed acyclic graph of constituent functions. The main assumption is that these functions and constraints may be evaluated for given parameter values, but they need not be known in closed form and could result from expensive or proprietary simulations. This setting lends itself to using sampling methods to gain an inner approximation of the feasible domain. To mitigate the curse of dimensionality, the paper contributes new methodology to leverage the graph structure for decomposing the problem into lower-dimensional subproblems defined on the respective nodes. The working hypothesis that the Cartesian product of the solution sets yielded by the subproblems will tighten the a priori parameter domain, before solving the full problem defined on the graph, is demonstrated through four case studies relevant to machine learning and engineering. Future research will extend this approach to cyclic graphs and account for parametric uncertainty.

Applying model-based design of experiments to compute maximally-informative campaigns with multiple parallel runs is challenging. Effort-based methods can overcome some of these challenges through discretizing the experimental space into a finite set of candidate experiments, then applying convex optimization techniques to determine the optimal Efforts for each candidate, and finally rounding the Efforts to integer numbers of runs for a target experimental campaign size. For small experiment campaigns in particular, the final rounding can result in large suboptimality. This paper presents an approach to solving the exact design problem, where the Effort variables being optimized are constrained to taking integer values. We consider model parametric uncertainty and formulate risk-inclined, risk-neutral and risk-averse exact design problems as mixed-integer nonlinear programs (MINLPs) with convex participating functions. We demonstrate the tractability of an outer-approximation algorithm to solve such MINLPs to global optimality on a case study involving the exothermic esterification of propionic anhydride with over 1000 experiment candidates and 100 uncertainty scenarios.

The industrial cultivation of microalgae has increased substantially over the past two decades. These microorganisms have the ability to adapt their photosynthetic pigments in response to the amount of light they experience. Herein, we investigate a dynamic model that describes pigment adaptation and its effect on microalgal productivity in a photobioreactor where light is shone onto the surface and attenuated as it traverses the culture medium. We consider two controls the light irradiance and the dilution rate of the photobioreactor under continuous operation and constant volume and analyze strategies for maximal production of microalgal biomass using Pontryagin's maximum principle. We also conduct a numerical investigation of turnpike properties in this context and discuss how self-shading within the culture could be exploited to increase productivity. Copyright (C)2024 The Authors. This is an open access article under the CC BY-NC-ND license (htips://creativecommons.org/licenses/by-nc-nd/4.0/)