A long-standing problem

Since completing my PhD in 1992, before that and more than a few times since, I have returned to the problem of modeling chemical reactions that are mixing-sensitive. I had the pleasure to meet and work with John Bourne through BHR Group in the late 1980s and early 1990s and to meet Jerzy Baldyga a few times including in Dublin with Steve Hearn in the late nineties, in the early years of PFD (which became Scale-up Systems). Building on practical findings by [^1] Ed Paul of Merck & Co., Bourne and Baldyga laid a great foundation in the field, developed interesting mathematical models, established the kinetics of liquid-phase test reactions and linked the results of mixing-sensitive reactions with fluid mechanics and energy dissipation rates. The field was developed separately in the [^2] combustion world where there was an early emphasis on CFD and closure models dealt primarily with steady-state problems in gases, with less at that time on chemical kinetics and more on energy output and turbulence modeling.

Modeling these systems allows ideas about how nature works to be tested against experimental data and to establish whether the model and the theory behind it are substantially correct. For successful models that leads to intuitions of what variables affect the result and those guide problem-solving and design of experiments to solve problems. A challenge to going further and predicting real industrially useful system behaviour quantitatively has always been the availability of chemical kinetic parameters; because at least some of the chemistry is fast but not instantaneous, special experimental setups are required. Computer power also remains a limiter even in 2026, especially when dealing with transient systems and using CFD with large computational grids.

In these posts I will discuss some practical modeling methods available, especially those from Baldyga’s group, current gaps where I know them and describe some computational research I have been doing in semi-retirement to clarify details and explore solutions. The material should be interesting for anyone making fine/ specialty chemicals or pharmaceutical active ingredients using organic synthesis and to the broad community of system modelers and CFD users. In a spoiler alert, I will not be announcing that AI can now replace existing first principles models in this area; however I have certainly found AI tools to be a force multiplier when doing this work.

Types of model

The original Lagrangian (follow the flow) models of Bourne and Balydga (including the EDD and E models) rely on a simple flow path and are not immediately suitable for integration with CFD. Dynochem’s extensive [^3] model library includes implementations of these models, assuming plug flow in a reaction zone. These models can be very effective, although their representation of the statistical state of the mixture is somewhat simplified, with typically only two or three environments with different concentrations at any time.

Balydga began to publish models formally named “closures” starting in [^4] 1989; these were immediately usable in CFD programs (via user subroutines). In addition to a multi-stage decay of concentration variance and a continuous probability density function to represent the distribution of fed materials, the concept of a stoichiometric mixture is central to Baldyga’s closure ideas. A crucial question in these models is how to estimate the instantaneous concentration of each reacting species, discussed in more detail in subsequent posts in this blog. Closures were developed for specific chemical reaction schemes, first a single reaction, later azo-coupling type reactions (competitive-consecutive), then parallel reactions and finally in [^5] 2009 a combination of competitive-consecutive and competitive-parallel reactions. Users with other types of reaction scheme could choose to apply these models as a surrogate of their real system, but did not know how to adapt the closure model to their actual reaction system. A very thorough extension of the 2009 work was published by Baldyga’a colleagues in [^6] 2025 and it was actually reading that paper early in 2026 that rekindled my personal interest in these closure type models.

I am sure we would have seen many more high impact publications from Jerzy Baldyga if he had not passed away too soon in 2019. I will refer both to his work and recent work by his colleagues as being from “Baldyga’s group”. I am glad to be in contact with and able to discuss their work with his colleague and co-author on the 2009 and 2025 papers, Magdalena Jasińska.

Considering fed batch reactions with fast kinetics, the reaction is highly localized near the feed and CFD models would waste considerable energy integrating equations throughout the bulk where no reaction is actually happening. About 20 years ago, I developed a zone model for such systems with time constants from Baldyga and Bourne but representing the reaction zone as a single separate compartment from the bulk, in which stoichiometric flows of feed and bulk reactants enter and mix. The dimensions of the zone depend on the mixing rate and it grows or shrinks as needed to reflect changing conditions. That type of model is also available in the Dynochem library [^3].

This series of posts will get technical quickly and will describe my explorations when thinking about closures again in 2026. I will use the 2025 paper’s 5-reaction azo-coupling chemistry as my main illustration and start in the next post with more detail, equations and code. After considerable effort and a lot of reading, thinking and coding, I have found a new and I believe generally applicable closure method. I will also illustrate application of the new closure to other reaction schemes for which no closure has been published to date.

Footnotes

[^1] Ed Paul

[^2] combustion world

[^3] model library

[^4] 1989

[^5] 2009

[^6] 2025