Chemical reactions can produce or absorb heat, which is quantified using the enthalpy of the reaction.
Reaction rates and total energy in the fluid can be used to develop a set of coupled Navier-Stokes equations describing fluid flow with chemical reactions.
The resulting equations and the thermal characteristics of the fluid depend on the flow regime and compression regime in the flow.
Fluid flow problems involving chemical reactions are the trifecta of CFD simulation workloads. They involve thermal simulations, chemical reactions that produce or remove heat, and, of course, they could involve laminar or turbulent flows. Problems involving fluid mixing add another dimension of complexity once reaction kinetics are included, as this will add or remove heat to the fluid, thus changing its total energy. For some fluids, like mixtures of gasses, changes in temperature may change the material properties and flow behavior of the fluid. Reaction engineering focuses on managing the flow rate in these situations with mixing and reactions to produce the desired species and possibly control its flow.
The main equations in CFD are now taken on a constituent basis when we examine the reaction kinetics of gasses or liquids in a flow. In this article, we’ll look briefly at mixtures of gasses as an example of how the main equations in fluid dynamics are affected by the enthalpy of reaction in fluid flow. By treating reacting fluids as mixtures and considering their reaction kinetics alongside the enthalpy of reactions, it’s possible to accurately model flows involving reacting fluids.
Fluid Flow Models Involving Enthalpy of Reaction
Models involving fluid flow wherein a chemical reaction occurs must consider certain thermodynamic characteristics of the fluid as well as the thermodynamics of the reaction itself. Reactions between mixtures of fluids could occur in the gas phase, liquid phase, or in a multiphase fluid (gasses and microscopic solids submerged in fluids). These fluid flows can be mathematically complex, depending on the constituents in the fluid, the forces acting on them (e.g., buoyant force acting on gaseous reactants/products), and the enthalpy of the reaction that leads to heat generation/dissipation.
Enthalpy of reaction is defined using a thermodynamic relation:
∆ H = ∆ U + P ∆ V
If we revisit the primary equations of motion in fluid dynamics, we can see that these equations already consider the thermal behavior of fluids and are statements about the conservation of energy. Therefore, modeling multi-fluid flows wherein a chemical reaction occurs requires developing a set of equations that describe:
- How the enthalpy of reaction affects the total energy (thermal + kinetic)
- The flow behavior for each fluid in the mixture
Both points are relatively simple for gasses; the first point relies on modeling with the ideal gas law (or considering adiabatic compression), while the second point relies on developing coupled equations for multiple gasses in the flow. Let’s look at how these equations are developed.
Coupled Navier-Stokes Equations for Reacting Gasses
The case of mixed reacting gasses in laminar or turbulent flow is relatively easy to model. A complete look at the example shown below can be found in the following journal article:
- Yu, S. T., Chang, S. C., Jorgenson, P., Park, S. J., & Lai, M. C. (1997). Basic equations of chemically reactive flows for computational fluid dynamics. In 36th AIAA aerospace sciences meeting and exhibit (p. 1051). [Link]
In this example, we consider N gasses involved in an inviscid unsteady flow. The total energy (Q) as being the sum of kinetic (E) and potential (S) terms, the latter of which may be driven by chemical factors. The Navier-Stokes equation would be written as:
The three matrices Q, E, and S are defined below:
Here, the symbols have their usual meanings. The terms in the S matrix are source terms, which must sum to zero if the continuity equation is to hold (i.e., conservation of mass). It is these source terms that become related to the enthalpy of reaction for the inter-reacting gasses.
Next, we define the source terms based on the rate of change in the concentration from species j into species i via a chemical reaction with known rate constants:
Each C term is based on the reaction progress and the rate of the reaction, both of which are known from simple reaction kinetics and Arrhenius equations, respectively. The M term is the molar fraction of species i. If we assume the gasses involved in the flow are ideal, we can rewrite the enthalpy of reaction for the entire fluid in terms of the summed enthalpies of reaction for each gas in the mixture. Using the ideal gas law, we have:
With these relations and Dalton’s law relating the partial pressure of each gas component, we now have the complete Navier-Stokes equation for the gas mixture.
Solving These Equations
For a set of N gasses, we have N + 2 with N + 2 unknowns, so the problem is in principle solvable with a standard numerical scheme. Typically, one would approach this with something like FDTD, possibly with dimension reduction to reduce the complexity of the problem. The sheer number of equations and the lack of resemblance to other models for turbulent flow has motivated a specialized set of techniques for analyzing these types of flows involving chemical reactions. However, commercial CFD simulation tools can also do the job, as these problems are generally well-posed, as we described above.
Simulation engineers from any discipline will find all the tools they need to account for the enthalpy of reaction complete set of CFD simulation applications from Cadence. The meshing features in Pointwise include everything needed to build highly accurate meshes for complex systems and the Omnis simulation suite implements modern numerical approaches to solve the main fluid dynamics equations in 3D.