What Governing Equations Are Used To Simulate Non-Isothermal Gas-Solid Reactions In Chemical Engineering Pilot Plants?

Simulating a non-isothermal gas-solid reaction in a pilot plant requires solving a system of tightly coupled mass and heat conservation equations. The core governing equations are a gas-phase mass balance that tracks reactant diffusion and consumption, a solid-phase mass balance that describes the depletion of the solid reactant, and a thermal balance that accounts for heat conduction, reaction heat, and external heat transfer. These equations are linked through a temperature-dependent Arrhenius rate term and are solved simultaneously with boundary conditions that capture external mass and heat transfer resistances.

A reliable simulation of a non-isothermal gas-solid reaction hinges on solving three coupled partial differential equations: a gas-phase mass balance (diffusion + reaction), a solid-phase mass balance (depletion kinetics), and a pellet-scale heat balance (conduction + reaction heat). Their interaction, driven by the Arrhenius law and the heat of reaction, captures the strong feedback between temperature and conversion.

Mass Balance for the Gaseous Reactant

The gas-phase equation describes how the concentration of the gaseous reactant changes over time at any point inside a porous catalyst or reactant pellet.

The Diffusion-Reaction Equation

The rate of change of gas concentration is balanced against two competing processes: internal diffusion and chemical consumption.

Internal diffusion is modeled using the Laplacian operator (∇²C), which represents the net flux of gas molecules moving through the pellet’s pores by Fickian diffusion.

The consumption term is the temperature-dependent reaction rate. It is nearly always expressed using an Arrhenius-type law, where the rate constant k = k₀ exp(–Ea/RT).

Thus, the governing equation for the gaseous reactant A takes the form:

∂Ca/∂t = De ∇²Ca – ν r(Ca, Cs, T)

Here, De is the effective diffusivity inside the pellet, ν is the stoichiometric coefficient, and r is the intrinsic reaction rate per unit volume, which depends on gas concentration, solid conversion, and local temperature.

Why This Coupling Matters

Because the reaction rate r explodes exponentially with temperature, any small temperature rise inside the pellet accelerates consumption, which in turn feeds back into the heat balance—making the system strongly non-linear.

Mass Balance for the Solid Reactant

The solid reactant mass balance tracks the rate of depletion of the non-volatile solid that is being consumed or transformed.

A Shrinking or Reacting Solid Phase

For a gas-solid reaction, the solid’s local conversion X changes at a rate proportional to the reaction term, adjusted for the stoichiometric solid coefficient.

A common simple form is:

∂Cs/∂t = – r(Ca, Cs, T)

where Cs is the local concentration or amount of reactive solid. In many models, the solid is assumed to be stationary within a pellet, so no convective or diffusive term appears—only the sink due to reaction.

In more detailed models (e.g., shrinking core or uniform conversion), the solid mass balance is coupled to a geometric description of the reaction front, but the core principle remains: solid consumption rate is dictated by the same temperature-sensitive r that appears in the gas balance.

Thermal Balance

The thermal balance is what makes the model non-isothermal. It captures how heat is conducted through the solid, generated by the reaction, and exchanged with the surrounding gas film.

Conduction and Reaction Heat

The temperature T inside the pellet follows a heat conduction equation that mirrors the mass balance structure:

ρ Cp ∂T/∂t = λ ∇²T + (–ΔHr) r(Ca, Cs, T)

The term λ ∇²T accounts for heat conduction through the pellet matrix. The source term (–ΔHr) r represents the heat released (exothermic) or absorbed (endothermic) by the reaction.

Because r is multiplied by the heat of reaction, a strong exothermic reaction can drive large internal temperature gradients, even if the external gas temperature is constant.

External Heat Transfer and Boundary Conditions

At the pellet surface, the heat flux leaving the pellet must equal the heat transferred through the external gas film. This is modeled using the Nusselt number (Nu), which quantifies the convective heat transfer coefficient.

Similarly, the gas concentration at the surface is determined by a Sherwood number (Sh) that sets the external mass transfer resistance.

These boundary conditions link the pellet-scale equations to the reactor-scale bulk gas temperature and composition, making them essential for accurate pilot plant simulations.

Understanding the Trade-offs

The fully coupled, non-isothermal model provides physical fidelity but introduces practical challenges.

Numerical Complexity

Solving three coupled nonlinear partial differential equations—especially when two or three spatial dimensions are involved—requires robust numerical solvers and careful grid resolution. Transient simulations can become computationally expensive, which matters when you need to rapidly explore operating conditions.

Parameter Sensitivity

The model’s predictions are only as good as the input parameters. The activation energy and effective diffusivity have an exponential impact on temperature and concentration profiles. Small uncertainties in these parameters can lead to large prediction errors, particularly near ignition or extinction temperatures.

Simplifications You Might Tempt

Isothermal pellet assumption (valid only for very fast heat conduction or very low heat of reaction) removes the temperature equation but can miss critical hot-spot formation.
Pseudo-steady-state assumption for the gas phase eliminates the time derivative, simplifying solution, but loses the ability to capture rapid transients.

The right level of detail depends on your pilot plant’s objective: if safety and thermal runaway are concerns, solving the full thermal balance is non-negotiable.

Making the Right Choice for Your Pilot Plant Goal

Use these guidelines to decide how to apply the governing equations to your pilot plant simulation.

If your primary focus is predicting hot-spot temperatures and preventing runaway: Solve the fully coupled transient model with proper Nu boundary conditions. This captures the thermal feedback that drives dangerous excursions.
If your primary focus is estimating steady-state conversion under mild temperature variations: You can often reduce the complexity by assuming a constant effective temperature for the pellet, using only the mass balances, but validate that assumption with a few spot checks from the full thermal model.
If your primary focus is sizing the reactor bed or designing external heat exchangers: A 1D or pseudo-homogeneous reactor-scale model that embeds pellet-level effectiveness factors derived from the coupled equations provides a good balance between accuracy and simulation speed.
If your primary focus is teaching or rapid parameter screening: Use dimensionless forms of the equations to identify controlling groups (Thiele modulus, Prater number) and run parametric studies before committing to full PDE simulations.

Accurate simulation of non-isothermal gas-solid reactions is fundamentally about respecting the tight coupling between mass and energy—and choosing the level of model detail that gives you actionable insight without drowning in unnecessary computational cost.

Summary Table:

Equation Type	Key Physical Phenomena	Primary Terms & Driving Forces
Gas-Phase Mass Balance	Internal pore diffusion & reactant consumption	Fickian diffusion (Laplacian operator) & Arrhenius reaction rate
Solid-Phase Mass Balance	Solid reactant depletion & reaction front movement	Consumption sink term proportional to local reaction kinetics
Thermal Balance	Heat conduction & reaction heat generation/absorption	Thermal conductivity (Fourier's law) & heat of reaction ($\Delta H_r$)
Boundary Conditions	External mass & heat transfer resistances	Sherwood ($Sh$) and Nusselt ($Nu$) numbers at the pellet surface

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