The mathematical distinction is the location and units of the reaction source term. For a homogeneous reaction, the rate appears directly in the differential species conservation equation as a volumetric source, (R_{Vi}), with units of (\mathrm{mol,m^{-3},s^{-1}}). For a heterogeneous reaction occurring at a surface or phase interface, the rate is applied as a boundary flux, (R_{Si}), with units of (\mathrm{mol,m^{-2},s^{-1}}), or converted into an equivalent volumetric source when using a reactor-scale averaged model.

Homogeneous reactions modify species concentrations throughout a phase volume; heterogeneous reactions modify species fluxes at surfaces or interfaces. The correct mathematical form depends on whether the reaction is resolved locally at the surface or represented as an averaged source inside the pilot-scale reactor model.

How the Species Balance Represents Reaction

The general conservation equation

For species (i), a molar species balance can be written as

[ \frac{\partial C_i}{\partial t}

-\nabla \cdot \mathbf{N}i + R{Vi} ]

where (C_i) is the species concentration, (\mathbf{N}i) is its molar flux, and (R{Vi}) is the net rate of generation per unit volume.

The divergence term accounts for transport into or out of a differential control volume. The reaction term accounts for chemical generation or consumption within that volume.

Homogeneous reaction rates

A homogeneous reaction occurs throughout a single fluid phase, such as a liquid solution or gas mixture. Its rate enters directly into the governing differential equation:

[ \frac{\partial C_i}{\partial t}

-\nabla \cdot \mathbf{N}i + R{Vi} ]

For a reaction mechanism with reactions (j=1,\ldots,N_r),

[ R_{Vi}

\sum_{j=1}^{N_r} \nu_{ij} r_{Vj} ]

where (\nu_{ij}) is the stoichiometric coefficient of species (i) in reaction (j), and (r_{Vj}) is the reaction rate per unit fluid volume.

Typical units are

[ r_{Vj}, R_{Vi} ;:; \mathrm{mol,m^{-3},s^{-1}} ]

A negative value represents net consumption, while a positive value represents net generation.

How Heterogeneous Reactions Enter the Balance

Surface reaction as a boundary condition

When a reaction occurs only at a catalyst surface or phase boundary, it does not act throughout the adjacent fluid volume. Instead, it changes the species flux at that surface.

At an interface (\Gamma), the boundary condition can be expressed as

[ -\mathbf{n}\cdot\mathbf{N}_i

R_{Si} ]

where (\mathbf{n}) is the outward unit normal and (R_{Si}) is the surface generation or consumption rate.

For multiple surface reactions,

[ R_{Si}

\sum_{j=1}^{N_s} \nu_{ij} r_{Sj} ]

with typical units

[ r_{Sj}, R_{Si} ;:; \mathrm{mol,m^{-2},s^{-1}} ]

The reaction therefore appears in the boundary condition, rather than as a volumetric source in the bulk fluid equation.

Interfacial flux jump

For a reaction at the interface between two phases, the more general condition is a flux discontinuity:

[ \mathbf{n}\cdot \left( \mathbf{N}{i,2}-\mathbf{N}{i,1} \right)

R_{Si} ]

This states that the difference between the species fluxes leaving and entering the interface equals the amount generated or consumed at the interface.

This form is useful for gas-liquid, gas-solid, or liquid-solid pilot reactors where both phases have separately modeled transport equations.

Converting Catalyst-Based Rates for Reactor Models

Rate per unit catalyst mass

Heterogeneous catalytic kinetics are often reported per unit mass of catalyst:

[ r_{i,\mathrm{cat}} ;:; \mathrm{mol,kg_{cat}^{-1},s^{-1}} ]

A fixed-bed reactor model may require a source term per unit reactor volume. The conversion is

[ R_{Vi,\mathrm{eff}}

\rho_{b,\mathrm{cat}},r_{i,\mathrm{cat}} ]

where (\rho_{b,\mathrm{cat}}) is the bulk catalyst loading per unit reactor volume.

The resulting effective source has units of

[ R_{Vi,\mathrm{eff}} ;:; \mathrm{mol,m^{-3}_{reactor},s^{-1}} ]

This lets a heterogeneous reaction be inserted into a one-dimensional reactor balance while preserving the original catalyst-based kinetics.

Rate per unit catalyst surface area

If the rate is expressed per unit catalyst surface area,

[ r_{i,\mathrm{surf}} ;:; \mathrm{mol,m^{-2}_{cat},s^{-1}} ]

then the corresponding reactor-volume source is

[ R_{Vi,\mathrm{eff}}

a_s r_{i,\mathrm{surf}} ]

where (a_s) is the catalyst surface area per unit reactor volume, with units of (\mathrm{m^2_{cat},m^{-3}_{reactor}}).

The distinction is important: a surface rate cannot be added directly to a volumetric balance without multiplying by the relevant interfacial area density.

Pilot-Scale Modeling Choices

Explicit heterogeneous model

An explicit heterogeneous model solves separate conservation equations for the fluid and solid phases. The fluid equation contains transport and interphase exchange terms, while the solid or catalyst equation accounts for surface reaction, intraparticle diffusion, or both.

For example, a fluid-phase balance may be represented schematically as

[ \frac{\partial C_{i,f}}{\partial t}

-\nabla\cdot\mathbf{N}{i,f} + a_s k{i,m}(C_{i,f}-C_{i,s}) ]

with the surface reaction represented through a separate catalyst or interface condition.

This approach can resolve film resistance, differences between fluid and solid concentrations, and pore-diffusion effects.

Pseudohomogeneous model

A pseudohomogeneous fixed-bed model treats the fluid and catalyst as one averaged continuum. The heterogeneous reaction is converted into an effective volumetric source:

[ \frac{\partial C_i}{\partial t}

-\nabla\cdot\mathbf{N}i + a_s r{Si} ]

or, for a catalyst-mass basis,

[ \frac{\partial C_i}{\partial t}

-\nabla\cdot\mathbf{N}i + \rho{b,\mathrm{cat}}r_{i,\mathrm{cat}} ]

This is computationally simpler and is often suitable when local fluid-solid gradients are not central to the design question.

Choosing the representation

The appropriate form depends on the required level of detail. A reactor sizing calculation may use an averaged volumetric source, while catalyst design or thermal management may require explicit surface and intraparticle transport equations.

The central modeling question is whether the catalyst surface and phase-specific gradients are being resolved or averaged into a reactor-scale source term.

Understanding the Trade-offs

Avoid mixing surface and volume units

A common error is inserting (\mathrm{mol,m^{-2},s^{-1}}) directly into an equation whose source term requires (\mathrm{mol,m^{-3},s^{-1}}). The rate must first be multiplied by interfacial area per reactor volume, or applied as a boundary flux.

The same issue occurs when using a rate per kilogram of catalyst. It must be multiplied by catalyst mass per reactor volume before entering a volumetric species balance.

Do not confuse heterogeneous chemistry with heterogeneous modeling

A chemically heterogeneous reaction may be represented by a pseudohomogeneous model. In that case, the chemistry still occurs on catalyst surfaces, but its net effect is averaged into an effective volumetric rate.

Conversely, a multiphase reactor model does not automatically require a surface reaction term. Phase-specific equations may be needed because of interphase transport even when the chemical reaction occurs in a bulk phase.

Account for internal catalyst surfaces

Catalyst reactions may occur on internal pore surfaces rather than only on the external pellet boundary. If those surfaces are not resolved geometrically, their contribution is usually represented through an effective surface area, catalyst-mass rate, effectiveness factor, or volume-averaged source.

Treating internal catalytic activity as an external boundary condition without accounting for the available internal area can substantially misrepresent the reaction rate.

Preserve stoichiometric consistency

Whether the rate is volumetric, surface-based, or catalyst-mass-based, all species source terms must use the same reaction progress rate and stoichiometric coefficients:

[ R_i=\sum_j \nu_{ij}r_j ]

Changing the normalization basis changes the units and scaling, but not the underlying stoichiometry.

How to Apply This to Your Project

The correct implementation depends on the physical location of the reaction and the basis of the available kinetic data.

If your primary focus is homogeneous reactor design: Use (R_{Vi}) directly in the species conservation equation, with units of (\mathrm{mol,m^{-3},s^{-1}}).
If your primary focus is resolved surface kinetics: Apply (R_{Si}) as an interfacial flux or boundary condition, with units of (\mathrm{mol,m^{-2},s^{-1}}).
If your primary focus is fixed-bed reactor scale-up: Convert catalyst-mass or surface-area rates into an effective volumetric source using (\rho_{b,\mathrm{cat}}) or (a_s).
If your primary focus is catalyst or transport limitations: Use separate fluid and solid equations to resolve film transfer, temperature differences, and intraparticle diffusion.
If your primary focus is rapid pilot-plant simulation: Use a pseudohomogeneous model with a carefully defined effective volumetric reaction rate.

A reaction rate is mathematically correct in a pilot-scale species balance only when its physical location, normalization basis, and units match the conservation equation or boundary condition where it is applied.

Summary Table:

Reaction or model type	Mathematical representation	Typical units	Pilot-scale use
Homogeneous reaction	Volumetric source, `Rᵥᵢ = Σνᵢⱼrᵥⱼ`	mol·m⁻³·s⁻¹	Bulk liquid or gas reactions
Resolved heterogeneous reaction	Surface flux or interfacial flux jump, `Rₛᵢ`	mol·m⁻²·s⁻¹	Surface and phase-interface modeling
Catalyst-mass basis	`Rᵥᵢ,eff = ρᵦ,cat rᵢ,cat`	mol·m⁻³ reactor·s⁻¹	Fixed-bed reactor models
Catalyst-surface basis	`Rᵥᵢ,eff = aₛ rᵢ,surf`	mol·m⁻³ reactor·s⁻¹	Porous catalyst and averaged models
Pseudohomogeneous model	Effective volumetric source in the balance	mol·m⁻³·s⁻¹	Faster reactor simulation and scale-up
Explicit multiphase model	Separate phase balances with interphase transfer and surface conditions	Basis-dependent	Film, pore, and thermal limitation analysis

Design the Right Pilot-Scale Reactor Model

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