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The purpose of this article is to provide a practical, expert-level guide to using Damköhler number scaling analysis to achieve reactor similarity during scale-up, while clarifying when additional dimensionless groups must be matched for reliable performance transfer.
1. Why Damköhler number matters in reactor similarity.
Reactor scale-up fails most often when the dominant controlling time scale changes between scales, even when geometry and operating conditions look similar on paper.
The Damköhler number is designed to prevent that failure by comparing how fast chemistry proceeds relative to how fast material is transported or renewed inside the reactor.
When Damköhler number is kept similar between two systems, the balance between reaction and transport is preserved, which is a central requirement for reactor similarity.
However, Damköhler number is not a universal scale-up key by itself, because transport can mean different things depending on the process, such as bulk flow, mixing, diffusion, interphase mass transfer, or heat removal.
1.1 Core definition as a time-scale ratio.
A widely used interpretation is a time-scale ratio.
Damköhler number equals a characteristic flow or transport time divided by a characteristic chemical reaction time.
Large Damköhler number implies chemistry is fast compared with transport, which pushes the system toward transport limitation and high conversion if the reactor provides sufficient contact.
Small Damköhler number implies chemistry is slow compared with transport, which pushes the system toward kinetic limitation and low conversion at a given residence time.
1.2 Common forms used in chemical reactor engineering.
For homogeneous kinetics under constant-density assumptions, a common convective Damköhler number uses the mean residence time or space time.
For a power-law rate expression with overall order n, a typical form is Da = k · C0^(n−1) · τ.
Here k is the appropriate rate constant at operating temperature, C0 is a chosen reference concentration scale, n is the apparent reaction order used in the design model, and τ is space time for flow reactors.
When transport is dominated by diffusion, interphase transfer, or mixing, the Damköhler number must be defined using that transport rate scale instead of τ alone.
Note : Always state which physical process sets the transport time scale in your Damköhler number definition, because different Damköhler numbers can be correct for the same reactor depending on what limits performance.
2. Similarity requirements for reactor scale-up.
Scale-up by similarity aims to make the dimensionless governing equations identical between the model scale and the target scale.
In reactor engineering, similarity is usually described as geometric similarity, kinematic similarity, and dynamic similarity, plus thermal and chemical similarity when reactions and heat effects are present.
2.1 Geometric similarity.
Geometric similarity means all relevant length ratios are preserved, such as diameter-to-height, impeller-to-tank diameter, baffle proportions, and catalyst pellet size distributions where applicable.
Geometric similarity is often necessary but rarely sufficient, because it does not force the same flow regime, mixing pattern, or heat-transfer capability.
2.2 Kinematic and dynamic similarity.
Kinematic similarity means velocity field similarity after scaling, which typically requires matching Reynolds number and sometimes Froude number when free surfaces or buoyancy effects dominate.
Dynamic similarity means force balance similarity, which can require matching Reynolds, Froude, Weber, Euler, and other groups depending on the flow physics.
2.3 Chemical similarity through Damköhler number.
Chemical similarity means the competition between reaction and transport remains the same at both scales.
Damköhler number is a compact way to enforce chemical similarity, provided it is defined consistently with the controlling transport mechanism.
2.4 Thermal similarity in exothermic or endothermic systems.
For heat-releasing or heat-absorbing reactions, thermal similarity can dominate outcomes such as selectivity, runaway risk, hot spots, and catalyst deactivation.
In these cases, Damköhler number must be complemented by heat-transfer and heat-generation dimensionless measures, such as relationships involving heat removal coefficients, Nusselt number, and adiabatic temperature rise scales.
Note : In strongly exothermic systems, matching Damköhler number without matching heat removal capacity can produce the same nominal conversion model but a completely different temperature profile and product distribution.
3. How Damköhler number emerges from nondimensional reactor balances.
A rigorous way to select the correct Damköhler number is to nondimensionalize the governing species balance that represents the reactor model you are using.
This step forces you to choose the correct reference scales and exposes which dimensionless groups actually control behavior for your assumptions.
3.1 Plug flow reactor form.
For a steady plug flow reactor with axial coordinate z and superficial velocity u, a simplified species balance can be written as u · dCA/dz = rA.
Define a dimensionless concentration c = CA/C0 and a dimensionless axial coordinate ξ = z/L.
If the rate is rA = −k · CA^n, substitution yields u · (C0/L) · dc/dξ = −k · (C0^n) · c^n.
Rearranging yields dc/dξ = −(k · C0^(n−1) · L/u) · c^n.
The factor (k · C0^(n−1) · L/u) is the Damköhler number for a plug-flow convection time scale, where L/u is a convective residence time scale.
3.2 CSTR form.
For a steady CSTR with volumetric flow rate Q and volume V, the design form is F_A0 − F_A + rA · V = 0.
For constant density, CA0Q − CAQ + rA · V = 0, which becomes CA0 − CA + rA · (V/Q) = 0.
With τ = V/Q and rA = −k · CA^n, the balance becomes CA0 − CA − k · CA^n · τ = 0.
With conversion X where CA = CA0(1−X), a dimensionless grouping appears as k · CA0^(n−1) · τ, which is the common CSTR Damköhler number for power-law kinetics.
3.3 When dispersion or diffusion must be included.
If axial dispersion, turbulent mixing, or diffusion matters, the model includes additional transport terms that introduce Péclet number, Schmidt number, Sherwood number, or effective diffusivity terms.
In such models, Damköhler number typically multiplies the reaction term while Péclet number multiplies the dispersion term, so both must be considered for similarity.
Typical convective Damköhler number choices. PFR with power-law kinetics: Da = k * C0**(n-1) * (L/u). CSTR with power-law kinetics: Da = k * C0**(n-1) * (V/Q). Diffusion-reaction in porous media often introduces a reaction-to-diffusion group that is not identical to the flow Da definition.4. Practical Damköhler scaling rules for common reactor types.
Once you have the correct Damköhler number definition for your model, similarity-based scale-up becomes a controlled exercise in keeping that group constant while meeting mechanical and safety constraints.
4.1 Rule for single-phase flow reactors dominated by residence time.
If conversion is primarily controlled by residence time versus intrinsic kinetics, keep Da constant between scales.
For fixed chemistry and feed, Da proportionality reduces to τ scaling, meaning you must preserve the product k · τ for first-order systems or k · C0^(n−1) · τ for n-th order systems.
Since k depends strongly on temperature, temperature control and accurate kinetics become the main knobs for similarity.
4.2 Rule for mixing-sensitive reactions.
When micromixing competes with fast reactions, the relevant transport time is a mixing time rather than the mean residence time.
In that case, define Damköhler number as a ratio of mixing time to chemical time, and keep that value constant.
This usually requires matching impeller-based power input per volume, Reynolds number in the stirred region, and sometimes a mixing time correlation calibrated to the specific geometry.
4.3 Rule for gas–liquid or liquid–solid mass transfer limited systems.
If reaction occurs in one phase but is limited by interphase transport, a Damköhler number based on mass transfer is appropriate.
A common structure is Da = (reaction rate capacity)/(mass transfer capacity), which often appears as k · C0^(n−1) divided by kLa for gas–liquid systems, or divided by an external mass-transfer coefficient times interfacial area for packed beds and slurries.
Similarity then requires keeping both Damköhler number and the dimensionless correlations that determine kLa or external mass transfer, which usually depend on Reynolds, Schmidt, and geometry.
4.4 Rule for catalytic porous solids where intraparticle diffusion matters.
If the observed rate is limited by diffusion inside porous catalyst particles, internal diffusion similarity must be preserved.
This is often addressed by a diffusion–reaction dimensionless group associated with pellet behavior, and by ensuring catalyst particle size, pore structure, and effective diffusivity scaling are consistent.
If you change pellet size during scale-up for pressure drop reasons, you can unintentionally change internal diffusion limitation even if bulk Damköhler number appears matched.
Note : For heterogeneous catalysis scale-up, treat bulk-flow Damköhler number and intraparticle diffusion similarity as separate requirements unless you have strong evidence that internal diffusion is negligible.
5. Damköhler number is necessary but not sufficient.
Matching Damköhler number alone can still yield different conversions, selectivities, and stability if other controlling groups shift with scale.
The most common missing groups are Reynolds number for flow regime, Péclet number for axial mixing or dispersion, and heat-transfer measures that control temperature rise.
| Dimensionless group. | What it controls in reactors. | Typical scaling levers. | When it must be matched for similarity. |
|---|---|---|---|
| Damköhler number Da. | Reaction versus transport renewal rate balance, conversion sensitivity to kinetics versus transport. | Residence time, temperature via k, concentration scale, catalyst loading. | Always, but only with a definition consistent with the dominant transport process. |
| Reynolds number Re. | Flow regime, turbulence level, mixing, heat and mass transfer correlations. | Velocity, diameter, viscosity, impeller speed, circulation rate. | Whenever turbulence or transfer coefficients depend strongly on flow regime. |
| Péclet number Pe. | Axial dispersion versus convection, RTD broadening, deviation from ideal PFR behavior. | Velocity, length scale, dispersion coefficient, packing structure. | Whenever axial mixing affects conversion or selectivity, especially in tubular reactors. |
| Schmidt number Sc and Sherwood number Sh. | External mass transfer behavior and boundary layer thickness relations. | Fluid properties, characteristic length, velocity, interfacial area design. | When interphase or external mass transfer competes with reaction rate. |
| Nusselt number Nu. | Heat transfer coefficient scaling, wall heat removal, hot-spot suppression. | Velocity, diameter, thermal conductivity, heat exchanger design. | When temperature profile influences k, selectivity, safety, or catalyst stability. |
6. A step-by-step workflow for Damköhler-based similarity analysis.
6.1 Step 1, define the reactor model that is valid at both scales.
Choose the simplest model that still captures the controlling physics, such as ideal CSTR, ideal PFR, axial dispersion model, two-phase mass transfer model, or catalytic packed-bed model.
Do not attempt similarity on a model that is known to be invalid at one scale, such as assuming perfect mixing in a very large tank where segregation is observed.
6.2 Step 2, identify the controlling transport process and define Da accordingly.
Use evidence such as sensitivity of conversion to flow rate, sensitivity to agitation speed, sensitivity to gas rate, or sensitivity to particle size.
Define the Damköhler number to compare reaction capacity with the controlling transport capacity, such as residence time, mixing time, mass transfer coefficient, or diffusion time.
6.3 Step 3, list all additional dimensionless groups that enter the same nondimensional equations.
If your nondimensional form includes Pe, Re, Nu, Sh, or other groups, treat them as co-equal similarity targets rather than afterthoughts.
In many practical scale-ups, you choose one primary similarity target, then hold the others within an acceptable similarity band based on sensitivity analysis and safety margins.
6.4 Step 4, compute scale-up knobs that keep Da constant.
For homogeneous kinetics, the main knobs are τ, T, and C0 scaling.
For transfer-limited systems, the main knobs are velocity, interfacial area, and agitation or sparging intensity because these change kLa or external coefficients.
6.5 Step 5, validate with a dimensionless performance plot.
A robust validation method is to plot conversion or yield versus Damköhler number across multiple operating points at the small scale and confirm the same curve is obtained at the larger scale within uncertainty.
This approach detects hidden similarity breaks, such as unmodeled heat loss or regime shifts, because the data will not collapse onto a single curve.
Note : Data collapse of performance versus Damköhler number is stronger evidence of similarity than a single matched operating point.
7. Worked scale-up example using Damköhler number matching.
Consider a liquid-phase tubular reactor modeled as plug flow with an apparent first-order rate expression rA = −kCA.
A lab unit operates at temperature T, with length L1 and velocity u1, and achieves a target conversion Xtarget.
For first-order plug flow, conversion relates to Damköhler number by X = 1 − exp(−Da), where Da = k · L/u.
Suppose the lab unit achieves Xtarget = 0.80.
Then Da_target = −ln(1−0.80) = −ln(0.20) = 1.609.
If you build a pilot unit with different diameter but similar hydrodynamics and you want the same conversion at the same temperature, you must keep k constant and keep L/u such that k · L/u = 1.609.
If the pilot length is increased to L2 = 3 · L1, then u2 must be increased to u2 = 3 · u1 to keep L/u constant, which keeps Da constant.
If instead you cannot increase velocity due to pressure drop limits, you must adjust temperature to change k, or accept a different conversion target.
Example calculations. Given Xtarget = 0.80 for first-order PFR. Da_target = -ln(1 - Xtarget) = -ln(0.20) = 1.609. To match conversion at same chemistry and temperature. Keep Da = k * (L/u) = 1.609. If L2 = 3*L1, then u2 = 3*u1 keeps L/u constant.7.1 Extending the example to n-th order kinetics.
For n-th order power-law kinetics, Damköhler number depends on C0^(n−1), so similarity also requires matching the chosen reference concentration scale.
If feed concentration changes at the larger scale due to upstream constraints, then keeping τ constant will not keep Da constant, and conversion will shift even if flow pattern is preserved.
In that case, use Da = k · C0^(n−1) · τ as the similarity condition and adjust τ or T accordingly.
8. Common pitfalls and how to avoid them.
8.1 Using the wrong Damköhler definition.
Using Da = k · τ for a system that is mixing-limited or mass-transfer-limited will produce misleading similarity targets.
Always tie the Damköhler number to the transport mechanism that actually competes with reaction.
8.2 Ignoring regime changes with scale.
Flow regime can change as diameter increases, even at similar superficial velocities, because practical constraints shift viscosity, temperature, or roughness, and because matching Re may be impossible.
When Re changes significantly, mass and heat transfer correlations change, and the effective Damköhler number based on transfer changes even if kinetic Da appears matched.
8.3 Ignoring temperature gradients and heat removal.
If temperature is not uniform, the effective rate constant varies in space, and a single k value does not represent the system.
In such cases, treat Damköhler matching as a local requirement and use coupled heat and mass balance similarity, including heat-transfer dimensionless groups and realistic boundary conditions.
8.4 Treating selectivity as automatically similar.
For parallel or consecutive reactions, selectivity depends on relative Damköhler numbers for each pathway and on mixing and temperature fields.
Similarity should be defined using a set of dimensionless groups that preserve the ratios of key reaction time scales, not only the overall conversion Da.
9. Reactor similarity checklist you can use immediately.
| Checklist item. | What to compute or verify. | Acceptance target for similarity. |
|---|---|---|
| Define controlling transport mechanism. | Residence time, mixing time, external mass transfer, internal diffusion, or heat removal control. | One dominant mechanism identified and justified by data or sensitivity tests. |
| Choose correct Damköhler definition. | Da based on the controlling transport rate scale and the relevant kinetics expression. | Same definition used at both scales with consistent reference scales. |
| Match Damköhler number. | Compute Da at both scales using measured or validated parameters. | Da within a defined band such as ±10% when feasible, tighter for sensitive selectivity systems. |
| Check Reynolds and flow regime. | Re, regime indicators, and whether correlations used remain valid. | Same regime, or proven insensitivity of performance to Re changes in the relevant range. |
| Check axial mixing or dispersion. | Pe or RTD measures for tubular systems, mixing indices for stirred systems. | Comparable dispersion behavior, or validated correction via a nonideal model. |
| Check heat transfer capability. | Heat removal coefficient scaling, Nu, coolant approach, surface-to-volume ratio changes. | No unacceptable temperature rise, and temperature profile similarity for sensitive systems. |
| Validate with dimensionless performance collapse. | Plot performance versus Da across multiple operating points at each scale. | Curves collapse within uncertainty, and deviations are explained by additional groups. |
FAQ
What is the most useful Damköhler number definition for basic reactor scale-up.
For many single-phase flow reactors where conversion is controlled mainly by residence time and intrinsic kinetics, Da = k · C0^(n−1) · τ is the most practical definition.
For plug flow, an equivalent form is Da = k · C0^(n−1) · L/u when L/u represents the convective time scale.
If mixing or interphase transport competes with reaction, redefine Da using mixing time or mass-transfer capacity rather than τ alone.
Can I scale up by matching Damköhler number only.
Matching Damköhler number alone can work for systems that remain in the same flow regime, remain close to isothermal, and are not controlled by mass transfer or diffusion limitations.
In many industrial cases, Reynolds number, heat-transfer limits, and dispersion effects shift with scale, so Damköhler number must be matched together with the additional groups that enter the same governing equations.
How do I keep Damköhler number constant if I cannot match residence time.
If τ cannot be matched due to throughput constraints, you can adjust k by changing temperature, catalyst loading, or effective activity, provided the kinetic model remains valid.
For n-th order kinetics, changing concentration scale also changes Da through C0^(n−1), so feed concentration control can be a similarity knob when chemically acceptable.
How does Damköhler number relate to selectivity in multiple reaction networks.
In networks, selectivity depends on relative rates and relative transport competition for each pathway.
A single overall Da is often insufficient, so define pathway-specific or lumped Damköhler numbers that preserve the ratios of characteristic reaction time scales, and ensure thermal and mixing similarity if selectivity is sensitive.
How should I use Damköhler number in experimental planning.
Plan experiments to sweep Damköhler number over a meaningful range by varying residence time, temperature, or concentration while monitoring conversion and selectivity.
Then scale up by targeting the same Damköhler number and verifying that performance data collapse versus Da, which is strong evidence that the controlling balance is preserved.