- Get link
- X
- Other Apps
Hatta Number Guide for Gas–Liquid Reactive Absorption: Film Theory Regimes, Enhancement Factor, and Design Workflow
The purpose of this article is to provide a practical, calculation-ready method to select the correct Hatta number regime for mass transfer with chemical reaction in a liquid film, and to convert that regime choice into the appropriate enhancement factor and absorption rate expressions for design and troubleshooting.
1. Why the Hatta number matters in reactive mass transfer.
In gas–liquid absorption with reaction, the observed uptake rate is controlled by a coupled competition between diffusion through the liquid-side mass-transfer zone and chemical consumption in that same zone or in the bulk liquid.
The Hatta number compresses that competition into one dimensionless indicator, allowing a defensible regime selection that determines which formulas are valid, which assumptions are safe, and what is actually limiting the rate.
2. Core film-theory framework and rate expressions.
2.1 Liquid-side flux with reaction expressed using an enhancement factor.
A common engineering form for the liquid-side molar flux of solute A from the interface into the liquid is written as an enhanced physical-absorption flux.
For a liquid-side controlled situation, the flux can be expressed as.
N_A = E · k_L · (C_{A,i} - C_{A,b}).Where N_A is the interfacial molar flux of A into the liquid, k_L is the physical liquid-side mass-transfer coefficient for A, C_{A,i} is the interfacial concentration of A in the liquid (often from equilibrium with the gas), C_{A,b} is the bulk concentration of A in the liquid, and E is the enhancement factor caused by chemical reaction.
When E = 1, the situation reduces to physical absorption with no meaningful reaction effect in the mass-transfer zone.
2.2 Pseudo-first-order reduction for A + B reactions.
Many practical absorptions involve an irreversible reaction A + νB → products in which B is present at sufficiently high concentration that it is nearly constant throughout the liquid film.
Under that condition, the intrinsic rate can be approximated as pseudo-first-order in A.
r_A = k_2 C_A C_B ≈ k' C_A, k' = k_2 C_{B,b}.This reduction is the most common entry point for Hatta number regime selection because it produces a single effective rate constant k' that can be compared to diffusion and mass transfer.
Note : Do not apply the pseudo-first-order form unless B is effectively in excess in the mass-transfer zone, meaning B does not significantly deplete within the film during absorption. If B depletes, you must switch to a two-reactant formulation and use the instantaneous-reaction limit or a full numerical solution depending on the severity of depletion.
3. Hatta number definitions you will actually use.
3.1 Pseudo-first-order Hatta number (most common in practice).
For pseudo-first-order reaction in the liquid film, a widely used definition is.
Ha = sqrt(k' D_A) / k_L.Where D_A is the diffusivity of A in the liquid.
Interpretation is direct because sqrt(k' D_A) has dimensions of velocity and represents the characteristic “reaction–diffusion” speed in the film, while k_L represents the physical mass-transfer speed.
3.2 Second-order Hatta number written in measurable quantities.
For the irreversible second-order form r_A = k_2 C_A C_B with B approximately constant at C_{B,b}, the squared Hatta number is often written as.
Ha^2 = (k_2 C_{B,b} D_A) / k_L^2.This is equivalent to the pseudo-first-order definition by recognizing k' = k_2 C_{B,b}.
4. Regime map: slow, intermediate, fast, and instantaneous.
The practical regime boundaries used in design are intentionally simple because they are meant for fast, defensible decisions rather than perfect classification.
| Regime label. | Typical criterion. | Where reaction mainly occurs. | What limits the overall rate. | Engineering consequence for E. |
|---|---|---|---|---|
| Slow reaction. | Ha < 0.3. | Mostly in the bulk liquid, not in the film. | Liquid-side diffusion behaves like physical absorption. | E ≈ 1. |
| Intermediate reaction. | 0.3 ≤ Ha ≤ 3. | Both in the film and in the bulk. | Coupled diffusion and reaction in the film. | Use a full E(Ha) expression. |
| Fast reaction (finite rate). | Ha > 3 and Ha < E∞. | Primarily in the film. | Diffusion with strong consumption in the film. | E ≈ Ha (pseudo-first-order fast limit). |
| Instantaneous reaction (diffusion-controlled by B supply). | Ha ≥ E∞. | At a reaction plane inside the film. | Delivery of reactant B to the reaction plane. | E ≈ E∞ (upper bound). |
5. Enhancement factor formulas for regime selection.
5.1 Intermediate regime: use the full pseudo-first-order enhancement.
For pseudo-first-order reaction, a widely used closed-form relationship is.
E = Ha / tanh(Ha).This expression has the correct limiting behavior, because as Ha → 0, tanh(Ha) → Ha and E → 1, and as Ha becomes large, tanh(Ha) → 1 and E → Ha.
In engineering workflows, this single relationship can cover slow, intermediate, and fast finite-rate regimes if pseudo-first-order assumptions remain valid, but it must still be capped by the instantaneous limit E∞ when B supply becomes limiting.
5.2 Instantaneous upper bound: the infinite enhancement factor.
The enhancement factor cannot increase without bound because the reaction ultimately requires a finite diffusive supply of the liquid reactant B to the zone where A is being consumed.
A common film-theory expression for the infinite enhancement factor for A + νB → products is.
E∞ = 1 + (D_B C_{B,b}) / (ν D_A C_{A,i}).Where D_B is the diffusivity of B in the liquid and ν is the stoichiometric coefficient of B.
This expression reflects that when reaction is instantaneous, the interfacial flux of A is supported by a counter-diffusion supply of B, and the rate becomes limited by how effectively B can reach the reaction plane in the film.
Note : E∞ requires C_{A,i}, which depends on interfacial equilibrium. If C_{A,i} is estimated incorrectly, the regime boundary Ha ≥ E∞ may be misclassified, and the predicted rate can be wrong even if the rest of the calculations are correct.
6. Step-by-step regime selection workflow you can reuse.
The following workflow avoids common misclassifications and keeps assumptions explicit.
6.1 Inputs to gather.
| Symbol. | Meaning. | Where it comes from in practice. |
|---|---|---|
| k_L. | Physical liquid-side mass-transfer coefficient for A. | Correlations or pilot data for the contacting device. |
| D_A, D_B. | Liquid diffusivities. | Property estimation or literature data for the solvent system. |
| k_2. | Intrinsic second-order rate constant. | Kinetic experiments under relevant temperature and ionic strength. |
| C_{B,b}. | Bulk concentration of B. | Solvent recipe, speciation model, or measured alkalinity. |
| C_{A,i}. | Interfacial concentration of dissolved A. | Equilibrium relationship with gas-side conditions. |
| ν. | Stoichiometric coefficient of B. | Reaction stoichiometry. |
6.2 Computation steps.
Step 1: Decide if pseudo-first-order is defensible. If B is in large excess and does not deplete in the film, set k' = k2 * C_B,b.
Step 2: Compute the Hatta number.
Ha = sqrt(k' * D_A) / k_L.
Step 3: Compute the instantaneous upper bound.
E_inf = 1 + (D_B * C_B,b) / (nu * D_A * C_A,i).
Step 4: Select regime and E.
If Ha < 0.3:
E = 1.
Else if 0.3 <= Ha <= 3:
E = Ha / tanh(Ha).
Else:
E_candidate = Ha / tanh(Ha) (or approximately Ha for very large Ha).
E = min(E_candidate, E_inf).
Step 5: Compute flux.
N_A = E * k_L * (C_A,i - C_A,b).7. Interpreting results and diagnosing “wrong-regime” symptoms.
7.1 What the regime tells you physically.
If Ha is small, increasing kinetics will not noticeably increase absorption because the reaction mostly happens after A reaches the bulk, so k_L and interfacial driving force dominate.
If Ha is moderate, both mixing and kinetics matter, and E is sensitive to changes in k_2, temperature, ionic strength, and solvent composition.
If Ha is large but still below E∞, the rate becomes strongly “reaction-enhanced,” and improvements often come from increasing k_L or increasing the effective reaction rate k'.
If Ha exceeds E∞, increasing kinetic rate constants no longer increases absorption, because the limiting factor becomes the supply of B to the reaction plane and the interfacial equilibrium that sets C_{A,i}.
7.2 Common misclassifications and how to correct them.
| Symptom in calculations or plant data. | Likely cause. | Correction action. |
|---|---|---|
| Predicted rate rises without bound as kinetics increase. | E∞ cap not applied or C_{A,i} omitted. | Compute E∞ and cap E, and ensure interfacial equilibrium is modeled. |
| Pseudo-first-order predicts high E, but measured rate is much lower. | B depletes in the film or parallel equilibria reduce reactive B. | Use effective C_{B,b} from speciation and consider instantaneous or two-reactant analysis. |
| Rate insensitive to agitation although k_L should improve. | Gas-side resistance or equilibrium limitation dominates, not liquid-side diffusion. | Re-check two-film resistances and interfacial driving force assumptions. |
| Regime flips with small parameter changes. | Ha near boundaries and uncertainties in k_L or k_2 are large. | Perform sensitivity on k_L, k_2, and C_{B,b}, and classify with ranges. |
8. Practical tips for engineering use in absorbers and scrubbers.
Use regime selection to guide what data to invest in, because the limiting mechanism dictates which measurement uncertainty matters most.
In slow regimes, prioritize k_L correlation validity, hydrodynamics, and interfacial area, because kinetics will not rescue poor contacting.
In intermediate regimes, prioritize both kinetics and mass transfer characterization, because E responds to both and the design margin is sensitive to parameter drift.
In fast or instantaneous regimes, prioritize accurate equilibrium at the interface and liquid composition, because the rate is often capped by E∞ and speciation-controlled availability of B.
Note : If you use electrolyte solvents, the “bulk concentration” C_{B,b} in k' = k_2 C_{B,b} should represent the kinetically active species, not simply the analytical concentration. When speciation is strong, using the wrong B definition can move Ha across regime boundaries and invalidate E.
FAQ
Is the Hatta number the same as a Damköhler number.
It is closely related. In reactive absorption, the Hatta number is commonly treated as the square root of a Damköhler-type ratio comparing reaction in the liquid film to diffusion or mass transfer through that film. Using Ha rather than a Damköhler number is convenient because the most useful enhancement factor formulas are naturally written in terms of Ha.
Why are the boundaries often written as 0.3 and 3.
They are practical engineering thresholds that separate clearly distinct limiting behaviors. Below the lower threshold, reaction in the film is negligible and E is effectively unity. Above the upper threshold, reaction in the film is strong enough that the enhancement approaches the fast-reaction limit. Between them, the full coupled expression E(Ha) is needed because neither limiting approximation is reliable.
When should I cap E using E∞.
You should cap E whenever the reaction consumes a liquid reactant B that must diffuse to the reaction zone, because the supply of B imposes a maximum achievable enhancement. A practical rule is to compute both Ha and E∞ and use E = min(Ha / tanh(Ha), E∞), provided the underlying assumptions match your chemistry and stoichiometry.
Can I use the same approach for reversible reactions or multiple reactions.
The concept of regime selection remains valid, but E(Ha) formulas may change because equilibrium, parallel pathways, or regeneration of reactants can reshape concentration profiles in the film. For complex reaction networks, you typically define an effective pseudo-first-order rate constant for the controlling consumption pathway or use generalized enhancement models or numerical film calculations.
What is the quickest way to avoid a wrong regime choice.
Compute both Ha and E∞ using consistent definitions of concentrations and diffusivities, then apply the slow threshold Ha < 0.3, the intermediate form E = Ha / tanh(Ha), and the cap E ≤ E∞. If your predicted E is very sensitive to uncertain parameters, classify the regime using ranges rather than a single point estimate.