Hatta number¶
Dimensionless comparison of liquid-phase reaction with liquid-side transport in a specified reactive gas–liquid absorption model.
Core Idea¶
The Hatta number (\(\mathrm{Ha}\)) compares reaction of an absorbed gas in the liquid with liquid-side transport in a specified gas–liquid contact and kinetic model. In a classical film treatment, \(\mathrm{Ha}^2\) expresses reaction capacity in the liquid film relative to transport through it. The comparison is dimensionless: it makes a kinetic timescale and a transport scale jointly interpretable without equating either to an observed absorption flux.[1][2]
For the first-order film form given in Perry's Chemical Engineers' Handbook, \(\mathrm{Ha}=\sqrt{k_1D_A}/k_L^0\). For its second-order reaction treated as pseudo-first-order in absorbed species \(A\), \(\mathrm{Ha}=\sqrt{k_2B^0D_A}/k_L^0\). Here \(D_A\) is the liquid diffusivity of \(A\), \(B^0\) is bulk concentration of liquid reactant \(B\), and \(k_L^0\) is the physical liquid-side mass-transfer coefficient. These equations carry their kinetic and transport assumptions; \(B^0\) is not an input to every Hatta calculation.[2]
A larger \(\mathrm{Ha}\) indicates a stronger reaction influence relative to transport within the declared model. It does not by itself report the absorption enhancement factor \(E\) or establish that the reaction is instantaneous. The available reactant, reaction law, and limiting enhancement \(E_\infty\) can change that interpretation.[3]
Structural Signature¶
- Absorbing species and reactive liquid system. Identify the gas entering the liquid and the liquid chemistry that consumes it. Without this pair, there is no reactive gas–liquid Hatta comparison.
- Declared liquid-side transport model. Specify the film or other contact-model convention and its physical transport reference. A coefficient from an incompatible model cannot silently replace \(k_L^0\) in a film expression.[2][3]
- Absorbed-species diffusivity. An explicit film calculation uses the liquid diffusivity of the gas species. A different model must state how its transport scale is constructed.[2]
- Specified kinetic law and conditional inputs. Reaction order and rate convention determine the formula. A first-order form uses \(k_1\); the shown pseudo-first-order second-order form also needs \(B^0\). Other reaction laws can need additional concentration inputs not present in either simple form.[2][3]
- Dimensionless reaction-to-transport calculation. Combine those declared inputs into \(\mathrm{Ha}\). A rate constant, diffusivity, or transfer coefficient alone is a component, not the named metric.
- Qualified regime inference. Use \(\mathrm{Ha}\) to assess whether reaction affects the liquid-side process, then consider reactant availability and \(E_\infty\) before inferring an enhancement or a reaction regime. \(E\) remains a separate calculated or measured quantity.[3]
What It Is Not¶
Not the enhancement factor. \(E\) describes reactive absorption relative to physical absorption. In a CO2–DEA membrane-contactor study, \(E\approx\mathrm{Ha}\) only in a particular region where \(E_\infty\) greatly exceeds \(\mathrm{Ha}\) and \(\mathrm{Ha}>1\); the authors do not find equality along the entire reaction path.[3]
Not a universal threshold test. Some film calculations use approximate numerical guides for slow and fast reaction, but those guides are conditioned on the model. Large \(\mathrm{Ha}\) alone does not certify an instantaneous reaction, because finite supply of liquid reactant can limit enhancement.[2][3]
Not a generic reaction rate or gas uptake measurement. A physical oxygen-transfer coefficient, observed flux, or kinetic constant can help construct or interpret the calculation, but none by itself compares reaction with the declared liquid-side transport scale.
Scope of Application¶
The metric is used in reactive gas–liquid mass transfer, including packed CO2 absorption into aqueous MEA,[4] CO2 absorption into DEA in a membrane contactor,[3] and oxygen absorption during S(IV) oxidation in a bubbling apparatus.[5] These cases differ in gas, liquid reactant, contactor, and engineering decision. Their common content is the declared comparison of reaction with liquid-side transport.
This scope does not make every reported value directly comparable. A film coefficient, a penetration-model parameter, and a third-order reaction expression may place different inputs into a Hatta calculation. Before comparing systems, state which model was used, which inputs were measured or assumed, and what range of conditions the inference covers.[3]
Clarity¶
The number separates three questions that are often compressed into “fast absorption”: How reactive is the dissolved gas relative to liquid-side transport? is the Hatta question; how much greater is absorption than physical absorption? is the enhancement question; what was the observed gas uptake? is an experimental flux question. A model may connect all three, but the answers are not interchangeable.[2][3]
It also forces a kinetic qualifier. Replacing \(k_1\) with \(k_2B^0\) is legitimate only under the stated pseudo-first-order treatment. A different reaction law may still yield a Hatta metric, but one cannot copy the simple square-root expression without checking its assumptions.[2][3]
Manages Complexity¶
Within one declared model, \(\mathrm{Ha}\) compresses a reaction constant, liquid diffusion, and physical transfer into one dimensionless comparison. This helps screen whether a calculation can neglect reaction in the liquid film, whether liquid-film detail matters, or whether enhancement must be evaluated more carefully. Kale and colleagues explicitly use Hatta numbers to examine the need for liquid-film discretisation in their CO2–MEA packed-column model.[2][4]
That compression does not eliminate chemistry. The chosen reaction law, solvent concentration, transport coefficient, and availability limit remain necessary to interpret the number. A scalar \(\mathrm{Ha}\) without those assumptions can conceal the very regime distinction it was meant to expose.[3]
Abstract Reasoning¶
First identify the absorbed species and liquid reaction. Next fix the contact model and the physical liquid-side transfer reference. Then select the kinetic expression and only its required concentration inputs. Calculate \(\mathrm{Ha}\) under that convention. Finally, ask what inference the particular model allows, including whether a finite \(E_\infty\) limits the expected enhancement.[2][3]
For a first-order film model, increasing \(k_1\) at fixed \(D_A\) and \(k_L^0\) increases \(\mathrm{Ha}\); increasing the physical transfer coefficient decreases it. This is a consequence of the conditional formula, not a universal prediction about observed flux. Changing the contact model or allowing the liquid reactant to deplete requires a renewed calculation and regime check.[2][3]
Knowledge Transfer¶
Literal transfer occurs between different reactive absorption systems when each system supplies its own kinetic and liquid-side transport inputs. The CO2–MEA packed model uses the number for film discretisation;[4] the O2–S(IV) bubbling study uses it to check a proposed rapid-reaction regime in its oxidation model.[5] They use the same kind of reaction/transport comparison without sharing a reagent or vessel; their exact formulas are not assumed to be identical.
Outside reactive gas–liquid absorption, other dimensionless reaction-versus-transport comparisons can be useful analogies. The Hatta name and the film formulas do not automatically transfer to a catalyst pellet, a purely homogeneous reaction, or a general planning bottleneck. Those settings require their own defined metric and governing scales.
Examples¶
CO2 absorption into DEA in a membrane contactor¶
Pantoleontos and colleagues model CO2 entering an aqueous DEA shell-side solution during biogas upgrading. They use a resistance-in-series film model (\(n=0\)); a penetration-model form (\(n=1/2\)) is discussed as an alternative. Their reaction treatment uses a third-order expression, not the first-order handbook formula. The study calculates Hatta number and enhancement separately and finds that equating them along the entire shell path misstates the mass-transfer behavior.[3]
Mapped back: CO2 and DEA supply the reacting pair; the adopted shell-side film model supplies the transport reference; dissolved CO2 diffusion and the study's third-order kinetic law supply model inputs; the calculated \(\mathrm{Ha}\) is compared with transport; the regime judgment is checked against \(E\) and \(E_\infty\). The published equations are image-rendered in the accessible full HTML, so no case-specific symbolic formula is transcribed here.
O2 absorption during S(IV) oxidation¶
Chen and colleagues studied oxygen absorption from gas bubbles during oxidation of S(IV) in basic aluminum sulfate solution. They modeled intrinsic reaction and oxygen absorption as separate steps, used a liquid-side resistance and diffusivity model, and wrote their Hatta number as the positive square root of a kinetic/diffusive term divided by the liquid mass-transfer coefficient (their Eq. 9). They estimated Ha near 1.4 and used it to check their proposed rapid-reaction regime; that numerical judgment belongs to their model and conditions.[5]
Mapped back: O2 and dissolved S(IV) supply the reacting pair; the bubbling apparatus supplies the gas–liquid contact setting; liquid-side transfer coefficient, O2 diffusivity, reaction constant, and concentration terms supply the model's declared inputs. Its Ha calculation contains an ordered quotient with a nonzero transport denominator. The paper's enhancement expression and its overall oxidation rate are separate from that calculated number.[5]
Additional packed-absorber use¶
Kale and colleagues report a rate-based CO2–aqueous MEA model checked against pilot-scale packed-column data. Its abstract and highlights say Hatta numbers were used to assess whether liquid-film discretisation was needed for the studied operating window.[4] This is a third use, but its accessible publisher text does not verify the particular kinetic equation, diffusivity, transfer-coefficient convention, or Ha values.
Structural Tensions¶
No intrinsic opposed-pressure tradeoff is established for the metric itself. The CO2–DEA study's finding that \(E\) and \(\mathrm{Ha}\) differ along its reaction path is a model-sufficiency limit, handled by the interpretation test above rather than labeled a universal tension.[3]
Structural–Framed Character¶
The Hatta number is structural within chemical reaction engineering because a declared comparison of reaction with liquid-side transport persists across different chemicals and contactors. Its evaluative weight is conditional: high or low does not itself mean good or bad design. Human-practice dependence enters through model choice, parameter estimation, and the decision the number informs, while its mathematical value follows from the chosen inputs. The institutional origin of the name is conventional, but current use does not depend on one named plant or historical experiment. Its vocabulary travels literally across packed columns, membrane contactors, and bubble columns only while reactive gas–liquid roles are preserved. For import versus recognition, a new case is recognized by documenting its reaction, transport reference, and calculation; applying “Hatta” to an unrelated slow-versus-fast comparison merely borrows the label. Its character: a model-qualified, domain-specific reaction/transport index with portable use inside reactive absorption and guarded interpretation outside it.
Structural Core vs. Domain Accent¶
The broad skeletal idea is a dimensionless comparison of competing physical processes. The domain-bound mechanism is narrower: a dissolved gas reacts in liquid while it is transported from an interface, and the chosen kinetic and transfer model determines the evaluable Hatta form. That chemical and transport specificity is what the named metric adds; removing it leaves only the general possibility of constructing dimensionless groups.
Dimensionless Quantity supplies the broad idea of a physical comparison invariant under coherent unit changes. Mass Transfer names the wider species-transport process, and Diffusion names one movement mechanism. The Hatta number's specific kinetic and interfacial transport relation remains a chemical-engineering residual; none of these broader notions supplies its conditional formula by itself.
Future-Prime question: Could a reaction-versus-transport comparison retain the same constitutive roles and inference in two independently attested domains that do not involve a gas entering a reactive liquid? A genuine broader Prime would need native cases and a shared test that survives changed physical carriers, not just the observation that both fields can divide one timescale by another. The current evidence establishes literal Hatta transfer among reactive gas–liquid systems; elsewhere it supports analogy only.
Instantiates / Related Primes¶
This entry is part of Ratio.
Dimensionless Quantity describes the broader physical-quantity property that \(\mathrm{Ha}\) has dimension one and retains its numerical value under coherent changes of base units. It does not identify the reacting species, rate law, transport reference, or regime interpretation. Mass Transfer concerns movement of a species between locations or phases; the Hatta number compares one such liquid-side process with reaction, without measuring total transferred mass. Diffusion is a transport mechanism and contributes a model input, but a diffusivity alone does not instantiate the full reaction-versus-transport index.
Relationships to Other Abstractions¶
Current abstraction Hatta number Domain-specific
Parents (1) — more general patterns this builds on
-
Hatta number is part of Ratio Prime
An ordered reaction-to-transport ratio is an internal constituent of the model-defined Hatta calculation.A declared reactive gas–liquid model supplies a kinetic/diffusive reaction term and a nonzero physical liquid-side transfer reference. Their aligned ordered quotient is internal to the Hatta calculation; its positive root is reported as Ha. First-order and pseudo-first-order film forms in Perry and the unlike O2–S(IV) bubbling model in Chen et al. each explicitly contain this quotient, while differing in reagents and kinetic inputs. Ratio and its Comparison ancestor supply order, division, aligned scope and relative magnitude; the Hatta identity adds the liquid-side reactive-absorption setting, conditional kinetic law and qualified regime interpretation. Removing the quotient destroys the named comparison. A bare Ratio can exist without Hatta, and the Ratio is inside this calculation rather than an external precondition.
Hierarchy path (1) — routes to 1 parentless root
- Hatta number → Ratio → Comparison
Neighborhood in Abstraction Space¶
Hatta number sits in a sparse region of the domain-specific corpus (94th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Unclustered & Miscellaneous (2721 abstractions)
Nearest neighbors
- Mass transfer — 0.81
- Equimolar Counterdiffusion — 0.79
- Variational Transition-State Theory — 0.78
- Ideal Scalar Transmission Conditions — 0.77
- Molecularity — 0.77
Computed from structural-signature embeddings · 2026-10-17
Not to Be Confused With¶
- Enhancement factor \(E\): a ratio of reactive to physical absorption behavior, not the input reaction-versus-transport index. Their approximate equality has a restricted regime.[3]
- An instantaneous-reaction criterion: a large \(\mathrm{Ha}\) without reactant-availability conditions does not establish instantaneous behavior.[2][3]
- A physical mass-transfer coefficient \(k_L^0\): a dimensional transport input, not the dimensionless comparison.[2]
- A universal second-order formula: \(\sqrt{k_2B^0D_A}/k_L^0\) belongs to the stated pseudo-first-order treatment; other laws and models need their own expressions.[2][3]
References¶
[1] Alexei A. Lapkin and Pawel K. Plucinski, “Engineering Factors for Efficient Flow Processes in Chemical Industries,” in Chemical Reactions and Processes under Flow Conditions (Royal Society of Chemistry, 2009), §1.2.2, doi:10.1039/9781847559739-00001. https://books.rsc.org/books/edited-volume/165/chapter/88117/Engineering-Factors-for-Efficient-Flow-Processes . Engineering chapter, full HTML inspected; its displayed Table 1.1 has inconsistent low-Ha rows, so no table threshold is used here. registry ↩
[2] James R. Fair et al., “Gas Absorption and Gas–Liquid System Design,” §14 in Perry's Chemical Engineers' Handbook, 7th ed. (McGraw-Hill, 1997), printed pp.14-18, 14-21–14-22, Eqs. (14-65), (14-79), (14-83), Fig. 14-14. https://schneider.cheme.cmu.edu/Files/Perry%27s%20Chemical%20Engineers%27%20Handbook%207th%20Edition/14.%20Gas%20Absorption%20and%20Gas-Liquid%20System%20Design.pdf . Authoritative full chapter inspected; Eq. (14-79) gives an enhanced transfer coefficient, while (14-65) and (14-83) give conditional Ha forms. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j ↩k ↩l ↩m ↩n
[3] Grigorios Pantoleontos et al., “Modeling, Simulation, and Membrane Wetting Estimation in Gas–Liquid Contacting Processes Including Shell-Side Reaction: Biogas Upgrading Using DEA Solution,” Industrial & Engineering Chemistry Research 63 (2024): 9185–9202, doi:10.1021/acs.iecr.4c00525. https://pmc.ncbi.nlm.nih.gov/articles/PMC11190974/ . Original full HTML inspected, especially §3.3, Fig. 2 and results around Fig. 11; equations are image-rendered. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j ↩k ↩l ↩m ↩n ↩o ↩p ↩q ↩r
[4] Chinmay Kale et al., “Modelling of the Reactive Absorption of CO2 Using Mono-Ethanolamine,” International Journal of Greenhouse Gas Control 17 (2013): 294–308, doi:10.1016/j.ijggc.2013.05.019. https://www.sciencedirect.com/science/article/pii/S1750583613002351 . Original publisher abstract and highlights inspected; full methods and numerical Ha values not verified. registry ↩a ↩b ↩c ↩d
[5] M. Chen, X. H. Deng, and F. Q. He, “Study on the Kinetics of S(IV) Oxidation in the Basic Aluminum Sulfate Wet Flue Gas Desulfurization Process,” RSC Advances 7 (2017): 39341–39348, doi:10.1039/C7RA07518A. https://pubs.rsc.org/en/content/articlepdf/2017/ra/c7ra07518a . Original full paper inspected, especially §4.1–4.3, Eqs. (6)–(13), and the conclusion. Its Eq. (9) is used only for the paper's stated liquid-side model, not as a universal Hatta formula. registry ↩a ↩b ↩c ↩d