# Hatta number

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The **Hatta number** (**Ha**) was developed by Shirôji Hatta (1895-1973 [1]) in 1932,[2][3] who taught at [Tohoku University](/source/Tohoku_University) from 1925 to 1958.[1][2] It is a dimensionless parameter that compares the rate of reaction in a liquid film to the rate of diffusion through the film.[4] It is related to one of the many [Damköhler numbers](/source/Damk%C3%B6hler_numbers), Hatta being the square root of such a Damköhler number of the second type. Conceptually the Hatta number bears strong resemblance to the [Thiele modulus](/source/Thiele_modulus) for diffusion limitations in porous catalysts, which also is the square root of a Damköhler number. For a second order reaction (*r*A = *k*2*C*B*C*A) Hatta is defined via:

Ha^2 = {{k_{2} C_{A,i} C_{B,bulk} \delta_L} \over {\frac{D_A}{\delta_L}\ C_{A,i}}} = {{k_2 C_{B,bulk} D_A} \over ({\frac{D_A}{\delta_L}}) ^2} = {{k_2 C_{B,bulk} D_A} \over {{k_L} ^2}}

For a reaction *m*th order in *A* and *n*th order in *B*:

Ha = {{ \sqrt{{\frac{2}{{m} + 1}}k_{m,n} {C_{A,i}}^{m - 1} C_{B,bulk}^n {D}_A}} \over {{k}_L}}

For gas-liquid absorption with chemical reactions, a high Hatta number indicates the reaction is much faster than diffusion, usually referred to as the "fast reaction" or "chemically enhanced" regime. In this case, the reaction occurs within a thin (hypothetical) film, and the surface area and the Hatta number itself limit the overall rate.[5]

For Ha>2, with a large excess of B, the maximum rate of reaction assumes that the liquid film is saturated with gas at the interfacial (*C*A,i) and that the bulk concentration of A remains zero; the flux and hence the rate of reaction becomes proportional to the mass transfer coefficient *k*L and the Hatta number: *k*L*C*A,i*Ha*.

Conversely, a Hatta number smaller than unity suggests the reaction is the limiting factor, and the reaction takes place in the bulk fluid; the concentration of A needs to be calculated taking the mass transfer limitation - without enhancement - into account.[5]

## References

1. Bird, R. Byron; Stewart, Warren E.; Lightfoot, Edwin N. (2002). *Transport phenomena*. 2nd ed. New York: J. Wiley. p. 696. ISBN 978-0-471-41077-5.

1. S. Hatta, Technological Reports of Tôhoku University, 10, 613-622 (1932).

1. Conesa, Juan A. (2019-09-06). [*Chemical Reactor Design*](http://dx.doi.org/10.1002/9783527823376). Wiley. [doi:10.1002/9783527823376](https://doi.org/10.1002/9783527823376). ISBN 978-3-527-34630-1.

1. [R.B. Bird](/source/Robert_Byron_Bird), W.E. Stewart, [E.N. Lightfoot](/source/Edwin_N._Lightfoot), Transport Phenomena, 2nd ed. John Wiley & Sons, 2002

1. Ramachandran, P. A. (2014). *Advanced transport phenomena: analysis, modeling and computations*. Cambridge: Cambridge University Press. p. 369. ISBN 978-0-521-76261-8.

## See also

- [Dimensionless quantity](/source/Dimensionless_quantity)
- [Dimensional analysis](/source/Dimensional_analysis)

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Adapted from the Wikipedia article [Hatta number](https://en.wikipedia.org/wiki/Hatta_number) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Hatta_number?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.