Hatta number

The Hatta number (Ha) was developed by Shirôji Hatta (1895-1973 ^[1]) in 1932,^[2]^[3] who taught at 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] For a second order reaction (r_A = k₂C_BC_A), the maximum rate of reaction assumes that the liquid film is saturated with gas at the interfacial concentration (C_A,i); thus, the maximum rate of reaction is k₂C_B,bulkC_A,iδ_L.

$Ha^{2}={{k_{2}C_{A,i}C_{B,bulk}\delta _{L}} \over {{\frac {D_{A}}{\delta _{L}}}\ C_{A,i}}}={{k_{2}C_{B,bulk}D_{A}} \over ({\frac {D_{A}}{\delta _{L}}})^{2}}={{k_{2}C_{B,bulk}D_{A}} \over {{k_{L}}^{2}}}$

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. In this case, the reaction occurs within a thin film, and the surface area limits the overall rate.^[5] Conversely, a Hatta number smaller than unity suggests the reaction is the limiting factor, and the reaction takes place in the bulk fluid, requiring larger volumes.^[5]

References

^ ^a ^b 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.
^ ^a ^b S. Hatta, Technological Reports of Tôhoku University, 10, 613-622 (1932).
^ Conesa, Juan A. (2019-09-06). Chemical Reactor Design. Wiley. ISBN 978-3-527-34630-1.
^ R.B. Bird, W.E. Stewart, E.N. Lightfoot, Transport Phenomena, 2nd ed. John Wiley & Sons, 2002
^ ^a ^b Ramachandran, P. A. (2014). Advanced transport phenomena: analysis, modeling and computations. Cambridge: Cambridge University Press. p. 369. ISBN 978-0-521-76261-8.