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] It is related to one of the many Damköhler 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 for diffusion limitations in porous catalysts, which also is the square root of a Damköhler number. For a second order reaction (r_A = k₂C_BC_A) Hatta is defined via:

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

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

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

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_LC_A,iHa.

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]