Hatta number

Template:Short description The Hatta number (Ha) was developed by Shirôji Hatta (1895-1973 ) in 1932, who taught at Tohoku University from 1925 to 1958. It is a dimensionless parameter that compares the rate of reaction in a liquid film to the rate of diffusion through the film. 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:

$H{a}^2=\frac{k_2{C}_{A,i}{C}_{B,bulk}{\delta }_L}{\frac{D_A}{{\delta }_L}{C}_{A,i}}=\frac{k_2{C}_{B,bulk}{D}_A}{(\frac{D_A}{{\delta }_L}{)}^2}=\frac{k_2{C}_{B,bulk}{D}_A}{{k_L}^2}$

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

$Ha=\frac{\sqrt{\frac{2}{m+1}{k}_{m,n}{C_{A,i}}^{m-1}{C}_{B,bulk}^n{D}_A}}{k_L}$

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.

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.

See also

• Dimensionless quantity
• Dimensional analysis

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