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 (rA = k2CBCA) 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 mth order in A and nth 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 (CA,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 kL and the Hatta number: kLCA,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]

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.
  2. ^ S. Hatta, Technological Reports of Tôhoku University, 10, 613-622 (1932).
  3. ^ Conesa, Juan A. (2019-09-06). Chemical Reactor Design. Wiley. doi:10.1002/9783527823376. ISBN 978-3-527-34630-1.
  4. ^ R.B. Bird, W.E. Stewart, E.N. Lightfoot, Transport Phenomena, 2nd ed. John Wiley & Sons, 2002
  5. ^ 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