:Hatta number

{{Short description|Dimensionless number in chemistry}}

The Hatta number (Ha) was developed by Shirôji Hatta (1895-1973 {{Cite book |last1=Bird |first1=R. Byron |title=Transport phenomena |last2=Stewart |first2=Warren E. |last3=Lightfoot |first3=Edwin N. |date=2002 |publisher=J. Wiley |isbn=978-0-471-41077-5 |edition=2nd |location=New York |pages=696}}) in 1932,{{Cite book |last=Conesa |first=Juan A. |url=http://dx.doi.org/10.1002/9783527823376 |title=Chemical Reactor Design |date=2019-09-06 |publisher=Wiley |doi=10.1002/9783527823376 |isbn=978-3-527-34630-1}} who taught at Tohoku University from 1925 to 1958.S. Hatta, Technological Reports of Tôhoku University, 10, 613-622 (1932). It is a dimensionless parameter that compares the rate of reaction in a liquid film to the rate of diffusion through the film.R.B. Bird, W.E. Stewart, E.N. Lightfoot, Transport Phenomena, 2nd ed. John Wiley & Sons, 2002 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 ({{math|1=r_A = k₂C_BC_A}}) Hatta is defined via:

$Ha^2\; =\; \{\{k\_\{2\}\; C\_\{A,i\}\; C\_\{B,bulk\}\; \backslash delta\_L\}\; \backslash over\; \{\backslash frac\{D\_A\}\{\backslash delta\_L\}\backslash \; C\_\{A,i\}\}\}\; =\; \{\{k\_2\; C\_\{B,bulk\}\; D\_A\}\; \backslash over\; (\{\backslash frac\{D\_A\}\{\backslash delta\_L\}\})\; ^2\}\; =\; \{\{k\_2\; C\_\{B,bulk\}\; D\_A\}\; \backslash over\; \{\{k\_L\}\; ^2\}\}$

For a reaction {{math|m^th}} order in {{math|A}} and {{math|n^th}} order in {{math|B}}:

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

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.{{Cite book |last=Ramachandran |first=P. A. |title=Advanced transport phenomena: analysis, modeling and computations |date=2014 |publisher=Cambridge University Press |isbn=978-0-521-76261-8 |location=Cambridge |pages=369}}

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 {{math|(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 {{math|k_L}} and the Hatta number: {{math|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.