Hatta number

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. For a second order reaction ($r_{A} = k_{2}C_{B}C_{A}$), 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 $k_{2}C_{B,bulk}C_{A,i}δ_{L}$.

$$Ha^2 = {{k_{2} C_{A,i} C_{B,bulk} \delta_L} \over {\frac{D_A}{\delta_L}\ C_{A,i}}} = {{k_2 C_{B,bulk} D_A} \over ({\frac{D_A}{\delta_L}}) ^2} = {{k_2 C_{B,bulk} D_A} \over {{k_L} ^2}}$$

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. 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.