Hatta number

The Hatta number ( Ha ) is a dimensionless number from the field of chemical macrokinetics . It describes the interaction of mass transport phenomena and pure reaction kinetics ( microkinetics ) for liquid-liquid or gas - liquid systems. The counterpart to the Hatta number from heterogeneous catalysis (gas-solid or liquid-solid) is the Thiele module .

In general, the Hatta number is defined as

{\ displaystyle Ha = {\ frac {\ text {Reaction speed without mass transfer inhibition}} {\ text {pure mass transfer speed}}}}

.

The following definition is often found in the literature:

{\ displaystyle Ha = \ delta {\ sqrt {\ frac {k_ {mn} C_ {1} ^ {* n-1}} {D_ {A}}}}}

Or in more detail:

{\ displaystyle Ha = {\ frac {1} {k_ {L}}} {\ sqrt {{\ frac {2} {m + 1}} D_ {A} k_ {mn} C_ {A} ^ {* m -1} C_ {B_ {0}} ^ {n}}}}

With

{\ displaystyle k_ {L} = {\ frac {D_ {A}} {\ delta}}}

For a simple first-order reaction , the Hatta number is simplified to the often used, shortened form:

{\ displaystyle Ha = \ delta {\ sqrt {\ frac {k} {D_ {A}}}}}

,

${\ displaystyle \ delta}$ Layer thickness of the boundary layer,
${\ displaystyle k_ {mn}}$ Rate constant of the reaction,
${\ displaystyle k_ {L}}$ Mass transfer rate of component A
${\ displaystyle C_ {A} ^ {*}}$ Concentration of component A at the phase interface
${\ displaystyle m}$ Reaction order of component A.
${\ displaystyle n}$ Reaction order of component B
${\ displaystyle D_ {A}}$ Diffusion coefficient of component A in the fluid phase (explanation below in the text).

Classification of multiphase reactions

The Hatta number can be used to classify reactions that are preceded by material transport processes or that are coupled with material transport processes. Examples are chemical absorption or multiphase reactions.

First, consider a boundary layer between two phases (liquid-liquid or gas-liquid). The concentration profile of the diffusing component can be described with the two-film model (according to Lewis and Whitman).

The Hatta number now indicates the ratio of the reaction speed in the reaction phase to the mass transport speed through the phase boundary into the reaction phase.

There are several cases:

Slow response

This case applies to a value of the Hatta number less than 0.3.

The reaction is much slower than the mass transfer. The reaction only takes place in the reaction phase.

The reaction has no interaction with the mass transfer.

Medium response

This case applies to a Hatta number between 0.3 and 3.

The reaction is about as fast as the mass transfer. Part of the diffusing component already reacts in the boundary layer.

Fast reaction

This case applies to a Hatta number greater than 3.

The reaction is faster than the mass transfer. The diffusing component already reacts in the boundary layer.

This means that the reaction is constantly removing the passing component from the boundary film. The concentration profile through the film is no longer straight, but curved. This leads to an acceleration of the mass transfer compared to the non-reactive mass transfer.

The corresponding acceleration factor, called the gain factor, has the value

{\ displaystyle E = {\ frac {Ha} {\ tanh Ha}}}

Instant response

This is the case with Ha≫3

The reaction is much faster than the mass transfer. The diffusing component reacts shortly after entering the boundary layer within a plane parallel to the surface. The maximum achievable reaction speed is limited by the diffusion of the liquid reaction partner to the reaction level. In the vicinity of this level, the reactant concentrations are low and the product concentrations are high.

literature

Heinz M. Hiersig : Lexicon production engineering, process engineering . Springer, 1995, ISBN 3-18-401373-1 , p. 431. ( Google Books )
E. Fitzer, W. Fritz, G. Emig: Technical chemistry, introduction to chemical reaction engineering. Springer-Verlag, Berlin / Heidelberg 1995.
M. Baerns, H. Hofmann, A. Renken: Chemical reaction engineering. 2nd Edition. Georg-Thieme-Verlag, Stuttgart 1987.
LK Doraiswamy, MM Shrama: Heterogeneous Reactions: Analysis, Examples and Reactor Design. Volume 2, John Wiley & Sons, New York et al. 1984.