Damkohler Numbers

The Damköhler Numbers (Da) are dimensionless quantities used to relate reaction rate to mass transport. Damköhler numbers give an idea of which process may dominate in driving the chemical distribution profile. The concept was introduced by Gerhard Damköhler (1908-1944). The mass transport can be through convection or diffusion.

The Damköhler number is defined as

$D a = \frac{reaction rate}{mass transport}$

(1)

Damköhler numbers frequently show up in continuous flow irreversible reactions such as those occurring in continuously stirred tanks (CSTRs) as well as in transport processes such bubblers and aerators.

For bubbler/aerators the Damköhler number is given by:

$D a = \frac{k_{1}}{a_{c} k_{c}}$

(2)

where

k₁ is the kinetic reaction rate constant
k_c is the mass transfer coefficient
a_c is the interfacial area (of the bubble)

For reactions the Damköler number is given by:

First order irreversible reaction:

$D a = \frac{- r_{A 0} V}{F_{A 0}} = \frac{k_{1} C_{A 0} V}{v_{0} C_{A 0}} = τ k_{1}$

(3)

Second order irreversible reaction:

$D a = \frac{- r_{A 0} V}{F_{A 0}} = \frac{k_{2} C_{A 0}^{2} V}{v_{0} C_{A 0}} = τ k_{2} C_{A 0}$

(4)

where

r_A0 is the reaction rate of A
F_A0 is the flow of A into the reactor
k₁ is the first order kinetic reaction rate constant
k₂ is the second order kinetic reaction rate constant
V is the volume of the reactor
$v_{0}$ is the initial volumetric flow rate
C_A0 is the initial Concentration of A
$τ$ is the residence time in the reactor or space time $\frac{V}{v_{0}}$

References

Fogler Scott (2016) Elements of Chemical Reaction Engineering (5th ed) p 154-155. Pearson Education. ISBN 978-0-13-388751-8