Damköhler numbers

The Damköhler numbers (Da) are dimensionless numbers used in chemical engineering to relate chemical reaction timescale to other phenomena occurring in a system. It is named after German chemist Gerhard Damköhler (1908–1944).

There are several Damköhler numbers, and their definition varies according to the system under consideration.

For a general chemical reaction A → B of nth order, the Damköhler number is defined as

$Da = k C_0^{\ n-1} t$

where:

• k = kinetics reaction rate constant
• C₀ = initial concentration
• n = reaction order
• t = time

and it represents a dimensionless reaction time. It provides a quick estimate of the degree of conversion (X) that can be achieved in continuous flow reactors.

Generally, if Da < 0.1, then X < 0.1. Similarly, if Da > 10, then X > 0.9.^[1]

In continuous or semibatch chemical processes, the general definition of the Damköhler number is:

$Da = \dfrac{ \mbox{reaction rate} }{ \mbox{convective mass transport rate} }$

or as

$Da = \dfrac{ \mbox{characteristic fluid time} }{ \mbox{characteristic chemical reaction time} }$

For example, in a continuous reactor, the Damköhler number is:

$Da = \frac{k_{c}C_{0}^{n}}{C_0/\tau} = k_{c}C_{0}^{(n-1)}\tau$

where τ is the mean residence time or space time.

In reacting systems that include also interphase mass transport, the second Damköhler number (Da_II) is defined as the ratio of the chemical reaction rate to the mass transfer rate

$Da_{\mathrm{II}} = \frac{k C_0^{n-1}}{k_g a}$

where

• k_g is the global mass transport coefficient
• a is the interfacial area

References

1. ^ Fogler, Scott (2006). Elements of Chemical Reaction Engineering (4th ed.). Upper Saddle River, NJ: Pearson Education. ISBN 0130473944.