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# 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 n-th order, the Damköhler number is defined as

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

where:

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

Generally, if Da<0.1 then X<0.1 similarly if Da>10 then X>0.9

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

reaction rate / convective mass transport rate or characteristic time/reaction time

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

$Da = \frac{k_cC_0^n}{C_0/\tau} = k_cC_0^{(n-1)}\tau$

where τ is the mean residence time or space time.

In reacting systems that include also interphase mass transport, the second Damköhler number (DaII) 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

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