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 \(X>0.9\).

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 \(\tau\) is the mean residence time or space time.

In reacting systems that include also interphase mass transport, the second Damköhler number (\(Da_{\mathrm{II}}\)) 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

Related Links

Dimensionless quantity

References

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