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 $\tau$ 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

References

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

Dimensionless numbers in fluid dynamics

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Damköhler numbers

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