# Damköhler numbers

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The Damköhler numbers (Da) are dimensionless numbers used in chemical engineering to relate the chemical reaction timescale (reaction rate) to the transport phenomena rate occurring in a system. It is named after German chemist Gerhard Damköhler. The Karlovitz number (Ka) is related to the Damköhler number by Da = 1/Ka.

In its most commonly used form, the Damköhler number relates the reaction timescale to the convection time scale, flow rate, through the reactor for continuous (plug flow or stirred tank) or semibatch chemical processes:

${\displaystyle \mathrm {Da} ={\frac {\text{reaction rate}}{\text{convective mass transport rate}}}}$

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

${\displaystyle \mathrm {Da} _{\mathrm {II} }={\frac {\text{reaction rate}}{\text{diffusive mass transfer rate}}}}$

It is also defined as the ratio of the characteristic fluidic and chemical time scales:

${\displaystyle \mathrm {Da} ={\frac {\text{flow time scale}}{\text{chemical time scale}}}}$

Since the reaction timescale is determined by the reaction rate, the exact formula for the Damköhler number varies according to the rate law equation. For a general chemical reaction A → B of nth order, the Damköhler number for a convective flow system is defined as:

${\displaystyle \mathrm {Da} =kC_{0}^{\ n-1}\tau }$

where:

On the other hand, the second Damköhler number is defined as:

${\displaystyle \mathrm {Da} _{\mathrm {II} }={\frac {kC_{0}^{n-1}}{k_{g}a}}}$

where

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

The value of Da provides a quick estimate of the degree of conversion that can be achieved. As a rule of thumb, when Da is less than 0.1 a conversion of less than 10% is achieved,and when Da is greater than 10 a conversion of more than 90% is expected.[1]

## References

1. ^ Fogler, Scott (2006). Elements of Chemical Reaction Engineering (4th ed.). Upper Saddle River, NJ: Pearson Education. ISBN 0-13-047394-4.