Damköhler numbers

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 times scale, flow rate, through the reactor for continuous or semibatch chemical processes:

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

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

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

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

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

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

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

where:

• k = kinetics reaction rate constant
• C₀ = initial concentration
• n = reaction order
• $\tau$ = mean residence time or space time

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

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

where

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

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.