Editing Damköhler numbers (section)

{{Short description|Dimensionless numbers used in chemical engineering}}
The '''Damköhler numbers''' ('''Da''') are [[dimensionless number]]s 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]], who worked in chemical engineering, thermodynamics, and fluid dynamics.<ref name="2020Weiland">{{Cite journal |last=Weiland |first=Claus |date=2020 |title=Mechanics of Flow Similarities |url=https://doi.org/10.1007/978-3-030-42930-0 |journal=SpringerLink |language=en |doi=10.1007/978-3-030-42930-0|isbn=978-3-030-42929-4 |url-access=subscription }}</ref> 
The '''[[Karlovitz number]]''' ('''Ka''') is related to the Damköhler number by Da = 1/Ka.

In its most commonly used form, the first Damköhler number (Da<sub>I</sub>) relates particles' characteristic residence time scale in a fluid region to the reaction timescale. The residence time scale can take the form of a [[convection]] time scale, such as [[volumetric flow rate]] through the reactor for continuous ([[Plug flow reactor model|plug flow]] or [[Continuous stirred-tank reactor|stirred tank]]) or [[Semibatch reactor|semibatch]] chemical processes:
: <math>\mathrm{Da_{\mathrm{I}}} = \frac{ \text{reaction rate} }{ \text{convective mass transport rate} }</math>
<!--or as
: <math>\mathrm{Da_{\mathrm{I}}} = \frac{ \text{characteristic fluid time} }{ \text{characteristic chemical reaction time} }</math> -->

In reacting systems that include interphase mass transport, the first Damköhler number can be written as the ratio of the chemical reaction rate to the mass transfer rate
: <math>\mathrm{Da}_{\mathrm{I}} = \frac{ \text{reaction rate} }{ \text{diffusive mass transfer rate} }</math>

It is also defined as the ratio of the characteristic fluidic and chemical time scales:
: <math>\mathrm{Da_{\mathrm{I}}} = \frac{ \text{flow timescale} }{ \text{chemical timescale} }</math>

Since the reaction rate determines the reaction timescale, the exact formula for the Damköhler number varies according to the rate law equation. For a general chemical reaction A → B following the [[Power law]] kinetics of n-th [[Order of reaction|order]], the Damköhler number for a convective flow system is defined as:

: <math>\mathrm{Da_{\mathrm{I}}} = k C_0^{\ n-1}\tau</math>
where:
* ''k'' = [[chemical kinetics|kinetics]] [[reaction rate constant]]
* ''C''<sub>0</sub> = initial concentration
* ''n'' = [[reaction order]]
* <math>\tau</math> = mean [[Residence time distribution|residence time]] or '''space-time'''

On the other hand, the second Damköhler number (Da<sub>II</sub>) is defined in general as:
: <math>\mathrm{Da}_{\mathrm{II}} = \frac{kQ}{c_p \Delta T}</math>
It compares the process energy of a thermochemical reaction (such as the energy involved in a nonequilibrium gas process) with a related enthalpy difference (driving force).<ref name="2020Weiland"></ref>
 
In terms of reaction rates:
: <math>\mathrm{Da}_{\mathrm{II}} = \frac{k C_0^{n-1}}{k_g a}</math>
where
* ''k<sub>g</sub>'' is the global mass transport coefficient
* ''a'' is the interfacial area

The value of Da provides a quick estimate of the degree of [[Conversion (chemistry)|conversion]] that can be achieved. If Da<sub>I</sub> goes to infinity, the residence time greatly exceeds the reaction time, such that nearly all chemical reactions have taken place during the period of residency, this is the transport limited case, where the reaction is much faster than the diffusion. Otherwise if Da<sub>I</sub> goes to 0, the residence time is much shorter than the reaction time, so that no chemical reaction has taken place during the brief period when the fluid particles occupy the reaction location, this is the reaction limited case, where diffusion happens much faster than the reaction. Similarly, Da<sub>II</sub> goes to 0 implies that the energy of the chemical reaction is negligible compared to the energy of the flow. The limit of the Damköhler number going to infinity is called the [[Burke–Schumann limit]]. 

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.<ref name="Fogler">{{cite book |last=Fogler |first=Scott |title=Elements of Chemical Reaction Engineering |location=Upper Saddle River, NJ |publisher=Pearson Education |year=2006 |edition=4th |isbn=0-13-047394-4 }}</ref>