Damköhler number

This article or the following section is not adequately provided with supporting documents ( e.g. individual evidence ). Information without sufficient evidence could be removed soon. Please help Wikipedia by researching the information and including good evidence.

The Damköhler numbers ( ) (developed by Gerhard Damköhler , 1908–1944) are dimensionless indicators of chemical reaction engineering . There are four known various Damkohler Numbers ( , , , ) which as a Damköhler number n th order are known, as well as a turbulent Damköhler number ( ). ${\displaystyle Da}$ ${\displaystyle Da_{I}}$ ${\displaystyle Da_{II}}$ ${\displaystyle Da_{III}}$ ${\displaystyle Da_{IV}}$ ${\displaystyle Da_{t}}$

First order Damköhler number

The Damköhler number of the first order  describes the ratio of the rate constants of the reaction to the rate constants of the convective mass transfer: ${\displaystyle Da_{I}}$

${\displaystyle Da_{I}={\frac {k_{\text{reakt}}}{k_{\text{konvekt}}}}=k\cdot \tau \cdot c_{0}^{n-1}={\frac {k\cdot L\cdot c_{0}^{n-1}}{w}}}$,

With

• ${\displaystyle k}$ = Rate constant
• ${\displaystyle \tau }$= Dwell time or reaction time
• ${\displaystyle c_{0}}$= Initial concentration
• ${\displaystyle n}$= Reaction order
• ${\displaystyle L}$= characteristic length
• ${\displaystyle w}$= Flow velocity .

To describe discontinuous reactors , the residence time is replaced by the reaction time . The dimensionless mass balance of the ideal stirred tank reactor is thus obtained in a much clearer representation . ${\displaystyle \tau }$ ${\displaystyle t_{r}}$

Second order Damköhler number

The second-order Damköhler number is found in the description of internal material transport processes ( pore diffusion ) at interfaces , e.g. B. balls on catalyst . It is defined as the ratio of the reaction rate to the diffusion rate : ${\displaystyle Da_{II}}$

${\displaystyle Da_{II}={\frac {k\cdot L^{2}\cdot c^{n-1}}{D}}={\frac {k\cdot c^{n-1}}{k_{L}\cdot a}}}$

With

• ${\displaystyle k_{L}}$= volume-related mass transfer coefficient
• a = specific exchange area.

${\displaystyle Da_{II}}$can be seen as the ratio of the reaction rate to surface conditions to the diffusion rate through the outer surface of the catalyst pellet .

Third order and fourth order Damköhler number

The third-order Damköhler number and the fourth-order Damköhler number are used to estimate operating conditions in the polytropic operating mode of reactors. ${\displaystyle Da_{III}}$ ${\displaystyle Da_{IV}}$

Turbulent Damköhler number

The turbulent Damköhler number (usually just referred to as in combustion research ) describes the relationship between the macroscopic time scale of a turbulent flow and the time scale of a chemical reaction : ${\displaystyle Da_{t}}$ ${\displaystyle Da}$ ${\displaystyle \tau _{0}}$ ${\displaystyle \tau _{R}}$

${\displaystyle Da_{t}:={\frac {\tau _{0}}{\tau _{\text{R}}}}\approx {\frac {l_{0}\,v_{\text{R}}}{v'\,l_{\text{R}}}}}$

${\displaystyle l}$stands for the respective length scale, whereby an integral length scale is usually chosen as the macroscopic length scale . This serves as a measure of the diameter of the most energetic (and therefore usually also the largest) eddies in the flow. Their speed of rotation is approximately equal to the standard deviation of the flow speed. In combustion research, the characteristic speed of propagation for chemical reactions is usually the laminar flame speed, i.e. the speed at which the flame front propagates in the laminar case: Analogously, with regard to combustion processes, it is common to use the thickness of the laminar flame front as the reaction length scale: ${\displaystyle v'}$ ${\displaystyle v_{\text{R}}}$ ${\displaystyle s_{\text{L}}}$ ${\displaystyle v_{\text{R}}=s_{\text{L}}}$ ${\displaystyle l_{\text{L}}}$ ${\displaystyle l_{\text{R}}=l_{\text{L}}}$

Using the turbulent Damköhler number, statements can be made about the spatial structure and the temporal behavior of the reaction area in a turbulent reacting flow.

See also

• Karlovitz number

Individual evidence

1. ↑ Stephen B. Pope: Turbulent Flows . Cambridge University Press, 2010, pp. 197 .
2. ↑ Jürgen Warnatz, Ulrich Maas, Robert W. Dibble: Combustion: Physico-chemical basics, modeling and simulation, experiments, pollutant generation (3rd edition) . Springer, 2001, p. 221-224 .
3. ^ Norbert Peters: Turbulent Combustion . Cambridge University Press, 2000, pp. 78 .