# Bodenstein number

Physical key figure
Surname Bodenstein number
Formula symbol ${\ displaystyle {\ mathit {Bo}}}$
dimension dimensionless
definition ${\ displaystyle {\ mathit {Bo}} = {\ frac {u \ cdot L} {D _ {\ mathrm {ax}}}}}$
 ${\ displaystyle u}$ Flow velocity ${\ displaystyle L}$ Length of the reactor ${\ displaystyle D _ {\ mathrm {ax}}}$ axial dispersion coefficient
Named after Max Bodenstein
scope of application Chemical reaction engineering

The Bodenstein number (after Max Bodenstein ), Bo for short  , is a dimensionless number from reaction engineering that describes the ratio of the moles supplied by convection to those supplied by diffusion . The Bodenstein number thus characterizes the backmixing within a system (the larger the Bodenstein number, the lower the backmixing) and enables statements to be made about whether and how much volume elements or substances mix within a reactor due to the prevailing currents.

The Bodenstein number is defined as the ratio of the convection flow to the dispersion flow. It is part of the dispersion model and is therefore also referred to as the dimensionless dispersion coefficient.

Mathematically, two idealized borderline cases are obtained for the Bodenstein number, which, however, cannot be fully achieved in practice:

• if the Bodenstein number were zero, the state of total backmixing would have been achieved, which is ideally desired in a continuously operated stirred tank reactor .
• If the Bodenstein number were infinitely large, there would be no backmixing, but only a continuous flow that prevails in an ideal flow pipe.

By regulating the flow rate within a reactor, the Bodenstein number can be set to a previously calculated, desired value. In this way, the backmixing of the substance components desired within the respective reactor can be achieved.

## determination

The Bodenstein number is calculated through

${\ displaystyle {\ mathit {Bo}} = {\ frac {u \ cdot L} {D _ {\ mathrm {ax}}}}}$

With

• the flow velocity ${\ displaystyle u}$
• the length of the reactor${\ displaystyle L}$
• the axial dispersion coefficient in m² / s.${\ displaystyle D _ {\ mathrm {ax}}}$

The Bodenstein number can be obtained experimentally from the residence time distribution. Assuming an open system, the following applies:

${\ displaystyle \ sigma _ {\ theta} ^ {2} = {\ frac {\ sigma ^ {2}} {\ tau ^ {2}}} = {\ frac {2} {\ mathit {Bo}}} + {\ frac {8} {{\ mathit {Bo}} ^ {2}}}}$

With

• the dimensionless variance ${\ displaystyle \ sigma _ {\ theta} ^ {2}}$
• the variance around the mean residence time${\ displaystyle \ sigma ^ {2}}$
• the hydrodynamic residence time .${\ displaystyle \ tau}$

## Individual evidence

1. Matthias Bohnet (Ed.): Mechanical process engineering. Wiley-VCH, Weinheim 2004, ISBN 3-527-31099-1 , pp. 213-229.