# Dwell time (technical process)

In a process , the dwell time (Greek small dew ) is the time during which e.g. B. a defined volume of liquid “lingers” in a reactor or in the entire system. ${\ displaystyle \ tau}$

• In the case of a batch reactor , it is identical to the reaction time .
• For continuous reactors, the residence time is calculated as the quotient of the fluid volume in the reactor to the exiting volume flow :${\ displaystyle V _ {\ mathrm {R}}}$ ${\ displaystyle {\ dot {V}}}$
${\ displaystyle \ tau = {\ frac {V _ {\ mathrm {R}}} {\ dot {V}}}}$
In continuous reactors, the residence time describes the efficiency of the process and is also referred to there as space-time .
Strictly speaking, the mean residence time relates to the exiting volume flow, while the space-time relates to the entering flow. However, if the density does not change (which is the case with most liquid phase reactions ) and therefore the incoming and outgoing volume flow are the same, then the space time and mean residence time are also identical.
${\ displaystyle D = \ tau ^ {- 1} = {\ frac {\ dot {V}} {V _ {\ mathrm {R}}}}}$

The residence time of a chemical reactor is one of the most important reaction engineering parameters . The product of the rate constant of a first-order reaction and the average residence time is the first Damköhler number , which essentially determines the conversion of a simple reaction in a reactor. ${\ displaystyle k}$${\ displaystyle \ tau}$ ${\ displaystyle DaI}$

## determination

In experimental apparatus , the residence time is usually determined using a tracer , which is injected into the inlet of the apparatus. It should be possible to quantify the tracer as it flows through the apparatus.

A basic distinction is made between:

• the impact marking, in which only a small amount of the tracer is introduced in the shortest possible time interval
• the displacement marking, in which the original inlet is replaced by another.

If the tracer concentration at the outlet of the apparatus is measured over time, the dwell time density function  E (t) is obtained for the impact marking . The integral over this function is by definition equal to 1:

${\ displaystyle \ int _ {0} ^ {\ infty} E (t) \; dt = 1}$

In order to obtain the total residence time function  F (t) , the following must be integrated via the distribution density function:

${\ displaystyle F (t) = \ int _ {0} ^ {t} E (t ') \; dt' \ leq 1}$

It represents the proportion of those volume elements that left the reactor at time  t after the addition at time  0 .

## Residence time behavior of different reactors

Residence time density functions of various ideal reactors

Basically, a distinction is made between the following continuous ideal reactors , which also differ in their residence time behavior :

• In the ideal flow pipe , the distribution density function is a step function , since a plug flow prevails and thus no back-mixing takes place.
• In the ideal, continuous stirred tank , the tracer substance is immediately and completely distributed in the reactor. Due to the further inflow and outflow in the reactor, the concentration at the outlet decreases continuously.
• Stirred tank cascades can - depending on the number  N of their stirred tanks - be described by the following function:
${\ displaystyle E (\ theta) = {\ frac {N \ cdot (N \ cdot \ theta) ^ {N-1}} {(N-1)!}} \ cdot \ exp (-N \ cdot \ theta )}$
with the normalized dwell time .${\ displaystyle \ theta = {\ frac {t} {\ tau}}}$

## literature

• Octave Levenspiel: Chemical Reaction Engineering . 3. Edition. John Wiley & Sons, New York NY u. a. 1999, ISBN 0-471-25424-X .
• Erwin Müller-Erlwein: Chemical reaction engineering . 2nd, revised and expanded edition. Teubner, Wiesbaden 2007, ISBN 978-3-8351-0187-6 .