#### Residence Time Distribution: Screencasts

Introduces the idea that not all molecules spend the same time in a chemical reactor and explains how the residence time distribution can be measured with a tracer.

We suggest that after watching this screencast, you list the important points as a way to increase retention.

Solves mass balances for a tracer injected into the first CSTR in a series to determine the residence time distribution (RTD) for CSTRs in series. Note that for this screencast, V is the volume of each CSTR, so the total volume is nV and τ is the space time for each CSTR.

We suggest that after watching this screencast, you list the important points as a way to increase retention.

Shows plots of the residence time distribution for CSTRs in series. Note that for this screencast, V is the volume of each CSTR, so the total volume is nV.

We suggest that after watching this screencast, you list the important points as a way to increase retention.

##### Important Equations:

Residence time distribution \(p(t)\) or exit age distribution \(E(t)\)

\[p(t) = E(t) = \frac{c(t)}{\int_{0}^{\infty}c(t)dt}\]

\(p(t)\) = probability a molecule will reside in reactor for time \(t\)

\(p(t)dt\) = fraction of material that has a residence time between \(t\) and \(t + dt\)

\(c(t)\) = tracer concentration at reactor exit as a function of time for a pulse injection of tracer at reactor inlet

\(t\) = time

\[\overline{t} = \frac{\int_{0}^{\infty}t\,c(t)dt}{\int_{0}^{\infty}c(t)dt}\]

where \(\overline{t}\) = average residence time

For a single ideal CSTR, the residence time distribution is

\[p(t) = E(t) = \frac{e^{-t/\tau}}{\tau}\]

where \(\tau = \frac{V}{\nu}\) = space time

\(V\) = volume of the CSTR

\(\nu\) = volumetric flowrate leaving CSTR

For n CSTRs in series, where \(\tau\) is the space time for each CSTR, the residence time distribution is:

\[p(t) = E_n(t) = \frac{t^{n-1}\,e^{-t/\tau}}{(n-1)! \,\tau ^n}\]

The residence time distribution is often written using \(\tau\) as the space time for the system of n CSTRs; i.e.,

\[\tau_{total} = \frac{V_{total}}{\nu}\]

\[E_n(t) = \frac{t^{n-1}}{(n-1)!} \left( \frac{N}{\tau_{total}} \right)^n \,e^{-tN/\tau_{total}}\]