RTD of CSTRs in Series

This simulation displays the residence time distribution (RTD), which is measured by injecting a tracer pulse into the first continuously-stirred tank reactor (CSTR) in a series and detecting the tracer concentration at the outlet of the last CSTR. The outlet of each CSTR is the inlet to the next CSTR. The RTD for the system of CSTRs is a probability that a molecule will spend a certain amount of time in the reactor system. Use the sliders to change the number of CSTRs in series and the space time of the system (i.e., the space time $ \tau $ is the total volume of CSTRs divided by the volumetric flow rate).

A pulse of tracer is injected into the first CSTR in a series such that at time $ t = 0 $, all the tracer is in the CSTR and the material balance on the first CSTR is

$$ [1] \quad V \frac{dC_{1}}{dt} = - \upsilon \, C_{1} $$

with initial condition $ C_{1} = C_{0} $ at $ t = 0 $. In equation [1], $ V $ is the volume of the CSTR (L), $ C_{1} $ is the concentration of tracer in the CSTR (and in the outlet from the CSTR) at time $ t $ (s), and $ \upsilon $ is volumetric flowrate (L/s). Rearranging this equation yields

$$ [2] \quad \int_{C_{0}}^{C_{1}} \frac{d C_{1}}{C_{1}} = - \int_{0}^{t} \frac{\upsilon \, dt}{V} $$

Space time for a series of $ N $ CSTRs is defined as $ \tau = V_{total} / \upsilon $ where the total volume $ V_{total} = NV $. Thus, integrating equation 2 yeilds

$$ [3] \quad \mathrm{ln} \left( \frac{C_{1}}{C_{0}} \right) = - \frac{N t}{\tau} $$

or

$$ [4] \quad C_{1} = C_{0} e^{- N t / \tau} $$

The material balance for the second CSTR is

$$ [5] \quad \frac{V d C_{2}}{dt} = \upsilon C_{1} - \upsilon C_{2} $$

with initial condition $ C_{2} = 0 $ at $ t = 0 $. In equation [5], $ C_{2} $ is the concentration of tracer in the second CSTR (and thus leaving the second CSTR). Solving this equation yields

$$ [6] \quad C_{2} = \frac{C_{0} N t e^{-Nt / \tau}}{\tau} $$

Solving material balances for additional CSTRs leads to the general equation for the n^th CSTR

$$ [7] \quad C_{n} = (C_{0} t^{n - 1})\left( \frac{N}{\tau} \right)^{n - 1} \frac{e^{-N t / \tau}}{(n - 1)!} $$

The residence time distribution (RTD) for $ N $ CSTRs $ E_{N} (t) $ is a probability function so that

$$ [8] \quad \int_{0}^{\infty} E_{N} (t) dt = 1 $$

thus,

$$ [8] \quad E(t) = \frac{C_{N}}{\int_{0}^{\infty} C_{N} dt} $$

and since $ \int_{0}^{\infty} C_{N} dt = \int_{0}^{\infty} C_{1} = \frac{C_{0} \tau}{N} $, then for a series of $ N $ CSTRs,

$$ [9] \quad E_{N} (t) = \frac{t^{N - 1}}{(N - 1)!} \left( \frac{N}{\tau} \right)^{N} e^{- N t / \tau} $$