Directions

A nonideal reactor of total volume V and volumetric flow rate v₀ is modeled as a combination of ideal reactors: a plug flow reactor (PFR) and a continuous stirred-tank reactor (CSTR), connected in series (either one first) or in parallel. Part of the reactor volume can be a stagnant dead zone, and part of the feed can bypass the reactor entirely.

The residence time distributions (RTDs) are shown in the plot. The exit age distribution E(θ) is obtained by a tracer experiment with a pulse input. The cumulative residence time distribution F(θ) is the integral of E(θ) and can be measured by a tracer experiment with a step input. Tracer measurements are used to determine how nonideal a reactor is and can be used to better model the reactor to determine conversions expected for given feed conditions. For example, a CSTR might be nonideal because a small fraction of the feed flows almost directly to the outlet (bypass) or because some part of the reactor contents is not well mixed (dead zone).

Select first- or second-order kinetics and set the Damköhler number with the slider. Choose the configuration with the buttons, and use the sliders to set the fraction of the active volume that is PFR, the dead-zone volume fraction, and the fraction of the feed that bypasses. In the parallel configuration, a slider sets the fraction of the reactor flow that goes to the PFR.

The schematic shows the flow network; line widths are proportional to flow rates, the area of the dead-zone square is proportional to its volume, and the shading shows reactant concentration C_A/C_A0. The lower plot shows the exit age distribution E(θ) or the cumulative residence time distribution F(θ), where θ = t/τ and τ = V/v₀. Delta functions in E(θ) are drawn as arrows labeled with their areas. The third plot shows conversion versus Damköhler number for the nonideal reactor, ideal PFR, ideal CSTR, and segregation model.

The table compares the conversion in the nonideal reactor with ideal PFR and CSTR conversions for the same V and v₀, and with the segregation-model conversion calculated from the RTD.

Details

The space time based on the total reactor volume is , and dimensionless time is . The dead-zone volume fraction is , the bypass fraction is , the PFR fraction of the active volume is , and (for parallel reactors) the fraction of the reactor flow that goes to the PFR is . The active volume and the flow rate through the reactor are

Space times of the ideal reactors (relative to ) are:

series:

parallel:

where is the PFR space time and is the CSTR space time.

Kinetics. For products, (first order) or (second order). The Damköhler number is (first order) or (second order). With , the outlet of each ideal reactor for inlet concentration is

first order:

second order:

For reactors in series, the outlet of the first reactor is the inlet of the second. For parallel reactors, and for the bypass stream, the outlet streams mix:

For first-order kinetics, the conversion for reactors in series does not depend on which reactor is first. For second-order kinetics it does: when the PFR is first, the concentration is high in the PFR, where it is used most effectively, so conversion is higher than when the CSTR is first.

Residence time distribution. The bypass contributes . For a PFR followed by a CSTR (or a CSTR followed by a PFR, which has the same RTD):

where is the unit step function. For parallel reactors:

The cumulative distribution is . Because the dead zone does not exchange fluid with the flowing stream, the mean residence time is less than :

Segregation model. The segregation-model conversion treats each fluid element as a batch reactor with a residence time given by the RTD:

For first-order kinetics, the segregation model gives the same conversion as the combined-reactor model for any configuration, because conversion depends only on the RTD. For second-order kinetics, the conversion also depends on how early or late the fluid mixes, so different configurations with the same RTD (PFR → CSTR and CSTR → PFR) have different conversions, and the segregation model (latest possible mixing) gives the highest conversion for a given RTD.

The ideal PFR and ideal CSTR conversions in the table are for the same total volume and flow rate , with no dead zone or bypass (space time ).

Definitions of terms

	reactant
	concentration of A (mol/L)
	feed concentration of A (mol/L)
	dimensionless concentration of A
	dimensionless concentrations entering a reactor and leaving the system
	dimensionless concentrations leaving the PFR, the CSTR, and the reactors in series
	Damköhler number
	dimensionless exit age distribution (residence time distribution)
	cumulative residence time distribution
	fraction of the feed that bypasses the reactor
	fraction of the reactor volume that is dead zone
	unit step function
	rate constant (1/s for first order, L/(mol s) for second order)
	rate of disappearance of A (mol/(L s))
	time (s)
	mean residence time (s)
	total reactor volume (L)
	active (non-dead) reactor volume (L)
	PFR and CSTR volumes (L)
	feed volumetric flow rate (L/s)
	volumetric flow rate through the reactor (L/s)
	conversion of A leaving the system
	conversion in a batch reactor after time
	conversion predicted by the segregation model
	fraction of the active volume that is PFR
	fraction of the reactor flow that goes to the PFR (parallel configuration)
	Dirac delta function
	dimensionless time
	variance of the residence time distribution (s)
	space time based on total reactor volume (s)
	space times of the PFR and the CSTR (s)

About

This simulation was created in the Department of Chemical and Biological Engineering at University of Colorado Boulder for LearnChemE.com by John L. Falconer using Claude AI. It was prepared with financial support from the National Science Foundation (DUE 2336987 and 2336988). Address any questions or comments to LearnChemE@gmail.com.