A real tubular reactor operates exactly between an ideal plug-flow reactor (PFR) and a perfectly mixed tank (CSTR). Vary the Peclet and Damkohler numbers and see in real time how axial dispersion — back-mixing along the flow — degrades the first-order conversion through the Danckwerts closed-form solution.
Parameters
Peclet number Pe
Ratio of convection to axial dispersion, Pe = uL/D_ax. Larger means less dispersion, closer to PFR
Damkohler number Da
Ratio of reaction rate to residence time, Da = k·τ. Larger means more reaction
Pe→∞ means zero axial dispersion — an ideal plug-flow reactor (PFR). Pe→0 means infinite dispersion — a perfectly mixed continuous stirred-tank reactor (CSTR). A real tubular reactor has a finite Pe and a conversion between the two. Da measures how long and how fast the reactor lets the reaction proceed.
Results
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Conversion X
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Parameter q
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PFR conversion (ideal, ref.)
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CSTR conversion (ref.)
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Peclet number Pe
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Flow regime
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Tubular reactor — tracer flow animation
A tracer concentration front flows through the reactor. At high Pe the front stays sharp (plug flow); at low Pe it smears out (strong back-mixing). The lower curve is the decaying reactant concentration profile C/C₀.
Conversion X of the axial dispersion model for a first-order reaction under closed-closed (Danckwerts) boundary conditions. q is an auxiliary parameter, Pe: Peclet number, Da: Damkohler number. Pe→∞ recovers plug flow (PFR) and Pe→0 recovers perfect mixing (CSTR).
$$Pe=\dfrac{uL}{D_{ax}},\qquad Da=k\,\tau$$
The Peclet number Pe is the ratio of convection (velocity u, length L) to the axial dispersion coefficient D_ax. The Damkohler number Da is the product of the rate constant k and the residence time τ.
The two reference limits: the ideal plug-flow conversion X_PFR and the ideal perfectly-mixed conversion X_CSTR. The conversion of a real reactor with finite Pe lies between them.
What is the Axial Dispersion Model?
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What is the "axial dispersion model" actually for? If I'm sizing a reactor, can't I just use the PFR or the CSTR formula?
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Good question. The PFR (plug flow) and the CSTR (perfect mixing) are both idealised extreme cases. A PFR is a model where the flow never mixes and moves in a single file; a CSTR is a model where the contents are perfectly stirred. But a real tubular reactor is neither — it mixes a little in the flow direction. The axial dispersion model captures that "little bit of mixing" with one term that looks like Fickian diffusion. The reason it is so handy in practice is that it represents the non-ideality with a single parameter, without a full two-dimensional flow calculation.
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So that "single parameter" is the Peclet number. Why does raising Pe raise the conversion?
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The Peclet number Pe = uL/D_ax is a tug-of-war ratio between convection (the force pushing forward) and axial dispersion (the force stirring things back). A large Pe means the dispersion D_ax is small and the flow barely mixes — close to plug flow. For a positive-order reaction such as a first-order reaction, a PFR is most efficient because it keeps the concentrated inlet reactant concentrated all the way through. With back-mixing, the diluted fluid near the outlet leaks back toward the inlet and lowers the reaction rate. So the larger Pe is, the closer to PFR and the higher the conversion.
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I see. Then what role does Da, the Damkohler number, play?
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Da = k·τ is the product of "how fast the reaction is" and "how long things stay inside the reactor". Roughly, it tells you the total amount of work the reactor does. With a small Da, even a perfect PFR barely converts anything. With a large Da, the reaction proceeds vigorously. At the default Da = 2, the ideal PFR conversion is 1 − e⁻² ≈ 0.865, i.e. you can reach 86.5%. The interesting part: the larger Da is, the larger the gap between PFR and CSTR. The harder the reaction works, the more visible the penalty of back-mixing.
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At the default settings (Pe=20, Da=2), the conversion sits between the PFR's 86.5% and the CSTR's 66.7%. What does it actually come out to?
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Right — using the Danckwerts closed-form solution, it works out to about 84%. Pe = 20 is reasonably large, so it lands much closer to the PFR's 86.5%. The auxiliary parameter q = √(1 + 4·2/20) = √1.4 ≈ 1.183, and with it you get X in a single step. The beauty of this formula is that you don't have to numerically integrate a differential equation. In design you use it like this: "measure our reactor's Pe with a tracer test, plug it into this formula, and the actual conversion comes straight out."
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Measuring Pe with a tracer test is hard for me to picture concretely.
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You inject a "tracer" such as a dye or salt at the reactor inlet for just an instant, and measure how its concentration emerges over time at the outlet. That is the residence time distribution (RTD). A perfect PFR sends out the same sharp pulse you put in, just delayed. With back-mixing, that pulse comes out gently spread. Analysing the amount of spread (the variance σ²) gives D_ax, and then you compute Pe = uL/D_ax. When you lower Pe in the animation above and the front widens, that is exactly this RTD spreading at work.
Frequently Asked Questions
The axial dispersion model represents a real tubular reactor as an ideal plug-flow reactor with a single extra term that accounts for slight mixing along the flow direction. The mass balance has convection and reaction terms plus a dispersion term −D_ax·(d²C/dz²) that looks like Fickian diffusion. The strength of the dispersion is captured by the dimensionless Peclet number Pe = uL/D_ax: the larger Pe is, the smaller the dispersion and the closer the reactor is to an ideal PFR. It is a practical model because it represents non-ideality with one parameter, without a full two-dimensional flow analysis.
The Damkohler number Da = k·τ is the ratio of reaction rate to residence time; a larger Da means more reaction and higher conversion. The Peclet number Pe = uL/D_ax is the ratio of convection to axial dispersion. For a positive-order reaction, a large Pe (little dispersion) pushes the conversion toward the ideal PFR value, while a small Pe (strong dispersion, i.e. strong back-mixing) pulls it toward the CSTR value. In other words, axial dispersion degrades the conversion of a real reactor below the ideal plug-flow value.
Solving the axial dispersion model for a first-order reaction under closed-closed (Danckwerts) boundary conditions gives the conversion X as a closed formula. With q = √(1 + 4Da/Pe), the result is X = 1 − 4q·e^(Pe/2) / [ (1+q)²·e^(Pe·q/2) − (1−q)²·e^(−Pe·q/2) ]. As Pe→∞ it reduces to the PFR result X = 1 − e^(−Da), and as Pe→0 it reduces to the CSTR result X = Da/(1+Da). Conversion can be obtained from this single equation without numerically integrating a differential equation.
The Peclet number is usually obtained from tracer response tests (pulse or step injection). Measuring the residence time distribution (RTD) at the outlet and analysing its variance σ² yields the axial dispersion coefficient D_ax, from which Pe = uL/D_ax is computed. For packed beds, D_ax can also be estimated from correlations with the particle Reynolds and Schmidt numbers. Tubular reactors typically have Pe of order tens to hundreds: Pe > 100 is treated as nearly PFR, and Pe < 1 indicates strong back-mixing.
Real-World Applications
Design of tubular and packed-bed reactors: Fixed-bed catalytic reactors in petrochemical processes and multi-tubular reactors for ethylene oxide or oxidation reactions are treated on paper as PFRs, but in reality there is always a performance loss from axial dispersion. The axial dispersion model lets you quantify "how much lower the conversion is than an ideal PFR" from a Peclet number measured by a tracer test. Pe tends to shrink when a reactor is shortened or scaled up, so this assessment of the loss becomes especially important.
Off-gas and wastewater treatment units: Catalytic combustors and plug-flow biological treatment basins need enough residence time to meet a target treatment ratio (conversion). With axial dispersion, the same residence time gives a lower treatment ratio, so a design margin is required. The axial dispersion model provides the basis for deciding how large a safety factor to apply, accounting for Pe, on top of a reactor length obtained from the ideal PFR design equation.
Microreactors and continuous-flow synthesis: In continuous-flow synthesis, increasingly used in pharmaceuticals and fine chemicals, you want reactants to react reliably inside a narrow channel — but the parabolic velocity profile of the flow (Taylor dispersion) generates axial dispersion. A low Pe leads to scatter in the outlet concentration and reduced conversion, so channel design and segmentation (droplet flow) are used to keep Pe high. This model is the starting point for those design guidelines.
Teaching and validating residence time distribution (RTD) analysis: The axial dispersion model is the most basic non-ideal flow model, characterising RTD data from a tracer test with a single parameter Pe. In chemical engineering courses and labs, a standard exercise is to obtain D_ax from the variance of a measured RTD, compute the predicted conversion with a formula like the one in this tool, and compare it with the measured conversion. It is taught as a leading analysis method alongside the tanks-in-series model.
Common Misconceptions and Pitfalls
The biggest misconception is assuming the axial dispersion model can represent any non-ideal flow. This model corrects "a small deviation from PFR" with one term, and it is most reliable when Pe is fairly large (roughly Pe > tens) and the back-mixing is mild. For reactors with severe flow anomalies — channelling (flow maldistribution), dead zones (stagnant regions) or bypassing — a one-parameter axial dispersion model is not appropriate. If the RTD shows two peaks or a long tail, do not try to fit it with this model; switch to a model that explicitly includes dead zones and bypassing.
Next, the misconception that the Peclet number is a fixed constant of the reactor. Because Pe = uL/D_ax contains the velocity u and the reactor length L, it changes with the operating conditions. In the same reactor, lowering the flow rate lowers u, which lowers Pe, so the relative effect of back-mixing grows. When scaling up by making the reactor wider and shorter, L drives Pe down as well. "It was Pe = 300 at bench scale, so the full-scale unit can be treated as a PFR" does not necessarily hold. Pe must be re-evaluated at each operating point.
Finally, memorising "axial dispersion always lowers conversion" too simply. For positive-order reactions such as a first-order reaction, dispersion does lower conversion, but for autocatalytic reactions and some complex reaction systems back-mixing can be beneficial. The picture changes further for selectivity (the fraction of the desired product): in series reactions, back-mixing can reduce the yield of an intermediate. This tool is a conversion model that assumes a single positive-order first-order reaction; do not forget that the conclusions change if the reaction order or the reaction network is different.
How to Use
Input Peclet number (Pe) range: Pe values between 0.01 and 1000 define flow regime from completely mixed (Pe<0.1) to plug-flow behavior (Pe>100)
Set Damköhler number (Da) range: Da quantifies reaction kinetics relative to residence time; typical industrial values range 0.1 to 10 for first-order reactions
Run simulation to compare your axial dispersion reactor conversion against ideal PFR and CSTR benchmarks, observing how Pe and Da interact to shift performance
Worked Example
A 5 m tubular reactor (diameter 0.1 m, flow rate 10 L/min) processes a first-order decomposition with k=0.15 s⁻¹. Residence time τ=50 s gives Da=7.5. Dispersion coefficient D=0.002 m²/s and mean velocity u=0.021 m/s yield Pe=52.5 (intermediate regime). Simulator predicts conversion X=0.82, compared to ideal PFR X=0.92 and CSTR X=0.58, reflecting real back-mixing losses typical of industrial packed-bed or tubular reactors.
Practical Notes
Pe<1 indicates severe axial dispersion (CSTR-like behavior); check for channeling, dead zones, or very high inlet turbulence in your reactor vessel
Pe>100 approaches ideal PFR; validate by measuring tracer breakthrough curves or residence-time distributions (RTD) experimentally
Da>5 means reaction completion dominates; residence time becomes critical; Da<0.5 suggests kinetic limitation where conversion improves little with longer residence time
For exothermic syntheses (e.g., ammoniation), high Pe avoids hot spots; for slow catalytic processes, moderate Pe (10–50) balances mixing and conversion efficiency