Explore the dimensionless number that governs chemical-reactor performance — the Damköhler number Da. Adjust the rate constant, residence time and inlet concentration to see the conversion and exit concentration of a CSTR and a PFR update in real time, and grasp whether reaction or flow is rate-controlling.
Parameters
Reactor type
Perfectly mixed CSTR or plug-flow PFR
Reaction order
Concentration dependence of the rate
Rate constant k
Units: 1/s for first order, m³/(mol·s) for second order
Residence time τ
s
Reactor volume ÷ volumetric flow. Longer = more reaction
Inlet concentration C₀
mol/m³
Directly affects Da and conversion for second order
Results
—
Damköhler number Da
—
Conversion X (%)
—
Exit conc. C (mol/m³)
—
Inlet rate r₀ (mol/m³·s)
—
Other reactor's conversion (%)
—
Controlling regime
—
Flow and concentration in the reactor — animation
Feed (blue) enters from the left, reacts into product (orange) and flows to the right. Colour intensity tracks concentration: a CSTR is uniform, while a PFR fades from inlet to outlet.
The Damköhler number Da is the dimensionless ratio of the reaction rate to the rate of convective mass transport. k: rate constant, τ: residence time, C₀: inlet concentration.
The second-order CSTR conversion is found by solving the quadratic mass balance. The exit concentration is C = C₀(1−X).
What is the Damköhler Number?
🙋
I've never heard of the "Damköhler number." Is it one of those dimensionless numbers like the Reynolds number?
🎓
Exactly — it's the single most important dimensionless number in the world of chemical reactors. Roughly speaking, it is "how fast the reaction is" divided by "how fast the flow carries the fluid away." Da = kτ, where k is the speed of the reaction and τ is the residence time — how long the fluid lingers inside the reactor. The larger Da is, the more thoroughly the fluid can react before it leaves.
🙋
I see. So what happens when Da is small? I just dragged the τ slider way down.
🎓
You saw the conversion plunge, right? Da ≪ 1 is the "residence-time-controlled" regime — the fluid passes straight through with no time to react. The feed comes out of the outlet almost untouched. There are two fixes: make the reactor bigger to gain τ, or throttle the flow rate to stretch τ. Conversely, when Da ≫ 1 you are "reaction-controlled" and almost everything reacts. So Da tells you at a glance whether the reactor is too small.
🙋
When I switch the reactor type from CSTR to PFR, the conversion goes up even though the conditions are the same. Why is that?
🎓
Good observation. A CSTR is "perfectly mixed" — the tank contents blend instantly so everything sits at the outlet concentration. Since the reaction rate is faster at higher concentration, a CSTR always runs at its thinnest, slowest state. A PFR, on the other hand, moves through the tube in plug flow: concentrated and fast at the inlet, thinner toward the outlet. By passing through the high-concentration, high-speed zone, a PFR reaches a higher conversion at the same Da. The two curves on the chart show exactly that.
🙋
When I switch to a second-order reaction, just raising the inlet concentration C₀ changes the conversion. It didn't move for first order.
🎓
That is where the reaction order matters. For a first-order reaction Da = kτ, and C₀ doesn't appear — double the concentration and the rate doubles too, so the conversion, which is a ratio, stays the same. But for a second-order reaction Da = kC₀τ. The rate scales with the square of concentration, so raising C₀ raises Da and the conversion. That's why "feeding it more concentrated" is itself a genuine design lever for second-order reactions.
🙋
So I could just make Da as large as possible to build a good tool, right?
🎓
I understand the temptation, but it isn't that simple. Once Da exceeds about 10 the conversion plateaus in the 99% range — stretching τ further gives only marginal gains while the reactor grows ever larger and more expensive. And an overly long residence time can drive series reactions or side reactions that consume the desired product, lowering the selectivity. In practice many designs keep Da in the "competitive" range of about 1 to 10 — the sweet spot where the curve rises steeply.
Frequently Asked Questions
The Damköhler number Da is the dimensionless ratio of the reaction rate to the rate of convective (flow) mass transport. For a first-order reaction Da = kτ, and for a second-order reaction Da = kC₀τ, where k is the rate constant, τ is the residence time and C₀ is the inlet concentration. When Da ≪ 1, the fluid leaves the reactor before reacting much; when Da ≫ 1, it reacts almost completely. Da is the starting point for deciding whether to make the reactor larger or to slow the flow.
At the same Da a PFR (plug-flow reactor) always reaches a higher conversion than a CSTR (continuous stirred-tank reactor). Because a CSTR is perfectly mixed, the whole reactor operates at the low outlet concentration, so a concentration-dependent reaction runs at its slowest possible rate. A PFR, in contrast, has a concentration that falls continuously from inlet to outlet and passes through high-concentration (fast) regions, so it converts more for the same residence time. For first order, the CSTR gives X = Da/(1+Da) and the PFR gives X = 1−e^(−Da).
Conversion is set by Da, so raising Da = kτ (first order) improves it. Increasing the residence time τ means a larger reactor or a lower flow rate, which affects capital cost and throughput. Increasing the rate constant k means a higher temperature or a catalyst, which affects operating cost and selectivity. When Da ≪ 1 (residence-time-controlled), increasing τ is the more effective lever; once Da is already large, adding more τ barely raises the conversion, which is nearly saturated.
When Da ≪ 0.1 the conversion is tiny and large amounts of unreacted feed flow to the outlet, so the reactor must be enlarged or the flow slowed. When Da ≫ 10 the reaction is nearly complete and raising Da further gives almost no extra conversion. Worse, an excessively long residence time can drive side reactions and series reactions that lower the selectivity to the desired product, and it makes the equipment needlessly large. In practice many designs keep Da in the competitive range of about 1 to 10.
Real-World Applications
Chemical-plant reactor sizing: The Damköhler number is the first calculation in choosing a reactor volume for a continuous process. From the target conversion (say 90%) you back out the required Da, then from the rate constant k and the required throughput (the volumetric flow) you obtain the reactor volume V = τ·Q. The choice between a CSTR and a PFR, or whether to stage several PFRs, is also made by reading the Da-versus-conversion curve.
Choosing the reactor type: The fact that a PFR delivers a higher conversion than a CSTR at the same Da is a basic principle of equipment selection. If a high conversion is needed, a PFR is favoured; if a strongly exothermic reaction needs careful temperature control, a well-mixed CSTR is favoured — a trade-off you quantify on a Da basis. In practice a "CSTR cascade" of stirred tanks in series is often used to approach PFR performance.
Combustion and combustor design: In the combustors of gas turbines and industrial furnaces, the Damköhler number is used as the ratio of the chemical-reaction time scale to the time the fluid stays in the combustion chamber. A small Da (flow too fast) causes the flame to blow off, while a large Da gives stable combustion. Flame-holding design for combustors is, in essence, Damköhler-number management.
Environmental and wastewater treatment: The aeration tanks of the activated-sludge process and catalytic exhaust-gas treatment units also have their removal efficiency set by the balance of residence time and reaction rate. When treatment falls short, it is a sign that Da is too small — the tank is too small or the flow too high — and any expansion or flow adjustment is evaluated through Da. It is also a key design metric for bioreactors.
Common Misconceptions and Pitfalls
The most common pitfall is assuming there is only one definition of the Damköhler number. This tool deals with the Da that is "reaction rate ÷ convective transport rate" (often called Da_I), but there is also a Da_II that represents "reaction rate ÷ diffusive transport rate." In heterogeneous catalytic reactions or multiphase reactions where mass transfer is rate-controlling, Da_II is the one that matters. Whenever you see a Damköhler number in a paper or textbook, always check which transport process it is being compared against. Mistaking the definition leads to a completely different conclusion.
Next, treating the residence time τ only as an average value. The PFR in this tool assumes ideal plug flow and the CSTR assumes ideal perfect mixing. Real reactors, however, have flow maldistribution, short-circuiting and dead zones, so individual fluid elements stay in the reactor for varying amounts of time. Ignoring this "residence-time distribution (RTD)" and designing on the mean τ alone can make the real conversion fall below the calculated value. Evaluating non-ideal flow requires RTD measurement by a tracer test.
Finally, the belief that "a larger Da is always a better design." Raising Da raises the conversion, but that does not mean the yield or selectivity to the desired product goes up. In a series reaction where the desired product reacts further (A→B→C when B is wanted, for example), an overly long residence time converts B into C, so the optimum Da is a moderate value. Moreover, in an exothermic reaction, pushing Da too high can outpace heat removal and risk a thermal runaway. Remember that Da is an indicator of conversion — it is not an all-purpose metric that also guarantees yield, safety and cost.
How to Use
Enter the reaction rate constant k (s⁻¹) in the kNum field; typical values range 0.01–10 s⁻¹ for first-order reactions.
Set the residence time τ (seconds) in tauNum; industrial packed-bed reactors often operate at 0.5–5 s.
Input initial concentration C₀ (mol/m³) in c0Num; for a dilute aqueous stream, C₀ = 100–500 mol/m³.
The simulator calculates Da = k·τ and predicts conversion X (%), exit concentration, and kinetically/diffusion-controlled regime.
Worked Example
Consider a continuous stirred-tank reactor (CSTR) with k = 0.5 s⁻¹, τ = 2 s, C₀ = 200 mol/m³. Da = 0.5 × 2 = 1.0. Conversion X = Da/(1+Da) = 50%. Exit concentration C = 200 × (1−0.5) = 100 mol/m³. Inlet rate r₀ = k·C₀ = 0.5 × 200 = 100 mol/m³·s. At Da = 1, kinetics and residence time equally control conversion; switching to a plug-flow reactor (PFR) geometry with identical k and τ yields ~63% conversion because PFRs exploit concentration gradients more efficiently.
Practical Notes
Da < 0.1 indicates kinetic control: reaction is slow; increase k (higher T, catalyst) or τ (larger vessel).
Da > 10 indicates diffusion/mixing control: reaction is fast; residence time dominates; further k increases yield minimal gain.
For exothermic syntheses (e.g., sulfuric acid plant), verify that τ is sufficient to avoid runaway; Da ≈ 2–3 is typical.
CSTR and PFR conversions diverge most at intermediate Da (0.5–5); select reactor type based on selectivity, not just Da.