Engineering & Science
Damköhler Number (Da): Isothermal First-Order CSTR vs. PFR Conversion
Calculate the first-order reactor Damköhler number Da = k * tau and compare continuous stirred-tank conversion X_CSTR = Da / (1 + Da) against plug-flow conversion X_PFR = 1 - EXP(-Da).
Why a plug-flow tubular reactor beats an equal-volume CSTR at high conversion!
Gerhard Damköhler (1936) showed that isothermal first-order chemical reactor performance collapses onto a single dimensionless ratio of residence time tau = V / Q (min or s) to characteristic reaction time 1 / k: the first Damköhler Number Da = k * tau! In an ideal well-mixed Continuous Stirred-Tank Reactor (CSTR), fresh feed immediately dilutes down to the exit concentration, limiting conversion to X_CSTR = Da / (1 + Da).
In an ideal Plug-Flow Reactor (PFR) with zero axial back-mixing, concentration decays exponentially along the tube length, achieving X_PFR = 1 - EXP(-Da)! Compare an equal-volume reactor pair at k = 0.25 min^-1 and tau = 12 min (Da = 3.0): the CSTR reaches 75.00% conversion, whereas the PFR reaches 95.02% conversion—cutting unreacted effluent by more than fourfold!
| Residence Time Scenario (k = 0.25 min^-1) | Damköhler Number Da = k * tau | CSTR Conversion (%) | PFR Conversion (%) |
|---|---|---|---|
| Long Residence Time (tau = 12 min) | 3.00 | 75.00% | 95.02% |
| Short Residence Time (tau = 2 min) | 0.50 | 33.33% | 39.35% |
| Invalid Negative Rate or Residence Time | Guarded Parameter Error | "Invalid Reactor Inputs" | Requires k > 0 and tau >= 0 |
Isothermal First-Order CSTR Conversion (%) = 100 * (k * tau) / (1 + k * tau)
Evaluates steady-state back-mixed tank conversion from rate constant B2 and space time B3.
=IF(OR(B2 <= 0, B3 < 0), "Invalid Reactor Inputs", ROUND(100 * ((B2 * B3) / (1 + B2 * B3)), 2))Isothermal First-Order Plug-Flow (PFR) Conversion (%) = 100 * (1 - EXP(-k * tau))
Evaluates steady-state tubular reactor conversion at the identical Damköhler number Da = B2 * B3.
=IF(OR(B2 <= 0, B3 < 0), "Invalid Reactor Inputs", ROUND(100 * (1 - EXP(-(B2 * B3))), 2))Try the example
Use the original values or change them to test your understanding.