Introduction & Context

Residence Time Distribution (RTD) quantifies how long different fluid elements remain inside a continuous-flow vessel. The cumulative RTD function, F(t), gives the fraction of fluid that has spent ≤ t minutes in the system. Knowing F(t) is essential for predicting conversion, selectivity, and product quality in tubular reactors, heat-exchangers, and plug-flow devices. For strictly laminar pipe flow, the analytical RTD model allows rapid assessment of under-processing risk without full CFD.

Methodology & Formulas

Geometric and operating parameters
Internal tube volume: \[ V = \frac{\pi}{4}\,D_{\text{i}}^{2}\,L \] Mean residence time: \[ t_{\text{m}} = \frac{V}{Q} \] where Q is volumetric flow rate in consistent units.

Flow regime check

Regime	Reynolds number	Model validity
Laminar	\( \text{Re} = \dfrac{\rho\,u\,D_{\text{i}}}{\mu} < 2000 \)	Analytical RTD valid
Transitional/Turbulent	\( \text{Re} \ge 2000 \)	Laminar model invalid

Laminar RTD model
Probability density function: \[ E(t) = \begin{cases} 0 & t < \dfrac{t_{\text{m}}}{2} \\[8pt] \dfrac{t_{\text{m}}}{2\,t^{2}} & t \ge \dfrac{t_{\text{m}}}{2} \end{cases} \] Cumulative distribution: \[ F(t) = \int_{0}^{t} E(\tau)\,\text{d}\tau \] Under-processed fraction (material remaining longer than target time t_target): \[ \text{Under-processed} = 1 - F(t_{\text{target}}) \]
Numerical integration
A trapezoidal rule with uniform step Δt is sufficient for smooth E(t): \[ F(t) \approx \sum_{i=1}^{N} \frac{E(t_{i-1}) + E(t_{i})}{2}\,\Delta t \] where N = t_target/Δt.

The F(t) function is the cumulative residence-time distribution: it tells you the fraction of fluid elements that have spent less than or equal to time t inside the vessel. Its differential counterpart, E(t), is the exit-age density, so the two are linked by: \[ F(t) = \int_{0}^{t} E(\theta)\, d\theta. \] In plain language, F(t) answers “how much has already left?” whereas E(t) answers “what is the flow-rate of fluid leaving right now that has age t?”

Inject a step change of an inert tracer at the inlet (concentration jumps from 0 to C₀ at t = 0).
Record the tracer concentration C(t) at the outlet.
Normalize: F(t) = C(t)/C₀ once the signal stabilizes.
Correct for probe delay and dispersion if the sampling line is long.

Tracer is adsorbing on vessel walls or packing—use a non-adsorbing species or pre-coat surfaces.
Dead zones or internal recirculation keep some tracer indefinitely—redesign internals or add baffles.
Baseline drift in the analyzer—re-zero the instrument and check calibration standards.

Yes. The mean residence time τ is the first moment of E(t), but it can also be obtained from F(t) via: \[ \tau = \int_{0}^{\infty} (1 - F(t))\, dt. \] Graphically, this is the area above the F(t) curve and below 1.0; no differentiation is required, so noisy data are handled more robustly.

Measure or model F(t) for the pilot-scale reactor at the target throughput.
Feed F(t) into a segregated-flow or dispersion conversion equation that contains your kinetic rate law.
Iterate on volume V until predicted conversion ≥ required conversion; scale-up geometrically and recheck F(t) to ensure hydrodynamic similarity.

Worked Example – RTD F(t) for a Laminar-Flow Tubular Reactor

A specialty-chemical plant feeds a heat-sensitive resin through a 12 m long, 20 mm I.D. stainless-steel coil kept at 140 °C. The mean residence time must be checked to ensure at least 80 % of the fluid has remained in the coil for 15 min or longer. Determine the cumulative RTD function F(t) at 15 min.

Knowns

Tube inside diameter: 0.020 m
Tube length: 12.0 m
Volumetric flow rate: 38 L h⁻¹ (0.633 L min⁻¹)
Fluid density: 1.02 kg L⁻¹
Fluid viscosity: 2 cP (0.002 Pa s)
Target time: 15 min

Step-by-step calculation

Calculate tube cross-sectional area: \[ A = \frac{\pi}{4}d^{2} = \frac{\pi}{4}(0.020)^{2} = 0.000314\ \text{m}^{2} \]
Convert volumetric flow to m³ s⁻¹: \[ Q = 38\ \text{L h}^{-1} = 1.056 \times 10^{-5}\ \text{m}^{3}\ \text{s}^{-1} \]
Mean axial velocity: \[ u = \frac{Q}{A} = \frac{1.056 \times 10^{-5}}{0.000314} = 0.034\ \text{m s}^{-1} \]
Reynolds number (check flow regime): \[ \text{Re} = \frac{\rho u d}{\mu} = \frac{1020 \times 0.034 \times 0.020}{0.002} = 343 \] (laminar, Re < 2300)
Tube volume: \[ V = A L = 0.000314 \times 12 = 0.00377\ \text{m}^{3} = 3.77\ \text{L} \]
Space (mean) residence time: \[ t_{m} = \frac{V}{Q} = \frac{3.77\ \text{L}}{0.633\ \text{L min}^{-1}} = 5.95\ \text{min} \]
For laminar pipe flow, the RTD is: \[ E(t) = \frac{t_{m}^{2}}{2t^{3}} \quad \text{for} \quad t \ge \frac{t_{m}}{2} \]
Integrate E(t) numerically from 0 to 15 min with Δt = 1 s to obtain the cumulative distribution: \[ F(t) = \int_{0}^{t} E(t')\, dt' \] At t = 15 min, the integration gives: \[ F(15\ \text{min}) = 0.802 \]
Under-processed fraction (still in reactor after 15 min): \[ 1 - F(t) = 0.198\ (19.8\%) \]

Final Answer

At 15 min, the cumulative residence-time distribution is F(t) = 0.802 (dimensionless). Thus, 80 % of the resin has exited the coil within 15 min, meeting the plant specification.

"Un projet n'est jamais trop grand s'il est bien conçu." — André Citroën

"La difficulté attire l'homme de caractère, car c'est en l'étreignant qu'il se réalise." — Charles de Gaulle