Segregation, dispersion and tanks-in-series models

Turning an RTD into conversion with the segregation, tanks-in-series and axial dispersion models, including macro- and microfluids, with worked first-order predictions from tracer data.

Drafted with Aria, reviewed by the AiCanCode.org team. Spotted an error? Use Give Feedback at the bottom of the page.

Why it matters

An RTD by itself is a curve; to design or troubleshoot a reactor you need a model that turns that curve into a conversion. The segregation, tanks-in-series and dispersion models are the standard one-parameter tools: they let you predict how far a real tubular or stirred reactor falls short of the ideal, and scale up from pilot data.

Key ideas

Macrofluids and microfluids. In a macrofluid (segregated flow) fluid elements travel as closed packets that never exchange molecules; each packet behaves as a small batch reactor whose reaction time is its residence time. In a microfluid molecules mix freely with neighbours. Most liquids of normal viscosity and gases behave as microfluids; very viscous liquids, droplets and solid particles behave as macrofluids.

Segregation model. Average the batch conversion over the RTD: X̄ = ∫X_batch(t)·E(t) dt. It needs only the RTD and batch kinetics. For first-order reactions, conversion does not depend on the state of mixing, so the segregation model gives the exact answer for any vessel with a known RTD. For n > 1, segregation gives higher conversion than maximum mixedness; for n < 1, lower. Early versus late mixing therefore matters only for non-linear kinetics.

Tanks-in-series (TIS) model. Represent the vessel as N equal ideal CSTRs in series with total mean residence time t̄. The single parameter N is fitted from the RTD variance: N = 1/σθ² = t̄²/σ². N = 1 is a CSTR; N → ∞ is plug flow. N need not be an integer for fitting purposes. It suits stirred-tank cascades, and also tubular vessels as a simple alternative to the dispersion model.

Axial dispersion model. Plug flow with superimposed back-mixing described by a Fick-type axial dispersion coefficient D (m²/s). The dimensionless vessel dispersion number D/(u·L) — the inverse of the Péclet number Pe = u·L/D — measures the departure from plug flow:

  • D/uL → 0: plug flow; D/uL → ∞: mixed flow.
  • Small deviation (D/uL < 0.01): the RTD is nearly Gaussian, σθ² ≈ 2·(D/uL), independent of boundary conditions.
  • Larger deviations require boundary conditions; for a closed vessel σθ² = 2(D/uL) − 2(D/uL)²·[1 − exp(−uL/D)]. Typical use: packed beds, pipes, tubular reactors. D comes from correlations of the vessel dispersion number against Reynolds and Schmidt numbers — take these from your data book.

Matching the two one-parameter models. For small deviations, equate variances: 1/N ≈ 2·(D/uL).

Conversion with first-order kinetics.

  • TIS: CA/CA0 = 1/(1 + kτ/N)^N.
  • Small dispersion: CA/CA0 ≈ exp[−kτ + (kτ)²·(D/uL)].
  • Large dispersion: use the Wehner–Wilhelm solution (from the data book).

Formulas

X̄ = ∫(0→∞) X_batch(t)·E(t) dt; CA/CA0 (mean) = ∫ (CA/CA0)_batch·E dt (segregation)

  • X_batch(t): conversion of a batch reactor after time t.

First-order segregation: CA/CA0 = ∫ exp(−k·t)·E(t) dt

Eθ = N·(N·θ)^(N−1)·exp(−N·θ)/(N − 1)! (TIS, θ = t/t̄)

σθ² = 1/N, N = t̄²/σ²

CA/CA0 = 1/(1 + k·t̄/N)^N (TIS, first order)

Pe = u·L/D; σθ² ≈ 2·(D/uL) (small dispersion)

  • u: mean axial velocity (m/s); L: vessel length (m); D: axial dispersion coefficient (m²/s).

σθ² = 2(D/uL) − 2(D/uL)²·[1 − exp(−uL/D)] (closed vessel)

CA/CA0 ≈ exp[−kτ + (kτ)²·(D/uL)] (first order, small D/uL)

Worked examples

Example 1 (standard, tanks in series). The pulse test of the previous topic gave t̄ = 15 min and σθ² = 0.211. A first-order reaction with k = 0.2 min⁻¹ runs in this vessel. Predict the conversion with the TIS model and compare with ideal reactors.

  1. N = 1/σθ² = 1/0.211 = 4.74.
  2. k·t̄ = 0.2 × 15 = 3.
  3. CA/CA0 = 1/(1 + 3/4.74)^4.74 = 0.0979 → X = 0.902.
  4. PFR: X = 1 − e⁻³ = 0.950. CSTR: X = 3/4 = 0.75.

X ≈ 0.90 — between the two ideal limits, closer to plug flow.

Example 2 (GATE level, segregation with tracer data). For the same vessel (E = 0.03, 0.05, 0.05, 0.04, 0.02, 0.01 min⁻¹ at t = 5, 10, 15, 20, 25, 30 min, Δt = 5 min) find the conversion for a first-order reaction with k = 0.1 min⁻¹.

  1. CA/CA0 = Σ exp(−k·t)·E·Δt.
  2. Terms: e^(−0.5)(0.15) = 0.0910; e^(−1)(0.25) = 0.0920; e^(−1.5)(0.25) = 0.0558; e^(−2)(0.20) = 0.0271; e^(−2.5)(0.10) = 0.0082; e^(−3)(0.05) = 0.0025.
  3. Sum = 0.2765 → X = 1 − 0.2765 = 0.724.

X ≈ 0.72 (compared with 0.777 for a PFR and 0.60 for a CSTR at t̄ = 15 min). Because the reaction is first order, this answer is exact for this RTD.

Example 3 (small dispersion). A tubular reactor has D/uL = 0.01 and kτ = 2 for a first-order reaction. CA/CA0 ≈ exp(−2 + 4 × 0.01) = e^(−1.96) = 0.141, versus e^(−2) = 0.135 for plug flow: X ≈ 0.859 against 0.865 — a small penalty.

Common mistakes

  • Using σ² in min² instead of the dimensionless σθ² when finding N.
  • Applying the small-dispersion approximation at D/uL around 0.1 or more.
  • Assuming the segregation model is exact for second-order reactions — for n ≠ 1 the micromixing state matters.
  • Confusing D (dispersion, m²/s) with molecular diffusivity; dispersion is usually far larger.
  • Forgetting that each tank in the TIS model has τi = t̄/N, not t̄.

For GATE CH

Expect NAT questions on N from a tracer variance, conversion with N tanks or with the small-dispersion formula, the segregation integral for first order with simple E curves (CSTR or tabulated), Pe or D/uL calculations, and conceptual MCQs on macro- vs microfluids and when mixing state matters. Practise the discrete segregation sum and the TIS formula.

Quick check

  1. What does N = 1 correspond to in the TIS model?
  2. If σθ² = 0.25, what is N?
  3. For which reaction order is conversion independent of the degree of micromixing?
  4. Using the segregation model, what conversion does a CSTR give for first order? (Integrate e^(−kt)·e^(−t/τ)/τ.)

Answers: 1. a single CSTR; 2. N = 4; 3. first order; 4. X = kτ/(1 + kτ), identical to the CSTR design equation.

Interview questions

All Chemical Reaction Engineering interview questions

Try answering each one aloud before you open it.

  1. 1.What is the segregation model in chemical reaction engineering?Concept

    The segregation model treats the fluid as a macrofluid: it moves through the vessel in small packets that never exchange molecules with each other, so each packet behaves as a small batch reactor whose reaction time equals its residence time. The mean exit conversion is the batch conversion averaged over the RTD, X̄ = ∫X_batch(t)·E(t) dt. It needs only the RTD and the batch kinetics. For first-order reactions it gives the exact answer for any degree of mixing; for other orders it is one limit of micromixing (higher conversion than maximum mixedness when n > 1).

  2. 2.Explain the dispersion model and its significance in chemical reactors.Concept

    The dispersion model describes the spread of fluid elements in a reactor due to velocity gradients and molecular diffusion. It is significant because it helps in understanding how deviations from ideal plug flow occur. The model is characterized by the dispersion coefficient, which quantifies the extent of mixing along the flow path. This model is particularly useful for tubular reactors where axial mixing is a concern.

  3. 3.What is the tanks-in-series model, and how does it approximate real reactor behavior?Concept

    The tanks-in-series model approximates real reactor behavior by representing a reactor as a series of perfectly mixed tanks. Each tank represents a stage of mixing, and the number of tanks determines the degree of mixing. This model is useful for simulating non-ideal flow patterns and can approximate both plug flow and complete mixing by adjusting the number of tanks.

  4. 4.Why is the dispersion model preferred over the tanks-in-series model for some reactors?Application

    The axial dispersion model is physically natural for pipes, packed beds and tubular reactors, where the deviation from plug flow comes from velocity profiles and eddy mixing along the axis. Its parameter, the vessel dispersion number D/uL, can be estimated from correlations against Reynolds and Schmidt numbers before any tracer test, which helps in design and scale-up. For small deviations the two models are equivalent with 1/N ≈ 2(D/uL), so the choice is often one of convenience.

  5. 5.How does increasing the number of tanks in the tanks-in-series model affect reactor performance?Application

    Increasing the number of tanks in the tanks-in-series model generally leads to a behavior closer to plug flow. As the number of tanks increases, the residence time distribution narrows, reducing the variance and making the flow more uniform. This can improve reactor performance by minimizing back-mixing and enhancing conversion efficiency.

  6. 6.What happens if the dispersion coefficient is very high in a tubular reactor?Application

    If the dispersion coefficient is very high in a tubular reactor, it indicates significant axial mixing, leading to behavior closer to that of a completely mixed reactor. This can result in reduced conversion efficiency for reactions that benefit from plug flow conditions, as reactants may exit the reactor before fully reacting.

  7. 7.In what scenarios would you use the segregation model over the tanks-in-series model?Application

    Use the segregation model when you have a measured RTD and want conversion directly without fitting a flow model: it is exact for first-order reactions in any vessel, and it describes genuine macrofluids such as viscous polymer melts, droplets or solid particles that react independently. The tanks-in-series model is preferred when you need a simple one-parameter description for scale-up or when the vessel physically consists of stages. For non-first-order reactions in microfluids neither is exact, and the segregation result is only one bound on conversion.

  8. 8.A tracer test gives a mean residence time of 5 min and a variance of 2.5 min². How many tanks are needed in a tanks-in-series model?Numerical

    For the tanks-in-series model the dimensionless variance is σθ² = σ²/t̄² = 1/N. Here σθ² = 2.5/5² = 0.1, so N = 1/0.1 = 10 tanks. Each tank then has τi = t̄/N = 0.5 min. Always convert to the dimensionless variance first — mixing σ² in min² with the formula 1/N is a common error.

  9. 9.A reactor has a dispersion number (D/uL) of 0.01. Is this reactor closer to plug flow or mixed flow, and why?Application

    A dispersion number (D/uL) of 0.01 indicates that the reactor is closer to plug flow. The dispersion number quantifies the extent of axial mixing; a lower value suggests minimal mixing and behavior similar to plug flow. In contrast, a higher dispersion number would indicate more mixing and behavior closer to a completely mixed flow.

  10. 10.For a given reaction, how would you decide between using the dispersion model and the tanks-in-series model?Application

    The choice between the dispersion model and the tanks-in-series model depends on the reactor type and the flow characteristics. The dispersion model is suitable for tubular reactors where axial mixing is significant. The tanks-in-series model is more appropriate for systems where the flow can be approximated by a series of well-mixed stages, such as in stirred tank reactors. Consider the degree of mixing and the reactor design when making the decision.

Finished this topic? Mark it so your progress, study plan and readiness keep up.

Stuck on something here?

PreviousResidence time distribution: E and F curvesNext Kinetics of heterogeneous catalytic reactions