10.4 PFR Design and Reactor Selection

10.4 PFR Design and Reactor Selection

Quick Answer: PFR: plug flow, (V = F_{A0}\int_0^X dX/(-r_A)). For n > 0, V_PFR < V_CSTR at the same X. Many CSTRs in series → PFR. Pick reactors for conversion, selectivity, heat, and continuity—not by name alone.

Section 10.3 showed that a single CSTR pays a steep volume penalty at high conversion. The plug flow reactor (PFR)—and trains of tanks that mimic it—is how continuous plants recover efficient use of volume. This section states the PFR design integral, compares volumes, and gives a compact reactor selection map including a selectivity introduction.

Plug Flow Idealization

An ideal PFR (tubular reactor idealization) assumes:

1. Plug flow: flat velocity profile; fluid elements do not overtake each other.
2. No axial mixing (no dispersion): each slice of fluid is a small batch moving down the tube.
3. Perfect radial mixing (uniform properties in a cross-section).
4. Steady state for continuous operation.

Composition (and often T, if nonisothermal) varies along the length z. At the inlet, conditions match the feed; at the outlet, they match the product. Unlike a CSTR, most of the volume does not sit at the lean exit concentration.

Real tubular reactors have some axial dispersion; packed beds have channeling risks. Exam items use the ideal PFR unless dispersion is mentioned.

PFR Design Integral

Mole balance on a differential slice and integration give the classic design equation:

[ \frac{V}{F_{A0}} = \int_{0}^{X}\frac{dX}{(-r_A)} ]

or

[ V = F_{A0}\int_{0}^{X}\frac{dX}{(-r_A)} ]

Levenspiel interpretation: V/F_{A0} equals the area under the curve of 1/(−r_A) versus X from 0 to the final conversion—not a rectangle to the final 1/(−r_A).

For constant density first-order kinetics (−r_A = k C_{A0}(1−X)):

[ \tau = \frac{V}{v_0} = \frac{1}{k}\ln\frac{1}{1-X} ]

which matches the batch time expression for the same k and X. That is the deep analogy: PFR space time ↔ batch reaction time for identical kinetics and density assumptions.

| Kinetics (isothermal, const. density liquid) | PFR τ to conversion X | |---|---|---| | First order | (1/k) ln[1/(1−X)] | | Second order (A only) | (1/(k C_{A0})) [X/(1−X)] | | Zero order | (C_{A0}/k) X (while form valid) |

Worked PFR vs CSTR numbers (first order)

Same k = 0.5 h⁻¹, X = 0.80 as in Section 10.3:

• PFR: τ = (1/0.5) ln(1/0.20) = 2 ln 5 ≈ 3.22 h
• Single CSTR: τ = X/[k(1−X)] = 0.8/(0.5×0.2) = 8.0 h

At X = 0.95:

• PFR: τ = 2 ln 20 ≈ 6.0 h
• CSTR: τ = 0.95/(0.5×0.05) = 38 h

Volume ratio V_CSTR/V_PFR ≈ 38/6 ≈ 6.3 at 95% conversion—exam-scale illustration that tubular (or batch-equivalent) designs dominate when high X and positive order meet.

PFR vs CSTR Volume for Positive-Order Kinetics

General rule (Fogler/Levenspiel standard): when (−r_A) is a decreasing function of conversion (typical irreversible reactions with order > 0),

[ V_{\mathrm{PFR}} < V_{\mathrm{CSTR}} ]

for the same F_{A0}, X, and kinetics. Graphically, the area under 1/(−r_A) is less than the rectangle of height 1/(−r_A)|_final.

CSTR then PFR: sometimes a small CSTR (or recycle) handles startup of autocatalytic systems, then a PFR finishes conversion—advanced recognition only; most UPDA items stick to the positive-order default.

Series CSTRs Approaching a PFR

N equal-sized ideal CSTRs in series, total volume V_total, intermediate conversions X₁ < X₂ < … < X_N = X_final:

• Each tank uses V_i = F_{A0}(X_i − X_{i−1}) / (−r_A at X_i).
• Intermediate concentrations are higher than the final exit early in the train ⇒ higher rates in upstream tanks.
• As N → ∞, the staircase of concentrations approaches the smooth PFR profile and V_total → V_PFR.

Why industry still uses tanks: easier maintenance, staged heat removal, flexibility, multiphase contacting—not because one CSTR is kinetically optimal for high X.

Reactor Choice: Conversion, Selectivity, and Operations

Conversion / volume efficiency

Selectivity introduction (multiple reactions)

When more than one reaction occurs, mixing pattern changes product distribution.

Parallel reactions (A → D desired, A → U undesired):

• If both rates are positive order in A, keeping C_A high often helps the higher-order desired path—batch/PFR better than a dilute single CSTR.
• If undesired path has higher order in A, a CSTR (low C_A) can improve selectivity to D—classic qualitative reverse of the volume rule.

Series reactions (A → B desired → C undesired):

• Intermediate B is favored by short times and PFR/batch profiles that do not instantly dilute to low C_A with long holdup at conditions that destroy B.
• A single large CSTR holds everything at low C_A and can over-react intermediates—often poor for maximum B.

Heat and practicality (exam-level)

Putting Domain D Together

Quick decision tree for MCQs:

1. Identify kinetics order and whether rate falls as conversion rises.
2. Identify reactor type and where (−r_A) is evaluated (exit vs integral).
3. For high X and n > 0: PFR/batch/series CSTRs beat one CSTR on volume.
4. If selectivity/stem mentions intermediate or parallel orders, do not optimize volume alone.
5. Temperature and catalyst change k (and sometimes selectivity), not the form of the mole balance.

UPDA Exam Checklist for Section 10.4

1. PFR: V/F_{A0} = ∫ dX/(−r_A); area under Levenspiel curve.
2. First-order isothermal const. density: τ_PFR = (1/k) ln[1/(1−X)].
3. Positive order ⇒ V_PFR < V_one CSTR at same X.
4. Infinite CSTRs in series ↔ PFR.
5. Series intermediates: beware long residence in one mixed tank.
6. Catalyst/T still enter only through (−r_A); balances stay the same structure.

Master these four sections and you cover Domain D’s CRE core for the UPDA/MMUP Chemical exam. Domain E next turns to instrumentation, control, and process safety—how plants measure, regulate, and protect the reactors you just sized.