Recycle reactors and autocatalytic reactions
The recycle PFR as a bridge between plug and mixed flow, and the best reactor choices for autocatalytic reactions, with recycle and CSTR-plus-PFR examples.
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Why it matters
Returning part of a plug flow reactor's exit stream to its inlet gives a reactor whose behaviour can be tuned continuously between plug flow and mixed flow. That is exactly what autocatalytic reactions (fermentations, some oxidations and polymerisations) need: product must be present for the reaction to go, so pure plug flow starves at the inlet while pure mixed flow is inefficient at high conversion.
Key ideas
The recycle reactor. A PFR in which a volumetric flow v_R of the exit stream is mixed with the fresh feed v0. The recycle ratio R = v_R/v0 = (volume of fluid returned)/(volume leaving the system). R = 0 is a plain PFR; R → ∞ is a CSTR. Note that this is an internal recycle of the full product stream, not the separation-and-recycle of unreacted reactant used in flowsheets.
Inlet condition. The fresh feed (conversion 0) mixes with recycle at the exit conversion Xf, so the stream entering the tube has X1 = [R/(R + 1)]·Xf. The tube carries (R + 1) times the fresh flow, so the design equation has a factor (R + 1) in front of the integral, which runs only from X1 to Xf.
Graphical meaning. On a plot of 1/(−rA) vs XA, the recycle reactor's τ/CA0 equals a rectangle of width Xf whose height is the average of 1/(−rA) over the range X1 to Xf. As R rises, the averaging range widens and the reactor approaches a CSTR.
For normal kinetics (n > 0) recycle always increases the volume needed (the best R is 0). Recycle is useful for (a) autocatalytic reactions, (b) temperature control of exothermic reactions (dilution of the feed), and (c) kinetic studies (high-R "differential" recycle reactors behave as CSTRs while using catalyst packed in a tube).
Autocatalytic reactions. A product catalyses its own formation, e.g. A + R → R + R with −rA = k·CA·CR. With CA + CR = C0 constant, the rate is zero when no R is present, rises as R forms, passes through a maximum at CA = CR = C0/2, and falls as A is used up. A plot of 1/(−rA) vs XA is therefore U-shaped.
Best reactor choice for autocatalytic reactions.
- At low conversion (below the rate maximum), a CSTR is smaller than a PFR.
- At high conversion, a PFR is smaller.
- The best single reactor is often a PFR with the optimum recycle ratio, chosen so that 1/(−rA) at the tube inlet equals the average of 1/(−rA) over the tube.
- The best combination of all: a CSTR operating exactly at the rate maximum (CA = C0/2), followed by a PFR to the final conversion. If unreacted A can be separated and recycled, operate a single CSTR at the maximum-rate point.
Formulas
R = v_R / v0 (recycle ratio; 0 for PFR, ∞ for CSTR)
X1 = [R/(R + 1)]·Xf (conversion entering the tube); CA1 = (CA0 + R·CAf)/(R + 1) (constant density)
- CA0, CAf: fresh feed and exit concentrations (mol/m³).
τ = CA0·V/FA0 = (R + 1)·CA0·∫(X1→Xf) dXA/(−rA) (recycle PFR)
kτ/(R + 1) = ln[(CA0 + R·CAf)/((R + 1)·CAf)] (first order, constant density)
−rA = k·CA·CR, CA + CR = C0 (autocatalytic A + R → 2R)
CA,opt = C0/2 (maximum rate)
k·C0·τ = ln[(CA,in·CR,out)/(CA,out·CR,in)] (PFR for A + R → 2R, constant density)
τ = (CA,in − CA,out)/(k·CA,out·CR,out) (CSTR for A + R → 2R)
- k in m³/kmol·s; concentrations in kmol/m³.
Worked examples
Example 1 (standard). A first-order liquid reaction must reach Xf = 0.9 in a PFR with recycle ratio R = 1. Find kτ and compare with a plain PFR and a CSTR.
- CAf/CA0 = 0.1. Formula: kτ = (R + 1)·ln[(CA0 + R·CAf)/((R + 1)·CAf)].
- Inside the log: (1 + 1 × 0.1)/(2 × 0.1) = 1.1/0.2 = 5.5.
- kτ = 2 × ln 5.5 = 2 × 1.705 = 3.41.
- PFR (R = 0): kτ = ln 10 = 2.30. CSTR: kτ = 0.9/0.1 = 9.
kτ = 3.41 — between plug flow (2.30) and mixed flow (9), and it rises toward 9 as R grows (R = 3 gives 4.71).
Example 2 (GATE level, autocatalytic). A + R → 2R, −rA = k·CA·CR, k = 1 m³/kmol·min. Feed: CA0 = 0.99 kmol/m³, CR0 = 0.01 kmol/m³ (C0 = 1.0). Target CA = 0.1 kmol/m³. Find τ for (a) the best CSTR + PFR combination, (b) a single CSTR, (c) a single PFR.
(a) CSTR to the rate maximum CA = CR = 0.5: τ1 = (0.99 − 0.5)/(1 × 0.5 × 0.5) = 1.96 min. PFR from CA = 0.5 to 0.1 (CR from 0.5 to 0.9): k·C0·τ2 = ln[(0.5 × 0.9)/(0.1 × 0.5)] = ln 9 = 2.197 → τ2 = 2.20 min. Total = 4.16 min. (b) Single CSTR: τ = (0.99 − 0.1)/(1 × 0.1 × 0.9) = 9.89 min. (c) Single PFR: k·C0·τ = ln[(0.99 × 0.9)/(0.1 × 0.01)] = ln 891 = 6.79 → 6.79 min.
The combination needs less than half the space time of a single CSTR.
Common mistakes
- Using FA0 of the fresh feed but forgetting the (R + 1) factor, or integrating from 0 instead of X1.
- Confusing an internal recycle reactor with a separator recycle of unreacted reactant.
- Putting a CSTR after the rate maximum (or a PFR before it) for autocatalytic reactions.
- Forgetting that an autocatalytic reaction will not start in a PFR fed with pure A — some R must be present.
- Writing the autocatalytic PFR result with CA0 instead of C0 = CA + CR.
For GATE CH
Questions ask for the recycle-reactor τ for first-order kinetics at a given R, the limiting behaviour (R = 0 and R → ∞), the concentration of maximum rate for an autocatalytic reaction, and the space time of a CSTR, a PFR or the optimum combination for A + R → 2R. Practise both integrated forms and the inlet-mixing balance.
Quick check
- What does a recycle reactor become as R → ∞?
- With R = 2 and Xf = 0.9, what is X1?
- For A + R → 2R with C0 = 2 kmol/m³, at what CA is the rate maximum?
- For first-order kinetics, is any R > 0 beneficial for reactor size?
Answers: 1. a CSTR; 2. X1 = (2/3) × 0.9 = 0.6; 3. CA = 1 kmol/m³; 4. no — R = 0 (plain PFR) gives the smallest volume.
Interview questions
All Chemical Reaction Engineering interview questionsTry answering each one aloud before you open it.
1.What is a recycle reactor in chemical reaction engineering?Concept
A recycle reactor is a type of chemical reactor where a portion of the reactor's output is fed back into the reactor's input. This setup can help improve conversion rates, control temperature, and manage reaction kinetics. Recycle reactors are often used in processes where complete conversion in a single pass is not feasible.
2.Explain the concept of autocatalytic reactions.Concept
Autocatalytic reactions are chemical reactions in which one of the products acts as a catalyst for the reaction itself. This means that as the reaction proceeds, the rate of reaction increases because the concentration of the catalyst (which is also a product) increases. These reactions can lead to exponential growth in reaction rate under certain conditions.
3.Why are recycle reactors used in industrial chemical processes?Application
Recycling part of a PFR's exit stream to its inlet lets the designer move the flow pattern anywhere between plug flow (R = 0) and mixed flow (R → ∞). This is valuable for autocatalytic reactions, which need product present at the inlet, and for strongly exothermic reactions, where diluting the feed with product moderates the temperature rise. High-recycle reactors are also used in the laboratory to obtain CSTR-like kinetic data with packed catalyst. For ordinary positive-order kinetics, however, recycle increases the volume needed for a given conversion.
4.What happens if the recycle ratio in a recycle reactor is increased?Application
As R increases the reactor moves from plug-flow toward mixed-flow behaviour: the tube inlet conversion X1 = R·Xf/(R + 1) approaches the exit value and concentrations inside become more uniform. For normal positive-order kinetics this lowers the conversion achieved in a given volume (or raises the volume needed), tending to the CSTR value. For autocatalytic reactions conversion first improves because product is supplied to the inlet, and there is an optimum R. Higher R also means a larger flow through the tube, so more pumping and pressure drop.
5.How does an autocatalytic reaction differ from a typical catalytic reaction?Concept
In a typical catalytic reaction, the catalyst is a separate substance that is not consumed in the reaction. In contrast, an autocatalytic reaction involves a product of the reaction acting as the catalyst. This means that the reaction rate can increase as more product is formed, leading to a self-accelerating process.
6.What are the potential challenges in designing a reactor for an autocatalytic reaction?Application
The rate is near zero when little product is present, so a plain PFR fed with pure reactant barely reacts; the reactor must be seeded with product, use recycle, or start with a mixed section. The rate passes through a maximum (at CA = CR for A + R → 2R), so a single CSTR is good at low conversion but poor at high conversion while a PFR is the reverse. The designer must therefore choose an optimum recycle ratio or a CSTR at the maximum-rate point followed by a PFR. Start-up and stability also need care because the reaction accelerates as product builds up.
7.For a first-order liquid reaction in a PFR with recycle ratio R = 1, what dimensionless space time kτ is needed for 90% conversion?Numerical
For a recycle PFR at constant density, kτ/(R + 1) = ln[(CA0 + R·CAf)/((R + 1)·CAf)]. With CAf/CA0 = 0.1 and R = 1, the log argument is (1 + 0.1)/(2 × 0.1) = 5.5, so kτ = 2 × ln 5.5 = 3.41. This lies between a plain PFR (ln 10 = 2.30) and a CSTR (9), as expected.
8.Describe a scenario where a recycle reactor would be preferred over a plain plug flow reactor.Application
The classic case is an autocatalytic reaction such as A + R → 2R, or a fermentation, where the rate is negligible without product present: a plain PFR fed with pure A hardly reacts at the inlet, whereas recycling product seeds the reaction and an optimum recycle ratio minimises volume. Recycle is also used for strongly exothermic reactions to dilute the feed and limit the temperature rise. For ordinary positive-order reactions under isothermal conditions a plain PFR remains the smaller reactor.
9.If an autocatalytic reaction is carried out in a batch reactor, what considerations should be made for safety?Application
In a batch reactor, an autocatalytic reaction can lead to rapid increases in reaction rate and temperature. Safety considerations should include robust temperature control systems, pressure relief mechanisms, and careful monitoring of reaction progress. It may also be necessary to design the reactor to handle potential runaway conditions.
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