Size comparison of single reactors
How CSTR and PFR volumes compare for the same duty, and how the ratio depends on reaction order, conversion and density change, with ratio and equal-volume examples.
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Why it matters
Choosing between a mixed-flow reactor (CSTR) and a plug flow reactor (PFR) is often the first design decision, and the volume difference can be a factor of ten or more at high conversion. Knowing how the size ratio depends on reaction order, conversion and density change lets you choose the right reactor quickly and explains why many plants use PFRs or cascades of CSTRs.
Key ideas
Same duty, different volumes. Compare reactors that handle the same feed (v0, CA0) and reach the same exit conversion XA. The ratio V_CSTR/V_PFR = τ_CSTR/τ_PFR depends only on the rate form and XA.
Why the CSTR is usually larger. A CSTR operates entirely at the exit concentration, which is the lowest concentration in the system. For any rate that increases with concentration (order n > 0), the rate inside a CSTR is the lowest possible, so more volume is needed. In a PFR the concentration falls gradually from CA0 to CA, so most of the reactor works at higher rates.
Levenspiel-plot picture. Plot 1/(−rA) (or FA0/(−rA)) against XA. The PFR needs the area under the curve; the CSTR needs the rectangle at the exit value. For normal kinetics the curve rises with XA, so the rectangle exceeds the area. If the curve is flat (zero order) they are equal. If the curve falls with XA (negative order, or the early part of an autocatalytic reaction), the CSTR is smaller.
Effect of order. At a fixed conversion the ratio V_CSTR/V_PFR is 1 for n = 0 and grows with n: for first order at XA = 0.9 it is about 3.9; for second order it is 10. Higher order means stronger dependence of rate on concentration, so the penalty for running at the exit concentration is greater.
Effect of conversion. The ratio grows steeply as XA → 1. At low conversion (XA < 0.2 or so) the two reactors are nearly the same size and a CSTR's other advantages (temperature control, easy cleaning) may dominate. At XA = 0.99, first order, the CSTR is about 21 times larger.
Effect of density change. For gas-phase reactions, expansion (εA > 0) increases the ratio slightly because the CSTR sees the fully expanded, diluted exit stream; contraction (εA < 0) decreases it. The effect is secondary compared with order.
Batch vs PFR. An ideal batch reactor and a PFR give the same conversion when batch reaction time t equals PFR space time τ (constant density), because each fluid element in a PFR behaves as a tiny batch reactor. The batch reactor's real disadvantage is down-time between batches.
Other factors. Size is not everything. CSTRs are preferred for highly exothermic liquid reactions (uniform temperature, easy cooling), for slurries and for operation at fixed composition; PFRs (tubular, packed beds) for gas phase, high pressure and high conversion. Selectivity in multiple reactions can override size arguments (see the multiple-reactions topic).
Formulas
V_CSTR/V_PFR = [XA/(1 − XA)] / ln[1/(1 − XA)] (first order, constant density)
- XA: exit conversion (dimensionless).
V_CSTR/V_PFR = [XA/(1 − XA)²] / [XA/(1 − XA)] = 1/(1 − XA) (second order −rA = k·CA², constant density)
(τ·CA0^(n−1))_CSTR / (τ·CA0^(n−1))_PFR = [XA/(1 − XA)^n] / {[(1 − XA)^(1−n) − 1]/(n − 1)} (n-th order, n ≠ 1, constant density)
- τ: space time (s); CA0: feed concentration (mol/m³); n: order.
V_CSTR = V_PFR for zero order (rate independent of concentration).
Same-size comparison, first order: XA,PFR = 1 − exp(−kτ), XA,CSTR = kτ/(1 + kτ)
- k: rate constant (s⁻¹).
Worked examples
Example 1 (standard). A liquid-phase first-order reaction is to reach XA = 0.9. Find V_CSTR/V_PFR.
- CSTR: kτ = XA/(1 − XA) = 0.9/0.1 = 9.
- PFR: kτ = ln[1/(1 − 0.9)] = ln 10 = 2.303.
- Ratio = 9/2.303 = 3.91.
The CSTR must be about 3.9 times larger.
Example 2 (GATE level, equal volumes). A second-order liquid reaction (−rA = k·CA²) has k·CA0·τ = 4 in both a PFR and a CSTR of equal volume. Find the conversion in each.
- PFR: k·CA0·τ = XA/(1 − XA) → 4 = XA/(1 − XA) → XA = 4/5 = 0.80.
- CSTR: k·CA0·τ = XA/(1 − XA)² → 4(1 − XA)² = XA → 4XA² − 9XA + 4 = 0.
- XA = [9 − √(81 − 64)]/8 = (9 − 4.123)/8 = 0.610 (the other root, 1.64, is physically impossible).
PFR: XA = 0.80; CSTR: XA ≈ 0.61.
Example 3 (order effect). For second order at XA = 0.9: ratio = 1/(1 − 0.9) = 10, compared with 3.9 for first order and 1 for zero order.
Common mistakes
- Assuming the PFR is always smaller — not for zero order (equal) or for negative-order or autocatalytic kinetics at low conversion.
- Choosing the wrong root of the CSTR quadratic; conversion must lie between 0 and 1.
- Comparing reactors at different feed concentrations: τ·CA0^(n−1) must be compared, not τ alone, when n ≠ 1.
- Forgetting batch down-time when claiming batch and PFR are "the same size".
For GATE CH
Common question types: ratio of CSTR to PFR volume for given order and conversion (NAT), conversion in a CSTR and PFR of the same volume, recognising from a 1/(−rA) vs XA plot which reactor is smaller, and conceptual MCQs on the effect of order and conversion. Practise solving the CSTR quadratic for second order and reading areas versus rectangles from plots.
Quick check
- For a zero-order reaction, how do V_CSTR and V_PFR compare?
- For first order with kτ = 2 in each, find XA for a CSTR and a PFR.
- Does the CSTR/PFR size ratio increase or decrease with conversion for n > 0?
- When can a CSTR be smaller than a PFR?
Answers: 1. equal; 2. CSTR 0.667, PFR 0.865; 3. it increases; 4. when the rate increases with conversion — negative-order kinetics, or the early part of an autocatalytic reaction.
Interview questions
All Chemical Reaction Engineering interview questionsTry answering each one aloud before you open it.
1.What is a chemical reactor, and why is it important in chemical engineering?Concept
A chemical reactor is a vessel designed to contain and control chemical reactions. It is important in chemical engineering because it allows for the efficient conversion of raw materials into desired products under controlled conditions. Reactors are crucial for optimizing reaction rates, yields, and selectivity, which are essential for industrial production processes.
2.What are the advantages of using a plug flow reactor (PFR) over a CSTR?Concept
For any positive-order reaction a PFR needs less volume than a CSTR for the same conversion, because the concentration falls gradually along the tube instead of the whole vessel operating at the low exit concentration; the advantage grows with order and conversion. Absence of back-mixing also favours selectivity when the desired product is an intermediate in series reactions, or when the desired reaction has the higher order. Tubular reactors suit gas-phase and high-pressure service. The trade-off is that temperature control is harder in a PFR, so hot spots can occur with strongly exothermic reactions, where a CSTR is easier to control.
3.Why is reactor size important in chemical reaction engineering?Application
Reactor size is crucial because it affects the conversion, selectivity, and yield of the reaction. A properly sized reactor ensures optimal contact time between reactants, efficient heat and mass transfer, and cost-effectiveness. Oversized reactors may lead to unnecessary capital and operating costs, while undersized reactors may not achieve the desired conversion or selectivity.
4.What happens if a reactor is undersized for a given chemical process?Application
An undersized reactor gives too short a space time, so the exit conversion falls below the design value. More unreacted feed then has to be separated and recycled, which loads the downstream separation units and raises operating cost, and product specifications may be missed. For multiple reactions the product distribution also shifts, which can help or hurt selectivity. To recover conversion you would need to lower throughput or raise temperature, both of which have their own costs and limits.
5.How does the choice of reactor type affect the design and operation of a chemical process?Application
The choice of reactor type affects the design and operation by determining the flow pattern, mixing characteristics, heat and mass transfer rates, and residence time distribution. For example, a CSTR provides uniform mixing and is suitable for reactions requiring constant temperature, while a PFR offers a gradient of concentration and temperature along its length, which can be advantageous for certain reaction kinetics.
6.Why might a chemical engineer choose a series of smaller reactors instead of one large reactor?Application
A cascade of CSTRs approaches plug flow: each tank runs at a progressively lower concentration, so for positive-order kinetics the total volume is smaller than a single CSTR of the same conversion. With N tanks the total volume moves toward the PFR value as N increases. A series arrangement also allows different temperatures or intermediate cooling/feeding in each stage, and gives some operating flexibility. The cost is more vessels, agitators and instruments, so in practice 3–5 tanks capture most of the benefit.
7.Calculate the volume of a liquid-phase CSTR required to achieve 80% conversion for a first-order reaction with k = 0.5 s⁻¹ and a volumetric feed rate of 2 m³/s.Numerical
For first order at constant density in a CSTR, kτ = X/(1 − X) = 0.8/0.2 = 4, so τ = 4/0.5 = 8 s. The volume is V = v0·τ = 2 m³/s × 8 s = 16 m³. For comparison, a PFR would need τ = ln 5/0.5 = 3.22 s, i.e. about 6.4 m³.
8.Determine the space time for a PFR with a volume of 5 m³ and a volumetric feed rate of 1 m³/s.Numerical
Space time is τ = V/v0, where v0 is the volumetric feed rate measured at inlet conditions. τ = 5 m³ / 1 m³/s = 5 s, and the space velocity is 1/τ = 0.2 s⁻¹. For a constant-density system this also equals the mean residence time; for a gas reaction with volume change it does not.
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