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Qwestrum Engineering360 · Chemical Engineering · Chemical Reaction Engineering

Isothermal Reactor Design

Isothermal reactor sizing integrates the mole balance: a PFR needs the integral of 1/(−r_A) over conversion, while a CSTR uses the single outlet-rate value; for positive-order kinetics the PFR is always the smaller reactor.

Exam tip: keep SI units consistent end-to-end, write the governing relation symbolically before substituting, and sanity-check magnitude and sign.

Key formulas & points

Skim these first — then read the full notes below.

Levenspiel plot: 1/(−r_A) vs X — area gives V/F_A0 for a PFR
Multiple reactors in series: outlet of one is inlet of the next
Constant density: space time τ equals batch time t

Topic details

Introduction

This topic converts kinetics into reactor volume for the two ideal reactors. You write the design equation for a plug-flow or continuous-stirred reactor, use the Levenspiel plot of 1/(−r_A) versus conversion to visualise the required volume as an area (PFR) or a rectangle (CSTR), and compare or combine reactors in series to minimise total volume.

Key relations & formulas

V = F_{A0} \int _{0}^X \frac{dX}{(-r_{A})}

(PFR design equation)

V = F_{A0} \frac{X}{(-r_{A})}|_X

(CSTR, rate evaluated at outlet)

\tau = C_{A0} \frac{X}{(-r_{A})}|_X

(space time for a CSTR, constant density)

Notation and sign conventions

Relation 1 —

V = F_{A0} \int _{0}^X dX /

V = F_{A0} \int _{0}^X \frac{dX}{(-r_{A})}

(PFR design equation)

Write this relation with symbols exactly as in Chemical Reaction Engineering — Octave Levenspiel before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.

Relation 2 —

V = F_{A0} X /

V = F_{A0} \frac{X}{(-r_{A})}|_X

(CSTR, rate evaluated at outlet)

Relation 3 —

\tau = C_{A0} X /

\tau = C_{A0} \frac{X}{(-r_{A})}|_X

(space time for a CSTR, constant density)

Concept in depth

A plug-flow reactor experiences the whole range of concentrations from inlet to outlet, so its volume is the integral (area under the 1/(−r_A) curve); a CSTR operates entirely at the low outlet concentration, so its rate is small and its required volume is the rectangle at that single point. For positive-order kinetics the rate falls as conversion rises, making the CSTR rectangle larger than the PFR area — hence a PFR is more volume-efficient. Placing a CSTR first (to jump past the high-rate region cheaply) followed by a PFR, or several CSTRs in series, is how designers approximate PFR performance while keeping the mixing benefits of stirred tanks.

Assumptions and validity limits

State assumptions explicitly before using any relation for isothermal reactor design — steady state, uniform properties, linear elastic material, ideal gas, incompressible flow, etc., as applicable.

Wrong assumptions invalidate the entire solution even when the formula is correct. In Reaction Engineering viva and GATE descriptive questions, listing valid assumptions often earns separate marks.

Step-by-step problem approach

1. Read the question and list given data with SI units (common in Reaction Engineering papers).
2. Draw a neat labelled diagram where applicable — examiners in Indian universities award diagram marks even when arithmetic slips.
3. Identify which relation from this topic applies to isothermal reactor design.
4. Use equation 1:

V = F_{A0} \int _{0}^X dX /

.
5. Use equation 2:

V = F_{A0} X /

.
6. Substitute values, compute, and verify units and sign (direction).
7. State conclusion in one line — e.g. safe/unsafe, stable/unstable, feasible/infeasible.

Applications & exam relevance

Isothermal Reactor Design appears in chemical and pharma plants. In Indian chemical curricula this topic is tested because it connects theory to reactor design and kinetics.

GATE and semester exams often combine isothermal reactor design with earlier units — revise prerequisites before attempting mixed problems.

Industry interview panels sometimes ask: "Where did you use isothermal reactor design?" — answer with a lab, mini-project, or plant visit example if possible.

Common mistakes in exams

Students evaluate the CSTR rate at inlet rather than outlet conditions, forget to base conversions on the same limiting reactant when sizing reactors in series, and confuse space time τ with residence time when density changes. Integrating the PFR equation with a constant rate is a conceptual slip.

Quick revision checklist

Before attempting isothermal reactor design problems, confirm you can:

1. Levenspiel plot: 1/(−r_A) vs X — area gives V/F_A0 for a PFR
2. Multiple reactors in series: outlet of one is inlet of the next
3. Constant density: space time τ equals batch time t

Revise the solved examples in Chemical Reaction Engineering — Octave Levenspiel and one previous-year GATE or university paper for this unit.

Worked examples

Try the problem first — open the solution when you are ready to check.

CSTR volume for first order

Problem

A liquid first-order reaction (k = 0.2 min⁻¹) needs 80% conversion. Feed is 10 L/min, C_A0 = 2 mol/L. Find the CSTR volume.

Solution

−r_A at outlet = k·C_A0(1−X) = 0.2×2×0.2 = 0.08 mol/L·min. τ = C_A0·X/(−r_A) = 2×0.8/0.08 = 20 min. V = τ·Q = 20×10 = 200 L.

Conceptual check — Isothermal Reactor Design

Problem

In a Reaction Engineering semester or GATE paper you are asked: "State the main assumption, the governing relation, and one practical consequence of isothermal reactor design." What should a complete answer include?

Exams & GATE

Compare PFR vs CSTR volumes — PFR is smaller for n > 0 kinetics.

📖 Standard books (India)

Chemical Reaction Engineering — Octave Levenspiel
Read: Syllabus unit
Reactor design and kinetics

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📚Key formulas & points