Adiabatic Mixing Temperature Calculator: Temperature-Dependent Cp(T) Energy Balance Guide

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CSTR in Series Design Equations: CSTR Cascade vs PFR Conversion (Reactor Design Guide)

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The purpose of this article is to provide practical, design-ready equations for CSTR cascades (CSTR in series) and to quantify how their conversion compares with an ideal plug flow reactor (PFR) for common kinetics, including how to size stages using Levenspiel-plot logic and how many tanks are typically needed to approach PFR performance.

1. What “CSTR cascade” means in reactor design.

A CSTR cascade, also called “CSTRs in series,” is a sequence of perfectly mixed continuous stirred-tank reactors where the outlet of reactor i feeds reactor i+1, with no bypass and steady-state operation assumed.

The cascade is often used to approximate PFR behavior while retaining CSTR advantages such as easier temperature control, simpler mixing for multiphase systems, and modular fabrication.

2. Core definitions and assumptions used in the equations.

2.1 Symbols used consistently throughout the design equations.

Symbol Meaning Typical units
FA0 Molar flow rate of A entering the cascade. mol/s
CA Concentration of A at a reactor location (for CSTR, at the exit equals tank concentration). mol/m3
rA Rate of disappearance of A (negative by convention), evaluated at reactor exit conditions for a CSTR. mol/(m3·s)
V Reactor volume (for a stage i, Vi). m3
v̇ Volumetric flow rate (assumed constant in many liquid-phase designs). m3/s
τ Space time, τ = V / v̇ (for stage i, τi). s
X Conversion of A, X = (FA0 − FA) / FA0. dimensionless
k Rate constant for a chosen kinetic model. varies
Note : The design equations below are for steady-state, single-pass operation with no recycle, and they require you to evaluate the rate law at the exit conditions of each stage for a CSTR and along the axial coordinate for a PFR.

3. Single CSTR design equation in conversion form.

For a single CSTR with feed conversion Xin (often 0 for fresh feed) and exit conversion X, the steady-state mole balance on A leads to a direct sizing equation when written in terms of conversion.

Single CSTR design equation (general kinetics).
V = F_A0 * (X − X_in) / (−r_A)_exit.

Equivalent in space time form when volumetric flow is approximately constant:

τ = (C_A0 * (X − X_in)) / (−r_A)_exit.

The key feature is that (−rA) is evaluated at the exit composition and temperature because the CSTR is perfectly mixed.

4. N-stage CSTR cascade design equations (general kinetics).

4.1 Stage-by-stage recursion using conversion.

For stage i, define the inlet conversion as Xi−1 and the outlet conversion as Xi, with X0 being the overall feed conversion to the first tank.

Stage i sizing equation (general kinetics).
V_i = F_A0 * (X_i − X_{i−1}) / (−r_A(X_i, T_i)).

If constant volumetric flow applies:

τ_i = (C_A0 * (X_i − X_{i−1})) / (−r_A(X_i, T_i)).

Because the rate depends on the stage exit conditions, the cascade is naturally solved by stepping forward from i = 1 to N if a target set {Xi} is specified, or by iterating on {Xi} if volumes are fixed and the resulting conversions are needed.

4.2 Graphical interpretation via Levenspiel plot rectangles.

For a single reaction A → products at constant density, a Levenspiel plot is often drawn as FA0/(−rA) versus X.

For a PFR, the required volume is the area under the curve from Xin to X.

For a CSTR, the required volume is a rectangle with width (X − Xin) and height evaluated at the exit X.

Therefore, a CSTR cascade approximates the PFR area by a sum of rectangles, and increasing the number of stages improves the approximation.

5. Closed-form conversion comparison for first-order kinetics.

5.1 Kinetic model and dimensionless group.

For a first-order reaction in a constant-density system, (−rA) = k CA, and CA = CA0(1 − X).

A convenient dimensionless group is the Damköhler number Da = k τ, where τ is the total space time based on total volume and inlet volumetric flow.

5.2 PFR conversion for first-order kinetics.

PFR, first-order, total space time τ_total:
X_PFR = 1 − exp(−k τ_total) = 1 − exp(−Da).

5.3 CSTR cascade conversion for equal-size stages (first-order kinetics).

If N equal CSTRs share the total space time equally, then τi = τtotal/N for all i.

CSTR in series, first-order, equal stages:
X_N = 1 − (1 + k τ_total / N)^(−N)
= 1 − (1 + Da / N)^(−N).

This expression increases monotonically with N and approaches the PFR result as N becomes large, because (1 + Da/N)N approaches exp(Da).

5.4 Numerical comparison example for first-order kinetics.

Example conditions: k = 0.5 min−1 and τtotal = 10 min, so Da = 5.0.

Reactor type Stages N Conversion equation used Conversion X
Single CSTR 1 X = Da / (1 + Da) 0.8333.
CSTR cascade 2 X = 1 − (1 + Da/2)−2 0.9184.
CSTR cascade 3 X = 1 − (1 + Da/3)−3 0.9473.
CSTR cascade 4 X = 1 − (1 + Da/4)−4 0.9610.
CSTR cascade 5 X = 1 − (1 + Da/5)−5 0.9688.
PFR Ideal X = 1 − exp(−Da) 0.9933.
Note : For fixed total space time, a PFR always provides equal or higher conversion than a single CSTR for positive-order kinetics under the same kinetic law and thermodynamic state, and a CSTR cascade provides intermediate performance that improves with stage count.

6. How to size a CSTR cascade to match or approach a PFR.

6.1 Equal-volume design workflow (robust and common in practice).

This workflow is frequently used when identical tanks are preferred for cost, spares, and operability.

Equal-volume cascade workflow.
Choose N based on layout, control needs, and desired approach to PFR behavior.

Set τ_i = τ_total / N and V_i = v̇ * τ_i.

For given kinetics, compute X_i stage-by-stage using the CSTR balance.

Adjust τ_total (or N) until the target overall conversion X_N is met.

6.2 Targeting a specified overall conversion using Levenspiel rectangles.

If you know the target conversion XN, you can design stage exit conversions {Xi} to control where each rectangle is evaluated, then compute each Vi from the general stage equation.

Levenspiel-rectangle cascade sizing.
Given a target set X_0 < X_1 < ... < X_N:

V_i = F_A0 * (X_i − X_{i−1}) / (−r_A at X_i).

This method can intentionally allocate more volume to later stages where rates may be lower, depending on kinetics and temperature policy.

6.3 Unequal volumes and optimal distribution for conversion.

For many positive-order kinetics at fixed total volume, pushing the system toward plug flow behavior is beneficial, which in a cascade means reducing backmixing by using more stages and by placing more volume where it reduces the loss in driving force.

In practice, unequal volumes are used when you are constrained by a fixed number of vessels but can vary sizes, or when you retrofit existing tanks.

Note : When you change stage volumes, you must recompute each stage exit conversion because the exit rate (and therefore the required rectangle height in a Levenspiel plot) changes with X and often with temperature.

7. Beyond first-order: general kinetics comparison logic.

7.1 Why PFR tends to outperform backmixed reactors for positive-order reactions.

For positive-order kinetics, rate typically decreases as reactant concentration decreases, so maintaining higher concentration earlier in the reactor produces higher average rates.

A PFR preserves the highest possible concentration near the inlet and progressively decreases it along the flow path.

A CSTR immediately mixes inlet with lower-concentration fluid, reducing the driving force everywhere in the vessel, which reduces the average rate for a given exit concentration.

7.2 How to compute conversion for a cascade with arbitrary rate laws.

For each stage i, you solve the implicit equation that results from substituting your rate law into the CSTR balance.

For example, in a liquid-phase constant-density system with known v̇, given τi and inlet conversion Xi−1, solve for Xi from the equation τi = CA0(Xi − Xi−1)/(−rA(Xi)).

Generic cascade calculation algorithm.
Given: N, τ_i (or V_i), X_0, kinetics (−r_A as function of X and T).

For i = 1..N:
Solve: τ_i = (C_A0 * (X_i − X_{i−1})) / (−r_A(X_i, T_i)).
Return X_N.

7.3 Example: second-order kinetics in constant-density liquid phase.

For (−rA) = k CA2 and CA = CA0(1 − X), the PFR relation between τ and X is often available in closed form, while the CSTR relation becomes algebraic and must be solved for X.

PFR, second-order, constant density:
τ = (1 / (k C_A0)) * ( X / (1 − X) ).
So X_PFR = (k C_A0 τ) / (1 + k C_A0 τ).

Single CSTR, second-order, constant density:

τ = X / (k C_A0 (1 − X)^2).
This requires solving for X given τ.

A cascade with N stages typically requires stage-by-stage algebraic solutions, which are straightforward with a numerical solver because each stage depends only on the previous stage’s exit conversion.

8. Practical design considerations that affect the comparison.

8.1 Temperature policy and heat removal.

The conversion comparison above assumes the same kinetic parameters and thermodynamic state policy for each reactor type.

In real designs, a CSTR cascade can provide superior temperature control because each stage can have dedicated cooling, agitation, and instrumentation, which can improve selectivity and safety even when the ideal conversion is lower than PFR for the same space time.

8.2 Non-ideal mixing and residence time distribution (RTD).

Real reactors depart from ideal models, and the cascade model is often used as an RTD-based approximation of partial backmixing between the ideal CSTR and PFR limits.

When you fit RTD data with an equivalent number of tanks in series, the same cascade equations become a practical bridge from measured hydrodynamics to predicted conversion.

8.3 Pressure drop and gas-phase volumetric flow changes.

If volumetric flow changes significantly, for example in gas-phase reactions with pressure drop, changing total moles, or strong temperature gradients, then using a constant v̇ in τ = V/v̇ is not valid.

In those cases you should perform the design using molar-flow formulations and explicit relationships between composition, temperature, pressure, and volumetric flow, while still applying the same balance structure for CSTR exit evaluation and PFR differential evaluation.

9. Design heuristics for selecting the number of CSTR stages.

The number of CSTRs needed to approximate PFR depends strongly on the Damköhler number and kinetics sensitivity to concentration and temperature.

For first-order kinetics, the equal-stage formula XN = 1 − (1 + Da/N)−N provides a direct way to quantify diminishing returns as N increases.

In many practical first-order cases, N between 3 and 6 provides a large fraction of the PFR conversion at the same total space time, while higher N provides incremental improvements that must be weighed against capital cost and complexity.

FAQ

How do I directly compare CSTR cascade conversion to PFR conversion for first-order kinetics.

Use Da = k τ_total.

Compute X_PFR = 1 − exp(−Da).

For N equal CSTRs, compute X_N = 1 − (1 + Da/N)^(−N).

The difference X_PFR − X_N quantifies the conversion penalty due to backmixing at that N.

Can a CSTR cascade ever outperform a PFR in conversion at the same total space time.

Under the same kinetic law and the same thermodynamic state policy for positive-order kinetics, an ideal PFR is the upper bound and a cascade approaches it from below as N increases.

However, if reactor types enable different temperature trajectories, different heat removal limits, or different feasible operating conditions, the realized conversion and selectivity can differ because the state policy is no longer the same.

Should I use equal volumes or unequal volumes for the stages.

Equal volumes are common when standardization, fabrication, and spares dominate the decision.

Unequal volumes are used when a limited number of vessels must be used to approach plug flow behavior more closely, or when existing equipment sizes constrain the design.

In all cases, compute each stage using V_i = F_A0 (X_i − X_{i−1}) / (−r_A at X_i), and verify performance with the full stage-by-stage calculation.

How do I design a cascade when the rate law is complex or temperature dependent.

Write the stage equation τ_i = C_A0 (X_i − X_{i−1}) / (−r_A(X_i, T_i)).

For each stage, solve this equation for X_i numerically using the inlet conversion X_{i−1} and the chosen temperature policy T_i.

Repeat for i = 1..N to obtain X_N.

What is the fastest practical way to size volumes from a target conversion without heavy iteration.

Use the Levenspiel-rectangle method by selecting intermediate conversions X_1..X_{N−1} between X_0 and X_N, then compute each V_i from V_i = F_A0 (X_i − X_{i−1}) / (−r_A at X_i).

This gives a direct first design that can be refined by adjusting the intermediate conversions or by adding stages.

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