Continuous Stirred-Tank Reactor (CSTR) Conversion Calculator

Chemical Reaction Engineering & The Ideal CSTR Model

In chemical reaction engineering (CRE), the Continuous Stirred-Tank Reactor (CSTR)—historically designated as the continuous-flow stirred-tank reactor (CFSTR) or backmix reactor—serves as the foundational continuous reactor configuration for industrial liquid-phase syntheses. Developed systematically through the foundational works of Octave Levenspiel and H. Scott Fogler, the ideal CSTR is modeled upon the physical premise of perfect macromixing and micromixing: the fluid inside the agitated reaction vessel is completely homogeneous in composition, temperature, and physical properties, such that the effluent exiting the overflow pipe possesses properties identical to the fluid at every location inside the vessel.

CSTRs are widely selected across industrial polymerizations, pharmaceutical active ingredient fermentations, wastewater bioremediation, and saponification processes. Their primary advantages include superior temperature control due to the large agitated thermal mass, ease of installing internal cooling coils or external heat exchange jackets, and low operating labor costs in continuous operations. However, because reactants are instantly diluted upon entering the vessel, CSTRs suffer from lower average reaction rates than tubular Plug Flow Reactors (PFRs), requiring substantially higher reactor volumes to achieve high fractional conversion.

Mathematical Formulations and Derivations

1. General Steady-State Mole Balance

A conservation of mass balance on reactant chemical species $A$ inside a control volume $V$ at steady state dictates:

Because the reaction rate $r_A$ is spatially uniform throughout the perfectly backmixed volume $V$:

Expressing the exit molar flow in terms of fractional conversion $X = (F_{A0} - F_A) / F_{A0}$:

2. Space-Time ($tau$) and Space Velocity ($s$)

Dividing reactor volume $V$ by volumetric entering feed flow rate $v_0$ yields the mean residence time or Space-Time ($tau$):

3. Analytical Solutions by Reaction Order

• First-Order Irreversible Reaction ($-r_A = k C_A$): Since $C_A = C_{A0}(1 - X)$ for constant-density liquid systems:
  tau = frac{C_{A0} X}{k C_{A0} (1 - X)} = frac{X}{k (1 - X)}
  Solving explicitly for fractional conversion $X$:
  X = frac{k tau}{1 + k tau} = frac{Da}{1 + Da} C_A = C_{A0} (1 - X) = frac{C_{A0}}{1 + Da}
  where $Da = k tau$ is the dimensionless Damköhler number.
• Second-Order Irreversible Reaction ($-r_A = k C_A^2$): Substituting $C_A = C_{A0}(1 - X)$:
  tau = frac{X}{k C_{A0} (1 - X)^2} quad implies quad (k C_{A0} tau) X^2 - (2 k C_{A0} tau + 1) X + k C_{A0} tau = 0
  Letting $Da_2 = k C_{A0} tau$, the physical root ($0 le X le 1$) is:
  X = frac{(2 Da_2 + 1) - sqrt{4 Da_2 + 1}}{2 Da_2}
• Zero-Order Reaction ($-r_A = k$):
  X = minleft(1.0, frac{k tau}{C_{A0}}right)

4. Cascade of Equal-Volume CSTRs in Series

To overcome the kinetic rate suppression of a single large CSTR, industrial chemical plants frequently connect $N$ identical smaller stirred tanks in series. For a first-order reaction across $N$ equal-volume tanks with individual space time $tau_i = tau_{text{total}} / N$:

The total cumulative conversion across the entire cascade is:

As the number of tanks in series approaches infinity ($N to infty$):

Thus, staging just 3 to 5 CSTRs in series captures over 85% to 95% of the volumetric efficiency advantage of a true Plug Flow Reactor while preserving continuous agitation and temperature control.

5. Comparison with Plug Flow Reactor (PFR) & Levenspiel Plot

In an ideal tubular PFR with no axial dispersion, reactants proceed down the pipe as a series of non-mixing plugs. The design equation is:

The ratio of CSTR volume to PFR volume required for an identical conversion $X$ is:

Graphically on a Levenspiel Plot ($1 / (-r_A)$ vs $X$):

• CSTR Volume: Represents the area of a large rectangle of width $X$ and height $1 / (-r_A)_{text{exit}}$.
• PFR Volume: Represents the area under the curve from $0$ to $X$. Because $1 / (-r_A)$ rises steeply at high conversion, the CSTR rectangle is far larger than the area under the curve.

Step-by-Step Worked Engineering Example

A specialty chemicals producer synthesizes an organic intermediate via a first-order liquid isomerization reaction in a cascade of $N = 3$ equal-volume CSTRs:

• Volumetric Feed Rate: $v_0 = 25.0 text{ L/min}$ ($1.5 text{ m}^3/text{h}$)
• Inlet Feed Concentration: $C_{A0} = 2.0 text{ mol/L}$
• Reaction Rate Constant: $k = 0.25 text{ min}^{-1}$
• Vessel Volume per Tank: $V = 500 text{ L}$ ($V_{text{total}} = 1,500 text{ L}$)

Step 1: Compute Space Time and Damköhler Number per Tank

Step 2: Calculate Concentration and Conversion for Tank 1

Step 3: Calculate Concentration and Conversion for Tank 2

Step 4: Calculate Concentration and Final Effluent Conversion for Tank 3

Step 5: Compare Single CSTR vs 3-CSTR Cascade vs PFR

If a single CSTR were tasked with achieving this final conversion $X = 0.99537$:

In contrast, the 3-CSTR cascade requires only $1,500 text{ L}$—a massive 14.3-fold reduction in total reactor volume! For an ideal PFR achieving $99.54%$ conversion:

Performance Comparison Across Chemical Reactor Types