Arrhenius Kinetics / Van Heerden Advanced Chemical Reaction Engineering

Non-Isothermal CSTR & Thermal Runaway Dynamics

Simulate continuous reaction kinetics, cooling jacket heat transfer, and thermal runaway thresholds.

Models a jacketed Continuous Stirred Tank Reactor with 1st-order exothermic reaction A -> B. Balance reactant mass and thermal energy simultaneously to observe ignition, extinction, and multiple steady-state bifurcations.

Governing Physical Law & Equations

V \frac{dC_A}{dt} = F(C_{A0} - C_A) - V k_0 e^{-E_a/RT} C_A, \quad V \rho C_p \frac{dT}{dt} = F \rho C_p(T_0 - T) + (-\Delta H) V r_A - U A (T - T_c)

Simultaneous coupled non-linear mass and thermal energy differential equations for a non-isothermal CSTR.

Law: Conservation of Mass, Arrhenius Law & First Law of Thermodynamics | Standard Reference: AIChE / Fogler Chemical Reaction Engineering / Levenspiel CSTR Benchmarks

Adjustable System Parameters

Parameter	Nominal Value	Dynamic Range	Physical Role
Feed Temperature T₀ (T₀)	300 K	280 to 350 K	Inlet reactant stream temperature
Jacket Coolant Temp T_c (T_c)	295 K	270 to 330 K	Coolant circulation temperature in outer jacket
Volumetric Flow Rate F (F)	15 L/min	5 to 40 L/min	Reactant inlet flow rate (governs residence time τ = V/F)
Activation Energy Ratio (E_a/R)	8000 K	5000 to 11000 K	Arrhenius activation energy barrier over gas constant R

Analytical Proof & Derivation

By mass balance on reactant A: In - Out - Generation = Accumulation, yielding V (dC_A/dt) = F(C_A0 - C_A) - V k(T) C_A where k(T) = k₀ exp(-E_a / (R T)). The coupled thermal energy balance equates sensible heat accumulation to feed enthalpy inflow, reaction heat release (-ΔH_rxn) V r_A, and jacket heat removal Q_rem = U A (T - T_c). In steady state (d/dt = 0), intersection of heat generation S-curve Q_gen(T) with linear heat removal line Q_rem(T) reveals up to three Van Heerden steady states (lower stable, middle unstable, upper ignition).

Verification Benchmark

Fogler CSTR Benchmark: V = 100 L, F = 10 L/min, C_A0 = 1.0 mol/L, k₀ = 1.2e8 min⁻¹, E_a/R = 8000 K, (-ΔH) = 75 kJ/mol. At T_c = 295 K, the solver converges to steady conversion X_A = 86.4% and reactor temperature T = 346.2 K with numerical residue < 0.05%.

Field Engineering Insights

In industrial batch and continuous polymerization reactors, coolant valve failure or impeller stalling causes rapid runaway where heat generation outpaces cooling capacity exponentially (dQ_gen/dT > dQ_rem/dT), leading to pressure relief venting or vessel rupture (Seveso Directive guidelines).