Non-Isothermal CSTR

[MAL-Seeker Execution Record] Steady-State Multiplicity Analysis of Non-Isothermal CSTR

$$f_1(x_1, x_2) = 1 - x_1 - 0.05 \cdot x_1 \cdot \exp(x_2) = 0$$ $$f_2(x_1, x_2) = 0.4 \cdot x_1 \cdot \exp(x_2) - 1.3 \cdot x_2 = 0$$

Variables: $x_1$ (Dimensionless Concentration), $x_2$ (Dimensionless Temperature)
Search Bounds: $[-10.0, 10.0]$

1. Problem Definition (Thermal Runaway Risk in Reactors)

This analysis focuses on the mass and energy balance equations for a non-isothermal Continuous Stirred-Tank Reactor (CSTR), a world-standard benchmark in chemical engineering.

Due to the non-linear coupling between heat generation from exothermic reactions and removal by cooling systems, multiple steady states (solutions) can exist simultaneously under certain conditions—a phenomenon known as "steady-state multiplicity."

2. Execution Results

Final log of the all-solution search using MAL-Seeker (Project Bellatrix V1.0.0).

=========================================
  🏆 CSTR STEADY-STATE MULTIPLICITY 🏆
  Found 3 Unique Roots
  Execution Time: 1.926944 seconds
=========================================
Root 1: (0.9275770, 0.4456797)  [Extinguished State]
Root 2: (0.5378303, 2.8441215)  [Saddle Point]
Root 3: (0.0568933, 5.8037333)  [Ignited State]

All three solutions existing in the search space were successfully captured in approximately 1.9 seconds. Each solution represents the following physical states:

Root 1 (Low Temp, High Conc): Extinguished stable state. Reaction barely progresses, and reactants pass through the reactor almost unchanged.
Root 2 (Intermediate State): Saddle point (unstable). A precarious state where a slight temperature fluctuation causes the system to drop into either Root 1 or Root 3.
Root 3 (High Temp, Low Conc): Ignited stable state. The reaction progresses actively, but this high-energy state requires careful monitoring to avoid thermal runaway if cooling capacity is exceeded.

Dr.WataWata Insight:

The greatest concern in chemical plant design is "thermal runaway." To prevent this, it is essential to accurately identify the unstable saddle point (Root 2)—the critical boundary between the safe extinguished state (Root 1) and the high-energy ignited state (Root 3).
Since saddle points are physically unstable and cannot be maintained without control, they are notoriously difficult for standard solvers to capture. Successfully identifying all three states rapidly with this engine demonstrates significant potential for complex industrial safety analysis.

3. Analysis Environment

Analysis Engine: MAL-Seeker (Project Bellatrix V1.0.0 / C)
PC: MacBook
OS: macOS Monterey 12.7.6
CPU: 1.2GHz Dual-Core Intel Core m5
Memory: 8GB

1. Problem Definition (Thermal Runaway Risk in Reactors)

2. Execution Results

3. Analysis Environment

Related books

Pythonと実例で学ぶ微分方程式 ―はりの方程式から感染症の数理モデルまで―

Nonlinear Programming (Japanese edition)

Cによる理工系解析の数値計算 ―基礎からの展開―