A. K. COKER, AKC Technology, Loughborough, England, U.K.
As chemical processes grow in complexity and demand more robust control strategies, understanding the transient dynamics of reactors becomes increasingly critical. This article presents a practical approach to dynamic simulation of chemical reactors with a focus on continuous stirred tank reactors (CSTRs) through phase plane analysis, nullcline computation and system trajectory visualization using Python. By integrating rigorous mathematical modeling with clear graphical insight, the study reveals how nullclines help identify multiple steady-states, assess stability and anticipate unsafe conditions like thermal runaway. The methodology, which is extendable to plug flow reactors (PFRs), packed bed reactors (PBRs) and fluid catalytic cracking units (FCCUs), offers process engineers a powerful design and diagnostic tool grounded in modern computation.
Modern process industries are under pressure to optimize reactor performance not just under steady-state conditions, but across the full range of transient operations. This includes startup, shutdown, load changes and upset scenarios. Traditional simulation techniques often focus on steady-state outcomes, neglecting the rich and sometimes unstable dynamics that can emerge during transitions.
Chemical reactors are the heart of hydrocarbon processing facilities, operating under conditions where precise control is paramount. A small disturbance—whether in feed composition, temperature or flowrate—can initiate a chain of events that leads to unsafe conditions or economic loss. Particularly in exothermic reactions, where heat release accelerates reaction rates, the risk of thermal runaway is ever-present.
Traditional steady-state models fall short in capturing these risks. They provide snapshots but not the action. They tell us where the process might settle but not what happens on the way there. That is where transient dynamic simulation and phase plane analysis are utilized.
Techniques such as nullcline visualization, system‑trajectory mapping and bifurcation analysis offer a deeper lens into how reactor systems evolve over time. These methods allow process engineers and control system designers to understand not only where a reactor may settle, but how it behaves during the critical transitions that determine safety and operability. Specifically, they help engineers:
Identify hidden instabilities that steady‑state models miss. Nullcline plots expose regions where the direction of change in concentration or temperature reverses, revealing unstable operating zones that are invisible in steady‑state calculations. This early insight is essential to prevent thermal runaway or divergence during startup, shutdown or disturbances.
Operating regimes and basins of attraction. Because nullclines show where each state variable momentarily stops changing, their intersections reveal all possible steady‑states, including multiple equilibria common in exothermic CSTRs. The surrounding vector field clarifies which initial conditions lead the reactor toward each regime, helping define safe operating envelopes.
Predict oscillatory or divergent behavior before it appears in real units. Phase‑plane trajectories illustrate whether the system spirals toward a stable point, oscillates around it or diverges away. This predictive capability allows engineers to recognize problematic dynamics such as limit cycles or runaway long before they manifest in plant equipment.
Develop control strategies suited to nonlinear reactor behavior. The geometry of the nullclines captures the nonlinear coupling between concentration and temperature. Understanding these interactions enables the design of control strategies that remain effective across the full operating range, rather than relying on linearized approximations near a single point. This leads to more robust, resilient and safer reactor control.
Together, these tools provide a powerful framework for anticipating dynamic behavior, improving control performance and strengthening the safety and reliability of modern hydrocarbon‑processing reactors.
In particular, nullclines curves—where the time derivative of a system variable is zero—serve as natural boundaries in the system’s phase space. Their intersections mark equilibrium points, and their layout reveals the stability landscape. Combined with trajectory vectors, they help visualize whether the system will stabilize, oscillate or diverge. Incorporating these tools into the design and operation of reactors is not simply good practice, it is an essential component of modern process safety and resilient control architecture.
This article connects dynamic simulation techniques to real-world applications in hydrocarbon processing, such as hydrocracking, polymerization and alkylation. By grounding the analysis in practical reactor operations, it demonstrates how phase plane and nullcline tools can enhance safety, efficiency and conversion in industrial settings.
The modeling approach presented aligns with emerging trends in process automation, including digital twins, artificial intelligence (AI)-based control and model predictive control (MPC). These technologies are reshaping how reactors are monitored and optimized, and this work supports their implementation by offering a robust framework for dynamic analysis.
Phase plane plots and nullcline diagrams are used not only to identify steady-states but also to interpret system trajectories and stability. Clear graphical representations supported by annotations and a bifurcation map provide intuitive insight into reactor behavior under transient conditions.
To ensure broad accessibility, the article balances mathematical rigor with editorial clarity. Key equations are summarized for quick reference, terminology is standardized and structural elements such as an executive summary and conclusion guide the reader through the technical narrative and its practical implications.
This article introduces a dynamic simulation framework for chemical reactors, particularly CSTRs centered on phase plane nullcline analysis and system trajectory mapping. These tools allow engineers to:
Visualize how a reactor’s state evolves in real time
Understand how variables like concentration and temperature interact dynamically
Predict stable vs. unstable behavior under varying operating conditions.
While rooted in classical chemical engineering principles (mass and energy balances), the approach leverages Python’s numerical solvers (e.g., solve_ivp) and plotting libraries (e.g., Matplotlib) to build an intuitive, educational and operationally valuable model.
A case study of an exothermic reaction in a CSTR demonstrates how nullclines curves along which a system variable does not change serve as guideposts in the reactor’s phase space. The intersection of these curves reveals equilibrium points, while the surrounding trajectories show how the system will behave over time: whether it will safely settle or spiral into instability.
This kind of visual, systems-oriented analysis is not just academic, it is deeply practical. As operating envelopes tighten and safety margins shrink, dynamic simulation with nullcline and phase plane analysis is rapidly becoming an essential tool for engineers tasked with ensuring safety, efficiency and resilience in modern hydrocarbon processing.
CSTR DYNAMIC SIMULATION
Reaction kinetics. We assume the kinetics are dominated by a single first order reaction (Eq. 1):
A → ( k ) Products (1)
The reaction rate per unit volume is modeled as the product kCA , where CA is the concentration of A. The rate constant k(T) increases with temperature following the Arrhenius law, shown in Eqs. 2 and 3:
k = koe (–Ea /RT), for the first order rate expression, ( - rA ) = kCA (2)
rA = ko e ( –Ea /RT) CA (3)
where,
Ea = the activation energy
R = the gas constant
T = absolute temperature
Ko = the pre-exponential factor.
We can see the strong temperature dependence by plotting k(T) vs. temperature over typical operating conditions. FIG. 1 shows a typical CSTR.
What are nullclines? Nullclines, particularly in phase plane analysis, are curves where the rate of change of one or more state variables is zero (see Phase plane analysis section below). In transient response, the nullclines represent the boundaries between regions of increasing and decreasing values of these variables, influencing the trajectory of the system's state towards equilibrium. Understanding nullcline profiles helps to visualize system behavior during transitions and predict how it will settle into a steady-state.
Furthermore, nullclines are graphical representations (e.g., lines or curves in a phase space) of where the time derivatives of state variables in a system of differential equations are zero. For example, a plot of variables like concentration and temperature where the rate of change of one variable is zero.
For a system of two variables (say x and y):
The x-nullcline is the set of points where dx/dt = 0
The y-nullcline is where dy/dt = 0
The intersections of the nullclines represent steady-states (where both derivatives are zero), and their shape helps us understand how the system behaves over time.
Transient response and nullclines. In a transient response, the system's state evolves over time, and its trajectory is influenced by the nullclines. The direction of motion of the system's state in the phase plane is determined by the vector field, which is derived from the derivatives. The nullclines act as "attractors" or "repellers" for the trajectory, guiding it towards or away from equilibrium points.
FIG. 2 shows nullclines and vector fields calculated for a two-dimensional (2D) ordinary differential equation (ODE) system that are derived from a 5D ODE system using a quasi-steady-state approximation.1 Circles show stable steady-states, while triangles represent unstable steady-states. Red and blue curves are nullclines for variables x1 and x2, respectively. The green line represents trajectories of limit cycles projected from the original 5D system to 2D space of x1 and x2.2 FIG. 3 shows nullclines in a transient model.
Phase plane analysis. Nullclines are powerful tools for analyzing the behavior of CSTRs. By plotting them on the phase plane, we can visualize the different flow patterns and identify potential steady-states and their stability.
Stability. The relative positions of the nullclines and the direction of the flow vectors in different regions of the phase plane can be used to determine the stability of the steady-states. For example, a steady-state where trajectories converge towards it is considered stable, while one where trajectories move away is unstable.
Nullclines in a transient CSTR. In a CSTR that is not at steady-state, concentrations and temperature change over time. The governing equations are typically ODEs from mass and energy balances. If we are analyzing two variables (e.g., CA: concentration of a reactant; and T: temperature), then:
The (dCA) /dt = 0 line is the concentration nullcline
The dT/dt = 0 line is the temperature nullcline
These nullclines show where the respective variables stop changing momentarily. By plotting them, you can visualize how the system is evolving and where it might settle.
Analyzing nullclines in CSTRs. Analyzing nullclines in CSTRs provides a compact yet powerful window into reactor behavior. By mapping where concentration and temperature rates of change vanish, engineers can pinpoint steady‑states including the multiple equilibria that often arise in nonlinear reaction systems while also gaining insight into their stability. This perspective clarifies how the reactor will respond to shifts in feed conditions, helps anticipate dynamic phenomena such as oscillations or thermal runaway, and ultimately supports the design of robust control strategies that keep operations safe, predictable, and efficient.
TABLE 1 summarizes the key practical implications of applying nullcline analysis across major engineering domains. By clarifying how concentration and temperature interactions shape system behavior, nullclines provide valuable insight for operation, simulation, design, control and safety, enabling engineers to anticipate instability, improve dynamic understanding and support inherently safer and more robust reactor performance.
Example: Exothermic reaction in a CSTR. In reactions that release heat, like many industrial chemical processes:
The temperature nullcline might intersect the concentration nullcline in a zone where the reaction rate is high.
This can create an unstable operating point, leading to thermal runaway if not controlled.
Nullcline analysis reveals these risks early in design or simulation.
Nullclines provide a powerful visual and analytical way to understand how a reactor behaves dynamically, helping engineers design safer and more reliable systems. FIG. 4 shows a schematic of CSTR with heat supply (Q) and notations.
STEP-BY-STEP COMPUTATION: CSTR DYNAMIC SIMULATION
1. Problem setup (mathematical modeling).4 A CSTR model is simulated dynamically. The simulation analyzes how concentration (CA) and T evolve over time. Governing ODEs account for the mass balance for A (reactant) and the energy balance for T.
2. Governing Equations [mass balance (A concentration, CA)]. Input by flow = output by flow + (disappearance by reaction) + accumulation (Eq. 4).
uCAO = uCA + (-rA ) VR + VR dCA ∕dt (4)
Rearranging Eq. 4 and dividing by VR gives Eq. 5:
(dCA ) / dt = u / VR (CAO - CA ) - ( - rA ) (5)
Mean residence time is calculated using Eq. 6:
t – = VR / u, ( - rA ) = ko e ( - Ea ) / RT) CA (6)
Substituting Eq. 6 into Eq. 5 gives Eq. 7:
(dCA ) / dt = 1 / t ̄ (CAO - CA ) - koe( - Ea ) / RT) CA (7)
Energy balance (T). For steady-state, Eq. 8 is used:
-Gδh + Q = 0 (8)
where (Eq. 9):
G = ρu = (Mass/Volume) x (Volume/Time) = Massflowrate (9)
For transient state, Eqs. 10–15 are used:
-ρu { CpΔT + ((-ΔΗR )/a) (δm ̄ A )/ MA } + Q = mCp dT/dt (10)
ρ (δm ̄ A )/ MA ) moles / mass x (Mass / Volume) =Concentration = dCA (11)
- { ρuCpΔT + u ((-ΔΗR )/a) dCA } + Q = mCp dT/dt (12)
{ - ρuCp (T - To )-u ((-ΔΗR )/a) dCA } + Q = mCp dT/dt (13)
{ - ρuCp (T - To )-u ((-ΔΗR )/a) (CA - CAO )} + Q = mCp dT/dt (14)
{ ρuCp (T - To )-u ((-ΔΗR )/a) (CAO - CA )}+ Q = mCp dT/dt (15)
From the mass balance, Eqs. 16–19 are used:
u (CAO - CA )=(-rA ) VR (16)
{ ρuCp (To - T) + ((-ΔΗR )/a) (-rA ) VR } + Q = mCp dT/dt (17)
m = ρVR (density x Volume) = Mass (18)
ρuCp (To - T)+((-ΔΗR )/a) (-rA ) VR +Q = ρVR Cp dT/dt (19)
Rearranging Eq. 19 and dividing by ρVR Cp gives (Eqs. 17 and 18):
dT/dt = (ρuCp (To - T)) / (ρVR Cp ) + ((-ΔΗR )/a) (-rA ) VR ) / (ρVR Cp ) + UA (Tc - T) / (ρVR Cp ) (20)
dT/dt = u (To - T)/ VR + ((-ΔΗR )/a) (((-rA )) / (ρCp )) + UA (Tc - T) / (ρVR Cp ) (21)
3. Dynamic simulation via solve_ivp.
Initial conditions: CA = 1.0 mol/L, T = 300 K
Simulates from t = 0 to 200 sec
Solution gives the trajectory (green curve) in CA vs. T space.
4. Nullcline calculations.
CA-nullcline (blue curve): Points where dCA/dt = 0
T-nullcline (red curve): Points where dT/dt = 0
These help visualize steady-state behavior and reactor stability.
TABLE 2 defines the color scheme used in the dynamic reactor plots. Each color highlights a distinct analytical feature: the green curve traces the reactor’s time‑dependent trajectory, the blue line marks the (dCA/dt = 0) nullcline, the red line represents the (dT/dt = 0) nullcline, and the black point identifies the final steady state where both nullclines intersect.
The full Python program used to generate the dynamic simulation and nullcline plots has been omitted for brevity. Readers who would like to access the complete code may contact the author or request it through Hydrocarbon Processing.
The result of the Python code is:
PS C:\Users\Kayode Coker\Desktop\Python_Folder> & "C:/Users/Kayode Coker/AppData/Local/Programs/Python/Python313/python.exe" "c:/Users/Kayode Coker/Desktop/Python_Folder/Continuous_Flow_Stirred_Tank_Reactor_Dynamic_Simulation_akc6c.py"
The final steady-state results are:
Final CA: 1.0000 mol/L
Final T: 299.95 K
PLOT DESCRIPTIONS
1. CSTR phase plane: Nullclines and system trajectory. The CSTR phase‑plane plot illustrates the reactor’s dynamic behavior in terms of concentration and temperature. The vector field streamlines depict the local direction of change, while the blue (dCA/dt = 0) nullcline and red (dT/dt = 0) nullcline identify conditions where concentration and temperature cease to change. Their intersections indicate potential steady states. The green trajectory traces the reactor’s actual dynamic path from its initial condition toward equilibrium, culminating in the black point that marks the final steady‑state operating condition.
Implications. These implications highlight how nullcline and streamline analysis support reactor understanding and design. Nullclines pinpoint equilibrium conditions and provide a basis for assessing the stability of operating points, while stream plots reveal whether the system naturally moves toward or away from these steady states. Together, they offer valuable insight for startup planning and for anticipating potential instabilities, including conditions that may lead to thermal runaway.
2. CSTR dynamics: CA and T vs. time. The time‑dependent CSTR plot illustrates how concentration and temperature evolve as the reactor progresses toward steady‑state. The blue and red curves capture the transient response, providing insight into the time required to reach equilibrium and offering a basis for assessing operational safety during dynamic transitions.
Implications. These implications emphasize how time‑dependent concentration and temperature profiles deepen understanding of reactor performance. The transient curves help distinguish whether the system exhibits stable, unstable or oscillatory behavior, providing essential insight for real‑time controller tuning. They also inform decisions related to thermal load management and overall reactant conversion efficiency during dynamic operation.
Design and control implications. These design and control implications underscore the importance of understanding CSTR dynamics when developing reliable reactor systems. Insight into the transient response enables engineers to anticipate how the reactor will behave under disturbances, maintain precise temperature control to prevent undesired side reactions and select appropriate heat‑exchange capacity, reactor volume and flowrate. This understanding also supports the formulation of effective feedback and feedforward control strategies. Overall, the modeling approach provides both valuable educational insight and practical guidance for real‑world reactor operation.
Two plots are shown in FIGS. 5 and 6.
DISCUSSION
1. CSTR phase plane: Nullclines and system trajectory. FIG. 5 shows that the CSTR phase plane plot offers a powerful visual tool for understanding the reactor’s dynamic behavior in terms of CA and T. Streamlines represent the direction field, showing how these variables evolve over time under the influence of reaction kinetics and heat exchange. The blue CA-nullcline marks the locus where the concentration change rate is zero (dCA/dt = 0), while the red T-nullcline indicates where the temperature change rate vanishes (dT/dt = 0).
Intersections of these nullclines reveal potential steady-states, which may be stable or unstable depending on the surrounding vector field. The green trajectory traces the actual path the reactor takes from its initial condition toward equilibrium, shaped by the interplay of mass and energy balances. The black dot marks the final operating point of a steady-state where both CA and T remain constant. This visualization not only helps identify feasible operating regimes but also highlights the sensitivity of the system to initial conditions and disturbances. It is especially valuable in detecting bi-stability, oscillations or runaway scenarios, guiding engineers in designing robust control strategies to ensure safe and efficient reactor performance.
Implications. Analyzing nullclines in the CSTR phase plane provides critical insight into the system’s equilibrium behavior and stability. By identifying points where the rates of change in CA and T vanish, engineers can pinpoint steady-states and evaluate their stability whether the reactor will naturally settle into these states or diverge from them. The accompanying stream plots further illuminate the system’s dynamic tendencies, revealing whether trajectories converge toward equilibrium or veer away, potentially indicating unstable or metastable behavior. This type of analysis is especially valuable during reactor startup, where initial conditions must be carefully chosen to avoid undesirable outcomes. It also plays a key role in diagnosing and preventing thermal runaway reactions, which can occur when exothermic reactions accelerate uncontrollably due to positive feedback between T and reaction rate. Overall, phase plane and nullcline analysis serve as essential tools in designing safe, efficient and resilient reactor systems.
2. CSTR dynamics: CA and T vs. time. FIG. 6 shows the time-series plot of CSTR dynamics and offers a clear view of how the reactant (CA) and temperature (T) evolve as the reactor progresses toward steady-state. The blue curve represents the CA profile, while the red curve tracks temperature changes over time, together illustrating the system’s transient behavior. These trajectories reveal how quickly the reactor responds to initial conditions or disturbances, making the plot a valuable tool for estimating the time required to reach equilibrium. Moreover, it provides critical information for operational safety, especially in scenarios involving exothermic reactions, where temperature spikes could signal potential runaway conditions. Engineers can use this dynamic data to fine-tune control strategies, optimize startup procedures and ensure that the reactor remains within safe operating limits throughout its transition to steady-state.
Implications. FIG. 6 shows the dynamic profiles of CA and T over time provide essential clues about the nature of the reactor’s behavior whether it is stable, oscillatory or prone to instability. Recognizing these patterns is crucial for designing and tuning process controllers, which must respond effectively to transients and maintain desired operating conditions in real time. For instance, a damped response suggests a stable system, while sustained oscillations or divergence may indicate the need for tighter control or design adjustments. Additionally, these time-based insights support thermal load management, helping engineers anticipate and mitigate excessive heat buildup, especially in highly exothermic systems. They also inform decisions around reactant conversion efficiency, as the rate and extent of concentration change directly impact product yield and process economics. Overall, this analysis is indispensable for ensuring both safety and performance in continuous reactor operations.
Conclusion. In summary, the phase‑plane and nullcline analysis of CSTR dynamics presented here delivers several important contributions. The modeling framework demonstrates clear industrial relevance, supporting reactor design and optimization in processes such as hydrocracking, alkylation and polymerization, where dynamic behavior strongly influences safety and product yield. As modern plants increasingly adopt digital twins and AI‑enabled control architectures, this approach provides a robust foundation for predictive modeling and real‑time operational decision‑making.
The use of annotated phase‑plane plots and nullcline diagrams enhances visualization, enabling engineers to intuitively interpret system trajectories, steady‑states and potential bifurcations. By combining mathematical rigor with clear editorial presentation, the work remains accessible to both technical specialists and broader stakeholders. Collectively, these insights deepen understanding of reactor dynamics and contribute to the development of smarter, safer and more efficient chemical processes.
Takeaways. In the transient state of a CSTR, understanding system dynamics is essential for effective design and control. Engineers rely on this insight to predict how the reactor responds to disturbances, ensuring stable operation and avoiding runaway reactions.5 Accurate temperature control is critical to suppress undesired side reactions, which is achieved by selecting appropriate heat exchange parameters (UA), reactor volume (V) and inlet flowrate (u). A powerful analytical tool in this context is the use of nullclines curves in phase space where the rate of change of a state variable (e.g., T or CA) is zero. By plotting nullclines, one can visualize equilibrium points and assess system stability. Their intersections indicate steady-states, while their geometry reveals whether perturbations will decay or amplify. This helps engineers detect potential bifurcations or oscillatory behavior and informs the design of feedback or feedforward control strategies to guide the reactor safely through transient regimes and maintain optimal performance.
Nullclines play a central role in CSTR design and control by revealing multiple steady‑states, particularly in exothermic systems through the presence of multiple intersection points. The geometric structure around these intersections provides insight into whether each steady‑state is stable or unstable, an essential consideration for maintaining safe operation. By visualizing how trajectories evolve in the vicinity of the nullclines, engineers can formulate effective feedback or feedforward control strategies that guide the reactor toward desired operating conditions. Nullcline analysis also offers predictive value by indicating how the system will evolve from any initial condition, whether it will converge, oscillate or diverge over time. HP
LITERATURE CITED
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FURTHER READING
Strogatz, S. H., Nonlinear dynamics and chaos, 2nd Ed., CRC Press, 2018.
Biegler, L., I. Grossmann and A. Westerberg, Systematic methods of chemical process design, 1st Ed., Prentice Hall, 1997.
Ortega, J. M. and W. C. Rheinboldt, Iterative solution of nonlinear equations in several variables, SIAM, 1970.
Yang, C. C. and V. N. A Naikan, Optimum tolerance design for complex assemblies using hierarchical interval constraint networks,” Computers & Industrial Engineering, October 2003, online: https://www.researchgate.net/publication/223101258
Python program reference, online: https://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.solve_ivp.html
A. Kayode Coker is an Engineering Consultant for AKC Technology; an Honorary Research Fellow at the University of Wolverhampton, U.K.; a former Engineering Coordinator at Saudi Aramco Shell Refinery Company (SASREF); and Chairman of the department of Chemical Engineering Technology at Jubail Industrial College, Saudi Arabia. Dr. Coker has been a chartered chemical engineer for more than 40 yrs. He is a Fellow of the Institution of Chemical Engineers, U.K. (C. Eng., FIChemE), and a senior member of the American Institute of Chemical Engineers (AIChE). He earned a B.Sc. honors degree in chemical engineering, an MS degree in process analysis and development and a PhD in chemical engineering, all from Aston University, Birmingham, U.K., as well as a Teacher’s Certificate in Education at the University of London, U.K.
Dr. Coker has directed and conducted short courses extensively around the world, has been a lecturer at the university level, and his articles have been published in several international journals. He is an author of 14 books in chemical and petroleum engineering, a contributor to the Encyclopedia of Chemical Processing and Design, Vol. 61, and is certified as a “train – the mentor trainer.”
Dr. Coker is a Technical Report Assessor and Interviewer for chartered chemical engineers (IChemE) in the U.K., and is a member of the International Biographical Centre in Cambridge, U.K. (IBC) as one of the Leading Engineers of the World for 2008. Also, he is a member of the International Who’s Who for Professionals™ and Madison’s Who’s Who in the U.S. He recently received > 35,000 reads of his research activities in Researchgate.net and > 3,000 citations of his works in Academia.edu. His forthcoming book, The Soft Skills Edge-Engineering Your Path to Professional Success will be published by Tellwell Publishing Co. in 2026. The author can be reached at kcoker1@hotmail.com.