Triangular Triple Loop Thermosyphon System

• The paper’s main contribution is the detailed mathematical modeling using reduced-order Fourier modes to capture the dynamics of the triangularly coupled thermosyphon system.
• Numerical experiments demonstrate that decentralized adaptive control with active disturbance rejection effectively stabilizes complex thermal–fluid interactions.
• Explicit stability conditions and feedback gain thresholds are derived, enabling robust design in energy conversion systems and passive safety applications.

A triangularly coupled triple loop thermosyphon system consists of three natural circulation loops arranged so that each exchanges both momentum and heat with its neighbors at coupling points, forming a closed triangular interaction topology. Such configurations are of particular technical relevance in advanced energy conversion systems, process engineering, and passive safety systems, where robust, decentralized thermal management and disturbance rejection are required. The system involves a combination of nonlinear thermal–fluid dynamics, distributed heat exchange, and complex coupled stability phenomena, motivating both detailed mathematical modeling and advanced control strategies.

1. Physical Configuration and System Modeling

The triangularly coupled triple loop thermosyphon system is constructed from three discrete thermosyphon (natural circulation) loops, each typically containing a working fluid subjected to cyclic heating and cooling. The loops are coupled at common nodes, allowing for both momentum and heat transfer. The physical model is governed by a set of nonlinear ordinary differential equations (ODEs) representing the bulk variables—often velocity ($x_i$), temperature difference ($y_i$), and secondary temperature ($z_i$)—for each loop $i = 1, 2, 3$.

Each loop’s dynamics include:

• Momentum balance: Incorporates buoyancy driving (from temperature differences), distributed viscous friction, and coupling terms parameterized by momentum coupling parameters $\gamma_i$.
• Energy balance: Modeled through temperature states with coupling via heat exchangers; incorporates thermal inputs, convective heat transfer terms, and thermal coupling parameters $\eta_i$.
• Heat exchanger boundary: The effective heat flux at each exchanger interface is a function of the inter-loop temperature differences and the local overall heat transfer coefficients $U$.

In spectral or reduced-order models, the spatially-dependent variables (such as temperature profiles $T_i(x, t)$) are expanded via truncated Fourier series:

$$T_i(x, t) = \sum_n Y_n^{(i)}(t) \cos\left(\frac{n\pi x}{L+L_i}\right)$$

where $L$ is the loop length and $Y_n^{(i)}(t)$ are time-dependent Fourier coefficients. This approach results in a finite-dimensional but stiff ODE system describing the evolution of dominant modes.

The canonical coupled ODE system for the three-loop configuration (see (Dey et al., 16 Oct 2025, Subramaniyan et al., 2023)):

$$\begin{align*} \dot{x}_i &= p [ (y_i - x_i) - \sum_j (\gamma_{ij} (x_i - x_j)) ] \ \dot{y}_i &= R_i x_i - x_i z_i - k_i y_i \ \dot{z}_i &= x_i y_i - z_i - \sum_j ( \eta_{ij} (z_i - z_j) ) \end{align*}$$

with $p$ the system constant, $R_i$ the Rayleigh numbers, $k_i$ feedback gains, and summations taken over neighbor loops.

2. Effects of Thermal and Momentum Coupling

Coupling in the triangular system has both thermal (via heat exchange at common plate/coupling points) and momentum (via fluid inertia/transient pressure) aspects:

• The heat transfer at each coupling point is quantified by:

$Q_{HX} = U \int (T_i(x, t) - T_j(x, t))\,dx$

where $U$ is the overall heat transfer coefficient, a critical parameter for the strength of coupling.

• Momentum coupling is described by parameters $\gamma_i$, which enter directly into the ODEs and affect the linear and nonlinear stability.
• The interplay between these couplings can stabilize, destabilize, or produce neutrally stable or chaotic dynamic regimes, depending on the magnitudes and functional forms.

A key finding is that, within feedback control configurations acting only on the $y$-states (temperature differences), the critical stability bounds on feedback gains depend on momentum coupling parameters ($\gamma_i$) and Rayleigh numbers ($R_i$), but are independent of the thermal coupling parameters ($\eta_i$) (Dey et al., 16 Oct 2025). This suggests that while heat transfer alters system dynamics, stability margins for this feedback structure are dictated primarily by momentum exchange.

3. Stability Conditions and Parametric Sensitivity

The stability of the triangularly coupled system is established through the construction of a symmetric matrix $A$ embedding coupling and feedback gain terms. Positive definiteness of $A$, verified via Sylvester’s criterion (positivity of all leading principal minors), provides explicit stability thresholds:

• Example lower bound for loop 1:

$k_1 > \frac{R_1}{1 + \gamma_1 + \gamma_2}$

• Higher order minors involve auxiliary quantities such as

$$\psi_1 = R_1R_2 \big[(1+\gamma_1+\gamma_2)(1+\gamma_1+\gamma_3) - \gamma_1^2 \big] - \frac{(R_1 - R_2)^2 \gamma_1^2}{4}$$

and similar analytic expressions, which combine all coupling parameters.

These explicit conditions offer precise criteria for feedback design. The stability boundaries reveal that, for fixed $R_i$ and $\gamma_i$, increasing the feedback gains $k_i$ above certain thresholds ensures global asymptotic stability, regardless of heat transfer strength (for $y$-state-only feedback). Furthermore, the system displays significant sensitivity to the geometric configuration (e.g., channel cross-sectional area), with oscillatory or chaotic behavior possible when design parameters push the system near or beyond these thresholds (Subramaniyan et al., 2023).

4. Decentralized Adaptive Control and Disturbance Rejection

A decentralized adaptive control scheme addresses the practically relevant case where parameters such as $R_i$ or $\gamma_i$ are unknown or subject to uncertainty:

• Each local controller applies the update law:

$\frac{dk_i}{dt} = \alpha_i y_i^2,\quad i = 1,2,3$

where $\alpha_i$ is a tunable learning rate. This ensures the local gain $k_i$ increases monotonically, automatically seeking a value exceeding the theoretically required bound exposed in the stability analysis.

• The controller employs only locally available measurements ($y_i$), rendering the design fully decentralized and suitable for large-scale implementations or networks where centralized coordination is infeasible.
• In the presence of additive or state-dependent uncertainties (e.g., from higher-order unmodeled dynamics or disturbances $f_i$), the system augments each $y$-channel with an extended state observer (ESO), as in the active disturbance rejection control (ADRC) framework:

$$\frac{d\hat{y}_{i,1}}{dt} = \hat{y}_{i,2} + \beta_{i,1}(y_i - \hat{y}_{i,1}) + u_i$$

$$\frac{d\hat{y}_{i,2}}{dt} = \beta_{i,2}(y_i - \hat{y}_{i,1}) + g_i(\hat{y}_i, \zeta_i)$$

The estimated lumped disturbance $\hat{f}_i$ is then canceled in real time via the feedback law $u_i = -k_i y_i - \hat{f}_i$.

The theoretical analysis demonstrates convergence of the observer error $|f_i - \hat{f}_i| \rightarrow 0$ under smoothness and Lipschitz conditions, thereby enabling robust rejection of both modeled and unmodeled disturbances (Dey et al., 16 Oct 2025).

5. Numerical Experiments and Dynamic Regimes

Simulations validate and illustrate the theoretical developments:

• In undisturbed scenarios, adaptive gains with low learning rates yield gradual stabilization; increasing the learning rate accelerates convergence to steady-state.
• When artificially introduced disturbances (e.g., sinusoidal functions of the state) are present, absence of the ADRC/ESO leads to instability even under high feedback gains. With disturbance rejection activated, the system is stabilized and feedback gains remain within practical bounds.
• The transition between stable, chaotic, and neutrally stable regimes is well captured by the mathematical analysis and confirmed in the simulation results; parameter sets outside the derived bounds invariably lead to persistent or amplifying oscillations, even when each loop is independently stable.

6. Extensions and Practical Relevance

The modeling, control design, and stability analysis techniques developed for triangularly coupled triple loop thermosyphon systems generalize key insights from studies of two-loop CNCLs (Subramaniyan et al., 2023):

• The spectral ODE approach, employing reduced Fourier modes, offers computational tractability when extended to three-loop (or larger) networks, supporting system-level design and optimization.
• Critical dependence on system geometry, fluid properties (especially through temperature-dependent density, viscosity, and specific heat), and coupling parameters is preserved and even amplified in triangular configurations.
• The decoupling of feedback stabilization from the thermal coupling rate (when using $y$-state feedback) allows direct controller synthesis without requiring precise knowledge or estimation of heat transfer coefficients—a crucial advantage for process and safety-critical applications.
• Observed phenomena and control strategies inform robust decentralized design in broader contexts, such as nuclear reactor passive heat removal, modular energy systems, and advanced process industry cooling architectures.

In summary, the triangularly coupled triple loop thermosyphon system presents a mathematically and operationally rich prototype for multi-agent, interconnected thermal-fluidic systems. The integration of explicit stability conditions, adaptive decentralized controllers, and active disturbance rejection establishes a framework for practical stabilization, diagnosis, and optimal operation under uncertainty and disturbance. These findings have immediate relevance for the engineering of scalable, robust, and efficient passive heat transfer systems (Subramaniyan et al., 2023, Dey et al., 16 Oct 2025).