Low-Order Bessel-Type PID Dynamics in Lithium-Based Tritium Breeding and Heat-Removal Systems

Abstract

Lithium plays a dual role in deuterium-tritium fusion systems by enabling tritium breeding in blankets and providing an efficient heat-removal medium in liquid-metal components. Here, we combine nuclear data for deuterium-tritium and lithium reactions with a reduced thermohydraulic model of a liquid lithium jet and an operator-theoretic formulation of feedback control. We derive a low-order model for jet thermal expansion under deuteron-beam loading and show that a continuous-time proportional-integral-derivative controller, written in operator form, can be locally embedded in a family of Bessel-type differential operators acting on the tritium-inventory error. The results suggest that lithium-based breeding and heat-removal systems admit low-order, proportional-integral-derivative controllable dynamics that can be interpreted in terms of localized Bessel modes, providing a compact analytical framework for guiding future controller design and blanket/jet optimization.

Figures (5)

• Figure 1: Total cross section for D–T collisions as a function of the center-of-mass energy $E_{CM}$. The red circle marks the maximum $\sigma \approx 5.0$ barns at $E_{CM} \approx 65$ keV, a range compatible with ion temperatures envisaged for near-term D–T reactors. Experimental data are from Ref. bosch_hale_1992.
• Figure 2: Schematic interaction between a deuteron beam and a free-surface liquid Li jet in IFMIF-DONES-type facilities. The configuration reproduces fusion-relevant neutron irradiation conditions for test materials while imposing stringent constraints on jet stability, heat removal, and surface integrity.
• Figure 3: Different values for the maximum velocity $v_z$ (in logarithmic scale) toward the front surface.
• Figure 4: Time evolution of the $^{6}\mathrm{Li}$ enrichment ratio $R(t)$ (top left) and the corresponding tritium breeding ratio $\mathrm{TBR}(t)$ (top right) obtained from the linear surrogate model \ref{['eq:tbr']}, together with the target $\mathrm{TBR}_{\text{target}} = 1.0$. The bottom panel shows the second derivative of the tritium-inventory error $E"(t)$ (solid curves) and the corresponding least-squares second-order fits $E"(t)+a_1E'(t)+a_0E(t)\simeq 0$ (dashed curves) for several Bessel orders $\nu$. For all cases, the controller maintains small, bounded deviations $0.90 \lesssim \mathrm{TBR}(t) \lesssim 1.10$, and the error dynamics are well described by a low-order LTI model.
• Figure 5: Numerically computed second derivative of the tritium‑inventory error $E(t)$ (left panel) from the reduced jet/blanket–PID model (solid curves) and corresponding least‑squares fits to the second‑order approximation $E"(t)+a_1E'(t)+a_0E(t)\simeq0$ (dashed curves) over the approximately linear time window. The close agreement indicates that the closed‑loop error dynamics are well captured by a low‑order model compatible with the localized Bessel‑type operator. The right panel shows that, for different $\nu$ (Bessel orders), the closed‑loop jet temperature always relaxes smoothly to the same target value with only mild oscillations.