Finite-Time Thermodynamics Perspective into Nuclear Power Plant Heat Cycle

TL;DR

This work reframes nuclear power plant optimization through finite-time thermodynamics, moving beyond quasi-static Carnot limits to power-focused performance. It first derives the Curzon-Ahlborn efficiency for a Brayton-like cycle and then extends the framework to Rankine cycles that include liquid-vapor phase transitions, highlighting how latent heat shapes achievable power and efficiency. The analysis shows that maximum power and efficiency increase with latent heat and depend critically on operating parameters such as the high-temperature pressure and mass flow rate, with HTR-PM-based examples illustrating practical implications. The findings offer design guidance for selecting working substances and operating regimes to improve nuclear power plant performance under varying load demands, particularly for Rankine-type cycles with phase changes.

Abstract

Nuclear power plants are prominent examples of heat-to-work conversion systems, and optimizing their thermodynamic performance offers significant potential for enhancing energy efficiency. With a development history of less than a century, optimization trends in nuclear power plants indicate that classical thermodynamics alone may be insufficient, particularly when maximizing output power rather than efficiency becomes the primary focus. This paper re-examines nuclear power plant thermodynamic cycles through the lens of finite-time thermodynamics, an approach specifically developed to address the practical requirement of enhancing power output. Beginning with the simpler Brayton cycle without phase transitions, we obtain the famous Curzon-Ahlborn formula for efficiency at maximum power. Subsequently we analyze the more complex Rankine cycle, which incorporates phase transitions. By explicitly considering the working fluid undergoing phase transitions within the cycle, we uncover the inherent trade-off between output power and efficiency. Additionally, we demonstrate that both the maximum attainable power and efficiency increase as latent heat rises. These findings shall provide insights and methodologies for future thermodynamic optimization of nuclear power plants and other Rankine-type cycle systems.

Figures (8)

• Figure 1: Nuclear power plant structure and the energy transfer loops. The power generated in the reactor core is delivered to grid by the two separate closed-loop circuits. The first loop (gray dotted box) circulates primary coolant through the reactor core and the steam generator, transferring heat from the core to the secondary side. The second loop (brown dotted box) carries the working substance---typically steam---which absorbs heat in the steam generator and then expands through the turbine to produce mechanical work. After the turbine, the exhaust steam is condensed at approximately constant pressure in the condenser to form feedwater. The feedwater is then pumped back to the steam generator, closing the thermodynamic cycle (the condenser is cooled by a separate cooling-water system, often via a cooling tower). $P_{\mathrm{th}}$ is the power generated by the nuclear reactor. $P_{\mathrm{el}}$ is the output power of the generator. $\dot{W}$ is the work output by turbine. $\dot{Q}_{h}$ is the heat power absorbed by the steam generator and $\dot{Q}_{c}$ is the heat power released to the condenser. Auxiliary power is generated by the turbine and used inside the steam cycle such as supplying the pump. Net electrical power is the power delivered to the electrical grid.
• Figure 2: Efficiency varies with generations. Different colors represent the different generations and different shapes distinguish the coolants applied by these nuclear power plants. Transparent ellipses are used to show the efficiency concentration zone of different generations. The black dotted line represents the Carnot efficiency. The green line represents the CA efficiency. The data is shown in Appendix A.
• Figure 3: Schematic diagrams of the Brayton cycle. (a) Temperature--volume ( $T-V$ ) diagram illustrating both quasi-static and finite-time Brayton cycles. The light-colored dashed lines with the endpoints $1^{\prime}\rightarrow2^{\prime}\rightarrow3^{\prime}\rightarrow4^{\prime}$ represents the quasi-static Brayton cycle. Processes $1^{\prime}\rightarrow2^{\prime}$ and $3^{\prime}\rightarrow4^{\prime}$ are quasi-static isobaric expansion and the contraction processes. Processes $2^{\prime}\rightarrow3^{\prime}$ and $4^{\prime}\rightarrow1^{\prime}$ are the two quasi-static adiabatic processes. The finite-time Brayton cycle is represented the solid black lines with the ending points $1\rightarrow2\rightarrow3\rightarrow4$. The carmine line indicates the temperature changes of the heat reservoir during the isobaric expansion, while the blue line represents the temperature changes of the cold reservoir during the isobaric contraction. (b) Diagram of heat-work conversion in the quasi-static Brayton cycle. The variations of color represents the changes of the temperature for both working substance and the reservoirs. (c) Transverse cross-sectional view of the heat exchanger facilitating the isobaric processes. The diameter of the heat exchanger is $d$. Temperatures of the heat reservoir ($T_{s}$) and the working substance ($T_{w}$) are assumed homogeneous within this cross-section. (d) Longitudinal cross-sectional view along the cylinder axis of the heat exchanger. It is assumed that the fluid within the heat exchanger reaches steady-state conditions with a stable temperature gradient. Over incremental distances, heat absorbed by the working substance equals heat released by the heat reservoir, with heat transfer rates governed by Newton's cooling law.
• Figure 4: Power as the function of the temperature ratio $r_{t}$ and mass flow rate $\dot{m}_{w}$. Parameters used in calculations: hot and cold reservoir temperatures $T_{H}=1023.15\textrm{K}$ and $T_{C}=298.15\textrm{K}$; coefficients of heat conduction in hot and cold side $k_{H}=k_{C}=3000\textrm{W}\textrm{m}^{-2}\textrm{K}^{-1}$; specific heat capacities $c_{pw}=c_{ps}^{H}=c_{ps}^{C}=1.004\times10^{3}\textrm{J}\textrm{kg}^{-1}\textrm{K}^{-1}$; hot and cold side heat exchanger lengths $L_{H}=L_{C}=2\textrm{m}$; hot and cold side heat exchanger diameters $d_{H}=d_{C}=0.6\textrm{m}$. The power attains a maximum when the temperature ratio and flow rate are optimized independently. The black solid line marks the optimal value $r_{t}=\sqrt{T_{C}/T_{H}}$.
• Figure 5: Heat exchanger model with phase transition. (a) The $T-V$ diagram of the Rankine cycle. The black lines with arrows show the Rankine cycle of the working substance. In the isobaric expansion ($1\rightarrow2$) and isobaric expansion ($3\rightarrow4$ ), the working substance undergoes the phase transition at the phase transition temperatures $T_{p}^{H}$ and $T_{p}^{C}$. In phase transition processes, the temperature maintains and the volume changes which is represented by the gap in the black lines. The carmine (blue) line represents the temperature changes of the hot (cold) reservoir. Due to the assumption of phase transition of the working substance occurring in one position, the temperatures of the reservoirs have a sudden change. (b) The heat exchanger model with phase transition in the isobaric expansion process. The outer cylinder in red color represents the hot reservoir transferring heat with stable temperature distribution. The inner cylinder represents the working substance with the navy(wathet) blue part representing the liquid (gas) phase. $L$ is the total length of the cylinder and $T_{H}$ is the inlet temperature of hot reservoir. $T_{w_{l}}^{H}(0)=T_{1}$ is the inlet temperature of the working substance. $L_{p}^{H}$ is the phase transition position, where the temperature of hot reservoir has a sudden changes from $T_{s_{p}}^{H}$ to $T_{s_{l}}^{H}.$