Heat pipe-cooled nuclear-powered system

After the micronuclear reactor Kilopower was designed and successfully tested for long-term deep-space exploration, the Los Alamos National Lab (LANL) recently proposed a similar but scaled-up reactor called Megapower [43]. This novel Megapower design offers many advantages. A 5 MW solid-state micro nuclear reactor was used as the power source, and then the heat was delivered to a power conversion system at the other end by potassium heat pipes. These heat pipes operated at approximately 950 K and demonstrated high heat transfer efficiency.

Fig. 1 shows the detailed structure of the reactor core. Many holes were drilled in a SS316 monolith to contain stacked dime-sized nuclear fuel pellets, resulting in a gap of 0.0065 cm between the fuel and monolith; hence, helium fed by the bottom gas plenum filled these gaps. The extra heat resistance caused by these gaps could be critical when determining the heat-transfer performance and spatial temperature distributions of the solid-state core during operation. In particular, the size of the assembly gaps changed because of the thermal expansion of the structural materials or release of fission gas from the fuel, making the gap behavior more unpredictable. An alkali metal working medium was filled into the other holes; thus the heat pipe evaporation section was integral to the SS316 monolith and no gap existed around it. The fuel pin and heat pipe had a hexagonal layout, and the ratio of their total numbers in the core was set to approximately 2:1, to balance the compact dimensions, neutronics, and heat transfer design requirements. Twelve control drums (CDs), which functioned as the main means of power control, were placed lateral to the core [44]. The drums made of B₄C are good at neutron absorption; therefore, the inward rotation of the CDs decreased the reactor power and vice versa.

Fig. 1Full-screen View

(Color online) Conceptual diagram of the Megapower nuclear reactor: (a) whole system including the power conversion component and (b) radial and axial layout of the 1/6 reactor core.

Reactor dynamic model

The point kinetic equation (PKE) is a simplified dynamic model in which a nuclear system is regarded as a single point in space, such that the power level is time-dependent. PKE is a set of ordinary differential equations (ODEs) whose numerical solutions can be easily obtained. The mathematical description of the PKE is as follows: $\frac{\text{d}P}{\text{d}t}=\frac{\rho -\beta }{\text{Λ}}P+{\displaystyle \sum _{i=1}^6{\lambda }_i{C}_i},$ (1) $\frac{\text{d}{C}_i}{\text{d}t}=\frac{{\beta }_i}{\Lambda }P-{\lambda }_i{C}_i,$ (2)

Here, P, $\rho$ and $\text{Λ}$ represent the normalized power, net reactivity, and prompt neutron generation time, respectively. ${\beta }_i$, ${\lambda }_i$ and $C_i$ represent the delayed neutron fraction, decay constant, and precursor concentration, respectively, in the i-th group. Although most of the prompt neutrons were released after the reaction, a small portion of delayed neutrons were still produced, making the nuclear power controllable. Based on the PKE, when $\rho$ is equal to $\beta$, it is defined as 1 $, and $\rho =0$ represents the steady state of the investigated system.

The other variables presented in the PKE are called point kinetic parameters and are determined by the reactor configuration. For this research, 1/6 of the Megapower reactor was modeled and simulated using the Monte Carlo code OpenMC [45], as shown in Fig. 1(b), to solve these parameters. These values are listed in Table 1.

The net reactivity is divided into two parts, as shown in the following equation: $\rho ={\rho }_{\text{e}}+{\rho }_{\text{f}}\left(T_{\text{core}}\right),$ (3) where ${\rho }_{\text{e}}$ is the external reactivity caused by the mentioned active CD rotation, and ${\rho }_{\text{f}}$ is the feedback reactivity related to the temperature variation of the reactor core. In this study, Doppler-broadening feedback was considered.

Based on a pre-conducted Monte Carlo simulation, the correlation between the reactor temperature and inherent reactivity was also obtained. The result suggests a linear relation between $\text{ln}\left(1-{\rho }_{\text{inh}}\right)$ and $\text{ln}T$, as shown in the following equation[46]: ${\alpha }_{\text{T}}=\frac{\text{ln}\left(1-{\rho }_{\text{inh}}\right)-\text{ln}\left(1-{\rho }_{\text{inh},\text{ref}}\right)}{\text{ln}T-\text{ln}{T}_{\text{ref}}}.$ (4)

Here, the subscript ref means the reference value, and the slope ${\alpha }_{\text{T}}$ is the Doppler temperature feedback coefficient. These three parameters were constant, and their fitting values are listed in Table 2. Consequently, the feedback reactivity at a certain core operating temperature $T_{\text{core}}$ can be solved as follows: ${\rho }_{\text{f}}\left(T_{\text{core}}\right)={\rho }_{\text{inh}}\left(T_{\text{core}}\right)-{\rho }_{\text{inh}}\left(T_0\right),$ (5) where $T_0$ is the room temperature at the beginning of the system operation.

Nonnuclear experimental setup and its evaluation

A schematic of the thermal test is shown in Fig. 2, and the key test parameters are listed in Table 3. Electrical heating rods and water heat pipes were used to imitate the heat released from the nuclear fuel and that removed by the alkali high-temperature heat pipe, respectively. Aluminum is lighter and softer and has a lower processing cost than stainless steel; furthermore, it has a high heat conductivity. Hence, an aluminum block with an edge of 10 cm was fabricated to provide a uniform heating environment. Many holes were drilled to place the heat pipes and heating rods. Eighteen heat pipes and four heating rods were placed in the block in a hexagonal layout; however, the ratio of their numbers was not the same as that of the Megapower reactor. This is because the water heat pipe-cooled system worked at a lower temperature; hence, more heat pipes were required to improve the maximum heat transfer capability of the test bench.

Fig. 2Full-screen View

(Color online) Structure and temperature measurement point of the mHPR prototype.

The diameters of the holes in the block were slightly larger than those of the heating rods and heat pipes, which resulted in small gaps of 0.3 mm. Thermocouples were inserted in these gaps to measure the internal core temperature. These gaps can deteriorate the heat transfer between the block and pipes. Hence, these gaps were filled with silicone grease to address this issue. A layer of aluminum silicate was packaged outside the entire facility for thermal isolation, leaving only the top of the heat pipes exposed to the environment for heat exchange and cooling.

The total length of the heat pipe was 34 cm, and the lengths of the evaporator, adiabatic section, and condenser were 10, 14, and 10 cm, respectively. This implies that the reactor and power conversion systems had the same length. These heat pipes were arranged in gravity-assisted orientations to improve the working circulation of the internal medium and effective heat conductivity. The temperature measurement points are shown in Fig. 2. The distances from points 1–6 to the evaporator end were 3, 7, 14, 20, 27, and 31 cm, respectively. We deployed 16 K-type thermocouples on the surfaces of heating rods #a and #b and heat pipes #c and #d. Another four thermocouples were glued to the lateral side of the block to measure the surface temperature. The sheaths of these thermocouples were made of Neoflon PFA, which has good insulation characteristics below 530 K.

In addition, a fan was installed to ensure air convection around the condenser. Two NI 9213 modules operating with NI c-DAQ 9174 chassis were used for temperature data logging. A programmable DC power supply with four independent channels was connected to the heating rod. The output power of these heaters can be controlled using the LabVIEW program. A photograph of the test bench is displayed in Fig. 3.

Fig. 3Full-screen View

(Color online) Photographic of the HIL-based mHPR prototype.

The thermal resistance of the heat pipe can be defined by the following equation: $R_{\text{HP}}=\frac{T_{\text{eva}}-T_{\text{con}}}{P_{\text{input}}/N},$ (6) where $T_{\text{eva}}$ and $T_{\text{con}}$ denote the average temperatures of the evaporator and condenser of the heat pipe, respectively. $P_{\text{input}}$ is the total input power. Based on [27], the use of the nominal heat input to evaluate $R_{\text{HP}}$ is reasonable because the trends are similar to those of the real $R_{\text{HP}}$, calculated using the actual heat transferred from the heat pipe ($P_{\text{trans}}$). In Eq. (6), $N$ represents the number of heat pipes and has a value of 18.

According to the specifications of the K-type thermocouple, it has an error of 2.2 K. The temperature measurement uncertainty can be calculated using Eq. (7) where $C=\sqrt{3}$ denotes the confidence coefficient. As the minimum temperature during the experimental process was approximately 300 K, the relative error of the temperature measurement was below 0.5%. $U_{\text{B},\text{T}}=\frac{\text{Δ}}{C}=\frac{2.2}{\sqrt{3}}=1.27 \text{K}$ (7)

The error in the supply voltage was 0.05% + 0.1% FS, whereas the error in the current was 0.1% + 0.1% FS. The rated power source range was 60 V/15 A. During the experiments, the maximum value was 32 V/8 A. The relative error of the power can be evaluated using Eq. (8) as follows: $\frac{\text{Δ}P}{P}=\sqrt{{\left(\frac{\text{Δ}V}{V}\right)}^2+{\left(\frac{\text{Δ}I}{I}\right)}^2}$ (8)

Hence, the uncertainty of the input power is near 0.37%.

The uncertainty of the heat pipe thermal resistance can be deduced as follows: $\frac{\text{Δ}{R}_{\text{HP}}}{R_{\text{HP}}}=\sqrt{{\left(\frac{\text{Δ}P}{P}\right)}^2+{\left(\frac{\text{Δ}\left(\text{Δ}T\right)}{\text{Δ}T}\right)}^2}.$ (9)

Here, $\text{Δ}T$ is the temperature difference between the evaporator and condenser. When the input power is low for a heat pipe, $\text{Δ}T$ will also be small, resulting in a relatively large error in the calculated thermal conductivity.

After the test bench was built, a series of pretests were conducted to evaluate the heat loss of the entire experimental installation. To achieve this, we closed the fan near the condenser, creating a natural convection environment for cooling. This allowed us to simplify the heat transfer boundary condition of the condenser. The heat loss of the test rig in the environment was evaluated as follows: $\text{Heat loss}=1-\frac{P_{\text{trans}}}{P_{\text{input}}} \left(\%\right),$ (10) where the heat input ($P_{\text{input}}$) was obtained directly from the DC power source, and the total actual power transferred by the heat pipes ($P_{\text{trans}}$) was evaluated using the radiation and natural convection of the condenser. $P_{\text{trans}}=A\epsilon {\sigma }_{\text{B}}\left(T_{\text{con}}^4-T_{\text{air}}^4\right)+Ah\left(T_{\text{con}}-T_{\text{air}}\right),$ (11) where $\epsilon$ denotes surface emissivity and has a value of 0.8 [47], ${\sigma }_{\text{B}}$ is the radiation constant, and $A$ is the condenser area of all these heat pipes. The following equation can be used to evaluate the Nusselt number of a vertical pipe undergoing natural convection, when $Gr$ is in the range of 1.43×10⁴–3×10⁹ [48]. $Nu=0.59{\left(Gr\cdot Pr\right)}^{\frac{1}{4}}$ (12)

The heat transfer coefficient $h$ in Eq. (11) can be obtained as follows. The thermal properties of air in Eq. (12) was evaluated using the average values of $T_{\text{con}}$ and $T_{\text{air}}$.

To protect the used thermocouples (< 530 K), the power input of the series of heat loss pretests was set within the range of ~60–180 W, which resulted in a steady-state core temperature $T_{\text{core},s}$ of ~368.73–464.54 K. Based on the results of the pretests shown in Table 4, the heat loss is within 20% when $T_{\text{core},s}$ is within ~368.73–464.54 K. The range of the core temperature for the following formal tests with the fan opened (Table 5) is similar to those in the pretests. Because the heat loss of the system is primarily dependent on the system temperature, the heat loss for the formal tests was also regarded as ~20%.

Table 5Full-screen View

Detailed test matrix for both steady-state and transient operations.

No	State	Inserted reactivity ($)	Power (W)
1	Steady-state	-	100
2			200
3			300
4			400
5			500
6	Transient		Open-loop start-up: initial power is 1 W and updated according to the inserted positive external reactivity and the temperature feedback effect.
7
8
9
10
11
12	Transient		Open-loop power-down: initial power is 400 W and updated according to the inserted negative external reactivity and the temperature feedback effect.
13
14
15	Transient	Closed-loop: determine ${\rho }_e$ based on the on-the-fly power and the PID controller.	Goal PSP equals the final power of Case 7.
16			Goal PSP equals the final power of Case 13.

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System architecture

In the present work, we use a HIL to study the dynamic response of a solid-state reactor, such as the Megapower reactor; the proposed architecture is shown in Fig. 4. As mentioned in Section 2, the nuclear system introduction, dynamic model, and scaled-down thermal test setup are demonstrated. Therefore, their combination should be designed and implemented as a functional HIL platform.

Fig. 4Full-screen View

HIL architecture applied for dynamic evaluation of the mHPR.

The test bench, serving as the hardware, plays a crucial role in sampling temperature data from the real world is a vital component of the dynamic simulation. Ensuring synchronous data exchange between the hardware and software components, especially under transient situations, has been thoroughly considered. The primary components of the hardware side include the main test section, power supply, and data acquisition unit are the hardware parts, as shown in Fig. 2. A LabVIEW program, running in real time on a laptop, served as the software component and was capable of reading the measured data, solving the reactor dynamic equation, and displaying the results, among other functions. The data exchange between the hardware and software was facilitated through the simultaneous handling of the power input and temperature measurement. The measured core temperature was used to update the feedback reactivity using Eq. (4).

Power controller

This program was designed as an optional controller. If a controller is added to the system, a closed-loop simulation is activated, which implies that the external reactivity appearing in Eq. (1) is adjusted by the process power to maintain the power output at the set point. Else, the external reactivity is set as a fixed value to test the power and temperature response features under open-loop operation [49].

A classic PID controller was used in this study because of its stable, effective, and model-free features. Because we built a test bench to determine the thermal response of the nuclear-powered system, the temperature data was periodically sampled from the real world; hence, a digital PID control algorithm was employed. After discretization of the proportional, integral, and derivative actions, the output of the positional PID controller was obtained as a linear combination of these actions, and it can be generally expressed by Eq. (13) [50]. $\begin{array}{c}u\left(k\right)=K_{\text{p}}e\left(k\right)+u_{\text{I}}\left(k-1\right)+\frac{K_{\text{p}}}{T_i}\text{Δ}T\left(\frac{e\left(k\right)+e\left(k-1\right)}{2}\right)\\ -K_{\text{p}}\frac{T_{\text{d}}}{\text{Δ}T}\left(PV\left(k\right)-PV\left(k-1\right)\right),\end{array}$ (13) where $k$ is one of the sampling points, $\text{Δ}T$ is the sample rate, and $u_{\text{I}}\left(k-1\right)$ is the past output of the integral action. $u$ and $e$ are the output and input of the controller, respectively, and $e$ represents the system error. $e\left(k\right)=SP\left(k\right)-PV\left(k\right),$ (14) where $SP$ and $PV$ are the setpoint and process variables, respectively. In Eq. (13), $K_{\text{p}}$, $T_{\text{i}}$, and $T_{\text{d}}$ represent the controller gain, integral time, and derivative time, respectively. These parameters work together to determine the controller performance; thus, the parameters should be appropriately tuned before the formal application of the PID controller. Based on the tuning results in this study, $K_{\text{p}}$, $T_{\text{i}}$, and $T_{\text{d}}$ were set to 0.012, 0.2 min and 0, respectively.

In this study, a setpoint power (P_sp) was applied as the input to the entire system. This error was produced by subtracting the process power (P_pv) from P_sp. We used this error as the controller input. The reactivity control mechanism of a real nuclear-powered system is significantly more complicated. Therefore, a PID controller was used to simplify the control process and produce the rate of external reactivity insertion as the output $u$, as shown in Fig. 4.

As shown in Fig. 5(a), with the rotation of the CDs and increase in its angle, the spatial location of the neutron absorption material B₄C gradually moved away from the reactor core, resulting in an increase in the neutron population, external reactivity, and power level. Based on the OpenMC simulation results, the relationship between the calculated reactivity and CD angle is shown in Fig. 5(a). A total external reactivity of approximately 9 $ is added into the nuclear system as the CDs rotate from 0° to 180°. However, the rate of the external reactivity increase is uneven; for instance, this process is faster when the CD angle is approximately 90°. According to past mechanical designs of small solid-state nuclear reactors and CDs, the maximum rotating speed of CDs should be 1°/s [51]. Therefore, the mean rate of external reactivity insertion among different ranges of the CD angle is shown in Fig. 5(b). Here, the maximum value is less than 0.1 $/s. It is easy to manipulate the rotation direction of these CDs while running. Hence, the output upper limit of the PID controller used here was set as ±0.1 $/s to improve the stability of the controller.

Fig. 5Full-screen View

(Color online) Determination of the upper limit of the rate of external reactivity insertion as the PID controller output, considering a maximum CDs rotation speed of 1°/s: (a) reactivity vs. CD angle and (b) rate of external reactivity insertion.

Real-time implementation

Assuming the HIL time step to be 250 ms, the working flow of the software and temperature data treatment during one step are displayed in Fig. 6. The sampling rate of the measurement module was set to 40 Hz, indicating 10 measurement points within one step. These data can be stored in a buffer zone located in the memory of a desktop PC. When the program runs, it removes data from the buffer by following the FIFO principle. An algebraic operation was used to calculate the average value, which represents the average temperature from $t_n$ to $t_{n+1}$. Under the above treatment, the temperature data can be used in time, and no more than 10 points will be stored in the buffer zone within one step, resulting in a good match between the data measurement and program calculation.

Fig. 6Full-screen View

Real-time feature of the HIL platform for a time step of 250 ms.

During a single data exchange period, the LabVIEW program mainly had 4 tasks. The first task was preprocessing for the dynamic model simulation, which includes using the measured temperature to update the feedback temperature or adjusting the external reactivity if the PID controller is activated. The remaining three tasks involved data presentation, application of the PKE solution from $t_n$ to $t_{n+1}$, and delivering the power output to the DC power supply. The numerical solution step for the PKE was set to 1*10^-4 s to ensure accuracy. For a general operating system such as Windows, the execution time for each task is indeterminate. Fig. 6 shows a typical physical time distribution when the step is chosen as 250 ms. These four tasks required 1, 4, 133, and 67 ms. For the first two tasks, the internal calculation in the LabVIEW environment does not involve time cost, whereas the remaining two are relatively time-intensive owing to the potential external communication with the MATLAB ODE solver and DC power supply. In short, the sum of the time spent on the above four tasks should never exceed the set time step to maintain the synchronous performance of the simulation and real-world times. Thus, a wait time of 45 ms was added at the end of the step to account for the discrepancy between these two time-concepts.

Fig. 7 shows the front configuration of the program during operation. The top-left part is the control panel, where essential simulation parameters, such as the time step and point kinetic parameters, can be set before running the simulation. The real-time monitor is in the top center zone, and it displays the plot of the time duration for each of the five working tasks in Fig. 6 during a single iteration, along with their summation. The power level, core temperature, and total operating time are listed in the top-right position for notification, in bold. With regard to these key results, their evolutions are presented at the bottom, including the power, reactivity, heat pipe mean temperature, core temperature, and room temperature. In all these plots, the x-coordinate represents the iteration number of the LabVIEW working loop, and the corresponding running time is calculated as the product of the iteration number and HIL time step Δt.

Fig. 7Full-screen View

Screenshot sample of the LabVIEW front panel while running the simulation.

Performance evaluation of the HIL platform

A sensitivity analysis of the time step was conducted to determine the best level for the current mHPR prototype. As shown in Fig. 6, the time spent solving the PKE was nearly 100+ ms. Thus, time steps of 250, 500, and 1000 ms were used in this sensitivity analysis, and their performance was evaluated independently.

First, a test scenario was set, as shown in Fig. 8(a). In this scenario, an external reactivity of 0.1 $ was suddenly inserted at 20 s, resulting in the power gradually increasing from the initial value of 1 W. The remaining three subplots in Fig. 8 show the physical time features of the HIL simulation for different time steps. During the first 20 s, the time required to solve the PKE was costless because the reactivity was zero, and the power was constant. However, for the first iteration after the reactivity was inserted, the solving time jumped to 300 ms, which was mainly attributed to the power prompt jump [49] and activation of the MATLAB node. Subsequently, the solving PKE time was stabilized at 100–150 ms. For all these tests, the time spent writing power to the DC power supply within a single iteration was in the range of 100–200 ms. The accurate value of the execution time was unpredictable because of the nature of the Windows OS. Therefore, the waiting time changed continuously to maintain the total execution time of each iteration within a set time step. As shown in Fig. 8(b), a step of 250 ms was inadequate because, at some points, the total execution time exceeded 250 ms because of disturbances, and thus, the real-time feature was not met. For a time step of 500 or 1000 ms, the total time step was always at the set value, suggesting that a larger time step has a better real-time performance for the HIL platform.

Fig. 8Full-screen View

Real-time performances under different time steps: (a) power and reactivity, (b) 250 ms, (c) 500 ms, and (d) 1000 ms.

Fig. 9 shows another test scenario used in the time-step sensitivity analysis, which considers the controller performance. The initial power was 1 W, and the setpoint of the power exhibited a linear increase in 20–50 s, finally reaching a new steady state at 100 W. The initial external reactivity was zero. From 20 s to 30 s, the rate of reactivity insertion quickly reached its upper limit of +0.1 $/s, indicating that the reactivity increased rapidly. During this period, the actual power did not follow the setpoint well as the initial power was low, and the neutron population grew exponentially. After 30 s, the processing power was consistent with that at the setpoint. However, as shown in Fig. 9(b), the controller output at 1000 ms showed an oscillation at approximately 50 s, but the results at 250 and 500 ms were similar. This test revealed that the time step should be reduced to obtain a better control performance.

Fig. 9Full-screen View

Power controller performances under different time steps: (a) power and (b) rate of reactivity insertion.

Based on the test results, if the time step is too small, the real-time features of the HIL platform can be influenced by the unpredictable execution time of program tasks, such as solving the PKE and writing power. However, if the time step is excessively large, the output of the PID controller becomes unstable because the reactivity and power information cannot be updated in a timely manner. Hence, we chose a time step of 500 ms in the subsequent scenarios, as it represents a good compromise between the real-time and control requirements.

Steady state

Fig. 10 shows the experimental results for Cases 1–5 and core temperature evolution process. The core temperature is defined as the mean temperature obtained from the thermocouples located inside the heated block. During the first 6000 s, the core temperature gradually increases to its maximum value, and this period is regarded as the heating period. Then, until 10,000 s, the core temperature remains stable with a disturbance of ±0.5 K, establishing the steady state. Finally, the external power supply was cut off, and the system temperature decreased to match the environment. The long test period was primarily due to the large thermal capacity of the solid block.

Fig. 10Full-screen View

Temperature variation under a constant power input (the heating was stopped at approximately 180 min).

The relationship between the steady-state temperature and power, shown in Fig. 11(a), exhibits a strong linear correlation. The average core temperature reached 500 K at 500 W. The results reveal a correspondence between the steady-state temperature of the power and system. For the heating and cooling processes, the concept of the temperature time constant can be applied to describe the speed of the temperature response of the system, whose mathematical expressions are shown in Eqs. (15) and (16) as follows: $\text{Δ}T\left(t\right)=\text{Δ}{T}_{\text{ss}}\left(1-e^{-\frac{t-t_0}{{\tau }_{\text{th,h}}}}\right)$ (15) $\text{Δ}T\left(t\right)=\text{Δ}{T}_{\text{ss}}\left(e^{-\frac{t-t_0}{{\tau }_{\text{th,c}}}}\right)$ (16)

Fig. 11Full-screen View

(Color online) Time response of the heat pipe cooled system under different input power: (a) steady-state temperature and (b) temperature time constant.

Here, ΔT(t) is the difference between the current and initial temperatures, and ΔT_ss is the difference between the final and initial temperatures. ${\tau }_{\text{th},\text{h}}$ and ${\tau }_{\text{th},\text{c}}$ are the time constants for the heating and cooling processes, respectively. Based on the experimental data shown in Fig. 10, the process temperature difference can be fitted using Eqs. (15) and (16), and the resulting values of ${\tau }_{\text{th},\text{h}}$ and ${\tau }_{\text{th},\text{c}}$ under different powers are plotted in Fig. 11(b). This indicates that the temperature–time constant of this system is in the range 1000–1200 s. Owing to the form of the temperature response described by Eqs. (15) and (16), the total heating and cooling times for all these cases were similar, as shown in Fig. 10. Hence, it can be concluded that the solid-state system temperature variation behavior is hardly affected by the constant power, and the thermal response of the system is relatively slow owing to the large thermal capacity with a temperature time constant of approximately 1100 s.

The operating performance of the water heat pipe under steady-state conditions is shown in Fig. 12. The axial temperature distribution was even for all cases. The temperature difference between the evaporator and condenser was approximately 10 K for Case 5, with a power input of 500 W. As shown in Fig. 12(b), the thermal resistance of one of the heat pipes was ~0.4 K/W, based on Eq. (6). Furthermore, the thermal resistance decreased as the power increased. This phenomenon is consistent with previous studies ^[52], which suggested that higher power improves the heat removal capability of the heat pipe. The uncertainty of the thermal resistance at a lower power level is much larger owing to the reduced temperature difference between the evaporator and condenser. In short, these heat pipes have an excellent the heat transfer capability, with an effective thermal conductivity of up to 4000 W/m-K.

Fig. 12Full-screen View

Heat pipe heat transfer evaluation under different power inputs: (a) axial temperature distribution and (b) thermal resistance of a heat pipe.

Start-up transient

Fig. 13 shows the startup process of Case 7, where we insert a step external reactivity of +0.1 $ at 300 s. Fig. 13(a) shows the variations in power and temperature. This positive external reactivity is compensated by the negative feedback reactivity caused by the temperature growth, such that a new steady state can eventually be established under a net reactivity of zero. As shown in Fig. 13(a), the intermediate process exhibits relatively complicated power and temperature oscillations, which can be explained by the unmatched time transient of the neutronics and thermal dynamics. A similar phenomenon has been previously reported [53]. The slow temperature increase at the beginning caused a significant power overshoot. The peak power rapidly emerged because of the exponential neutron growth. The high power accelerated temperature growth, resulting in a temperature overshoot. After passing through another much lower peak, the oscillation gradually vanished, indicating the establishment of a dynamic balance among reactivity, power, and temperature.

Fig. 13Full-screen View

Results of Case 7 with an external reactivity step of +0.1 $ inserted: (a) power and temperature and (b) reactivity.

The variation in the wall-temperature distribution of heat pipe #c in Case 7 is shown in Fig. 14. The wall temperature difference between the evaporator and condenser is within 3.5 K during the start-up transient. Based on [24], the variation in the axial vapor temperature is within 0.9 K for water heat pipes; thus, the wall temperature has a positive relationship with the wall heat flux. In this study, only the evaporator heating power was controlled and varied, whereas the condenser was cooled by natural convection and radiation. It can be found from the plots that the transient variation of ΔT_a-c has a similar oscillation trend with the power, meaning that heat pipes have better isothermal characteristics and the temperature difference between the reactor and the power conversion system is flatten when the heating power is decreasing.

Fig. 14Full-screen View

Variation of the wall temperature distribution of the heat pipe #c in Case 7.

During the sensitivity analysis of Cases 6–11, similar power and temperature oscillations were observed using the HIL test bench. In the theory of neutronics, the stable exponential power growth after a step of external reactivity is inserted can be approximated using Eq. (17). $P\left(t\right)=A{P}_0{e}^{\frac{t}{{\tau }_{\text{n}}}},$ (17) where $A$ is a proportionality coefficient, and $P_0$ is the initial power. ${\tau }_{\text{n}}$ is defined as the neutronics period, which means the time needed for the power to change by $e$ times. Table 6 lists the calculated neutronics period of Cases 6–11. As the external reactivity increases, the neutronics period decreases rapidly, and the power increases in the nuclear-powered system. For Case 11, a neutronics period of 31.61 s means that it only needs 72.8 s for the power to increase by an order of magnitude. However, the temperature time constant, shown in Fig. 11(b), is approximately 1100 s, which is much lower than the rate of power increase. This discrepancy could offer a quantitative interpretation of the oscillations observed in solid-state nuclear systems.

To investigate the nature of the oscillations, we utilize the characteristic times ${\tau }_1$ and ${\tau }_2$, along with the amplitudes of power and temperature, as defined in Fig. 13(a). This allows for a comprehensive description of both the duration and strength of the oscillations. The relationship between the external reactivity and neutronics period is displayed in Table 6 for open-loop Cases 6–11. As the net reactivity changes with temperature, the inserted external reactivity is used to represent the transient features of the system. The initial external reactivity reflects the rate of neutron multiplication. The trends in these characteristic parameters, as functions of the neutronics period, are shown in Fig. 15. As the external reactivity increases and neutronics period shortens, the first overshoot (${\tau }_1$) seems to occur earlier, and the amplitude of oscillation increases. In addition, the oscillations during long neutronic periods lasted longer. Fig. 15(c) demonstrates a strong linear correlation between the characteristic oscillation time and neutronics period. Notably, the power overshoot was much more severe than the temperature overshoot. Taking Case 11 as an example, the amplitude of the power reached 1200 W. Consequently, the oscillations following a substantial external reactivity insertion must be carefully monitored in the rapid start-up scenarios to protect the reactor core from the impact of intense power and thermal transients.

Fig. 15Full-screen View

Oscillation phenomena of temperature and power as functions of neutronics period: (a) amplitude of temperature, (b) amplitude of power, and (c) characteristic time.

To eliminate the oscillations observed in Cases 6–11 during the power-up process, a closed-loop operation and controlled external reactivity insertion were adopted in Case 15, which resulted in the key parameter evolution shown in Fig. 16. As mentioned in Section 3, the time step of the data exchange and operation of the PID controller was set as 500 ms. The power setpoint was chosen to linearly increase from 1 W to 94 W during the time period from 300 s to 330 s. Hence, the rate of power increase was 3.1 W/s. As seen in Fig. 16(a), the process power can generally follow the setpoint well, thereby affirming the effectiveness of the controller. As seen in Figs. 16 (a–c), the rate of reactivity insertion quickly reached its upper limit of +0.1 $/s at 300 s, resulting in an increase in the external reactivity to 0.7 $, indicating that no prompt criticality occurred and that the nuclear system was safe. The power also increased from 1 W to ~35 W under this large external reactivity. Subsequently, the rate of reactivity insertion suddenly decreased to its lower limit of –0.1 $/s to ensure that the power followed the setpoint. This sudden change in the controller output indicates a variation in the direction of the CD rotation during real-time operation. Subsequently, over the course of 330 s, the CD slowly rotated inward to maintain the power linearly at 95 W. During the initial power-up period, the temperature fluctuations were not pronounced; therefore, the net reactivity was almost equal to the external reactivity. After 400 s, the CDs were controlled to gradually insert an external reactivity to compensate for the negative feedback reactivity caused by the core temperature growth; thus, the power could remain at 95 W under a zero net reactivity. During this period, the rotation of the CD was quite slow; therefore, a dead zone should be applied to the controller to prevent frequent movement of the CDs and protect its actuator instrument under real-world applications. During the entire process, the core temperature increased smoothly to its steady state, and no oscillations appeared owing to the contribution of the controller.

Fig. 16Full-screen View

Results of Case 15, with a PID controller. The target steady-state power matches that obtained in Case 7. (a) Power and temperature, (b) reactivity, and (c) rate of external reactivity insertion.

Shut-down transient

In addition to the start-on transient, a series of shut-down analyses was conducted in this study because small nuclear reactors often need to accommodate flexible power output requirements. Fig. 17 shows the HIL simulation results for the open-loop power-down processes for Cases 12–14. The initial power was 400 W, and the preheating process lasted for 7500 s until the core temperature stabilized at approximately 450 K. During the first 1000 s after the negative step, external reactivity was inserted, causing a rapid decrease in the power, which dropped below 50 W. However, a significant power occurred in all these cases because the of the rapid temperature drop, leading to a positive net reactivity at 8000–9000 s. The power peak reached 450 W, which was even higher than the initial power, was attributed to the large initial neutron density. Oscillations also emerged during the power-down process when considering the temperature feedback. In different cases, the amplitudes of these oscillations were quite similar, but the oscillation duration significantly increased as the inserted negative external reactivity increased. After 11000 s, this oscillation gradually vanished, and the system reached a new steady state.

Fig. 17Full-screen View

Power and temperature overshoot in Cases 12–14: (a) temperature and (b) power.

The variation in the wall temperature distribution of heat pipe #c in Case 14 during the shutdown process is shown in Fig. 18. Similar to the start-up process, the axial temperature difference of the heat pipe varies with the heating power, and ΔT_a-c drops from 10 K to 6 K. The temperature distribution between the reactor and power conversion system appeared to be more uniform after the heating power, and the heat flux outside the evaporator decreased.

Fig. 18Full-screen View

Variation of the wall temperature distribution of the heat pipe #c in Case 14.

The final stable power of Case 13 was 205 W, which was used as the target set point for Case 16. In Case 16, the PID controller was adopted, and a closed-loop simulation was conducted. The evolution of these key parameters in Case 16 is shown in Fig. 19. Over a span of 7500 s, the power setpoint decreased at a rate of -6.5 W/s, and the controller output promptly adjusted to -0.01 $/s, resulting in an external reactivity of -0.2 $. Subsequently, the CD reversed and rotated outside, providing a positive external reactivity of +0.012 $/s. Thus, the reactivity were controlled to maintain the process power following the setpoint. As per the results shown in Fig. 19(a), the controller demonstrated excellent performance, and no temperature oscillation was observed, in contrast to the results of Case 13. This test revealed the effectiveness of the closed-loop operation in regulating the power-down process and emphasized the importance of precise control over the external reactivity by adjusting the angles of the drums. This outcome is consistent with that discussed in the startup simulation in Section 4.2.

Fig. 19Full-screen View

Results of Case 16 with a PID controller involved. The target steady-state power was set based on the result obtained in Case 13. (a) Power and temperature, (b) reactivity, and (c) rate of external reactivity insertion.