Numerical investigation on the startup performance of high-temperature heat pipes for heat pipe cooled reactor application

Numerical investigation on the startup performance of high-temperature heat pipes for heat pipe cooled reactor application 1 first-author https://orcid.org/0000-0001-7799-403X 1 1 corresp lizeguang@mail.tsinghua.edu.cn lizeguang@mail.tsinghua.edu.cn lizeguang@mail.tsinghua.edu.cn https://orcid.org/0000-0003-0069-5538 1 Department of Engineering Physics, Tsinghua University, Beijing 100084, China Department of Engineering Physics, Tsinghua University, Beijing 100084, China A suitable model for high-temperature heat pipe startup is a prerequisite to realizing the numerical simulation for the heat pipe cooled reactor startup from the cold state. It is required that this model not only describe the transient behavior during the startup period, but also reduce the computing resources of the heat pipe cooled reactor simulation in the simplest way. In this study, a simplified model that integrates the two-zone and network models is proposed. In this model, vapor flow in the vapor space, evaporation, and condensation in the vapor-liquid interface are decoupled with heat conduction to achieve a fast calculation of the transient characteristics of the heat pipe. An experimental system for a high-temperature heat pipe was developed to validate the proposed model. A potassium heat pipe was utilized as the experimental material. Startup experiments were performed with different heating powers. Compared with the experimental results, the accuracy of the proposed model was verified. Moreover, the proposed model can predict the vapor flow, pressure drop, and temperature drop in the vapor space. As indicated by the analysis results, the essential requirements for successful startup are also determined. The heat pipe cannot achieve a successful startup until the heating power satisfies these requirements. All the discussions indicate the capability of the proposed model for the simulation of a high-temperature heat pipe startup from the frozen state; hence, can act as a basic tool for the heat pipe cooled reactor simulation. High-temperature heat pipe Startup Two-zone model Network model 1.　Introduction 1 High-temperature heat pipes can be applied to nuclear reactor systems owing to their advantages of long-term stable operation at high temperatures, and the ability to achieve high-efficiency heat transfer capability. Typical heat pipe cooled reactor systems include the heat pipe-operated Mars exploration reactor (HOMER) [ 1 1 ] , heat pipe-segmented thermoelectric module converter (HP-STMC) space reactor [ 2 2 ] , and the scalable AMTEC-integrated reactor space power (SAIRS) systems [ 3 3 ] . For a heat pipe cooled reactor, heat transport through the heat pipes is the sole method of fission heat export from the core. To simulate the transient behavior of the reactor core, it is necessary to develop a suitable model to describe the characteristics of the high-temperature heat pipe. On one hand, limited by the heat transfer capability of a single heat pipe, there are a significant number of heat pipes in the reactor core. To achieve an accurate simulation of the heat pipe cooled reactor, it is necessary to model each heat pipe independently. On the other hand, for reactor analysis, it is not necessary to know the detailed flow state of the vapor, but to know the startup state, heat absorption, and safety state of the heat pipe. Therefore, it is hoped that the heat pipe model can not only predict the transient behavior of the heat pipe, but also be simple enough to reduce the difficulty of reactor modeling and the computing resources of reactor simulation. The operation of a high-temperature heat pipe can be divided into the startup and stable operation stages. During the startup stage, the heat pipe gradually increases from the frozen state to the high-temperature state, and the working medium in the heat pipe will change to the molten state. The working medium may still be in a frozen state, and the temperature of the heat pipe is too low to completely form a continuous vapor flow in the vapor space. During the stable operation stage, stable flow circulation is established to achieve efficient heat transport. In this study, a numerical investigation of the high-temperature heat pipe startup from the frozen state is performed to lay the foundation for the analysis of the heat pipe cooled reactor system startup from the cold state. Thus far, numerous studies have been conducted on the transient behavior of high-temperature heat pipe startups. The research method comprise the numerical, approximate analysis, and experimental analysis methods. A few researchers have presented numerical methods for heat pipe startups. Jang et al. [ 4 4 ] proposed a numerical method for startups. They ignored the heat transfer of the free molecular flow in the vapor space during the early startup period, and modeled the continuum vapor flow by employing the one-dimensional, transient, compressible, laminar flow N-S Stokes equation. Cao et al. [ 5 5 ] utilized the rarefied vapor self-diffusion model to simulate the early startup period of the high-temperature heat pipe. At the preliminary stage of the startup, the vapor flow was considered as the self-diffusion flow, owing to the large density gradient along the axial direction of the heat pipe. For the porous wick structure, a temperature-based fixed grid formulation [ 6 6 ] was proposed to consider the phase change of the frozen working medium. Moreover, it was assumed that only heat conductance existed in the wick. Under a sufficiently high vapor density, Cao et al. [ 7 7 , 8 8 ] utilized the compressible laminar flow N-S equation to express the vapor flow. The Knudsen number was adopted to determine the flow state of the vapor. All the governing equations were discretized using the control-volume finite-difference approach [ 9 9 ] , and the Gauss-Seidel method was employed to solve the discretization equations. Tournier et al. [ 10 10 ] presented a two-dimensional heat pipe transient analysis model that further considered the transition vapor flow during startup. The second approximation of the dusty gas model for the vapor flow was employed to express the transition flow. The numerical method adopts a range of conservation equations to express the physical phenomena in different regions. It is capable of simulating the transient performance of a heat pipe to a certain extent. However, this method consumes considerable computing resources as impacted by the fluid-solid coupling iterative calculation, and the accuracy of the calculation results is determined by the quality of mesh generation. Furthermore, numerous experimental studies have been conducted on heat pipe startup. Faghri et al. [ 11 11 , 12 12 ] conducted experiments on sodium/stainless steel heat pipes. They reported that the startup behavior of a sodium heat pipe was significantly determined by the heat rejection rate in the condenser. Supersonic vapor velocities could occur in the condenser section during startup. Guo et al. [ 13 13 ] experimentally investigated the effect of the inclination angle on the startup performance of the potassium heat pipes. They reported that the startup and heat transfer performances were improved, and the temperature difference decreased with an increase in the inclination angle. In addition, Wang et al. [ 14 14 ] experimentally studied the effect of the inclination angle and input power on the startup performance of high-temperature alkali heat pipes. They determined that the inclination angle and input power could play critical roles during startup under a constant heat flux, and the inclination was conducive to reducing the startup time. Experimental methods can directly obtain the transient response of the heat pipe; therefore, it is suitable for the performance test of the heat pipe. However, compared with numerical methods, experimental methods require high reliability of the equipment and considerable cost. For the approximate analytical method, Sockol and Forman [ 15 15 ] investigated the high-temperature heat pipe with lithium. They divided the heat pipe into the “hot zone” and the “cold zone”. In the hot zone, the wall temperature was assumed to be uniform, whereas it remained at the initial temperature in the cold zone. Moreover, at the interface between both zones, the vapor velocity was treated as sonic. Accordingly, two first-order differential energy equations were built for the evaporator and condenser regions. Combined with the one-dimensional flow model, the temperature variation with time was calculated. Cao and Faghri [ 16 16 ] presented a simple analytical closed-form solution based on a flat-front startup model. In addition, they divided the heat pipe into the hot zone and the cold zone. The temperature at the interfaces was treated as a continuum-rarefied transition temperature. Compared with the experimentally obtained results, the proposed model was proven to be capable of simulating the temperature variation for the heat pipe startup from the frozen state. Overall, the approximate analytical method can effectively simplify the calculation of the heat pipe startup, and can obtain the temperature change and distribution during the transient. However, it neglects the axial heat conduction in the wall and the wick region; hence, the temperature in the cold zone is always the initial temperature. The aforementioned results obviously deviate from reality. Furthermore, for the startup performance simulation of the heat pipe, it is insufficient to know only the change in the characteristic temperature over time. To overcome the shortcomings of existing methods, a new model combining the network and two-zone models is proposed. This model can realize the fast calculation of the heat pipe performance and can simulate the startup statement, as well as the movement of the melting interface. Heating experiments on the high-temperature potassium heat pipe were performed to verify the proposed model. After verifying the accuracy of this model, the characteristics of the heat transfer and flow of the heat pipe during the startup are discussed, as well as the conditions for a successful startup. 2.　Mathematical method 1 2.1.　Two-zone model 2 As indicated by the existing experimental and numerical results [ 7 7 , 8 8 , 17 17 ] , at sufficiently high-power inputs, the wall temperature of the high-temperature heat pipe increases to a characteristic temperature, and remains approximately constant at a steep temperature front ( Fig.1a Fig.1a ). Subsequently, the temperature front will move to the condenser with continuous heating. Phenomenally, the heat pipe is divided into hot and cold zones. Cao et al. [ 16 16 ] considered the transition temperature as a characteristic temperature. The flow regime changes from the free molecular flow/self-diffusion flow to the continuum flow when the vapor temperature reaches the transition temperature. As impacted by the flow characteristics of the two flow regimes, the heat pipe acted as a "two-zone" mode ( Fig.1b–Fig.1c Fig.1b–Fig.1c ). As illustrated in Fig. 1c Fig. 1c , although the two-zone model is easy to calculate, the calculated results deviate far from reality, which means that this model can be directly utilized for reactor simulations. (a) Wall temperature along the axial direction using plat-front model [ 7 7 ] (b) Description of the two-zone model (c) Calculated wall temperature by two-zone model [ 16 16 ] For the transition temperature, it is expressed as [ 7 7 ] : <math display="block" id="M1"><mrow><mi>ln</mi><mrow><mo>(</mo><mrow><mfrac><mrow><msub><mi>T</mi><mrow><mtext>tr</mtext></mrow></msub><msub><mi>ρ</mi><mi>v</mi></msub><mi>R</mi></mrow><mrow><msub><mi>p</mi><mrow><mtext>rf</mtext></mrow></msub></mrow></mfrac></mrow><mo>)</mo></mrow><mo>+</mo><mfrac><mrow><msub><mi>h</mi><mrow><mtext>fg</mtext></mrow></msub></mrow><mi>R</mi></mfrac><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mrow><msub><mi>T</mi><mrow><mtext>tr</mtext></mrow></msub></mrow></mfrac><mo>−</mo><mfrac><mn>1</mn><mrow><msub><mi>T</mi><mrow><mtext>rf</mtext></mrow></msub></mrow></mfrac></mrow><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mrow></math> In Eq. (1), the vapor density is expressed as: <math display="block" id="M2"><mrow><msub><mi>ρ</mi><mi>v</mi></msub><mo>=</mo><mfrac><mrow><mn>1.051</mn><mi>k</mi></mrow><mrow><msqrt><mn>2</mn></msqrt><mi>π</mi><msup><mi>σ</mi><mn>2</mn></msup><mi>R</mi><mi>d</mi><mi>K</mi><mi>n</mi></mrow></mfrac></mrow></math> 2.2.　Network model 2 When the cold zone disappears, a stable continuous flow is formed in the vapor space. Based on the operating characteristics of the heat pipe in this stage, Zuo et al. [ 18 18 ] proposed the network model ( Fig. 2 Fig. 2 ). It ignored the temperature drop caused by vapor flow, simplified the heat transport of the heat pipe to heat conduction, and developed the energy conservation equations in terms of six subregions. Because only six temperature variables are solved in this model, a fast calculation of the transient performance of the heat pipe can be achieved. However, this model can only calculate the temperature variation of the subregions, whereas it cannot simulate the fluid flow and pressure drop of the fluid. Moreover, it can only be applied to the operating conditions in the absence of the cold zone in heat pipe. 2.3.　Description of this model 2 Combined with the two-zone model and the network model, a new model for the heat pipe startup is presented. The node division of the proposed model is illustrated in Fig. 3a Fig. 3a . Along the axial direction, the heat pipe is divided into the evaporator, the adiabatic section and the condenser. In the radial direction, it includes the vapor space, the wick, the wall and the insulating layer. Specifically, the actual number of temperature nodes can be regulated flexibly based on the geometric dimensions of the heat pipe and the calculation requirements. (a) Node division of the proposed model (b) Main route of heat transport during the startup (c) Vapor flow in the vapor space In this model, the cold zone and the hot zone are existed in the heat pipe. In the cold zone, it is assumed that the vapor temperature is identical to that of the corresponding node temperature in the wick. For the hot zone, there is evaporation and condensation of the working medium, and the excess vapor generated by evaporation is condensed at the interface between the hot zone and the cold zone. There are also other assumptions to simplify the modeling difficulty which include: 1) Heat conduction in a solid region is two-dimensional; 2) Vapor flow in a vapor space is one-dimensional; 3) The interface temperature between two zones is its transition temperature; 4) Vapor flow in a hot zone is a continuous fluid flow; During startup, the main route of heat transport is presented in Fig. 3b Fig. 3b . First, the heat absorbed from the heating source is transferred to the wick by heat conductance. Subsequently, it is removed by the evaporation of the working medium. Then, the generated vapor flows to the interface between the two zones, and finally, it is condensed. Heat transfer leads to a temperature difference. The total temperature difference between the hot zone and zone interface is <math display="block" id="M3"><mrow><mi>Δ</mi><msub><mi>T</mi><mrow><mtext>total</mtext></mrow></msub><mo>=</mo><mi>Δ</mi><msub><mi>T</mi><mn>1</mn></msub><mo>+</mo><mi>Δ</mi><msub><mi>T</mi><mn>2</mn></msub><mo>+</mo><mi>Δ</mi><msub><mi>T</mi><mn>3</mn></msub><mo>,</mo></mrow></math> where Δ T 1 is the temperature difference attributed to heat conduction, Δ T 2 is the temperature difference attributed to evaporation, and Δ T 3 is the temperature difference attributed to heat convection. Because the interface temperature is known, the wall temperature in the hot zone can be obtained as follows: <math display="block" id="M4"><mrow><msub><mi>T</mi><mtext>w</mtext></msub><mo>=</mo><mi>Δ</mi><msub><mi>T</mi><mn>1</mn></msub><mo>+</mo><mi>Δ</mi><msub><mi>T</mi><mn>2</mn></msub><mo>+</mo><mi>Δ</mi><msub><mi>T</mi><mn>3</mn></msub><mo>+</mo><msub><mi>T</mi><mrow><mtext>tr</mtext></mrow></msub><mo>=</mo><mrow><mo>(</mo><mrow><mi>Δ</mi><msub><mi>T</mi><mn>1</mn></msub><mo>+</mo><msub><mi>T</mi><mrow><mtext>tr</mtext></mrow></msub></mrow><mo>)</mo></mrow><mo>+</mo><mi>Δ</mi><msub><mi>T</mi><mn>2</mn></msub><mo>+</mo><mi>Δ</mi><msub><mi>T</mi><mn>3</mn></msub><mo>.</mo></mrow></math> In Eq. (4), it is assumed that: <math display="block" id="M5"><mrow><msubsup><mi>T</mi><mtext>w</mtext><mo>'</mo></msubsup><mo>=</mo><mi>Δ</mi><msub><mi>T</mi><mn>1</mn></msub><mo>+</mo><msub><mi>T</mi><mrow><mtext>tr</mtext></mrow></msub><mo>.</mo></mrow></math> Therefore, it is rewritten as: <math display="block" id="M6"><mrow><msub><mi>T</mi><mtext>w</mtext></msub><mo>=</mo><msubsup><mi>T</mi><mtext>w</mtext><mo>'</mo></msubsup><mo>+</mo><mi>Δ</mi><msub><mi>T</mi><mn>2</mn></msub><mo>+</mo><mi>Δ</mi><msub><mi>T</mi><mn>3</mn></msub><mo>.</mo></mrow></math> For the heat transport of the heat pipe, the temperature difference is solely related to the heat flux, and it is independent of the actual temperature of the heat pipe. In this model, it is assumed that once the subregion enters the hot zone, the wick temperature remains at the transition temperature T tr ; therefore, <math id="M7"><mrow><msubsup><mi>T</mi><mtext>w</mtext><mtext>"</mtext></msubsup></mrow></math> can be obtained. Based on energy conservation, the vapor flow attributed to evaporation or condensation can be calculated. Δ T 2 and Δ T 3 can be obtained using simplified equations. Specifically, the vapor flow and evaporation/condensation are decoupled with the heat conduction in this model (for detailed discussions, kindly refer to Sect. 2.6 and Sect. 2.7). 2.4.　Modeling in wall region 2 There are radial and axial heat conductance in the wall region, and adjacent temperature nodes are connected by the thermal resistance: <math display="block" id="M8"><mrow><mi>ρ</mi><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo><mi>C</mi><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo><mfrac><mrow><mtext>d</mtext><mi>T</mi><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo></mrow><mrow><mtext>d</mtext><mi>t</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>T</mi><mo stretchy="false">(</mo><mi>i</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo><mo>−</mo><mi>T</mi><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo></mrow><mrow><mi>R</mi><mo stretchy="false">(</mo><mi>i</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>+</mo><mfrac><mrow><mi>T</mi><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>−</mo><mi>T</mi><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo></mrow><mrow><mi>R</mi><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mfrac><mo>−</mo><mfrac><mrow><mi>T</mi><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo><mo>−</mo><mi>T</mi><mo stretchy="false">(</mo><mi>i</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo></mrow><mrow><mi>R</mi><mo stretchy="false">(</mo><mi>i</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>−</mo><mfrac><mrow><mi>T</mi><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo><mo>−</mo><mi>T</mi><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mrow><mi>R</mi><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mfrac><mo>+</mo><mi>Q</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></math> In Eq. (7), the thermal resistance is defined as: Radial thermal resistance: <math display="block" id="M9"><mrow><msub><mi>R</mi><mrow><mtext>radial</mtext></mrow></msub><mo>=</mo><mfrac><mrow><mi>ln</mi><mrow><mo>(</mo><mrow><mrow><mrow><msub><mi>r</mi><mrow><mtext>out</mtext></mrow></msub></mrow><mo>/</mo><mrow><msub><mi>r</mi><mrow><mtext>in</mtext></mrow></msub></mrow></mrow></mrow><mo>)</mo></mrow></mrow><mrow><mn>2</mn><mi>π</mi><mi>k</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo><mi>L</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>.</mo></mrow></math> Axial thermal resistance: <math display="block" id="M10"><mrow><msub><mi>R</mi><mrow><mtext>axial</mtext></mrow></msub><mo>=</mo><mfrac><mrow><mi>L</mi><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo></mrow><mrow><mi>π</mi><mrow><mo>(</mo><mrow><msubsup><mi>r</mi><mrow><mtext>out</mtext></mrow><mtext>2</mtext></msubsup><mo>−</mo><msubsup><mi>r</mi><mrow><mtext>in</mtext></mrow><mtext>2</mtext></msubsup></mrow><mo>)</mo></mrow><mi>k</mi><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>.</mo></mrow></math> In the wall, Q ( i ) denotes the heat transfer from outside to the wall, and has different expressions under different boundary conditions. Dirichlet boundary condition: <math display="block" id="M11"><mrow><mi>Q</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi>T</mi><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo><mo>−</mo><msub><mi>T</mi><mrow><mtext>out</mtext></mrow></msub></mrow><mrow><msub><mi>R</mi><mrow><mtext>conduct</mtext></mrow></msub></mrow></mfrac><mo>.</mo></mrow></math> Neumann boundary condition: <math display="block" id="M12"><mrow><mi>Q</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>Q</mi><mrow><mtext>heating</mtext></mrow></msub><mo>.</mo></mrow></math> Robin boundary condition: <math display="block" id="M13"><mrow><mi>Q</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi>T</mi><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo><mo>−</mo><msub><mi>T</mi><mtext>f</mtext></msub></mrow><mrow><msub><mi>R</mi><mrow><mtext>conduct</mtext></mrow></msub><mo>+</mo><msub><mi>R</mi><mrow><mtext>f</mtext><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></msub></mrow></mfrac><mo>.</mo></mrow></math> R f expresses the convective thermal resistance: <math display="block" id="M14"><mrow><msub><mi>R</mi><mrow><mtext>f</mtext><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></msub><mo>=</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>π</mi><msub><mi>r</mi><mrow><mtext>out</mtext></mrow></msub><mi>L</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo><mi>λ</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>.</mo></mrow></math> 2.5.　Modeling in the wick region 2 2.5.1.　energy equation 3 This model ignores the local flow in the wick region during startup. Therefore, Eq. (7) was utilized to express the heat transfer in the wick region. Moreover, Q ( ij ) represents the heat transfer between the wick and vapor space. The wick, comprises the working medium, and the porous structure, is treated as an equivalent material. The physical properties (e.g., density, specific heat capacity, and thermal conductivity coefficient) can be calculated as follows [ 19 19 ] : <math display="block" id="M15"><mrow><msub><mi>ρ</mi><mrow><mtext>equ</mtext></mrow></msub><mo>=</mo><mi>ε</mi><msub><mi>ρ</mi><mtext>f</mtext></msub><mo>+</mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><mi>ε</mi></mrow><mo>)</mo></mrow><msub><mi>ρ</mi><mrow><mtext>wc</mtext></mrow></msub><mo>,</mo></mrow></math> <math display="block" id="M16"><mrow><msub><mi>C</mi><mrow><mtext>equ</mtext></mrow></msub><mo>=</mo><mi>ε</mi><msub><mi>C</mi><mtext>f</mtext></msub><mo>+</mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><mi>ε</mi></mrow><mo>)</mo></mrow><msub><mi>C</mi><mrow><mtext>wc</mtext></mrow></msub><mo>,</mo></mrow></math> <math display="block" id="M17"><mrow><msub><mi>k</mi><mrow><mtext>equ</mtext></mrow></msub><mo>=</mo><msub><mi>k</mi><mtext>l</mtext></msub><mfrac><mrow><mrow><mo>(</mo><mrow><msub><mi>k</mi><mtext>l</mtext></msub><mo>+</mo><msub><mi>k</mi><mrow><mtext>wc</mtext></mrow></msub></mrow><mo>)</mo></mrow><mo>−</mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><mi>ε</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><msub><mi>k</mi><mtext>l</mtext></msub><mo>−</mo><msub><mi>k</mi><mrow><mtext>wc</mtext></mrow></msub></mrow><mo>)</mo></mrow></mrow><mrow><mrow><mo>(</mo><mrow><msub><mi>k</mi><mtext>l</mtext></msub><mo>+</mo><msub><mi>k</mi><mrow><mtext>wc</mtext></mrow></msub></mrow><mo>)</mo></mrow><mo>+</mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><mi>ε</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><msub><mi>k</mi><mtext>l</mtext></msub><mo>−</mo><msub><mi>k</mi><mrow><mtext>wc</mtext></mrow></msub></mrow><mo>)</mo></mrow></mrow></mfrac><mo>.</mo></mrow></math> 2.5.2.　Melting equation 3 During heating, the working medium changes from a solid state to a liquid state. Inconsistent with the temperature-based fixed grid formulation adopted by Cao et al. [ 7 7 ] , the following method was utilized to determine the working medium state. If the calculated temperature is less than the melting temperature, the working medium is in the solid state. If the temperature is higher than the melting temperature, while the specific enthalpy is lower than the minimum value of the liquid working medium, it is identified as the mixed state. According to the results, the specific enthalpy of the working medium is corrected (Eq. (18)), the temperature is forced to the melting temperature (Eq. (19)). If the two aforementioned conditions are not satisfied, the working medium is completely melted. When the working medium is in a mixed state, the energy conservation equation with a time step of Δ t is: <math display="block" id="M18"><mrow><msub><mi>V</mi><mi>i</mi></msub><msub><mi>C</mi><mrow><mtext>equ</mtext></mrow></msub><msub><mi>ρ</mi><mrow><mtext>equ</mtext></mrow></msub><mrow><mo>[</mo><mrow><msub><mi>T</mi><mrow><mi>t</mi><mi>n</mi><mo>+</mo><mi>Δ</mi><mi>t</mi></mrow></msub><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo><mo>−</mo><msub><mi>T</mi><mrow><mi>t</mi><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo></mrow><mo>]</mo></mrow><mo>=</mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><mi>ε</mi></mrow><mo>)</mo></mrow><msub><mi>V</mi><mi>i</mi></msub><msub><mi>ρ</mi><mi>m</mi></msub><mrow><mo>[</mo><mrow><msub><mi>h</mi><mi>m</mi></msub><mo stretchy="false">(</mo><mi>T</mi><mi>m</mi><mi>p</mi><mo stretchy="false">)</mo><mo>−</mo><msub><mi>h</mi><mi>m</mi></msub><mo stretchy="false">(</mo><msub><mi>T</mi><mrow><mi>t</mi><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mo>]</mo></mrow><mo>+</mo><mi>ε</mi><msub><mi>V</mi><mi>i</mi></msub><msub><mi>ρ</mi><mi>l</mi></msub><mrow><mo>[</mo><mrow><msub><mi>h</mi><mrow><mi>t</mi><mi>n</mi><mo>+</mo><mi>Δ</mi><mi>t</mi></mrow></msub><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo><mo>−</mo><msub><mi>h</mi><mrow><mi>t</mi><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo></mrow><mo>]</mo></mrow></mrow></math> So: <math display="block" id="M19"><mrow><msub><mi>h</mi><mrow><mi>t</mi><mi>n</mi><mo>+</mo><mi>Δ</mi><mi>t</mi></mrow></msub><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>h</mi><mrow><mi>t</mi><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo><mo>+</mo><mrow><mrow><mrow><mo>{</mo><mrow><msub><mi>C</mi><mrow><mtext>equ</mtext></mrow></msub><msub><mi>ρ</mi><mrow><mtext>equ</mtext></mrow></msub><mrow><mo>[</mo><mrow><msub><mi>T</mi><mrow><mi>t</mi><mi>n</mi><mo>+</mo><mi>Δ</mi><mi>t</mi></mrow></msub><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo><mo>−</mo><msub><mi>T</mi><mrow><mi>t</mi><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo></mrow><mo>]</mo></mrow><mo>−</mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><mi>ε</mi></mrow><mo>)</mo></mrow><msub><mi>ρ</mi><mi>m</mi></msub><mrow><mo>[</mo><mrow><msub><mi>h</mi><mi>m</mi></msub><mo stretchy="false">(</mo><mi>T</mi><mi>m</mi><mi>p</mi><mo stretchy="false">)</mo><mo>−</mo><msub><mi>h</mi><mi>m</mi></msub><mo stretchy="false">(</mo><msub><mi>T</mi><mrow><mi>t</mi><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mo>]</mo></mrow></mrow><mo>}</mo></mrow></mrow><mo>/</mo><mrow><mi>ε</mi><msub><mi>ρ</mi><mi>l</mi></msub></mrow></mrow></mrow></math> Regarding the working medium temperature: <math display="block" id="M20"><mrow><msub><mi>T</mi><mrow><mi>t</mi><mi>n</mi><mo>+</mo><mi>Δ</mi><mi>t</mi></mrow></msub><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo><mo>=</mo><mi>T</mi><mi>m</mi><mi>p</mi></mrow></math> 2.5.3.　Boundary condition 3 As discussed in Sect. 2.3, this model decouples the vapor flow, evaporation, condensation, and heat conduction. Regarding heat transfer calculation for the wick, the key is to judge whether this subregion enters the hot zone, and update the boundary condition based on the calculated results. It is hypothesized that if the subregion is in the cold zone, no vapor will be generated. When it enters the hot zone, the temperature of the wick always remains at the transition temperature. The excess heat generated by heat conduction in time step Δ t is evaporated. <math display="block" id="M21"><mrow><mi>Q</mi><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mtext> in cold zone</mtext></mrow></math> <math display="block" id="M22"><mrow><mi>T</mi><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>T</mi><mrow><mtext>tr</mtext></mrow></msub><mtext> in hot zone</mtext></mrow></math> Accordingly, the wall temperature <math id="M23"><mrow><msubsup><mi>T</mi><mi>w</mi><mtext>"</mtext></msubsup><mtext> </mtext></mrow></math> in the hot zone can be preliminarily obtained. 2.6　Δ T 2 calculation 2 In Sect. 2.5.3, it is assumed that the wick temperature will be constant when the subregion enters the hot zone. Thus, Q ( i ) of the wick region can be defined as: <math display="block" id="M24"><mrow><mi>Q</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi>T</mi><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo><mo>−</mo><mi>T</mi><mo stretchy="false">(</mo><mi>i</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo></mrow><mrow><mi>R</mi><mo stretchy="false">(</mo><mi>i</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>+</mo><mfrac><mrow><mi>T</mi><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo><mo>−</mo><mi>T</mi><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mrow><mi>R</mi><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mfrac><mo>−</mo><mfrac><mrow><mi>T</mi><mo stretchy="false">(</mo><mi>i</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo><mo>−</mo><mi>T</mi><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo></mrow><mrow><mi>R</mi><mo stretchy="false">(</mo><mi>i</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>−</mo><mfrac><mrow><mi>T</mi><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>−</mo><mi>T</mi><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo></mrow><mrow><mi>R</mi><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mfrac><mo>.</mo></mrow></math> The heat transfer between the wick and vapor space is in the form of latent heat of vaporization of the working medium. The mass flow rate of the evaporation/condensation is <math display="block" id="M25"><mrow><msub><mi>q</mi><mrow><mtext>iv</mtext></mrow></msub><mo>=</mo><mrow><mrow><mi>Q</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow><mo>/</mo><mrow><msub><mi>h</mi><mrow><mtext>fg</mtext></mrow></msub></mrow></mrow><mo>.</mo></mrow></math> The thermal resistance is adopted to connect two temperature nodes adjacent to the wick and the vapor space [ 20 20 ] : <math display="block" id="M26"><mrow><mi>Δ</mi><msub><mi>T</mi><mn>2</mn></msub><mo>=</mo><mrow><mrow><mi>Q</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow><mo>/</mo><mrow><mi>R</mi><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mrow></mrow></math> 2.7　Δ T 3 calculation 2 Vapor flow in the vapor space is considered to be a one-dimensional, steady-state, viscous laminar flow. Based on the evaporation or condensation at different nodes, the vapor flow in the vapor space can be calculated as: <math display="block" id="M27"><mrow><mi>q</mi><msub><mrow><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow><mrow><mtext>mv</mtext></mrow></msub><mo>=</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>i</mi></munderover><mrow><mi>q</mi><msub><mrow><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo></mrow><mtext>v</mtext></msub></mrow></mstyle><mo>.</mo></mrow></math> The pressure drop for the vapor flow includes the frictional pressure drop and accelerated pressure drop: <math display="block" id="M28"><mrow><mi>Δ</mi><mi>P</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mfrac><mi>L</mi><mi>D</mi></mfrac><mi>f</mi><mi>ρ</mi><msup><mi>v</mi><mn>2</mn></msup><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mrow><mo>(</mo><mrow><mi>ρ</mi><msubsup><mi>v</mi><mn>2</mn><mn>2</mn></msubsup><mo>−</mo><mi>ρ</mi><msubsup><mi>v</mi><mn>1</mn><mn>2</mn></msubsup></mrow><mo>)</mo></mrow></mrow></math> For isothermal laminar flow in a circular tube: <math display="block" id="M29"><mrow><mi>f</mi><mo>=</mo><mfrac><mrow><mn>64</mn></mrow><mrow><mi>Re</mi></mrow></mfrac></mrow></math> Owing to the evaporation and condensation, Eq. (26) can be written as follows ( Fig. 3c Fig. 3c ). <math display="block" id="M30"><mrow><mi>Δ</mi><mi>P</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mn>8</mn><mi>μ</mi><mi>L</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow><mrow><mi>π</mi><msup><mi>R</mi><mn>4</mn></msup><mi>ρ</mi><msub><mrow><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow><mtext>v</mtext></msub></mrow></mfrac><mrow><mo>(</mo><mrow><mi>q</mi><msub><mrow><mo stretchy="false">(</mo><mi>i</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mrow><mtext>mv</mtext></mrow></msub><mo>+</mo><mn>0.5</mn><mi>ρ</mi><msub><mrow><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow><mtext>v</mtext></msub></mrow><mo>)</mo></mrow><mo>+</mo><mfrac><mn>1</mn><mrow><mn>2</mn><msup><mi>π</mi><mn>2</mn></msup><msup><mi>R</mi><mn>4</mn></msup><mi>ρ</mi><msub><mrow><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow><mtext>v</mtext></msub></mrow></mfrac><mrow><mo>(</mo><mrow><mi>q</mi><msubsup><mrow><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow><mrow><mtext>mv</mtext></mrow><mtext>2</mtext></msubsup><mo>−</mo><mi>q</mi><msubsup><mrow><mo stretchy="false">(</mo><mi>i</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mrow><mtext>mv</mtext></mrow><mtext>2</mtext></msubsup></mrow><mo>)</mo></mrow></mrow></math> It is assumed that the vapor exhibits an equilibrium condition, and the Clausius-Clapeyron equation is applied: <math display="block" id="M31"><mrow><mfrac><mrow><mtext>d</mtext><mi>P</mi></mrow><mrow><mtext>d</mtext><mi>T</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><msub><mi>ρ</mi><mtext>v</mtext></msub><msub><mi>h</mi><mrow><mtext>fg</mtext></mrow></msub></mrow><mi>T</mi></mfrac><mo>.</mo></mrow></math> Changing it into a different form: <math display="block" id="M32"><mrow><mi>Δ</mi><mi>T</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mrow><mo>(</mo><mrow><mfrac><mi>T</mi><mrow><msub><mi>ρ</mi><mtext>v</mtext></msub><msub><mi>h</mi><mrow><mtext>fg</mtext></mrow></msub></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mrow><mtext>av</mtext></mrow></msub><mi>Δ</mi><mi>P</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo><mo>.</mo></mrow></math> Δ T 3 can be determined: <math display="block" id="M33"><mrow><mi>Δ</mi><msub><mi>T</mi><mn>3</mn></msub><mo>=</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>t</mi><mi>r</mi></mrow></munderover><mrow><mi>Δ</mi><mi>T</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></mstyle><mo>.</mo></mrow></math> Combining Eq. (5), Eq. (24) and Eq. (31), the wall temperature in the hot zone is: <math display="block" id="M34"><mrow><msub><mi>T</mi><mrow><mtext>wall</mtext></mrow></msub><mo>=</mo><mrow><mo>(</mo><mrow><mi>Δ</mi><msub><mi>T</mi><mn>1</mn></msub><mo>+</mo><msub><mi>T</mi><mrow><mtext>tr</mtext></mrow></msub></mrow><mo>)</mo></mrow><mo>+</mo><mrow><mrow><mi>Q</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow><mo>/</mo><mi>R</mi></mrow><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo>+</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mtext>tr</mtext></mrow></munderover><mrow><mi>Δ</mi><mi>T</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></mstyle><mo>.</mo></mrow></math> 3　Experimental system for heat pipe 1 3.1.　System description 2 To verify the proposed model, a heating system for a high-temperature heat pipe was developed ( Fig. 4a Fig. 4a ). It comprises an iron-chrome wire heater, holding derive, detachable cooling jacket, angle controlling derive, temperature data collection, and other assistive devices. The evaporator and adiabatic section were wrapped by the insulating layer with aluminum silicate wool. The iron-chrome wire heater provided a maximum heating power of 3000 W. By regulating the angle-controlling device, an inclination range of <math id="M35"><mrow><mo> </mo><mo>−</mo><msup><mrow><mn>90</mn></mrow><mo>°</mo></msup><mo>–</mo><mo>+</mo><msup><mrow><mn>90</mn></mrow><mo>°</mo></msup></mrow></math> can be set. (a) Schematic diagram of the heat pipe heating system (b) Heat pipe division and thermocouples fixation A potassium heat pipe was utilized as the experimental material. A basic description of this heat pipe is presented in table 1 table 1 . It is uniformly divided into evaporator, adiabatic section, and condenser with dimensions of 200 mm, respectively. Eight K-type armored thermocouples were fixed on the wall surface ( Fig. 4b Fig. 4b ). Each thermocouple was first pasted on the surface with a high-temperature adhesive, and fixed with a high-temperature adhesive tape. Finally, a clamp was utilized to ensure complete fixation of the thermocouple. In particular, for the thermocouples in the evaporator to reduce the effect of heating on the temperature measurement, when winding the iron-chrome wire, all the measuring points are bypassed. 1 1 Description 1 1 Value 1 1 Length of evaporator <math id="M38"><mrow><mtext> </mtext><mrow><mo>(</mo><mrow><mtext>mm</mtext></mrow><mo>)</mo></mrow></mrow></math>   1 1 200.0 1 1 Length of adiabatic section <math id="M39"><mrow><mrow><mo>(</mo><mrow><mtext>mm</mtext></mrow><mo>)</mo></mrow></mrow></math>   1 1 200.0 1 1 Length of condenser <math id="M40"><mrow><mtext> </mtext><mrow><mo>(</mo><mrow><mtext>mm</mtext></mrow><mo>)</mo></mrow></mrow></math>   1 1 200.0 1 1 Outer diameter of heat pipe <math id="M41"><mrow><mrow><mo>(</mo><mrow><mtext>mm</mtext></mrow><mo>)</mo></mrow></mrow></math>   1 1 17.0 1 1 Inner diameter of vapor space <math id="M42"><mrow><mrow><mo>(</mo><mrow><mtext>mm</mtext></mrow><mo>)</mo></mrow></mrow></math>   1 1 15.5 1 1 Wall thickness <math id="M43"><mrow><mrow><mo>(</mo><mrow><mtext>mm</mtext></mrow><mo>)</mo></mrow></mrow></math>   1 1 0.4 1 1 Wick thickness <math id="M44"><mrow><mrow><mo>(</mo><mrow><mtext>mm</mtext></mrow><mo>)</mo></mrow></mrow></math>   1 1 0.35 1 1 Mass of potassium (g) 1 1 18 1 1 Mesh number 1 1 300 1 1 Wick structure 1 1 Screen wick 1 1 Wall material 1 1 316 L stainless steel 1 1 Wick material 1 1 316 L stainless steel The purpose of the experiments was to verify the proposed model; hence, the experimental system was simplified. The heat pipe was kept horizontal and cooled by the natural convection of air. The natural convection coefficient is calculated as follows: <math display="block" id="M36"><mrow><mi>N</mi><mi>u</mi><mo>=</mo><mi>C</mi><msup><mrow><mrow><mo>(</mo><mrow><mi>G</mi><mi>r</mi><mo>·</mo><mi>Pr</mi></mrow><mo>)</mo></mrow></mrow><mi>n</mi></msup><mrow><mo>{</mo><mtable columnalign="left"><mtr><mtd><mi>C</mi><mo>=</mo><mn>0.85</mn><mo>,</mo><mi>n</mi><mo>=</mo><msup><mn>0.188....10</mn><mn>2</mn></msup><mo>≤</mo><mi>G</mi><mi>r</mi><mo>·</mo><mi>Pr</mi><mo><</mo><msup><mn>10</mn><mn>4</mn></msup></mtd></mtr><mtr><mtd><mi>C</mi><mo>=</mo><mn>0.48</mn><mo>,</mo><mi>n</mi><mo>=</mo><mrow><mn>1</mn><mo>/</mo><mn>4</mn></mrow><msup><mn>.......10</mn><mn>4</mn></msup><mo>≤</mo><mi>G</mi><mi>r</mi><mo>·</mo><mi>Pr</mi><mo><</mo><msup><mn>10</mn><mn>7</mn></msup></mtd></mtr><mtr><mtd><mi>C</mi><mo>=</mo><mn>0.48</mn><mo>,</mo><mi>n</mi><mo>=</mo><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow><msup><mn>.......10</mn><mn>7</mn></msup><mo>≤</mo><mi>G</mi><mi>r</mi><mo>·</mo><mi>Pr</mi><mo><</mo><msup><mn>10</mn><mrow><mn>12</mn></mrow></msup></mtd></mtr></mtable></mrow></mrow></math> <math display="block" id="M37"><mrow><mi>N</mi><mi>u</mi><mo>=</mo><mfrac><mrow><mi>λ</mi><mi>d</mi></mrow><mi>k</mi></mfrac><mtext> </mtext><mi>G</mi><mi>r</mi><mo>=</mo><mfrac><mrow><mi>g</mi><mi>α</mi><mrow><mo>(</mo><mrow><msub><mi>T</mi><mtext>w</mtext></msub><mo>−</mo><msub><mi>T</mi><mi>∞</mi></msub></mrow><mo>)</mo></mrow><msup><mi>d</mi><mn>3</mn></msup></mrow><mrow><msup><mi>ν</mi><mn>2</mn></msup></mrow></mfrac><mtext> </mtext><mi>Pr</mi><mo>=</mo><mfrac><mi>ν</mi><mi>a</mi></mfrac></mrow></math> 3.2.　Uncertainty analysis 2 Errors are inevitable during the experimental investigation. However, as long as the value is negligible, and its existence does not affect the conclusions, it is considered that such an error is acceptable. Based on the type-B evaluation of uncertainty, assuming errors with uniform distribution, the uncertainty of the data can be obtained by Eq. (35): <math display="block" id="M45"><mrow><msub><mi>U</mi><mtext>B</mtext></msub><mo>=</mo><mrow><mi>Δ</mi><mo>/</mo><mi>C</mi></mrow><mo>,</mo></mrow></math> where Δ is the error of the instrument itself, and C is the confidence coefficient of confidence probability, P=0.863. Regarding the assumption of an error with a uniform distribution, <math id="M46"><mrow><mi>C</mi><mo>=</mo><msqrt><mn>3</mn></msqrt></mrow></math> . During the experiment, a K-type thermocouple is utilized to measure the temperature, and its error is 1.5 K. the uncertainty of the temperature measurement is 0.866 K (Eq. (36)). <math display="block" id="M47"><mrow><msub><mi>U</mi><mrow><mi>B</mi><mo>,</mo><mi>T</mi></mrow></msub><mo>=</mo><mfrac><mrow><msub><mi>Δ</mi><mrow><mtext>instrument</mtext></mrow></msub></mrow><mrow><msqrt><mn>3</mn></msqrt></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>1.5</mn></mrow><mrow><msqrt><mn>3</mn></msqrt></mrow></mfrac><mo>=</mo><mn>0.866</mn><mi>K</mi></mrow></math> The temperature measured in the experiments ranges from 300 K to 700 K, such that an uncertainty of 0.866k is negligible, and the experimental results can be utilized for model validation. 4　Results and discussion 1 Three cases with different heating powers of 150 W, 200 W, and 250 W were set to perform the high-temperature heat pipe startup experiments. The natural convective coefficient between the heat pipe and environment was calculated using Eq. (33) and Eq. (34). In Section 4.1, the simulated results are compared with the experimentally obtained results to verify this model. In Sect. 4.2, more calculated results are discussed to demonstrate the capability of this model to simulate the startup process of a high-temperature heat pipe. 4.1.　Comparison and validation 2 Figure 5a–c Figure 5a–c illustrate the change in wall temperature distribution along the axial direction under different heating powers. In these figures, the curve represents the calculated results of the proposed model, and the scatter represents the wall temperature measured with the thermocouples at different positions. It is noteworthy that the curve in the figures is plotted by interpolating the calculated values at different temperature nodes. The more the temperature nodes set, the more realistic the temperature axial distribution will be. (a) Analytical wall temperature compared with experimental results (150 W) (b) Analytical wall temperature compared with experimental results (200 W) (c) Analytical wall temperature compared with experimental results (250 W) From Fig.5a–c Fig.5a–c , the calculated temperature distribution using the proposed model is concluded to be consistent with the experimental results. This model can simulate the startup characteristics of a high-temperature heat pipe properly. Because it combines the network and two-zone models , it can realize the fast simulation of the heat pipe startup, as well as consider the axial heat transport, owing to the heat conductance. As indicated from these figures, the higher the heating power, the faster the startup of the heat pipe, and the higher the wall temperature of the evaporator. This is because a larger temperature gradient is required to achieve higher heat transfer. Moreover, the larger the heating power, the larger the vapor flow in the vapor space, which causes a significant pressure drop during the flow, thereby resulting in a more obvious temperature difference in the vapor flow. However, it can also be observed that the calculated results are slightly different from the experimental results. On one hand, the calculated temperature in the evaporator deviates from the experimental results when the heating power is sufficient. This is mainly because the vapor flow is assumed to be a one-dimensional, incompressible, and stable laminar flow. Bowman and Hitchcock [ 21 21 ] and Issacci et al. [ 22 22 ] determined that a one-dimensional assumption could be inadequate because it could not accurately predict the axial mass transfer, axial heat transfer, and axial pressure drop. This model is a simplified model that can provide a fast calculation of heat pipe transient characteristics during startup. It preliminarily considers the vapor flow and pressure drop, and the difference between the two results is acceptable. It can be considered that the model is valid. On the other hand, the calculated temperature in the cold zone was slightly lower than that of the experimental measure. This is because the proposed model considers that there is no vapor flow in the cold zone. In fact, the rarefied vapor remains to transport heat. In comparison, although the computational accuracy of the proposed model is not as good as that of the plant-front model ( Fig. 1a Fig. 1a ), it is better than the two-zone model ( Fig. 1c Fig. 1c ). In contrast to the plant-front model, which utilizes the continuous flow and self-diffusion equations to calculate the vapor flow, the proposed model adopts the pressure drop and Clausius-Clapeyron equations to perform a preliminary analysis of vapor flow. It should be noted that the purpose of this model is to provide a method for the numerical analysis of heat pipe cooled reactor startups. In this stage, compared with the temperature prediction in the cold zone and in the vapor space, it is more important to simulate the startup state of the heat pipe. The difference between the model and the experiments is acceptable. 4.2.　Further discussion 2 Compared with the two-zone model, the proposed model can achieve a more realistic simulation of the working medium melting interface movement and startup state of the heat pipe. In the two-zone model, it is assumed that the cold zone always maintains the initial temperature, that is, the melting interface is completely coincident with the interface between the two zones. In fact, owing to the axial heat conduction and heat transport of the rarefied vapor, the melting interface always moves faster than that of the hot zone ( Fig. 3b Fig. 3b ). Table 2 Table 2 and Table 3 Table 3 present the interface movement over time using the node division presented in Fig. 3a Fig. 3a . When t = 200 s, the working medium in the evaporator melted completely, and part of the working medium in the adiabatic section also melted. However, the temperature of the wick remained lower than the transition temperature, and it did not enter the hot zone. With the continuous absorption of heat, the melting interface and hot zone moved to the condensation section. * Melting fraction in different nodes. 1 1 Time (s) 1 1 Node 1 1 1 1 1 2 1 1 3 1 1 4 1 1 5 1 1 6 1 1 7 1 1 8 1 1 9 1 1 10 1 1 11 1 1 12 1 1 200 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0.994 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 400 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 600 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 0 1 1 0 1 1 800 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1000 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 * “1” means in the hot zone, “0” means in the cold zone 1 1 Time (s) 1 1 Node 1 1 1 1 1 2 1 1 3 1 1 4 1 1 5 1 1 6 1 1 7 1 1 8 1 1 9 1 1 10 1 1 11 1 1 12 1 1 200 1 1 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 400 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 600 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 0 1 1 0 1 1 0 1 1 800 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 0 1 1 1000 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 Fig.6 Fig.6 illustrates the vapor flow distribution in the vapor space at different times. Because this model considers that there is no vapor flow in the cold zone, it can only flow to the interface between the two zones, and vapor is forced to be condensed. The evaporator is always heated, and it first enters the hot zone. Along the axial direction, vapor continuously accumulated, indicating an increase in the flow rate. The adiabatic section was wrapped with aluminum silicate wool to reduce the heat leakage. However, because the heat pipe temperature is always higher than the environmental temperature, the vapor will be partially condensed when flowing to the adiabatic section. It can be predicted that if the insulation effect of the heat pipe is not good, a considerable part of the vapor will be condensed, which might cause startup failure of the high-temperature heat pipe. When the vapor flows to the interface, it is considered to be completely condensed, owing to the significant temperature difference between the vapor and the wick. The vaporization latent heat released by vapor condensation is absorbed by the wick, and the temperature gradually rises and finally enters the hot zone. Moreover, it can also be determined that the closer the region to the end of the condenser, the more difficult it will be for successful startups. This is because the vapor will be partially condensed as it flows, resulting in the reduction of heat that can be utilized for the startup. For instance, when t = 400 s, the steam flow to the interface is 0.039 g/s. When t = 1000 s, the flow rate that can be condensed at the interface decreases to 0.0248 g/s. Figure 7 Figure 7 illustrates the time required for the successful startup of this heat pipe under different heating powers. As indicated in this figure, the lower the heating power, the longer the startup time. However, when the heating power is less than 83 W, the heat pipe cannot successfully startup even if it is heated for a long time. For this heat pipe, the value of 83 W is the lower limit of the heating power. For the heat pipe startup, the relationship between the heat absorption and release should be satisfied: <math display="block" id="M48"><mrow><msub><mi>Q</mi><mrow><mtext>heating</mtext></mrow></msub><mo>≥</mo><msub><mi>Q</mi><mrow><mtext>leakage</mtext></mrow></msub><mo>+</mo><msub><mi>Q</mi><mrow><mtext>convect</mtext></mrow></msub><mo>.</mo></mrow></math> Q leakage denotes the heat leakage through the insulating layer, and Q convect denotes the heat transport through the condenser. The condition of a successful startup is that the temperature of each subregion reaches the transition temperature. Therefore, the following can be obtained: <math display="block" id="M49"><mrow><msub><mi>Q</mi><mrow><mtext>leakage</mtext></mrow></msub><mo>≥</mo><mfrac><mrow><msub><mi>T</mi><mrow><mtext>tr</mtext></mrow></msub><mo>−</mo><msub><mi>T</mi><mi>∞</mi></msub></mrow><mrow><msub><mi>R</mi><mrow><mtext>insulate</mtext></mrow></msub><mo>+</mo><msub><mi>R</mi><mrow><mtext>f</mtext><mn>1</mn></mrow></msub></mrow></mfrac><mo>,</mo><mtext> </mtext><msub><mi>Q</mi><mrow><mtext>convect</mtext></mrow></msub><mo>≥</mo><mfrac><mrow><msub><mi>T</mi><mrow><mtext>tr</mtext></mrow></msub><mo>−</mo><msub><mi>T</mi><mi>∞</mi></msub></mrow><mrow><msub><mi>R</mi><mrow><mtext>condensor</mtext></mrow></msub><mo>+</mo><msub><mi>R</mi><mrow><mtext>f</mtext><mn>2</mn></mrow></msub></mrow></mfrac><mo>.</mo></mrow></math> By combining Eq. (36) and Eq. (37), the lower limit of the heating power can be determined as <math display="block" id="M50"><mrow><msub><mi>Q</mi><mrow><mtext>heating</mtext></mrow></msub><mo>≥</mo><msub><mi>Q</mi><mrow><mtext>lower</mtext></mrow></msub><mo>=</mo><mfrac><mrow><msub><mi>T</mi><mrow><mtext>tr</mtext></mrow></msub><mo>−</mo><msub><mi>T</mi><mi>∞</mi></msub></mrow><mrow><msub><mi>R</mi><mrow><mtext>insulate</mtext></mrow></msub><mo>+</mo><msub><mi>R</mi><mrow><mi>f</mi><mn>1</mn></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><msub><mi>T</mi><mrow><mtext>tr</mtext></mrow></msub><mo>−</mo><msub><mi>T</mi><mi>∞</mi></msub></mrow><mrow><msub><mi>R</mi><mrow><mtext>condensor</mtext></mrow></msub><mo>+</mo><msub><mi>R</mi><mrow><mi>f</mi><mn>2</mn></mrow></msub></mrow></mfrac><mo>.</mo></mrow></math> Meanwhile, there is also an upper limit for the heat absorbed by the high-temperature heat pipe. When the heat absorption exceeds this limit, the heat pipe may dry locally and become damaged. During the startup period, the possible heat transfer limitations include the viscous limit, the sonic limit, the entrainment limit and the capillary limit. Accordingly, the upper limit of heating power is: <math display="block" id="M51"><mrow><msub><mi>Q</mi><mrow><mtext>heating</mtext></mrow></msub><mo>≤</mo><msub><mi>Q</mi><mrow><mtext>upper</mtext></mrow></msub><mo>=</mo><mi>min</mi><mrow><mo>(</mo><mrow><msub><mi>Q</mi><mrow><mtext>viscous</mtext></mrow></msub><mo>,</mo><msub><mi>Q</mi><mrow><mtext>sonic</mtext></mrow></msub><mo>,</mo><msub><mi>Q</mi><mrow><mtext>entrain</mtext></mrow></msub><mo>,</mo><msub><mi>Q</mi><mrow><mtext>capillary</mtext></mrow></msub></mrow><mo>)</mo></mrow><mo>.</mo></mrow></math> For heat transfer limitations, the following equations can be adopted: Viscous limit: <math display="block" id="M52"><mrow><msub><mi>Q</mi><mrow><mtext>vi</mtext></mrow></msub><mo>=</mo><mfrac><mrow><msubsup><mi>D</mi><mtext>v</mtext><mtext>2</mtext></msubsup><msub><mi>h</mi><mrow><mtext>fg</mtext></mrow></msub></mrow><mrow><mn>64</mn><msub><mi>μ</mi><mtext>v</mtext></msub><msub><mi>L</mi><mrow><mtext>eff</mtext></mrow></msub></mrow></mfrac><msub><mi>ρ</mi><mtext>v</mtext></msub><msub><mi>P</mi><mi>v</mi></msub><msub><mi>A</mi><mtext>v</mtext></msub></mrow></math> Sonic limit: <math display="block" id="M53"><mrow><msub><mi>Q</mi><mtext>s</mtext></msub><mo>=</mo><msub><mi>A</mi><mtext>v</mtext></msub><msub><mi>ρ</mi><mtext>v</mtext></msub><msub><mi>h</mi><mrow><mtext>fg</mtext></mrow></msub><msup><mrow><mrow><mo>[</mo><mrow><mfrac><mrow><msub><mi>γ</mi><mtext>v</mtext></msub><msub><mi>R</mi><mtext>v</mtext></msub><msub><mi>T</mi><mtext>v</mtext></msub></mrow><mrow><mn>2</mn><mrow><mo>(</mo><mrow><msub><mi>γ</mi><mtext>v</mtext></msub><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mfrac></mrow><mo>]</mo></mrow></mrow><mrow><mn>0.5</mn></mrow></msup></mrow></math> Entrainment limit: <math display="block" id="M54"><mrow><msub><mi>Q</mi><mtext>e</mtext></msub><mo>=</mo><msub><mi>A</mi><mtext>v</mtext></msub><msub><mi>h</mi><mrow><mtext>fg</mtext></mrow></msub><msup><mrow><mrow><mo>[</mo><mrow><mfrac><mrow><msub><mi>ρ</mi><mtext>v</mtext></msub><mi>σ</mi></mrow><mrow><mn>2</mn><msub><mi>r</mi><mtext>h</mtext></msub></mrow></mfrac></mrow><mo>]</mo></mrow></mrow><mrow><mn>0.5</mn></mrow></msup></mrow></math> Capillary limit: <math display="block" id="M55"><mrow><msub><mi>Q</mi><mtext>c</mtext></msub><mo>=</mo><mfrac><mrow><mfrac><mrow><mn>2</mn><mi>σ</mi></mrow><mrow><msub><mi>r</mi><mtext>c</mtext></msub></mrow></mfrac><mo>−</mo><msub><mi>ρ</mi><mtext>l</mtext></msub><mi>g</mi><msub><mi>D</mi><mtext>v</mtext></msub><mi>cos</mi><mi>ϕ</mi><mo>+</mo><msub><mi>ρ</mi><mtext>l</mtext></msub><mi>g</mi><mi>L</mi><mi>sin</mi><mi>ϕ</mi></mrow><mrow><mrow><mo>(</mo><mrow><msub><mi>F</mi><mtext>l</mtext></msub><mo>+</mo><msub><mi>F</mi><mtext>v</mtext></msub></mrow><mo>)</mo></mrow><msub><mi>L</mi><mrow><mtext>eff</mtext></mrow></msub></mrow></mfrac></mrow></math> Until the heating power satisfies Eq. (39) and Eq. (40) simultaneously, the heat pipe cannot achieve a successful startup. Moreover, it can be observed that there is a certain relationship between the heating power and the startup time. The resulting interpolation function is expressed by Eq. (44). It can be utilized to calculate the starting time for a potassium heat pipe. <math display="block" id="M56"><mrow><msub><mi>t</mi><mrow><mtext>startup</mtext></mrow></msub><mo>=</mo><mi>a</mi><mo>·</mo><mi>exp</mi><mrow><mo>(</mo><mrow><mi>b</mi><mo>·</mo><msub><mi>Q</mi><mrow><mtext>heating</mtext></mrow></msub></mrow><mo>)</mo></mrow><mo>+</mo><mi>c</mi><mo>·</mo><mi>exp</mi><mrow><mo>(</mo><mrow><mi>d</mi><mo>·</mo><msub><mi>Q</mi><mrow><mtext>heating</mtext></mrow></msub></mrow><mo>)</mo></mrow></mrow></math> <math display="block" id="M57"><mrow><mtext>a</mtext><mo>=</mo><mn>7911</mn><mo>,</mo><mtext> b=-0</mtext><mtext>.02358, c=1286, d=0</mtext><mtext>.003861</mtext></mrow></math> Notably, the coefficients in Eq. (45) will change once the insulating condition or cooling condition changes. However, regarding the proposed model, adopting the basic idea of the network and the two-zone models can realize a fast calculation for startup performance. The values of the coefficients can be easily determined. 5　Conclusion 1 In this study, a simplified model for a high-temperature heat pipe was proposed to simulate the operating characteristics during startup from the frozen state. To achieve fast calculation of startup, this model decoupled the vapor flow, evaporation, condensation, and heat conductance. To verify this model, a heating experimental system for high-temperature heat pipes was established. A potassium heat pipe with a length of 600 mm was selected as the experimental object. Startup experiments with heating powers of 150 W, 200 W, and 250 W were executed. By comparing the calculated results with the experimentally obtained results, it was concluded that this model could effectively simulate the operating characteristics of a high-temperature heat pipe startup from the frozen state. Moreover, this model could also predict the phenomena of evaporation/condensation of the working medium, melting interface movement, and startup statement of the heat pipe. According to the discussion, the conditions for a successful startup were determined. The heat pipe cannot achieve startup from the frozen state if the heating power does not satisfy the aforementioned conditions. All these discussions revealed the capability of the proposed model to simulate the transient characteristics of a high-temperature heat pipe. The model acted as a powerful tool for the analysis of heat pipe cooled reactors. 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The first draft of the manuscript was written by Yu-Chuan Guo, and all authors commented on the previous versions of the manuscript. All authors read and approved the final manuscript. 10.1007/s41365-021-00947-2 This work was supported by the National Key Research and Development Project of China (No. 2020YFB1901700) , Science Challenge Project (No. TZ2018001) , the National Natural Science Foundation of China (Nos. 11775126 and 11775127) , and the Tsinghua University Initiative Scientific Research Program. 2021-05-10 2021-08-04 2021-08-31 2021-09-29 2021-10-01 2021-11-29 © China Science Publishing & Media Ltd. (Science Press), Shanghai Institute of Applied Physics, the Chinese Academy of Sciences, ChineseNuclear Society and Springer Nature Singapore Pte Ltd. 2021 This is a post-peer-review, pre-copyedit version of an article published in Nuclear Science and Techniques. The final authenticated version is available online at: http://dx.doi.org/10.1007/s41365-021-00947-2 http://dx.doi.org/10.1007/s41365-021-00947-2 . 2021 Abbreviations a Thermal diffusion coefficient (m 2 /s) A Area (m 2 ) C Specific heat capacity (J/(kg·K)) D Diameter (m) f Friction coefficient g Gravitational acceleration (m/s 2 ) h Specific enthalpy (J/kg) k Heat conductivity (W/(m k))/Boltzmann constant L Length (m) P Pressure (Pa) q Mass flow rate (kg/s) Q Transport power (W) r Radius (m) R Thermal resistance (K/W)/gas constant (J/mol K) t Time (s) T Temperature (K) v Velocity (m/s) V Volume (m 3 ) α Volume expansion coefficient(K -1 ) ρ Density (kg/m 3 ) μ Dynamic viscosity (Pa.m) υ Kinematic viscosity (m 2 /s) ε Porosity λ Convective coefficient (W/(m 2 ·K)) γ Specific heat ratio φ Inclined angle hl Minimum specific enthalpy of liquid (J/kg) hs Maximum specific enthalpy of solid (J/kg) Tmp Melting temperature (K) Kn Knudsen number Re Reynolds number Nu Nusselt number Gr Grashof number Pr Prandtl number Subscripts c Capillary e Entrainment f Fluid h Hydraulic l Liquid m Mesh s Sonic limit w Wall region v Vapor fg Phases change between fluid and gas tr Transition temperature in Inner radius wc Wick region mv Mass flow rate of vapor vi Viscous limit equ Equivalent parameter out Outer radius eff Effective length radial Radial direction axial Axial direction conduct Heat conduction Nuclear Science and Techniques 10 32