Analysis and Optimization of the Opening Dynamic Characteristics of Molten Salt Check Valves for Concentrating Solar Power

Abstract

The poor opening dynamic characteristics of molten salt check valves, used in concentrating solar thermal systems, constitute the main cause of valve disc oscillation and low pressure difference difficulty in opening during molten salt delivery. A molten salt swing check valve is designed to meet the requirements of high-temperature and high-pressure sealing and anti-crystallization flow channels. A transient dynamics model of the valve motion components is established, dynamic mesh and UDF (user-defined function) techniques are used to simulate the non-constant flow of hot molten salt and the opening process of the check valve and to analyze the dynamic characteristics of the opening process. Topological optimization of the valve motion components is proposed for the first time in order to improve the opening performance of the check valve, and the topological optimization of the valve motion components is based on the solid isotropic material penalty (SIMP) model with the variable density method and thermal–fluid–mechanical coupling method. The design is also verified for the dangerous working condition of a molten salt hammer. The results show that the mass of the valve motion component is reduced by 57.76% after optimization while meeting the requirements of strength and stiffness. The optimized molten salt check valve achieves a larger angle and faster opening, the full opening angle is increased by 6°, the positive resting pressure difference of the valve is reduced by 5 kPa, the minimum opening pressure difference is reduced by 8.9 kPa, the optimized flow characteristics are smoother, and the valve disc oscillation problem is avoided. The study provides a method for researchers to use to optimize the design of a molten salt swing check valve and its dynamic characteristics for concentrating solar power, which is of great significance in efforts to improve the stability of the molten salt transport system.

1. Introduction

Solar energy is a clean and renewable energy source. Its utilization can reduce greenhouse gas emissions, which significantly contributes to the achievement of sustainable development goals. Concentrating solar power (CSP) is a promising renewable and sustainable energy technology [1]. The integration of CSP with thermal storage systems can address the shortcomings of low renewable energy conversion efficiency, stochastic volatility, and difficulty of consumption [2,3,4]. Considering many factors such as solar thermal efficiency, thermal storage capacity and scalable commercial operation, tower-type thermal power technology using molten salt as the heat absorption and storage mass is the most promising technological route [5,6]. The improvement of the operational efficiency and stability of concentrated solar thermal systems is of great significance to the promotion and application of solar thermal power generation.

To improve the efficiency and stability of the concentrating solar thermal systems, researchers have carried out relevant research on solar plants and their accompanying accessories. Al-Maliki et al. [7] performed a dynamic system analysis and optimization of a thermal salt storage facility to identify carbonate as the optimal storage fluid to maximize capacity and efficiency and stabilize and relieve pressure on the local grid. Said et al. [8] described the use of data-driven machine learning methodologies to model and optimize a solar-assisted shell and tube heat exchanger using multi-wall carbon nanotubes and water nanofluids, ultimately improving the overall efficiency of the device. Luo et al. [9] analyzed the temperature and thermal stresses of a solar tower molten salt receiver under multi-source uncertainty conditions to provide additional reliability in the temperature and thermal stress evaluation processes of the solar receiver tube. Some researchers have analyzed the molten salt pump flow and impeller modalities through numerical simulations to improve the stability of the molten salt pump by improving it structure [10,11]. As one of the important pieces of equipment of the molten salt transport system, the safety and reliability of the molten salt check valve are related to the safe operation of the molten salt pump and the entire power station [12,13].

Several researchers have studied the opening and closing process of various types of check valves. Mao et al. [14] optimized the structure, fabrication, and bonding process of the microfluidic check valve by an experimental comparison in order to improve the movability of the cantilever and achieve large forward flow and small reverse leakage. Using the dynamic mesh method to simulate the transient flow characteristics of the check valve opening and closing process is a complex but effective method to determine the flow state, velocity distribution, and pressure distribution at each angle, monitor and study the dynamic process of angle, valve disc fluid force and velocity and other parameters with time, and to provide convenience for the design and selection of check valves [15,16,17]. Gao et al. [18] compared the opening dynamic characteristics of the swing check valve and tilting check valve and proposed that the opening process of the swing check valve is more stable. Bi et al. [19] simulated different minimum velocities at which the check valve starts to close, and discussed the closing characteristic time when the check valve closes at a certain velocity and so achieves better control of backflow. Liu et al. [20] constructed an analytical model for the transient motion of the check valve, based on the UDF dynamic mesh technique, to simulate the dynamic opening process of the valve and studied the vibration characteristics of the valve with different spring stiffness. Ye et al. [21], based on the transient CFD method and dynamic mesh technology, established a transient hydrogen flow model of the check valve, and the effect of different spool head angles on the closing shock was studied to obtain the optimized design value of the spool head angle.

Check valve poor opening dynamic characteristics will occur during valve oscillation and thus cause system pressure fluctuations and valve wear damage, but also will happen during low pressure difference operating conditions that make the valve disc difficult to open [22]. This seriously affects the safety and efficiency of the molten salt conveying system and so it is necessary to study the dynamic characteristics optimization method. Check valve opening characteristics usually include three indicators: minimum opening differential pressure, minimum full opening flow rate, and flow characteristics. The existing literature lacks discussions of the optimization of dynamic characteristics of check valves, but relevant papers on the optimization of product strength and flow properties can provide references. Li et al. [23] optimized the profile line of the valve core variable opening area of the dynamic flow balance valve by using the pressure drop compensation coefficient revising method, which improved the flow control accuracy. Sun et al. [24] optimized the dimensional parameters of the unloader and suction valve based on the optimization proxy model and NSGA-II algorithm, which effectively improved the movement characteristics and the comprehensive working performance of the stepless flow control system. Yu et al. [25] designed a three-dimensional heat sink with a load-bearing capacity based on a topological optimization method for the heat-flow structure problem. Tang et al. [26] proposed an effective method to eliminate cavitation from a throttling orifice plate, based on genetic algorithm and topology optimization, which ultimately reduces the cavitation area and orifice plate mass.

There is a lack of research on the opening dynamic characteristics of molten salt swing check valves for concentrating solar power. The molten salt properties and boundary conditions in molten salt transport systems have a large impact on the opening dynamic characteristics of molten salt check valves. At the same time, the limitations of molten salt test conditions make the relevant tests difficult to perform, and so the numerical simulation study of the dynamic characteristics of high-temperature molten salt swing check valve is an important step in the performance testing of check valves. More importantly, the existing research on the dynamic characteristics of valves mainly analyzes the dynamic characteristics of valves under different working parameters or different structural parameters to facilitate the valve design and selection process, but there is a lack of research on how to optimize the dynamic characteristics through structural improvement.

In this study, a high-temperature molten salt swing check valve is designed to meet the working requirements of a molten salt swing check valve in high-temperature and high-pressure conditions. The product is designed to possess anti-crystallization properties and overall exhibit reduced leakage. Based on the moment balance of the valve and considering the molten salt density, a transient dynamics model of the valve motion components is established. After this, the three-dimensional passive dynamic mesh and UDF techniques are used to study the opening dynamic characteristics. A method is proposed for optimizing the topology of the valve motion components to improve the opening performance of the check valve, analyze the effect of mass retention rate u on the topology optimization results, and determine a new valve disc structure that meets the requirements of strength, stiffness and reverses sealing. A comparative analysis of the opening dynamic characteristics of the molten salt swing check valve before and after optimization verifies the effectiveness of the topological optimization method of the disc structure in improving the dynamic characteristics of the swing check valve. This will allow researchers to ensure the molten salt swing check valve achieves low pressure difference opening and opening process stability, to ensure the safety and stability of the molten salt pump and the whole system, and to promote the solar thermal power generation industry to enhance its capacities, each of which has important significance as an objective.

2. Design of Molten Salt Swing Check Valve

2.1. Structural Design of Molten Salt Swing Check Valve

The combination of tower technology and molten salt heat absorption and storage technology gives the tower-type molten salt solar thermal power plant the comprehensive advantages of high solar thermal efficiency, 24 h continuous power generation, and commercial operation on a large scale [27]. Figure 1 illustrates the tower-type collector solar thermal power generation process, which consists of three parts: the heliostat field, the molten salt system, and the power generation system.

In the molten salt system, the molten salt pump transports the high-temperature molten salt to the steam generation system. The high-temperature molten salt check valve is used at the outlet end of the molten salt pump to prevent the backflow of molten salt, protect the molten salt pump, piping, and related supporting facilities, and ensure the safe and reliable operation of the system.

Molten salt has properties such as high operating temperature, high freezing point, large operating temperature variation and high corrosiveness. These characteristics place high demands on the design, manufacturing, and production of molten salt valves [28].

The molten salt valve is designed according to the molten salt system circuit parameters, in particular the DN100 PN100 high-temperature molten salt swing check valve structure, as shown in the schematic diagram in Figure 2, mainly in reference to the valve body, valve seat, valve motion components, sealing components and other components. According to the special needs of high-temperature molten salt media, the valve design in the valve body material uses wear-resistant, high-temperature austenitic stainless steel ASTM A351-CF8C. The valve seat uses 5° inclined sealing surface and valve self-weight to increase the closing sealing force.

The pressure seal structure is designed so that the higher the internal pressure is, the greater the sealing force will be. Easy dismantling is made possible by dropping the Bonnet assembly into the body cavity and driving out the four-segmental thrust rings by means of a push pin. An internal hanging structure and butt welding connection can effectively reduce the leakage point.

The “herringbone” valve body flow channel structure is designed to have a good one-way flow which does not easily cause molten salt accumulation at the bottom groove and inflection point. The integrated design of the valve body avoids thermal deformation and thermal stress concentration caused by uneven heating.

2.2. Dynamics Model of the Opening Process of Molten Salt Swing Check Valve

2.2.1. Opening Process Dynamics Model

To address the fact that a high-temperature molten salt transport system molten salt swing check valve opening process will appear difficult to open under conditions of low pressure difference, and to deal with the opening process valve oscillation, pressure fluctuations and other problems, an analysis of the opening process valve motion components force can be performed, as shown in Figure 3. Based on the valve moment balance and considering the molten salt density, this model will allow the researchers to establish a molten salt check valve opening process dynamics model, to make predictions and, in actual operations, to avoid the above problems.

When the molten salt medium flows through the check valve, the disc rotates about the rotating shaft under the action of the resultant force or is in a state of equilibrium. The main torque affecting disc rotation are torque by jet impingement on the disc and the torque by pressure difference, gravity moment, buoyancy moment, and friction moment, among which the friction moment at the rotating axis is usually very small and can be ignored. The dynamic model of the valve opening process is established as follows, and the opening characteristic parameters of the molten salt check valve are analyzed.

The angular momentum equation of the opening process can be expressed as:

$$I_{disc}\frac{d\omega }{dt}=T_V+T_P+T_W$$

(1)

where I_disc is the rotational inertia of the valve motion components with respect to the rotating shaft; ω is the angular velocity of the valve motion components opening; T_V is the torque induced by jet impingement; T_P is the torque induced by pressure difference; T_W is the combined moment of gravity and buoyancy of the valve motion components.

The torque induced by jet impingement T_V is caused by the relative velocity of the valve motion components to the molten salt medium. Define the relative flow velocity as [29]:

$$v_{rel}=v-L_c\omega \cos \theta$$

(2)

where v is the molten salt inlet flow velocity; L_c is the distance from the valve shape center to the rotating axis; θ is the valve opening angle, taking the value range of 0°~61.5°.

The medium impact moment can be expressed as follows [30]:

$$T_V={\displaystyle \int R\times }\mathrm{d}F=\rho A_{disc}{v}_{rel}\left|v_{rel}\right|L_c\cos \theta$$

(3)

The pressure difference moment, caused by the inlet and outlet pressure difference, can be defined as:

$$T_P=\Delta p{A}_{disc}{L}_c\cos \theta$$

(4)

where Δp is the upstream and downstream pressure difference of the valve motion components; A_disc is the area of the valve disc.

The combined moment of gravity and buoyancy of the valve motion components can be defined as:

$$T_W=-V_{disc}\left({\rho }_{disc}-{\rho }_{fluid}\right)g{L}_g\sin \theta$$

(5)

where V_disc is the volume of valve motion components; ρ_disc and ρ_fluid are valve motion component material density and molten salt medium density, respectively; L_g is the distance from the center of gravity of the valve motion components to the rotating axis.

2.2.2. Minimum Opening Pressure Difference

The opening process of molten salt swing check valves is influenced by the properties of the fluid medium and the operating environment, such as fluid viscosity, pressure, temperature, and fluid velocity. To avoid the problem of incurring difficulty opening the check valve at a low pressure difference, it is critical to determine the minimum opening pressure difference of the valve for the molten salt pump and system stability [31].

Assuming that the downstream gauge pressure is zero, the critical opening pressure of the valve is determined by assessing the torque around the rotating shaft, i.e., the minimum opening pressure. Using dynamic mesh and UDF technology to simulate the opening process of the swing check valve, the minimum opening pressure difference of the molten salt swing check valve is determined.

2.2.3. Minimum Full Opening Velocity

During the operation of the molten salt system, the check valve should remain stable after opening the valve flap, thus avoiding bringing about fluctuations in system pressure and reducing wear damage to the valve [32].

The minimum full opening velocity is defined as the minimum velocity when the valve disc is fully open and stable without motion. Using dynamic mesh and UDF techniques to simulate the opening process of the swing check valve, the flow rate, and the corresponding minimum full opening velocity are obtained when the valve disc is fully opened and stable.

3. Opening Process Dynamic Characteristics Simulation Analysis

3.1. Computational Domain Model and Meshing

Three-dimensional modeling software is used to establish the three-dimensional model of the PN 100 DN 100 molten salt swing check valve. In order to fully develop the turbulent flow before entering the valve body and improve the simulation accuracy, straight pipe sections are added before and after the valve body. The straight pipe section in front of the valve is 200 mm long, which is 2 times the nominal diameter of the valve, and the rear valve pipe is 600 mm long, which is 6 times the nominal diameter of the valve.

The 3D model is imported into the CFD software to generate the flow channel model. To ensure the continuity of the fluid domain before and after the valve disc, the initial opening angle of the valve is set to 1°, and the mesh is divided to realize a discrete fluid domain discrete. The use of an unstructured mesh can better represent the irregular shape of the fluid domain within the check valve, and the tetrahedral mesh can maximize the integrity of individual meshes and result in the strongest mesh deformation adaptation capability, improving the convergence and computational accuracy of the dynamic mesh calculation. The schematic diagram of the volumetric mesh of the studied check valve and the cross-sectional view of the mesh near the valve motion components are shown in Figure 4. The mesh in the valve motion components region is dense to ensure high mesh quality despite the valve disc motion. The valve opening degree of 30% and inlet pressure of 100 kPa are used as the verification conditions, and the mesh independence is verified by the mass flow rate. When the mesh number increases to more than 1.08 million, the variation of the flow rate is within 0.2%, and the mesh number is finally determined to be 1,082,692 in order to reduce the computational cost while ensuring computational accuracy.

3.2. Numerical Method and Boundary Conditions

This issue is solved using the finite volume method, with the pressure term discretized in a standard format and the rest in a second-order windward format, are the coupled equations are solved by the coupled algorithm. The turbulence model is chosen as k-ω SST. The k-ω SST model combines a robust formulation of k-ω for the near-wall region and k-ε for the far-wall region. A hybrid function ensures a smooth transition between the two models. A shear stress limiter helps the k-ω model to avoid excessive turbulent kinetic energy levels near the stagnation point [33,34].

The medium temperature in the hot molten salt circuit is 565 °C, its density is 1730.66 kg/m³, and the dynamic viscosity is 1.14 mPa·s. Assuming that the operating pressure is 101,325 Pa, the inlet pressure increases linearly from 0 to 10 MPa in 2 s, the outlet gauge pressure remains at 0 MPa, the time step is set to 0.0001 s, and the calculation time is 2 s.

3.3. Dynamic Mesh Setup and UDF Programming

Dynamic mesh technology is a computational model that emerges to adapt to changes in the fluid computational domain by stretching and compressing the mesh, adding and subtracting from the mesh, and locally constructing a new mesh to adapt to changes in the computational domain. The dynamic mesh is divided into active and passive types. The active dynamic mesh realizes the active deformation of the mesh by predefining the rigid body motion mode. The molten salt swing check valve is a self-acting valve, its motion process cannot be predefined, the motion is determined by the calculation result of the previous step, and the mesh update is done automatically according to the boundary change in each iteration step. Additionally, using passive-type dynamic mesh can express the real state of the valve motion component of the molten salt swing check valve.

The automatic grid update is based on the change of the boundary position at each time increment and is expressed as follows:

$$\frac{\mathrm{d}}{\mathrm{d}t}{\displaystyle \underset{V}{\int }\rho \phi \mathrm{d}V}+{\displaystyle \underset{\partial V}{\int }\rho \phi \left(u-u_s\right)}\mathrm{d}A={\displaystyle \underset{\partial V}{\int }G}\nabla \phi \mathrm{d}A+{\displaystyle \underset{V}{\int }{S}_{\phi }}\mathrm{d}V$$

(6)

where ρ is the molten salt density; u is the molten salt velocity vector; u_s is the mesh deformation velocity of the 3D dynamic mesh; G is the diffusion coefficient; A is the surface area of the control body; S_φ is the source term of the flux; ∂V is the boundary of the control body V.

The flow chart of the dynamic opening process of the molten salt swing check valve, simulated by CFD 3D passive-type dynamic mesh and UDF technology, is shown in Figure 5. The opening process of the swing check valve is a force balance process, which is drawn from Newton’s second law, as shown in Equation (1). According to the SolidWorks software, the inertia of valve rotation I_disc = 0.01866 kg·m² is calculated, and the angular acceleration can be found by using UDF custom macros to find out the pressure and torque around the rotation axis of the valve motion components during the opening process.

Based on the definition of angular acceleration, the amount of angular velocity change for one time step and the new angular velocity can be obtained. Then, based on the definition of angular velocity, the amount of angular change for each time step is calculated, and finally the new angle for the discrete time step is obtained.

As Figure 6 shows in its depiction of the opening process 3D passive-type dynamic mesh update schematic, the tetrahedral mesh can ensure the integrity of individual meshes and provide the strongest mesh deformation adaptation capability to the greatest extent during the dynamic deformation pulling process of the main mesh edges in the basin until the single-side mesh reaches the maximum deformation. Spring smoothing and local remeshing are used to achieve the mesh deformation process to ensure better mesh convergence.

The changing inlet boundary conditions are written using the macro DEFINE_PROFILE, and the rigid body (valve motion component) motion is controlled by compiling the DE-FINE_CG_MOTION macro in the dynamic mesh UDF to complete the node position update in each time step until the valve is fully opened.

3.4. Opening Dynamic Characteristic Analysis

Attaining stable opening characteristics for the check valve is the key to ensuring the safe operation of the system. Throughout the time taken to establish the connection between the parameters, the dynamic characteristics of the opening process of the molten salt swing check valve are analyzed to provide a basis for the optimization of its opening dynamic characteristics.

3.4.1. 40% Opening Flow Analysis

Taking a typical opening of 40% opening as an example, the relevant velocity flow line diagram is shown in Figure 7. The flow line in the valve is more uniformly distributed. Due to the throttling effect of the valve disc, the media flow through the gap between the valve disc and the valve body media flow rate increases rapidly. There is a low flow velocity region behind the valve disc and this produced a more unevenly distributed vortex. The main flow channel flow is uniform, and the bottom of the valve body also did not produce the phenomenon of vortex reflux as molten salt struggled to accumulate here, helping to avoid the accumulation of molten salt crystallization.

3.4.2. Opening Process Angle Change Analysis

The molten salt swing check valve opening process valve opening angle versus time is shown in Figure 8.

Before 0.01 s, the force on the valve disc is small, which is not enough to overcome the friction and the gravity of the valve motion components, and the valve disc is in a static state. In the time period ranging from 0.01 s to 0.0288 s, the check valve opened slowly to 5° with a small angular velocity. In the time period from 0.029 s to 0.073 s, the valve disc angle increased linearly, with an average angular velocity of 15.629 rad/s and a valve opening angle of 44.5°.

3.4.3. Analysis of the Minimum Opening Pressure Difference

The relationship between the opening angle of the valve disc and the pressure difference during the opening of the molten salt swing check valve is shown in Figure 9.

When the pressure is small, the check valve has the problem of having difficult opening. Additionally, the figure shows that when the pressure difference is less than 15 kPa, the valve disc opening angle is close to 0 degrees, which is called the valve resting pressure difference. The opening angle of 1° is defined as the effective opening angle of the molten salt swing check valve, which corresponds to the minimum opening pressure difference of 84.8 kPa. When the pressure difference is less than 145.5 kPa, the opening angle of the check valve is less than 5°.

In the partial enlargement of Figure 9, in the pressure difference range from 424 kPa to 429 kPa, the valve disc opening angle curve appears at the crossed part of the valve disc in the opening process oscillation from 61.5° back to 61.1°, accompanied by pressure fluctuations, and then the valve opens up to the maximum opening angle.

3.4.4. Analysis of Minimum Full Opening Velocity

The relationship between the opening angle of the disc and the molten salt inlet velocity during the opening of the molten salt swing check valve is shown in Figure 10.

In the process from 0 to A, the valve opening angle changes rapidly as molten salt inlet velocity increases, with the coordinates of point A being (5.75, 43.5). The valve opening angle changes slowly from point A to point B, with the coordinates of point B being (15.37, 61.5). Point B corresponds to the valve just opening to full valve opening, with the corresponding inlet velocity being v_open = 15.37 m/s. However, the valve is not stable at this point due to the BC section valve disc motion and tapping. At point C, the valve reaches a stable state, and the corresponding velocity is the minimum fully open velocity, v_min = 23.8 m/s.

4. Topology Optimization of Valve Motion Components Based on Thermal–Fluid–Mechanical Coupling

The performance of the molten salt swing check valve opening process is critical for the stable operation of molten salt-concentrating solar power. In checking valve performance evaluation, it can be seen that the weight, shape and size of the valve motion components have a significant impact on opening performance [35]. In order to optimize the minimum opening pressure difference and the minimum full opening flow velocity, and in order to attenuate the oscillation of the disc during the opening, the topology optimization technique is used to change the material distribution, i.e., to change the disc mass and the center of gravity position. This is performed in order to optimize the opening dynamic characteristics of the swing check valve from the perspective of disc mass and shape improvement.

Considering that the molten salt swing check valve will be impacted by high-temperature molten salt reverse flow in the closed state of the valve disc, the optimized valve disc model in the closed state of the valve is subjected to thermal–fluid–mechanical coupling analysis to evaluate the high-temperature thermal stress, deformation and reverse-sealing performance to ensure structural safety and reliability while optimizing the valve disc.

4.1. Topology Optimization Model

The main continuum topology optimization methods include solid isotropic material with a penalty (SIMP), evolutionary structure optimization (ESO), and level sets. The most popular method in continuum topology optimization is the isotropic material with the penalty method. Its main advantages are that it is robust, well understood and easy to implement [36].

The variable density method assumes a mapping relationship between the elastic modulus of the material and its relative density, obtains the relative density values of the cells when the objective function has an optimal solution through optimal iterative calculations, and updates the topological configuration of the structure by adding or deleting cells according to the magnitude of the relative density of each cell. To minimize the existence of intermediate density cells, a material penalty model is artificially introduced.

The SIMP model assumes that the modulus of elasticity of a material as a function of its relative density can be expressed as:

$$E\left(\rho \right)=E_{\min }+{\rho }^p\left(E_0-E_{\min }\right),\rho \in \left[0,1\right]$$

(7)

where ρ is the relative density of the cell, p is the penalty factor, E₀ is the modulus of elasticity of the material before being penalized, and Emin is the elastic model of the material out of the hole in the structure, whose value is a very small value approximately equal to 0.

Simplify Equation (7) and get the following equation:

$$E\left(\rho \right)=E_0{\rho }^p,\rho \in \left[0,1\right]$$

(8)

The main role of the penalty factor p is to penalize the intermediate densities so that they better converge to 0 or 1 and eventually redistribute the material.

The goal of topology optimization is to redistribute the material and provide maximum stiffness for various mass constraints. In this study, based on the SIMP model, the optimization objective is to minimize the structural compliance under the given load and boundary conditions, the constraints are the masses of the valve motion components, and the established mathematical model for topology optimization is shown in Equation (9).

$$\{\begin{array}{l}\min C\left(\rho \right)=U^{\mathrm{T}}KU\\ \mathrm{s}.\mathrm{t}. K\left(\rho \right)U\left(\rho \right)=F\\ {\displaystyle \sum _{c=1}^n{V}_c{\rho }_c-V}\le 0\\ 0<{\rho }_{\min }\le {\rho }_c\le 1\\ {\sigma }_{\max }\le \left[\sigma \right]\end{array}$$

(9)

where C is the structural compliance; ρ is the discrete unit density, ρ = [ρ₁, ρ₂, …, ρ_n]^T; K(ρ) is the stiffness matrix of the overall structure; U(ρ) and F are the displacements and loads of the discrete model, respectively; V_c is the volume of the discrete unit; V is the target volume of the discrete unit; ρ_min is the lower limit set to prevent the singularity of the overall stiffness matrix due to 0 density; σ_max is the maximum stress of the model structure and [σ] is the allowable stress of the model structure.

4.2. Check Valve Reverse Flow State Thermal–Fluid–Mechanical Coupling Simulation Conditions

4.2.1. Geometric Model

Figure 11 is the molten salt swing check valve fluid backflow state model schematic diagram. When the molten salt pump suddenly stops working, the pressure inside the pump disappears and the medium flows back, and then the check valve immediately closes to prevent the high-pressure molten salt from flowing back into the pump. More importantly, the molten salt hammer causes a large number of shock waves, and the check valve prevents the molten salt hammer from damaging the pipeline and the molten salt pump.

The material of valve motion components is A351 CF8, and the specific parameters are shown in

Table 1. Material parameters of valve motion components.

Temperature	ρ/(kg/m3)	E/(GPa)	α/(10−6 °C−1)	λ/(W/(m·°C))	σs/(MPa)
22 °C	8027	192	15.3	14.8	207
565 °C	8027	153	17.4	23.6	98

.

4.2.2. Mesh Division

The check valve reverse flow state model in Figure 11 is reasonably simplified and the flow channels are extracted. The model mesh division and mesh independence check were carried out to obtain the thermal–fluid–mechanical coupling model mesh, as shown in Figure 12. This contains two parts, fluid domain mesh and solid mesh, and is used for fluent and static structural calculations, respectively.

Considering the influence of mesh quantity and mesh quality on the calculation results, the mesh is continuously refined to ensure a small numerical analysis error between adjacent mesh densities. According to the respective advantages of tetrahedral and hexahedral meshes, the flow channel model mesh is divided by a mixed tetrahedral/hexahedral mesh. At the same time, a reasonable grid height is chosen for the first layer of the boundary layer so that the grid nodes fall within the logarithmic law region to avoid the differences in numerical calculation results caused by the grid nodes falling on the viscous bottom layer. Considering the influence of boundary layers, five boundary layers are divided on the surface of the valve body and pipe. The grid height of the first layer is taken to be 0.8 mm, and the internal growth rate is 1.2.

4.2.3. Loads and Constraints

At the moment of pump stoppage, the molten salt flow decreases. Thus, the valve disc closes, and the valve disc is subjected to the impact pressure generated by the molten salt reverse flow, representing the maximum pressure that the valve disc is subjected to. Assuming a maximum pressure head H of 200 m, the molten salt hammer pressure is given in the following equation [37].

$$P_{\mathrm{MShammer}}=\gamma \cdot H$$

(10)

where γ is the specific weight of molten salt. According to Equation (10), the molten salt hammer pressure of the valve disc is 3.391 MPa.

The flow and heat transfer numerical methods and media properties are the same as in Section 2.2, the time step is set to 0.01 s, and the calculation time is 5 s. The original outlet of the check valve is set as a counterflow inlet, the inlet imposes a pressure boundary condition of 3.391 MPa, and the inlet media temperature is 565 °C.

The calculated fluid pressure and fluid temperature data in the flow field are imported into static structural calculations.

4.3. Topology Optimization Design

4.3.1. Definition of the Optimal Design Domain

Topology optimization should ensure the valve sealing reliability and the quality continuity of the optimization result, which should be kept intact, with the sealing surface, the inner surface of the rotating shaft bore and rocker connection part as the retention area. Setting the topology optimization domain as shown in Figure 13, blue is the topology optimization area and red is the retention area.

4.3.2. Optimization Parameters

The structural discontinuity problem that easily arises in the calculation process of the SIMP method and the influence of mass retention rate u on the topology optimization results are considered next. The mass retention rate as a design variable, if set too small, will make the topology optimized model distorted into a discontinuity to lose the optimization meaning, although it will achieve the effect of mass reduction. As such. the mass retention rate should be finally determined through multiple simulations and comparisons. In order to obtain a better material topology configuration, the mass retention rate u is set to 60%, 50%, 40% and 30% of the original valve motion components, and the optimization goal is the maximum overall structural stiffness. The OC optimization criterion is chosen and the convergence tolerance is defined as 0.0001.

4.3.3. Optimization Calculations

The iterative accuracy of the topology optimization calculation is 0.1%. By comparing the convergence curves of structural compliance, the optimization convergence under each mass retention rate u is judged. As shown in Figure 14, the iterative convergence curve of the objective function (i.e., structural compliance) is shown. The horizontal axis represents the number of iterations, and the vertical axis represents the optimization objective function of the valve motion components.

As the number of iterations increases, the objective function decreases, and the structural compliance value decreases the fastest in steps 0 to 16 before gradually stabilizing from step 17. According to the mass retention rate u from large to small, the calculation results converge after 24, 22, 27, and 38 iterations of update, respectively, indicating that the stiffness of the valve motion assembly reaches the maximum under satisfying the mass constraint, at which time the unit density of the design area is the optimal force transmission path and material distribution of the valve motion components.

The retention threshold is set to 0.5. Figure 15 and Figure 16 show the optimization results of the valve motion components at each mass retention rate u. The red part comprises the cells with density 0~0.4, which can be deleted, and the gray part consists of the cells with density 0.6~1, which should be retained.

From the view of the structural optimization process, the objective functions under the four groups of mass retention rate u gradually converge to the minimum value (that is, the structural stiffness reaches the maximum value). Among them, when u = 50% and u = 30% are optimized, they are not as smooth as when u = 60% and u = 40% are optimized. The main reason for this is that during the optimization process, when u = 50% is optimized, there are material discontinuities at the fixed connection part of the valve motion components nut. When u = 30% is optimized, there are material discontinuities at the center of the valve disc because of the large proportion of material removal. As a result, the force transmission path of the structure changes, the stress distribution of the structure also changes, and the structure redistributes units. Elements are added at places with high stress and deleted at places with low stress, and so there will be some ups and downs in the optimization process. When u = 60% and u = 40% are optimized, there are no intermittent and sudden changes in the structural force transmission path and stress distribution, and so the optimization process will be relatively stable and easier to converge.

The materials that can be removed from the back of the valve disc, rocker and connection parts are clearly seen in Figure 15. When u is reduced, the original valve motion components’ nut fixed connection parts have larger optimization spaces under each mass retention rate distribution, while a larger optimization area are generated in the back of the valve disc and rocker parts.

Figure 16 shows the effective area of material retention. It can be seen that the front side of the valve has a similar optimization trend at each mass retention rate. Additionally, the material is reduced in a spherical shape, forming a spherical surface that reduces the mass and ensures the pressure tolerance. The incompleteness of the valve disc appears at u = 30%, u = 60% has more topology optimization potential at the back of the valve disc and the connection area, and u = 50% also has more topological optimization potential at the connection area. In contrast, the optimized solution of u = 40% has a reasonable material distribution after optimization, which can effectively reduce the structural compliance but also smoothly and efficiently optimize the structure. Finally, the scheme with u = 40% is selected as the optimal optimization solution.

4.3.4. Optimization Results Model Reconstruction

The topology optimization calculation results are mesh data with different cell densities which are not smooth, discontinuous, and unclear. According to the topology optimization results shown in Figure 16 with u = 40%, the calculated results were surface fitted and the model reconstruction was carried out using SolidWorks software to obtain an engineering feasible structural optimization model of the disc, as shown in Figure 17.

The structural topology optimization undergoes a big change at the valve disc and the connection between the valve disc and the rocker. The valve disc is optimized from the original model’s flat plate structure to a ball crown structure. The connection method is changed from the original fixed valve disc and rocker through the nut to the optimized valve integrated structure. The mass of the valve motion components before optimization is 2.2847 kg, and after optimization, the mass is 0.965 kg, with a weight reduction of 57.76%.

4.3.5. Finite Element Analysis of Valve Disc for Molten Salt Hammer Conditions

Based on the basic equations of elastodynamics, the finite element analysis of the optimized valve disc with molten salt reverse flow and molten salt hammer occurs is carried out using the thermal–fluid–mechanical coupling method to evaluate the high-temperature thermal stress and deformation for structural safety. The mesh division and mesh irrelevance check are carried out, the same load and constraint conditions as in Section 4.2.3 are applied, and the cloud diagram of the valve stress and deformation distribution under thermal–fluid–mechanical coupling is obtained via finite element calculation, as shown in Figure 18.

The maximum stress of the valve flap is distributed in the rotating shaft and rocker connection part, with a maximum stress value of 85.122 MPa, less than the material at 565 °C allowable stress of 98 MPa, to meet the structural strength requirements. The maximum deformation of the valve disc is distributed in the middle part of the rocker; with a maximum deformation of 0.030638 mm, the deformation is small.

The distal displacement constraint is applied to the valve body inlet pipe end face, the axial free constraint is applied to the valve body outlet pipe end face, and the sealing finite element analysis is carried out by using the extended Lagrange contact algorithm to obtain the molten salt swing check valve sealing surface contact stress distribution cloud, as shown in Figure 19.

Under molten salt hammer conditions, the contact stress on the sealing surface is greater than the required specific pressure of 18.37 MPa, which can form a completely closed sealing specific pressure ring belt. The maximum contact stress is 136.78 MPa. This is less than the allowable specific pressure of the sealing surface material of 250 MPa and ensures the sealing and that phenomenon of seal sub-crush will not occur the.

4.4. Comparison Analysis of the Opening Dynamic Characteristics before and after Optimization

Figure 20 shows the relationship between the opening angle and time of the check valve before and after optimization. The optimized molten salt swing check valve avoids valve disc oscillation, reduces valve collision and wear between valve disc and pin, and weakens system pressure fluctuation.

Figure 21 shows the opening flow characteristics curve of check valve before and after optimization. When the opening degree is less than 30°, the opening flow characteristics are similar before and after optimization. Conversely, the process from 30° to full opening, the optimized flow characteristics are smoother, and the optimized opening angle shows a larger mass flow rate.

The relationship between the opening angle of the molten salt swing check valve and the pressure difference is shown in Figure 22, revealing that the valve does not open when the pressure difference is less than 10 kPa. After optimization, the minimum opening pressure difference of the molten salt swing check valve is 75.9 kPa, and the minimum opening pressure difference is reduced by 8.9 kPa.

5. Conclusions

Based on the dynamic mesh and UDF technology, the dynamic characteristics of the opening process of the molten salt swing check valve are simulated and analyzed, and a topological optimization method based on the thermal–fluid–mechanical coupling is proposed to optimize the disc and achieve improvement in the dynamic characteristics of the check valve. The specific conclusions can be summarized as follows:

(1)

The flow analysis reveals that the special flow channel structure design of the molten salt swing check valve reduces the flow dead zone and avoids the accumulation of molten salt crystals.

(2)

The spherical valve structure, obtained by the topology optimization method, reduces the mass of the valve motion components by 57.76% while meeting the strength and stiffness requirements, and has a sufficient reverse-sealing performance.

(3)

The opening dynamic characteristics of the molten salt swing check valve have been significantly improved with the optimization of the valve disc. The full opening angle is increased from 61.5° to 67.5°, and the time to reach the steady state is shortened from 0.18 s to 0.139 s. The positive resting pressure difference of the valve is reduced by 5 kPa, the minimum opening differential pressure is reduced by 8.9 kPa, and the flow characteristics are smoother after optimization while avoiding valve oscillation during the opening.

(4)

The topology optimization method is used to improve the mass and shape of the check valve disc, which has a significant effect on the optimization of the dynamic characteristics of the check valve. It provides new ideas for the industry to solve undesirable problems such as check valve disc oscillation, valve disc knocking on the valve cavity, and strange noises during operation.