Multi-objective optimisation of K-shape notch multi-way spool valve using CFD analysis, discharge area parameter model, and NSGA-II algorithm

Abstract

Multi-way spool valves (MWSVs) with K-shape notches (KSNs) provide advantages such as improvement of the actuator speed control, micro-action, and response performance. However, the pressure drop (PD) associated with the energy consumption is larger of MWSVs under the high-pressure and large-flow condition. Meanwhile, extremely complex flow coefficient and multi-parameter, highly coupled KSNs are the chief factors restricting the flow-pressure characteristics exploration and optimisation design of MWSVs. To address these problems, complete numerical research and experiment are performed in this study, especially concerning the method of MWSV flow-pressure characteristics modelling, and surrogated model-based optimal design of KSN structures. First, the relationships between KSN structure parameters and flow-pressure properties of MWSV are modelled on the innovative discharge area parameter model (DAPM) and response surface methodology (RSM) (RSM-DPAM) using the CFD dataset. Second, to reduce the PD combining the flow control performance, surrogate model-based optimisation design is addressed. During optimisation, six KSN structure parameters are chosen as design variables, PD and flow area relative deviation (FARD) are selected as objective functions, and the RSM is employed as the projected model. Depended on the created surrogate model, the non-dominated sorting genetic algorithm (NSGA-II) is established to search for the optimal KSN structure. To certify the performance of the optimisation, flow field characteristics are analysed. The results demonstrate that the proposed RSM-DAPM-RSM model achieves reliable prediction for PD and FARD with a great correlation coefficient (0.9757 and 0.9946). The average PD reduces as much as 7.23% while the FARD is only 1.28%. Moreover, the region of low-pressure, high-velocity, and high-turbulent kinetic energy in the flow field are reduced. The proposed framework enhances the performance in KSN spool optimisation and could be applied to other kinds of notches.

Abbreviations: BOI: Body of influence; BSV: Bucket spool valve; CFD: Computational fluid dynamics; CM: Construction machinery; DAPM: Discharge area parameter model; FARD: Flow area relative deviation; FCC: Flow control characteristic; GSA: Global sensitivity analysis; KSN: K-shape notch; MHAs: Meta-heuristic algorithms; MODE: Multi-objective differential evolution; MOGA: Multi-objective genetic algorithm; MOO: Multi-objective optimisation; MOPSO: Multi-objective particle swarm optimisation; MWSV: Multi-way spool valve; NRMS: Non-road mobile source; NSGA-II: Non-dominated sorting genetic algorithm; OLHD: Optimal Latin hypercube design; PD: Pressure drop; PDCs: Pressure drop characteristics; RSM: Response surface methodology; SA: Sensitivity analysis; TKE: Turbulent kinetic energy; TOPSIS: Technique for order preference by similarity to ideal solution; VFR: Volume flow rate

KEYWORDS:

• Multi-way spool valve
• K-shape notch
• discharge area parameter model
• computational fluid dynamics (CFD)
• multi-objective optimisation (MOO)
• NSGA-II

1. Introduction

Lately, it is non-road mobile sources (NRMSs) that have drawn gradual attention due to their dedication to pollution of air and climatic variation (Campbell et al., 2018). Construction machinery (CM) plays a significant role in NRMS air pollution. It has been assumed that CM drainages of NO_x and PM (2021, China) represent 30.0% and 32.1% of the overall NRMS drainages, respectively (X. Li et al., 2023). As concerns the global warming (Ribeiro et al., 2022), energy crisis (Umar et al., 2022), and tightened regulations on exhaust gas, the requirements for pollution reduction and energy conservation of CM are gradually stringent (Lin et al., 2020).

The MWSV is constituted of more than two directional valves and other valves such as safety valves and check valves. It is widely used in various CMs such as loaders and excavators as a result of its advantages of compact structure and multi-position communication (Do et al., 2021). Taking the loader as an example, the MWSV is often installed in the front frame of the loader to manipulate multiple actuator movements to achieve the functions of shovelling, loading, and unloading materials well. As the core component of loader hydraulic systems, the MWSV adjusts the valve opening to control the action of actuators. However, its PD associated with the energy consumption is larger under the high-pressure and large-flow condition (Xu et al., 2021), which dissipate as the form of heat energy and reduces the machine efficiency. Based on the national ‘carbon peaking and carbon neutrality’ policy (P. Yang et al., 2022), an in-depth investigation on the flow-pressure characteristics of MWSVs and design optimisation of high energy efficiency are greatly significant.

Flow coefficient is a crucial indicator of the flow control characteristics (FCCs) of a hydraulic spool valve. During engineering practice or some simulation software such as LMS Imagine. Lab AMESim, flow coefficients were often treated as a fixed value (Hua et al., 2018), with significant errors in small valve opening or low volume flow rate (VFR) (Afatsun & Tuna Balkan, 2019; Edward Lisowski et al., 2018). Actually, it is comprehensively determined by valve opening, VFR, and spool structure parameters. To accurately calculate the flow for any regime, the flow coefficient should be further investigated. When the VFR is large, the flow coefficient tends to the saturation value. The concept of ultimate saturation of VFR were proposed, at which time the flow coefficient was only associated with the spool structure and the valve opening (X. Zhang et al., 2020). The approximating functions of the flow coefficient varying with the valve opening were obtained by curve-fitting method (E. Lisowski & Filo, 2017; Valdés et al., 2014). However, the function is only suitable for specific spool structures. The relation between the flow coefficient of different kinds of spools and the opening is significantly complicated and is hard to find a simple function to fit (Y. Ye et al., 2014). Succinctly, the capacity to precisely predict the flow coefficient of the MWSV is a prerequisite for consideration of the valve FCCs, and this is also where the challenge exists.

The FCCs of the MWSV essentially depend upon the shape of notches. The shape of notches is diverse, such as U-shape, K-shape, Spheroid-shape, and so on (Lu et al., 2022). Notches with diverse throttling grooves provide the advantages of a wide range of flow rates, favourable stable in the condition of low flow rates, and enriched FCCs. Therefore, notches are broadly applied to highly efficient hydraulic valves where the finest accuracy and stable are needed. Many professors had developed numbers of innovative investigations to discuss the relation between the notch shape and the flow characteristics and flow phenomena using CFD simulations and experiments (Lu et al., 2022; Y. Ye et al., 2014; X. Zhang et al., 2020). Moreover, the spool structure parameters of the hydraulic valve such as the butterfly valve (Corbera et al., 2016), subsea gate valve (Liu et al., 2019), and throttling valve in managed pressure drilling (Z. Zhang et al., 2023) were optimised to obtain better working performances. The impact of KSNs on flow characteristics and notch design and optimisation, however, have received limited attention. In addition, the characteristics of FCCs and Pressure drop characteristics (PDCs) are seldom taken into account simultaneously.

As the structure parameters of the MWSV spool are powerfully coupled, the FCCs and PDCs of the MWSV are highly complex and collide with each other, which belongs to a MOO problem. The conventional experiment or analysis methods for optimisation is often expensive and ineffective. Many researchers have started to use CFD and computational optimisation to solve the problem in recent years, and have presented significant advantages, for example higher precision and effectiveness of flow details, which lend a great foundation for the comprehensive research of valves. Especially, several meta-heuristic algorithms (MHAs) were triumphantly applied to solve MOO questions (S. Li et al., 2020). Through a group of intelligent policies, MHAs enhance the efficiency of heuristic procedures. Extensively applied MOO algorithms encompass multi-objective particle swarm optimisation (MOPSO) (Hong et al., 2022; Z. Sun et al., 2020), multi-objective differential evolution (MODE) (Y. Wang et al., 2023; Xue & Wang, 2023; Yu et al., 2022), and multi-objective genetic algorithm (MOGA) (Fan et al., 2022; Jiang et al., 2018; M. Yang et al., 2023). These algorithms always depend on swarm-intelligence.

However, for MWSVs, which possess highly complicated flow paths and multi-parameter notch features, directed modelling needs considerable computational sources and time, which results in great difficulties to the analysis and optimisation design in the actual engineering application. In order to enhance the efficiency and credibility of the system, there exists a pressing demand to explore approaches for the effective and high-accuracy modelling of valves and programmes for multi-parameter-based optimisation design.

For further recognise previous researches dealing with the FCCs and PDCs of hydraulic spool valves, the literature survey was investigated. From these researches, the next three points can be consolidated. (1) Most of these studies only investigate the flow characteristics and flow phenomena of simple notches such as U, V, U-U and so on, whereas few studies have taken the impacts of KSNs on flow characteristics into consideration. Moreover, the immediate design methods for the valve seldom consider both PDCs and FCCs. (2) As the key indicator of the FCCs of a hydraulic spool valve, flow coefficient is comprehensively determined by valve opening, VFR, and spool structure parameters. The capacity to precisely predict the flow coefficient considering these three factors comprehensively is the basis for investigation of the valve FCCs. But few studies specially against this field have been published. (3) Finally, CFD is a strong method to handle the flow field characteristics of MWSVs; but constraints in computing sources and time mean that utilising significantly repeat CFD modelling is inefficient. Few studies have specifically paid attention to this kind of CFD model simplification.

In the present research, both numerical simulation analysis and surrogate model-based optimisation design are carried out on a MWSV with KSNs. For FCCs and PDCs analysis, the RSM-DAPM model combined parametric CFD simulation under the saturated satisfaction flow is developed, where the mapping relationship between the notch structure parameters and the valve opening, VFR, and PD is obtained. This method realises the reliable prediction of flow-pressure characteristics and provides a reference for the model establishment of valves with parallel notch structures. Therefore, to minimise the energy consumption combining the flow control performance, a surrogate-model based optimisation is performed on the KSN structures.

In the following sections, we firstly establish the physical and mathematical model of the MWSV in Section 2. Section 3 conducts CFD analysis and experimental verification; In Section 4, the DAPM-RSM is modelled, the PD characteristic is analysed, and the KSN structure parameters are optimised. In addition, the flow field distribution (pressure, velocity, turbulent kinetic energy (TKE)) of the notch valve port before and after optimisation is compared.

2. Methodology

2.1. Physical model

The loader is a kind of self-propelled CM involving multiple technical fields. It is mainly used for loading and unloading piles of bulk materials, and can also handle a certain strength of shovel excavation and ground traction work. The loader is driven by the movement of the work unit to achieve the function of shovelling, loading, and unloading materials. The work unit of the loader is shown in Figure 1(a). The working device is primarily composed of a bucket, boom, rocker arm, boom cylinder, and bucket cylinder. Both ends of the boom are respectively connected with the bucket and the frame. The boom cylinder drives the boom up and down to lift the bucket. The bucket is used to scoop and load materials. The bucket cylinder makes the bucket turn through the connecting rod mechanism.

Figure 1. Loader work unit and BSV working principle: (a) Loader work unit; (b) Hydraulic control system; (c) Working principle.

The MWSV is a combined reversing valve which is constituted of more than two directional valves and other valves such as safety valves and check valves. Due to its advantages of compact structure and multi-position commutation, the MWSV is often installed in the front frame of the CM to manipulate multiple actuator movements.

As the key component of the MWSV hydraulic system, the bucket coupling of a loader MWSV is taken as the research object. Figure 1(b) shows its hydraulic control system. Component 1 is the bucket spool valve (BSV). The BSV is driven by the pilot pressure a and b to adjust its spool position. Then the functions of flow adjustment and commutation are achieved. Component 2 is a check valve, mainly used to prevent oil backflow caused by load pulsation, so as to avoid the bucket ‘nodding’. Component 3 represents the main safety valve to protect the main oil circuit from overpressure. Components 4 and 5 are overload valves to prevent chambers A and B of bucket cylinder 7 from overloading. Component 6 is the oil refill valve, which functions as oil refilling when the bucket is uploading.

The working principle of the BSV could be referred to as Figure 1(c). The BSV is a six-way three-position (6/3) valve, which possesses three positions: bucket uploading, neutral position, and bucket receiving. When working, the BSV spool is driven to move around by manipulating the handle. The specific working principle is described as follows:

• When there is no pilot pressure at both ends of the BSV spool, the spool maintains the neutral position. Then the $\text{P}\to {\text{T}}_{\text{1}},{\text{T}}_{\text{2}}$ manifold is created. At this point, the bucket cylinder remains motionless.

• While manipulating the handle to control the bucket upload, the BSV spool is driven left by the pilot pressure. Then the $\text{P}\to \text{B}$ and $\text{A}\to \text{T}$ manifolds are established. At this time, the bucket cylinder retracts.

• As manipulating the handle to control the bucket receive, the BSV spool is motivated right under the pilot pressure. The $\text{P}\to \text{A}$ and $\text{B}\to \text{T}$ manifolds are developed. Accordingly, the bucket cylinder extends.

For the current manifold (Figure 1(c)), $\text{P}\to \text{A}$ or $\text{P}\to \text{B}$ manifold is termed no-notch flow-out status, while $\text{A}\to \text{T}$ or $\text{B}\to \text{T}$ manifold is declared as KSN flow-in status. The KSN flow-in state is usually complex. Since the notch structure of the $\text{A}\to \text{T}$ or $\text{B}\to \text{T}$ manifold is similar, this paper only concentrates on the flow area, discharge area, pressure drop characteristics, flow field characteristics, and structure optimisation of the $\text{B}\to \text{T}$ manifold KSN valve port.

Figure 2(a) shows a BSV spool prototype with various types of KSNs. The KSN is made by cutting the shoulder with a rectangular milling cutter along a circular arc of R (Figure 2(b)). The KSN spool valve has a good flow area gradient and FCCs, which effectively improves the speed control, micro-action, and response performance of the actuator. Although KSN spools have so many advantages, the design of notch structures in engineering practice is often based on experience and trial-and-error, and rarely takes into account energy consumption characteristics. Consequently, it is of certain engineering significance to develop the spool structure with low energy consumption and high energy efficiency.

Figure 2. BSV spool prototype and KSN structure: (a) BSV spool prototype; (b) KSN structure.

2.2. Mathematic model of KSN valves

2.2.1. Flow area of KSN valve port

Ji et al. (2003) proposed a principle of calculating the flow area of gradually expanding notches. The flow area $A_{\text{k}}$ of the KSN is shown in Figure 3(a). The specific formula is derived as Equations (1)–(3), (1) $\begin{array}{rl}L& =\sqrt{R^2-{\left(R-D\right)}^2}\end{array}$(1) (2) $\begin{array}{rl}k& =R-\sqrt{{\left(R-D\right)}^2+{\left(L-x\right)}^2}\end{array}$(2) (3) $\begin{array}{rl}{A}_{\text{k}}& =n\cdot k\cdot W\end{array}$(3)

Figure 3. Flow area and KSN parameters: (a) Flow area; (b) KSN parameters.

Additionally, the KSN flow area could be characterised as Equation (4), (4) $A_{\text{k}}\left(x,R\text{,}W,D,n\right)$(4) where x is the valve opening, $R$ represents the radius of the notch, W denotes the width of the notch, $D$ is the depth of the notch, and n indicates the number of notches on the shoulder.

2.2.2. Ultimate saturation flow coefficient

The flow-pressure drop characteristic curve can be approximated by a quadratic function (Valdés et al., 2014). The fitting relationship between VFR Q and PD $\mathrm{\Delta }p$ is expressed as Equation (5), (5) $\mathrm{\Delta }p=a{Q}^2+bQ$(5) where a and b are the common factors obtained from the function fitting.

In the operating phase, the fluid through the notch is described using Equation (6), (6) $Q=C_{\text{d}}A\sqrt{\frac{2\mathrm{\Delta }p}{\rho }}$(6) where $C_{\text{d}}$ denotes the flow coefficient, A represents the overflow area, and ρ is the working fluid density.

Further, the ratio of the $\mathrm{\Delta }p$ to $Q^2$ is given in Equation (7), (7) $\frac{\mathrm{\Delta }p}{Q^2}=\frac{\rho }{2{C_{\text{d}}}^2{A}^2}$(7) From Equation (5), it follows that, (8) $\frac{\mathrm{\Delta }p}{Q^2}=a+\frac{b}{Q}$(8) Further, by combining Equations (7) and (8), we can get, (9) $C_d=\frac{1}{A}\sqrt{\frac{\rho }{2\left(a+b/Q\right)}}$(9) When the Q continues to increase to infinity, the $C_{\text{d}}$ will increasingly approach a steady value, namely the ultimate saturated flow coefficient $C_{\text{dus}}$, (10) $C_{\text{dus}}=\underset{Q\to \mathrm{\infty }}{lim}{C}_{\text{d}}=\frac{1}{A}\sqrt{\frac{\rho }{2a}}$(10) Substituting $C_{\text{dus}}$ into Equation (9), we can get, (11) $C_{\text{d}}=C_{\text{dus}}\sqrt{\frac{1}{1+b/\left(aQ\right)}}$(11) Equation (11) shows the association between $C_{\text{d}}$ and Q. However, Q can only describe the volume change of fluid within a unit of time, and cannot reflect the fluid flow status. Therefore, it is challenging for Equation (11) to interpret the variation mechanism of the flow coefficient with fluid boundary conditions. Consequently, Reynolds number Re, which represents the fluid flow state, is introduced. Reynolds number can be formulated as, (12) $Re=\frac{d_{\text{H}}\nu }{\upsilon }$(12) where $d_{\text{H}}$ denotes the hydraulic diameter, $d_{\text{H}}=4A/\chi$, $\chi$ represents the perimeter length of A; $\upsilon$ shows the average flow velocity, $\upsilon =Q/A$; $\nu$ is the kinematic viscosity of the fluid, $\nu =\mu /\rho$, and μ denotes the dynamic viscosity of the fluid.

Reynolds number represents the ratio of the inertia force to the viscous force. If the Reynolds number is large, the influence of inertial force to the flow field is greater than the viscous force, and the fluid flow is less stable. Figure 4 presents the flow coefficient achieved from the check valve CFD simulations (Valdés et al., 2014). A fast rise could be investigated at low Reynolds numbers, then the progressive increase towards the steady value classical of high Reynolds number flow. It is the regular tendency that could be found in the literature (X. Zhang et al., 2020): a linear growth with Reynolds numbers for purely viscous flow, and a progressive increase in the transient condition towards the steady value typical of completely turbulent status.

Figure 4. Flow coefficient varies with Reynolds number for a ball check valve.

Further, Equation (12) can be written as Reynolds number and volume flow, (13) $Q=\frac{\mu \chi }{4\rho }Re$(13) By substituting Equation (13) into Equation (11), we can get, (14) $C_{\text{d}}=C_{\text{dus}}\sqrt{\frac{1}{1+4\rho b/\left(a\mu \chi Re\right)}}$(14) Let $4\rho b/a\mu \chi =k_{\text{us}}$, then Equation (14) can be written as, (15) $C_{\text{d}}=C_{\text{dus}}\sqrt{\frac{Re}{Re+k_{\text{us}}}}$(15) From Equations (10), (12), and, (14), the flow coefficient $C_{\text{d}}$ is a function of notch geometry, flow condition, fitting coefficients and its stable value. Substitute the values of A, a, b, $\mu$, $\rho$, $\chi$ into Equations (15), plots of the dependences of the $C_{\text{d}}$ on the Re can be obtained for different notches at the feature openings. The association between the C_d and Re can be expressed by Equation (15). From Equation (6), the flow coefficient C_d represents one of the main parameters of the flow calculation block, along with the flow area and the fluid density. The Re characterises the fluid flow condition. Furthermore Equation (15) describe the relationship between the orifice throttling characteristics and the fluid flow status.

The ultimate saturation number $k_{\text{us}}$ measures the closeness of $C_{\text{d}}$ to $C_{\text{dus}}$. When the notch structure is definite, the ultimate saturation number k_us is a constant value. In terms of unsaturated flow conditions, the VFR is small, and the Re is low accordingly. From Equation (15), the $C_{\text{d}}$ is more correlated with Re. Under saturated flow conditions, the Re is larger, thus $Re\gg k_{\text{us}}$. At this time the $C_{\text{d}}$ tends to be close to $C_{\text{dus}}$ and less correlated with Re.

2.2.3. Discharge area of KSN valve port

In the operating phase, the throttling equation of the MWSV port is shown as Equation (16), (16) $Q_{\text{k}}=C_{\text{d}}{A}_{\text{k}}\sqrt{\frac{2\mathrm{\Delta }p}{\rho }}$(16) where $Q_{\text{k}}$ denotes the VFR through KSNs, $C_{\text{d}}$ represents the flow coefficient, $A_{\text{k}}$ indicates the flow area of KSNs; $\mathrm{\Delta }p$ denotes the PD; ρ is the working fluid density.

$A_{\text{k}}$ is an inherent structural parameter that varies with the valve opening. $C_{\text{d}}$ is a comprehensive index of throttling characteristics, which is jointly dependent on the VFR Q, the valve opening x, and the notch structure. Under saturated flow conditions, the effect of Q can be ignored. $C_{\text{d}}$ is only related to the spool structure and the valve opening. The flow coefficient distribution with the opening of three kinds of notches is shown in Figure 5 (Y. Ye et al., 2014). Take the triangle-shape notch as an example. As x<0.8 mm, $C_{\text{d}}$ rises rapidly from 0.64 to 0.71 because of flow clogging under the status of minor flow area and large PD. Conversely, it remains declining when x > 1 mm. This phenomenon is chiefly due to the generation of cavitation in some low-pressure zones of the orifice cross-section, which minimises the real flow area. The effect is more obvious when the valve opening grows under a fixed PD which gives rise to a consistent reduction of $C_{\text{d}}$.

Figure 5. Flow coefficient distribution with the valve opening.

Three notches could be applied to different hydraulic systems depending on their relevant flow characteristics. At small openings, $C_{\text{d}}$ of the spheroid-shape notch is good, which does well in rapidly developing the system working pressure. $C_{\text{d}}$ of the triangle-shape notch decreases softly with the valve opening after its original growth, which is valuable to the flow fine tuning and stable running. Regarding the divergent U-shape notch, the $C_{\text{d}}$ is large as the opening is minor and followed by a speedy droop, which is good at preventing the vibration from happening during start-up step.

During engineering practice, the accuracy requirement is sometimes not very high. The relation between C_d and the VFR, the valve opening, the notch structure is often complicated. Consequently, sometimes the dependency on Re is overlooked and a fixed C_d is used to calculate the VFR in valves and other hydraulic components for low Reynolds number flows and even lightly compressible flows (Valdés et al., 2014). From Figure 5, different kinds of notches under different valve openings have various C_d values. Taking the spheroid-shape notch as an example, The C_d for the valve opening of 2.8 mm is 26.7% smaller than the valve opening of 0.4 mm. As a consequence, the method of using a fixed value for C_d lacks reliability and shows poor accuracy.

The C_d distributions with the valve opening of three different kinds of KSNs are shown in Figure 6, the C_d significantly varies with the valve opening x. Meanwhile, it is difficult to find a simple function to fit the relationship between the C_d and the valve opening x. However, the curve of the C_d × A_k varying with the x has the characteristics of smooth and continuous change, and the relationship between the C_d × A_k and the x can be fitted by a simple function (see Figure 7). To reflect the flow control performance of the KSN valve more specific and comprehensive, the ‘Discharge area $C_{\text{A}}$’ is introduced as the parameter to characterise the FCCs. According to Equation (16), the discharge area can be considered as the intermediate variable to reflect the relation of the VFR and PD. Under the fixed pressure drop, the KSN with higher discharge area could passes more flow. The discharge area is defined as Equation (17), (17) $C_{\text{A}}=C_{\text{d}}{A}_{\text{k}}$(17) Assuming that the number of KSNs on the shoulder is n, the flow rates through notches are $Q_{\text{k1}}$, $Q_{\text{k2}}$, … , $Q_{\text{k}n}$, the flow coefficient of each notch is $C_{\text{d1}}$, $C_{\text{d2}}$, … , $C_{\text{d}n}$, the flow area of each notch is $A_{\text{k}1}$, $A_{\text{k2}}$, … , $A_{\text{k}n}$, the discharge area of each notch is $C_{\text{d1}}{A}_{\text{k}1}$, $C_{\text{d2}}{A}_{\text{k2}}$, … , $C_{\text{d}n}{A}_{\text{k}n}$. From Equation (16), it is obtained as Equation (18), (18) $\{\begin{array}{l}{Q}_{\text{1}}=C_{\text{d1}}{A}_{\text{k1}}\sqrt{{\displaystyle \frac{2\mathrm{\Delta }p}{\rho }}}\\ Q_2=C_{\text{d2}}{A}_{\text{k2}}\sqrt{{\displaystyle \frac{2\mathrm{\Delta }p}{\rho }}}\\ ⋮\\ Q_n=C_{\text{d}n}{A}_{\text{k}n}\sqrt{{\displaystyle \frac{2\mathrm{\Delta }p}{\rho }}}\end{array}$(18) From the principle of parallel connection of liquid resistance, we know that $Q_{\text{k}}=Q_{\text{k1}}+Q_{\text{k2}}+\cdots +Q_{\text{k}n}$. Thus, the entire discharge area $C_{\text{d}}{A}_{\text{k}}$ can be linearly superimposed by individual KSN, which is shown in Equation (19), (19) $\begin{array}{rl}{C}_{\text{d}}{A}_{\text{k}}& =C_{\text{d1}}{A}_{\text{k1}}+C_{\text{d2}}{A}_{\text{k2}}+\cdots +C_{\text{d}n}{A}_{\text{k}n}\\ & =\sum _{i=1}^n{C}_{\text{d}i}{A}_{\text{k}i}\end{array}$(19) If the mathematical model of each notch discharge area $C_{\text{d}i}{A}_{\text{k}i}$ regarding notch structure is established, the entire discharge area $C_{\text{d}}{A}_{\text{k}}$ mathematical model could be calculated by Equation (19). Then according to Equation (16), the mapping relationship of VFR Q with PD $\mathrm{\Delta }p$ can be obtained by regarding the discharge area $C_{\text{d}}{A}_{\text{k}}$ as an intermediate variable.

Figure 6. KSN flow coefficient distribution with the valve opening.

Figure 7. KSN C_d × A_k distribution with the valve opening.

2.3. Mathematic model of computational optimisation methods

2.3.1. Response surface methodology

RSM was described and evolved by Box and Wilson (Shanmugam & Sirisha Maganti, 2023), which utilises a powerful statistical method using the least square method that fits the values obtained from experiment or simulation to the quadratic polynomial model (Park et al., 2020). RSM is an aggregation of statistical and mathematical techniques used to develop, improve, and optimise processes (T. Sun et al., 2021). Through a mathematical approach and statistical method, RSM is often used to seek the association between design variables ($x_1,x_2,\dots x_k$) and objective function G using an establishing regression model. The connection between $x_1,x_2,\dots x_k$ and G can be written as, (20) $G=f\left(x_1,x_2,\dots x_k\right)+ϵ$(20) where $ϵ$ is residual error among the approximate value f and error value. The nonlinear association between $x_1,x_2,\dots x_k$ and G can be characterised using quadratic polynomial equations. The nonlinear association is also named as approximate function. The quadratic polynomial terms mainly encompass linear, square and interaction terms, which is formulated as follows, (21) $G=a_0+\sum _{i=1}^k{a}_i{x}_i+\sum _{i=1}^k{a}_{ii}{x_i}^2+{\sum }_{i<j}{a}_{ij}{x}_i{x}_j+ϵ$(21) where $a_0$ is the intercept coefficient; $a_i$ denotes the line impact of $x_i$; $a_{ii}$ represents the quadratic impact of ${x_i}^2$; $a_{ij}$ shows the linear association impact between $x_i$ and $x_j$.

2.3.2. Sensitivity analysis

In order to appreciate each variable value inputted on the PD and FARD, sensitivity analysis (SA) is used (H. Wang et al., 2023). Suppose that $\hat{y}$ denotes the PD value anticipated by RSM for a data set (x), and $x_a$ shows an input variable with L levels. Global sensitivity analysis (GSA) transforms all variables inputted considering input and output space simultaneously. GSA delivers a holistic vision on the effect of inputs on outputs while the SA provides a local view. Accordingly, GSA was utilised in this study to evaluate the input variable sensitivity to PD and FARD. Firstly, the respective sensitivity response is calculated for each variable inputted by the developed RSM. Then the sensitivity measure should be used to balance the output-produced changes. The sensitivity measure mainly includes range, gradient, variance, average absolute deviation and so on. Nearly all sensitivity measures perfectly capture the theoretical importance values (Cortez & Embrechts, 2013). Especially, the gradient metric measure performs well to understand the importance of the inputs on the output variables in many researches (Huang et al., 2020; J. Zhang et al., 2021). Consequently, the gradient metric is used to balance the output-produced changes, (22) $g_{\text{a}}=\sum _{j=2}^L|{\hat{y}}_{\text{a},j}-{\hat{y}}_{\text{a},j-1}|/\left(L-1\right)$(22) where ${\hat{y}}_{\text{a},j}$ denotes the corresponding sensitivity response for the jth level inputted. Furthermore, the relative value $R_{\text{a}}$ is written by, (23) $R_{\text{a}}=g_{\text{a}}/\sum _{j=2}^M{g}_i$(23) where $g_{\text{a}}$ indicates the sensitivity measurement for $x_{\text{a}}$ and M is the input variable number.

2.3.3. Multi-objective optimisation

1. Decision variables

Figure 8 is a schematic diagram of the bucket valve spool notch structure. In order to minimise the influence of radial imbalance force on the spool notch, notches are usually symmetrically distributed in the circumferential direction on the spool. Consequently, the shoulder of the BSV spool has two kinds of symmetrical KSNs I and II. $D_{\text{1,2}}$ are the depths of KSNs I and II; $R_{\text{1,2}}$ are the radiuses of KSNs I and II; $W_{\text{1,2}}$ are the widths of KSNs I and II; $L_{\text{1,2}}$ are the lengths of KSNs I and II.

Figure 8. BSV spool notch structure parameters.

Among the above eight design variables, the valve opening x is the difference of the spool displacement x_d and the oil sealing lengths x_{01, 2}. According the relative position of spool and valve body, the sum of the spool lengths L_{1, 2} and the oil sealing lengths x_{01, 2} is a constant value. That is, the changes of L_{1, 2} will directly affect the lengths of x_{01, 2}. Furthermore, the relationship between x and x_d will change accordingly. Therefore, $L_{\text{1}}$ and $L_{\text{2}}$ are fixed to the original value, and the remaining six variables are used as decision variables. The range of the decision variables is shown in Table 1.

1. Objective functions

Table 1. Range of decision variables.

Design variable	D1	D2	R1	R2	W1	W2
Lower limit /mm	7.6	7.6	8	8	6.4	6.4
Upper limit /mm	11.4	11.4	12	12	9.6	9.6
Original value /mm	9.5	9.5	10	10	8	8

The destination of optimising the spool notch structures is to reduce throttle losses while balancing control capability. The flow area of the notch valve port reflects the FCCs of the MWSV. Since this paper uses the original flow area characteristics as the basis for optimisation, it is expected that the optimised notch valve port has a similar flow area. That is, the sum of the flow area error square is minor before and after optimisation, which is correspond to the mind of the least square method. The least square method is a mathematical tool extensively applied in the field of data processing such as error estimation, uncertainty, system identification and prediction, and its basic principle is to find the best function match of data by minimising the sum of errors squares, which has the advantages of simple operation and high efficiency of problem optimisation. Relied on the least square method mind, the FARD of the optimised and original spool structure is given by Equation (24). The FARD g₁ shows the degree of flow area deviation between the optimised and original notch spool. The smaller the g₁ is, the more similar the flow area before and after optimisation is. (24) $g_{\text{1}}=\frac{1}{n}\sqrt{\sum _{i=1}^n{\left(\frac{{A^{\mathrm{\prime }}}_{\text{k}1i}-A_{\text{k}1i}}{A_{\text{k}1i}}\right)}^2}$(24) where $A_{\text{k}1i}$ and $A_{\text{k}1i}^{\mathrm{\prime }}$ are the flow areas at the corresponding valve port opening before and after optimisation; n is the number of valve port opening, n = 10.

The boundary conditions are set to velocity inlet and pressure outlet, and the inlet VFR of the spool valve is 250 L/min while the outlet pressure is a constant value of 0.1 MPa. The inlet pressure reflects the change of the PD, if the PD is too large, the energy dissipation is serious. Therefore, it is hoped that the optimised notch valve will have a low inlet pressure so that energy saving can be achieved while taking into account the control performance. The PD relative superposition average is shown in Equation (25). The PD relative superposition average g₂ indicates the degree of pressure deviation between the inlet and outlet under various valve openings. A lower g₂ corresponds to a lower PD. Consequently, g₂ can be used as an evaluation index of parameter optimisation. (25) $g_2=\frac{1}{n}\sqrt{\sum _{i=1}^n{\left(\frac{p_{\text{1},i}-p_0}{p_0}\right)}^2}$(25) where $p_{\text{1},i}$ is the inlet pressure at the corresponding valve opening, and $p_0$ is the outlet pressure, $p_0$ = 0.1 MPa; n is the number of valve port opening, n = 10.

In conclusion, the FARD g₁ and PD relative superposition average g₂ were regarded as the optimisation objectives to reflect the characteristics of flow control and energy consumption.

1. Range constraints

The decision variables are assumed to extend within the disassembled dataset in the MWSV optimisation problem, which is formulated as, (26) $\{\begin{array}{l}{D}_{imin}\le D_i\le D_{imax}\\ R_{imin}\le R_i\le R_{imax}\\ W_{imin}\le W_i\le W_{imax}\\ D_i<R_i\end{array}$(26) where $D_i$ is the ith decision variable with the minimum value $D_{imin}$ and maximum value $D_{imax}$; $R_i$ is the ith decision variable with the minimum value $R_{imin}$ and maximum value $R_{imax}$;$W_i$ is the ith decision variable with the minimum value $W_{imin}$ and maximum value $W_{imax}$.

1. MOO model of the MWSV optimisation

With the g₁ and g₂ as the objective function and the KSN structure parameters $D_{\text{1,2}}$, $R_{\text{1,2}}$, and $W_{\text{1,2}}$ as the decision variables, the mathematical model for MOO can be described as Equation (27), (27) $\begin{array}{rl}& \text{find}x=[D_{1,2},R_{1,2},W_{1,2}{]}^T\\ & \{\begin{array}{l}min\left\{{\text{g}}_1\left(D_{1,2},R_{1,2},W_{1,2}\right),{\text{g}}_2\left(D_{1,2},R_{1,2},W_{1,2}\right)\right\}\\ \begin{array}{ll}\text{s}\text{.t}\text{.}& D_{imin}\le D_i\le D_{imax}\\ & R_{imin}\le R_i\le R_{imax}\\ & W_{imin}\le W_i\le W_{imax}\\ & D_i<R_i\end{array}\end{array}\end{array}$(27)

2.3.4. NSGA-II and Pareto theory

The NSGA-II based on the Pareto theory solves MOO problems efficiently (Afshari et al., 2019). Its workflow and corresponding description are shown in Figure 9(a). Pareto theory declares the relation between two solutions, utilising domination as the fundamental notion (Dong et al., 2022).

1. Initialise a primary population.

2. The scale of the primary parent population is stepped up through mutation and crossover.

3. The solutions of the generated group are classified into various ranks in accordance with the Pareto dominating regulation. The Pareto set is created (Figure 9(b)), subsequently being segregated to seek the following rank of the Pareto set till solutions are all sequenced.

4. The crowding distance (Manhattan distance in the objective space between the proximate double solutions sharing a similar Pareto rank) of the solution is evaluated one by one. A great crowding distance indicates a great space for group biodiversity.

5. The solution (greater Pareto rank and crowding distance) is selected to generate the parent group of offspring. NSGA-II increases the optional pressure using a mating optional strategic of binary contention.

6. The algorithm will circulate from step II to V till the maximum number of circulations is achieved.

Figure 9. NSGA-II algorithm mechanism: (a) One iteration procedure; (b) Pareto theory-based genetic evolution. (Dong et al., 2022).

2.3.5. Decision-making method

For the sake of seeking the final optimal percentage in the MOO question, a multi-criteria decision analysis method is necessary. The technique for order preference by similarity to the ideal solution (TOPSIS) (Wang & Xie, 2023) is extensively applied. In the TOPSIS approach, a solution farthest from the negative ideal point and nearest the positive ideal point is chosen. The positive ($d_{i+}$) and negative ($d_{i-}$) ideal points are given by, (28) $\begin{array}{rl}{d}_{i+}& =\sqrt{{\sum }_{j=1}^n{\left(F_{ij}-F_j^{\text{ideal}}\right)}^2}\end{array}$(28) (29) $\begin{array}{rl}{d}_{i-}& =\sqrt{{\sum }_{j=1}^n{\left(F_{ij}-F_j^{\text{non - ideal}}\right)}^2}\end{array}$(29) where $d_{i-}$ and $d_{i+}$ are the distances to the negative and positive ideal points, respectively; n denotes the number of objectives; i is a solution point in the Pareto front set; and the closeness coefficient is given by the following equation, (30) $C_i=\frac{d_{i-}}{d_{i-}+d_{i+}}$(30) The Pareto solution with the highest $C_i$ is chosen as the final optimal solution. Finally, the $C_i$ is normalised, and the final score $y_i$ is taken between 0 and 1, and the solution with a score of 1 is the optimal solution, as shown in the following equation: (31) $y_i=\frac{C_i-C_{min}}{C_{max}-C_{min}}$(31)

3. Simulation and experimental setup

3.1. Simulation

Fluent is used for numerical simulation of the complicated flow of high-pressure turbulent flow. Models of different notch structure and spool displacements, repeated mesh generations, and duplicate Fluent simulation parameter settings are necessary. Using the parametric simulation approach, key simulation characteristics can be defined as parameters. Then, parameters can be managed under the parameter set interface in Workbench, and quickly change the geometry and topology parameters, mesh parameters, boundary conditions and other settings of the simulation model through parameter drive. By the way of improving the efficiency of simulation calculation and data acquisition, the parametric simulation approach is applied to simulate the flow field of the BSV.

3.1.1. Solution strategy

The Reynolds time-averaging method is applied to simplify the control equations to reduce the computational difficulty and resource consumption. Meanwhile, the Realizable k-ε model is introduced to supplement the Reynolds stress term. The pressure-based solution algorithms consist of two categories. One type is separated solution algorithms, including SIMPLE, SIMPLEC, and so on (Lu et al., 2022). The separated solution algorithm is mainly applied to micro-pressure flow. The other type is coupled solution algorithm, for example, Coupled algorithm. The coupled solution algorithm is suitable for high-speed non-pressure flow. Usually, the coupled solution algorithm is more accurate in the calculation of velocity and pressure values. As a consequence, the Coupled algorithm is chosen as the solver.

In order to decline the effect of irrelevant factors on the simulation, the following assumptions are made. First, the fluid medium is an incompressible Newtonian fluid. Second, the internal flow field of the valve is well hermetic and there is no leakage. Finally, the effect of temperature on the internal flow field is overlooked. The mass flow rate is obtained by converting the VFR and density. Thus, the inlet is set as mass flow rate inlet while the outlet is expressed as pressure outlet, and the remaining boundaries are non-slip stationary walls. The detailed simulation parameters are shown in Table 2.

Table 2. Simulation parameters.

Solution methods
Solver	Coupled algorithm
Spatial Discretization of Gradient	Least Squares Cell Based
Spatial Discretization of Pressure	PRESTO!
Spatial Discretization of Momentum	Second order upwind
Spatial Discretization of Turbulent Kinetic Energy	Second order upwind
Spatial Discretization of Turbulent Dissipation Rate	Second order upwind
Monitor residuals
Continuity	1.00E-06
X-velocity	1.00E-06
Y-velocity	1.00E-06
Z-velocity	1.00E-06
Turbulence	1.00E-06
Calculation activities
Number of iterations	1000
Boundary conditions
Mass flow rate inlet min	0.741–6.668 kg/s
Pressure outlet pout	0.1 MPa
Flow field parameters
Liquid oil density ρ	889 kg/m^3
Liquid oil viscosity μ	0.045 kg/ (m · s)

3.1.2. Grid generation and independence verification

The 3D model of the MWSV is imported into ANSYS SpaceClaim 2021R1 for simplification. The flow path $\text{B}\to \text{T}$ of BSV is extracted and the fluid domain is partitioned. Due to the large variation of velocity and pressure gradient at the valve port, a Body of Influence (BOI) is set to greater emulate the flow state. Accordingly, the BOI is encrypted in the mesh module.

The region of large pressure and velocity gradients often appears at the valve port. Consequently, the fluid domain can be divided into three parts as shown in Figure 10. The region III represents the KSN valve port, which is the main domain of PD. The regions II and IV connect with the inlet and outlet of KSN, and there may also be PD when the PD of region III is severe. The rest regions I and V possess no PD basically. According to the degree of PD, the grid density in the region III should be greater than in regions II and IV than in regions I and V. Fluent Meshing 2021 R1 is applied to classify the unstructured grid of the flow path model to better fit the boundary. The poly-hex-core method based on the ‘mosaic’ technology can meet co-node connection of hexahedral grids to polyhedral grids without any additional manual meshing. Meanwhile, the mosaic technology also supports the division of boundary layer grids, so it can realise the state of calculating area filling for layered prismatic grids near the wall, pure polyhedral grids in the transition area, hexahedral grids in the core area. The method greatly improves the overall mesh quality and effectively reduces the overall mesh quantity and solution time.

Figure 10. Grid of flow domain.

Poly-hex-core is chosen as the body meshing method. The method follows an octopus meshing scheme using discretization of cell sizes in the core area. It results in multi-level isotropic cartesian hexahedral grids with cell sizes varying by a factor of two in each adjacent level. Consequently, the grid of region III is densest, the number of grids gradually decreases in other regions following the octopus meshing scheme.

The grid independence analysis is performed to prevent the impact of the grid quantity on the simulation results. The fluid domain with a valve opening of 2 mm is selected as the verification example. The inlet mass flow rate is set as 3.704 kg/s while the outlet pressure is 0.1 MPa. The PD is the reference quantity for calculation. The grid independence analysis is shown in Table 3. It can be found that the PD remains basically unchanged after Grid 2. It seems that the application of the Grid 2 condition is appropriate for the current simulation. The final determination of the grid number is 837,000 to assure the quality of the grid and the best allocation of computational resources. At this point, the maximum skewness value of the surface grid is 0.40, and the minimum orthogonal quality value of the body grid is 0.33, which indicates the grid quality is good.

Table 3. Grid independence analysis.

Grid type	Grid 1	Grid 2	Grid 3	Grid 4
Number of grids	497,000	837,000	1,653,000	3,269,000
Maximum Skewness (<0.95)	0.42	0.40	0.39	0.34
Minimum Orthogonal Quality (>0.05)	0.29	0.33	0.31	0.35
PD (MPa)	3.92	3.46	3.48	3.45

3.2. Experimental setup

The Fluent was used in Section 3.1 to get the PD data sample and flow field of the MVSV. To examine the accuracy of CFD simulation, a static performance test bench of the MWSV was set up. The test bench relied on the State Key Laboratory of Intelligent Manufacturing of Advanced Construction Machinery of China. Figure 11(a,b) shows the MWSV test bench and monitor.

Figure 11. Test bench for the static performance of MWSV: (a) Test bench; (b) Monitor; (c) Upper control system; (d) Data acquisition system; (e) Hydraulic power system.

The MWSV test bench is mainly composed of the upper control system, data acquisition system, and hydraulic power system. The upper control system is shown in Figure 11(c). A computer was visually applied to adjust the VFR, pilot pressure, and so on. The data acquisition system is shown in Figure 11(d). The pressure values of each port are measured by pressure gauges, the VFR is measured by the flow metre, and the displacement of the bucket valve spool is measured by the displacement sensor. The hydraulic power system is shown in Figure 11(e). The fluid is accessed through a pump from the tank with a filter, the relief valve restricts the inlet pressure to 20.5 MPa, and the outlet pressure is contained using a throttling valve in 0.1 MPa. The specific experimental parameters are presented in Table 4.

Table 4. Experimental parameters.

Test oil parameters
Liquid oil category	46^# anti-wear hydraulic oil
Liquid oil density ρL	889 kg/m^3
Liquid oil viscosity μL	0.045 kg/ (m · s)
Notch Structure parameters
Notch number n	4
Notch length L	6 mm
Notch depth D	9.5 mm
Notch width W	8 mm
Notch radius R	10 mm
Valve port opening x	0–11 mm
Flow field parameters
Inlet volume flow rate	250, 350, 450 L/min
Outlet pressure	0.1 MPa
Experimental condition
Main relief valve pressure	20.5 MPa
Maximum pressure of power system	25 MPa
Maximum flow of power system	365 L/min
Oil temperature	−20–100°C

The test bench converts the physical quantities of the system into voltage and current signals using components such as pressure sensors and flowmeters. The data collected is transmitted to the computer via the data cable to realise the real-time storage of the pressure and flow data in the test system. Taking the BSV as the test object, the specific steps of the PD test are as follows:

1. Hydraulic hoses are used with quick coupling to connect different oil ports of the BSV with the corresponding interface of the comprehensive test bench. Specifically, the oil inlet P is connected to the pump pressure port, the working oil ports A and B are connected to the loading port, the oil return port T is connected to the oil return port. The BSV pipeline connection is further checked safe and reliable, and ensure that there is no leakage in the connection joints everywhere.

2. The pressure sensor is installed at the corresponding port (P, T, A, B) position of the BSV by means of pressure measuring flanges.

3. During the test, according to the test circuit of $\text{P}\to \text{A}$ and $\text{B}\to \text{T}$, the bucket spool valve under test is reversed to the corresponding working position (0–15 mm). The back pressure of the return port of the multi-way valve is adjusted, the hydraulic pump is started, and the opening degree of the relief valve is adjusted to gradually increase the VFR from 0 L/min to 250, 350, and 450 L/min.

4. The data of each pressure sensor is collected by the computer, the pressure of the working port B and the oil return port T of the multi-way valve under test are recorded. Accordingly, the PD value is calculated.

Figure 12 expresses the comparison between the experimental and simulated data of PD characteristics, where Figure 12(a–c), respectively show the effects of inlet VFR. In this case, the experimental and simulated PDs mutate in a similar tendency with a deviation of no more than 15%, which indicates the appropriate credibility of the simulation model.

Figure 12. Comparison of simulation and experiment: (a) Q = 250 L/min; (b) Q = 350 L/min; (c) Q = 450 L/min.

4. Results and discussion

4.1. Pressure drop characteristics

4.1.1. Flow saturation verification

There are four same KSNs on the shoulder of the BSV connected to the manifold $\text{B}\to \text{T}$ (see Figure 2). The main structural parameters of the KSN are R = 10 mm, D = 9.5 mm, and W = 8 mm. The flow area A_k is computed by Equations (2)–(6). The curve of flow area A_k versus opening is shown in Figure 13. It can be recognised that the flow area grows linearly with the change of valve port opening.

Figure 13. Flow area characteristics.

From Equation (15), the relation between the flow coefficient and the Reynolds number is that, (32) $C_{\text{d}}=C_{\text{dus}}\sqrt{\frac{Re}{Re+k_{\text{us}}}}$(32) where $C_{\text{dus}}$ is the ultimate saturation flow coefficient; $k_{\text{us}}$ is the ultimate saturation number. Chen et al. (2019) proposed that when the spool displacement is maximum, the $k_{\text{us}}$ value is maximum and the Reynolds number required for the flow coefficient to reach the ultimate value is maximum.

According to the manifold $\text{B}\to \text{T}$ notch structure, CFD simulation models under the maximum orifice opening are established, and the simulation parameters are set in accordance with Section 3.1. Set a wide range of flow rate values to reflect variations in Reynolds number. The simulation results are curve-fitted using Equation (21), and the evaluation function is used to generate the fit parameters $C_{\text{dus}}$ and $k_{\text{us}}$ using the least squares method. Figure 14 shows the simulation results and the fitted curve of the flow coefficient when the valve port is fully open, where the coefficient of determination of the goodness of fit $R^2$ is 0.9893, and it can be found that the flow coefficient gradually reaches the limit value with the increase of Reynolds number. In engineering practice, the flow coefficient is often taken to the limit, and it is then necessary to determine under what conditions this approximation will not cause large errors.

Figure 14. Flow coefficient with the valve port is fully open.

Set $C_{\text{d}}=\eta C_{\text{dus}}$, $\eta \in \left(0,1\right)$, Substitute it into the Equation (32), (33) $Re=k_{\text{us}}{\eta }^2/\left(1-{\eta }^2\right)$(33) In this example, when the valve port is fully open, $k_{\text{us}}=159.03$, set η for 0.98, that is, the assumption that the flow coefficient reaches the limit value of 98% for the saturation flow state. At this time, through the Equations (13) and (33), Q = 74.36 L/min, the meaning is: when the 2% error is allowed, the flow coefficient could be taken to the limit value of $C_{\text{dus}}$ if the inlet VFR is greater than 74.36 L/min.

When the VFR of the KSN valve port meets the saturated flow condition, the influence of the flow rate on the flow coefficient can be ignored. Then it is considered that the flow coefficient is only relevant to the valve port opening and notch structure (X. Zhang et al., 2020). To further verify the VFR in this paper meets the flow saturation condition, the CFD simulation models are established. The inlet VFR is set to 80, 100, and 150 L/min respectively. Figure 15(a) shows the PD under different VFRs. According to Equation (16), the error rod histogram of the discharge area is expressed in Figure 15(b). In the diagram, the cylinder represents the average value of PDs under three VFRs while the error bar expresses the standard deviation.

Figure 15. Flow saturation verification: (a) Pressure drop of the valve port varies with valve opening; (b) Discharge area varies with valve opening.

With the change in the inlet volume flow (see Figure 15(a)), the changing trend of the PD is similar. The PD is greater when the valve port opening is small, and it is gradually constant with the increase of the opening. According to Figure 15(b), the discharge area of the valve port grows linearly with the change of valve port opening. The alteration of the inlet volume flow has little effect on the discharge area, that is, the flow saturation condition is satisfied when the VFR is greater than 80 L/min.

4.1.2. Discharge area parameter model

Section 4.1.1 establishes the model of ultimate saturation flow model. It is considered that the flow coefficient is only associated to the valve port opening and notch structure when the VFR is greater than 80 L/min. If the mathematical model of each notch discharge area $C_{\text{d}i}{A}_{\text{k}i}$ regarding notch structure is established, the entire discharge area $C_{\text{d}}{A}_{\text{k}}$ mathematical model could be calculated by Equation (19). Then according to Equation (16), the mapping relationship of VFR with PD can be obtained by regarding the discharge area $C_{\text{d}}{A}_{\text{k}}$ as an intermediate variable.

The depth $D$, radius $R$, and width W of the KSN are used as design variables to establish a three-dimensional space of $L=\left\{D,R,W\right\}$. It represents the set formed by such structures in different size combinations. Due to the sample points are unlimited in the overall design space, quantitative sample points are essential to be chosen to represent properties of the whole design space. To ensure the uniformity and continuity of sampling, the optimal latin hypercube design (OLHD) is applied to collect appropriate sample points from the variable space (S. Zhang et al., 2018). The value of each variable is taken as the real structure size, and it floats within the range of no more than 20% above and below. The range of values is shown in Table 5.

Table 5. Range of design variables.

Design Variable	D (mm)	R (mm)	W (mm)
Lower limit	7.6	8	6.4
Upper limit	11.4	12	9.6
Initial value	9.5	10	8

For the sake of the possibility of manufacturing and processing, the depth $D$ should be smaller than the radius R. Then each variable is sampled according to the OLHD and finally 59 sets of CFD simulation samples are obtained. The sample points are shown in Figure 16.

Figure 16. OLHD sampling points: (a) D vs W and R; (b) D vs W; (c) D vs R.

Table 6 expresses the statistical analysis of parameters. The correlation between input variables is described in a correlation matrix (Figure 17). It can be seen that the majority of the Pearson correlation coefficients between two various variables are no more than 0.5, indicating that input variables will not produce multicollinearity issues.

Table 6. Statistics of the dataset.

Variables	Min	Max	Mean	Standard deviation
D (mm)	7.6	11.4	8.9	0.98
R (mm)	8	12	10.5	1.11
W (mm)	6.4	9.6	8.1	0.94

Figure 17. Correction matrix of variables.

The sample points obtained from the experimental design are simulated. The inlet VFR is set to 100 L/min and the outlet pressure to 0.1 MPa. The PDs under different valve openings are calculated by Fluent, and further evaluating the corresponding discharge area. Figure 18 shows the discharge area and curve fitting results of six sampling points. Taking the notch $L=\left\{7.8,11.5,8.6\right\}$ as an example, the discharge area varies essentially linearly with the valve port opening. Two parameters are obtained by fitting: a = −0.184, b = 5.735, and the coefficient of determination of goodness of fit R² = 0.998 (the closer to 1, the higher the accuracy of the fit). Thus, the parameter model of the discharge area is transformed into the mapping relationship of the notch structure parameters on the curve fitting parameters a and b.

Figure 18. Discharge area and curve fitting.

Through fitting complicated equations for the design variables $L=\left\{D,R,W\right\}$ and parameters a and b, an approximate model is established. Thus, the engineering question is transferred into a mathematical question, which greatly improves the efficiency of the simulation. RSM could be applied to fit complicated relations and has proven to be efficient and accurate (Park et al., 2020; T. Sun et al., 2021). Accordingly, an approximate model is built by a second-order RSM. The process of the surrogate model based on RSM is shown in Figure 19. The minimum quantity of sample points required for the second-order RSM is 10, and 59 sample points created by OLHD satisfy the requirements. Discrete points of the notch structure parameters about a and b are obtained by 59 times curve fitting. Then the approximate model of the notch structure parameters about a and b is further established by RSM, which is expressed in Figure 20. The mapping relationship between the notch structure and the flow area curve is obtained. The error analysis is shown in Table 7, and all errors are within the allowed range.

Figure 19. Process of surrogate modelling.

Figure 20. RSM analysis: (a), (b), (c) Contour of D, R, and W interaction of a; (d), (e), (f) Contour of D, R, and W interaction of b.

Table 7. Error analysis.

Error term	a	b	Allowable range
Average error	0.0551	0.0168	<0.3
Root mean square	0.0709	0.0198	<0.2
R-squared	0.9788	0.9958	>0.9

To further verify the accuracy of the approximate model, three groups of structure parameters other than 59 groups of samples are selected, and the CFD simulation model is established according to the same parameter settings in Section 3.1. The performance of RSM is compared with simulation results by a Taylor diagram (Figure 21). The Taylor diagram utilises three indexes: Correlation coefficient (R), RMSE, and standard deviation (SD) to characterise the association between the predictive and CFD ones. The model owning greater R, and poor RMSE and SD is close to the CFD value, illustrating it has good prediction capability. It can be seen that the three models are all accessible to CFD values, demonstrating the high prediction performance of RSM.

Figure 21. Taylor diagram for comparison of three test models: (a) L = {10.5, 11.5, 7}; (b) L = {9.5, 11, 7.5}; (c) L = {8.5, 10.5, 8}.

4.1.3. Analysis of pressure drop characteristics

Section 4.1.2 establishes the mathematical model of the discharge area regarding the KSN structure. Then the mathematical model of the valve port flow area of the manifold $\text{B}\to \text{T}$ is obtained by linearly superimposes according to Equation (19). To verify the accuracy of the manifold $\text{B}\to \text{T}$ discharge area mathematical model, the corresponding flow path of the bucket spool valve is extracted and the CFD simulation model is established. Figure 22 shows the calculated and simulated values of the valve port discharge area of the bucket spool valve flow loop $\text{B}\to \text{T}$. The max deviation of the calculated (Cal.) and simulated (Sim.) value of the discharge area is 4.02%. From the Taylor diagram for comparison of DAPM (Figure 23), the Simulation value is situated close to the Calculation value. It suggested that the established mathematical model of the discharge area can better express the mapping relationship between the discharge area characteristics of the valve port and the structure of the KSN structure.

Figure 22. Discharge area comparison of predicted and simulated values.

Figure 23. Taylor diagram for comparison of DAPM.

Depending on the mathematical model of the valve port discharge area characteristics and the structure of KSNs, the PD at different inlet VFRs can be obtained from Equation (16). To establish the mapping relationship between the inlet VFR, the PD of the manifold $\text{B}\to \text{T}$ valve port, and the notch structure parameters. The inlet VFRs are 250, 350, and 450 L/min, and the PDs with valve opening are shown in Figure 24. It is found that when the valve port opening is small, the PD is extremely large and the energy dissipation is serious. With the addition of the valve opening, the PD decreases gradually with the trend of the exponential function.

Figure 24. Pressure drop with different inlet VFRs: (a) Q = 250 L/min; (b) Q = 350 L/min; (c) Q = 450 L/min.

4.2. Optimisation design

As mentioned above, the PD characteristics of conventional MWSVs are unsatisfactory. In this study, integrated with the superiors of the less time-consuming and low cost CFD simulation (Gao et al., 2016), the optimisation design of the bucket spool valve is carried out.

4.2.1. Sensitivity analysis

SA is an analytical method applied to study the effects of decision variables on objective functions (Jankovic et al., 2021). The OLHD is used to continuously and uniformly sample the decision variables. The range of the decision variables is shown in Table 1. Finally, 71 sets of eligible sample points are extracted. The correlation between decision variables is shown in a correlation matrix (Figure 25). It is evident that most Pearson correlation coefficients between variables are less than 0.5.

Figure 25. Correction matrix of decision variables.

The graphs generated by SA are shown in Figure 26, which reflect the percentage contribution of each design variable to the optimisation objectives $g_1$ and $g_2$. W₁ has the greatest impact on g₁ and g₂. D₂ and R₂ possesses the minimum influence on g₁ and g₂ respectively. From Figure 26(a), it can be found that $W_{\text{1}}$, $W_{\text{2}}$, $D_{\text{1}}$, and $D_{\text{2}}$ are negatively correlated with $g_2$; $R_{\text{1}}$ and $R_{\text{2}}$ are positively correlated with $g_2$. To reduce the PD, $R_{\text{1}}$ and $R_{\text{2}}$ must be reduced while $W_{\text{1}}$, $W_{\text{2}}$, $D_{\text{1}}$, and $D_{\text{2}}$ increased. However, it can be found that reducing $R_{\text{1}}$ and $R_{\text{2}}$, and increasing $D_{\text{1}}$ and $D_{\text{2}}$ lead to an increase in the value of $g_1$ (Figure 26(b)), which results in a deterioration of the flow area characteristics. Therefore, reasonable design variable values must be determined to minimise both $g_1$ and $g_2$. Moreover, it is noted that $D_{\text{1}}$, $D_{\text{2}}$, $R_{\text{1}}$, and $R_{\text{2}}$ generate minor effects on g₁ and g₂ compared to W₁ and W₂, which is possible due to the small scopes of these design parameters. Consequently, the ranges of these design parameters should be adequately expanded instead of choosing a conserved range.

Figure 26. Sensitivity analysis: (a) g₁; (b) g₂.

4.2.2. Optimised design based on TOPSIS

The second-order RSM was used to construct the approximate model. The minimum sample points requested for the second-order RSM of six decision variables were 28, and the 71 sets of sample points acquired by OLHD met the demand. The scatter distribution of the error analysis generated by the RSM is shown in Figure 27. The scatter points used for error analysis are close to the isoline between the predicted and actual values, and the goodness-of-fit R-squared values of $g_1$ and $g_2$ are 0.97571 and 0.99462. It suggested that DAPM-RSM models show excellent prediction accuracy. The CFD simulation model (D₁= 8.92 mm, R₁= 9.45 mm, W₁= 8.06 mm, D₂= 7.66 mm, R₂= 8.48 mm, W₂= 8.67 mm) is established other than 71 groups of samples. The CFD simulation values of $g_1$ and $g_2$ are 0.0128 and 32.23 respectively. The errors are 1.6% and 0.68% compared to the DAPM-RSM prediction values 0.013 and 32.01. The simulation results further verify the reliability of the DAPM-RSM model.

Figure 27. Comparison of predicted and actual value: (a) g₁; (b) g₂.

The NSGA-II is appropriate for the optimisation of consecutive questions and possesses the advantages of fast running speed and good convergence (Fan et al., 2022; Jiang et al., 2018). The approximate model is solved using NSGA-II, and Table 8 presents the setting condition for NSGA-II (Park et al., 2020). The corresponding Pareto front is obtained after iterative convergence, as shown in Figure 28.

Table 8. Genetic algorithm setup.

Population type	Double vector
Population size	100
Max number of function evaluations	10,000
Crowding distance fraction	0.35
Crossover fraction	0.8
Crossover operation	Intermediate crossover with the default value of 1.0

Figure 28. Pareto front of NSGA-II.

The values of g₁ and g₂ in the Pareto front are extensively distributed within an appropriate range, demonstrating that the MOO model has great effectiveness and generalisation abilities. Especially, it can be found that the whole actual mixtures are distributed above the Pareto front, indicating that the PD of each mixture in the data set is greater than necessary for reaching the required flow area and that the MOO model is efficient in declining the PD. From Figure 28, it can be found that as $g_1$ increases, $g_2$ decreases. This may be attributed to the fact the PD decreases with increasing flow area. It could be seen that almost every Pareto solution varies within a logical scope. For example, as $g_1$ varies from 0.0052 to 0.0388, $g_2$ changes from 37.94 to 27.81. In addition, except for the data in the ignored area (such as point A), the Pareto front for $g_1$ and $g_2$ is located below the original $g_2$ value of 34.88. It indicated that the presented MOO model is efficient for searching for the optimal set. What should be paid attention to is that the NSGA-II algorithm is not successful to acquire an efficient solution at g₁ smaller than 0.0052 and greater than 0.0388. It is possible attracted by the constraint of the sampled dataset. Accordingly, next work should pay attention to a collection of a wider dataset to enhance the robustness of the MOO model.

With respect to the optimisation of $g_1$ and $g_2$ (Figure 28), the NSGA-II algorithm discovers 53 nondominant solutions. The TOPSIS assessment approach is operated to obtain the best solution. The best solution (point D) with the supreme score (y = 1) is suggested as the best solution. Additionally, the best programme could be selected in accordance with the specific engineering demand. For instance, point B possesses fewer $g_1$ with greater $g_2$, which is suitable for the condition that focuses on the FCCs, while point D enjoys a greater $g_1$ with lower $g_2$, which could meet the request for energy conservation. It is noteworthy that point C (with more balanced $g_1$ and $g_2$) could be applied in the majority regular construction items. In this paper, priority is given to determining the PD characteristics from the perspective of the valve port flow area. Point C (0.0133, 32.43) on the Pareto front obtained is finally determined as the optimal solution, at which the average PD is reduced as much as 7.02% and the FARD is only 1.33%. The values of the decision variables are shown in Figure 29.

Figure 29. Comparison of notch structure before and after optimisation.

4.2.3. Analysis of flow area and pressure drop characteristics

The comparison of the KSN structure before and after optimisation is shown in Figure 29. The optimised valve port is numerically simulated by Fluent. The inlet VFR is set to 250 L/min, the outlet pressure is 0.1 MPa, and other parameters are the same as in Section 3.1. After optimisation, the average PD of the throttle orifice is reduced by 7.23%, and the relative deviation of the flow area is 1.28%, which is very similar to the results of the NSGA-II optimisation algorithm.

The PDs and valve port flow area before and after optimisation are compared as expressed in Figure 30. From Figure 30, it can be found that the valve port flow area changes little, and when the valve port opening is 5.4 mm, the valve port area transforms the most, and the error is 3.7%, that is, there is no significant difference in flow control performance before and after optimisation. Under the identical valve port opening, the PD of the optimised valve port is reduced by 8.63% on average. When the valve port opening is 0.6 mm, the PD is reduced from 33.48 to 30.97 MPa, which is optimised by 7.50%. When the valve port opening is 6mm, the PD is reduced from 1.22 to 1.11 MPa, reducing by 9.02%. To sum up, the optimised spool valve not only takes into account the control performance but also realises energy saving. This indicates the superiority and necessity of the optimisation. Moreover, the MOO proposed in this study is of considerable reference value for practical engineering applications.

Figure 30. Comparison of pressure drop and valve port flow area before and after optimisation: (a) Comparison of pressure drop and valve port area; (b) Local magnification A; (c) Local magnification B.

4.2.4. Analysis of flow field characteristics

In order to compare the flow field distribution characteristics of the KSN spool valve with different port openings and inlet VFRs before and after optimisation, the pressure, velocity, and turbulent kinetic energy distribution at the port of the spool valve are analysed. Under the same valve port opening, the flow field exhibits similar distribution characteristics. Therefore, the flow field on the symmetrical surface with the valve port opening of 1.2, 2.4, and 3.6 mm are selected to reflect the flow field distribution of low, medium, and large openings, respectively.

1. Pressure analysis

Figure 31 shows pressure counters on the symmetrical surface at different openings and inlet VFRs before and after optimisation. It could be seen from Figure 31 that as fluid flows through the KSN, the pressure decreases increasingly, and the PD is mainly concentrated at the KSN. Similar phenomenon was also observed in the simulation of spool valve with U-shape notch (Lu et al., 2022; Y. Ye et al., 2014). The pressure gradient changes significantly at the small opening (x = 1.2 mm). There are local low-pressure regions at the entrance and outlet of the KSN, which is easy to produce cavitation, noise, and other phenomena (B. Zhang et al., 2017). With the increase of valve opening and inlet VFR, the flow area of the valve port gradually enlarges, the pressure changing regions gradually expand to the notch entrance, and the pressure change gradient and the local low-pressure region area at the notch entrance gradually decrease.

Figure 31. Pressure counters on the symmetrical surface at different openings and inlet VFRs before and after optimisation: (a) Q = 250 L/min, x = 1.2 mm; (b) Q = 350 L/min, x = 2.4 mm; (c) Q = 450 L/min, x = 3.6 mm.

According to the local magnified pressure isoline at the entrance of the notch, the pressure of the valve port and chamber is generally lower after optimisation under the same inlet VFR and opening. The PD is reduced from 9.48, 4.77, and 3.68 MPa before optimisation to 8.69, 4.39, and 3.31 MPa after optimisation, respectively, with an optimisation ratio of 8.33%, 7.97%, and 10.05%, reducing the PD and realising energy saving. The local low-pressure region area at the notch is obviously reduced after optimisation. Especially when the valve port opening is 3.6mm and the inlet volume flow is 450 L/min, as shown in Figure 31(c), the local low-pressure area at the notch entrance disappears, reducing the probability of cavitation, noise, and other phenomena.

1. Velocity analysis

Figure 32 shows the velocity counters on the symmetrical surface at different openings and inlet VFRs before and after optimisation. It can be found that the fluid flow is complex near the notch and the velocity change gradient is large. As the high-pressure fluid flows through the notch, the fluid velocity increases gradually, and a high-speed region appears at the notch. This is due to the sharp reduction of the valve port flow area, while the inlet VFR remains unchanged, giving rise to a speedy increase in velocity. This velocity impact will have a great scouring effect on the spool, particularly as the fluid comprises some impurities, it shall intensify the spool wear (J. Ye et al., 2022). Therefore, reducing the maximum flow velocity and the range of high-speed region is significant to improve the personality of the spool valve.

Figure 32. Velocity counters on the symmetrical surface at different openings and inlet VFRs before and after optimisation: (a) Q = 250 L/min, x = 1.2 mm; (b) Q = 350 L/min, x = 2.4 mm; (c) Q = 450 L/min, x = 3.6 mm.

According to the isoline map of the local magnifying velocity at the entrance of the notch, with the addition of the opening and the inlet VFR, the velocity change gradient decreases gradually. Under the same inlet VFR and opening, the high-speed area at the valve port after optimisation reduces. The maximum flow velocity is decreased from 140.0, 95.7, 87.1 m/s to 130.7, 87.1, 78.6 m/s, and the optimisation ratio is 6.64%, 8.99%, 9.76%, which reduces the influence of fluid on the components in the valve. It is more facilitated to the permanent reliable operation of the KSN spool.

1. Turbulent kinetic energy analysis

Figure 33 shows the TKE counters on the symmetrical surface at different openings and inlet VFRs before and after optimisation. TKE is a physical quantity that denotes the significance of turbulence. The greater the TKE value is, the greater the time and length scope of the turbulent pulse. Similarly, a bigger degree of turbulence represents a larger degree of energy consumption (Choudhury & Anupindi, 2022). It can be found from Figure 33 that there are high TKE regions at the entrance of the KSN and the junction of the notch outlet and the valve neck. As the valve opening is small (x = 1.2 mm), the throttling impact of the KSNs gives rise to abrupt variation in pressure and velocity, and the turbulent flow at the portal is also the most significant. While the valve opening is greater (x = 3.6 mm), the flow area is great enough, and the throttling effect is no longer apparent, so the turbulence degree decreases. However, the high-velocity area where the notch outlet connecting to the valve neck still exists, so the maximum TKE value transfers here.

Figure 33. Turbulent kinetic energy counters on the symmetrical surface at different openings and inlet VFRs before and after optimisation: (a) Q = 250 L/min, x = 1.2 mm; (b) Q = 350 L/min, x = 2.4 mm; (c) Q = 450 L/min, x = 3.6 mm.

According to the isoline map of the local magnification velocity at the entrance of the notch and the junction of the notch outlet and the valve neck, under the same inlet VFR and valve opening, the area of high TKE area decreases after optimisation. Especially when the valve opening is 1.2 mm and the inlet VFR is 250 L/min, the maximum TKE value at the inlet of the notch is reduced from 800.0 to 678.6 m²/s² as shown in Figure 33(a), with an optimisation ratio of 15.18%. The maximum TKE value at the junction of the notch outlet and the valve neck is reduced from 621.4 to 564.3 m²/s², with an optimisation ratio of 9.19%. That means a greatly lower degree of energy consumption. While the valve opening is 3.6 mm and the inlet VFR is 450 L/min, although the maximum TKE value does not decrease, the high TKE region shrinks significantly, reducing the energy loss to some extent.

To sum up, the optimised KSN spool valve can effectively reduce the pressure gradient, velocity gradient, PD, maximum velocity, and maximum turbulent kinetic energy in the valve. And the area of local low-pressure, high-velocity, and high TKE region at the notch port is decreased. The energy utilisation efficiency is improved while reducing the influence of fluid on the valve components.

5. Conclusion

To solve the problems of serious thermal energy dissipation of MWSVs. This research innovatively developed the RSM-DAPM-RSM model to predict the relationship between KSN structure and VFR & PD. The impact of various input variables was also analysed by SA. A MOO model was developed using NSGA-II and TOPSIS for optimisation of KSV spools considering PD, and FARD. Based on the analysis, the following conclusions are drawn:

1. Discharge area was introduced as the parameter to characterise the FCCs. The DAPM model of the KSN valve port was also established to predict the mapping relationship between the KSN structure parameters and VFR and PD. The RSM-DAPM-RSM model achieves high accuracy for prediction of discharge area with a max relative deviation of 4.02% compared to CFD simulation.

2. A MOO model depended on NSGA-II integrated with TOPSIS was established to intelligently optimise the KSN structure parameters. The optimisation model minimised PD and FARD simultaneously. The Pareto solution demonstrated greater weight than real experimental data. Compared with the traditional MWSV, the mean PD of the optimised valve was decreased by 7.23%, and the FARD was only 1.28%, which improved the energy efficiency of the system.

3. Based on the counters of pressure, velocity, and TKE distribution, it is observed that the notch structure affected the FFCs, and further the energy dissipation. The pressure gradient, velocity gradient, PD, maximum velocity, and maximum TKE after optimisation are reduced. The area of the local low-pressure region, high-velocity region, and high-TKE region shrinks. The energy utilisation efficiency is improved while reducing the negative influence of fluid on the valve components. At the same point, the chance of cavitation, noise, and other phenomena is dropped, which is more favourable to the permanent reliable operation of the KSN valve spool.

This study innovatively gives a novel framework to optimise KSN structure parameters considering PD and FARD. The developed RSM-DAPM-RSM model obtains credible and precise prediction capacity. The MOO model promotes engineers and researchers to rapidly design and optimise the design parameters of KSN spools. Meanwhile, there are several limitations in this study: (1) restricted by the test conditions, experimental tests after optimisation are absent, which may weaken the credibility of the KSN optimisation, although the method of CFD modelling after optimisation and experimental tests before optimisation have been verified; (2) although the DAPM-RSM model showed good performance in PD and FARD, its predictive ability is worth of further optimisation using more innovative algorithms; and (3) the influence of the optimised spool structure on flow field characteristics such as steady flow force also need to be investigated. These limitations are supposed to be addressed carefully in future studies. In addition, the optimisation design of higher-level combined KSNs such as K-K or K-K-K deserves further attention.

Acknowledgements

The authors would like to thank Liu Xianhang, a Master's student, for his support in data processing.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by National Natural Science Foundation of China [grant number 52275066]; Key R&D projects of Hebei Province [grant number 20314402D]; Natural Science Foundation of Xinjiang Uygur Autonomous Region [grant number 2021D01A63]; Science and Technology Research Project of Colleges and Universities of Hebei Province [grant number ZD2021340].