Analysis of Energy Loss on a Tunable Check Valve through the Numerical Simulation

Abstract

The article presents a study of the flow through a tunable check valve used as a hydraulic lock in a system with an actuator. Special attention is given to energy losses of the liquid stream in the poppet gap. In the first stage of the research, CFD methods were used to determine the speed and pressure distributions for the fixed values of the input flow rate and the poppet position. The values of the jet angle and pressures determined based on the CFD results were used to build a simulation model of the entire hydraulic system in Matlab/Simulink environment. The influence of the spring parameters pressing the poppet against the valve seat on the pressure drop and thus on the amount of energy lost was investigated. In particular, the spring stiffness and initial tension were studied. The obtained results were used to develop guidelines for constructing a valve prototype. Finally, the results of simulation tests were verified based on the actual valve characteristic obtained on a test bench.

1. Introduction

Check valves are widely used as flow control elements in current hydraulic and pneumatic drives. They are the key elements controlling the working medium flow direction, exposed to high pressures, and thus subjected to high forces. In hydraulic systems, especially those containing actuators subjected to long-term, continuous loads, check valves are applied as hydraulic locks, the function of which is to quickly shut off the flow and maintain complete tightness in the event of a power failure or the need to keep the current position. Such requirements apply to construction equipment, such as cranes, load-handling devices, including forklifts, conveyors, stock pickers, elevators, and manufacturing equipment, e.g., industrial presses. Studies on determining the characteristics of these valves, and thus estimating and minimizing energy losses during the flow of the working fluid through valve channels and gaps, are conducted in leading research centers. The subject of research is often related to the transient state during flow opening and the appropriate selection of parameters of simulation models. Ye et al. [1] studied transient flow characteristics of a high-pressure hydrogen check valve. In the first step, fluid flow properties during the opening process were obtained, including the velocity changes and pressure distribution. Next, fluid forces, valve spool acceleration and velocity were determined. Finally, the influence of inlet pressure on the operating characteristics of the spool was analyzed. In further work, Ye presented check valve spool-head angle optimization results in the high-pressure hydrogen storage system [2]. The transient CFD method and the moving mesh technology were used to determine flow characteristics against the spool-head angle during the opening process. In contrast, Kim and Yeong [3] provided recommendations related to the choice of an appropriate CFD mesh moving method in the check valve closing behavior modeling. Four different techniques were modeled and tested. Then the obtained results were compared, and practical conclusions were formulated.

The flow characteristics of a butterfly check valve in a small modular reactor by means of CFD method were determined by Wen et al. [4]. The authors obtained the valve loss coefficient against the valve opening angle using five different turbulence models. The research showed that the best compliance of the simulation results with experiments was obtained for the $k-\epsilon$ model. The transient flow of check valves installed in the nuclear industry systems was also investigated by Lai et al. [5], using the unsteady CFD method. Lai analyzed a multi-receiver system in the parallel arrangement and used check valves to reduce energy consumption in the case when not all branches are put into operation. Another analysis of the characteristics of a directional poppet valve using the CFD method was presented by Gomez in [6]. The influence of the poppet cone angle on pressure and flow forces was investigated. Special attention was paid to predicting the valve behavior operating at high frequencies. It was found that with the assumed operational conditions, the cone angle has a negligible effect on pressure loss, while the flow force rises as the angle decreases. Similarly, Deng et al. [7] took up the transient opening state of a check valve under multiphase flow conditions. Deng built several candidate CFD transient models with different turbulence models, which were next subjected to probabilistic modeling to predict the check valve opening height more accurately. Moreover, Ling et al. [8] focused on the dynamics and transient states of a novel, high-speed, precise piezoelectric flow control valve.

In the field of research on the check valve design modifications, one can mention the publication by Shin et al. [9], in which the authors proposed to change the pressure valve opening control to an active driving force, which significantly improved the reliability of the studied system. In turn, Morgan et al. [10] introduced the concept of a tunable check valve for low flow, which can be used for marine nutrient sensing. The use of a permanent magnet eliminated the need for an additional power supply. Gao et al. [11] conducted a comparative analysis of two design solutions: swing and tilting check valves with systems containing compressible fluids. Based on the CFD tests performed, recommendations were formulated regarding the advantages and disadvantages of both tested solutions. Furthermore, Baluyev et al. [12] presented a study on the functional characteristics of a check valve installed in the safety system of a liquid metal cooled reactor. The research included calculations in Ansys CFX based on a scaled model and laboratory experiments. As a result, the hydrodynamic characteristics of the valve against the selected design and operational parameters were obtained. In the field of research on check valves in specific systems, Yang [13] provided a method of integrating check valves in the flapping wing membrane of a micro air vehicle. Schickofer [14] used high-order FSI simulations and low-order analytical models to examine the stability and non-linear dynamics as well as the opening and closing characteristics of a check valve in a shock absorber.

This work is a continuation of previous research conducted by the authors using the CFD method and laboratory experiments on check valves. The studies included an analysis of flow through a directional control valve consisting of four poppet seat valves [15] and a proposal of geometric modifications of an adjustable check valve [16]. This paper concerns research on a check valve designed for larger flows, with a nominal volumetric flow rate $Q_{nom}=360$ dm³ min⁻¹. The greatest emphasis is placed on the analysis of flow parameters, such as velocity and pressure, with special attention put on pressure losses occurring in the poppet gap.

2. Working Principle of a Tunable Check Valve

The subject of the analysis is a tunable check valve whose default design is shown in Figure 1. The flow channels are made in the body (1) closed with covers (2) and (3). The flow can occur in both directions, A-B and B-A, through the annular slot of the poppet (4). The valve opens the flow in the A-B direction against the spring (7) force and the flow forces. In contrast, unblocking the flow through the valve in the B-A direction is possible after giving an appropriate pressure control signal acting on the pilot surface (8). The signal generates a force sufficiently large for displacing the poppet. A relatively high nominal flow characterizes the analyzed valve; therefore, it was proposed to use an initial valve, including the tapered hole in the poppet, the ball (5) and the disc (6). This solution makes it possible to compensate for the pressures on both sides of the poppet before the main flow is opened.

Essential, both for the initial phase of valve opening and the total pressure losses during the flow, is the appropriate selection of the spring (7) stiffness, which generates the force of pressing the operating surface of the poppet against the edge of the seat. The second parameter affecting the flow forces, and thus the valve-opening characteristic, is the initial tension imposed on the spring.

3. Mathematical Model of the System

The modeled hydraulic system includes a fixed speed pump, a control valve, the above-mentioned check valve, and a hydraulic actuator with a maximum load of 500 kN (Figure 2). The model was created according to the methodology proposed by Novak et al. [17], considering both flow directions through the check valve, A-B and B-A, respectively. The input signal is the pump flow rate. Next, equations of fluid mass conservation are written for the distinguished volumes, and equations of motion based on force balances are formulated for the valve poppet and the hydraulic cylinder piston. Flow equations through throttle slots are based on Bernoulli’s law and geometrical relations.

The input flow rate was determined, assuming the use of a piston pump with z pistons, the piston stroke rate per revolution $q_{rev}$ and the angular speed ${\omega }_0$, as follows:

$$Q_0\left(t\right)=q_{rev}·{\omega }_0\left(t\right)·\sum _{k=0}^{z/2-1}sin({\omega }_0·t+\frac{2\pi k}{z}),$$

(1)

where the angular speed ${\omega }_0\left(t\right)$ increases linearly at the start-up time $t_{start}$ and then remains constant:

$${\omega }_0\left(t\right)=\left\{\begin{array}{cc}{\omega }_{nom}·t/t_{start}\hfill & \mathrm{for}t<t_{start}\hfill \\ {\omega }_{nom}\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

(2)

The equation of mass conservation formulated for the supply line takes into account the division of the fluid stream into control valve and relief valve flow rates, $Q_1$ and $Q_{ret}$, respectively:

$$\frac{d{p}_0\left(t\right)}{dt}=B_f/V_0·(Q_0\left(t\right)-Q_1\left(t\right)-Q_{ret}\left(t\right)).$$

(3)

Control valve flow rate $Q_1\left(t\right)$ is determined based on the characteristics obtained during previous studies [18], as a function of the pressure difference $\Delta p=p_0-p_1$ and the control signal U:

$$Q_1\left(t\right)=f(p_0\left(t\right),p_1\left(t\right),U),$$

(4)

while the relief valve opens in the case when the $p_0$ pressure exceeds the allowable range:

$$Q_{ret}\left(t\right)=f\left(p_0\left(t\right)\right).$$

(5)

Pressures $p_1\left(t\right)$ and $p_2\left(t\right)$ depend on the control valve setting. Depending on whether the pump is connected to port A (A-B flow direction) or port B (B-A flow direction), the mass conservation equations have the following form:

$$\frac{d{p}_1\left(t\right)}{dt}=\left\{\begin{array}{cc}{B}_f/V_1·(Q_1\left(t\right)-Q_2\left(t\right))\hfill & (\mathrm{A}\text{-}\mathrm{B}\mathrm{direction})\hfill \\ B_f/V_1·(Q_1\left(t\right)-A_{cyl,r}·v_{cyl})\hfill & (\mathrm{B}\text{-}\mathrm{A}\mathrm{direction})\hfill \end{array}\right.$$

(6)

$$\frac{d{p}_2\left(t\right)}{dt}=\left\{\begin{array}{cc}{B}_f/V_2·(Q_2\left(t\right)-A_{cyl}·v_{cyl})\hfill & (\mathrm{A}\text{-}\mathrm{B}\mathrm{direction})\hfill \\ B_f/V_2·(A_{cyl}·v_{cyl}-Q_2\left(t\right))\hfill & (\mathrm{B}\text{-}\mathrm{A}\mathrm{direction}).\hfill \end{array}\right.$$

(7)

Similarly, the equation of the hydraulic cylinder piston motion in both directions can be written as follows:

$$m_{cyl}\frac{d^2{x}_{cyl}\left(t\right)}{d{t}^2}=\left\{\begin{array}{cc}{p}_2\left(t\right)·A_{cyl}-p_{ret}\left(t\right)·A_{cyl,r}-\phi ·\frac{d{x}_{cyl}\left(t\right)}{dt}-F_{load}\hfill & (\mathrm{A}\text{-}\mathrm{B}\mathrm{direction})\hfill \\ p_1\left(t\right)·A_{cyl,r}-p_2\left(t\right)·A_{cyl}-\phi ·\frac{d{x}_{cyl}\left(t\right)}{dt}-F_{load}\hfill & (\mathrm{B}\text{-}\mathrm{A}\mathrm{direction}).\hfill \end{array}\right.$$

(8)

The flow rate through the check valve depends on the poppet position and, therefore, the width of the annular gap. The force balances in the A-B and B-A flow directions are shown in Figure 3a,b, respectively.

For the A-B direction, the poppet motion equation was written as follows:

$$m_p\frac{d^2{x}_{p(A-B)}\left(t\right)}{d{t}^2}=F_{hsA1}-F_{hsA2}-F_{spr}-F_f-F_{hd}.$$

(9)

where hydrostatic forces acting on both sides of the poppet $F_{hsA1}\left(t\right)=p_1\left(t\right)·A_{pA}$, $F_{hsA2}\left(t\right)=p_2\left(t\right)·A_{pB}$. Spring force includes the initial spring tension and deflection due to poppet displacement: $F_{spr}\left(t\right)=(x_{s0}+x_p\left(t\right))·k_{spr}$. The friction force, which is proportional to poppet velocity, depends on the geometry and fluid dynamic viscosity: $F_f\left(t\right)=\eta ·{\mathrm{\Phi }}_g·\frac{d{x}_p}{dt}$, where the geometrical factor ${\mathrm{\Phi }}_g=\pi ·d_p·l_p·{\left(c_{rp}\sqrt{1-(e_p/c_{rp})}\right)}^{-1}$. Hydrodynamic force acting on the spool: $F_{hd}=\rho ·Q_2\left(t\right)·v_{avg}·cos\left(\lambda \right)$, where the average fluid velocity $v_{avg}=\sqrt{2/\rho ·(p_1\left(t\right)-p_2\left(t\right))}$. The methodology for determining the $\lambda$ angle based on CFD results was presented in [19,20,21]. In the B-A flow direction, the equation of the poppet motion has the form:

$$m_p\frac{d^2{x}_{p(B-A)}\left(t\right)}{d{t}^2}=F_{hsC}+F_{hsA1}-F_{hsA2}-F_{spr}-F_f+F_{hd}.$$

(10)

Compared to the A-B direction, there are no differences in the hydrostatic force $F_{hsA2}$, spring force $F_{spr}$ and friction force $F_f$ formulas. The hydrodynamic force equation is also the same, except for the difference in determining the average velocity: $v_{avg}=\sqrt{2/\rho ·(p_2\left(t\right)-p_{ret})}$. The hydrostatic force moving the valve pilot comes from the pressure transmitted through the X port: $F_{hsC}\left(t\right)=p_2\left(t\right)·A_{pC}$, while the force acting on the left side of the poppet comes from the return line pressure: $F_{hsA1}\left(t\right)=p_{ret}\left(t\right)·A_{pA}$.

4. Check Valve Discrete Model and CFD Analysis

The CFD analyses were carried out utilizing a university research package of ANSYS/Fluent software, while the check valve geometric model was built using PTC Creo. Next, boundary conditions and the turbulence model in ANSYS/Fluent were assigned. Based on the geometric model, the initial meshing process was conducted and preliminary CFD calculations were made in order to assess the mesh quality. After the mesh parameters were finally selected, complete valve flow tests were performed for the set flow rate values.

4.1. Boundary Conditions and Turbulence Model

The following boundary conditions were set: the inlet average fluid velocity based on the input flow rate and the outlet pressure. The input velocity was assigned as perpendicular to the boundary with the Boundary Conditions/Velocity Specification Method option. The Outlet Condition/Gauge Pressure option was used to define the outlet pressure $p_{out}$. Values of the physical model parameters are shown in

Table 1. Physical model parameters.

Kinematic Viscosity	Density	Temperature	Min Inlet FLOW Rate	Max Inlet Flow Rate	Outlet Pressure
$\nu$	$\rho$	T	$Q_{min}$	$Q_{max}$	$p_{ret}$
$41\times {10}^{-6}$	870	50.0	100	900	0.1
m2 s−1	kg m−3	°C	dm3 min−1	dm3 min−1	MPa

.

The value of the Reynolds number determined for the given flow channel geometry, fluid viscosity $\nu$ and the flow rate range $Q_{min}-Q_{max}$ varies from $R{e}_{min}=2.8\times {10}^3$ to $R{e}_{max}=2.5\times {10}^4$. The value of $Re$ indicates the transient or turbulent flow; thus, the turbulence model needs to be assigned. Based on the literature related to CFD studies on hydraulic and pneumatic elements, such as a pump [22], a turbine [23], and valves [4,24,25], the standard $k-\epsilon$ model was chosen. ANSYS/Fluent provides several variants of the $k-\epsilon$ model. However, earlier studies of the authors on a similar, smaller-size check valve [16] have shown that the standard $k-\epsilon$ model gives almost identical results to the RNG and realizable variants. On the other hand, the computation time, which is a crucial parameter in long-term calculations, is shorter by about 10%.

The transport equations used for determining the main factors, such as kinetic energy of the turbulence k, kinetic energy dissipation $\epsilon$ and turbulent viscosity ${\mu }_t$, can be found in many publications, e.g., [16,24,26].

Table 2. Turbulence model parameters.

Kinetic Energy Constant	Kinetic Energy Dissipation Constants			Turbulent Viscosity Constant	Turbulence Intensity	Length Scale
$s_k$	$s_{\epsilon }$	$C_{1\epsilon }$	$C_{2\epsilon }$	$C_{\mu }$	I	ℓ
1.0	1.3	1.44	1.92	0.09	4.5–5.9	1.53
-	-	-	-	-	%	mm

shows the parameters defining the specificity of turbulence, including intensity I (Equation (11)), length scale ℓ (Equation (12)) and other constants with values recommended by ANSYS [27].

$$I=0.16·{Re}^{-0.125},$$

(11)

$$\ell =0.07·D_H,$$

(12)

where $D_H$ is the relevant hydraulic diameter, $Re$ is the Reynolds number, and $\rho$ is the fluid density.

4.2. Meshing and Assessing Mesh Quality

The mesh was created using irregular elements. The boundary layers were made of prisms, while the bulk flow was modeled with tetrahedrons. The pressure–velocity coupling was achieved using the pressure-based solver. The condition for convergence was to obtain values less than ${10}^{-4}$ for both mass and momentum residuals.

The mesh quality was assessed by comparing the results of the CFD simulation with the results of actual laboratory tests of the valve, carried out for the A-B direction at the fixed flow rate $Q=300$ dm³ min⁻¹. The pressure drop measured at the test bench was $\Delta p_{lab}=0.66$ MPa. The steady-state CFD simulations performed using the $k-\epsilon$ turbulence model allowed us to obtain the value of pressure losses, following ISO 4411 [28].

In the beginning, the sizes of mesh elements were adopted under the general recommendations of ANSYS, obtaining values significantly different from the laboratory results. After that, using the progressive mesh refinement technique, the mesh density was gradually increased, especially in the critical areas near the valve poppet head and the seat. These activities significantly brought the CFD simulation outcomes closer to the results of laboratory experiments. However, the computation time of a single test was simultaneously extended from 30 minutes to almost 28 hours. Three representative meshes are shown in Figure 4, while the comparison of results is summarized in (

Table 3. Mesh quality assessment.

Mesh Case	No. of Nodes	No. of Elements	Comput. Time	Pressure Drop	Difference from Laboratory Result
Initial (I)	$3.0\times {10}^3$	$1.3\times {10}^4$	30 min	$0.566$ MPa	$14.21$%
Dense, refined (II)	$9.1\times {10}^4$	$4.6\times {10}^5$	240 min	$0.644$ MPa	$2.36$%
Double dense, refined (III)	$2.5\times {10}^5$	$1.4\times {10}^6$	1660 min	$0.645$ MPa	$2.32$%

).

The results obtained for cases II and III were almost identical and highly consistent with the results of laboratory tests. Since the computation time of the case II mesh was approximately 7 times shorter, this mesh was adopted for further tests. The following mesh quality parameter values were read with the Statistics/Mesh Metrics option: maximum aspect ratio $8.9$, maximum skewness $0.92$ with the average value below $0.30$, the average orthogonal quality $0.92$ with the minimum value of $0.68$.

4.3. CFD Simulation Results

The CFD test plan included simulation with the nominal flow rate $Q=300$ dm³ min⁻¹ in the A-B direction and different poppet positions that define the flow gap widths: $x_p=1.0,2.0,3.0,4.0,5.0$ mm, respectively. In each case, the pressure and velocity distributions were obtained. The example results for $x_p=1.0,3.0\mathrm{and}5.0$ mm are shown in Figure 5, Figure 6 and Figure 7, while jet angle values are summarized in

Table 4. Jet angle against poppet position from CFD simulation.

Input	Q (dm3 min−1)	300
	$x_p$ (mm)	1.0	2.0	3.0	4.0	5.0
Output	$\lambda$ (${}^{\circ }$)	67.8	65.0	62.1	58.9	56.0

.

Based on the obtained results, a linear function of jet angle against the poppet position $\lambda =f\left(x_p\right)$ was determined using linear interpolation (Equation (13)), achieving the correlation coefficient $R=0.997$ and the residual sum of squares $RSS=1.4\times {10}^{-4}$. The function is valid for $x_p$ given in mm, and the resulting $\lambda$ value is in radians.

$$\lambda \left(x_p\right)=-0.0502·x_p+1.235.$$

(13)

In contrast, there is no need to determine the $\lambda \left(x_p\right)$ function for the B-A direction, as the valve is designed to fully open when a pressure signal is applied to the X port. The obtained $\lambda \left(x_p\right)$ dependency was next used to build a complete simulation model in Matlab/Simulink.

5. Simulations in Matlab/Simulink Environment

A general view of the Simulink model for the A-B flow direction, including blocks representing the main hydraulic components and reduced volumes, is shown in Figure 8. The difference in the block arrangement for the B-A direction is displayed in Figure 9.

The model includes the following components: pump, relief valve, control valve, tunable check valve and hydraulic cylinder. The reduced volumes represent lines between the elements: pump and control valve (index 0), control valve and check valve (index 1), as well as a check valve and hydraulic cylinder (index 2), respectively. The system operates at a maximum load force exerted on the actuator $F_{load}=500$ kN. Figure 10 shows the input signal values in the form of a piston pump flow rate. The mean values were $Q_0=100,200,300,600,900$ dm³ min⁻¹, marked as (1), (2) … (5), respectively. In the graph, one can observe the pulsation of approximately 6%.

5.1. Determining the Valve Gap Width and Actuator Piston Displacement

After assigning nominal spring parameters, including stiffness of 41 N mm⁻¹ and initial tension of 7 mm, the hydraulic system was simulated with the actuator movement in both A-B and B-A directions. The obtained courses of the check valve opening width and the actuator displacement against time for both flow directions are shown in Figure 11.

The displacement of the actuator piston in the B-A direction takes place at a higher speed due to the fluid supply to the cylinder from the piston rod side. For the same reason, the valve opening width $x_p$ is larger for the B-A flow direction. It obtains the maximum value of $x_p=5.0$ mm for $Q_0=600$ dm³ min⁻¹, and for $Q_0=300$ dm³ min⁻¹ is equals to $x_p=3.8$ mm. In contrast, in the A-B direction $x_p=3.95$ mm for $Q_0=600$ dm³ min⁻¹ and reaches full opening for $Q_0=900$ dm³ min⁻¹.

In two subsequent tests, the valve springs with the modified stiffness were applied. Figure 12 shows the results obtained with a softer spring (stiffness reduced by 40%, $k_{spr}=k_1=24.6$ N mm⁻¹). The analogous results achieved with a 40% stiffer spring ($k_{spr}=k_2=57.4$ N mm⁻¹) are presented in Figure 13.

The results show that the spring parameters are important for the width of the flow gap. For example, at a flow rate of 300 dm³ min⁻¹ in the A-B direction, the gap width $x_p$ varies from $2.0$ mm (stiffness $k_2$) to $2.85$ mm (stiffness $k_1$). In the B-A direction the difference in the opening width is even greater, from $3.1$ mm to $4.8$ mm. As a result, the pressure drop across the valve changes, which leads to different energy losses.

5.2. Determination of the Pressure Drop across the Valve

In the next stage, the value of the pressure drop on the check valve was determined as a function of the flow rate depending on the stiffness and initial spring tension. Figure 14 and Figure 15 and

Table 5. Check valve pressure drop (MPa) against flow rate Q and spring stiffness $k_{spr}$.

${\mathit{x}}_{\mathit{s}0}$ mm	Q dm3 min−1	${\mathit{k}}_{\mathbf{spr}}=24.6$ N/mm		${\mathit{k}}_{\mathbf{spr}}=41.0$ N/mm		${\mathit{k}}_{\mathbf{spr}}=57.4$ N/mm
		A-B	B-A	A-B	B-A	A-B	B-A
7.0	100	0.34	0.06	0.48	0.12	0.65	0.17
	200	0.41	0.10	0.56	0.15	0.74	0.21
	300	0.47	0.15	0.65	0.19	0.83	0.25
	600	0.70	0.39	0.92	0.42	1.15	0.43
	900	1.26	0.89	1.37	0.92	1.51	0.92

show the pressure drop against time for different stiffness values in the A-B and B-A direction, respectively. Figure 16 and Figure 17 and

Table 6. Check valve pressure drop (MPa) against flow rate Q and initial tension $x_{s0}$.

${\mathit{k}}_{\mathbf{spr}}$ N/mm	Q dm3 min−1	${\mathit{x}}_{\mathit{s}0}=2.0$ mm		${\mathit{x}}_{\mathit{s}0}=7.0$ mm		${\mathit{x}}_{\mathit{s}0}=12.0$ mm
		A-B	B-A	A-B	B-A	A-B	B-A
41.0	100	0.24	0.05	0.48	0.12	0.77	0.21
	200	0.32	0.08	0.56	0.15	0.85	0.24
	300	0.41	0.11	0.65	0.19	0.93	0.26
	600	0.67	0.39	0.92	0.42	1.21	0.43
	900	1.26	0.90	1.37	0.92	1.56	0.92

present the results obtained for the nominal spring with the different initial tension values. Depending on the flow rate and spring parameters, the obtained pressure drop varies from $0.25$ MPa to $1.59$ MPa in the A-B direction and from $0.08$ MPa to $0.95$ MPa in the B-A, respectively.

5.3. Estimation of the Check Valve Energy Efficiency

With the valve flow rate and pressure drop determined, the power required to open the valve and the total energy loss during the working cycle were estimated. The summary of the required power is shown in Figure 18, while the energy consumption is presented in Figure 19. As can be seen from the results, energy losses across the valve depend mainly on the flow rate and, to a lesser extent, on the spring stiffness. When operating at a nominal flow rate of $Q=300$ dm³ min⁻¹, the power requirements do not exceed 7 kW for the A-B flow direction and 3 kW for the opposite. This results in energy consumption during the working cycle of approximately 31 kJ and 12 kJ, respectively. Increasing the flow rate to 900 dm³ min⁻¹ causes an over 4-fold increase in power demand, which leads to the energy consumption of the working cycle being 122 kJ (A-B) and 59 kJ (B-A).

The percentage share of energy consumed by the check valve and the effective cylinder piston displacement compared to the total energy necessary to complete the hydraulic system working cycle in both directions is presented in

Table 7. Percentage share of energy consumption by the check valve $E_{cv}$ and the hydraulic cylinder $E_{cyl}$ against flow rate Q and spring stiffness $k_{spr}$.

Direction	Q dm3 min−1	${\mathit{k}}_{\mathbf{spr}}=24.6$ N/mm		${\mathit{k}}_{\mathbf{spr}}=41.0$ N/mm		${\mathit{k}}_{\mathbf{spr}}=57.4$ N/mm
		${\mathbf{E}}_{\mathbf{cv}}\%$	${\mathbf{E}}_{\mathbf{cyl}}\%$	${\mathbf{E}}_{\mathbf{cv}}\%$	${\mathbf{E}}_{\mathbf{cyl}}\%$	${\mathbf{E}}_{\mathbf{cv}}\%$	${\mathbf{E}}_{\mathbf{cyl}}\%$
A-B	100	2.1	95.4	2.9	94.6	3.8	93.7
	200	2.4	95.1	3.4	94.2	4.3	93.2
	300	2.8	94.3	3.8	93.3	4.9	92.3
	600	4.1	91.5	5.2	90.4	6.4	89.3
	900	7.0	88.3	7.0	86.7	8.2	85.6
B-A	100	0.9	98.1	1.2	97.9	1.6	97.7
	200	0.8	98.0	1.2	97.7	1.5	97.5
	300	0.9	97.6	1.3	97.3	1.7	97.1
	600	2.5	95.5	2.5	95.5	2.6	95.4
	900	5.2	91.5	5.2	91.5	5.2	91.5

.

The obtained results show that the energy losses on the valve at nominal flow 300 dm³ min⁻¹ do not exceed 3–5% of the total energy consumption during the flow in the A-B direction and 1–2% in the B-A direction, depending on the spring stiffness. Increasing the flow rate to 900 dm³ min⁻¹ results in a greater percentage of valve loss; however, in the worst case, it is 7–8% and 5%, respectively. At the same time, the share of energy effectively used to move the actuator decreases from 92.3–94.3% to 85.6–88.3% for A-B and from 97.1–97.6% to 91.5% for B-A.

6. Comparison of Results with Laboratory Experiments

The last stage of the research was verifying the simulation results using the laboratory tests of the valve prototype provided by the manufacturer. Due to technical limitations, the bench tests were carried out for flow rates up to 350 dm³ min⁻¹. A valve spring with a nominal rate of $k_{spr}=41.0$ N mm⁻¹ was used. The test bench scheme and the valve prototype during the tests are shown in Figure 20.

Each test consisted of the following steps: setting the minimum pump flow, opening the flow by overriding the control valve, and measuring the pressures while gradually increasing the pump flow to the maximum value and then back to zero. The pressures were measured with Trafag NAT sensors with a range of 32 MPa, an accuracy class of $\pm 0.2\%$ and voltage transducers 0–10 V. A Kracht TM turbine flow meter with a range of 2–500 dm³ min⁻¹ and an accuracy class of $\pm 0.5\%$ and analogous voltage output signal was used to measure the flow rate. A 16-bit NI PCIe-6321 DAQ card with an SCB-68A terminal acquired pressure and flow rate signals from transducers. The measured pressure drop on the check valve was compared with the results of the numerical calculations, shown in Figure 21.

A high level of convergence of simulation and laboratory results was obtained. The value of the Pearson’s correlation coefficient determined for the A-B flow direction is $R_{A-B}=0.95$, while for B-A, it is $R_{B-A}=0.98$.

7. Summary and Conclusions

The subject of the analysis presented in this paper was a tunable check valve with a nominal flow $Q=300$ dm³ min⁻¹ and maximum working pressure $p_{max}=35$ MPa. As part of the research, CFD analyses and simulations in the Matlab/Simulink environment were carried out to obtain the flow characteristics of the valve. In particular, the values of the flow gap opening width and pressure drop were obtained as a function of the flow rate and the parameters of a spring pressing the valve plug against the seat, including spring stiffness and initial tension. A system with a one-sided hydraulic cylinder and a control valve was adopted to carry out the tests, ensuring the possibility of moving the piston rod in both directions. The hydraulic cylinder rod was loaded with the force $F_{cyl}=500$ kN, which ensured that the pressure in the system was close to the maximum value. Pressure drop and the related energy losses across the valve were estimated at the flow rates up to $Q=900$ dm³ min⁻¹. The simulation outcomes show a high level of compliance with the results of laboratory tests carried out on the valve prototype by the manufacturer.

The detailed conclusions are as follows:

• The energy losses across the valve vary substantially depending on flow direction. Although the same connection diameters are used, the flow direction significantly impacts the amount of loss.
• The spring stiffness and the initial tension, which is necessary to maintain tightness, are of significant importance for the level of energy loss, especially at lower flows. On the other hand, energy losses at the flow that guarantees full valve opening no longer depend on the spring parameters.
• With the flow rate not exceeding the nominal value $Q=300$ dm³ min⁻¹, the determined energy losses are at the maximum level of 2–4% of the total system demand. However, the results show that the valve is not fully open. On the one hand, this makes the spring parameters more affecting the pressure drop, while on the other, it encourages research on expanding the flow rate range.
• Doubled nominal flow $Q=600$ dm³ min⁻¹ causes the valve to open fully in the B-A direction and almost entirely in the A-B when using a softer spring. The share of energy losses, in this case, was 2.5% (B-A) and 4.1% (A-B), which can be considered acceptable values, as they are close to those obtained by a valve with a standard spring at nominal flow.
• The maximum flow rate $Q=900$ dm³ min⁻¹ resulted in full opening in both directions. Depending on the flow direction, energy losses were estimated at 5.2–8.2%. These are not disqualifying values; however, taking into account the additional losses on the other components of the system, they should be included in the overall energy balance of the system.