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

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## Abstract

**:**

## 1. Introduction

^{3}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

## 3. Mathematical Model of the System

## 4. Check Valve Discrete Model and CFD Analysis

#### 4.1. Boundary Conditions and Turbulence Model

#### 4.2. Meshing and Assessing Mesh Quality

^{3}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].

#### 4.3. CFD Simulation Results

^{3}min

^{−1}in the A-B direction and different poppet positions that define the flow gap widths: ${x}_{p}=1.0,\phantom{\rule{0.222222em}{0ex}}2.0,\phantom{\rule{0.222222em}{0ex}}3.0,\phantom{\rule{0.222222em}{0ex}}4.0,\phantom{\rule{0.222222em}{0ex}}5.0$ mm, respectively. In each case, the pressure and velocity distributions were obtained. The example results for ${x}_{p}=1.0,\phantom{\rule{0.222222em}{0ex}}3.0\phantom{\rule{0.277778em}{0ex}}\mathrm{and}\phantom{\rule{0.277778em}{0ex}}5.0$ mm are shown in Figure 5, Figure 6 and Figure 7, while jet angle values are summarized in Table 4.

## 5. Simulations in Matlab/Simulink Environment

^{3}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

^{−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.

^{3}min

^{−1}, and for ${Q}_{0}=300$ dm

^{3}min

^{−1}is equals to ${x}_{p}=3.8$ mm. In contrast, in the A-B direction ${x}_{p}=3.95$ mm for ${Q}_{0}=600$ dm

^{3}min

^{−1}and reaches full opening for ${Q}_{0}=900$ dm

^{3}min

^{−1}.

^{−1}). The analogous results achieved with a 40% stiffer spring (${k}_{spr}={k}_{2}=57.4$ N mm

^{−1}) are presented in Figure 13.

^{3}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

#### 5.3. Estimation of the Check Valve Energy Efficiency

^{3}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 dm

^{3}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).

^{3}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 dm

^{3}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

^{3}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.

^{3}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.

## 7. Summary and Conclusions

^{3}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 $Q=900$ dm

^{3}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 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 $Q=300$ dm
^{3}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 $Q=600$ dm
^{3}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 $Q=900$ dm
^{3}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.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

Indices | |

$0,1,2$ | flow line |

$cv$ | check valve |

$cyl$ | hydraulic cylinder |

p | poppet of the check valve |

$ret$ | return line |

Parameters | |

${A}_{cyl},\phantom{\rule{0.277778em}{0ex}}{A}_{cyl,r}$ | cylinder piston area, piston area on the rod side (m^{2}) |

${A}_{pA},\phantom{\rule{0.277778em}{0ex}}{A}_{pB}$ | valve poppet left/right sectional area (m^{2}) |

${A}_{pC}$ | valve pilot sectional area (m^{2}) |

${B}_{f}$ | fluid bulk modulus (Pa) |

${C}_{1\epsilon},\phantom{\rule{0.222222em}{0ex}}{C}_{2\epsilon},\phantom{\rule{0.222222em}{0ex}}{C}_{\mu}$ | turbulence model constants (-) |

${E}_{total}$ | total energy consumption per operating cycle (kJ) |

${F}_{load}$ | external force exerted on the hydraulic cylinder (N) |

${F}_{hs},\phantom{\rule{0.277778em}{0ex}}{F}_{hd},\phantom{\rule{0.277778em}{0ex}}{F}_{spr},\phantom{\rule{0.277778em}{0ex}}{F}_{f}$ | force: hydrostatic, hydrodynamic, spring, friction (N) |

$I,\ell $ | turbulence model factors: intensity, length scale (-, m) |

${Q}_{nom}$ | nominal flow rate (dm^{3} min^{−1}) |

${Q}_{0},\phantom{\rule{0.222222em}{0ex}}{Q}_{1},\phantom{\rule{0.222222em}{0ex}}{Q}_{2}\phantom{\rule{0.222222em}{0ex}}{Q}_{ret}$ | flow rate: pump, line 1, 2, return (dm^{3} min^{−1}) |

U | control valve control signal (percentage) (%) |

${V}_{0},\phantom{\rule{0.277778em}{0ex}}{V}_{1},\phantom{\rule{0.277778em}{0ex}}{V}_{2}$ | volume: supply line, line 1,2 (m^{3}) |

${d}_{p},\phantom{\rule{0.277778em}{0ex}}{l}_{p},\phantom{\rule{0.277778em}{0ex}}{c}_{rp},\phantom{\rule{0.277778em}{0ex}}{e}_{p}$ | poppet dimensions: diameter, length, clearance, eccentricity (m) |

${k}_{spr}$ | valve spring stiffness (N m^{−1}) |

${m}_{p},\phantom{\rule{0.277778em}{0ex}}{m}_{p2},\phantom{\rule{0.277778em}{0ex}}{m}_{cyl}$ | mass: valve poppet, valve pilot, cylinder piston with rod (kg) |

${p}_{0},\phantom{\rule{0.277778em}{0ex}}{p}_{1},\phantom{\rule{0.277778em}{0ex}}{p}_{2},\phantom{\rule{0.277778em}{0ex}}{p}_{ret}$ | pressure: supply line, 1, 2, return line (MPa) |

$\Delta p$ | pressure drop (MPa) |

${q}_{rev}$ | pump piston stroke rate per revolution (m^{3}/rev) |

${s}_{k},\phantom{\rule{0.222222em}{0ex}}{s}_{\epsilon}$ | turbulence model constants (-) |

$t,\phantom{\rule{0.277778em}{0ex}}{t}_{start}$ | time, start-up time (s) |

${v}_{p},\phantom{\rule{0.277778em}{0ex}}{v}_{cyl},\phantom{\rule{0.277778em}{0ex}}{v}_{avg}$ | velocity: poppet, cylinder piston, fluid (average) (m s^{−1}) |

${x}_{p},\phantom{\rule{0.277778em}{0ex}}{x}_{cyl}$ | position: valve poppet, hydraulic cylinder piston (m) |

${x}_{s0}$ | initial valve spring deflection (m) |

z | number of pistons (pump) (-) |

$\alpha $ | poppet head opening angle (°) |

$\eta $ | fluid dynamic viscosity (Pa s) |

$\lambda $ | check valve jet angle coefficient (-) |

$\nu $ | fluid kinematic viscosity (m^{2} s^{−1}) |

${\mu}_{t}$ | turbulent viscosity (m^{2} s^{−1}) |

$\rho $ | fluid density (kg m^{−3}) |

$\phi $ | damping coefficient (N s m^{−1}) |

${\Phi}_{g}$ | valve poppet friction force geometrical factor (m) |

${\omega}_{0},\phantom{\rule{0.277778em}{0ex}}{\omega}_{nom}$ | pump rotational speed, pump nominal speed (rev/s) |

## References

- Ye, J.; Zhao, Z.; Zheng, J.; Salem, S.; Yu, J.; Cui, J.; Jiao, X. Transient Flow Characteristic of High-Pressure Hydrogen Gas in Check Valve during the Opening Process. Energies
**2020**, 13, 4222. [Google Scholar] [CrossRef] - Ye, J.; Zhao, Z.; Cui, J.; Hua, Z.; Peng, W.; Jiang, P. Transient flow behaviors of the check valve with different spool-head angle in high-pressure hydrogen storage systems. J. Energy Storage
**2022**, 46, 103761. [Google Scholar] [CrossRef] - Kim, N.S.; Jeong, Y.H. An investigation of pressure build-up effects due to check valve’s closing characteristics using dynamic mesh techniques of CFD. Ann. Nucl. Energy
**2021**, 152, 107996. [Google Scholar] [CrossRef] - Wen, Q.; Liu, Y.; Chen, Z.; Wang, W. Numerical simulation and experimental validation of flow characteristics for a butterfly check valve in small modular reactor. Nucl. Eng. Des.
**2022**, 391, 111732. [Google Scholar] [CrossRef] - Lai, Z.; Karney, B.; Yang, S.; Wu, D.; Zhang, F. Transient performance of a dual disc check valve during the opening period. Ann. Nucl. Energy
**2017**, 101, 15–22. [Google Scholar] [CrossRef] - Gomez, I.; Gonzalez-Mancera, A.; Newell, B.; Garcia-Bravo, J. Analysis of the Design of a Poppet Valve by Transitory Simulation. Energies
**2019**, 12, 889. [Google Scholar] [CrossRef] [Green Version] - Deng, H.; Liu, Y.; Li, P.; Ma, Y.; Zhang, S. Integrated probabilistic modeling method for transient opening height prediction of check valves in oil-gas multiphase pumps. Adv. Eng. Softw.
**2018**, 118, 18–26. [Google Scholar] [CrossRef] - Ling, M.; He, X.; Wu, M.; Cao, L. Dynamic Design of a Novel High-Speed Piezoelectric Flow Control Valve Based on Compliant Mechanism. IEEE/ASME Trans. Mechatron.
**2022**, 1–9. [Google Scholar] [CrossRef] - Shin, S.M.; Kim, D.S.; Kang, H.G. Power-operated check valve in abnormal situations. Nucl. Eng. Des.
**2018**, 330, 28–35. [Google Scholar] [CrossRef] - Morgan, S.C.; Hendricks, A.D.; Seto, M.L.; Sieben, V.J. A Magnetically Tunable Check Valve Applied to a Lab-on-Chip Nitrite Sensor. Sensors
**2019**, 19, 4619. [Google Scholar] [CrossRef] [Green Version] - Gao, Z.x.; Liu, P.; Yue, Y.; Li, J.y.; Wu, H. Comparison of Swing and Tilting Check Valves Flowing Compressible Fluids. Micromachines
**2020**, 11, 758. [Google Scholar] [CrossRef] [PubMed] - Baluyev, D.; Gusev, D.; Meshkov, S.; Nikanorov, O.; Osipov, S.; Rogozhkin, S.; Rukhlin, S.; Shepelev, S. Study of functional characteristics for safety system check valve using scaled model. Nucl. Energy Technol.
**2015**, 1, 99–102. [Google Scholar] [CrossRef] [Green Version] - Yang, L.J.; Waikhom, R.; Wang, W.C.; Jabaraj Joseph, V.; Esakki, B.; Kumar Unnam, N.; Li, X.H.; Lee, C.Y. Check-Valve Design in Enhancing Aerodynamic Performance of Flapping Wings. Appl. Sci.
**2021**, 11, 3416. [Google Scholar] [CrossRef] - Schickhofer, L.; Wimmer, J. Fluid–structure interaction and dynamic stability of shock absorber check valves. J. Fluids Struct.
**2022**, 110, 103536. [Google Scholar] [CrossRef] - Filo, G.; Lisowski, E.; Rajda, J. Pressure Loss Reduction in an Innovative Directional Poppet Control Valve. Energies
**2020**, 13, 3149. [Google Scholar] [CrossRef] - Filo, G.; Lisowski, E.; Rajda, J. Design and Flow Analysis of an Adjustable Check Valve by Means of CFD Method. Energies
**2021**, 14, 2237. [Google Scholar] [CrossRef] - Novak, P.; Guinot, V.; Jeffrey, A.; Reeve, D.E. Hydraulic Modelling—An Introduction: Principles, Methods and Applications; CRC Press, Taylor & Francis Group: Boca Raton, FL, USA, 2010. [Google Scholar]
- Lisowski, E.; Filo, G. CFD analysis of the characteristics of a proportional flow control valve with an innovative opening shape. Energy Convers. Manag.
**2016**, 123, 15–28. [Google Scholar] [CrossRef] - Valdés, J.R.; Miana, M.J.; Núñez, J.L.; Pütz, T. Reduced order model for estimation of fluid flow and flow forces in hydraulic proportional valves. Energy Convers. Manag.
**2008**, 49, 1517–1529. [Google Scholar] [CrossRef] - Posa, A.; Oresta, P.; Lippolis, A. Analysis of a directional hydraulic valve by a Direct Numerical Simulation using an immersed-boundary method. Energy Convers. Manag.
**2013**, 65, 497–506. [Google Scholar] [CrossRef] - Lisowski, E.; Filo, G.; Rajda, J. Pressure compensation using flow forces in a multi-section proportional directional control valve. Energy Convers. Manag.
**2015**, 103, 1052–1064. [Google Scholar] [CrossRef] - Anh, P.N.; Bae, J.S.; Hwang, J.H. Computational Fluid Dynamic Analysis of Flow Rate Performance of a Small Piezoelectric-Hydraulic Pump. Appl. Sci.
**2021**, 11, 4888. [Google Scholar] [CrossRef] - Woldemariam, E.T.; Lemu, H.G.; Wang, G.G. CFD-Driven Valve Shape Optimization for Performance Improvement of a Micro Cross-Flow Turbine. Energies
**2018**, 11, 248. [Google Scholar] [CrossRef] [Green Version] - Chen, F.; Qian, J.; Chen, M.; Zhang, M.; Chen, L.; Jin, Z. Turbulent compressible flow analysis on multi-stage high pressure reducing valve. Flow Meas. Instrum.
**2018**, 61, 26–37. [Google Scholar] [CrossRef] - Jin-yuan, Q.; Lin, W.; Zhi-jiang, J.; Jian-kai, W.; Han, Z.; An-le, L. CFD analysis on the dynamic flow characteristics of the pilot-control globe valve. Energy Convers. Manag.
**2014**, 87, 220–226. [Google Scholar] [CrossRef] - Scuro, N.; Angelo, E.; Angelo, G.; Andrade, D. A CFD analysis of the flow dynamics of a directly-operated safety relief valve. Nucl. Eng. Des.
**2018**, 328, 321–332. [Google Scholar] [CrossRef] - ANSYS Fluent Tutorial Guide, 18.0 ed.; ANSYS Inc.: Canonsburg, PA, USA, 2017; Available online: http://www.ansys.com (accessed on 6 June 2022).
- ISO:4411:2019; Hydraulic Fluid Power—Valves—Determination of Differential Pressure/Flow Rate Characteristics. ISO: Geneva, Switzerland, 2019.

**Figure 1.**Scheme of the analyzed tunable check valve: 1—body, 2, 3—cover, 4—poppet, 5—ball, 6—disc, 7—spring, 8—pilot spool; A,B—connection ports.

**Figure 2.**Scheme of hydraulic system: 1—pump, 2—relief valve, 3—control valve, 4—check valve, 5—actuator; (A-B), (B-A)—flow directions.

**Figure 3.**Balance of forces acting on the check valve poppet; (

**a**) A-B flow direction, (

**b**) B-A flow direction.

**Figure 4.**Mesh model with refined areas marked: (

**a**) initial, (

**b**) dense, refined poppet head and seat, (

**c**) double dense, refined; A,B—connection ports.

**Figure 5.**Results of CFD simulation for poppet position ${x}_{p}=1.0$ mm: (

**a**) pressure distribution, (

**b**) velocity distribution; $\lambda =67.{8}^{\circ}$.

**Figure 6.**Results of CFD simulation for poppet position ${x}_{p}=3.0$ mm: (

**a**) pressure distribution, (

**b**) velocity distribution; $\lambda =62.{1}^{\circ}$.

**Figure 7.**Results of CFD simulation for poppet position ${x}_{p}=5.0$ mm: (

**a**) pressure distribution, (

**b**) velocity distribution; $\lambda =56.{0}^{\circ}$.

**Figure 10.**Piston pump flow rate with an average value of: $100\phantom{\rule{0.222222em}{0ex}}\left(1\right)$, $200\phantom{\rule{0.222222em}{0ex}}\left(2\right)$, $300\phantom{\rule{0.222222em}{0ex}}\left(3\right)$, $600\phantom{\rule{0.222222em}{0ex}}\left(4\right)$, and $900\phantom{\rule{0.222222em}{0ex}}\left(5\right)$ dm

^{3}min

^{−1}.

**Figure 11.**Check valve gap width ${x}_{p}\left(t\right)$ and hydraulic actuator displacement ${c}_{cyl}\left(t\right)$ for the nominal stiffness ${k}_{spr}={k}_{nom}=41.0$ N mm

^{−1}; (

**a**) A-B flow direction, (

**b**) B-A flow direction.

**Figure 12.**Check valve gap width ${x}_{p}\left(t\right)$ and hydraulic actuator displacement ${c}_{cyl}\left(t\right)$ for the reduced stiffness ${k}_{spr}={k}_{1}=24.6$ N mm

^{−1}; (

**a**) A-B flow direction, (

**b**) B-A flow direction.

**Figure 13.**Check valve gap width ${x}_{p}\left(t\right)$ and hydraulic actuator displacement ${c}_{cyl}\left(t\right)$ for the increased stiffness ${k}_{spr}={k}_{2}=57.4$ N mm

^{−1}; (

**a**) A-B flow direction, (

**b**) B-A flow direction.

**Figure 14.**Pressure drop on check valve for different spring rates: (

**a**) reduced, (

**b**) nominal, (

**c**) increased; flow direction A-B.

**Figure 15.**Pressure drop on check valve for different spring rates: (

**a**) reduced, (

**b**) nominal, (

**c**) increased; flow direction B-A.

**Figure 16.**Pressure drop on check valve for different initial spring tension values: (

**a**) reduced, (

**b**) nominal, (

**c**) increased; flow direction A-B.

**Figure 17.**Pressure drop on check valve for different initial spring tension values: (

**a**) reduced, (

**b**) nominal, (

**c**) increased; flow direction B-A.

**Figure 19.**Energy consumption of the valve during the working cycle, flow direction: (

**a**) A-B, (

**b**) B-A.

**Figure 20.**Test bench: (

**a**) scheme, (

**b**) valve; 1—pump, 2—relief valve, 3—tested check valve, 4—control valve, 5—throttle valve, 6—filter, 7—flow meter, 8—temperature gauge, 9,10—pressure transducers, 11—computer DAQ system.

**Figure 21.**Comparison of the pressure drop on the check valve from numerical simulations and laboratory tests as a function of flow rate; vertical markers—standard deviation.

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 |

m^{2} s^{−1} | kg m^{−3} | °C | dm^{3} min^{−1} | dm^{3} min^{−1} | MPa |

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 |

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 | $2.32$% |

Input | Q (dm^{3} 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 |

${\mathit{x}}_{\mathit{s}0}$ mm | Q dm ^{3} 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 |

${\mathit{k}}_{\mathbf{spr}}$ N/mm | Q dm ^{3} 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 |

**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 dm ^{3} 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 | |||
---|---|---|---|---|---|---|---|

${\mathbf{E}}_{\mathbf{cv}}\phantom{\rule{0.277778em}{0ex}}\%$ | ${\mathbf{E}}_{\mathbf{cyl}}\phantom{\rule{0.277778em}{0ex}}\%$ | ${\mathbf{E}}_{\mathbf{cv}}\phantom{\rule{0.277778em}{0ex}}\%$ | ${\mathbf{E}}_{\mathbf{cyl}}\phantom{\rule{0.277778em}{0ex}}\%$ | ${\mathbf{E}}_{\mathbf{cv}}\phantom{\rule{0.277778em}{0ex}}\%$ | ${\mathbf{E}}_{\mathbf{cyl}}\phantom{\rule{0.277778em}{0ex}}\%$ | ||

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 |

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## Share and Cite

**MDPI and ACS Style**

Lisowski, E.; Filo, G.; Rajda, J.
Analysis of Energy Loss on a Tunable Check Valve through the Numerical Simulation. *Energies* **2022**, *15*, 5740.
https://doi.org/10.3390/en15155740

**AMA Style**

Lisowski E, Filo G, Rajda J.
Analysis of Energy Loss on a Tunable Check Valve through the Numerical Simulation. *Energies*. 2022; 15(15):5740.
https://doi.org/10.3390/en15155740

**Chicago/Turabian Style**

Lisowski, Edward, Grzegorz Filo, and Janusz Rajda.
2022. "Analysis of Energy Loss on a Tunable Check Valve through the Numerical Simulation" *Energies* 15, no. 15: 5740.
https://doi.org/10.3390/en15155740