Modeling and Experimental Validation of a Reed Check Valve for Hydraulic Applications

Reed check valves have been studied extensively in the context of two-stroke internal combustion engines [1–7], compressors [8–10], and general air systems [11]. The relatively low mass of the reed makes this valve desirable for use in systems that operate at high frequencies. A range of high-frequency hydraulic circuits, including piston pumps and soft-switched switch-mode circuits [12,13], make use of check valves. In many cases, the performance of such systems is limited by the response time of the check valves. During valve transition, backflow and throttling create leakage and energy losses. Implementing reed valves in hydraulic circuits presents an opportunity to increase efficiency and power density by decreasing check valve switching time.

A common approach for modeling check valves is to develop equations of motion describing the valve dynamics, the fluid domain, and their interaction. Such models typically contain several parameters which can be estimated using analytical derivation [14], experiments [15], or numerical simulation [16]. The advantage of an analytical approach is reduced cost both in terms of constructing and running experiments as well as computational resources. However, experiments and numerical simulation can provide increased accuracy by accounting for more physical phenomena with fewer assumptions. The use of numerical simulation, such as finite element analysis (FEA) and computational fluid dynamics (CFD), is becoming increasingly feasible for the design and optimization of dynamic systems containing check valves.

In this paper, a novel reed valve model is presented. The model is experimentally validated in a piston pump system using direct measurement of reed position and pump delivery. The reed valve and pump circuit model are then derived including empirical parameters. Next, the experimental setup is described. Finally, the model and experimental results are compared, demonstrating validation of the analytical model.

Before beginning model development, a preliminary reed valve was designed to test the concept and guide model development. The reed valve assembly, shown in Fig. 1, is comprised of three parts: the reed, a stop to limit the maximum opening, and a seat to provide the reed with support during large negative pressure differentials. The cut-out feature in the stop is used to minimize stiction between the reed and the stop as well as provide optical access to measure the reed position experimentally. The design of the seat slots is an important consideration. Larger slots allow for a larger flow area, while smaller slots allow for greater support of the reed, thus lowering its stress. Similarly, for the stop, a larger maximum opening allows for a smaller pressure drop across the reed while a smaller maximum opening helps prevent plastic deformation of the reed and allows it to close faster, resulting in less backflow.

An iterative process was used to design the seat. First, a seat was designed to provide sufficient support for the reed at the highest negative pressure differential expected. Next, the maximum pressure drop across the seat was estimated. After several iterations, the design shown in Fig. 2 was found to provide a good compromise between stress in the reed and pressure drop across the seat. The reed and stop geometric design parameters were primarily determined by the seat geometry. Note that the iterative design process was not optimized. Custom parts were designed based on rough estimates to be conservative and performance was not a priority as the ultimate goal of this work is modeling.

In this study, a 1.35 mm thick Delrin reed, shown in Fig. 3, was used. Grooves 1.59 mm wide and 0.51 mm deep were added to the reed. The purpose of these grooves is twofold. First, by reducing the contact area between the reed and the stop, stiction is reduced. Second, the area of the reed that is directly exposed to upstream pressure while the valve is closed is increased. Both stiction and a reduced reed area directly exposed to upstream pressure delay valve opening in response to a positive pressure differential. Corresponding dimensions of the reed valve are given in Table 1.

To ensure the reed would not experience any plastic deformation, a 3D finite element analysis was performed. At the maximum tip opening, 1.9 mm, the maximum von Mises stress in the reed is 26.2 MPa compared to a yield stress of 63.0 MPa for Delrin.

This paper seeks to develop a modeling approach for a reed of arbitrary shape to allow for evaluation across a wide range of geometries for the purpose of design space exploration and optimization. First, Euler–Bernoulli beam theory is applied to a uniform reed to obtain an equation of motion. Then, the equation of motion is modified to predict the behavior of a nonuniform reed. Finally, the valve is modeled as an orifice to predict the flowrate through it.

Implicit in the modeling approach is the assumption that the reed shares the properties of an Euler–Bernoulli beam, namely, that shear and rotational inertia are negligible. The relative difference in the fundamental eigenvalue of a beam, $r1=(λEB1−λFEA1)/λFEA1$⁠, computed with Euler–Bernoulli beam theory and FEA, is the one measure used to determine the applicability of Euler–Bernoulli beam theory [17]. To compute the relative difference in the fundamental eigenvalue of the reed, the presence of grooves and a rounded tip are neglected and it is assumed that these features do not affect whether or not Euler–Bernoulli beam theory is applicable. The effects of these features are, however, captured later during modeling. A method of computing the eigenvalues of a beam with a step change in width, according to Euler–Bernoulli beam theory, has been developed by Naguleswaran [18]. Commercial fea software is used to compute the actual eigenvalues of a beam. Applying these two methods to the present reed yields a value of r₁ = 0.032. For practical purposes, such a small relative difference is sufficient to conclude the reed may be modeled as an Euler–Bernoulli beam.

where m is the effective reed mass, c is the damping coefficient, γ₁ is the first mode function parameter, k is the reed stiffness, x₀ is initial tip deflection, $ΔPu−d$ is the pressure differential between upstream and downstream, A is the reed face area, and X is the pressure load multiplier, described in Sec. 5.2.

A lumped parameter model is used, which does not account for the nonuniform pressure distribution acting on the reed without some type of correction. The pressure load multiplier acts as such a correction to allow for modeling of the pump with only information about the pressure at some location upstream and downstream of the valve. In this paper, for the purpose of calculating $ΔPu−d$⁠, the pressure is taken at the location of the pressure transducers which are used to experimentally determine the pressure load multiplier. The exact location used to measure the upstream and downstream pressure is a free choice, as long as there is consistency between the calculation of $ΔPu−d$ and X.

For a reed valve operating in hydraulic oil, damping is provided both by internal damping of the reed as it dissipates energy during deformation and fluid damping as the reed does work on the fluid. If viscous fluid damping is assumed, the damping coefficient may be defined to include both sources of damping in series. Also, fluid in the vicinity of the reed is accelerated with the reed, creating an added mass effect. For the cases presented in this paper, inclusion of added mass in the model did not make a significant difference and therefore it was neglected.

The first mode function parameter, γ₁, is defined in Ref. [11]. γ₁ is a function of the mode shape, ψ, which is effectively the nondimensional deflection of the reed as a function of s. For the purpose of this paper, γ₁ can be understood as a parameter that accounts for the deflection shape in order to model the reed motion at a single point, namely, the tip. The deflection shape and therefore mode function of a static cantilever subjected to a variety of load cases can be found using simple beam deflection formulas that assume a uniform, homogenous, linearly elastic, slender beam experiencing small deflection. The mode function of a freely vibrating Euler–Bernoulli reed with a uniform or step change in cross section can be determined using the method developed by Naguleswaran [18]. Mode functions for four such cases are plotted in Fig. 4.

The mode function is relatively insensitive to the particular load case, suggesting it is possible to assume a single mode shape without introducing significant error. To quantitatively justify this assumption, a range of loading cases are considered. In the nonuniform loading case, the reed is subjected to a maximum pressure differential at $s/L=0$ that linearly decreases to zero at $s/L=1$⁠. The stepped reed increases in width by a factor of 2 at $s/L=R1$⁠. The value of γ₁ for each case shown in Fig. 4 is listed in Table 2.

A reed in a hydraulic system will be subjected to a range of loading cases so γ₁ is not necessarily a constant. However, as shown in Table 2, γ₁ only varies by a 1.7% across the range of cases considered, so it is reasonable to approximate it as constant for a given reed shape. Therefore, an appropriate value of γ₁ for the prototype reed valve is 1.549.

From Eq. (3), the effective reed mass is 5.18 g compared to a nominal reed mass of 5.78 g. The final two properties that need to be determined are the damping coefficient and pressure load multiplier. Due to the difficulty in calculating these analytically, they are experimentally determined. The damping coefficient is adjusted to best fit the experimental data. The authors acknowledge tuning of the damping coefficient as the most significant limitation of the current model. Further work is needed to develop a more rigorous damping model which would likely require experiments or numerical simulation using coupled FEA and CFD. Measurement of the pressure load multiplier is outlined in Sec. 5.2.

A travel limit on the read at the valve seat is modeled by setting position, velocity, and acceleration to zero if a position of less than zero is calculated. Similarly, for the stop if a position greater than x_stop is calculated, x is set equal to x_stop, and velocity and acceleration are a set to zero.

In this paper, stiction and flow forces are assumed to be negligible. At low speeds, it is expected that steady flow forces are negligible. Further, transient flow forces are typically low compared to other forces [14]. Previous research of disk style check valves operating under similar conditions to those described in the present work has shown that the exclusion of stiction and flow forces from a disk style check valve model do not adversely affect its agreement with experimental measurements [15]. The authors did observe stiction when using reeds without grooves. Indeed, this was the motivating factor behind cutting grooves to reduce stiction, a measure which proved very effective. The addition of grooves markedly reduced the stiction effect to a level negligible compared to other forces. It is possible, however, that these terms may become significant in reed valves operating at higher pressures and/or speeds or with different geometries. Therefore, the inclusion of these terms should be carefully considered when modeled check valves.

A lumped parameter pump model based on the simplified circuit shown in Fig. 5, was constructed to provide a system in which the reed valve model could be validated. A crank-slider driven piston displaces fluid in the cylinder at a rate of $dVcyl/dt$⁠, pumping from the tank through the inlet check valve at a rate of Q_v and to the load at constant pressure P_load through the delivery check valve at a rate of $Qv,del$⁠.

Note that Eq. (10) has been modified according to Van de Ven [22] such that P_cyl is absolute pressure rather than gauge. It is assumed that there is no leakage or cavitation.

The discharge coefficient and pressure load multiplier of the reed valve were measured experimentally using a hydraulic power unit to create flow through the reed valve. Pressure transducers measured the pressure drop across the valve while a gear flowmeter measured the volumetric flowrate through the valve. A laser triangulation sensor (LTS) measured the reed tip displacement as shown in Fig. 6. The flowrate was varied from approximately zero to $1.3×10−4 m3/s$⁠. Steady-state measurements were taken and averaged over a 10 second sampling period.

To measure the reed valve opening, an acrylic sight glass was installed in the check valve manifolds to allow for optical access as shown in Fig. 6. A technique for measuring position through multiple interfaces was developed by Peterson and Peterson to measure film thickness [23]. Yudell and Van de Ven [24] showed that an LTS could be used to measure solenoid valve position through an acrylic glass in a hydraulic circuit. Building on their work, Knutson and Van de Ven [15] developed a mathematical method to relate the positon of a check valve to the LTS voltage output through a corrected scale factor. The corrected scale factor differs from the manufacturer's scale factor which assumes use in air. For the experimental setup described in this paper, the corrected scale factor was calculated to be 3.746 mm/V compared to the manufacturer's scale factor of 2.5 mm/V in air. Factors including the valve manifold geometry and acrylic thickness determine the corrected scale factor. The LTS measurement system is nearly identical to that used by Knutson and Van de Ven [15]. As a result, the position measurement uncertainty is expected to be approximately the same value of ±1.12% [15].

The experimental discharge coefficient measurements and empirical correlation are shown in Fig. 7. The maximum flowrate measured was approximately $1.6×10−4 m3/s$⁠. Considering the hydraulic diameter of the groove to be the characteristic length, this corresponds to a Reynolds number of 108.

where Q_o has units $m3/s$⁠.

The experimental pressure load multiplier measurements and empirical correlation are shown in Fig. 8.

where x has units of meters. Experimental measurements of X suggest there is a maximum reed opening, for which additional pressure differential does cause further deflection. Since this value of approximately 1.4 mm is less than the maximum opening allowed by the stop of 1.9 mm, the reed never contacts the stop during the following experiments. As the reed valve closes, the pressure load multiplier approaches a value of 0.86. Measurements below 0.3 mm could not be made presumably due to lack of fine control of the needle valve. There are many geometric factors that likely have an effect on the pressure load multiplier such as the reed, seat, and valve manifold geometry.

Experiments were performed using a single-piston linkage-driven pump with a displacement of 2.22 cc/rev. The experimental setup is shown in Fig. 9, and the pump experimental parameters are provided in Table 3.

Pump speed was controlled with a variable frequency drive (VFD) and the load pressure was controlled with a needle valve. The pressure ripple from the single-cylinder pump is smoothed with a 0.95 l accumulator precharged to 2.41 MPa connected between the delivery check valve and the needle valve. To begin an experiment, the needle valve was opened and the variable frequency drive was set to the desired speed. Once the pump speed reached steady-state, the needle valve was gradually closed until the accumulator pressure reached approximately 2.76 MPa. Variability in the load pressure was due to coarse adjustment of the needle valve. The pump was allowed to run until it reached cyclic steady-state before data was taken. Cyclic steady-state was identified by a constant flowrate through the gear flowmeter, indicating no net flow into or out of the accumulator over a cycle. Experiments were conducted at three speeds—595 rpm, 743, rpm, and 893 rpm.

Due to the short duration of the experiments and low flowrate relative to tank volume, the effects of the fluid temperature were not considered. The duration of each experiment was approximately 1–2 min with sufficient time between to prevent a rise in temperature over the course of the experiments. Therefore, oil properties were selected at room temperature and the instrumentation did not require any temperature corrections to their calibration.

Measurements of piston position, cylinder and load pressure, volumetric flowrate, and reed opening were taken at a sampling frequency of 10 kHz. A linear variable differential transformer was connected to the piston to measure its position. Silicon on sapphire pressure transducers with a response time of 0.2 ms and accuracy of 0.25% were placed at the cylinder and downstream of the needle valve. The gear flowmeter with an accuracy of 0.5% and repeatability of 0.1% was placed downstream of the needle valve to measure mean volumetric flowrate. A LTS measured the reed position.

Since access was limited to a single LTS, two experiments were conducted at each pumping speed to measure both the inlet and delivery reed valve opening for a total of six sets of experimental data. Modeled and measured reed-opening data were aligned at top dead center using piston position data. Figures presented in this section include vertical lines at the top (TDC) and bottom dead center (BDC) positions of the piston.

A comparison of the modeled and measured cylinder pressure is shown in Fig. 10. The other cases have similar trends so only a single representative cylinder pressure plot is shown here. After the piston reaches bottom dead center, there is a slight delay before the cylinder pressure rapidly increases. There are several reasons for this. First, there is a delay in closing of the inlet valve as will be shown shortly. This allows for backflow into the tank. Second, there was a small amount of clearance in the plain bearings, allowing the linkages to deflect slightly under load such that compression of the cylinder fluid by the piston was delayed. Similarly, a delay in closing of the delivery valve causes the cylinder pressure to remain at the load pressure past top dead center, allowing for backflow into the piston cylinder. The delays in valve opening and closing are the result of many factors including stiction, valve and fluid inertia, fluid compressibility, and linkage deflection. Cylinder pressure drives the valve motion but is also affected by it, illustrating the highly coupled nature of the piston pump and the need to model it accordingly.

The results for the inlet reed valve opening, x, are shown in Figs. 11 and 12 and for the delivery reed in Figs. 13 and 14. Reed-opening plots are not included for 743 rpm as they do not provide addition insightful information. Agreement between the model and experiments agreement at 743 rpm is similar to what is observed at 893 rpm. Experimental data are very repeatable from cycle to cycle as observed in Figs. 11–14.

Agreement between the modeled and measured reed opening is remarkably good in all four cases presented with the possible exception of the inlet valve at 893 rpm, although the timing of the opening and closing transition events are still captured quite well in that case. There are several possible sources, both in the model and experiments, for the discrepancies observed.

First, the pressure and flow dynamics in a pump are quite complex. A lumped parameter approach, while favorable due to its simplicity and low computational expense, cannot be expected to capture all the physics of the system. For example, the effect of pressure waves induced in the hydraulic lines are not captured by the present model. A more accurate pump model could be obtained by including the effects of inertance, resistance, and capacitance in the transmission lines [26]; however, this is beyond the scope of the present work. Nonetheless, it can predict several metrics relevant to design and optimization including as reed transition timing, maximum opening, and volumetric efficiency.

Second, consider the LTS measurements and how they may differ from the actual reed valve opening. During its operation, the single-cylinder pump-induced moderate vibration of the experimental setup. The effects of vibrations in the test setup are not captured by the model and therefore have a bearing on model agreement with experimental results. Nonetheless, experiments have shown that a simple, semi-empirical reed valve model can capture the behavior of reed valves in a pumping circuit with a high degree of accuracy.

This paper also seeks to show that the model presented can predict the pump delivery, i.e., backflow, which depends strongly on check valve behavior. Because the pump has a significant dead volume and the oil contains an estimated 0.9% air content, the cylinder volume is expected to be quite compressible. Coupled with the fact that the valve is nonoptimized, significant backflow is to be expected. While this is not desirable from a design standpoint, it is useful for validating the check valve model. Table 4 shows the error between measured and modeled pump delivery. Measured delivery is the average value over 5 s of pumping.

Error in the pump delivery was less than 2.2% in all cases. This reveals several things about the model. First is that the reed position is not so tightly coupled to the average flowrate that error in one will necessarily induce error in the other. For example, the worst agreement in reed opening was for the inlet reed at 893 rpm which corresponded to excellent agreement in the pump delivery. Second, application of the steady-state orifice equation is a good approximation across the range of conditions in this study. It seems likely that careful, experimental measurement of the discharge coefficient contributes significantly to the accuracy of the flowrate model.

In this paper, a mathematical model of a reed style check valve based on Euler–Bernoulli beam theory was constructed and experimentally validated in the context of a hydraulic piston pump. A novel approach was used to account for the highly nonuniform pressure distribution acting on the reed by experimentally determining the pressure load multiplier by using the reed itself as a force transducer. Flowrate through the valve was characterized by determining its effective orifice area and fluid displacement per unit reed tip opening and applying the steady orifice equation.

The reed valve model was experimentally validated in the context of a 2.22 cc/rev single-cylinder piston pump using two metrics—pump delivery and reed tip displacement. Results show excellent agreement between the model and experimentally measured reed opening across the range of conditions studied. Furthermore, the error between predicted and measured pump delivery remained less than 2.2% for all cases.

The primary motivation in developing a lumped parameter is that the analytical model was allowed for design space exploration and optimization toward realization of the potential advantages of reed valves in hydraulic circuits. This paper presents a modeling approach that should be extended to a wider range of operating conditions so that the performance of an optimized reed valve design can be measured against current check valve technology. To generalize the reed valve model, stiction and flow forces should be included in addition to a predictive damping model.

• National Science Foundation (Grant No. 1414053; Funder ID: 10.13039/100000001).

A =

reed valve face area

A_o =

valve orifice area

A_p =

piston area

c =

fluid-reed damping coefficient

C_d =

discharge coefficient

h =

reed thickness

k =

effective reed stiffness

l =

connecting rod length

L =

reed length

m =

effective reed mass

P_load =

load pressure

P_cyl =

cylinder pressure

P₀ =

atmospheric pressure

r =

crank radius

R =

volume air fraction

Q_v =

orifice volumetric flowrate

Q_v =

valve volumetric flowrate

V_cyl =

cylinder volume

V_tdc =

cylinder volume at top dead center

W =

reed width

x =

reed tip opening

X =

pressure load multiplier

x₀ =

initial reed tip opening

β_eff =

effective fluid bulk modulus

γ =

ratio of specific heats of air

γ₁ =

reed deflection parameter

$ΔPu−d$ =

valve pressure differential

ρ =

fluid density

ψ =

reed mode shape