Keywords

1 Introduction

Reciprocating diaphragm pump consists of power end [1], hydraulic end, drive device, control system, hydraulic auxiliary system, feed and discharge compensation system, etc. The diaphragm is used to separate the transported pulp from the power end drive system, and the conveying granular pulp medium cannot contact with the reciprocating power end parts, avoiding the serious wear caused by solid particles to the pump [2]. As one of the core mechanical components of reciprocating diaphragm pump, the pump valve is susceptible to the particle size grading of conveying minerals, rheological characteristics and other factors of pulp, and is more prone to failure than other parts of the pump [3, 4].

With the increasing application of reciprocating pump in petrochemical industry, the basic theory research of reciprocating diaphragm pump has developed rapidly, but the research of domestic and foreign scholars focuses on the inhalation performance and volume efficiency and other issues, because the pump valve movement involves many factors, and the related basic theory is not perfect [5]. Scholars at home and abroad obtain the lift curve characteristics of the valve disc by establishing the method of the surface pressure difference and flow function of the valve disc, and verify the accuracy of the function [6,7,8,9,10,11] through experimental methods. For a long time, scholars who study pump valves generally derive according to the classical theory when studying the motion law of pump valves [12,13,14].

Because the problems involved in the pump operation are very complex, scholars at home and abroad are often limited to study the dynamic characteristics of the hydraulic end pump valve core under certain assumptions. This leads to a serious lag in the study of the valve core of the diaphragm pump. In 1968, German U. Adolph [15] established the second-order nonlinear inhomogeneous motion differential equation of lift with mass valve. The defect of this equation is that although it can be solved by numerical solution method, there is no method to solve the motion singularity. In 1977, The Japanese scholar [16] simulated the results of Adolf’s equation by numerical method. Although this method is relatively accurate, the object of the pump valve studied is relatively single, because the design parameters of the pump valve cannot be involved in the calculation, and must be designed before the calculation. Li Xiangyang [17] using the classical reciprocating pump theory, a new three-port valve plate structure is designed and optimized to adjust the angle of the non-dead point transition zone to obtain the optimal structure. Wang Teng [18] et al. studied the mechanism of the movement of the pump valve plate under the condition of ultra-high pressure, and established a set of special real-time measurement detection system, in order to optimize the life of the pump valve. Jiang’ao Zhao [19] et al. found the contact surface of cylinder block and valve plate as the key design problem to determine the pump performance and service life, conducted an experimental study, and summarized several typical structural optimizations, surface forming and reinforcement methods.

About all, the research on the mechanism of the pump valve of the reciprocating diaphragm pump has lasted for quite a long time. The mechanism of the pump valve studied by scholars at home and abroad is basically using different methods to establish equations of motion or using simulation software to try to avoid a series of mathematical problems that cannot be solved. There is no deep pump valve core in the process of movement dynamics and valve gap fluid flow rate problems, resulting in the lack of impact damage and erosion damage foundation. Therefore, the impact dynamic mechanics and valve gap flow rate are necessary in the design and optimization of pump valve structure.

According to the basic principles of dynamics and fluid mechanism. The method of numerical calculation is applied, and the dynamic characteristic motion curve of the diaphragm pump valve is obtained. The flow curve flowing through the valve gap and the measured data curve are compared to verify the accuracy of the applied method in this paper. For the practical problems of closer joint engineering, the relationship between the geometrical parameters of pump valve and the characteristic index of diaphragm pump is also calculated and discussed in detail.

2 Cone Valve Nonlinear Motion Model

2.1 Simplified Model of the Cone Valve

The working principle of diaphragm pump is shown in Fig. 1. The motor drives the crankshaft rotation through the reducer, and then transforms the rotating motion into linear motion through the connecting rod and crosshead, which drives the piston to do reciprocating motion. When the piston moves to the left end, the indoor pressure of the hydraulic oil chamber at the front of the piston decreases to/or below the atmospheric pressure. Under the action of atmospheric pressure and the positive pressure of the front pump of the inlet, the diaphragm in the diaphragm chamber moves to the left, and the slurry enters the chamber at the larger end of the diaphragm under the positive pressure until the piston reaches the extreme position on the left side. While in this process the discharge valve closes and the suction valve opens. When the piston moves to the right end, the suction valve is closed, and the piston pushes the hydraulic oil to push the rubber diaphragm in the diaphragm chamber to the right direction. At the same time. The discharge valve is opened with the help of pressure to deliver the slurry to the pipe.

Fig. 1.
figure 1

Schematic of the working principle of the diaphragm pump

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In the schematic, (1) motor (2) crankshaft (3) connecting rod (4) crosshead (5) piston liner (6) piston (7) guide rod (8) probe (9) diaphragm (10) diaphragm chamber (11) discharge valve (12) suction valve.

Due to the slurry does not contact the moving parts such as the piston during the work process, the abrasion of these parts is avoided. Therefore, the maintenance times and operation costs are reduced. As the part of the reciprocating diaphragm pump, the pump valve is the important link responsible for the pulp transportation. The design of diaphragm pump valve has an main impact on the flow rate and volume efficiency of pulp conveying. Through the reciprocating movement of diaphragm pump piston, the pump valve realizes periodic closed and opened. So, the diaphragm chamber realizes the alternating contact between inhalation, discharge and pipe cycle, as the way to complete the process of pulp inhalation and discharge of diaphragm pump circulation.

In Fig. 2, a simplified model diagram of the conical valve structure of a new type of diaphragm pump used in the CQPI. This type of conical valve is also commonly used in the domestic diaphragm pump series products.

Fig. 2.
figure 2

Schematic of the conical valve

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Among them, vk-seat hole slurry flow rate, dk-seat hole diameter, h-valve core rise, vx-valve gap slurry flow rate, df-valve core diameter, θ-valve core cone angle.

In the actual operation of the diaphragm pump, the movement law of the pump valve is very complex. Therefore, some small factors in the pump valve work are assumed in the practical engineering. According to the classical pump valve theory, the basic assumptions of diaphragm pump cone valve are as follows:

  1. 1)

    Liquid is incompressible;

  2. 2)

    Pump body parts are rigid;

  3. 3)

    The valve core quality is negligible;

  4. 4)

    The liquid cylinder block is always fully filled during inhalation;

  5. 5)

    The link is infinitely long.

2.2 Movement Law of the Cone Valve in the Steady State

Due to the complex movement law of the diaphragm pump valve, the stable state of the pump valve is generally analyzed and discussed firstly. And the stable state of the pump valve is defined: the speed of the liquid flowing through the valve gap is steady with time. And then, the valve core is in a certain fixed lift h, which is called the stable state. Therefore, in this state, the force analysis of the valve core is shown in Fig. 3:

Fig. 3.
figure 3

Loading analysis of valve core

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In which: G - The gravity of the cell and spring in the media, \(G = G_{0} \left( {1 - {\rho \mathord{\left/ {\vphantom {\rho {\rho_{0} }}} \right. \kern-0pt} {\rho_{0} }}} \right)\), G0, ρ0 and ρ respectively are the weight, density and dielectric density of the core spring in the air;

Fs-spring force, k corresponds to the spring stiffness coefficient;

P1, P2-indicates the pressure on the lower and upper surface of the diaphragm pump valve respectively.

According to the balance equation of the force on the valve cell of the diaphragm pump, and the area difference between the upper and lower surfaces of the valve core is not enough to affect the movement law of the valve core. Therefore, the balance equation of the valve core movement can be obtained as

$$ \frac{{P_{2} - P_{1} }}{\rho g} = \frac{{G + F_{s} }}{{\rho gA{}_{k}}} $$
(1)

where, Ak-the cross-sectional area of the seat hole. Equation (1) means that the valve cell of the diaphragm pump maintains a stable state through the pressure difference between its upper and lower surfaces.

Considering that the upper and lower surfaces of the hydraulic end valve core of the diaphragm pump are two different liquid levels, the Bernoulli equation when the liquid flows through the valve gap can be listed as follows:

$$ \frac{{P_{2} - P_{1} }}{\rho g} = (z_{1} - z_{2} ) + \frac{{v_{1}^{2} - v_{2}^{2} }}{2g} + \Delta h_{L} $$
(2)

where z1 and z2 indicate the liquid height on the upper and lower surfaces of the valve, respectively. ΔhL represents the hydraulic loss. According to the actual working state of the diaphragm pump, compared with the hydraulic loss of the valve, the height difference \(z_{1} - z_{2}\) and the velocity difference head \({{v_{1}^{2} - v_{2}^{2} } \mathord{\left/ {\vphantom {{v_{1}^{2} - v_{2}^{2} } {2g}}} \right. \kern-0pt} {2g}}\) of the upper and lower surface of the valve core are very small. Therefore, negligible compared with the hydraulic loss of the valve, the specific load \(H_{v}\) of the diaphragm pump valve can be obtained

$$ H_{v} = \frac{{G + F_{s} }}{{\rho gA_{k} }} $$
(3)

Based on the assumptions of the classical pump valve theory, the piston discharge flow is equal to the flow through the valve gap or seat. Therefore available based on the continuity conditions between the flows,

$$ q = \alpha l_{f} h\cos \theta v_{x} y(t) $$
(4)

where, q represents the flow rate through the valve gap. α shows the shrinkage coefficient of the orifice section.

From the hole outlet equation, the calculated expression for the valve gap flow velocity vx can be obtained,

$$ v_{x} = \frac{1}{{\sqrt {1 + \xi } }}\sqrt {2\frac{{P_{2} - P_{1} }}{\rho }} $$
(5)

According to the design of reciprocating diaphragm pump, the theoretical flow calculation formula of valve can be obtained as

$$ q = AR\omega \left[ {\sin \phi + \frac{\lambda }{2}\sin 2\phi } \right] $$
(6)

where, ω is the crank speed, φ is the crank rotation angle, and \(\lambda = {R \mathord{\left/ {\vphantom {R L}} \right. \kern-0pt} L}\) is the link ratio.

Replacing Eq. (6) and (5) into Eq. (3) and considering the link length L is much greater than R, ignoring the influence of the link ratio \(\lambda = {R \mathord{\left/ {\vphantom {R L}} \right. \kern-0pt} L}\). The calculation expression of the dynamic displacement of the valve core of the diaphragm pump valve in stable state is,

$$ y(t) = \frac{AR\omega \sin \phi }{{\mu l_{f} \cos \theta \sqrt {2g\frac{{G + F_{s} }}{{\rho gA_{f} }}} }} $$
(7)

where, μ - the flow coefficient of the pump valve; The lf - the perimeter of the pump valve cell.

Further, we can obtain the lift speed calculation expression of the diaphragm pump valve,

$$ \dot{y}(t) = \frac{{AR\omega^{2} \cos \phi }}{{\mu l_{f} \cos \theta \sqrt {2g\frac{{G + F_{s} }}{{\rho gA_{f} }}} }} $$
(8)

When estimating core displacement, the flow factor μ and specific load Hv can be assumed unchanged. The maximum core rise when the crank angle is \(\phi = 90^\circ\).

2.3 Movement of Pump Valve in Unstable State

As a special case in the steady state is only suitable for flow field calculation of non-automatic valves. In the flow field calculation of the non-automatic valve, the valve core is added to the flow field as a boundary condition to calculate the flow field characteristics. The diaphragm pump cone valve, as a spring automatic valve, needs to calculate its impact dynamic performance and its flow characteristics through the valve gap. Therefore, it is necessary to calculate the valve core in an unstable state. The movement law of the pump valve in the stable state is not applicable in practical engineering. When the diaphragm pump is working in practice, the valve core rise is changed and is non-stable state. In the process of valve core movement, the liquid involved in the movement will rise less and fall more, which is called Westphal phenomenon. Considered the Westphal phenomenon of the pump valve, the continuous equation of motion of the core will become,

$$ q = \alpha l_{f} \cos \theta v_{x} y \pm A_{f} \frac{dy}{{dt}} $$
(9)

Considering the characteristics of the Westphal phenomenon and the direction of the core movement speed, the negative sign in the formula can be removed, and the core speed is replaced into the type (7), so the calculation expression of the core lift in the unstable state is obtained as,

$$ y(t) = \frac{AR\omega \sin \phi }{{\mu l_{f} \cos \theta \sqrt {2g\frac{{G + F_{s} }}{{\rho gA_{f} }}} }} - \frac{{A_{f} AR\omega^{2} \cos \phi }}{{\left( {\mu l_{f} \cos \theta \sqrt {2g\frac{{G + F_{s} }}{{\rho gA_{f} }}} } \right)^{2} }} $$
(10)

Because the spring force changes with the expansion of the spring during the movement of the valve stem, it needs to be considered as a variable rather than a constant amount when calculating the motion lift curve. So put the spring force Fs = G + k(y0 + y) into type (10), and arrange the motion control equation of the valve core lift y is,

$$ y^{4} + \frac{{2F_{sv} }}{k}y^{3} + \left( {A\cos \phi + B} \right)y^{2} + \left( {C\cos \phi - D\sin^{2} \phi } \right) + E\cos^{2} \phi - F = 0 $$
(11)

The coefficients of the terms are as follows:

$$ A = \frac{{\gamma A_{f}^{2} AR\omega^{3} }}{{\mu gkl_{f}^{2} \cos^{2} \theta }},\;\;B = \frac{{F_{sv}^{2} }}{{k^{2} l_{f}^{2} }},\;\;C = \frac{{\gamma A_{f}^{2} AR\omega^{2} F_{sv} }}{{\mu^{2} l_{f}^{2} k^{2} \cos^{2} \theta }}, $$
$$ D = \frac{{\gamma A_{f} A^{2} R^{2} \omega^{2} }}{{2\mu^{2} gkl_{f}^{2} \cos^{2} \theta }},\;\;E = \frac{{A^{2} A_{f}^{2} \gamma^{2} R^{2} \omega^{4} }}{{4g^{2} k^{2} l_{f}^{4} \mu^{4} \cos^{4} \theta }},\;\;F = \frac{{\gamma A_{f} A^{2} R^{2} \omega^{2} F_{sv} }}{{2\mu gk^{2} l_{f}^{2} \cos^{2} \theta }}, $$
$$ \gamma = \rho g,\;\;\;\Delta = \frac{\omega }{{\mu l_{f} \cos \theta }}. $$

Equation (11) is the time domain control equation of the dynamic lift y(t) of a new diaphragm pump cone valve, which is a quartic nonlinear inhomogeneous equation. When the diaphragm pump performance and valve structure parameters are determined A, B, C, D, E, γ, ε and Δ of Eq. (11) are constant. Since Eq. (11) is a fourth-order nonlinear equation, it is difficult to solve by analytical or semi-analytical method. In this paper, we will use the ode45 algorithm in MATLAB to solve Eq. (11), and compare with the existing experimental test results to verify the correctness of the diaphragm pump cone valve motion model established in this paper. On this basis, the dynamic lift performance, valve gap flow rate and its optimization performance will be further discussed.

3 Results and Analysis and Discussion

3.1 Influence of Impact Speed and Spring Stiffness

Due to the Westphal phenomenon, the core displacement is not 0 when the crank angle is at 0° or 180°. According to type (13), the lag angle and lag height of the massless valve can be deduced.

The angle obtained when h = 180° is the closed lag angle:

$$ \tan \phi_{0} = \frac{{A_{f} \omega }}{{\mu l_{f} \cos \theta \sqrt {2g\frac{{G + F_{s} }}{{\rho gA_{f} }}} }} $$
(12)

At the \(\phi = 180^\circ\) time, the rise value of the valve core is the lag height:

$$ h_{d} = \frac{{A_{f} AR\omega^{2} }}{{\left( {\mu l_{f} \cos \theta \sqrt {2g\frac{{G + F_{s} }}{{\rho gA_{f} }}} } \right)^{2} }} $$
(13)

Combined the continuity Eq. (12). When the core is closed, the valve gap flow is 0. So the end speed can obtained:

$$ \dot{h}_{d} = \frac{{A_{f} AR\omega^{2} }}{{\mu l_{f} \cos \theta \sqrt {2g\frac{{G + F_{s} }}{{\rho gA_{f} }}} }} $$
(14)

Figure 4 shows the valve core lift and Westphal phenomenon flow curves under different strokes. It can be seen from (a) that in the process of impulse change, the difference between the opening lag angle under the three impulse is only 0.33°, while the difference between the closing lag angle is 1.23°. The maximum value can be clearly seen that the higher the impulse, the greater the lift maximum. Since the stroke of the piston is determined by the pump, the time of discharging the same flow rate is changed within a cycle. (b) reflects the flow curve contained in the Westphal phenomenon. The volumetric efficiency of the diaphragm pump is a very important index in the design. The larger the flow rate, the higher the lag height of the valve core. According to the pump valve mechanism, the greater the leakage flow caused by the lag height, the lower the volume efficiency of the diaphragm pump.

Fig. 4.
figure 4

Two curves under different stroke

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Fig. 5.
figure 5

Two kind of curves under different spring stiffness

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Figure 5 shows the flow curve of the core rise and Westphal phenomenon with different spring stiffness. When the characteristics of the diaphragm pump have been determined, the greater the spring stiffness, the smaller the lift maximum value. The difference between the core opening lag angle for k = 7000 N/m and k = 5000 N/m is 1.53°, but the interpolation between the lag height is only 0.0848 mm. Westphal The difference between flow leakage (i.e., the integral of the curve in (b)) is only 0.0057%. Therefore, it can be considered that the largest factor affecting the volume efficiency of the diaphragm pump is the liquid leakage caused by the opening lag Angle.

3.2 Effect on the Overall Performance of the Hydraulic End of the Diaphragm Pump

This subsection mainly uses the calculation method of the massless valve in the unstable state to discuss the relationship between the above parameters for the diaphragm pump volume efficiency, pump valve performance and structure. The pump valve is the only key component in the diaphragm pump except the diaphragm. Its performance such as lift maximum value, lag Angle, lag height, closing speed, closing speed, valve gap flow rate, opening resistance and hydraulic loss when the fluid passes through the valve are important indicators to ensure the normal operation of the diaphragm pump. Some of these performance indicators are related to volumetric efficiency, while others are related to the service life of the pump valve itself, such as the erosion damage caused by the flow rate of the valve gap and the collision damage caused by the closing speed. The design of the pump valve and its optimization are discussed in a table below.

Table 1. Performance index of and pump valve with different stroke
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As shown in Table 1, the opening pressure is constant, it is not related to the change of impulse. The higher the lag height, the larger the lag angle, the lower the volume efficiency of the diaphragm pump. Volume efficiency includes three factors, namely, seal failure leakage, lag height and lag angle caused by lag leakage and liquid compression. In the case of the determination of the performance of diaphragm pump, the leakage caused by the hysteresis effect becomes the main factor affecting the volume efficiency of diaphragm pump. Closing speed is an important test of the life of the pump valve. The closing speed increases from 45 to 55 times/min by about 0.043 m/s per minute. The greater the closing speed, the stronger the collision with the base when the core is closed, and the easier the sealing performance of the pump valve is to fail. The erosion effect caused by the fluid of the valve gap is the main factor of seat damage. The greater the flow rate, the more likely the seat is to damage. The hydraulic loss also increased by about 0.115m with the increase of the impulse. This increase also brings the suction pressure burden of the pump, which should be mainly considered when designing and optimizing the performance of the diaphragm pump.

Table 2. Performance index of pump valve with spring stiffness
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Table 2 clearly shows that the change of hydraulic loss in the spring stiffness is a fixed value, adjusting the spring stiffness does not improve the suction performance of the diaphragm pump. With the increase of spring stiffness, the lag height and lag Angle gradually decrease, the resulting lag effect leakage is reduced, and the volume efficiency of the diaphragm pump is also improved. The closing speed decreases, the damage caused by the collision effect decreases, and the service life of the pump valve also increases. The failure caused by the pulp scour effect becomes stronger with the increase of the maximum gap flow rate, and the erosion damage of the valve seat becomes stronger. It can be seen that in the process of the design and optimization of the pump valve, the spring stiffness can improve the volume efficiency of the diaphragm pump, but also consider the damage of the erosion effect.

4 Conclusion

This paper uses the classic massless valve method, to establish the nonlinear equation of the movement of the valve core. Using the motion curve, analyze the lag angle, lag height and lift maximum. The theoretical flow curve of 10 cycles is obtained by the peak superposition method, and is compared with the measured data. Through the comparison of two curves, the calculation method is correct. Through the basic theory of reciprocating diaphragm pump, derived the performance of the cone valve, such as valve flow rate maximum, sitting speed and hydraulic loss, and calculate and analyze the relationship between the impulse, spring stiffness and diaphragm pump volume efficiency, further analysis of the following conclusions in this paper, can provide reference value for the design of pump valve in practical engineering.

  1. 1.

    After the cone valve equation of motion derived from the classical plate valve, the theoretical flow curve and the motion curve integral obtained by the numerical analytical method are compared with the measured flow curve, which can be considered to be correct within the range of the error allowed;

  2. 2.

    The higher the opening of the diaphragm pump, the greater the leakage caused by the hysteresis effect, and the lower the volume efficiency of the pump in the process of the design optimization of the diaphragm pump. At the same time, the negative impact of the additional suction pressure of the diaphragm pump should also be considered;

  3. 3.

    The greater the spring stiffness, the smaller the leakage caused by the hysteresis effect, and the higher the volumetric efficiency of the pump. In the working process of the diaphragm pump design and optimization, the spring stiffness can be adjusted to improve the volume efficiency. But at the same time, the maximum flow rate of the valve gap increases, and the erosion and damage effect is also an important indicator.