Original Article
Investigation on dynamic
characteristics of a plate-type
discharge valve in a diaphragm pump
for SCR system by two-way FSI model
Youcheng Shi , Shudong Yang, Xiwei Pan
Proc IMechE Part D:
J Automobile Engineering
1–13
Ó IMechE 2019
Article reuse guidelines:
sagepub.com/journals-permissions
DOI: 10.1177/0954407019862168
journals.sagepub.com/home/pid
and Yinshui Liu
Abstract
A plate-type port valve with rubber valve plate in a miniature diaphragm pump is presented. To investigate the dynamic
characteristics of this type of discharge valve, a two-way fluid structure interaction model is proposed. The interaction
between the dynamic behavior of the fluid and rubber is considered in the fluid structure interaction model. Based on
the fluid structure interaction model, the internal flow of the pump, the deflection of the diaphragm and discharge valve
plate is calculated. To verify the validity of the numerical model, a prototype pump is fabricated and tested. The experimental pressures in the working chamber of the pump show the same overall trends with the numerical results. The
deviations between the numerical and experimental flowrates are less than 7.2%. The experimental results prove that
the numerical model is effective in predicting a complete discharge process of the pump. There is a big difference
between the deflection of the center of the valve plate and the edge of the valve plate. The oscillation period of the pressure in the working chamber of the pump is approximately double that of the discharge valve plate. When the pump
speed is lower than 2500 r/min, it has little influence on the lag angles of the discharge valve under rated pressure. The
lag angles at rated pump speed increase when the backpressures increase. The stress of the discharge valve plate reaches
a peak when the valve plate impact on the valve limiter or valve seat.
Keywords
Plate-type valve, diaphragm pump, fluid structure interaction (FSI), rubber, dynamic characteristics, selective catalytic
reduction (SCR) system
Date received: 25 February 2019; accepted: 5 June 2019
Introduction
Diaphragm pump is widely used in urea selective catalytic reduction (Urea-SCR) system which is one of the
most promising after-treatment technologies that can
reduce nitrogen oxides (NOx) emission efficiently.1–5 The
pump that is widely used in non-air-assisted Urea-SCR
system is one of the miniature diaphragm pumps.6,7
The disadvantage of diaphragm pump is the pressure pulsation that occurs due to the periodic change of
pump working chamber volume and opening of the
port valves, which can lead to vibration of piping system.8,9 In addition, the interaction between the diaphragm and fluid, check valve and fluid make it very
difficult to precisely predict the dynamic performance
of diaphragm pump.
Many researchers have employed different methods
and models to study the dynamic characteristics of a
diaphragm pump. Xu et al. developed an electroactive
polymer for a micro diaphragm pump. The displacement strokes and profiles as a function of amplifier and
frequency of electric field had been characterized
through the theory of small deflection of thin-plate.
The volume stroke rates as function of electric field,
driving frequency had been theoretically evaluated, too.
The experimental results are close to the theoretical calculations.10 Knutson and Van used Laser Triangulation
Sensor measuring the displacement of a reed valve and
plate valve of hydraulic pump. A mathematical model
and test model are developed. Agreement between
School of Mechanical Science and Engineering, Huazhong University of
Science and Technology, Wuhan, China
Corresponding author:
Shudong Yang, School of Mechanical Science and Engineering, Huazhong
University of Science and Technology, Wuhan 430074, China.
Email: yangsd@mail.hust.edu.cn
2
modeled and experimentally measured valve opening is
demonstrated. They provide a valuable way for measuring the check valve in pump.11,12 Pei et al. presented an
experiment system for the valve motion of a triplex single acting reciprocating pump. They directly acquired
the valve disk motion parameters (acceleration, velocity, and displacement) under actual conditions. The
test results were compared with the calculation results
based on U. Adolph theory and approximation theory.
The comparison results show that U. Adolph theory is
more suitable for the determination of valve disk
motion parameters during different strokes and can
explain the jumping and hysteresis phenomena of the
valve well.13 Leati et al. investigated the dynamic behavior of a plate type check valve in a high-frequency
oscillation pump. The valve consists of a disk, housing
and a wave spring. They analyzed the parameters contributing to the dynamics of the valve. An agreement
between simulation and experiment results is achieved.
The results show the valve contact surfaces have important consequences on the dynamics of the pump system.14,15 Ma et al. studied the influence of valve’s lag
characteristic on pressure pulsation and performance of
reciprocating multiphase pump by a computational
fluid dynamics (CFD) model. By the comparisons of
input power, the simulated results in accordance with
the test results. The study provides a theoretical basis
for the design of high-efficiency multiphase pump and
valve.16,17 Van et al. proposed a novel fluid structure
interaction (FSI) model for a piston diaphragm pump
by immersed boundary methods. They used a combination of two different types of immersed boundary methods, a feedback forcing approach for interaction with
the flexible structure, and a direct forcing approach for
the interaction with the rigid structure. The calculated
diaphragm deformation results were in accordance with
those from experiments. Analysis results show that diaphragm thickness and rigid inclusion size affect the
flowrate, backpressure, and output energy coefficient.18
Wang et al. analyzed the influence of spring stiffness
and valve core mass on the motion behaviors of a reciprocating plunger pump (2000-fracturing pump) discharge valve by FSI simulation. The experimental
results indicate that the FSI model could provide an
effective approach for the improvement of valve reliability, volumetric efficiency, and life span.19
In this article, a diaphragm pump with plate-type
valves used in SCR system is presented. The diaphragm
and valve plates of the pump are all made of rubber
materials. There are few literature studies on the
dynamic behaviors of this type of diaphragm pump
and valve. The primary goal of this work is to establish
a numerical model of the diaphragm pump and understand the dynamic characteristics of this type of valve.
We wish the numerical model and calculation results
can provide a reference for the design and optimization
of this type of valve. A three-dimensional (3D) twoway FSI model of the pump is established and calculated. Based on the proposed FSI model, the dynamic
Proc IMechE Part D: J Automobile Engineering 00(0)
characteristics of the discharge valve, the dynamic pressure and flow of the pump are investigated. The validation test of a prototype pump is also conducted, and
test results match well with numerical values.
Numerical methods
The numerical model of the pump concludes structure
domain and fluid domain.
Pump configuration
Figure 1 shows the general configuration of the diaphragm pump assembly. The pump consists of pump
shaft, eccentric wheel, connecting rod, diaphragm, port
valves, and so on. The port valves are plate-type valves,
and the structure of the port valves are the same.
Figure 2 depicts the configuration of the plate-type
valve. The edge of the valve plate is clamped, and valve
plate can deform under fluid pressure. When the valve
plate deflects under pressure and leaves the valve seat,
the valve is open.
As shown in Figure 1, the pump shaft and the outer
rotor of motor are fixed together. The rotation motion
of the pump shaft can be transformed into the linear
motion of the diaphragm through the eccentric wheel,
bearing, and connecting rod. When the working
Figure 1. Configuration of the diaphragm pump assembly. 1connecting rod, 2-pump shaft, 3-eccentric wheel, 4-pump body,
5-lower valve body, 6-discharge valve, 7-upper valve body, 8suction valve, 9-rubber part of diaphragm, 10-rigid inclusion of
diaphragm, and 11-outer rotor of motor.
Shi et al.
3
Figure 3. Fluid domain.
mechanics. In a fixed Cartesian coordinate frame of
reference, they can be expressed in conservative forms
as follows
Figure 2. Configuration of the plate-type valve.
Table 1. Parameters of the diaphragm pump.
Rated pressure (MPa)
Rated speed (r/min)
Eccentricity of eccentric wheel (mm)
Connecting rod length (mm)
Effective radius of diaphragm (mm)
Radius of rigid inclusion (mm)
ri (mm)
rs (mm)
0.9
3000
0.55
23
10.75
5.5
2.25
2.6
chamber of the pump expands, the suction valve is
open and fluid in the inlet pipe can be sucked into the
working chamber. When the working chamber is compressed, the discharge valve is open and the fluid can
be discharged into the outlet pipe. It is obvious that the
height of the pump assembly can be significantly
decreased by adopting this type of valve. Several parameters of the pump are listed in Table 1.
As the diaphragm and valve plate are flexible, they
will largely deform under the 0.9 MPa pressure.
Meanwhile, the diaphragm deformation will significantly change the volume of the pump working chamber and influence the pressure in the pump working
chamber during a working cycle. The interaction
between the flexible body and fluid is an important
characteristic of this type of pump. To analyze the
dynamic behavior of the pump, a 3D FSI model is proposed based on ADINA 9.3Ò. For FSI analysis, the
fluid domain and solid domain are defined respectively,
through their material data, boundary conditions, and
so on.
Fluid model
Theoretical basis. The motion of a continuous fluid
medium is governed by the principles of classical
∂r
+ r ðrVÞ = 0
∂t
∂rV
+ r ðrVV tÞ = fb
∂t
∂rE
+ r ðrVE t V + qÞ = fb V + qb
∂t
ð1Þ
ð2Þ
ð3Þ
where t is the time, r is the density of the fluid medium,
V is the velocity vector of the fluid medium, fb is the
body force vector of the fluid medium, t is the stress
tensor, E is the specific total energy, q is the heat flux
and qb is the specific rate of heat generation.20
Mesh model and boundary conditions. To carry out the FSI
calculation, it is important to deal with nodal coincidence when modelling bodies with interfaces. The moving mesh technology is also adopted to simulate the
large deflection of the diaphragm and valve plate. It is
easy to get a mesh model using triangular mesh but that
can dramatically increase the number of elements and
computational cost. Therefore, a high-quality hexahedral mesh is necessary. There is a symmetry plane in
the computational domain. The half model of computational domain is also sufficient to conduct the calculation. The simplified mesh model of fluid domain with
high-quality hexahedral cells is shown in Figure 3. The
fluid domain consists of the working chamber of the
pump, and the working chamber of the port valves and
pipes. The point 1 in Figure 3 is the monitor point of
the pressure in the working chamber of the pump. Grid
cells in the fluid model are 3D 8-node elements.
The Spalart-Allmaras (SA) turbulence model has
been adopted to calculate this specific problem. The SA
turbulence model is a one-equation model and has proven to be numerically well behaved in most cases.21,22
Some authors had already used the SA model to conduct similar analyses and confirmed the accuracy of the
solutions.23,24
4
Proc IMechE Part D: J Automobile Engineering 00(0)
€ + CU_ + KU = R F
MU
ð4Þ
where M is the constant mass matrix, C is the constant
damping matrix, K is the constant stiffness matrix, R is
the external load vector, F is the nodal point force vector equivalent to the element stresses and U is the vector
of nodal point displacement increments.
Figure 4. GAPBC of the diaphragm pump.
GAP boundary condition (GAPBC) is used for
simulating the opening and closing of the port valves.
As the structure of the two port valves is the same, the
GAPBC applied on the two valves are the same. As
presented in Figure 4, a small gap of 0.05 mm is kept
on the flow channel between the valve seat and valve
plate at initial condition. GAPBC is applied on the
specified cylindrical surfaces. The GAPBC is a switch
between two fluid fields. When the GAPBC is open,
the fluid can flow across the specified interfaces and the
fluid variables of the two fields are continuous across
the interfaces. When the GAPBC is close, the two fields
are disconnected and the fluid cannot flow across the
interfaces. Boundary condition of fluid structure interface (FSIBC) is also applied on all interfaces between
the fluid and structure domain. The pressure condition
has applied on the inlet and outlet face in the fluid
model. The influence of GAPBC parameters and pressure boundary conditions will be discussed latter.
Structure model
Theoretical basis. In nonlinear analysis, the incremental
finite element equilibrium equations are used, which are
shown in equation (4) as follows
Figure 5. Structure domain.
Mesh model and boundary conditions. It is necessary to do
some simplification of the pump structure to complete
the calculation, such as ignore the chamfer and fillet.
The simplified mesh model of the structure domain is
presented in Figure 5. Grid cells in the structure model
are also 3D 8-node elements. The solid domain consists
of the diaphragm assembly, plate valves and pump
body. Points 2, 3, and 4 are the monitor points of the
valve plate lift. The pump body, valve seat, upper valve
body, and limiters are all fixed. Outer edges of the diaphragm are clamped. A displacement function is
applied on the rigid inclusion. FSIBC is also applied on
the interfaces between the fluid and structure domain in
this model. Contact conditions are set to simulate the
impacts between the valve plate and valve seat. Four
contact pairs are defined in the structure model. The
contact conditions defined in discharge valve model are
shown in Figure 6, which are Contact Pair 1 between
the valve seat and the valve plate, Contact Pair 2
between the valve plate and valve limiter, Contact Pair
3 between the valve plate and the upper valve body. A
contact condition between the diaphragm and the
pump body is also proposed. It is defined as Contact
Pair 4 in Figure 6.
The stress–strain curve of rubber is shown in
Figure 7. The Ogden model is adopted to simulate the
characteristics of rubber in this analysis. The Ogden
model is widely used for predicting the nonlinear
stress–strain behavior of rubber. The strain energy density function for an Ogden material is as follows
Shi et al.
5
Figure 6. The contact conditions applied on structure domain.
Table 2. Setting parameters in the numerical model.
Elasticity modulus of steel (GPa)
Passion’s ratio of steel
Elasticity modulus of aluminum alloy (GPa)
Passion’s ratio of aluminum alloy
Elasticity modulus of nylon (GPa)
Passion’s ratio of nylon
Bulk modulus of water (GPa)
Dynamic viscosity of water (Pas)
Density of water (kg/m3)
Rubber density (kg/m3)
Time step (s)
200
0.3
69
0.32
1.0
0.34
2.2
1 3 1023
1000
2000
5 3 1026
Figure 7. Data of rubber material.
W=
n
X
m
i
a
i=1 i
la1 i + la2 i + la3 i 3
ð5Þ
where, lj (j = 1,2,3) is the principal stretch ratio, and
mi and ai are empirically determined material
constants.25
The material of the fabric and rigid inclusion of the
diaphragm are nylon and steel, respectively. The material of the pump body and valve body is aluminum
alloy. They are considered as isotropic. The setting
parameters in the numerical model are listed in Table 2.
As presented in Figure 8, a simplified model of the
pump is built to analyze the displacement of the rigid
inclusion.
Assume that point T is the top dead center and point
B is the bottom dead center. The motion of the rigid
inclusion in the direction of the z axis and y axis can be
expressed as follows
wz ¼ e½ð1 cosuÞ þ ðl=4Þ3ð1 cos2uÞ
ð6Þ
Figure 8. Schematic diagram of diaphragm pump.
wy = esinuð0 \ u \ p=2, p \ u \ p3=2Þ
wy = esinuðp=2 \ u \ p, p3=2 \ u \ 2pÞ
ð7Þ
where w is the displacement of rigid inclusion; u is the
phase angle; e is the eccentricity; l is the length of connecting rod, l= e/l; v is the angular velocity; b is the
radius of rigid inclusion; R is the effective outer radius
6
Proc IMechE Part D: J Automobile Engineering 00(0)
of diaphragm; and S is the distance between the top
and bottom dead center, S = 2e. The displacement
function applied on the rigid inclusion is described as
equations (6) and (7).
FSI model
The basic conditions applied on the FSIBCs are the
kinematic condition (or displacement compatibility).
This can be described as follows
df = ds
ð8Þ
and the dynamic condition (or traction equilibrium)
can be described as follows
n tf = n ts
ð9Þ
where df and ds are the fluid and solid displacements,
respectively. t f and ts are the fluid and solid stresses,
respectively.
The fluid velocity condition is derived from the kinematic condition and can be described as follows
V = d_s
ð10Þ
Based on the kinematic conditions, the fluid nodal
positions on the FSIBCs can be calculated. The displacements of the other fluid nodes are determined by the
program to keep the initial mesh quality. Then, the governing equations of fluid flow in the ALE formulations
are solved. The fluid velocities on the FSIBC are zero.
Based on the dynamic conditions, the fluid traction
is integrated into fluid force along FSIBC and exerted
onto the structure node. Therefore, the objective function can be described by equation (11)
ð
FðtÞ = h tf dS
ð11Þ
where h is the virtual quantity of the solid
displacement.20
Although the two models are defined, respectively,
they are solved simultaneously by one FSI solver. The
iterative computing methods are used to solve the twoway coupling system. The flow chart of FSI calculation
is shown in Figure 9. The initial solution of the calculation is ds21 = ds0 = tds and t s0 = tt s.
Mesh sensitivity analysis
A mesh sensitivity analysis is conducted on the model
by changing the size of the mesh grid. Six different mesh
models are generated by changing the grid parameters.
The mass flowrate of the diaphragm pump calculated
from the models is presented in Table 3. From Table 3,
it is clear that only with a grid above of 51,400 cells
would make the solution stable. In order to achieve the
best compromise between the accuracy and computational time, the final model consists of 51,400 cells.
Simulations have been run on IntelÒ XeonÒ E3-1230
CPU @3.30 GHz with 12 GB RAM. To complete a
Figure 9. The flow chart of FSI calculation.
Table 3. The results of mesh sensitivity test.
Item
Test 1
Test 2
Test 3
Test 4
Test 5
Test 6
Mesh number (104)
Fluid domain
Structure domain
6.55
6.55
4.35
4.35
3.9
3.9
1.06
0.79
1.06
0.79
1.06
0.78
Mass flowrate (kg/h)
25.59
25.45
25.47
25.51
25.91
26.13
calculation, such as test 4, at least 310 h is needed. With
the increase in the maximum material deformation and
grid cells in numerical model, the total calculation time
increases as well.
Boundary conditions analysis
The GAPBC has already been introduced in section
‘‘Mesh model and boundary conditions.’’ When the
valve is close, there is no gap between the valve plate
and valve seat in reality. Therefore, the GAPBC should
be small enough. The available minimum gap value in
ADINA is 0.02 mm. The maximum lift of the valve
plate depends on the upper valve body. The maximum
Shi et al.
7
Table 4. Calculation results with different GAPBC.
GAPBC
size (mm)
Valve
opening
lag angle (°)
Pressure
drop across
GAPBC (kPa)
Maximum
pressure in the
pump working
chamber (MPa)
0.02
0.05
0.1
67.5
67.5
67.5
25.1
23.5
20.9
1.26
1.26
1.26
GAPBC: GAP boundary condition.
lift of the valve plate is 1 mm in this pump model. For
comparison, 5% and 10% of the maximum valve lift
has also been set as the gap in the GAPBC. The calculation result with three different gaps under rated condition is presented in Table 4.
From Table 4, we can see that the GAPBC size has
little influence on the valve opening lag angle and maximum pressure in the pump working chamber. The
deviation between the pressure drop under 0.02 mm
GAPBC and 0.05 mm GAPBC is 6.4%. Besides, a gap
of 0.02 mm is too small and can influence the grid quality. Therefore, a gap of 0.05 mm is reasonable.
A chamber with 1 mL in volume (nearly 5 times of
the pump displacement) is designed before the inlet
valve. The inlet pipe length in the numerical model is
same as the distance between the chamber and inlet
valve. Therefore, the inlet pressure of the pump can be
assumed as atmospheric pressure. The outlet pressure
of the pump has significant influence on the pump performance. The influence of different outlet pressure will
be discussed later in section ‘‘Validation of the FSI
model.’’
Experiment setup
To validate the numerical model, a prototype pump
and test rig is designed and fabricated. The schematic
diagram and component photos of the test rig are
shown in Figure 10.
The parameters of the components of the test rig are
shown in Table 5. The flowrates are measured by the
weight increasing method. The pressures in the working
chamber of the pump are monitored by the pressure
sensor. The data from the pressure sensor are collected
by the data acquisition card and displayed on the monitor. The water is used for conducting the experiments.
Results and discussion
Validation of the FSI model
To validate the FSI model, the simulation results of the
dynamic pressure in the working chamber of the pump
and the flowrate are compared with the experimental
results.
Pressure in the working chamber. In reality, the pump outlet pressure mainly depends on the flow system. A
Figure 10. Test rig: (a) schematic diagram and (b) component
photos. 1-tank, 2-pressure sensor, 3-diaphragm pump assembly,
4-filter and accumulator, 5-pressure gauge, 6-flow control valve,
7- data acquisition card, and 8-monitor.
Table 5. Parameters of the components of the test rig.
Sampling rate of data acquisition card (kHz)
Sampling rate of pressure sensor (kHz)
Range of pressure sensor (MPa)
Accuracy of pressure sensor (%)
Range of pressure gauge (MPa)
Accuracy of pressure gauge (%)
48
20
20.1~2.0
0.1
0~1.6
0.25
constant outlet pressure does not exist in actual conditions. The outlet pressure tested from the test system is
applied on the pump outlet to conduct the simulation.
The outlet pressure of the diaphragm pump in the
test system is shown in Figure 11. They are collected by
the pressure sensor that is arranged before the accumulator. The outlet pressure is significantly influenced by
the fluid system component, especially the accumulator.
The numerical and experimental pressure in the working chamber of the pump during the discharge process
is compared and the results are shown in Figure 12.
8
Figure 11. Outlet pressure of the diaphragm pump in the test
system.
Proc IMechE Part D: J Automobile Engineering 00(0)
Figure 13. Relationship between the pump speed and flowrate.
Define rs-ri as the sealing surface width. The maximum value of the pressure mainly depends on the sealing surface width of the discharge valve. Based on the
parameters in Table 1, the maximum pressure in the
pump working chamber can be calculated as follows
2
rs
pmax =
30:9 = 1:2 MPa
ð12Þ
ri
Obviously, the maximum pressure in the pump
working chamber will decrease when the sealing surface
width decreases. But the contact stress of the valve
plate will increase with the decrease of sealing surface
width. If the outlet pressure of the diaphragm pump is
a constant value, the pressure in the working chamber
will sharply drop to the outlet pressure after the valve
opens. In the real-world flow system, the accumulator
will significantly change the trend and the periods of
pressure oscillation. The dynamic characteristics of the
valve will also be influenced by fluid system component. The above result means that the FSI model is reliable to simulate a complete discharge process.
Figure 12. Dynamic pressure in the working chamber during
the discharge process.
The numerical data shown in Figure 12 are collected
from point 1 at rated condition. The experimental pressure data in the pump working chamber are also collected by pressure sensor at the same location of point
1. The pressure changes with time show almost the
same overall trends between the numerical results and
experimental values. The deviation between the maximum experimental pressure and numerical pressure is
5.3%. The phase angle when the numerical pressure
reaches the maximum value is nearly equal to experimental value. The oscillation period of the pressure is
approximately equal to 0.0026 s (46.8°).
Flowrate. The numerical and experimental flowrates
under different pump speeds are presented in Figure 13.
The numerical and experimental flowrates are nearly
proportional to the pump speed, when the pump speed
changed from 0 to 3000 r/min. The deviations between
the experimental and numerical flowrate are less than
7.2%. The above results mean that the numerical
method is effective.
Lift of discharge valve plate
The displacement contours of structure domain at different time are shown in Figure 14. To accurately
describe the deflection of the valve plate, three points
Shi et al.
9
The relationship between the oscillation period and
pump speed are shown in Figure 16. The period
decreases when the pump speed increases at rated backpressure. The oscillation periods of the discharge valve
plate under different backpressure are depicted in
Figure 17. The period remains nearly unchanged when
the backpressure is lower than 0.7 MPa, but it decreases
when the backpressure is higher than 0.7 MPa. It is
probably because of the difference of the valve plate
deformation under different pressure.
Lag angle of the discharge valve
Figure 14. Displacement contours of discharge valve plate at:
(a) j = 281° and (b) j = 342°.
on the valve plate are monitored. The three points are
marked in Figure 5. The deflection data of the three
points during the working process are shown in Figure
15. It is found that the motion of point 2 and point 4
consisted of several periodic oscillations. The oscillation period of the valve plate is approximately equal to
0.0013 s (23.4°). From Figure 12, we can see that the
oscillation period of the pressure in the working chamber is approximately equal to 0.0026 s (46.8°). The
oscillation period of the pressure is double that of the
valve plate. It is obvious that the motion of point 3 is
significantly different to the other points. The
max displacement of point 3 is determined by the
valve limiter. The max displacement of point 3 is
0.3 mm.
Figure 15. Lift of discharge valve plate.
The opening and closing lag angles of the discharge
valve under different pump speeds at rated pressure are
presented in Figure 18. The opening and closing lag
angles of the discharge valve under different backpressure at rated speed are showed in Figure 19. As shown
in Figure 18, the closing lag angle of the discharge valve
decreases when the pump speed increases. The opening
lag angles remain unchanged when the pump speeds are
lower than 2500 r/min. As depicted in Figure 19, the lag
angle of the discharge valve increases when the backpressure increases.
Stress of the discharge valve plate
To ensure the whole FSI calculation, we have to simplify the lower valve seat to a plate. Therefore, it is not
very accurate to calculate the valve plate stress at the
contact zone between the valve plate and sealing surface by this numerical model. But the valve plate stress
at the contact zone between the valve plate and valve
limiter can be exactly accurate. Limited by the article
length, this article mainly focuses on the valve plate
stress at the contact zone between the valve plate and
valve limiter. The stress of the three monitor points at
rated condition are presented in Figure 20. The results
show that the max stress of the point 1 at rated conditions is 0.9 MPa. As shown in Figure 20, the stress of
10
Proc IMechE Part D: J Automobile Engineering 00(0)
Figure 16. Oscillation periods of the discharge valve plate
under different pump speed.
Figure 19. Lag angles of the discharge valve under different
backpressure.
discharge valve plate before the valve open is equal to
zero, it is because that the lower valve seat is simplified.
The stress reaches a peak at the moment that the valve
plate reaches the maximum deformation and impacts
on the valve seat or valve limiter.
Three kinds of valve plate limiters are designed for
the FSI calculation. Their configurations are presented
in Figure 21. The maximum stress of the discharge
valve plate under different conditions is listed in
Table 6. Obviously, a rounded limiter is the best choice.
With the increase in the limiter width, the maximum
stress of the discharge valve plate will decrease. A
rounded and wider valve limiter is the better choice.
Figure 17. Oscillation periods of the discharge valve plate
under different backpressure.
Internal flow of the pump
The pressure contours and velocity contours in two different directions of the fluid domain at rated conditions
are presented in Figures 22 and 23, respectively. From
the figures, we can see the total pressure change and
velocity change across the valve chamber. The pressure
drop of this type of valve is relatively less than that with
a reset spring. The max pressure of point 1 at 264.5°
and 310.5° are 1.26 and 1.05 MPa, respectively. The
max velocity at 264.5° and 310.5° are 6.573 m/s and
7.2 m/s, respectively. Obviously, the fluid in the corner
of the valve chamber almost not flows. The position
and structure of the outlet port of the valve will also
influence the deformation of the valve plate. A uniformly distributed outlet valve port is the better choice.
Conclusion
Figure 18. Lag angles of the discharge valve under different
pump speed.
A miniature diaphragm pump with a plate-type valve is
presented and investigated. The materials of valve plate
and diaphragm are rubber. To study the dynamic characteristics of the discharge valve, the interaction
between the fluid and rubber is considered and a two-
Shi et al.
11
Figure 20. Stress of the discharge valve plate.
Figure 21. Configuration of three different valve limiters: (a) rectangular, (b) trapezoid and (c) rounded.
Table 6. Maximum stress of the discharge valve plate with
different limiters (Unit: MPa).
Limiter structure
Rectangular
Trapezoid
Rounded
Width of the limiter (mm)
1.5
2.5
3.5
4.5
1.45
1.1
0.75
–
–
0.52
–
–
0.41
–
–
0.3
way FSI model is proposed. A prototype pump is manufactured and tested on a test rig.
Numerical analysis suggests that the change trends
of the pressure in the working chamber are similar to
the experimental results. The deviations between the
numerical and experimental flowrate under different
pump speed are less than 7.2%. The experimental
results demonstrate that the proposed numerical model
is effective to simulate a complete discharge process of
the diaphragm pump. The discharge valve plate lifts of
the three monitor point show significantly different
change trends. The center of the discharge valve plate
keeps the max lift until the discharge process ends, but
the edge of the discharge valve plate oscillates with a
approximated constant period. The oscillation period
of the pressure in the working chamber is approximately double that of the discharge valve plate. The
Figure 22. Pressure contours of the fluid domain at: (a)
j = 264.5 and (b) j = 310.5°.
closing lag angle decreases when the pump speed
increases. The opening angle is nearly unchanged when
the pump speeds are lower than 2500 r/min. The stress
12
Figure 23. Velocity contours of the fluid domain at:
(a) j = 264.5 and (b) j = 310.5°.
of discharge valve plate reaches a peak at the moment
of valve plate impact on the valve limiter or valve seat.
The max stress of the discharge valve plate at rated
condition is 0.9 MPa. It shows that the FSI model is
effective to predict the dynamic characteristics of the
plate-type valve in diaphragm pump, and the results
will be helpful in the design and development of the
similar diaphragm pump and port valves.
Declaration of conflicting interests
The author(s) declared no potential conflicts of interest
with respect to the research, authorship, and/or publication of this article.
Funding
The author(s) disclosed receipt of the following
financial support for the research, authorship, and/or
publication of this article: The authors would like to
acknowledge the financial support from Science and
Technology Planning Projects of Jiangsu Province
(BY2016025-01).
ORCID iDs
https://orcid.org/0000-0002-0326Youcheng Shi
9297
https://orcid.org/0000-0001-9113-6170
Xiwei Pan
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Appendix 1
Notation
b
C
radius of rigid inclusion (m)
damping matrix
V
r
w
fluid displacement
solid displacement
specific total energy
eccentricity of eccentric wheel (m)
nodal point force vector equivalent to the
element stresses
body force vector of the fluid medium
stiffness matrix
connecting rod length (m)
mass matrix
maximum pressure in the working
chamber (MPa)
heat flux
specific rate of heat generation
effective radius of diaphragm (m)
external load vector
radius of valve inlet port (m)
outer radius of sealing surface (m)
distance between the top and bottom dead
center (m)
time (s)
vector of nodal point displacement
increments
velocity vector of fluid medium
density (kg/m3)
rigid inclusion displacement (m)
u
v
lj
t
tf
ts
phase angle of pump shaft (rad)
angular velocity of pump shaft (rad/s)
principal stretch ratio
stress tensor
fluid and solid stress
solid stress
df
ds
E
e
F
fb
K
l
M
pmax
Q
qb
R
R
ri
rs
S
t
U