Theoretical model and characteristics analysis of deflector-jet servo valve’s pilot stage

3.1. Main cross sections of flow distribution

In this work, the oil is considered as incompressible viscous fluid. Four cross sections are selected in the flow distribution, as shown in Fig. 3. The velocity and pressure of the flow for each section need to be analyzed.

In Fig. 3, $A_s$ is the area of the jet pan entrance, $p_s$ is the supply oil pressure of servo valve,$u_s$is the original velocity when oil flows into jet pan and $\beta$ is the gradual contractile angle of the jet pan entrance. Section 0, 1 and 2 respectively represent the virtual jet origin, the jet collision location on the lateral wall of the V-shaped guiding groove and the nozzle on the deflector plate. Correspondingly, the sectional areas are $A_0$, $A_1$, $A_2$. Besides, the pressure and velocity of each cross section are defined as $p_i$ and $u_i\left(i=\text{0, 1, 2}\right)$. Section 3 is determined concerning the location where the flow velocity is 0 when the deflector plate is at the neutral position. The two sectional areas are respectively $A_{3m}$ and $A_{4m}$, and the restoring pressures in them are defined as $p_{3m}$, $p_{4m}$.

Fig. 3Definition of the pilot stage pressure section

3.2. Initial part of pilot stage’s flow

As shown in Fig. 3, the oil comes into the jet pan and flows into Section 0. The equation derived from the ideal Bernoulli equation can be written as:

where $\rho$ is the oil density, $g$ is the gravity, $z_s$ is the potential energy in the jet pan entrance, $z_0$ is the potential energy in cross Section 0, and $\mathrm{\Delta }{h}_0$ is the loss of tube friction resistance.

When the servo valve works normally, the oil original velocity is very slow, so $u_s\approx 0$. Because the pilot stage is in horizontal direction, the influence of potential energy can be ignored and $z_s=z_0$ can be assumed. The loss of tube friction is $\mathrm{\Delta }{h}_0={\zeta }_0\cdot u_0^2/2g$ and ${\zeta }_0$ is the friction factor of a square gradual contractile tube in the jet pan entrance. So, Eq. (1) can be expressed as Eq. (2):

Suppose that $\mathrm{\Delta }p=$ 16 MPa, from Eq. (2) the Reynolds number’s calculation result of the nozzle in the jet pan is 3066.6, which is much bigger than the critical Reynolds number [17]. So, the jet flow in pilot stage is turbulent.

3.3. Attachment property description of pilot stage’s flow

Observing the structure from Section 0 to Section 1, the jet flow happens in the thin jet pan that is covered by top and bottom sealing cover, so the jet can be regard as 2-D jet [18]. When the oil comes into the limited space of the V-shaped guiding groove, the interference of the lateral wall and jet flow happens with an entrainment effect on ambient fluid. Eventually, the jet flow attaches on the wall which is called offset jet attachment effect or coanda effect. Based on the below assumed conditions [19], the jet attachment effect of the pilot stage’s flow can be described.

(1) The flow velocity is identical when the oil comes out from the jet pan. And the flow trajectory is approximately regarded as part of an arc between the jet central line of the nozzle and the collision central point on the lateral wall of the deflector plate;

(2) Ignoring the friction loss of the sealing covers and the energy loss of collision with lateral wall, the fluid momentum is invariant in the process of fluid’s jet before collision;

(3) The static pressure in the jet flow is equal everywhere.

The schematic diagram of the wall attachment jet in a pilot stage’s flow is shown in Fig. 4.

Fig. 4Geometry model of the wall attached jet in pilot stage

In Fig. 4, $a$ is the width of the jet pan nozzle and $d$ is the position difference which is the distance between the nozzle and the intersection point B of the lateral wall’s extended line and the jet pan’s boundary. Assuming that the virtual jet original point is $O$, the distance between the point $O$ and the nozzle is $s_0$, and the inclination angle of the lateral wall is $\alpha$. From condition Eq. (1), the curvature radius of the jet middle line $O_{1}A$ can be supposed to $R$ and the circle center is located at $O\text{'}$. The collision angle between jet and the lateral wall is $\theta$ and the collision point between the jet bottom line and the lateral wall is in point C. The arc length of the jet middle line is $s_c$ and the vertical distance between the jet central line and the jet boundary is $y_c$. According to condition Eq. (2), it is assumed that the original jet momentum is $J$. After collision with the lateral wall, the oil on the right of point C continues flowing to the deflector plate along the lateral wall and the momentum is $J_1$. In the meantime, the oil on the left of point C flows back to the low pressure vortex zone $O_{1}AB{O}_1$ and its momentum is $J_2$.

3.3.3. Jet collision angle

Because Point C is the boundary point of the flow’s direction, the reflux rate of flow can be expressed as:

13

$Q_r={\left.\underset{y_c}{\overset{\infty }{\int }}udy\right|}_{s=s_c}={\left(\frac{3J\left(s_c+s_0\right)}{4\rho \sigma }\right)}^{1/2}\left(1-\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\frac{\sigma y_c}{s_c+s_0}\right).$

On the other hand, the entrainment rate of flow at the arbitrary point of the jet flow line is:

14

$Q_e={\left.\underset{0}{\overset{\infty }{\int }}udy\right|}_{s=s\text{'}}-{\left.\underset{0}{\overset{a/2}{\int }}udy\right|}_{s=0},$

where $s^{\text{'}}$ is the arc length from the nozzle to the arbitrary point on the jet flow line. Without considering other exits in the vortex zone, the entrainment rate of flow at the length of $s_c$ and the reflux rate of flow are equal, that is:

15

${\left.Q_e\right|}_{s=s_c}=Q_r.$

From Eqs. (13-15) and Eq. (6), we can obtain:

16

$s_c=\frac{\sigma a}{3}\left(\mathrm{t}\mathrm{a}\mathrm{n}{\mathrm{h}}^{-2}\frac{\sigma y_c}{s_c+s_0}-1\right).$

According to Fig. 4, $s_c$ and $R$ can be expressed as:

17

$s_c=R\left(\theta -\alpha \right).$

18

$R=\frac{d+\frac{a}{2}}{\left(1-\frac{\mathrm{c}\mathrm{o}\mathrm{s}\theta }{\mathrm{c}\mathrm{o}\mathrm{s}\alpha }\right)}.$

Thus, from Eq. (12), Eq. (16) and Eqs. (17-18), the jet collision angle can be acquired by:

19

$\left(d+\frac{a}{2}\right)\frac{\left(\theta -\alpha \right)\cdot \mathrm{c}\mathrm{o}\mathrm{s}\alpha }{\mathrm{c}\mathrm{o}\mathrm{s}\alpha -\mathrm{c}\mathrm{o}\mathrm{s}\theta }=\frac{\sigma a}{3}\left(\frac{1}{4\mathrm{c}\mathrm{o}{\mathrm{s}}^2\frac{\theta +\pi }{3}}-1\right).$

Then the arc length of jet central line is:

20

$s_c=\frac{\sigma a}{3}\left(\frac{1}{4\mathrm{c}\mathrm{o}{\mathrm{s}}^2\frac{\theta +\pi }{3}}-1\right).$

And the vertical distance between jet middle line and jet inner boundary is:

21

$y_c=\frac{a}{3}\cdot \frac{1}{4\mathrm{c}\mathrm{o}{\mathrm{s}}^2\frac{\theta +\pi }{3}}\cdot \mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left(2\mathrm{c}\mathrm{o}\mathrm{s}\frac{\theta +\pi }{3}\right).$

Obviously, jet attachment position difference $d$ in Eq. (19) is relevant to the displacement of the deflector plate and the distance between the deflector plate and the jet pan nozzle, which is shown in Fig. 5.

Fig. 5Schematic diagram of deflector offsets

In Fig. 5, $h_0$ is the original distance between the deflector plate and the jet pan nozzle, and the middle position difference is $d_0$.When the servo valve works. $\mathrm{\Delta }x$ is the displacement of the deflector plate and $\mathrm{\Delta }h$ is the vertical distance variety between the deflector plate and the jet pan caused by servo valve assembly, so the position difference can be given by:

22

$d=d_0-\mathrm{\Delta }x+\frac{\mathrm{\Delta }h}{\mathrm{t}\mathrm{a}\mathrm{n}\alpha }.$

According to Eq. (19), the jet collision angle $\theta$ is calculated by:

23

$\left(\frac{d_0}{a}-\frac{\mathrm{\Delta }x}{a}+\frac{\mathrm{\Delta }h}{a\cdot \mathrm{t}\mathrm{a}\mathrm{n}\alpha }+\frac{1}{2}\right)\frac{\mathrm{c}\mathrm{o}\mathrm{s}\alpha \left(\theta -\alpha \right)}{\mathrm{c}\mathrm{o}\mathrm{s}\alpha -\mathrm{c}\mathrm{o}\mathrm{s}\theta }=\frac{\sigma }{3}\left(\frac{1}{4\mathrm{c}\mathrm{o}{\mathrm{s}}^2\frac{\theta +\pi }{3}}-1\right),$

where $d_0/a$ is the middle position difference’s dimensionless value, $\mathrm{\Delta }x/a$ is the deflector plate displacement’s, and $\mathrm{\Delta }h/\left(a\cdot \mathrm{t}\mathrm{a}\mathrm{n}\alpha \right)$ is the vertical distance variety’s. Consequently, the pilot stage jet collision angle $\theta$ is affected by the width of the jet pan nozzle, the middle position difference, the displacement of the deflector plate, the inclination angle of lateral walls and the vertical distance between the deflector plate and the jet pan nozzle. And the original jet momentum is irrelevant to the collision angle.

3.4. Pilot stage pressure gain

After collision with the deflector plate’s lateral wall, part of oil continues to flow along the lateral wall from cross Section 1 to 2 and reaches the deflector plate’s nozzle, which is shown in Fig. 3. Supposing that the principal part of the oil jet has the same pressure, the pressure at Section 1 can be express as below:

According to Eq. (3), the velocity at Section 1 can be given by:

Considering the square gradual contractile loss of the V-shaped guiding groove [20], ignoring the factor of potential energy and applying Bernoulli equation, we can learn the description of the pressure variety that is expressed as:

where ${\zeta }_2$ is the friction factor of the lateral wall. $A_1$ can be calculated from Eqs. (22-26). And $u_2$ can be obtained from the fluid’s continuity condition, then $P_2$ is achieved from Eqs. (3-4), Eq. (6), Eq. (20), Eq. (27) and Eq. (29).

Through the flow continuous theory, $u_2$ is obtained from Eq. (28) and Eq. (30):

By analyzing the process from Section 2 to 3 in Fig. 3, the inner flow situation in the pilot valve is shown in Fig. 6.

Fig. 6Schematic diagram of the flow distribution in pilot stage

In Fig. 6 $Q_3$, $Q_4$, $Q_5$ and $Q_6$ are respectively the rates of flow in the receivers of the jet pan and in the gaps between the deflector plate and the jet pan. $p_3$ and $p_4$ are the entrance pressures of the two receivers, while $x_f$ is the displacement of the deflector plate. Assuming that the valve is symmetric and the oil supply’s pressure, temperature and density are constant, the rates of flow in the receivers and in the gaps can be shown as follows:

where $A_3$, $A_4$, $A_5$, and $A_6$ are these flow rates’ corresponding throttling areas and $C_d$ is the throttling coefficient. Based on the actual pilot valve structure, the relative position between the nozzle of the deflector plate and the jet pan receivers is shown as Fig. 7.

Fig. 7Relative position between deflector plate nozzle and receivers

In Fig. 7, $b$ is the length of receiver entrance, $c$is the width of receiver entrance, $e$is the distance between the two receivers, $B$ is the width of the deflector plate nozzle and $Q_L$ is the load flow rate. According to Fig. 7, when the deflector plate turns to the right, the calculation of each throttling area can be shown as:

And the load flow rate $Q_L$ can be calculated by:

When the deflector plate is at neutral position and the entrance pressures of the two receivers are equal, then according to Eqs. (31-39), we can obtain:

where $K=2c/\left(B-e\right)$, indicating the relative position between the deflector plate nozzle and the jet pan receivers. Because each receiver is closed, the flow velocities in the receivers drop significantly while the pressures increase to the recovery pressure, which can be given by:

According to Bernoulli equation, where ${\zeta }_3$ and ${\zeta }_4$ are the square divergent friction factors of the lateral walls of the receivers. The load pressure is the difference of the two receivers’ recovery pressures, i.e.:

Obviously, the load flow rate $Q_L$ is the function of the deflector plate displacement $x_f$ and the load pressure $p_L$. Then the load rate can be approximated with a Maclaurin’s expansion of the first order, which is written as:

Since $Q_L$ can be calculated by Eqs. (31-43), according to Eq. (44), we can get:

Thus, according to Eq. (45), the pressure gain can be calculated by:

From Eq. (46), we can learn that the pressure gain is affected by the nozzle pressure of deflector plate, the receivers’ width, the receivers’ distance, and the size of the deflector plate nozzle, but it is irrelevant to the thickness of the jet pan. When the load flow rate is zero, the load pressure is shown as below:

Considering Eq. (47), the larger the pressure gain, the better the valve’s pressure sensitivity.

4.1. Discussion on theoretical model

Based on the derivation in the previous chapter, calculation and analysis for attachment properties can be made according to the structure parameters shown in Table 1.

Based on Eq. (6), the distance between the jet origin and the jet pan nozzle is calculated as $s_0=$ 0.33 mm. Considering the case that deflector plate moves to the left and the offset range is 0-0.04 mm, the collision angle $\theta$ and the collision distance $x_c$ affected by deflector plate offset are indicated in Figs. 8-9.

Fig. 8Jet’s impact angle

Fig. 9Jet’s collision distance

Fig. 10Deflector plate nozzle’s pressure

Fig. 11Deflector plate nozzle’s flow speed

Hence the deflector plate’s offset has slight influence on the collision angle and distance, because the offset is so small and the guiding groove’s V-shaped structure restrains the free jet’s length and diverging effect. Meanwhile, there is a zone without attachment around the deflector plate’s neutral position, since the deflector plate and the jet pan nozzle have symmetric structures. This non-attachment zone can be observed from the numerical simulation, though it is very small and difficult to be described analytically.

The deflector plate’s nozzle pressure $p_2$ and flow velocity $u_2$ are indicated in Figs. 10-11. It is shown that when the deflector plate is in the center, the nozzle pressure is 7.74 MPa. As the deflector plate offsets in the left direction, the deflector plate nozzle’s pressure decreases by the speed of 0.067 MPa/0.01 mm and the deflector plate nozzle’s flow velocity also declines.

In accordance with Table 1 and Eq. (47), we can calculate the average pressure gain under 0.04 mm displacement, with the result that $K_p=$ 122 MPa/mm. However, with the deflector plate’s movement, the pressure gain decreases gradually, as shown in Fig. 12. Additionally, this theory shows that there is an approximately linear relation between the deflector plate’s displacement and the pressure gain.

Fig. 12Pressure gain of the pilot valve

Fig. 13Mesh of the jet deflector servo valve pilot stage

4.2. Numerical simulation of pilot valve

Considering that the deflector plate’s offsets are respectively 0 mm, 0.02 mm and 0.04 mm, the 2-D mesh models of pilot valve’s flow distribution are built. For example, mesh dividing of 0 mm offset is displayed in Fig. 13.

With the supply pressure of 21 MPa, the oil density of 870 kg/m³ and the kinematic viscosity of 0.0087 kg/ms, the pressure and velocity distribution contours can be listed in Fig. 14, when the deflector plate’s offsets are 0 mm, 0.02 mm and 0.04 mm.

After collision, small part of oil flows back to the return-oil outlet on both sides by the gaps between the deflector and the jet pan, and other oil continues to flow along the lateral wall. The flow pressure increases because of the constriction of the guiding groove’s lateral wall and then the jet comes out from the deflector plate nozzle again. In fact, the lateral wall can significantly improve the stability of turbulent jets. The numerical simulation result is in agreement with the description of the offset jet attachment model in Section 2.

From the numerical simulation result, the deflector plate’s nozzle pressure, nozzle velocity, the jet’s collision angle and the collision distance are listed in Table 2.

From Table 2, with the offset’s increasing, the jet collision angle has no significant growth, while the collision point is moving down to the deflector plate nozzle. The deflector nozzle’s pressure and flow velocity decrease slightly. Comparing the theoretical calculation with the numerical simulation, the difference is shown in Table 3.

In addition, the pressure in the two receivers can be attained by Fig. 14, listed in Table 4.

So, the DJSV’s average pressure gain is 126 MPa/mm according to above data, which is consistent with the model-based calculation.

Fig. 14Pressure and velocity contours of the flow distribution with different deflector offsets

a) Pressure contour (offset from 0 mm to 0.04 mm in order)

b) Velocity contour (offset from 0 mm to 0.04 mm in order)

5.1. Experiment design of pressure gain

The equipment is designed to achieve micro-displacement’s precise control and measurement. Aiming at a special servo valve with an extension outside the housing, a high-precision electric push rod is set to drive the torque motor and a laser sensor is used for detecting the movement, which is showed in Fig. 15 and Fig. 16. Thus, the deflector plate’s displacement $x_f$ can be calculated according to the structural dimensions while the restoring pressure $p_{3m}$, $p_{4m}$ can be detected directly. Consequently, the real pressure gain will be obtained by Eq. (47).

5.2. Experiment results analysis

Applying the supply pressure of 21 MPa and driving the deflector plate from 0 mm to 0.04 mm, the restoring pressures in the two receivers are shown in Fig. 17 and Table 5.

The result shows that when the deflector is in the center, the pressure testing value in each receiver is 6.7 MPa which has an error of 6.16 % with regard to the calculating result. Based on the experiment data and Eq. (47), the pressure gain average value is 118 MPa/mm, which has an error of 3.28 % with regard to the calculating result 122 MPa/mm. So, it has been showed that the theoretical calculations and the experiment results are essentially coincident, which verifies correctness and rationality of the proposed theoretical model. In fact, the testing pressure gain will decline with the deflector plate offset’s increasing, which is also consistent with the calculation and simulation, as shown in Fig. 18.

Fig. 15Driving and detecting method of deflector plate

a)

b)

Fig. 16Test device for pressure gain

Fig. 17Recovery pressures at the jet-pan receivers

Fig. 18Pressure gain contrast