Spool Orifice (P)#

The Spool Orifice (P) block simulates changes in energy and pressure of the liquid when component of thermal hydraulic system passes - holes of variable area between the coil and the sleeve with holes. The sleeve holes can be a series of circular or rectangular holes. The flow rate depends on the total area of hole between the sleeve, holes and the spool, which is extended or retracted depending on the signal received at port S.

The force acting on the coil is calculated as (1):

(1)#\[F=\frac{\dot{-m_A^2}}{\rho\cdot A}\cdot cos(\alpha)\cdot \epsilon\]

where:

\(\dot{m_A}\) - mass flow rate in port A, [kg/s];

\( ho\) – fluid density, [kg/m\(^3\)];

A - open area of the hole, which is determined by the spindle position and the parameterization of the hole, [m\(^2\)];

\(\epsilon\) - hole orientation, which indicates the hole’s hole that is associated with a positive or negative signal at point S.

\(\alpha\) - jet angle, which is calculated by approximation of von Mises formula (2):

(2)#\[\alpha_{jet}=0.3663+0.8373\cdot (1-e^{\frac{-h}{1.848\cdot c}})\]

where:

c - radial clearance;

h - height of the hole.

1. Hole area of the hole#

Set the Open orientation to Positive for adjustable holes indicates that the hole opens when the control is extended, and Negative indicates that the hole opens when the control is retracted.

The leakage area is considered to be a small area opened to flow when the hole is closed, which provides numerical continuity. In addition, the non-zero Smoothing factor can provide increased numerical stability when the hole is in an almost closed or almost opened position.

Round holes

Set Orifice geometry to Round uniformly distribute a user-defined number of holes around the perimeter of the sleeve with identical diameters and centers located in the same plane, Fig. 1 :

The open area is calculated as (3):

(3)#\[A_{orifice}=n_0 \cdot \frac{d_0^2}{8}\cdot(\theta-sin\theta)+A_{leak}\]

Opening angle of hole θ is set by control signal in S (4):

(4)#\[\theta=2 \cdot cos^{-1} (1-\frac{2\cdot \Delta S}{d_0})\]

where:

\(n_0\) - number of holes;

\(d_0\) - diameter of holes;

\(A_{leak}\) = c \(\cdot\) d\(_0\)\(\cdot\) n\(_0\)

\(\Delta S\) – distance of control movement from initial position.

The maximum open area is (5):

(5)#\[A_{max}=\frac{П}{4} \cdot d_0^2 \cdot n_0 + A_{leak}\]

Rectangular hole

Setting Orifice geometry for Rectangle slot simulation of one rectangular slot in the pipe coupling, Fig. 2 :

Open area for a hole with a slot in a rectangular sleeve is equal to (6):

(6)#\[A_{orifice}=w\cdot \Delta S+A_{leak}\]

where w — hole width.

The maximum opening distance between the sleeve and the casing is (7):

(7)#\[A_{max}=w\cdot \Delta S_{max}+A_{leak}\]

where \(\Delta S_{max}\) – distance between the slot hole and the distance between closed and opened hole.

The leakage area at the minimum hole area of the hole is (8):

(8)#\[A_{leak}=c\cdot w\]

2. Numerically smoothed displacement#

You can maintain numerical stability in your simulation by adjusting the block at extremes of the hole opening range Smoothing factor. The block applies a smoothing function to each calculated displacement, but primarily affects the simulation at extremes of this range.

If Smoothing factor is not zero, the block smoothly saturates the hole between 0 and \(\Delta S_{max}\), where \(\Delta S_{max}\):

The value of the Orifice diameter parameter when the Orifice geometry is set to Round.
The value of the Spool offset parameter when Orifice geometry is set to Rectangle.

Mass flow calculation

The block calculates the mass flow rate as (9):

(9)#\[ \dot m=C_d \cdot S_r \cdot \sqrt {\frac{2 \cdot \gamma}{\gamma - 1}\cdot p_{in} \cdot \rho_{in} \cdot (\frac{p_{out}}{p_{in}})^{\frac {2}{\gamma}}\cdot(\frac{1-(\frac{p_{out}}{p_{in}})^{\frac {\gamma-1}{\gamma}}}{1-(\frac{S_R}{S})^2\cdot(\frac{p_{out}}{p_{in}})^{\frac {2}{\gamma}}})}\]

where:

\(C_d\) - discharge coefficient;

\(S_r\) – area of the hole, [m\(^2\)];

S - cross-sectional area of ports A and B, [m\(^2\)];

\( ho\) - gas density, [kg/m\(^3\)];

p - gas pressure, [Pa];

\(\gamma\) - the adiabatic index.

Linearization of the equation for \(\frac{p_{out}}{p_{in}}\)>B\(_{lam}\) (10):

(10)#\[ \dot m=C_d \cdot S_r \cdot \sqrt {\frac{2 \cdot \gamma}{\gamma - 1}\cdot p_{avg}^{\frac {2 - \gamma}{\gamma}} \cdot \rho_{avg} \cdot B_{lam}^{\frac {2}{\gamma}}\cdot(\frac{1- B_{lam}^{\frac {\gamma - 1}{\gamma}}}{1-(\frac{S_r}{S})^2\cdot B_{lam}^{\frac {2}{\gamma}}}})\cdot (\frac {p_{in}^{\frac {\gamma - 1}{\gamma}}-p_{out}^{\frac {\gamma - 1}{\gamma}}}{1-B_{lam}^{\frac {\gamma - 1}{\gamma}}})\]

When the pressure ratio is less than the ratio \(\frac{p_{out}}{p_{in}}\) < \((\frac{2}{\gamma +1})^\frac {\gamma}{\gamma-1}\) the equation assumes (11):

(11)#\[ \dot m=C_d \cdot S_r \cdot \sqrt {\frac{2 \cdot \gamma}{\gamma + 1}\cdot p_{in} \cdot \rho_{in} \cdot (\frac{1}{(\frac{\gamma +1}{2})^{\frac{2}{\gamma -1}}-(\frac{S_r}{S})^2}})\]

Energy conservation law (12):

(12)#\[ {\phi}_{A}+{\phi}_{B}=0 \]

where:

\(\phi_А\) - energy flow at the input, [W];

\(\phi_B\) - energy flow at the output, [W].

3. Properties#

Property	Default value	Description	Units of measurement
Orifice geometry: Round; Rectangle	Round	-	-
Open orientation: Positive; Negative	Positive	-	-
Pressure recovery	Off	-	-
Orifice diameter, available when `Orifice geometry` - `Round`	5e-3	-	m
Orifice quantity, available when `Orifice geometry` - `Round`	6.0	-	pcs
Rectangle width, available when `Orifice geometry` - `Rectangle`	0.01	-	m
Spool travel, available when `Orifice geometry` - `Rectangle`	0.005	-	m
Spool offset	0.0	-	m
Discharge coefficient, available when `Pressure recovery` - `On`	0.64	-	-
Leakage area	1e-10	-	m\(^2\)
Smoothing factor	0.01	-	-
Laminar flow pressure ratio	0.999	-	-

4. Inputs and outputs#

Parameters	Input/Output	Description	Units of measurement	Port type
Spool offset	Input	-	m	Automation
Input	Input	-	-	Pneumatics
Output	Output	-	-	Pneumatics

In order to see the connection capabilities of the block, you need to go to page Description of block ports