Spool Orifice (TL)#

The Spool Orifice (TL) 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];

\(\rho\) – 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#

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 coefficient can provide increased numerical stability when the hole is in an almost closed or almost opened position.

Round holes

Set Orifice parametrization 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 (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 parametrization 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 \(\Delta S_{max}\):

Value for Orifice diameterr when Orifice parametrization is set to Round orifices.
Value for Spool travel when Orifice parametrization is set to Rectangle orifices.

3. Mass flow rate equation#

The mass is retained by the valve (9):

(9)#\[\dot m_{A}+\dot m_{B}=0\]

The mass flow rate through the valve is calculated as (10):

(10)#\[\dot m\:=\:\frac{C_{d}\:\cdot A_{orifice}\:\cdot \:\sqrt{2\:\cdot \:\bar\rho}}{\sqrt{PR_{loss}\:\cdot \:(1-(\frac{A_{orifice}}{A_{port}})^2)}}\:\cdot \frac {\Delta p}{(\Delta p^2+\Delta p_{crit})^{1/4}}\]

where:

\(C_d\) - discharge coefficient;

\(A_{orifice}\) - open area of the hole, except cases, when:

Opening is greater or equal to the area spindle movement between closed and opened distance between holes. Then the opening area will be \(A_{max}\).
Hole opening is less than or equal to minimum opening distance. Then the area of the hole is equal to the leakage.

\(A_{port}\) - cross-section area of ports A and B;

\(\bar\rho\) - average fluid density;

\(\Delta p\) - pressure difference at valve \(p_A\) - \(p_B\)

The critical pressure drop \(\Delta p_{crit}\), is the pressure drop associated with critical Reynolds number, Re\(_{crit}\), the transition point of the flow mode between laminar and turbulent flow (11):

(11)#\[\Delta p_{crit}\:=\:\frac{\Pi\:\cdot \:\bar\rho}{8\cdot A_{orifice}}\cdot \:(\frac{\nu\:\cdot Re_{crit}}{C_{d}})^2\]

The pressure loss describes a decrease in pressure in the valve due to a decrease in area. PR losses are calculated as (12):

(12)#\[PR_{loss}\:=\:\frac{\sqrt{1-(\frac{A_{orifice}}{A_{port}})^2\cdot (1-C_{d}^2)}-C_{d}\cdot \:\frac{A_{orifice}}{A_{port}}}{\sqrt{1-(\frac{A_{orifice}}{A_{port}})^2\cdot (1-C_{d}^2)}+C_{d}\cdot \:\frac{A_{orifice}}{A_{port}}}\]

Pressure recovery describes a positive pressure change in the valve due to area increase. If you do not want to record this pressure increase, remove Pressure recovery flag. In this case, \(PR_{loss}\) is 1.

Law of conservation of energy (13):

(13)#\[\phi_А+\phi_B=0\]

where:

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

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

4. Properties#

Property	Default value	Description	Units of measurement
Orifice parametrization: Round; Rectangle	Round	-	-
Open orientation: Positive; Negative	Positive	-	-
Pressure recovery	Off	-	-
Rectangle width	0.01	Available, when `Orifice parametrization` - `Rectangle`	m
Spool travel	0.005	Available, when `Orifice parametrization` - `Rectangle`	m
Orifice diameter	5e-3	Available, when `Orifice parametrization` - `Round`	m
Orifice quantity	6.0	Available, when `Orifice parametrization` - `Round`	pcs
Spool offset	0.0	-	m
Port area	0.07	-	m\(^2\)
Leakage area	1e-10	-	m\(^2\)
Discharge coefficient	0.64	Available, when `Pressure recovery` - `On`	-
Critical Reynolds number	150.0	The Reynolds number at which the transition from laminar to turbulent flow occurs	-
Smoothing factor	0.01	-	-

5. Inputs and outputs#

Parameters	Input/Output	Description	Units of measurement	Port type
Auto	Input	-	m	Automation
Input	Input	-	-	Thermohydraulics
Output	Output	-	-	Thermohydraulics

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