Y-junction (P)#

The Y-junction (P) block models the intersection of three branches of a pneumatic network at an arbitrary angle. To calculate the loss coefficients at a flow intersection, one of three correlation options can be selected in accordance with the values of the Loss coefficient model property:

According to Idelchik;
Custom.

Branch and flow type symbols:

Hint

The AIB branch is the main one, the CI branch is the side one.

1. Idelchik’s Loss Calculation#

The loss coefficients for converging and diverging flows are equal and are defined for the main and side branches as follows (1), (2), (3), (6):

(1)#\[K_{side,conv}=(1+(\frac{\dot{m_C}}{\dot{m_B}}\cdot \frac{A_B}{A_C})^2-2 \cdot(\frac{A_B}{A_A})\cdot (1-\frac{\dot{m_C}}{\dot{m_B}})^2-2 \cdot cos \alpha \cdot \frac{A_B}{A_C}\cdot (\frac{\dot{m_C}}{\dot{m_B}})^2)\cdot (\frac{\dot{m_C}}{\dot{m_B}}\cdot \frac{A_C}{A_B})^{-2}\]

(2)#\[K_{main,conv}=(1-(1-\frac{\dot{m_C}}{\dot{m_B}})^2-2 \cdot cos \alpha \cdot \frac{A_B}{A_C}\cdot (\frac{\dot{m_C}}{\dot{m_B}})^2)\cdot (1-(\frac{\dot{m_C}}{\dot{m_B}})^2)^{-1}\]

(3)#\[K_{side,div}=A'\cdot(1+(\frac{\dot{m_C}}{\dot{m_B}}\cdot \frac{A_B}{A_C})^2-2 \cdot cos \alpha \cdot \frac{A_B}{A_C}\cdot \frac{\dot{m_C}}{\dot{m_B}})\cdot (\frac{\dot{m_C}}{\dot{m_B}}\cdot \frac{A_C}{A_B})^{-2}\]

where \(A'\) coefficient is defined as (4) :

(4)#\[A'=\frac{1}{2}\cdot (1+tanh(\frac{4\cdot (R_{A'}-0.8)}{0.8}))+\frac{0.9}{2}\cdot (1-tanh(\frac{4\cdot (R_{A'}-0.8)}{0.8}))\]

where \(R_{A'}\) coefficient is defined as (5) :

(5)#\[R_{A'}=\frac{\sqrt{\dot{m^2_C}+\dot{m^2_{thresh}}}}{\sqrt{\dot{m^2_B}+\dot{m^2_{thresh}}}}\]

(6)#\[K_{main,div}=0.4 \cdot (1-\frac{\dot{m_A}}{\dot{m_B}}\cdot\frac{A_B}{A_A})^2\cdot (\frac{\dot{m_A}}{\dot{m_B}})^{-2}\]

User Loss Factors:

With this setting, the user directly sets dimensionless loss coefficients for converging and diverging flows on the main and side branches:

\(K_{main,conv}\) – divergent flow on the main branch;

\(K_{main,div}\) – converging flow on the main branch;

\(K_{side,conv}\) – divergent flow on the side branch;

\(K_{side,div}\) – converging flow on the side branch.

Mass conservation equation (7) :

(7)#\[m_A+m_B+m_C=0\]

Pressure drop calculation (8), (9), (10) :

(8)#\[p_A-p_I=I_A+\frac{K_A}{2\cdot \bar \rho \cdot A^2_{main}}\cdot \dot{m_A} \cdot \sqrt{\dot{m_A}^2+\dot{m_{thresh}}^2}\]

(9)#\[p_B-p_I=I_B+\frac{K_B}{2\cdot \bar \rho \cdot A^2_{main}}\cdot \dot{m_B} \cdot \sqrt{\dot{m_B}^2+\dot{m_{thresh}}^2}\]

(10)#\[p_C-p_I=I_C+\frac{K_C}{2\cdot \bar \rho \cdot A^2_{side}}\cdot \dot{m_C} \cdot \sqrt{\dot{m_C}^2+\dot{m_{thresh}}^2}\]

where:

\(A_{main}\) – main branch cross-sectional area, [m\(^2\)];

\(A_{side}\) – side branch cross-sectional area, [m\(^2\)];

\(K_A, K_B, K_C\) – dimensionless loss coefficients on branches AI, BI, CI respectively;

\(I_A, I_B, I_C\) – gas inertia on branches AI, BI, CI respectively;

\(m_A, m_B, m_C\) – mass flow rate of gas through ports A, B, C respectively, [kg/s].

\(m_{thresh}\) – threshold flow rate, below which the flow is considered stagnant and cannot be defined as converging or diverging. Calculated using the following formula (11) :

(11)#\[\dot{m_{thresh}}=Re_{crit} \cdot \nu \cdot \bar \rho \cdot \sqrt{\frac{\pi}{4}\cdot A_{min}}\]

where:

\(A_{min}\) – minimum area of the side and main branches, [m\(^2\)];

\( ho\) – gas density, [kg/m\(^3\)];

\( u\) – kinematic viscosity of gas, [mm\(^2\)/s].

Calculation of inertia (12), (13), (14) :

(12)#\[I_A=\ddot{m_A}\cdot \frac{\sqrt{\pi \cdot A_{side}}}{A_{main}}\]

(13)#\[I_B=\ddot{m_B}\cdot \frac{\sqrt{\pi \cdot A_{side}}}{A_{main}}\]

(14)#\[I_C=\ddot{m_C}\cdot \frac{\sqrt{\pi \cdot A_{main}}}{A_{side}}\]

2. Properties#

Property	Default Value	Description	Units of measurement
Loss coefficient model: Idelchik; Custom	Idelchik	-	-
Main branch area	0.01	-	m\(^2\)
Side branch area	0.01	-	m\(^2\)
Junction angle, available when selecting `Loss coefficient model` - `Idelchik`	45.0	-	\(^о\)
Critical Reynolds number	150.0	-	-
Loss coefficient for main branch (convergence flow), available when selecting `Loss coefficient model` - `Custom`	1.12	-	-
Loss coefficient for main branch (divergence flow), available when selecting `Loss coefficient model` - `Custom`	1.12	-	-
Loss coefficient for side branch (convergence flow), available when selecting `Loss coefficient model` - `Custom`	1.12	-	-
Loss coefficient for side branch (divergence flow), available when selecting `Loss coefficient model` - `Custom`	1.12	-	-

3. Inputs and outputs#

Parameters	Input/Output	Description	Units of measurement	Port type
Port A	Input	-	-	Pneumatics
Port B	Input	-	-	Pneumatics
Port C	Input	-	-	Pneumatics

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