T-junction (P)#

The T-junction (P) block models the intersection of three branches of a pneumatic network at a 90° angle. To calculate the loss coefficients at the intersection of flows, one of three correlation options can be selected in accordance with the values of the Loss coefficient model property:

According to Crane;
According to Rennel;
Custom.

Branch and flow type symbols:

Hint

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

1. Crane’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) :

(1)#\[K_{main,div}=K_{main,conv}=20 \cdot f_{T,main}\]

(2)#\[K_{side,div}=K_{side,conv}=60 \cdot f_{T,said}\]

where the dimensionless coefficients \(f_T\) for the side and main branch depend on the corresponding diameters and are calculated as per the table by interpolation method:

Nominal size, mm	5	10	15	20	25	32	40	50	72.5	100	125	150	225	350	609.5
Friction coefficient, \(f_T\)	0.035	0.029	0.027	0.025	0.023	0.022	0.021	0.019	0.018	0.017	0.016	0.015	0.014	0.013	0.012

2. Rennel’s Loss Calculation#

The loss coefficients for converging and diverging flows on the main and side branch are defined as (3), (4), (6), (9), (11), (12) :

(3)#\[K_{main,div}=0.62 - 0.98 \cdot \frac{\dot{m_1}}{\dot{m_2}}+0.36\cdot( \frac{\dot{m_1}}{\dot{m_2}})^2+0.03\cdot (\frac{\dot{m_2}}{\dot{m_1}})^6\]

(4)#\[K_{side,div}=(0.81 - 1.13 \cdot \frac{\dot{m_1}}{\dot{m_3}}+( \frac{\dot{m_1}}{\dot{m_3}})^2)\cdot (\frac{d_3}{d_1})^4+1.12 \cdot \frac{d_3}{d_1}-1.08 \cdot (\frac{d_3}{d_1})^2+K_*\]

where \(K_*\) is calculated as (5) :

(5)#\[K_*=0.57-1.07 \cdot(\frac{r}{d_3})^{1/2}-2.13 \cdot \frac{r}{d_3}+8.24 \cdot (\frac{r}{d_3})^{3/2}-8.48 \cdot (\frac{r}{d_3})^2+2.9 \cdot (\frac{r}{d_3})^{5/2}\]

(6)#\[K_{main,conv}=(\frac{\dot{m_1}}{\dot{m_2}})^2-0.95-2\cdot C_{xC} \cdot ( \frac{\dot{m_1}}{\dot{m_2}}-1)-2\cdot C_M ((\frac{\dot{m_1}}{\dot{m_2}})^2-\frac{\dot{m_1}}{\dot{m_2}})\]

where \(С_M\) and \(C_{xC}\) coefficients are calculated as (7), (8) :

(7)#\[C_M=0.23+1.46 \cdot (\frac{r}{d_3})-2.75 \cdot(\frac{r}{d_3})^2+1.65 \cdot(\frac{r}{d_3})^3\]

(8)#\[C_{xC}=0.08+0.56\cdot (\frac{r}{d_3})-1.75 \cdot(\frac{r}{d_3})^2+1.83 \cdot(\frac{r}{d_3})^3\]

(9)#\[K_{side,conv}=(2\cdot C_{yC}-1)+(\frac{d_3}{d_1})^4\cdot(2\cdot(C_{xC}-1)+2\cdot (2-C_{xC}-C_M)\cdot \frac{\dot{m_1}}{\dot{m_3}}-0.92 \cdot (\frac{\dot{m_1}}{\dot{m_3}})^2)\]

where coefficient \(C_{yC}\) is calculated as (10) :

(10)#\[C_{yC}=1-0.25 \cdot (\frac{d_3}{d_1})^{1.3}-(0.11 \cdot \frac{r}{d_3}-0.65 \cdot (\frac{r}{d_3})^2+0.83 \cdot(\frac{r}{d_3})^3)\cdot (\frac{d_3}{d_1})^2\]

(11)#\[K_{convtoside}=(0.81-1.16 \cdot \sqrt{\frac{r}{d}}+0.5 \cdot \frac{r}{d})\cdot (\frac{\dot{m_1}}{\dot{m_2}})^2-(0.95-1.65 \cdot \frac{r}{d})\cdot \frac{\dot{m_1}}{\dot{m_2}}+1.34-1.69 \cdot \frac{r}{d}\]

(12)#\[K_{divfromside}=0.59\cdot (\frac{\dot{m_1}}{\dot{m_2}})^2+(1.18-1.84\cdot \sqrt{\frac{r}{d}}+1.16 \cdot \frac{r}{d})\cdot \frac{\dot{m_1}}{\dot{m_2}}-0.68+1.04 \cdot \sqrt{\frac{r}{d}}-1.16 \cdot \frac{r}{d}\]

Hint

All the above coefficients are dimensionless.

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(13) :

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

Pressure drop calculation (14), (15), (16) :

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

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

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

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 (17) :

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

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 (18), (19), (20) :

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

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

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

Law of conservation of energy (21) :

(21)#\[\phi_A+\phi_B=0\]

where:

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

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

3. Properties#

Property	Default Value	Description	Units of measurement
Loss coefficient model: Crane; Rennel; Custom	Crane	-	-
Main branch area	0.01	-	m\(^2\)
Side branch area	0.01	-	m\(^2\)
Junction radius, available when selecting `Loss coefficient model` - `Rennel`	0.5	-	m
Critical Reynolds number	150.0	-	-
Minimum flow ratio, available when selecting `Loss coefficient model` - `Rennel`	0.2	-	-
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	-	-

4. 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