T-Junction (IL)

Three-way junction in an isothermal liquid system

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• Simscape / Fluids / Isothermal Liquid / Pipes & Fittings

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Description

The T-Junction (IL) block models a three-way pipe junction with a branch line at port C connected at a 90^o angle to the main pipe line, between ports A and B. You can specify a custom or standard junction type. When Three-way junction type is set to Custom, you can specify the loss coefficients of each pipe segment for converging and diverging flows. The standard model applies industry-standard loss coefficients to the momentum equations.

Flow Direction

The flow is converging when the branch flow, the flow through port C, merges into the main flow. The flow is diverging when the branch flow splits from the main flow. The flow direction between A and I, the point where the branch meets the main, and B and I must be consistent for all loss coefficients to be applied. If they are not, as shown in the last two diagrams in the figure below, the losses in the junction are approximated with the main branch loss coefficient for converging or diverging flows.

Flow Scenarios

Standard T-Junction

When Three-way junction type is set to Standard, the pipe loss coefficients, K_main and K_side, and the pipe friction factor, f_T, are calculated according to Crane [1]:

$$K_{main}=20f_{T,main},$$

$$K_{side}=60f_{T,side}.$$

In contrast to the custom junction type, the standard junction loss coefficient is the same for both converging and diverging flows. K_A, K_B, and K_C are then calculated in the same manner as custom junctions.

Friction Factor per Nominal Pipe Diameter

Custom T-Junction

When Three-way junction type is set to Custom, the pipe loss coefficient at each port, K, is calculated based on the user-defined loss parameters for converging and diverging flow and mass flow rate at each port. The coefficients are defined generally for positive and negative flows:

$$K_A=m_A^{+}\left({\dot{m}}_B^{+}{\dot{m}}_C^{-}\frac{K_{main,conv}}{2}+{\dot{m}}_B^{-}{\dot{m}}_C^{+}{K}_{main,conv}\right)+m_A^{-}\left({\dot{m}}_B^{+}{\dot{m}}_C^{-}{K}_{main,div}+{\dot{m}}_B^{-}{\dot{m}}_C^{+}\frac{K_{main,div}}{2}\right),$$

where

• K_main,conv is the Main branch converging loss coefficient.

• K_main,div is the Main branch diverging loss coefficient.

$$K_B=m_A^{+}\left({\dot{m}}_B^{+}{\dot{m}}_C^{-}\frac{K_{main,conv}}{2}+{\dot{m}}_B^{-}{\dot{m}}_C^{-}{K}_{main,div}\right)+m_A^{-}\left({\dot{m}}_B^{+}{\dot{m}}_C^{+}{K}_{main,conv}+{\dot{m}}_B^{-}{\dot{m}}_C^{+}\frac{K_{main,div}}{2}\right).$$

$$K_C=\left(m_A^{+}{\dot{m}}_B^{-}+{\dot{m}}_A^{-}{\dot{m}}_B^{+}\right)\left({\dot{m}}_C^{+}{K}_{side,conv}+{\dot{m}}_C^{-}{K}_{side,conv}\right),$$

where:

• K_side,conv is the Side branch converging loss coefficient.

• K_side,div is the Side branch diverging loss coefficient.

The positive mass flow direction at each port, when the flow direction is from A to B, from A to C, and from C to B, is defined as:

$${\dot{m}}_{port}^{+}=\frac{1+\tanh \left(\frac{4{\dot{m}}_{port}}{{\dot{m}}_{thresh}}\right)}{2}.$$

The negative mass flow direction is defined as:

$${\dot{m}}_{port}^{-}=\frac{1-\tanh \left(\frac{4{\dot{m}}_{port}}{{\dot{m}}_{thresh}}\right)}{2}.$$

The mass flow rate threshold, which is the point at which the flow in the pipe begins to reverse direction, is calculated as:

$${\dot{m}}_{thresh}={\mathrm{Re}}_c\upsilon \overline{\rho }\sqrt{\frac{\pi }{4}{A}_{\min }},$$

where:

• Re_c is the Critical Reynolds number, beyond which the transitional flow regime begins.

• ν is the fluid viscosity.

• $$\overline{\rho }$$ is the average fluid density.

• A_min is the smallest cross-sectional area in the pipe junction.

Momentum Balance

Mass is conserved in the pipe segment:

$${\dot{m}}_A+{\dot{m}}_B+{\dot{m}}_C=0.$$

Flow through the pipe junction is calculated from momentum conservation equations between ports A, B, and C:

$$p_A-p_I=\frac{K_A}{2\overline{\rho }{A}_{{}_{main}}^2}{\dot{m}}_A\sqrt{{\dot{m}}_A^2+{\dot{m}}_{thresh}^2}$$

$$p_B-p_I=\frac{K_B}{2\overline{\rho }{A}_{{}_{main}}^2}{\dot{m}}_B\sqrt{{\dot{m}}_B^2+{\dot{m}}_{thresh}^2}$$

$$p_C-p_I=\frac{K_C}{2\overline{\rho }{A}_{{}_{side}}^2}{\dot{m}}_C\sqrt{{\dot{m}}_C^2+{\dot{m}}_{thresh}^2}$$

where A_main is the Main branch area (A-B) and A_side is the Side branch area (A-C, B-C).

Ports

Conserving

A — Liquid port

isothermal liquid

Liquid entry or exit port.

B — Liquid port

isothermal liquid

Liquid entry or exit port.

C — Liquid port

isothermal liquid

Liquid entry or exit port.

Parameters

Main branch area (A-B) — Cross-sectional area between ports A and B

0.01 m^2 (default) | positive scalar

Area of connecting pipe between ports A and B.

Side branch area (A-C, B-C) — Cross-sectional area of the side branch

0.01 m^2 (default) | positive scalar

Area of connecting pipe between ports A and C and between ports B and C.

Three-way junction type — Junction loss coefficient type

Standard (default) | Custom

The junction loss coefficient type. Set this parameter to Custom to specify individual diverging and converging loss coefficients for each flow path segment.

Main branch converging loss coefficient — Loss coefficient between A and B for custom junctions

1.12 (default) | positive scalar

Loss coefficient for pressure loss calculations between ports A and B for converging flow.

Dependencies

To enable this parameter, set Three-way junction type to Custom.

Main branch diverging loss coefficient — Loss coefficient between A and B for custom junctions

1.12 (default) | positive scalar

Loss coefficient for pressure loss calculations between ports A and B for diverging flow.

Dependencies

To enable this parameter, set Three-way junction type to Custom.

Side branch converging loss coefficient — Branch loss coefficient for custom junctions

1.12 (default) | positive scalar

Loss coefficient for pressure loss calculations between port C and the main line for converging flow.

Dependencies

To enable this parameter, set Three-way junction type to Custom.

Side branch diverging loss coefficient — Branch loss coefficient for custom junctions

1.12 (default) | positive scalar

Loss coefficient for pressure loss calculations between port C and the main line for diverging flow.

Dependencies

To enable this parameter, set Three-way junction type to Custom.

Critical Reynolds number — Upper Reynolds number limit for laminar flow

150 (default) | positive scalar

Upper Reynolds number limit for laminar flow through the junction.

Model Examples

Lubrication System

A simplified version of a lubrication system fed with the centrifugal pump. The system consists of five major units: Pump Unit, Scavenge Unit, Heat Exchanger Manifold, Nozzle Manifold, and Control Unit. Both the pump and the scavenge unit are built around the centrifugal pump. The Scavenge Unit collects fluid discharged by nozzles and pumps it back into the reservoir of the Pump Unit. The Control Unit generates commands to bypass either the heat exchanger, represented as a local resistance, or the nozzles block. In a real system, these commands are generated by temperature sensors installed in lubrication cavities.

References

[1] Crane Co. Flow of Fluids Through Valves, Fittings, and Pipe TP-410. Crane Co., 1981.