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Three-way junction in an isothermal liquid system

**Library:**Simscape / Fluids / Isothermal Liquid / Pipes & Fittings

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.

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**

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]:

$${K}_{main}=20{f}_{T,main},$$

$${K}_{side}=60{f}_{T,side}.$$

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**

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:

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

The negative mass flow direction is defined as:

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

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

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

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.

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)**.

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

Area Change (IL) | Elbow (IL) | Local Resistance (IL) | Pipe Bend (IL)