# T-Junction (TL)

Three-way junction in a thermal liquid system

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

## Description

The T-Junction (TL) block represents a three-way pipe
junction with a branch line at port **C** connected at a 90° 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**

The block uses mode charts to determine each loss coefficient for a given flow configuration. This table describes the conditions and coefficients for each operational mode.

Flow Scenario | ṁ_{A} | ṁ_{B} | ṁ_{C} | K_{A} | K_{B} | K_{C} |
---|---|---|---|---|---|---|

Converging to node B | >ṁ_{thresh} | <-ṁ_{thresh} | >ṁ_{thresh} | K_{main,conv} | 0 | K_{main,conv} |

Converging to node A | <-ṁ_{thresh} | >ṁ_{thresh} | >ṁ_{thresh} | 0 | K_{main,conv} | K_{side,conv} |

Diverging from node A | >ṁ_{thresh} | <-ṁ_{thresh} | <-ṁ_{thresh} | 0 | K_{main,div} | K_{side,div} |

Diverging from node B | <-ṁ_{thresh} | >ṁ_{thresh} | <-ṁ_{thresh} | K_{main,div} | 0 | K_{side,div} |

Converging to node C (branch) | >ṁ_{thresh} | >ṁ_{thresh} | <-ṁ_{thresh} | (K +
_{main,conv}K)/2_{branch,conv} | (K +
_{main,conv}K)/2_{branch,conv} | 0 |

Diverging from node C (branch) | <-ṁ_{thresh} | <-ṁ_{thresh} | >ṁ_{thresh} | (K +
_{main,div}K)/2_{branch,div} | (K +
_{main,div}K)/2_{branch,div} | 0 |

Stagnant | – | – | – | 1 or last valid | 1 or last valid | 1 or last valid |

The flow is stagnant when the mass flow rate conditions do not match any defined flow scenario. 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*is the smallest cross-sectional area in the pipe junction._{min}

### 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]:

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

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

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**

### 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. You must specify
*K _{main,conv}*,

*K*,

_{main,div}*K*, and

_{side,conv}*K*as the

_{side,div}**Main branch converging loss coefficient**,

**Main branch diverging loss coefficient**,

**Side branch converging loss coefficient**, and

**Side branch diverging loss coefficient**parameters, respectively.

### Mass and Momentum Balance

The block conserves mass in the junction such that

$${\dot{m}}_{A}+{\dot{m}}_{B}+{\dot{m}}_{C}=0.$$

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

$$\begin{array}{l}{p}_{A}-{p}_{I}={I}_{A}+\frac{{K}_{A}}{2\overline{\rho}{A}_{{}_{main}}^{2}}{\dot{m}}_{A}\sqrt{{\dot{m}}_{A}^{2}+{\dot{m}}_{thresh}^{2}}\\ {p}_{B}-{p}_{I}={I}_{B}+\frac{{K}_{B}}{2\overline{\rho}{A}_{{}_{main}}^{2}}{\dot{m}}_{B}\sqrt{{\dot{m}}_{B}^{2}+{\dot{m}}_{thresh}^{2}}\\ {p}_{C}-{p}_{I}={I}_{C}+\frac{{K}_{C}}{2\overline{\rho}{A}_{{}_{side}}^{2}}{\dot{m}}_{C}\sqrt{{\dot{m}}_{C}^{2}+{\dot{m}}_{thresh}^{2}}\end{array}$$

where *I* represents the fluid inertia, and

$$\begin{array}{l}{I}_{A}={\ddot{m}}_{A}\frac{\sqrt{\pi \cdot {A}_{side}}}{{A}_{main}}\\ {I}_{B}={\ddot{m}}_{B}\frac{\sqrt{\pi \cdot {A}_{side}}}{{A}_{main}}\\ {I}_{C}={\ddot{m}}_{C}\frac{\sqrt{\pi \cdot {A}_{main}}}{{A}_{side}}\end{array}$$

*A _{main}* is the

**Main branch area (A-B)**parameter and

*A*is the

_{side}**Side branch area (A-C, B-C)**parameter.

### Energy Balance

The block balances energy such that

$${\varphi}_{A}+{\varphi}_{B}+{\varphi}_{C}=0,$$

where:

*ϕ*is the energy flow rate at port_{A}**A**.*ϕ*is the energy flow rate at port_{B}**B**.*ϕ*is the energy flow rate at port_{C}**C**.

### Variables

To set the priority and initial target values for the block variables prior to simulation, use
the **Initial Targets** section in the block dialog box or
Property Inspector. For more information, see Set Priority and Initial Target for Block Variables.

Nominal values provide a way to specify the expected magnitude of a variable in a model.
Using system scaling based on nominal values increases the simulation robustness. Nominal
values can come from different sources, one of which is the **Nominal
Values** section in the block dialog box or Property Inspector. For more
information, see Modify Nominal Values for a Block Variable.

## Ports

### Conserving

## Parameters

## References

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

## Extended Capabilities

## Version History

**Introduced in R2022a**