# T-Junction (TL)

**Libraries:**

Simscape /
Fluids /
Thermal 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 junction or a junction that uses a Rennels correlation or Crane
correlation loss coefficient. When **Loss coefficient model** is set to
`Custom`

, you can specify the loss coefficients of each
pipe segment for converging and diverging flows.

### Flow Direction

The flow is *converging* when 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} |
---|---|---|---|---|---|---|

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

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

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

Converging to node C (branch) when the Loss
coefficient model parameter is ```
Crane
correlation
``` or
`Custom` | >ṁ_{thresh} | >ṁ_{thresh} | <-ṁ_{thresh} | (K +
_{main,conv}K)/2_{side,conv} | (K +
_{main,conv}K)/2_{side,conv} | 0 |

Diverging from node C (branch) when the Loss
coefficient model parameter is ```
Crane
correlation
``` or
`Custom` | <-ṁ_{thresh} | <-ṁ_{thresh} | >ṁ_{thresh} | (K +
_{main,div}K)/2_{side,div} | (K +
_{main,div}K)/2_{side,div} | 0 |

When the **Loss coefficient model** parameter ```
Rennels
correlation
```

, the values for converging to node C (branch) and
diverging from node C (branch) are calculated directly.

The flow is stagnant when the mass flow rate conditions do not match any defined flow scenario. Stagnant flow is permitted at the start of the simulation, but the block does not revert to stagnant flow after it has achieved another mode. The mass flow rate threshold, which is the point at which the flow in the pipe begins to reverse direction, is

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

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}

### Crane Correlation Coefficient Model

When you set the **Loss coefficient model** parameter to
`Crane correlation`

, the pipe loss coefficients,
*K _{main}* and

*K*, and the pipe friction factor,

_{side}*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**

### Rennels Correlation Coefficient Model

When you set the **Loss coefficient model** parameter to
`Rennels correlation`

, the block calculates the pipe
loss coefficients according to [2].

**Diverging Flow on Main Branch**

The main branch diverging loss coefficient is

$${\text{K}}_{main,div}=0.62-0.98\frac{{\dot{m}}_{1}}{{\dot{m}}_{2}}+0.36{\left(\frac{{\dot{m}}_{1}}{{\dot{m}}_{2}}\right)}^{2}+0.03{\left(\frac{{\dot{m}}_{2}}{{\dot{m}}_{1}}\right)}^{6},$$

where:

$${\dot{m}}_{1}$$ is the mass flow rate at the inflow of the main branch.

$${\dot{m}}_{2}$$ is the mass flow rate at the outflow of the main branch.

The value of *K _{main,div}* saturates
when $${\dot{m}}_{2}/{\dot{m}}_{1}$$ is equal to the value of the

**Minimum valid flow ratio for coefficient calculation**parameter.

The side branch diverging loss coefficient is

$${K}_{side,div}=\left(0.81-1.13\frac{{\dot{m}}_{1}}{{\dot{m}}_{3}}+{\left(\frac{{\dot{m}}_{1}}{{\dot{m}}_{3}}\right)}^{2}\right){\left(\frac{{d}_{3}}{{d}_{1}}\right)}^{4}+1.12\frac{{d}_{3}}{{d}_{1}}-1.08{\left(\frac{{d}_{3}}{{d}_{1}}\right)}^{3}+{K}_{*},$$

where

$${\dot{m}}_{3}$$ is the mass flow rate at the outflow of the side branch.

*d*is the diameter of the main branch._{1}*d*is the diameter of the side branch._{3}$${K}_{*}=0.57-1.07{\left(\frac{r}{{d}_{3}}\right)}^{1/2}-2.13\frac{r}{{d}_{3}}+8.24{\left(\frac{r}{{d}_{3}}\right)}^{3/2}-8.48{\left(\frac{r}{{d}_{3}}\right)}^{2}+2.90{\left(\frac{r}{{d}_{3}}\right)}^{5/2}.$$

*r*is the value of the**Junction radius of curvature**parameter.

The value of *K _{side,div}* saturates
when $${\dot{m}}_{3}/{\dot{m}}_{1}$$ is equal to the value of the

**Minimum valid flow ratio for coefficient calculation**parameter.

**Converging Flow on Main Branch**

The main branch converging loss coefficient is

$${\text{K}}_{main,conv}={\left(\frac{{\dot{m}}_{1}}{{\dot{m}}_{2}}\right)}^{2}-0.95-2{C}_{xC}\left(\frac{{\dot{m}}_{1}}{{\dot{m}}_{2}}-1\right)-2{C}_{M}\left({\left(\frac{{\dot{m}}_{1}}{{\dot{m}}_{2}}\right)}^{2}-\frac{{\dot{m}}_{1}}{{\dot{m}}_{2}}\right),$$

where

$$\begin{array}{l}{C}_{M}=0.23+1.46\left(\frac{r}{{d}_{3}}\right)-2.75{\left(\frac{r}{{d}_{3}}\right)}^{2}+1.65{\left(\frac{r}{{d}_{3}}\right)}^{3}\\ {C}_{xC}=0.08+0.56\left(\frac{r}{{d}_{3}}\right)-1.75{\left(\frac{r}{{d}_{3}}\right)}^{2}+1.83{\left(\frac{r}{{d}_{3}}\right)}^{3}\end{array}$$

The value of *K _{main,conv}* saturates
when $${\dot{m}}_{2}/{\dot{m}}_{1}$$ is equal to the value of the

**Minimum valid flow ratio for coefficient calculation**parameter.

The side branch converging loss coefficient is

$${\text{K}}_{side,conv}=(2{C}_{yC}-1)+{\left(\frac{{d}_{3}}{{d}_{1}}\right)}^{4}\left[2\left({C}_{xC}-1\right)+2\left(2-{C}_{xC}-{C}_{M}\right)\frac{{\dot{m}}_{1}}{{\dot{m}}_{3}}-0.92{\left(\frac{{\dot{m}}_{1}}{{\dot{m}}_{3}}\right)}^{2}\right],$$

where:

$${\text{C}}_{yC}=1-0.25{\left(\frac{{d}_{3}}{{d}_{1}}\right)}^{1.3}-\left[0.11\frac{r}{{d}_{3}}-0.65\left(\frac{r}{{d}_{3}}\right)+0.83{\left(\frac{r}{{d}_{3}}\right)}^{3}\right]{\left(\frac{{d}_{3}}{{d}_{1}}\right)}^{2}.$$

The value of *K _{side,conv}* saturates
when $${\dot{m}}_{3}/{\dot{m}}_{1}$$ is equal to the value of the

**Minimum valid flow ratio for coefficient calculation**parameter.

**Converging or Diverging flow from Side Branch**

The loss coefficient when the flow is converging to the side branch is

$${K}_{convtoside}=\left(0.81-1.16\sqrt{\frac{r}{d}}+0.5\frac{r}{d}\right){\left(\frac{{\dot{m}}_{1}}{{\dot{m}}_{2}}\right)}^{2}-\left(0.95-1.65\frac{r}{d}\right)\frac{{\dot{m}}_{1}}{{\dot{m}}_{2}}+1.34-1.69\frac{r}{d},$$

where *d* is the diameter of the side branch. The value of
*K _{conv to side}* saturates when $${\dot{m}}_{2}/{\dot{m}}_{1}$$ is equal to the value of the

**Minimum valid flow ratio for coefficient calculation**parameter.

The loss coefficient when the flow is diverging from the side branch is

$${K}_{divfromside}=0.59{\left(\frac{{\dot{m}}_{1}}{{\dot{m}}_{2}}\right)}^{2}+\left(1.18-1.84\sqrt{\frac{r}{d}}+1.16\frac{r}{d}\right)\frac{{\dot{m}}_{1}}{{\dot{m}}_{2}}-0.68+1.04\sqrt{\frac{r}{d}}-1.16\frac{r}{d}.$$

The value of *K _{div from side}*
saturates when $${\dot{m}}_{2}/{\dot{m}}_{1}$$ is equal to the value of the

**Minimum valid flow ratio for coefficient calculation**parameter.

### Custom T-Junction

When you set the **Loss coefficient model** parameter to
`Custom`

, the block calculates the pipe loss
coefficient at each port, *K*, 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

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

The block calculates the flow through the pipe junction using the
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.

[2] Rennels, D. C., & Hudson,
H. M. *Pipe flow: A practical and comprehensive guide*.
Hoboken, N.J: John Wiley & Sons., 2012.

## Extended Capabilities

## Version History

**Introduced in R2022a**