# T-Junction (TL)

Three-way junction in a thermal liquid system

Since R2022a

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 ScenarioABCKAKBKC
Stagnant1 or last valid value1 or last valid value1 or last valid value
Diverging from node A>thresh<-ṁthresh<-ṁthresh0Kmain,divKside,div
Diverging from node B<-ṁthresh>thresh<-ṁthreshKmain,div0Kside,div
Converging to node A<-ṁthresh>thresh>thresh0Kmain,convKside,conv
Converging to node B>thresh<-ṁthresh>threshKmain,conv0Kside,conv
Converging to node C (branch) when the Loss coefficient model parameter is ```Crane correlation``` or `Custom`>thresh>thresh<-ṁthresh(Kmain,conv + Kside,conv)/2(Kmain,conv + Kside,conv)/20
Diverging from node C (branch) when the Loss coefficient model parameter is ```Crane correlation``` or `Custom`<-ṁthresh<-ṁthresh>thresh(Kmain,div + Kside,div)/2(Kmain,div + Kside,div)/20

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

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

where:

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

• ν is the fluid viscosity.

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

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

### Crane Correlation Coefficient Model

When you set the Loss coefficient model parameter to `Crane correlation`, the pipe loss coefficients, Kmain and Kside, and the pipe friction factor, fT, 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. KA, KB, and KC 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

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

where:

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

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

The value of Kmain,div saturates when ${\stackrel{˙}{m}}_{2}/{\stackrel{˙}{m}}_{1}$ is equal to the value of the Minimum valid flow ratio for coefficient calculation parameter.

The side branch diverging loss coefficient is

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

where

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

• d1 is the diameter of the main branch.

• d3 is the diameter of the side branch.

• `${K}_{*}=0.57-1.07{\left(\frac{r}{{d}_{3}}\right)}^{1/2}-2.13\frac{r}{{d}_{3}}+8.24{\left(\frac{r}{{d}_{3}}\right)}^{3/2}-8.48{\left(\frac{r}{{d}_{3}}\right)}^{2}+2.90{\left(\frac{r}{{d}_{3}}\right)}^{5/2}.$`

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

The value of Kside,div saturates when ${\stackrel{˙}{m}}_{3}/{\stackrel{˙}{m}}_{1}$ 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

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

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 Kmain,conv saturates when ${\stackrel{˙}{m}}_{2}/{\stackrel{˙}{m}}_{1}$ is equal to the value of the Minimum valid flow ratio for coefficient calculation parameter.

The side branch converging loss coefficient is

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

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 Kside,conv saturates when ${\stackrel{˙}{m}}_{3}/{\stackrel{˙}{m}}_{1}$ 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

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

where d is the diameter of the side branch. The value of Kconv to side saturates when ${\stackrel{˙}{m}}_{2}/{\stackrel{˙}{m}}_{1}$ 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

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

The value of Kdiv from side saturates when ${\stackrel{˙}{m}}_{2}/{\stackrel{˙}{m}}_{1}$ 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 Kmain,conv, Kmain,div, Kside,conv, and Kside,div as the 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

`${\stackrel{˙}{m}}_{A}+{\stackrel{˙}{m}}_{B}+{\stackrel{˙}{m}}_{C}=0.$`

The block calculates the flow through the pipe junction using the momentum conservation equations between ports A, B, and C:

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

where I represents the fluid inertia, and

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

Amain is the Main branch area (A-B) parameter and Aside is the Side branch area (A-C, B-C) parameter.

### Energy Balance

The block balances energy such that

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

where:

• ϕA is the energy flow rate at port A.

• ϕB is the energy flow rate at port B.

• ϕC is the energy flow rate at port 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

expand all

Thermal liquid conserving port associated with the liquid entrance or exit of the junction.

Thermal liquid conserving port associated with the liquid entrance or exit of the junction.

Thermal liquid conserving port associated with the liquid entrance or exit of the junction.

## Parameters

expand all

Area of connecting pipe between ports and B.

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

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

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

#### Dependencies

To enable this parameter, set Loss coefficient model to `Custom`.

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

#### Dependencies

To enable this parameter, set Loss coefficient model to `Custom`.

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

#### Dependencies

To enable this parameter, set Loss coefficient model to `Custom`.

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

#### Dependencies

To enable this parameter, set Loss coefficient model to `Custom`.

Radius of the curvature of the junction used to calculate the loss coefficients when Loss coefficient model is set to `Rennels correlation`.

#### Dependencies

To enable this parameter, set Loss coefficient model to ```Rennels correlation```.

Minimum flow ratio allowed when calculating the loss coefficients in the Rennels correlation parameterization. If the flow ratio is below this limit, it will saturate at this value.

#### Dependencies

To enable this parameter, set Loss coefficient model to ```Rennels correlation```.

Upper Reynolds number limit for laminar flow through the junction.

## 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.

## Version History

Introduced in R2022a

expand all