# Y-Junction (TL)

Y-junction in a thermal liquid system

Since R2023a

Libraries:
Simscape / Fluids / Thermal Liquid / Pipes & Fittings

## Description

The Y-Junction (TL) block models a Y-junction consisting of a main branch and a side branch in an thermal liquid network. The path between ports A and B is the main branch. The path between ports A and C or between ports B and C is the side branch. The side branch extends from the main branch at an angle specified by the value of the Junction angle between (A-C) parameter.

### 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 block uses mode charts to determine each loss coefficient for a given flow configuration. This table describes the conditions and coefficients when the Loss coefficient model parameter is `Custom`.

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) >thresh>thresh<-ṁthresh(Kmain,conv + Kside,conv)/2(Kmain,conv + Kside,conv)/20
Diverging from node C (branch)<-ṁthresh<-ṁthresh>thresh(Kmain,div + Kside,div)/2(Kmain,div + Kside,div)/20

This table describes the conditions and coefficients when the Loss coefficient model parameter is ```Idel'chik correlation```.

Flow ScenarioABCKAKBKC
Stagnant1 or last valid value1 or last valid value1 or last valid value
Diverging from node A — Invalid>thresh<-ṁthresh<-ṁthresh011
Diverging from node B<-ṁthresh>thresh<-ṁthreshKmain,div0Kside,div
Converging to node A — Invalid<-ṁthresh>thresh>thresh011
Converging to node B>thresh<-ṁthresh>threshKmain,conv0Kside,conv
Converging to node C (branch) — Invalid>thresh>thresh<-ṁthresh11 0
Diverging from node C (branch) — Invalid<-ṁthresh<-ṁthresh>thresh110

In stagnant flow, 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 parameter, 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.

### Idel'chik Correlation Coefficient Model

When you set the Loss coefficient model parameter to `Idel'chik correlation`, the block calculates the pipe loss coefficients according to [2].

Flow Configuration

The block supports two flow configurations between the main branch from ports A and B and the side branch from port C. The side branch is offset from the main branch at an angle α, which is the value of the Junction angle between (A-C) parameter.

In a converging flow, the flow enters at ports A and C and exits at port B.

In a diverging flow, the flow enters at port B and exits at ports A and C.

You can control the block behavior in a prohibited flow configuration by using the Report when flow configuration is invalid parameter. If the Report when flow configuration is invalid parameter is set to `None` or `Warning`, the model continues to run in the prohibited flow configuration, but the results may not be correct.

Converging Flow

For a converging flow, the block calculates the loss coefficient of the side branch between ports C and B using a simplified version of Idel'chik that assumes the area of the main branch is constant. The loss coefficient of the side branch is

`${K}_{side,conv}=\left[1+{\left(\frac{{\stackrel{˙}{m}}_{C}}{{\stackrel{˙}{m}}_{B}}\frac{{A}_{B}}{{A}_{C}}\right)}^{2}-2\left(\frac{{A}_{B}}{{A}_{A}}\right){\left(1-\frac{{\stackrel{˙}{m}}_{C}}{{\stackrel{˙}{m}}_{B}}\right)}^{2}-2\ast \mathrm{cos}\alpha \ast \frac{{A}_{B}}{{A}_{C}}{\left(\frac{{\stackrel{˙}{m}}_{C}}{{\stackrel{˙}{m}}_{B}}\right)}^{2}\right]{\left[\frac{{\stackrel{˙}{m}}_{C}}{{\stackrel{˙}{m}}_{B}}\frac{{A}_{C}}{{A}_{B}}\right]}^{-2},$`

where:

• ${\stackrel{˙}{m}}_{A},{\stackrel{˙}{m}}_{B},{\stackrel{˙}{m}}_{C}$ are the mass flow rates from rates at ports A, B, and C, respectively.

• AA, AB, AC are fitting coefficients for ports A, B, and C, respectively.

The block calculates the loss coefficient of the main branch between ports A and B using a simplified version of Idel'chik that assumes the main branch area is constant and ${\text{Q}}_{C}={Q}_{B}+{Q}_{S}$, where Q is the volumetric flow rate through the specified port. The main branch loss coefficient is

`${K}_{main,conv}=\left[1-{\left(1-\frac{{\stackrel{˙}{m}}_{C}}{{\stackrel{˙}{m}}_{B}}\right)}^{2}-2\ast \mathrm{cos}\alpha \ast \frac{{A}_{B}}{{A}_{C}}{\left(\frac{{\stackrel{˙}{m}}_{C}}{{\stackrel{˙}{m}}_{B}}\right)}^{2}\right]{\left[1-{\left(\frac{{\stackrel{˙}{m}}_{C}}{{\stackrel{˙}{m}}_{B}}\right)}^{2}\right]}^{-1}.$`
Diverging Flow

For a diverging flow, the block calculates the loss coefficient of the side branch between ports C and B as

`${K}_{side,div}=A\prime \left[1+{\left(\frac{{\stackrel{˙}{m}}_{C}}{{\stackrel{˙}{m}}_{B}}\frac{{A}_{B}}{{A}_{C}}\right)}^{2}-2\ast \mathrm{cos}\alpha \ast \frac{{\stackrel{˙}{m}}_{C}}{{\stackrel{˙}{m}}_{B}}\frac{{A}_{B}}{{A}_{C}}\right]{\left[\frac{{\stackrel{˙}{m}}_{C}}{{\stackrel{˙}{m}}_{B}}\frac{{A}_{C}}{{A}_{B}}\right]}^{-2},$`

where

`$\text{A'=}\frac{1}{2}\left[1+\mathrm{tanh}\left(\frac{4\left({R}_{{A}^{\prime }}-0.8\right)}{0.8}\right)\right]+\frac{0.9}{2}\left[1-\mathrm{tanh}\left(\frac{4\left({R}_{{A}^{\prime }}-0.8\right)}{0.8}\right)\right]$`

and

`${R}_{{A}^{\prime }}=\frac{\sqrt{{\stackrel{˙}{m}}_{C}^{2}+{\stackrel{˙}{m}}_{thresh}^{2}}}{\sqrt{{\stackrel{˙}{m}}_{B}^{2}+{\stackrel{˙}{m}}_{thresh}^{2}}}.$`

The block calculates the loss coefficient of the main branch between ports A and B as

`${K}_{main,div}=0.4{\left(1-\frac{{\stackrel{˙}{m}}_{A}}{{\stackrel{˙}{m}}_{B}}\frac{{A}_{B}}{{A}_{A}}\right)}^{2}{\left(\frac{{\stackrel{˙}{m}}_{A}}{{\stackrel{˙}{m}}_{B}}\right)}^{-2}.$`

### Custom Y-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. The custom loss coefficient model behavior for the Y-junction is the same as for the custom T-junction.

### Mass and Momentum Balance

The block conserves mass in the junction such that

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

Flow through the pipe junction behaves according to 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 (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

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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

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Area of connecting pipe between ports and B.

Area of connecting pipe between port C and the main branch.

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

Time scale of the smoothing function. Use this parameter to ensure smooth transitions by tuning the block dynamic behavior during flow reversals.

Upper Reynolds number limit for laminar flow through the junction.

Reynolds number that the block uses to calculate the threshold beneath which the flow is stagnant. The block uses this threshold to reduce numerical chatter during simulation. This parameter does not have a physical meaning, but must be small in comparison to the expected mass flow rate values.

Angle of the side branch offset with respect to the main branch.

#### Dependencies

To enable this parameter, set Loss coefficient model to ```Idel'chik correlation```.

Minimum value for any flow ratio that the block uses to calculate loss coefficients. If the flow ratio is below this limit, it saturates at this value. The block uses this parameter to limit the impact of differences in flow magnitude between branches and to increase numerical stability if one branch has significantly less flow than others.

Only adjust this setting if your model has numerical stability problems.

#### Dependencies

To enable this parameter, set Loss coefficient model to ```Idel'chik correlation```.

Continuous smoothing factor that introduces a layer of gradual change to the flow response when it approaches the limit specified by the Minimum valid flow ratio for coefficient calculation parameter. Set this parameter to a nonzero value less than one to increase the stability of your simulation.

Only adjust this setting if your model has numerical stability problems.

#### Dependencies

To enable this parameter, set Loss coefficient model to ```Idel'chik correlation```.

Whether to model fluid inertia, which lowers the risk of a block numerical issue during flow reversals. Avoid modeling fluid inertia unless it is numerically necessary, because it increases the computational cost.

#### Dependencies

To enable this parameter, set Loss coefficient model to ```Idel'chik correlation```.

Simulation warning mode for operating conditions outside the valid flow configurations. If you select `Warning`, the block generates a warning for conditions outside the valid flow configurations. The warning does not cause simulation to stop. If you select `Error`, the simulation will stop for conditions outside the valid flow configurations.

#### Dependencies

To enable this parameter, set Loss coefficient model to ```Idel'chik correlation```.

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`.

## References

[1] Idel’chik, I. E. Handbook of hydraulic resistance: Coefficients of local resistance and of friction. Jerusalem: Israel Program for Scientific Translations, 1966.

## Version History

Introduced in R2023a

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