Cross-Junction (MA)

Cross-junction in a moist air network

Since R2024b

• 

Libraries:
Simscape / Fluids / Moist Air / Pipes & Fittings

Description

The Cross-Junction (MA) block represents a four-way junction in a moist air network. The block abstracts the fluid interactions depending on the direction of flow at each port, where ports A and C are the main line and ports B and D are the branch line.

Mass, Momentum, and Energy Balance

The block conserves mass such that

$$\begin{array}{l}{\dot{m}}_A+{\dot{m}}_B+{\dot{m}}_C+{\dot{m}}_D=0\\ {\dot{m}}_{wA}+{\dot{m}}_{wB}+{\dot{m}}_{wC}+{\dot{m}}_{wD}=0\\ {\dot{m}}_{gA}+{\dot{m}}_{gB}+{\dot{m}}_{gC}+{\dot{m}}_{gD}=0\\ {\dot{m}}_{dA}+{\dot{m}}_{dB}+{\dot{m}}_{dC}+{\dot{m}}_{dD}=0\end{array}$$

• ṁ_i is the mixture mass flow rate for a given port, i.

• $$\dot{m}$$_wi is the water vapor mass flow rate for a given port, i.

• $$\dot{m}$$_gi is the trace gas mass flow rate for a given port, i.

• $$\dot{m}$$_di is the water droplet mass flow rate for a given port, i.

The block determines which coefficient and vector element to use, then computes the pressure difference to conserve momentum such that

$$p_i-p_1=I_i+\frac{\raisebox{1ex}{$k_i$}\!\left/ \!\raisebox{-1ex}{$2$}\right.{\dot{m}}_i\sqrt{{\dot{m}}_i^2+{\dot{m}}_{threshold}^2}}{\overline{\rho }{A}_i^2},$$

where:

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

• p_i is the pressure for a given port, i.

• p₁ is the pressure at node 1.

• k_i is the flow coefficient that the block assigns to a given port, i. The Loss coefficient model parameter specifies the block method used to calculate k_i.

• I_i is the fluid inertia for a given port, i.

The block calculates the inertia for each port as

$$\begin{array}{l}{I}_A={\ddot{m}}_A\frac{\sqrt{\pi A_{side}}}{A_{main}}\\ I_B={\ddot{m}}_B\frac{\sqrt{\pi A_{main}}}{A_{side}}\\ I_C={\ddot{m}}_C\frac{\sqrt{\pi A_{side}}}{A_{main}}\\ I_D={\ddot{m}}_D\frac{\sqrt{\pi A_{main}}}{A_{side}}\end{array}$$

where A_main is the value of the Main line area (A,C) parameter and A_side is the value of the Branch line area (B,D) parameter.

The block conserves energy such that

$${\phi }_A+{\phi }_B+{\phi }_C+{\phi }_D=0.$$

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 [1]. The Idel'chik model only supports flow converging to port C or flow diverging from port C. To achieve expected results, orient the block appropriately to the expected flow configuration of the model fluid network.

The block treats the junction as symmetrical about both the horizontal and vertical planes such that A_A = A_C is the value of the Main line area (A,C) parameter, and A_B = A_D is the value of the Branch line area (B,D) parameter, where A_A,B,C,D is the area of the corresponding port.

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

Converging Flow to Port C

When the flow is converging to port C, the loss coefficients for ports A and C are

$$\begin{array}{l}{k}_A=\left(1+{\left(\frac{{\dot{m}}_A}{{\dot{m}}_C}\right)}^2-{\left(\frac{{\dot{m}}_A}{{\dot{m}}_C}\right)}^2\frac{1+\frac{{\dot{m}}_A}{{\dot{m}}_C}}{{\left(0.75+0.25\frac{{\dot{m}}_A}{{\dot{m}}_C}\right)}^2}\right){\left(\frac{{\dot{m}}_A}{{\dot{m}}_C}\frac{A_C}{A_A}\right)}^{-2}\\ k_C=0\end{array}$$

The converging flow loss coefficients for ports B and D are

$$\begin{array}{l}{k}_B=\left(1+{\left(\frac{{\dot{m}}_B}{{\dot{m}}_C}\frac{A_C}{A_B}\right)}^2-8{\left(\frac{{\dot{m}}_B}{{\dot{m}}_C}\right)}^2\frac{{\left[\frac{{\dot{m}}_C}{{\dot{m}}_B}-\left(1+\frac{{\dot{m}}_D}{{\dot{m}}_B}\right)\right]}^2}{4-\left(1+\frac{{\dot{m}}_D}{{\dot{m}}_B}\right)\frac{{\dot{m}}_B}{{\dot{m}}_C}}\right){\left(\frac{{\dot{m}}_B}{{\dot{m}}_C}\frac{A_C}{A_B}\right)}^{-2}\\ k_D=\left(1+{\left(\frac{{\dot{m}}_D}{{\dot{m}}_C}\frac{A_C}{A_D}\right)}^2-8{\left(\frac{{\dot{m}}_D}{{\dot{m}}_C}\right)}^2\frac{{\left[\frac{{\dot{m}}_C}{{\dot{m}}_D}-\left(1+\frac{{\dot{m}}_B}{{\dot{m}}_D}\right)\right]}^2}{4-\left(1+\frac{{\dot{m}}_B}{{\dot{m}}_D}\right)\frac{{\dot{m}}_D}{{\dot{m}}_C}}\right){\left(\frac{{\dot{m}}_D}{{\dot{m}}_C}\frac{A_C}{A_D}\right)}^{-2}\end{array}$$

Diverging Flow From Port C

When the flow diverges from port C, the loss coefficients for ports A and C are

$$\begin{array}{l}{k}_A=0.4{\left(\frac{{\dot{m}}_C}{{\dot{m}}_A}-1\right)}^2\\ k_C=0\end{array}$$

The diverging flow loss coefficients for ports B and D are

$$\begin{array}{l}{k}_B=A\text{'}{\left(\frac{{\dot{m}}_C}{{\dot{m}}_B}\frac{A_B}{A_C}+1\right)}^2\\ k_D=A\text{'}{\left(\frac{{\dot{m}}_C}{{\dot{m}}_D}\frac{A_D}{A_C}+1\right)}^2\end{array}$$

where A'=1 when $$\frac{{\dot{m}}_B}{{\dot{m}}_C}\frac{A_C}{A_B}\le 0.8\text{ or }\frac{{\dot{m}}_D}{{\dot{m}}_C}\frac{A_C}{A_D}\le 0.8$$ and A'≈0.9 when $$\frac{{\dot{m}}_B}{{\dot{m}}_C}\frac{A_C}{A_B}>0.8\text{ or }\frac{{\dot{m}}_D}{{\dot{m}}_C}\frac{A_C}{A_D}>0.8$$. The block smooths the transition between A' = 1 and A' = 0.9 using a hyperbolic tangent function.

According to [1], these loss coefficients are only applicable for diverging flow when the ratio of the side branch diameter to the main branch diameter is less than or equal to 2/3. However, the expected error for ratios above 2/3 is generally low enough that the block continues to use the above loss coefficients.

Custom Cross-Junction

When you set the Loss coefficient model parameter to Custom, the block uses the user-defined loss parameters to describe the pipe loss coefficient at each port.

For a custom cross-junction, the block allows four configurations diverging flow, converging flow, perpendicular flow, and colliding flow:

• Diverging flow — Flow enters at node 1 and exits at nodes 2, 3, and 4.

• Converging flow — Flow enters at nodes 1, 2, and 4 and exits at node 3.

• Perpendicular flow — Flow enters at nodes 1 and 2 and exits at nodes 3 and 4.

• Colliding flow — Flow enters at nodes 1 and 3 and exits at nodes 2 and 4.

This figure demonstrates these configurations.

The block treats the junction as symmetrical about both the horizontal and vertical planes such that A₁ = A₃ and A₂ = A₄, where A_1,2,3,4 is the area of the corresponding port in the figure. Using this symmetry, the block assumes the loss coefficients from 1 to 2 and from 1 to 4 are identical during diverging, converging, and colliding flow.

During simulation, the block continually checks the direction of flow at each port and compares the result to the four possible flow configurations. When the block determines the flow configuration, it adjusts the node that each port corresponds to. For example, when the block experiences diverging flow where the fluid enters at port A and exits at ports B, C, and D, it assigns node 1 to port A as in the first panel of the figure. The other panels show the assigned nodes for diverging flow entering at ports other than A.

For the custom cross-junction, all four flow configurations can occur in the same block during a simulation. You only need to supply parameters for each condition that the block experiences. If you use two-element vectors to specify the flow coefficients, the block uses the first or second element depending on whether node 1 aligns with the main line or the branch line. The first element corresponds to node 1 at port A or C, and the second element corresponds to node 1 at port B or D.

This table describes the conditions and coefficients for each operational mode used to determine the loss coefficients.

Flow Scenario	ṁA	ṁB	ṁC	ṁD	kA	kB	kC	kD
Diverging from node A	>ṁthresh	<-ṁthresh	<-ṁthresh	<-ṁthresh	0	kdiv,turning,main	kdiv,straight,main	kdiv,turning,main
Diverging from node B	<-ṁthresh	>ṁthresh	<-ṁthresh	<-ṁthresh	kdiv,turning,side	0	kdiv,turning,side	kdiv,straight,side
Diverging from node C	<-ṁthresh	<-ṁthresh	>ṁthresh	<-ṁthresh	kdiv,straight,main	kdiv,turning,main	0	kdiv,turning,main
Diverging from node D	<-ṁthresh	<-ṁthresh	<-ṁthresh	>ṁthresh	kdiv,turning,side	kdiv,straight,side	kdiv,turning,side	0
Converging to node A	<-ṁthresh	>ṁthresh	>ṁthresh	>ṁthresh	0	kconv,turning,main	kconv,straight,main	kconv,turning,main
Converging to node B	>ṁthresh	<-ṁthresh	>ṁthresh	>ṁthresh	kconv,turing,side	0	kconv,turing,side	kconv,straight,side
Converging to node C	>ṁthresh	>ṁthresh	<-ṁthresh	>ṁthresh	kconv,straight,main	kconv,turning,main	0	kconv,turning,main
Converging to node D	>ṁthresh	>ṁthresh	>ṁthresh	<-ṁthresh	kconv,turing,side	kconv,straight,side	kconv,turing,side	0
Perpendicular with main entry A	>ṁthresh	>ṁthresh	<-ṁthresh	<-ṁthresh	0	kperp,turning in,main	kperp,straight,main	kperp,turning out,main
Perpendicular with main entry B	<-ṁthresh	>ṁthresh	>ṁthresh	<-ṁthresh	kperp,turning out,side	0	kperp,turning in,side	kperp,turning straight,side
Perpendicular with main entry C	<-ṁthresh	<-ṁthresh	>ṁthresh	>ṁthresh	kperp,straight,main	kperp,turning out,main	0	kperp,turning in,main
Perpendicular with main entry D	>ṁthresh	<-ṁthresh	<-ṁthresh	>ṁthresh	kperp,turning in,side	kperp,turning straight,side	kperp,turning out,side	0
Colliding from main to branch	>ṁthresh	<-ṁthresh	>ṁthresh	<-ṁthresh	0	kcoll,turning,main	kcoll,straight,main	kcoll,turning,main
Colliding from branch to main	<-ṁthresh	>ṁthresh	<-ṁthresh	>ṁthresh	kcoll,turning,side	0	kcoll,turning,side	kcoll,straight,side
Stagnant	–	–	–	–	1 or last valid	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 block uses these parameters to calculate the flow coefficients:

• k_{div,straight,main} and k_{div,straight,side} are the first and second elements of the Diverging flow straight loss coefficient parameter, respectively.

• k_{div,turning,main} and k_{div,turning,side} are the first and second elements of the Diverging flow turning loss coefficient parameter, respectively.

• k_{conv,straight,main} and k_{conv,straight,side} are the first and second elements of the Converging flow straight loss coefficient parameter, respectively.

• k_{conv,turning,main} and k_{conv,turning,side} are the first and second elements of the Converging flow turning loss coefficient parameter, respectively.

• k_{perp,straight,main} and k_{perp,straight,side} are the first and second elements of the Perpendicular flow straight loss coefficient parameter, respectively.

• k_{perp,turning in,main} and k_{perp,turning in,side} are the first and second elements of the Perpendicular flow straight loss coefficient parameter, respectively.

• k_{perp,turning out,main} and k_{perp,turning out,side} are the first and second elements of the Perpendicular flow straight loss coefficient parameter, respectively.

• k_{coll,straight,main} and k_{coll,straight,side} are the first and second elements of the Colliding flow straight loss coefficient parameter, respectively.

• k_{coll,turning,main} and k_{coll,turning,side} are the first and second elements of the Colliding flow turning loss coefficient parameter, respectively.

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

A — Moist air port

moist air

Moist air conserving port associated with the liquid port.

B — Moist air port

moist air

Moist air conserving port associated with the liquid port.

C — Moist air port

moist air

Moist air conserving port associated with the liquid port.

D — Moist air port

moist air

Moist air conserving port associated with the liquid port.

Parameters

Junction Properties

Main line area (A,C) — Area of main path

0.01 m^2 (default) | positive scalar

Cross-sectional area of the main flow path from port A to port C.

Branch line area (B,D) — Area of branch path

0.01 m^2 (default) | positive scalar

Cross-sectional area of the branch flow path from port B to port D.

Laminar flow pressure ratio — Pressure ratio at which flow transitions between laminar and turbulent regimes

0.999 (default) | positive scalar

Pressure ratio at which the moist air flow transitions between the laminar and turbulent regimes.

Zero-flow threshold pressure ratio — Pressure ratio that corresponds to stagnant flow

0.999999 (default) | positive scalar

Pressure ratio 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.

Dependencies

To enable this parameter, set Loss coefficient model to either Crane correlation, Rennels correlation, or Custom.

Loss coefficient configuration change time constant — Time scale of smoothing function

0.0001 s (default) | positive scalar

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

Dependencies

To enable this parameter, set Loss coefficient model to either Crane correlation, Rennels correlation, or Custom.

Minimum valid flow ratio for coefficient calculation — Limit on flow ratios for coefficient calculations

0.2 (default) | positive scalar

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.

Smoothing factor — Numerical smoothing factor

0.01 (default) | scalar between 0 and 1

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.

Fluid inertia — Option to model fluid inertia

off (default) | on

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.

Loss coefficient model — Loss coefficient type

Idel'chik correlation (default) | Custom

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

Report when flow configuration is invalid — Simulation warning mode for invalid flow configurations

None (default) | Warning | Error

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.

Diverging flow loss coefficients — Option to specify loss coefficients for diverging flow

on (default) | off

Whether to specify loss coefficients for diverging flow.

Dependencies

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

Converging flow loss coefficients — Option to specify loss coefficients for converging flow

on (default) | off

Whether to specify loss coefficients for converging flow.

Dependencies

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

Perpendicular flow loss coefficients — Option to specify loss coefficients for perpendicular flow

off (default) | on

Whether to specify loss coefficients for perpendicular flow.

Dependencies

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

Colliding flow loss coefficients — Option to specify loss coefficients for colliding flow

off (default) | on

Whether to specify loss coefficients for colliding flow.

Dependencies

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

Diverging Flow Coefficients

Diverging flow straight loss coefficient — Diverging flow straight loss coefficient

[1.12, 1.12] (default) | positive scalar or vector

Loss coefficient for the straight portion of the diverging flow. When you specify this parameter as a vector, the first element represents the loss coefficient when ports A or C align to node 1. The second element represents when ports B or D align to node 1. The block behaves the same for all orientations when you use a scalar.

Dependencies

To enable this parameter, set Loss coefficient model to Custom and select Diverging flow loss coefficients.

Diverging flow turning loss coefficient — Diverging flow turning loss coefficient

[1.12, 1.12] (default) | positive scalar or vector

Loss coefficient for the turning portions of the diverging flow. When you specify this parameter as a vector, the first element represents the loss coefficient when ports A or C align to node 1. The second element represents when ports B or D align to node 1. The block behaves the same for all orientations when you use a scalar.

Dependencies

To enable this parameter, set Loss coefficient model to Custom and select Diverging flow loss coefficients.

Converging Flow Coefficients

Converging flow straight loss coefficient — Converging flow straight loss coefficient

[1.12, 1.12] (default) | positive scalar or vector

Loss coefficient for the straight portion of the converging flow. When you specify this parameter as a vector, the first element represents the loss coefficient when ports A or C align to node 1. The second element represents when ports B or D align to node 1. The block behaves the same for all orientations when you use a scalar.

Dependencies

To enable this parameter, set Loss coefficient model to Custom and select Converging flow loss coefficients.

Converging flow turning loss coefficient — Converging flow turning loss coefficient

[1.12, 1.12] (default) | positive scalar or vector

Loss coefficient for the turning portions of the converging flow. When you specify this parameter as a vector, the first element represents the loss coefficient when ports A or C align to node 1. The second element represents when ports B or D align to node 1. The block behaves the same for all orientations when you use a scalar.

Dependencies

To enable this parameter, set Loss coefficient model to Custom and select Converging flow loss coefficients.

Perpendicular Flow Coefficients

Perpendicular flow straight loss coefficient — Perpendicular flow straight loss coefficient

[1.12, 1.12] (default) | positive scalar or vector

Loss coefficient for the straight portion of the perpendicular flow. When you specify this parameter as a vector, the first element represents the loss coefficient when ports A or C align to node 1. The second element represents when ports B or D align to node 1. The block behaves the same for all orientations when you use a scalar.

Dependencies

To enable this parameter, set Loss coefficient model to Custom and select Perpendicular flow loss coefficients.

Perpendicular flow turning to inflow loss coefficient — Perpendicular flow turning to inflow loss coefficient

[1.12, 1.12] (default) | positive scalar or vector

Loss coefficient for the flow turning to inflow in the perpendicular flow. When you specify this parameter as a vector, the first element represents the loss coefficient when ports A or C align to node 1. The second element represents when ports B or D align to node 1. The block behaves the same for all orientations when you use a scalar.

Dependencies

To enable this parameter, set Loss coefficient model to Custom and select Perpendicular flow loss coefficients.

Perpendicular flow turning to outflow loss coefficient — Perpendicular flow turning to outflow loss coefficient

[1.12, 1.12] (default) | positive scalar or vector

Loss coefficient for the flow turning to outflow in the perpendicular flow. When you specify this parameter as a vector, the first element represents the loss coefficient when ports A or C align to node 1. The second element represents when ports B or D align to node 1. The block behaves the same for all orientations when you use a scalar.

Dependencies

To enable this parameter, set Loss coefficient model to Custom and select Perpendicular flow loss coefficients.

Colliding Flow Coefficients

Colliding flow straight loss coefficient — Colliding flow straight loss coefficient

[1.12, 1.12] (default) | positive scalar or vector

Loss coefficient for the straight portion of the colliding flow. When you specify this parameter as a vector, the first element represents the loss coefficient when ports A or C align to node 1. The second element represents when ports B or D align to node 1. The block behaves the same for all orientations when you use a scalar.

Dependencies

To enable this parameter, set Loss coefficient model to Custom and select Colliding flow loss coefficients.

Colliding flow turning loss coefficient — Colliding flow turning loss coefficient

[1.12, 1.12] (default) | positive scalar or vector

Loss coefficient for the turning portion of the colliding flow. When you specify this parameter as a vector, the first element represents the loss coefficient when ports A or C align to node 1. The second element represents when ports B or D align to node 1. The block behaves the same for all orientations when you use a scalar.

Dependencies

To enable this parameter, set Loss coefficient model to Custom and select Colliding flow loss coefficients.

References

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

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.

Version History

Introduced in R2024b