Area Change (TL)

Area expansion or contraction in a thermal liquid network

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

Description

The Area Change (TL) block models a sudden or a gradual area change that has fixed areas and variable flow direction. When the fluid moves from port A to port B, it experiences an area contraction. When the fluid flows from port B to port A, it experiences an area expansion. The inlet and outlet areas can be equal. You can use semi-empirical or tabular parameterizations to model losses.

Semi-Empirical Parameterizations

If you set Local loss parameterization to either of the analytical semi-empirical correlation settings, the hydraulic loss coefficient, K, characterizes losses in pressure and velocity based on the Contraction correction factor, C_contraction, and Expansion correction factor, C_expansion, parameters [1]. The block calculates the area change coefficient from both expansion and contraction loss factors and based on the flow rate through the block.

When Local loss parameterization is Semi-empirical correlation - gradual area change, the loss factor depends on the value of the Cone angle parameter.

For area contractions where the value of the Cone angle parameter, θ, is between 0 and 45 degrees, the contraction loss factor is

$$K_{contraction}=0.8C_{contraction}\sin \left(\frac{\theta }{2}\right)\left(1-R\right),$$

where R is the port area ratio $$\frac{A_{smaller}}{A_{bigger}}.$$ For area contractions where the value of the Cone angle parameter is between 45 and 180 degrees, the contraction loss factor is

$$K_{contraction}=\frac{C_{contraction}}{2}\sqrt{\sin \left(\frac{\theta }{2}\right)}\left(1-R\right).$$

For area expansions where the value of the Cone angle parameter is between 0 and 45 degrees, the expansion loss factor is

$$K_{expansion}=2.6C_{expansion}\sin \left(\frac{\theta }{2}\right){\left(1-R\right)}^2,$$

and for expansions where the value of the Cone angle parameter is between 45 and 180 degrees, the expansion loss factor is

$$K_{expansion}=C_{expansion}{\left(1-R\right)}^2.$$

When Local loss parameterization is Semi-empirical correlation - sudden area change, the contraction loss factor is $$K_{contraction}=\frac{C_{contraction}}{2}\left(1-R\right).$$The expansion loss factor is $$K_{expansion}=C_{expansion}{\left(1-R\right)}^2.$$

For both semi-empirical settings, the hydraulic loss coefficient is

$$K=K_{expansion}+\frac{K_{contraction}-K_{expansion}}{2}\left(\tanh \left(\frac{3{\dot{m}}_A}{{\dot{m}}_{th}}\right)+1\right),$$

where:

• $$\dot{m}$$_A is the mass flow rate through port A. Mass is conserved through the segment: $${\dot{m}}_A+{\dot{m}}_B=0.$$

• $${\dot{m}}_{th}$$ is the threshold mass flow rate for flow reversal, which is calculated from the Critical Reynolds number parameter, Re_c,

  $${\dot{m}}_{th}=\frac{{\mathrm{Re}}_c{A}_R\nu \overline{\rho }}{D_h},$$

  where:

  • A_R is the smaller of the values of the Cross-sectional area at port A or Cross-sectional area at port B parameters.

  • ν is the fluid kinematic viscosity.

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

  • D_h is the hydraulic diameter at A_R, $$D_h=\sqrt{\frac{4A_R}{\pi }}.$$

Tabulated Data Parameterization

Set Local loss parameterization to Tabulated data - loss coefficient vs. Reynolds number to parameterize the loss factor by using data interpolated from the Reynolds number at the smallest area, which is a function of the Critical Reynolds number parameter,

$$K=TLU\left(\mathrm{Re}\right).$$

The block uses linear interpolation between data points, and nearest-neighbor extrapolation beyond the table boundaries.

Pressure Differential

The pressure differential over the area change is

$$p_A-p_B=\frac{{\dot{m}}^2}{2\overline{\rho }{A}_R^2}\left(1-R^2\right)+\Delta p_{loss},$$

where the pressure loss is

$$\Delta p_{loss}=\frac{K}{2\overline{\rho }{A}_R^2}{\dot{m}}_A\sqrt{{\dot{m}}_A^2+{\dot{m}}_{th}^2}.$$

Energy Balance

The energy conservation equation in the sudden area change is

$${\varphi }_A+{\varphi }_B=0,$$

where:

• Φ_A and Φ_B are the energy flow rates into the block through ports A and B.

Assumptions and Limitations

• The flow is incompressible. The fluid density is constant in the sudden area change.

• This component is adiabatic. It does not exchange heat with its surroundings.

Ports

Conserving

A — Liquid entry or exit port

thermal liquid

Thermal liquid conserving port associated with the liquid entry or exit port.

B — Liquid entry or exit port

thermal liquid

Thermal liquid conserving port associated with the liquid entry or exit port.

Parameters

Local loss parameterization — Method to use to model hydraulic losses

Semi-empirical correlation - sudden area change (default) | Semi-empirical correlation - gradual area change | Tabulated data - loss coefficient vs. Reynolds number

Method to use to model the hydraulic losses due to area change. You can choose from two semi-empirical formulations or provide your own data by selecting Tabulated data - loss coefficient vs. Reynolds number.

Cone angle — Gradual area change angle

30 deg (default) | scalar between 0 and 180

Angle of expansion for a gradual area change. The block models the gradual area change as an increasing cone from the smaller port to the larger port.

Dependencies

To enable this parameter, set Local loss parameterization to Semi-empirical correlation - gradual area change.

Cross-sectional area at port A — Area at port A

0.02 m^2 (default) | positive scalar

Area at port A.

Cross-sectional area at port B — Area at port B

0.01 m^2 (default) | positive scalar

Area at port B.

Reynolds number vector — Reynolds number values for tabular parameterization

[10, 20, 30, 40, 50, 100, 200, 500, 1000, 2000] (default) | 1-by-n vector

Reynolds number values for the tabular parameterization of the area change. The elements must correspond one-to-one with the elements of the Contraction loss coefficient vector and the Expansion loss coefficient vector parameters. The vector values must be in ascending order.

Dependencies

To enable this parameter, set Local loss parameterization to Tabulated data - loss coefficient vs. Reynolds number.

Contraction loss coefficient vector — Loss coefficients for area contraction

[5, 2.7, 1.8, 1.46, 1.3, .9, .65, .42, .3, .2] (default) | 1-by-n vector

Loss coefficients for an area contraction that corresponds to the Reynolds number vector parameter. The elements must be in descending order and greater than zero.

Dependencies

To enable this parameter, set Local loss parameterization to Tabulated data - loss coefficient vs. Reynolds number.

Expansion loss coefficient vector — Loss coefficients for area expansion

[3.1, 2.3, 1.65, 1.35, 1.15, .9, .75, .65, .9, .65] (default) | 1-by-n vector

Loss coefficients for an area expansion that corresponds to the Reynolds number vector parameter. The elements must be in descending order and greater than zero.

Dependencies

To enable this parameter, set Local loss parameterization to Tabulated data - loss coefficient vs. Reynolds number.

Expansion correction factor — Loss factor coefficient for area expansion

1 (default) | positive scalar

Coefficient the block uses in the semi-empirical calculation of the area expansion loss factor.

Dependencies

To enable this parameter, set Local loss parameterization to either:

• Semi-empirical correlation - sudden area change

• Semi-empirical correlation - gradual area change

Contraction correction factor — Loss factor coefficient for area contraction

1 (default) | positive scalar

Coefficient the block uses in the semi-empirical calculation of the area contraction loss factor.

Dependencies

To enable this parameter, set Local loss parameterization to either:

• Semi-empirical correlation - sudden area change

• Semi-empirical correlation - gradual area change

Critical Reynolds number — Upper Reynolds number limit for laminar flow

150 (default) | positive scalar

Upper Reynolds number limit for laminar flow through the orifice.

Extended Capabilities

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

Version History

Introduced in R2016a

R2026a: New block name and functionality

The name of the Sudden Area Change (TL) is now Area Change (TL). You can also now model a sudden or gradual area change.