Area Change (IL)

Area expansion or contraction along a pipe in an isothermal liquid network

Since R2020a

• 

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

Description

The Area Change (IL) block models a sudden or a gradual area change along a pipe with 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.

Both semi-empirical and tabular formulations are available for correlating flows to losses.

Semi-Empirical Formulation

In the semi-empirical, analytical formulation, losses in pressure and velocity are characterized by a Hydraulic loss coefficient, K, in terms of a user-defined Contraction correction factor, C_contraction, and Expansion correction factor, C_expansion, from Crane [1]. The coefficient that characterizes area change is calculated from both expansion and contraction loss factors and based on the flow rate through the block.

For gradual conical area contractions 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 gradual area contraction between 45 and 180 degrees:

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

where θ is the Cone angle. A sudden area change has an angle of 180 degrees. In this case, the loss factor is calculated as $$K_{contraction}=\frac{C}{2}\left(1-R\right).$$

For gradual conical area expansions 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 gradual area expansion between 45 and 180 degrees:

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

The hydraulic loss coefficient for the pipe change segment is calculated from these values as:

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

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 based on the Critical Reynolds number, Re_c:

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

  where:

  • A_R is the smallest segment area (either the Cross-sectional area at port A or the Cross-sectional area at port B.)

  • 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

The loss factor can also be parameterized with user-provided data interpolated from the Reynolds number at the smallest area, which in turn is a function of the Critical Reynolds number:

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

Linear interpolation is employed between data points, and nearest-neighbor extrapolation is employed 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:

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

Examples

Lubrication System

A simplified version of a lubrication system fed with the centrifugal pump. The system consists of five major units: Pump Unit, Scavenge Unit, Heat Exchanger Manifold, Nozzle Manifold, and Control Unit. Both the pump and the scavenge unit are built around the centrifugal pump. The Scavenge Unit collects fluid discharged by nozzles and pumps it back into the reservoir of the Pump Unit. The Control Unit generates commands to bypass either the heat exchanger, represented as a local resistance, or the nozzles block. In a real system, these commands are generated by temperature sensors installed in lubrication cavities.

• Open Model

Ports

Conserving

A — Liquid port

isothermal liquid

Liquid entry port.

B — Liquid port

isothermal liquid

Liquid exit port.

Parameters

Cone angle — Gradual area change angle

30 (default) | positive scalar

Angle of expansion for a gradual area change parameterized as a cone.

Dependencies

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

Local loss parameterization — Model of hydraulic losses

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

Model of hydraulic losses due to area change. You can choose from two analytical, semi-empirical formulations or you can provide your own data by selecting Tabulated data - loss coefficient vs. Reynolds number.

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

0.02 m^2 (default) | positive scalar

Area at the section entrance, port A.

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

0.01 m^2 (default) | positive scalar

Area at the section exit, port B.

Reynolds number vector — Vector of Reynolds number values for tabular parameterization

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

Vector of 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 are listed in ascending order.

Dependencies

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

Contraction loss coefficient vector — Vector of loss coefficients for an area contraction

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

Vector of loss coefficients for an area contraction corresponding to the Reynolds number vector parameter, where n is the length of the Reynolds number vector. The elements are listed in descending order and must be greater than 0.

Dependencies

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

Expansion loss coefficient vector — Vector of loss coefficients for an area expansion

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

Vector of loss coefficients for an area expansion corresponding to the Reynolds number vector parameter, where n is the length of the Reynolds number vector. The elements are listed in descending order and must be greater than 0.

Dependencies

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

Expansion correction factor — Coefficient in semi-empirical loss factor equation

1 (default) | positive scalar

Coefficient used 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 — Coefficient in semi-empirical loss factor equation

1 (default) | positive scalar

Coefficient used 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.

References

[1] Flow of Fluids Through Valves, Fittings, and Pipe, Crane Valves North America, Technical Paper No. 410M

[2] Idel'chik, I.E., Handbook of Hydraulic Resistance, CRC Begell House, 1994

Extended Capabilities

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

Version History

Introduced in R2020a