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

**Library:**Simscape / Fluids / Isothermal Liquid / Pipes & Fittings

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.

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:

$${K}_{contraction}=0.8{C}_{contraction}\mathrm{sin}\left(\frac{\theta}{2}\right)\left(1-R\right),$$

where *R* is the port area ratio $$\frac{{A}_{smaller}}{{A}_{bigger}}.$$ For gradual area contraction between 45 and 180 degrees:

$${K}_{contraction}=\frac{{C}_{contraction}}{2}\sqrt{\mathrm{sin}\left(\frac{\theta}{2}\right)}\left(1-R\right),$$

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:

$${K}_{expansion}=2.6{C}_{expansion}\mathrm{sin}\left(\frac{\theta}{2}\right){\left(1-R\right)}^{2},$$

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:

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

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}: $${D}_{h}=\sqrt{\frac{4{A}_{R}}{\pi}}.$$

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.

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:

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

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

[2] Idelchik, I.E.,
*Handbook of Hydraulic Resistance*, CRC Begell House,
1994

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