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Pipe bend segment in an isothermal liquid network

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

The Pipe Bend (IL) block models a curved pipe in an isothermal liquid network. You can define the pipe characteristics to calculate losses due to friction and pipe curvature and optionally model fluid compressibility.

The coefficient for pressure losses due to geometry changes comprises an angle
correction factor, *C*_{angle}, and a bend
coefficient, *C*_{bend}:

$${K}_{loss}={C}_{angle}{C}_{bend}.$$

*C*_{angle} is calculated
as:

$${C}_{angle}=0.0148\theta -3.9716\cdot {10}^{-5}{\theta}^{2},$$

where *θ* is the **Bend
angle**, in degrees.

*C*_{bend} is calculated from the tabulated
ratio of bend radius to pipe diameter for 90^{o} bends from
Crane [1]:

The friction factor, *f*_{T}, for
clean commercial steel is interpolated from tabular data based on pipe diameter [1]:

Note that the correction factor is valid for a ratio of bend radius to diameter between 1 and 24. Beyond this range, nearest-neighbor extrapolation is employed.

The pressure loss formulations are the same for the flow at ports
**A** and **B**.

When the flow in the pipe is fully laminar, or below *Re* = 2000,
the pressure loss over the bend is:

$$\Delta {p}_{loss}=\frac{\mu \lambda}{2{\rho}_{I}{d}^{2}A}\frac{L}{2}{\dot{m}}_{port},$$

where:

*μ*is the fluid dynamic viscosity.*λ*is the Darcy friction factor constant, which is 64 for laminar flow.*ρ*_{I}is the internal fluid density.*d*is the pipe diameter.*L*is the bend length segment, the product of the**Bend radius**and the**Bend angle**: $${L}_{bend}={r}_{bend}\theta .$$.*A*is the pipe cross-sectional area, $$\frac{\pi}{4}{d}^{2}.$$$${\dot{m}}_{port}$$ is the mass flow rate at the respective port.

When the flow is fully turbulent, or greater than *Re* = 4000, the
pressure loss in the pipe is:

$$\Delta {p}_{loss}=\left(\frac{{f}_{D}L}{2d}+\frac{{K}_{loss}}{2}\right)\frac{{\dot{m}}_{port}\left|{\dot{m}}_{port}\right|}{2{\rho}_{I}{A}^{2}},$$

where *f*_{D} is the Darcy
friction factor. This is approximated by the empirical Haaland equation and is based
on the **Internal surface absolute roughness**. The differential is
taken over half of the pipe segment, between port **A** to an
internal node, and between the internal node and port **B**.

When the flow is incompressible, the pressure loss over the bend is:

$${p}_{A}-{p}_{B}=\Delta {p}_{loss,A}-\Delta {p}_{loss,B}.$$

When the flow is compressible, the pressure loss over the bend is calculated based
on the internal fluid volume pressure, *p*_{I}:

$${p}_{A}-{p}_{I}=\Delta {p}_{loss,A},$$

$${p}_{B}-{p}_{I}=\Delta {p}_{loss,B}.$$

For an incompressible fluid, the mass flow into the pipe equals the mass flow out of the pipe:

$${\dot{m}}_{A}+{\dot{m}}_{B}=0.$$

When the fluid is compressible, the difference between the mass flow into and out of the pipe depends on the fluid density change due to compressibility:

$${\dot{m}}_{A}+{\dot{m}}_{B}={\dot{p}}_{I}\frac{d{\rho}_{I}}{d{p}_{I}}V,$$

where *V* is the product of the pipe
cross-sectional area and bend length, *AL*.

[1] Crane Co. *Flow of
Fluids Through Valves, Fittings, and Pipe TP-410*. Crane Co.,
1981.

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