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Pipe turn in an isothermal liquid network

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

The Elbow (IL) block models flow in a pipe turn in an isothermal liquid network. Pressure losses due to pipe turns are calculated, but the effect of viscous friction is omitted.

Two **Elbow type** settings are available:
`Smoothly-curved`

and ```
Sharp-edged
(Miter)
```

. For a smooth pipe with a 90^{o} bend
and modeled losses due to friction, you can also use the Pipe Bend (IL) block.

For smoothly-curved pipe segments, the loss coefficient is calculated as:

$$K=30{f}_{T}{C}_{angle}.$$

*C*_{angle}, the angle
correction factor, is calculated from Keller [2] as:

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

where *θ* is the **Bend
angle** in degrees. The friction factor,
*f*_{T}, is defined for clean commercial
steel. The values are interpolated from tabular data based on the internal elbow
diameter for *f*_{T} provided by Crane [1]:

The values provided by Crane are valid for diameters up to 600 mm. The friction factor for larger diameters or for wall roughness beyond this range is calculated by nearest-neighbor extrapolation.

For sharp-edged pipe segments, the loss coefficient *K* is
calculated for the bend angle, *α*, according to Crane [1]:

Mass is conserved through the pipe segment:

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

The mass flow rate through the elbow is calculated as:

$$\dot{m}=A\sqrt{\frac{2\overline{\rho}}{K}}\frac{\Delta p}{{\left[\Delta {p}^{2}+\Delta {p}_{crit}^{2}\right]}^{1/4}},$$

where:

*A*is the flow area.$$\overline{\rho}$$ is the average fluid density.

*Δp*is the pipe segment pressure difference,*p*_{A}–*p*_{B}.

The critical pressure difference,
*Δp*_{crit}, is the pressure differential
associated with the **Critical Reynolds number**,
*Re*_{crit}, the flow regime transition
point between laminar and turbulent flow:

$$\Delta {p}_{crit}=\frac{\overline{\rho}}{2}K{\left(\frac{\nu {\mathrm{Re}}_{crit}}{D}\right)}^{2},$$

where

*ν*is the fluid kinematic viscosity.*D*is the elbow internal diameter.

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

[2] Keller, G. R.
*Hydraulic System Analysis*. Penton, 1985.

Area Change (IL) | Local Resistance (IL) | Partially Filled Pipe (IL) | Pipe (IL) | Pipe Bend (IL) | T-Junction (IL)