# Elbow (TL)

Pipe turn in a thermal liquid network

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• Simscape / Fluids / Isothermal Liquid / Pipes & Fittings

## Description

The Elbow (TL) block represents flow in a pipe turn in a thermal liquid network. Pressure losses due to pipe turns are calculated, but the block omits the effects of viscous friction.

Two Elbow type settings are available: `Smoothly-curved` and ```Sharp-edged (Miter)```. For a smooth pipe with a 90° bend and losses due to friction, you can also use the Pipe Bend (TL) block.

### Loss Coefficients

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

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

Cangle, 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, fT, is defined for clean commercial steel. The values are interpolated from tabular data based on the internal elbow diameter for fT 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 Flow Rate

Mass is conserved through the pipe segment:

`${\stackrel{˙}{m}}_{A}+{\stackrel{˙}{m}}_{B}=0.$`

The mass flow rate through the elbow is calculated as:

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

where:

• A is the flow area.

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

• Δp is the pipe segment pressure difference, pApB.

The critical pressure difference, Δpcrit, is the pressure differential associated with the Critical Reynolds number, Recrit, 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.

### Energy Balance

The block balances energy such that

`${\Phi }_{A}+{\Phi }_{B}=0,$`

where:

• ϕA is the energy flow rate at port A.

• ϕB is the energy flow rate at port B.

## Ports

### Conserving

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Thermal liquid conserving port associated with the liquid entrance or exit of the pipe bend.

Thermal liquid conserving port associated with the liquid entrance or exit of the pipe bend.

## Parameters

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Bend specification of the pipe segment. A sharp-edged, or mitre, bend introduces a sharp change in flow direction, such as at a pipe joint, and the flow losses are modeled by a separate set of empirical data from gradually-turning pipe segments.

Internal diameter of the pipe elbow segment.

Angle of the swept pipe curve.

Upper Reynolds number limit for laminar flow through the pipe segment.

## References

[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.

## Version History

Introduced in R2022a