# Elbow (TL)

**Libraries:**

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

When the **Elbow type** parameter is ```
Smoothly
curved
```

, the block calculates the loss coefficient as:

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

The block calculates
*C _{angle}*, the angle correction factor,
from Keller [2] as

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

where *θ* is the value of the **Bend
angle** parameter in degrees. The block defines the friction factor,
*f*_{T}, as the value for clean commercial
steel. The block then interpolates the values from tabular data based on the
internal elbow diameter for *f _{T}* based on
Crane [1]. This table contains the pipe friction data for clean commercial steel
pipe with flow in the zone of complete turbulence.

r/d | 1 | 1.5 | 2 | 3 | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 20 | 24 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

K | 20 f_{T} | 14 f_{T} | 12 f_{T} | 12 f_{T} | 14 f_{T} | 17 f_{T} | 24 f_{T} | 30 f_{T} | 34 f_{T} | 38 f_{T} | 42 f_{T} | 50 f_{T} | 58 f_{T} |

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

When the **Elbow type** parameter is ```
Sharp-edged
(Miter)
```

, the block calculates the loss coefficient
*K* for the bend angle, *α*, according to
Crane [1].

α | 0° | 15° | 30° | 45° | 60° | 75° | 90° |
---|---|---|---|---|---|---|---|

K | 2 f_{T} | 4 f_{T} | 8 f_{T} | 15 f_{T} | 25 f_{T} | 40 f_{T} | 60 f_{T} |

### Mass Flow Rate

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.

### Energy Balance

The block balances energy such that

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

where:

*ϕ*is the energy flow rate at port_{A}**A**.*ϕ*is the energy flow rate at port_{B}**B**.

## Ports

### Conserving

## Parameters

## 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.

## Extended Capabilities

## Version History

**Introduced in R2022a**