# Elbow

Hydraulic resistance in elbow

## Library

Local Hydraulic Resistances

## Description

The Elbow block represents an elbow as a local hydraulic resistance. The pressure loss is computed with the semi-empirical formula based on pressure loss coefficient, which is determined in accordance with the Crane Co. recommendations (see [1], p. A-29). Two types of elbow are considered: smoothly curved (standard) and sharp-edged (miter). The block covers elbows in the 5–100 mm and 0–90 degrees range.

The block is based on the Local Resistance block. It computes the pressure loss coefficient and passes its value, as well as the critical Reynolds number value, to the Local Resistance block, which computes the pressure loss according to the formulas explained in the reference documentation for that block.

The pressure loss for turbulent flow regime is determined according to the following formula:

$p=K\frac{\rho }{2{A}^{2}}q|q|$

where

 `q` Flow rate `p` Pressure loss `K` Pressure loss coefficient `A` Elbow cross-sectional area ρ Fluid density

The flow regime is checked in the underlying Local Resistance block by comparing the Reynolds number to the specified critical Reynolds number value. For laminar flow regime, the formula for pressure loss computation is modified, as described in the reference documentation for the Local Resistance block.

The core data for the pressure loss coefficient computation is the table-specified relationship between the friction factor `fT` and the internal diameter for clean commercial steel pipes, with flow in the zone of complete turbulence (see [1], p. A-26). For smoothly curved, standard 90o elbows, the pressure loss coefficient is determined with the formula

$K=30{f}_{T}$

For elbows with different angles, the coefficient is corrected with the relationship presented in [2], Fig.4.6:

${K}_{corr}=\alpha \left(0.0142-3.703·{10}^{-5}\alpha \right)$

where α is the elbow angle in degrees (0 ≤ α ≤ 90).

Therefore, the pressure loss coefficient for smoothly curved, standard elbows is determined with the formula

${K}_{SCE}=30{f}_{T}·\alpha \left(0.0142-3.703·{10}^{-5}\alpha \right)$

For sharp-edged, miter bends the pressure loss coefficient is determined according to the table provided in [1], p. A-29, as a function of the elbow diameter and angle

${K}_{ME}=f\left(d,\alpha \right)$

where 5 ≤ `d` ≤ 100 mm and 0 ≤ α ≤ 90 degrees.

Connections A and B are conserving hydraulic ports associated with the block inlet and outlet, respectively.

The block positive direction is from port A to port B. This means that the flow rate is positive if fluid flows from A to B, and the pressure differential is determined as $p={p}_{A}-{p}_{B}$.

 Warning   The formulas used in the Elbow block are very approximate, especially in the laminar and transient flow regions. For more accurate results, use the Local Resistance block with a table-specified `K=f(Re)` relationship.

## Basic Assumptions and Limitations

• Fluid inertia is not taken into account.

• The transition between laminar and turbulent regimes is assumed to be sharp and taking place exactly at `Re`=`Recr`.

• The elbow is assumed to be made of a clean commercial steel pipe.

## Dialog Box and Parameters

Elbow internal diameter

The internal diameter of the pipe. The value must be in the range between 5 and 100 mm. The default value is `0.01` m.

Elbow angle

The angle of the bend. The value must be in the range between 0 and 90 degrees. The default value is `90` deg.

Elbow type

The parameter can have one of two values: ```Smoothly curved elbow``` or `Miter bend`. The default value is `Smoothly curved elbow`.

Critical Reynolds number

The maximum Reynolds number for laminar flow. The transition from laminar to turbulent regime is assumed to take place when the Reynolds number reaches this value. The value of the parameter depends on the orifice geometrical profile. You can find recommendations on the parameter value in hydraulics textbooks. The default value is `80`.

## Global Parameters

Parameters determined by the type of working fluid:

• Fluid density

• Fluid kinematic viscosity

Use the Hydraulic Fluid block or the Custom Hydraulic Fluid block to specify the fluid properties.

## Ports

The block has the following ports:

`A`

Hydraulic conserving port associated with the elbow inlet.

`B`

Hydraulic conserving port associated with the elbow outlet.

## References

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

[2] George R. Keller, Hydraulic System Analysis, Published by the Editors of Hydraulics & Pneumatics Magazine, 1970