Elbow (TL)

Pipe turn in a thermal liquid network

Since R2022a

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:

$K = 30 f_{T} C_{a n g l e} .$

The block calculates C_angle, the angle correction factor, from Keller [2] as

$C_{a n g l e} = 0.0148 θ - 3.9716 \cdot 10^{- 5} θ^{2},$

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

Diagram displaying Mitre bends and a table for K values and corresponding bend angles.

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:

$\dot{m} = A \sqrt{\frac{2 \bar{ρ}}{K}} \frac{Δ p}{{[Δ p^{2} + Δ p_{c r i t}^{2}]}^{1 / 4}},$

where:

A is the flow area.
$\bar{ρ}$ 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:

$Δ p_{c r i t} = \frac{\bar{ρ}}{2} K {(\frac{ν {Re}_{c r i t}}{D})}^{2},$

where

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

Energy Balance

The block balances energy such that

$Φ_{A} + Φ_{B} = 0,$

where:

ϕ_A is the energy flow rate at port A.
ϕ_B is the energy flow rate at port B.

Ports

Conserving

expand all

A — Pipe bend entrance or exit
thermal liquid

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

B — Pipe bend entrance or exit
thermal liquid

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

Parameters

expand all

Elbow type — Bend geometry
`Smoothly curved` (default) | `Sharp-edged (Miter)`

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.

Elbow internal diameter — Pipe internal diameter
`0.01 m` (default) | positive scalar

Internal diameter of the pipe elbow segment.

Elbow angle — Pipe curve angle
`90 deg` (default) | positive scalar

Angle of the swept pipe curve.

Critical Reynolds number — Upper Reynolds number limit for laminar flow
`150` (default) | positive scalar

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.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.

Version History

Introduced in R2022a