Pipe Bend (IL)

Pipe bend segment in an isothermal liquid network

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Libraries:
Simscape / Fluids / Isothermal Liquid / Pipes & Fittings

Description

The Pipe Bend (IL) block models a curved pipe in an isothermal liquid network. You can define the pipe characteristics to calculate losses due to friction and pipe curvature and optionally model fluid compressibility.

Pipe Curvature Loss Coefficient

The coefficient for pressure losses due to geometry changes comprises an angle correction factor, C_angle, and a bend coefficient, C_bend:

$$K_{loss}=C_{angle}{C}_{bend}.$$

The block calculates C_angle 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 calculates C_bend from the tabulated ratio of the bend radius, r, to the pipe diameter, d, for 90° bends from data based on Crane [1]:

r/d	1	1.5	2	3	4	6	8	10	12	14	16	20	24
K	20 fT	14 fT	12 fT	12 fT	14 fT	17 fT	24 fT	30 fT	34 fT	38 fT	42 fT	50 fT	58 fT

The block interpolates the friction factor, f_T, for clean commercial steel from tabular data based on the pipe diameter [1]. This table contains the pipe friction data for clean commercial steel pipe with flow in the zone of complete turbulence.

Nominal size (mm)	5	10	15	20	25	32	40	50	72.5	100	125	150	225	350	609.5
Friction factor, fT	.035	.029	.027	.025	.023	.022	.021	.019	.018	.017	.016	.015	.014	.013	.012

The correction factor is valid for a ratio of bend radius to diameter between 1 and 24. Beyond this range, the block employs nearest-neighbor extrapolation.

Losses Due to Friction in Laminar Flows

The pressure loss formulations are the same for the flow at ports A and B.

When the flow in the pipe is fully laminar, or below Re = 2000, the pressure loss over the bend is:

$$\Delta p_{loss}=\frac{\mu \lambda }{2{\rho }_I{d}^{2}A}\frac{L}{2}{\dot{m}}_{port},$$

where:

• μ is the fluid dynamic viscosity.

• λ is the Darcy friction factor constant, which is 64 for laminar flow.

• ρ_I is the internal fluid density.

• d is the pipe diameter.

• L is the bend length segment, the product of the Bend radius and the Bend angle: $$L_{bend}=r_{bend}\theta .$$

• A is the pipe cross-sectional area, $$\frac{\pi }{4}{d}^2.$$

• $${\dot{m}}_{port}$$ is the mass flow rate at the respective port.

Losses due to Friction in Turbulent Flows

When the flow is fully turbulent, or greater than Re = 4000, the pressure loss in the pipe is:

$$\Delta p_{loss}=\left(\frac{f_{D}L}{2d}+\frac{K_{loss}}{2}\right)\frac{{\dot{m}}_{port}\left|{\dot{m}}_{port}\right|}{2{\rho }_I{A}^2},$$

where f_D is the Darcy friction factor. This is approximated by the empirical Haaland equation and is based on the Internal surface absolute roughness. The differential is taken over half of the pipe segment, between port A to an internal node, and between the internal node and port B.

Pressure Differential for Incompressible Fluids

When the flow is incompressible, the pressure loss over the bend is

$$p_A-p_B=\Delta p_{loss,A}-\Delta p_{loss,B}+{\rho }_{I}g\Delta z,$$

where g is the value of the Gravitational acceleration parameter and Δz is the value of the Elevation gain from port A to port B parameter.

Pressure Differential for Compressible Fluids

When the flow is compressible, the block calculates the pressure loss over the bend based on the internal fluid volume pressure, p_I,

$$\begin{array}{l}{p}_A-p_I=\Delta p_{loss,A}+{\rho }_{I}g\frac{\Delta z}{2}\\ p_B-p_I=\Delta p_{loss,B}-{\rho }_{I}g\frac{\Delta z}{2}\end{array}$$

Mass Conservation

When you clear the Enable dynamic compressibility checkbox, the mass flow into the pipe equals the mass flow out of the pipe

$${\dot{m}}_A+{\dot{m}}_B=0.$$

When you select Enable dynamic compressibility, the difference between the mass flow into and out of the pipe depends on the fluid density change due to compressibility

$${\dot{m}}_A+{\dot{m}}_B={\dot{p}}_I\frac{d{\rho }_I}{d{p}_I}V,$$

where V is the product of the pipe cross-sectional area and bend length, AL.

Ports

Conserving

A — Liquid port

isothermal liquid

Liquid entry or exit port.

B — Liquid port

isothermal liquid

Liquid entry or exit port.

Parameters

Pipe diameter — Pipe diameter

0.01 m (default) | positive scalar

Diameter of the pipe.

Bend radius — Bend circle radius

0.04 m (default) | positive scalar

Radius of the circle formed by the pipe bend.

Bend angle — Bend angle

90 deg (default) | positive scalar

Swept degree of the pipe bend.

Elevation gain from port A to port B — Pipe elevation change

0 m (default) | scalar

Elevation change along the length of the pipe from port A to port B.

Gravitational acceleration — Gravitational acceleration

9.81 m/s^2 (default) | scalar

Gravitational acceleration. The block only uses this parameter if the value of the Elevation gain from port A to port B parameter is not 0.

Internal surface absolute roughness — Pipe wall roughness

15e-6 m (default) | positive scalar

Pipe wall absolute roughness. This parameter is used to determine the Darcy friction factor, which contributes to pressure loss in the pipe.

Enable dynamic compressibility — Whether to model fluid compressibility

off (default) | on

Whether to model any change in fluid mass due to fluid compressibility. When you select Fluid compressibility, mass changes due to varying fluid density in the segment are calculated. The fluid volume in the pipe remains constant. In the Isothermal Liquid library, all blocks calculate density as a function of pressure.

Initial liquid pressure — Pipe pressure at beginning of simulation

0.101325 MPa (default) | positive scalar

Pipe pressure at the beginning of the simulation.

Dependencies

To enable this parameter, select Enable dynamic compressibility.

References

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

Extended Capabilities

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

Version History

Introduced in R2020a

R2025a: Model changes in pipe bend elevation

Use the Elevation gain from port A to port B parameter to specify an elevation change along the length of the pipe bend.