Pipe Bend (G)

Pipe bend segment in a gas network

Since R2023a

• 

Libraries:
Simscape / Fluids / Gas / Pipes & Fittings

Description

The Pipe Bend (G) block represents a curved pipe in a gas network. You can define the pipe characteristics to calculate losses due to friction and pipe curvature. The block models losses with hydraulic loss coefficients and the results may be more accurate for lower speed flows where you can neglect compressible flow effects.

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 relative humidity.

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

• 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. The block approximates this value by using the empirical Haaland equation and the Internal surface absolute roughness parameter. The block takes the differential over each half of the pipe segment, between port A to an internal node, and between the internal node and port B.

Pressure Differential

The block calculates the pressure loss over the bend based on the internal fluid volume pressure, p_I :

$$p_A-p_I=\Delta p_{loss,A}$$

$$p_B-p_I=\Delta p_{loss,B}$$

Mass Balance

Mass conservation relates the mass flow rates to the dynamics of the pressure and temperature of the internal node representing the gas volume. When you select Enable dynamic compressibility, the mass balance is

$$\frac{\partial M}{\partial p}\frac{d{p}_I}{dt}+\frac{\partial M}{\partial T}\frac{d{T}_I}{dt}={\dot{m}}_A+{\dot{m}}_B,$$

where:

• $$\frac{\partial M}{\partial p}$$ is the partial derivative of the mass of the gas volume with respect to pressure at constant temperature and volume.

• $$\frac{\partial M}{\partial T}$$ is the partial derivative of the mass of the gas volume with respect to temperature at constant pressure and volume.

• p_I is the pressure of the gas volume.

• T_I is the temperature of the gas volume.

• t is time.

• $$\dot{m}$$_A and $$\dot{m}$$_B are the mass flow rates at ports A and B, respectively. The flow rate at a port is positive when gas flows into the block through that port.

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

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

Energy Balance

Energy conservation relates the energy and heat flow rates to the dynamics of the pressure and temperature of the internal node that represents the gas volume. When you select Enable dynamic compressibility, the energy balance is

$$\frac{\partial U}{\partial p}\frac{d{p}_I}{dt}+\frac{\partial U}{\partial T}\frac{d{T}_I}{dt}={\Phi }_A+{\Phi }_B,$$

where:

• $$\frac{\partial U}{\partial p}$$ is the partial derivative of the internal energy of the gas volume with respect to pressure at constant temperature and volume.

• $$\frac{\partial U}{\partial T}$$ is the partial derivative of the internal energy of the gas volume with respect to temperature at constant pressure and volume.

• Φ_A and Φ_B are the energy flow rates at ports A and B, respectively.

When you clear the Enable dynamic compressibility check box, the energy balance is

$${\rho }_0{c}_{p}V\frac{dT}{dt}={\Phi }_A+{\Phi }_B,$$

where:

• c_p is the gas specific heat.

• V is the pipe volume.

• ρ₀ is the gas density. The block calculates this constant value from the Nominal liquid temperature and Nominal liquid pressure parameters.

Ports

Conserving

A — Liquid port

gas

Gas conserving port associated with the liquid entry or exit port

B — Liquid port

gas

Gas conserving port associated with the 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.

Internal surface absolute roughness — Pipe wall roughness

15e-6 m (default) | positive scalar

Pipe wall absolute roughness. The block uses this parameter to determine the Darcy friction factor, which contributes to pressure loss in the pipe.

Enable dynamic compressibility — Option to model pressure dynamics in the gas channel

On (default) | Off

Option to model the pressure dynamics in the gas channel. If you clear this checkbox, the block removes the pressure derivative terms from the component energy and mass conservation equations.

Nominal temperature — Temperature at nominal operating conditions

293.15 K (default) | scalar

Gas temperature at nominal operating conditions. The block uses this value to calculate the nominal density to use in the mass and energy conservation equation when dynamic compressibility is disabled.

Dependencies

To enable this parameter, clear the Enable dynamic compressibility checkbox.

Nominal pressure — Pressure at nominal operating conditions

0.101325 MPa (default) | scalar

Gas pressure at nominal operating conditions. The block uses this value to calculate the nominal density to use in the mass and energy conservation equation when dynamic compressibility is disabled.

Dependencies

To enable this parameter, clear the Enable dynamic compressibility checkbox.

Initial pressure — Pressure at the start of simulation

0.101325 MPa (default) | scalar

Pressure in the gas channel at the start of simulation.

Dependencies

To enable this parameter, select Enable dynamic compressibility.

Initial pressure priority — Pressure parameter priority

High (default) | Low | None

Priority the solver assigns to the Initial pressure parameter when initializing the block.

Set this parameter to High to define your initial conditions. You may need to set this parameter to Low or None if this initial condition conflicts with the initial conditions of another block.

Dependencies

To enable this parameter, select Enable dynamic compressibility.

Initial temperature — Temperature at the start of simulation

293.15 K (default) | scalar

Temperature in the gas channel at the start of simulation.

Initial temperature priority — Temperature parameter priority

High (default) | Low | None

Priority the solver assigns to the Initial temperature parameter when initializing the block.

Set this parameter to High to define your initial conditions. You may need to set this parameter to Low or None if this initial condition conflicts with the initial conditions of another block.

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 R2023a

R2025a: Optionally model dynamic compressibility

The Pipe Bend (G) block can now optionally model dynamic compressibility by using the Enable dynamic compressibility parameter. Previously, this block always modeled pressure dynamics, which can cause models to take longer to achieve steady state. Clear the Enable dynamic compressibility parameter to prevent dynamic compressibility modeling.