Pipe Bend (TL)

Pipe bend segment in a thermal liquid network

Since R2022a

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

Description

The Pipe Bend (TL) block models a curved pipe in a thermal 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 select Enable dynamic compressibility, the pipe mass conservation equation is

$${\dot{m}}_{\text{A}}+{\dot{m}}_{\text{B}}={\rho }_{\text{I}}\dot{V}+{\rho }_{\text{I}}V\left(\frac{{\dot{p}}_{\text{I}}}{{\beta }_{\text{I}}}-{\alpha }_{\text{I}}{\dot{T}}_{\text{I}}\right),$$

where:

• p_I is the thermal liquid pressure at the internal node I.

• $$\dot{T}$$_I is the rate of change of the thermal liquid temperature at the internal node I.

• β_I is the thermal liquid bulk modulus.

• α is the liquid thermal expansion coefficient.

Energy Conservation

The energy conservation equation for the block is

$$\dot{E}={\varphi }_A+{\varphi }_B-{\dot{m}}_{avg}g\Delta z,$$

where:

• ϕ_A is the energy flow rate at port A.

• ϕ_B is the energy flow rate at port B.

If the fluid is incompressible, the energy accumulation rate is

$$\dot{E}={\rho }_0{c}_{p_I}V\frac{d{T}_I}{dt},$$

where:

• c_pI is the fluid specific heat at the internal node of the block.

• V is the pipe volume.

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

If the fluid is compressible, the energy accumulation rate is

$$\dot{E}=\frac{d{p}_I}{dt}\frac{dU}{dp}+\frac{dT}{dt}\frac{dU}{dT}.$$

Examples

Residential Ground Source Heat Pump

Models a ground source heat pump system that is used to heat a residential building having hot-water radiators for heat distribution. The ground source heat pump uses R410a, a two-phase fluid refrigerant, as the working fluid. The heat pump takes up the naturally existing heat stored in the ground and transfers the heat to the hot-water radiators. The compressor drives the refrigerant through a condenser, a thermostatic expansion valve, and an evaporator. An accumulator ensures that only vapor returns to the compressor. A receiver ensures that only liquid returns to the thermostatic expansion valve.

• Open Model

Ports

Conserving

A — Liquid port

thermal liquid

Thermal liquid conserving port associated with one end of the pipe bend.

B — Liquid port

thermal liquid

Thermal liquid conserving port associated with one end of the pipe bend.

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 Enable dynamic compressibility, mass changes due to varying fluid density in the segment are calculated. The fluid volume in the pipe remains constant.

Initial liquid temperature — Liquid temperature at start of simulation

293.15 K (default)

Liquid temperature at the beginning of the simulation.

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, set select Enable dynamic compressibility.

Nominal liquid temperature — Temperature at nominal operating conditions

293.15 K (default) | scalar

Liquid 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 liquid pressure — Pressure at nominal operating conditions

0.101325 MPa (default) | scalar

Liquid 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.

Extended Capabilities

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

Version History

Introduced in R2022a

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.