Pipe Bend (TL)

Pipe bend segment in a thermal liquid network

Since R2022a

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:

$K_{l o s s} = C_{a n g l e} C_{b e n d} .$

The block calculates C_angle 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 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]:

Diagram displaying 90° pipe bend

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 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, f_T	.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:

$Δ p_{l o s s} = \frac{μ λ}{2 ρ_{I} d^{2} A} \frac{L}{2} {\dot{m}}_{p o r t},$

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: $L_{b e n d} = r_{b e n d} θ .$ .
A is the pipe cross-sectional area, $\frac{π}{4} d^{2} .$
${\dot{m}}_{p o r t}$ 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:

$Δ p_{l o s s} = (\frac{f_{D} L}{2 d} + \frac{K_{l o s s}}{2}) \frac{{\dot{m}}_{p o r t} | {\dot{m}}_{p o r t} |}{2 ρ_{I} A^{2}},$

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:

$p_{A} - p_{B} = Δ p_{l o s s, A} - Δ p_{l o s s, B} .$

Pressure Differential for Compressible Fluids

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

$p_{A} - p_{I} = Δ p_{l o s s, A}$

$p_{B} - p_{I} = Δ p_{l o s s, B}$

Mass Conservation

For an incompressible fluid, the mass flow into the pipe equals the mass flow out of the pipe:

${\dot{m}}_{A} + {\dot{m}}_{B} = 0.$

For a flexible pipe with a compressible fluid, the pipe mass conservation equation is: This dependence is captured by the bulk modulus and thermal expansion coefficient of the thermal liquid:

${\dot{m}}_{A} + {\dot{m}}_{B} = ρ_{I} \dot{V} + ρ_{I} V (\frac{{\dot{p}}_{I}}{β_{I}} - α_{I} {\dot{T}}_{I}),$

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

$V \frac{d (ρ u)}{d t} = ϕ_{A} + ϕ_{B,}$

where:

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

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

expand all

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

expand all

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

Fluid 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 dynamic 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 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 Fluid dynamic compressibility.

Extended Capabilities

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

Version History

Introduced in R2022a

Pipe Bend (TL)

Description

Pipe Curvature Loss Coefficient

Losses Due to Friction in Laminar Flows

Losses due to Friction in Turbulent Flows

Pressure Differential for Incompressible Fluids

Pressure Differential for Compressible Fluids

Mass Conservation

Energy Conservation

Examples

Residential Ground Source Heat Pump

Ports

Conserving

A — Liquid port thermal liquid

B — Liquid port thermal liquid

Parameters

Pipe diameter — Pipe diameter 0.01 m (default) | positive scalar

Bend radius — Bend circle radius 0.04 m (default) | positive scalar

Bend angle — Bend angle 90 deg (default) | positive scalar

Internal surface absolute roughness — Pipe wall roughness 15e-6 m (default) | positive scalar

Fluid dynamic compressibility — Whether to model fluid compressibility off (default) | on

Initial liquid temperature — Liquid temperature at start of simulation 293.15 K (default)

Initial liquid pressure — Pipe pressure at beginning of simulation 0.101325 MPa (default) | positive scalar

Dependencies

Extended Capabilities

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

Version History

See Also

A — Liquid port
thermal liquid

B — Liquid port
thermal liquid

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

Bend radius — Bend circle radius
`0.04 m` (default) | positive scalar

Bend angle — Bend angle
`90 deg` (default) | positive scalar

Internal surface absolute roughness — Pipe wall roughness
`15e-6 m` (default) | positive scalar

Fluid dynamic compressibility — Whether to model fluid compressibility
`off` (default) | `on`

Initial liquid temperature — Liquid temperature at start of simulation
`293.15` `K` (default)

Initial liquid pressure — Pipe pressure at beginning of simulation
`0.101325 MPa` (default) | positive scalar

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