# Pipe Bend (TL)

Pipe bend segment in a thermal liquid network

**Library:**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}:

$${K}_{loss}={C}_{angle}{C}_{bend}.$$

*C*_{angle} is calculated
as:

$${C}_{angle}=0.0148\theta -3.9716\cdot {10}^{-5}{\theta}^{2},$$

where *θ* is the **Bend
angle**, in degrees.

*C*_{bend} is calculated from the tabulated
ratio of bend radius to pipe diameter for 90^{o} bends from
Crane [1]:

The friction factor, *f*_{T}, for
clean commercial steel is interpolated from tabular data based on pipe diameter [1]:

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

### 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:

$${p}_{A}-{p}_{B}=\Delta {p}_{loss,A}-\Delta {p}_{loss,B}.$$

### 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}:

$${p}_{A}-{p}_{I}=\Delta {p}_{loss,A},$$

$${p}_{B}-{p}_{I}=\Delta {p}_{loss,B}.$$

### 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:

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

where:

*p*is the thermal liquid pressure at the internal node I._{I}$$\dot{T}$$

_{I}is the rate of change of the thermal liquid temperature at the internal node I.*β*is the thermal liquid bulk modulus._{I}*α*is the liquid thermal expansion coefficient.

### Energy Conservation

The energy conservation equation for the block is

$$V\frac{d(\rho u)}{dt}={\varphi}_{A}+{\varphi}_{B,}$$

where:

*ϕ*is the energy flow rate at port_{A}**A**.*ϕ*is the energy flow rate at port_{B}**B**.

## Ports

### Conserving

## Parameters

## Extended Capabilities

## Version History

**Introduced in R2022a**