# 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, Cangle, and a bend coefficient, Cbend:

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

Cangle is calculated as:

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

where θ is the Bend angle, in degrees.

Cbend is calculated from the tabulated ratio of bend radius to pipe diameter for 90o bends from Crane [1]:

The friction factor, fT, 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:

`$\Delta {p}_{loss}=\frac{\mu \lambda }{2{\rho }_{I}{d}^{2}A}\frac{L}{2}{\stackrel{˙}{m}}_{port},$`

where:

• μ is the fluid dynamic viscosity.

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

• ρI 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, $\frac{\pi }{4}{d}^{2}.$

• ${\stackrel{˙}{m}}_{port}$ 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:

`$\Delta {p}_{loss}=\left(\frac{{f}_{D}L}{2d}+\frac{{K}_{loss}}{2}\right)\frac{{\stackrel{˙}{m}}_{port}|{\stackrel{˙}{m}}_{port}|}{2{\rho }_{I}{A}^{2}},$`

where fD 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, pI:

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

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

### Mass Conservation

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

`${\stackrel{˙}{m}}_{A}+{\stackrel{˙}{m}}_{B}=0.$`

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:

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

where:

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

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

• βI is the thermal liquid bulk modulus.

• α is the liquid thermal expansion coefficient.

### Energy Conservation

The energy conservation equation for the block is

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

where:

• ϕA is the energy flow rate at port A.

• ϕB is the energy flow rate at port B.

## Ports

### Conserving

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

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

## Parameters

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Diameter of the pipe.

Radius of the circle formed by the pipe bend.

Swept degree of the pipe bend.

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

Whether to model any change in fluid mass due to fluid compressibility. When Fluid compressibility is set to `On`, 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.

Liquid temperature at the beginning of the simulation.

Pipe pressure at the beginning of the simulation.

#### Dependencies

To enable this parameter, set Fluid dynamic compressibility to `On`.

## Version History

Introduced in R2022a