# Pipe Bend (MA)

Pipe bend segment in a moist air network

Since R2023a

Libraries:
Simscape / Fluids / Moist Air / Pipes & Fittings

## Description

The Pipe Bend (MA) block represents a curved pipe in a moist air network. You can define the pipe characteristics to calculate losses due to friction and pipe curvature.

### Pipe Curvature Loss Coefficient

The coefficient for pressure losses due to geometry changes comprises an angle correction factor, Cangle, and a bend coefficient, Cbend:

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

The block calculates Cangle 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 Cbend from the tabulated ratio of the bend radius, r, to the pipe diameter, d, for 90° bends from data based on Crane [1]:

r/d11.523468101214162024
K20 fT14 fT12 fT12 fT14 fT17 fT24 fT30 fT34 fT38 fT42 fT50 fT58 fT

The block interpolates the friction factor, fT, 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)51015202532405072.5100125150225350609.5
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

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

where:

• μ is the relative humidity.

• 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, 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, $\frac{\pi }{4}{d}^{2}.$

• ${\stackrel{˙}{m}}_{port}$ 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. 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, pI :

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

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

### Mass and Energy Balance

The net flow rates into the moist air volume inside the pipe bend are

`$\begin{array}{l}{\stackrel{˙}{m}}_{total}={\stackrel{˙}{m}}_{A}+{\stackrel{˙}{m}}_{B}\\ {\Phi }_{total}={\Phi }_{A}+{\Phi }_{B}\\ {\stackrel{˙}{m}}_{w,total}={\stackrel{˙}{m}}_{wA}+{\stackrel{˙}{m}}_{wB}\\ {\stackrel{˙}{m}}_{g,total}={\stackrel{˙}{m}}_{gA}+{\stackrel{˙}{m}}_{gB}\end{array}$`

where:

• $\stackrel{˙}{m}$ is the mass flow rate. The subscript w denotes water vapor and the subscript g denotes trace gas.

• Φ is the energy flow rate.

Mixture mass conservation relates the mixture mass flow rate to the dynamics of the pressure, temperature, and mass fractions of the internal moist air volume

`$\left(\frac{1}{{p}_{I}}\frac{d{p}_{I}}{dt}-\frac{1}{{T}_{I}}\frac{d{T}_{I}}{dt}\right){\rho }_{I}V+\frac{{R}_{a}-{R}_{w}}{{R}_{I}}\left({\stackrel{˙}{m}}_{w,total}-{x}_{w}{\stackrel{˙}{m}}_{total}\right)+\frac{{R}_{a}-{R}_{g}}{{R}_{I}}\left({\stackrel{˙}{m}}_{g,total}-{x}_{g}{\stackrel{˙}{m}}_{total}\right)={\stackrel{˙}{m}}_{total},$`

where:

• pI is the pressure of the internal volume.

• ρI is the density of the internal volume.

• TI is the temperature of the internal volume.

• V is the volume.

• Ra, Rg, Rw, and RI are the specific gas constants of the air, gas, water vapor, and internal volume, respectively.

• xw is the specific humidity.

• xg is the trace gas mass fraction.

Energy conservation relates the energy flow rate to the dynamics of the pressure, temperature, and mass fractions of the internal moist air volume

`${\rho }_{I}\left({c}_{pI}-{R}_{I}\right)V\frac{d{T}_{I}}{dt}+\left({u}_{wI}-{u}_{aI}\right)\left({\stackrel{˙}{m}}_{w,total}-{x}_{w}{\stackrel{˙}{m}}_{total}\right)+\left({u}_{gI}-{u}_{aI}\right)\left({\stackrel{˙}{m}}_{g,total}-{x}_{g}{\stackrel{˙}{m}}_{total}\right)+{u}_{I}{\stackrel{˙}{m}}_{total}={\Phi }_{total},$`

where:

• uI is the internal energy.

• cpI is the specific heat.

Water vapor mass conservation relates the water vapor mass flow rate to the dynamics of the moisture level in the internal moist air volume

`$\frac{d{x}_{wI}}{dt}{\rho }_{I}V+{x}_{wI}{\stackrel{˙}{m}}_{total}={\stackrel{˙}{m}}_{w,total}.$`

Similarly, trace gas mass conservation relates the trace gas mass flow rate to the dynamics of the trace gas level in the internal moist air volume

`$\frac{d{x}_{gI}}{dt}{\rho }_{I}V+{x}_{gI}{\stackrel{˙}{m}}_{total}={\stackrel{˙}{m}}_{g,total}.$`

### Variables

To set the priority and initial target values for the block variables prior to simulation, use the Initial Targets section in the block dialog box or Property Inspector. For more information, see Set Priority and Initial Target for Block Variables.

Nominal values provide a way to specify the expected magnitude of a variable in a model. Using system scaling based on nominal values increases the simulation robustness. Nominal values can come from different sources, one of which is the Nominal Values section in the block dialog box or Property Inspector. For more information, see Modify Nominal Values for a Block Variable.

## Ports

### Conserving

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Moist air conserving port associated with the liquid entry or exit port.

Moist air conserving port associated with the liquid entry or exit port.

## 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. The block uses this parameter to determine the Darcy friction factor, which contributes to pressure loss in the pipe.

## References

[1] Crane Co. Flow of Fluids Through Valves, Fittings, and Pipe TP-410. Crane Co., 1981.

## Version History

Introduced in R2023a