# Pipe Bend (MA)

**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, *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 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

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

where:

*μ*is the relative humidity.*λ*is the Darcy friction factor constant, which is 64 for laminar flow.*ρ*is the internal fluid density._{I}*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, $$\frac{\pi}{4}{d}^{2}.$$$${\dot{m}}_{port}$$ 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. 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, *p _{I}* :

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

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

### Mass and Energy Balance

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

$$\begin{array}{l}{\dot{m}}_{total}={\dot{m}}_{A}+{\dot{m}}_{B}-{\dot{m}}_{w,cond}-{\dot{m}}_{w,conv}+{\dot{m}}_{d,evap}\\ {\dot{m}}_{w,total}={\dot{m}}_{wA}+{\dot{m}}_{wB}-{\dot{m}}_{w,cond}-{\dot{m}}_{w,conv}+{\dot{m}}_{d,evap}\\ {\dot{m}}_{g,total}={\dot{m}}_{gA}+{\dot{m}}_{gB}\\ {\dot{m}}_{d,total}={\dot{m}}_{dA}+{\dot{m}}_{dB}+{\lambda}_{d}\left({\dot{m}}_{w,cond}+{\dot{m}}_{w,conv}\right)-{\dot{m}}_{d,evap}\end{array}$$

where:

$$\dot{m}$$ is the mass flow rate. The subscript

*w*denotes water vapor, the subscript*g*denotes trace gas, and the subscript*d*denotes water droplets.*λ*is the value of the_{d}**Fraction of condensate entrained as water droplets**parameter.$$\dot{m}$$

_{w,cond}is the rate of water vapor condensation due to a saturated fluid volume.$$\dot{m}$$

_{w,conv}is the rate of condensation on the wall surface.$$\dot{m}$$

_{d,evap}is the rate of water droplet evaporation.*Φ*is the energy flow rate.

The mass conservation equations for the mixture relate the pressure, temperature, and internal moist air volume mass fractions

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

where:

*p*is the pressure of the internal volume._{I}*ρ*is the density of the internal volume._{I}*T*is the temperature of the internal volume._{I}*V*is the volume.*R*,_{a}*R*,_{g}*R*, and_{w}*R*are the specific gas constants of the air, gas, water vapor, and internal volume, respectively._{I}*x*is the specific humidity._{w}*x*is the trace gas mass fraction._{g}

The energy conservation equation relates the energy flow rate to the pressure, temperature, and internal moist air volume mass fractions

$$\begin{array}{l}\left({c}_{pI}-{R}_{I}+{r}_{d}{c}_{{p}_{dI}}\right)V{\rho}_{I}\frac{d{T}_{I}}{dt}+{u}_{aI}{\dot{m}}_{MA,total}+\left({u}_{wI}-{u}_{aI}\right){\dot{m}}_{w,total}+\left({u}_{gI}-{u}_{aI}\right){\dot{m}}_{g,total}+{h}_{dI}{\dot{m}}_{d,total}=\text{\hspace{0.05em}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.05em}}{\varphi}_{\text{A}}+{\varphi}_{\text{B}}-\left(1-{\lambda}_{d}\right)\left({\dot{m}}_{\text{w,cond}}{h}_{dI}\right),\end{array}$$

where:

*u*is the internal energy._{I}*c*is the specific heat._{pI}*r*is the mass ratio of water droplets to moist air._{d}*c*is the water droplet specific heat._{pdI}$${\dot{m}}_{d,total}$$ is the total water droplet mass flow rate.

*h*is the water droplet specific enthalpy._{dI}*ϕ*is the energy flow rate at port_{A}**A**.*ϕ*is the energy flow rate at port_{B}**B**.$$\dot{m}$$

_{w,cond}is the rate of water vapor condensation due to a saturated fluid volume.*λ*is the value of the_{d}**Fraction of condensate entrained as water droplets**parameter.*ϕ*is the energy flow rate at port_{A}**A**.*ϕ*is the energy flow rate at port_{B}**B**.

The mass conservation equation for the water droplets relates the water vapor mass flow rate to the internal moist air volume moisture level

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

The trace gas mass conservation equation relates the trace gas mass flow rate to the internal moist air volume trace gas level

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

The water droplets mass conservation equation relates the water droplet mass flow rate to the entrained water droplet dynamics in the internal moist air volume

$$\frac{d{r}_{dI}}{dt}{\rho}_{I}V+{r}_{dI}{\dot{m}}_{total}={\dot{m}}_{d,total}.$$

## Ports

### Conserving

## Parameters

## References

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

## Extended Capabilities

## Version History

**Introduced in R2023a**