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, 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 relative humidity.
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, which is the product of the Bend radius and the Bend angle parameters: $L_{b e n d} = r_{b e n d} θ .$
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. 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 :

$p_{A} - p_{I} = Δ p_{l o s s, A}$

$p_{B} - p_{I} = Δ p_{l o s s, B}$

Mass and Energy Balance

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

$\begin{array}{l} {\dot{m}}_{t o t a l} = {\dot{m}}_{A} + {\dot{m}}_{B} - {\dot{m}}_{w, c o n d} - {\dot{m}}_{w, c o n v} + {\dot{m}}_{d, e v a p} \\ {\dot{m}}_{w, t o t a l} = {\dot{m}}_{w A} + {\dot{m}}_{w B} - {\dot{m}}_{w, c o n d} - {\dot{m}}_{w, c o n v} + {\dot{m}}_{d, e v a p} \\ {\dot{m}}_{g, t o t a l} = {\dot{m}}_{g A} + {\dot{m}}_{g B} \\ {\dot{m}}_{d, t o t a l} = {\dot{m}}_{d A} + {\dot{m}}_{d B} + λ_{d} ({\dot{m}}_{w, c o n d} + {\dot{m}}_{w, c o n v}) - {\dot{m}}_{d, e v a p} \end{array}$

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_d is the value of the 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

$(\frac{1}{p_{I}} \frac{d p_{I}}{d t} - \frac{1}{T_{I}} \frac{d T_{I}}{d t}) ρ_{I} V + \frac{R_{a} - R_{w}}{R_{I}} ({\dot{m}}_{w, t o t a l} - x_{w} {\dot{m}}_{t o t a l}) + \frac{R_{a} - R_{g}}{R_{I}} ({\dot{m}}_{g, t o t a l} - x_{g} {\dot{m}}_{t o t a l}) = {\dot{m}}_{t o t a l},$

where:

p_I is the pressure of the internal volume.
ρ_I is the density of the internal volume.
T_I is the temperature of the internal volume.
V is the volume.
R_a, R_g, R_w, and R_I are the specific gas constants of the air, gas, water vapor, and internal volume, respectively.
x_w is the specific humidity.
x_g is the trace gas mass fraction.

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

$\begin{array}{l} (c_{p I} - R_{I} + r_{d} c_{p_{d I}}) V ρ_{I} \frac{d T_{I}}{d t} + u_{a I} {\dot{m}}_{M A, t o t a l} + (u_{w I} - u_{a I}) {\dot{m}}_{w, t o t a l} + (u_{g I} - u_{a I}) {\dot{m}}_{g, t o t a l} + h_{d I} {\dot{m}}_{d, t o t a l} = \\ ϕ_{A} + ϕ_{B} - (1 - λ_{d}) ({\dot{m}}_{w,cond} h_{d I}), \end{array}$

where:

u_I is the internal energy.
c_pI is the specific heat.
r_d is the mass ratio of water droplets to moist air.
c_{p_dI} is the water droplet specific heat.
${\dot{m}}_{d, t o t a l}$ is the total water droplet mass flow rate.
h_dI is the water droplet specific enthalpy.
ϕ_A is the energy flow rate at port A.
ϕ_B is the energy flow rate at port B.
$\dot{m}$ _w,cond is the rate of water vapor condensation due to a saturated fluid volume.
λ_d is the value of the Fraction of condensate entrained as water droplets parameter.
ϕ_A is the energy flow rate at port A.
ϕ_B is the energy flow rate at port B.

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

$\frac{d x_{w I}}{d t} ρ_{I} V + x_{w I} {\dot{m}}_{t o t a l} = {\dot{m}}_{w, t o t a l} .$

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

$\frac{d x_{g I}}{d t} ρ_{I} V + x_{g I} {\dot{m}}_{t o t a l} = {\dot{m}}_{g, t o t a l} .$

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

$\frac{d r_{d I}}{d t} ρ_{I} V + r_{d I} {\dot{m}}_{t o t a l} = {\dot{m}}_{d, t o t a l} .$

Ports

Conserving

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A — Liquid port
moist air

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

B — Liquid port
moist air

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

Parameters

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Main

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

Effects and Initial Conditions

Initial pressure — Moist air pressure at the start of simulation
`0.101325 MPa` (default) | positive scalar

Moist air pressure at the start of the simulation.

Initial pressure priority — Pressure parameter priority
`High` (default) | `Low` | `None`

Priority the solver assigns to the Initial pressure parameter when initializing the block.

Set this parameter to High to define your initial conditions. You may need to set this parameter to Low or None if this initial condition conflicts with the initial conditions of another block.

Initial temperature — Temperature at the start of simulation
`293.15 K` (default) | positive scalar

Initial moist air temperature.

Initial temperature priority — Temperature parameter priority
`High` (default) | `Low` | `None`

Priority the solver assigns to the Initial temperature parameter when initializing the block.

Set this parameter to High to define your initial conditions. You may need to set this parameter to Low or None if this initial condition conflicts with the initial conditions of another block.

Initial humidity specification — Humidity specification
`Relative humidity` (default) | `Specific humidity` | `Mole fraction` | `Humidity ratio` | `Wet-bulb temperature`

Method to specify the initial moist air humidity.

Initial relative humidity — Relative humidity at the start of simulation
`0.5` (default) | scalar in the range of [0,1]

Relative humidity in the moist air at the start of the simulation. The relative humidity is the ratio of the water vapor partial pressure to the water vapor saturation pressure, or the ratio of the water vapor mole fraction to the water vapor mole fraction at saturation.

Dependencies

To enable this parameter, set Initial humidity specification to Relative humidity.

Initial relative humidity priority — Relative humidity parameter priority
`High` (default) | `Low` | `None`

Priority the solver assigns to the Initial relative humidity parameter when initializing the block.

Set this parameter to High to define your initial conditions. You may need to set this parameter to Low or None if this initial condition conflicts with the initial conditions of another block.

Dependencies

To enable this parameter, set Initial humidity specification to Relative humidity.

Initial specific humidity — Specific humidity at the start of simulation
`0.01` (default) | scalar in the range of [0,1]

Specific humidity in the moist air at the start of simulation. The specific humidity is the mass fraction of water vapor to the combined total mass of water vapor, trace gas, and dry air.

Dependencies

To enable this parameter, set Initial humidity specification to Specific humidity.

Initial specific humidity priority — Specific humidity parameter priority
`High` (default) | `Low` | `None`

Priority the solver assigns to the Initial specific humidity parameter when initializing the block.

Set this parameter to High to define your initial conditions. You may need to set this parameter to Low or None if this initial condition conflicts with the initial conditions of another block.

Dependencies

To enable this parameter, set Initial humidity specification to Specific humidity.

Initial water vapor mole fraction — Mole fraction of water vapor at the start of simulation
`0.01` (default) | scalar in the range [0,1]

Mole fraction of the water vapor in the moist air channel at the start of simulation. The water vapor mole fraction is relative to the combined molar quantity of water vapor, trace species, and dry air.

Dependencies

To enable this parameter, set Initial humidity specification to Mole fraction.

Initial water vapor mole fraction priority — Water vapor mole fraction humidity parameter priority
`High` (default) | `Low` | `None`

Priority the solver assigns to the Initial water vapor mole fraction parameter when initializing the block.

Set this parameter to High to define your initial conditions. You may need to set this parameter to Low or None if this initial condition conflicts with the initial conditions of another block.

Dependencies

To enable this parameter, set Initial humidity specification to Mole fraction.

Initial humidity ratio — Humidity ratio at the start of simulation
`0.01` (default) | scalar in the range [0,1]

Humidity ratio in the moist air channel at the start of the simulation. The humidity ratio is the ratio of the mass of water vapor to the mass of dry air and trace gas.

Dependencies

To enable this parameter, set Initial humidity specification to Humidity ratio.

Initial humidity ratio priority — Humidity ratio parameter priority
`High` (default) | `Low` | `None`

Priority the solver assigns to the Initial humidity ratio humidity parameter when initializing the block.

Set this parameter to High to define your initial conditions. You may need to set this parameter to Low or None if this initial condition conflicts with the initial conditions of another block.

Dependencies

To enable this parameter, set Initial humidity specification to Humidity ratio.

Initial wet-bulb temperature — Wet-bulb temperature at the start of simulation
`287 K` (default) | scalar in the range [0,1]

Wet-bulb temperature at the start of the simulation. The block uses this value to calculate moisture.

Dependencies

To enable this parameter, set Initial humidity specification to Wet-bulb temperature.

Initial wet-bulb temperature priority — Wet-bulb temperature parameter priority
`High` (default) | `Low` | `None`

Priority the solver assigns to the Initial wet-bulb temperature parameter when initializing the block.

Set this parameter to High to define your initial conditions. You may need to set this parameter to Low or None if this initial condition conflicts with the initial conditions of another block.

Dependencies

To enable this parameter, set Initial humidity specification to Wet-bulb temperature.

Initial trace gas specification — Measurement type of trace gas
`Mass fraction` (default) | `Mole fraction`

Measurement type of trace gas.

Initial trace gas mass fraction — Amount of trace gas in the moist air channel
`0.001` (default) | scalar in the range [0,1]

Amount of trace gas in the moist air by mass fraction at the start of the simulation. The mass fraction is relative to the combined total mass of water vapor, trace gas, and dry air.

The block ignores this parameter if the Trace gas model parameter in the Moist Air Properties (MA) block is None.

Dependencies

To enable this parameter, set Initial trace gas specification to Mass fraction.

Initial trace gas mass fraction priority — Trace gas mass fraction parameter priority
`High` (default) | `Low` | `None`

Priority the solver assigns to the Initial trace gas mass fraction priority parameter when initializing the block.

Set this parameter to High to define your initial conditions. You may need to set this parameter to Low or None if this initial condition conflicts with the initial conditions of another block.

Dependencies

To enable this parameter, set Initial trace gas specification to Mass fraction.

Initial trace gas mole fraction — Mole fraction of trace gas at the start of simulation
`0.001` (default) | scalar in the range [0,1]

Amount of trace gas in the moist air channel by mole fraction at the start of the simulation. The mole fraction is relative to the combined molar total of water vapor, trace gas, and dry air.

The block ignores this parameter if the Trace gas model parameter in the Moist Air Properties (MA) block is None.

Dependencies

To enable this parameter, set Initial trace gas specification to Mole fraction.

Initial trace gas mole fraction priority — Trace gas mole fraction parameter priority
`High` (default) | `Low` | `None`

Priority the solver assigns to the Initial trace gas mole fraction priority parameter when initializing the block.

Set this parameter to High to define your initial conditions. You may need to set this parameter to Low or None if this initial condition conflicts with the initial conditions of another block.

Dependencies

To enable this parameter, set Initial trace gas specification to Mole fraction.

Initial mass ratio of water droplets to moist air — Ratio of water droplets to moist air
`0` (default) | positive scalar

Initial mass ratio of water droplets to moist air.

Initial mass ratio of water droplets to moist air priority — Mass ratio of water droplets to moist air parameter priority
`High` (default) | `Low` | `None`

Priority the solver assigns to the Initial mass ratio of water droplets to moist air priority parameter when initializing the block.

Set this parameter to High to define your initial conditions. You may need to set this parameter to Low or None if this initial condition conflicts with the initial conditions of another block.

Relative humidity at saturation — Relative humidity point of condensation
`1` (default) | scalar in the range [0,1]

Relative humidity point of condensation. Condensation occurs above this value. A value greater than 1 indicates a supersaturated vapor.

Water vapor condensation time constant — Time scale for condensation
`1e-3 s` (default) | positive scalar

Characteristic time scale at which an oversaturated moist air volume returns to saturation by condensing out excess moisture.

Water droplets evaporation time constant — Time scale for evaporation
`1e-3 s` (default) | positive scalar

Characteristic time scale at which water droplets evaporate to vapor.

Fraction of condensate entrained as water droplets — Fraction of condensate in water droplets
`1` (default) | scalar in the range [0,1]

Fraction of the condensate in the moist air that is entrained as water droplets.

References

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

Extended Capabilities

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

Version History

Introduced in R2023a

expand all

R2024b: Model water droplets suspended in moist air flow

Blocks in the moist air domain can now model water droplets suspended in a moist air flow. To model water droplets, select Enable entrained water droplets in the Moist Air Properties (MA) block connected to your moist air network.

R2024b: Updated options for initial conditions

You can now specify initial conditions for the block as parameters.

Pipe Bend (MA)

Description

Pipe Curvature Loss Coefficient

Losses Due to Friction in Laminar Flows

Losses due to Friction in Turbulent Flows

Pressure Differential

Mass and Energy Balance

Ports

Conserving

A — Liquid port moist air

B — Liquid port moist air

Parameters

Main

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

Effects and Initial Conditions

Initial pressure — Moist air pressure at the start of simulation 0.101325 MPa (default) | positive scalar

Initial pressure priority — Pressure parameter priority High (default) | Low | None

Initial temperature — Temperature at the start of simulation 293.15 K (default) | positive scalar

Initial temperature priority — Temperature parameter priority High (default) | Low | None

Initial humidity specification — Humidity specification Relative humidity (default) | Specific humidity | Mole fraction | Humidity ratio | Wet-bulb temperature

Initial relative humidity — Relative humidity at the start of simulation 0.5 (default) | scalar in the range of [0,1]

Dependencies

Initial relative humidity priority — Relative humidity parameter priority High (default) | Low | None

Dependencies

Initial specific humidity — Specific humidity at the start of simulation 0.01 (default) | scalar in the range of [0,1]

Dependencies

Initial specific humidity priority — Specific humidity parameter priority High (default) | Low | None

Dependencies

Initial water vapor mole fraction — Mole fraction of water vapor at the start of simulation 0.01 (default) | scalar in the range [0,1]

Dependencies

Initial water vapor mole fraction priority — Water vapor mole fraction humidity parameter priority High (default) | Low | None

Dependencies

Initial humidity ratio — Humidity ratio at the start of simulation 0.01 (default) | scalar in the range [0,1]

Dependencies

Initial humidity ratio priority — Humidity ratio parameter priority High (default) | Low | None

Dependencies

Initial wet-bulb temperature — Wet-bulb temperature at the start of simulation 287 K (default) | scalar in the range [0,1]

Dependencies

Initial wet-bulb temperature priority — Wet-bulb temperature parameter priority High (default) | Low | None

Dependencies

Initial trace gas specification — Measurement type of trace gas Mass fraction (default) | Mole fraction

Initial trace gas mass fraction — Amount of trace gas in the moist air channel 0.001 (default) | scalar in the range [0,1]

Dependencies

Initial trace gas mass fraction priority — Trace gas mass fraction parameter priority High (default) | Low | None

Dependencies

Initial trace gas mole fraction — Mole fraction of trace gas at the start of simulation 0.001 (default) | scalar in the range [0,1]

Dependencies

Initial trace gas mole fraction priority — Trace gas mole fraction parameter priority High (default) | Low | None

Dependencies

Initial mass ratio of water droplets to moist air — Ratio of water droplets to moist air 0 (default) | positive scalar

Initial mass ratio of water droplets to moist air priority — Mass ratio of water droplets to moist air parameter priority High (default) | Low | None

Relative humidity at saturation — Relative humidity point of condensation 1 (default) | scalar in the range [0,1]

Water vapor condensation time constant — Time scale for condensation 1e-3 s (default) | positive scalar

Water droplets evaporation time constant — Time scale for evaporation 1e-3 s (default) | positive scalar

Fraction of condensate entrained as water droplets — Fraction of condensate in water droplets 1 (default) | scalar in the range [0,1]

References

Extended Capabilities

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

Version History

R2024b: Model water droplets suspended in moist air flow

R2024b: Updated options for initial conditions

A — Liquid port
moist air

B — Liquid port
moist air

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

Initial pressure — Moist air pressure at the start of simulation
`0.101325 MPa` (default) | positive scalar

Initial pressure priority — Pressure parameter priority
`High` (default) | `Low` | `None`

Initial temperature — Temperature at the start of simulation
`293.15 K` (default) | positive scalar

Initial temperature priority — Temperature parameter priority
`High` (default) | `Low` | `None`

Initial humidity specification — Humidity specification
`Relative humidity` (default) | `Specific humidity` | `Mole fraction` | `Humidity ratio` | `Wet-bulb temperature`

Initial relative humidity — Relative humidity at the start of simulation
`0.5` (default) | scalar in the range of [0,1]

Initial relative humidity priority — Relative humidity parameter priority
`High` (default) | `Low` | `None`

Initial specific humidity — Specific humidity at the start of simulation
`0.01` (default) | scalar in the range of [0,1]

Initial specific humidity priority — Specific humidity parameter priority
`High` (default) | `Low` | `None`

Initial water vapor mole fraction — Mole fraction of water vapor at the start of simulation
`0.01` (default) | scalar in the range [0,1]

Initial water vapor mole fraction priority — Water vapor mole fraction humidity parameter priority
`High` (default) | `Low` | `None`

Initial humidity ratio — Humidity ratio at the start of simulation
`0.01` (default) | scalar in the range [0,1]

Initial humidity ratio priority — Humidity ratio parameter priority
`High` (default) | `Low` | `None`

Initial wet-bulb temperature — Wet-bulb temperature at the start of simulation
`287 K` (default) | scalar in the range [0,1]

Initial wet-bulb temperature priority — Wet-bulb temperature parameter priority
`High` (default) | `Low` | `None`

Initial trace gas specification — Measurement type of trace gas
`Mass fraction` (default) | `Mole fraction`

Initial trace gas mass fraction — Amount of trace gas in the moist air channel
`0.001` (default) | scalar in the range [0,1]

Initial trace gas mass fraction priority — Trace gas mass fraction parameter priority
`High` (default) | `Low` | `None`

Initial trace gas mole fraction — Mole fraction of trace gas at the start of simulation
`0.001` (default) | scalar in the range [0,1]

Initial trace gas mole fraction priority — Trace gas mole fraction parameter priority
`High` (default) | `Low` | `None`

Initial mass ratio of water droplets to moist air — Ratio of water droplets to moist air
`0` (default) | positive scalar

Initial mass ratio of water droplets to moist air priority — Mass ratio of water droplets to moist air parameter priority
`High` (default) | `Low` | `None`

Relative humidity at saturation — Relative humidity point of condensation
`1` (default) | scalar in the range [0,1]

Water vapor condensation time constant — Time scale for condensation
`1e-3 s` (default) | positive scalar

Water droplets evaporation time constant — Time scale for evaporation
`1e-3 s` (default) | positive scalar

Fraction of condensate entrained as water droplets — Fraction of condensate in water droplets
`1` (default) | scalar in the range [0,1]

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