Pipe (IL)

Rigid conduit for fluid flow in isothermal liquid systems

Since R2020a

Libraries:
Simscape / Foundation Library / Isothermal Liquid / Elements

Description

The Pipe (IL) block models pipe flow dynamics in an isothermal liquid network. The block accounts for viscous friction losses, and can also account for dynamic compressibility and fluid inertia.

The pipe contains a constant volume of liquid. The liquid experiences pressure losses due to viscous friction, following the Darcy-Weisbach equation.

Pipe Effects

The block lets you include dynamic compressibility and fluid inertia effects. Turning on each of these effects can improve model fidelity at the cost of increased equation complexity and potentially increased simulation cost:

When dynamic compressibility is off, the liquid is assumed to spend negligible time in the pipe volume. Therefore, there is no accumulation of mass in the pipe, and mass inflow equals mass outflow. This is the simplest option. It is appropriate when the liquid mass in the pipe is a negligible fraction of the total liquid mass in the system.
When dynamic compressibility is on, an imbalance of mass inflow and mass outflow can cause liquid to accumulate or diminish in the pipe. As a result, pressure in the pipe volume can rise and fall dynamically, which provides some compliance to the system and modulates rapid pressure changes. This is the default option.
If dynamic compressibility is on, you can also turn on fluid inertia. This effect results in additional flow resistance, besides the resistance due to friction. This additional resistance is proportional to the rate of change of mass flow rate. Accounting for fluid inertia slows down rapid changes in flow rate but can also cause the flow rate to overshoot and oscillate. This option is appropriate in a very long pipe. Turn on fluid inertia and connect multiple pipe segments in series to model the propagation of pressure waves along the pipe, such as in the water hammer phenomenon.

Mass Balance

The mass conservation equation for the pipe is

${\dot{m}}_{A} + {\dot{m}}_{B} = {\begin{matrix} 0, & if fluid dynamic compressibility is off \\ \frac{{\dot{p}}_{I}}{β_{I}} ρ_{I} V, & if fluid dynamic compressibility is on \end{matrix}$

where:

${\dot{m}}_{A}$ and ${\dot{m}}_{B}$ are the mass flow rates through ports A and B.
V is the pipe fluid volume.
p_I is the pressure inside the pipe.
ρ_I is the fluid density inside the pipe.
β_I is the fluid bulk modulus inside the pipe.

The fluid can be a mixture of pure liquid and a small amount of entrained air, as specified by the Isothermal Liquid Properties (IL) block connected to the circuit. Equations used to compute ρ_I and β_I, as well as port densities ρ_A and ρ_B in the viscous friction pressure loss equations for each half pipe, depend on the selected isothermal liquid model. For detailed information, see Isothermal Liquid Modeling Options.

Momentum Balance

The table shows the momentum conservation equations for each half pipe.

For half pipe adjacent to port A	$p_{A} - p_{I} = {\begin{matrix} Δ p_{v,A}, & if fluid inertia is off \\ Δ p_{v,A} + \frac{L}{2 S} {\ddot{m}}_{A}, & if fluid inertia is on \end{matrix}$
For half pipe adjacent to port B	$p_{B} - p_{I} = {\begin{matrix} Δ p_{v,B}, & if fluid inertia is off \\ Δ p_{v,B} + \frac{L}{2 S} {\ddot{m}}_{B}, & if fluid inertia is on \end{matrix}$

In the equations:

p, p_A, and p_B are the liquid pressures at port A and port B, respectively.
Δp_v,A and Δp_v,B are the viscous friction pressure losses between the pipe volume center and ports A and B.
L is the pipe length.
S is the pipe cross-sectional area.

Viscous Friction Pressure Losses

The table shows the viscous friction pressure loss equations for each half pipe.

For half pipe adjacent to port A	$Δ p_{v,A} = {\begin{matrix} λ μ (\frac{L + L_{eq}}{2}) \frac{{\dot{m}}_{A}}{2 ρ_{I} D_{h}^{2} S}, & if {Re}_{A} < {Re}_{lam} \\ f_{A} (\frac{L + L_{eq}}{2}) \frac{{\dot{m}}_{A} \| {\dot{m}}_{A} \|}{2 ρ_{I} D_{h} S^{2}}, & {if Re}_{A} > {Re}_{tur} \end{matrix}$
For half pipe adjacent to port B	$Δ p_{v,B} = {\begin{matrix} λ μ (\frac{L + L_{eq}}{2}) \frac{{\dot{m}}_{B}}{2 ρ_{I} D_{h}^{2} S}, & if {Re}_{B} < {Re}_{lam} \\ f_{B} (\frac{L + L_{eq}}{2}) \frac{{\dot{m}}_{B} \| {\dot{m}}_{B} \|}{2 ρ_{I} D_{h} S^{2}}, & {if Re}_{B} \geq {Re}_{tur} \end{matrix}$

In the equations:

λ is the pipe shape factor, used to calculate the Darcy friction factor in the laminar regime.
μ is the dynamic viscosity of the liquid in the pipe.
L_eq is the aggregate equivalent length of the local pipe resistances.
D_h is the hydraulic diameter of the pipe.
f_A and f_B are the Darcy friction factors in the pipe halves adjacent to ports A and B.
Re_A and Re_B are the Reynolds numbers at ports A and B.
Re_lam is the Reynolds number above which the flow transitions to turbulent.
Re_tur is the Reynolds number below which the flow transitions to laminar.

When the Reynolds number is between Re_lam and Re_tur, the flow is in transition between laminar flow and turbulent flow. The pressure losses due to viscous friction during the transition region follow a smooth connection between those in the laminar flow regime and those in the turbulent flow regime.

The block computes the Reynolds numbers at ports A and B based on the mass flow rate through the appropriate port:

$Re = \frac{| \dot{m} | D_{h}}{μ S} .$

The Darcy friction factors follow from the Haaland approximation for the turbulent regime:

$f = \frac{1}{{(- 1.8 \log_{10} (\frac{6.9}{Re} + {(\frac{1}{3.7} \frac{r}{D_{h}})}^{1.11}))}^{2}},$

where:

f is the Darcy friction factor.
r is the pipe surface roughness.

Assumptions and Limitations

The pipe wall is rigid.
The flow is fully developed.
The effect of gravity is negligible.

Ports

Conserving

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A — Inlet or outlet
isothermal liquid

Isothermal liquid conserving port associated with the inlet or outlet of the pipe. This block has no intrinsic directionality.

B — Inlet or outlet
isothermal liquid

Isothermal liquid conserving port associated with the inlet or outlet of the pipe. This block has no intrinsic directionality.

Parameters

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Geometry

Pipe length — The length of the pipe
`5 m` (default) | positive scalar

The length of the pipe along the direction of flow.

Cross-sectional area — The internal area of the pipe
`0.01 m^2` (default) | positive scalar

The internal area of the pipe normal to the direction of the flow.

Hydraulic diameter — Diameter of an equivalent cylindrical pipe with the same cross-sectional area
`0.1128 m` (default) | positive scalar

Diameter of an equivalent cylindrical pipe with the same cross-sectional area.

Friction

Aggregate equivalent length of local resistances — The combined length of all local resistances present in the pipe
`1 m` (default) | positive scalar

The combined length of all local resistances present in the pipe. Local resistances include bends, fittings, armatures, and pipe inlets and outlets. The effect of the local resistances is to increase the effective length of the pipe segment. This length is added to the geometrical pipe length only for friction calculations. The liquid volume inside the pipe depends only on the pipe geometrical length, defined by the Pipe length parameter.

Internal surface absolute roughness — Average depth of all surface defects on the internal surface of the pipe
`15e-6 m` (default) | positive scalar

Average depth of all surface defects on the internal surface of the pipe, which affects the pressure loss in the turbulent flow regime.

Laminar flow upper Reynolds number limit — The Reynolds number above which flow begins to transition from laminar to turbulent
`2000` (default) | scalar greater than 1

The Reynolds number above which flow begins to transition from laminar to turbulent. This number equals the maximum Reynolds number corresponding to fully developed laminar flow.

Turbulent flow lower Reynolds number limit — The Reynolds number below which flow begins to transition from turbulent to laminar
`4000` (default) | scalar greater than Laminar flow upper Reynolds number limit value

The Reynolds number below which flow begins to transition from turbulent to laminar. This number equals to the minimum Reynolds number corresponding to fully developed turbulent flow.

Laminar friction constant for Darcy friction factor — Effect of pipe geometry on the viscous friction losses
`64` (default) | positive scalar

Dimensionless factor that encodes the effect of pipe cross-sectional geometry on the viscous friction losses in the laminar flow regime. Typical values are 64 for a circular cross section, 57 for a square cross section, 62 for a rectangular cross section with an aspect ratio of 2, and 96 for a thin annular cross section [1].

Effects and Initial Conditions

Fluid dynamic compressibility — Select whether to model fluid dynamic compressibility
`On` (default) | `Off`

Select whether to account for the dynamic compressibility of the liquid. Dynamic compressibility gives the liquid density a dependence on pressure, impacting the transient response of the system at small time scales.

Fluid inertia — Select whether to model fluid inertia
`Off` (default) | `On`

Select whether to account for the flow inertia of the liquid. Flow inertia gives the liquid a resistance to changes in mass flow rate.

Dependencies

To enable this check box, select the Fluid dynamic compressibility check box.

Initial liquid pressure — Liquid pressure at time zero
`0.101325 MPa` (default) | positive scalar

Liquid pressure in the pipe at the start of simulation.

Dependencies

Enabled when the Fluid dynamic compressibility parameter is set to On.

Initial mass flow rate from port A to port B — Mass flow rate at time zero
`0 kg/s` (default) | scalar

Mass flow rate from port A to port B at time zero.

Dependencies

Enabled when the Fluid inertia parameter is set to On.

References

[1] White, F. M., Fluid Mechanics. 7th Ed, Section 6.8. McGraw-Hill, 2011.

Extended Capabilities

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

Version History

Introduced in R2020a

Pipe (IL)

Description

Pipe Effects

Mass Balance

Momentum Balance

Viscous Friction Pressure Losses

Assumptions and Limitations

Ports

Conserving

A — Inlet or outlet
isothermal liquid

B — Inlet or outlet
isothermal liquid

Parameters

Geometry

Pipe length — The length of the pipe
`5 m` (default) | positive scalar

Cross-sectional area — The internal area of the pipe
`0.01 m^2` (default) | positive scalar

Hydraulic diameter — Diameter of an equivalent cylindrical pipe with the same cross-sectional area
`0.1128 m` (default) | positive scalar

Friction

Aggregate equivalent length of local resistances — The combined length of all local resistances present in the pipe
`1 m` (default) | positive scalar

Internal surface absolute roughness — Average depth of all surface defects on the internal surface of the pipe
`15e-6 m` (default) | positive scalar

Laminar flow upper Reynolds number limit — The Reynolds number above which flow begins to transition from laminar to turbulent
`2000` (default) | scalar greater than 1

Turbulent flow lower Reynolds number limit — The Reynolds number below which flow begins to transition from turbulent to laminar
`4000` (default) | scalar greater than Laminar flow upper Reynolds number limit value

Laminar friction constant for Darcy friction factor — Effect of pipe geometry on the viscous friction losses
`64` (default) | positive scalar

Effects and Initial Conditions

Fluid dynamic compressibility — Select whether to model fluid dynamic compressibility
`On` (default) | `Off`

Fluid inertia — Select whether to model fluid inertia
`Off` (default) | `On`

Dependencies

Initial liquid pressure — Liquid pressure at time zero
`0.101325 MPa` (default) | positive scalar

Dependencies

Initial mass flow rate from port A to port B — Mass flow rate at time zero
`0 kg/s` (default) | scalar

Dependencies

References

Extended Capabilities

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

Version History

See Also

Topics

Pipe (IL)

Description

Pipe Effects

Mass Balance

Momentum Balance

Viscous Friction Pressure Losses

Assumptions and Limitations

Ports

Conserving

A — Inlet or outlet isothermal liquid

B — Inlet or outlet isothermal liquid

Parameters

Geometry

Pipe length — The length of the pipe 5 m (default) | positive scalar

Cross-sectional area — The internal area of the pipe 0.01 m^2 (default) | positive scalar

Hydraulic diameter — Diameter of an equivalent cylindrical pipe with the same cross-sectional area 0.1128 m (default) | positive scalar

Friction

Aggregate equivalent length of local resistances — The combined length of all local resistances present in the pipe 1 m (default) | positive scalar

Internal surface absolute roughness — Average depth of all surface defects on the internal surface of the pipe 15e-6 m (default) | positive scalar

Laminar flow upper Reynolds number limit — The Reynolds number above which flow begins to transition from laminar to turbulent 2000 (default) | scalar greater than 1

Turbulent flow lower Reynolds number limit — The Reynolds number below which flow begins to transition from turbulent to laminar 4000 (default) | scalar greater than Laminar flow upper Reynolds number limit value

Laminar friction constant for Darcy friction factor — Effect of pipe geometry on the viscous friction losses 64 (default) | positive scalar

Effects and Initial Conditions

Fluid dynamic compressibility — Select whether to model fluid dynamic compressibility On (default) | Off

Fluid inertia — Select whether to model fluid inertia Off (default) | On

Dependencies

Initial liquid pressure — Liquid pressure at time zero 0.101325 MPa (default) | positive scalar

Dependencies

Initial mass flow rate from port A to port B — Mass flow rate at time zero 0 kg/s (default) | scalar

Dependencies

References

Extended Capabilities

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

Version History

See Also

Topics

A — Inlet or outlet
isothermal liquid

B — Inlet or outlet
isothermal liquid

Pipe length — The length of the pipe
`5 m` (default) | positive scalar

Cross-sectional area — The internal area of the pipe
`0.01 m^2` (default) | positive scalar

Hydraulic diameter — Diameter of an equivalent cylindrical pipe with the same cross-sectional area
`0.1128 m` (default) | positive scalar

Aggregate equivalent length of local resistances — The combined length of all local resistances present in the pipe
`1 m` (default) | positive scalar

Internal surface absolute roughness — Average depth of all surface defects on the internal surface of the pipe
`15e-6 m` (default) | positive scalar

Laminar flow upper Reynolds number limit — The Reynolds number above which flow begins to transition from laminar to turbulent
`2000` (default) | scalar greater than 1

Turbulent flow lower Reynolds number limit — The Reynolds number below which flow begins to transition from turbulent to laminar
`4000` (default) | scalar greater than Laminar flow upper Reynolds number limit value

Laminar friction constant for Darcy friction factor — Effect of pipe geometry on the viscous friction losses
`64` (default) | positive scalar

Fluid dynamic compressibility — Select whether to model fluid dynamic compressibility
`On` (default) | `Off`

Fluid inertia — Select whether to model fluid inertia
`Off` (default) | `On`

Initial liquid pressure — Liquid pressure at time zero
`0.101325 MPa` (default) | positive scalar

Initial mass flow rate from port A to port B — Mass flow rate at time zero
`0 kg/s` (default) | scalar

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