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# Pipe (IL)

Rigid conduit for fluid flow in isothermal liquid systems

• Library:
• 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

`${\stackrel{˙}{m}}_{\text{A}}+{\stackrel{˙}{m}}_{\text{B}}=\left\{\begin{array}{cc}0,& \text{if}\text{\hspace{0.17em}}\text{fluid}\text{\hspace{0.17em}}\text{dynamic}\text{\hspace{0.17em}}\text{compressibility}\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{off}\\ \frac{{\stackrel{˙}{p}}_{I}}{{\beta }_{I}}{\rho }_{I}V,& \text{if}\text{\hspace{0.17em}}\text{fluid}\text{\hspace{0.17em}}\text{dynamic}\text{\hspace{0.17em}}\text{compressibility}\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{on}\end{array}$`

where:

• ${\stackrel{˙}{m}}_{\text{A}}$ and ${\stackrel{˙}{m}}_{\text{B}}$ are the mass flow rates through ports A and B.

• V is the pipe fluid volume.

• pI is the pressure inside the pipe.

• ρI is the fluid density inside the pipe.

• βI 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 ρI and βI, as well as port densities ρA and ρB 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 `${p}_{A}-{p}_{I}=\left\{\begin{array}{cc}\Delta {p}_{\text{v,A}},& \text{if}\text{\hspace{0.17em}}\text{fluid}\text{\hspace{0.17em}}\text{inertia}\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{off}\\ \Delta {p}_{\text{v,A}}+\frac{L}{2S}{\stackrel{¨}{m}}_{\text{A}},& \text{if}\text{\hspace{0.17em}}\text{fluid}\text{\hspace{0.17em}}\text{inertia}\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{on}\end{array}$` For half pipe adjacent to port B `${p}_{B}-{p}_{I}=\left\{\begin{array}{cc}\Delta {p}_{\text{v,B}},& \text{if}\text{\hspace{0.17em}}\text{fluid}\text{\hspace{0.17em}}\text{inertia}\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{off}\\ \Delta {p}_{\text{v,B}}+\frac{L}{2S}{\stackrel{¨}{m}}_{\text{B}},& \text{if}\text{\hspace{0.17em}}\text{fluid}\text{\hspace{0.17em}}\text{inertia}\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{on}\end{array}$`

In the equations:

• p, pA, and pB are the liquid pressures at port A and port B, respectively.

• Δpv,A and Δpv,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 For half pipe adjacent to port B

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.

• Leq is the aggregate equivalent length of the local pipe resistances.

• Dh is the hydraulic diameter of the pipe.

• fA and fB are the Darcy friction factors in the pipe halves adjacent to ports A and B.

• ReA and ReB are the Reynolds numbers at ports A and B.

• Relam is the Reynolds number above which the flow transitions to turbulent.

• Retur is the Reynolds number below which the flow transitions to laminar.

When the Reynolds number is between Relam and Retur, 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:

`$\mathrm{Re}=\frac{|\stackrel{˙}{m}|{D}_{h}}{\mu S}.$`

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

`$f=\frac{1}{{\left(-1.8{\mathrm{log}}_{10}\left(\frac{6.9}{\mathrm{Re}}+{\left(\frac{1}{3.7}\frac{r}{{D}_{h}}\right)}^{1.11}\right)\right)}^{2}},$`

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

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

The length of the pipe along the direction of flow.

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

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

### Friction

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.

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

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.

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.

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

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.

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

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

Liquid pressure in the pipe at the start of simulation.

#### Dependencies

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

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.

## Version History

Introduced in R2020a