# Pipe (IL)

Pipe segment in an isothermal liquid network

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• Simscape / Fluids / Isothermal Liquid / Pipes & Fittings

## Description

The Pipe (IL) block models flow in a rigid or flexible-walled pipe with losses due to wall friction. The effects of dynamic compressibility, fluid inertia, and pipe elevation can be optionally modeled. You can define multiple pipe segments and set the liquid pressure for each segment. By segmenting the pipe and setting Fluid inertia to `On`, you can model events such as water hammer in your system.

### Pipe Characteristics

The pipe block can be divided into segments with the Number of segments parameter. When the pipe is composed of a number of segments, the pressure in each segment is calculated based on the inlet pressure and the effect on the segment mass flow rate of the fluid compressibility and wall flexibility, if applicable. The fluid volume in each segment remains fixed. For a two-segment pipe, the pressure evolves linearly with respect to the pressure defined at ports A and B. For a pipe with three or more segments, you can specify the fluid pressure in each segment in vector or scalar form in the Initial liquid pressure parameter. The scalar form will apply a constant value over all segments.

Flexible Walls

You can model flexible walls for all cross-sectional geometries. When you set Pipe wall specification to `Flexible`, the block assumes uniform expansion along all directions and preserves the defined cross-sectional shape. This may not result in physical results for noncircular cross-sectional areas undergoing high pressure relative to atmospheric pressure. Two options are available for modeling the volumetric expansion of a cross-sectional area:

1. `Cross-sectional area vs. pressure`, where the change in volume is modeled by:

`$\stackrel{˙}{V}=L\left(\frac{{S}_{N}+{K}_{ps}\left(p-{p}_{atm}\right)-S}{\tau }\right),$`

where:

• L is the Pipe length.

• Snom is the nominal pipe cross-sectional area defined for each shape.

• S is the current pipe cross-sectional area.

• p is the internal pipe pressure.

• patm is the atmospheric pressure.

• Kps is the Static pressure-cross sectional area gain.

Assuming uniform elastic deformation of a thin-walled, open-ended cylindrical pipe, you can calculate Kps as:

`${K}_{ps}=\frac{\Delta D}{\Delta p}=\frac{\pi {D}_{N}^{3}}{4tE},$`

where t is the pipe wall thickness and E is Young's modulus.

• τ is the Volumetric expansion time constant.

2. `Hydraulic diameter vs. pressure`, where the change in volume is modeled by:

`$\stackrel{˙}{V}=\frac{\pi }{2}DL\left(\frac{{D}_{N}+{K}_{pd}\left(p-{p}_{atm}\right)-D}{\tau }\right),$`

where:

• DN is the nominal hydraulic diameter defined for each shape.

• D is the current pipe hydraulic diameter.

• Kpd is the Static pressure-hydraulic diameter gain. Assuming uniform elastic deformation of a thin-walled, open-ended cylindrical pipe, you can calculate Kpd as:

`${K}_{pd}=\frac{\Delta D}{\Delta p}=\frac{{D}_{N}^{2}}{2tE}.$`

When flexible walls are not modeled, SN = S and DN = D.

Circular

The nominal hydraulic diameter and the Pipe diameter, dcircle, are the same. The pipe cross sectional area is: ${S}_{N}=\frac{\pi }{4}{d}_{circle}^{2}.$

Annular

The nominal hydraulic diameter, Dh,nom, is the difference between the Pipe outer diameter and Pipe inner diameter, dodi. The pipe cross sectional area is ${S}_{N}=\frac{\pi }{4}\left({d}_{{}_{o}}^{2}-{d}_{{}_{i}}^{2}\right).$

Rectangular

The nominal hydraulic diameter is:

`${D}_{N}=\frac{2hw}{h+w},$`

where:

• h is the Pipe height.

• w is the Pipe width.

The pipe cross sectional area is ${S}_{N}=wh.$

Elliptical

The nominal hydraulic diameter is:

`${D}_{N}=2{a}_{maj}{b}_{\mathrm{min}}\frac{\left(64-16{\left(\frac{{a}_{maj}-{b}_{\mathrm{min}}}{{a}_{maj}+{b}_{\mathrm{min}}}\right)}^{2}\right)}{\left({a}_{maj}+{b}_{\mathrm{min}}\right)\left(64-3{\left(\frac{{a}_{maj}-{b}_{\mathrm{min}}}{{a}_{maj}+{b}_{\mathrm{min}}}\right)}^{4}\right)},$`

where:

• amaj is the Pipe major axis.

• bmin is the Pipe minor axis.

The pipe cross sectional area is ${S}_{N}=\frac{\pi }{4}{a}_{maj}{b}_{\mathrm{min}}.$

Isosceles Triangular

The nominal hydraulic diameter is:

`${D}_{N}={l}_{side}\frac{\mathrm{sin}\left(\theta \right)}{1+\mathrm{sin}\left(\frac{\theta }{2}\right)}$`

where:

• lside is the Pipe side length.

• θ is the Pipe vertex angle.

The pipe cross sectional area is ${S}_{N}=\frac{{l}_{side}^{2}}{2}\mathrm{sin}\left(\theta \right).$

### Pressure Loss Due to Friction

Haaland Correlation

The analytical Haaland correlation models losses due to wall friction either by aggregate equivalent length, which accounts for resistances due to nonuniformities as an added straight-pipe length that results in equivalent losses, or by local loss coefficient, which directly applies a loss coefficient for pipe nonuniformities.

When the Local resistances specification parameter is set to `Aggregate equivalent length` and the flow in the pipe is lower than the Laminar flow upper Reynolds number limit, the pressure loss over all pipe segments is:

`$\Delta {p}_{f,A}=\frac{\upsilon \lambda }{2{D}^{2}S}\frac{L+{L}_{add}}{2}{\stackrel{˙}{m}}_{A},$`

`$\Delta {p}_{f,B}=\frac{\upsilon \lambda }{2{D}^{2}S}\frac{L+{L}_{add}}{2}{\stackrel{˙}{m}}_{B},$`

where:

• ν is the fluid kinematic viscosity.

• λ is the , which you can define when Cross-sectional geometry is set to `Custom` and is otherwise equal to 64.

• D is the pipe hydraulic diameter.

• Ladd is the Aggregate equivalent length of local resistances.

• $\stackrel{˙}{m}$A is the mass flow rate at port A.

• $\stackrel{˙}{m}$B is the mass flow rate at port B.

When the Reynolds number is greater than the Turbulent flow lower Reynolds number limit, the pressure loss in the pipe is:

`$\Delta {p}_{f,A}=\frac{f}{2{\rho }_{I}{S}^{2}}\frac{L+{L}_{add}}{2}{\stackrel{˙}{m}}_{A}|{\stackrel{˙}{m}}_{A}|,$`

`$\Delta {p}_{f,B}=\frac{f}{2{\rho }_{I}{S}^{2}}\frac{L+{L}_{add}}{2}{\stackrel{˙}{m}}_{B}|{\stackrel{˙}{m}}_{B}|,$`

where:

• f is the Darcy friction factor. This is approximated by the empirical Haaland equation and is based on the Surface roughness specification, ε, and pipe hydraulic diameter:

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

Pipe roughness for brass, lead, copper, plastic, steel, wrought iron, and galvanized steel or iron are provided as ASHRAE standard values. You can also supply your own Internal surface absolute roughness with the `Custom` setting.

• ρI is the internal fluid density.

When the Local resistances specification parameter is set to `Local loss coefficient` and the flow in the pipe is lower than the Laminar flow upper Reynolds number limit, the pressure loss over all pipe segments is:

`$\Delta {p}_{f,A}=\frac{\upsilon \lambda }{2{D}^{2}S}\frac{L}{2}{\stackrel{˙}{m}}_{A}.$`

`$\Delta {p}_{f,B}=\frac{\upsilon \lambda }{2{D}^{2}S}\frac{L}{2}{\stackrel{˙}{m}}_{B}.$`

When the Reynolds number is greater than the Turbulent flow lower Reynolds number limit, the pressure loss in the pipe is:

`$\Delta {p}_{f,A}=\left(\frac{f\frac{L}{2}}{D}+{C}_{loss,total}\right)\frac{1}{2{\rho }_{I}{S}^{2}}{\stackrel{˙}{m}}_{A}|{\stackrel{˙}{m}}_{A}|,$`

`$\Delta {p}_{f,B}=\left(\frac{f\frac{L}{2}}{D}+{C}_{loss,total}\right)\frac{1}{2{\rho }_{I}{S}^{2}}{\stackrel{˙}{m}}_{B}|{\stackrel{˙}{m}}_{B}|,$`

where Closs,total is the loss coefficient, which can be defined in the Total local loss coefficient parameter as either a single coefficient or the sum of all loss coefficients along the pipe.

Nominal Pressure Drop vs. Nominal Mass Flow Rate

The Nominal Pressure Drop vs. Nominal Mass Flow Rate parameterization characterizes losses with a loss coefficient for rigid or flexible walls. When the fluid is incompressible, the pressure loss over the entire pipe due to wall friction is:

`$\Delta {p}_{f,A}={K}_{p}{\stackrel{˙}{m}}_{A}\sqrt{{\stackrel{˙}{m}}_{A}^{2}+{\stackrel{˙}{m}}_{th}^{2}},$`

where Kp is:

`${K}_{p}=\frac{\Delta {p}_{N}}{{\stackrel{˙}{m}}_{N}^{2}},$`

where:

• ΔpN is the Nominal pressure drop, which can be defined either as a scalar or a vector.

• ${\stackrel{˙}{m}}_{N}$ is the Nominal mass flow rate, which can be defined either as a scalar or a vector.

When the Nominal pressure drop and Nominal mass flow rate parameters are supplied as vectors, the scalar value Kp is determined from a least-squares fit of the vector elements.

Tabulated Data – Darcy Friction Factor vs. Reynolds Number

Pressure losses due to viscous friction can also be determined from user-provided tabulated data of the Darcy friction factor vector and the Reynolds number vector for turbulent Darcy friction factor parameters. Linear interpolation is employed between data points.

### Momentum Balance

For an incompressible fluid, the mass flow into the pipe equals the mass flow out of the pipe:

`${\stackrel{˙}{m}}_{A}+{\stackrel{˙}{m}}_{B}=0.$`

When the fluid is compressible and pipe walls are rigid, the difference between the mass flow into and out of the pipe depends on the fluid density change due to compressibility:

`${\stackrel{˙}{m}}_{A}+{\stackrel{˙}{m}}_{B}={\stackrel{˙}{p}}_{I}\frac{d{\rho }_{I}}{d{p}_{I}}V,$`

When the fluid is compressible and the pipe walls are flexible, the difference between the mass flow into and out of the pipe is based on the change in fluid density due to compressibility, and the amount of fluid accumulated in the newly deformed regions of the pipe:

`${\stackrel{˙}{m}}_{A}+{\stackrel{˙}{m}}_{B}={\stackrel{˙}{p}}_{I}\frac{d{\rho }_{I}}{d{p}_{I}}V+{\rho }_{I}\stackrel{˙}{V}.$`

The changes in momentum between the pipe inlet and outlet comprises the changes in pressure due to pipe wall friction, which is modeled according to the Viscous friction parameterization and pipe elevation. For a pipe that does not model fluid inertia, the momentum balance is:

`${p}_{A}-{p}_{I}=\Delta {p}_{f,A}+{\rho }_{I}\frac{\Delta z}{2}g,$`

`${p}_{B}-{p}_{I}=\Delta {p}_{f,B}-{\rho }_{I}\frac{\Delta z}{2}g,$`

where:

• pA is the pressure at port A.

• pI is the fluid volume internal pressure.

• pB is the pressure at port B.

• Δpf is the pressure loss due to wall friction, parameterized by the Viscous friction losses specification according to the respective port.

• Δz is the pipe elevation. In the case of constant-elevation pipes, this is the Elevation gain from port A to port B parameter; otherwise, it is received as a physical signal at port EL.

• g is the gravitational acceleration. In the case of a fixed gravitational constant, this is the Gravitational acceleration parameter; otherwise, it is received as a physical signal at port G.

For a pipe with modeled fluid inertia, the momentum balance is:

`${p}_{A}-{p}_{I}=\Delta {p}_{f,A}+{p}_{I}\frac{\Delta z}{2}g+{\stackrel{¨}{m}}_{A}\frac{L}{2S},$`

`${p}_{B}-{p}_{I}=\Delta {p}_{f,B}-{p}_{I}\frac{\Delta z}{2}g+{\stackrel{¨}{m}}_{B}\frac{L}{2S},$`

where:

• $\stackrel{¨}{m}$ is the fluid acceleration at its respective port.

• S is the pipe cross-sectional area.

## Ports

### Conserving

expand all

Liquid entry or exit port.

Liquid entry or exit port.

### Inputs

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Variable elevation from port A to port B, specified as a physical signal. The elevation magnitude must be less than or equal to the pipe length. If the signal falls below the value of –Pipe length, the value at EL is maintained at ```–pipe length```. If the signal exceeds the value of Pipe length, the value at EL is maintained at `pipe length`.

#### Dependencies

To enable this port, set Elevation gain specification to `Variable`.

Variable gravitational acceleration, specified as a physical signal.

#### Dependencies

To enable this port, set Gravitational acceleration specification to `Variable`.

## Parameters

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### Configuration

Total pipe length across all pipe segments.

Number of pipe divisions. Each division represents an individual segment for which pressure is calculated, depending on the pipe inlet pressure, fluid compressibility, and wall flexibility, if applicable. The fluid volume in each segment remains fixed.

Cross-sectional pipe geometry. A nominal hydraulic diameter and nominal cross-sectional area is calculated based on the cross-sectional geometry.

Diameter for circular cross-sectional pipes.

#### Dependencies

To enable this parameter, set Cross-sectional geometry to `Circular`.

Inner diameter for annular pipe flow, or flow between two concentric pipes.

#### Dependencies

To enable this parameter, set Cross-sectional geometry to `Annular`.

Outer diameter for annular pipe flow, or flow between two concentric pipes.

#### Dependencies

To enable this parameter, set Cross-sectional geometry to `Annular`.

Width of rectangular pipe.

#### Dependencies

To enable this parameter, set Cross-sectional geometry to `Rectangular`.

Height of rectangular pipe.

#### Dependencies

To enable this parameter, set Cross-sectional geometry to `Rectangular`.

Major axis for ellipsoidal pipes.

#### Dependencies

To enable this parameter, set Cross-sectional geometry to `Elliptical`.

Minor axis for ellipsoidal pipes.

#### Dependencies

To enable this parameter, set Cross-sectional geometry to `Elliptical`.

Length of the two equal sides of isosceles-triangular pipes.

#### Dependencies

To enable this parameter, set Cross-sectional geometry to ```Isosceles triangular```.

Vertex angle for triangular pipes. The value must be less than 180 degrees.

#### Dependencies

To enable this parameter, set Cross-sectional geometry to ```Isosceles triangular```.

Hydraulic diameter used in calculations of the pipe Reynolds number. For noncircular pipes, the hydraulic diameter is the effective diameter of the fluid in the pipe. For circular pipes, the hydraulic diameter and pipe diameter are the same.

#### Dependencies

To enable this parameter, set Cross-sectional geometry to `Custom`.

Pipe cross-sectional area for a custom pipe geometry.

#### Dependencies

To enable this parameter, set Cross-sectional geometry to `Custom`.

Whether to model any change in fluid density due to fluid compressibility. When Fluid compressibility is set to `On`, changes due to the mass flow rate into the block are calculated in addition to density changes due to changes in pressure. In the Isothermal Liquid Library, all blocks calculate density as a function of pressure.

Whether to account for resistance to changes in the flow rate due to the fluid mass.

#### Dependencies

To enable this parameter, set Fluid dynamic compressibility to `On`.

Whether the pipe elevation remains constant or variable from port A to B.

Elevation differential for constant-elevation pipes. The elevation gain must be less than or equal to the Pipe length.

#### Dependencies

To enable this parameter, set Elevation gain specification to `Constant`.

Whether the gravitational constant is constant or variable.

Gravitational acceleration for environments with constant gravitational acceleration.

#### Dependencies

To enable this parameter, set Gravitational acceleration specification to `Constant`.

### Viscous Friction

Parameterization of pressure losses due to wall friction. Both analytical and tabular formulations are available.

Method for quantifying pressure losses due to pipe nonuniformities.

#### Dependencies

To enable this parameter, set Viscous friction parameterization to ```Haaland correlation```.

Loss coefficient associated with each pipe nonuniformity. You can input a single loss coefficient or the sum of all loss coefficients along the pipe.

#### Dependencies

To enable this parameter, set Viscous friction parameterization to ```Haaland correlation``` and Local resistance specifications to ```Local loss coefficient```.

Length of pipe that would produce the equivalent hydraulic losses as would a pipe with bends, area changes, or other nonuniform attributes. The effective length of the pipe is the sum of the Pipe length and the Aggregate equivalent length of local resistances.

#### Dependencies

To enable this parameter, set Viscous friction parameterization to ```Haaland correlation``` and Local resistance specifications to ```Aggregate equivalent length```.

Absolute surface roughness based on pipe material. The provided values are ASHRAE standard roughness values. You can also input your own value by setting Surface roughness specification to `Custom`.

#### Dependencies

To enable this parameter, set Viscous friction parameterization to ```Haaland correlation```.

Pipe wall absolute roughness. This parameter is used to determine the Darcy friction factor, which contributes to pressure loss in the pipe.

#### Dependencies

To enable this parameter, set Viscous friction parameterization to ```Haaland correlation``` and Surface roughness specification to `Custom`.

Upper Reynolds number limit to laminar flow. Beyond this number, the fluid regime is transitional, approaches the turbulent regime, and becomes fully turbulent at the Turbulent flow lower Reynolds number limit.

#### Dependencies

To enable this parameter, set Viscous friction parameterization to either:

• `Haaland correlation`

• ```Tabulated data - Darcy friction factor vs. Reynolds number```

Lower Reynolds number limit for turbulent flow. Below this number, the flow regime is transitional, approaches laminar flow, and becomes fully laminar at the Laminar flow upper Reynolds number limit.

#### Dependencies

To enable this parameter, set Viscous friction parameterization to either:

• `Haaland correlation`

• ```Tabulated data - Darcy friction factor vs. Reynolds number```

Nominal mass flow rate used for calculating the pressure loss coefficient for rigid and flexible pipes, specified as a scalar or a vector. All nominal values must be greater than 0 and have the same number of elements as the Nominal pressure drop parameter. When this parameter is supplied as a vector, the scalar value Kloss is determined as a least-squares fit of the vector elements.

#### Dependencies

To enable this parameter, set Viscous friction parameterization to ```Nominal pressure drop vs. nominal mass flow rate```.

Nominal pressure drop used for calculating the pressure loss coefficient for rigid and flexible pipes, specified as a scalar or vector. All nominal values must be greater than 0 and must have the same number of elements as the Nominal mass flow rate parameter. When this parameter is supplied as a vector, the scalar value Kloss is determined as a least-squares fit of the vector elements.

#### Dependencies

To enable this parameter, set Viscous friction parameterization to ```Nominal pressure drop vs. nominal mass flow rate```.

Mass flow rate threshold for reversed flow. A transition region is defined around 0 kg/s between the positive and negative values of the mass flow rate threshold. Within this transition region, numerical smoothing is applied to the flow response. The threshold value must be greater than 0.

#### Dependencies

To enable this parameter, set Viscous friction parameterization to ```Nominal pressure drop vs. nominal mass flow rate```.

Vector of Reynolds numbers for the tabular parameterization of the Darcy friction factor. The vector elements form an independent axis with the Darcy friction factor vector parameter. The vector elements must be listed in ascending order. A positive Reynolds number corresponds to flow from port A to port B. A negative Reynolds number corresponds to flow from port B to A. If you provide a vector with only positive or only negative elements, the same data set is extended for flows in the opposite direction.

#### Dependencies

To enable this parameter, set Viscous friction parameterization to ```Tabulated data - Darcy friction factor vs. Reynolds number```.

Vector of Darcy friction factors for the tabular parameterization of the Darcy friction factor. The vector elements must correspond one-to-one with the elements in the Reynolds number vector for turbulent Darcy friction factor parameter, and must be unique and greater than or equal to 0.

#### Dependencies

To enable this parameter, set Viscous friction parameterization to ```Tabulated data - Darcy friction factor vs. Reynolds number```.

Friction constant for laminar flows. The Darcy friction factor captures the contribution of wall friction in pressure loss calculations. If Cross-sectional geometry is not set to `Custom`, this parameter is internally set to 64.

#### Dependencies

To enable this parameter, set Viscous friction parameterization to either:

• `Haaland correlation`

• ```Tabulated data - Darcy friction factor vs. Reynolds number```

and Cross-sectional geometry to `Custom`.

### Pipe Wall

Specifies wall flexibility. This parameter is independent of pipe cross-sectional geometry. The `Flexible` setting preserves the initial pipe shape and applies equal expansion of the cross-sectional area. It may not be accurate for non-circular cross-sectional geometry under high deformation.

#### Dependencies

To enable this parameter, set Fluid dynamic compressibility to `On`.

Linear expansion correlation. The settings correlate the new cross-sectional area or hydraulic diameter to the pipe pressure.

#### Dependencies

To enable this parameter, set Fluid dynamic compressibility to `On` and Pipe wall specification to `Flexible`.

Coefficient for calculating pipe deformation for the `Cross-sectional area vs. pressure` setting. The gain is multiplied by the pressure differential between the segment pressure and atmospheric pressure.

#### Dependencies

To enable this parameter, set Fluid dynamic compressibility to `On`, Pipe wall specification to `Flexible`, and Volumetric expansion specification to ```Cross-sectional area vs. pressure```.

Coefficient for calculating pipe deformation for the `Hydraulic diameter vs. pressure` setting. The gain is multiplied by the pressure differential between the segment pressure and atmospheric pressure.

#### Dependencies

To enable this parameter, set Fluid dynamic compressibility to `On`, Pipe wall specification to `Flexible`, and Volumetric expansion specification to ```Hydraulic diameter vs. pressure```.

Time required for the wall to reach steady-state after pipe deformation. This parameter impacts the dynamic change in pipe volume.

#### Dependencies

To enable this parameter, set Fluid dynamic compressibility to `On` and Pipe wall specification to `Flexible`.

### Initial Conditions

Initial liquid pressure, specified as a scalar or vector. A vector n elements long defines the liquid pressure for each of n pipe segments. If the vector is two elements long, the pressure along the pipe is linearly distributed between the two element values. If the vector is three or more elements long, the initial pressure in the nth segment is set by the nth element of the vector.

#### Dependencies

To enable this parameter, set Fluid dynamic compressibility to `On`.

Initial mass flow rate for pipes with simulated fluid inertia.

#### Dependencies

To enable this parameter, set Fluid dynamic compressibility to `On` and Fluid inertia to `On`.