Hydraulic pipeline which accounts for friction losses and variable port elevations

Low-Pressure Blocks

The Resistive Pipe LP with Variable Elevation block models hydraulic pipelines with circular and noncircular cross sections and accounts for resistive property only. Use this block for low-pressure system simulation in which the pipe ends change their positions with respect to the reference plane. The elevations are provided through respective physical signal inputs.

To reduce model complexity, you can use this block to simulate not only a pipe itself, but also a combination of pipes and local resistances such as bends, fittings, inlet and outlet losses, associated with the pipe. You must convert the resistances into their equivalent lengths, and then sum up all the resistances to obtain their aggregate length. Then add this length to the pipe geometrical length.

Pressure loss due to friction is computed with the Darcy equation, in which losses are proportional to the flow regime-dependable friction factor and the square of the flow rate. The friction factor in turbulent regime is determined with the Haaland approximation (see [1]). The friction factor during transition from laminar to turbulent regimes is determined with the linear interpolation between extreme points of the regimes. As a result of these assumptions, the tube is simulated according to the following equations:

$$p=f\frac{\left(L+{L}_{eq}\right)}{{D}_{H}}\frac{\rho}{2{A}^{2}}q\xb7\left|q\right|+\rho \xb7g\left({z}_{B}-{z}_{A}\right)$$

$$f=\{\begin{array}{ll}{K}_{s}/Re\hfill & \text{for}Re=R{e}_{L}\hfill \\ {f}_{L}+\frac{{f}_{T}-{f}_{L}}{R{e}_{T}-R{e}_{L}}\left(Re-R{e}_{L}\right)\hfill & \text{for}R{e}_{L}ReR{e}_{T}\hfill \\ \frac{1}{{\left(-1.8{\mathrm{log}}_{10}\left(\frac{6.9}{Re}+{\left(\frac{r/{D}_{H}}{3.7}\right)}^{1.11}\right)\right)}^{2}}\hfill & \text{for}Re=R{e}_{T}\hfill \end{array}$$

$$\mathrm{Re}=\frac{q\cdot {D}_{H}}{A\cdot \nu}$$

where

p | Pressure loss along the pipe due to friction |

q | Flow rate through the pipe |

Re | Reynolds number |

Re_{L} | Maximum Reynolds number at laminar flow |

Re_{T} | Minimum Reynolds number at turbulent flow |

K_{s} | Shape factor that characterizes the pipe cross section |

f_{L} | Friction factor at laminar border |

f_{T} | Friction factor at turbulent border |

A | Pipe cross-sectional area |

D_{H} | Pipe hydraulic diameter |

L | Pipe geometrical length |

L_{eq} | Aggregate equivalent length of local resistances |

r | Height of the roughness on the pipe internal surface |

ν | Fluid kinematic viscosity |

z_{A}, z_{B} | Elevations of the pipe port A and port B, respectively |

g | Gravity acceleration |

The block positive direction is from port A to port B. This means that the flow rate is positive if it flows from A to B, and the pressure loss is determined as $$p={p}_{A}-{p}_{B}$$.

Flow is assumed to be fully developed along the pipe length.

Fluid inertia, fluid compressibility, and wall compliance are not taken into account.

**Pipe cross section type**The type of pipe cross section:

`Circular`

or`Noncircular`

. For a circular pipe, you specify its internal diameter. For a noncircular pipe, you specify its hydraulic diameter and pipe cross-sectional area. The default value of the parameter is`Circular`

.**Internal diameter**Pipe internal diameter. The parameter is used if

**Pipe cross section type**is set to`Circular`

. The default value is`0.01`

m.**Noncircular pipe cross-sectional area**Pipe cross-sectional area. The parameter is used if

**Pipe cross section type**is set to`Noncircular`

. The default value is`1e-4`

m^2.**Noncircular pipe hydraulic diameter**Hydraulic diameter of the pipe cross section. The parameter is used if

**Pipe cross section type**is set to`Noncircular`

. The default value is`0.0112`

m.**Geometrical shape factor**Used for computing friction factor at laminar flow. The shape of the pipe cross section determines the value. For a pipe with a noncircular cross section, set the factor to an appropriate value, for example, 56 for a square, 96 for concentric annulus, 62 for rectangle (2:1), and so on [1]. The default value is

`64`

, which corresponds to a pipe with a circular cross section.**Pipe length**Pipe geometrical length. The default value is

`5`

m.**Aggregate equivalent length of local resistances**This parameter represents total equivalent length of all local resistances associated with the pipe. You can account for the pressure loss caused by local resistances, such as bends, fittings, armature, inlet/outlet losses, and so on, by adding to the pipe geometrical length an aggregate equivalent length of all the local resistances. The default value is

`1`

m.**Internal surface roughness height**Roughness height on the pipe internal surface. The parameter is typically provided in data sheets or manufacturer’s catalogs. The default value is

`1.5e-5`

m, which corresponds to drawn tubing.**Laminar flow upper margin**Specifies the Reynolds number at which the laminar flow regime is assumed to start converting into turbulent. Mathematically, this is the maximum Reynolds number at fully developed laminar flow. The default value is

`2000`

.**Turbulent flow lower margin**Specifies the Reynolds number at which the turbulent flow regime is assumed to be fully developed. Mathematically, this is the minimum Reynolds number at turbulent flow. The default value is

`4000`

.**Gravitational acceleration**Value of the gravitational acceleration constant (

*g*). The block uses this parameter to calculate time changes in pressure due to time changes in elevation. The default value is`9.80655`

m/s^2.

**Pressure drop from port A to port B**Value at time zero of the pressure drop between the ports. Simscape™ software uses this parameter to guide the initial configuration of this component and of the model. Initial variables that conflict with each other or that are incompatible with the model may be ignored. Set the

**Priority**column to`High`

to prioritize this variable over other, lower-priority, variables.**Flow rate from port A to port B**Value at time zero of the flow rate through the ports. Simscape software uses this parameter to guide the initial configuration of this component and of the model. Initial variables that conflict with each other or that are incompatible with the model may be ignored. Set the

**Priority**column to`High`

to prioritize this variable over other, lower-priority, variables.

Parameters determined by the type of working fluid:

**Fluid density****Fluid kinematic viscosity**

Use the Hydraulic Fluid block or the Custom Hydraulic Fluid block to specify the fluid properties.

The block has the following ports:

`A`

Hydraulic conserving port associated with the pipe inlet.

`B`

Hydraulic conserving port associated with the pipe outlet.

`el_A`

Physical signal input port that controls pipe elevation at port A.

`el_B`

Physical signal input port that controls pipe elevation at port B.

[1] White, F.M., *Viscous Fluid Flow*, McGraw-Hill,
1991

Hydraulic Pipe LP with Variable Elevation | Hydraulic Pipe LP | Hydraulic Pipeline | Hydraulic Resistive Tube | Linear Hydraulic Resistance | Resistive Pipe LP | Segmented Pipe LP | Segmented Pipeline