Hydraulic pipeline with resistive, fluid inertia, fluid compressibility, and elevation properties

Low-Pressure Blocks

The Segmented Pipe LP block models hydraulic pipelines with circular cross sections. Hydraulic pipelines, which are inherently distributed parameter elements, are represented with sets of identical, connected in series, lumped parameter segments. It is assumed that the larger the number of segments, the closer the lumped parameter model becomes to its distributed parameter counterpart. The equivalent circuit of a pipeline adopted in the block is shown below, along with the segment configuration.

**Pipeline Equivalent Circuit**

**Segment Configuration**

The model contains as many Constant Volume Hydraulic Chamber blocks as there are segments. The chamber lumps fluid volume equal to

$$V=\frac{\pi \xb7{d}^{2}}{4}\frac{L}{N}$$

where

`V` | Fluid volume |

`d` | Pipe diameter |

`L` | Pipe length |

`N` | Number of segments |

The Constant Volume Hydraulic Chamber block is
placed between two branches, each consisting of a Hydraulic
Resistive Tube block and a Fluid Inertia block.
Every Hydraulic Resistive Tube block lumps `(`

-th
portion of the pipe length, while Fluid Inertia block
has * L*+

`L_ad`

`N`

`L`

/(`N`

+1)

length
(`L_ad`

The nodes to which Constant Volume Hydraulic Chamber blocks
are connected are assigned names `N_1`

, `N_2`

,
…, `N_`

(`n`

* n* is
the number of segments). Pressures at these nodes are assumed to be
equal to average pressure of the segment. Intermediate nodes between Hydraulic
Resistive Tube and Fluid Inertia blocks are
assigned names

`nn_0`

, `nn_1`

, `nn_2`

,
…, `nn_``n`

. The Constant
Volume Hydraulic Chamber blocks are named `ch_1`

, `ch_2`

,
…, `ch_``n`

, Hydraulic
Resistive Tube blocks are named `tb_0`

, `tb_1`

, `tb_2`

,
…, `tb_``n`

, and Fluid
Inertia blocks are named `fl_in_0`

, `fl_in_1`

, `fl_in_2`

,
…, `fl_in_``n`

.The number of segments is the block parameter. In determining the number of segments needed, you have to find a compromise between the accuracy and computational burden for a particular application. It is practically impossible to determine analytically how many elements are necessary to get the results with a specified accuracy. The golden rule is to use as many elements as possible based on computational considerations, and an experimental assessment is perhaps the only reliable way to make any conclusions. As an approximate estimate, you can use the following formula:

$$N>\frac{4L}{\pi \xb7c}\omega $$

where

`N` | Number of segments |

`L` | Pipe length |

`c` | Speed of sound in the fluid |

ω | Maximum frequency to be observed in the pipe response |

The table below contains an example of simulation of a pipeline where the first four true eigenfrequencies are 89.1 Hz, 267 Hz, 446 Hz, and 624 Hz.

Number of Segments | 1st Mode | 2nd Mode | 3rd Mode | 4th Mode |
---|---|---|---|---|

1 | 112.3 | – | – | – |

2 | 107.2 | 271.8 | – | – |

4 | 97.7 | 284.4 | 432.9 | 689 |

8 | 93.2 | 271.9 | 435.5 | 628 |

As you can see, the error is less than 5% if an eight-segmented version is used.

The difference in elevation between ports A and B is distributed evenly between pipe segments.

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.

**Pipe internal diameter**Internal diameter of the pipe. The default value is

`0.01`

m.**Pipe length**Pipe geometrical length. The default value is

`5`

m.**Number of segments**Number of lumped parameter segments in the pipeline model. The default value is

`1`

.**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. This length is added to the geometrical pipe length only for hydraulic resistance computation. Both the fluid volume and fluid inertia are determined based on pipe geometrical length only. 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`

.

**Pipe wall type**The parameter can have one of two values:

`Rigid`

or`Flexible`

. If the parameter is set to`Rigid`

, wall compliance is not taken into account, which can improve computational efficiency. The value`Flexible`

is recommended for hoses and metal pipes where wall compliance can affect the system behavior. The default value is`Rigid`

.**Static pressure-diameter coefficient**Coefficient that establishes relationship between the pressure and the internal diameter at steady-state conditions. This coefficient can be determined analytically for cylindrical metal pipes or experimentally for hoses. The parameter is used if the

**Pipe wall type**parameter is set to`Flexible`

. The default value is`2e-12`

m/Pa.**Viscoelastic process time constant**Time constant in the transfer function that relates pipe internal diameter to pressure variations. By using this parameter, the simulated elastic or viscoelastic process is approximated with the first-order lag. The value is determined experimentally or provided by the manufacturer. The parameter is used if the

**Pipe wall type**parameter is set to`Flexible`

. The default value is`0.01`

s.**Specific heat ratio**Gas-specific heat ratio for the Constant Volume Hydraulic Chamber block. The default value is

`1.4`

.

**Port A elevation wrt reference plane**The parameter specifies vertical position of the pipe port A with respect to the reference plane. The default value is

`0`

.**Port B elevation wrt reference plane**The parameter specifies vertical position of the pipe port B with respect to the reference plane. The default value is

`0`

.

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.

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

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