## Modeling Low-Pressure Fluid Transportation Systems

### How Fluid Transportation Systems Differ from Power and Control Systems

In hydraulics, the steady uniform flow in a component with one entrance and one exit is characterized by the following energy equation

$$-\frac{{\dot{W}}_{s}}{\dot{m}g}=\frac{{V}_{2}^{2}-{V}_{1}^{2}}{2g}+\frac{{p}_{2}-{p}_{1}}{\rho g}+{z}_{2}-{z}_{1}+{h}_{L}$$ | (1) |

where

$${\dot{W}}_{s}$$ | Work rate performed by fluid |

$$\dot{m}$$ | Mass flow rate |

V_{2} | Fluid velocity at the exit |

V_{1} | Fluid velocity at the entrance |

p,
_{1}p_{2} | Static pressure at the entrance and the exit, respectively |

g | Gravity acceleration |

ρ | Fluid density |

z,
_{1}z_{2} | Elevation above a reference plane (datum) at the entrance and the exit, respectively |

h_{L} | Hydraulic loss |

Subscripts 1 and 2 refer to the entrance and exit, respectively. All the terms in Equation 1 have dimensions of height and are named kinematic head, piezometric head, geometric head, and loss head, respectively. For a variety of reasons, analysis of hydraulic power and control systems is performed with respect to pressures, rather than to heads, and Equation 1 for a typical passive component is presented in the form

$$\frac{\rho}{2}{V}_{1}^{2}+{p}_{1}+\rho g{z}_{1}=\frac{\rho}{2}{V}_{2}^{2}+{p}_{2}+\rho g{z}_{2}+{p}_{L}$$ | (2) |

where

V,
_{1}p,
_{1}z_{1} | Velocity, static pressure, and elevation at the entrance, respectively |

V,
_{2}p,
_{2}z_{2} | Velocity, static pressure, and elevation at the exit, respectively |

p_{L} | Pressure loss |

Term $$\frac{\rho}{2}{V}^{2}$$ is frequently referred to as kinematic, or dynamic, pressure, and $$\rho gz$$ as piezometric pressure. Dynamic pressure terms are usually neglected because they are very small, and Equation 2 takes the form

$${p}_{1}+\rho g{z}_{1}={p}_{2}+\rho g{z}_{2}+{p}_{L}$$ | (3) |

The size of a typical power and control system is usually small and rarely exceeds 1.5 – 2 m. To add to this, these systems operate at pressures in the range 50 – 300 bar. Therefore, $$\rho gz$$ terms are negligibly small compared to static pressures. As a result, Simscape™ Fluids™ components (with the exception of the ones designed specifically for low-pressure simulation, described in Available Blocks and How to Use Them) have been developed with respect to static pressures, with the following equations

$$\begin{array}{l}p={p}_{L}=f(q)\\ q=f({p}_{1},{p}_{2})\end{array}$$ | (4) |

where

p | Pressure difference between component ports |

q | Flow rate through the component |

Fluid transportation systems usually operate at low pressures (about 2-4 bar), and the difference in component elevation with respect to reference plane can be very large. Therefore, geometrical head becomes an essential part of the energy balance and must be accounted for. In other words, the low-pressure fluid transportation systems must be simulated with respect to piezometric pressures $${p}_{pz}=p+\rho gz$$, rather than static pressures. This requirement is reflected in the component equations

$$\begin{array}{l}p={p}_{L}=f(q,{z}_{1},{z}_{2})\\ q=f({p}_{1},{p}_{2},{z}_{1},{z}_{2})\end{array}$$ | (5) |

Equations in the form Equation 5 must be applied to describe a hydraulic component with significant difference between port elevations. In hydraulic systems, there is only one type of such components: hydraulic pipes. The models of pipes intended to be used in low pressure systems must account for difference in elevation of their ports. The dimensions of the rest of the components are too small to contribute noticeably to energy balance, and their models can be built with the constant elevation assumption, like all the other Simscape Fluids blocks. To sum it up:

You can build models of low-pressure systems with difference in elevations of their components using regular Simscape Fluids blocks, with the exception of the pipes. Use low-pressure pipes, described in Available Blocks and How to Use Them.

When modeling low-pressure systems, you must use low-pressure pipe blocks to connect all nodes with difference in elevation, because these are the only blocks that provide information about the vertical locations of the system parts. Nodes connected with any other blocks, such as valves, orifices, actuators, and so on, will be treated as if they have the same elevation.

### Available Blocks and How to Use Them

When modeling low-pressure hydraulic systems, use the pipe blocks from the Low-Pressure Blocks library instead of the regular pipe blocks. These blocks account for the port elevation above reference plane and differ in the extent of idealization, just like their high-pressure counterparts:

Resistive Pipe LP — Models hydraulic pipe with circular and noncircular cross sections and accounts for friction loss only, similar to the Resistive Tube block, available in the Simscape Foundation library.

Resistive Pipe LP with Variable Elevation — Models hydraulic pipe with circular and noncircular cross sections and accounts for friction losses and variable port elevations. Use this block for low-pressure system simulation in which the pipe ends change their positions with respect to the reference plane.

Hydraulic Pipe LP — Models hydraulic pipe with circular and noncircular cross sections and accounts for friction loss along the pipe length and for fluid compressibility, similar to the Hydraulic Pipeline block in the Pipelines library.

Hydraulic Pipe LP with Variable Elevation — Models hydraulic pipe with circular and noncircular cross sections and accounts for friction loss along the pipe length and for fluid compressibility, as well as variable port elevations. Use this block for low-pressure system simulation in which the pipe ends change their positions with respect to the reference plane.

Segmented Pipe LP — Models circular hydraulic pipe and accounts for friction loss, fluid compressibility, and fluid inertia, similar to the Segmented Pipe block in the Pipelines library.

Use these low-pressure pipe blocks to connect all Hydraulic nodes in your model with difference in elevation, because these are the only blocks that provide information about the vertical location of the ports. Nodes connected with any other blocks, such as valves, orifices, actuators, and so on, will be treated as if they have the same elevation.

The additional models of pressurized tanks available for low-pressure system simulation include:

Constant Head Tank — Represents a pressurized hydraulic reservoir, in which fluid is stored under a specified pressure. The size of the tank is assumed to be large enough to neglect the pressurization and fluid level change due to fluid volume. The block accounts for the fluid level elevation with respect to the tank bottom, as well as for pressure loss in the connecting pipe that can be caused by a filter, fittings, or some other local resistance. The loss is specified with the pressure loss coefficient. The block computes the volume of fluid in the tank and exports it outside through the physical signal port V.

Variable Head Tank — Represents a pressurized hydraulic reservoir, in which fluid is stored under a specified pressure. The pressurization remains constant regardless of volume change. The block accounts for the fluid level change caused by the volume variation, as well as for pressure loss in the connecting pipe that can be caused by a filter, fittings, or some other local resistance. The loss is specified with the pressure loss coefficient. The block computes the volume of fluid in the tank and exports it outside through the physical signal port V.

Variable Head Two-Arm Tank — Represents a two-arm pressurized tank, in which fluid is stored under a specified pressure. The pressurization remains constant regardless of volume change. The block accounts for the fluid level change caused by the volume variation, as well as for pressure loss in the connecting pipes that can be caused by a filter, fittings, or some other local resistance. The loss is specified with the pressure loss coefficient at each outlet. The block computes the volume of fluid in the tank and exports it outside through the physical signal port V.

Variable Head Three-Arm Tank — Represents a three-arm pressurized tank, in which fluid is stored under a specified pressure. The pressurization remains constant regardless of volume change. The block accounts for the fluid level change caused by the volume variation, as well as for pressure loss in the connecting pipes that can be caused by a filter, fittings, or some other local resistance. The loss is specified with the pressure loss coefficient at each outlet. The block computes the volume of fluid in the tank and exports it outside through the physical signal port V.

### Low-Pressure Fluid Transportation System

The following illustration shows a simple system consisting of three tanks whose bottom surfaces are located at heights H1, H2, and H3, respectively, from the reference plane. The tanks are connected by pipes to a hydraulic manifold, which may contain any hydraulic elements, such as valves, orifices, pumps, accumulators, other pipes, and so on, but these elements have one feature in common – their elevations are all the same and equal to H4.

The models of tanks account for the fluid level heights F1, F2, and F3, respectively, and represent pressure at their bottoms as

$$\begin{array}{ll}{p}_{i}=\rho g{F}_{i}\hfill & \text{for}i=1,2,3\hfill \end{array}$$

The components inside the manifold can be simulated with regular Simscape Fluids blocks, like you would use for hydraulic power and control systems simulation. The pipes must be simulated with one of the low-pressure pipe models: Resistive Pipe LP, Hydraulic Pipe LP, or Segmented Pipe LP, depending on the required extent of idealization. Use the Constant Head Tank or Variable Head Tank blocks to simulate the tanks. For details of implementation, see the Water Supply System and the Three Constant Head Tanks examples.