# Resistive Pipe LP with Variable Elevation

Hydraulic pipeline which accounts for friction losses and variable port elevations

## Library

Low-Pressure Blocks

## Description

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:

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

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

where

 p Pressure loss along the pipe due to friction q Flow rate through the pipe Re Reynolds number ReL Maximum Reynolds number at laminar flow ReT Minimum Reynolds number at turbulent flow Ks Shape factor that characterizes the pipe cross section fL Friction factor at laminar border fT Friction factor at turbulent border A Pipe cross-sectional area DH Pipe hydraulic diameter L Pipe geometrical length Leq Aggregate equivalent length of local resistances r Height of the roughness on the pipe internal surface ν Fluid kinematic viscosity zA, zB 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 $p={p}_{A}-{p}_{B}$.

## Basic Assumptions and Limitations

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

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

## Parameters

### Basic Parameters Tab

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.

### Variables Tab

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.

## Global Parameters

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.

## Ports

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.

## References

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