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Hydraulic pipeline which accounts for friction losses and variable port elevations
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.
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.
Pipe internal diameter. The parameter is used if Pipe cross section type is set to Circular. The default value is 0.01 m.
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.
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.
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 geometrical length. The default value is 5 m.
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.
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.
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.
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.
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.