Flexible Line $—$ Flexible line with laminar and turbulent flow

The Flexible Line component describes the dynamics of a long line with compressibility, inductance and frequency dependent viscosity. This model uses n elements for the line, therefore:

${\ell }_{\mathrm{element}}=\frac{{\ell }_{\mathrm{line}}}{n}$

The oil in flexible lines is compressible, has a mass, and a resistance. The model includes the frequency dependent friction, compressibility, and inductance. The dynamic response of flexible lines is a function of time and the spatial coordinate (that is, described by partial differential equations). For the library, a lumped parameter model is used that breaks the whole line into $n$ short elements. Flexible Line uses an entrance and exit elements that have only half the length of the $n-1$ middle elements, leading to an effective element length of $\frac{1}{n}$ of the whole line length. The model is adequate if:

${\ell }_{\mathrm{element}}<\frac{1}{10}\frac{\mathrm{aSound}}{\mathrm{fmax}}$

Variables used in the above equations

 $\mathrm{length}$ length of line segment $\left[m\right]$ $\mathrm{aSound}$ speed of sound in fluid $\left[\frac{m}{s}\right]$ $\mathrm{fmax}$ highest frequency of interest $\left[\mathrm{Hz}\right]$

The steady state flow rate q is positive if oil enters the line at port_A; the pressure drop (port_A.p - port_B.p) is then positive.

The model is valid if the flow is laminar. There is no warning if this condition is not met. The Reynolds number is computed for the entrance and the exit elements (to reduce the necessary computations).

Related Components

 Losses of a rigid line as a function of Reynolds number

 Equations $\mathrm{msim/FOR}\left(\mathrm{msim/IN}\left(\mathrm{i#1},1..n-2\right),\mathrm{msim/CONNECT}\left(\mathrm{LMiddle\left[i#1\right].port_B},\mathrm{LMiddle\left[i#1 + 1\right].port_A}\right)\right)$

Variables

 Name Value Units Description Modelica ID ${p}_{\mathrm{sat}}$ [1] $\mathrm{Pa}$ Gas saturation pressure p_sat $T$ [2] $K$ Local temperature T $\mathrm{\rho }$ [3] $\frac{\mathrm{kg}}{{m}^{3}}$ Density of oil rho ${p}_{A\left(\mathrm{summary}\right)}$ ${p}_{A}$ $\mathrm{Pa}$ Pressure at port A summary_pA ${p}_{B\left(\mathrm{summary}\right)}$ ${p}_{B}$ $\mathrm{Pa}$ Pressure at port B summary_pB ${\mathrm{Δp}}_{\mathrm{summary}}$ ${p}_{A}-{p}_{B}$ $\mathrm{Pa}$ Pressure drop summary_dp ${q}_{\mathrm{summary}}$ $\frac{\mathrm{port_A.m_flow}}{\mathrm{\rho }}$ $\frac{{m}^{3}}{s}$ Flow rate flowing into port_A summary_q ${P}_{\mathrm{hyd}\left(\mathrm{summary}\right)}$ [4] $W$ Hydraulic Power summary_HP $\mathrm{LMiddle}$ LMiddle $\mathrm{LEntrance}$ LEntrance $\mathrm{LExit}$ LExit

[1] $\mathrm{oil.gasSaturationPressure}\left(T=T,{v}_{\mathrm{gas}}={\mathrm{oil.v}}_{\mathrm{gas}}\right)$

[2] $\mathrm{oil.T0}+{\mathrm{ΔT}}_{\mathrm{system}}$

[3] $\mathrm{Modelica.Media.Air.MoistAir.Utilities.spliceFunction}\left(x={\mathrm{Δp}}_{\mathrm{summary}},\mathrm{pos}=\mathrm{oil.density}\left(p={p}_{A}+\mathrm{oil.p_atm},T=T,{v}_{\mathrm{air}}=\mathrm{oil.v_gas},{p}_{\mathrm{sat}}={p}_{\mathrm{sat}}\right),\mathrm{neg}=\mathrm{oil.density}\left(p={p}_{B}+\mathrm{oil.p_atm},T=T,{v}_{\mathrm{air}}=\mathrm{oil.v_gas},{p}_{\mathrm{sat}}={p}_{\mathrm{sat}}\right),\mathrm{Δx}=100\right)$

[4] $-{\mathrm{Δp}}_{\mathrm{summary}}{q}_{\mathrm{summary}}$

Connections

 Name Description Modelica ID ${\mathrm{port}}_{A}$ Layout of port where oil flows into an element ($0<{m}_{\mathrm{flow}}$, ${p}_{B}<{p}_{A}$ means $0<\mathrm{Δp}$) port_A ${\mathrm{port}}_{B}$ Hydraulic port where oil leaves the component (${m}_{\mathrm{flow}}<0$, ${p}_{B}<{p}_{A}$ means $0<\mathrm{Δp}$) port_B $\mathrm{oil}$ oil

Parameters

General Parameters

 Name Default Units Description Modelica ID ${\mathrm{ΔT}}_{\mathrm{system}}$ $0$ $K$ Temperature offset from system temperature dT_system ${\ell }_{\mathrm{line}}$ $1$ $m$ Total length of line LineLength $d$ $0.01$ $m$ Line diameter diameter constant properties $\mathrm{true}$ If true, properties are treated as constant over time constantProperties elastic wall $\mathrm{true}$ If true, pipe properties affect speed of sound in oil elasticWall ${\mathrm{\beta }}_{0}$ [1] $\mathrm{Pa}$ Bulk modulus (for non-elastic line) beta0 ${\mathrm{\rho }}_{0}$ [2] $\frac{\mathrm{kg}}{{m}^{3}}$ Density of oil rho0 ${a}_{0}$ $\sqrt{\frac{{\mathrm{\beta }}_{0}}{{\mathrm{\rho }}_{0}}}$ $\frac{m}{s}$ Speed of sound in oil (for non-elastic line) a0 ${\mathrm{\nu }}_{0}$ [3] $\frac{{m}^{2}}{s}$ Kinematic viscosity nu0 $n$ $10$ Number of line segments n $\mathrm{Δh}$ $0$ $m$ Height difference, if positive then portA higher than portB heightDiff friction type [4] Type of flow model frictionType steady-state init $\mathrm{false}$ If true, initialize in steady state steadyStateInit dynamic friction $\mathrm{true}$ If true, dynamic friction dynFriction $\mathrm{\nu }$ $\frac{3}{10}$ Poisson ratio nu roughness $0$ $m$ Roughness of line material roughness $\mathrm{thickness}$ $0.003$ $m$ Line wall thickness thickness $E$ $2.·{10}^{11}$ $\mathrm{Pa}$ Modulus of elasticiy of the wall E

[1] $\mathrm{oil.bulkModulus}\left(\mathrm{oil.p0},\mathrm{oil.T0}+{\mathrm{ΔT}}_{\mathrm{system}}\right)$

[2] $\mathrm{oil.density}\left(\mathrm{oil.p0},\mathrm{oil.T0}+{\mathrm{ΔT}}_{\mathrm{system}}\right)$

[3] $\mathrm{oil.kinematicViscosity}\left(\mathrm{oil.p0},\mathrm{oil.T0}+{\mathrm{ΔT}}_{\mathrm{system}}\right)$

[4] $\mathrm{Hydraulics.Lines.FrictionTypes.Laminar}$

Constant Parameters

 Name Default Units Description Modelica ID $A$ [1] ${m}^{2}$ Line cross-section area A ${\ell }_{\mathrm{element}}$ $\frac{{\ell }_{\mathrm{line}}}{n}$ $m$ Length of one segment of line ElementLength ${K}_{\mathrm{pipe}}$ [2] bulkModPipe

[1] $\frac{1}{4}\mathrm{\pi }{d}^{2}$

[2] if elasticWall then $\frac{E\mathrm{thickness}\left(2d+2\mathrm{thickness}\right)}{\left(1+\mathrm{\nu }\right){\left(d+2\mathrm{thickness}\right)}^{2}+\left(1-\mathrm{\nu }\right){d}^{2}}$ else 10^15