Line Middle $—$ Line element between entrance and exit

The LineMiddle component contains the equations for an element between the entrance and the exit element. The connectors are used to connect this element to another middle element of long line.

 Equations $\left\{\begin{array}{cc}\left\{\left\{\begin{array}{cc}\mathrm{\beta }=\frac{1}{\frac{1}{{\mathrm{\beta }}_{0}}+\frac{1}{{K}_{\mathrm{pipe}}}}& \mathrm{elasticWall}\\ \mathrm{\beta }={\mathrm{\beta }}_{0}& \mathrm{otherwise}\end{array}\right\,\mathrm{\nu }={\mathrm{\nu }}_{0},\mathrm{\rho }={\mathrm{\rho }}_{0}\right\}& \mathrm{constantProperties}\\ \left\{\left\{\begin{array}{cc}\mathrm{\beta }=\frac{1}{\frac{1}{\mathrm{oil.bulkModulus}\left(p=\mathrm{pk}+\mathrm{oil.p_atm},T=T,{v}_{\mathrm{air}}=\mathrm{oil.v_gas},{p}_{\mathrm{sat}}={p}_{\mathrm{sat}}\right)}+\frac{1}{{K}_{\mathrm{pipe}}}}& \mathrm{elasticWall}\\ \mathrm{\beta }=\mathrm{oil.bulkModulus}\left(p=\mathrm{pk}+\mathrm{oil.p_atm},T=T\right)& \mathrm{otherwise}\end{array}\right\,\mathrm{\nu }=\mathrm{oil.kinematicViscosity}\left(p=\mathrm{pk}+\mathrm{oil.p_atm},T=T,{v}_{\mathrm{air}}=\mathrm{oil.v_gas},{p}_{\mathrm{sat}}={p}_{\mathrm{sat}}\right),\mathrm{\rho }=\mathrm{oil.density}\left(p=\mathrm{pk}+\mathrm{oil.p_atm},T=T,{v}_{\mathrm{air}}=\mathrm{oil.v_gas},{p}_{\mathrm{sat}}={p}_{\mathrm{sat}}\right)\right\}& \mathrm{otherwise}\end{array}\right\$ $\left\{\begin{array}{cc}\left\{\mathrm{dx}=\mathrm{noEvent}\left(\left\{\begin{array}{cc}0& {\mathrm{Re}}_{\mathrm{friction}}<\mathrm{Re1}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{or}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{Re2}<{\mathrm{Re}}_{\mathrm{friction}}\\ \frac{\mathrm{ln}\left(\frac{{\mathrm{Re}}_{\mathrm{friction}}}{\mathrm{Re1}}\right)}{\mathrm{ln}\left(10\right)}& \mathrm{otherwise}\end{array}\right\\right),\mathrm{\lambda }=\mathrm{noEvent}\left(\left\{\begin{array}{cc}1& {\mathrm{Re}}_{\mathrm{friction}}<64\\ \frac{{\mathrm{\lambda }}_{2}}{{\mathrm{Re}}_{\mathrm{friction}}^{2}}& \mathrm{otherwise}\end{array}\right\\right),{\mathrm{\lambda }}_{2}=\mathrm{noEvent}\left(\left\{\begin{array}{cc}64{\mathrm{Re}}_{\mathrm{friction}}& {\mathrm{Re}}_{\mathrm{friction}}\le \mathrm{Re1}\\ \left\{\begin{array}{cc}\frac{{\mathrm{Re}}_{\mathrm{friction}}^{2}{\mathrm{ln}\left(10\right)}^{2}}{4{\mathrm{ln}\left(\frac{10{\mathrm{\Delta }}_{\mathrm{rel}}}{37}+\frac{287}{50{\mathrm{Re}}_{\mathrm{friction}}^{9}{10}}}\right)}^{2}}& \mathrm{Re2}\le {\mathrm{Re}}_{\mathrm{friction}}\\ 64\mathrm{Re1}{\left(\frac{{\mathrm{Re}}_{\mathrm{friction}}}{\mathrm{Re1}}\right)}^{1+\mathrm{dx}\left({c}_{2}+\mathrm{dx}\mathrm{c3}\right)}& \mathrm{otherwise}\end{array}\right\& \mathrm{otherwise}\end{array}\right\\right)\right\}& \mathrm{frictionType}=\mathrm{Hydraulics.Lines.FrictionTypes.LaminarTurbulent}\\ \left\{\mathrm{dx}=0,\mathrm{\lambda }=1,{\mathrm{\lambda }}_{2}=64{\mathrm{Re}}_{\mathrm{friction}}\right\}& \mathrm{otherwise}\end{array}\right\$ $\left\{\begin{array}{cc}\left\{{\mathrm{Δp}}_{\mathrm{dynfriction}}=\left(-\frac{\left(\frac{128{k}_{1}}{{\mathrm{\tau }}_{1}}+\frac{128{k}_{2}}{{\mathrm{\tau }}_{2}}+\frac{128{k}_{3}}{{\mathrm{\tau }}_{3}}\right)\mathrm{\nu }{q}_{\mathrm{k+}}}{{d}^{4}\mathrm{\pi }}+\frac{\left(32{\mathrm{wk}}_{1}+32{\mathrm{wk}}_{2}+32{\mathrm{wk}}_{3}\right)\mathrm{\nu }}{{d}^{2}\mathrm{\rho }{a}_{\mathrm{sound}}}\right){\ell }_{\mathrm{element}}\mathrm{\rho },{\partial }_{t}\left({\mathrm{wk}}_{1}\right)=\frac{\mathrm{\alpha }{k}_{1}\mathrm{Zc}{q}_{\mathrm{k+}}}{{\mathrm{\tau }}_{1}^{2}}-\frac{\mathrm{\alpha }{\mathrm{wk}}_{1}}{{\mathrm{\tau }}_{1}},{\partial }_{t}\left({\mathrm{wk}}_{2}\right)=\frac{\mathrm{\alpha }{k}_{2}\mathrm{Zc}{q}_{\mathrm{k+}}}{{\mathrm{\tau }}_{2}^{2}}-\frac{\mathrm{\alpha }{\mathrm{wk}}_{2}}{{\mathrm{\tau }}_{2}},{\partial }_{t}\left({\mathrm{wk}}_{3}\right)=\frac{\mathrm{\alpha }{k}_{3}\mathrm{Zc}{q}_{\mathrm{k+}}}{{\mathrm{\tau }}_{3}^{2}}-\frac{\mathrm{\alpha }{\mathrm{wk}}_{3}}{{\mathrm{\tau }}_{3}}\right\}& \mathrm{dynFriction}=\mathrm{true}\\ \left\{{\mathrm{wk}}_{1}=0,{\mathrm{wk}}_{2}=0,{\mathrm{wk}}_{3}=0,{\mathrm{Δp}}_{\mathrm{dynfriction}}=0\right\}& \mathrm{otherwise}\end{array}\right\$ $\mathrm{\alpha }=\frac{32\mathrm{\nu }}{{d}^{2}}$ $T={T}_{0\left(\mathrm{oil}\right)}+{\mathrm{ΔT}}_{\mathrm{system}}$ $\mathrm{Zc}=\frac{4{a}_{\mathrm{sound}}\mathrm{\rho }}{{d}^{2}\mathrm{\pi }}$ $\mathrm{pk}={p}_{A}$ ${\Re }_{\mathrm{friction}}=\mathrm{noEvent}\left(\frac{4\left|{q}_{\mathrm{k+}}\right|}{\mathrm{π}d\mathrm{ν}}\right)$ ${a}_{\mathrm{sound}}=\sqrt{\frac{\mathrm{\beta }}{\mathrm{\rho }}}$ ${p}_{\mathrm{k+}}={p}_{B}$ ${q}_{\mathrm{k+}}=-\frac{{m}_{\mathrm{flow}\left(B\right)}}{\mathrm{\rho }}$ ${q}_{\mathrm{k−}}=\frac{{m}_{\mathrm{flow}\left(A\right)}}{\mathrm{\rho }}$ ${\mathrm{Δp}}_{\mathrm{friction}}=\mathrm{noEvent}\left(\frac{1}{2}\frac{{\ell }_{\mathrm{element}}{\mathrm{lambda}}_{2}{\mathrm{\nu }}^{2}\mathrm{\rho }\left\{\begin{array}{cc}1& 0\le {q}_{\mathrm{k+}}\\ -1& \mathrm{otherwise}\end{array}\right\}{{d}^{3}}\right)$ ${\mathrm{Δp}}_{\mathrm{statichead}}=\frac{981}{100}\mathrm{\rho }\mathrm{Δh}$ $\mathrm{\rho }{\ell }_{\mathrm{element}}{\partial }_{t}\left({q}_{\mathrm{k+}}\right)=\frac{1}{4}{d}^{2}\mathrm{\pi }\left(\mathrm{pk}-{p}_{\mathrm{k+}}-{\mathrm{Δp}}_{\mathrm{friction}}+{\mathrm{Δp}}_{\mathrm{dynfriction}}+{\mathrm{Δp}}_{\mathrm{statichead}}\right)$ ${\partial }_{t}\left(\mathrm{pk}\right)=\frac{{a}_{\mathrm{sound}}\mathrm{Zc}\left({q}_{\mathrm{k−}}-{q}_{\mathrm{k+}}\right)}{{\ell }_{\mathrm{element}}}$

Variables

 Name Value Units Description Modelica ID $\mathrm{\alpha }$ alpha $\mathrm{Zc}$ Zc ${\mathrm{wk}}_{1}$ wk1 ${\mathrm{wk}}_{2}$ wk2 ${\mathrm{wk}}_{3}$ wk3 $\mathrm{pk}$ $\mathrm{Pa}$ pk ${p}_{\mathrm{k+}}$ $\mathrm{Pa}$ pkplus ${q}_{\mathrm{k−}}$ $\frac{{m}^{3}}{s}$ qkminus ${q}_{\mathrm{k+}}$ $\frac{{m}^{3}}{s}$ qkplus ${p}_{\mathrm{sat}}$ [1] $\mathrm{Pa}$ Gas saturation pressure p_sat ${\Re }_{\mathrm{friction}}$ $1$ Dummy or Reynolds number of flow, if frictionType = DetailedFriction Re_friction $\mathrm{\lambda }$ Dummy or friction coefficient, if frictionType = DetailedFriction lambda ${\mathrm{\lambda }}_{2}$ Dummy or non-standard friction coefficient, if frictionType = DetailedFriction (= lambdaRe^2) lambda2 ${\mathrm{Δp}}_{\mathrm{friction}}$ Stationary friction loss dp_friction ${\mathrm{Δp}}_{\mathrm{dynfriction}}$ Dynamic friction term dp_dynfriction ${\mathrm{Δp}}_{\mathrm{statichead}}$ Static head dp_statichead ${a}_{\mathrm{sound}}$ $\frac{m}{s}$ Speed of sound in oil aSound $\mathrm{\nu }$ $\frac{{m}^{2}}{s}$ Viscosity, if constantProperties is false nu $\mathrm{\rho }$ $\frac{\mathrm{kg}}{{m}^{3}}$ Density  if constantProperties is false rho $\mathrm{\beta }$ $\mathrm{Pa}$ Bulk modulus if constantProperties is false beta $T$ $K$ Local temperature T

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

Connections

 Name Description Modelica ID $\mathrm{oil}$ oil ${\mathrm{port}}_{A}$ Port A, where oil flows into the component ($0, ${p}_{B}<{p}_{A}$ means $0<\mathrm{Δp}$) port_A ${\mathrm{port}}_{B}$ Port A, where oil leaves the component ($q<0$, ${p}_{B}<{p}_{A}$ means $0<\mathrm{Δp}$) port_B

Parameters

General Parameters

 Name Default Units Description Modelica ID steady-state init $\mathrm{false}$ If true, initialize in steady state steadyStateInit constant properties $\mathrm{true}$ If true, properties are treated as constant over time constantProperties elastic wall $\mathrm{false}$ If true, pipe properties affect speed of sound in oil elasticWall dynamic friction $\mathrm{false}$ If true, dynamic friction dynFriction ${\ell }_{\mathrm{element}}$ $m$ Length of one element ElementLength $d$ $m$ Line diameter diameter $\mathrm{Δh}$ $0$ $m$ Height difference heightDiff friction type [1] Type of flow model frictionType ${K}_{\mathrm{pipe}}$ Pipe bulk modulus of elasticity bulkModPipe roughness $0$ $m$ Roughness of pipe roughness ${\mathrm{\rho }}_{0}$ $\frac{\mathrm{kg}}{{m}^{3}}$ Constant density, if constantProperties is true rho0 ${\mathrm{\nu }}_{0}$ $\frac{{m}^{2}}{s}$ Constant viscosity, if constantProperties is true nu0 ${\mathrm{\beta }}_{0}$ $\mathrm{Pa}$ Bulk Modulus beta0 ${\mathrm{ΔT}}_{\mathrm{system}}$ $0$ $K$ Temperature offset from system temperature dT_system

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

Constant Parameters

 Name Default Units Description Modelica ID ${\mathrm{\Delta }}_{\mathrm{rel}}$ $\frac{\mathrm{roughness}}{d}$ Relative roughness Delta

Constants

 Name Value Units Description Modelica ID ${k}_{1}$ $0.192$ k1 ${k}_{2}$ $0.0948$ k2 ${k}_{3}$ $0.0407$ k3 ${\mathrm{\tau }}_{1}$ $0.25$ tau1 ${\mathrm{\tau }}_{2}$ $0.0352$ tau2 ${\mathrm{\tau }}_{3}$ $0.0024$ tau3