Id Var Pump $—$ Ideal pump with variable displacement

This component describes an ideal pump with variable displacement.

Implementation

The flow rate (q) is given by:

q = Dpump $\cdot$ w $\cdot$ sig $\cdot$ factor / (2 $\cdot$ $\mathrm{\pi }$)

The commanded relative displacement volume of the pump is input at commandSignal, [-1 ... 1]. If the inlet pressure is lower than the atmospheric pressure, the delivered flow rate becomes smaller (that is, the value of factor decreases). Flow from port_A to port_B is considered positive.

Variables used in the above equation

 q Flow rate at the outlet port $\left[\frac{{m}^{3}}{s}\right]$. Dpump Maximum displacement per revolution $\left[{m}^{3}\right]$. w Angular velocity $\left[\frac{\mathrm{rad}}{s}\right]$. sig $\left\{\begin{array}{cc}-1& \mathrm{commandSignal}<-1\\ 1& \mathrm{commandSignal}>1\\ \mathrm{commandSignal}& \mathrm{otherwise}\end{array}\right\$ factor Pressure dependent variable. Equals 1 at normal running conditions and 0 at vapor pressure.

Assumptions

Ideal pump (that is, no losses).

 Equations $T={T}_{0\left(\mathrm{oil}\right)}+{\mathrm{ΔT}}_{\mathrm{system}}$ $\mathrm{factor}=\left\{\begin{array}{cc}0& 0<\mathrm{sig}w\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{and}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{port_A.p}<{p}_{\mathrm{vapour}\left(g\right)}\\ \left\{\begin{array}{cc}1-{\left(1+\frac{\mathrm{port_A.p}-{p}_{\mathrm{vapour}\left(g\right)}}{{p}_{\mathrm{vapour}\left(g\right)}}\right)}^{2}& \mathrm{port_A.p}<0<\mathrm{sig}w\\ \left\{\begin{array}{cc}0& \mathrm{sig}w<0\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{and}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{port_B.p}<{p}_{\mathrm{vapour}\left(g\right)}\\ \left\{\begin{array}{cc}1-{\left(1+\frac{\mathrm{port_B.p}-{p}_{\mathrm{vapour}\left(g\right)}}{{p}_{\mathrm{vapour}\left(g\right)}}\right)}^{2}& \mathrm{sig}w<0\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{and}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{port_B.p}<0\\ 1& \mathrm{otherwise}\end{array}\right\& \mathrm{otherwise}\end{array}\right\& \mathrm{otherwise}\end{array}\right\& \mathrm{otherwise}\end{array}\right\$ $q=\frac{{m}_{\mathrm{flow}\left(A\right)}}{{\mathrm{\rho }}_{\mathrm{oil}}\left({p}_{A\left(\mathrm{abs}\right)},T,{v}_{\mathrm{air}}={v}_{\mathrm{gas}\left(\mathrm{oil}\right)},{p}_{\mathrm{sat}}={p}_{\mathrm{sat}}\right)}$ $q=\frac{1}{2}\frac{{\mathrm{D}}_{\mathrm{pump}}w\mathrm{sig}\mathrm{factor}}{\mathrm{\pi }}$ $\mathrm{sig}=\left\{\begin{array}{cc}-1& \mathrm{commandSignal}<-1\\ \left\{\begin{array}{cc}1& 1<\mathrm{commandSignal}\\ \mathrm{commandSignal}& \mathrm{otherwise}\end{array}\right\& \mathrm{otherwise}\end{array}\right\$ $w={\partial }_{t}\left({\mathrm{\phi }}_{a}\right)$ $\mathrm{Δp}={p}_{A\left(\mathrm{limited}\right)}-{p}_{B\left(\mathrm{limited}\right)}$ ${\mathrm{\phi }}_{a}={\mathrm{\phi }}_{b}$ ${p}_{A\left(\mathrm{abs}\right)}={p}_{A}+{p}_{\mathrm{atm}\left(\mathrm{oil}\right)}$ ${p}_{A\left(\mathrm{limited}\right)}=\mathrm{max}\left({p}_{A},{p}_{\mathrm{vapour}\left(\mathrm{oil}\right)}-{p}_{\mathrm{atm}\left(\mathrm{oil}\right)}\right)$ ${p}_{B\left(\mathrm{abs}\right)}={p}_{B}+{p}_{\mathrm{atm}\left(\mathrm{oil}\right)}$ ${p}_{B\left(\mathrm{limited}\right)}=\mathrm{max}\left({p}_{B},{p}_{\mathrm{vapour}\left(\mathrm{oil}\right)}-{p}_{\mathrm{atm}\left(\mathrm{oil}\right)}\right)$ $-\frac{1}{2}\frac{{\mathrm{D}}_{\mathrm{pump}}\mathrm{sig}\mathrm{Δp}\mathrm{factor}}{\mathrm{\pi }}=\mathrm{flange_a.tau}+\mathrm{flange_b.tau}$ ${m}_{\mathrm{flow}\left(A\right)}+{m}_{\mathrm{flow}\left(B\right)}=0$

Variables

 Name Value Units Description Modelica ID $\mathrm{Δp}$ $\mathrm{Pa}$ Pressure drop dp ${p}_{A\left(\mathrm{limited}\right)}$ $\mathrm{Pa}$ Limited gauge pressure pA_limited ${p}_{B\left(\mathrm{limited}\right)}$ $\mathrm{Pa}$ Limited gauge pressure pB_limited ${p}_{A\left(\mathrm{abs}\right)}$ $\mathrm{Pa}$ Absolute pressure pA_abs ${p}_{B\left(\mathrm{abs}\right)}$ $\mathrm{Pa}$ Absolute pressure pB_abs $q$ $\frac{{m}^{3}}{s}$ Flow rate at connector A q $w$ $\frac{\mathrm{rad}}{s}$ Angular velocity of pump shaft w $T$ $K$ Local temperature T ${p}_{A\left(\mathrm{summary}\right)}$ ${p}_{A}$ $\mathrm{Pa}$ Pressure at port A summary_pA ${p}_{B\left(\mathrm{summary}\right)}$ ${p}_{B}$ $\mathrm{Pa}$ Pressure at port B summary_pB ${\mathrm{Δp}}_{\mathrm{summary}}$ $\mathrm{Δp}$ $\mathrm{Pa}$ Pressure drop summary_dp ${q}_{\mathrm{summary}}$ $q$ $\frac{{m}^{3}}{s}$ Flow rate flowing into port_A summary_q ${P}_{\mathrm{mech}\left(\mathrm{summary}\right)}$ [1] $W$ Mechanical Rotational Power summary_MP ${P}_{\mathrm{hyd}\left(\mathrm{summary}\right)}$ $-\mathrm{Δp}q$ $W$ Hydraulic Power summary_HP ${p}_{\mathrm{sat}}$ [2] $\mathrm{Pa}$ Gas saturation pressure p_sat ${p}_{\mathrm{vapour}\left(g\right)}$ [3] $\mathrm{Pa}$ Vapour pressure as gauge pressure pvapour_g $\mathrm{sig}$ sig $\mathrm{factor}$ factor

[1] ${\stackrel{.}{\varphi }}_{a}{\mathrm{\tau }}_{a}+{\stackrel{.}{\varphi }}_{b}{\mathrm{\tau }}_{b}$

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

[3] ${\mathrm{oil.p}}_{\mathrm{vapour}}-{\mathrm{oil.p}}_{\mathrm{atm}}$

Connections

 Name Description Modelica ID ${\mathrm{port}}_{A}$ Port A, where oil flows into the component ($0, ${p}_{B}<{p}_{A}$ means $0<\mathrm{Δp}$) port_A ${\mathrm{port}}_{B}$ Port B, where oil leaves the component ($q<0$, ${p}_{B}<{p}_{A}$ means $0<\mathrm{Δp}$) port_B ${\mathrm{flange}}_{a}$ flange_a ${\mathrm{flange}}_{b}$ flange_b $\mathrm{oil}$ oil $\mathrm{commandSignal}$ Input signal commands relative displacement volume commandSignal

Parameters

 Name Default Units Description Modelica ID ${\mathrm{ΔT}}_{\mathrm{system}}$ $0$ $K$ Temperature offset from system temperature dT_system ${\mathrm{D}}_{\mathrm{pump}}$ ${10}^{-4}$ ${m}^{3}$ Displacement per revolution Dpump