Gas Volume BB $—$ Refined model of charge gas

The Gas Volume BB component describes the behavior of the charge gas in an accumulator according to the Beattie-Bridgeman equation.

 Equations $\mathrm{\rho }={\mathrm{\rho }}_{\mathrm{oil}}\left(p={p}_{\mathrm{abs}},T={T}_{\mathrm{oil}},{v}_{\mathrm{air}}={v}_{\mathrm{gas}\left(\mathrm{oil}\right)},{p}_{\mathrm{sat}}={p}_{\mathrm{sat}}\right)$ $\mathrm{Cv}={\mathrm{pgas}}^{2}{\mathrm{temp}}^{2}{C}_{33}+{\mathrm{pgas}}^{2}\mathrm{temp}{C}_{32}+\mathrm{pgas}{\mathrm{temp}}^{2}{C}_{23}+{\mathrm{pgas}}^{2}{C}_{31}+\mathrm{pgas}\mathrm{temp}{C}_{22}+{\mathrm{temp}}^{2}{C}_{13}+\mathrm{pgas}{C}_{21}+\mathrm{temp}{C}_{12}+{C}_{11}$ $\mathrm{pgas}=\frac{100000R\mathrm{temp}\left(1-\frac{C}{{\mathrm{temp}}^{3}\mathrm{spezVol}}\right)\left(\mathrm{spezVol}+{B}_{0}\left(1-\frac{B}{\mathrm{spezVol}}\right)\right)}{{\mathrm{spezVol}}^{2}}-\frac{100000{A}_{0}\left(1-\frac{\mathrm{AA}}{\mathrm{spezVol}}\right)}{{\mathrm{spezVol}}^{2}}$ $\mathrm{spezVol}=\frac{\mathrm{GasVol}}{{m}_{\mathrm{gas}}}$ $\mathrm{GasVolDot}=-\frac{{m}_{\mathrm{flow}\left(A\right)}}{\mathrm{\rho }}$ $\mathrm{spezVolDot}=\frac{\mathrm{GasVolDot}}{{m}_{\mathrm{gas}}}$ ${T}_{\mathrm{oil}}={T}_{0\left(\mathrm{oil}\right)}+{\mathrm{ΔT}}_{\mathrm{system}}$ ${p}_{\mathrm{abs}}=\mathrm{pgas}$ ${p}_{\mathrm{abs}}={p}_{A}+{p}_{\mathrm{atm}\left(\mathrm{oil}\right)}$ ${\partial }_{t}\left(\mathrm{GasVol}\right)=\mathrm{GasVolDot}$ ${\partial }_{t}\left(\mathrm{temp}\right)=-\frac{\left(\mathrm{pgas}\mathrm{Τ}{\mathrm{temp}}^{2}{\mathrm{spezVol}}^{4}+\left(\mathrm{Τ}F{\mathrm{temp}}^{2}{A}_{0}+3\mathrm{Τ}CFR\right){\mathrm{spezVol}}^{2}+\left(-\mathrm{Τ}\mathrm{AA}F{\mathrm{temp}}^{2}{A}_{0}+3\mathrm{Τ}CFR{B}_{0}\right)\mathrm{spezVol}-3B{B}_{0}CFR\mathrm{Τ}\right)\mathrm{spezVolDot}+\left(-\mathrm{\Theta }{\mathrm{temp}}^{2}\overline{cv}+{\mathrm{temp}}^{3}\overline{cv}\right){\mathrm{spezVol}}^{4}}{\mathrm{Cv}\mathrm{Τ}{\mathrm{temp}}^{2}{\mathrm{spezVol}}^{4}}$

Variables

 Name Value Units Description Modelica ID ${p}_{\mathrm{abs}}$ $\mathrm{Pa}$ Absolute pressure p_abs $\mathrm{\rho }$ $\frac{\mathrm{kg}}{{m}^{3}}$ Gas Density rho ${T}_{\mathrm{oil}}$ $K$ Local temperature of oil T_oil ${p}_{\mathrm{sat}}$ [1] $\mathrm{Pa}$ Gas saturation pressure p_sat $\mathrm{temp}$ $K$ Gas temperature in K, initial condition temp $\mathrm{GasVol}$ ${m}^{3}$ Gas volume in m^3, initial condition GasVol $\mathrm{GasVolDot}$ Change of gas volume, d gasvolume / d time in m^3 / s GasVolDot $\mathrm{pgas}$ $\mathrm{Pa}$ Gas pressure in Pa pgas $\mathrm{Cv}$ spec. thermal capacity, constant volume Cv $\mathrm{spezVol}$ Gas volume / gm spezVol $\mathrm{spezVolDot}$ spezVolDot

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

Connections

 Name Description Modelica ID ${\mathrm{port}}_{A}$ Hydraulic port where oil enters the component port_A $\mathrm{oil}$ oil

Parameters

 Name Default Units Description Modelica ID ${\mathrm{ΔT}}_{\mathrm{system}}$ $0$ $K$ Temperature offset from system temperature dT_system ${m}_{\mathrm{gas}}$ $0.0394$ $\mathrm{kg}$ Gas mass (default is for 1e-3 m3 und 3.5 MPa) gm ${V}_{\mathrm{gas}\left(\mathrm{T0}\right)}$ $0.001$ ${m}^{3}$ Gas volume at t = 0 GasVol_T0 $\mathrm{Τ}$ $5$ $s$ Thermal timeconstant Tau $\mathrm{\Theta }$ $300$ $K$ Temperature of environment Theta $\overline{cv}$ $750$ Mean value of cv CvBar

Constants

 Name Value Units Description Modelica ID $R$ $0.00297$ R ${A}_{0}$ $0.00174$ A0 ${B}_{0}$ $0.0018$ B0 $\mathrm{AA}$ $9.34·{10}^{-4}$ AA $B$ $-2.47·{10}^{-4}$ B $C$ $5.09·{10}^{-8}$ C $F$ ${10}^{5}$ F ${C}_{11}$ $745.005$ C11 ${C}_{12}$ $-0.0576$ C12 ${C}_{13}$ $1.11·{10}^{-4}$ C13 ${C}_{21}$ $9.15·{10}^{-6}$ C21 ${C}_{22}$ $-3.3·{10}^{-8}$ C22 ${C}_{23}$ $3.08·{10}^{-11}$ C23 ${C}_{31}$ $-5.61·{10}^{-14}$ C31 ${C}_{32}$ $2.15·{10}^{-16}$ C32 ${C}_{33}$ $-2.03·{10}^{-19}$ C33