Oil $—$ Oil with undissolved air (Gibson model)

A typical hydraulic oil with undissolved air (Gibson model).

Equations

 Density $\mathrm{\rho }=\frac{{v}_{\mathrm{gas}\left(\mathrm{lim}\right)}{\mathrm{\rho }}_{\mathrm{gas}\left(\mathrm{ref}\right)}+\left(1-{v}_{\mathrm{gas}\left(\mathrm{lim}\right)}\right){\mathrm{\rho }}_{\mathrm{hyd}\left(\mathrm{ref}\right)}}{\frac{\left(1-{v}_{\mathrm{gas}\left(\mathrm{lim}\right)}\right){\mathrm{\rho }}_{\mathrm{hyd}\left(\mathrm{ref}\right)}}{{\mathrm{\rho }}_{\mathrm{hyd}}}+\frac{{v}_{\mathrm{gas}\left(\mathrm{lim}\right)}T{p}_{\mathrm{ref}}y}{{T}_{\mathrm{ref}}{p}_{\mathrm{lim}}}}$ ${\mathrm{\rho }}_{\mathrm{hyd}}={\mathrm{\rho }}_{0}\mathrm{max}\left(\mathrm{gain},1.{10}^{-9}\right)$ ${\mathrm{\rho }}_{\mathrm{hyd}\left(\mathrm{ref}\right)}={\mathrm{\rho }}_{0}$ $\mathrm{gain}={\begin{array}{cc}{\left(\frac{{p}_{\mathrm{atm}}}{{p}_{\mathrm{lim}}}\right)}^{\mathrm{\alpha }}\cdot ⅇxp\left(\left({p}_{\mathrm{lim}}-{p}_{\mathrm{atm}}\right)\left(\frac{1}{\mathrm{\beta }}+\frac{{v}_{\mathrm{air}}}{{p}_{\mathrm{lim}}}\right)\right)& \mathrm{\gamma }\le 1\\ ⅇxp\left(\frac{{p}_{\mathrm{lim}}-{p}_{\mathrm{atm}}}{\mathrm{\beta }}\right)\cdot ⅇxp\left(\frac{{v}_{\mathrm{air}}\left({p}_{\mathrm{lim}}^{\left(\frac{1}{\mathrm{\gamma }}\right)}-{p}_{\mathrm{atm}}^{\left(\frac{1}{\mathrm{\gamma }}\right)}\right)}{{p}_{\mathrm{lim}}^{\left(\frac{1}{\mathrm{\gamma }}\right)}}\right)\cdot ⅇxp\left(\frac{\mathrm{\gamma }{v}_{\mathrm{air}}{p}_{\mathrm{atm}}^{\left(\frac{1}{\mathrm{\gamma }}\right)}\left({p}_{\mathrm{lim}}^{\left(\frac{\mathrm{\gamma }-1}{\mathrm{\gamma }}\right)}-{p}_{\mathrm{atm}}^{\left(\frac{\mathrm{\gamma }-1}{\mathrm{\gamma }}\right)}\right)}{\left(\mathrm{\gamma }-1\right)\mathrm{\beta }}\right)& \mathrm{otherwise}\end{array}$
 Bulk Modulus ${\mathrm{\beta }}_{\mathrm{mix}}=\frac{\frac{{C}_{1}}{{\mathrm{\rho }}_{\mathrm{hyd}}}+\frac{{C}_{2}y}{{p}_{\mathrm{lim}}}}{\frac{{C}_{1}}{{\mathrm{\rho }}_{\mathrm{hyd}}{\mathrm{\beta }}_{\mathrm{hyd}}}+\frac{{C}_{2}\left(\frac{y}{{p}_{\mathrm{lim}}}-\left(\frac{{\partial }}{{\partial }p}y\right)\right)}{{p}_{\mathrm{lim}}}}$ ${\mathrm{\beta }}_{\mathrm{hyd}}=\frac{\mathrm{\gamma }{\mathrm{\beta }}_{0}{p}_{\mathrm{lim}}^{2}}{\mathrm{\gamma }{p}_{\mathrm{lim}}^{2}-{v}_{\mathrm{air}}{p}_{\mathrm{atm}}^{\left(\frac{1}{\mathrm{\gamma }}\right)}{p}_{\mathrm{lim}}^{\left(2-\frac{1}{\mathrm{\gamma }}\right)}+{\mathrm{\beta }}_{0}{v}_{\mathrm{air}}{p}_{\mathrm{atm}}^{\left(\frac{1}{\mathrm{\gamma }}\right)}{p}_{\mathrm{lim}}^{\left(\frac{\mathrm{\gamma }-1}{\mathrm{\gamma }}\right)}}$ ${C}_{1}={\mathrm{\rho }}_{\mathrm{hyd}}\left(1-{v}_{\mathrm{gas}\left(\mathrm{lim}\right)}\right)$ ${C}_{2}=\frac{{v}_{\mathrm{gas}\left(\mathrm{lim}\right)}T{p}_{\mathrm{ref}}}{{T}_{\mathrm{ref}}}$
 Dynamic Viscosity The dynamic viscosity is $\mathrm{\eta }=\mathrm{\nu }\mathrm{\rho }$.
 Kinematic Viscosity ${\mathrm{\nu }}_{\mathrm{mix}}=\frac{{\mathrm{\nu }}_{\mathrm{gas}}{V}_{\mathrm{gasFree}\left(\mathrm{nom}\right)}+{\mathrm{\nu }}_{\mathrm{hyd}}{V}_{\mathrm{hyd}\left(\mathrm{nom}\right)}}{{V}_{\mathrm{tot}\left(\mathrm{nom}\right)}}$ ${V}_{\mathrm{tot}\left(\mathrm{nom}\right)}={V}_{\mathrm{gasFree}\left(\mathrm{nom}\right)}+{V}_{\mathrm{hyd}\left(\mathrm{nom}\right)}$ ${V}_{\mathrm{hyd}\left(\mathrm{nom}\right)}=\frac{\left(1-{v}_{\mathrm{gas}\left(\mathrm{lim}\right)}\right){\mathrm{\rho }}_{\mathrm{hyd}\left(\mathrm{ref}\right)}}{{\mathrm{\rho }}_{\mathrm{hyd}}}$ ${V}_{\mathrm{gasFree}\left(\mathrm{nom}\right)}=\frac{{v}_{\mathrm{gas}\left(\mathrm{lim}\right)}T{p}_{\mathrm{ref}}y}{{T}_{\mathrm{ref}}{p}_{\mathrm{lim}}}$ ${\mathrm{\nu }}_{\mathrm{hyd}}={\mathrm{\nu }}_{T}{w}_{p}$ ${\mathrm{\nu }}_{T}={10}^{\left({T}^{A}{10}^{B}-6.7\right)}$ ${w}_{p}=ⅇxp\left(\mathrm{\alpha }\left({p}_{\mathrm{abs}}-{p}_{\mathrm{atm}}\right){10}^{\left(-5\right)}\right)$
 Miscellaneous ${p}_{\mathrm{lim}}=\mathrm{max}\left({p}_{\mathrm{abs}},{p}_{abs\left(\mathrm{min}\right)}\right)$ ${p}_{abs\left(\mathrm{min}\right)}=0.001$ ${p}_{\mathrm{sat}}=\frac{{v}_{\mathrm{gas}}{p}_{\mathrm{ref}}}{\mathrm{av}}$ ${v}_{\mathrm{gas}\left(\mathrm{lim}\right)}=\mathrm{min}\left(\mathrm{max}\left(0.000001,{v}_{\mathrm{gas}}\right),1\right)$ $y={\left(1-z\right)}^{5}\left(70{z}^{4}+35{z}^{3}+15{z}^{2}+5z+1\right)$ $z={\begin{array}{cc}{\begin{array}{cc}1& {p}_{\mathrm{sat}}<{p}_{\mathrm{abs}}\\ 0& {p}_{\mathrm{abs}}<{p}_{\mathrm{vap}}\\ \mathrm{max}\left(0,\frac{{p}_{\mathrm{abs}}-{p}_{\mathrm{vap}}}{{p}_{\mathrm{sat}}-{p}_{\mathrm{vap}}}\right)& \mathrm{otherwise}\end{array}& {p}_{\mathrm{vap}}<{p}_{\mathrm{sat}}\\ {\begin{array}{cc}1& {p}_{\mathrm{sat}}<{p}_{\mathrm{abs}}\\ \mathrm{max}\left(0,\frac{{p}_{\mathrm{abs}}}{\mathrm{max}\left(0.0001,{p}_{\mathrm{sat}}\right)}\right)& \mathrm{otherwise}\end{array}& \mathrm{otherwise}\end{array}$ The variables $y$ and $z$ are, respectively, the fractions of undissolved and dissolved gas in the mixture.

Parameters

General Parameters

 Name Default Units Description Modelica ID ${T}_{0}$ $293.15$ $K$ Working temperature T0 ${p}_{0}$ $1·{10}^{7}$ $\mathrm{Pa}$ Reference pressure p0 ${p}_{\mathrm{vapour}}$ $100$ $\mathrm{Pa}$ Absolute vapour pressure p_vapour ${p}_{\mathrm{atm}}$ $1·{10}^{5}$ $\mathrm{Pa}$ Atmospheric pressure: in case change is wanted for high altitudes p_atm display labels $\mathrm{false}$ Display labels display_labels ${\mathrm{\beta }}_{0}$ $1.8·{10}^{9}$ $\mathrm{Pa}$ Bulk modulus beta of liquid at atmospheric pressure beta_0 ${\mathrm{\rho }}_{0}$ $865$ $\frac{\mathrm{kg}}{{m}^{3}}$ Oil density at atmospheric pressure rho_0

Gas Parameters

 Name Default Units Description Modelica ID ${v}_{\mathrm{gas}}$ $1·{10}^{-6}$ $1$ Gas/(hydraulic medium) volume fraction at atmospheric pressure and 0 degC v_gas $\mathrm{\gamma }$ $1.4$ 1.0 => bubble formation is isotherm; 1.4 => bubble formation is adiabatic gamma $\mathrm{av}$ $6.8$ Bunsen coefficient av ${\mathrm{\rho }}_{\mathrm{gas}\left(0\right)}$ $1.18$ $\frac{\mathrm{kg}}{{m}^{3}}$ Gas density at atmospheric pressure and 0 degC rho_gas_0 ${\mathrm{\nu }}_{\mathrm{gas}}$ $1.5·{10}^{-5}$ $\frac{{m}^{2}}{s}$ Gas kinematic viscosity nu_gas

Viscosity Parameters

 Name Default Units Description Modelica ID $A$ $-3.36858$ Coefficient for temperature dependent viscosity A $B$ $8.6289$ Coefficient for temperature dependent viscosity B $\mathrm{\alpha }$ $0.0017$ Coefficient for pressure dependent viscosity alpha

Constants

 Name Value Units Description Modelica ID ${T}_{\mathrm{ref}}$ $273.15$ $K$ Reference temperature for v_air T_ref ${p}_{\mathrm{ref}}$ $1·{10}^{5}$ $\mathrm{Pa}$ Reference pressure for v_air p_ref