## Isothermal Liquid Modeling Options

In the isothermal liquid domain, the working fluid is a mixture of liquid and a small amount of entrained air. Entrained air is the relative amount of nondissolved gas trapped in the fluid. You can control the liquid and air properties separately:

• You can specify zero amount of entrained air. Fluid with zero entrained air is ideal, that is, it represents pure liquid.

• Mixture bulk modulus can be either constant or a linear function of pressure.

• If the mixture contains nonzero amount of entrained air, then you can select the air dissolution model. If air dissolution is off, the amount of entrained air is constant. If air dissolution is on, entrained air can dissolve into liquid.

Equations used to compute various fluid properties depend on the selected model.

Use the Isothermal Liquid Properties (IL) block to select the appropriate modeling options.

### Common Equation Symbols

The equations use these symbols:

 p Liquid pressure p0 Reference pressure pmin Minimum valid pressure pc Critical pressure βmix Mixture isothermal bulk modulus βL Pure liquid bulk modulus βL0 Pure liquid bulk modulus at reference pressure p0 Kβp Proportionality coefficient when bulk modulus is a linear function of pressure ρmix Mixture density ρL Pure liquid density ρL0 Pure liquid density at reference pressure p0 ρg Air density ρg0 Air density at reference pressure p0 θ(p) Fraction of entrained air as a function of pressure α Volumetric fraction of entrained air in the fluid mixture α0 Volumetric fraction of entrained air in the fluid mixture at reference pressure p0 V Total mixture volume VL Pure liquid volume VL0 Pure liquid volume at reference pressure p0 Vg Air volume Vg0 Air volume at reference pressure p0 M Total mixture mass ML Pure liquid mass ML0 Pure liquid mass at reference pressure p0 Mg Air mass Mg0 Air mass at reference pressure p0 n Air polytropic index

### Ideal Fluid

Fluid with zero entrained air is ideal, that is, it represents pure liquid.

#### Constant Bulk Modulus

For this model, the defining equations are:

• Mixture density

`${\rho }_{mix}={\rho }_{L0}\cdot {e}^{\left(p-{p}_{0}\right)/{\beta }_{L}}$`
• Mixture density partial derivative

`$\frac{\partial {\rho }_{mix}}{\partial p}=\frac{{\rho }_{L0}}{{\beta }_{L}}{e}^{\left(p-{p}_{0}\right)/{\beta }_{L}}$`
• Mixture bulk modulus

`${\beta }_{mix}={\beta }_{L}$`

#### Bulk Modulus Is a Linear Function of Pressure

For this model, the defining equations are:

• Mixture density

`${\rho }_{mix}={\rho }_{L0}{\left(1+\frac{{K}_{\beta p}}{{\beta }_{L0}}\left(p-{p}_{0}\right)\right)}^{1/{K}_{\beta p}}$`
• Mixture density partial derivative

`$\frac{\partial {\rho }_{mix}}{\partial p}=\frac{{\rho }_{L0}}{{\beta }_{L0}}{\left(1+\frac{{K}_{\beta p}}{{\beta }_{L0}}\left(p-{p}_{0}\right)\right)}^{1/{K}_{\beta p}-1}$`
• Mixture bulk modulus

`${\beta }_{mix}={\beta }_{L0}+{K}_{\beta p}\left(p-{p}_{0}\right)$`

### Constant Amount of Entrained Air

In practice, the working fluid contains a small amount of entrained air. This set of models assumes that the amount of entrained air remains constant during simulation.

#### Constant Bulk Modulus

For this model, the defining equations are:

• Mixture density

`${\rho }_{mix}=\frac{{\rho }_{L0}+\left(\frac{{\alpha }_{0}}{1-{\alpha }_{0}}\right){\rho }_{g0}}{{e}^{-\left(p-{p}_{0}\right)/{\beta }_{L}}+\left(\frac{{\alpha }_{0}}{1-{\alpha }_{0}}\right){\left(\frac{{p}_{0}}{p}\right)}^{1/n}}$`
• Mixture density partial derivative

`$\frac{\partial {\rho }_{mix}}{\partial p}=\frac{\left({\rho }_{L0}+\left(\frac{{\alpha }_{0}}{1-{\alpha }_{0}}\right){\rho }_{g0}\right)\left(\frac{1}{{\beta }_{L}}{e}^{-\left(p-{p}_{0}\right)/{\beta }_{L}}+\frac{1}{n}\left(\frac{{\alpha }_{0}}{1-{\alpha }_{0}}\right)\left(\frac{{p}_{0}^{1/n}}{{p}^{1/n+1}}\right)\right)}{{\left({e}^{-\left(p-{p}_{0}\right)/{\beta }_{L}}+\left(\frac{{\alpha }_{0}}{1-{\alpha }_{0}}\right){\left(\frac{{p}_{0}}{p}\right)}^{1/n}\right)}^{2}}$`
• Mixture bulk modulus

`${\beta }_{mix}={\beta }_{L}\frac{{e}^{-\left(p-{p}_{0}\right)/{\beta }_{L}}+\left(\frac{{\alpha }_{0}}{1-{\alpha }_{0}}\right){\left(\frac{{p}_{0}}{p}\right)}^{1/n}}{{e}^{-\left(p-{p}_{0}\right)/{\beta }_{L}}+{\beta }_{L}\frac{1}{n}\left(\frac{{\alpha }_{0}}{1-{\alpha }_{0}}\right)\left(\frac{{p}_{0}^{1/n}}{{p}^{1/n+1}}\right)}$`

#### Bulk Modulus Is a Linear Function of Pressure

For this model, the defining equations are:

• Mixture density

`${\rho }_{mix}=\frac{{\rho }_{L0}+\left(\frac{{\alpha }_{0}}{1-{\alpha }_{0}}\right){\rho }_{g0}}{{\left(1+\frac{{K}_{\beta p}}{{\beta }_{L0}}\left(p-{p}_{0}\right)\right)}^{-1/{K}_{\beta p}}+\left(\frac{{\alpha }_{0}}{1-{\alpha }_{0}}\right){\left(\frac{{p}_{0}}{p}\right)}^{1/n}}$`
• Mixture density partial derivative

`$\frac{\partial {\rho }_{mix}}{\partial p}=\frac{\left({\rho }_{L0}+\left(\frac{{\alpha }_{0}}{1-{\alpha }_{0}}\right){\rho }_{g0}\right)\left(\frac{1}{{\beta }_{L}}{\left(1+\frac{{K}_{\beta p}}{{\beta }_{L0}}\left(p-{p}_{0}\right)\right)}^{-1/{K}_{\beta p}-1}+\frac{1}{n}\left(\frac{{\alpha }_{0}}{1-{\alpha }_{0}}\right)\left(\frac{{p}_{0}^{1/n}}{{p}^{1/n+1}}\right)\right)}{{\left({\left(1+\frac{{K}_{\beta p}}{{\beta }_{L0}}\left(p-{p}_{0}\right)\right)}^{-1/{K}_{\beta p}}+\left(\frac{{\alpha }_{0}}{1-{\alpha }_{0}}\right){\left(\frac{{p}_{0}}{p}\right)}^{1/n}\right)}^{2}}$`
• Mixture bulk modulus

`${\beta }_{mix}={\beta }_{L}\frac{{\left(1+\frac{{K}_{\beta p}}{{\beta }_{L0}}\left(p-{p}_{0}\right)\right)}^{-1/{K}_{\beta p}}+\left(\frac{{\alpha }_{0}}{1-{\alpha }_{0}}\right){\left(\frac{{p}_{0}}{p}\right)}^{1/n}}{{\left(1+\frac{{K}_{\beta p}}{{\beta }_{L0}}\left(p-{p}_{0}\right)\right)}^{-1/{K}_{\beta p}-1}+{\beta }_{L}\frac{1}{n}\left(\frac{{\alpha }_{0}}{1-{\alpha }_{0}}\right)\left(\frac{{p}_{0}^{1/n}}{{p}^{1/n+1}}\right)}$`

### Air Dissolution Is On

This set of models lets you account for the air dissolution effects during simulation:

• At pressures less than or equal to the reference pressure, p0 (which is assumed to be equal to atmospheric pressure), all the air is assumed to be entrained.

• At pressures equal or higher than pressure pc, all the entrained air has been dissolved into the liquid.

• At pressures between p0 and pc, the volumetric fraction of entrained air that is not lost to dissolution, θ(p), is a function of the pressure.

#### Constant Bulk Modulus

For this model, the defining equations are:

• Mixture density

`${\rho }_{mix}=\frac{{\rho }_{L0}+\left(\frac{{\alpha }_{0}}{1-{\alpha }_{0}}\right){\rho }_{g0}}{{e}^{-\left(p-{p}_{0}\right)/{\beta }_{L}}+\left(\frac{{\alpha }_{0}}{1-{\alpha }_{0}}\right){\left(\frac{{p}_{0}}{p}\right)}^{1/n}\theta \left(p\right)}$`
• Mixture density partial derivative

`$\frac{\partial {\rho }_{mix}}{\partial p}=\frac{\left({\rho }_{L0}+\left(\frac{{\alpha }_{0}}{1-{\alpha }_{0}}\right){\rho }_{g0}\right)\left(\frac{1}{{\beta }_{L}}{e}^{-\left(p-{p}_{0}\right)/{\beta }_{L}}+\left(\frac{{\alpha }_{0}}{1-{\alpha }_{0}}\right){\left(\frac{{p}_{0}}{p}\right)}^{1/n}\left(\frac{\theta \left(p\right)}{np}-\frac{d\theta \left(p\right)}{dp}\right)\right)}{{\left({e}^{-\left(p-{p}_{0}\right)/{\beta }_{L}}+\left(\frac{{\alpha }_{0}}{1-{\alpha }_{0}}\right){\left(\frac{{p}_{0}}{p}\right)}^{1/n}\theta \left(p\right)\right)}^{2}}$`
• Mixture bulk modulus

`${\beta }_{mix}={\beta }_{L}\frac{{e}^{-\left(p-{p}_{0}\right)/{\beta }_{L}}+\left(\frac{{\alpha }_{0}}{1-{\alpha }_{0}}\right){\left(\frac{{p}_{0}}{p}\right)}^{1/n}\theta \left(p\right)}{{e}^{-\left(p-{p}_{0}\right)/{\beta }_{L}}+{\beta }_{L}\left(\frac{{\alpha }_{0}}{1-{\alpha }_{0}}\right){\left(\frac{{p}_{0}}{p}\right)}^{1/n}\left(\frac{\theta \left(p\right)}{np}-\frac{d\theta \left(p\right)}{dp}\right)}$`

#### Bulk Modulus Is a Linear Function of Pressure

For this model, the defining equations are:

• Mixture density

`${\rho }_{mix}=\frac{{\rho }_{L0}+\left(\frac{{\alpha }_{0}}{1-{\alpha }_{0}}\right){\rho }_{g0}}{{\left(1+\frac{{K}_{\beta p}}{{\beta }_{L0}}\left(p-{p}_{0}\right)\right)}^{-1/{K}_{\beta p}}+\left(\frac{{\alpha }_{0}}{1-{\alpha }_{0}}\right){\left(\frac{{p}_{0}}{p}\right)}^{1/n}\theta \left(p\right)}$`
• Mixture density partial derivative

`$\frac{\partial {\rho }_{mix}}{\partial p}=\frac{\left({\rho }_{L0}+\left(\frac{{\alpha }_{0}}{1-{\alpha }_{0}}\right){\rho }_{g0}\right)\left(\frac{1}{{\beta }_{L}}{\left(1+\frac{{K}_{\beta p}}{{\beta }_{L0}}\left(p-{p}_{0}\right)\right)}^{-1/{K}_{\beta p}-1}+\left(\frac{{\alpha }_{0}}{1-{\alpha }_{0}}\right){\left(\frac{{p}_{0}}{p}\right)}^{1/n}\left(\frac{\theta \left(p\right)}{np}-\frac{d\theta \left(p\right)}{dp}\right)\right)}{{\left({\left(1+\frac{{K}_{\beta p}}{{\beta }_{L0}}\left(p-{p}_{0}\right)\right)}^{-1/{K}_{\beta p}}+\left(\frac{{\alpha }_{0}}{1-{\alpha }_{0}}\right){\left(\frac{{p}_{0}}{p}\right)}^{1/n}\theta \left(p\right)\right)}^{2}}$`
• Mixture bulk modulus

`${\beta }_{mix}={\beta }_{L}\frac{{\left(1+\frac{{K}_{\beta p}}{{\beta }_{L0}}\left(p-{p}_{0}\right)\right)}^{-1/{K}_{\beta p}}+\left(\frac{{\alpha }_{0}}{1-{\alpha }_{0}}\right){\left(\frac{{p}_{0}}{p}\right)}^{1/n}\theta \left(p\right)}{{\left(1+\frac{{K}_{\beta p}}{{\beta }_{L0}}\left(p-{p}_{0}\right)\right)}^{-1/{K}_{\beta p}-1}+{\beta }_{L}\left(\frac{{\alpha }_{0}}{1-{\alpha }_{0}}\right){\left(\frac{{p}_{0}}{p}\right)}^{1/n}\left(\frac{\theta \left(p\right)}{np}-\frac{d\theta \left(p\right)}{dp}\right)}$`

## References

[1] Gholizadeh, Hossein, Richard Burton, and Greg Schoenau. “Fluid Bulk Modulus: Comparison of Low Pressure Models.” International Journal of Fluid Power 13, no. 1 (January 2012): 7–16. https://doi.org/10.1080/14399776.2012.10781042.