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.

Zero Entrained Air	Constant Entrained Air	Air Dissolution Is On
Constant Bulk Modulus	Constant Bulk Modulus	Constant Bulk Modulus
Bulk Modulus Is a Linear Function of Pressure	Bulk Modulus Is a Linear Function of Pressure	Bulk Modulus Is a Linear Function of Pressure

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 p_c, all the entrained air has been dissolved into the liquid.

• At pressures between p₀ and p_c, 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

  $${\rho }_{mix}=\frac{{\rho }_{L0}+\left(\frac{{\alpha }_0}{1-{\alpha }_0}\right){\rho }_{g0}}{e^{-\left(p-p_0\right)/{\beta }_L}+\left(\frac{{\alpha }_0}{1-{\alpha }_0}\right){\left(\frac{p_0}{p}\right)}^{1/n}\theta (p)}$$

• Mixture density partial derivative

  $$\frac{\partial {\rho }_{mix}}{\partial p}=\frac{\left({\rho }_{L0}+\left(\frac{{\alpha }_0}{1-{\alpha }_0}\right){\rho }_{g0}\right)\left(\frac{1}{{\beta }_L}{e}^{-\left(p-p_0\right)/{\beta }_L}+\left(\frac{{\alpha }_0}{1-{\alpha }_0}\right){\left(\frac{p_0}{p}\right)}^{1/n}\left(\frac{\theta (p)}{np}-\frac{d\theta (p)}{dp}\right)\right)}{{\left(e^{-\left(p-p_0\right)/{\beta }_L}+\left(\frac{{\alpha }_0}{1-{\alpha }_0}\right){\left(\frac{p_0}{p}\right)}^{1/n}\theta (p)\right)}^2}$$

• Mixture bulk modulus

  $${\beta }_{mix}={\beta }_L\frac{e^{-\left(p-p_0\right)/{\beta }_L}+\left(\frac{{\alpha }_0}{1-{\alpha }_0}\right){\left(\frac{p_0}{p}\right)}^{1/n}\theta (p)}{e^{-\left(p-p_0\right)/{\beta }_L}+{\beta }_L\left(\frac{{\alpha }_0}{1-{\alpha }_0}\right){\left(\frac{p_0}{p}\right)}^{1/n}\left(\frac{\theta (p)}{np}-\frac{d\theta (p)}{dp}\right)}$$

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

  $$\frac{\partial {\rho }_{mix}}{\partial p}=\frac{\left({\rho }_{L0}+\left(\frac{{\alpha }_0}{1-{\alpha }_0}\right){\rho }_{g0}\right)\left(\frac{1}{{\beta }_L}{\left(1+\frac{K_{\beta p}}{{\beta }_{L0}}\left(p-p_0\right)\right)}^{-1/K_{\beta p}-1}+\left(\frac{{\alpha }_0}{1-{\alpha }_0}\right){\left(\frac{p_0}{p}\right)}^{1/n}\left(\frac{\theta (p)}{np}-\frac{d\theta (p)}{dp}\right)\right)}{{\left({\left(1+\frac{K_{\beta p}}{{\beta }_{L0}}\left(p-p_0\right)\right)}^{-1/K_{\beta p}}+\left(\frac{{\alpha }_0}{1-{\alpha }_0}\right){\left(\frac{p_0}{p}\right)}^{1/n}\theta (p)\right)}^2}$$

• Mixture bulk modulus

  $${\beta }_{mix}={\beta }_L\frac{{\left(1+\frac{K_{\beta p}}{{\beta }_{L0}}\left(p-p_0\right)\right)}^{-1/K_{\beta p}}+\left(\frac{{\alpha }_0}{1-{\alpha }_0}\right){\left(\frac{p_0}{p}\right)}^{1/n}\theta (p)}{{\left(1+\frac{K_{\beta p}}{{\beta }_{L0}}\left(p-p_0\right)\right)}^{-1/K_{\beta p}-1}+{\beta }_L\left(\frac{{\alpha }_0}{1-{\alpha }_0}\right){\left(\frac{p_0}{p}\right)}^{1/n}\left(\frac{\theta (p)}{np}-\frac{d\theta (p)}{dp}\right)}$$

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.