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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
p_min	Minimum valid pressure
p_c	Critical pressure
β_mix	Mixture isothermal bulk modulus
β_L	Pure liquid bulk modulus
β_L0	Pure liquid bulk modulus at reference pressure p₀
K_βp	Proportionality coefficient when bulk modulus is a linear function of pressure
ρ_mix	Mixture density
ρ_L	Pure liquid density
ρ_L0	Pure liquid density at reference pressure p₀
ρ_g	Air density
ρ_g0	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
V_L	Pure liquid volume
V_L0	Pure liquid volume at reference pressure p₀
V_g	Air volume
V_g0	Air volume at reference pressure p₀
M	Total mixture mass
M_L	Pure liquid mass
M_L0	Pure liquid mass at reference pressure p₀
M_g	Air mass
M_g0	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
$ρ_{m i x} = ρ_{L 0} \cdot e^{(p - p_{0}) / β_{L}}$
Mixture density partial derivative
$\frac{\partial ρ_{m i x}}{\partial p} = \frac{ρ_{L 0}}{β_{L}} e^{(p - p_{0}) / β_{L}}$
Mixture bulk modulus
$β_{m i x} = β_{L}$

Bulk Modulus Is a Linear Function of Pressure

For this model, the defining equations are:

Mixture density
$ρ_{m i x} = ρ_{L 0} {(1 + \frac{K_{β p}}{β_{L 0}} (p - p_{0}))}^{1 / K_{β p}}$
Mixture density partial derivative
$\frac{\partial ρ_{m i x}}{\partial p} = \frac{ρ_{L 0}}{β_{L 0}} {(1 + \frac{K_{β p}}{β_{L 0}} (p - p_{0}))}^{1 / K_{β p} - 1}$
Mixture bulk modulus
$β_{m i x} = β_{L 0} + K_{β p} (p - p_{0})$

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
$ρ_{m i x} = \frac{ρ_{L 0} + (\frac{α_{0}}{1 - α_{0}}) ρ_{g 0}}{e^{- (p - p_{0}) / β_{L}} + (\frac{α_{0}}{1 - α_{0}}) {(\frac{p_{0}}{p})}^{1 / n}}$
Mixture density partial derivative
$\frac{\partial ρ_{m i x}}{\partial p} = \frac{(ρ_{L 0} + (\frac{α_{0}}{1 - α_{0}}) ρ_{g 0}) (\frac{1}{β_{L}} e^{- (p - p_{0}) / β_{L}} + \frac{1}{n} (\frac{α_{0}}{1 - α_{0}}) (\frac{p_{0}^{1 / n}}{p^{1 / n + 1}}))}{{(e^{- (p - p_{0}) / β_{L}} + (\frac{α_{0}}{1 - α_{0}}) {(\frac{p_{0}}{p})}^{1 / n})}^{2}}$
Mixture bulk modulus
$β_{m i x} = β_{L} \frac{e^{- (p - p_{0}) / β_{L}} + (\frac{α_{0}}{1 - α_{0}}) {(\frac{p_{0}}{p})}^{1 / n}}{e^{- (p - p_{0}) / β_{L}} + β_{L} \frac{1}{n} (\frac{α_{0}}{1 - α_{0}}) (\frac{p_{0}^{1 / n}}{p^{1 / n + 1}})}$

Bulk Modulus Is a Linear Function of Pressure

For this model, the defining equations are:

Mixture density
$ρ_{m i x} = \frac{ρ_{L 0} + (\frac{α_{0}}{1 - α_{0}}) ρ_{g 0}}{{(1 + \frac{K_{β p}}{β_{L 0}} (p - p_{0}))}^{- 1 / K_{β p}} + (\frac{α_{0}}{1 - α_{0}}) {(\frac{p_{0}}{p})}^{1 / n}}$
Mixture density partial derivative
$\frac{\partial ρ_{m i x}}{\partial p} = \frac{(ρ_{L 0} + (\frac{α_{0}}{1 - α_{0}}) ρ_{g 0}) (\frac{1}{β_{L}} {(1 + \frac{K_{β p}}{β_{L 0}} (p - p_{0}))}^{- 1 / K_{β p} - 1} + \frac{1}{n} (\frac{α_{0}}{1 - α_{0}}) (\frac{p_{0}^{1 / n}}{p^{1 / n + 1}}))}{{({(1 + \frac{K_{β p}}{β_{L 0}} (p - p_{0}))}^{- 1 / K_{β p}} + (\frac{α_{0}}{1 - α_{0}}) {(\frac{p_{0}}{p})}^{1 / n})}^{2}}$
Mixture bulk modulus
$β_{m i x} = β_{L} \frac{{(1 + \frac{K_{β p}}{β_{L 0}} (p - p_{0}))}^{- 1 / K_{β p}} + (\frac{α_{0}}{1 - α_{0}}) {(\frac{p_{0}}{p})}^{1 / n}}{{(1 + \frac{K_{β p}}{β_{L 0}} (p - p_{0}))}^{- 1 / K_{β p} - 1} + β_{L} \frac{1}{n} (\frac{α_{0}}{1 - α_{0}}) (\frac{p_{0}^{1 / n}}{p^{1 / n + 1}})}$

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
$ρ_{m i x} = \frac{ρ_{L 0} + (\frac{α_{0}}{1 - α_{0}}) ρ_{g 0}}{e^{- (p - p_{0}) / β_{L}} + (\frac{α_{0}}{1 - α_{0}}) {(\frac{p_{0}}{p})}^{1 / n} θ (p)}$
Mixture density partial derivative
$\frac{\partial ρ_{m i x}}{\partial p} = \frac{(ρ_{L 0} + (\frac{α_{0}}{1 - α_{0}}) ρ_{g 0}) (\frac{1}{β_{L}} e^{- (p - p_{0}) / β_{L}} + (\frac{α_{0}}{1 - α_{0}}) {(\frac{p_{0}}{p})}^{1 / n} (\frac{θ (p)}{n p} - \frac{d θ (p)}{d p}))}{{(e^{- (p - p_{0}) / β_{L}} + (\frac{α_{0}}{1 - α_{0}}) {(\frac{p_{0}}{p})}^{1 / n} θ (p))}^{2}}$
Mixture bulk modulus
$β_{m i x} = β_{L} \frac{e^{- (p - p_{0}) / β_{L}} + (\frac{α_{0}}{1 - α_{0}}) {(\frac{p_{0}}{p})}^{1 / n} θ (p)}{e^{- (p - p_{0}) / β_{L}} + β_{L} (\frac{α_{0}}{1 - α_{0}}) {(\frac{p_{0}}{p})}^{1 / n} (\frac{θ (p)}{n p} - \frac{d θ (p)}{d p})}$

Bulk Modulus Is a Linear Function of Pressure

For this model, the defining equations are:

Mixture density
$ρ_{m i x} = \frac{ρ_{L 0} + (\frac{α_{0}}{1 - α_{0}}) ρ_{g 0}}{{(1 + \frac{K_{β p}}{β_{L 0}} (p - p_{0}))}^{- 1 / K_{β p}} + (\frac{α_{0}}{1 - α_{0}}) {(\frac{p_{0}}{p})}^{1 / n} θ (p)}$
Mixture density partial derivative
$\frac{\partial ρ_{m i x}}{\partial p} = \frac{(ρ_{L 0} + (\frac{α_{0}}{1 - α_{0}}) ρ_{g 0}) (\frac{1}{β_{L}} {(1 + \frac{K_{β p}}{β_{L 0}} (p - p_{0}))}^{- 1 / K_{β p} - 1} + (\frac{α_{0}}{1 - α_{0}}) {(\frac{p_{0}}{p})}^{1 / n} (\frac{θ (p)}{n p} - \frac{d θ (p)}{d p}))}{{({(1 + \frac{K_{β p}}{β_{L 0}} (p - p_{0}))}^{- 1 / K_{β p}} + (\frac{α_{0}}{1 - α_{0}}) {(\frac{p_{0}}{p})}^{1 / n} θ (p))}^{2}}$
Mixture bulk modulus
$β_{m i x} = β_{L} \frac{{(1 + \frac{K_{β p}}{β_{L 0}} (p - p_{0}))}^{- 1 / K_{β p}} + (\frac{α_{0}}{1 - α_{0}}) {(\frac{p_{0}}{p})}^{1 / n} θ (p)}{{(1 + \frac{K_{β p}}{β_{L 0}} (p - p_{0}))}^{- 1 / K_{β p} - 1} + β_{L} (\frac{α_{0}}{1 - α_{0}}) {(\frac{p_{0}}{p})}^{1 / n} (\frac{θ (p)}{n p} - \frac{d θ (p)}{d p})}$

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.

See Also

Isothermal Liquid Properties (IL)

Related Topics

Modeling Isothermal Liquid Systems