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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.

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

Common Equation Symbols

The equations use these symbols:

pLiquid pressure
p0Reference pressure
pminMinimum valid pressure
pcCritical pressure
βmixMixture isothermal bulk modulus
βLPure liquid bulk modulus
βL0Pure liquid bulk modulus at reference pressure p0
KβpProportionality coefficient when bulk modulus is a linear function of pressure
ρmixMixture density
ρLPure liquid density
ρL0Pure liquid density at reference pressure p0
ρgAir density
ρg0Air density at reference pressure p0
θ(p)Fraction of entrained air as a function of pressure
αVolumetric fraction of entrained air in the fluid mixture
α0Volumetric fraction of entrained air in the fluid mixture at reference pressure p0
VTotal mixture volume
VLPure liquid volume
VL0Pure liquid volume at reference pressure p0
VgAir volume
Vg0Air volume at reference pressure p0
MTotal mixture mass
MLPure liquid mass
ML0Pure liquid mass at reference pressure p0
MgAir mass
Mg0Air mass at reference pressure p0
nAir 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

    ρmix=ρL0e(pp0)/βL

  • Mixture density partial derivative

    ρmixp=ρL0βLe(pp0)/βL

  • Mixture bulk modulus

    βmix=βL

Bulk Modulus Is a Linear Function of Pressure

For this model, the defining equations are:

  • Mixture density

    ρmix=ρL0(1+KβpβL0(pp0))1/Kβp

  • Mixture density partial derivative

    ρmixp=ρL0βL0(1+KβpβL0(pp0))1/Kβp1

  • Mixture bulk modulus

    βmix=βL0+Kβp(pp0)

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

    ρmix=ρL0+(α01α0)ρg0e(pp0)/βL+(α01α0)(p0p)1/n

  • Mixture density partial derivative

    ρmixp=(ρL0+(α01α0)ρg0)(1βLe(pp0)/βL+1n(α01α0)(p01/np1/n+1))(e(pp0)/βL+(α01α0)(p0p)1/n)2

  • Mixture bulk modulus

    βmix=βLe(pp0)/βL+(α01α0)(p0p)1/ne(pp0)/βL+βL1n(α01α0)(p01/np1/n+1)

Bulk Modulus Is a Linear Function of Pressure

For this model, the defining equations are:

  • Mixture density

    ρmix=ρL0+(α01α0)ρg0(1+KβpβL0(pp0))1/Kβp+(α01α0)(p0p)1/n

  • Mixture density partial derivative

    ρmixp=(ρL0+(α01α0)ρg0)(1βL(1+KβpβL0(pp0))1/Kβp1+1n(α01α0)(p01/np1/n+1))((1+KβpβL0(pp0))1/Kβp+(α01α0)(p0p)1/n)2

  • Mixture bulk modulus

    βmix=βL(1+KβpβL0(pp0))1/Kβp+(α01α0)(p0p)1/n(1+KβpβL0(pp0))1/Kβp1+βL1n(α01α0)(p01/np1/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 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

    ρmix=ρL0+(α01α0)ρg0e(pp0)/βL+(α01α0)(p0p)1/nθ(p)

  • Mixture density partial derivative

    ρmixp=(ρL0+(α01α0)ρg0)(1βLe(pp0)/βL+(α01α0)(p0p)1/n(θ(p)npdθ(p)dp))(e(pp0)/βL+(α01α0)(p0p)1/nθ(p))2

  • Mixture bulk modulus

    βmix=βLe(pp0)/βL+(α01α0)(p0p)1/nθ(p)e(pp0)/βL+βL(α01α0)(p0p)1/n(θ(p)npdθ(p)dp)

Bulk Modulus Is a Linear Function of Pressure

For this model, the defining equations are:

  • Mixture density

    ρmix=ρL0+(α01α0)ρg0(1+KβpβL0(pp0))1/Kβp+(α01α0)(p0p)1/nθ(p)

  • Mixture density partial derivative

    ρmixp=(ρL0+(α01α0)ρg0)(1βL(1+KβpβL0(pp0))1/Kβp1+(α01α0)(p0p)1/n(θ(p)npdθ(p)dp))((1+KβpβL0(pp0))1/Kβp+(α01α0)(p0p)1/nθ(p))2

  • Mixture bulk modulus

    βmix=βL(1+KβpβL0(pp0))1/Kβp+(α01α0)(p0p)1/nθ(p)(1+KβpβL0(pp0))1/Kβp1+βL(α01α0)(p0p)1/n(θ(p)npdθ(p)dp)

See Also

Related Topics