Parameterizing Entrained Air in a Thermal Liquid

In the Simscape™ thermal 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 specify how the entrained air in a thermal liquid network works by using the Thermal Liquid Settings (TL) or Thermal Liquid Properties (TL) (Simscape Fluids) blocks. You can specify the liquid and air properties separately:

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

• If the mixture contains nonzero amount of entrained air, then you can model air dissolution by enabling the Model Air Dissolution. If you clear the parameter, the amount of entrained air is constant. If you enable the parameter, the entrained air can dissolve into liquid.

How Simscape Models Entrained Air

The Thermal Liquid Settings (TL) and Thermal Liquid Properties (TL) blocks determine the properties of the thermal liquids in their networks, including the properties for entrained air. Both blocks model the fluid density of the entrained air by using the equation

$${\rho }_{mix}=\frac{f_0\frac{p_{g0}}{R{T}_{g0}}+(1-f_0){\rho }_{L0}}{f_0\frac{p_{g0}}{T_{g0}}\frac{T}{p}\theta {\rho }_L+(1-f_0){\rho }_{L0}}{\rho }_L,$$

where:

• R is the value of the Specific gas constant parameter.

• f₀ is the value of the Volumetric fraction of entrained air in the fluid mixture at atmospheric conditions parameter.

• θ is the volumetric fraction of entrained air that is not lost to dissolution. When you clear the Model air dissolution check box, the value of θ is 1.

• ρ is the density.

• p is the pressure.

• T is the temperature.

• The subscripts L0, g0, and L denote the liquid alone at reference conditions, the gas alone at reference conditions, and the liquid alone at operating conditions, respectively.

The kinematic viscosity of the liquid with entrained air is

$${\nu }_{mix}={\nu }_L\frac{{\rho }_L}{{\rho }_{mix}},$$

where ν_L is the kinematic viscosity of the thermal liquid alone.

The isobaric thermal expansion coefficient is

$${\alpha }_{mix}=\frac{\frac{p_{g0}}{p}\frac{T}{T_{g0}}{f}_0}{\theta \frac{p_{g0}}{p}\frac{T}{T_{g0}}{f}_0+\frac{{\rho }_{L0}}{{\rho }_L}(1-f_0)}\frac{\theta }{T}+\frac{\frac{{\rho }_{L0}}{{\rho }_L}(1-f_0)}{\theta \frac{p_{g0}}{p}\frac{T}{T_{g0}}{f}_0+\frac{{\rho }_{L0}}{{\rho }_L}(1-f_0)}{\alpha }_L,$$

where α_L is the isobaric thermal expansion coefficient of the thermal liquid alone.

When you select Model air dissolution, the bulk modulus of the liquid with entrained air is

$$K_{mix}=K_L\frac{f_0\frac{p_{g0}T}{T_{g0}p}\theta {\rho }_L+(1-f_0){\rho }_{L0}}{f_0\frac{p_{g0}}{T_{g0}}T\left(\frac{\theta }{p^2}-\frac{1}{p}\frac{\partial \theta }{\partial p}\right){\rho }_L{K}_L+(1-f_0){\rho }_{L0}},$$

where K_L is the bulk modulus of the thermal liquid alone.

Otherwise, the bulk modulus is

$$K_{mix}=K_L\frac{f_0{\rho }_L\frac{p_{g0}T}{T_{g0}p}+(1-f_0){\rho }_{L0}}{f_0{K}_L{\rho }_L\frac{p_{g0}T}{T_{g0}{p}^2}+(1-f_0){\rho }_{L0}}.$$

The thermal liquid specific internal energy, constant-pressure specific heat, thermal conductivity, dynamic viscosity, and Prandtl number do not account for entrained air. If the volumetric fraction of entrained air is very large, these calculations may not be accurate.

Air Dissolution

To model the air dissolution into the fluid, select Model air dissolution in the Thermal Liquid Settings or Thermal Liquid Properties block. The calculation for the volumetric fraction of entrained air that is not lost to dissolution depends on the fluid pressure. All the air is entrained at pressures less than or equal to the reference pressure, p₀, which the block assumes is equal to atmospheric pressure. At pressures equal or higher than p_crit, all the entrained air is dissolved into the liquid. At pressures between p₀ and p_crit, θ is a linear function of the pressure, but is approximated by a third-order polynomial function to smoothly connect the density and bulk modulus values between the three pressure regions,

$$\theta \left(p\right)=\{\begin{array}{c}1,p\le p_0\\ 1-\left(3{\left(\frac{p_{crit}-p}{p_{crit}-p_0}\right)}^2-2{\left(\frac{p_{crit}-p}{p_{crit}-p_0}\right)}^3\right),p_0<p<p_{crit}\\ 0,p_{crit}\le p\end{array}$$

where p_crit is the value of the Full dissolution pressure parameter.