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Physical properties of isothermal liquid

**Library:**Simscape / Foundation Library / Isothermal Liquid / Utilities

The Isothermal Liquid Properties (IL) block defines the liquid properties that act as global parameters for all the blocks connected to a circuit. The default liquid is water.

Each topologically distinct isothermal liquid circuit in a diagram can have a Isothermal Liquid Properties (IL) block connected to it. If no Isothermal Liquid Properties (IL) block is attached to a circuit, the blocks in this circuit use the properties corresponding to the default Isothermal Liquid Properties (IL) block parameter values.

The Isothermal Liquid Properties (IL) block provides a choice of modeling options:

Mixture bulk modulus: either constant or a linear function of pressure

Entrained air: zero, constant, or a linear function of pressure

Equations used to compute various liquid properties depend on the selected isothermal liquid model. For detailed information, see Isothermal Liquid Modeling Options.

Entrained air is the relative amount of nondissolved gas trapped in the fluid. Fluid with zero entrained air is ideal, that is, it represents pure liquid.

In its default configuration, the Isothermal Liquid Properties (IL) block models an ideal fluid with a constant bulk modulus:

**Isothermal bulk modulus model**is`Constant`

**Entrained air model**is`Constant`

**Entrained air-to-liquid volumetric ratio at atmospheric pressure**is`0`

In this model, the liquid bulk modulus is assumed to be constant and therefore the liquid density increases exponentially with the liquid pressure:

$${\rho}_{L}={\rho}_{L0}\cdot {e}^{\frac{p-{p}_{0}}{{\beta}_{L}}},$$

where:

*β*_{L}is liquid bulk modulus.*ρ*_{L}is liquid density.*ρ*_{L0}is liquid density at reference pressure.*p*is liquid pressure.*p*_{0}is reference pressure. By default, the block assumes reference pressure to be the atmospheric pressure,`0.101325 MPa`

, but you can specify a different value.

In systems where the liquid pressure can change over a wide range, and the assumption of
constant bulk modulus is not valid anymore, you can use the **Isothermal bulk
modulus model** parameter to define the liquid bulk modulus as a linear function
of pressure:

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

where:

*β*_{L0}is liquid bulk modulus at reference pressure.*K*_{βp}is the coefficient of proportionality between the bulk modulus and pressure increase.

If liquid pressure decreases below the reference pressure
*p*_{0}, the liquid bulk modulus value in the
previous equation can become negative, which is nonphysical. To ensure that the liquid bulk
modulus always stays positive, use the **Minimum valid pressure** parameter
to specify the minimum valid pressure, *p*_{min}:

$${p}_{\mathrm{min}}>{p}_{0}-\frac{{\beta}_{L0}}{{K}_{\beta p}}.$$

If liquid pressure drops below the **Minimum valid pressure** parameter
value, simulation issues an error.

In practice, working fluid is a mixture of liquid and a small amount of entrained air.
To model this type of fluid, specify a nonzero value for the **Entrained
air-to-liquid volumetric ratio at atmospheric pressure** parameter, but keep the
**Entrained air model** as `Constant`

.

The mixture density at a given pressure is defined as the total mass of liquid and entrained air over the total volume of the liquid and the entrained air at that pressure. While the total mass of the mixture is conserved as the pressure changes, the mixture volume does not stay constant. The entrained air is given by a volumetric fraction:

$${\alpha}_{0}=\frac{{V}_{g0}}{{V}_{g0}+{V}_{L0}},$$

where:

*α*_{0}is entrained air-to-liquid volumetric ratio at reference (atmospheric) pressure.*V*_{g0}is air volume at reference pressure.*V*_{L0}is pure liquid volume at reference pressure.

The entrained air is assumed to follow the ideal gas law. The compression or expansion of air in the liquid is a polytropic process, in which the air pressure and liquid pressure are identical:

$${V}_{g}={V}_{g0}{\left(\frac{{p}_{0}}{p}\right)}^{1/n},$$

where:

*V*_{g}is air volume.*n*is air polytropic index.

To model the air dissolution effects into the fluid, set the **Entrained air
model** parameter to `Linear function of pressure`

.

The process of dissolving air into the fluid is described by Henry’s law. At pressures
less than or equal to the reference pressure, *p*_{0}
(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 *p*_{0} and
*p*_{c}, the volumetric fraction of entrained air
that is not lost to dissolution, *θ(p)*, is a linear function of the
pressure, and is approximated by a third-order polynomial function to smoothly connect the
density and bulk modulus values between the three pressure regions:

$$\theta (p)=\{\begin{array}{ll}1,\hfill & p\le {p}_{0}\hfill \\ 0,\hfill & p\ge {p}_{c}\hfill \\ 1-\left(3{\left(\frac{{p}_{c}-p}{{p}_{c}-{p}_{0}}\right)}^{2}-2{\left(\frac{{p}_{c}-p}{{p}_{c}-{p}_{0}}\right)}^{3}\right),\hfill & {p}_{0}<p<{p}_{c}\hfill \end{array}.$$

The block provides the option to plot the specified fluid properties (density and isothermal bulk modulus) as a function of pressure. Plotting the properties lets you visualize the data before simulating the model.

To plot the data, right-click the Isothermal Liquid Properties
(IL) block in your model and, from the context menu, select
**Foundation Library** > **Plot Fluid
Properties**. Use the drop-down list located at the top of the plot to select
the fluid property to visualize. Click the **Reload** button to regenerate
a plot following a block parameter update.

**Isothermal Liquid Properties Plot**