Fluid properties for two-phase fluid network

Description

The Two-Phase Fluid Properties (2P) block sets the thermophysical properties of a fluid in a two-phase fluid network. These properties, which include density, viscosity, and specific heat, among others, can extend into the supercritical region of the fluid (water, by default, with a supercritical region extending up to 100 MPa in pressure).

The properties apply to the whole of the two-phase fluid network (the group of continuously connected two-phase fluid blocks). A network can have only one working fluid, and therefore only one instance of this block. If the network is modeled without this block, the fluid is set to water and its properties are obtained from the domain definition.

The block parameterizes the fluid properties in terms of pressure and normalized internal energy—a linear transformation of the specific internal energy. In a subcooled liquid, the normalized internal energy definition is

$\begin{matrix} \bar{u} = \frac{u - u_{m i n}}{u_{s a t}^{L} (p) - u_{m i n}} - 1, & u_{m i n} \leq u < u^{L}_{s a t} \end{matrix} (p),$

where:

$\bar{u}$ is the normalized internal energy of the fluid.
u is the specific internal energy of the fluid.
u_min is the lowest specific internal energy allowed in the two-phase fluid network.
u^L_sat is the specific internal energy of the liquid phase at saturation.

In a superheated vapor, the normalized internal energy definition is

$\begin{matrix} \bar{u} = \frac{u - u_{m a x}}{u_{m a x} - u_{s a t}^{V} (p)} + 2, & u^{V}_{s a t} (p) < u \leq u_{\max} \end{matrix},$

where:

u_max is the highest specific internal energy allowed in the two-phase fluid network.
u^V_sat is the specific internal energy of the vapor phase at saturation.

In a two-phase mixture, the normalized internal energy definition is

$\begin{matrix} \bar{u} = \frac{u - u_{s a t}^{L} (p)}{u_{s a t}^{V} (p) - u_{s a t}^{L} (p)}, & u^{L}_{s a t} (p) \leq u \leq u^{V}_{s a t} (p) \end{matrix} .$

These expressions correspond to a normalized internal energy that is at all pressures -1 at the minimum valid specific internal energy, 0 at the liquid saturation boundary, +1 at the vapor saturation boundary, and +2 at the maximum valid specific internal energy.

In a two-phase mixture, the normalized internal energy ranges in value from 0 to 1 and is therefore equivalent to vapor quality—the mass fraction of the vapor phase in a two-phase mixture. In subcooled liquid and superheated vapor, the normalized internal energy behaves as an extension of vapor quality.

The normalized internal energy provides an advantage over the specific internal energy. It transforms the p-u phase diagram so that the subcooled liquid and superheated vapor phases occupy distinct rectangular regions. This transformation, shown in the figure, enables you to specify the fluid properties on separate rectangular p- $\bar{u}$ grids, one for each phase.

A pressure vector, of length N, and two normalized internal energy vectors, of lengths M_L and M_V, provide the (p, $\bar{u}$ ) coordinates of the two grids. The pressure vector is common to both grids. The subcooled liquid grid is M_L-by-N in size and the superheated vapor grid M_V-by-N.

Two-way lookup tables provide the fluid property values on the (p, $\bar{u}$ ) grids. The table rows correspond to different normalized internal energies and the table columns to pressures. Fluid properties in the p- $\bar{u}$ continuum are computed using linear interpolation between the p- $\bar{u}$ data points.

Two-Way Property Lookup Table

Saturated specific internal energy vectors provide the phase boundaries in the (p, $\bar{u}$ ) phase diagram. These separate the different regions of the phase diagram—subcooled liquid, two-phase mixture, and superheated vapor.

Along with the minimum and maximum valid specific internal energy values, the saturated specific internal energy vectors enable the Two-Phase Fluid blocks to convert the normalized internal energies specified in this block into the specific internal energies they use for calculation purposes.

Supercritical Fluids

The fluid can become supercritical. Liquid and vapor then cease to exist as separate phases. Their saturation boundaries merge, forming what is known as the pseudocritical line (in the figure, the segment extending upward from the hump of the p–u phase diagram).

The pseudocritical line divides the fluid into liquid-like and vapor-like regions. As the phases are no longer distinct, however, the transition between the regions is gradual rather than abrupt. This means that at the pseudocritical line, the liquid-like and vapor-like fluids must share the same physical properties.

Property Tables

In this block, the liquid-like portion of the supercritical region is conceived as an extension of the liquid phase. Likewise, the vapor-like portion is conceived as an extension of the vapor phase. In other words, the properties of the supercritical fluid are divided into the liquid and vapor property tables of the block.

The distinction between the phases is, however, superfluous in the supercritical region. For breakpoints over the critical pressure, the liquid and vapor property tables must agree at the saturated boundaries (or, more properly, at the pseudocritical line). The supercritical portions of the tables then behave as a continuous whole.

Consider the kinematic viscosity tables as an example. The bottom row of the Liquid kinematic viscosity table parameter, as it corresponds to a normalized specific internal energy of 1, gives the data for the saturated liquid. The elements in that row corresponding to pressures at or above the critical point give the data over the pseudocritical line.

Likewise, the upper row of the Vapor kinematic viscosity table parameter, as it corresponds to a normalized specific internal energy of 1, gives the data for the saturated vapor. The elements in that row corresponding to pressures at or above the critical point therefore give the data over the pseudocritical line. To be valid, this data must be exactly the same as that given over the pseudocritical line in the liquid table.

Pseudocritical Line

The pseudocritical line is encoded in the saturation boundaries of the liquid and vapor phases. These are specified in the block via the Saturated liquid specific internal energy vector and Saturated vapor internal energy vector parameters. At pressures below the critical point, the values in one vector will generally differ from those in the other; above the critical point, however, one must always equal the other.

The pseudocritical line can assume any shape. For a simple approximation, it often suffices to extend the specific internal energy of the critical point upward along the pressure axis. This means setting the supercritical portions of the saturated specific internal energy vectors to that critical value. If the exact shape of the pseudocritical line is known, the specific internal energies along that line can be used instead.

Thermal Transport Properties

The data for the thermal transport properties—the specific heat, thermal conductivity, and Prandtl number—often shows a large peak near the critical point. The scale and size of the peak can, in some cases, cause numerical problems during simulation. To avoid such problems, the block allows the peak to be clipped. This option is specified in the Thermal transport properties near critical point block parameter (by setting it to Clip peak values).

The clipping is limited to a small pressure range around the critical point. That range is specified as a fraction of the critical pressure (f in the figure), and it extends in both directions (by the same amount) from the critical pressure. Within that range, the thermal properties are each limited to a maximum given by the greater of its two end values. The figure shows a plot of the Prandtl number for water with clipping.

Data Visualization

The block lets you plot the specified two-phase fluid properties as a function of pressure and specific internal energy. Plotting the properties lets you visualize the data before simulating the model.

To plot the data, right-click a Two-Phase Fluid Properties (2P) block in your model and, from the context menu, select Foundation Library > Plot Fluid Properties (3D) or Foundation Library > Plot Fluid Properties (Contours). Use the drop-down list located at the top of the plot to select the property to visualize. Click the Reload button to regenerate a plot following a block parameter update.

Fluid Properties (3D) Plot

Fluid Properties (Contours) Plot

Parameters

Parameters Tab

Minimum valid specific internal energy

Lowest specific internal energy allowed in the two-phase fluid network. The default value is 0 kJ/kg.

Maximum valid specific internal energy

Highest specific internal energy allowed in the two-phase fluid network. The default value is 4000 kJ/kg.

Pressure vector

Vector of length N containing the pressure values corresponding to the columns of the fluid property tables. The default vector is a logarithmically spaced 100-element vector ranging from 1e-3 to 100 MPa (spanning both subcritical and supercritical regions of the fluid).

Critical pressure

Pressure at which liquid and vapor cease to exist as distinct phases. Set this parameter to inf to model a fluid that is always subcritical.

Thermal transport properties near critical point

Select whether to trim the peaks of the thermal transport properties near the critical point. (The peaks are often sharp and large, which may lead to numerical problems during simulation.) The width of the clipping region is specified in the Fraction above and below critical pressure for clipping block parameter (exposed when Clip peak values is selected here). Set this parameter to Do not clip peak values to leave the thermal transport properties as is.

Fraction above and below critical pressure for clipping

Pressure range above and below the critical pressure, expressed as a fraction of the same, over which to clip the peaks of the thermal transport properties. This parameter is exposed when the Thermal transport properties near critical point parameter is set to Clip peak values.

Atmospheric pressure

Absolute pressure of the two-phase fluid system surroundings. The default value, 0.101325 Pa, is the atmospheric pressure at mean sea level.

Dynamic pressure threshold for flow reversal

Dynamic pressure at which the flow at a port begins to reverse in direction. Simscape™. The dynamic pressure is the difference between the total pressure and the static pressure. Treat this parameter as a means to smooth the flow reversal. Larger values correspond to smoother reversals and smaller values to sharper reversals. The default value is 0.01 Pa.

The Two-Phase Fluid domain uses an upwind scheme that derives the flow rate at a port from its value just upwind of the port. During flow reversals, the source of the flow rate value changes abruptly and the flow rate can become discontinuous. To prevent discontinuities and improve simulation robustness, flow reversals are smoothed out, with the dynamic pressure threshold marking the beginning of the smoothed transitions.

Liquid Properties Tab

Normalized liquid internal energy vector: Vector of length M_L containing the normalized internal energy values corresponding to the rows of the liquid property tables. The vector must begin at -1 and end at 0 (the saturated liquid state). The default is a uniformly spaced 25-element vector.
Liquid specific volume table: M_L × N matrix containing the liquid specific volume values corresponding to the normalized liquid internal energy and pressure vectors. Elements at the saturation boundary and at or above the critical pressure must equal their counterparts in the Vapor specific volume table parameter. The default matrix is a 25 × 100 table for water.
Liquid specific entropy table: M_L × N matrix containing the liquid specific entropy values corresponding to the normalized liquid internal energy and pressure vectors. Elements at the saturation boundary and at or above the critical pressure must equal their counterparts in the Vapor specific entropy table parameter. The default matrix is a 25 × 100 table for water.
Liquid temperature table: M_L × N matrix containing the liquid temperature values corresponding to the normalized liquid internal energy and pressure vectors. Elements at the saturation boundary and at or above the critical pressure must equal their counterparts in the Vapor temperature table parameter. The default matrix is a 25 × 100 table for water.
Liquid kinematic viscosity table: M_L × N matrix containing the liquid kinematic viscosity values corresponding to the normalized liquid internal energy and pressure vectors. Elements at the saturation boundary and at or above the critical pressure must equal their counterparts in the Vapor kinematic viscosity table parameter. The default matrix is a 25 × 100 table for water.
Liquid thermal conductivity table: M_L × N matrix containing the liquid thermal conductivity values corresponding to the normalized liquid internal energy and pressure vectors. Elements at the saturation boundary and at or above the critical pressure must equal their counterparts in the Vapor thermal conductivity table parameter. The default matrix is a 25 × 100 table for water.
Liquid Prandtl number table: M_L × N matrix containing the liquid Prandtl number values corresponding to the normalized liquid internal energy and pressure vectors. Elements at the saturation boundary and at or above the critical pressure must equal their counterparts in the Vapor Prandtl number table parameter. The default matrix is a 25 × 100 table for water.
Saturated liquid specific internal energy vector: Vector of length N containing the saturated liquid specific internal energy values corresponding to the pressure vector. Elements at or above the critical pressure must equal their counterparts in the Saturated vapor specific internal energy vector block parameter. The default is a 100-element vector for water.

Vapor Properties Tab

Normalized vapor internal energy vector: Vector of length M_V containing the normalized internal energy values corresponding to the rows of the vapor property tables. The vector must begin at 1 (the saturated vapor state) and end at 2. The default is a uniformly spaced 25-element vector.
Vapor specific volume table: M_V × N matrix containing the vapor specific volume values corresponding to the normalized vapor internal energy and pressure vectors. Elements at the saturation boundary and at or above the critical pressure must equal their counterparts in the Liquid specific volume table parameter. The default matrix is a 25 × 100 table for water.
Vapor specific entropy table: M_V × N matrix containing the vapor specific entropy values corresponding to the normalized vapor internal energy and pressure vectors. Elements at the saturation boundary and at or above the critical pressure must equal their counterparts in the Liquid specific entropy table parameter. The default matrix is a 25 × 100 table for water.
Vapor temperature table: M_V × N matrix containing the vapor temperature values corresponding to the normalized vapor internal energy and pressure vectors. Elements at the saturation boundary and at or above the critical pressure must equal their counterparts in the Liquid temperature table parameter. The default matrix is a 25 × 100 table for water.
Vapor kinematic viscosity table: M_V × N matrix containing the vapor kinematic viscosity values corresponding to the normalized vapor internal energy and pressure vectors. Elements at the saturation boundary and at or above the critical pressure must equal their counterparts in the Liquid kinematic viscosity table parameter. The default matrix is a 25 × 100 table for water.
Vapor thermal conductivity table: M_V × N matrix containing the vapor thermal conductivity values corresponding to the normalized vapor internal energy and pressure vectors. Elements at the saturation boundary and at or above the critical pressure must equal their counterparts in the Liquid thermal conductivity table parameter. The default matrix is a 25 × 100 table for water.
Vapor Prandtl number table: M_V × N matrix containing the vapor Prandtl number values corresponding to the normalized vapor internal energy and pressure vectors. Elements at the saturation boundary and at or above the critical pressure must equal their counterparts in the Liquid Prandtl number table parameter. The default matrix is a 25 × 100 table for water.
Saturated vapor specific internal energy vector: Vector of length N containing the saturated vapor specific internal energy values corresponding to the pressure vector. Elements at or above the critical pressure must equal their counterparts in the Saturated liquid specific internal energy vector block parameter. The default is a 100-element vector for water.

Ports

The block has a two-phase fluid conserving port. This port identifies the two-phase fluid network whose fluid properties the block provides.

Two-Phase Fluid Properties (2P)

Library