packagePartialLinearFluid

Information

Linear Compressibility Fluid Model

This linear compressibility fluid model is based on the assumptions that:

• The specific heat capacity at constant pressure (cp) is constant
• The isobaric expansion coefficient (beta) is constant
• The isothermal compressibility (kappa) is constant
• Pressure and temperature are used as states
• The influence of density on specific enthalpy (h), entropy (s), inner energy (u) and heat capacity (cv) at constant volume is neglected.

That means that the density is a linear function in temperature and in pressure. In order to define the complete model, a number of constant reference values are needed which are computed at the reference values of the states pressure p and temperature T. The model can be interpreted as a linearization of a full non-linear fluid model (but it is not linear in all thermodynamic coordinates). Reference values are needed for

1. the density (reference_d),
2. the specific enthalpy (reference_h),
3. the specific entropy (reference_s).

Apart from that, a user needs to define the molar mass, MM_const. Note that it is possible to define a fluid by computing the reference values from a full non-linear fluid model by computing the package constants using the standard functions defined in a fluid package (see example in liquids package).

In order to avoid numerical inversion of the temperature in the T_ph and T_ps functions, the density is always taken to be the reference density in the computation of h, s, u and cv. For liquids (and this model is intended only for liquids) the relative error of doing so is 1e-3 to 1e-4 at most. The model would be more "correct" based on the other assumptions, if occurrences of reference_d in the computations of h,s,u and cv would be replaced by a call to density(state). That would require a numerical solution for T_ps, while T_ph can be solved symbolically from a quadratic function. Errors from this approximation are small because liquid density varies little.

Efficiency considerations

One of the main reasons to use a simple, linear fluid model is to achieve high performance in simulations. There are a number of possible compromises and possibilities to improve performance. Some of them can be influenced by a flag. The following rules where used in this model:

• All forward evaluations (using the ThermodynamicState record as input) are exactly following the assumptions above.
• If the flag constantJacobian is set to true in the package, all functions that typically appear in thermodynamic Jacobians (specificHeatCapacityCv, density_derp_h, density_derh_p, density_derp_T, density_derT_p) are evaluated at reference conditions (that means using the reference density) instead of the density of the current pressure and temperature. This makes it possible to evaluate the thermodynamic Jacobian at compile time.
• For inverse functions using other inputs than the states (e.g pressure p and specific enthalpy h), the inversion is using the reference state whenever that is necessary to achieve a symbolic inversion.
• If constantJacobian is set to false, the above list of functions is computed exactly according to the above list of assumptions

Authors:

Francesco Casella
Dipartimento di Elettronica e Informazione
Politecnico di Milano
Via Ponzio 34/5
I-20133 Milano, Italy
email: casella@elet.polimi.it
and
Hubertus Tummescheit, previously at
Modelon AB
Ideon Science Park
SE-22730 Lund, Sweden

Parameters

Contents

Name	Description
ThermodynamicState	A selection of variables that uniquely defines the thermodynamic state
BaseProperties	Base properties of medium
setState_pTX	Set the thermodynamic state record from p and T (X not needed)
setState_phX	Set the thermodynamic state record from p and h (X not needed)
setState_psX	Set the thermodynamic state record from p and s (X not needed)
setState_dTX	Set the thermodynamic state record from d and T (X not needed)
setSmoothState	Return thermodynamic state so that it smoothly approximates: if x > 0 then state_a else state_b
pressure	Return the pressure from the thermodynamic state
temperature	Return the temperature from the thermodynamic state
density	Return the density from the thermodynamic state
specificEnthalpy	Return the specific enthalpy from the thermodynamic state
specificEntropy	Return the specific entropy from the thermodynamic state
specificInternalEnergy	Return the specific internal energy from the thermodynamic state
specificGibbsEnergy	Return specific Gibbs energy from the thermodynamic state
specificHelmholtzEnergy	Return specific Helmholtz energy from the thermodynamic state
velocityOfSound	Return velocity of sound from the thermodynamic state
isentropicExponent	Return isentropic exponent from the thermodynamic state
isentropicEnthalpy	Return isentropic enthalpy
specificHeatCapacityCp	Return specific heat capacity at constant volume
specificHeatCapacityCv	Return specific heat capacity at constant volume from the thermodynamic state
isothermalCompressibility	Return the isothermal compressibility kappa
isobaricExpansionCoefficient	Return the isobaric expansion coefficient
density_derp_h	Return density derivative w.r.t. pressure at constant specific enthalpy
density_derh_p	Return density derivative w.r.t. specific enthalpy at constant pressure
density_derp_T	Return density derivative w.r.t. pressure at constant temperature
density_derT_p	Return density derivative w.r.t. temperature at constant pressure
density_derX	Returns the partial derivative of density with respect to mass fractions at constant pressure and temperature
molarMass	Return molar mass
T_ph	Return temperature from pressure and specific enthalpy
T_ps	Return temperature from pressure and specific entropy