Thermodynamic Equations of State

Chapter 4. Physical Models for Thermo-Hydraulics

4.5 Thermodynamic Equations of State stresses, most often the assumption of a Newtonian fluid, and a thermo-dynamic equation of state (EOS). The latter of these relations originates from the assumption that the transients of the system under consider-ation are much slower than the time that a fluid needs to reach ther-modynamic equilibrium. The assumption of therther-modynamic equilibrium is very common in thermodynamic models. It holds very well for single phase systems, but for two phase systems with fast dynamics, thermody-namic non-equilibrium is sometimes taken into account. The deviation of the phases from equilibrium is the physical driving force for the phase change mass transfer.

The existence of an EOS can be derived directly from the First and Sec-ond Law of thermodynamics, written in differential form in (4.23–4.24), when the reversible limit is taken for system changes between equilib-rium states in the Second Law². When the First and Second Law for simple systems consisting of n distinct chemical species are integrated over an infinitesimal time interval dt:

dU = dQ − dW + Xn

i=1

hidMi (4.23)

dS_gen= dS − dQ T₀ −

Xn i=1

sidMi ≥ 0 (4.24) and reversible changes are assumed(dSgen= 0 and dW = dWrev= pdV), the equations can be combined into the following form:

dU = TdS − pdV + Xn

i=1

(h − Ts)dMi (4.25)

If a molar description and basis is used for all variables, the dependence of the energy on three different potentials is even more obvious:

dU = TdS − pdV + Xn

i=1

µidN_i (4.26)

whereµi is the chemical potential of species i. This form of the combined law proclaims the existence of a function of(n + 2) variables

U =

U

2A rigorous derivation is given in[Bejan, 1997], chapter 4. The definition of thermody-namic equilibrium is closely connected to the derivation of an equation of state and the assumption of reversibility.

Chapter 4. Physical Models for Thermo-Hydraulics

T= V U V S 

V,,N1,N2,...,Nn

−p = V U V V 

S,N1,N2,...,Nn

µi= V U V Ni

S,V,N1,...,Ni−1,Ni+1,...,Nn

From this fundamental relation in energy representation, a variety of other forms can be derived. Equations are called fundamental because they contain the complete thermodynamic property information of the fluid. Using this derivation of an equation of state, it is now possible to give a simple explanation of thermodynamic equilibrium: In a system in thermodynamic equilibrium, the values of the potential variables p, T and µiare homogeneous throughout the system. Thermodynamic equilibrium is thus equivalent to the “perfect mixing” assumption.

In order to get a reasonable parameterization, fundamental relations are calculated for pure chemical species and mixing rules are applied to construct multi-component relations. These material laws give an alge-braic relation between the thermodynamic variables pressure p, temper-ature T, densityρ, specific energy u, enthalpy h, entropy s, free energy f and free enthalpyg. As can be seen from (4.27), T and p can be calculated by symbolically differentiating the EOS. The enthalpy is then obtained from h = u + pv, other properties follow from further differentiation or non-linear combinations of known variables.

Three further fundamental equations can be derived by applying the Legendre transformation [Bejan, 1997] to the above combined First and Second Law of thermodynamics. Loss of information through the coordi-nate change is avoided when applying the Legendre transformation, de-tails about how to apply the transformation to thermodynamic functions are given in Appendix B. The further fundamental equations are:

• the Helmholtz free energy F=

F

• the Gibbs free enthalpy G=

G

• the enthalpy fundamental equation H=

H

Most often these fundamental equations are derived for single species properties and intensive variables as u(s,v),f(T,v),g(T,p) and h(s,p).

For numerical reasons they are also normalized, mostly with the critical values of pressure, temperature or density as normalization factors. For high-accuracy EOS, the Helmholtz energy function is the most popular form closely followed by the Gibbs free energy g(T,p) because temper-ature T, density ρ = 1/v and pressure p are the most easily measured

4.5 Thermodynamic Equations of State thermodynamic variables. The basic u(s,v) or h(s,p) which use the non-measurable entropy as one of their base variables are more of theoretical interest. Equations of state are derived by adapting coefficients in high-order two-dimensional polynomials, sometimes augmented with nonlinear terms, to measurement data, see[Span, 2000].

A practical approach with a long history which is very popular in pro-cess engineering is to use pressure – volume – temperature correlations, mostly cubic equations of state of the general form:

p= RT

V− b− ΘV −η

(V − b)V²+δV +ε ^(4.28) where R is the gas constant, T is the temperature and V is the volume, Θ,b> V,η,δ andεmay be constants including 0 or they may vary with T and/or composition. They are also based on easily measurable quantities but the cubic EOS have the disadvantage that they are not fundamental equations. In order to calculate all fluid properties, the heat capacity in the ideal gas state is needed in addition. It can be shown that the combi-nation of a p(V,T) surface and a function for the heat capacity cp= cp(T) in the ideal gas state are equivalent to a fundamental equation. The at-tractive feature of cubic EOS is that they allow relatively good property estimates with few parameters, namely the critical values of temperature Tc, pressure pcand specific volume vc. Numerical features of cubic EOS in combination with dynamic simulation are discussed in Section 4.6.

The simplest form of an EOS is the Ideal Gas Law pV = N R T

with the general gas constant R. In the simplest form it is complemented with a constant cpand sometimes called Perfect Gas Law. For ideal gases cp and h are functions of temperature only. The following relations are used to calculate the caloric properties:

c_p(T) = f (T) (4.29)

h(T) = h0+ Z T

T0

c_p(T)dT (4.30)

s(T,p) = s0+ Z _T

T0

cp(T)dT

T − R ln p

p₀ (4.31)

Note that the pressure dependent part of the specific entropy is identical for all ideal gases, only the temperature dependent part depends on the gas.

Chapter 4. Physical Models for Thermo-Hydraulics

Implemented forms of these material laws in the ThermoFluidlibrary are the Ideal Gas Law, the steam tables for water and high accuracy property functions for some refrigerants. The steam tables use both Gibbs functions g(T,p) and Helmholtz functions f (T,d) in different regions of the input coordinates T and p. The development of these equations of state is a complex task, using non-linear optimization techniques to fit the thermodynamic surface to available measurements, see[Span, 2000].

The generation of such property models on numerically efficient form is often the most troublesome area in simulation of thermodynamic flow problems, especially when high accuracy models are required. This prob-lem is also well known in process industry, where the quality of fluid property routines is a limiting factor for simulation accuracy in spite of a large number of commercially available medium property databases. The optimal situation for modelers and control engineers would be to have a spectrum of options ranging from simple, but computationally efficient to complex, high accuracy EOS models with a free choice of primary vari-ables.

For steam power plants the situation has improved recently with the standardization of the IAPWS/IF97³. The IF97 splits the region of validity of the steam tables into 5 regions, three of them use a Gibbs equation, one uses a Helmholtz equation and the two-phase region is defined by a saturation pressure curve and the adjoining fundamental equations. Even for the steam tables, an alternative with lower computational cost for online computations for control tasks is not available, but highly desirable.

Another, related problem with the available models is that they are not designed to be used in dynamic simulation. For dynamic computa-tions, efficiency is of higher priority than for steady state calculation or generation of tables. The main issues are:

• Computational efficiency is improved a lot if iterative routines are replaced by explicit computations. One type of iterations is avoided when the dynamic states of the thermodynamic model are identical to the input variables of the equation of state. Usually, there is no choice on the side of the equation of state (the only exception are the IF97 steam tables), but it is possible to apply a non-linear trans-formation to the mass and energy balance, see Section 4.6, to use the transformed version with any pair of intensive thermodynamic variables. The transformations require knowledge of certain thermo-dynamic derivatives used in the Jacobian matrix. These derivatives can easily be computed analytically from the equation of state, but are not included in standard implementations.

3The standard of the “Industrial Formulation of the properties of Water and Steam” is described in detail in[Wagner and Kruse, 1998].