PURE SUBSTANCE PROPERTIES

Last modified: 09/13/2011 06:21 PM

REPRESENTATION OF PURE SUBSTANCE PROPERTIES

SOLID, LIQUID, GASEOUS PHASES

A pure substance can be in one or more of three phases: solid, liquid or gaseous. Solid state may even include several varieties known as allotropic, which reflect the different possible arrangements of the crystal lattice.

These three phases are distinguished, at the microscopic level, by the intensity of intermolecular forces. In the solid state, they allow atoms only to oscillate around fixed positions randomly distributed or ordered (crystal).

Their intensity decreases in liquids, which have no proper form, but remain slightly compressible. This is called a short distance order and disorder at long range. In a gas, intermolecular forces are very weak and the molecules move in an erratic motion.

When heating a solid at a well chosen constant pressure, it turns into liquid, and we talk of fusion. If we continue to provide heat, the liquid turns to vapor, and we talk of vaporization. It is also possible that a solid turns directly into vapor, which is called sublimation. The temperature at which these changes are realized depends on the pressure exerted on the substance considered. For example at atmospheric pressure, the CO2 sublimes, that is to say, goes directly from solid to gaseous state, while water boils at 100 °C.

When a given mass of a pure substance is present in a single phase, its state is defined by two variables, for example its pressure and temperature. In the (P, T) plane, the three phases correspond to three areas, separated by three saturation curves (sublimation, vaporization and fusion) joining at the triple point T (Figure below).

Phases of a substance

Each curve corresponds to a two-phase equilibrium. For example, the rightmost curve is the set of points representing the equilibrium of a liquid with its vapor. The two-phase equilibrium assumes that the pressure and temperature satisfy a relationship characteristic of the nature of the fluid.

For each of these phase changes to happen, it is necessary to provide or absorb energy, called latent heat of change of state. During the change of state, there are significant variations in the specific volume, except for fusion-solidification. This is particularly the case during vaporization, vapor being about 600 to 1000 times less dense than the liquid. This change in specific volume occurs at constant pressure and temperature.

Let us give some examples illustrating either the practical use of phase changes, or the constraints induced by the presence of a liquid-vapor equilibrium:

When adding ice cubes to a warm drink, we provide heat which melts them, which cools it. As the latent heat of fusion of ice is much larger than the heat capacity of the drink, we get the desired cooling effect without bringing too much water dilution;
To transport methane over long distances by sea, it is liquefied at a temperature of - 160 °C, reducing its specific volume 600 times with respect to gas. It is thus possible to maintain atmospheric pressure in the tanks of the LNG ship. Although these tanks are very well insulated, you cannot avoid some heat exchange with the surroundings, which has the effect of vaporizing a small amount of gas which is used for propulsion;
In contrast, butane or propane gas distributed for culinary purposes is confined in a liquid state at room temperature in thick metal cylinders, in order to resist the inside pressure of a few hundred psi or tens of bar;
All cooking done in boiling water takes place at 100 °C if the pressure is equal to 1 atmosphere, and this irrespective of the thermal heat supplied to the cooking. Thus we can define the precise duration for cooking a recipe, for example, a boiled egg;
The principle of the pressure cooker is to overcome this limit of 100 °C by doing the cooking in a chamber at a pressure exceeding 1 atm. It can reach 110 °C and 120 °C, in order to cook food more quickly;
An example of condensation is that which is deposited on cold surfaces in contact with moist air, like mist on a window, or the morning dew on leaves.

The triple point corresponds to the state where it is possible to simultaneously maintain equilibrium between all three phases. The critical point represents the state where the phase of pure steam has the same properties as the pure liquid phase. At higher temperatures and pressures (supercritical), it is not possible to observe a separation between liquid and gas phases: the disk surface which separates the liquid and vapor phases disappears at the critical point.

In practice, in heat engines, the working fluid is most often in the gaseous or liquid state, or as a mixture of gaseous and liquid phases. To calculate their properties, one is led to distinguish two broad categories of fluids: the ideal gas, which can be pure or compound, which includes the perfect gas, and condensable real fluids, which can also be pure or compound.

In Thermoptim all these types of fluids are represented, with the exception of mixtures of real fluids, which can however be taken into account through external mixtures. In what follows, we will successively deal with ideal gases, their mixtures, liquids and solids, properties of a mixture of phases in liquid-vapor equilibrium and condensable real fluids. We conclude with the study of moist mixtures of dry gas with water vapor.

PERFECT AND IDEAL GASES

Many thermodynamic fluids in the vapor phase may be treated as ideal gases in a wide range of temperatures and pressures. This requires that the temperature-pressure combination deviates from the condensation zone as much possible (that is to say that the pressure is not "too" high or the temperature "too" low). Such conditions are commonly the case for gases known as "permanent" at ambient temperature and pressure, such as hydrogen, oxygen, nitrogen, the oxygen-nitrogen mixture that is dry air etc. Even the water vapor in the atmosphere behaves almost like an ideal gas as its partial pressure remains moderate.

The ideal gas model is based on the assumption that the molecular interactions in the gas can be neglected, except for collisions between them. The kinetic theory of gases can then explain the gas macroscopic behavior from mechanical considerations, and statistics on the movements of its molecules.

The fundamental assumption of ideal gases is that their internal energy (and their enthalpy) is independent of pressure. Given that all real gases can be liquefied, there is rigorously no ideal or perfect gas. These concepts are fundamental, however, because the practical determination of the state of a real fluid is always made by reference to the corresponding ideal or perfect gas, which approximates the behavior at very low pressure and/or high temperature.

Specifically, to represent the state of a fluid, a cascade of increasingly complex models is used depending on the desired accuracy, the simplest being that of the ideal gas, the most elaborate corresponding to real fluids.

Let us recall that Diapason session S04aEn is devoted to the properties of pure substances.

Equation of state of ideal gases

The equations of perfect and ideal gases are very close, the first being in fact a special case of the latter. The equation of state of an ideal gas can be written:

Pv = rT (1)

with r = R/M (kJ/kg/K)

R is the universal constant = 8,314 (kJ/kmol/K)

M is the molar mass of the gas (kg/kmol)

According to the units used, equation (1) takes different forms:

in mass units: P v = r T
in molar units: P vm = R T

Based on the total volume V occupied by the fluid, n being the number of kilomol:

in mass units: P V = m r T
in molar units: P V = n R T

We can prove that equation (1) implies in particular that the internal energy and enthalpy of an ideal gas depend only on its temperature, and that:

r = cp - cv

We thus have:

cv = du/dT (J/kg/K)
cp = dh/dT (J/kg/K)

We call a "perfect" gas an ideal gas whose specific heat capacities cp and cv are constant.

For such a gas, the internal energy and enthalpy are linear functions of temperature. Note that other authors call perfect gas what we call an ideal gas. In this case, they must each time tell whether the heat capacity of gas depends or not on the temperature.

The assumption of perfect gas (cp and cv constant) is rigorously met for monatomic gases (which have no rotation or molecular vibration mode The larger the number of atoms in the gas molecule (and thus possible vibration modes), the less this assumption is valid.

Statistical thermodynamics allows us to determine the values of molar heat capacity of monatomic and diatomic gases..

For the former, we get:

Cp = 5/2 R = 20.785 kJ/kmol Cv = 3/2 R = 12.471 kJ/kmol

For diatomic gases usually, at room temperature, we obtain:

Cp = 7/2 R = 29.1 kJ/kmole Cv = 5/2 R = 20.785 kJ/kmole

The figure below shows the changes in Cp for some typical mono-, bi-and tri-atomic gases.

Molar heat capacity of some gases

Practical determination of the state of a perfect gas

Two parameters are sufficient to define an ideal gas: either its heat capacities at constant pressure and volume, or one of them and the value of its molar mass M, or the values of M and g, the ratio of cp to cv.

Under the assumptions, Pv = RT, cp - cv = r, and constant cp and cv, we can easily calculate the internal energy u, enthalpy h of the gas from any reference state T0.

u = u0 + cv (T - T0) et h = h0 + cp (T - T0) (2)

Tds = du + Pdv

ds = 1/T du + p/T dv

which allows us to easily calculate the entropy of the fluid: ds = cp/T dT + r/v dv

s = s0 + cp ln(P/P0) + cv ln(v/v0)

s = s0 + cv ln(T/T0) + r ln(v/v0) (3)

s = s0 + cp ln(T/T0) - r ln(P/P0)

The choice of the reference point is arbitrary and depends on conventions.

The characteristics (M, cp, cv, r, g) of some substances are given in the table below.

	M	cp	cv	r	g
	kg/kmole	J/kg/K	J/kg/K	J/kg/K
Air	28.97	1005	718	287.1	1.40
Hydrogen (H2)	2.016	14320	10170	4127	1.41
Helium (He)	4.003	5234	3140	2078	1.66
Methane (CH4)	16.04	2227	1687	518.7	1.32
steam (H2O)	18.02	1867	1406	461.4	1.33
Neon (Ne)	20.18	1030	618	411.9	1.67
Acetylene (C2H2)	26.04	1712	1394	319.6	1.23
Carbon monoxyde (CO)	28.01	1043	745	296.6	1.40
Nitrogen (N2)	28.02	1038	741	296.6	1.40
Ethylene (C2H4)	28.05	1548	1252	296.4	1.24
Ethane (C2H6)	30.07	1767	1495	276	1.18
Oxygen (O2)	32.00	917	653	259.6	1.40
Argon (Ar)	39.94	515	310	208	1.67
Carbon dioxyde (CO2)	44.01	846	653	188.9	1.30
Propane (C3H8)	44.09	1692	1507	188.3	1.12
Isobutane (C4H10)	58.12	1758	1620	143.1	1.09
Octane (C8H18)	114.23	1711	1638	72.8	1.04

Practical determination of the state of an ideal gas

An ideal gas differs from a perfect gas because its thermal capacity is not constant, but depends solely on temperature.

Most often, Cp is represented by a polynomial fit of order n in T, as (either in molar units, as below, or in mass units):

Cp = S Cpi Ti (4)

The solution chosen in Thermoptim is a 7-term development of the following type:

Cv is deduced by Cv = Cp - R, and specific variables cp and cv are obtained by dividing these values by the molar mass of the substance.

In practice, if we do not have a software application to calculate the properties of substances, and if we intend to determine the evolution of an ideal gas over a limited range of temperature, it is possible to assimilate it to a perfect gas, which enables us to use all the results established for them, provided its specific heat capacity is calculated at the mean temperature of the process. Of course, this is only valid in first approximation, but the loss of accuracy is compensated by a simplification of the calculations.

From a polynomial cp or cv = f (T), it is easy to calculate u, h or s by integrating their differential relations, which leads to:

Equation of isentropic process

In many compressors or turbines, the fluid undergoes an evolution close to the isentropic, which forms the reference against which the actual process is calculated. The isentropic equations are therefore of particular importance.

By posing s = s0 = Const, we find, for perfect gas

or, in differential form:

dP/P + g dv/v = 0 (9)

For an ideal gas, the isentropic equation does not take as simple a form as for a perfect gas. The formalism is more complex, but it remains quite usable in practice, as the pressure and temperature variables can be separated.

IDEAL GAS MIXTURES

In many practical applications, we are dealing not with pure gases, but gas mixtures, whose composition may vary. This is particularly the case in an internal combustion engine cylinder: the composition of flue gas is evolving gradually as the combustion unfolds.

Dalton's law states an important result: a mixture of ideal gases behaves itself as an ideal gas.

The composition of a mixture is usually determined from either mole fractions or mass fractions of the constituents. In this section we establish the expressions for various usual thermodynamic properties in terms of these quantities.

Mole fractions and mass fractions

The total number of moles n of the mixture equals the sum of the numbers of moles of each component:

n = n1 + n2 + n3 + ... nn = S ni

The mole fraction of a component is defined as the ratio of the number of moles of this constituent to the total number of moles in the mixture:
xi = ni/n et S xi = 1

Moreover, the law of mass conservation implies that the total mass is equal to the sum of the masses of the constituents:

m = m1 + m2 + m3 + ... mn = S mi

The mass fraction of a component is defined as the ratio of the mass of the component to the total mass of the mixture:

yi = mi/m and S yi = 1

Dalton law of ideal gases

We define the partial pressure Pi of a component as the pressure exerted by this component if it occupied alone the volume V of the mixture, its temperature being equal to that of the mixture

Dalton's law postulates that the pressure, internal energy, enthalpy and entropy of a mixture of ideal gases at temperature T and pressure P are respectively the sum of the pressures, internal energies, enthalpies and partial entropies of gas constituents, that is to say taken separately at temperature T and their partial pressures.

Each component behaves as if it existed at the temperature T of the mixture and was alone in the volume V.

Physically, this means that the fields of molecular forces of the individual components do not interfere with each other.

Mathematically, Dalton's law translates into the following two laws, the total pressure being P:

Pi = xi P (10)

cp = 1/m S cpi mi = S yi cpi (11)

The mixture heat capacity at constant pressure equals the sum of the products of the specific heat capacity of the constituents by their mass fractions.

Dalton's law means that a mixture of ideal gases behaves itself as an ideal gas, whose fictitious mole molar mass would be M = S xi Mi.

This means that the results established for ideal gases can be used to calculate the evolution of mixtures of these gases, which is of paramount importance in practice.

We have: yi = xi Mi/M

We can also define an equivalent ideal gas constant by:

r = R/M (kJ/kg/K)

In mass notations, the ideal gas constant of the mixture is expressed in a very simple form:

r = S yi ri (12)

Moreover, H = S Hi et m cp T = S cpi mi T

We deduce (11).

In molar notations, (11) becomes:

Cp = S xi Cpi (13)

Energy properties of ideal gas mixtures

Enthalpy of a mixture

According to Dalton's law, the enthalpy of a mixture of ideal gases is equal to the sum of the enthalpies of each component.

mass notations: h = S yi hi (14)
molar notations: H = S xi Hi (15)

The calculation of internal energy would be done in the same way.

Entropy of a mixture

It is when calculating the entropy of the mixture that the profound significance of Dalton's law appears, i.e. that the value of the entropy of the mixture shall be calculated by summing the entropies of the constituents taken at temperature T and their partial pressures Pi.

Indeed, the entropy of the mixture is greater than the sum of the entropies of the constituents before mixing by a value km = - r S xi ln xi > 0. Physically, this is explained by the irreversibility of the mixing operation. It is important to note that the km gap remains constant (independent of T and P) as long as the concentrations do not vary. This gap must be taken into account once and for all when calculating the entropy of reference, but then, the mixture behaving itself as an ideal gas, we no longer need to worry about it.

Of course, this does not remain true if the composition of the mixture comes to vary during the process, in which case it is necessary to calculate the different values of km to determine the variations of entropy.

Thermoptim uses two types of ideal gases:

pure gases, whose properties are predetermined in the software, not user-modifiable; they number about 20;
compound gases, constructed by the user at will from pure gases included in the database. Their properties are calculated by the software by applying Dalton’s law.

LIQUIDS AND SOLIDS

Liquids and solids are described as ideal when compressibility is negligible (v = Const).

Since an "ideal" liquid or solid cannot be subjected to any form of reversible work, a well chosen single variable is enough to represent its thermodynamic state.

One can indeed consider that the internal energy of an ideal liquid or solid depends only on its temperature:

du/dT = c, as for an ideal gas.

However the enthalpy of the ideal liquid (that of the solid has no physical sense) is still a function of pressure:

dh = cdT + vdP

We can still define the entropy: ds = 1/T du + P/T dv, which gives:

ds = c dT/T

As for gas, a liquid or solid whose heat capacity does not vary significantly is called "perfect": c = Const.

Then:

u - u0 = c (T - T0)

h - h0 = c ( T - T0) + v ( P - P0)

s - s0 = c ln(T/T0)

In practice, characteristics of liquids are often identified at the saturation pressure. However, as the correction v (P - P0) is generally small, this convention has little importance.

For an isobaric evolution, the calculation of thermodynamic properties of liquids involves only their heat capacity.

LIQUID - VAPOR EQUILIBRIUM OF A PURE SUBSTANCE

In compressible fluid machines it is often necessary to study the processes bringing the fluid into the liquid state. The ideal gas to zero does not exist, all fluids being condensable, and it is necessary to know their properties in the liquid state.

The study of vapor-liquid equilibrium is based on the law of phase mixture or lever rule that merely reflects the extensiveness of state functions with the assumption that the interfacial energy is negligible, which reads: volume, internal energy, enthalpy, entropy of a phase mixture, at pressure P and temperature T, are respectively the sums of these properties in the different phases constituting the mixture, taken in isolation at the same pressure and at the same temperature.

On various thermodynamic charts presented below, the vaporization or vapor-liquid equilibrium area is evident for temperatures and pressures lower than the critical point. This area is bounded on the left by the saturated liquid curve, and on the right by the dry saturated vapor curve. These two curves define the saturation curve, whose shape is characteristic. Between these two curves, pressure and temperature are no longer independent: they are connected by a relationship known as saturation pressure law or vapor pressure law, and the system is mono-variant.

Saturation pressure law

Many formulas have been proposed to algebraically represent the saturation pressure law. One of the most used is that of Antoine:

ln(Ps) = A - B/(C+T)

where A, B and C are characteristic parameters of the fluid, and Ps the saturated vapor pressure.

With Ps in bar and T in K, for example for water: A = 11.783 B = 3895.65 C = - 42.1387

This however is not very precise, and in Thermoptim, the following development was selected:

Vapor quality

In the middle part of the vapor-liquid equilibrium zone, fluid is present in both liquid and vapor phases. In this central zone, isobars and isotherms are combined, the liquid-vapor change taking place at constant temperature and pressure. The composition of the mixture is defined by its quality x, ratio of vapor mass mg to the total mass (mg plus the mass of liquid ml).

x = mg/(mg + ml) (16)

Enthalpy of vaporization

The length of the vaporization line gives the enthalpy (or heat) of vaporization L for the fluid conditions P and T considered. It is proportional to it in the entropy (s, T) and Mollier (s, h) charts, and equal to it in the (h, ln(P)) chart:

hg - hl = L
sg - sl = L/T

The above relationships can be demonstrated from relationship 2.4.20 expressing that the free energy is minimal at equilibrium: during the vaporization process, the Gibbs energy evolves from gl to gg.

If evolution is reversible, we have: dg = 0, or gl = gg:

hl - T sl = hg - T sg

We thus find the relationship hlg = (hg - hl) = T (sg - sl) = T slg.

L is a decreasing function of temperature, zero for T above the critical temperature. A formula due to Clapeyron allows us to estimate L from the saturation pressure law:

L = T (vg- vl) dPs/dT

In this formula, the gas specific volume vg is obtained from the vapor equation of state, and the liquid specific volume vl from a proper relationship.

In Thermoptim, we opted for a direct relationship giving L as a function of reduced temperature Tr = T/Tc :

Calculation of pure two-phase substance properties

By applying the law of phase mixture, we have:

v = (1 - x) vl + x vg

u = (1 - x) ul + x ug

h = (1 - x) hl + x hg = hl + x L

s = (1 - x) sl + x sg = sl + x L/T

Values of critical points and vaporization enthalpies for some common substances are given in table below.

	Ts (1bar)	r à Ts	L	cp (Ts)	Pc	Tc
	K	kg/liter	MJ/kg/K	kJ/kg/K	bar	K
air	80.2	0.860			37.70	132.6
oxygen	90.2	1.120	0.211	1.699	50.40	154.4
nitrogen	77.4	0.812	0.197	2.038	33.96	126.3
CO2 (sublimation)	194.7	0.793	0.369		73.50	304.2
CO	81.7	0.799	0.215		34.90	133.0
water	373.2	0.958	2.260	4.185	221.00	620.4
hydrogen	20.4	0.070	0.467	9.794	12.96	33.3
helium	4.3	0.122	0.023	4.604	2.28	5.3
argon	87.3	1.420	0.163	1.130	48.59	150.8
methane	111.5	0.424	0.503	3.474	46.27	190.7
ethane	184.6	0.546	0.489	2.427	49.80	305.4
ethylene	169.7	0.610	0.467	2.637	51.33	282.7
propane	230.6	0.582	0.410	2.511	42.52	370.0

REPRESENTATIONS OF REAL FLUIDS

The increase in pressure and/or lowering of the temperature can justify a reconsideration of the ideal gas equation (1). This is especially the case near the liquid-vapor equilibrium zone.

When the fluid no longer satisfies the ideal gas equation, its internal energy and enthalpy are no longer based solely on temperature.

This behavior is shown in Figurebelow, for superheated steam. It is clear that the heat capacity cp of this substance is the more affected by the pressure as it is higher and the temperature is lower (that is to say, especially in the immediate vicinity of the saturation curve).

To determine the state of a real fluid, we use most often a thermodynamic chart, a table of thermodynamic property values, or a set of equations of state covering the various zones necessary.

Traditionally, thermodynamic charts are the most used. There has however been a marked evolution of the practice, the development of micro-computers making possible the direct calculation of fluid thermodynamic properties in a wide range of variation of state variables. Thermoptim allows us to make such calculations precisely.

However, even if one has a fluid properties computer, such as the Thermoptim applet calulator, charts retain a strong interest to provide education, because they can easily view the real gas properties, including the liquid-vapor zone.