Download Thermophysical Properties of High-Temperature Reacting Mixtures

 Thermophysical
Properties
of
High-Temperature
Reacting
Mixtures of Carbon and Water in the Range 400–30 000 K and
0.1–10 atm. Part 1: Equilibrium Composition and Thermodynamic
Properties
Wei-Zong Wang1,2*, A. B. Murphy3, J. D. Yan2, Ming-Zhe Rong1, J.W. Spencer 2,M. T. C. Fang 2
1. State Key Lab of Electrical Insulation for Power Equipment, Xi’an Jiaotong University, China
2. Department of Electrical Engineering and Electronics, University of Liverpool, UK
3. CSIRO Materials Science and Engineering, PO Box 218, Lindfield NSW 2070, Australia
Abstract
This paper is devoted to the calculation of the chemical equilibrium composition and thermodynamic
properties of reacting mixtures of carbon and water at high temperature. Equilibrium particle concentrations and
thermodynamic properties including mass density, molar weight, entropy, enthalpy and specific heat at constant
pressure, sonic velocity, and heat capacity ratio are determined by the method of Gibbs free energy
minimization, using species data from standard thermodynamic tables. The calculations, which assume local
thermodynamic equilibrium, are performed in the temperature range from 400 to 30 000 K for pressures of 0.10,
1.0, 3.0, 5.0 and 10.0 atm. The properties of the reacting mixture are affected by the possible occurrence of
solid carbon formation at low temperature, and therefore attention is paid to the influence of the carbon phase
transition by comparing the results obtained with and without considering solid carbon formation. The results
presented here clarify some basic chemical process and are reliable reference data for use in the simulation of
plasmas in reacting carbon and water mixtures together with the need of transport coefficients computation.
Keyword
Carbon, Water, thermal plasmas, Gibbs free energy minimization, equilibrium composition, thermodynamic
properties, phase transition
PACS: 52.25.Kn, Thermodynamics of plasmas
Submitted to Plasma Chemistry and Plasma Processing
1. Introduction
Electric arcs, which are classified as thermal plasmas under most conditions, are generated when electric
current passes through electrodes and an intervening gas. Due to their typical characteristics of high
temperature, high energy intensity and high reaction activity, electric arcs are widely used in many
1 applications, including arc welding, plasma spraying, current interruption and waste and biomass gasification.
The use of arc discharges in water with carbon (graphite) electrodes to produce carbon nanomaterials and
hydrocarbons under different conditions has raised a great deal of interest in the scientific literature and in
industry.
Carbon nanotubes (CNTs) and fullerenes are recognized as promising materials for many potential
applications in nanotechnology, electronics, optics, and other fields of materials science, as well as potential
uses in architectural fields [1], owing to their unique structures and novel physical properties. The preparation
of CNTs and fullerenes using an arc discharge with graphite electrodes in liquid water has emerged recently
[2]-[7]. It is noted that during the arc discharge process, bubbles forms at the liquid–gas interface and provide
the gaseous atmosphere for the arc channel [8]. The graphite electrodes are consumed and react with water
vapour in the gaseous discharge environment. To improve the quality of the CNTs and fullerenes that are
produced, it is important to understand the characteristics of the arc plasma. However, it is very difficult to
determine the properties of the arc plasma experimentally because of the liquid environment in which it is
formed.
Many processes have been developed that use high-intensity arcs to convert feed materials to
hydrocarbons. The gaseous or solid feed materials pass through the arc discharge channel and are heated to
high temperature. In one such process, steam (water vapour) is transferred into the chamber and reacts with
graphite (carbon) electrodes in the high current arc plasmas [9].
A different approach to producing gaseous
hydrocarbons is to induce arcing in bubbles formed at the interface between carbon electrodes and liquid
water [10]-[11] . Such processes are of particular importance because of the increasingly serious problems that
are associated with the consumption of large quantities of fossil fuels.
Numerical modelling is an important means of obtaining the information required to improve our
understanding of the processes by which carbon nanostructures and hydrocarbon fuels are formed. There have
however been very few models developed of carbon arcs in water and steam environments. An important
reason for this is the lack of available data for the thermodynamic properties and transport coefficients of the
arc plasmas reaction system; these data are necessary prerequisites for reliable numerical simulation studies.
To address this problem, we have undertaken a theoretical study of the composition, thermodynamic
properties and transport coefficients of high-temperature reacting mixtures of carbon and water.
The water–carbon mixture can be described by the formula Cq (H 2 O)1-q , where the carbon molar amount
q was chosen to range from 0 to 1 in steps of 0.1. The plasmas are assumed to be in local thermodynamic
equilibrium (LTE). Calculations are presented of thermodynamic properties and transport coefficients of these
mixtures of carbon and water vapour for the temperature range 400–30 000K, conditions which will satisfy
most requirements of thermal plasma modelling. Moreover, the influence of pressure on the properties is also
considered. The study is divided into two separate parts, which are presented as separate papers:
2 (1) Calculation of the equilibrium composition and the thermodynamic properties, including mass density,
molar weight, sonic velocity, heat capacity ratio, enthalpy, entropy, and specific heat, taking into account the
influence of the formation of solid carbon.
(2) Calculation of the transport coefficients, namely diffusion coefficients, thermal conductivities,
electrical conductivity and viscosity, including the collision integrals and the intermolecular potentials for
binary interactions used in their evaluation, as well as a comparison of the calculated properties with available
data in the literature.
The calculation of the equilibrium composition and the thermodynamic properties is the subject of this
paper. To our knowledge, the only comparable investigation was the early work by Tremblay et al. [12], who
studied the reaction of water vapour with carbon vapour in a high-intensity arc reactor, and gave the chemical
equilibrium composition in the limited temperature range from 2000 K to 5500 K at atmospheric pressure. We
note that some of the thermochemical data of the species that were estimated in Tremblay et al.’s paper have
since been updated. We have performed extensive calculations of the equilibrium composition of the C-H-O
system in high-temperature reacting mixtures with a larger set of species, for a wide range of pressures and
temperature and mixtures of carbon and water, in which the phase transition, dissociation and ionization all
occurs These data are of great use in understanding the reactions that occur.
We present in the next section the computational techniques and the database for a large set of species
produced from the reaction of carbon and water, which together are used to determine the equilibrium
composition and the thermodynamic properties at constant pressure and temperature. In Section 3, we describe
the calculated equilibrium composition for different mixtures of carbon and water. In the following section,
the calculated thermodynamic properties for the different mixtures are presented at atmospheric pressure. The
influence of pressure on the composition and thermodynamic properties as well as the carbon sublimation
temperature are described in Section 5. Finally, we give our conclusions in Section 6.
2. Method of calculation of equilibrium composition and thermodynamic properties
2.1. Data for the individual species
A prerequisite to determining the thermodynamic properties of gas mixtures is the computation of the
equilibrium composition, which is also the starting point for obtaining the transport coefficients [13]. We first
consider the dominant species of the Cq (H 2 O)1-q system. Earlier studies have revealed that the importance
of the concentrations of Cn H radicals in the temperature range between 2000 K and 6000 K, as well as the
carbon molecules and atom Cn and the hydrocarbons C x H y [12]. The species of the CHO system are
mostly reactive [14]-[15] and therefore, a large number of chemical species (69 heavy species in all),
including relevant atoms, ions and molecules, are taken into account as listed in table I. Solid carbon in the
form of graphite is taken into account and electrons are also considered when ionization occurs. Note that
3 other carbon particles such as clusters Cn (n  6) will form at temperatures under 2500 K; however, they have
negligible influence on the properties owing to the condensation of most of the pure carbon and are therefore
ignored. The large number of species allows a precise determination of the equilibrium concentrations, even if
the small concentration of some species does not have a significant influence on the transport coefficients
[16].
Table 1 Heavy species considered in the equilibrium composition calculation
Elements Species
C
C(s) , C, C+, C-, C2, C2-, C3, C4, C5, C2+, C++, C+++
H
H , H+, H-, H2, H2−, H2+, H3+
O
O2, O2-, O2+, O, O3, O+, O++, O+++, O-
C,H
CH, CH2, CH3, CH4, C2H, C3H, C4H, C5H, C6H, C2H2, C2H4, CH+, CH-
C,O
CO, CO2, C2O, C3O2, CO+, CO2+, CO-, CO2-
O,H
OH, OH+, OH-, HO2+, HO2-, H3O+, H2O2, H2O, H2O+
C,H,O
CH2O, CHO, C2H4O, CHO+, HCO3-, CH4O, CH2OH, CH3O, COOH, HCOOH, C2H6O
The required thermodynamic data for Cn H radicals are calculated from reported values of rotational
constants, vibrational fundamentals, excited electronic state energies, and enthalpies of formation [17]-[19],
and are determined by standard statistical mechanical methods using rigid-rotor and harmonic-oscillator
approximations. The coefficients are fitted using a least-squares method, constructing the three
thermodynamic functions of heat capacity, enthalpy and entropy as functions of temperature in a form
compatible to most computational plasma aerodynamics codes by means of the NASA computer program
CEA (Chemical Equilibrium with Applications) [20] ; they are presented in table 2. The standard form of
Gordon and McBride is used [21] .
Heat capacity C0P  a1T -2  a 2 T -1  a 3  a 4 T+a 5T 2  a 6 T 3 +a 7 T 4
(1)
Enthalpy H 0T / RT  a1T -2  a 2 T -1  a 3  a 4 T/2+a 5T 2 / 3  a 6 T3 /4+a 7 T 4 / 5  b1 /T
(2)
Entropy
S0T / R  a1T -2 / 2  a 2 T -1  a 3 ln T  a 4 T+a 5T 2 / 2  a 6 T3 /3+a 7 T 4 / 4  b 2
(3)
where R is the ideal gas constant and T is temperature.
The required thermodynamic data for neutral atoms and positively-charged atomic ions were calculated
from the internal partition functions, which were derived from the electronic energy levels tabulated by Moore
[22]-[23]. Data for other species such as the gaseous diatomic and polyatomic species and the solid phase
carbon were taken from the JANAF Thermochemical Tables in the form of polynomials whose coefficients
were fitted using a least-squares method [24] and extended to higher temperature by means of the formulas
and the molecular data given with the tables.
4 Table 2 Thermodynamic function coefficients for Cn H radicals
Species
C3 H
C4 H
C5 H
C6 H
a1
1.26154125E+05
1.07904731E+05
1.56558808E+05
1.94243816E+05
a2
-1.39329586E+03 -1.48234930E+03 -2.09064742E+03 -2.54570932E+03
a3
9.14529400E+00
1.08302384E+01
1.41882632E+01
1.71705356E+01
a4
1.53153382E-03
2.73836115E-03
2.62941322E-03
2.86242089E-03
a5
-5.67452769E-07
-1.00764917E-06
-9.91002460E-07
-1.10577072E-06
a6
9.18694217E-11
1.63796299E-10
1.62737449E-10
1.84100162E-10
a7
-5.47985566E-15
-9.82461300E-15
-9.81420989E-15
-1.12019598E-14
b1
8.13428050E+04
9.92774797E+04
1.11531739E+05
1.30190379E+05
b2
-2.71993054E+01 -3.51276387E+01 -5.29585619E+01 -6.84144584E+01
2.2. Method of calculation of plasma composition
The composition of the water steam–carbon plasma is defined as a stoichiometric combination of molar
amounts Cq (H 2 O)1-q , as was done for the carbon–argon mixture system considered in Ref. [25]. The
calculations presented in this paper are based on the assumption local thermodynamic equilibrium (LTE). We
take into account the influence of the presence of solid carbon in most of our calculations, although results
neglecting solid carbon are also given for comparison. The chemical equilibrium composition is determined as
a function of temperature and pressure by finding the composition that minimizes the Gibbs free energy,
which is a standard technique in equilibrium chemistry and described comprehensively in Refs. [26]-[29].
This method is applicable to closed isothermal and isobaric systems.
The thermodynamic properties of the system are completely determined by the total Gibbs free energy in
terms of the chemical potentials. w
G   ni i
(4)
i 1
where G is the Gibbs free energy of system; ni and i are the number density and chemical potential of
species i .
The chemical potential of species i can be evaluated as follows:
g
wmax
i  i0  RT ln( P / P0 )  RT ln(ni /  ni )
i 1
i  i0 for gaseous and plasmas species;
for condensed species
(5)
g
where i0 and P0 are the chemical potential and reference pressure in the standard state respectively; wmax
5 is the total number of gaseous and plasmas species. For gaseous species, the standard state is a hypothetical
state in which the gas is behaving ideally at unit fugacity and obeying the ideal gas equation whereas the
standard state of a pure condensed substance is its most stable for under a total pressure of 1atm.
These equilibrium rules must be combined with the following conditions: electrically quasi-neutrality,
Dalton’s Law and the law of conservation of stoichiometric equilibrium [30]. The Debye–Hückel correction,
which is an adjustment to the usual expression for the Gibbs free energy representing the additional effect of
electrostatic interactions between the charged particles, is taken into account in determining the chemical
potential of charged species and the total number density of the plasma, as described by Kovitya [31]. This
requires correction terms to be added to the expression given in Eq. (5) for the gaseous species and to Dalton’s
Law, i.e. the equation of state, as presented in Eq. (6) and (7) respectively:
i 
i0
g
wmax
 RT ln( P / P0 )  RT ln(ni /  ni )-RT (eZ i ) 2 / (8 0 kT D )
i 1
3
g
wmax
P  kT / (24 D )   ni kT
(6)
(7)
i 1
where Z i is the charge number of species i , e and  0 are respectively the electron charge and permittivity
of free space, and D is the Debye length calculated including only the contribution from electrons.
The relations used to determine the composition are not all linear in the independent variables and
therefore an iteration procedure is generally required for their solution. The minimization method of Gordon
and McBride [26] using Lagrange multipliers along with steepest descent Newton–Raphson iteration is
utilized to solve the non-linear equation system numerically.
2.3. Method of calculation of thermodynamic properties
The thermodynamic properties of gaseous mixtures, such as mass density, molar weight, enthalpy and
specific heat at constant pressure can be determined relatively simply once the chemical equilibrium
composition is known, using the standard formulas:
Mass density
g
wmax
   mi ni
(8)
i 1
where mi is the mass of species i .
Molar weight
g
wmax
M   xi M i
i 1
with
g
wmax
xi  ni /  ni
(9)
i 1
where xi and M i are the mole fraction and molar mass of species i .
6 Specific enthalpy
g
wmax
h   xi H i0,T / M
(10)
i 1
where H i0,T is the standard state molar enthalpy of species i .
Specific heat at constant pressure
Cp  (
h
)p
T
(11)
Specific entropy
g
wmax
s   xi Si / M
i 1
Si  ST0,i  R ln( xi )  R ln( P / P0 )
with
(12)
where ST0,i is the standard-state molar entropy of species i .
Heat capacity ratio

Cp
(13)
Cp  R / M
Sonic velocity
vs   RT / M
(14)
Corrections to perfect gas behaviour have been determined using the Debye-Hückel theory as described
in the literature [31]-[32].
3. Chemical equilibrium composition
The temperature dependence of the electron and carbon monoxide mole fractions at atmospheric pressure
depends on molar amount of carbon q, as shown in Fig. 1. The addition of carbon to water increases the
electron mole fraction at a given temperature due to the lower ionization energy of carbon atoms compared to
those of oxygen and hydrogen atoms. The behaviour of the mole fraction of carbon monoxide depends
critically on whether q is greater or less than 0.5. For q  0.5 , the mole fraction of CO is very low below
600 K, while for q  0.5 (i.e. high proportions of carbon in the mixture) its mole fraction is at least 0.3.
Clearly its mole fraction is highly sensitive to the proportion of carbon in the system for q between 0.5 and 0.6.
Further, for q  0.5 , increasing the carbon proportion leads increases the CO mole fraction at a given
temperature, while for q  0.5 the trend is reversed.
7 (a)
(b)
Fig. 1. Temperature dependence of equilibrium electron (a) and carbon monoxide (b) mole fractions for molar
amounts of carbon q from 0 to 1 in steps of 0.1 at atmospheric pressure, with the presence of solid carbon
taken into account.
The equilibrium composition of mixtures of carbon and water, taking into account the presence of solid
carbon, are presented for q = 0.2, 0.5 and 0.8 in Fig. 2. The fraction of carbon in the mixture strongly
influences the chemical reactions and hence the equilibrium species composition, especially in the low
temperature range for which dissociation reactions dominate.
8 9 Fig 2. Temperature dependence of equilibrium composition of carbon and water mixtures at atmospheric
pressure for q = 0.2, 0.5, and 0.8, taking into account the presence of solid carbon.
10 The following general trends in composition, as illustrated by the typical results shown in Fig. 2, can be
noted.
(1) Below 800 K, the reaction between carbon and water produces CO2 and CH4. As the temperature
increases, the mole fractions of H2O, CO2 and CH4 decrease quickly due to their rapid dissociation, mainly
into CO and H2 at about 1000 K.
(2) CO and H2 are typically present in appreciable quantities, and dissociate into carbon and oxygen
atoms and hydrogen atoms respectively at higher temperature, i.e. at around 7000 K for CO and 3500 K for
H2.
(3) For temperatures exceeding 10 000 K, the main reactions are ionization of the carbon, oxygen and
hydrogen atoms, which occurs at around 15 000 K. Note that the ionization of carbon begins at lower
temperature owing to its lower ionization energy, and therefore electrical neutrality below 11 000 K
corresponds to a balance between the mole fractions of carbon ions and electrons. From 20 000 K, multiple
ionizations start to occur and the plasma is completely ionized.
While the gaseous phase is always present in the system, substances in the condensed state, such as solid
carbon in present work, may also be present in some cases. When the temperature is increased through the
sublimation temperature, which has here been arbitrarily defined as the temperature corresponding to a solid
carbon mole fraction of 10-5 , this solid carbon enters the gaseous phase, and the addition of carbon into the
system has a substantial impact on the species composition.
(1)
For q = 0.2, which corresponds to a water-rich environment, we do not observe the formation of
solid carbon in the temperature range considered in present work. The concentrations of H2 reaches its peak
value at around 1000 K. Further temperature increases result in the dissociation of H2 into H atoms and the
production of O atoms, which mainly occurs at around 4000 K. The OH radical is present in appreciable
quantities, as is the case for pure water plasmas.
(2) For a C/H2O ratio equal to 1 (q = 0.5), solid carbon is produced, disappearing at around 1500 K in
accordance with previous results [12]. The only species existing in appreciable proportions in the temperature
range 1500 to 8000 K are CO and H2, and the hydrogen, carbon and oxygen atoms resulting from their
dissociation. For higher temperatures, carbon, oxygen and hydrogen are predominantly found in the form of
atoms and ions.
(3) For a relatively carbon-rich environment (q = 0.8), a series of stable substances appears in the
system in appreciable quantities. The phase transition of carbon occurs at about 4000 K. For carbon-rich
environments, hydrogenated radicals are formed when the solid carbon sublimates. The neutral species with
mole fraction above 0.01% (sufficient to influence the values of the transports coefficients) include C2H, CH,
C2H2, CH4, H2, H, O, carbon molecules Cn (n  2 ~ 5) , Cn H (n  1 ~ 6) radicals and the oxygenated
compounds CO and CO2. The dissociation and conversion of carbon molecules and carbon hydrogen radicals
mainly takes place in the temperature range from 2000 K to 6000 K.
11 Fig. 3. Temperature dependence of equilibrium composition of carbon and water mixtures for q = 0.8 at
atmospheric pressure, calculated without considering solid carbon.
By comparing the three cases, it can be seen that the appearance of solid carbon appear depends on the
proportion of carbon. For a fixed pressure, the higher the value of q, the higher the sublimation temperature
and thus the proportion of solid carbon.
The equilibrium composition of carbon and water mixtures for q = 0.8, calculated neglecting the formation
of solid carbon, are presented in Fig. 3. Comparing Figs. 2 and 3, it can be seen the composition is strongly
influence by the presence of solid carbon below the sublimation temperature. When solid carbon is not
considered, there are no water molecules present below 600 K, but mainly CO, C6H, C2H2, C2H4 and C3O2 (in
increasing order of importance). The carbon–water vapour mixture is not thermodynamically stable.
(1) CO and C2H2 are the dominant species in the temperature range 600 to 2500 K. The species C2H4
and C3O2 disappear suddenly and the C6H radical dissociates to form the C5H radical as the temperature
increases from 600 K. This in turn dissociates to form C4H and C3H or C5 molecules at around 2200 K, and
subsequent dissociation reactions form C2H and C3, and to a lesser extent C4, at about 2500 K. Above 2500 K,
C2H2 starts to dissociate, yielding chiefly C2H radicals and subsequently H atoms.
(2) The concentration of the C2H radical reaches a maximum at around 3800 K and the mole fraction of
C3 reaches its peak at around 4200 K. At higher temperatures the C2H radical begins to dissociate, leading to
the appearance of C2 and an increase in the H atom concentration. The mole fraction of these two species is
highest at around 4800 K.
(3) Above the sublimation temperature, the calculated chemical composition considering and neglecting
the presence of solid carbon is identical.
4. Thermodynamic properties
12 Using the chemical equilibrium composition data presented in the previous section, we evaluated the
thermodynamic properties of different carbon and water mixtures, in particular the mass density, molar weight,
enthalpy, specific heat at constant pressure, entropy, sonic velocity and specific heat, using Eqs. (8) to (14).
4.1. Mass density and molar weight
The formation of solid carbon clearly affects the composition and hence other properties below the
temperature of the phase transition. The variation of mass density and molar weight as a function of
temperature at atmospheric pressure is shown in Figs. 4 and 5 respectively. The two quantities are closely
associated.
If it was not for the chemical reactions such as dissociation and ionization, the mass density ρ would vary
in inverse proportion to temperature and the molar weight would remain constant according to the state
equation of an ideal gas. However, carbon, oxygen and hydrogen are highly reactive, even at lower
temperatures. Dissociation and ionization reactions cause both the mass density and molar weight of the gas
mixtures to decrease monotonically as the temperature increases when solid carbon is neglected, as shown in
Figs. 4b and 5b respectively. When solid carbon is considered, the mass density and molar weight change
rapidly in the temperature range in which the phase transition occurs. As the carbon to water ratio increases,
more carbon enters the gaseous phase in the form of molecules containing multiple carbon atoms, which
increases the molar weight as well as the mass density. The consequent jump in the molar weight and mass
density associated with the phase transition is clearly seen in Figs. 4 a and 5 a.
(a)
13 (b)
Fig. 4. Temperature dependence of equilibrium mass density of carbon and water mixtures for molar amounts
of carbon q from 0 to 1 in steps of 0.1 at atmospheric pressure considering (a) and neglecting (b) solid carbon.
For q = 1, the behaviour of pure carbon neglecting the formation of solid carbon is also shown in (a), with the
line marked with crosses, for comparison.
(a)
14 (b)
Fig. 5. Temperature dependence of equilibrium molar weight of carbon and water mixtures for molar amounts
of carbon q from 0 to 1 in steps of 0.1 at atmospheric pressure considering (a) and neglecting (b) solid carbon.
For q = 1, the behaviour of pure carbon neglecting the formation of solid carbon is also shown in (a), with the
line marked with crosses, for comparison.
4.2. Enthalpy and specific heat at constant pressure and entropy
(a)
15 (b)
Fig. 6. Temperature dependence of equilibrium enthalpy of carbon and water mixtures for molar amounts of
carbon q from 0 to 1 in steps of 0.1 at atmospheric pressure taking into account solid carbon in the
temperature range 400 to 30 000 K (a), and a comparison of enthalpy calculated considering (solid lines) and
neglecting (dotted lines) solid carbon (b). For q = 1, the behaviour of pure carbon neglecting the formation of
solid carbon is also shown in (a), with the line marked with crosses, for comparison.
The specific enthalpy and specific heat at constant pressure are shown in Figs. 6 and 7, respectively. As
for the molar weight and mass density, chemical reactions also contribute strongly to these quantities. Above 14 000 K, addition of carbon decreases the enthalpy. In contrast, from 8000 K to 14 000 K,
increasing the proportion of carbon has the opposite effect. At temperatures between the carbon sublimation
temperature and 8000 K, the enthalpy increases with increasing carbon proportion for q  0.5 , and decreases
for q > 0.5. As can be seen in Fig. 6a, the enthalpy at a given pressure and temperature is independent of the
proportion of carbon below the carbon sublimation temperature for q > 0.2, since the equilibrium carbon
concentration in the gas phase is governed by the phase transition. In the case of q = 0, 0.1, and 0.2, no solid
carbon appears and the addition of carbon causes the enthalpy to increase at temperatures below around
3000 K, and then decrease until 8000 K. It is noted that an “anomalous” behaviour occurs for temperatures
from 4000 K to 7500 K: the enthalpy of carbon–water mixtures is greater than that of pure water and pure
carbon.
16 (a)
(b)
17 (c)
Fig. 7. Temperature dependence of equilibrium specific heat at constant pressure of carbon and water
mixtures for molar amounts of carbon q from 0 to 1 in steps of 0.1 at atmospheric pressure considering (a)-(b),
and neglecting (c), solid carbon. For q = 1, the behaviour of pure carbon neglecting the formation of solid
carbon is also shown in (a) and (b), with the line marked with crosses, for comparison.
The steep changes in enthalpy corresponding to the respective peaks of the specific heat at constant
pressure are essentially a consequence of the heats of reaction. As shown in Figs. 7a and b, the specific heat at
constant pressure has multiple peaks. These are associated with, respectively, the dissociation reactions of the
polyatomic molecules CH4, CO2 and H2O at around 900 K, the dissociation of H2 at around 3800 K, the
dissociation of carbon molecules and hydrocarbon radicals at around 5000 K, the dissociation of CO at around
7000 K, the first-order ionization of carbon, hydrogen and oxygen atoms at around 15 000 K and the second
ionization of carbon atoms at around 28 000 K.
The mixing ratio of carbon and water affects the temperatures at which the peak values of the specific heat
occur. For the cases in which carbon sublimation occurs, the specific heat at constant pressure has the same
value below the sublimation temperature due to carbon condensation. For q  0.5 , the dissociation reactions
of carbon molecules and hydrocarbon radicals do not occur in sufficient quantity to produce a peak in the
specific heat. As more carbon is added, these reactions become more important and a peak occurs. In contrast,
the maximum of the specific heat associated with hydrogen dissociation becomes smaller as q increases.
The specific heat peak due to first ionization of carbon atoms is superimposed on the one caused by the
first ionization of oxygen and hydrogen atoms, even though the ionization energy of carbon atoms is lower, as
demonstrated in Fig. 7a. The different ionization energies (first ionization: 11.26 eV, 13.62 eV, and 13.60 eV
for C, O and H respectively; second ionization: 24.38 eV and 35.12 eV for C and O respectively) manifest
themselves through shifts in the temperature at which the first and second ionization peaks occur as the
18 mixing ratio changes.
If solid carbon is not taken into account, the specific heat, shown in Fig. 7 c, behaves differently due to
the different chemical equilibrium composition. The first two maxima in the specific heat are connected with
the dissociation reactions of the polyatomic molecules CH4, CO2 and H2O at around 900 K and the
dissociation of H2 at around 3800 K in the case of q  0.5 . As q increases, the height of the first peak falls
and that of the other rises, and the peaks moves to higher temperature. For q > 0.5, increasing q leads to a
larger number of species appearing in appreciable quantities, especially in the temperature range from 2500 K
to 5000 K, as was shown in Fig. 3. However, the peak associated with the dissociation of hydrogen is not as
significant as for q  0.5 .
Fig. 8 illustrates the entropy evolution of various mixtures of carbon and water.
As in the case of the
enthalpy, the observed drastic variations of the entropy strongly depend on dissociation and ionization
phenomena. The addition of carbon to the reacting mixtures drastically modifies the entropy for temperatures
higher than 4000 K. The change becomes more sensitive to the carbon concentration for q > 0.5, irrespective
of whether solid carbon is included. Below 4000 K, the dependence of entropy increases on q depends
strongly on whether q is below or above 0.5.
(a)
19 (b)
Fig. 8. Temperature dependence of equilibrium entropy of carbon and water mixtures for molar amounts of
carbon q from 0 to 1 in steps of 0.1 at atmospheric pressure, considering (a) and neglecting (b) solid carbon.
For q = 1, the behaviour of pure carbon neglecting the formation of solid carbon is also shown in (a) with the
line marked with crosses, for comparison.
4.3. Sonic velocity and heat capacity ratio
The sonic velocity and the heat capacity ratio are parameters that characterize a compressed fluid. The
sonic velocity is the distance travelled per unit time by a sound wave propagating through an elastic medium.
The heat capacity ratio, also known as the isentropic expansion factor, is the ratio of the specific heat at constant
pressure to the specific heat at constant volume.
The calculated values of sonic velocity and heat capacity ratio are shown in Figs. 9 and 10. The sonic
velocity increases with temperature. Dissociation and ionization are be reflected in changes in the slope.
Anomalous behaviour takes place in the temperature intervals 6200 to 8000 K and 1000 to 3500 K
respectively, for the cases considering and neglecting solid carbon.
The heat capacity ratio shows several characteristic peaks, which are related to the chemical reactions that
have been mentioned above. A discontinuity caused by the phase transition occurs q > 0.5, as can be seen by
comparing Fig. 10a with the results obtained without solid carbon shown in Fig. 10b.
20 (a)
(b)
Fig. 9. Temperature dependence of equilibrium sonic velocity of carbon and water mixtures for molar
amounts of carbon q from 0 to 1 in steps of 0.1 at atmospheric pressure considering (a) and neglecting (b)
soild carbon. For q = 1, the behaviour of pure carbon neglecting the formation of solid carbon is also shown in
(a) with the line marked with crosses, for comparison.
21 (a)
(b)
Fig. 10. Temperature dependence of equilibrium heat capacity ratio of carbon and water mixtures for molar
amounts of carbon q from 0 to 1 in steps of 0.1 at atmospheric pressure considering (a) and neglecting (b)
soild carbon. For q = 1, the behaviour of pure carbon neglecting the formation of solid carbon is also shown in
(a) with the line marked with crosses, for comparison.
5. Influence of pressure on equilibrium composition and thermodynamic properties
22 Fig. 11. Temperature dependence of equilibrium electron (a) and carbon monoxide (b) mole fractions at
pressures of 0.1, 1, 2, 3, 5 and 10 atm with q = 0.8, taking into account solid carbon.
23 Fig. 12. Carbon sublimation temperature at pressures of 0.1, 1, 2, 3, 5 and 10 atm for different molar ratios of
carbon and water.
Fig. 13. Temperature dependence of the specific heat at constant pressure at pressures of 0.1, 1, 2, 3, 5 and
10 atm with q = 0.8, taking into account solid carbon.
According to Le Chatelier’s principle, an increase in pressure tends to suppress dissociation and
ionization reactions, as well as the phase transition. Taking the case of q = 0.8 as an example, the influence of
pressure on chemical equilibrium composition and thermodynamic properties is illustrated in Figs. 11 to 13.
It is found that liquid water appears in the reacting system at 400 K, due to the increased boiling point,
when the pressure is increased to 10 atm. Changes in the equilibrium electron and carbon monoxide mole
fractions are shifted to higher temperatures as the pressure increases. The rapid increase of the carbon
monoxide mole fractions at temperatures above 3000 K shows the significant influence of the carbon phase
transition, i.e. carbon sublimation. At higher pressures, this increase becomes more intense. The decrease in
the mole fraction of electrons at a given temperature as pressure increases, shown in Fig. 11b, can be
explained by the suppression of successive ionization reactions.
Fig. 12 shows the sublimation temperature of carbon as a function of the initial proportions of carbon and
water vapour for different pressures. Increasing the pressure favours the production of solid carbon, increasing
the carbon sublimation temperature. The sublimation temperature is most sensitive to the molar ratio of
carbon and water for a ratio of about 1. For lower proportions of carbon, the change of sublimation
temperature resulting from a change in pressure is lower than that for molar ratios greater than 1.
The evolution of the specific heat at constant pressure depends on chemical reactions that are influenced
by the pressure. Figure 13 shows the influence of pressure on the position and height of the peaks in the
specific heat. Firstly, the temperature at which a given peak occurs is shifted to higher temperature due to the
suppression of chemical reactions caused by the increasing pressure, in accordance with Le Chatelier’s
principle. In addition, as the pressure increases, the rate of change of the degree of dissociation or ionization
24 with temperature decreases, and the height of the peaks in the specific heat becomes smaller.
6. Discussion and Conclusion
In this paper, we have presented the results of calculations of the chemical equilibrium composition and
thermodynamic properties of carbon and water plasmas with molar ratios of carbon from 0 to 1 for
temperatures from 400 to 30 000 K and pressures from 0.1 to 10 atm, which are conditions relevant to a wide
range of applications. A total of 69 heavy species including solid carbon, as well as electrons, have been taken
into account. The method of the Gibbs free energy minimization was used to determine the plasma
composition. Thermodynamic properties, including mass density, mole weight, enthalpy, entropy, specific
heat at constant pressure, sonic velocity and heat capacity ratio were presented in detail. The influence of the
carbon phase transition on the composition and thermodynamic properties has been described by comparing
the results considering and neglecting the formation of solid carbon. The important role of pressure in
determining composition and thermodynamic properties has also been demonstrated.
Some important results of the present study are as follows.
1. The molar ratio of carbon and water has a strong effect on the chemical reactions that occur, as
well as the species generated, and hence the thermodynamic properties. The influence of the
molar ratio depends strongly on whether the molar ratio is less than or greater than one.
2. Solid carbon is no longer present for low proportions of carbon in the mixtures.
3. Anomalous behaviour of some thermodynamic properties, with the value for the mixture not
falling between those for pure carbon and pure water, occurs due to the high reactivity of the
chemical elements present.
4. Increasing the pressure suppresses dissociation and ionization reactions and the phase transition,
and thus has a significant influence on the composition and thermodynamic properties, as well as
increasing the carbon sublimation temperature.
5. Rapid changes and in some cases discontinuities of species mole fractions and thermodynamic
properties with temperature occur near the carbon sublimation temperature.
6. Below the carbon sublimation temperature, the mass density is independent of the molar ratio of
carbon and water, since the relative mole fractions of the different gaseous species are constant
for a given pressure and temperature.
The results presented here clarify some basic chemical processes, and are reliable reference data for use in
simulation of plasmas in carbon and water mixtures. Transport coefficients, which are also required for such
simulations, will be given in Part 2.
Acknowledgments
This work was supported by the Chinese Government Scholarship program for postgraduates and Dual
25 Collaborative PhD Degree Program between Xi’an Jiaotong University and University of Liverpool.
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