we examine several thermodynamic concepts that are important in the study of combustion. We first briefly review basic property relations for ideal gases and ideal-gas mixtures and the first law of thermodynamics. Although these concepts are likely to be familiar to you from a previous study of thermodynamics, we present them here since they are an integral part of our study of combustion. We next focus on thermodynamic topics related specifically to combustion and reacting systems: concepts and definitions related to element conservation; a definition of enthalpy that accounts for chemical bonds; and first-law concepts defining heat of reaction, heating values, etc., and adiabatic flame temperatures. Chemical equilibrium, a second-law concept, is developed and applied to combustion-product mixtures. We emphasize equilibrium because, in many combustion devices, a knowledge of equilibrium states is sufficient to define many performance parameters of the device; for example, the temperature and major species at the outlet of a steady-flow combustor are likely to be governed by equilibrium considerations. Several examples are presented to illustrate these principles.
REVIEW OF PROPERTY RELATIONS
EXTENSIVE AN INTENSIVE PROPERTIES
The numerical value of an extensive property depends on the amount (mass or number of moles) of the substance considered. Extensive properties are usually denoted with capital letters; for example, V (m3) for volume, U (J) for internal energy, H (J) (= U + PV) for enthalpy, etc. An intensive property, on the other hand, is expressed per unit mass (or per mole), and its numerical value is independent of the amount of substance present. Mass-based intensive proper-ties are generally denoted with lower-case letters; for example, v (m3/kg) for specific volume, u (J/kg) for specific internal energy, h (J/kg) (= u + Pv) for specific enthalpy, etc. Important exceptions to this lower-case convention are the intensive properties temperature T and pressure P. Molar-based intensive properties are indicated in this book with an overbar, e.g., u and h (J/kmol). Extensive properties are obtained simply from the corresponding intensive properties by multiplying the property value per unit mass (or mole) by the amount of mass (or number of moles); i.e.,
(2.1)
In the following developments, we will use either mass- or molar-based intensive properties, depending on which is most appropriate to a particular situation.
EQUATION OF STATE
An equation of state provides the relationship among the pressure, P, temperature, T, and volume V (or specific volume v) of a substance. For ideal-gas behavior, i.e., a gas that can be modeled by neglecting intermolecular forces and the volume of the molecules, the following equivalent forms of the equation of state apply:
(2.2a)
(2.2b)
(2.2c)
or
(2.2d)
where the specific gas constant R is related to the universal gas constant Ru(= 8315 J/kmol-K) and the gas molecular weight MW by
(2.3)
The density ρ in Eqn. 2.2d is the reciprocal of the specific volume (ρ =1/v = m/ V). Throughout this blog, we assume ideal-gas behavior for all gaseous species and gas mixtures. This assumption is appropriate for nearly all of the systems we wish to consider since the high temperatures associated with combustion generally result in sufficiently low densities for ideal-gas behavior to be a reasonable approximation.
CALORIFIC EQUATIONS OF STATE
Expressions relating internal energy (or enthalpy) to pressure and temperature are called calorific equations of state, i.e.,
(2.4a)
(2.4b)
The word "calorific" relates to expressing energy in units of calories, which has been superseded by the use of joules in the SI system. General expressions for a differential change in u or h can be expressed by differentiating Eqns. 2.4a and b:
(2.5a)
(2.5b)
In the above, we recognize the partial derivatives with respect to temperature to be the constant-volume and constant-pressure specific heats, respectively, i.e.,
(2.6a)
(2.6b)
For an ideal gas, the partial derivatives with respect to specific volume, (∂u/∂v)T, and pressure, (∂h/∂P)T, are zero. Using this knowledge, we integrate Eqn. 2.5, substituting Eqn. 2.6 to provide the following ideal-gas calorific equations of state:
(2.7a)
(2.7b)
In a subsequent section, we will define an appropriate reference state that accounts for the different bond energies of various compounds. For both real and ideal gases, the specific heats cv, and cp, are generally functions of temperature. This is a consequence of the internal energy of a molecule consisting of three components: translational, vibrational, and rotational; and the fact that, as a consequence of quantum theory, the vibrational and rotational energy storage modes become increasingly active as temperature increases. Figure 2.1 schematically illustrates these three energy storage modes by contrasting a monatomic species, whose internal energy consists solely of translational kinetic energy, and a diatomic molecule, which stores energy in a vibrating chemical bond, represented as a spring between the two nuclei, and by rotation about two orthogonal axes, as well as possessing kinetic energy from translation.
Figure 2.1 (a) The internal energy of monatomic species consist
only of translational (kinetic) energy, while (b) a diatomic species
internal energy results from translation together with energy from vibration
(potencial and kinetic) and rotation (kinetic).
With these simple models (Fig. 2.1), we would expect the specific heats of diatomic molecules to be greater than monatomic species. In general, the more complex the molecule, the greater its molar specific heat. This can be seen clearly in Fig. 2.2, where molar specific heats for a number of combustion product species are shown as functions of temperature. As a group, the triatomics have the greatest specific heats, followed by the dia-tomics, and, lastly, the monatomics. Note that the triatomic molecules also have a greater temperature dependence than the diatomics, a consequence of the greater number of vibrational and rotational modes that are available to become activated as temperature is increased. In comparison, the monatomic species have nearly constant specific heats over a wide range of temperatures; in fact, the H-atom specific heat is constant (Ep = 20.786 kJ/kmol-K) from 200 K to 5000 K.
Constant-pressure molar specific heats are tabulated as a function of temperature for various species in Tables A.1 to A.12 in Appendix A. Also provided in Appendix A are the curvefit coefficients, taken from the Chemkin thermodynamic database [1], which were used to generate the tables. These coefficients can be easily used with spreadsheet software to obtain cp values at any temperature within the given temperature range.
Figure 2.2 Molar Constant-Pressure specific heats as functions of temperature
for monatomic (H, N, and O), diatomic (CO, H2 and O2) and
triatomic (CO2, H2O and NO2) species. Values are from Appendix A.
Ideal-Gas Mixtures
Two important and useful concepts used to characterize the composition of a mixture are the constituent mole fractions and mass fractions. Consider a multicomponent mixture of gases composed of N1 moles of species 1, N2 moles of species 2, etc. The mole fraction of species i, Xi, is defined as the fraction of the total number of moles in the system that are species i; i.e.,
(2.8)
Similalry, The mass fraction of species i, Yi , is the amount of mass of species i compared with the total mixture mass:
(2.9)
Note that, by definition, the sum of all the constitutent mole (or mass)fractions must be unity,i.e.,
(2.10a)
(2.10b)
Mole fractions and mass fractions are readily converted from one to another using the molecular weights of the species of interest and of the mixture:
(2.11a)
(2.11b)
The mixture molecular weight, MWmix, is easily calculated from a knowledge of either the species mole or mass fractions:
(2.12a)
(2.12b)
Species mole fractions are also used to determine corresponding species partial pressures. The partial pressure of the ith species, Pi, is the pressure of the ith species if it were isolated from the mixture at the same temperature and volume as the mixture. For ideal gases, the mixture pressure is the sum of the constituent partial pressures:
(2.13)
The partial pressure can be related to the mixture composition and total pressure as
LATENT HEAT OF VAPORIZATION
(2.14)
For ideal-gas mixtures, many mass- (or molar-) specific mixture properties are calculated simply as mass (or mole) fraction weighted sums of the individual species-specific properties. For example, mixture enthalpies are calculated as
(2.15a)
(2.15b)
Other frequently used properties that can be treated in this same manner are internal energies, u and ū. Note that, with our ideal-gas assumption, neither the pure-species properties (ui, ūi hi, ħi) nor the mixture properties depend on pres-sure. The mixture entropy also is calculated as a weighted sum of the constituents:
(2.16a)
(2.16b)
In this case, however, the pure-species entropies (si and ŝi) depend on the species partial pressures as indicated in Eqn. 2.16. The constituent entropies in Eqn. 2.16 can be evaluated from standard-state (Pref= Pº = 1 atm) values as
(2.17a)
(2.17b)
Standard-state molar specific entropies are tabulated in Appendix A for many species of interest to combustion.
LATENT HEAT OF VAPORIZATION
In many combustion processes, a liquid-vapor phase change is important. For example, a liquid fuel droplet must first vaporize before it can burn; and, if cooled sufficiently, water vapor can condense from combustion products. Formally, we define the latent heat vaporization, km, as the heat required in a constant-pressure process to completely vaporize a unit mass of liquid at a given temperature, i.e.,
(2.18)
where T and P are the corresponding saturation temperature and pressure, respectively. The latent heat of vaporization is also known as the enthalpy of vaporization. Latent heats of vaporization for various fuels at their normal (1 atm) boiling points are tabulated in Table B.1 (Appendix B). The latent heat of vaporization at a given saturation temperature and pressure is frequently used with the Clausius-Clapeyron equation to estimate saturation pressure variation with temperature:
(2.19)
This equation assumes that the specific volume of the liquid phase is negligible compared with that of the vapor and that the vapor behaves as an ideal gas. Assuming hfg is constant, Eqn. 2.19 can be integrated from (Psatl,Tsatl) to (Psat2, Tsat2) in order to permit, for example, Psat2 to be estimated from a knowledge of Psatl, Tsat1 and Tsat2. We will employ this approach in our discussion of droplet evaporation and combustion.
FIRST LAW OF THERMODYNAMICS
FIRST LAW - FIXED MASS
Conservation of energy is the fundamental principle embodied in the first law of thermodynamics, for a fixed mass i.e., a system, fig 2.3a, energy conservation is expressed for a finite change bet ween two states, 1 and 2, as
(2.20)
Both 1Q2 and 1W2 are path functions and occur only at the system boundaries; ∆E1-2 (≡ E2 - E1) is the change in the total energy of the system, wich is the sum of the internal, kinetic, and potential energies, i.e.,
(2.21)
Figure 2.3 (a) Schematic of fixed-mass system with moving boundary above piston.
(b) Control volume with fixed boundaries and steady flow.
The system energy is a state variable and, as such, ∆E does not depend on the path taken to execute a change in state. Equation 2.20 can be converted to unit mass basis or expressed to represent an instant in time. These forms are
(2.22)
and
(2.22)
or
(2.24)
where lowercase letters are used to denotate mass-specific quantities, e.g., e ≡ E/m
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