9781441979421-c1 - Chapter 2 Thermodynamics of Combustion...

Chapter 2
Thermodynamics of Combustion
2.1 Properties of Mixtures
The thermal properties of a pure substance are described by quantities including
internal energy, u , enthalpy, h , specific heat, c
p
, etc. Combustion systems consist of
many different gases, so the thermodynamic properties of a mixture result from a
combination of the properties of all of the individual gas species. The ideal gas law
is assumed for gaseous mixtures, allowing the ideal gas relations to be applied to
each gas component. Starting with a mixture of K different gases, the total mass, m ,
of the system is
m ¼
X
K
i ¼ 1
m
i
; (2.1)
where m
i
is the mass of species i . The total number of moles in the system, N , is
N ¼
X
K
i ¼ 1
N
i
; (2.2)
where N
i
is the number of moles of species i in the system. Mass fraction, y
i
, and
mole fraction, x
i
, describe the relative amount of a given species. Their definitions
are given by
y
i
±
m
i
m
and x
i
±
N
i
N
; (2.3)
where i ¼ 1,2, . . . , K . By definition,
X
K
i ¼ 1
y
i
¼ 1 and
X
K
i ¼ 1
x
i
¼ 1 :
S. McAllister et al., Fundamentals of Combustion Processes ,
Mechanical Engineering Series, DOI 10.1007/978-1-4419-7943-8_2,
#
Springer Science+Business Media, LLC 2011
15

With M
i
denoting the molecular mass of species i , the average molecular mass, M,
of the mixture is determined by
M ¼
m
N
¼
P
i
N
i
M
i
N
¼
X
i
x
i
M
i
: (2.4)
From Dalton’s law of additive pressures and Amagat’s law of additive volumes
along with the ideal gas law, the mole fraction of a species in a mixture can be found
from the partial pressure of that species as
P
i
P
¼
N
i
N
¼
V
i
V
¼ x
i
; (2.5)
where P
i
is the partial pressure of species i , P is the total pressure of the gaseous
mixture, V
i
the partial volume of species i, and V is the total volume of the mixture.
The average intrinsic properties of a mixture can be classified using either a molar
base or a mass base. For instance, the internal energy per unit mass of a mixture, u,
is determined by summing the internal energy per unit mass for each species
weighted by the mass fraction of the species.
u ¼
U
m
¼
P
i
m
i
u
i
m
¼
X
i
y
i
u
i
; (2.6)
where U is the total internal energy of the mixture and u
i
is the internal energy per
mass of species i. Similarly, enthalpy per unit mass of mixture is
h ¼
X
i
y
i
h
i
and specific heat at constant pressure per unit mass of mixture is
c
p
¼
X
i
y
i
c
p ; i
:
A molar base property, often denoted with a
^
over bar, is determined by the sum
of the species property per mole for each species weighted by the species mole
fraction, such as internal energy per mole of mixture
^ u ¼
X
i
x
i
^ u
i
;
enthalpy per mole of mixture
^
h ¼
X
i
x
i
^
h
i
;
16 2 Thermodynamics of Combustion

and entropy per mole of mixture
^ s ¼
X
i
x
i
^ s
i
:
Assuming constant specific heats during a thermodynamic process, changes of energy,
enthalpy, and entropy of an individual species per unit mass are described as follows:
D u
i
¼ c
v ; i
ð T
2
² T
1
Þ (2.7)
D h
i
¼ c
p ; i
ð T
2
² T
1
Þ (2.8)
D s
i
¼ c
p ; i
ln
T
2
T
1
² R
i
ln
P
i ; 2
P
i ; 1
(2.9)
P
i,1
and P
i,2
denote the partial pressures of species i at state 1 and state 2, respectively.