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# Lecture 2

## Adiabatic Flame Temperature

and
Chemical Equilibrium

2.-1

## First law of thermodynamics - balance between different forms of energy

change of specific internal energy:

du

wrev=pdv, v=1/

## specific frictional work:

wR

*)

Related quantities
specific enthalpy (general definition):
specific enthalpy for an ideal gas:
Energy balance:
*)

The list of energies is not exhaustive, f. e. work of forces and kinetic or potential energies
are missing, but the most important which will appear in a balance are considered here.

2.-2

Multicomponent system
The specific internal energy and specific enthalpy are the mass weighted sums
of the specific quantities of all species

For an ideal gas the partial specific enthalpy is related to the partial specific
internal energy by

2.-3

For an ideal gas the inner energy and enthalpy depend on temperature alone.

If cpi is the specific heat capacity at constant pressure and hi,ref is the reference
enthalpy at the reference temperature Tref ,
the temperature dependence of the partial specific enthalpy is given by

## The reference temperature may be arbitrarily chosen, most frequently used:

Tref = 0 K

or Tref = 298.15 K

2.-4

the mixture is

2.-5

## The molar reference enthalpies at reference temperature of species are

listed in tables.
It should be noted that the reference enthalpies of H2, O2, N2 and solid carbon Cs
were chosen as zero, because they represent the chemical elements.
Reference enthalpies of combustion products such as CO2 and H2O are typically
negative.

2.-6

The temperature dependence of molar enthalpy, molar entropy and molar heat
capacities may be calculated from the NASA polynomials.

## The constants aj for each species i are listed in tables.

2.-7a

NASA Polynomials for two temperature ranges and standard pressure p = 1 atm

2.-7b

## First law of thermodynamics for an adiabatic system at constant pressure

(q = 0, dp = 0)
we neglect the work done by friction (wR = 0).
From
we then have:

Integrated from the unburnt, index u, to the burnt state, index b, it follows:
or

2.-8

## this may be written as

Here the specific heat capacities are those of the mixture, to be calculated with
the mass fractions of the burnt and unburnt gases

2.-9

which gives

such that

2.-10

## The heat of combustion changes very little with temperature.

It is often set equal to:

## Simplification: Tu = Tref and assume cp,b approximately constant

For combustion in air, the contribution of nitrogen is dominant in calculating cp,b.

2.-11

The value of cpi is somewhat larger for CO2 and somewhat smaller for O2 while that for
H2O is twice as large.
A first approximation for the specific heat of the burnt gas for
lean and stoichiometric mixtures:
cp = 1.40 kJ/kg/K
Assuming cp constant and Q = Qref , the adiabatic flame temperature for a lean mixture
(YF,b = 0) is calculated from
and

with F = - 'F as
2.-12

## For a rich mixture

must be replaced by

2.-13

Equations

and

Introducing

and

## and specifying the temperature of the unburnt mixture by

where T2 is the temperature of the oxidizer stream and T1 that of the fuel stream.
This equation describes mixing of the two streams with cp assumed to be constant.

2.-14

Equations

and

## The maximum temperature appears at Z = Zst :

2.-15

2.-16

In the table for combustion of a pure fuels (YF,1 = 1) in air (YO2,2 = 0.232)
with Tu,st = 300 K and cp = 1.4 kJ/kg/K
stoichiometric mixture fractions and
stoichiometric flame temperatures for
some hydrocarbon-air mixtures are
shown.

2.-17

Chemical Equilibrium
The assumption of complete combustion is an approximation because it disregards
the possibility of dissociation of combustion products.
A more general formulation is the assumption of chemical equilibrium.
Complete combustion represents the limit of an infinite equilibrium constant (see
below).
Chemical equilibrium and complete combustion are valid in the limit of infinitely
fast reaction rates only, which will seldom be valid in combustion systems.
We will consider finite rate chemical kinetics in a later lecture.
2.-18

## Only for hydrogen diffusion flames complete chemical equilibrium is a good

approximation, while for hydrocarbon diffusion flames finite kinetic rates are
needed.
In hydrocarbon diffusion flames the fast chemistry assumption overpredicts the
formation of intermediates such as CO and H2 due to the dissociation of fuel on the
rich side by large amounts.
Nevertheless, since the equilibrium assumption represents an exact thermodynamic
limit, it shall be considered here.

2.-19

## Chemical potential and the law of mass action

Differently from the enthalpy, the partial molar entropy Si of a chemical species in a
mixture of ideal gases depends on the partial pressure

## depends only on temperature.

Values for the reference entropy Si,ref are also listed in tables.

2.-20

The partial molar entropy may now be used to define the chemical potential

where

## is the chemical potential at 1 atm.

The condition for chemical equilibrium for the l-th reaction is given by

2.-21

Using
in

2.-22

2.-23

## An approximation of equilibrium constants may be derived by introducing the

quantity

For constant Cpi the second term in this expression would yield a logarithm of the
temperature, while the last term does not vary much if

2.-24

one obtains

where

2.-25

## Values for iA and iB were obtained by linear interpolation in terms of ln T

for the values given in the JANAF-Tables at T = 300 K and T = 2000 K.
For some species, which are important in combustion, values for iA and iB
are listed in Tab. 2.1 of the lecture notes.

2.-26

2.-27

## An example: Equilibrium Calculation of the H2-air system

Using the law of mass action one obtains for the reaction 2 H2 +O2 = 2 H2O
the relation between partial pressures

where
was approximated using

2.-28

## the partial pressures

are written with as

2.-29

2.-30

## These equations lead to the following nonlinear equation for H2O,b

2.-31

This equation has one root between H2O,b = 0 and the maximum values
H2O,b = ZH/2WH and H2O,b = ZO/WO
which correspond to complete combustion for lean and rich conditions in the limit

## The solution, which is a function of the temperature, may be found by successively

bracketing the solution within this range.
The temperature is then calculated by employing a Newton iteration on

2.-32

## The iteration converges readily following

where i is the iteration index.
The solution is plotted here for a hydrogen-air flame as a function of the mixture
fraction for Tu = 300 K.

2.-33

## The table shows equilibrium mass fractions of H2, O2 and H2O

for p=1 bar and p=10 bar and different temperatures

2.-34

Equilibrium temperature profiles for lean methane, acetylene and propane flames as
a function of the equivalence ratio for Tu = 300 K

2.-35

## The heterogeneous equilibrium

A reaction is called heterogeneous, if it occurs for instance at the gas-to-solid
interface.
Since the chemical potential of the solid is independent of pressure

only the partial pressures of the gaseous components will appear in the
law of mass action.

2.-36

## The ratio of partial pressures of CO2 and O2 becomes

Here the molar enthalpy HCs, ref of solid carbon is zero per definition,
while A,Cs = - 9.979 and B,Cs = 1.719

2.-37

Example
Calculate the equilibrium mole fraction of NO in air at T = 1000 K and T = 1500 K
by assuming that the mole fractions of O2 (XO2 = 0.21) and N2 (XN2 = 0.79) remain
unchanged.

Solution
The equilibrium constant of the reaction N2 + O2 = 2 NO
is with the values of the Janaf Table

2.-38

## Neglecting the consumption of N2 and O2 as a first approximation, their partial

pressures in air may be approximated as

## The equilibrium mole fraction of NO is then

At T = 1000 K one obtains 38 ppv (parts per volume = Xi 10-6) and at T = 1500 K
230 ppv.

2.-39

## This indicates that at high temperatures equilibrium NO-levels exceed by far

those that are accepted by modern emission standards which are around 100 ppv or
lower.
Equilibrium considerations therefore suggest that in low temperature exhaust gases
NO is above the equilibrium value and can be removed by catalysts.

2.-40
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