In the study of combustion, there are two types of adiabatic flame temperature depending on how the process is completed, constant volume and constant pressure, describing the temperature, the combustion products theoretically reach if no energy is lost to the outside environment.

The constant volume adiabatic flame temperature is the temperature that results from a complete combustion process that occurs without any work, heat transfer or changes in kinetic or potential energy. The constant pressure adiabatic flame temperature is the temperature that results from a complete combustion process that occurs without any heat transfer or changes in kinetic or potential energy. Its temperature is lower than the constant volume process because some of the energy is utilized to change the volume of the system (i.e., generate work).

## Common flames

In daily life, the vast majority of flames one encounters are those of organic compounds including wood, wax, fat, common plastics, propane, and gasoline. The constant-pressure adiabatic flame temperature of such substances in air is in a relatively narrow range around 1950 °C. This is because, in terms of stoichiometry, the combustion of an organic compound with n carbons involves breaking roughly 2n C–H bonds, n C–C bonds, and 1.5n O2 bonds to form roughly n CO2 molecules and n H2O molecules.

Because most combustion processes that happen naturally occur in the open air, there is nothing that confines the gas to a particular volume like the cylinder in an engine. As a result, these substances will burn at a constant pressure allowing the gas to expand during the process.

## Common flame temperatures

Assuming initial atmospheric conditions (1 bar and 20 °C), the following table lists the adiabatic flame temperature for various gases under constant pressure conditions. The temperatures mentioned here are for a stoichiometric fuel-oxidizer mixture (i.e. equivalence ratio φ = 1).

Note these are theoretical, not actual, flame temperatures produced by a flame that loses no heat. The closest will be the hottest part of a flame, where the combustion reaction is most efficient. This also assumes complete combustion (e.g. perfectly balanced, non-smokey, usually bluish flame)

Adiabatic flame temperature (constant pressure) of common gases/materials
Fuel Oxidizer ${\displaystyle T_{ad}}$ (°C) ${\displaystyle T_{ad}}$ (°F)
Acetylene (C2H2) Air 2500 4532
Acetylene (C2H2) Oxygen 3480 6296
Butane (C4H10) Air 1970 3578
Cyanogen (C2N2) Oxygen 4525 8177
Dicyanoacetylene (C4N2) Oxygen 4990 9010
Ethane (C2H6) Air 1955 3551
Ethanol (C
2
H
5
OH
)
Air 2082 3779 [1]
Gasoline Air 2138 3880 [1]
Hydrogen (H2) Air 2254 4089 [1]
Hydrogen (H2) Oxygen 3200 5792 [2]
Methane (CH4) Air 1963 3565 [3]
Methanol (CH4O) Air 1949 3540 [3]
Natural gas Air 1960 3562 [4]
Pentane (C5H12) Air 1977 3591 [3]
Propane (C3H8) Air 1980 3596 [5]
Propane (C3H8) Oxygen 2526 4579
MAPP gas Methylacetylene (C3H4) Air 2010 3650
MAPP gas Methylacetylene (C3H4) Oxygen 2927 5301
Toluene (C7H8) Air 2071 3760 [3]
Wood Air 1980 3596
Kerosene Air 2093 [6] 3801
Light fuel oil Air 2104 [6] 3820
Medium fuel oil Air 2101 [6] 3815
Heavy fuel oil Air 2102 [6] 3817
Bituminous Coal Air 2172 [6] 3943
Anthracite Air 2180 [6] 3957
Anthracite Oxygen ≈2900 [see 1] ≈5255
Aluminum Oxygen 3732 6750 [3]
Lithium Oxygen 2438 4420 [3]
Phosphorus (white) Oxygen 2969 5376 [3]
Zirconium Oxygen 4005 7241 [3]
1. ^ The temperature equal to ≈3200 K corresponds to 50 % of chemical dissociation for CO2 at pressure 1 atm. The latter one stays invariant for adiabatic flame and the carbon dioxide constitutes 97 % of total gas output in the case of anthracite burning in oxygen. Higher temperatures will occur for reaction output while it going under higher pressure (up to 3800 K and above, see e.g. Jongsup Hong et al, p.8).

## Thermodynamics

First law of thermodynamics for a closed reacting system

From the first law of thermodynamics for a closed reacting system we have,

${\displaystyle {}_{R}Q_{P}-{}_{R}W_{P}=U_{P}-U_{R}}$

where, ${\displaystyle {}_{R}Q_{P}}$ and ${\displaystyle {}_{R}W_{P}}$ are the heat and work transferred from the system to the surroundings during the process respectively, and ${\displaystyle U_{R}}$ and ${\displaystyle U_{P}}$ are the internal energy of the reactants and products respectively. In the constant volume adiabatic flame temperature case, the volume of the system is held constant hence there is no work occurring,

${\displaystyle {}_{R}W_{P}=\int \limits _{R}^{P}{pdV}=0}$

and there is no heat transfer because the process is defined to be adiabatic: ${\displaystyle {}_{R}Q_{P}=0}$. As a result, the internal energy of the products is equal to the internal energy of the reactants: ${\displaystyle U_{P}=U_{R}}$. Because this is a closed system, the mass of the products and reactants is constant and the first law can be written on a mass basis,

${\displaystyle U_{P}=U_{R}\Rightarrow m_{P}u_{P}=m_{R}u_{R}\Rightarrow u_{P}=u_{R}}$.
Internal energy versus temperature diagram illustrating closed system calculation
Enthalpy versus temperature diagram illustrating closed system calculation

In the constant pressure adiabatic flame temperature case, the pressure of the system is held constant which results in the following equation for the work,

${\displaystyle {}_{R}W_{P}=\int \limits _{R}^{P}{pdV}=p\left({V_{P}-V_{R}}\right)}$

Again there is no heat transfer occurring because the process is defined to be adiabatic: ${\displaystyle {}_{R}Q_{P}=0}$. From the first law, we find that,

${\displaystyle -p\left({V_{P}-V_{R}}\right)=U_{P}-U_{R}\Rightarrow U_{P}+pV_{P}=U_{R}+pV_{R}}$

Recalling the definition of enthalpy we recover: ${\displaystyle H_{P}=H_{R}}$. Because this is a closed system, the mass of the products and reactants is constant and the first law can be written on a mass basis,

${\displaystyle H_{P}=H_{R}\Rightarrow m_{P}h_{P}=m_{R}h_{R}\Rightarrow h_{P}=h_{R}}$.

We see that the adiabatic flame temperature of the constant pressure process is lower than that of the constant volume process. This is because some of the energy released during combustion goes into changing the volume of the control system. One analogy that is commonly made between the two processes is through combustion in an internal combustion engine. For the constant volume adiabatic process, combustion is thought to occur instantaneously when the piston reaches the top of its apex (Otto cycle or constant volume cycle). For the constant pressure adiabatic process, while combustion is occurring the piston is moving in order to keep the pressure constant (Diesel cycle or constant pressure cycle).

Adiabatic flame temperatures and pressures as a function of stoichiometry (relative ratio of fuel and air)
Constant volume flame temperature of a number of fuels

If we make the assumption that combustion goes to completion (i.e. ${\displaystyle CO_{2}}$ and ${\displaystyle H_{2}O}$), we can calculate the adiabatic flame temperature by hand either at stoichiometric conditions or lean of stoichiometry (excess air). This is because there are enough variables and molar equations to balance the left and right hand sides,

${\displaystyle {\rm {C}}_{\alpha }{\rm {H}}_{\beta }{\rm {O}}_{\gamma }{\rm {N}}_{\delta }+\left({a{\rm {O}}_{\rm {2}}+b{\rm {N}}_{\rm {2}}}\right)\to \nu _{1}{\rm {CO}}_{\rm {2}}+\nu _{2}{\rm {H}}_{\rm {2}}{\rm {O}}+\nu _{3}{\rm {N}}_{\rm {2}}+\nu _{4}{\rm {O}}_{\rm {2}}}$

Rich of stoichiometry there are not enough variables because combustion cannot go to completion with at least ${\displaystyle CO}$ and ${\displaystyle H_{2}}$ needed for the molar balance (these are the most common incomplete products of combustion),

${\displaystyle {\rm {C}}_{\alpha }{\rm {H}}_{\beta }{\rm {O}}_{\gamma }{\rm {N}}_{\delta }+\left({a{\rm {O}}_{\rm {2}}+b{\rm {N}}_{\rm {2}}}\right)\to \nu _{1}{\rm {CO}}_{\rm {2}}+\nu _{2}{\rm {H}}_{\rm {2}}{\rm {O}}+\nu _{3}{\rm {N}}_{\rm {2}}+\nu _{5}{\rm {CO}}+\nu _{6}{\rm {H}}_{\rm {2}}}$

However, if we include the Water gas shift reaction,

${\displaystyle {\rm {CO}}_{\rm {2}}+H_{2}\Leftrightarrow {\rm {CO}}+{\rm {H}}_{\rm {2}}{\rm {O}}}$

and use the equilibrium constant for this reaction, we will have enough variables to complete the calculation.

Different fuels with different levels of energy and molar constituents will have different adiabatic flame temperatures.

Constant pressure flame temperature of a number of fuels
Nitromethane versus isooctane flame temperature and pressure

We can see by the following figure why nitromethane (CH3NO2) is often used as a power boost for cars. Since each mole of nitromethane contains two moles of oxygen, it can burn much hotter because it provides its own oxidant along with fuel. This in turn allows it to build up more pressure during a constant volume process. The higher the pressure, the more force upon the piston creating more work and more power in the engine. It is interesting to note that it stays relatively hot rich of stoichiometry because it contains its own oxidant. However, continual running of an engine on nitromethane will eventually melt the piston and/or cylinder because of this higher temperature.

Effects of dissociation on adiabatic flame temperature

In real world applications, complete combustion does not typically occur. Chemistry dictates that dissociation and kinetics will change the relative constituents of the products. There are a number of programs available that can calculate the adiabatic flame temperature taking into account dissociation through equilibrium constants (Stanjan, NASA CEA, AFTP). The following figure illustrates that the effects of dissociation tend to lower the adiabatic flame temperature. This result can be explained through Le Chatelier's principle.

## References

1. ^ a b c Flame Temperature Analysis and NOx Emissions for Different Fuels
2. ^ Flame temperatures
3. CRC Handbook of Chemistry and Physics, 96th Edition, p. 15-51
4. ^ North American Combustion Handbook, Volume 1, 3rd edition, North American Mfg Co., 1986.
5. ^ [1]
6. Power Point Presentation: Flame Temperature, Hsin Chu, Department of Environmental Engineering, National Cheng Kung University, Taiwan