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8.5 · Estimate adiabatic flame temperature under stated assumptions

Learn to estimate adiabatic flame temperature under stated assumptions through clear examples and targeted practice.

University of Alberta MEC E 340: Applied Thermodynamics

Chemical Reactions and Combustion

A steady-flow energy balance for introductory combustion estimates

Begin with a steady combustion control volume: reactants enter at state 1, and gaseous products leave at state 2. The working material is a reacting gas mixture. The target is the state-2 temperature when heat transfer is zero. In this lesson, each estimate assumes no shaft work and negligible changes in kinetic and potential energy. Numerical heating values and heat capacities are supplied as problem data; they are not universal values. The estimates assume complete combustion and neglect dissociation. They are energy-balance estimates, not predictions of reaction speed or chemical equilibrium.

What you will learn

  • Define a reacting control volume and state the assumptions used in an adiabatic flame-temperature estimate.
  • Balance a complete-combustion reaction before applying the energy balance.
  • Estimate product temperature from supplied heating values and heat capacities.
  • Explain how excess air and reactant preheating affect the estimate.

1. Define the reaction and energy basis

Adiabatic means that heat does not cross the control-volume boundary. It does not mean that the gas temperature stays constant. Fuel energy and any sensible enthalpy carried in by preheated reactants raise the product enthalpy.
First balance the reaction atoms. State whether the air is stoichiometric or in excess, whether combustion is complete, and whether product water is vapour or liquid. These choices determine the product amounts and the energy-data basis. The examples use gaseous product water and a supplied lower heating value (LHV), consistent with that stated product phase.
The examples use constant molar heat capacities only because each problem supplies them. A different supplied set would give a different estimate. No unlisted property-table values or temperature-dependent heat capacities are assumed.
\dot Q-\dot W+\sum_{in}\dot n h=\sum_{out}\dot n h
  • Specify the fuel, air amount, combustion assumption, product-water phase, and heating-value basis.
  • Balance atoms before writing the energy balance.
  • Use only data supplied in the problem or an explicitly named source.

2. Apply the steady-flow energy balance

For a steady reacting control volume, inlet enthalpy and the fuel-energy release must account for outlet enthalpy, subject to the stated assumptions. With zero heat transfer, no shaft work, and negligible kinetic- and potential-energy changes, the balance equates the fuel energy plus reactant sensible enthalpy to product sensible enthalpy.
Use one mole of fuel as the basis unless another basis is more convenient. For complete combustion of methane with stoichiometric air, one mole of methane reacts with two moles of oxygen. Represent air as oxygen plus 3.76 moles of nitrogen per mole of oxygen; the reactants then contain 7.52 moles of nitrogen. Nitrogen does not react in this model, but it warms along with the products and must be included in their sensible enthalpy.
With constant molar heat capacities, a species’ sensible enthalpy change is its amount multiplied by its molar heat capacity and temperature change. When reactants enter at the reference temperature, the supplied LHV provides the product sensible-enthalpy rise. For preheated reactants, add their sensible enthalpy above that same reference temperature.
Keep the basis and units consistent. An LHV in kilojoules per mole of fuel and a summed product heat capacity in kilojoules per kelvin per mole of fuel give a temperature rise in kelvins. Temperature differences in kelvins and degrees Celsius have the same numerical value.
nfLHV+∑reactantsnicˉp,i(Ti−Tref)=∑productsnjcˉp,j(Tad−Tref)n_f\mathrm{LHV}+\sum_{reactants}n_i\bar c_{p,i}(T_i-T_{ref})=\sum_{products}n_j\bar c_{p,j}(T_{ad}-T_{ref})
  • Assume steady flow, no heat transfer, no shaft work, and negligible kinetic- and potential-energy changes.
  • Include every product species in the product heat-capacity sum, including nitrogen and unused oxygen.
  • Preheating adds reactant sensible enthalpy to the energy available for raising product temperature.

3. Check the estimate and its limits

Check that the reaction conserves each atom, all species amounts use the same fuel basis, and units reduce to temperature. With positive fuel-energy release and reactants at or above the reference temperature, the estimated product temperature should exceed that reference.
In this simplified model, excess air generally lowers the estimate because additional oxygen and nitrogen must be heated without adding fuel energy. Reactant preheating generally raises it because the inlet mixture already carries sensible enthalpy. These trends are useful checks, not replacements for evaluating the balance.
Report the assumptions with the temperature. Changing the product-water phase requires consistent energy data and enthalpy treatment. Do not present a constant-heat-capacity result as an exact flame temperature.
  • Check atom balance, energy balance, units, and product amounts.
  • Excess air tends to lower the estimate; preheating tends to raise it.
  • State the heat capacities, phase assumption, and fuel-energy basis with the answer.

Worked example

Stoichiometric methane with reactants at the reference temperature

Estimate the adiabatic flame temperature for one mole of methane burned completely with stoichiometric air. Reactants enter at 298 K. Use gaseous water, a supplied LHV of 802 kJ/mol methane, and constant product molar heat capacities of 0.037 kJ/(mol·K) for carbon dioxide, 0.036 kJ/(mol·K) for water vapour, and 0.035 kJ/(mol·K) for nitrogen. Neglect dissociation.
Combustion control volume
Combustion control volumeCombustion chamberState 1: methane and stoichiometric airState 2: gaseous products, including carbon dControl-volume schematic

Steady reacting-gas control volume; schematic device view, not to scale. State 1 is the reactant inlet and state 2 the product outlet. Heat transfer and shaft work are zero by assumption.

  1. Balance the reaction
    For one mole of methane, complete combustion requires two moles of oxygen. Stoichiometric air supplies 7.52 moles of nitrogen with that oxygen. Nitrogen is unchanged chemically, so it appears in the gaseous products.
    CH4+2(O2+3.76N2)→CO2+2H2O+7.52N2\mathrm{CH_4+2(O_2+3.76N_2)\rightarrow CO_2+2H_2O+7.52N_2}
  2. Sum the product heat capacities
    Add the heat-capacity contributions of every product mole. Include nitrogen because it also warms as the product mixture leaves at the estimated flame temperature.
    Cp,products=0.037+2(0.036)+7.52(0.035)=0.3722 kJ/KC_{p,products}=0.037+2(0.036)+7.52(0.035)=0.3722\ \mathrm{kJ/K}
  3. Estimate the temperature
    The reactants enter at the 298 K reference temperature, so they contribute no sensible enthalpy above that reference. Under the stated assumptions, the supplied LHV becomes product sensible enthalpy.
    Tad=298 K+802 kJ0.3722 kJ/K≈2453 KT_{ad}=298\ \mathrm{K}+\frac{802\ \mathrm{kJ}}{0.3722\ \mathrm{kJ/K}}\approx2453\ \mathrm{K}
Answer: The estimated adiabatic flame temperature is approximately 2453 K for the supplied constant heat capacities and assumptions.
Check: The result exceeds the 298 K inlet temperature, as expected. The energy rise is 802 kJ on the one-mole methane basis. This is an estimate, not an exact flame temperature.

Worked example

Methane with 50% excess air

Estimate the adiabatic flame temperature for one mole of methane burned completely with 50% excess air. Reactants enter at 298 K. Use an LHV of 802 kJ/mol methane and constant product molar heat capacities supplied for this problem: carbon dioxide 0.037, water vapour 0.036, oxygen 0.032, and nitrogen 0.035 kJ/(mol·K). Assume gaseous water and neglect dissociation.
Combustion with excess air
Combustion with excess airCombustion chamberState 1: methane and 50% excess airState 2: gaseous products, including unused oControl-volume schematic

Steady reacting-gas control volume; schematic device view, not to scale. State 1 is methane with excess air; state 2 includes unused oxygen. Heat transfer and shaft work are zero by assumption.

  1. Find the product amounts
    Fifty percent excess air means supplying 1.5 times the stoichiometric oxygen and nitrogen. Three moles of oxygen enter; complete combustion consumes two, leaving one mole in the products. The corresponding nitrogen amount is 11.28 moles.
    CH4+3(O2+3.76N2)→CO2+2H2O+O2+11.28N2\mathrm{CH_4+3(O_2+3.76N_2)\rightarrow CO_2+2H_2O+O_2+11.28N_2}
  2. Sum the product heat capacities
    Use the heat capacities supplied for this example, including unused oxygen and all nitrogen. The specified oxygen value is needed to reproduce the estimate.
    Cp,products=0.037+2(0.036)+0.032+11.28(0.035)=0.5358 kJ/KC_{p,products}=0.037+2(0.036)+0.032+11.28(0.035)=0.5358\ \mathrm{kJ/K}
  3. Estimate the temperature
    Because the reactants enter at the reference temperature, balance the fuel LHV against the sensible-enthalpy rise of the larger product mixture.
    Tad=298 K+802 kJ0.5358 kJ/K≈1795 KT_{ad}=298\ \mathrm{K}+\frac{802\ \mathrm{kJ}}{0.5358\ \mathrm{kJ/K}}\approx1795\ \mathrm{K}
Answer: The estimated adiabatic flame temperature is approximately 1795 K using the heat capacities supplied for this example.
Check: This is lower than the stoichiometric-air estimate because more product gas must be heated by the same fuel-energy input. The supplied oxygen heat capacity is included.

Worked example

Preheated stoichiometric methane–air reactants

Estimate the adiabatic flame temperature for stoichiometric methane–air reactants entering at 600 K. Use the complete-combustion reaction, gaseous-water assumption, LHV of 802 kJ/mol methane, and product heat capacities from Example 1. For the reactants, use supplied constant molar heat capacities of 0.035 kJ/(mol·K) for methane, 0.030 kJ/(mol·K) for oxygen, and 0.029 kJ/(mol·K) for nitrogen. Use 298 K as the reference temperature and neglect dissociation.
Combustion with preheated reactants
Combustion with preheated reactantsCombustion chamberState 1: preheated methane and stoichiometricState 2: gaseous products, including carbon dControl-volume schematic

Steady reacting-gas control volume; schematic device view, not to scale. State 1 is preheated methane and stoichiometric air; state 2 is gaseous combustion products. Heat transfer and shaft work are zero by assumption.

  1. Calculate inlet sensible enthalpy
    On a one-mole methane basis, the reactants contain one mole of methane, two moles of oxygen, and 7.52 moles of nitrogen. Use their supplied heat capacities to find sensible enthalpy above 298 K.
    Hreactants,sens=[0.035+2(0.030)+7.52(0.029)](600−298)=94.55 kJH_{reactants,sens}=[0.035+2(0.030)+7.52(0.029)](600-298)=94.55\ \mathrm{kJ}
  2. Apply the energy balance
    The products receive both the fuel-energy release and the reactants’ incoming sensible enthalpy. Their summed heat capacity is the value calculated from Example 1’s supplied product data.
    Tad=298 K+802+94.55 kJ0.3722 kJ/K≈2707 KT_{ad}=298\ \mathrm{K}+\frac{802+94.55\ \mathrm{kJ}}{0.3722\ \mathrm{kJ/K}}\approx2707\ \mathrm{K}
Answer: The estimated adiabatic flame temperature is approximately 2707 K under the stated assumptions and supplied heat capacities.
Check: Preheating raises the estimate above Example 1 because the reactants carry 94.55 kJ of sensible enthalpy above 298 K on the stated basis.

Common mistakes and how to avoid them

Using stoichiometric products when the problem specifies excess air.
Correction: Calculate supplied oxygen and nitrogen, then include unused oxygen and all nitrogen among the products.
Leaving nitrogen out because it does not react.
Correction: Include nitrogen in the product sensible-enthalpy sum because its temperature rises.
Combining gaseous product water with an inconsistent fuel-energy basis.
Correction: State the water phase and use the given heating value consistently with it.
Reporting a constant-heat-capacity estimate as an exact flame temperature.
Correction: Identify the supplied heat capacities and describe the result as an estimate.

Lesson summary

  • Define the reacting control volume, states, reaction assumptions, product-water phase, and fuel-energy basis.
  • Balance atoms before applying the steady adiabatic energy balance.
  • Include sensible enthalpy for each entering reactant and each leaving product.
  • Check units and trends: excess air tends to lower the estimate, while reactant preheating tends to raise it.

Check your understanding

Question 1

In the simplified model, why does adding excess air usually lower the estimated flame temperature?
  1. The additional nitrogen and unused oxygen increase the amount of product gas that must be heated.
  2. Excess air increases the methane LHV per mole of methane.
  3. Nitrogen removes heat from an adiabatic control volume.
  4. The balanced reaction releases no energy when oxygen is in excess.
Show answer and explanation
The additional nitrogen and unused oxygen increase the amount of product gas that must be heated.
For the same fuel-energy input, more product gas must be heated, so the estimated temperature rise is smaller.

Question 2

For reactants entering above the reference temperature, what changes in the energy balance?
  1. Add their sensible enthalpy to the energy available to heat the products.
  2. Subtract the fuel LHV from the product sensible enthalpy.
  3. Ignore reactant temperature because the control volume is adiabatic.
  4. Remove nitrogen from the product heat-capacity sum.
Show answer and explanation
Add their sensible enthalpy to the energy available to heat the products.
Preheated reactants carry sensible enthalpy into the control volume, and that energy contributes to product enthalpy.

Key terms

Adiabatic flame temperature
The estimated combustion-product temperature when the reacting system exchanges no heat with its surroundings.
Lower heating value (LHV)
Fuel energy released on a basis that leaves product water as vapour.
Excess air
Air supplied in an amount greater than the stoichiometric amount required for complete combustion.
Sensible enthalpy
The enthalpy change associated with a temperature change, estimated here using a stated heat capacity.

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