Question 14.8: With initial temperature and pressure being 1 atm and 298K, ......

Given: (1) Combustion reactants: Air and Methane, (2) Initial condition: 1 atm pressure and 298 K temperature.
Find out: Adiabatic flame temperature at constant pressure.
Properties: Molar mass of C,  H_2 and O_2 and air are: 12 kg/kmol, 2 kg/kmol, 32 kg/kmol and 29 kg/kmol
respectively.
Assumption: (1) Combustion is complete, (2) Air and combustion gases are ideal (3) Kinetic and potential energies are negligible, (4) There is no work interaction, (5) Combustion chamber is isolated.
Chemical reaction for the combustion is:

CH_4 + 2 (O_2 + 3.78  N_2)\longrightarrow 1 CO_2 + 2H_2O + 7.56  N_2

Here, N_{CH_{4}} = 1,  N_{O_2} = 2,  N_{N_2} = 7.56,  N_{CO_2} = 1, N_{H_2O} = 2.
Constant pressure adiabatic flame temperature is given by the relation:

H_{products} = H_{reactants}

⇒                    \sum{H_R(T_{in} , p)} = \sum{H_p} (T, p)
⇒                    \sum{H_p(T_{i} , p)} = \sum{N_i} \int_{T_{ref}}^{T}{C_{p, i}} dT + Q_L

H_{reactants} = \sum{N_p} ( H^0_f + H – H_0)_p = N_{CH_4}(H^0_f + H – H_0)_{CH_4} + N_{O_2}(H^0_f + H – H_0)_{O_2} + N_{N_2} (H^0_f + H – H_0)_{N_2}
= 1 × (-74,831) + 2 × 0 + 7.56 × 0 = – 74,831 kJ / kmol

H_{products} = \sum{N_i \int_{T_{ref}}^{T}{C_{p, i}} dT + Q_L}

Assuming c_p to be constant,

H_{products} = \sum{N_i} (H^0_f + C_{p, i} (T_{adi} – 298))_{N_2}
= 1 \times (- 3,93,546 + 56.21 (T_{adi} – 298)) + 2x (-2,41,845 + 43.87 (T_{adi} – 298)) + 7.52 x (0 + 33.71 (T_{adi} – 298))

H_{products} = H_{reactants}
1 \times (-3,93,546 + 56.21 (T_{adi} – 298)) + 2 \times (-2,41,845 + 43.87 (T_{adi} – 298)) + 7.52 \times (0 + 33.71 (T_{adi} – 298)) = -74,831

Solving it for T_{adi},

T_{adi} = 2,318  K