Adiabatic Combustion

2. Adiabatic Combustion

In several applications, such as gas turbines, boilers, and residential gas burners, combustion occurs at constant pressure. The higher the combustion temperature, the larger is the availability. Therefore, when excess air is used (which induces smaller equivalence ratios φ), the combustion product temperatures decrease, which increases irreversibility and lowers

the availability. This is illustrated in Figure 1 in terms of the irreversibility (= T o σ). (Recall that the overall combustion reaction, say, of methane, in terms of the equivalence ratio can be

represented by the reaction equation CH 4 + (2/ φ) (O 2 + 3.76N 2 ) → CO 2 + 2H 2 O + 2(1/ φ–1) O 2 + (2/ φ) 3.76N 2 .) Figure 2 presents the entropy generated per unit amount of heat released for hydrocarbon fuels under adiabatic combustion. Hence even though enthalpy is conserved,

availability is lost during irreversible chemical reactions.

e. Example 5 Consider the adiabatic and stoichiometric combustion of molecular hydrogen with air. The inlet conditions to the burner are at 298 K and 1 bar (state 1). The products leave the combustor at state 2. Determine the irreversibility. If the products are cooled back

to 298 K (state 3), determine the optimum work. Assume that combustion of H 2 is

complete and H 2 O exists as a gas at 298 K, 1bar.

Solution The overall chemical reaction is

H 2 + 1/2 O 2 + 1.88 N 2 →H 2 O + 1.88 N 2 .

Recall that σ=W opt /T o , where

W opt = Ψ 1 – Ψ 2 .

With the values

X H 2 =1 ÷(1+0.5+1.88) = 0.296, X O 2 = 0.148, X N 2 = 0.556, Ψ 1 =N H 2 ψ ˆ H 2 +N O 2 ψ ˆ O 2 +N N 2 ψ ˆ N 2 , where, for instance

ψ ˆ H 2 = h H 2 (T,p H 2 )–T o s H 2 and p H 2 =X H 2 P,

Ψ 1 = (0 – 298 × (130.57 – 8.314 × ln(1 × 0.296÷1))) + × (0 – 298 × (205.03 – 8.314 × ln(1 × 0.148÷1))) 1/2

1.88 × (0 – 209 × (191.5 – 8.314 × ln(1 × 0.556÷1))) = –184920 kJ

The adiabatic flame temperature is obtained by considering the energy balance

dE cv /dt = ˙ Q cv – W ˙ cv + Σ k,i N ˙ k ¯e T,k – Σ k,e N ˙ k ¯e T,k . Under ideal gas conditions ˆh k = h , and assuming negligible contributions from the k potential and kinetic energies, steady state, and a single step overall chemical reac-

tion, and in the absence of work

( Σ ν k h T,k ) i = ( Σ ν k h T,k ) e , or H 1 =H 2 , where

H 1 = h H 2 ,i + ν O 2 ,i h O 2 ,i + ν N 2 h N 2 ,i = 0 (elemental species), and

H 1 = ν N 2 ,e h N 2 ,e + ν HO 2 ,e h HO 2 ,e = 0. Solving iteratively, T 2 =T ad = 2528.7 K.

Ψ 2 =N HO 2 ,e ψ ˆ HO 2 ,e +N N 2 ,e ψ ˆ N 2 ,e , i.e.,

Ψ 2 =1 × (–241820 + 99704 – 298 × (276.4 – 8.314 × ln (1×0.347÷1))) + 1.88 (0 + 75597 – 298 × (260.7 – 8.314 × ln(1 × 0.653÷1))) =

(– 233079) kJ. W opt,12 = –184920 – (–233079) = 48159 kJ per kmole of fuel burnt.

i 12 = 48159 kJ per kmole of fuel. σ –1

12 = 48159 ÷298 = 162 kJ K per kmole of fuel. If the gases are cooled from T 2 = 2528.7 to T 3 =T o = 298 K, but with the same com-

position,

W opt,23 = Ψ 2 – Ψ 3 , where

Ψ 3 =1 × (–241820 +298× (188.72 – 8.314 × ln (1 × 0.347÷1))) + 1.88 × (0 –298×(191.5–8.314×ln (1×0.653÷1))) =

(–408259) kJ per kmole of fuel, i.e.,

W = –233079– (–408259) = 175180 kJ per kmole of fuel. opt,23 Remarks The stream exergy at state 1 is Ψ 1 - Ψ 0 = Ψ 1 - Ψ 3 =W opt 13 = -184,920 – (-408259) = 223,339 kJ; Even though state 2 has the same enthalpy as state 1, the stream exergy at state 2 is Ψ 2 - Ψ 0 = Ψ 2 - Ψ 3 =W opt,23 = 175,180. Defining the availability frac- tion at state 2 as ξ opt,23 = W opt,23 /W opt,13 = 0.78, which indicates that approximately 22% of maximum possible optimum work that was otherwise available at state 1 is lost during the irreversible adiabatic combustion.

The calculations can be repeated for a stoichiometric mixture of methane and air en- tering the burner at 298 K and 1 atm. In this case, the flame temperature, optimum

work and availability fraction at the flame temperature are T ad = 2325 K, W loss =236

MJ/kmol, and ξ opt,23 = 0.72 (i.e., the availability loss fraction is 0.28). The combustion temperature of 2325 K is achieved when the combustion proceeds without any loss of

heat to the surroundings (H products -H reactants ). The only irreversibility is due to the chemical reaction. We see, therefore, that a combustion reaction is highly irreversible and wasteful of availability, even when it is conducted so that enthalpy of combustion is retained in the combustion products. An availability analysis reveals that chemical processes may be degrading the energy quality while conserving its quantity.