Adiabatic Flame Temperature and Composition of the Combustion Products of Hydrogen and Oxygen

Introduction


Consider the combustion of hydrogen and oxygen.
H_{2} (g) + 0.5 O_{2} (g) ! H_{2}O (g)
The application calculates the adiabatic flame temperature and equilibrium composition of the combustion products.
• 
The equilibrium composition is calculated by minimizing the Gibbs Free Energy of the system

• 
The flame temperature is found by equating the enthalpy of the feed and combustion products.

Both relationships are solved simultaneously using fsolve.
Temperaturedependent thermodynamic data are calculated with the ThermophysicalData:Chemicals package.


Physical Properties


> 
with(ThermophysicalData:Chemicals):

Enthalpies
> 
h_H2O := Property("Hmolar", "H2O(g)", "temperature" = T): h_O2 := Property("Hmolar", "O2(g)", "temperature" = T): h_H2 := Property("Hmolar", "H2(g)", "temperature" = T):

Entropies
> 
s_H2O := Property("Smolar", "H2O(g)", "temperature" = T): s_O2 := Property("Smolar", "O2(g)", "temperature" = T): s_H2 := Property("Smolar", "H2(g)", "temperature" = T):

Heat of formation
> 
h_f_O2 := Property("HeatOfFormation", "O2(g)", useunits): h_f_H2 := Property("HeatOfFormation", "H2(g)", useunits): h_f_H2O := Property("HeatOfFormation", "H2O(g)", useunits):

Reference enthalpies
> 
h_r_O2 := eval(h_O2, T = 298.15 * Unit(K)): h_r_H2 := eval(h_H2, T = 298.15 * Unit(K)): h_r_H2O:= eval(h_H2O, T = 298.15 * Unit(K)):



Equilibrium Composition as a Function of Temperature


Gibbs free energy of H_{2}O, O_{2} and H_{2}
> 
G_H2O := proc(temp) local DeltaH, DeltaS, DeltaG_H2O: DeltaH := eval(h_H2O  (h_H2 + 0.5 * h_O2), T = temp): DeltaS := eval(s_H2O  (s_H2 + 0.5 * s_O2), T = temp); DeltaG_H2O := DeltaH  DeltaS * temp: return DeltaG_H2O: end proc: G_O2:= proc(temp) return 0 end proc: G_H2:= proc(temp) return 0 end proc:



Constraints


With 1 mole of H_{2} and 0.5 mole of O_{2} in the initial mixture, balancing the amount of H_{2}, O_{2} and H_{2}O before and after combustion gives
H_{2} + 0.5 O_{2} = n1 H_{2}O + n2 H_{2} + n3 O_{2}
Balancing the H atoms gives
2 = 2 n1 + 2 n2
Balancing the O atoms gives
0.5 x 2 = n1+ 2 n3
> 
con1 := n1 + n2 = 1 * Unit(mol): con2 := n1 + 2*n3 = 1 * Unit(mol):

Total number of moles in products


Equilibrium Composition


Gibbs energy of the combustion product
> 
gibbs := n1 * (G_H2O(T) + 8.3144 * Unit(J/mol/K) * T * ln(n1/nt)) + n2 * (G_H2(T) + 8.3144 * Unit(J/mol/K) * T * ln(n2/nt)) + n3 * (G_O2(T) + 8.3144 * Unit(J/mol/K) * T * ln(n3/nt)):

For a given temperature, minimizing the Gibbs Energy of the combustion products will give the equilibrium composition. Hence the values of n1, n2, n3, and n4 are given by the numeric solution of these equations, where L1 and L2 are the Lagrange multipliers.
> 
eqComposition := L1 * diff(lhs(con1), n1) + L2 * diff(lhs(con2), n1) = diff(gibbs, n1), L1 * diff(lhs(con1), n2) + L2 * diff(lhs(con2), n2) = diff(gibbs, n2), L1 * diff(lhs(con1), n3) + L2 * diff(lhs(con2), n3) = diff(gibbs, n3):



Adiabatic Flame Temperature as a Function of Composition


This function gives the adiabatic flame temperature as a function of the moles of H_{2}O in the product, n1
> 
H_reactants := 0: H_products := n1*(h_f_H2O + (h_H2O  h_r_H2O)) + n2*(h_f_H2 + (h_H2  h_r_H2)) + n3*(h_f_O2 + (h_O2  h_r_O2)):

> 
flameTemp := H_reactants = H_products:



Adiabatic Flame Temperature and Composition of Combustion Products


> 
res := fsolve({eqComposition, flameTemp, con1, con2}, {L1 = 30000 * Unit(J/mol), L2 = 30000 * Unit(J/mol), T = 3500 * Unit(K), n1 = 0.1 * Unit(mol), n2 = 0.1 * Unit(mol), n3 = 0.1 * Unit(mol)})


(7.1) 
The products consist of
0.663 H_{2}O + 0.337 H_{2} + 0.168 O_{2}
with a flame temperature of

(7.2) 

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