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Adiabatic Flame Temperature and Equilibrium Composition of the Combustion of Carbon Monoxide
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Introduction
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One mole of carbon monoxide and 0.5 moles of oxygen are ignited at 298.15 K and 1 atm. The reaction is
CO + 0.5 O2 → CO2
The combustion products undergo dissociation and contain CO2, CO, O and O2.
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The equilibrium composition is found by minimizing the Gibbs Energy of the combustion products (formulated as a series of equations constructed via the method of Lagrange multipliers)
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The adiabatic flame temperature is found by balancing the enthalpy of the reactants against the enthalpy of the products
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The resulting equations are solved numerically to give the adiabatic flame temperature and equilibrium composition of the mixture.
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Physical Properties
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restart:
with(ThermophysicalData:-Chemicals):
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Enthalpies of formation
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h_f_CO := Property("HeatOfFormation", "CO");
h_f_C := Property("HeatOfFormation", "C(gr)");
h_f_O2 := Property("HeatOfFormation", "O2");
h_f_CO2 := Property("HeatOfFormation", "CO2");
h_f_O := Property("HeatOfFormation", "O");
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(2.1) |
Enthalpies as a function of temperature
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h_CO := Property("Hmolar", "CO", "temperature" = T):
h_C := Property("Hmolar", "C(gr)", "temperature" = T):
h_O2 := Property("Hmolar", "O2", "temperature" = T):
h_O := Property("Hmolar", "O", "temperature" = T):
h_CO2 := Property("Hmolar", "CO2", "temperature" = T):
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Reference enthalpies
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h_r_CO2 := eval(h_CO2, T = 298.15);
h_r_CO := eval(h_CO, T = 298.15);
h_r_O2 := eval(h_O2, T = 298.15);
h_r_O := eval(h_O, T = 298.15);
h_r_C := eval(h_C, T = 298.15)
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(2.2) |
Entropy as a function of temperature
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s_CO := Property("Smolar", "CO", "temperature" = T):
s_C := Property("Smolar", "C(gr)", "temperature" = T):
s_O2 := Property("Smolar", "O2", "temperature" = T):
s_O := Property("Smolar", "O", "temperature" = T):
s_CO2 := Property("Smolar", "CO2", "temperature" = T):
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Gibbs Energy
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CO
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C (gr) + 0.5 O2 (g) → CO (g)
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G_CO := proc(temp)
local DeltaH, DeltaS, DeltaG_CO:
DeltaH := eval(h_CO - (h_C + 0.5 * h_O2), T = temp):
DeltaS := eval(s_CO - (s_C + 0.5 * s_O2), T = temp);
DeltaG_CO := DeltaH - DeltaS * temp
end proc:
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CO2
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C (gr) + 1 O2 (g) → CO2 (g)
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G_CO2 := proc(temp)
local DeltaH, DeltaS, DeltaG_CO:
DeltaH := eval(h_CO2 - (h_C + h_O2), T = temp):
DeltaS := eval(s_CO2 - (s_C + s_O2), T = temp);
DeltaG_CO := DeltaH - DeltaS * temp
end proc:
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O2
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O (g)+ O (g) → O2 (g)
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G_O2 := proc(temp)
return 0:
end proc:
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O
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0.5 O2 (g) →O (g)
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G_O := proc(temp)
local DeltaH, DeltaS, DeltaG_O:
DeltaH := eval(h_O - 0.5 * h_O2, T = temp):
DeltaS := eval(s_O - 0.5 * s_O2, T = temp);
DeltaG_O := DeltaH - DeltaS * temp:
end proc:
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Constraints
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Balancing the reactants and products gives
CO + 0.5 O2 = n1 CO2 + n2 CO + n3 O + n4 O2
This results in the following constraint on the carbon atoms...
n1 + n2 = 1
...and this constraint on the oxygen atoms
2 n1 + n2 + n3 + 2 n4 = 2
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con1 := 2 * n1 + n2 + n3 + 2 * n4 = 2:
con2 := n1 + n2 = 1:
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Equilibrium Composition
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Total number of moles in products
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nt := n1 + n2 + n3 + n4:
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Gibbs Energy of the combustion product
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gibbs := n1 * (G_CO2(T) + 8.3144 * T * ln(n1/nt))
+ n2 * (G_CO(T) + 8.3144 * T * ln(n2/nt))
+ n3 * (G_O(T) + 8.3144 * T * ln(n3/nt))
+ n4 * (G_O2(T) + 8.3144 * T * ln(n4/nt)):
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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.
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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),
L1 * diff(lhs(con1), n4) + L2 * diff(lhs(con2), n4) = diff(gibbs, n4):
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Adiabatic Flame Temperature
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Equating the enthalpy of the reactants and products gives
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H_reactants := h_f_CO + 0.5 * h_f_O2:
H_products :=
n1 * (h_f_CO2 + h_CO2 - h_r_CO2)
+ n2 * (h_f_CO + h_CO - h_r_CO)
+ n3 * (h_f_O + h_O - h_r_O)
+ n4 * (h_f_O2 + h_O2 - h_r_O2):
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flameTemp := H_reactants = H_products:
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Numerical solution
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Hence the adiabatic flame temperature and equilibrium composition of the products are
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fsolve({eqComposition, flameTemp, con1, con2}, {L1 = -1000, L2 = -1000, T = 4000, n1 = 0.1, n2 = 0.1, n3 = 0.1, n4 = 0.1})
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(6.1) |
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