Week 5.1 - Compact Notation Derivation for a simple Mechanism
Combustion of fuel involves burning/oxidation of fuel which results in release of heat. This heat is used to perform work.
Below equation represents a combustion reaction for methane.
CH4 + 2(O2 + 3.76N2) -> CO2 + 2H2O + 7.52 * N2
However, such reactions are not single step and goes through many sequence of elementary reactions.These elementary reactions are usually short lived and hence the species of reactions are also short lived.
According to reaction order, reactions are of three types:
1. First order reactions: First order reactions serve to model decomposition reactions A+M→B+M when the concentration of the unspecified molecule M is much larger than the reactive species A
2. Second order reactions serve to model the majority of elementary reactions (formation of activated complex, isomerisation(A <-> B), double decomposition(A <-> B + C)) since bimolecular reactions are the most common.
3. Third order reactions serve to model ter-molecular reactions. Ter-molecular collisions, although not as probable as bimolecular collisions, are much more fertile and thus many third-order reactions appear in reaction mechanisms.
According to their effect reactions are of following types:
1. Chain initiation and chain termination
2. Chain propogation
3. Chain branching
To find out the reaction path ways of all the intermediate reaction that takes place during combusion we need two important inputs,
1. Operating conditions - pressure and temperature
2. Rate of reaction
The rate of each elementary reaction is determined by the reactant concentrations and the rate coffecient, `k`.
`frac{d[X]}{dt}` = Total rate of forming X - total rate of removing X
However, a typical real life combustion reaction would have elementary reactions in order of 1000s with similar amount of species involved and hence, calculation of reaction rates manully would be a time consuming task.
Considering following set of reactions:
`CO + O_2 -> CO_2 + O ` -(1)
`O + H_2O -> OH + OH` -(2)
`CO + OH -> CO_2 + H` -(3)
`H + O_2 -> OH + O` -(4)
| Species | j |
| `CO` | 1 |
| `O_2` | 2 |
| `CO_2` | 3 |
| `O` | 4 |
| `H_2O` | 5 |
| `OH` | 6 |
| `H` | 7 |
| Reaction | Name |
| 1 | R1 |
| 2 | R2 |
| 3 | R3 |
| 4 | R4 |
No. of reactions, `i` = 4
No. of species, `j` = 8
Forming a stoichiometric cofficient matrix for reactants and products, we get
Reactants = `v_(ji)^(')` = `[[1,1,0,0,0,0,0],[0,0,0,1,1,0,0],[1,0,0,0,0,1,0],[0,1,0,0,0,0,1]]`
Products = `v_(ji)^('')` = `[[0,0,1,1,0,0,0],[0,0,0,0,0,2,0],[0,0,1,0,0,0,1],[0,0,0,1,0,1,0]]`
`v_(ji)` = `v_(ji)^('')` - `v_(ji)^(')`
`v_(ji)` = `[[-1,-1,1,1,0,0,0],[0,0,0,-1,-1,2,0],[-1,0,1,0,0,-1,1],[0,-1,0,1,0,1,-1]]`
Now , the generic form of Net Reaction Rate can be written as :
`q_r` = `k_(f_(r))[X_1]^(v_(11)^('))[X_2]^(v_(21)^('))[X_3]^(v_(31)^('))...[X_n]^(v_(n1)^(')) - k_(r_(r))[X_1]^(v_(11)^(''))[X_2]^(v_(21)^(''))[X_3]^(v_(31)^(''))...[X_n]^(v_(n1)^(''))`
A positive magnitude of `q_r` implies that that reaction moves in forward reaction while a negative magnitude will imply the reaction is favoured in reverse direction.
Applying above form to given set of chemical reactions we get,
`q_1` = `k_(f_(1))[CO]^(1) [O_2]^(1) [CO_2]^(0) [O]^(0) [H_2O]^(0) [OH]^(0) [H]^(0) - k_(r_(1)) [CO]^(0) [O_2]^(0) [CO_2]^(1) [O]^(1) [H_2O]^(0) [OH]^(0) [H]^(0)`
`q_1`= `k_(f_(1))[CO]^(1) [O_2]^(1) - k_(r_(1))[CO_2]^(1) [O]^(1)` -(5)
`q_2` = `k_(f_(2))[CO]^(0) [O_2]^(0) [CO_2]^(0) [O]^(1) [H_2O]^(1) [OH]^(0) [H]^(0) - k_(r_(2)) [CO]^(0) [O_2]^(0) [CO_2]^(0) [O]^(1) [H_2O]^(0) [OH]^(2) [H]^(0) `
`q_2`= `k_(f_(2))[O]^(1) [H_2O]^(1) - k_(r_(2))[OH]^(2)` -(6)
`q_3` = `k_(f_(3))[CO]^(1) [O_2]^(0) [CO_2]^(0) [O]^(0) [H_2O]^(0) [OH]^(1) [H]^(0) - k_(r_(3)) [CO]^(0) [O_2]^(0) [CO_2]^(1) [O]^(0) [H_2O]^(0) [OH]^(0) [H]^(1) `
`q_3`= `k_(f_(3))[CO]^(1) [OH]^(1) - k_(r_(3))[CO_2]^(1) [H]^(1)` -(7)
`q_4` = `k_(f_(4))[CO]^(0) [O_2]^(1) [CO_2]^(0) [O]^(0) [H_2O]^(0) [OH]^(0) [H]^(1) - k_(r_(4)) [CO]^(0) [O_2]^(0) [CO_2]^(1) [O]^(1) [H_2O]^(0) [OH]^(1) [H]^(0)`
`q_4`= `k_(f_(4))[H]^(1) [O_2]^(1) - k_(r_(4))[OH]^(1) [O]^(1)` -(8)
The rate of forward and backward reactions `k_(fi)` and `k_(ri)` are the only unknowns left in calculation. The cofficient of forward reaction rate is calculated using Arrhenius form given as :
`k_(fi) = A_i T^(beta_i) exp(frac{-E_i}{R_c T})` -(9)
where
`A_i` is exponential factor (known value and constant for each chemical equation),
`T` is the absolute operating temperature (known value)
`beta` temperature exponent (known value )
`E_i` activation energy for a chemical reaction (known value)
`R_c` characteristic gas constant
To calculate reverse reaction rate following set of equations are solved sequentially.
`K_(p i) = exp(frac{(DeltaS^circ)_i}{R} - frac{(DeltaH^circ)_i}{RT})` -(10)
where `S` and `H` are calculated from thermodynamic file
Output of above equation is used in below equation:
`K_(ci) = K_(p i)(frac{P_(atm)}{RT})^(sum_(k=1)^k nu_(ki))` -(11)
where `K_(ci)` and `K_(p i) ` are equilibrium constants.
And finally the value of `K_(ci)` is plugged into following relation:
`k_(ri) = frac{k_(fi)}{K_(ci)}` -(12)
where `k_(fi)` is forward reaction rate calculated using Arrhenius form and `K_(ci)` using above relation (11) for each reaction `i`
Now net production rate of each species can be defined
Production Rate for specie `j` , `omega_j` = `sum_(i=1)^r v_(ji)q_i`
`omega_(CO)` =`omega_1` = `sum_(i=1)^r v_(1i)q_i` = `v_(11)q_1` + `v_(12)q_2` + `v_(13)q_3` + `v_(14)q_4` = `-q_1 - q_3`
`omega_(O_2)` =`omega_2` = `sum_(i=1)^r v_(2i)q_i` = `v_(21)q_1` + `v_(22)q_2` + `v_(23)q_3` + `v_(24)q_4` = `-q_1 - q_4`
`omega_(CO_2)` =`omega_3` = `sum_(i=1)^r v_(3i)q_i` = `v_(31)q_1` + `v_(32)q_2` + `v_(33)q_3` + `v_(34)q_4` = `q_1 + q_3`
`omega_(O)` =`omega_4` = `sum_(i=1)^r v_(4i)q_i` = `v_(41)q_1` + `v_(42)q_2` + `v_(43)q_3` + `v_(44)q_4` = `q_1 - q_2 + q_4`
`omega_(H_2O)` =`omega_5` = `sum_(i=1)^r v_(5i)q_i` = `v_(51)q_1` + `v_(52)q_2` + `v_(53)q_3` + `v_(54)q_4` = `-q_2`
`omega_(OH)` =`omega_6` = `sum_(i=1)^r v_(6i)q_i` = `v_(61)q_1` + `v_(62)q_2` + `v_(63)q_3` + `v_(64)q_4` = `2q_2 - q_3 + q_4`
`omega_(H)` =`omega_7` = `sum_(i=1)^r v_(7i)q_i` = `v_(71)q_1` + `v_(72)q_2` + `v_(73)q_3` + `v_(74)q_4` = `q_3 - q_4`
Above set of equations are coupled ODE equations as the rate of a specie which is determined using particular equation appears on both the sides of equation as well in other equations.
These equations can also be converted to programmatic functions and a suitable integration package can be used to solve them simultaneously. The advantage of using such compact notation is that even if the number of reactions, `i` and number of species , `j` increase the programmatic structure will not change , only more compute power will be used for obtaining accurate results.
While creating programs for such equations care should be taken as ODE for most combustion system tends to be STIFF in nature, i.e. the reaction for a specific specie might increase rapidly as compared to another specie. This might to lead to unstability. Hence, a proper step size should be chosen before finalising the combustion model.