Week 5 - Compact Notation Derivation for a simple Mechanism
In order to find the thermodynamic properties of species after the combustion, rate of reaction of each reaction in the reaction mechanism has to be obtained.
The given reactions are
`CO + O_2 <=> CO_2 + O` ---> R1`
`O + H_2O <=> OH + OH` ---> R2
`CO + OH <=> CO_2 + H` ---> R3
`H + O_2 <=> OH + O` ---> R4
The above reaction mechanism has 4 reversible reactions and 7 species.
Species:
| 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| CO | `O_2` | `CO_2` | O | `H_2O` | OH |
H |
The rate at which the each species is consumed is directly proportional to the molar concentration of reactant species.
For example, in R1,
`d/dt([CO]) prop [CO][O_2]`
`d/dt[CO] = -K_(f)*[CO][O_2]`
where `K_f` is the forward reaction rate coefficient and [CO] and [O_2] are the molar concentrations of carbon monoxide and oxygen respectively expressed in `(k-mol)/m^3`
'-' sign indicates that the concentration of CO is decreasing.
As this is a reversible reaction,
`d/dt[CO] = -K_(f)*[CO][O_2] + K_r[CO_2][O]`
`K_r` is the backward reaction rate coefficient and is '+' ve since the concentration of CO increases during the backward reaction.
Each reaction has a forward reaction rate coefficient (`K_f`) and backward reaction rate coefficient (`K_r`). These coefficients are functions of temperature.
Let us say forward and backward reaction rate coefficients for
R1 be `K_(f1)`and `K_(r1)`
R2 be `K_(f2)` and `K_(r2)`
R3 be `K_(f3)` and `K_(r3)`
R4 be `K_(f4)` and `K_(r4)`
Likewise, rate of reaction of each species is determined which results in a set of coupled ordinary differential equations which are solved by numerical integration.
In general, there will be hundreds of elementary reactions and intermediate species forming during the combustion process. Solving the rate of reactions for each species manually is a extremely time taking process. So, these differential equations are solved by computer programs.
To solve for the net production rate of each species, a compact notation that represents both mechanism and individual species production rate is used.
Compact Notation:
lets denote 'j' for species index and 'i' for reaction index.
As there are 7 species, 'j' will be varied from 1 to 7 and as there are 4 reactions, 'i' will be varied from 1 to 4.
So,
| j1 | j2 | j3 | j4 | j5 | j6 | j7 |
| CO | `O_2` | `CO_2` | O | `H_2O` | OH |
H |
| i1 | i2 | i3 | i4 |
| R1 | R2 | R3 | R4 |
Net producion rate of each species, `w_j = sum_(i=1)^L (v_(ji)*q_i)` ---> (1)
where `v_(ji) = v_(ji)^('') - v_(ji)^'` ---> (2) and L = no. of reaction`
and `q_(i)`= Net Reaction rate of ith reaction = `K_(fi)prod_(j=1)^N[X_j]^(v_(ji)^') - K_(ri)prod_(j=1)^N[X_j]^(v_(ji)^('')` ---> (3)
where `K_(fi) = A_i*T^(beta_i)*e^(-E_i/(R_cT))`
and `K_(ri) = K_(fi)/K_(ci)`
`K_(ci) = K_(p i)*(P_(atm)/(RT))^(sum_(k=1)^K(v_(ki)^'-v_(ki)^(''))` where K = no. of species
`K_(p i) = e^(((deltaS_i^0)/R) - (deltaH_i^0)/(RT)` where `deltaS`= entropy change and `deltaH` = enthalpy change
`[X_j]` is the molar concentration of species 'j'.
In equation (2), `v_(ji)^'` and `v_(ji)^('')` are matrices of order i x j
`v_(ji)^'` = no. of moles of 'j'th species in 'i'th reaction of reactants
`v_(ji)^('')` = no. of moles of 'j'th species in 'i'th reaction of products
Calculating `v_(ji)`:
`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]]`
`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]]`
Therefore,
`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]]`
Calculating Net Reaction rate of each reaction:
Using eq. (3),
For Reaction 1 (R1):
`q_1 = K_(f1)*([CO]^1[O_2]^1[CO_2]^0[O]^0[H_2O]^0[OH]^0[H]^0) - K_(r1)*([CO]^0[O_2]^0[CO_2]^1[O]^1[H_2O]^0[OH]^0[H]^0)`
= `K_(f1)*([CO][O_2]) - K_(r1)*([CO_2][O])`
For Reaction 2 (R2):
`q_2 = K_(f2)*([CO]^0[O_2]^0[CO_2]^0[O]^1[H_2O]^1[OH]^0[H]^0) - K_(r2)*([CO]^0[O_2]^0[CO_2]^0[O]^0[H_2O]^0[OH]^2[H]^0)`
= `K_(f2)*([O][H_2O]) - K_(r2)*([OH]^2)`
For Reaction 3 (R3):
`q_3 = K_(f3)*([CO]^1[O_2]^0[CO_2]^0[O]^0[H_2O]^0[OH]^1[H]^0) - K_(r3)*([CO]^0[O_2]^0[CO_2]^1[O]^0[H_2O]^0[OH]^0[H]^1)`
= `K_(f3)*([CO][OH]) - K_(r3)*([CO_2][H])`
For Reaction 4 (R4):
`q_4 = K_(f4)*([CO]^0[O_2]^1[CO_2]^0[O]^0[H_2O]^0[OH]^0[H]^1) - K_(r4)*([CO]^0[O_2]^0[CO_2]^0[O]^1[H_2O]^0[OH]^1[H]^0)`
= `K_(f4)*([[O_2][H]) - K_(r4)*([O][OH])`
Calculating Net Production Rate of each species:
Net production rate of each species is the rate of change of concentration of that species.
`w_j = d/dt[X_j]`
Using eq. (1),
For j = 1 (CO): Rate of change of concentration of carbon monoxide :
`w_1 = v_(11)q_1 + v_(12)q_2 + v_(13)q_3 + v_(14)q_4`
= `(-1)q_1 + (0)q_2 + (-1)q_3 + (0)q_4`
= `-q_1 - q_3`
= `-K_(f1)*([CO][O_2]) + K_(r1)*([CO_2][O])` - `K_(f3)*([CO][OH]) + K_(r3)*([CO_2][H])`
For j = 2 (`O_2`): Rate of change of concentration of oxygen :
`w_2 = v_(21)q_1 + v_(22)q_2 + v_(23)q_3 + v_(24)q_4`
=`(-1)q_1 + (0)q_2 + (0)q_3 + (-1)q_4`
=`-q_1 - q_4`
= `-K_(f1)*([CO][O_2]) + K_(r1)*([CO_2][O])` - `K_(f4)*([O_2][H]) + K_(r4)*([O][OH])`
For j = 3 (`CO_2)`: Rate of change of concentration of carbon dioxide :
`w_3 = v_(31)q_1 + v_(32)q_2 + v_(33)q_3 + v_(34)q_4`
=`(1)q_1 + (0)q_2 + (1)q_3 + (0)q_4`
= `q_1 + q_3`
= `K_(f1)*([CO][O_2]) - K_(r1)*([CO_2][O])` + `K_(f3)*([CO][OH]) - K_(r3)*([CO_2][H])`
For j = 4 (O): Rate of change of concentration of atomic oxygen :
`w_4 = v_(41)q_1 + v_(42)q_2 + v_(43)q_3 + v_(44)q_4`
= `(1)q_1 + (-1)q_2 + (0)q_3 + (1)q_4`
= `q_1 - q_2 + q_4`
= `K_(f1)*([CO][O_2]) - K_(r1)*([CO_2][O])` - `K_(f2)*([O][H_2O]) + K_(r2)*([OH]^2)` + `K_(f4)*([[O_2][H]) - K_(r4)*([O][OH])`
For j = 5 (`H_2O`): Rate of change of concentration of water :
`w_5 = v_(51)q_1 + v_(52)q_2 + v_(53)q_3 + v_(54)q_4`
= `(0)q_1 + (-1)q_2 + (0)q_3 + (0)q_4`
= `-q_2`
= `-K_(f2)*([O][H_2O]) + K_(r2)*([OH]^2)`
For j = 6 (OH): Rate of change of concentration of hydroxyl :
`w_6 = v_(61)q_1 + v_(62)q_2 + v_(63)q_3 + v_(64)q_4`
= `(0)q_1 + (2)q_2 + (-1)q_3 + (1)q_4`
= `2q_2 - q_3 + q_4`
= `2K_(f2)*([O][H_2O]) - 2K_(r2)*([OH]^2)` - `K_(f3)*([CO][OH]) + K_(r3)*([CO_2][H])` + `K_(f4)*([[O_2][H]) - K_(r4)*([O][OH])`
For j = 7 (H): Rate of change of concentration of atomic hydrogen :
`w_ 7= v_(71)q_1 + v_(72)q_2 + v_(73)q_3 + v_(74)q_4`
= `(0)q_1 + (0)q_2 + (1)q_3 + (-1)q_4`
= `q_3 - q_4`
= `K_(f3)*([CO][OH]) - K_(r3)*([CO_2][H])` - `K_(f4)*([[O_2][H]) + K_(r4)*([O][OH])`