Week 5 - Compact Notation Derivation for a simple Mechanism
JORGE LOZANO
Computational Combustion using Python and Cantera
Compact Notation Derivation for a simple Mechanism
Objective
For this challenge, you will derive the reaction rate ODEs of a simple reaction mechanism. You will have to use the compact notation that was discussed in the last 3 lectures and write down the desired matrices. This is a pen and paper assignment. However, you must use the equation editor to type your answers.
Reaction Mechanism:
- CO + O2 -> CO2 + O
- O + H2O -> OH + OH
- CO + OH -> CO2 + H
- H + O2 -> OH + O
To determine the derivation for a simple mechanism as noted in the objective it's important to define the species and reactions proposed by which a simulation or analysis will be performing.:
Species:7
Reactions:4
1
`CO + O2 -> CO2 + O`
2
`O + H2O -> OH + OH`
3
`CO + OH -> CO2 + H`
4
`H + O2 -> OH + O`
1-7
`Species: CO, O2, CO2, O, H2O, OH, H`
Documentacion:
Reaction Mechanism [2 & 3]
A balanced equation for a chemical reaction indicates what is reacting and what is produced, but it reveals nothing about how the reaction actually takes place. The reaction mechanism (or reaction path) is the process, or pathway, by which a reaction occurs.
A chemical reaction usually occurs in steps, although it may not always be obvious to an observer. The decomposition of ozone, for example, appears to follow a mechanism with two steps:
`\begin{array}{r @{{}\longrightarrow{}} l} \text{O}_3(g) & \text{O}_2(g)\;+\;\text{O} \\[0.5em] \text{O}\;+\;\text{O}_3(g) & 2\text{O}_2(g) \end{array}`
We call each step in a reaction mechanism an elementary reaction. Elementary reactions occur exactly as they are written and cannot be broken down into simpler steps. Elementary reactions add up to the overall reaction, which, for the decomposition, is:
`2\text{O}_3(g)\rightarrow3\text{O}_2(g)`
Notice that the oxygen atom produced in the first step of this mechanism is consumed in the second step and therefore does not appear as a product in the overall reaction. Species that are produced in one step and consumed in a subsequent step are called intermediates.
While the overall reaction equation for the decomposition of ozone indicates that two molecules of ozone react to give three molecules of oxygen, the mechanism of the reaction does not involve the collision and reaction of two ozone molecules. Rather, it involves a molecule of ozone decomposing to an oxygen molecule and an intermediate oxygen atom; the oxygen atom then reacts with a second ozone molecule to give two oxygen molecules. These two elementary reactions occur exactly as they are shown in the reaction mechanism.
In a sample of C4H8, a few of the rapidly moving C4H8 molecules collide with other rapidly moving molecules and pick up additional energy. When the C4H8 molecules gain enough energy, they can transform into an activated complex, and the formation of ethylene molecules can occur. In effect, a particularly energetic collision knocks a C4H8 molecule into the geometry of the activated complex. However, only a small fraction of gas molecules travel at sufficiently high speeds with large enough kinetic energies to accomplish this. Hence, at any given moment, only a few molecules pick up enough energy from collisions to react.
The rate of decomposition of C4H8 is directly proportional to its concentration. Doubling the concentration of C4H8 in a sample gives twice as many molecules per liter. Although the fraction of molecules with enough energy to react remains the same, the total number of such molecules is twice as great. Consequently, there is twice as much C4H8 per liter, and the reaction rate is twice as fast:
A similar relationship applies to any unimolecular elementary reaction; the reaction rate is directly proportional to the concentration of the reactant, and the reaction exhibits first-order behavior. The proportionality constant is the rate constant for the particular unimolecular reaction.
Relating reaction mechanisms to rate laws
It’s often the case that one step in a multistep reaction mechanism is significantly slower than the others. Because a reaction cannot proceed faster than its slowest step, this step will limit the rate at which the overall reaction occurs. The slowest step is therefore called the rate-limiting step (or rate-determining step).
As described earlier, rate laws may be derived directly from the chemical equations for elementary reactions. This is not the case, however, for ordinary chemical reactions. The balanced equations most often encountered represent the overall change for some chemical system, and very often this is the result of some multistep reaction mechanisms. In every case, we must determine the overall rate law from experimental data and deduce the mechanism from the rate law (and sometimes from other data).
The sequence of individual steps, or elementary reactions, by which reactants are converted into products during the course of a reaction is called the reaction mechanism. The overall rate of a reaction is determined by the rate of the slowest step, called the rate-determining step. Unimolecular elementary reactions have first-order rate laws, while bimolecular elementary reactions have second-order rate laws. By comparing the rate laws derived from a reaction mechanism to that determined experimentally, the mechanism may be deemed either incorrect or plausible.
Note the important difference between writing rate laws for elementary reactions and the balanced chemical equation of the overall reaction. Because the balanced chemical equation does not necessarily reveal the individual elementary reactions by which the reaction occurs, we cannot obtain the rate law for a reaction from the overall balanced chemical equation alone. In fact, it is the rate law for the slowest overall reaction, which is the same as the rate law for the slowest step in the reaction mechanism, the rate-determining step, that must give the experimentally determined rate law for the overall reaction.This statement is true if one step is substantially slower than all the others, typically by a factor of 10 or more. If two or more slow steps have comparable rates, the experimentally determined rate laws can become complex. Our discussion is limited to reactions in which one step can be identified as being substantially slower than any other. The reason for this is that any process that occurs through a sequence of steps can take place no faster than the slowest step in the sequence. In an automotive assembly line, for example, a component cannot be used faster than it is produced. Similarly, blood pressure is regulated by the flow of blood through the smallest passages, the capillaries. Because movement through capillaries constitutes the rate-determining step in blood flow, blood pressure can be regulated by medications that cause the capillaries to contract or dilate. A chemical reaction that occurs via a series of elementary reactions can take place no faster than the slowest step in the series of reactions.
A balanced chemical reaction does not necessarily reveal either the individual elementary reactions by which a reaction occurs or its rate law. A reaction mechanism is the microscopic path by which reactants are transformed into products. Each step is an elementary reaction. Species that are formed in one step and consumed in another are intermediates. Each elementary reaction can be described in terms of its molecularity, the number of molecules that collide in that step. The slowest step in a reaction mechanism is the rate-determining step.
Process [1]
As the example from the videos course, species and the reactions are identified along the subindex "j", and 1-4 respectively, generating the matrices for reactants and products of the mechanism.
`(nu_{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)]`
`(nu_{ji})^('') = [(0,,0,,1,,1,,0,,0,,0),(0,,0,,0,,0,,0,,1,,0),(0,,0,,1,,0,,0,,0,,1),(0,,0,,1,,0,,0,,0,,1)]`
Meaning the following equation is in accordance:
`v_(ji)=v_(ji)^('') - v_(ji)^'`
`v_(ji)` is presented as:
`(nu_{ji}) = [(-1,,-1,,1,,1,,0,,0,,0),(0,,0,,0,,-1,,-1,,1,,0),(-1,,0,,1,,0,,0,,-1,,1),(0,,-1,,1,,0,,0,,0,,0)]`
Being N as the quantity of reactants:
`q_i = k_{fi}prod_(j=1)^N [X_j]^(nu'_{ji}) - k_{ri}prod_(j=1)^N [X_j]^(nu''_{ji})`
Having the species from the reactions extended as:
`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`
And having the partial separated mechanisms as:
`q_1 = k_{f1}[CO][O_2] - k_{r1}[CO_2][O]`
`q_2 = k_{f2}[O][H_2O] - k_{r2}[OH]^2`
`q_3 = k_{f3}[CO][OH] - k_{r3}[CO_2][H]`
`q_4 = k_{f4}[O_2][H] - k_{r4}[O][OH]`
The mechanism rates are represented by:
`omega_j = sum_(i=1)^L nu_(ji) q_i`
Meaning the following matrix is equal to the before presented values for each species:
`[(d[CO]"/"dt),(d[O_2]"/"dt),(d[CO_2]"/"dt),(d[O]"/"dt),(d[H_2O]"/"dt),(d[OH]"/"dt),(d[H]"/"dt)]=[(omega_1),(omega_2),(omega_3),(omega_4),(omega_5),(omega_6),(omega_7)]`
Having obtained each one of the species derivates that will determine the reaction of forward and backward reactions:
`{d[CO]}/{dt} = k_{r1}[CO_2][O] - k_{f1}[CO][O_2] + k_{r3}[CO_2][H] - k_{f3}[CO][OH]`
`{d[O_2]}/{dt} = k_{r1}[CO_2][O] - k_{f1}[CO][O_2] + k_{r4}[O][OH] - k_{f4}[O_2][H]`
`{d[CO_2]}/{dt} = k_{f1}[CO][O_2] - k_{r1}[CO_2][O] + k_{f3}[CO][OH] - k_{r3}[CO]`
`{d[O]}/{dt} = k_{f1}[CO][O_2]-k_{r1}[CO_2][O] + k_{r2}[OH]^2 - k_{f2}[O][H_2O] + k_{f4}[O_2][H] - k_{r4}[O][OH]`
`{d[H_2O]}/{dt} = k_{r2}[OH]^2 - k_{f2}[O][H_2O]`
`{d[OH]}/{dt} = 2k_{f2}[O][H_2O] - 2k_{r2}[OH]^2 + k_{r3}[CO_2][H] - k_{f3}[CO][OH] + k_{f4}[O_2][H] - k_{r4}[O][OH]`
`{d[H]}/{dt} = k_{f3}[CO][OH] - k_{r3}[CO_2][H] + k_{r4}[O][OH] - k_{f4}[O_2][H]`
Conclusion:
As defined in the video course, a simple mechanism's derivation for each species and reaction rates were defined and obtained, being able to use them in a computational analysis to compute several systems by using the according defined combustion and reactions proposed or obtained.
References: