Day3 - CE 561 Lecture Notes Fall 2009 Day 3: Matrix...

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CE 561 Lecture Notes Fall 2009 p. 1 of 8 Day 3: Matrix notation for reactions and rate equations; Classical matrix methods for systems of 1 st order reactions Matrix notation and linearly independent reaction sets As noted before, we may write a general set of reactions as M i A N j j ij , 1 , 0 1 = = = α ( α ij <0 for reactants, >0 for products) where N is the number of species, M is the number of reactions, and ij is the stoichiometric coefficient of species j in reaction i . This may be written in matrix notation as 0 = A where is the matrix of stoichiometric coefficients and A is the vector of chemical species. Example : For the set of reactions (note that these are not elementary reactions) CH 4 + 2 O 2 CO 2 + 2 H 2 O CH 4 + 3/2 O 2 CO + 2 H 2 O CO + H 2 O CO 2 + H 2 CO+1/2 O 2 CO 2 we can define the vector of species as = 2 2 2 2 4 H CO O H CO O CH A then the matrix of stoichiometric coefficients is = 0 1 0 1 0 1 1 1 1 0 0 0 1 2 0 1 0 0 2 1 2 1 2 1 2 3 In general, the reactions in a mechanism may not all be linearly independent. That is, some of the reactions may be linear combinations of others. The number of independent reactions is
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CE 561 Lecture Notes Fall 2009 p. 2 of 8 equal to the rank of the matrix of stoichiometric coefficients. The rank may be found by performing elementary row operations – as you may have learned in a linear algebra class. For the example above, there are 3 independent reactions. Prove this to yourself. For a thermodynamic analysis (like determining the equilibrium composition) or for performing material balances it is almost always best to consider the minimal set of linearly independent reactions. For kinetic analysis and reactor simulation it is often better to include more than the linearly independent set. This is because we generally know the rate parameters for particular reactions. If we were to take linear combinations of these reactions, we would also have to take linear combinations of their rate expressions. We may also define a vector of reaction rates r and a vector of species concentrations C so that the rate equations (as always during this first part of the course, for a well-mixed, constant volume system) are given by r dt C d T α = where T is the transpose of the stoichiometric matrix. For the example that we were just considering, this may be explicitly written as = 4 3 2 1 2 1 2 3 H CO O H CO O CH 0 1 0 0 1 1 1 0 0 1 2 2 1 1 0 1 0 2 0 0 1 1 C C C C C C 2 2 2 2 4 r r r r dt d We could perform row operations on the transpose of the stoichiometric matrix to obtain the number of linearly independent reactions. That is also the number of linearly independent ordinary differential equations.
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Day3 - CE 561 Lecture Notes Fall 2009 Day 3: Matrix...

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