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reaction_rate_equation [2014/03/21 11:11] 127.0.0.1 external edit |
reaction_rate_equation [2015/08/15 20:34] nikolaj |
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| | @#55CCEE: context | @#55CCEE: $ \nu^-,\nu^+\in\mathrm{Matrix}(R,J,\mathbb Q) $ | | | @#55CCEE: context | @#55CCEE: $ \nu^-,\nu^+\in\mathrm{Matrix}(R,J,\mathbb Q) $ | | ||
| | @#55CCEE: context | @#55CCEE: $ k\in \mathbb R^R $ | | | @#55CCEE: context | @#55CCEE: $ k\in \mathbb R^R $ | | ||
| - | |||
| | @#FFBB00: definiendum | @#FFBB00: $ [A] \in \mathrm{it} $ | | | @#FFBB00: definiendum | @#FFBB00: $ [A] \in \mathrm{it} $ | | ||
| - | |||
| | $j\in \text{range}(J)$ | | | $j\in \text{range}(J)$ | | ||
| - | |||
| | @#55EE55: postulate | @#55EE55: $ [A]:C(\mathbb R,\mathbb R^J) $ | | | @#55EE55: postulate | @#55EE55: $ [A]:C(\mathbb R,\mathbb R^J) $ | | ||
| - | |||
| | @#DDDDDD: range | @#DDDDDD: $ ::[A](t) $ | | | @#DDDDDD: range | @#DDDDDD: $ ::[A](t) $ | | ||
| - | |||
| | @#55EE55: postulate | @#55EE55: $ \frac{\partial}{\partial t}[A]_j=\sum_{r=1}^R k_r\cdot(\nu_{rj}^+-\nu_{rj}^-)\cdot\prod_{i=1}^J [A]_i^{\nu_{ri}^-} $ | | | @#55EE55: postulate | @#55EE55: $ \frac{\partial}{\partial t}[A]_j=\sum_{r=1}^R k_r\cdot(\nu_{rj}^+-\nu_{rj}^-)\cdot\prod_{i=1}^J [A]_i^{\nu_{ri}^-} $ | | ||
| - | ==== Discussion ==== | + | ----- |
| The quantities $R$ and $J$ denote the number of reactions and the number of different species. | The quantities $R$ and $J$ denote the number of reactions and the number of different species. | ||
| Then $\nu_{rj}^-$ and $\nu_{rj}^+$ are stochastic coefficients of the reactants and products and $k_r$ is the reaction rate coefficient of the $r$'s reaction. | Then $\nu_{rj}^-$ and $\nu_{rj}^+$ are stochastic coefficients of the reactants and products and $k_r$ is the reaction rate coefficient of the $r$'s reaction. | ||
| Line 23: | Line 18: | ||
| Non-time resolved, this reads for all $r$ | Non-time resolved, this reads for all $r$ | ||
| - | $\sum_{j=1}^J \nu_{rj}^{(e)} A_j \overset{k_r}{\longrightarrow} \sum_{j=1}^J \nu_{rj}^{(p)} A_j$ | + | $\sum_{j=1}^J \nu_{rj}^{(e)} A_j \overset{k_r}{\longrightarrow} \sum_{j=1}^J \nu_{rj}^{(p)} A_j.$ |
| - | e.g. the simplest carbon combustion process | + | For example, the simplest carbon combustion process: |
| + | $\mathrm{C}\mathrm{H}_4 + 2\ \mathrm{O}_2 \longrightarrow \mathrm{C}\mathrm{O}_2 + 2\ \mathrm{H}_2\mathrm{O}.$ | ||
| - | $CH_4 + 2\ O_2 \longrightarrow CO_2 + 2\ H_2O $ | + | (Or more explicitly: |
| + | $1\ \mathrm{C}\mathrm{H}_4 + 2\ \mathrm{O}_2 + 0\ \mathrm{C}\mathrm{O}_2 + 0\ \mathrm{H}_2\mathrm{O} \longrightarrow 0\ \mathrm{C}\mathrm{H}_4 + 0\ \mathrm{O}_2 + 1\ \mathrm{C}\mathrm{O}_2 + 2\ \mathrm{H}_2\mathrm{O}$.) | ||
| - | or more explicitly | + | In practice, $k$ depends on the temperature, which, through the equation of state, can again be a nonlinear function of the concentrations. |
| - | $1\ CH_4 + 2\ O_2 + 0\ CO_2 + 0\ H_2O \longrightarrow 0\ CH_4 + 0\ O_2 + 1\ CO_2 + 2\ H_2O $ | ||
| - | |||
| - | In practice, $k$ depends on the temperature, which, through the equation of state, can again be a nonlinear function of the concentrations. | ||
| === Reference === | === Reference === | ||
| Wikipedia: [[https://en.wikipedia.org/wiki/Rate_equation|Rate equation]] | Wikipedia: [[https://en.wikipedia.org/wiki/Rate_equation|Rate equation]] | ||
| - | ==== Parents ==== | + | ----- |
| === Subset of === | === Subset of === | ||
| [[ODE system]] | [[ODE system]] | ||
