Question 14.3: Consider a system in which the following reactions occur: CH...
Consider a system in which the following reactions occur:
\mathrm{CH}_4+\mathrm{H}_2 \mathrm{O} \rightarrow \mathrm{CO}+3 \mathrm{H}_2 (1)
\mathrm{CH}_4+2 \mathrm{H}_2 \mathrm{O} \rightarrow \mathrm{CO}_2+4 \mathrm{H}_2 (2)
where the numbers (1) and (2) indicate the value of j, the reaction index. If 2 mol CH_4 and 3 mol H_2O are initially present, determine expressions for the yi as functions of ε_1 and ε_2 .
The stoichiometric numbers \nu_{i, j} can be arrayed as follows:
| i = | CH_4 | H_2O | CO | CO_2 | H_2 | |
| j
|
\nu_j
|
|||||
| 1 | −1 | −1 | 1 | 0 | 3 | 2 |
| 2 | −1 | −2 | 0 | 1 | 4 | 2 |
Here, we have labeled the columns with the actual species names, rather than numbers, understanding that CH_4 is species 1, H_2O is species 2, and so on. Application of Eq. (14.7) now gives:
y_i=\frac{n_{i_0}+\sum_j \nu_{i, j} \varepsilon_j}{n_0+\sum_j \nu_j \varepsilon_j} \quad(i=1,2, \ldots, N) (14.7)
y_{\mathrm{CH}_4}=\frac{2-\varepsilon_1-\varepsilon_2}{5+2 \varepsilon_1+2 \varepsilon_2} \quad y_{\mathrm{CO}}=\frac{\varepsilon_1}{5+2 \varepsilon_1+2 \varepsilon_2}
y_{\mathrm{H}_2 \mathrm{O}}=\frac{3-\varepsilon_1-2 \varepsilon_2}{5+2 \varepsilon_1+2 \varepsilon_2} \quad y_{\mathrm{CO}_2}=\frac{\varepsilon_2}{5+2 \varepsilon_1+2 \varepsilon_2}
y_{\mathrm{H}_2}=\frac{3 \varepsilon_1+4 \varepsilon_2}{5+2 \varepsilon_1+2 \varepsilon_2}
The composition of the system is a function of two independent variables ε_1 and ε_2.