A chemical process is represented by the following set of equations: (i) x1 - sin(πx2) = 0; (ii) 2x2 - ln(x3) = 1; (iii) x2 - x4/(4x1) = 0; (iv) x4/2 + 2x5 - 4 = 0; (v) cos(π(x5 -... A chemical process is represented by the following set of equations: (i) x1 - sin(πx2) = 0; (ii) 2x2 - ln(x3) = 1; (iii) x2 - x4/(4x1) = 0; (iv) x4/2 + 2x5 - 4 = 0; (v) cos(π(x5 - x2)) + x1 = 0; (vi) ln(x6 - x1) = 0. Precedence order these equations using the SWS (Structural Analysis with Substitution) algorithm along with Stewart and Rudd algorithms. Read more

Steps to Solve

1. Identify Equations and Variables

List all equations with their respective variables:

• (i) $x_1 - \sin(\pi x_2) = 0$
• (ii) $2x_2 - \ln x_3 = 1$
• (iii) $x_2 - \frac{x_4}{4x_1} = 0$
• (iv) $\frac{x_4}{2} + 2x_5 - 4 = 0$
• (v) $\cos(\pi(x_5 - x_2)) + x_1 = 0$
• (vi) $\ln(x_6 - x_1) = 0$

2. Determine Dependencies

Analyze dependencies among the variables in each equation:

• Equation (i) depends on $x_2$ for $x_1$.
• Equation (ii) depends on $x_3$ for $x_2$.
• Equation (iii) depends on $x_1$ and $x_4$ for $x_2$.
• Equation (iv) depends on $x_4$ and $x_5$.
• Equation (v) relates $x_1$, $x_5$, and $x_2$.
• Equation (vi) depends on $x_1$ and $x_6$.

3. Create Dependency Graph

Draw a dependency graph, listing which equations depend on which variables.

4. Rank Equations by Dependencies

Using the graph:

• Start with the variables that have no dependencies.
• List equations based on their dependency requirements in reverse topological order.

5. Precedence Order

Determine the precedence order as follows:

• From the dependencies analyzed, create a precedence list.

This may be approximated as:

• (ii) → (i) → (iii) → (iv) → (v) → (vi)