Liquid n-pentane at 25°C is burned with 30% excess oxygen (not air) fed at 75°C. The adiabatic flame temperature is T_ad (°C).

(a) Take as a basis of calculation 1.00 mol C₅H₁₂ (l) burned and use an energy balance on the adiabatic reactor to derive an equation of the form f(T_ad) = 0 where f(T_ad) is a fourth-order polynomial [f (T_ad) = C₀ + C₁ T_ad + C₂T²_ad + C₃T³_ad + c₄T⁴_ad]. If your derivation is correct, the ratio c0, c4 should equal – 6.892 x 10¹⁴. Then solve the equation to determine Tad. (This solution is easily obtained using the goal seek tool on a spreadsheet.)

(b) Repeat the calculation of part (a) using successively the first two terms, the first three terms, and the first four terms of the fourth-order polynomial equation. If the solution of part (a) is taken to be exact, what percentage errors are associated with the linear (two-term), quadratic (three-term), and cubic (four-term) approximations?

(c) Determine the fourth-order solution using Newton’s method (Appendix A.2) taking the linear approximation as the first guess and stopping when |f (T_ad)| <0.01.

(d) Why is the fourth-order solution at best an approximation and quite possibly a poor one?