Chapter 9, Problem 9.76P

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).

1. 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 _{a d} ) = 0, where f(T _{a d} ) is a fourth-order polynomial $\left[f\left(T_{ad}\right)=c_0+c_1{T}_{ad}+c_2{T}_{ad}^2+c_3{T}_{ad}^3+c_4{T}_{ad}^4\right]$. If your derivation is correct, the ratio $\begin{array}{l}\\ c_0/c_4\end{array}$should equal -6.892 × 10 ^{1 4}. Use a spreadsheet program to determine $T_{ad}$ .

1. 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?
2. Why is the fourth-order solution at best an approximation and quite possibly a poor one? (Hint: Examine the conditions of applicability of the heat capacity formulas in Table B.2.)