Chapter 9, Problem 9.42P

The equilibrium constant for the ethane dehydrogenation reaction,

   $C_2{H}_6\left(g\right)\rightleftharpoons C_2{H}_4\left(g\right)+H_2\left(g\right)$

is defined as

   $K_p\left(atm\right)=\frac{y_{C_2{H}_1}{y}_{H_2}}{y_{C_2{H}_6}}P$

where P(atm)is the total pressure and y₁is the mole fraction of the ith substance in an equilibrium mixture. The equilibrium constant has been found experimentally to vary with temperature according to the formula

   $K_p\left(T\right)=7.28\times {10}^6\exp \left[-17,000/T\left(K\right)\right]$(1)

The heat of reaction at 1273 K is +145.6 kJ, and the heat capacities of the reactive species may be approximated by the formulas

   $\begin{array}{l}\left(C_p\right)C_2{H}_4-9.419+0.1147T\left(K\right)\\ \left(C_p\right)H_2=26.90+4.167\times {10}^{-3}T\left(K\right)\\ \left(C_p\right)C_2{H}_6=11.35+0.1392T\left(K\right)\end{array}\}\left[J/mol\cdot K\right]$

Suppose pure ethane is fed to a continuous constant-pressure adiabatic reactor at 1273 K and pressure P(atm), the products emerge at T_f(K)and P(atm), and the residence time of the reaction mixture in the reactor is large enough for the outlet stream to be considered an equilibrium mixture of ethane, ethylene, and hydrogen.

1. Prove that the fractional conversion of ethane in the reactor is

   $f={\left(\frac{K_p}{P+K_P}\right)}^{1/2}$

(2)

Write an energy balance on the reactor, and use it to prove that

   $f=\frac{1}{1+\varphi \left(T_f\right)}$

(3)

where

   $\varphi \left(T_f\right)=\frac{\Delta H_f\left(1273K\right)-{\displaystyle {\int }_{T_f}^{1273K}\left[{\left(v{C}_p\right)}_{C_2{H}_4}+\left(v{C}_p\right)H_2\right]dT}}{{\displaystyle {\int }_{T_f}^{1273K}{\left(v{C}_p\right)}_{C_2{H}_6}dT}}$

(4)

Finally, substitute for $\Delta H_f$and the heat capacities in Equation 4 to derive an explicit expression for $\varphi \left(T_f\right)$.

We now have two expressions for the fractional conversion f: Equation 2 and Equation 3. If these expressions are equated, K_pis replaced by the expression of equation 1 and $\varphi \left(T_f\right)$

1. is replace by the expression derived in Part(b), the result is one equation in one unknown, T_f.Derive this equation, and transpose the right side to obtain an expression of the form
$\psi \left(T_f\right)=0$

(5)

Prepare a spreadsheet to take P as input, solve equation 5 for T_f(use Goal Seek or Solver), and determine the final fractional conversion, f.(suggestion: Set up columns for P, T_f,f,K_p, $\varphi$,and $\psi$.) Run the program for P(atm)=0.01, 0.05, 0.10, 0.50, 1.0, 5.0 and 10.0. Plot T_fversus P and f versus p, using a logarithmic coordinate scale for P.