Chapter 11: Nonisothermal Reactor Design: The Steady State Energy Balance and Adiabatic PFR Applications

Second Order Reaction Carried Out Adiabatically in a CSTR

6. Solve the Mole Balance for \( X_{MB} \) as a function of T:

\( \tau k C_{A0} = \frac{X}{(1 - X)^2} \)

Rearranging gives:

\( \tau k C_{A0} - 2 \tau k C_{A0} X + \tau k C_{A0} X^2 = X \)

\( \tau k C_{A0} - (2 \tau k C_{A0} + 1) X + \tau k C_{A0} X^2 = 0 \)

Our equation for X has taken the form of a quadratic equation, so we solve for X accordingly:

\( X = \frac{(2 \tau k C_{A0} + 1) - \sqrt{(2 \tau k C_{A0} + 1)^2 - 4(\tau k C_{A0})^2}}{2 \tau k C_{A0}} \)

\( X = \frac{(2 \tau k C_{A0} + 1) - \sqrt{4(\tau k C_{A0})^2 + 4 \tau k C_{A0} + 1 - 4 (\tau k C_{A0})^2}}{2 \tau k C_{A0}} \)

After some final rearranging we get:

\( X_{MB} = \frac{(2 \tau k C_{A0} + 1) - \sqrt{4 \tau k C_{A0} + 1}}{2 \tau k C_{A0}} \)