Chapter 6: Isothermal Reactor Design: Molar Flow Rates

Stoichiometric Table – Measures other than Conversion

Let’s again consider the production of ethyl benzene (click back)

\( 2 \text{Ethylene} + \text{Toluene} \rightarrow \text{ethyl benzene} + \text{propylene} \)

The gas feed consists of 25% toluene and 75% ethylene. Write the rate of disappearance of toluene as a function of the molar flow rate. What are the coupled mole balances and rate laws?

Hint 1: What are the relative rates of reaction?  

Hint 2: Write the volumetric flow rate in terms of the molar flow rates  

Hint 3: Write the concentration of toluene and ethylene in terms of the molar flow rates  

Hint 4: Write the rate of disappearance of A, –r_A solely as a function of the molar flow rates  

Hint 5: Write combined rate laws and mole balances in a form to be solved using an ODE solver such as Polymath.

Hint 1.

\( A + 2B \rightarrow C + D \)

For relative rates for measures other than conversion, the stoichiometry comes in relating the relative rates.

For this elementary reaction

\( -r_A = k C_A C_B^2 \)

\( \frac{-r_A}{1} = \frac{-r_B}{2} = r_C = r_D \)

\( r_C = r_D = -r_A \)

\( r_B = 2r_A = -2k C_A C_B^2 \)

Back to problem

Hint 2.

Write the volumetric flow rate in terms of molar flow rates

\( v = v_0 \frac{F_T}{F_{T0}} \frac{P_0}{P} \frac{T}{T_0} \)

Isothermal operation and (T = T₀ and P = P₀)

\( v = \frac{F_T}{F_{T0}} \)

\( F_{T0} = 4F_{A0} \)

\( F_T = F_A + F_B + F_C + F_D \)

\( v = v_0 \left( \frac{F_A + F_B + F_C + F_D}{4F_{A0}} \right) \)

Back to problem

Hint 3.

In terms of molar flow rates

\( C_A = \frac{F_A}{v} \)

For a flow system at constant T and P

\( v = v_0 \frac{F_T}{F_{T0}} \)

\( C_A = \frac{F_{T0} F_A}{v_0 F_T} = C_{T0} \frac{F_A}{F_T} = 0.2 \frac{F_A}{F_T} \)

\( C_B = \frac{F_B}{v} = \frac{F_{T0} F_B}{v_0 F_T} = C_{T0} \frac{F_B}{F_T} = 0.2 \frac{F_B}{F_T} \)

\( F_T = F_A + F_B + F_C + F_D \)

Back to problem

Hint 4.

In terms of molar flow rates

\( -r_A = k C_A C_B^2 = k C_{T0} \left( \frac{F_A}{F_T} \right) \left( \frac{F_B}{F_T} \right)^2 \)

\( -r_A = k C^3 T_0 \frac{F_A F_B^2}{(F_A + F_B + F_C + F_D)^2} \)

\( k C^3 T_0 = \frac{250 \, \text{dm}^6}{\text{s} \cdot \text{mol}^2} \left( \frac{0.2 \, \text{mol}}{\text{dm}^3} \right)^3 = 2 \times 10^{-1} \left( \frac{\text{mol}}{\text{dm}^3} \right) \, \text{s}^{-1} \)

\( -r_A = 2 \times 10^{-1} \, \frac{\text{mol}}{\text{dm}^3} \frac{F_A F_B^2}{(F_A + F_B + F_C + F_D)^3} \)

\( -r_B = 2 \times 10^{-1} \, \frac{\text{mol}}{\text{dm}^3} \frac{F_A F_B^2}{F_T^3} \)

Back to problem

Hint 5.

\( -r_A = 2 \times 10^{-1} \, \frac{\text{mol}}{\text{dm}^3 \cdot \text{s}} \frac{F_A F_B^2}{(F_A + F_B + F_C + F_D)^3} \)

\( \frac{dF_A}{dV} = r_A \)

\( r_B = 2 r_A \)

\( \frac{dF_B}{dV} = 2 r_A = -2 \left( 2 \times 10^{-1} \right) \frac{F_A F_B^2}{(F_A + F_B + F_C + F_D)^3} \)

\( r_C = -r_A \)

\( \frac{dF_C}{dV} = -r_A \)

\( r_D = -r_A \)

\( \frac{dF_D}{dV} = -r_A \)

\( F_T = F_A + F_B + F_C + F_D \)

\( V = 0 \, F_A = F_{A0}, F_B = 3 F_{A0}, F_C = F_D = 0 \)

Use Polymath to solve these coupled ODEs

Back to problem

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