\[\require{mhchem}\]

HW1 (due noon Monday 9/5)

Catalyst deactivation in a batch reactor [50 pts]

Consider the irreversible, liquid-phase isomerization reaction carried out in a solvent containing dissolved catalyst at 25 C in a batch reactor:

\[\begin{align*} \ce{B ->[k_b] C} \end{align*}\]

The apparent first-order reaction rate constant \(k_b\) decreases with time because of catalyst deterioriation. A chemist friend of yours has studied the catalyst deactivation process and has proposed that it can be modeled with

\[\begin{align*} k_b = \frac{k}{1+k_dt} \end{align*}\]

in which \(k\) is the fresh catalyst rate constant and \(k_d\) is the deactivation rate constant.

We can derive the ODE to solve by starting with a mol balance on the entire system:

\[\begin{align*} \frac{dN_b}{dt} &=N_b^0 - Vr \\ \frac{dC_b}{dt} &= -k_bC_b=-\frac{kC_b}{1+k_dt} \end{align*}\]

with the initial condition \(C_b(t=0)=C_{b0}=5\) M.

Mol balance solve

Solve the mole balance for \(C_B(t)\) assuming \(k\)=0.6/hr and \(k_d\)=2/hr for the first two hours. Plot the conversion % for your solution (defined as \(1-C_B(t)/C_{B0}\)).

If it takes two hours to reach 50% conversion and the fresh catalyst has a rate constant of 0.6/hr what is the actual \(k_d\)?

Using \(k_d\) you found from the previous step, use solve_ivp events to determine how long it takes to reach 75% conversion in the reactor.

Catalyst refresh

Say that we can stop the batch process after 2 hours, filter the spent catalyst, and replace with fresh catalyst. \(C_B\) will start wherever the first reaction left off. Solve and plot for \(C_B(t)\) over 5 hours, and include the plot with the catalyst refresh. Include a legend on the plot to label each solution

Curve fitting with two species and multiple data (50 pt)

Consider the irreversible reaction:

\[\begin{align*} \ce{A + B -> Products} \end{align*}\]

with \(r=kC_A^nC_B^m\) taking place in an isothermal liquid-phase batch reactor. Measurements of \(C_A\) vs \(C_B\) are included in the attached file isothermal_liquidphase_batch_data.dat. We wish to determine from the data the rate constant and the order of the reaction with respect to A and B. We have data from two experiments.

Load the data from the file into a numpy array and plot the concentration of each species

You can use either the csv library https://docs.python.org/3/library/csv.html or pandas https://pandas.pydata.org/pandas-docs/stable/generated/pandas.read_csv.html.

The first column is time in minutes. The second and third column is C_A and C_B for the first experiment (in mol/L). The fourth and fifth column is C_A and C_B for the second experiment. Plot the data for \(C_A\) and \(C_B\) for each experiment (one experiment per figure).

Using lmfit, estimate rate parameters \(k, n, m\) and initial concentration \(C_{A0},C_{B0}\) from the data in the first experiment using a numerical solution for the concentrations of each species (standard mol balance + solve_ivp). Plot the final fit of the experiment along with the experimental data and calculate the uncertainty in each value

Consider the second experiment, estimate the parameters using only this data. Comment on how the values and confidence intervals are different than the first case

Estimate the parameters using both experiments simultaneously. Are the confidence intervals better? Comment on why or why not. Hint: For initial parameter guesses, consider the regions \(k,n,m\in [1.5, 2.5]\). For the initial concentrations, there are natural guesses based on the data set.

Using the noise-free data in isothermal_liquidphase_batch_errorfree.dat, repeat the fit using all of the data. Do the results agree with your initial trials?

Based on these results, to determine the order of the rate expression with respect to two different species, how should you choose the initial conditions of the two species?