These are a re-visitation of the kind of problems done in CE 561.  Hopefully your newly increased proficiency in Matlab and Maple will make them easier this time.

(1)  Laboratory Problem

Bhatnagar and Carr (Chem. Phys. Lett., 238, 1995) studied the reaction of CF₃CFHO₂ with NO. The peroxy radical, CF₃CFHO₂, is formed in the troposphere by reaction of HFC-134a (CF₃CFH₂) with OH and O₂. HFC-134a is the freon replacement that is currently used in automobile air conditioners. The reaction of CF₃CFHO₂ with NO is an important step in the degradation of HFC-134a in the troposphere. It is important that this compound is degraded in the troposphere so that it does not reach the stratosphere where it could release fluorine atoms that would contribute to ozone destruction. The rate parameters for the reaction studied in this work are needed for atmospheric models that include HFC-134a. Policy decisions about the use of different freon replacements are based (at least partially) on the predictions of these atmospheric models.

In their experiments, Bhatnagar and Carr initiated the reaction by photolyzing Cl₂ to produce Cl atoms that reacted with the HFC-134a to rapidly produce the peroxy radical (CF₃CFHO₂). The reaction under study

CF₃CFHO₂ + NO ® CF₃CFHO + NO₂

then occurred, followed by subsequent reactions of the oxy radical, CF₃CFHO. They followed the progress of the above reaction by measuring the concentration of NO₂ using time resolved mass spectrometry. The proposed reaction mechanism and rate coefficients at experimental conditions (for all reactions except the one under study) are
 
 

[Cl] = 4.32´ 10¹³ molecules cm^-3

[CF₃CFH₂] = 2.67´ 10¹⁷ molecules cm^-3

[NO] = 7.26´ 10¹³ molecules cm^-3

[O₂] = 2.3´ 10¹⁶ molecules cm^-3

Note that an initial concentration of Cl atoms is given, since the photolysis of Cl₂ produces these atoms on a time scale much shorter than that of the experiment.

Using the reaction mechanism presented above, apply a non-linear least squared analysis to obtain a value of the rate coefficient for reaction (4) (Bhatnagar and Carr got k₄ = 1.3´ 10^-11 cm³ molecule s^-1). Plot the resulting model prediction along with the experimental data. Compute the sensitivity of the NO₂ concentration profile to each of the rate constants in the mechanism. How does the sensitivity of the model results to the rate coefficient being measured compare to the sensitivity of the model results to the other rate coefficients? Comment on the effect of uncertainties in the other rate constants on the fitted value of the rate constant being measured.

(2)  Test Problem
Consider the network of 1^st order reactions given by

taking place in a fixed bed reactor with significant axial mixing.  In dimensionless form, the governing equations for this situation are given by the boundary value problem:

If the Peclet number is sufficiently large, then the diffusive terms can be neglected, and the equations become the initial value problem

Use Maple to find an analytical solution to the initial value problem that describes the reactor when the Peclet number is large.  Evaluate this solution for
C_1o= 1, Pe = 10, Da₁₂ = 5, Da₂₁ = 2, Da₂₃ = 10, Da₃₂ = 5.  Plot the result.

Use Matlab to solve the same equations numerically, for these values of of the parameters.  Confirm that you obtain the same result that you got using Maple.

Use Matlab to solve the full boundary-value-problem (including axial dispersion) for C_1o= 1, Da₁₂ = 5, Da₂₁ = 2, Da₂₃ = 10, Da₃₂ = 5, L=1, and values of the Peclet number ranging from Pe = 0.001 to Pe = 1000.  For Pe = 10, compare the result to the solution of the initial value problem.
Obtain this solution by discretizing the equations using finite differences, as discussed in CE 561.  The relevant lecture notes, including an example, are given here.