Unit 14. Differential Data Analysis

This website provides learning and teaching tools for a first course on kinetics and reaction engineering. The course is divided into four parts (I through IV). Here, in Part II of the course, the focus is on chemical reaction kinetics, and more specifically, on rate expressions, which are mathematical models of reaction rates. As you progress through Part II, you will learn how rate expressions are generated from experimental kinetics data.

Part II of the course concludes with Section C which describes how to test a rate expression (Section A) using experimental data (Section B). The testing of a rate expression entails its substitution into the model for the experimental reactor and the subsequent fitting of that model to the experimental data. The end result will reveal whether the selected rate expression offers a sufficiently accurate representation of the rate of the reaction under consideration. If it does, the fitting process also will yield the best values for the parameters that appear in the selected rate expression.

When only one reaction takes place in either an isothermal batch reactor or an isothermal, steady state PFR, the model for each experiment takes the form of a differential equation. Unit 14 describes an approach to fitting this model equation to experimental data wherein the derivative is treated like an experimentally measured variable. When this is done, the fitting process becomes the same as that described for a CSTR in Unit 13. This approach is known as differential data analysis, and the situations where it can be used are restricted. Unit 14 describes these restrictions.

Learning Resources

• Archive (.zip) - Contains all learning resources listed below for this unit
• Documents to Read:
  • Unit 14 Informational Reading (.pdf)
  • Example 14.1 (.pdf)
  • Example 14.2 (.pdf)
  • Example 14.3 (.pdf)
• Videos to Watch (please right-click and save, then play back locally on your computer):
  • Unit 14 Informational Video (.mov)
  • Example 14.1 Video (.mov)
  • Example 14.2 Video (.mov)
  • Example 14.3 Video (.mov)
• Reference Files:
  • Unit 14 Definitions, Nomenclature and Equations (.pdf)
• Other Files:
  • 14_How_To.pdf - How to test a rate expression using differential data analysis and linear least squares
  • Example_14_1.m - MATLAB file used in the solution of Example 14.1
  • Example_14_2.m and Example_14_2_polyfit.m - MATLAB files used in the solution of Example 14.2
  • Example_14_3.m - MATLAB file used in the solution of Example 14.3

Teaching Resources

• Archive (.zip) - Contains all teaching resources listed below for this unit
• Sample Class
  • Online Pre-class Quiz (.pdf file)
  • Redacted slides to make available to students before class
    • Keynote 09
    • PowerPoint
  • Learning Activity Files to make available to students before class
    • Activity 1 Problem Statement (.pdf)
    • Activity 1 Handout (.xlsx)
  • Class Lesson Plan (.pdf)
  • Complete slides to make available to students after class
    • Keynote 09
    • PowerPoint
  • Solution to the in-class activities to make available to students after class
    • Activity 14.1 Solution (.pdf)
    • Activity_14_1.m and Activity_14_1_polyfit.m - MATLAB files used in the solution of Activity 14.1

Practice Problems

1*. The data in the table below represent the conversion versus time behavior for reaction (1a) taking place in a 5 L batch reactor with an initial composition that was 1 M in A, 1.5 M in B and contained no Y or Z. Actually, the data in the table were calculated using an ideal batch reactor model with the rate expression given in equation (1b) and with k_1f = 0.0947 L mol^-1 min^-1 and k_1r = 0.0369 L mol^-1 min^-1. As such, these data do not contain any experimental noise. Perform a differential data analysis to find the values and uncertainties of the two rate coefficients using forward differences. Then repeat the analysis only using backward differences. Comment upon any differences among the actual rate coefficient values and the values obtained using the two different analysis methods.

(Problem Statement as .pdf file)

2. Suppose three preliminary experiments involving the aqueous reaction of acetic acid (A) with excess butanol (B) to form butyl acetate (Z) and water (W), reaction (2a) were performed using an agitated 500 mL round-bottomed flask as the reactor. All three experiments were performed at the same temperature; they differed in the initial concentrations of A and B. In each experiment the concentration of acetic acid was measured at increasing reaction times. The data are shown in this Excel© file. Using a differential data analysis, determine whether the second order rate expression in equation (2b) is acceptably accurate. If it is, determine the best value of the rate coefficient, including 95% confidence limits.

(Problem Statement as .pdf file)