Parameter Estimation

---

THE ESTIMATION PROCEDURE

Kinetic rate constants and rate expressions can be determined from reactor data. For batch reactor data you would employ an optimization algorithm, such as npsol to minimize the error in the objective function

where n is the number of data points and (t_i) is the experimental measurement of c_j and at time t_i. A rate expression is assumed and the batch reactor material balance is solved to give an analytical equation of the form:

c_j = f (t, k, Vr, initial conditions, stoichiometry)

The optimization routine will search for the parameter or parameters that best fit c_j(t_i) to (t_i). This is illustrated with the following example.

---

Example

The following reaction was studied in an isothermal batch reactor.

CH₃COCH₃ + HCN ---> (CH₃)₂(OH)CN

The reaction rate is known to be first order in the concentration of acetone and first order in the concentration of HCN. When the reaction was conducted in an aqueous solution and the initial concentrations were c_HCN,o = 0.0758 and c_acetone,o = 0.1164 gmoles/liter, the concentration data listed in the table below were recorded. Use these data to determine the magnitude of the reaction rate constant.

Solution

For this particular problem we integrate the design equation

where g = c_acetone,o - c_HCN,o. After integration between time t = 0 and t and between c_HCN,o and c_HCN, and sorting for c_HCN, we get

This equation can be used to calculate the c_j(t_i) in F . Fitting these data to a first order rate expression results in k = 0.00763 min^-1.

---

THE PROBLEM STATEMENT

The hydrolysis of ethylene oxide in the presence of H₂SO₄ to ethylene glycol has been studied at 55°C by mixing 500 mL of a 2M solution of ethylene oxide in water with 500 mL of water containing 0.9 wt% H₂SO₄ in a batch reactor. (H₂SO₄ is a catalyst and is not consumed during the reaction.)

CH2CH2O + H₂O ---> CH₂OHCH₂OH

The concentration versus time data are shown and you are to use these data to determine the reaction order for ethylene oxide and the value of the rate constant.

---

THE QUESTIONS

1. Note that the initial concentration of ethylene oxide is 1 gmole/liter and the initial concentration of water is 52.8 gmoles/liter. So as the concentration of ethylene oxide changes, do you think you will notice any change in the amount of water? What first guess of the rate expression seems most reasonable?

2. Develop equations for

C_{ethylene oxide} = f (t, k, V_r, initial conditions, stoichiometry)

when

(a)

r = kc_{ethylene oxide}

(b)



3. Click on the Simulate button to view the fits of the data to the assumed expressions. These plots were generated by using the optimum values for k in a numerical solution to

Note the optimization routine is forced to give a best fit to an assumed model. You still need to assess if this assumed model adequately represents the data. The plot of the data against a best fit to a second order model illustrates the poor quality of the fit and when compared against a better fit for the first order reaction, you would choose the first order reaction. The distinction between competing models is not always as straightforward. One way to help establish which model, of competing models, best represents the data is to view the residuals,

and select those models with randomly distributed residuals. Incorrect models will display trends in the residuals. The optimization program generates a plot of the residuals that you can view.