Main

Nonlinear confidence intervals can also be visualized as a function of 2 parameters. In this case, both parameters are simultaneously varied to find the confidence region. The confidence interval is determined with an F-test that specifies an upper limit to the deviation from the optimal solution

-> Attach:f-test_equation.gif

with p=2 (number of parameters), n=number of measurements, theta=[parameter 1, parameter 2] (parameters), theta'^*^' as the optimal parameters, SSE as the sum of squared errors, and the F statistic that has 3 arguments (alpha=confidence level, degrees of freedom 1, and degrees of freedom 2). For many problems, this creates a multi-dimensional nonlinear confidence region. In the case of 2 parameters, the nonlinear confidence region is a 2-dimensional space. Below is an example that shows the confidence region for the dye fading experiment confidence region for forward and reverse activation energies.

-> Attach:dye_2d_confidence_region.png

The optimal parameter values are in the 95% confidence region. This plot demonstrates that the 2D confidence region is not necessarily symmetric.
~~* [[Attach:dye_fading_multiple.zip | Fitting Multiple Dynamic Data Sets in Python (zip)]]~~

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!!!! Multiple Data Sets

* [[Attach:dye_fading_multiple.zip | Fitting Multiple Dynamic Data Sets in Python (zip)]]

* [[Attach:dye_fading_multiple.zip | Confidence Intervals with Multiple Dynamic Data Sets in Python (zip)]]
~~* [[Attach:dye_fading_conf_int.zip | Nonlinear Confidence Intervals in MATLAB and Python (zip)]]~~

## Estimate Kinetic Parameters from Dynamic Data

## Main.KineticModeling History

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Nonlinear confidence intervals can also be visualized as a function of 2 parameters. In this case, both parameters are simultaneously varied to find the confidence region. The confidence interval is determined with an F-test that specifies an upper limit to the deviation from the optimal solution

-> Attach:f-test_equation.gif

with p=2 (number of parameters), n=number of measurements, theta=[parameter 1, parameter 2] (parameters), theta'^*^' as the optimal parameters, SSE as the sum of squared errors, and the F statistic that has 3 arguments (alpha=confidence level, degrees of freedom 1, and degrees of freedom 2). For many problems, this creates a multi-dimensional nonlinear confidence region. In the case of 2 parameters, the nonlinear confidence region is a 2-dimensional space. Below is an example that shows the confidence region for the dye fading experiment confidence region for forward and reverse activation energies.

-> Attach:dye_2d_confidence_region.png

The optimal parameter values are in the 95% confidence region. This plot demonstrates that the 2D confidence region is not necessarily symmetric.

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* [[Attach:dye_fading_multiple.zip | Confidence Intervals with Multiple Dynamic Data Sets in Python (zip)]]

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* [[Attach:dye_fading_multiple_conf_int.zip | Confidence Intervals with Multiple Dynamic Data Sets in Python (zip)]]

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!!!! Nonlinear Confidence Intervals

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!!!! Multiple Data Sets

* [[Attach:dye_fading_multiple.zip | Fitting Multiple Dynamic Data Sets in Python (zip)]]

* [[Attach:dye_fading_multiple.zip | Confidence Intervals with Multiple Dynamic Data Sets in Python (zip)]]

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* [[Attach:dye_fading_conf_int.zip | Nonlinear Confidence Intervals in Excel, MATLAB, and Python (zip)]]

(:html:)

<iframe width="560" height="315" src="http://www.youtube.com/embed/rL7Mvl2-XIM" frameborder="0" allowfullscreen></iframe>

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* [[Attach:dye_fading_conf_int.zip | Nonlinear Confidence Intervals in Excel, MATLAB, and Python (zip)]]

(:html:)

<iframe width="560" height="315" src="http://www.youtube.com/embed/rL7Mvl2-XIM" frameborder="0" allowfullscreen></iframe>

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* [[Attach:dye_fading_multiple.zip | Fitting Multiple Dynamic Data Sets in Python (zip)]]

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* [[Attach:dye_fading_conf_int.zip | Nonlinear Confidence Intervals (zip)]]

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* [[Attach:dye_fading_conf_int.zip | Nonlinear Confidence Intervals in MATLAB and Python (zip)]]

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* [[Attach:dye_fading_experiment.zip | Dye Fading Experiment Files (zip)]]

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* [[Attach:dye_fading_experiment.zip | Dye Fading Experiment Files in MATLAB and Python (zip)]]

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* [[Attach:dye_fading_conf_int.zip | Nonlinear Confidence Intervals (zip)]]

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* [[Attach:dye_fading_experiment.zip | Dye Fading Experiment Files (zip)]]

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* [[Attach:dye_fading_experiment.zip | Dye Fading Experiment Files (zip)]]

(:html:)

<iframe width="560" height="315" src="http://www.youtube.com/embed/WIXeySSa1fk" frameborder="0" allowfullscreen></iframe>

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<iframe width="560" height="315" src="http://www.youtube.com/embed/WIXeySSa1fk" frameborder="0" allowfullscreen></iframe>

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An engineer collected time-varying data for a non-isothermal run and needs you to estimate the kinetic parameters. The temperature ranged from about 60 - 140 degF. In order to appropriately size the reactor vessel, ~~they~~ need the kinetic parameters with the associated 95% confidence intervals.

to:

An engineer collected time-varying data for a non-isothermal run and needs you to estimate the kinetic parameters. The temperature ranged from about 60 - 140 degF. In order to appropriately size the reactor vessel, engineers need the kinetic parameters with the associated 95% confidence intervals.

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* [[Attach:dye_fading_experiment.pdf | Dye Fading Experiment (pdf)]]

* [[Attach:dye_fading_experiment.zip | Dye Fading Experiment (zip)]]

* [[Attach:dye_fading_experiment.zip | Dye Fading Experiment (zip)]]

to:

* [[Attach:dye_fading_experiment.pdf | Dye Fading Experiment Information (pdf)]]

* [[Attach:dye_fading_experiment.zip | Dye Fading Experiment Files (zip)]]

* [[Attach:dye_fading_experiment.zip | Dye Fading Experiment Files (zip)]]

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Reaction 1 is very fast and can be assumed to be instantaneous when the phenolphthalein (Ph) is added to a hydroxide solution. We are considering using phenolphthalein as an indicator to determine the residence time of several large CSTR reactors in a pilot plant. Please determine the kinetic parameters for reaction 2 with respect to phenolphthalein associated with the fading of phenolphthalein in sodium hydroxide solutions (reaction ~~order~~, ~~k~~, ~~and Ea~~).

An engineer collected time-varying data for a non-isothermal run and needs you to estimate the kinetic parameters. The temperature ranged from about 60 - 140 degF. In order to appropriately size the reactor vessel, they need the~~heat of reaction within a~~ 95% confidence ~~level~~.

An engineer collected time-varying data for a non-isothermal run and needs you to estimate the kinetic parameters. The temperature ranged from about 60 - 140 degF. In order to appropriately size the reactor vessel, they need the

to:

Reaction 1 is very fast and can be assumed to be instantaneous when the phenolphthalein (Ph) is added to a hydroxide solution. We are considering using phenolphthalein as an indicator to determine the residence time of several large CSTR reactors in a pilot plant. Please determine the kinetic parameters for reaction 2 with respect to phenolphthalein associated with the fading of phenolphthalein in sodium hydroxide solutions (reaction orders, A1, A2, Ea1, and Ea2).

An engineer collected time-varying data for a non-isothermal run and needs you to estimate the kinetic parameters. The temperature ranged from about 60 - 140 degF. In order to appropriately size the reactor vessel, they need the kinetic parameters with the associated 95% confidence intervals.

An engineer collected time-varying data for a non-isothermal run and needs you to estimate the kinetic parameters. The temperature ranged from about 60 - 140 degF. In order to appropriately size the reactor vessel, they need the kinetic parameters with the associated 95% confidence intervals.

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* [[Attach:dye_fading_experiment.pdf | Dye Fading Experiment]]

to:

* [[Attach:dye_fading_experiment.pdf | Dye Fading Experiment (pdf)]]

* [[Attach:dye_fading_experiment.zip | Dye Fading Experiment (zip)]]

* [[Attach:dye_fading_experiment.zip | Dye Fading Experiment (zip)]]

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Attach:dye_fading_experiment.~~pdf~~

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* [[Attach:dye_fading_experiment.pdf | Dye Fading Experiment]]

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(:title Estimate Kinetic Parameters from Dynamic Data:)

(:keywords nonlinear, optimization, engineering optimization, dynamic estimation, interior point, active set, differential, algebraic, modeling language, university course:)

(:description Case study on dynamic reconciliation for kinetic modeling using optimization techniques in engineering:)

!!!! Case Study on Dynamic Parameter Estimation

The reaction of phenolphthalein with a base solution follows the 2 reaction sequence given below.

* Reaction 1:

** Irreversible reaction, fast reaction dynamics

** Ph + 2 OH- => Ph(2-) + 2 H2O

* Reaction 2:

** Reversible, slow reaction dynamics

** Ph(2-) + OH- <=> PhOH(3-)

Reaction 1 is very fast and can be assumed to be instantaneous when the phenolphthalein (Ph) is added to a hydroxide solution. We are considering using phenolphthalein as an indicator to determine the residence time of several large CSTR reactors in a pilot plant. Please determine the kinetic parameters for reaction 2 with respect to phenolphthalein associated with the fading of phenolphthalein in sodium hydroxide solutions (reaction order, k, and Ea).

An engineer collected time-varying data for a non-isothermal run and needs you to estimate the kinetic parameters. The temperature ranged from about 60 - 140 degF. In order to appropriately size the reactor vessel, they need the heat of reaction within a 95% confidence level.

Attach:dye_fading_experiment.pdf

(:keywords nonlinear, optimization, engineering optimization, dynamic estimation, interior point, active set, differential, algebraic, modeling language, university course:)

(:description Case study on dynamic reconciliation for kinetic modeling using optimization techniques in engineering:)

!!!! Case Study on Dynamic Parameter Estimation

The reaction of phenolphthalein with a base solution follows the 2 reaction sequence given below.

* Reaction 1:

** Irreversible reaction, fast reaction dynamics

** Ph + 2 OH- => Ph(2-) + 2 H2O

* Reaction 2:

** Reversible, slow reaction dynamics

** Ph(2-) + OH- <=> PhOH(3-)

Reaction 1 is very fast and can be assumed to be instantaneous when the phenolphthalein (Ph) is added to a hydroxide solution. We are considering using phenolphthalein as an indicator to determine the residence time of several large CSTR reactors in a pilot plant. Please determine the kinetic parameters for reaction 2 with respect to phenolphthalein associated with the fading of phenolphthalein in sodium hydroxide solutions (reaction order, k, and Ea).

An engineer collected time-varying data for a non-isothermal run and needs you to estimate the kinetic parameters. The temperature ranged from about 60 - 140 degF. In order to appropriately size the reactor vessel, they need the heat of reaction within a 95% confidence level.

Attach:dye_fading_experiment.pdf