## Main.LectureNotes15 History

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We derived a FOPDT model of the process using empirical fitting techniques. A first principles approach was also used to obtain a model from a material balance. A comparison of the two models is shown below:

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* %list list-page% [[Attach:Lecture15_worksheet.pdf | Lecture 15 Worksheet on Linearization]]

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!!! Lecture 15 - Linearization

Now that we've derived nonlinear models based on material and energy balances, we need to get them into a form for linear systems analysis. We do this through a process called linearization. In Lecture 15, we review the mathematics of how to linearize a nonlinear function. We have a couple exercises to help practice the linearization process.

* %list list-page% [[Attach:Lecture15_notes.pdf | Lecture 15 Class Notes]]

!!!! Linear vs. Nonlinear Models

We previously ran through an example of a gravity drained tank.

Attach:tank2.png

'''Fig 1:''' Diagram of the Gravity Drained Tank

Attach:tank_percent_open.png

'''Fig 2:''' Sequence of Valve Movements to Test Models

Half of the class derived a FOPDT model of the process using empirical fitting techniques. The other half of the class used a material balance to obtain a model. A comparison of the two models is shown below:

Attach:tank_linear.png

'''Fig 3:''' Linear Model (FOPDT). The linear response is easy to fit to the data but deviates, especially during the periods that are far from the steady state values.

Attach:tank_nonlinear.png

'''Fig 4:''' Nonlinear Model Based on a Material Balance. The nonlinear response is valid over a wider range of operation.

!!!! Homework

# Course reading for next class: 3.1-3.4 (PDC).

# Assignment due by the start of Lecture #16: [[Attach:sp9.pdf | SP9]]

Don't forget that the first part of each homework assignment is to write a sentence or two stating the purpose of the problem and what concept is reinforced.

Now that we've derived nonlinear models based on material and energy balances, we need to get them into a form for linear systems analysis. We do this through a process called linearization. In Lecture 15, we review the mathematics of how to linearize a nonlinear function. We have a couple exercises to help practice the linearization process.

* %list list-page% [[Attach:Lecture15_notes.pdf | Lecture 15 Class Notes]]

!!!! Linear vs. Nonlinear Models

We previously ran through an example of a gravity drained tank.

Attach:tank2.png

'''Fig 1:''' Diagram of the Gravity Drained Tank

Attach:tank_percent_open.png

'''Fig 2:''' Sequence of Valve Movements to Test Models

Half of the class derived a FOPDT model of the process using empirical fitting techniques. The other half of the class used a material balance to obtain a model. A comparison of the two models is shown below:

Attach:tank_linear.png

'''Fig 3:''' Linear Model (FOPDT). The linear response is easy to fit to the data but deviates, especially during the periods that are far from the steady state values.

Attach:tank_nonlinear.png

'''Fig 4:''' Nonlinear Model Based on a Material Balance. The nonlinear response is valid over a wider range of operation.

!!!! Homework

# Course reading for next class: 3.1-3.4 (PDC).

# Assignment due by the start of Lecture #16: [[Attach:sp9.pdf | SP9]]

Don't forget that the first part of each homework assignment is to write a sentence or two stating the purpose of the problem and what concept is reinforced.