### 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.

#### Linear vs. Nonlinear Models

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

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

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

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:

**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.

**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: 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.

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