Creating and Manipulating Models

Distillation Column Example

This example is adapted from [2] and illustrates the use of I/O delays in process modeling. The process of interest is the distillation column depicted by the figure below. This column is used to separate a mix of methanol and water (the feed) into bottom products (mostly water) and a methanol-saturated distillate.

Figure 1-2: Distillation Column

Schematically, the distillation process functions as follows:

• Steam flows into the reboiler and vaporizes the bottom liquid. This vapor is reinjected into the column and mixes with the feed
• Methanol, being more volatile than water, tends to concentrate in the vapor moving upward. Meanwhile, water tends to flow downward and accumulate as the bottom liquid
• The vapor exiting at the top of the column is condensed by a flow of cooling water. Part of this condensed vapor is extracted as the distillate, and the rest of the condensate (the reflux) is sent back to the column.
• Part of the bottom liquid is collected from the reboiler as bottom products (waste).

The regulated output variables are:

• Percentage of methanol in the distillate
• Percentage of methanol in the bottom products.

The goal is to maximize by adjusting the reflux flow rate and the steam flow rate in the reboiler.

To obtain a linearized model around the steady-state operating conditions, the transient responses to pulses in steam and reflux flow are fitted by first-order plus delay models. The resulting transfer function model is

Note the different time delays for each input/output pair.

You can specify this MIMO transfer function by typing

• H = tf({12.8 -18.9;6.6 -19.4},...
          {[16.7 1] [21 1];[10.9 1] [14.4 1]},...
          'iodelay',[1 3;7 3],...
          'inputname',{'R' , 'S'},...
          'outputname',{'Xd' , 'Xb'})
  

The resulting TF model is displayed as

• Transfer function from input "R" to output...
                       12.8
   Xd:  exp(-1*s) * ----------
                    16.7 s + 1
   
                       6.6
   Xb:  exp(-7*s) * ----------
                    10.9 s + 1
   
  Transfer function from input "S" to output...
                        -18.9
   Xd:  exp(-3*s) * --------
                    21 s + 1
   
                          -19.4
   Xb:  exp(-3*s) * ----------
                    14.4 s + 1
  

Specifying Input/Output Delays

Specifying Delays on the Inputs or Outputs

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