10182

Dynamic Behavior of Three Tanks in Series

Consider three identical tanks in a series subject to an input function . The heights of the liquid in the three tanks (i.e. , , ) obey the following equations:
,
,
,
where is the cross-sectional area of a tank and is related to the discharge coefficient for the exit pipes.
Suppose the height of tank 3 is sampled for a given input function to give the following data list: .
Then the constants and can be estimated using a least-squares optimization method. That is, we define the following objective function
.
Here is the height in tank 3 predicted by the model at time , and is the value of measured at time . The goal then is to determine and such that sum of squares is minimized for spanning the duration of the experiment.
One finds as shown in the second snapshot and . It is possible then to solve the governing equations shown above and determine the height of tanks 1 and 2. The second snapshot presents the height versus time for tanks 1, 2, and 3 in blue, magenta, and brown, respectively.
Once and have been determined, one can run simulations for various forms of the input function: impulse input, triangle input, square input, and staircase input. The subsequent snapshots show the responses for all the above mentioned special input functions, which are shown in red in a separate plot.

PERMANENT CITATION

 Share: Embed Interactive Demonstration New! Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details » Download Demonstration as CDF » Download Author Code »(preview ») Files require Wolfram CDF Player or Mathematica.

Related Topics

 RELATED RESOURCES
 The #1 tool for creating Demonstrations and anything technical. Explore anything with the first computational knowledge engine. The web's most extensive mathematics resource. An app for every course—right in the palm of your hand. Read our views on math,science, and technology. The format that makes Demonstrations (and any information) easy to share and interact with. Programs & resources for educators, schools & students. Join the initiative for modernizing math education. Walk through homework problems one step at a time, with hints to help along the way. Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet. Knowledge-based programming for everyone.