Dynamic Behavior of Three Tanks in Series

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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.


Contributed by: Housam Binous, Brian G. Higgins, and Ahmed Bellagi (March 2011)
Open content licensed under CC BY-NC-SA



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