Toucan Lab

7/30/2019 Toucan Lab

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Toucan Lab

This laboratory provides an opportunity for comparing theoretical and empiricalmodeling of a physical system. In the end, a measure of success will come from

evaluating how well the model(s) predict the actual behavior in a laboratoryexperiment. The system and components under study are very basic, yet provide a

basis for introducing:

• hydraulic resistive-capacitive systems,

• model-based experimentation,

• analytical solutions for nonlinear systems, and

• nonlinear simulation both for design and verification.

Specific Objectives

• Review specific fluid (hydraulic) systems modeling

• Apply simulation methods

• Design and conduct experiments and perform basic data analysis. • Practice state-equation derivation, and apply the concept of a state-

determined system in the practical study of a laboratory model. • Study how the model of a system can be validated with experimental data

and then used to design the system for a desired performance

specification

Project Toucan involves the study of a class of systems that can be used to

demonstrate and motivate concepts and techniques useful in system modeling anddesign. The basic ideas are introduced through a study of fluid flowing into and out

of one- and/or two-can systems (Note, the term "can" will be used to refer to thelaboratory realization of a "tank").

A basic (defining) experiment is shown below, where water flows out of a single canthat has an orifice centrally located in its base.



These systems form the basis for developing methods useful in modeling andanalyzing systems that have analogous resistive-capacitive (RC) combinations.

The resistive element in this system, however, has a nonlinear characteristic.Consequently, we can examine how such systems are distinguished from the

common linear RC systems.

An extensive discussion of the modeling of this basic system is given in theblackboard slides.

Historical Note: " Project Toucan", as it was referred to by Prof. H.M. Paynter, wasused by him for many years during his tenure as professor at MIT from the 1950's to

the 1980's.

Model-based One-Can Experiments

The experiments for a single can system can be designed using the results from analysis

of the basic model, .

where V is the volume in the can. Recall that the coefficient, K , in this equation is

assumed to include all geometric and material information. The ideal flow coefficientfor this system can be derived as shown in the ideal-flow coefficient document

on blackboard. The ideal coefficient will not include losses and other fluid dynamic

effects that are not easy to quantify. The K value can be empirically derivedbeginning by solving the equation given above through direct integration,

,

to find,

.

This relation suggests a design for a simple experiment to evaluate K . Theexperiment is described as follows:

• Fill a can (i.e., which has an exit hole at bottom, preferably centered) with an

initial volume of water, V o.

• Measure the time it takes for that volume to exit the can, T e • Observe that any water that remains at the time the experiment is completed

is subtracted from the total volume (or else measure the volume of water

collected; this will improve accuracy). • Repeat these measurements for several values of V o. • The first experiment is the (0,0) experiment. That is, it takes 0 seconds for

an empty can to empty! This data point is already recorded in the table given

below. Use the following table as a guide. Record volume in milliliters. Let D= diameter of the can, d = diameter of the orifice, H = height of the can.

• Construct a table for each can that will be used in the two-can experiments.



Can # X: Height = ____, D/d =____

TestNo.

InitialVolume, Vo

sqrt(Vo)Time to Empty,Te (sec)

1 0 0 0

2 -- -- --

-- -- -- --

N -- -- --

Analysis of One-Can Data to find K The linear relation between the square-root of initial volume and time-to-empty

suggests that plotting the data in this form will allow a direct determination of Kusing a linear regression. An example is shown below.

Note that a linear fit to this data should include a (0,0) point, so the line should beforced to go through this point. In the next section, we run a simple evaluation

experiment to test this hypothesis.

A set of data is included on the plot to the right that was collected using a slightly

curved funnel. You can see that you can find a correlation even for non-constantarea tanks in some cases that allows you to come up with a model similar to the one

used in this lab. For this funnel, it looks like a 3/4 exponent on volume will give alinear relation with time to empty.



Testing the K value

After determining K , it is simple to check how well it parameterizes our model. Youcan predict the time to empty for an independent experiment (i.e., you could either

check against the data you used to determine this parameter, or run an independenttest) using the relation,

.

In addition, you might also compare to what the ideal K value will predict. Do theresults seem to make physical sense? Is there a good explanation for the

relationship between the ideal and empirical K values?

Note that if the K value is too high, the flowrate out of the can would be

overestimated, and predictions would show that the can empties sooner thanobserved. When K is too low, the opposite occurs. Remember, the flow through the

orifice was modeled using the basic relation,

where V is the instantaneous volume in the can (and note K includes all the

geometric and material properties inherent to this relation).

Summary

This laboratory study should provide the basis for putting together a fully-

parameterized system model of the two-can system. This study should haveprovided experience in:

• designing a model-based experimental study,

• conducting experiments and collecting data, •

analyzing data to extract parameter values (and using computer-basedsoftware to conduct the analysis), and

• assessing the quality of the data through independent testing.

The second part of this study will examine additional concepts in system modeling,experimentation, and simulation. The performance evaluation of the system will also

provide a means for assessing the overall accuracy of our model.

Assignment

From the data obtained in the lab, determine the K value for each of the cans.

Utilize these values along with the equations of motion to simulate the response of the two-can system.

Determine the volume of water, and height of water needed in the top can to

completely fill the lower can without overflowing.

Write up the results in a report along with plots for next week.



The Two-Can System

Our interest now is in cascading the two cans that were used in the first part of thislab to form a two-can 'system'. A schematic of this experiment are shown below. Our

interest is in developing a model for the two-can system using the one-can modelsthat have been experimentally "validated" in Part 1 of the Toucan Lab.

For each can, a value for K should have been found that allows the prediction of flowrate, QR, as a function of the instantaneous volume in the can using the

equation,

.

Recall that that K -value incorporates all the non-ideal effects we can not model

without going to the laboratory.

The cascading of the two cans is similar to the cascading of two tanks studied in anearlier lab (see the two-tank simulation (matlab-based or LabVIEW-based)). Unlike

the two-tank system, however, the flow through the orifice in Can 1 does not depend on the conditions in Can 2. In this way, the dynamic behavior of Can 1 will

not depend on that of Can 2 (i.e., there is no back effect from Can 2 to Can 1).

Experiments for Model Evaluation

The two-can system equations are derived directly from continuity, giving,

.

http://www.me.utexas.edu/~lotario/me244L/labs/simulation/twotanks/twotanks.html

http://www.me.utexas.edu/~lotario/me244L/labs/lvsimulation/twotanks/twotanks.html

http://www.me.utexas.edu/~lotario/me244L/labs/simulation/twotanks/twotanks.html

http://www.me.utexas.edu/~lotario/me244L/labs/lvsimulation/twotanks/twotanks.html



Because of the relative simplicity of this system, it provides a good laboratory formodel evaluation. That is, the model can be evaluated to determine how well it can

predict the system performance. In this case, the model can be conducted by simpleobservation.

Metrics can be easily defined and determined to within reasonable accuracy. Of

particular interest are:

1. Peak Volume or Level Detection. If concerned with level prediction, one

way to check the validity of the model is to measure peak volume or level

during different experiments. These results can then be compared withnonlinear simulation results.

2. Time Events. If the concern is with "when" a can empties, or when a peak is

reached, we might be more concerned with measuring the time at which such

an event occurs.

A more complete evaluation might involve a combination of the two types of

observations listed above. Depending on the requirements, available equipment, and

other resources, an experiment should be designed that will be useful in measuringthe variables of interest to within an accuracy suitable for assessing the model. Inthis study, this can be accomplished by (careful) observation. In a later laboratory,

these cans can be instrumented with level sensors..

An easy way to evaluate the system model and the component models (i.e., the

models developed through the one-can studies) together is to run the followingexperiment:

Fill Can 1 to a certain volume, and let it discharge into an initially empty Can2. This experiment can be numerically simulated using the model equations

(and K values), and the results compared to those observed in this

experiment.

Without any measurements other than the level of Can 2, the model can be

effectively evaluated. The level in Can 2 will reach a maximum that is easily

observable and thus forms a basis for comparison with a model. Both the peak level and time to reach this peak can be observed.

The only measurements required, for a first approximation, are made by visual inspection.

Conduct this evaluation and record the results. The numerical simulation

should be run immediately after the experiment.

Performance Design Problem As a final test of your model and simulation capabilities, you will be asked to assessyour model in the following way:

Determine the initial volume required in Can 1 (the upper can), so that thevolume in Can 2 (lower can) reaches a pre-defined value (to be specified by

the lab instructor).



This test is much more accurate (and dramatic!) if it is possible to reach themaximum allowed level in Can 2. Spillage is then easily detected; as is

failure to reach the maximum.

Note that the point at which Can 2 reaches a maximum is a stationaryequilibrium. This concept can be applied directly to help devise a simulation-based

solution of the above stated problem.

Summary

The final experiments described here allow practice in developing a state-space

model of a simple nonlinear system. The final evaluation problem provides anopportunity for utilizing two system concepts:

1. equilibrium - this allows you to determine a condition for maximum attained

flow in the bottom can, and 2. state-space systems - where we see that a properly formulated model can

allow us to make predictions and design performance

While this is a very simple system, it provides an opportunity to conductexperimental studies together with simulations of a nonlinear system.

Toucan Lab

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