USE OF INTERNET TO DO EXPERIMENTS IN DYNAMICS AND CONTROL FROM

ZACATECAS MEXICO IN THE LABORATORY OF THE UNIVERSITY OF TENNESSEE

AT CHATANOOGAA.

Jim Henry1*

, José Alberto González Guerrero2, Benito Serrano Rosales

2.

1. The University of Tennessee at Chattanooga, 615 McCallie Ave., Chattanooga, TN 37403

2. Universidad Autonoma de Zacatecas, Unidad Academica de Ciencias Quimicas, Campus

Siglo XXI, Edificio 6, Carr. a Guadalajara Km. 6, Ejido la Escondida, Zacatecas Zac., 98160,

Mexico, beniserra@prodigy.net.mx Phone 52-492-925-6690 Ext 6137

Abstract

The internet was used to do remote experiments in Universidad Autonoma de Zacatecas Mexico,

using the laboratories in the University of Tennessee at Chattanooga, specifically, the system of

two non-interacting tanks. In the first part, step and pulse perturbations were applied, and in the

second part, a proportional controller was used and step perturbation on the set point was

applied. The mathematical models were deducted in all the cases, linearizing the expression for

the flow between tank 1 and 2.

In all the cases, the agreement between experimental and predicted results was very good,

indicating the models were conceptually correct, and the most important, the use of remote

laboratory was possible and allowed a relationship between two universities.

Introduction

Currently, the use of internet to do experiments in a remote laboratory is a reality. The

experiments were performed in the Program of Chemical Engineering, Faculty of Chemical

Sciences, Autonomous University of Zacatecas Mexico, at any day, any time, using the two non-

interacting tanks in the laboratory of the University of Tennessee at Chattanooga. Each tank was

6 inches in diameter and 12 inches in height. The feed from the first tank to the second tank is by

gravity (See figure 1). A mathematical model was constructed and it was necessary to linearize

it.

Experimentation

The first perturbation was to turn on the equipment of the two non-interacting tanks and to let it

to reach the steady state, which means a step perturbation. The implementation of an interface

through internet http://flow.engr.utc.edu:443/ allows access to this piece of equipment from

anywhere in the world. See figure 2.

The first part of the experimental section was to run some experiments for different values of the

pump. In the second part, one step perturbation was applied, with magnitude A, and in other

experiment, after the start of the equipment, a pulse perturbation was applied. For both cases, the

response of the height of the level of the second tank was registered, and the experimental results

were compared with the predictions of the model. In the third part, the experiments were

performed with step perturbation, regulated by a proportional controller.

Figure 1.- System of two non-interacting tanks.

Figure 2.- Screen to select the piece of equipment.

Click the link “2 Tank Water Level Control Constant Input or Step Input.”

Figure 3.- Screen to start the experiments with the system of two non interacting tanks.

A typical experiment for the start of the experiments is shown in figure 3, with 100% of the

capacity of the pump. It is observed that the steady state height of the level in tank 2, 22.5 cm, is

reached in 150 seconds.

Figure 4.- Level of tank 2, with 100 % capacity of the pump.

The next experiment was to work with the pump at its 80 % of capacity, and to apply a step

perturbation of 20 %. Figure 5 shows the results of this experiment.

Figure 5.- Response of the level of tank 2 to a step perturbation.

The next experiment was carried out with the capacity of the pump of 70 %. Once the stady state

was reached, a pulse perturbation was applied increasing the pump capacity in other 30%, for

100 seconds. After this period, the pump was returned to 70 %. The results are shown in figure 6.

Figure 6.- Response of the level of tank 2 under a pulse perturbation of 100 seconds.

In the next experiment, the pump was operated at 60 % of its capacity, using a proportional

controller with gain 0. After the steady state was reached, a step perturbation was applied

increasing another 40 %. Several values of the gain were used. The experimental results are

shown in Figure 7. It is observed that increasing the value of the gain, an unstability is generated,

as it is predicted by theory.

Figure 7.- Response of the level of tank 2 under a step perturbation for several values of the gain

of the proportional controller.

It is very important to note that all these experiments were done using internet, in the laboratory

of Unviersity of Tennessee from the Autonomous University of Zacatecas, Mexico.

Mathematical modeling.

The mass balances for tanks 1 and 2 are:

( ) ( )( )

dt

tdhAtqtq Ti

111 =−

( ) ( )( )

dt

tdhAtqtq T

2221 =− ,

Using the Taylor series, the linearized expression for q1(t), the flow leaving tank 1 is obtained:

( )( )( )

1

1111

R

hshqtq s

s

−+≅

The transfer function for this system of non interacting tanks for the liquid level of the second

tank is:

( )( )

+

+

=

21

21

211

τττ ssA

sQsH

T

i

Where:

cminDD 24.15621 === , cmh s 2.172 = , s

cmH

3

30= 2

21 415.182 cmAA TT ==

,6317.1111 sAR T ==τ ,1929.31222 sAR T ==τ2

1

1

1 008945.02

cm

s

C

hR

v

s==

2

2

2

2 171.02

cm

s

C

hR

v

s==

( )[ ]s

cm

g

grtpCC

c

vvTv

25

222 5.48== ρ

Cv1 is the secondary coefficient of valve in tank 1. Cv2 is the secondary coefficient of valve in

tank 2.

CvT1 coefficient of valve in tank 1

CvT2 coefficient of valve in tank 2

Pv1(t) = position of the valve opening in tank 1.

Pv2(t) = position of the valve opening in tank 2.

If we apply a step perturbation, with magnitude H, and with transfer function H/s. The response

of the second tank is:

Substituting the specific values, we obtain:

The comparison of the predictions of this equation with the experimental results obtained after

using the remote laboratory of the University of Tennessee campus Chattanooga, are shown in

Figure 1. In this case, the agreement is very good.

Figure 8.- Comparison between the mathematical model and experimental results for a step

perturbation.

Now, applying a rectangular pulse of magnitude H and duration T, The response is:

,

,

The numerical values are:

s

cmC

cm

sRsT

s

cmHcmh vs

25

221

3

2 21,008954.0,100,45,11 =====

Substituting these values in the pulse response:

A comparison between the expermiental and calculated results is shown in figure 2. The

agreement during the first stage are excellent, however, some diference is observed after the

perturbation.

Figure 9.- Comparison between the mathematical model and experimental results for a pulse

perturbation.

Experiments were also performed using a proportional controller. The block diagram is shown in

the following figure:

Where:

R(s) = Set point.

Qi(s) = Flow entering the first tank, or load variable.

C(s) = Controller transfer function.

P(s) = Transfer function for the process of two non-interacting tanks.

M(s) = Transfer function of the measurement device.

H2(s) = Transfer function of the level of the second tank, or exit variable.

The following transfer function is obtained:

( ) pKsC = , ( ) ( ) 1,11

1

21

21

=

+

+

= sM

ssA

sP

Tττ

τ

Performing algebra:

( )( )( ) ( )( )221

2

221121

2

1212121

2sssssA

H

sssssA

H

sssA

HsH

TTT −−+

−−+=

τττ

Performing the inverse transform:

( )( ) ( )

ts

T

ts

TT

esssA

He

sssA

H

ssA

HtH 21

21

2

22121

2

1212121

2−

+−

+=τττ

Where: H = 30 cm, pT KxsA 3

121 103597.30844.03224.0,6465.297 −−+−==τ

pKxs 3

1 103597.30844.03224.0 −−−−= , pKxxss33

21 103597.31054.19−−

+=

pp KxKxs 332

1 103597.3103597.30844.06448.018844.0 −− −−−=

pp KxKxs 332

1 103597.3103597.30844.06448.018844.0 −− −−+=

( ) pp KxKxsss 33

21

2

1 107194.6103597.30844.06448.01688.0 −− −−−=−

( ) pp KxKxsss 33

21

2

2 107194.6103597.30844.06448.01688.0 −− −−+=−

The comparison between the experimental and predicted results is done in figure 10.

Figure 10.- Comparison of experimental and predicted results using a step perturbation and a

proportional controller.

Certainly, the agreement is not very good but the model indicats very well the tendency of the

experimental results.

Conclusions.

1.- The agreement between the experimental and modelled results is good, and this validates the

mathematical model, which is able to predict the response of the system to any step perturbation,

whether at the start or after a steady state is reached.

2.- In general, the model using a pulse perturbation is conceptually correct and reproduces the

experimental data, in spite of after the perturbation, some diference between data is observed.

3.- The mathematical model using proportional controller predicts the response tendency when

the value of Kc is increased, indicating that the model is conceptually correct.

References.

1.- Donald R. Coughanowr, “Process system analysis and control”, Mc. Graw Hill, second

edition, 1991.

2.- Wodek K. Gawronski, “Advanced Structural Dynamics and Active Control of Structures”,

Spronger, 2004.