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CHAPTER 2
CONICAL TANK PROCESS EXPERIMENTAL SETUP AND
ITS MATHEMATICAL MODELING
2.1
INTRODUCTION
In this chapter, the lab scale non-linear conical tank level process in
which the level of liquid to be maintained at a constant value is described.
The time constant and gain of the selected process vary as a function of level.
The non-linear mathematical model, process input-output characteristics,
linearisation and process reaction curves obtained in various regions are
discussed.
2.2
DESCRIPTION OF THE LAB SCALE LEVEL PROCESS
SETUP
A conical tank laboratory level process whose parameters vary with
respect to process variable is considered for simulation and real time
implementation. Even though selected system is simple, it has non-linearity.
The system is a single input, single output process. The output of the process
is the level and can be measured easily and the input to the process is the
change in voltage to the motor which changes the inflow into the tank when
the voltage gets varied. A shift in the operating point towards the top of the
tank implies an increase in the time constant and a decrease in the static gain.
Similarly a shift in the operating point towards the bottom of the tank implies,
decrease in the time constant and increases in the static gain. Thus the conical
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tank level process, whose time constant and gain are functions of the process
variable, becomes suitable for the present work.
The schematic diagram of the hardware set-up is shown in the
Figure 2.1. The set-up consists of a process tank, submersible pump, float
sensor arrangement, overhead sump, inlet valve, outlet valve, level indicator
and interfacing card. The process tank is in the shape of an inverted cone
fabricated from a sheet metal. Provisions for liquid inflow and outflow are
provided at the top and bottom of the tank respectively. Metal rods are welded
around the circumference for support. The height of the process tank is 80 cm
and top radius of the tank is 20 cm. The submersible pump used here is
capable of discharging liquid at the rate of 300 cm3/seconds. The pump is
immersed in the overhead tank and a flexible hosepipe is connected to the
pump to supply the inflow into the tank. A gate valve is used in this path to
give disturbance input. The minimum voltage applied to the pump for
discharge is 104 volts (AC). A float sensor is used here to sense the water
Figure 2.1 Schematic diagram of the hardware setup
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level in the process tank. The arrangement consists of a float made up of light
stainless steel material. It is tied to a nylon thread through a pulley
arrangement. The other end of the thread is tied to a calculated weight. A
rotary potentiometer is attached to the center of the pulley shaft to obtain
electrical output.
The Bourns potentiometer used here has 10 turns of coil providing
an electrical output in the range of 0 to 5Volts DC. When liquid is pumped
into the tank, the liquid level in the tank rises. As a result, the position of the
float is changed and it causes a change in potentiometer output due to pulley
action. Thus, a suitable electrical output is generated with the movement of
the float. A thin transparent tube made of plastic, is provided externally to
view the actual level of the liquid in the tank. A graduated scale placed
parallel to the tube indicates the present level. Gate valves, one each at the
inflow and outflow of the process tank are connected to maintain the liquid
level. Also, gate valve at the inlet side is used for giving disturbance inputs
and the gate valve at the outlet side is used for varying process parameters.
The clockwise rotation ensures the closure of the valve, thus stopping the
flow of liquid and vice-versa.
Input and output Analog to Digital, Digital to Analog Modules
(ADAM) are used to interface the hardware setup with PC. The electrical
output generated from the potentiometer is acquired by the computer through
the input ADAM module. The controller output is sent to the process through
the output ADAM module. The fabricated lab scale experimental setup of
conical tank level process is shown in Figure 2.2 and the conical tank diagram
is shown in the Figure 2.3.
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Figure 2.2 Lab scale experimental setup
Figure 2.3 Conical tank diagram
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2.3
MATHEMATICAL MODELING
Using the law of conservation of mass, in conical tank diagram
shown in Figure 2.3,
1  dh
dA 
Fin  Fout   A  h 
3  dt
dt 
where
Fin
(2.1)
Inflow rate of the tank cm3/seconds =
-
0.000300 m3/seconds
Fout -
Outflow rate of the tank in cm3/seconds
R
-
Top radius of the tank (cm)
H
-
Total height of the tank (cm)
r
-
Radius at any height h i in cm
tan  
Also,
R r

H h
(2.2)
Fout  b h
(2.3)
where b, valve constant = 2.2
Hence,
1  dh
dA 
Fin  b h   A
h

3  dt
dt 
dh

dt


3 Fin  b h  h
R 2 h 2
H2
dA
dt
(2.4)
(2.5)
The time constant and gain of the process change as the level
changes. This is obvious from the mathematical model given by equation
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(2.5). The simulink model developed to simulate the conical tank level
process is shown in Figure 2.4.
The process steady state input-output
characteristic obtained from the real time setup with inflow variations from 7
cm 3 / seconds to 300 cm3 / seconds is shown in Figure 2.5. The characteristic
shows the non-linear behaviour of the process. To obtain a linear model the
characteristic is divided into six different linear regions as shown in Figure
2.6. A first order mathematical model is then obtained for each region using
process reaction curve method and the reaction curves for regions 1 to 4 are
shown in Figures 2.7 to 2.10. The gain (K), dead time (td) and time constant
(τ) are measured from the reaction curves and are given in the Table 2.1. It is
found practically that the dead time is 2 seconds in all the regions due to 2
seconds sampling time.
Figure 2.4 Simulink model of conical tank level process
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Figure 2.5 Process input-output characteristic
Figure 2.6 Piecewise linearised process input- output characteristic
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Figure 2.7
Reaction curve for first region when step change in inflow
from 10 cm3 / seconds to 25 cm3 / seconds is applied to the
process
Figure 2.8 Reaction curve for second region when step change in inflow
from 25 cm3 / seconds to 48 cm3 / seconds is applied to the
process
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Figure 2.9
Reaction curve for third region when step change in inflow
from 48 cm3 / seconds to 80 cm3 / seconds is applied to the
process
Figure 2.10 Reaction curve for fourth region when step change in inflow
from 80 cm3 / seconds to 140 cm3 / seconds is applied to the
process
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Table 2.1 Process parameters obtained from the reaction curves
A
(input
B
(output at
K
(Steady
τ
7-22
15
22
1.46
35
22-38
23
38
1.65
80
38-50
32
50
1.56
146
50-64
60
64
1.06
245
64-72
80
72
0.9
475
72-80
65
80
0.8
750
Level
Inflow range
range
(cm3/seconds)
(cm)
10-25
(I Region)
25-48
(II Region)
48-80
(III Region)
80-140
(IV Region)
140-210
(V Region)
210-275
(VI Region)
Time constant
change) steady state) state gain) (seconds)
From the Table 2.1, it is observed that, the gain of the system varies
from 1.46 to 0.8 and the time constant varies from 35 seconds to 750
seconds as the level increases from 0 to 80 cm.
During the normal operation of the process, the outlet valve is kept
in the middle position. The corresponding valve constant ‘b’ for that position
is obtained experimentally as b = 2.2. When the valve is nearly completely
closed, the respective valve co-efficient experimentally obtained is b = 1.32.
Similarly for the completely opened position, the valve co-efficient
experimentally obtained is b = 4.4. Hence, the valve co-efficient is changing
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from 4.4 to 1.32. when the valve motion is changed from completely opened
to merely closed position. The change in process characteristics for these
valve positions are shown in Figure 2.11.
Figure 2.11 Change in plant input-output characteristics for changes in
valve parameter
2.4
Conclusion
Thus the process parameters are obtained using the process
reaction curve method in various linearised regions. Using these
parameters of the linearised models and controller tuning methods ,
it is proposed to design conventional controller first for the control
of the level process, that is discussed in the next chapter.