process intensification

1. 1
Manipal University
Department of Chemical Engineering
CHEMICAL REACTION ENGINEERING LAB
IV YEAR, 7th SEMESTER, 2020
EXPERIMENT #2
To study the dynamic response of two non interacting tanks system subjected to step
INSTRUCTOR:
Dr. Anees Khan
MEMBER: Madhumita Kumar
Experiment carried on: March 18th 2021
Report submitted on: March 25th 2021
PRELAB (10) _______
EXECUTIVE SUMMARY (10) _______
INTRODUCTION/OBJECTIVES/SCOPE/PROCEDURE (30) _______
RESULTS & DISCUSSION (30) _______
CONCLUSIONS (5) _______
REFERENCES (5) _______
APPENDIX
a) Original data, sample calculations, other information (5) _______
GENERAL COMPLETENESS
a) Conciseness and neatness (5) _______
TOTAL (100) ______

2. 2
1. Executive Summary
We have performed this experiment in order to find the transfer function and study the
dynamic response of a first order two tank system. In this experiment, we measured the
change in height (output) of the level of water in both the tanks at constant time intervals
when the inlet flow rate of water into the tank (input) is changed. Then, we constructed
graphs of the change in output versus the time. Using the graphs, we calculated the time
constant 𝜏 and the steady state gain 𝐾𝑝 for both the tanks. This was done for three different
flow rate (input)- 30-40 LPH, 40-50 LPH, 60-70 LPH.
In this experiment, we have studied the dynamic response of a first order system in a two
tank series system. We have obtained the different time constant values and system gain
values for three different flow rates. These are shown in Table 1. There was a consistent time
lag between the first tank response and the second tank response. There is no visible trend
between the time constant and the increasein flow rate and there is no visibletrend between
the system gain and the increase in flow rate.

3. 3
Contents
1. Executive Summary ............................................................................................................2
2. Nomenclature ........................................................................................................................4
3. Introduction ...........................................................................................................................5
4. Objective................................................................................................................................7
5. Experimental set up............................................................................................................8
6. Apparatus required.............................................................................................................8
7. Experimental procedure .....................................................................................................8
8. Results and Discussion........................................................................................................9
9. Conclusion.........................................................................................................................12
10. References.....................................................................................................................13
11. Appendix A....................................................................................................................13

4. 4
2. Nomenclature
Symbol Quantity Units
𝜏 Time constant Seconds
𝐾𝑝 Steady state gain
𝑋(𝑡) Input
𝑌(𝑡) Output
𝑥𝑠 Steady state input
𝑦𝑠 Steady state output
𝐴 Magnitude of step change
𝑞𝑡 Input flow rate LPH
𝑞0 Output flow rate LPH

5. 5
3. Introduction
A control system is used to maintain process conditions at their required values by
manipulating process variables to adjust the magnitude of the variables of interest. For
example: A home hot water heater. The heater tries to maintain the temperature in the tank
at a required value by controlling the fuel flow to the burner or electrical input to the heater
against deviations and disturbances like the changing demand on the heater.
In general, a control system can control the process variable at its desired value despite the
disturbances that it may be subjected to. The control system also has the ability to move the
process variable from one setting to a new desired setting. It compares the measured signal
of the controlled variable to the set point (desired value of the controlled variable). The
difference between the values is the error. The generalized process control system is shown
in figure 1.
Depending on the magnitude of the error, the controller takes action by sending a signal to
the final control element, which further provides an input to the process to return the
controlled variable to the set point. The time behavior of a system is very important. Being
able to determine or predict the dynamic behavior of a process is crucial to being able to
design a control system for it.
Figure 1: Generalized process control system

6. 6
The order of a differential equation is the highest degree of derivative present in that
equation. A first order systemis asystemwhose dynamic behavior is described by afirst order
differential equation. It is a systemwhose input-output relationship is a first order differential
equation. A first order differential equation has a first order derivative but no derivative
higher than first order. First order systems have singlestorageelements. In general, the order
of the input output differential equation will be same as the number of independent energy
storage elements of the systems. First order systems are very important. Many practical
systems are first order systems such as a mass damper system or a mass heating system.
A transfer function is used to represent the relationship between the output signal of a
control system and the input signal of the control system for all input values. It is
mathematically defined as the ratio of the Laplace transform of the output variable to the
Laplace transport of the input variable assuming all initial conditions to be zero. In order to
determine the transfer function for a process: Firstly, we have to write the appropriate
balance equations, then linearize the terms, place the balance equations in deviation variable
form, Laplace transform the linear balance equations and then solve the resulting
transformed equations for the transfer functions (output divided by input).
The forcing function is a load disturbance that affects the process and causes it to deviate
from steady state.
In the non interacting tank in series system
shown in figure 2, it is assumed that the liquid
is of constant density and the tanks have
uniform cross sectional area and the flow
resistances are linear.
Performing a mass balance on tank 1 gives:
𝑞 − 𝑞1 = 𝐴1
𝑑ℎ
𝑑𝑡
(Eqn 1)
Performing a mass balance on tank 2 gives
𝑞 − 𝑞2 = 𝐴2
𝑑ℎ
𝑑𝑡
(Eqn 2)
Figure 2: Two tank non interacting liquid system

7. 7
The flow head relationships for the two linear resistances are given by:
𝑞
1
=
ℎ1
𝑅1
(Eqn 3)
𝑞
2
=
ℎ2
𝑅2
(Eqn 4)
Introducing deviation variables,
𝑄 = 𝑞 − 𝑞𝑠
𝑄1 = 𝑞1 − 𝑞1𝑠
Substituting the deviation variables and applying LaPlace transform , we get transform
function for tank 1:
𝑄1(𝑠)
𝑄(𝑠)
=
1
𝜏1𝑠+1
(Eqn 5)
Where,
𝜏1 = 𝑅1𝐴1
In the same way, we can combine Equation 2 and Equation 4 to get the transfer function for
tank 2:
𝐻2(𝑠)
𝑄1(𝑠)
=
𝑅2
𝜏2𝑠+1
(Eqn 6)
Where,
𝐻2 = ℎ2 − ℎ2𝑆
𝜏2 = 𝑅2𝐴2
The overall transfer function can be obtained by combining equation 5 and equation 6:
𝐻2(𝑠)
𝑄(𝑠)
=
1
𝜏1𝑠+1
𝑅2
𝜏2𝑠+1
(Eqn 7)
4. Objective
To study the dynamic response of two non interacting tanks system subjected to step

8. 8
5. Experimental set up
In this set up, there is a sump tank present that needs to be filled with water. The tank
contains a pump that allows the water to flow up. There is a three tank set up but in this
experiment, we will only be using Tank 1 and Tank 2. The level of water input into the first
tank can be controlled using the rotameter. The water level has to be set on the rotameter
and the pump will start moving pumping the water from the sump tank to the rotameter to
the tank. The tank outlet has valves to control the resistance as well.
6. Apparatus required
- Tank set up
- Water
- Stop watch
- Pen and paper (to take down readings)
7. Experimental procedure
1. Fill the sump tank with water
2. Set the rotameter at a starting flow rate (30 LPH). Make sure that the outlet and the
inlet valve of Tank 1 are open.
Figure 3: Experimental set up

9. 9
3. Let the flow of water through tank one reach steady state (the height of water in tank
1 is constant, no longer rising)
4. Note down the steady state height as the height of the water at time t=0
5. Now increase the rotameter flow rate to 40 LPH and start the timer
6. At the interval of every 10 seconds,note down the height of water. Repeat this pricess
untill the height has reached a steady state.
7. Repeat this process for Tank 2 as well
8. Repeat steps 2-7 for different rotameter flow rates (40-50 LPH, 50-60 LPH)
9. After the experiment is done, drain the tank and drain the sump tank as well.
8. Results and Discussion
Change in flow rate (LPH) 𝝉𝟏(s) 𝝉𝟐(s) kp1 kp2
30-40 71 104 3.8 3
40-50 90 120 2.8 2.2
50-60 75 135 6.9 3.6
Table 1 Processed data
Figure 4 H vs T for 30-40 LPH
0
5
10
15
20
25
30
35
40
0 50 100 150 200 250
H
(cm)
T (seconds)
H vs T (30-40 LPH)
H1 vs T
H2 vs T
Poly. (H1 vs T)
Poly. (H2 vs T)

10. 10
0
5
10
15
20
25
30
0 50 100 150 200 250
H
(cm)
T (seconds)
H vs T (40-50 LPH)
H1 vs T
H2 vs T
Poly. (H1 vs T)
Poly. (H2 vs T)
Figure 5 H vs T for 40-50 LPH
0
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
0 50 100 150 200 250
H
(cm)
T (seconds)
H vs T (50-60 LPH)
H1 vs T
H2 vs T
Poly. (H1 vs T)
Poly. (H2 vs T)
Figure 6 H vs T for 50-60 LPH

11. 11
The raw data for this experiment is presented in Appendix A. In this experiment, we have
calcultated the time constant 𝜏 and the steady state gain 𝐾𝑝 for a first order system in series
with different inlet flow rates of water. This is has been done by initially recording the change
in height of the level of water in Tank 1 and then in Tank 2 at uniform time intervals after the
flow rate of water has been increased. The graphs of the change in height of water versus
time has been plotted. In this experiment, the change in input is the change in the flow rate
of water being supplied to the tank (for example, 30 to 40 LPH will have a change in input of
10 LPH). The change in output is the change in the height of water from the steady state
height to the new height reached at the end of the experiment. The steady state height is the
height of the water at the start of the experiment (for example, the height of water in the
tank at 30 LPH before the flow rate has increased, the final height of the water is the height
of the water when it has reached steady state at 40 LPH). The steady state gain 𝐾𝑝 is the ratio
of the input change to the output change.
The value of the time constant 𝜏 has been calculated at the point where the change in height
is at 63.2%. It is essentially defined as the time required to implement a 63.2% change in the
output for a system’s step resoonse. This 63.2% is taken, because in radioactive decay, the
time constant is related to the decay constant and it represents the mean life time of a
decaying systembefore it decays. It represents the time it takes for allbut 36.8 % of the atoms
to decay. Hence because of this the time constant is 63.2% which is longer than half life.
There is no trend present between the time constant of tank 1 and 2 and the flow rate of the
water inlet. As the flow rate of the water entering the tank increases, the time constant also
increases. There is no trend to be seen in the steady state gain of tank 1 and 2 and the flow
rate of water.
The transfer function is of the format:
𝑋1(𝑠)
𝑋0(𝑠)
=
𝐾𝑝1
𝜏1𝑠 + 1
𝑋2(𝑠)
𝑋1(𝑠)
=
𝐾𝑝2
𝜏2𝑠 + 1

12. 12
𝑋2(𝑠)
𝑋0(𝑠)
=
𝐾𝑝2𝐾𝑝1
(𝜏2𝑠 + 1)(𝜏1𝑠 + 1)
The tranfer functions in this experiment are as follows:
1. For 30-40 LPH, the transfer function is
11.4
(71𝑠+1)(104𝑠 +1)
2. For 40-50 LPH, the transfer function is
6.16
(90𝑠+1)(120𝑠+1)
3. For 60-70 LPH, the transfer function is
24.84
(75𝑠+1)(135𝑠 +1)
The ultimate value of the transfer function is also the steady state gain.
The step response of system having two first order systems is S shaped and the respinse
changes very slowly after the introduction of the step input. This sluggishness or delay is
known as the transfer lag and is always present when two or more first order systems are
connected in series. This can be seen in figure 7 and it is imitated in the experimental results
as well, the reason the graph of the first tank response and the second tank response are so
far apart in all the flow rate changes is because of the phenomenon of transfer lag.
For a single first order system, there is no transfer lag. The response begins immediately after
the step change is applied and the rate of change of the response is maximum at t=0. The
overall transfer function for two non interacting first order stsrems connected in series is the
product of the individual transfer functions.
9. Conclusion
Figure 7 Step response of non interacting first order
system in series

13. 13
In this experiment, we have studied the dynamic response of a first order system in a two
tank series system. We have obtained the different time constant values and system gain
values for three different flow rates. These are shown in Table 1. There was a consistent time
lag between the first tank response and the second tank response. There is no visible trend
between the time constant and the increasein flow rate and there is no visibletrend between
the system gain and the increase in flow rate.
10. References
Coughanowr, Donald R.,and LowellB. Koppel. Process Systems Analysis and Control. McGraw-
Hill, 1991.
11. Appendix A
Time (s) h1 (cm) H1 (cm)2 h2 (cm) H2 (cm)3
0 40 0 30 0
10 46 6 32 2
20 52 12 33 3
30 54 14 36 6
40 58 18 38 8
50 60 20 40 10
60 63 23 42 12
70 66 26 44 14
80 67 27 45 15
90 68 28 47 17
100 68 28 49 19
110 70 30 50 20
120 73 33 51 21
130 74 34 52 22
140 74 34 53 23
150 75 35 54 24
160 75 35 56 26
170 76 36 57 27
180 77 37 58 28
190 77 37 58 28
200 78 38 60 30
210 78 38 60 30
220 78 38 60 30
230 78 38 60 30
Table 2 Raw data for 30-40 LPH

14. 14
Time (s) h1 (cm) H1 (cm)2 h2 (cm) H2 (cm)3
0 78 0 60 0
10 83 5 64 4
20 86 8 65 5
30 87 9 66 6
40 88 10 66 6
50 90 12 68 8
60 91 13 68 8
70 92 14 70 10
80 94 16 70 10
90 95 17 70 10
100 97 19 72 12
110 98 20 72 12
120 100 22 73 13
130 100 22 75 15
140 102 24 76 16
150 102 24 77 17
160 103 25 78 18
170 104 26 78 18
180 104 26 78 18
190 105 27 80 20
200 105 27 80 20
210 105 27 81 21
220 106 28 82 22
230 106 28 82 22
Table 3 Raw data for 40-50 LPH
Time (s) h1 (cm) H1 (cm)2 h2 (cm) H2 (cm)3
0 106 0 82 0
10 119 13 86 4
20 125 19 88 6
30 130 24 90 8
40 140 34 91 9
50 143 37 92 10
60 146 40 94 12
70 149 43 95 13
80 150 44 96 14
90 154 48 98 16
100 160 54 100 18
110 162 56 100 18
120 163 57 102 20
130 165 59 104 22
140 167 61 106 24
150 168 62 108 26
160 169 63 109 27
170 170 64 110 28

15. 15
180 171 65 112 30
190 172 66 114 32
200 173 67 115 33
210 173 67 116 34
220 174 68 117 35
230 175 69 118 36
Table 4 Raw data for 50-60 LPH