Cooling Tank Control Assignment - Desklib

This assignment focuses on the cooling tank control system and its design tasks. It covers topics such as steady state balance equations, linear deviation models, and reaction rate equations. The Simulink model for the system is also discussed. The assignment is available for study material at Desklib.

COOLING TANK CONTROL
ASSIGNMENT
INSTITUTIONAL AFFILIATION
LOCATION
INSTRUCTOR (PROFESSOR)
STUDENT NAME
STUDENT ID NUMBER
DATE OF SUBMISSION
1

TABLE OF CONTENTS
INTRODUCTION...........................................................................................................................2
PROBLEM STATEMENT..........................................................................................................2
ASSUMPTIONS..........................................................................................................................2
SYSTEM PARAMETERS..........................................................................................................2
COOLING TANK CONTROLLER DESIGN TASKS..................................................................3
PART I.........................................................................................................................................3
PART II........................................................................................................................................9
PART III....................................................................................................................................10
PART IV....................................................................................................................................12
PART V......................................................................................................................................15
PART VI....................................................................................................................................16
DISCUSSION................................................................................................................................17
Process modeling.......................................................................................................................17
Control systems..........................................................................................................................19
Heat transfer in Thermodynamics..............................................................................................21
CONCLUSION..............................................................................................................................22
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INTRODUCTION
PROBLEM STATEMENT
A hot liquid flow is coming to the tank with flow rate F and temperature T0. The liquid in
the tank is cooled by a jacket cooling water with flow rate Fc and inlet temperature Tcin. The
outlet temperature of cooling water is Tcout. By controlling the jacket cooling water flow rate, the
outlet temperature of the tank, T, can be maintained at a desired level.
ASSUMPTIONS
(i) The cooling tank is properly insulated to avoid heat transfer to the surroundings.
(ii) The amassing of energy in the tank walls and cooling jacket is considered negligible
compared to the accumulation of energy in the liquid.
(iii) It the liquid in the tank is well-mixed and the system is initially at steady state and the
physical properties of the liquid and tank environment are constant.
SYSTEM PARAMETERS
The system parameters of the cooling tank that define the inlet and the outlet are,
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COOLING TANK CONTROLLER DESIGN TASKS
PART I
DESIGN: COOLING TANK CONTROL
Steady state balance equations of the cooling water tank
For an unsteady state mass balance of the cooling water tank, the rate of accumulation of the
mass in the tank is equivalent to the rate of mass at the inflow minus the rate at which the water
flows out.
d ( Vρ )
d t =w1+ w2 −w
The unsteady state component balance is
d ( Vρx )
dt =w1 x1 + w2 x2−wx
For the corresponding steady-state model is given as,
w1+w2−w=0
w1 x1 +w2 x2−wx=0
In the steady state equation of the cooling tank, the laws of conservation hold following a
number of theoretical models of chemical processes. In thermodynamics, the rate of energy
accumulation is the difference between the convection energy rate of the incoming water and the
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convection energy rate of the water flowing out taking into consideration the mesh rate of head
addition to the system as a result of surrounding and the net amount of work accomplished on the
system by the ambiances. The total energy of a thermodynamic system is the sum of its internal
energy, the kinetic energy, and the potential energy. The energy balance is given as,
d U∫¿
dt =−∆ ( w ^H ) +Q ¿
Using molar quantities, the enthalpy per mole and molar flow rate are used such that,
d U∫¿
dt =−∆ ( ~w ~
H ) +Q ¿
Revisiting the cooling process,
ρ dV
dt =w1 + w2−w
ρ d ( Vx )
dt =w1 x1+ w2 x2−wx
The system is differentiated using the chain rule such that,
ρV dx
dt + px dV
dt =w1 x1+w2 x2 −wx
Further,
ρV dx
dt + x ( w1 +w2−w )=w1 x1 +w2 x2−wx
Simplifying the terms by eliminating the common terms in the equation,
dV
dt = 1
p ( w1+ w2−w )
dx
dt = w1
Vρ ( x1−x ) + w2
Vρ ( x2−x )
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In the cooling tank scenario, it is possible to determine the amassing of the internal energy such
that,
d U∫¿
dt =ρVC dT
dt ¿
For the cooling tank with a temperature T with an enthalpy H,
^H− ^H ref =C ( T −T ref )
^Href =0
^H=C ( T −T ref )
At the inlet valve section,
^Hi=C ( Ti −Tref )
The convection energy is given as,
−∆ ( w ^H ) =w [ C ( T i−T ref ) ] −w [ c ( T −T ref ) ]
VρC dT
dt =wC ( Ti−T ) +Q
At the outlet valve section,
^Hout =C ( T out−T ref )
The convection energy is given as,
−∆ ( w ^Hout )=w [ C (T out−T ref ) ]−w [ c ( Tout −Tref ) ]
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To determine the energy balance of the cooling tank system, the convection energy equations
discussed above are used in the process modeling. The enthalpy and entropy are considered state
functions which state that the path of integration is arbitrary. Using the tank temperature
modeled differential equation,
dT
dt
F
V ( T o−T )− Q
V ρ Cρ
The coil temperature is modeled as,
d T cout
dt = Fc
V c
( T cin−T cout ) + Q
V ρ Cρ
Q=UA ( T −T cout )
On the system output, a temperature sensor, in this case, the thermocouple coil is implemented. It
focuses on testing the process fluid by a metal sleeve and because there is a distance between the
plant and the sensor, a dynamic lag is experienced which is denoted using the transport lag.
VρC d T '
dt =wC ( T i
' −T' )+Q'
Performing the Laplace transform of the equation,
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VρC [ s T ' ( s ) −T ' ( 0 ) ] =wC [ T i
' ( s ) −T' ( s ) ] −Q' ( s )
For the initial steady state, T ' ( 0 ) =0
VρC [ s T ' ( s ) ] =wC [ T i
' ( s ) −T ' ( s ) ] −Q' ( s )
Rearranging the terms in the equation,
T ' ( s )= [ K
τs+1 ]Q' ( s ) +
[ 1
τs+1 ]Ti
' ( s )
K= 1
wC
τ =Vρ
w
T ' ( s )=G1 ( s ) Q' ( s ) +G2 ( s ) T i
' ( s)
The steady state of a transfer function is used in the calculation of the change in the steady state
value at the input. In a linear system, K is constant and for the nonlinear system, K depends on
the operating condition (Gene, J. Davied, & Abbas, 2010). The steady state gain is expressed
using the final value theorem such that,
K=lim
s →0
G( s)
G ( s )= Y ( s )
U ( s ) =
∑
i=0
m
bi si
∑
i=0
n
ai si
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When the sensor is placed at the temperature downstream from the heated tank with a reasonable
transport lag, the distance L is used in the plug flow. The dead time is given as,
θ= L
V
The tank’s transfer function is given as,
G1= T ( s )
U ( s ) = K1
1+τ1 s
The sensor is given as,
G2= T cout ( s )
Tcin ( s ) = K2 e−θs
1+τ2 s K2 ≤ 1
Neglecting the value of τ 2, the overall transfer function is given as,
Tcou t ( s )
U =G2 . G1= K 1 K2 e−θs
1+τ1 s
Replacing with actual values,
0= F'
V (T o
' −T ' )− Q'
V ρ Cp
Q'=ρ C p F' ( T o
' −T' )
Q'= ( 0.15 x ( 90−40 ) x 106 x 1 )
Q '=7.5 x 106 cal/min
0= F ❑c '
V❑c
(T cin
' −T cout
' )+ Q'
V ❑c ρc C p
T cout
' = V ❑c
F ❑c '∗
( Tcin
'
V ❑c
+ Q'
V ❑c
ρc C p )
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T cout
' = 0.5
1 ∗¿
T cout=27. 50 C
UA= Q
( T −T cout )
UA=7.5 x 106
40−27.5
UA=6.0 x 105 cal /min0 C
PART II
Simulink model for the system
The cooling tank is modeled as illustrated below using the MATLAB Simulink,
The scope output for the initial simulation is obtained as,
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PART III
Linear deviation models of the system
The first section of the analysis has provided the transfer function of the cooling tank
control system. There is need to linearize the nonlinear model to gain a good approximation
which is obtained near the given operating point. It is possible to obtain gain, time constant
which may change the operating point. Using the first order of Taylor series,
dy
dt =f ( y , u )
f ( y , u ) ≅ f ( y , u ) + ∂ f
dy |y ,u
( y− y ) + ∂ f
du |y ,u
( u−u )
The steady-state equation is subtracted from the dynamic equation,
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d y'
dt = ∂ f
dy |s
y'+ ∂ f
du |s
u'
The control and disturbance parameters are used to assess the linear time variables,
As H' ( s ) =qi
' ( s ) −q0
' ( s ) … deviation variables
The control variable is constant, hence, q0
' =0
As H' ( s )=qi
' ( s ) , H' ( s )
qi
' ( s ) = 1
As
The pure integrator in ramp form is used to obtain the step change in qi. The control variable is
given as shown for the non-linear element,
q0=Cv √h
The linear model is given as,
q'= 1
R h'
A d h'
dt =qi
' − 1
R h'
If q0=Cv √h
A d h'
dt =qi
'−Cv √ h
Performing Taylor series on the right hand side of the equation,
A d h'
dt =qi
' −Cv h0.5+ ∂ f
d qi
( qi−qi ) + ∂ f
dh ( h−h )
A d h'
dt =0+ ( qi −qi ) −1
2 Cv h−0.5 ( h−h )
qi
' − 1
2 Cv h−0.5 h'
A d h'
dt =qi
' − 1
R h'
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R=2 h0.5
Cv
The linearized transfer function based on the Taylor series expansion of the expressions above is,
Goverall (s )= 0.09
s2+3.575 s+0.84
PART IV
The reaction rate for the disappearance of energy from one state to another is given as,
r =k c A
The rate constant has an Arrhenius temperature dependence such that,
k =k0 e− ( E
RT )
k 0−frequency factor , E−activation energy , R−gas constant
In modeling, the updraft capacitances of the coolant and the chilling coil walls are
negligible as associated to the updraft capacitance of the liquid in the tank. All the coolant is at a
uniform temperature such that the increase in the coolant temperature as the coolant passes
through the coil is neglected. The rate of the heat transfer from the reactor contents to the coolant
is given by,
Q=U A ( T −T cout )
As a result the CSTR energy balance equation is obtained for analysis as,
VρC dT
dt =wC ( Ti−T ) + ( −∆ HR ) V k CA+UA (T c−T )
An open loop has no feedback loop hence the system output is not bounded which is
quite harmful. To control the system using a PID controller, it is important to determine the
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proportional parameter, the integral parameter, and the derivative parameter for the controller
(Adian, 2009). Using MATLAB to determine the controller in the Simulink model that achieves
the control over the cooling process in the tank,
(i) To plot the reaction curve on MATLAB using the step response and the linearized
transfer function of the system.
function [model,controller]=ReactionCurve(t,y,u)
if nargin<3
du=1;
t0=0;
u=1;
elseif isscalar(u)
du=u;
t0=0;
else
du=u(end)-u(1);
t0=find(diff(u));
end
gain=(y(end)-y(1))/du;
dy=diff(y);
dt=diff(t);
[mdy,I]=max(abs(dy)./dt);
time_constant=abs(y(end)-y(1))/mdy;
time_delay=t(I)-abs(y(I)-y(1))/mdy-t0;
subplot(211)
plot(t,y,[t0+time_delay t0+time_delay+time_constant],[y(1)
y(end)],...
[t(1) t(end)],[y(1) y(1)],'--',[t(1) t(end)],[y(end)
y(end)],'--');
title('output')
subplot(212)
if isscalar(u)
plot([t0 t(end)],[u u])
else
plot(t,u)
end
title('input')
model.gain=gain;
model.time_constant=time_constant;
model.time_delay=time_delay;
time_delay=max(time_delay,time_constant/10);
controller.P=time_constant/time_delay/gain;
controller.PI=0.9*time_constant/time_delay/
gain*tf([3.3*time_delay 1],[3.3*time_delay 0]);
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controller.PID=1.2*time_constant/time_delay/
gain*tf([time_delay^2 2*time_delay 1],[2*time_delay 0]);
Using the MATLAB software, it was possible to obtain the steady state gain parameter
from the linear plot of the system as well as the gradient of the line, K. When the reaction curve
is obtained, the control parameters are as listed below
(i) Process gain, K = -5.3398
(ii) Time constant, τ = 5.5664
(iii) Time delay, td = 0.8517
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One of the most prevalent PID controller tuning methods is the Ziegler-Nichols method.
The method allows the PID controller to have its parameters tuned on an individual level such
that the system obtains the most suitable parameters for the controller design. For an open loop,
the controller is given as PID using
(i) KpKc =7.84
(ii) τ1 = 1.70
(iii) τd = 0.445
(iv) tau=5.567
(v) td = 0.852
To obtain the value of the proportional gain parameter,
Kc (1+ 1
τ1 s + τd s )
K= T ( s )
Fc ( s )
To obtain the integral component of the controller,
Ki= Kc
τ1
PART V
The demerit of the open loop systems, even in the case of an installed PID controller, is
that the system output is not bounded. The output is represented using a large overshoot and a
very long settling time. A thermocouple acts as the temperature sensor in this scenario, cooling
tank system. The closed loop controller is preferred to overcome the caveat introduced by the
open loop system. In this case, the values chosen and tuned for the proportional, integral, and
derivative components are fit to ensure the stability of the system and efficient control of the
system.
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PART VI
Using the feed-forward PID feedback approach,
The derivative parameter is given as -16.5745
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DISCUSSION
Transfer delays are as a result of the resistance and capacitance of the plant elements and
the dead time is the transport delay cause by a distance-velocity lag. The energy is carried by a
flow of water. The temperature sensor is positioned a long way downstream from the throttling
valve that constitutes the final control element for the temperature control, and the pipe is
oversized relative to the actual flow rate, a significant time will be required for the fluid at an
increased or decreased temperature to travel from the valve to the sensor. This period of delay
constitutes the dead time where the sensor is unable to detect the fluid during the outflow and the
valve section. The delays are observed in the first order RC delay of the initial cascading units of
the system. The energy transfer is not easily processed using the cooling coil. The linearized
model is used to detect relatively small changes in the flow rate and in water depth (Åström & E,
2005). Based on the principle of superposition, the operations of one linear element are not
affected by others within the systems so as to be attributed in the same way and there are
mathematical solutions constructed by adding the simple solutions that are obtained by the
response of the system to the complex disturbance by summing the responses to its component.
There is an assumption that the fluid motion is laminar and the Reynolds number is Re<2000.
The system transfer function is obtained on the linear system and the ratio is converted to
Laplace transforms of the output and input considering that all the initial conditions are null. The
steady state of the Laplace transform is the limit of the time approaching infinity is represented
by settling the Laplace operator equal to zero.
Process modeling
In thermodynamics, there are different forms of energy and the total quantity of energy is
constant. The energy disappears from one form to another according to the energy law of
conservation. There is no loss of energy in physical environments. Process models are used to
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develop deeper understanding of a process considering the physical and thermodynamic
properties of the process. In this research paper, the process under research is the cooling tank
which takes in hot water and yields cold water. It is important to use the model to understand the
cooling process of the tank and how the energy is transferred from one form to another (Ling,
Rahmat, Husain, & Ghazali, 2012). The most common process model implemented is the
mathematical model which shows the change in energy. These models guarantee an
understanding of the process, train a plant operating personnel on how to manage the system,
determine the controller settings as well as the control law that will govern the cooling process.
The mathematical models in process modeling are useful to optimize the process operation
conditions as well as in the design of a control strategy.
Industries that perform a lot of chemical reactions in different vessels are used to contain
chemical reactions. These vessels perform a lot of heat exchange which needs to be removed
from the reactors in a very safe manner to avoid explosion in the factory or plant. Most of these
reactions that emit heat to their surroundings are known as exothermic reactions. There are heat
cooling systems also known as jackets. These are used to liberate or absorb heat during the
processing (ATCS, 2007). This research work uses the continuous stirred-tank reactor model that
has an impeller or a rotating pump component. For the tank, it is observed that the temperature of
the water in the tank is uniform as a result of the impeller action. The system demonstrates that
the steady state of the flow rate is equal to the mass flow rate of the tank during the transient
state. The use of the impeller in the tank ensures that there is perfect mixing in the tank. In an
open loop system, when the temperature in the tank increases, there is more heat that needs to be
removed (Ali & Vahid, 2006). The change in temperature between the inflow and the outflow
water describes the change in temperature in the tank. An open loop system is not desired in
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design or implementation in industrial applications as it provides unbounded outputs. The output
may be amplified by noise or disturbances in the environment of the cooling tank.
Control systems
In advanced control systems, the model predictive control is a class of the model based
multivariable discrete time control algorithms. The MPC uses the dynamic model to perform an
online prediction of future behavior of the process and for planning the future control moves.
The model does not specify a specific control strategy rather it impacts a myriad of control
methods that use models to define the process required to minimize an objective function.
Models are used to predict the process outputs for future values. The MPC is used in the industry
applications to determine the advanced control methodologies and provide a number of
capabilities. The control strategies that adopt MPC ensure that the operating constraints are
explicitly handled. It is easily implemented using complex multivariable systems and there are
difficult dynamics such as time delay, analysis delay, and inverse response behavior which can
be handled with ease. The MPC improves the robustness when the different control strategies
and methods are tuned (Baogang, Mann, & Gosine, 2009).
The MPC improves the efficiency of tuning for the highly trained control personnel. The
MPC is used to control a number of processes with relatively simple dynamics (Noor, Hamid,
Mahanijah, & Faieza, 2009). The linear control law can be implemented on the resulting
controller. The methodology is based on certain basic principles that allow for future extensions.
It is easy to handle scenarios that have multivariable cases to deal with and the systems are easily
implemented using future references. The model is implemented as,
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Implementing the MPC assures the designer that the system can be controlled or tuned in
different stages as required. The system is commonly implemented in chemical reactor systems
as well as in the cooling tank system. The MPC algorithms are based on a number of principles
such as the predictive model, objective function, and obtaining the control law. The prediction
model shows the necessary mechanisms of obtaining the most important model. The transfer
function model is used to obtain the system structure. The state space model is formed based on
the state space model as well as the free and forced model response.
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Heat transfer in Thermodynamics
The heat can be transferred from one section to another in the cooling tank in a number of
days. The energy is transferred from one object to another, while in contact, by conduction. The
molecules are cooler when the energy is transferred through currents in a fluid such as a gas or a
liquid through the convention. The objects also perform temperature transfer through radiation
by the electromagnetic radiation (Venkatesh & and Seshagiri, 2012). These objects tend to
absorb the radiation as well and so much is absorbed during radiation. At the thermal
equilibrium, the energy is absorbed as much as it radiates. The cooling tank loses energy through
convection. The impeller ensures that the water in the tank is uniformly warm. The convection
currents transfer heat from the tank to the air through the water using the convection medium. It
is used to determine global weather patterns (Chanin, 2008).
The laws of thermodynamics are used to define the heat transfer modes of the objects.
According to the laws of thermodynamics,
“Energy is always conserved. It can change forms from the kinetic, potential, internal,
etc. but the total energy is constant and there is a way to change the thermal energy of a system is
equal to the sum of the work done on it and the amount of heat energy transferred to it.”
(Thermodynamics, 2008)
“During any natural process the total amount of entropy in the universe always increases.
The entropy is defined informally as a measure of the randomness or disorder in a system. The
heat flows naturally from a hot to cooler surroundings as a consequence of the second
law.”(Thermodynamics, 2008)
System modeling intrinsically involves a concession between model accuracy and
complexity as well as the cost and effort required to make simplifying assumptions that results in
an appropriate model. The dynamic models of chemical processes consist of ordinary differential
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equations, partial differential equations, and the related algebraic equations. In the development
of this model, there is a single, irreversible reaction and perfect mixing using the impeller
(Zhang, Fang, Lu, Zhang, & Li, 2009). The liquid volume is maintained constant by an overflow
line. The mass densities of the feed and product streams are equal and constant. The heat losses
are negligible.
CONCLUSION
In a nutshell, the cascaded system with a feed-forward loop properly implements the PID
controller for both the master and slave PID with the aim of correcting disturbance. The inner
loop of the controller is given such that the gain parameter is 0.15 of the total controller
parameter values. The design task used the Ziegler-Nichols tuning to obtain the most preferred
proportional, derivative, and integral gain. In an open loop system cooling may not be controlled
and the output may vary in temperature due to the system disturbances in the surrounding.
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References
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Åström, & E, H. K. (2005). PID controllers: Theory, Design and Tuning.
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Control and signal processing.
Baogang, H., Mann, G., & Gosine, R. (2009). New Methodology for analytical and optimal
design of fuzzy PID controllers. IEEE Transactions on Fuzzy Systems.
Chanin, P. (2008). Feedback linearization controller design for continuous stirred-tank reactor
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Engineering/Electronics Computer Telecommunications and Information Technology, 5-
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Gene, F. F., J. Davied, P., & Abbas, E.-N. (2010). Feedback control of Dynamic systems.
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