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International Journal of Engineering, Applied and Management Sciences Paradigms, Vol. 45, Issue 01
Publishing Month: March 2017
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Digital Control of Continuous Fluidized Bed Dryers for
Pharmaceutical Products
Gurashi Abdullah Gasmelseed1 and Mahdi Mohammed2
1Department of Chemical Engineering, University of Science and Technology, Khartoum, Sudan
2Department of Chemical Engineering, University of Science and Technology, Khartoum, Sudan
Publishing Date: March 05, 2017
Abstract For continuous system, we know that certain behaviors results
from different pole location in the s-plane. For instance, a
system in unstable when any pole is located to the right of the
imaginary axis. For discrete system, we can analyze the system
behaviors from different pole location in the z-plane. The
characteristic in the z-pole can be related to those in the s-plane
by the expression:
Z=e ST
T = Sampling time (Sec \ Sample)
S = Location in the s-plane
Z = Location in the z-plane
The stability of any system is determined by the location of the
roots its characteristic equation of the transfer function. The
characteristic equation of the continuous system is a polynomial
in the complex variable S. If all the roots of the polynomial lie
in the LHP of the S-plane, the system is stable. The stability of a
sampled-data system is determined by the location of the roots
of the characteristic equation that is polynomial in the complex
variable Z. The region of stability in the Z-plane can be found
directly from the region of stability in the s-plane using the basic
relationship between the complex variable S and Z: Z = e -TS Keywords: Continuous System, Discrete System, Stability
of any System, Stability and Transient Response,
Fluidized Bed Dryers, Drying, Moisture Content, The
Computer System, Discrete-Time Response of Dynamic
Systems, Digital Control.
Introduction
Drying means the removal of relatively small amounts of
water from wet material by the application of heat.
Drying is an energy-intensive operation that accounts for
up to 15% of the industrial energy usage. Moreover, conventional dryers often operate at low thermal
efficiency, typically between 25% and 50%, but it may be
as low as 10%.
Fluidized bed dryer is used widely in food, metallurgical, chemical and pharmaceutical industry,
because of the shorter drying time required and simple
maintenance and operation. This type of dryers is based
on the phenomena of fluidization. Fluidization is the operation by which solid
particles are transformed into fluid-like state through
suspension in gas or liquid. When a gas is passed through
a layer of particles supported by a grid at low flow rate,
the fluid percolates through the void spaces between
stationary particles. As the fluid velocity increased, the void age increases, this resulting in an increase in pressure
drop on the particles.
The pressure drop across the particle layer will
continue to increase in proportion to the gas velocity till
the pressure drop reaches a constant value that is
equivalent to the weight of the particles in the bed divided
by the area of the bed, at this point the frictional force between particles and fluid counterbalances the weight of
the particles. At this stage the bed is to be incipiently
fluidized. Fluid velocity at this point is known as
minimum fluidization velocity. With an increase in flow
rates beyond minimum fluidization, large instabilities
with bubbling, channeling of gas and decrease in pressure
drop are observed. Fluidized bed dryers have some drawbacks.
Material with a wide particle-size distribution cannot be
handled satisfactorily, while at high temperatures the
melting and fusing of the material on the grid plate can
become a problem. To circumvent these difficulties,
dryers, originally developed for grain drying, have been
made with conical bottom sections which give a spouted
bed rather than a fully fluidized one.
Deriving mathematical models can be done by
utilizing physical laws to derive a mathematical model,
this model must be rigorous enough to give an accurate
description of the process. In most cases the obtained
models are set of ordinary differential equations. Prior to control system design, control synthesis
must be performed. The synthesis of control
configurations for multivariable system involves selection
of controlled and manipulated variables, pairing
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manipulated inputs and controlled outputs (loop pairing), and selection of the best control configuration. Generally,
input variables can be classified into manipulated
variables and disturbances or load variables. The most
desirable drying process output variable to control is
product moisture content, but this is difficult to measure
directly
Figure 1: Flow sheet of Fluidized Bed Dryer
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Often, the moisture content of the dried product can be
inferred from the temperature and humidity of the exhaust
gas. However, due to the weak correlation between the
temperature and the actual product moisture content, using
indirect control usually results in poor control of the
drying process. Multivariable control design must be
considered for fluidized bed dryers in order to account for
dynamic interactions between the control loops.
Research Objectives:
1/ Design of fluidized bed dryer for drying pharmaceutical
products.
2/ Design of conventional and digital control of fluidized
bed dryer.
3/ Comparison of performance between conventional and
digital control of fluidized bed dryer.
Methodology
Digital Control:
The computer system collects data from the process
measurements and calculates the values of the manipulated
variable and implements the control action on the process,
based on the control algorithm that is already programmed
and stored in the memory of the computer. Signals are
converted by digital to analog (D/A) and analog to digital
(A/D) converters.
Z-Transforms play the same role for discrete-time
systems as that played by Laplace transforms for dynamic
analysis and design of continuous open or closed loops
systems.
Block Diagram:
Tuning of the level in the continuous fluidized bed
dryer from the physical diagram:
Figure 2: Block diagram of the discrete-digital control loop (DDC)
Discrete-time Response of Dynamic Systems:
In continuous analog systems, we get the overall
transfer in Laplace or time-domain and we make our
analysis of the system such as stability, controller settings
and design. The above methodology cannot be used for
dynamic analysis of digital control-loop as these poses
discrete element (Digital-Control algorithm) and discrete
time signals. These are two primary distinct components
whose responses are very important in the dynamic
analysis of control systems:
This is a discrete element of the DDC-Loop with discrete-time input and output signal.
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Conversion to z-transform (Using c2d):
There is a MATLAB function called c2d that converts a
given continuous system (either in transfer or state – space
form) to a discrete system using the zero order hold
operation explained above. The basic command for this in
MATLAB is the sampling time (Ts in sec /sample) should
be smaller than 1/30*BW), where BW is the closed- loop
bandwidth frequency.
Stability and transient response:
For continuous system , we know that certain behaviors
results from different pole location in the s-plane .For
instance , a system is un stable when any pole is located to
the right of the imaginary axis .For discrete system , we
can analyze the system behaviors from different pole
location in z-plane . The characteristics in the z-plane can
be related to those in s-plane by the expression:
z = eST
T = Sampling time (sec /sample)
S = Location in the s-plane
Z = Location in the z-plane
Stability in the Z-plane:
The stability of any system is determined by the location
of the roots of its characteristic equation of its transfer
function. The characteristic equation of the continuous
system is a polynomial in the complex variable S. If all the
roots of this polynomial lie in the LHP of the S-plane, the
system is stable. The stability of a sampled-data system is
determined by the location of the roots of a characteristic
equation that is polynomial in the complex variable Z. The
region of stability in the Z-plane can be found directly
from the region of stability in the s-plane using the basic
relationship between the complex variable S and Z:
Z = e-TS
Tuning:
Discrete Root Locus:
The root Locus is the locus of points where roots of
characteristic equation can be found as a single gain is
varied from zero to infinity. The characteristic equation of
a unity feedback system is:
1 + KG(z) Hzoh (z) = 0
Where G(z) is the compensator implemented in the digital
controller and Hzoh (z) is the plant transfer function in z .
Bode Plot:
It represents the amplitude ratio and phase angle of the
response of the system as the function of the frequency. It
shows the variation of the logarithm of the amplitude
ratios with the frequency and the variation of the phase
shift with the frequency. To determine the Amplitude ratio
value AR should be converted from decibels to absolute.
This can be done using one of the following:
1. Double click on bode diagram window.
A ‘property editor’ window will appear. Select ‘units’
option and then change AR from (db) to (abs). Then
find the value of AR at from the new curve.
2. Use the formula: Reading in db = 20 log AR
𝑲𝒖 =𝟏
𝑨𝑹
Results and discussion
Control Strategy with Digital Control:
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Figure 3: Control strategy of the Fluidized Bed Dryer with Digital Control
Control of the Air Temperature (Loop 1):
Conversion to z-transform:
For P-controller, the closed loop transfer function in
Laplace domain is:
G(s) =7.874𝑠 + 39.37
0.1s3 + 1.52s2 + 5.3s + 40.37
The closed loop transfer function in z-domain is:
𝐺(𝑧) =0.2859𝑧2 + 0.03319𝑧 + 0.1253
𝑧3 − 1.834𝑧2 + 1.251𝑧 − 0.2187
Stability and transient response:
From the overall transfer function:
𝐺(𝑧) =0.2859𝑧2 + 0.03319𝑧 + 0.1253
𝑧3 − 1.834𝑧2 + 1.251𝑧 − 0.2187
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Figure 4: Stability and transient response using MATLAB
From the above figure:
𝜔𝑛 = 0.17 𝜋/𝑇, where T is the sampling time.
For 𝑇 =1
10 𝑠𝑒𝑐, 𝜔𝑛 = 5.341 𝑟𝑎𝑑/𝑠𝑒𝑐
And 𝜉 = 0.15
Step response in digital control system:
The overall transfer function for P-controller:
𝐺(𝑧) =0.2859𝑧2 + 0.03319𝑧 + 0.1253
𝑧3 − 1.834𝑧2 + 1.251𝑧 − 0.2187
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Figure 5: Step response in the set point of the digital control system
Tuning:
Discrete Root Locus:
𝐺(𝑧) =0.2859𝑧2 + 0.03319𝑧 + 0.1253
𝑧3 − 1.834𝑧2 + 1.251𝑧 − 0.2187
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Figure 6: Root locus for the discrete system
From the plot, the system is stable because all poles are
located inside the unit circle, and Ku = 0.373
Bode plot:
𝐺(𝑧) =0.2859𝑧2 + 0.03319𝑧 + 0.1253
𝑧3 − 1.834𝑧2 + 1.251𝑧 − 0.2187
Figure 7: Bode plot for the discrete system
𝐴𝑅 = 2.8, 𝐾𝑢 =1
𝐴𝑅=
1
2.8= 0.360
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Table 1: Comparison between the methods of tuning used for the discrete system
The method Ku
Root Locus 0.373
Bode 0.360
Average 0.367
The two methods are in agreement.
Effect of sampling time on overall T.F in z-domain and on system response:
The overall T.F for the system at different sampling time were determined as following:
Table 2: Overall T.F's and (OV) at different sampling time
Sampling Time Overall T.F's G(z) Overshoot(%)
0.01 0.003806 z^2 + 5.846e-005 z - 0.0035
------------------------------------
z^3 - 2.854 z^2 + 2.713 z - 0.859
88.2
0.1 0.2859 z^2 + 0.03319 z - 0.1253
----------------------------------
z^3 - 1.834 z^2 + 1.251 z - 0.2187
88
1 0.4894 z^2 + 0.1811 z - 0.04745
----------------------------------------
z^3 - 0.5419 z^2 + 0.1808 z - 2.505e-007
6.81
1.5 1.184 z^2 + 0.02086 z - 0.02017
-----------------------------------------
z^3 + 0.1377 z^2 + 0.07687 z - 1.253e-010
21.4
2 0.9357 z^2 + 0.1462 z - 0.008577
------------------------------------------
z^3 + 0.06793 z^2 + 0.03268 z - 6.273e-014
4.42
2.5 0.9332 z^2 - 0.07602 z - 0.003647
----------------------------------------
z^3 - 0.1387 z^2 + 0.0139 z - 3.127e-017
1.17
By changing the values of Ts from 0.01 to 0.1, 1 , 1.5 , 2 and 2.5, the response plots are:
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Figure 8: system response at Ts = 0.01 sec
Figure 9: system response at Ts = 0.1 sec
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Figure 10: System response at Ts = 1 sec
Figure 11: system response at Ts = 1.5 sec
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Figure 12: system response at Ts = 2 sec
Figure 13: System response at Ts = 2.5 sec
The values of overshoot decreases as Ts increases.
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Control of the fluidized bed pressure (Loop 2):
Conversion to z-transform:
For P-controller, the closed loop transfer function in
Laplace domain is:
G(s) =8.703𝑠+43.52
0.6s3+5.06s2+10.5s+44.52
The closed loop transfer function in z-domain is:
𝐺(𝑧) =0.0648𝑧2 + 0.01863𝑧 − 0.03514
𝑧3 − 2.286𝑧2 + 1.765𝑧 − 0.4303
Stability and transient response:
From the overall transfer function:
𝐺(𝑧) =0.0648𝑧2 + 0.01863𝑧 − 0.03514
𝑧3 − 2.286𝑧2 + 1.765𝑧 − 0.4303
Figure 14: Stability and transient response using MATLAB
From the above figure:
𝜔𝑛 = 0.1 𝜋/𝑇, where T is the sampling time.
For 𝑇 =1
10 𝑠𝑒𝑐, 𝜔𝑛 = 3.142 𝑟𝑎𝑑/𝑠𝑒𝑐
And 𝜉 = 0.17
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Step response in digital control system:
The overall transfer function for P-controller:
𝐺(𝑧) =0.0648𝑧2 + 0.01863𝑧 − 0.03514
𝑧3 − 2.286𝑧2 + 1.765𝑧 − 0.4303
Figure 15: Step response in the set point of the digital control system
Tuning
Table 3: Comparison between the methods of tuning used for the discrete system
The method Ku
Root Locus 3.59
Bode 3.50
Average 3.545
The two methods are in agreement.
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Effect of sampling time on overall T.F in z-domain and on system response:
The overall T.F for the system at different sampling time were determined as following:
Table 4: Overall T.F's and (OV) at different sampling time
Sampling Time Overall T.F's G(z) Overshoot (%)
0.01 0.000717 z^2 + 2.682e-005 z - 0.0006743
--------------------------------------- z^3 - 2.917 z^2 + 2.837 z - 0.9191
66.3
0.1 0.0648 z^2 + 0.01863 z - 0.03514
---------------------------------- z^3 - 2.286 z^2 + 1.765 z - 0.4303
65.5
1 1.617 z^2 + 0.9244 z - 0.0305
-------------------------------------- z^3 + 1.206 z^2 + 0.3632 z - 0.0002175
65.4
1.5 0.9816 z^2 + 0.2567 z - 0.01804
----------------------------------------- z^3 + 0.02879 z^2 + 0.2195 z - 3.208e-006
23.8
2 0.5919 z^2 - 0.1849 z - 0.01088
--------------------------------------- z^3 - 0.7272 z^2 + 0.1324 z - 4.73e-008
0
2.5 0.9689 z^2 + 0.06496 z - 0.006565
------------------------------------------ z^3 - 0.02895 z^2 + 0.07988 z - 6.976e-010
8.63
Control of the outlet air humidity (Loop 3):
Conversion to z-transform:
For P-controller, the closed loop transfer function in Laplace domain is:
G(s) =1.198𝑠+11.98
0.1s3+1.2s2+2.1s+12.98
The closed loop transfer function in z-domain is:
𝐺(𝑧) =0.05596𝑧2 + 0.03194𝑧 − 0.01934
𝑧3 − 2.137𝑧2 + 1.512𝑧 − 0.3012
Stability and transient response:
The overall transfer function:
𝐺(𝑧) =0.05596𝑧2 + 0.03194𝑧 − 0.01934
𝑧3 − 2.137𝑧2 + 1.512𝑧 − 0.3012
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Figure 16: Stability and transient response using MATLAB
From the above figure:
𝜔𝑛 = 0.11 𝜋/𝑇, where T is the sampling time.
For 𝑇 =1
10 𝑠𝑒𝑐, 𝜔𝑛 = 3.456 𝑟𝑎𝑑/𝑠𝑒𝑐
And 𝜉 = 0.12
Step response in digital control system:
The overall transfer function for P-controller:
𝐺(𝑧) =0.05596𝑧2 + 0.03194𝑧 − 0.01934
𝑧3 − 2.137𝑧2 + 1.512𝑧 − 0.3012
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Tuning:
Table 5: Comparison between the methods of tuning used for the discrete system
The method Ku
Root Locus 1.88
Bode 1.87
Average 1.875
The two methods are in agreement
Effect of sampling time on overall T.F in z-domain and on system response:
The overall T.F for the system at different sampling time were determined as following:
Table 6: Overall T.F's and (OV) at different sampling time
Sampling
Time(sec)
Overall T.F's G(z) Overshoot(%)
0.01 0.000595 z^2 + 5.267e-005 z - 0.0005348
--------------------------------------- z^3 - 2.885 z^2 + 2.772 z - 0.8869
68.5
0.1 0.05596 z^2 + 0.03194 z - 0.01934
---------------------------------- z^3 - 2.137 z^2 + 1.512 z - 0.3012
68.4
1 1.531 z^2 + 0.9722 z - 0.004256
--------------------------------------- z^3 + 1.276 z^2 + 0.4319 z - 6.144e-006
65.9
1.5 0.7874 z^2 + 0.05001 z - 0.002788 23.1
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---------------------------------------- z^3 - 0.3796 z^2 + 0.2838 z - 1.523e-008
2 0.5499 z^2 - 0.158 z - 0.001832
--------------------------------------- z^3 - 0.764 z^2 + 0.1865 z - 3.775e-011
1.06
2.5 1.085 z^2 + 0.3207 z - 0.001204
---------------------------------------- z^3 + 0.3994 z^2 + 0.1226 z - 9.357e-014
17.6
Conclusions
The stability and tuning are different giving different
parameters, the root locus and bode plots are also different
with different parameters and stability limits. It may be
concluded that the digital controller (PLC) itself tune to
stable perform.
Recommendations
There for it is recommended that continuous control
system should be replaced by discrete control system.
Acknowledgement
The authors wish to thank the graduate college of the
Karary University for Help and registration of this work
for PhD in chemical engineering.
References
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"Control of Industrial Dryers", Handbook of Industrial
Drying, 2nded, (A.S.Mujumdar, ed.), Marcel Dekker,
New York, (1995), pp. 1343-1368. [4] Strumillo, C., Jones, P., and Zulla, R., "Energy
Aspects in Drying", Handbook of Industrial Drying,
2nded, (A.S.Mujumdar, ed.), Marcel Dekker, New
York, (1995), pp.1343-1368.