Thư viện tri thức trực tuyến
Kho tài liệu với 50,000+ tài liệu học thuật
© 2023 Siêu thị PDF - Kho tài liệu học thuật hàng đầu Việt Nam
Tài liệu Process Dynamics, Operations, and Control pptx
Nội dung xem thử
Mô tả chi tiết
Spring 2006 Process Dynamics, Operations, and Control 10.450
Lesson 1: Processes and Systems
1.0 context and direction
Process control is an application area of chemical engineering - an
identifiable specialty for the ChE. It combines chemical process
knowledge (how physics, chemistry, and biology work in operating
equipment) and an understanding of dynamic systems, a topic important to
many fields of engineering. Thus study of process control allows
chemical engineers to span their own field, as well as form a useful
acquaintance with allied fields. Practitioners of process control find their
skills useful in design, operation, and troubleshooting - major categories of
chemical engineering practice.
Process control, like any coherent topic, is an integrated body of
knowledge - it hangs together on a multidimensional framework, and
practitioners draw from many parts of the framework in doing their work.
Yet in learning, we must receive information in sequence - following a
path through multidimensional space. It is like entering a large building
with unlighted rooms, holding a dim flashlight and clutching a vague map
that omits some of the stairways and passages. How best to learn one’s
way around?
In these lessons we will attempt to move through a significant portion of
the structure - say, half a textbook - in about two weeks. Then we will
repeat the journey several times, each time inspecting the rooms more
thoroughly. By this means we hope to gain, from the start, a sense of
doing an entire process control job, as well as approach each new topic in
the context of a familiar path.
1.1 the job we will do, over and over
We encounter a process, learn how it behaves, specify how we wish to
control it, choose appropriate equipment, and then explore the behavior
under control to see if we have improved things.
1.2 introducing a simple process
A large tank must be filled with liquid from a supply line. One operator
stands at ground level to operate the feed valve. Another stands on the
tank, gauging its level with a dipstick. When the tank is near full, the stick
operator will instruct the other to start closing the valve. Overfilling can
cause spills, but underfilling will cause later process problems.
revised 2006 Jan 30 1
Spring 2006 Process Dynamics, Operations, and Control 10.450
Lesson 1: Processes and Systems
To learn how the process works, we write an overall material balance on
the tank.
FV i dt
d
ρ=ρ (1.2-1)
The tank volume V can be expressed in terms of the liquid level h. The
inlet volumetric flow rate Fi may vary with time due to supply pressure
fluctuations and valve manipulations by the operator. The liquid density
depends on the temperature, but will usually not vary significantly with
time during the course of filling. Thus (1.2-1) becomes
)t(F dt
dh A = i (1.2-2)
We integrate (1.2-2) to find the liquid level as a function of time.
∫ += t
0 i dt)t(F A
1 )0(hh (1.2-3)
1.3 planning a control scheme
Clearly the liquid level h is important, and we will call it the controlled
variable. Our control objective is to bring h quickly to its target value hr
and not exceed it. (To be realistic, we would specify allowable limits ± δh
on hr.) We will call the volumetric flow Fi the manipulated variable,
because we adjust it to achieve our objective for the controlled variable.
The existing control scheme is to measure the controlled variable via
dipstick, decide when the controlled variable is near target, and instruct
revised 2006 Jan 30 2
Spring 2006 Process Dynamics, Operations, and Control 10.450
Lesson 1: Processes and Systems
the valve operator to change the manipulated variable. The scheme suffers
from
• delay in measurement. Overfilling can occur if the stick operator
cannot complete the measurement in time.
• performance variations. Both stick and valve operators may vary in
attentiveness and speed of execution.
• resources required. There are better uses for operating personnel.
• unsafe conditions. There is too much potential for chemical exposure.
A new scheme is proposed: put a timer on the valve. Calculate the time
required for filling from (1.2-3). Close the valve when time has expired.
The timing scheme would no longer require an operator to be on the tank
top, and with a motor-driven valve actuator the entire operation could be
directed from a control room. These are indeed improvements. However,
the timing scheme abandons a crucial virtue of the existing scheme: by
measuring the controlled variable, the operators can react to unexpected
disturbances, such as changes in the filling rate. Using knowledge of the
controlled variable to motivate changes to the manipulated variable is a
fundamental control structure, known as feedback control. The proposed
timing scheme has no feedback mechanism, and thus cannot accommodate
changes to h(0) and Fi(t) in (1.2-3).
An alternative is to build on the feedback already inherent in the twooperator scheme, but to improve its operation. We propose an automatic
controller that behaves according to the following controller algorithm:
near i max
r
near i max
r near
hh F F
h h hh FF
h h
< =
− > = −
(1.3-1)
Algorithm (1.3-1) is an idealization of what the operators are already
doing: filling occurs at maximum flow until the level reaches a value hnear.
Beyond this point, the flow decreases linearly, reaching zero when h
reaches the target hr. The setting of hnear may be adjusted to tune the
control performance.
1.4 choosing equipment
We need a sensor to replace the dipstick, a valve actuator to replace the
valve operator, and a controller mechanism to replace the stick operator.
We imagine a buoyant object floating on the liquid surface. The float is
linked to a lever that drives the valve stem. When the liquid level is low,
the float rests above it on a structure so that the valve is fully open.
revised 2006 Jan 30 3
Spring 2006 Process Dynamics, Operations, and Control 10.450
Lesson 1: Processes and Systems
1.5 process behavior under automatic control
Typically these things work quite well. We predict its performance by
combining our process model (1.2-2) with the controller algorithm (1.3-1),
which eliminates the manipulated variable between the equations. We
take the simple case in which Fmax does not vary during filling due to
pressure fluctuations, etc. For h less than hnear,
t
A
F )0(hh
known)0(hF dt
dh A
max
max
+=
= =
(1.5-1)
Equation (1.5-1) can be used to calculate tnear, the time at which h reaches
hnear. For h greater than hnear,
( ) ( )
r
max near near
r near
r near
r r near
fill r near
dh h h A F h(t ) h dt h h
h (t t ) h h h h exp thh
− = = −
⎡ − − =− − ⎢ ⎥ − ⎣ ⎦
⎤
(1.5-2)
revised 2006 Jan 30 4
Content removed due to copyright restrictions.
(To see a cut-away diagram of a toilet, go to
http://www.toiletology.com/lg-views.shtml#cutaway2x)
Spring 2006 Process Dynamics, Operations, and Control 10.450
Lesson 1: Processes and Systems
where the parameter tfill is the time required for the level to reach hr at
flow Fmax, starting from an empty tank.
r
fill
max
Ah
t
F = (1.5-3)
The plot shows the filling profile from h(0) = 0.10hr with several values of
hnear/hr. Certainly the filling goes faster if the flow can go instantaneously
from Fmax to zero at hr; however this will not be practical, so that hnear will
be less than hr.
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1 1.2 1.4
t/tfill
h/hr
hnear/hr = 0.95
0.75
0.50
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1 1.2 1.4
t/tfill
h/hr
hnear/hr = 0.95
0.75
0.50
1.6 defining ‘system’
In Section 1.2, we introduced a process - a tank with feed piping - whose
inventory varied in time. We thought of the process as a collection of
equipment and other material, marked off by a boundary in space,
communicating with its environment by energy and material streams.
'Process' is a good notion, important to chemical engineers. Another
useful notion is that of 'system'. A system is some collection of equipment
and operations, usually with a boundary, communicating with its
environment by a set of input and output signals. By these definitions, a
process is a type of system, but system is more abstract and general. For
example, the system boundary is often tenuous: suppose that our system
comprises the equipment in the plant and the controller in the central
control room, with radio communication between the two. A physical
boundary would be in two pieces, at least; perhaps we should regard this
revised 2006 Jan 30 5
Spring 2006 Process Dynamics, Operations, and Control 10.450
Lesson 1: Processes and Systems
system boundary as partly physical (around the chemical process) and
partly conceptual (around the controller).
Furthermore, the inputs and outputs of a system need not be material and
energy streams, as they are for a process. System inputs are "things that
cause" or “stimuli”; outputs are "things that are affected" or “responses”.
system inputs
(causes)
outputs
(responses)
To approach the problem of controlling our filling process in Section 1.3,
we thought of it in system terms: the primary output was the liquid level h
-- not a stream, certainly, but an important response variable of the system
-- and inlet stream Fi was an input. And peculiar as it first seems, if the
tank had an outlet flow Fo, it would also be an input signal, because it
influences the liquid level, just as does Fi.
The point of all this is to look at a single schematic and know how to view
it as a process, and as a system. View it as a process (Fo as an outlet
stream) to write the material balance and make fluid mechanics
calculations. View it as a system (Fo as an input) to analyze the dynamic
behavior implied by that material balance and make control calculations.
System dynamics is an engineering science useful to mechanical,
electrical, and chemical engineers, as well as others. This is because
transient behavior, for all the variety of systems in nature and technology,
can be described by a very few elements. To do our job well, we must
understand more about system dynamics -- how systems behave in time.
That is, we must be able to describe how important output variables react
to arbitrary disturbances.
1.7 systems within systems
We call something a system and identify its inputs and outputs as a first
step toward understanding, predicting, and influencing its behavior. In
some cases it may help to determine some of the structure within the
system boundaries; that is, if we identify some component systems. Each
of these, of course, would have inputs and outputs, too.
system
inputs outputs 1
2
revised 2006 Jan 30 6
Spring 2006 Process Dynamics, Operations, and Control 10.450
Lesson 1: Processes and Systems
Considering the relationship of these component systems, we recognize
the existence of intermediate variables within a system. Neither inputs
nor outputs of the main system, they connect the component systems.
Intermediate variables may be useful in understanding and influencing
overall system behavior.
1.8 the system of single-loop feedback control
When we add a controller to a process, we create a single time-varying
system; however, it is useful to keep process and controller conceptually
distinct as component systems. This is because a repertoire of relatively
few control schemes (relationships between process and controller)
suffices for myriad process applications. Using the terms we defined in
Section 1.3, we represent a control scheme called single-loop feedback
control in this fashion:
process
controller
final control
element
sensor
set point
manipulated
variable
other inputs other
outputs
controlled
variable
system
process
controller
final control
element
sensor
set point
manipulated
variable
other inputs other
outputs
controlled
variable
system
Figure 1.8-1 The single-loop feedback control system and its
subsystems
We will see this structure repeatedly. Inside the block called "process" is
the physical process, whatever it might be, and the block is the boundary
we would draw if we were doing an overall material or energy balance.
HOWEVER, we remember that the inputs and outputs are NOT
necessarily the same as the material and energy streams that cross the
process boundary. From among the outputs, we may select a controlled
variable (often a pressure, temperature, flow rate, liquid level, or
composition) and provide a suitable sensor to measure it. From the inputs,
we choose a manipulated variable (often a flow rate) and install an
appropriate final control element (often a valve). The measurement is fed
to the controller, which decides how to adjust the manipulated variable to
keep the controlled variable at the desired condition: the set point. The
revised 2006 Jan 30 7
Spring 2006 Process Dynamics, Operations, and Control 10.450
Lesson 1: Processes and Systems
other inputs are potential disturbances that affect the controlled variable,
and so require action by the controller.
1.9 conclusion
Think of a chemical process as a dynamic system that responds in
particular ways to its inputs. We attach other dynamic systems (sensor,
controller, etc.) to that process in a single-loop feedback structure and
arrive at a new dynamic system that responds in different ways to the
inputs. If we do our job well, it responds in better ways, so to justify all
the trouble.
revised 2006 Jan 30 8
Spring 2006 Process Dynamics, Operations, and Control 10.450
Lesson 2: Mathematics Review
2.0 context and direction
Imagine a system that varies in time; we might plot its output vs. time. A
plot might imply an equation, and the equation is usually an ODE
(ordinary differential equation). Therefore, we will review the math of the
first-order ODE while emphasizing how it can represent a dynamic
system. We examine how the system is affected by its initial condition
and by disturbances, where the disturbances may be non-smooth, multiple,
or delayed.
2.1 first-order, linear, variable-coefficient ODE
The dependent variable y(t) depends on its first derivative and forcing
function x(t). When the independent variable t is t0, y is y0.
y)t(y)t(Kx)t(y 00 dt
dy )t(a =+ = (2.1-1)
In writing (2.1-1) we have arranged a coefficient of +1 for y. Therefore
a(t) must have dimensions of independent variable t, and K has
dimensions of y/x. We solve (2.1-1) by defining the integrating factor p(t)
∫ = )( exp)(
ta
dt
tp (2.1-2)
Notice that p(t) is dimensionless, as is the quotient under the integral. The
solution
∫ = +
t
t
00
0
dt
)t(a
)t(x)t(p
)t(p
K
)t(p
)t(y)t(p y )t( (2.1-3)
comprises contributions from the initial condition y(t0) and the forcing
function Kx(t). These are known as the homogeneous (as if the right-hand
side were zero) and particular (depends on the right-hand side) solutions.
In the language of dynamic systems, we can think of y(t) as the response
of the system to input disturbances Kx(t) and y(t0).
2.2 first-order ODE, special case for process control applications
The independent variable t will represent time. For many process control
applications, a(t) in (2.1-1) will be a positive constant; we call it the time
constant τ.
00 y)t(y)t(Kx)t(y dt
dy =+τ = (2.2-1)
The integrating factor (2.1-2) is
revised 2005 Jan 11 1
Spring 2006 Process Dynamics, Operations, and Control 10.450
Lesson 2: Mathematics Review
τ = τ = ∫ t
e
dt exp)t(p (2.2-2)
and the solution (2.1-3) becomes
( )
dt)t(xee K ey)t(y
t
t
tt tt
0
0
0
∫ ττ − τ
−−
τ = + (2.2-3)
The initial condition affects the system response from the beginning, but
its effect decays to zero according to the magnitude of the time constant -
larger time constants represent slower decay. If not further disturbed by
some x(t), the first order system reaches equilibrium at zero.
However, most practical systems are disturbed. K is a property of the
system, called the gain. By its magnitude and sign, the gain influences
how strongly y responds to x. The form of the response depends on the
nature of the disturbance.
Example: suppose x is a unit step function at time t1. Before we proceed
formally, let us think intuitively. From (2.2-3) we expect the response y to
decay toward zero from IC y0. At time t1, the system will respond to being
hit with a step disturbance. After a long time, there will be no memory of
the initial condition, and the system will respond only to the disturbance
input. Because this is constant after the step, we guess that the response
will also become constant.
Now the math: from (2.2-3)
( )
( ) ( )
⎟
⎠
⎞ ⎜
⎝
⎛ = −−+
− τ = +
τ
−−
τ
−−
ττ − τ
−−
∫
0 1
0
0
tt
1
tt
0
t
t
1
tt tt
0
ey e1)tt(KU
dt)tt(Uee K ey)t(y
(2.2-4)
Figure 2.2-1 shows the solution. Notice that the particular solution makes
no contribution before time t1. The initial condition decays, and with no
disturbance would continue to zero. At t1, however, the system responds
to the step disturbance, approaching constant value K as time becomes
large. This immediate response, followed by asymptotic approach to the
new steady state, is characteristic of first-order systems. Because the
response does not track the step input faithfully, the response is said to lag
behind the input; the first-order system is sometimes called a first-order
lag.
revised 2005 Jan 11 2
Spring 2006 Process Dynamics, Operations, and Control 10.450
Lesson 2: Mathematics Review
0
0.5
1
012345
disturbance
6
0
0.5
1
1.5
2
2.5
0123456
time
response
t0 t1
y0
K
0
0.5
1
012345
disturbance
6
0
0.5
1
1.5
2
2.5
0123456
time
response
t0 t1
y0
K
Figure 2.2-1 first-order response to initial condition and step
disturbance
2.3 piecewise integration of non-smooth disturbances
The solution (2.2-3) is applied over succeeding time intervals, each
featuring an initial condition (from the preceding interval) and disturbance
input.
( )
( )
⎪
⎪
⎪
⎪
⎩
⎪
⎪
⎪
⎪
⎨
⎧
<<
τ
+
<<
τ
+
= ∫
∫
ττ − τ
−−
ττ − τ
−−
.etc
tttdt)t(xee K e)t(y
tttdt)t(xee K e)t(y
)t(y 1 2
t
t
tt tt
1
0 1
t
t
tt tt
0
1
1
0
0
(2.3-1)
Example: suppose
revised 2005 Jan 11 3
Spring 2006 Process Dynamics, Operations, and Control 10.450
Lesson 2: Mathematics Review
( ) ⎪
⎩
⎪
⎨
⎧
<
<<−
<<
=
=+ =
t20
2t11t2
1t00
x
0)0(yxy dt
dy
(2.3-2)
In this problem, variables t, x, and y should be presumed to have
appropriate, if unstated, units; in these units, both gain and time constant
are of magnitude 1. From (2.3-1),
( ) ( ) ⎪ ( ) ⎩
⎪
⎨
⎧
<
+− <<
<<
=
−−−
−−
ee2 t2
2t1e2t2
0 1t0
)t(y
2t1
1t (2.3-3)
With a zero initial condition and no disturbance, the system remains at
equilibrium until the ramp disturbance begins at t = 1. Then the output
immediately rises in response, lagging behind the linear ramp. At t = 2,
the disturbance ceases, and the output decays back toward equilibrium.
revised 2005 Jan 11 4
Spring 2006 Process Dynamics, Operations, and Control 10.450
Lesson 2: Mathematics Review
0
0.5
1
1.5
2
2.5
012345
disturbance
6
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0123456
time
response
0
0.5
1
1.5
2
2.5
012345
disturbance
6
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0123456
time
response
2.4 multiple disturbances and superimposition
Systems can have more than one input. Consider a first-order system with
two disturbance functions.
11 22 00 y)t(y)t(xK)t(xK)t(y dt
dy +=+τ = (2.4-1)
Applying (2.2-3) and distributing the integral across the disturbances, we
find that the effects of the disturbances on y are additive.
( )
dt)t(xee K dt)t(xee K ey)t(y 2
t
t
tt 2
1
t
t
tt 1 tt
0
0 0
0
∫ ∫ ττ − ττ − τ
−−
τ
+
τ = + (2.4-2)
This additive behavior is a happy characteristic of linear systems. Thus
another way to view problem (2.4-1) is to decompose it into component
problems. That is, define
21H ++= yyyy (2.4-3)
revised 2005 Jan 11 5
Spring 2006 Process Dynamics, Operations, and Control 10.450
Lesson 2: Mathematics Review
and write (2.4-1) in three equations. We put the initial condition with no
disturbances, and each disturbance with a zero initial condition.
0)t(y)t(xK)t(y dt
dy
0)t(y)t(xK)t(y dt
dy
0)t(y y)t(y dt
dy
2 22 02
2
1 11 01
1
H 00H
H
=+τ =
=+τ =
=+τ =
(2.4-4)
Equations and initial conditions (2.4-4) can be summed to recover the
original problem specification (2.4-1). The solutions are
( )
dt)t(xee K )t(y
dt)t(xee K )t(y
ey)t(y
2
t
t
tt 2
2
1
t
t
tt 1
1
tt
H 0
0
0
0
∫
∫
ττ −
ττ −
τ
−−
τ =
τ =
=
(2.4-5)
and of course these solutions can be added to recover original solution
(2.4-2). Thus we can view the problem of multiple disturbances as a
system responding to the sum of the disturbances, or as the sum of
responses from several identical systems, each responding to a single
disturbance.
Example: consider
2)0(y)3t(U)1t(U4
3
y
4
1
dt
dy
4
1
−−−=+ = (2.4-6)
We first place the equation in standard form, in which the coefficient of y
is +1.
2)0(y)3t(U4)1t(U3y dt
dy −−−=+ = (2.4-7)
Equation (2.4-7) shows us that the time constant is 1, and that the system
responds to the first disturbance with a gain of 3, and to the second with a
gain of -4. The solution is
( ) ( ) t )1t( )3t( e1)3t(U4e1)1t(U3e2y − −− −− −−−−−+= (2.4-8)
revised 2005 Jan 11 6
Spring 2006 Process Dynamics, Operations, and Control 10.450
Lesson 2: Mathematics Review
In Figure 2.4-1, the individual solution components are plotted as solid
traces; their sum, which is the system response, is a dashed trace. Notice
how the first-order lag responds to each new disturbance as it occurs.
0
0.5
1
0246
disturbances
8
-5
-4
-3
-2
-1
0
1
2
3
4
0246
time
response
8
0
0.5
1
0246
disturbances
8
-5
-4
-3
-2
-1
0
1
2
3
4
0246
time
response
8
)
Figure 2.4-1 first-order response to multiple disturbances
Writing the step functions explicitly in solution (2.4-8) emphasizes that
particular disturbances do not influence the solution until the time of their
occurrence. For example, if they were omitted, some deceptively correct
but inappropriate rearrangement would lead to errors.
( )(
)3t()1t(t
t )1t( )3t(
t )1t( )3t(
e4e3e21
e44e33e2
e14e13e2y
−−− −−
− −− −−
− −− −−
+−+−=
+−−+=
−−−+=
(do not do this!) (2.4-9)
This notation at least implies that two of the exponential functions have
delayed onsets. However, further correct-but-inappropriate rearrangement
makes things even worse.
revised 2005 Jan 11 7