PROCESS

AUTOMATION
Process Dynamics
Basics of theoretical
modelling
BASICS IN PROCESS AUTOMATION

Learning outcomes?
BASICS IN PROCESS AUTOMATION

 Why we automate processes? What are the benefits?

 Basic principle for plantwide automation design
 Automation hierarchy
 Technical implementation of the plantwide automation
 Automation systems
 networks
 How is the automation system operating?
 Distributed automation system
 Signal flow from the field to the operator room and back
BASICS IN PROCESS AUTOMATION
 The automation system operates the process via
controllers. How is the controller operating ?
 There are different types of controllers and they
have different parameters. How to select the
right type of the controller?
 The controller type is decided based on the
process characteristics = Process dynamics
 How we know the process characteristics?
 The dynamics defines the process
characteristics. What kind of modelling?
BASICS IN PROCESS AUTOMATION
 State space model or diff equations or s-
space transfer function model?
 Process model+controller. How we know we have
the controller?
 Stability analysis
 A CONTROLLED PROCESS

disturbances

State variables

Input variables Output variables

In process dynamics a unit process ( fex reactor) is considered as a system.

The state of the system is described by state variables e g temperature, pressure

A CONTROLLED PROCESS
 The input variables of a system describe the
states of the environment; change in these causes
a change in the state of the system.
 There are two kinds of input variables:
 Control variables and disturbances variables
 Relationships between input, state and output
variables of the system are examined by
mathematical models
1. PROCESS CHARACTERISTICS
Process Gain
Process order
Dead time
Forcing functions
Stable Process
1.1 PROCESS GAIN K
 Ratio between the steady state response
difference in an output variable delta y to
an input delta u
 F. Ex how much we have to increase the
heat to the reactor that its temperature
increases by 2 degrees
u y
PROCESS

y
Process Gain K
u
 1.2 PROCESS ORDER AND TIME
CONSTANT

 Changes in the state variables due to the

input changes are taking place with their
characteristic speeds and are linked to so
called accumulation phenomena.
 In the tank liquid will be accumulated
depending on the ratio of input/output flows
 Material and energy are usually considered as
accumulating quantity
 1.2 PROCESS ORDER AND TIME
CONSTANT

 One capacity system has one material

or energy storage
 Two capacity system has two

 n. capacity n material- and energy

storages
 Zero capacity system has no
storaging characteristic
A ) 0. order process b) 1. order process and c) 2. order process
 1.2 PROCESS ORDER AND TIME
CONSTANT

 Accumulating phenomena belongs to the

process dynamic behavior and it is system
specific.
 Process time constant describes the
speed, by which the material- and
energy storages are changing, when
there are changes in input variables.
 The bigger the time constant, the slower the
process.

Video aikavakiosta
1.3 DELAY OR DEAD TIME
 time between the response and the
change in the input variable

a) Process response b) process response + dead time

1.4 STABLE PROCESS

 Process is stable, if it will settle down to the

specific value

a) Stable b) unstable c) oscillating system

2. BASIC ELEMENTS OF DYNAMIC ANALYSIS

 Processdynamics concerns studies on

the dynamic behavior of processes in
response to different types of inputs

Part
II:
Proc
ess
Dyn
amic
s
2.1 TESTS WITH THE REAL PROCESS AND
THE RESPONSES

 Study on process dynamics by forcing

functions

 Forcing functions:
 Step function
 Ramp function
 Sine function
 impulsefunction
 step ramp

impulse

sine
 TESTS WITH THE REAL PROCESS AND
THEIR RESPONSES

 It is not always possible to perform the

tests in the real industrial processes
 Disturbs the process
 expensive

 we have to take the mathematical tools in

use and imitate ( simulate) the process.
2.2 TOOLS OF DYNAMIC ANALYSIS
 Therefore dynamical analysis is done using
a mathematical model of the process
 We also need to have the inputs well
defined
 Typically we work with differential
equations (ODE, PDE, linear, nonlinear)
 One important tool: Laplace transform

 The Laplace transform leads to models in

transfer function form Part
II:
Proc
ess
Dyn
amic
s
Modelling (fysical-chemical)

L-transform
Laplace-
Transfer
Time space L-1 -transform space function
s-space
s = j 
F-transform

Part II: Process Dynamics

Frequency
F-1-transform
Response
Fourier-space
(f-space)

State Space Discretization Discrete-time z = eTS

x = Ax + Bu responses

Z-transform Z-space
2.3 CHEMICAL AND PHYSICAL BASIS OF
MODELLING

 Mass balance (continuity equation)

 Total continuity equation
 Component balance
 Energy equation
 Impulse balance (equation of motion)
 Transport equations
 Equations of state
 Equilibrium
 Chemical equilibrium
 Phase equilibrium
 Kinetics
 WHEN MODELLING
… WRITE DOWN SYSTEMATICALLY
 Picture/figure and variables
 Definition of process
 Choose variables
 A sketch helps
 Assumptions
 Simplifications
 Area of qualification for model
 Equations of model
 Principle of constancy:
 MASS, ENERGY, IMPULSE
 Other dependencies
 WHEN MODELLING
… WRITE DOWN SYSTEMATICALLY
 Degrees of freedom vs. equations of model
 Is model solvable or is there some equations missing?
 Solution of model/Usage of model
2.3.1 A) TOTAL CONTINUITY EQUATION
(MASS BALANCE)

Time rate of
Mass flow _ Mass flow
change of mass =
into system out of system
inside system
 EXAMPLE 1

F0(t)
0(t)

V(t) F(t)
(t) (t)
 EXAMPLE 1
PROCESS DESCRIPTION

 The tank of perfectly mixed liquid into which flows a

stream F0.
 The volumetric hold-up of liquid in the tank is V and
its density .
 The volumetric flow rate from the tank is F and
density .

 Task: Write continuity equation for liquid phase in the

tank.
 EXAMPLE 1
ASSUMPTIONS
 Liquid is perfectly mixed

Density is the same everywhere in the tank.

Macroscopic system description

Only one independent variable t

 EXAMPLE 1
EQUATIONS OF MODEL
 Mass balance (continuity equation)

d ( V )
 F0  0  F
dt
2.3.2 B) COMPONENT CONTINUITY
EQUATIONS

Time rate of change of Flow of moles of

moles of jth component = jth component __
inside system into system

Flow of moles of Rate of formation of

jth component + moles of jth component
out of system from chemical reactions
 EXAMPLE 2

F0 , 0

CA0,CB0

F, , CA, CB
V, , CA, CB
 EXAMPLE 2
PROCESS DESCRIPTION
 Tank of perfectly mixed liquid where a simple
first-order isothermal reaction takes place
k
 A--->B
 Concentration of component A in the inflowing
feed stream is CA0.
 Inflow stream is F0
 Other parameters are in the picture above.

 Task: Write component continuity equations and

total mass balance
 EXAMPLE 2
ASSUMPTIONS
 First-order reaction, the rate of consumption of
reactant A per unit volume is directly
proportional to the concentration of A in the
tank.
 -VkCA
 EXAMPLE 2
EQUATIONS OF MODEL
 Time rate of change of A inside tank:

d (VC A )
 F0C A0  FC A  VkCA
dt

 Time rate of change of B inside tank:

d (VC B )
 F0CB 0  FC B  VkCA
dt
 EXAMPLE 3

F0, 0, CA0,CB0,CC0

V, ,
CA, CB, CC F, , CA, CB, CC
 EXAMPLE 3
PROCESS DESCRIPTION

 The macroscopic system is the same as above

except that consecutive reactions occur
 A---> B ------>C
k1 k2

 Task: Write component continuity equations and

mass balance
 EXAMPLE 3
ASSUMPTIONS

 Ideally mixed reactor

--> macroscopic system
 EXAMPLE 3
EQUATIONS OF MODEL

d (VC A )
dt  F0C A0  FC A  Vk1C A
d (VC B )
 F0C B 0  FC B  Vk1C A  Vk 2C B
dt
d (VCc )
 F0CC 0  FCC  Vk 2C B
dt
If only two component balances are used in the solution, as a third
equation can be used: C

M
j A
j Cj  
 2.3.3 Energy Balance

Time rate of change Flow of energy into

of total energy = system by convection __
inside system or diffusion

Flow of energy out of Heat added to system

__
system by convection + by conduction,
or diffusion radiation, and reaction

Work done by system

Total energy=internal energy+

kinetic energy+potential energy
 Example 4

F0,CA0,0,T0

F,CA,,T F,CA,,T
-Q
 EXAMPLE 4
PROCESS DESCRIPTION
 Stirred-tank, that removes the heat.
 In the tank a simple first-order isothermal
reaction takes place, heat of reaction 
 Let
 U internal energy /m.y.
 K kinetic energy /m.y.
  potential energy/ m.y.
 W shaft work done by system
 P pressure of system
 P0 pressure of feed stream
 EXAMPLE 4
ASSUMPTIONS
 Usually shaft work done by system is zero, W=0
 Kinetic energy almost 0, if the inlet and outlet flow
velocities are not very high.
 If the elevations of the inlet and outlet flows are
about the same, = 0.
 Enthalpy is the product of temperature and
average heat capacity.
 Density of liquid is constant.
 EXAMPLE 4
EQUATIONS OF MODEL

d
(U  K  )V   F0 0 (U 0  K 0   0 ) 
dt
F (U  K  )  (QG  Q)  (W  FP  F0 P0 )

Simplifying:

d
(U  K  )V   F0 0 (U 0  K 0   0 ) 
dt
F (U  K  )  (QG  Q)  (W  FP  F0 P0 )
 EXAMPLE 4
EQUATIONS OF MODEL

d ( VU ) P P0
 F0  0U 0  FU  QG  Q  F  F0  0
dt  0
 F0  0 (U 0  P0V0 )  F (U  PV )  QG  Q

V is the specific volume, the reciprocal of the density

 EXAMPLE 4
EQUATIONS OF MODEL

Enthalpy H  U  pV ( gas )
h  U  pV (liquid )
d ( VU )
 F0  0 h0  Fh  Q  VkCA
dt

pV  U
d ( Vh )
 F0  0 h0  Fh  Q  VkCA
dt
 EXAMPLE 4
EQUATIONS OF MODEL
 Enthalpies are functions of composition,
temperature and pressure.
 Assume, that enthalpy of liquid is mainly the
product of absolute temperature and mean heat
capacity
 h=CpT

 Assume the densities of liquids to be constant

 EXAMPLE 4
EQUATIONS OF MODEL

d (VT )
CP  C p ( F0T0  FT )  Q  VkCA
dt
h=CpT
Density constant
t
i
h
DYNAMIC SYSTEMS ANALYSIS USING
THE LAPLACE TRANFORM
 Tranfer function model
L-transform
Laplace-space
Time space
s-space
L-1-transform
Physical
chemical
model
Easier to
Diff. Eq
analysis
solve
models
 Dynamic systems analysis via Laplace-
transforms
 Laplace-transformas and transfer
function
 Transfer function characteristics

(poles and zeros)

 Dynamic behavior of the 1. and 2. order
processes
LAPLACE-TRANSFORMS AND TRANSFER
FUNCTION

 Laplace-transformis used as a tool in

analysis of dynamic systems
 Forcing functions vs responses
 Lin diff equations can be transformed to
the algebraic functions and their solving
is easy
 In Laplace-space we can study quickly the
filtering and stability properties of the
systems
 Tranfer function model
L-transform
Laplace-space
Time space
s-space
L-1-transform
Physical
chemical
model
Easier to
Diff. Eq
analysis
solve
models
DEFINITION

 Laplace transform from the funktion f(t) is

marked L{f(t)} or F(s)
 Transform is defined integral from function f(t)
multiplied by the term e-st


Lf (t )   F ( s)   e  st f (t ) dt
0

f (t )  0, when t  0
FROM DIFF EQUATION TO THE TRANSFER
FUNCTION AND BLOCK DIAGRAMS

 It
is assumed that the diff equation of the
process is as follows

dy (t )
T  y (t )  Kx(t )
dt
where x(t) on input or manipulated variable
y(t) on output variable
T ja K are constants
T = time constant
K = gain
Laplace-transformation with the following initial
values t = 0, y(0) = 0 and x(0)= 0

=> s T Y (s) + Y (s) = K X (s)

And the ratio (output/input)

Y ( s) K
  G( s)
X ( s ) 1  Ts
G(s) is the transfer function of the process. (
The most common form of the G(s)
im

 b i si
bm s ...b1s b 0
m 1
G(s)  i0
in  an sn ... a1s1  a 0
 a i si
i0
G(s) is the ration of two polynomials, in which the
numerator order m is smaller or equal than
denominator order n.

Numerator zeros are called zeros and denominator

zeros are called poles.
 RATIONAL TRANSFER FUNCTION G(S)
CHARACTERISTICS

 System stability can be defined from the so called

characteristic function, by equating the
denominator of the transfer function to the value
zero.
 If all the roots of the denominator ( system poles)
have negative real parts, the system is
asymptotically stable.
In the pole-zero map the poles and zeros of the system
are presented graphically in the complex plane
(imaginary axel vs. real axel)

Im

i
Re
-3 -2 -1

-i
pole-zero-map
THE EFFECTS OF THE POLES AND ZEROS
TO THE SYSTEM DYNAMIC BEHAVIOR

 The system is asymptotically stable, if the poles

of the transfer function are in the lefthand side
of the complex plane (negative)
 System is unstable, if one (or more) pole is in the
righthand (positive) plane
THE EFFECTS OF THE POLES AND ZEROS
TO THE SYSTEM DYNAMIC BEHAVIOR

 The system is stable if the poles of the

transfer function are in the lefthand side
plane and the poles in the imaginary axel
are single.
THE EFFECTS OF THE POLES AND ZEROS TO
THE SYSTEM DYNAMIC BEHAVIOR

 The system response does not oscillate , if

the poles are real
 The system response oscillates, if one of
the poles is complex
THE EFFECTS OF THE POLES AND ZEROS
TO THE SYSTEM DYNAMIC BEHAVIOR

 The system is a minimum phase, if all its

zeros are in the negative half-plane and it
soesn not have dead time.
THE EFFECTS OF THE POLES AND ZEROS
TO THE SYSTEM DYNAMIC BEHAVIOR

 The further a system’s poles are from the

origin the faster the system.
THE EFFECTS OF THE POLES AND ZEROS
TO THE SYSTEM DYNAMIC BEHAVIOR

 Poles define the most important

characteristics of he system-stability,
oscillations, rate.
 Zeros are affecting a system’s initial
trajectory and their effect on the
response is comparable to one of
eigenvalues.
GENERAL TRANSFER FUNCTIONS

 1. order dynamics:
 Time space: dy(t)/dt+y(t)=Ku(t)

 Laplace-space:

K
GI ( s ) 
s  1
K = process static gain
 = process time constant
First order process
never oscillates.
1) τ < 0 --> process unstable
2) τ > 0 --> process is asymptotic stable

Im

X Re
-1/
2.ORDER DYNAMICS

Time space:
d 2 y (t ) dy (t )
 2 0   0 y (t )  K 0 u(t )
2 2

dt dt
Laplace-space:
K 2
G( s)  2 0
s  2 0 s   02
missä: K = process gain
ω0 = process frequency
 process damping coefficient
Kun :ω0 >0
1) <0 process unstable
>0 process stable

2) ll<1 process oscillates

ll>1 process does not oscillates
STABLE PROCESSES

= 0 process is a harmonic oscillator

0<<1 process is under-damped

=1 process is critically damped

>1 process in over-damped.