Transcript Chapter 3

Chapter 3
Dynamic Modeling
Overall Course Objectives
• Develop the skills necessary to function as an
industrial process control engineer.
– Skills
•
•
•
•
Tuning loops
Control loop design
Control loop troubleshooting
Command of the terminology
– Fundamental understanding
• Process dynamics
• Feedback control
Process Dynamics
• Chemical Engineering courses are generally
taught from a steady-state point-of-view.
• Dynamics is the time varying behavior of
processes.
• Chemical processes are dynamically
changing continuously.
• Steady-state change indicates where the
process is going and the dynamic
characteristics of a system indicates what
dynamic path it will take.
Uses of Dynamic Process Models
• Evaluation of process control configurations
– For analysis of difficult control systems for
both existing facilities and new projects
• Process design of batch processes
• Operator Training
• Start-up/shut-down strategy development
Classification of Models
• Lumped parameter models- assume that
the dependent variable does not change with
spatial location within the process, e.g., a
perfectly well mixed vessel.
• Distributed parameter models- consider
that the dependent variable changes with
spatial location within the process.
Example of a Lumped Parameter
Process
FC
(F1)spec
FT
F1
F2
T1
T2
TT
T
Example of a Distributed
Parameter Process
S te am
PC
PT
Toutlet TT
Fe e d
C on de n sate
Modeling Approaches
• Lumped parameter processes- Macroscopic
balances are typically applied for
conservation of mass, moles, or energy and
result in ODE’s.
• Distributed parameter processesMicroscopic balances are typically applied
yielding differential equations for
conservation of mass, moles, or energy for a
single point in the process which result in
PDE’s.
Conservation Equations:
Mass, Moles, or Energy Balances
Rateof
 RateEntering
Accumulation   theSystem  

 

RateLeaving Rateof Generationby

 theSystem    Reaction within theSystem

 

Mass Balance Equation
Rate of mass Rate of mass
Rateof accumulation  
 _ leaving the 

ent
ering
the
of mass in the syst em 
 


 syst em
 syst em


Accumulation Term
dm
d ( V )
Total mass balance :
or
dt
dt
dmi
d ( xi m)
Component mass balance:
or
dt
dt
Other Terms in Mass Balance Eq.
Mass entering or leaving the system:
xi F (component balance) or
F (overall mass balance)
Mole Balance Equation
Rate of moles Rate of moles
Rate of accumulation  
 _ leaving the 

entering
the
of moles in the system 
 


 system
 system


Rate of generation  Rate of consumption 
 



of
moles
by
reaction
of
moles
by
reaction

 

Accumulation Term
Component mole balance:
dni
d (VCi )
or
dt
dt
Other Terms in Mass Balance Eq.
Moles of component i entering
or leaving the system:
xi N (based on molar flow rate, N ) or
Ci FV (based on volumetric flow rate (FV )
Generation or consumption of
component i by reaction :
Vri  V  i r
Thermal Energy Balance
Equation
Rate of convective Rate of convective
Rate of accumulation 
  heat transfer


heat
transfer
of thermalenergy  
 


 enteringthe system leaving the system

 

 Net rate of
  Net rate of heat transfer
 heat generation   through the boundaries 
by reaction  of the system

Accumulation Term
Energy Balance Equation :
dT
MCv
dt
Other Terms in Energy Balance Eq.
Convective Heat Transfer :
FC p (Ts  Tref )
Heat Generation by reaction :
Vri H rxn (T )
Constitutive Relationships
• Usually in the form of algebraic equations.
• Used with the balance equations to model
chemical engineering processes.
• Examples include:
–
–
–
–
Reaction kinetic expressions
Equations of state
Heat transfer correlation functions
Vapor/liquid equilibrium relationships
Degree of Freedom Analysis
N f  Nv  Ne
• The number of degrees of freedom (DOF) is equal
to the number of unknowns minus the number of
equations.
• When DOF is zero, the equations are exactly
specified.
• When DOF is negative, the system is
overspecified.
• When DOF is positive, it is underspecified.
Different Types of Modeling
Terms
• Dependent variables are calculated from
the solution of the model equations.
• Independent variables require
specification by the user or by an
optimization algorithm and represent extra
degrees of freedom.
• Parameters, such as densities or rate
constants, are constants used in the model
equations.
Dynamic Models of Control
Systems
• Control systems affect the process through the
actuator system which has its own dynamics.
• The process responds dynamically to the
change in the manipulated variable.
• The response of the process is measured by
sensor system which has its own dynamics.
• There are many control systems for which the
dynamics of the actuator and sensor systems
are important.
Dynamic Modeling Approach for
Process Control Systems
c
y
u
Actuator
Process
Sensor
ys
Dynamic Model for Actuators




dF
1

Fspec  F
dt  v
dQ
1

Qspec  Q
dt  H
• These equations
assume that the
actuator behaves as a
first order process.
• The dynamic behavior
of the actuator is
described by the time
constant since the gain
is unity
Heat addition as a Manipulated
Variable
• Consider a steam heated reboiler as an example.
• A flow control loop makes an increase in the
flow rate of steam to the reboiler.
• The temperature of the metal tubes in the
reboiler increases in a lagged manner.
• The flow rate of vapor leaving the reboiler
begins to increase.
• The entire process is lumped together into one
first order dynamic model.
Dynamic Response of an
Actuator (First Order System)
Fspec
F
0
2
4
6
Time (seconds)
8
10
Dynamic Model for Sensors
dTs
1
T  Ts 

dt
 Ts
dLs
1
L  Ls 

dt
 Ls
• These equations
assume that the
sensors behave as a
first order system.
• The dynamic behavior
of the sensor is
described by the time
constant since the gain
is unity
• T and L are the actual
temperature and level.
Dynamic Model for an Analyzer
Cs (t )  C(t  A )
• This equation assumes
that the analyzer
behaves as a pure
deadtime element.
• The dynamic behavior
of the sensor is
described by the
analyzer deadtime
since the gain is unity
Dynamic Comparison of the
Actual and Measured
Composition
C(t)
Cmeas (t)
A
Time
Model for Product Composition
for CSTR with a Series Reaction

dF
1

Fspec  F
dt  v
FC
FT
Fe e d
Product

dCA F
Vr
 C A0  C A   Vr k1C A2
dt

AT
Vr
r1
r2
A

B 
C
r1  k1 C A2 r2  k 2 C B
dCB
FCB

 Vr k1 C A2  Vr k 2 CB
dt

CBs (t )  CB (t  A )
Model for Cell Growth in a FedBatch Reactor
Glucose
Feed
Variable
E-1
Speed
Pump
AT
FV  FV , spec
dV
d (Vx)
 FV ;
 Vrx  V max x
dt
dt
d (VS )
V
 FV S0  max x
dt
YxS
xs (t )  x(t   x )
Class Exercise: Dynamic Model
of a Level in a Tank
Fin
LT
L
Fout
dL
 Ac
 Fin  Fout
dt
• Model equation is
based on dynamic
conservation of mass,
i.e., accumulation of
mass in the tank is
equal to the mass flow
rate into the tank
minus the mass flow
rate out.
Class Exercise Solution:
Dynamic Model for Tank Level

dFout
1

Fout , spec  Fout
dt
v

dL
 Ac
 Fin  Fout
dt
Ls  L
• Actuator on flow out
of the tank.
• Process model
• Level sensor since the
level sensor is much
faster than the process
and the actuator
Sensor Noise
• Noise is the variation in a measurement of a
process variable that does not reflect real
changes in the process variable.
• Noise is caused by electrical interference,
mechanical vibrations, or fluctuations
within the process.
• Noise affects the measured value of the
controlled variable; therefore, it should be
included when modeling process dynamics.
Modeling Sensor Noise
• Select standard deviation (s) of noise. s is
equal to 50% of repeatability.
• Generate random number.
• Use random number in a correlation for the
Gaussian distribution which uses s. This
result is the noise on the measurement.
• Add the noise to the noise free measurement
of the controlled variable.
Numerical Integration of ODE’s
• Accuracy and stability are key issues.
• Reducing integration step size improves
accuracy and stability of explicit integrators
• The ODE’s that represent the dynamic
behavior of control systems in the CPI are
not usually very stiff.
• As a result, a Euler integrator is usually the
easiest and most effective integrator to use.
Development of Dynamic
Process Models for Process
Control Analysis
• It is expensive, time consuming, and
requires a specific expertise.
• It is typically only used in special cases for
particularly difficult and important
processes.
Overview
• Dynamic modeling for process control
analysis should consider the dynamics of
the actuator, the process, and the sensor as
well as sensor noise.