Chemical Engineering Department
Master of Science Course
Advanced Process Control
Course Description
This course is designed to improve the ability of master candidates'
chemical students to design and analysis of more advanced complex
control systems in order to achieve a high degree of automatic control
system. Emphasis will be on generating alternative control configuration.
Students will learn process control using digital computer and will use
computer software such as MATLab.
Objectives
At the end of this course, students are expected to:
 Be able to apply the theory of the advanced process control system
to engineering applications, especially the chemical engineering
processes involving the digital controller.
 Be able to derive the process models using conservation equations
such as mass, energy and momentum balance for different models
such as lumped and distributed.
 Be able to use computer in solving control problems.
1.
Course Detail
 Introduction to Control Engineering
 System Modeling
 Mathematical Models
 Time Domain Analysis
 Classical Design in The s-Plane
 Classical Design in The Frequency Domain
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 Analysis of State Space Models
 The State-Space Approach
 Matrix Transfer function

Eigenvalues and Eigenvectors
 Multivariable Control System
 Multiple Loops Control System
 Controllability and Observability
 Interaction analysis
 Relative Gain Array
 Decoupling Method
 Advanced Control System
 Ratio controller
 Override controller
 Feedforward controller
 Cascade controller
 Intelligent Control System
 Fuzzy Control System
 Neural Network Control System
 Model Predictive Control system
 Digital Control System Design
 The z-Transform
 Digital Control Systems
References
 Roland S. Burns, "Advanced Control Engineering", 2001
 Coughanowr, D. R., "Process systems analysis and control",
McGraw-Hill, 1991.
 Wayne, B., "Process Dynamics", Prentice-Hall, 1998. systems
analysis and control", 1997.
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Introduction of Control Engineering
Why Advanced Process Control
 Multivariable System
 Performance
 Cost
 Constraints
 Environmental Constraints
 Computer and modern control theory
Modern Control Theory
These theories are dealing with the following equations:
 Linear, variable coefficient differential equations.
 Nonlinear differential equations.
 Differential –difference equations.
 Partial differential and integral equations.
Types of the Modern Control Theory
 Optimal Control Theory
 Process Identification
 State Estimation
Classical Control Theory (SISO)
The development of a control strategy consists of formulating or identifying the
following.
1. Control objective(s).
2. Input variables—classify these as (a) manipulated or (b) disturbance variables;
inputs may change continuously, or at discrete intervals of time.
3. Output variables—classify these as (a) measured or (b) unmeasured variables;
measurements may be made continuously or at discrete intervals of time.
4. Constraints—classify these as (a) hard or (b) soft.
5. Operating characteristics—classify these as (a) continuous, (b) batch, or (c) semicontinuous
(or semibatch).
6. Safety, environmental, and economic considerations.
7. Control structure—the controllers can be feedback or feed forward in nature. Here we
discuss each of the steps in formulating a control problem in more detail.
1. The first step of developing a control strategy is to formulate the control objectives.
Chemical-process operating unit often consists of several unit operations. The control of an
operating unit is generally reduced to considering the control of each unit operation
separately. Even so, each unit operation may have multiple, sometimes conflicting objectives,
so the development of control objectives is not a trivial problem.
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Figure.1 Conceptual process input/output block diagram.
2. Input
variables can be classified as manipulated or disturbance variables. A
manipulated input is one that can be adjusted by the control system (or process
operator). A disturbance input is a variable that affects the process outputs but
that cannot be adjusted by the control system. Inputs may change continuously
or at discrete intervals of time.
3. Output variables can be classified as measured or unmeasured variables. Measurements
may be made continuously or at discrete intervals of time.
4. Any process has certain operating constraints, which are classified as hard or soft. An
example of a hard constraint is a minimum or maximum flow rate—a valve operates between
the extremes of fully closed or fully open. An example of a soft constraint is a product
composition—it may be desirable to specify a composition between certain values to sell a
product, but it is possible to violate this specification without posing a safety or environmental
hazard.
5. Operating characteristics are usually classified as continuous, batch, or semicontinuous
(semibatch). Continuous processes operate for long periods of time under relatively constant
operating conditions before being ―shut down‖ for cleaning, catalyst regeneration, and so
forth. For example, some processes in the oil-refining industry operate for 18 months between
shutdowns. Batch processes are dynamic in nature—that is, they generally operate for a short
period of time and the operating conditions may vary quite a bit during that period of time.
Example batch processes include beer or wine fermentation, as well as many specialty
chemical processes. For a batch reactor, an initial charge is made to the reactor, and
conditions (temperature, pressure) are varied to produce a desired product at the end of the
batch time. A typical semibatch process may have an initial charge to the reactor, but feed
components may be added to the reactor during the course of the batch run. Another
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important consideration is the dominant timescale of a process. For continuous processes this
is very often related to the residence time of the vessel.
6. Safety, environmental, and economic considerations are all very important. In a sense,
economics is the ultimate driving force—an unsafe or environmentally hazardous process will
ultimately cost more to operate, through fines paid, insurance costs, and so forth. In many
industries (petroleum refining, for example), it is important to minimize energy costs while
producing products that meet certain specifications. Better process automation and control
allows processes to operate closer to ―optimum‖ conditions and to produce products where
variability specifications are satisfied. The concept of ―fail-safe‖ is always important in the
selection of instrumentation. For example, a control valve needs an energy source to move the
valve stem and change the flow; most often this is a pneumatic signal (usually 3–15 psig). If
the signal is lost, then the valve stem will go to the 3-psig limit. If the valve is air-to-open,
then the loss of instrument air will cause the valve to close; this is known as a fail-closed
valve. If, on the other hand, a valve is air to- close, when instrument air is lost the valve will
go to its fully open state; this is known as a fail-open valve.
7. The two standard control types are feed forward and feedback. A feed-forward controller
measures the disturbance variable and sends this value to a controller, which adjusts the
manipulated variable. A feedback control system measures the output variable, compares that
value to the desired output value, and uses this information to adjust the manipulated variable.
For the first part of this textbook, we emphasize feedback control of single-input
(manipulated) and single-output (measured) systems. Determining the feedback control
structure for these systems consists of deciding which manipulated variable will be adjusted to
control which measured variable. The desired value of the measured process output is called
the setpoint.
Feedback Control System
The control and instrumentation diagram for a feedback control strategy for scenario 1 is
shown in Figure (2). Notice that the level transmitter (LT) sends the measured height of liquid
in the tank (hm) to the level controller (LC). The LC compares the measured level with the
desired level (hsp, the height setpoint) and sends a pressure signal (Pv) to the valve. This
valve top pressure moves the valve stem up and down, changing the flow rate through the
valve (F1). If the controller is designed properly, the flow rate changes to bring the tank
height close to the desired setpoint. In this process and instrumentation diagram we use
dashed lines to indicate signals between different pieces of instrumentation. A simplified
block diagram representing this system is shown in Figure (3). Each signal and device (or
process) is shown on the block diagram. We use a slightly different form for block diagrams
when we use transfer function notation for control system analysis. Note that each block
represents a dynamic element. We expect that the valve and LT dynamics will be much faster
than the process dynamics. We also see clearly from the block diagram why this is known as a
feedback control ―loop.‖ The controller ―decides‖ on the valve position, which affects the
inlet flow rate (the manipulated input), which affects the level; the outlet flow rate (the
disturbance input) also affects the level.
The level is measured, and that value is fed back to the controller [which compares the
measured level with the desired level (setpoint)]. Notice that the control valve should be
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specified as fail-closed or air-to-open, so that the tank will not overflow on loss of instrument
air or other valve failure.
Figure 2 Feedback control strategy 1. The level is measured and the inlet flow rate (valve
position) is manipulated.
Figure 3 Feedback control schematic (block diagram) for scenario 1. F1 is manipulated and F2
is a disturbance.
Scenario 2 Process 1 regulates flow rate F1. This could happen, for example, if process 1 is
producing a chemical compound that must be processed by process 2. Perhaps process 1 is set
to produce F1 at a certain rate. F1 is then considered ―wild‖ (a disturbance) by the tank
process. In this case we would adjust F2 to maintain the tank height. Notice that the control
valve should be specified as fail-open or air-to-close, so that the tank will not overflow on loss
of instrument air or other valve failure.
The process and instrumentation diagram for this scenario is shown in Figure (4). The only
difference between this and the previous instrumentation diagram (Figure 1–3) is that F2
rather than F1 is manipulated.
The simplified block diagram shown in Figure (5)differs from the previous case (Figure 3)
only because F2 rather than F1 is manipulated. F1 is a disturbance input.
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Figure 4 Feedback control strategy 2. Outlet flow rate is manipulated.
Figure 5 Feedback control schematic (block diagram) for scenario 2. F2 is manipulated and F1
is a disturbance.
Instrumentation
The example level-control problem had three critical pieces of instrumentation: a sensor
(measurement device), actuator (manipulated input device), and controller. The sensor
measured the tank level, the actuator changed the flow rate, and the controller determined
how much to vary the actuator, based on the sensor signal. There are many common sensors
used for chemical processes. These include temperature, level, pressure, flow, composition,
and pH. The most common manipulated input is the valve actuator signal (usually pneumatic).
Each device in a control loop must supply or receive a signal from another device. When
these signals are continuous, such as electrical current or voltage, we use the term analog. If
the signals are communicated at discrete intervals of time, we use the term digital.
Analog
Analog or continuous signals provided the foundation for control theory and design and
analysis. A common measurement device might supply either a 4- to 20-mA or 0- to 5-V
signal as a function of time. Pneumatic analog controllers (developed primarily in the 1930s,
but used in some plants today) would use instrument air, as well as a bellows-andsprings
arrangement to ―calculate‖ a controller output based on an input from a measurement device
(typically supplied as a 3- to 15-psig pneumatic signal). The controller output of 3–15 psig
would be sent to an actuator, typically a control valve where the pneumatic signal would
move the valve stem. For large valves, the 3- to 15-psig signal might be ―amplified‖ to supply
enough pressure to move the valve stem. Electronic analog controllers typically receive a 4- to
20-mA or 0- to 5-V signal from a measurement device, and use an electronic circuit to
determine the controller output, which is usually a 4- to 20-mA or 0- to 5-V signal. Again, the
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controller output is often sent to a control valve that may require a 3- to 15-psig signal for
valve stem actuation. In this case the 4- to 20-mA current signal is converted to the 3- to 15psig signal using an I/P (current-to-pneumatic) converter.
Digital
Many devices and controllers are now based on digital communication technology. A sensor
may send a digital signal to a controller, which then does a discrete computation and sends a
digital output to the actuator. Very often, the actuator is a valve, so there is usually a D/A
(digital-to-electronic analog) converter involved. Indeed, if the valve stem is moved by a
pneumatic actuator rather than electronic, then an I/P converter may also be used. In the past
few decades, digital control-system design techniques that explicitly account for the discrete
(rather than continuous) nature of the control computations have been developed. If small
sample times are used, the tuning and performance of the digital controllers is nearly equal to
that of analog controllers.
Figure 6 Room temperature control system
Figure 7 Block diagram of Room temperature control system
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Mathematical Model
Objectaves
1.
2.
3.
4.
Improve understanding of the process
Train Plant operating personnel
Develop control strategy for new plant
Optimize process operating conditions
Type of Models
1.
2.
Theoretical model
 Developed using the principles of chemistry, physic and biology.
 applicable over wide ranges of conditions
 expensive and time consuming
 some model parameters such as reaction rate coefficients
 physical properties, or heat transfer coefficients are unknown
Empirical models




obtained fitting experimental data
range of the data is typically quite small compared to
the whole range of process operating conditions
do not extrapolate well
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3.
Semi-empirical models
 combination of 1 and 2
 overcome the previously mentioned limitations to many
 extents
 widely used in industry
A Systematic Approach for Developing Dynamic Models
1.
State the modeling objectives and the end use of the model. They determine the
required levels of model detail and model accuracy.
2.
Draw a schematic diagram of the process and label all process variables.
3.
List all of the assumptions that are involved in developing the model. Try for
parsimony; the model should be no more complicated than necessary to meet the modeling
objectives.
4.
Determine whether spatial variations of process variables are important. If so, a partial
differential equation model will be required.
5.
Write appropriate conservation equations (mass, component, energy, and so forth).
6.
Introduce equilibrium relations and other algebraic equations (from thermodynamics,
transport phenomena, chemical kinetics, equipment geometry, etc.).
7.
Perform a degrees of freedom analysis (Section 2.3) to ensure that the model equations
can be solved.
8.
Simplify the model. It is often possible to arrange the equations so that the dependent
variables (outputs) appear on the left side and the independent variables (inputs) appear on the
right side. This model form is convenient for computer simulation and subsequent analysis.
9.
Classify inputs as disturbance variables or as manipulated variables.
Degrees of Freedom Analysis
1.
List all quantities in the model that are known constants (or parameters that can be
specified) on the basis of equipment dimensions, known physical properties, etc.
2.
Determine the number of equations NE and the number of process variables, NV .
Note that time t is not considered to be a process variable because it is neither a process input
nor a process output.
3.
Calculate the number of degrees of freedom, NF = NV - NE .
4.
Identify the NE output variables that will be obtained by solving the process model.
5.
Identify the NF input variables that must be specified as either disturbance variables or
manipulated variables, in order to utilize the NF degrees of freedom.
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Concept of the system
Figure 8 Concept of the system
Modeling and Simulation procedure
Translating the description of a physical system into an appropriate mathematical
form.
•Selecting a suitable computational technique.
•Implementing the computational technique in the form of a computer program.
General Process Unit Analysis
1.Define system variables.
2.Write simulation equations.
3.Check degrees of freedom.
4.Choose design variables.
5.Choose appropriate math solver.
APPLICATIONS OF DYNAMIC SIMULATION
Dynamic simulation is useful throughout the entire
lifetime of a plant: from conception to decommissioning.
Plant design:
a) A priori assessment of intrinsic operability and controllability of a plant - especially for
highly integrated plants.
b) Design and testing of regulatory control systems - selection of control structures, control
algorithms, and initial tuning of loops.
c) Design and testing of operating procedures - startup, shut-down, feed stock changeover,
etc.
d) Hazard/safety studies - answer important questions raised by HAZOP study.
e) Design and testing of protective and relief devices - study plant behaviour and performance
of protective system during major deviations from steady-state (need good models!)
f) Environmental studies - predict emissions generated during plant upsets/failures.
g) Analysis of intrinsically dynamic processes: batch/semi-continuous processes, periodic
processes (e.g., pressure swing adsorption)
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APPLICATIONS OF DYNAMIC SIMULATION
Plant operation: typically a dynamic simulation is linked to the plant's real time
control and monitoring software:
(a) computer based operator training:
 Dynamic simulation runs in real time and mimics behaviour of real plant.
• Operator interfaces to control system (not simulator) — more realistic.
• Instructor monitors and creates scenarios (e.g.,disturbances, failures)
(b) Validation of operating and safety procedures — test control software by
dynamic simulation before implementing on the real plant!
(c) On-line operator decision support tool (simulation runs in parallel with real
plant):
 Update current state (initial condition) directly from measurements.
• Dynamic predictor — run simulation faster than real time to predict hazards or
problems (so that early corrective or preventative action can be taken).
• Experimental tool — predict consequences of proposed actions.
APPLICATIONS OF THE MODEL


steady-state simulation:
 steady-state is just one initial condition for a dynamic model: always get steady-state
model for
free in dynamic study!
 optimize plant for different feed concentrations and loads
 identify which equipment must be revamped to increase capacity
plant must be regulated at new optimal operating points - use dynamic model for
identification and controllability studies:
 sensitivity studies to identify necessary process measurements
 sensitivity studies to define appropriate control structure (matching of sensors to
control elements) and tuning of control parameters
 validation of control system by predicting plant response to a variety of load and
setpoint changes.
PRELUDE TO MODELLING
Because of this fundamental nature of models, before and during any modeling activity it is
important to clarify and document the following information:
(a) Identify the system for which a model is required
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Figure (9) system
Identify:
• Boundaries (function of system) - a system is defined by its boundary
• Constraints
• Quantities describing system behaviour: inputs, states, outputs
• Assume: inputs ≠ ƒ(outputs)
Notes:
• We are developing a model for the system. Everything else is the environment.
• Inputs define the influence of the environment on the system.
• While in real life the inputs will be further subdivided into:
(a) Controls — those inputs that can be manipulated in order to control the system behaviour.
(b) Disturbances — those inputs over which we have no control.
. . . from the point of view of modelling: we must define the time variation of all the inputs in
order to pose a fully determined simulation problem.
• Inputs ≠ ƒ(outputs):
— More precisely: we can sufficiently decouple the influence of the outputs on the inputs
(feedback via the environment) for the purposes of the current exercise.
— Pragmatic view: can only model so much at a given level of detail.
(b) What is the intended application of the model?
What questions will be asked about the system?
Begin to identify:
• What phenomena are of interest?
• What quantities describe the system behaviour?
• How detailed should the model be?
• What assumptions can be made?
(c) what data concerning the system is available?
or can be obtained . . . imposes constraints on the phenomena that can be modeled and the
accuracy of the simulation results.
Typically relates to parameters and empirical correlations:
• What is available?
• In what range of process conditions are the predictions valid?
• How much uncertainty is there in the predictions?
Examples: non-equilibrium distillation tray models, reaction kinetics for novel synthesis.
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STEP 2 — MODELLING THE COMPONENTS
Each component model is itself a system (recursion):
(a) Identify the boundaries.
• The model may represent a physical artifact — e.g., a plant section, a unit operation, a
vessel, or a section of a vessel.
• The model may represent some phenomenological abstraction — e.g., the set of equations
defining a mixture enthalpy, or a set of reaction rate expressions.
(b) Identify the connections of the model to its environment
- This will enable us to connect the model to others in a larger structure. Connections will
typically represent either:
(i) Fluxes of extensive properties, such as:
• A diffusive flux - e.g., heat transfer through a vessel wall
• A convective flux - e.g., a pipe connecting two vessels (ignoring the holdup of material in
the pipe) Note fluxes will have a direction implied.
(ii) Information transfer - e.g., pressure and/or voltage signals for control instruments, or
intensive properties (e.g., temperature, pressure) to determine driving forces for fluxes
structures are defined by establishing equivalence (merging) between the connection of one
component model and the connection of another component model (obviously, the two
connections must be compatible - e.g., must represent a convective flux)
(c) Define the internal behaviour of the component model
- what describes the "state" of this system, and how is it related to both the inputs and the
outputs.
INTERNAL BEHAVIOUR
Ultimately internal behaviour is represented by a set of:
VARIABLES - e.g. Inputs, States, Outputs ...and these variables are related by a set of:
EQUATIONS - e.g. Mass Balances, Energy Balances, Physical Constraints, Thermodynamic
Models etc.
However, there are several options as to how these two sets can be defined:
(a) The model is primitive — e.g., just define a set of variables and a set of equations
(b) Decompose the model further — e.g., define a set components and their connections. The
set of variables is then the union of the sets of variables in the sub models, and similarly the
set of equations is the union of the sets of equations in the sub models and the equations
implied by the connections
(c) a hybrid of (a) and (b) — e.g., both variables and sub models.
DERIVING THE EQUATIONS
Once a suitable decomposition has been established, it is necessary to develop all the
primitive models - e.g., derive a set of variables and equations that describe the dynamic
behaviour of that section of the overall system.
Basically, we must utilize our knowledge of chemical engineering science at this point control volume analysis, mass conservation, energy conservation, etc. Experience with
teaching this material suggests that it is worthwhile to review the principles frequently used in
dynamic modeling - but this discussion is by no means exhaustive (nor is my experience!).
The approach is relatively systematic and will allow derivation of component models for a
system that are consistent.
The following information should be developed at this stage:
1. Define the control volume(s) for balance equations (again. . .identify the boundaries)
2. List the assumptions employed — under what conditions is the model valid? Example:
phase transitions
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3. List the set of variables required to describe the system — e.g., symbol, verbal
description, units (always try to use a consistent set of units).
4. Derive the set of equations — always check that the units of each term in an equation are
consistent.
5. Perform a degree of freedom analysis:
(a) Identify the natural set of input variables to the system.
(b) Given values for this subset of the variables it should be possible to calculate values for all
the other variables — e.g. it is necessary that the number of equations equal the number of
remaining unknown variables (ignore time derivatives of variables in this analysis).
We will be applying one or more conservation principles to a macroscopic control volume of
definite size and shape containing a fluid. Further, we will assume that the contents of the
control volume are well mixed, so intensive properties are uniform throughout the control
volume and do not vary with spatial position (or with another independent variables such as
polymer chain length).
=> One or more ordinary differential equations (ODEs) with time as the independent
variable.
These ODEs will be augmented with algebraic equations (e.g., equations not involving time
derivatives) that relate variables or model phenomena that take place on a much faster time
scale than that of interest, for example:
• A pseudo steady-state assumption
• Phase equilibrium models
Conservation Laws
Theoretical models of chemical processes are based on conservation laws.
Conservation of Mass
Conservation of Component i
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Conservation of Energy
The general law of energy conservation is also called the First Law of
Thermodynamics. It can be expressed as:
. . (1)
The total energy of a thermodynamic system, Utot , is the sum of its internal energy,
kinetic energy, and potential energy:Examples of Models
.
For the processes and examples considered in this case, it is appropriate to make two
assumptions:
.
1. Changes in potential energy and kinetic energy can be neglected because they
are small in comparison with changes in internal energy.
2. The net rate of work can be neglected because it is small compared to the rates
of heat transfer and convection.
For these reasonable assumptions, the energy balance in Eq. 1 can be written as
. . . . (2)
Uint:the internal energy of the system
H: enthalpy per unit mass
W:mass flow rate
Q: rate of heat transfer to the system
Δ: denotes the difference between outlet and inlet conditions of the flowing streams;
therefore
- Δ(WH): rate of enthalpy of the inlet stream(s) - the enthalpy of the outlet stream(s)
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The analogous equation for molar quantities is,
…..(3)
where H is the enthalpy per mole and is the molar flow rate.
In order to derive dynamic models of processes from the general energy balances in
Eqs. 2 and 3, expressions for Uint and H or H are required, which can be derived from
thermodynamics.
MASS CONSERVATION — EXAMPLE
Buffer tank open to the atmosphere:
Figure(10): Liquid level system
Control volume: liquid in tank — e.g., boundary is not rigid, so size and shape of control
volume will change as the level rises and sinks
Assumptions:
• isothermal system => no need for energy balance
• single chemical species => fluid density constant
• inlet flow defined as an input (or by an upstream model) — e.g., FIN (t) known.
• vessel has uniform cross sectional area
• model valid for interval 0 ≤h ≤hmax
Variables (what quantities are we interested in?):
 M* mass of fluid in vessel [kg]
 density of fluid in vessel [kg m-3]
 V: volume of fluid in vessel [m3]
 FIN volumetric flow of inlet stream [m3s-1]
 FOUT volumetric flow of outlet stream [m3s-1]
 P0 atmospheric pressure [Nm-2]
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 P1 pressure at bottom of vessel [Nm-2]
 A cross sectional area of vessel [m2]
 h* liquid level in vessel [m]
Note (*): in steady-state models we do not usually worry about quantities describing the
"state" of the control
volume — we are only concerned with the quantities crossing the boundary and the fact that
they must balance at steady-state. This is a major reason why dynamic models are more
complex — we now have to relate how this state changes to the quantities crossing the
boundary. Clearly, the whole point of doing dynamic simulation is to determine how this state
changes with time.
Equations:
Mass Conservation
Relate volume and mass
Relate volume and liquid level (constant cross sectional area)
Hydrostatic pressure
g: gravitational acceleration (9.81ms-2) Flow pressure relationship — flow out driven
by hydrostatic pressure in vessel
k: loss coefficient (value known)
Degree of freedom analysis:
Total number of quantities = 11
Time invariant quantities (model parameters):
A, ρk,g (= 4)
Natural input set:
FIN (t), P0 (t) (= 2)
Note: these functions are defined by the "environment" which could be either other
models, or the engineer.
Remaining variables:
M, V, h, P1, FOUT (= 5)
Equations = 5
=> degrees of freedom are satisfied given specification of input set above.
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Note: due to the assumptions made above, equation (2) could be substituted into (1) to
eliminate M - leading to a "volume balance." This is an error that is frequently made
- volume is not a conserved quantity (e.g., consider a non isothermal and/or multicomponent system) whereas mass is.
SPECIES BALANCES
Unlike mass, chemical species are not conserved: if a reaction takes place inside a
control volume, reactants will be consumed and products generated.
species balance can be written for each chemical species in the system (note we are
now using moles rather than mass):
Further, we can sum these NC(NC will be used to denote the number of chemical
species in a system) species balances to derive a total mole balance.
Therefore, we can derive three different balance equations:
(a) NC species balances
(b) a total mole balance
(c) a mass balance
. . . clearly these are not independent — the total number of moles and the mass in the
system can be related algebraically to the number of moles of each species, e.g.:
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In general, it is best to derive a model in terms of the minimal number of independent
species balances (usually NC) and derive other quantities via algebraic equations.
The rationale behind this statement should become clearer when we discuss numerical
solution of dynamic simulation problems.
EX. Consider the same tank as before, but the following liquid phase first order
irreversible isomerization reaction takes place (i.e. isothermal CSTR):
Species A and B are present in dilute solution (i.e. NC = 3; A, B, and the solvent).
Assumptions:
• Vessel contents well mixed
• Isothermal, species A and B present in dilute solution
=> Fluid density constant (e.g., neglect density changes due to presence of A and B)
• Model valid for interval 0 < h ≤ hMAX (note difference)
• Otherwise, same as above
Variables:
V volume of fluid in vessel [m3]
ρ density of fluid in vessel [kg m-3]
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FIN volumetric flow of inlet stream [m3 s-1]
FOUT volumetric flow of outlet stream [m3 s-1]
P0 atmospheric pressure [N m-2]
P1 pressure at bottom of vessel [N m-2]
A cross sectional area of vessel [m2]
h liquid level in vessel [m]
CA concentration of species A in vessel [mol m-3]
CB concentration of species B in vessel [mol m-3]
CA,IN concentration of species A in inlet stream [mol m-3]
CB,IN concentration of species B in inlet stream [mol m-3]
g gravitational acceleration [m s-2]
k loss coefficient
kR reaction rate constant [s-1]
r reaction rate [mol m-3 s-1]
Equations:
Mass Conservation
Species balances for A and B
Notes:
• Use of volume in mass balance is only valid for the assumptions above
• Concentrations in outlet stream equal to bulk concentration because vessel contents
well mixed
• We have derived NC (=3) mass and species balances
— This is sufficient to define the state of the system: all other quantities can be
related to {V, CA,CB} via algebraic relationships. Note that an alternative would
have been to derive a species balance for the solvent instead of the mass balance.
Relate volume and liquid level
Hydrostatic pressure
Flow pressure relationship
21
Reaction rate
. . . and additional equations to define NA (= CAV),
NB (= CB V), M (= ρV) if desired.
Note: if significant density changes occur due to reaction (e.g., not sufficiently dilute)
then it is better to derive NC species balances and derive the volume from the molar
volumes in solution.
Degree of freedom analysis:
Total number of quantities = 16
Time invariant parameters:
A,ρ,g, k, kR (= 5)
Natural input set:
FIN (t),CA,IN (t),CB, IN (t ),P0 (t) (=4)
Remaining variables:
V,FOUT ,P1,h,CA,CB,r (=7)
Equations = 7
=> degrees of freedom satisfied given specification of input set above
ENERGY CONSERVATION
According to the First Law of Thermodynamics, energy is a conserved quantity. For
an open system, this can be expressed as:
Warning: always start from the First Law when deriving energy balances!!
Here, energy is the summation of internal energy (e.g., that associated with translation,
rotation and vibration of molecules), kinetic energy, and potential energy. In most process
simulation applications, it is usually reasonable to neglect kinetic and potential energy (or to
perform separate balances for internal energy and these other forms of energy) — but always
check this assumption.
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EX.Given this assumption, the differential form of the First Law of Thermodynamics for the
following "general" open system :
...where we have introduced time as the independent variable.
Note that the conserved quantity is the internal energy of the control volume contents
(not the enthalpy!). The work term is composed of two contributions:
...the summation of the shaft work done on the system (e.g., an impeller to keep the
contents well mixed) and the PV work due to changes in volume of the system.
Variables:
U internal energy of control volume contents [J]
Q˙ rate of heat addition to control volume from environment [J s-1]
W˙ rate of work done on the control volume by the environment [J s-1]
Fi rate of material addition/discharge [kg s-1] or [mol s-1]
hi specific enthalpy of material stream [J kg-1] or [J mol-1]
The energy balance can therefore be expressed in two equivalent forms:
or du/ dt can be eliminated by substitution of the differential form of the definition of
enthalpy (H =U + PV):
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Of course, there are many situations in which neither the volume or the pressure
remains constant over the time horizon of interest. In this case, we must be able to
express the rate of change of the volume or pressure explicitly, or have this implied by
the algebraic equations (which leads to difficulties - see discussion of "high index"
problems later).
ENERGY BALANCE — CONSTANT PRESSURE
EXAMPLE
Consider the CSTR again — the reaction is now exothermic and heat is removed by a
jacket containing a vaporizing medium, so an energy balance is necessary.
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Assumptions:
• Vessel contents well mixed
• Species A and B present in dilute solution
• Atmospheric pressure is constant (e.g., P0 ≠f (t))
• Neglect shaft work of impeller
• Neglect heat interaction with atmosphere
• Otherwise, same as above.
Variables:
M mass of fluid in vessel [kg]
V volume of fluid in vessel [m3]
NA number of moles of species A in vessel [mol]
NB number of moles of species B in vessel [mol]
CA concentration of species A in vessel [mol m-3]
CB concentration of species B in vessel [mol m-3]
ρ density of fluid in vessel [kg m-3]
A cross sectional area of vessel [m2]
AJ heat transfer area of jacket [m2]
UJ overall heat transfer coefficient for jacket [J m-2 K-1 s-1]
H enthalpy of vessel contents [J]
CA,IN concentration of species A in inlet stream [mol m-3]
CB,IN concentration of species B in inlet stream [mol m-3]
FIN volumetric flow of inlet stream [m3 s-1]
FOUT volumetric flow of outlet stream [m3 s-1]
TIN temperature of inlet stream [K]
TOUT temperature of outlet stream [K]
P0 atmospheric pressure [Nm-2]
P1 pressure at bottom of vessel [Nm-2]
g gravitational acceleration [m s-1]
k loss co-efficient
kR reaction rate constant [s-1]
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r reaction rate [mol m-3 s-1]
ρIN density of inlet stream [kg m-3]
ρOUT density of outlet stream [kg m-3]
Q˙ heat transfer to fluid in vessel from jacket [J s-1]
hIN specific enthalpy of inlet stream [J kg-1]
hOUT specific enthalpy of outlet stream [J kg-1]
ER activation energy [J mol-1]
R universal gas constant [J mol-1 K-1]
h liquid level in vessel [m]
T temperature of vessel contents [K]
TJ temperature of vaporizing medium
[K]
Equations:
Mass conservation
...note we are unable to assume ρ is constant
Species balances for A and B
Energy balance (constant pressure formulation — e.g., it is most convenient to use
enthalpy in the accumulation term)
Relate volume and mass
Relate volume and liquid level
Relate mole numbers and concentration
26
Hydrostatic pressure
Flow pressure relationship
Contents well mixed
Define enthalpy holdup (implies temperature of contents)
Physical Properties (abstract functions)
Heat transfer
A commonly asked question is: how is the temperature in the reactor determined? In fact, the
temperature is determined by the simultaneous solution of the complete set of implicit
relationships above. One can view equation (4) as determining the extensive enthalpy H of the
vessel contents, equation (14) determining the intensive enthalpy hOUT, and equation (18)
implicitly determining T given hOUT.
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Degree of freedom analysis:
Note: in many textbooks it is common practice to include a reaction term in the
energy balance. The control volume approach (equation (E1) or (E2)) clearly shows
that no energy crosses the system boundary due to a reaction taking place, so a
reaction term should not appear in the energy balance. If a reaction is taking place in
an isolated system, the total energy of the system remains unchanged, but the
distribution of energy between "heat of formation" energy and "sensible heat" energy
changes as the reaction progresses (e.g., the temperature will rise or drop if the
reaction is exothermic or endothermic). To reflect this, the constitutive equation
defining the specific enthalpy (18) must include the contribution of both the heat of
formation and the sensible heat of each species to the total system energy. So, the zero
energy reference state for equation (18) must be defined as the elements making up
the chemical species in their standard states at some temperature and pressure. If heats
of formation are not included in equation (18), a heat of reaction term must be added
to the energy balance, and the heat of reaction must be calculated as a function of
temperature. Overall, I consider it clearer and simpler to work with the first law and
include heats of formation in species enthalpies (obviously, none of this is necessary
if no chemical reactions occur in the system of interest).
MOMENTUM BALANCE
It is sometimes necessary to model the velocity (or momentum) of the contents of a
control volume. As flows can in general be three dimensional, velocity is a vector
quantity with components corresponding to the velocity resolved into the coordinate
directions of the chosen coordinate system. So, in principle we can formulate a
momentum balance for each co-ordinate direction (three balances).
Applying Newton's Second Law to a control volume, we obtain:
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. . . for each direction i in which material is flowing.
Note: momentum is defined as the product of mass and velocity. Care should be taken
if both the mass and the velocity of the control volume are changing with respect to
time.
MOMENTUM BALANCE EXAMPLE
Consider the buffer tank open to the atmosphere, but now the fluid flows out into a
long pipeline:
Control volumes: (a) liquid in tank, (b) liquid in pipeline
Assumptions:
• Same as for original buffer tank example
• One dimensional plug flow in pipeline and incompressible liquid => velocity uniform
throughout pipeline (macroscopic control volume)
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Variables:
M mass of fluid in vessel [kg]
ρdensity of fluid in vessel [kg m-3]
FIN volumetric flow of inlet stream [m3s-1]
FOUT volumetric flow of inlet stream [m3s-1]
AP cross sectional area of pipeline [m2]
L length of pipeline [m]
v velocity of fluid in pipeline (uniform) [ms-1]
FH hydraulic force on fluid [N]
FF frictional force resisting flow [N]
g gravitational acceleration [ms-2]
h level of liquid in vessel [m]
V volume of liquid in vessel [m3]
kF constant related to Fanning friction factor
A cross sectional area of vessel [m2]
Equations:
Mass conservation in tank
. . . mass balance on pipeline unnecessary – incompressible liquid in fixed volume (Flow in =
Flow out) Momentum conservation on pipeline - axial direction only
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Degree of freedom analysis:
Total number of quantities = 14
Time invariant parameters: ρ, Ap ,L,g, kF, A (=6)
Natural input set: FIN (t) (=1)
Remaining variables: M,FOUT,v, FH,FF,h,V (=7)
Equations = 7
=> degrees of freedom satisfied
Non-Isothermal CSTR
We reconsider the previous CSTR example, but for non-isothermal conditions. The reaction A
B is
exothermic and the heat generated in the reactor is removed via a cooling system as shown in figure
below. The effluent temperature is different from the inlet temperature due to heat generation by the
exothermic reaction.
Figure Non-isothermal CSTR
Assuming constant density, the macroscopic total mass balance (Eq. 2.6) and mass component
balance remain the same as before. However, one more ODE will be produced from the applying
the conservation law for total energy balance. The dependence of the rate constant on the
temperature:
The generation term is zero since the mass is conserved. The balance equation yields:
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….(1)
Under isothermal conditions we can further assume that the density of the liquid is constant i.e.
ρf = ρo=ρ. In this case Eq. 1 is reduced to:
…(2)
The general energy balance for macroscopic systems applied to the CSTR yields, assuming
constant density and average heat capacity:
(3)
where Qr (J/s) is the heat generated by the reaction, and Qe (J/s) the rate of heat
removed by the cooling system. Assuming Tref = 0 for simplicity and using the
differentiation principles, equation (3) can be written as follows:
Substituting Equation (1) into the last equation and rearranging yields:
…(4)
The rate of heat exchanged Qr due to reaction is given by:
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where ΔHr (J/mole) is the heat of reaction (has negative value for exothermic reaction
and positive value for endothermic reaction). The non-isothermal CSTR is therefore
modeled by three ODE's:
where the rate (r) is given by:
The system can be solved if the system is exactly specified and if the initial conditions
are given:
Degrees of freedom analysis
Parameter of constant values: ρ, E, R, Cp, ΔHr and ko
(Forced variable): Ff , CAf and Tf
Remaining variables: V, Fo, T, CA and Qe
• Number of equations: 3 (Eq. 1, 2 and 4)
The degree of freedom is 5−3 = 2. Following the analysis of this example, the two extra relations
are between the effluent stream (Fo) and the volume (V) on one hand and between the rate of
heat exchanged (Qe) and temperature (T) on the other hand, in either open loop or closed loop
operations.
A more elaborate model of the CSTR would include the dynamic of the cooling jacket.
Assuming the jacket to be perfectly mixed with constant volume Vj, density ρj and constant
average thermal capacity Cpj, the dynamic of the cooling jacket temperature can be modeled by
simply applying the macroscopic energy balance on the whole jacket:
Since Vj, ρj, Cpj and Tjf are constant or known, the addition of this
equation introduces only one variable (Tj). The system is still exactly specified.
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Jacketed Non-isothermal CSTR
Single Stage Heterogeneous Systems: Multi-component flash drum
The previous treated examples have discussed processes that occur in one single phase.
There are several chemical unit operations that are characterized with more than one
phase. These processes are known as heterogeneous systems. In the following we cover
some examples of these processes. Under suitable simplifying assumptions, each phase
can be modeled individually by a macroscopic balance.
A multi-component liquid-vapor separator is shown in figure 12. The feed consists of Nc
components with the molar fraction zi (i=1,2… Nc). The feed at high temperature and
pressure passes through a throttling valve where its pressure is reduced substantially. As a
result, part of the liquid feed vaporizes. The two phases are assumed to be in phase
equilibrium. xi and yi represent the mole fraction of component i in the liquid and vapor
phase respectively. The formed vapor is drawn off the top of the vessel while the liquid
comes off the bottom of the tank. Taking the whole tank as our system of interest, a
model of the system would consist in writing separate balances for vapor and liquid
phase. However since the vapor volume is generally small we could neglect the dynamics
of the vapor phase and concentrate only on the liquid phase.
Multicomponent Flash Drum
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For liquid phase:
Total mass balance:
Component balance:
Energy balance:
where h~ and H~ are the specific enthalpies of liquid and vapor phase respectively.
In addition to the balance equations, the following supporting thermodynamic
relations can be written:
Liquid-vapor Equilibrium:
Raoult's law can be assumed for the phase equilibrium
Together with the consistency relationships:
Physical Properties:
The densities and enthalpies are related to the mole fractions, temperature and pressure
through the following relations:
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Note that physical properties are not included in the degrees of freedom since they are
specified through given relations. The degrees of freedom is therefore (2Nc+5)(2Nc+3)=2. Generally the liquid holdup (VL) is controlled by the liquid outlet flow rate (FL)
while the pressure is controlled by FV. In this case, the problem becomes well defined for
a solution.
Multi-component Distillation Column
Distillation columns are important units in petrochemical industries. These units process their
feed, which is a mixture of many components, into two valuable fractions namely the top
product which rich in the light components and bottom product which is rich in the heavier
components. A typical distillation column is shown in Figure 13. The column consists of n
trays excluding the re-boiler and the total condenser. The convention is to number the stages
from the bottom upward starting with the re-boiler as the 0 stage and the condenser as the n+1
stage.
Description of the process
The feed containing nc components is fed at specific location known as the feed tray (labeled
f) where it mixes with the vapor and liquid in that tray. The vapor produced from the re-boiler
flows upward. While flowing up, the vapor gains more fraction of the light component and
loses fraction of the heavy components. The vapor leaves the column at the top where it
condenses and is split into the product (distillate) and reflux which returned into the column
as liquid. The liquid flows down gaining more fraction of the heavy component and loses
fraction of the light components. The liquid leaves the column at the bottom where it is
evaporated in the re-boiler. Part of the liquid is drawn as bottom product and the rest is
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recycled to the column. The loss and gain of materials occur at each stage where the two
phases are brought into intimate phase equilibrium.
Distillation Column
Modeling the unit:
We are interested in developing the unsteady state model for the unit using the flowing
assumptions:

100% tray efficiency

Well mixed condenser drum and re-boiler.

Liquids are well mixed in each tray.

Negligible vapor holdups.

liquid-vapor thermal equilibrium
Since the vapor-phase has negligible holdups, then conservation laws will only be written
for the liquid phase as follows:
Stage n+1 (Condenser),:
Total mass balance:
37
Stage n
Total Mass balance:
Component balance:
Energy balance:
Stage i
Total Mass balance:
38
Stage f (Feed stage)
Energy balance
Stage 1
Total Mass balance:
39
Stage 0 (Re-boiler)
Total Mass balance:
Additional given relations:
41
Phase equilibrium: yj = f (xj, T,P)
Liquid holdup: Mi = f (Li)
Enthalpies: Hi = f (Ti, yi,j), hi = f (Ti, xi,j)
Vapor rates: Vi = f (P)
Degrees of freedom analysis
Variables
41
Equations:
42
Constants: P, F, Z
Therefore; the degree of freedom is 4
To well define the model for solution we include four relations imported from inclusion
of four feedback control loops as follows:
• Use B, and D to control the liquid level in the condenser drum and in the re-boiler.
 Use VB and R to control the end compositions i.e., xB, xD
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