chapter 2: theoretical models of chemical processes
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introduction to process controlTRANSCRIPT
Introduction to Process Control Romagnoli & Palazoglu
Chapter 2: Theoretical Models of Chemical Processes (cont’d)
Inputs ),( uxfdtdx
=
The knowledge of the process is contained in the model of
the process.
Introduction
Principles of modeling:
Basis AssumptionsMathematical consistencySolution of the modelVerification
Introduction to Process Control Romagnoli & Palazoglu
A model will help us create an abstraction of the process and, in turn, will be the source of additional insight into its behavior.
Basis
The basis for mathematical models is the fundamental physical and chemical laws, such as:
Law of conservation of massLaw of conservation of energyLaw of conservation of momentum.
Introduction to Process Control Romagnoli & Palazoglu
Assumptions
Probably the most vital role that the engineer plays in modeling is in exercising his/her judgment as to what valid assumptions can be made.
“An engineering compromise between a rigorous description and
getting an answer that is good enough is always required.”
Introduction to Process Control Romagnoli & Palazoglu
Consistency
Once all the equations of the mathematical model have been written, we need to make sure that the number of variables equals the number of equations.
degrees of freedom of the system must be zero in order to
obtain a solution.
Introduction to Process Control Romagnoli & Palazoglu
Solution
The available solution techniques andtools must be kept in mind as themathematical model is developed.
“An equation without a way to solve it is not worth much.”
Introduction to Process Control Romagnoli & Palazoglu
Verification
An important but often neglected part of developing a mathematical model is proving that the model describes the actual process behavior.
“Model validation is performed using plant
data..”
Introduction to Process Control Romagnoli & Palazoglu
Analytical Models
To characterize a processing system and its behavior, we need:
A set of fundamental dependent quantities whose values describe the natural state of the system State Variables
A set of equations which will describe how the natural state of the given system changes over time State Equations
Introduction to Process Control Romagnoli & Palazoglu
Analytical Models
The state equations are derived from application of the balance equations:
TOTAL MASS BALANCECOMPONENT MASS BALANCEENERGY BALANCE MOMENTUM BALANCE
The balance equation for a quantity S states that:
ACCUMULATION = IN - OUT + AMOUNT OF S GENERATED
- AMOUNT OF S CONSUMED
Introduction to Process Control Romagnoli & Palazoglu
Additional Elements
In addition to the state variables, there are other type of variables that can be classified into two groups:
Parameters: For example cp, ρ, (- ΔH), α, etc.Input Variables: Manipulated and disturbance variables
To solve the system of equations, we must specify both the values of the parameters as well as the values of the input variables.
Introduction to Process Control Romagnoli & Palazoglu
Additional Elements
In the same way, in addition to balance equations, we need other relationships such as:
Transport equations: Transport of energy, mass and momentum. For example,
Kinetic rate equations: For example,
Reaction and phase equilibrium relationships: For example, phase equilibrium, chemical equilibrium, etc.
Equations of state: Relationships between the intensive variables. For example, ideal gas law, Van der Waals equation, etc.
Introduction to Process Control Romagnoli & Palazoglu
ARTE
oA cekr /−=
)( TTUAQ s −=
Example - Tank Level
Fin
FV=A h
A : Cross sectional areah: Height of the liquid levelV : Volume of the tank
Introduction to Process Control Romagnoli & Palazoglu
The fundamental quantity that provides theinformation about the dynamics of the tank is:
The total mass of the liquid in the tank
ρ : density of the liquidV: volume of the liquidA: cross sectional area of the tankh: height of the liquid level
AhVMassTotal ρρ == AhVMassTotal ρρ ==
Example - Tank Level
Introduction to Process Control Romagnoli & Palazoglu
State Variable: hConstant parameters: ρ and A
Note: It is assumed that the density ρ isconstant and independent of temperature.Conservation principle on the fundamentalquantity: the total mass
[ ] [ ] [ ]timetimetime
mass total of Outputmass total of Inputmass total of onAccumulati−=
Example - Tank Level
Introduction to Process Control Romagnoli & Palazoglu
( ) FFdtAhd
i ρρρ−=
Fi : inlet volumetric flow rate (m3/min)
F : outlet volumetric flow rate (m3/min)
FFdtdhA i −=
ρ: constant
Additional Equation:
hF α=h : State Variable and Output
Example - Tank Level
Introduction to Process Control Romagnoli & Palazoglu
Steady-State Condition
A very important concept associated with the model for control is the notion or condition of steady-state of the system.
Steady-State is the state at which the state variables no longer change with time. In this case, the accumulation term is equal to zero.
The application of the steady-state condition leads to a set of algebraic equations
The solution of this set of equations gives us the values of the state variables at steady-state, also called the equilibrium state.
Introduction to Process Control Romagnoli & Palazoglu
Example - Tank Level
Steady-State Equation
FFdtdhA i −== 0
hFi α−=0
or
Introduction to Process Control Romagnoli & Palazoglu
It may happen that a system has more than one steady-state solution. In this case, we say
that the system has multiplicity or multiple steady-states.
Introduction to Process Control Romagnoli & Palazoglu
Steady-State Condition
Introduction to Process Control Romagnoli & Palazoglu
Multiple Steady-State Condition
Introduction to Process Control Romagnoli & Palazoglu
Multiple Steady-State ConditionO’all Mass Balance:
At Steady state and assuming constant density,
Component A mass balance:
Energy Balance equation:
First-order reaction:
Introduction to Process Control Romagnoli & Palazoglu
Multiple Steady-State ConditionAt steady state, solving for CA:
is the steady state concentration for any given reactor temperature Ts
Solving for Ts from the steady-state energy balance equation:
Energy Removed by flow and heat exchange
= Energy generated by reaction
Introduction to Process Control Romagnoli & Palazoglu
Multiple Steady-State ConditionNote the form of the Energy removal term:
Notice that this is an equation for a line, where the independent variable is reactor tempeature (Ts). The slope of the line is and the intercept is . Changes in jacket or feed temperature shift the intercept, but not the slope. Changes in UA or F effect both the slope and intercept. Now, consider the Energy generation term:
substituting for
Introduction to Process Control Romagnoli & Palazoglu
Multiple Steady-State Condition
A steady-state solution exists when there is an intersection of the Qrem [or
R(T)] and Qgen [or G(T)] curves.
increasingTfs
Steady-StateReactor Temperature
s
Ste
ady-
Sta
teR
eact
or T
empe
ratu
re
Inlet-feed TemperatureTfs
Ignitionpoint
Extinctionpoint
Role of Process Simulators
Process simulation packages like Dynsim, HYSYS, ASPEN PLUS, gProms and PRO/II provide a valuable basis for the design and overall evaluation of advanced process control applications. This comes with obvious economical, environmental and operational benefits.
Introduction to Process Control Romagnoli & Palazoglu
Role of Process Simulators
Introduction to Process Control Romagnoli & Palazoglu
Role of Process Simulators
Dynamic simulation offers many benefits over steady-state simulation where it allows the design and process engineers to investigate the time-dependent behavior of a process.
All variables can be observed regardless of instrumentation, The process model can be taken well beyond the safe limits of actual operation.
Using a dynamic model, individual and plantwide control strategies can be designed and tested.
Even the control loops can be tuned before choosing one that may be suitable for final implementation.
Introduction to Process Control Romagnoli & Palazoglu
Open Modeling Architectures
There are many modeling problems which may need the concurrent use of two or more software packages to arrive at a meaningful solution. Each program offers unique advantages and functions, but there is no single program that incorporates all of the data and methods, which may possibly be used to solve every problem. Nor will there ever be an all-encompassing process solverthat would be a solution to all engineering modeling situations. A recent trend is to develop techniques to interconnect different software packages, enabling data communication across established links.
Introduction to Process Control Romagnoli & Palazoglu
The vision is to create an open system whereby different programs can import and export data and communicate with each other freely.
Open Modeling Architectures
The open software vision
Introduction to Process Control Romagnoli & Palazoglu
Example - Pilot-Scale Crystallization
HYSYS is responsible for the modeling of the utilities section, including all heating and cooling, temperatures and pressures.
gPROMS provides solutions for the crystallizer temperature and more significantly it provides the crystal size distribution (CSD).
Excel, while acting as a data link and used for data management and visualization, also allows the execution of the crystallizer jacket heat transfer model.
Introduction to Process Control Romagnoli & Palazoglu
HYSYS flowsheet for the utilities section of the crystallization facility.
Introduction to Process Control Romagnoli & Palazoglu
Example - Pilot-Scale Crystallization
The Graphical User Interface (GUI) where connection and simulation execution can be undertaken.
Introduction to Process Control Romagnoli & Palazoglu
Example - Pilot-Scale Crystallization
Summary
Dynamic process models are developed by paying close attention to the modeling principles.The starting point is the fundamental laws of conservation for the quantity of interest.Steady-state condition arises when the state variables no longer change with time (equilibrium).The algebraic equations describing the steady-state condition may have multiple solutions (multiplicity).Process simulators like HYSYS, Dynsim and Aspen Plus help process designers and control engineers visualize the dynamic behavior of complete plants with control systems.Open modeling architectures enable the use of multiple modeling technologies to work together.
Introduction to Process Control Romagnoli & Palazoglu