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Phn 3: Ti u haModeling, simulation and optimization for chemical process
Instructor: Hoang Ngoc HaEmail: [email protected]
B mn QT&TBT. F. Edgar, D. M. Himmelblau. Optimization of chemical Processes.
Second edition.
Bi Minh Tr. Tiu ha (l thuyt v bi tp). NXB KHKT, H Ni, 2005.
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Introduction
The chemical industry has undergone
significant changes during the past 25 yearsdue to the
increased cost of energy
increasingly stringent environmental regulations global competition in product pricing and quality
One of the most important engineering tools foraddressing these issues is optimization
Decision-making process
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Introduction
As the power ofcomputers has increased,
the size and complexity of problems that canbe solved by optimization techniques have
correspondingly expanded
The necessary tools for solving problem
We will focus on those techniques and discusssoftware that offers the most potential for
success and gives reliable results
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Outline
Problem formulation
Nature and organization of Optimization problems Developing models for optimization (constraints or
process model)
Formulation of the objective function
Optimization theory and methods Optimization of unconstrained functions
Linear programming with constraints
Nonlinear programming with constraints
Multi-objective optimization
Applications of Optimization
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Optimization
OPTIMIZATION IS THE use of specific methods to
determine the most cost-effective and efficientsolution to a problem or design for a process
This technique is one of the major quantitative toolsin industrial decision making
A wide variety of problems in the design,
construction, operation, and analysis of chemicalplants (as well as many other industrial processes)can be resolved by optimization
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Problem formulation
Formulating the problem is perhaps the most crucial
step in optimization (from verbal statement of agiven application and organizing them into aprescribed mathematical form) The objective function (economic criterion)
The process model (constraints)
The objective function represents such factors asprofit, cost, energy, and yield in terms of the keyvariables of the process being analyzed
The process model and constraints describe theinterrelationships of the key variables
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Problem formulation
What optimization is all about
Optimization is concerned with selecting the bestvalue by efficient quantitative methods
Why optimize?
Largest production
Greatest profit
Minimum cost
The least energy usage
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Problem formulation
Examples of applications of optimization
Determining the best sites for plant location
Routing tankers for the distribution of crude and refined products
Sizing and layout of a pipeline
Designing equipment and an entire plant
Scheduling maintenance and equipment replacement
Operating equipment, such as tubular reactors, columns, andabsorbers
Evaluating plant data to construct a model of a process
Minimizing inventory charges
Allocating resources or services among several processes
Planning and scheduling construction
Example: See ref.
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Problem formulation
Main features of optimization problems
At least one objective function to be optimized
Equality constraints (equations)
Inequality constraints (inequalities)
Economic model
}Model of process
or equipment
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Problem formulation
Main features of optimization problems
Feasible solution/Feasible region
Underdetermined
Optimal solution
Overdetermined
Degrees of freedom
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Problem formulation
An optimization problem:
Minimize: f(x) objective function
Subject to: h(x) = 0 equality constraints
h(x) is a vector of equations of dim. m1g(x) is a vector of equations of dim. m2
where x = (x1 xn) X Rn
D =nx X|h(x) = 0, g(x) 0
o
g(x) 0 inequality constraint
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Problem formulation
Example: optimal scheduling
tA1 tA2
tB1 tB2
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Problem formulation
What is the objective function?
tA1 + tA2 = 365
tB1 + tB2 = 365
tAi 0
tBi 0
f(t) = tA1MA1SA1 + tA2MA2SA2+tB1MB1SB1 + tB2MB2SB2
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Problem formulation
Cc loi bi ton ti u (quy hoch ton hc)
Quy hoch tuyn tnh (QHTT)
V d thuc dng ny c Bi Ton Vn Ti
Quy hoch tham s (QHTS) l QHTT m cc h s trong
Quy hoch ng (QH): L qu trnh c nhiu giai on ni chung, hay cc qu trnh
pht trin theo thi gian ni ring
l tuyn tnhf(x), g(x), h(x)
ph thuc tham sf(x), g(x), h(x)
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Problem formulation
Cc loi bi ton ti u (quy hoch ton hc)
Quy hoch phi tuyn (QHPT)
Quy hoch ri rc (QHRR)
Quy hoch a mc tiu (QHMT) Nu trn cng mt min rng buc D ta xt nhiu hm
mc tiu khc nhau
l cc hm phi tuynf(x) hoc g(x) hoch(x)
DNu min rng buc l tp ri rc
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Formulation of the objective function
Translate a verbal statement or concept of
the desired objective into mathematical terms Example
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Formulation of the objective function
Example
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Formulation of the objective function
Example
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Formulation of the objective function
Example
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Problem formulation
The six steps used to solve optimization
problems Make a list of all of the process variables
Determine the criterion for optimization, and
specify the objective function in terms of thevariables defined in step 1 together withcoefficients (Economic model)
Using mathematical expressions, develop a valid
process or equipment model (Process model)that relates the input-output variables of theprocess and associated coefficients
Problem formulation
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Problem formulation
The six steps used to solve optimization
problems If the problem formulation is too large in scope
Break it up into manageable parts or
Simplify the objective function and model
Apply a suitable optimization technique to the
mathematical statement of the problem
Check the answers, and examine the sensitivity ofthe result to changes in the coefficients in the
problem and the assumptions
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Outline
Problem formulation
Nature and organization of Optimization problems Developing models for optimization (constraints or
process model)
Formulation of the objective function
Optimization theory and methods Optimization of unconstrained functions
Linear programming with constraints
Nonlinear programming with constraints
Multi-objective optimization
Applications of Optimization
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Scope of course
Optimization problems
OPTIMIZATION OF UNCONSTRAINED
FUNCTIONS: ONE-DIMENSIONAL SEARCHUNCONSTRAINED
MULTIVARIABLE OPTIMIZATION
NON LINEAR PROGRAMMINGLINEAR PROGRAMMING
MULTI-OBJECTIVE OPTIMIZATION
(OR MULTI-OBJECTIVE PROGRAMMING)