Process Optimization

By

Dr : Mona Ossman

What is optimization?• Optimization is derived from the Latin word “optimus”, the

best.

• Optimization characterizes the activities involved to find “the best”.

• The goal of optimization is to find the values of the variables in the process that yield the best value of the performance

• Optimization is a mathematical discipline that concerns the finding of minima and maxima of functions, subject to so-

called constraints.

minimize: C = 1,000P +4*10^9/P*R + 2.5*10^5Rsubject to: P*R = 9000

Applications of optimization• people use optimization, often without actually realizing,

for simple things such as traveling from one place to another and time management,

• • optimization finds many applications in engineering,

science, business, economics, etc.

• Optimization has many applications in chemical, mineral processing, oil and gas, petroleum, pharmaceuticals and related industries.

• Optimization of chemical and related processes requires a mathematical model that describes and predicts the process behavior.

EXAMPLES OF APPLICATIONS OF OPTIMIZATION

• Optimization can be applied in numerous ways to chemical processes and plants.

• Typical projects in which optimization has been used include:

1. Determining the best sites for plant location. 2. Routing tankers for the distribution of crude

and refined products. 3. Sizing and layout of a pipeline. 4. Designing equipment and an entire plant.

5. Scheduling maintenance and equipment replacement. 6. Operating equipment, such as tubular reactors,

columns, and absorbers. 7. Evaluating plant data to construct a model of a

process. 8. Minimizing inventory charges. 9. Allocating resources or services among several

processes. 10. Planning and scheduling construction.

EXAMPLES OF APPLICATIONS OF OPTIMIZATION

THE ESSENTIAL FEATURES OF OPTIMIZATION PROBLEMS

• The formulation of an optimization problem must use mathematical expressions. Such expressions do not necessarily need to be very complex.

• Every optimization problem contains three essential categories:

1. At least one objective function to be optimized (profit function, cost function,etc.).

2. Equality constraints (equations).

3. Inequality constraints (inequalities).

optimization basicsPROBLEM FORMULATION

• Formulating the problem is the most crucial step in optimization.

• Problem formulation requires identifying the essential elements of a conceptual or verbal statement of a given application and organizing them into a prescribed mathematical form, namely,

1. The objective function (economic model)The objective function represents such factors as profit, cost, energy, and yield in terms of the key variables of the process being analyzed. sometimes called the economic model.

2. The process model The process model describe the interrelationships of the key variables whish is either

a- Equality constraints (equations).b- Inequality constraints (inequalities).

• In this text the following notation will be used for each category of the optimization

problem:

Example:• In this example we illustrate the formulation of the components of an

optimization problem.

• We want to schedule the production in two plants, A and B, each of which can manufacture two products: 1 and 2. How should the scheduling take place to maximize profits while meeting the market requirements based on the following data:

• How many days per year (365 days) should each plant operate processing each kind of material? Hints: Does the table contain the variables to be optimized? How do you use the information mathematically to formulate the optimization problem? What other factors must you consider?

Solution.

• How should we start to convert the words of the problem into mathematical

statements?

First, let us define the variables.

• There will be four variables whish are tA1,tA2,tB1,and tB2 representing, respectively, the number of days per year each plant operates on each material.

What is the objective function?

• We select the annual profit so that

Do any equality constraints evolve from the problem statement or from

implicit assumptions? If each plant runs 365 days per year, two equality constraints arise

Do any inequality constraints evolve from the problem statement or implicit assumptions?

• On first glance it may appear that there are none, but further thought indicates t must be nonnegative since negative values of t have no physical meaning.

• Other inequality constraints might be added after further analysis, such as a limitation on the total amount of material 2 that can be sold which is (L1):

or a limitation on production rate for each product at each plant, namely

To find the optimal t, we need to optimize (a) subject to constraints (b) to (g).

Example to be discussed in class• Suppose the flow rates entering

and leaving a process are measured periodically.

• Determine the best value for stream A in kg/h for the process shown from the three hourly measurements indicated of B and C in next figure, assuming steady-state operation at a fixed operating point.

• The process model is

• where M is the mass per unit time of throughput.