The purpose of this work is to summarize and organize successful approaches that can be used by the chemical process engineer when faced with optimization problems. In general, optimization is making any process better; common goals are to increase efficiency or maximize the economic potential of a process. Improvements made to a process during optimization are generally measured in terms of an objective function.
Since the goal of any industrial chemical process is to be profitable, objective functions mostly have units of dollars. The objective function depends on changes in decision variables, those variables over which the engineer has control. The goals of each approach are the same: maximize the profitability of a process within the given constraints.
The information presented in this work is not intended for professionals in the field, but rather for a general audience seeking to organize their thoughts on chemical process optimization and provide a basis from which to address process optimization problems.
Introduction and Purpose
Background and Terms
Objective Functions
Focusing on individual cost or savings areas rather than overall costs or savings is called incremental analysis. Changing the process continues as long as the return on incremental investment is greater than the MARR.1. Not all objective functions are directly based on economics, but the objective function should be quantitative.1 For example, the objective may be to maximize the production of a chemical or to minimize the concentration of a pollutant in order to comply with environmental regulations.
If maximizing profit, not revenue, is the goal, then maximizing production may not be desirable. Likewise, if the goal is to cause the least damage to the environment, then reducing the concentration of contaminant, rather than the total flow rate, may not be the best approach.1.
Decision Variables
Thus, high overall feedstock conversion and recovery of unreacted feedstock is an essential goal in optimization. Since there are an infinite number of combinations of decision variables within a process, knowing the sensitivity of the objective function to changes in the decision variables provides useful insight into where one should focus one's efforts. For example, if the goal is to increase conversion with one pass through a reactor for which temperature, pressure, and volume are the decision variables, then elaborate mathematical models can be constructed to estimate the sensitivity of the objective function to each variable.
However, the most efficient technique is to evaluate the objective function at the boundaries of each variable. If there is little effect on the objective function over the range of possible pressures, another variable, such as temperature, should be chosen. Many process simulation software tools can evaluate an objective function or dependent decision variable, such as single-pass conversion, over a range of pressures, temperatures, and reactor volumes.
Graphs can be generated that show clear trends, or no trend at all if the dependent variable is not very sensitive to changes in the independent variable.
Constraints and the Process Optimum
In general, an equality constraint reduces the number of truly independent decision variables, while an inequality constraint limits the range over which a decision variable can be evaluated.1 Constraints can be imposed by environmental regulations, industry standards, or consumer preferences. Often times, constraints simplify the optimization process by limiting the possibilities to be evaluated. In general, the goal of an optimization problem is to find the extreme value of the objective function for a process.
A situation where the objective function has been minimized or maximized, whichever is desirable, is called the local optimum. In other words, no small, permissible change in decision variables in any direction will improve the objective function once the local optimum is reached.1 It is worth mentioning that almost all optimization problems of any complexity have local optima at the extrema of at least one restriction. The global optimum, on the other hand, is a situation where the best objective function exists for all allowed values of the decision variables.1 A true global optimum will almost certainly defy process constraints.
The global optimum is not an achievable goal, however there are a number of approaches to optimization that will guide the chemical process engineer in that direction.
Approaches to Optimization
If the objective function includes capital and operating costs, the base case analysis should include calculations of equipment sizes and prices and material and energy balances to determine utility costs. The analysis should clearly show the effect of changes in all relevant decision variables on the objective function.1 Sensitivity analysis, discussed later, is an effective visual aid in determining which decision variables have the greatest effect on the objective function. Almost every change in the terms of the process has some downstream effect that needs to be taken into account.
Topological Optimization
Reducing the per-through conversion of limiting reactant through a reactor can suppress side reactions by reducing the concentration of the reactants that produce unwanted by-products. This is accomplished via a recycling loop that flows from the reactor effluent to the reactor inlet. If A is fed to the reactor in sufficient excess than B, the concentration of B in the reactor will be lowered so that the molecular collisions that cause the second reaction will be minimized.
This results in the suppression of the undesired side reaction and thus the reduced concentration of the undesired by-product in the downstream direction. If the resulting concentration of an undesired byproduct is below the highest specification that could be set by the customer or environmental regulation, the separation of the final product could be facilitated, possibly allowing the separation of the other. Do the easiest separation first—that is, the one that requires the least trays and reflux—and leave the most difficult for last.
There is an abundance of separation technologies available to the chemical process engineer for the separation of raw materials and products. Single pass conversion is the ratio of reactant consumed in the reactor to the reactant fed to the reactor. Selectivity is the ratio of the production rate of the desired product to the production rate of unwanted by-products.
The conversion of the limiting reactant to the desired product is limited by competition from undesired reactions.1 Yield is the ratio of moles of reactant reacted to produce the desired product to moles of limiting reactant reacted. Heat must be supplied to the reaction efficiently enough for the reaction to proceed. In both cases, the rate of heat transfer depends on the reactor and heat transfer configurations, the properties of the reacting stream and the heat transfer medium, and the temperature driving force, which can be affected by temperature gradients.1 4.
The basic premise of process heat integration optimization is to take heat generated in one section of a process, perhaps from an exothermic reaction, and use it in another section of the process. One method of achieving this is to use heat to produce steam from the boiler feedwater, which can be used in another part of the process that requires heat, perhaps to vaporize the contents within the reboiler of a distillation column. To minimize the rate of utility current flow, a minimum approach temperature should be used, generally around 10°C.
This means that the temperature difference between the inlet of the usable stream and the outlet of the process stream in the heat exchanger must be at least 10 °C.
Parametric Optimization
There are methods by which the minimum number of heat exchangers needed in a process can be found, but those methods are beyond the scope of this work. Selectivity of the desired products is a function of the single-door conversion, which in turn is a function of the parameters mentioned in (1). This is true even for recycling loops, where any change in operating conditions within the loop affects the operation of all pieces of equipment in the loop.
Non-loop process equipment, such as a distillation column that separates a binary mixture of two products, can be considered independently once the upstream process has been optimized. During parametric optimization, it is common for the plant topology to remain unchanged unless the result of optimizing a process condition allows the elimination of process equipment, as previously mentioned. It is worth remembering that utilities account for a large portion of a plant's annual costs, so time is well spent minimizing their effect on the target function.
To minimize the amount of time spent in process simulation, the number of points evaluated for each variable must be chosen wisely. The case study function allows an independent variable, such as reactor temperature, to be varied over a specified interval and the effect on the dependent variable, or objective function, is represented by graphs or tables which can clearly show local maxima or minima. Another optimization tool that can help guide where to spend more time is a sensitivity analysis, which measures the sensitivity of the objective function to changes in the decision variables.
A sensitivity analysis can be constructed by changing one decision variable while holding all others constant and observing the effect on the objective function. This can be a very effective visual tool to convey which decision variable should receive the most attention. Note that the greatest effect on the estimated NPV occurs when the selling price of the product is varied.
Raw material costs, on the other hand, can be effectively lowered by optimization, either by achieving better conversion and selectivity of the desired product, or by recovering more unconverted raw materials. For the arbitrary case, the costs of consumables and equipment have little effect on NPV, primarily because they represent only a small part of the cost of the process.
Conclusions
Example: Utility Optimization
Many constraints must be met and some assumptions made, but in the end it is possible to fairly compare the options to choose the one that provides the greatest increase in the economic potential of the process.
