numerical Optimisation of one and two variables

Optimization

1. Unit - 6 OPTIMUM DESIGN AND DESIGN STRATEGY

2. An optimum design is based on the best or most favorable conditions. In almost every case, these optimum conditions can ultimately be reduced to a consideration of costs or profits . Thus, an optimum economic design could be based on conditions giving the least cost per unit of time or the maximum profit per unit of production . When one design variable is changed, it is often found that some costs increase and others decrease. Under these conditions, the total cost may go through a minimum at one value of the particular design variable, and this value would be considered as an optimum

3. An example illustrating the principles of an optimum economic design is presented in Fig. 11-1. In this simple case, the problem is to determine the optimum thickness of insulation for a given steam-pipe installation. As the insulation thickness is increased, the annual fixed costs increase, the cost of heat loss decreases, and all other costs remain constant. Therefore, as shown in Fig. 11-1, the sum of the costs must go through a minimum at the optimum insulation thickness.

4. INCREMENTAL COSTS Consideration of incremental costs shows that a final recommended design does not need to correspond to the optimum economic design, because the incremental return on the added investment may become unacceptable before the optimum point is reached. However, the optimum values can be used as a basis for starting the incremental-cost analyses. This chapter deals with methods for determining optimum conditions, and it is assumed that the reader understands the role of incremental costs in establishing a final recommended design.

5. INTANGIBLE AND PRACTICAL CONSIDERATIONS The various mathematical methods for determining optimum conditions, as presented in this chapter, represent on a theoretical basis the conditions that best meet the requirements. However, factors that cannot easily be quantitized or practical considerations may change the final recommendation to other than the theoretically correct optimum condition. Thus, a determination of an “optimum condition,” as described in this chapter, serves as a base point for a cost or design analysis, and it can often be quantitized in specific mathematical form. From this point, the engineer must apply judgment to take into account other important practical factors, such as return on investment or the fact that commercial equipment is often available in discrete intervals of size.

6. As an example, consider the case where an engineer has made an estimation of the optimum pipe diameter necessary to handle a given flow stream based on minimizing the costs due to fixed charges and frictional pumping costs. The mathematical result shows that the optimum inside pipe diameter is 2.54 in. based on costs for standard (schedule 40 steel pipe. Nominal pipe diameters available commercially in this range are 2; in. (ID of 2.469 in.) and 3 in. (ID of 3.069 in.). The practical engineer would probably immediately recommend a nominal pipe diameter of 2 in. without going to the extra effort of calculating return on investment for the various sizes available. This approach would normally be acceptable because of the estimations necessarily involved in the optimum calculation and because of the fact that an investment for pipe represents only a small portion of the total investment.

7. Intangible factors may have an effect on the degree of faith that can be placed on calculated results for optimum conditions. Perhaps the optimum is based on an assumed selling price for the product from the process, or it might be that a preliminary evaluation is involved in which the location of the plant is not final obviously, for cases of this type, an analysis for optimum conditions can give only a general idea of the actual results that will be obtained in the final plant, and it is not reasonable to go to extreme limits of precision and accuracy in making recommendations. Even for the case of a detailed and firm design, intangibles, such as the final bid from various contractors for the construction, may make it impractical to waste a large amount of effort in bringing too many refinements into the estimation of optimum conditions.

8. GENERAL PROCEDURE FOR DETERMINING OPTIMUM CONDITIONS The first step in the development of an optimum design is to determine what factor is to be optimized . Typical factors would be total cost per unit of production or per unit of time , profit, amount of final product per unit of time, and percent conversion. Once the basis is determined, it is necessary to develop relationships showing how the different variables involved affect the chosen factor. Finally, these relationships are combined graphically or analytically to give the desired optimum conditions.

9. PROCEDURE WITH ONE VARIARLE There are many cases in which the factor being optimized is a function of a single variable. The procedure then becomes very simple. Consider the example presented in Fig. 11-1, where it is necessary to obtain the insulation thickness which gives the least total cost. The primary variable involved is the thickness of the insulation, and relationships can be developed showing how this variable affects all costs. Cost data for the purchase and installation of the insulation are available, and the length of service life can be estimated. Therefore, a relationship giving the effect of insulation thickness on fixed charges can be developed. Similarly, a relationship showing the cost of heat lost as a function of insulation thickness can be obtained from data on the value of steam, properties of the insulation, and heat-transfer considerations. All other costs, such as maintenance and plant expenses, can be assumed to be independent of the insulation thickness. The two cost relationships obtained might be expressed in a simplified form similar to the following

10. The graphical method for determining the optimum insulation thickness is shown in Fig. 11-1. The optimum thickness of insulation is found at the minimum point on the curve obtained by plotting total variable cost versus insulation thickness. The slope of the total-variable-cost curve is zero at the point of optimum insulation thickness. Therefore, if Eq. (3) applies, the optimum value can be found an analytically by merely setting the derivative of C, with respect to x equal to zero and solving for X.

11. PROCEDURE WITH TWO OR MORE VARIABLES When two or more independent variables affect the factor being optimized, the procedure for determining the optimum conditions may become rather tedious; however, the general approach is the same as when only one variables involved.

12. GRAPHICAL PROCEDURE. The relationship among C T, x, and y could be shown as a curved surface in a three-dimensional plot, with a minimum value of C, occurring at the optimum values of x and y. However, the use of a three-dimensional plot is not practical for most engineering determinations. The optimum values of x and y in Eq. (8) can be found graphically on a two-dimensional plot by using the method indicated in Fig. 11-2. In this figure, the factor being optimized is plotted against one of the independent variables (x), with the second variable (y) held at a constant value. A series of such plots is made with each dashed curve representing a different constant value of the second variable. As shown in Fig. 11-2, each of the curves (A, B, C, D, and E) gives one value of the first variable x at the point where the total cost is a minimum. The curve NM represents the locus of all these minimum points, and the optimum value of x and y occurs at the minimum point on curve NM.

13. Similar graphical procedures can be used when there are more than two independent variables. For example, if a third variable z were included in Eq. (8), the first step would be to make a plot similar to Fig. 11-2 at one constant value of z. Similar plots would then be made at other constant values of z. Each plot would give an optimum value of x, y, and C, for a particular z. Finally, as shown in the insert in Fig. 11-2, the overall optimum value of x, y, z, and C, could be obtained by plotting z versus the individual optimum values of CT.

14. At the optimum conditions, both of these partial derivatives must be equal to zero; thus, Eqs . (9) and (10) can be set equal to zero and the optimum values of x = ( cb /a 2 )1/3 and y = ( ab /c 2 )1/3 can be obtained by solving the two simultaneous equations. If more than two independent variables were involved, the same procedure would be followed, with the number of simultaneous equations being equal to the number of independent variables.

15. THE BREAK-EVEN CHART FOR PRODUCTION SCHEDULE AND ITS SIGNIFICANCE FOR OPTIMUM ANALYSIS In considering the overall costs or profits in a plant operation, one of the factors that has an important effect on the economic results is the fraction of total available time during which the plant is in operation. If the plant stands idle or operates at low capacity, certain costs, such as those for raw materials and labor, are reduced, but costs for depreciation and maintenance continue at essentially the same rate even though the plant is not in full use.

16. There is a close relationship among operating time, rate of production, and selling price. It is desirable to operate at a schedule which will permit maximum utilization of fixed costs while simultaneously meeting market sales demand and using the capacity of the plant production to give the best economic results. Figure 11-3 shows graphically how production rate affects costs and profits. The fixed costs remain constant while the total product cost, as

17. OPTIMUM PRODUCTION RATES IN PLANT OPERATION The same principles used for developing an optimum design can be applied when determining the most favorable conditions in the operation of a manufacturing plant. One of the most important variables in any plant operation is the amount of product produced per unit of time . The production rate depends on many factors, such as the number of hours in operation per day, per week, or per month; the load placed on the equipment; and the sales market available. From an analysis of the costs involved under different situations and consideration of other factors affecting the particular plant, it is possible to determine an optimum rate of production or a so-called economic lot size.

18. When a design engineer submits a complete plant design, the study ordinarily is based on a given production capacity for the plant. After the plant is put into operation, however, some of the original design factors will have changed, and the optimum rate of production may vary considerably from the “designed capacity.” For example, suppose a plant had been designed originally for the batch wise production of an organic chemical on the basis of one batch every 8 hours . After the plant has been put into operation, cost data on the actual process are obtained, and tests with various operating procedures are conducted. It is found that more total production per month can be obtained if the time per batch is reduced. However, when the shorter batch time is used, more labor is required, the percent conversion of raw materials is reduced, and steam and power costs increase . Here is an obvious case in which an economic balance can be used to find the optimum production rate. Although the design engineer may have based the original recommendations on a similar type of economic balance, price and market conditions do not remain constant, and the operations engineer now has actual results on which to base an economic balance. The following analysis indicates the general method for determining economic production rates or lot sizes.

19. The total product cost per unit of time may be divided into the two classifications of 1. O perating costs 2. Organization costs. Operating costs depend on the rate of production and include expenses for direct labor, raw materials, power, heat, supplies and similar items which are a function of the amount of material produced. Organization costs are due to expenses for directive personnel, physical equipment, and other services or facilities which must be maintained irrespective of the amount of material produced. Organization costs are independent of the rate of production.

20. It is convenient to consider operating costs on the basis of one unit of production. When this is done, the operating costs can be divided into two types of expenses as follows: (1) Minimum expenses for raw materials, labor, power, etc., that remain constant and must be paid for each unit of production as long as any amount of material is produced; (2) Extra expenses due to increasing the rate of production. These extra expenses are known as super production costs. They become particularly important at high rates of production. Examples of super production costs are extra expenses caused by overload on power facilities, additional labor requirements, or decreased efficiency of conversion. Super production costs can often be represented as follows:

21. OPTIMUM PRODUCTION RATE FOR MINIMUM COST PER UNIT OF PRODUCTION It is often necessary to know the rate of production which will give the least cost on the basis of one unit of material produced. This information shows the selling price at which the company would be forced to cease operation or else operate at a loss. At this particular optimum rate, a plot of the total product cost per unit of production versus the production rate shows a minimum product cost; therefore, the optimum production rate must occur where dC T / dP = 0. An analytical solution for this case may be obtained from Eq. (121, and the optimum rate P o giving the minimum cost per unit of production is found as follows:

22. OPTIMUM PRODUCTION RATE FOR MAXIMUM TOTAL PROFIT PER UNIT OF TIME In most business concerns, the amount of money earned over a given time period is much more important than the amount of money earned for each unit of product sold. Therefore, it is necessary to recognize that the production rate for maximum profit per unit of time may differ considerably from the production rate for minimum cost per unit of production. Equation (15) presents the basic relationship between costs and profits. A plot of profit per unit of time versus production rate goes through a maximum. Equation (19), therefore, can be used to find an analytical value of the optimum production rate. When the selling price remains constant, the optimum rate giving the maximum profit per unit of time is

23. OPTIMUM CONDITIONS IN CYCLIC OPERATIONS Many processes are carried out by the use of cyclic operations which involve periodic shutdowns for discharging, cleanout, or reactivation . This type of operation occurs when the product is produced by a batch process or when the rate of production decreases with time , as in the operation of a plate-and-frame filtration unit . In a true batch operation, no product is obtained until the unit is shut down for discharging. In semi continuous cyclic operations, product is delivered continuously while the unit is in operation, but the rate of delivery decreases with time.

24. Thus, in batch or semi continuous cyclic operations, the variable of total time required per cycle must be considered when determining optimum conditions . Analyses of cyclic operations can be carried out conveniently by using the time for one cycle as a basis. When this is done, relationships similar to the following can be developed to express overall factors, such as total annual cost or annual rate of production:

25. SEMICONTINUOUS CYCLIC OPERATIONS Semicontinuous cyclic operations are often encountered in the chemical industry, and the design engineer should understand the methods for determining optimum cycle times in this type of operation. Although product is delivered continuously, the rate of delivery decreases with time owing to scaling, collection of side product, reduction in conversion efficiency, or similar causes . It becomes necessary, therefore, to shut down the operation periodically in order to restore the original conditions for high production rates . The optimum cycle time can be determined for conditions such as maximum amount of production per unit of time or minimum cost per unit of production.

26. Scale Formation in Evaporation During the time an evaporator is in operation, solids often deposit on the heat-transfer surfaces, forming a scale. The continuous formation of the scale causes a gradual increase in the resistance to the flow of heat and, consequently, a reduction in the rate of heat transfer and rate of evaporation if the same temperature-difference driving forces are maintained. Under these conditions, the evaporation unit must be shut down and cleaned after an optimum operation time, and the cycle is then repeated.

27. Scale formation occurs to some extent in all types of evaporators, but it is of particular importance when the feed mixture contains a dissolved material that has an inverted solubility. The expression inverted solubility means the solubility decreases as the temperature of the solution is increased . For a material of this type, the solubility is least near the heat-transfer surface where the temperature is the greatest. Thus, any solid crystallizing out of the solution does so near the heat-transfer surface and is quite likely to form a scale on this surface. The most common scale-forming substances are calcium sulfate, calcium hydroxide, sodium carbonate, sodium sulfate, and calcium salts of certain organic acids. When true scale formation occurs, the overall heat-transfer coefficient may be related to the time the evaporator has been in operation by the straight-line equation

28. CYCLE TIME FOR MINIMUM COST PER UNIT OF HEAT TRANSFER There are many different circumstances which may affect the minimum cost per unit of heat transferred in an evaporation operation. One simple and commonly occurring case will be considered. It may be assumed that an evaporation unit of fixed capacity is available, and a definite amount of feed and evaporation must be handled each day. The total cost for one cleaning and inventory charge is assumed to be constant no matter how much boiling time is used. The problem is to determine the cycle time which will permit operation at the least total cost

29. The total cost includes (1) fixed charges on the equipment and fixed overhead expenses, (2) steam, materials, and storage costs which are proportional to the amount of feed and evaporation, (3) expenses for direct labor during the actual evaporation operation, and (4) cost of cleaning. Since the size of the equipment and the amounts of feed and evaporation are fixed, the costs included in (1) and (2) are independent of the cycle time. The optimum cycle time, therefore, can be found by minimizing the sum of the costs for cleaning and for direct labor during the evaporation. If C, represents the cost for one cleaning and S, is the direct labor cost per hour during operation, the total variable costs during H of operating and cleaning time must be

30. The optimum cycle time determined by the preceding methods may not fit into convenient operating schedules. Fortunately, as shown in Figs. 11-4 and 11-5, the optimum points usually occur where a considerable variation in the cycle time has little effect on the factor that is being optimized. It is possible, therefore, to adjust the cycle time to fit a convenient operating schedule without causing much change in the final results. The approach described in the preceding sections can be applied to many different types of semicontinuous cyclic operations. An illustration showing how the same reasoning is used for determining optimum cycle times in filter-press operations is presented in Example 4.

31. FLUID DYNAMICS (OPTIMUM ECONOMIC PIPE DIAMETER) The investment for piping and pipe fittings can amount to an important part of the total investment for a chemical plant. It is necessary, therefore, to choose pipe sizes which give close to a minimum total cost for pumping and fixed charges. For any given set of flow conditions, the use of an increased pipe diameter will cause an increase in the fixed charges for the piping system and a decrease in the pumping or blowing charges. Therefore, an optimum economic pipe diameter must exist. The value of this optimum diameter can be determined by combining the principles of fluid dynamics with cost considerations. The optimum economic pipe diameter is found at the point at which the sum of pumping or blowing costs and fixed charges based on the cost of the piping system is a minimum.

32. Pumping or Blowing Costs For any given operating conditions involving the flow of a noncompressible fluid through a pipe of constant diameter, the total mechanical-energy balance can be reduced to the following form:

33. THE STRATEGY OF LINEARIZATION FOR OPTIMIZATION ANALYSIS In the preceding analyses for optimum conditions, the general strategy has been to establish a partial derivative of the dependent variable from which the absolute optimum conditions are determined. This procedure assumes that an absolute maximum or minimum occurs within attainable operating limits and is restricted to relatively simple conditions in which limiting constraints are not exceeded. However, practical industrial problems often involve establishing the best possible program to satisfy existing conditions under circumstances where the optimum may be at a boundary or limiting condition rather than at a true maximum or minimum point.

34. A typical example is that of a manufacturer who must determine how to blend various raw materials into a final mix that will meet basic specifications while simultaneously giving maximum profit or least cost. In this case, the basic limitations or constraints are available raw materials, product specifications, and production schedule, while the overall objective (or objective function) is to maximize profit.