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Search Methods 547 (CSD) method, the method of feasible directions, the gradient projection method, and the generalized reduced gradient (GRG) method. Many efcient algorithms have been developed to obtain the optimum with the least number of trials or iterations. Some of these are available in the public domain, while others are available commercially. The difference between all these methods lies in decid- ing on the direction of the move and the scheme used to return to the constraint. The major problem remains the calculation of the gradients. Linearization of the nonlinear optimization problem is also carried out in some cases, and linear pro- gramming techniques can then be used for the solution. For details on these and other methods, see Arora (2004). 9.5 EXAMPLES OF THERMAL SYSTEMS We have discussed a wide range of search methods and their application to ther- mal systems in Chapter 7 and in this chapter. A few examples are given here for illustration of the application of these methods to practical thermal systems. Optimization of the optical ber drawing furnace, as shown in Figure 1.10(c), can be carried out based on the numerical simulation of the process. Because of the dominant interest in ber quality, the objective function can be based on the ten- sion, defect concentration, and velocity difference across the ber, all these being the main contributors to lack of quality. These are then scaled by the maximum values obtained over the design domain to obtain similar ranges of variation. The objective function U is taken as the square root of the sum of the squares of these three quantities and is minimized. The two main process variables are taken as the furnace temperature, representing the maximum in a parabolic distribution, and the draw speed. The univariate search method is applied, using the golden section search for each variable and alternating from one variable to the other. Figure 9.15 shows typical results from this search strategy for the optimal draw temperature and draw speed. The results from the rst search are used in the sec- ond search, following the univariate search strategy, to obtain optimal design in terms of these two variables. The optimization process can be continued though additional iterations to narrow the domain further. However, each iteration is time consuming and expensive. Several other results have been obtained on this com- plicated problem, particularly on furnace dimensions and operating conditions to achieve optimal drawing. Another problem that is considered for illustration is a chemical vapor deposi- tion (CVD) system, shown in Figure 9.16(a), for the deposition of materials such as silicon and titanium nitride (TiN) on a given surface, known as the substrate, to fabricate electronic devices or to provide a coating on a given part. The main quantities of interest include product quality, production rate, and operating cost. These three may be incorporated into one possible objective function, such as that given by U (coating nonuniformity) r (operating cost)/production rate. The objective function represents equal weighting for each of these characteristics. A minimum value in U implies greater lm thickness uniformity. Operating costs are represented by heat input and gas ow rate. The production rate is expressed 548 Design and Optimization of Thermal Systems in terms of the deposition rate. All these quantities are normalized to provide uni- form ranges of variation. Obviously, many different formulations of the objective function can be used. A detailed study of the design space is carried out to deter- mine the domain of acceptable designs and the effects of various parameters on the objective function. Using the steepest ascent method, with univariate search, 4.0 3.0 2.0 1.0 Objective function 0.0 2200.0 2400.0 Draw temperature T F (K) 2600.0 2800.0 3000.0 2.0 1.5 1.0 0.5 Objective function 0.0 5.0 10.0 15.0 20.0 25.0 30.0 Draw speed V f (m/s) FIGURE 9.15 Optimization of the optical ber drawing process: Evaluation of optimal furnace draw temperature at a draw speed of 15 m/s and the optimal draw speed at a draw temperature of 2489.78 K by using the golden section search method. The objective func- tion U is chosen to represent ber quality. Search Methods 549 the optimal design is obtained. Some typical results are shown in Figure 9.16, indicating the minimization of the objective function with the inlet velocity V inlet and the susceptor temperature T sus . Again, other objective functions, design parameters, and operating conditions can be considered to optimize the system and the process (Jaluria, 2003). A problem that is of considerable interest in the cooling of electronic equip- ment is one pertaining to heat transfer from isolated heat sources, representing electronic components, located in a channel, as shown in Figure 9.17(a). A vortex generator is placed in the channel to oscillate the ow and thus enhance the heat transfer. The main quantities of interest are the pressure head $P and the heat transfer rates Q 1 and Q 2 from the two sources. It is desirable to maximize the heat transfer rates from the two sources, to accommodate more electronic com- ponents in a given space, and to minimize the pressure head, which affects the cost of the cooling system. These three quantities can be considered separately FIGURE 9.16 Optimization of a chemical vapor deposition (CVD) system, sketched in: (a) Variation of the objective function U with (b) inlet velocity and (c) susceptor tem- perature. The objective function U is dened as U (coating nonuniformity r operating cost)/production rate. 14 (a) ( b )( c ) 12 10 8 6 4 2 0 0.01 0.02 0.03 0.04 V inlet (m/s) 0.05 0.06 0.07 O b jective f unction U 8 6 4 2 0 723 773 823 873 T sus (K) 923 973 1023 Objective function U Hot susceptor γ Cooled walls Flow Horizontal Reactor 550 Design and Optimization of Thermal Systems as a multi-objective function problem, or they can be combined, in their normal- ized forms, to form a single objective function, F, which is then maximized. One such objective function, with the normalized quantities indicated by overbars is FWQ WQ WP 11 2 2 3 $ (9.30) where the W’s are the weights of the three individual objective functions. The choice of the weights strongly depends on the design priorities. The responses for two different objective functions, as obtained from different values of the W’s, are presented in Figure 9.17. Here, both experimental and numerical results are employed for the data points. For the rst case, the optimal Reynolds num- ber is obtained as 5600 and the height of the vortex promoter h p /H as 0.12. For the second case, the Reynolds number is obtained as 5460 and the vortex promoter height as 0. Thus, a greater emphasis on pressure in the second case leads to a better solution without a promoter. If the weight W 3 for pressure is made half of the weights for the heat transfer rates, the optimal promoter height ! FIGURE 9.17 (a) A simple system for cooling of electronic equipment, consisting of two heat sources representing electronic components and a vortex promoter. (b) Response surface for the objective function FWQWQ WP 11 2 2 3 $ for W 1 W 2 W 3 . (c) Response surface for the objective function F for W 1 W 2 W 3 / 2. Search Methods 551 is obtained as 0.26. Similarly, other weights and promoter geometries can be considered (Icoz and Jaluria, 2006). The preceding electronic cooling system can also be considered without the vortex promoter. The total heat transfer rate and the pressure head are taken as the two main objective functions. Response surfaces can be drawn from these to investigate the optimum. Multi-objective function optimization can also be used, as discussed in Chapter 7. As was done for the preceding problem, both experi- mental and computation data are used to build the database for the response sur- faces in terms of length dimensions L 1 and L 2 . Second-order, third-order, and higher-order regression models are considered. Comparing the second order with the third order, it was observed that the third-order tting was substantially a better choice because it had higher correlation coefcients. The difference between third- order and fourth-order models was small. Hence, the third-order model, based on computational and experimental data, was employed as the regression model for the multi-objective design optimization problem. The response surfaces obtained from this regression model for $P and the total heat transfer rate, given in terms of the Stanton number, St, where St is the Nusselt number divided by the Reyn- olds and Prandtl numbers, are shown in Figures 9.18(a) and (b), respectively. After the regression model is obtained for dimensionless $P and St, the Pareto Set is obtained for the multi-objective design optimization problem (Zhao et al., 2007). The resulting Pareto Set is plotted in Figure 9.18(c). From the gure, it is observed that if the pressure drop is decreased, implying a lower pumping cost, the Stanton number is also decreased, and vice versa. The maximum Stanton number and the minimum pressure drop cannot be obtained at the same time. This is expected from the discussion of the physical problem given earlier. A higher heat transfer rate requires a greater ow rate, which in turn needs a greater pressure head. How- ever, interest lies in maximizing heat transfer and minimizing the pressure head. Thus, for decision-making, other considerations have to be added to select the proper solution from the Pareto Set, as outlined earlier in Chapter 7. 9.6 SUMMARY This chapter presents search methods, which constitute one of the most impor- tant, versatile, and widely used approaches for optimizing thermal systems. Search methods can be used if the objective function and constraints are con- tinuous functions as well as if these take on discrete values. In many circum- stances, combinations of the components and other design variables yield a nite number of feasible designs. Search methods are ideally suited for such problems to determine the best or optimum design. Both constrained and unconstrained optimizations can be carried out using search methods. The simplest problem of single-variable unconstrained optimization is consid- ered rst. Such circumstances are of limited practical interest, but are illustrative of the optimization techniques for more complicated problems. In addition, mul- tivariable problems are often broken down into simpler single-variable problems for which these methods can be used. An exhaustive search for the optimum in the 552 Design and Optimization of Thermal Systems feasible design domain is sometimes used because of its simplicity and to deter- mine subdomains containing the optimum, even though it is not very efcient. In addition, experience and information available on the system can frequently be used with an unsystematic search to focus on a particular subdomain to extract the optimum. Efcient elimination methods, such as Fibonacci and dichotomous schemes, are presented next. The efciency of these methods in reducing the 0.026 0.024 0.022 0.02 0.018 0.016 0.014 0.012 0.01 0.008 0.2 0.3 0.4 (c) Pressure drop Stanton number 0.5 0.6 0 10 0 –10 DP –20 0 0 0 –0.5 0.5 0St 5 5 5 5 L2 L1 L2 L1 10 (a) (b) 10 2 1 0 –1 –2 –3 –5 –4 –6 –7 –8 –9 –10 –11 0.1 0.05 0 –0.05 –0.1 –0.15 –0.2 –0.25 –0.3 –0.35 –0.4 –0.45 FIGURE 9.18 Optimization of the system shown in Figure 9.17(a), without the vortex promoter. (a) Third-order response surface for $P. (b) Third-order response surface for Stanton number, St. (c) Pareto front for multi-objective design optimization. Search Methods 553 interval of uncertainty for a given number of iterative designs is discussed. These schemes are quite commonly used for the optimization of thermal systems. Multivariable unconstrained problems are discussed next. A lattice search, which is relatively easy to use but is an inefcient method, is considered, followed by an univariate search strategy, which breaks the problem down into alternating searches with a single variable. This is an important approach because it allows the use of efcient methods, such as Fibonacci and calculus methods, to solve the problem as a series of single-variable problems. Hill-climbing techniques, such as steepest ascent, are very efcient for multivariable unconstrained problems. How- ever, this approach requires the determination of the derivatives of the objective function. These derivatives are obtained analytically in relatively simple cases and numerically in cases that are more complicated. However, this does limit the use of the method to problems that can be represented by continuous functions and expressions. Constrained multivariable problems are the most complicated ones encountered in the optimization of thermal systems. Because of their complexity, efforts are made to include the constraints in the objective function, thus obtaining an unconstrained problem. The inequality constraints generally dene the feasible domain and the equality constraints often arise from conservation principles. In the simulation of most thermal systems, the equations stemming from conservation laws are gener- ally part of the solution and do not result in equality constraints. However, there are problems that have to be solved as constrained problems. Two main approaches are presented in this chapter. The rst is the penalty function method, which denes a new objective function, with the constraints included, and imposes a penalty if the constraints are not satised. The second approach is based on searching along the constraint. Derivatives are needed for the implementation of this method, thus restricting its applicability to continuous and differentiable functions. Examples of the application of search methods to practical thermal systems are nally outlined. REFERENCES Arora, J.S. (2004) Introduction to Optimum Design, 2nd ed., Academic Press, New York. Dieter, G.E. (2000) Engineering Design: A Materials and Processing Approach, 3rd ed., McGraw-Hill, New York. Haug, E.J. and Arora, J.S. (1979) Applied Optimal Design, Wiley, New York. Icoz, T. and Jaluria, Y. (2006) Design optimization of size and geometry of vortex pro- moter in a two-dimensional channel, ASME J. Heat Transfer, 128:1081–1092. Jaluria, Y. (2003) Thermal processing of materials: From basic research to engineering, J. Heat Transfer, 125:957–979. Rao, S.S. (1996) Engineering Optimization: Theory and Practice, 3rd ed., Wiley, New York. Ravindran, A., Ragsdell, K.M., and Reklaitis, G.V. (2006) Engineering Optimization, Wiley, New York. Reklaitis, G.V., Ravindran, A., and Ragsdell, K.M. (1983) Engineering Optimization Methods and Applications, Wiley, New York. Siddall, J.N. (1982) Optimal Engineering Design, Marcel Dekker, New York. 554 Design and Optimization of Thermal Systems Stoecker, W.F. (1989) Design of Thermal Systems, 3rd ed., McGraw-Hill, New York. Vanderplaats, G.N. (1984) Numerical Optimization Techniques for Engineering Design, McGraw-Hill, New York. Zhao, H., Icoz, T., Jaluria, Y., and Knight, D. (2007) Application of data driven design optimization methodology to a multi-objective design optimization problem, J. Eng. Design, 18:343–359. PROBLEMS 9.1. Use a Fibonacci search to nd the minimum of the function U(x), where Ux xx x () [()]sin( /) ln 2 5 23 Obtain a nal interval of uncertainty in x of 0.1 or less. 9.2. Reduce the cylindrical storage tank problem considered in Example 8.3 to its unconstrained form and determine the optimal dimensions using the following search methods: (a) Uniform exhaustive search (b) Dichotomous search (c) Fibonacci search Compare the number of trial runs needed in the three cases and the nal solution obtained. Take the desired nal interval of uncertainty for the radius as 5 cm. 9.3. The amount of ammonia produced in the chemical reactor considered in Example 4.6 is to be optimized by varying the bleed over the range of 0 to 40 moles/s. Using any search method, with the numerical model given earlier, determine if an optimum in ammonia productions exists in this range and obtain the applicable bleed rate if it does. 9.4. An optimum ow rate is to be achieved in the fan and duct system con- sidered in Example 4.7 by varying the constants 15 and 80, which rep- resent the zero pressure and the zero ow parameters. Use any suitable search method to determine if the ow can be optimized by varying these two parameters over the range o30% of the given base values. 9.5. Use an univariate search to nd the optimum of the unconstrained objective function U(x,y) given by Ux y x xy y (, ) 2 20 3 Show that you have obtained a minimum. Also, use calculus methods to obtain the minimum and compare the results from the two approaches. Search Methods 555 9.6. The cost C of a storage chamber is given in terms of its three dimen- sions as C 12x 2 2y 2 5z 2 with the volume given as 10 units, i.e., xyz 10. Recast this problem as an unconstrained optimization problem with two independent vari- ables. Applying an univariate search, determine the dimensions that minimize the cost. 9.7. We wish to minimize the cost U of a system, where U is given in terms of the three independent variables x, y, and z as Uxy xz yz 1 16 2 Starting with the initial point (1, 0.5, 0.5), in x, y, and z, respectively, obtain the optimum by the univariate search method as well as by the steepest descent method with \$x\ 1.0. Compare the results and number of trial runs in the two cases. Is the given value of \$x\ satisfactory? 9.8. Apply any search method to solve the optimization problem for a solar energy system considered in Example 8.6. Employ the area A and the volume V as the two independent variables. Compare the results obtained with those presented in the example and the computational effort needed to obtain the solution. 9.9. In Example 5.1, an acceptable design of a refrigeration system was obtained to achieve the desired cooling. As seen earlier, an accept- able design may be selected from a wide domain. Considering the evaporator and condenser temperatures as the only design variables, formulate the optimization problem for maximizing the coefcient of performance. Using any suitable search method, determine the optimal design of the system. 9.10. The heat transfer Q from a spherical reactor of diameter D is given by the equation Q hT A, where h is the heat transfer coefcient, T is the temperature difference from the ambient, and A(PD 2 ) the surface area of the sphere. Here, h is given by the expression h 2 0.5T 0.2 D 1 A constraint also arises from material limitations as DT 20 Set up the optimization problem for minimizing the total heat transfer Q. Using the method of steepest ascent, obtain the optimum, starting at the initial point D 0.1 and T 50, with step size in T equal to 10. Also, 556 Design and Optimization of Thermal Systems obtain the minimum by simple differentiation of the unconstrained objective function and compare the results from the two approaches. 9.11. In a water ow system, the total ow rate Y is given in terms of two variables x and y as Y 8.5x 2 7.1y 3 25 with a constraint due to mass balance as x y 1.75 32 Solve this optimization problem both as a constrained problem and as an unconstrained problem, using any appropriate search method for the purpose. 9.12. The heat loss Q from a furnace depends on the temperature a and the wall thickness b as Q a ab b 2 2 4 2 Starting with the initial point (1,1), use the univariate search method to obtain the optimum value of Q. Also, apply the method of steepest ascent to obtain the optimum. Is it a maximum or a minimum? 9.13. The cost of a thermal system is given by the expression Cxy xy ¤ ¦ ¥ ³ µ ´ (3.3 4 ) 22 1400 1500 where x and y are the sizes of two components. The terms within the rst parentheses represent the capital costs and the terms within the second parentheses quantify the maintenance costs. Using the method of steepest ascent, calculate the values of x and y that optimize the cost. 9.14. In an extrusion process, the diameter ratio x, the velocity ratio y, and the temperature z are the main design variables. The cost function is obtained after including the constraints as C x yxz xy z58 3 4 305 Using any suitable optimization technique, obtain the optimal cost or show a few steps toward the minimum. 9.15. Apply the method of searching along a constraint to solve the con- strained two-variable problem given in Example 9.5. Compare the [...]... C* 3. 5 w1 w1 14. 8 w2 w2 5 64 Design and Optimization of Thermal Systems with w2 w1 1 and 1.4w1 2.2w2 0 From these equations, w1 0.611 and w2 0 .38 9 Therefore, C* (3. 5/0.611)0.611 ( 14. 8/0 .38 9)0 .38 9 11.965 Then, from Equation (10. 13) and Equation (10. 14) , 3. 5 m1 .4 0.611 C* which gives m (0.611 C* /3. 5)1/1 .4 1.692 These values are close to those obtained earlier in Example 8.1 by using a calculus-based optimization. .. w1 30 0 w2 w2 30 0 w3 w3 1, 000 w4 w4 with w2 w3 w4 1 w1 w2 w4 0 w2 w3 w4 0 w1 w1 w3 w4 0 The last three equations ensure that the independent variables x, y, and z are eliminated from the objective function at the optimum, resulting in the preceding expression for U*, and the first equation ensures that the sum of the w’s is unity 570 Design and Optimization of Thermal Systems This system of linear equations... value of the objective function is given by w1 10.5 HT 0.5 w1 U* w2 12 H 2 w2 0.15 HT 1.25 w3 w3 with w1 w2 1 2w2 w1 w3 w3 0 1.25w3 0 0.5w1 The last two equations are obtained by setting the sum of the exponents of H and T equal to zero, respectively, to eliminate these variables from the expression for U* This gives w1 10 /3, w2 –1, and w3 4 /3 Therefore, U* 10.5 10 /3 10 / 3 12 1 1 0.15 4 /3 4 /3 70 .30 ... along with the rate of net energy supply 566 Design and Optimization of Thermal Systems The operating temperature T is obtained from the equation for w1, i.e., w1 6.75T 0.5 U* 1.667 which gives 1.667 32 .31 6.75 T 2 63. 68 We can now add the constant dropped from the original objective function to give the maximum rate of energy supplied as 30 32 .31 62 .31 kW Similarly, the rate of energy loss El can... easily to yield w1 w2 w3 1/5, and w4 2/5 This implies that the last term is twice as important as each of the other terms Therefore, the optimum value of the objective function is U* 150 1/5 1/5 30 0 1/5 1/5 30 0 1/5 1/5 2/5 1, 000 2/5 $1601.86 The independent variables x, y, and z may be obtained from the equations 150xz w1U* w2U* 30 0xy 30 0yz w3U* which give x 1 .46 m, y 0. 73 m, and z 1 .46 m at the optimum... optimization technique Since thermal systems are often nonlinear in character, geometric programming is a useful method for optimizing these systems The method, as presented here, 559 560 Design and Optimization of Thermal Systems is very easy to apply, since it involves the solution of linear equations, rather than nonlinear equations that have to be solved for the calculus methods of optimization It first... UNCONSTRAINED OPTIMIZATION Let us first consider the application of geometric programming to unconstrained optimization problems Since we are interested in problems with degree of difficulty zero, the number of terms must be greater than the number of variables by one 562 Design and Optimization of Thermal Systems Single Independent Variable The objective function U may be written in terms of the independent... by means of examples, which follow 568 Design and Optimization of Thermal Systems Example 10 .3 In Example 10.2, if the height H of the system is also included as an additional independent variable, the thermal efficiency, in percent, is represented by the expression 100(0.2 0.07H T 0.5 – 0.08 H 2) and the rate of energy loss by 0.15H T 1.25 If the power input is still 150 kW, formulate the optimization. .. function and the constraints can be expressed as sums of polynomials of the independent variables The exponents of the variables can be integer or noninteger, positive or negative, quantities A few examples of the objective function, U(x1, x2, , xn), in unconstrained problems, which can be treated by geometric programming are U 3 2 x2 / 2 U U U 7 x1x1/ 3 2 1 4 3 x1 2 x1 6 550 105 x1 2 x1x2 2 x1 3 x1 1 x3 .3. .. (10.1) (10.2) 120, 000 1 10 x1 4 3 x1 (10 .3) 2 2 3. 7 x2 x3 8 x2 2 (10 .4) Similarly, for constrained problems, the following form is suitable for geometric programming: U 2 3 0 4 x1 x2 8 x1 1/ 2 x2 1/ 3 6 x1 6 3 x2 (10.5) with the constraint x1x1.2 2 20 (10.6) Geometric, Linear, and Dynamic Programming 561 Therefore, fractional or integral, positive or negative exponents and coefficients may be considered . Design and Optimization of Thermal Systems with w 1 w 2 1 and 1.4w 1 2.2w 2 0 From these equations, w 1 0.611 and w 2 0 .38 9. Therefore, C * (3. 5/0.611) 0.611 ( 14. 8/0 .38 9) 0 .38 9 . Optimization of Thermal Systems Stoecker, W.F. (1989) Design of Thermal Systems, 3rd ed., McGraw-Hill, New York. Vanderplaats, G.N. (19 84) Numerical Optimization Techniques for Engineering Design, McGraw-Hill,. T., Jaluria, Y., and Knight, D. (2007) Application of data driven design optimization methodology to a multi-objective design optimization problem, J. Eng. Design, 18 : 34 3 35 9. PROBLEMS 9.1.
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