Design and Optimization of Thermal Systems Episode 3 Part 4 ppsx

The results from the first search are used in the sec-ond search, following the univariate search strategy, to obtain optimal design in terms of these two variables.. Using the steepest

(CSD) method, the method of feasible directions, the gradient projection method, and the generalized reduced gradient (GRG) method Many efficient 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 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 fiber drawing furnace, as shown in Figure 1.10(c), can

ther-be carried out based on the numerical simulation of the process Because of the dominant interest in fiber quality, the objective function can be based on the ten-sion, defect concentration, and velocity difference across the fiber, 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 first 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 tion (CVD) system, shown in Figure 9.16(a), for the deposition of materials such

deposi-as silicon and titanium nitride (TiN) on a given surface, known deposi-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 film thickness uniformity Operating costs

are represented by heat input and gas flow rate The production rate is expressed

in terms of the deposition rate All these quantities are normalized to provide 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,

FIGURE 9.15 Optimization of the optical fiber 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 fiber quality.

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

Vinletand the susceptor temperature Tsus 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 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 flow and thus enhance the heat transfer The main quantities of interest are the pressure head $P and the heat

equip-transfer rates Q1 and Q2 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 perature The objective function U is defined as U (coating nonuniformity r operating cost)/production rate.

Cooled walls Flow

Horizontal Reactor

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 maximnormal-ized One

such objective function, with the normalized quantities indicated by overbars is

FW Q1 1W Q2 2 W P3$ (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 first 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 W3 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 FW Q1 1W Q2 2 W P3$ for W123 (c) Response

surface for the objective function F for W1 2 3/ 2.

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 mental and computation data are used to build the database for the response sur-

experi-faces in terms of length dimensions L1 and L2 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 fitting was substantially a better choice because it had higher correlation coefficients 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 figure, 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 flow 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 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 finite number of feasible designs Search methods are ideally suited for such problems

impor-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 ered first Such circumstances are of limited practical interest, but are illustrative

consid-of the optimization techniques for more complicated problems In addition, 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

mul-feasible design domain is sometimes used because of its simplicity and to mine subdomains containing the optimum, even though it is not very efficient 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 Efficient elimination methods, such as Fibonacci and dichotomous schemes, are presented next The efficiency of these methods in reducing the

deter-0.026 0.024 0.022 0.02 0.018 0.016 0.014 0.012 0.01 0.008

(c) Pressure drop

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.

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 inefficient 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 efficient 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 efficient 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 define 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 first is the penalty function method, which defines

a new objective function, with the constraints included, and imposes a penalty if the constraints are not satisfied 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 finally 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.

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

Obtain a final 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 final solution obtained Take the desired final 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 flow rate is to be achieved in the fan and duct system sidered in Example 4.7 by varying the constants 15 and 80, which rep-resent the zero pressure and the zero flow parameters Use any suitable search method to determine if the flow can be optimized by varying these two parameters over the range o30% of the given base values.9.5 Use an univariate search to find the optimum of the unconstrained

con-objective function U(x,y) given by

9.6 The cost C of a storage chamber is given in terms of its three

dimen-sions as

C  12x2 2y2 5z2

with the volume given as 10 units, i.e., xyz 10 Recast this problem

as an unconstrained optimization problem with two independent ables Applying an univariate search, determine the dimensions that minimize the cost

vari-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

 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 coefficient 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  h • T • A, where h is the heat transfer coefficient, T is the temperature difference from the ambient, and A(  PD2) the surface

area of the sphere Here, h is given by the expression

h  2 0.5T0.2D

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,

obtain the minimum by simple differentiation of the unconstrained objective function and compare the results from the two approaches.

9.11 In a water flow system, the total flow rate Y is given in terms of two variables x and y as

Y  8.5x2 7.1y3 25 with a constraint due to mass balance as

x y1.75 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

 2 2

42

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

(3.3 2 4 )2 1400 1500

where x and y are the sizes of two components The terms within the

first 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

con-results obtained with those from the penalty function method given in the example Also, present the trial runs needed to obtain the solution.9.16 Solve the constrained optimization problem considered in Example 9.6

by the penalty function method Compare the results obtained with those given in the example Also, compare the computational effort needed by the two methods

9.17 Solve the constrained optimization problem given in Problem 8.12 by any appropriate search method given is this chapter

9.18 Solve Problem 8.6 as a constrained optimization problem by the stitching method