Design and Optimization of Thermal Systems Episode 2 Part 9 pps

The loan has to be paid off in 10 years with payments starting at the end of the first year.. facil-424 Design and Optimization of Thermal Systemsyear and starting at the end of the firs

422 Design and Optimization of Thermal Systems

Dieter, G.E (2000) Engineering Design, 3rd ed., McGraw-Hill, New York.

Newnan, D.G., Eschenbach, T.G., and Lavelle, J.P (2004) Engineering Economic Analysis,

9th ed., Oxford University Press, Oxford, U.K.

Park, C.S (2004) Fundamentals of Engineering Economics, Prentice-Hall, Upper Saddle

River, NJ.

Riggs, J.L and West, T (1986) Engineering Economics, 3rd ed., McGraw-Hill, New York Stoecker, W.F (1989) Design of Thermal Systems, 3rd ed., McGraw-Hill, New York Sullivan, W.G., Wicks, E.M., and Luxhoj, J (2005) Engineering Economy, 13th ed.,

Prentice-Hall, Upper Saddle River, NJ.

Thuesen, G.J and Fabrycky, W.J (1993) Engineering Economy, 8th ed., Prentice-Hall,

Englewood Cliffs, NJ.

White, J.A., Agee, M.H., and Case, K.E (2001) Principles of Engineering Economic Analysis, 4th ed., Wiley, New York.

PROBLEMS

6.1 A steel plant has a hot-rolling facility for steel sheets that is to be sold

to a smaller company at $15,000 after 10 years What is the present worth of this salvage price if the interest is 8%, compounded annu-ally? Also, calculate the present worth for an interest rate of 12% with annual compounding Will the present worth be larger or smaller if the compounding frequency was increased to monthly? Explain the observed behavior

6.2 A chemical company wants to replace its hot water heating and storage system One buyer offers $10,000 for the old system, payable immedi-ately on delivery Another buyer offers $15,000, which is to be paid five years after delivery of the old system If the current interest rate is 10%, compounded monthly, which offer is better financially?

6.3 A company wants to put aside $150,000 to meet its expenditure on repair and maintenance of equipment Considering yearly, quarterly, monthly, and daily compounding, determine the total annual interest the company will get in these cases if the nominal interest rate is 7.5%

6.4 For nominal interest rates of 8 and 12%, calculate the effective interest rates for yearly, quarterly, monthly, daily, and continuous compounding.6.5 A company acquires a manufacturing facility by borrowing $750,000

at 8% nominal interest, compounded daily The loan has to be paid off

in 10 years with payments starting at the end of the first year late the effective annual rate of interest and the amount of the annual payment

Calcu-6.6 In the preceding problem, calculate the amount of the loan left after four and after eight payments Also, calculate the total amount of inter-est paid by the company over the duration of the loan

6.7 A food processing company wants to buy a facility that costs $500,000

It can obtain a loan for 10 years at 10% interest or for 15 years at 15% interest In both cases, yearly payments are to be made starting at the end of the first year

Economic Considerations 423

(a) Which alternative has a lower yearly payment?

(b) What is the loan amount paid off after 5 years for the two cases? What are the amounts needed to pay off the entire loan at this time?6.8 A company makes a profit of 10% Calculate the real profit in terms of buying power for inflation rates of 4, 6, and 8%

6.9 A firm wants to have an actual profit of 8% in terms of buying power

If the inflation rate is 11%, calculate the profit that must be achieved by the firm in order to achieve its goal

6.10 A small chemical company wants to obtain a loan of $120,000 to buy a plastic recycling machine It has the option of a loan at 6% interest for

10 years or a loan at 8% for 8 years, with monthly compounding and payment in both the cases Calculate the monthly payments in the two cases, assuming that the first payment is made at the end of the first month Also, calculate the total interest paid in the two options

6.11 A $1000 bond has 4 years to maturity and pays 8% interest twice a year If the current interest is 6% compounded annually, calculate the sale price of the bond Repeat the problem if the current interest is compounded daily

6.12 A $5000 bond has 5 years to maturity and it pays 7% interest at the end of each year If it is sold at $4500, calculate the current nominal interest.6.13 A pharmaceutical company wants to acquire a packaging machine It can buy it at the current price of $100,000 or rent it at $18,000 per year The rental payments are to be made at the beginning of each year, starting on the date the machine is delivered If the interest rate is 10%, compounded annually, and if the machine becomes the property of the company after

10 yearly payments, which option is better economically?

6.14 In the preceding problem, if the machine has a salvage value of $15,000

at the end of 10 years for the option of buying the facility, will the clusions change? If the rate is 20%, with salvage, how will the results change?

con-6.15 An industrial concern wants to procure a manufacturing facility It can buy an old machine by paying $50,000 now and 10 yearly payments of

$2,000 each, starting at the end of the first year It can also buy a new machine by paying $100,000 now and 5 yearly payments of $1000 each, starting at the end of the sixth year The salvage value is $10,000 and $20,000 in the two cases, respectively The nominal interest rate is 10% Which is the better option, assuming that the performance of the two machines is the same?

6.16 As a project engineer involved in the design of a manufacturing ity, you need to acquire a polymer injection-molding machine Two options are available from two different companies The first one, option A, requires 15 payments of $8000 per year, paid at the begin-ning of each year and starting immediately The second one, option B, requires eight payments of $15,000 per year, paid at the end of each

facil-424 Design and Optimization of Thermal Systems

year and starting at the end of the first year Determine which option

is better economically if the interest rate is 8% Also, calculate the amounts needed to pay off the loan after half the number of payments have been made in the two options

6.17 A company needs 1000 thermostats a year for a factory that tures heating equipment It can buy these at $10 each from a subcon-tractor, with payment made at the beginning of each year for the annual demand It can also procure a facility at $75,000, with $2000 needed for maintenance at the end of each year, to manufacture these If the facility has a life of 10 years and a salvage value of $10,000 at the end

manufac-of its life, which option is more economical? Take the interest rate as 8% compounded annually

6.18 In the preceding problem, calculate the annual demand for thermostats

at which the two options will incur the same expense

6.19 You have designed a thermal system that needs a plastic part in the assembly You can either buy the required number of parts from a man-ufacturer or buy an injection-molding machine to produce these items yourself The number of items needed is 2000 every year In the first option, you have to pay $12 per item for the yearly consumption at the beginning of each year The chosen life of the project is 10 years For the other option, you can lease a machine for $20,000 each year, paid

at the end of each year for 10 years The maintenance of the machine and raw materials cost $1000 at the end of the first year, $2000 at the end of the second year, and increasing by $1000 each year, until the last payment of $9000 is made at the end of the ninth year Pro-vide the payment schedule for the second option and determine which option is better financially Take the interest rate as 10%, compounded annually

6.20 A manufacturer of electronic equipment needs 10,000 cooling fans over a year The company can buy these for $20 each, payable on deliv-ery at the beginning of each year, or at $24, payable two years after delivery Which is the better financial alternative if the interest rate is 9% compounded daily? Also, calculate the results if the interest rate drops to 8%

6.21 A gas burner needed for a furnace can be purchased from three ferent suppliers The first one wants $100 for each burner, payable on delivery The second supplier is willing to take payments of $55 each

dif-at the end of six months and the year The third supplier claims thdif-at his deal is the best and asks for $110 at the end of the year The current interest rate is 8.5%, compounded continuously Since a large number

of burners are to be bought, it is important to get the best financial deal Whom would you recommend? Would your recommendation change if the interest rate were to go up, say to 12%?

6.22 A company acquires a manufacturing facility for $300,000, to be paid in

15 equal annual payments starting at the end of the first year The rate of

Economic Considerations 425

interest is 8%, compounded annually After six payments, the company

is in good financial condition and wants to pay off the loan in four more equal annual payments, starting with the end of the seventh year, as shown in Figure P6.22 Calculate the first and the last payment (at the end of the tenth year) made by the company

6.23 An industry takes a loan of $200,000 for a machine, to be paid off in

10 years by annual payments beginning at the end of the first year The rate of interest is 10%, compounded monthly At the end of five payments, the company finds itself in a good financial situation and management decides to pay off the loan in the following year, as shown

in Figure P6.23 How much does it have to pay at the end of the sixth year to end the debt? Also, calculate the amount of the annual payment

in the first 5 years

6.24 A company is planning to buy a machine, which requires a down ment of $150,000 and has a salvage value of $30,000 after 10 years The cost of maintenance is covered by the manufacturer up to the end

pay-of 3 years For the fourth year, the maintenance cost is $1000, paid

at the end of the year These costs increase by $1000 each year until the end of the tenth year, when the company pays for the maintenance

of the facility and sells it, as shown in Figure P6.24 The rate of est is 10%, compounded annually Find the present worth of buying

FIGURE P6.23

426 Design and Optimization of Thermal Systems

and maintaining the machine over 10 years If the company wants to take out a fixed amount annually from its income to cover the entire expense, calculate this amount, starting at the end of the first year

6.25 A manufacturing company wants to buy a welding machine, which costs $10,000 The cost of maintenance is zero in the first year, $500

in the second year, and increases by $500 each year until the eighth year when the company pays the maintenance expense and sells the facility for $2000 The maintenance expense is paid at the end of each year The rate of interest is 9%, compounded annually Find the present worth of acquiring and maintaining this machine over 8 years

6.26 A company is considering the purchase and operation of a ing system The initial cost of the system is $200,000 and the main-tenance costs are zero at the end of the first year, $5000 at the end of the second year, $10,000 at the end of the third year, and continue to increase by $5000 each year If the life of the system is 15 years, find the present worth of buying and maintaining it over this period Also, find the uniform annual amount that the system costs the company each year, starting after the first year Take the interest rate as 10% compounded annually

manufactur-6.27 An industrial firm wants to acquire a laser-cutting machine It can buy

a new one by paying $150,000 now and six yearly payments of $20,000 each, starting at the end of the fifth year It can also buy an old machine by paying $100,000 now and 10 yearly payments of $15,000, starting at the end of the first year At the end of 10 years, the salvage value of the new machine is $80,000 and that of the old one is $60,000 Which is the better purchase for the firm, if the interest rate is 12% compounded annually? Use lifecycle savings Repeat the calculation for a 10% interest rate.6.28 Using the data given in Example 6.7, choose between the two machines for interest rates of 4, 6, and 10% Compare the results obtained with

$150,000

Down payment

$30,000 Salvage

10 9 8 7 6 5 4 3 2 1

Years

FIGURE P6.24

6.30 Calculate the rates of return for the two facilities given in Example 6.8

as functions of the useful lives of the facilities Take the life as 4, 6, and

8 years, and calculate the corresponding rates of return with and out taxes at the rate of 50% of the profit taken into account Compare these with the earlier results and comment on their significance in the design process

with-6.31 A loan of $5000 is taken from a bank that charges a nominal interest

rate i, compounded monthly If a monthly payment of $200, starting at

the end of the first month, is needed for 36 months to pay off the loan,

calculate the value of i.

for Optimization

7.1 INTRODUCTION

In the preceding chapters, we focused our attention on obtaining a workable, feasible,

or acceptable design of a system Such a design satisfies the requirements for the given application, without violating any imposed constraints A system fabricated

or assembled because of this design is expected to perform the appropriate tasks for which the effort was undertaken However, the design would generally not be the

best design, where the definition of best is based on cost, performance, efficiency,

or some other such measure In actual practice, we are usually interested in ing the best quality or performance per unit cost, with acceptable environmental effects This brings in the concept of optimization, which minimizes or maximizes quantities and characteristics of particular interest to a given application

obtain-Optimization is by no means a new concept In our daily lives, we attempt

to optimize by seeking to obtain the largest amount of goods or output per unit expenditure, this being the main idea behind clearance sales and competition In the academic world, most students try to achieve the best grades with the least amount of work, hopefully without violating the constraints imposed by ethics and regulations The value of various items, including consumer products like televisions, automobiles, cameras, vacation trips, advertisements, and even edu-cation, per dollar spent, is often quoted to indicate the cost effectiveness of these items Different measures of quality, such as durability, finish, dependability, corrosion resistance, strength, and speed, are included in these considerations, often based on actual consumer inputs, as is the case with publications such as

Consumer Reports Thus, a buyer, who may be a student (or a parent) seeking an

appropriate college for higher education, a couple looking for a cruise, or a young professional searching for his first dream car may use information available on the best value for their money to make their choice

7.1.1 O PTIMIZATION IN D ESIGN

The need to optimize is similarly very important in design and has become ticularly crucial in recent times due to growing global competition It is no longer enough to obtain a workable system that performs the desired tasks and meets the given constraints At the very least, several workable designs should be generated and the final design, which minimizes or maximizes an appropriately chosen quantity, selected from these In general, many parameters affect the performance and cost of a system Therefore, if the parameters are varied, an optimum can

par-430 Design and Optimization of Thermal Systems

often be obtained in quantities such as power per unit fuel input, cost, efficiency, energy consumption per unit output, and other features of the system Different product characteristics may be of particular interest in different applications and the most important and relevant ones may be employed for optimization For instance, weight is particularly important in aerospace and aeronautical applica-tions, acceleration in automobiles, energy consumption in refrigerators, and flow rate in a water pumping system Thus, these characteristics may be chosen for minimization or maximization

Workable designs are obtained over the allowable ranges of the design ables in order to satisfy the given requirements and constraints A unique solu-tion is generally not obtained and different system designs may be generated for

vari-a given vari-applicvari-ation We mvari-ay cvari-all the region over which vari-acceptvari-able designs vari-are

obtained the domain of workable designs, given in terms of the physical variables

in the problem Figure 7.1 shows, qualitatively, a sketch of such a domain in terms

of variables x1 and x2, where these may be physical quantities such as the diameter and length of the shell in a shell-and-tube heat exchanger Then, any design in this domain is an acceptable or workable design and may be selected for the problem

at hand Optimization, on the other hand, tries to find the best solution, one that minimizes or maximizes a feature or quantity of particular interest in the appli-cation under consideration Local extrema may be present at different points in the domain of acceptable designs However, only one global optimal point, which yields the minimum or maximum in the entire domain, is found to arise in most applications, as sketched in the figure It is this optimal design that is sought in the optimization process

Domain of acceptable designs

Optimum design

x2

x1

FIGURE 7.1 The optimum design in a domain of acceptable designs.

Problem Formulation for Optimization 431

7.1.2 F INAL O PTIMIZED D ESIGN

The optimization process is expected to yield an optimal design or a subdomain

in which the optimum lies, and the final system design is obtained on the basis

of this solution The design variables are generally not taken as exactly equal

to those obtained from the optimal solution, but are changed somewhat to use more convenient sizes, dimensions, and standard items available in the market For instance, an optimal dimension of 4.65 m may be taken as 5.0 m, a 8.34 kW motor as a 10 kW motor, or a 1.8 kW heater as a 2.0 kW heater, because items with these specifications may be readily available, rather than having the exact values custom made An important concept that is used at this stage to finalize the design

variables is sensitivity, which indicates the effect of changing a given variable on

the output or performance of the system In addition, safety factors are employed

to account for inaccuracies and uncertainties in the modeling, simulation, and design, as well as for fluctuations in operating conditions and other unforeseen circumstances Some changes may also be made due to fabrication or material limitations Based on all these considerations, the final system design is obtained and communicated to various interested parties, particularly those involved in fabrication and prototype development

Generally, optimization of a system refers to its hardware, i.e., to the etry, dimensions, materials, and components As discussed in Chapter 1, the hardware refers to the fixed parts of the system, components that cannot be easily varied and items that determine the overall specifications of the system However, the system performance is also dependent on operating conditions, such as tem-perature, pressure, flow rate, heat input, etc These conditions can generally be varied quite easily, over ranges that are determined by the hardware Therefore, the output of the system, as well as the costs incurred, may also be optimized with respect to the operating conditions Such an optimum may be given in terms

geom-of the conditions for obtaining the highest efficiency or output For instance, the settings for optimal output from an air conditioner or a refrigerator may be given

as functions of the ambient conditions

This chapter presents the important considerations that govern the zation of a system The formulation of the optimization problem and different methods that are employed to solve it are outlined, with detailed discussion of these methods taken up in subsequent chapters It will be assumed that we have been successful in obtaining a domain of acceptable designs and are now seek-ing an optimal design The modeling and simulation effort that has been used

optimi-to obtain a workable design is also assumed optimi-to be available for optimization Therefore, the optimization process is a continuation of the design process, which started with the formulation of the design problem and involved modeling, simu-lation, and design as presented in the preceding chapters The conceptual design

is generally kept unchanged during optimization However, for a true optimum, even the concept should be varied

This chapter also considers special considerations that arise for thermal tems, such as the thermal efficiency, energy losses, and heat input rate, that are

sys-432 Design and Optimization of Thermal Systems

associated with thermal processes Important questions regarding the tation of the optimal solution, such as sensitivity analysis, dependence on the model, effect of quantity chosen for optimization, and selection of design vari-ables for the final design, are considered Many specialized books are available

implemen-on optimizatiimplemen-on in design, for instance, those by Fox (1971), Vanderplaats (1984), Stoecker (1989), Rao (1996), Papalambros and Wilde (2003), Arora (2004), and Ravindran et al (2006) Books are also available on the basic aspects of optimiza-tion, such as those by Beveridge and Schechter (1970), Beightler et al (1979), and Miller (2000) These books may be consulted for further details on optimization techniques and their application to design

7.2 BASIC CONCEPTS

We can now proceed to formulate the basic problem for the optimization of a thermal system Since the optimal design must satisfy the given requirements and constraints, the designs considered as possible candidates must be acceptable or workable ones This implies that the search for an optimal design is carried out

in the domain of acceptable designs The conceptual design is kept fixed so that optimization is carried out within a given concept Generally, different concepts are considered at the early stages of the design process and a particular concep-tual design is selected based on prior experience, environmental impact, material availability, etc., as discussed in Chapter 2 However, if a satisfactory design is not obtained with a particular conceptual design, the design process may be repeated, starting with a different conceptual design

7.2.1 O BJECTIVE F UNCTION

Any optimization process requires specification of a quantity or function that is

to be minimized or maximized This function is known as the objective function,

and it represents the aspect or feature that is of particular interest in a given circumstance Though the cost, including initial and maintenance costs, and profit are the most commonly used quantities to be optimized, many others aspects are employed for optimization, depending on the system and the application The objective functions that are optimized for thermal systems are frequently based

on the following characteristics:

1 Weight

2 Size or volume

3 Rate of energy consumption

4 Heat transfer rate

Problem Formulation for Optimization 433

10 Durability and dependability

is also of particular interest in many thermal systems, such as manufacturing processes and automobiles However, even if one wishes to maximize the thrust, torque, or power delivered by a motor vehicle, cost is still a very important con-sideration Therefore, in many cases, the objective function is based on the output per unit cost Similarly, other relevant measures of performance are considered

in terms of the costs involved Environmental effects, safety, product quality, and several other such aspects are important in various applications and may also be considered for optimization

Let us denote the objective function that is to be optimized by U, where U is a function of the n independent variables in the problem x1, x2, x3, , x n Then the objective function and the optimization process may be expressed as

U  U (x1, x2, x3, , x n)l Uopt (7.1)

where Uopt denotes the optimal value of U The x’s represent the design variables

as well as the operating conditions, which may be changed to obtain a workable

or optimal design Physical variables such as height, thickness, material ties, heat flux, temperature, pressure, and flow rate may be varied over allowable ranges to obtain an optimum design, if such an optimum exists A minimum

proper-or a maximum in U may be sought, depending on the nature of the objective

function

The process of optimization involves finding the values of the different design variables for which the objective function is minimized or maximized, without violating the constraints Figure 7.2 shows a sketch of a typical variation

of the objective function U with a design variable x1, over its acceptable range

It is seen that though there is an overall, or global, maximum in U(x), there are