Design and Optimization of Thermal Systems Episode 2 Part 10 ppsx

A useful objective function is the rate of energy consumption per unit output, where the output may be power delivered, heat removed, products manufactured, and so on.. Since energy cons

Similarly, a third-order model for the response, z, is

zB0B1xB2yB3xyB4x2B y B x yB xy B x

5 2 6 2 7 2 8 3 B y9 3 (7.9)where the B’s are coefficients to be determined from the data and x, y are the two independent design variables Once the response surface has been generated, visual inspection can be used to locate the region where the optimum is located and a closer inspection can then be used to accurately determine the location of the optimum Calculus can also be used to identify the minimum or maximum Both local and global optimum locations can generally be identified However, since a limited number of data points are used in order to generate the response surface, the surface approximates the actual behavior and the results are similarly approximate, though for many practical problems this is quite adequate

7.4 OPTIMIZATION OF THERMAL SYSTEMS

We have considered the basic formulation for optimization, as well as different methods that are available for solving these problems Several physical problems have been mentioned as examples to illustrate the general approach Let us now briefly consider these aspects as related to the optimization of thermal systems

7.4.1 I MPORTANT C ONSIDERATIONS

Thermal systems are mainly concerned with energy and fluid flow Therefore, the objective function is frequently based on energy consumption, which involves con-siderations of energy transport and losses, efficiency of the system and its compo-nents, energy exchange with the environment, fuel consumed, etc A useful objective function is the rate of energy consumption per unit output, where the output may be power delivered, heat removed, products manufactured, and so on The design that requires the least amount of energy per unit output is then the optimum Similarly, the system that delivers the largest output per unit energy consumption is optimum Since energy consumption can be expressed in terms of cost, this objective function can also be considered as the output per unit cost

Similar considerations often apply to fluid flow, where again it is important to minimize the energy consumed This frequently implies minimizing the flow rate, pressure head, and fluid leakage or loss, particularly if a closed system is needed for preserving the purity and if the fluid is expensive A lower pressure head generally translates into lower cost of the pumping system and is desirable Therefore, some of

the physical quantities that are often maximized in thermal systems may be listed as

1 Efficiency

2 Output per unit energy, or fuel, consumption

3 Output per unit cost

4 Heat removal rate in electronic systems

5 Heat exchange rate

whereas the quantities that are minimized may be listed as

1 Energy losses

2 Energy input for cooling systems

3 Pressure head for fluid flow

4 Flow rate of fluid

5 Fluid leakage or loss

6 Rate of energy or fuel consumed per unit output

In thermal systems, the constraints arise largely from the conservation laws for mass, momentum, and energy, and from limitations of the material, space, and equipment being used, as discussed earlier However, these usually lead to nonlinear, multiple, coupled, partial differential equations, with complicated geometries and boundary conditions in typical systems of practical interest Other complexities may also arise due to the material characteristics, combined thermal transport mechanisms, etc., as discussed in earlier chapters The main problem that arises due to these complexities is that the simulation of the system for each set of conditions requires a considerable amount of time and effort Therefore,

it is usually necessary to minimize the number of simulation runs needed for optimization For relatively simple thermal systems, numerical or experimental simulation results may be used, with curve fitting, to obtain algebraic expressions and equations to characterize the behavior of the system Then the optimiza-tion problem becomes straightforward and many of the available methods can be used to extract the optimum Unfortunately, this approach is possible in only a few simple, and often impractical, circumstances For common practical systems, numerical modeling is employed to obtain the simulation results, as needed, to obtain the optimum Experimental data are also used if a prototype is available, but again such data are limited because experimental runs are generally expen-sive and time consuming

7.4.2 D IFFERENT A PPROACHES

Several different optimization methods have been mentioned earlier and will be discussed in detail in later chapters Some of these have only limited applicability with respect to thermal systems Calculus methods require continuous functions that can be differentiated, and geometric programming requires sums of polyno-mials to characterize the system Therefore, both of these methods can be used only if the system behavior can be represented by explicit, closed-form expres-sions This is possible only for simple systems, with few components and ideal-ized behavior, or for cases where curve fitting is employed to obtain representative equations The latter approach is used for many systems, particularly for thermo-dynamic systems such as engines heating and cooling systems, where curve fitting can be effectively employed to represent material and process characteristics.Linear programming is of little interest in the optimization of thermal systems because it is rare to obtain linear equations to represent the constraints and the

objective function Because of the intrinsic nonlinear behavior of most thermal processes, it is generally not feasible to linearize the governing equations while maintaining the accuracy and validity of the representation Dynamic program-ming is of greater interest in thermal systems, particularly in networking of chan-nels for material and fluid flow However, it is still of limited use in the optimization

of common thermal systems

Search methods constitute the most important optimization strategy for mal systems Many different approaches have been developed and are particularly appropriate for different problems However, the underlying idea is to generate a

ther-number of designs, which are also called trials or iterations, and to select the best

among these Effort is made to keep the number of trials small, often going to the next iteration only if necessary This is a very desirable feature with respect to thermal systems because each trial may take a considerable amount of computa-tional effort The same consideration applies to experimental data

Search methods may also be combined with other methods in order to erate convergence or approach to the optimum For instance, calculus methods may be used at certain stages to narrow the domain in which the optimum lies Trials for the search method are then used to provide information for extracting the derivatives and other relevant quantities Prior knowledge on the optimum

accel-for similar systems may also be used to develop heuristic rules to accelerate the

search Many such strategies are discussed in later chapters

7.4.3 D IFFERENT T YPES OF T HERMAL S YSTEMS

As we have seen in the preceding chapters, thermal systems cover a very wide range of applications Different concerns, constraints, and requirements arise in different types of systems Therefore, the objective function and the nature of the constraints would generally vary with the application Though costs and overall profit or return are frequently optimized, other quantities are also of interest and are used Let us consider some of the common types of thermal systems and dis-cuss the corresponding optimization problems

1 Manufacturing systems The objective function is typically the number

of items produced per unit cost It could also be the amount of rial processed in heat treatment, casting, crystal growing, extrusion,

mate-or fmate-orming The number of solder mate-or welding joints made, length of material cut in gas or laser cutting, or the length of optical fiber drawn may also be used, depending on the application Again, the output per unit cost or the cost for a given output may also be used as the objective function The constraints are often given on the temperature and pres-sure due to material limitations Conservation principles and equip-ment limitations restrict the flow rates, cutting speed, draw speed, etc

2 Energy systems The amount of power produced per unit cost is the

most important measure of success in energy systems and is, therefore,

an appropriate quantity to be optimized The overall thermal efficiency

is another important variable that may be optimized Most of the straints arise from conservation laws However, environmental and safety considerations also lead to important limitations on items such

con-as the water outlet temperature and flow rate from the condensers of a power plant to a cooling pond or lake Material and space limitations will also provide some constraints on the design variables

3 Electronic systems The rate of thermal energy removed from the system

as well as this quantity per unit cost are important design requirements and may, thus, be used as objective functions The cost of the system may also be minimized while ensuring that the temperature requirements of the components are satisfied The weight and volume are important con-siderations in portable systems and in systems used in planes and rockets These may also be chosen for optimization Besides the constraints due to conservation principles, space and material limitations generally restrict the temperatures, fluid flow rates, and dimensions in the system

4 Transportation systems The torque, thrust, or power delivered are important

considerations in these systems Therefore, these quantities, or these taken per unit cost, may be maximized This feature may also be taken as the out-put per unit fuel consumed The costs for a given output in thrust, accelera-tion, etc., may also be chosen for minimization The thermal efficiency of the system is another important aspect that may be maximized The constraints are largely due to material, weight, and size limitations, besides those due

to conservation laws Thus, the temperature, pressure, dimensions, and fuel consumption rate may be restricted within specified limits

5 Heating and cooling systems The amount of heat removed or provided

per unit cost is a good measure of the effectiveness of these systems and may be chosen for maximization The system cost as well as the operat-ing cost, which largely includes the energy costs, may be minimized while satisfying the requirements The thermal efficiency of the system may be maximized for optimum performance Besides those due to con-servation laws, most of the constraints arise due to space limitations Weight constraints are important in mobile systems Fluid properties lead to constraints on the temperature and pressure in the system

6 Heat transfer and fluid flow equipment The rate of heat transfer and the

total flow rate are important considerations in these systems These tities may be used for optimization The heat transfer or flow rate per unit equipment, or operating, costs may also be considered The resulting tem-perature of a fluid being heated or cooled, the efficiency of the equipment, energy losses, etc., may also be chosen as objective functions Space limi-tations often provide the main constraints on dimensions Constraints due

quan-to weight are also important in many cases, particularly in auquan-tomobiles Conservation laws provide constraints on temperatures and flow rates.The foregoing discussion serves to illustrate the diversity of the objective func-tion and the constraints in the wide range of applications that involve thermal

systems Even though costs and profit are important concerns in engineering tems, other quantities such as output, efficiency, environmental effect, etc., also provide important considerations that may be used effectively in the optimization process Clearly, the preceding list is not exhaustive Many other objective func-tions, constraints, and applications can be considered, depending on the nature and type of thermal system being optimized.

charged into the lake can recirculate to the outflow, raising the temperature there This effect increases the temperature of the cooling water entering the condensers

of the power plant This, in turn, raises the temperature at which heat rejection occurs and thus lowers the thermal efficiency of the plant, as is well known from

thermodynamics Therefore, an increase in x increases the cost of the piping and pumps, while a decrease in x increases the cost of power generation by lowering the thermal efficiency If the objective function U is taken as cost per unit of generated

power, we may write

where F1(x) and F2(x) are costs related to piping and efficiency of the system, tively This implies that an optimum distance x may be obtained for minimum costs

respec-per unit output.

This is actually a very complicated problem because the model involves lent, multidimensional flow, complex geometries, varying ambient conditions, and

turbu-Outflow

Inflow

Cooling pond

B

A

x

FIGURE 7.9 Heat rejection from a power plant to a cooling pond, with x as the distance

between the inflow and outflow.

several combined modes of heat transfer The problem has to be solved cally, with many simplifications, to obtain the desired inputs for design and opti- mization Some simple problems were considered in Chapter 5 Constraints due to conservation principles are already taken into account in the numerical simulation

numeri-However, limitations on x due to the shape and size of the pond define an

accept-able design domain If the numerical simulation results are curve fitted to yield expressions of the form

where A, B, C, a, b, and c are constants obtained from curve fitting, calculus

meth-ods can easily be applied to determine the optimum However, this is a suming process because adequate data points are needed and a more appropriate

time-con-approach would be search methods where x is varied over the given domain and

selective simulation runs are carried out at chosen locations to determine the mum, as discussed in Chapter 9 This has been an important problem for the power industry for many years and has resulted in many different designs to obtain the highest efficiency-to-cost ratio.

opti-Example 7.2

In an automobile, the drag force on the vehicle due to its motion in air increases

with its speed V The engine efficiency H also varies with the speed due to the higher

revolutions per minute of the engine and increased fuel flow rate The efficiency

initially increases and then decreases at large V due to the effect on the combustion

process These two variations are sketched qualitatively in Figure 7.10 If the cost

Speed, V Speed, V

FIGURE 7.10 Dependence of engine efficiency and drag force on the speed V of an

automobile.

per mile of travel is taken as the objective function U, then we may write

where F1(V) represents the drag force and F2(V) represents the engine efficiency

An increase in drag force increases the cost, and an increase in efficiency reduces

the cost The constants A and B represent the effect of these quantities on the cost.

Again, this is a complicated numerical simulation problem because of the transient, three-dimensional problem involving turbulent flow and combustion The con- straints due to the conservation principles are already accounted for in the simula-

tion The physical limitations on the speed V, say, from 0 to 200 km/h for common

vehicles, may be used to define the domain If the simulation results are curve fitted

with algebraic expressions, we may use calculus methods, as dU/dV 0, to obtain the optimum Search methods are more appropriate because only a limited number

of simulations are needed at chosen values of V to extract the optimum.

Example 7.3

In a metal extrusion process, the total cost for a given amount of extruded material

may be taken as the objective function U This cost includes the capital or equipment cost A, the cost of the die subsystem, and the cost of the arrangement for applying

the extrusion force For the metal extrusion process sketched in Figure 7.11, the

independent variables are taken as x1  d/D and x2  V2/V1 Then, the objective

function may be written as

Extruded material

d D

FIGURE 7.11 A metal extrusion process.

n, and m as constants A constraint that arises from mass balance is given by

This constraint may be included in the analysis or may have to be brought in

sepa-rately if an expression, such as Equation (7.13), is given for U The ranges of x1 and

x2 due to limitations on the forces exerted are used to define the design domain This problem can be solved by calculus methods as well as by geometric program-

ming The effect of temperature T on the process may also be included in the

opti-mization process.

Example 7.4

In many processes, such as optical fiber drawing, hot rolling, continuous casting, and extrusion, the material is cooled by the flow of a cooling fluid, such as inert

gases in optical fiber drawing, at velocity V1, while the material moves at velocity V2,

as shown in Figure 7.12 Numerical simulation may be used to obtain the temperature

D d L

Cooling region

Heated rod

Temperature T o

FIGURE 7.12 Cooling of a heated moving rod by the flow of inert gases.

decay with distance for different values of these variables, as shown qualitatively in

Figure 7.13 The temperature decay increases with increasing V1 because of erated cooling, but decreases with increasing V2 since the time available for heat

accel-removal in the cooling section of length L decreases at higher speed The exit perature must drop below a given value T2.

tem-Numerical simulation may be used to solve this combined conduction tion problem and obtain the inputs needed for design and optimization of the cool- ing system If the cost per unit length of processed material is taken as the objective

convec-function U, we may write

where the function F1 represents the costs for feeding and pickup of the material, F2 represents the productivity, and F3 represents the cost of the inert gas and the flow arrangement Limitations on V1 and V2 due to physical considerations define the

domain Constraints due to mass and energy balances are part of the model Search

methods can be used for obtaining the optimum values of V1 and V2 Calculus ods and geometric programming may be applicable if the simulation results are curve fitted to obtain closed-form expressions for the preceding functions.

meth-7.4.5 C ONSIDERATION OF T HE S ECOND L AW OF T HERMODYNAMICS

We have already considered the first law of thermodynamics, which states that energy cannot be created or destroyed, leading to the conservation of energy How-ever, in dealing with thermal systems, an important consideration is the second law of thermodynamics, which brings in the concepts of entropy and maximum

(b) (a)

Distance, x Distance, x

FIGURE 7.13 Dependence of temperature decay with distance x on (a) the velocity V1 of

inert gases and (b) velocity V2 of the heated moving rod.

useful work that can be extracted from a system Entropy is used extensively in analyzing thermal processes and systems, and in defining ideal processes that are isentropic, i.e., in which the entropy does not change Isentropic efficiencies are based on this ideal behavior, as has been mentioned earlier and as is well known from a study of thermodynamics However, a concept that is finding increasing use in recent years for the analysis, design, and optimization of thermal processes

and systems is that of exergy.

Exergy is defined as the maximum theoretical useful work, involving shaft

or electrical work, that can be obtained from a system as it exchanges heat with the surroundings to attain equilibrium Similarly, it is the minimum theoretical useful work needed to change the state of matter, as in a refrigerator Therefore, exergy is a measure of the availability of energy from a thermal system Exergy is generally not conserved and can be destroyed, e.g., in the uncontrolled expansion

of a pressurized gas For a specified environment, exergy may be treated as an extensive property of the system, which can thus be characterized by the exergy contained by the system Exergy can also be transferred between systems The main purpose for an exergy analysis is to determine where and how losses occur

so that energy may be used most effectively This leads to an optimization of the process and thus of the system

Several recent papers have focused on exergy analysis and the use of the second law of thermodynamics for the optimization of thermal systems; see, for instance, Bejan (1982, 1995) and Bejan et al (1996) Similar to the conservation

of mass and energy, exergy balance equations may be written for closed systems and control volumes The destruction of exergy due to friction and heat transfer

is included in the balance An efficiency, known as exergetic efficiency and based

on the second law, may then be employed to give a true measure of the behavior

of a thermal system Such an efficiency can be defined for compressors, pumps, fans, turbines, heat exchangers, and other components of thermal systems Then a maximization of this efficiency would result in the optimization of the system in order to extract the maximum amount of useful work from it Thus, exergy may also be used as a basis for optimization and for obtaining the most cost-effective system for a given application

The second law aspects can also be included in the analysis and design of thermal systems by considering irreversibilities that arise due to heat transfer and friction As just mentioned, these effects lead to the destruction of exergy, which may also be looked on as the generation of entropy Therefore, the local and over-all generation of entropy may be determined This can be done for different types

of flows and heat transfer mechanisms, finally obtaining the entropy generation

in a given process or system A minimization of the generated entropy leads to

an optimum system based on thermal aspects alone These considerations may

be linked with other aspects, particularly economic considerations, to obtain an optimal design As discussed earlier, this involves trade-offs to obtain a satisfac-tory system design For further details on this approach, the references given in the preceding paragraph may be consulted

7.5 PRACTICAL ASPECTS IN OPTIMAL DESIGN

There are several important aspects associated with the optimization process and with the implementation of the optimal design obtained These considerations are common to all the different approaches and address the practical issues involved

in optimization Because our interest lies in an optimum design that is both sible and practical, it is necessary to include the following aspects in the overall design and optimization of thermal systems

fea-7.5.1 C HOICE OF V ARIABLES FOR O PTIMIZATION

Several independent variables are generally encountered in the design of a mal system A workable design is obtained when the design, as represented by

ther-a selection of vther-alues for these vther-arither-ables, sther-atisfies the given requirements ther-and constraints The same variables, considered over their allowable ranges, indicate the boundaries of the domain in which the optimal design is sought If only two

design variables are considered, the objective function U(x1, x2) may be plotted

as the elevation over an x1 – x2 coordinate plane to yield a surface, as discussed earlier and as shown in Figure 7.14(a) Then, depending on the problem, the

FIGURE 7.14 Optimum value of the objective function U(x1, x2), shown on a sional elevation plot and on graphs for each of the independent variables.

maximum or minimum value of U on this surface gives the desired optimum

Because of the difficulty of drawing such three-dimensional representations on

a two-dimensional drawing surface, the variation of U with x1 and with x2

may be plotted separately to determine the corresponding optima, as shown

in Figures 7.14(b) and (c)

Clearly, it is much easier to deal with a relatively small number of dent variables, as compared to the full set of variables With just one or two variables, it is possible to visualize the variation of the objective function and it is easy to extract the optimum Therefore, it is best to focus on the most important variables, as judged from a physical understanding of the system or as derived

indepen-from a sensitivity analysis outlined in the next section One may start with a

workable design and vary just one or two dominant design variables to obtain the optimum For instance, after a feasible design of a power plant is obtained, the boiler pressure may be considered as the most important design variable to seek

an optimum in the power output per unit cost As the pressure is increased, the objective function increases, with local decreases resulting from the need to go to

a larger boiler or one with a different design An overall maximum may arise, as

shown in Figure 7.15, with a decrease in U beyond this value due to the higher

material and construction costs at large pressures Thus, an optimum boiler sure may be determined Other variables, such as condenser pressure, may also

pres-be considered to seek the optimal design

Similarly, Figure 7.16 shows the variation of the objective function with a dominant design variable in two other cases In the first, the objective function

is the productivity per unit cost in an optical fiber drawing process and the fiber speed is the dominant variable In the second case, the heat removal rate per unit cost for an electronic system is the objective function and the fan size or rating is the main design variable Therefore, the optimum fiber speed and fan size may be determined by applying optimization techniques In all such cases, effort is made

to use the smallest number of variables, considering only the most crucial ones in the optimization process

Change in boiler

Boiler pressure

FIGURE 7.15 Variation of power output/cost ratio for a power plant as a function of the

boiler pressure, showing the effect of changing the boiler size and a global maximum.