Mechanical Engineers’ Handbook: Energy and Power, Volume 4, Third Edition. Edited by Myer Kutz Copyright  2006 by John Wiley & Sons, Inc.
CHAPTER 15 THERMAL SYSTEMS OPTIMIZATION Reinhard Radermacher University of Maryland College Park, Maryland
1
1
INTRODUCTION
554
2
OPTIMIZATION TOOLBOX 2.1 System Evaluation 2.2 Optimization Drivers
554 555 559
3
METHODOLOGY 3.1 Case Studies
562 562
SYMBOLS
571
BIBLIOGRAPHY
571
INTRODUCTION Thermal systems include all functional groups of equipment and working fluids that are designed to manage temperature and humidity conditions inside various spaces or materials. Thermal management systems provide comfort, establish and maintain conditions necessary for the functionality of other equipment, or utilize the change of thermo-physical properties of materials for energy conversion. Applications run from thermal management of electronic systems (electronic cooling) to space conditioning and power generation. Optimization is the systematic procedure that guides system designers in their choice of processes and components such that all requirements for the system are balanced in the best fashion possible. In most applications the designer has to balance several contradicting demands, such as high efficiency and reliability versus low costs and emissions. To keep the design time and associated costs as low as possible, it is essential to take all requirements of the thermal system into account at the earliest possible design stage. Often a great amount of time and costs can be saved if the design engineer has means to evaluate the approximate costs of a design in the early stages of the development. The optimization of thermal systems usually includes a mixture of technology decisions and the optimization of specific properties of selected components. One example is the decision between tube-fin and microchannel technology for an air-refrigerant heat exchanger in the air-conditioning system of a commercial building and the subsequent optimization of tube diameter / channel geometry and fin spacing. The system designer should find the least expensive designs for each technology that provides the required performance, in this example heat load, while minimizing the fan power consumption. An informed decision can be reached only if the best options of all feasible technologies are compared. Additionally, other factors have to be considered: The best microchannel heat exchanger may be more expensive than the best fin-tube heat exchanger, but may require a smaller fan and thus lead to savings at other system components. This shows that the optimization of thermal systems requires the evaluation of entire system performance and costs. The system designer must conduct the component optimization and selection with the system perspective in mind.
2
OPTIMIZATION TOOLBOX Optimizing thermal systems requires evaluating, comparing, and modifying large numbers of design options. The system evaluation usually includes engineering factors such as effi-
554
2
Optimization Toolbox
555
ciency and reliability, as well as accounting factors such as first and operating costs. To evaluate a large number of design options it is helpful to employ a computer-simulation tool (or a collection of tools) that is capable of predicting the system performance and system costs with sufficient accuracy. Section 2.1 describes a formulation of the simulation of general energy-conversion systems. The system designer must develop a basis of comparison for the various design options. This can be a single parameter of the system performance or costs, but usually is a combination of many parameters. If the relative importance of the significant parameters is known in advance, the designer can formulate a weighted penalty function, which assigns a characteristic value to each design option. If the relative importance is not known beforehand, the design task becomes a multiobjective optimization problem. In this case the optimization procedure should determine a Pareto-optimum set of solutions (see Section 2.2). The optimization driver derives new design options based on the comparison of evaluated options. Section 2.2 illustrates a number of optimization drivers with their advantages and disadvantages. The selection of the appropriate driver for the optimization problem is essential for the success of the design process.
2.1
System Evaluation In most design optimization problems the designer has to take into consideration engineeringlevel parameters of the thermal system as well as accounting level parameters. The engineering-level evaluation includes parameters such as system efficiency, reliability, noise, vibration, and emissions. This level of evaluation requires a physics-based simulation of the system, which is sufficiently detailed to reflect the effects of relevant component variations on the system performance. The accounting-level evaluation requires a cost model for the system and its components that is sufficiently detailed to reflect the effect of component variations on the overall costs. Engineering-Level System Simulation Thermal systems can generally be described as networks of components and their interaction with the environment. System components are connected by junctions, through which they exchange flow rates as a result of driving forces imposed by the states of the junctions. A large number of component models typically encountered in thermal systems are presented throughout this volume of the handbook. The component models should be physics-based descriptions of the components with a level of detail that allows for evaluating the effect of changes in optimization parameters on the component and system performance. For instance, if the heat-transfer area of a heat exchanger is an optimization parameter, the model of the heat exchanger must reflect the effect of a change in heat-transfer area on the performance (as opposed to using a constant heat exchanger effectiveness, for instance). An appropriate mathematical description of the system results in a system of residual equations, which has to be solved numerically. The residuals can be formulated by the differences of flow rates (mass flow rates, heat flow rates etc.) entering and leaving a junction. Figure 1 illustrates this on the example of a vapor compression system at steady-state conditions. Table 1 lists the residual equations. Flow rates leaving a component are associated with a negative sign; flow rates entering a component have a positive sign. Table 2 gives examples for the forms of the component equations for steady-state conditions. The residual equations can be passed to an equation solver. A list of solvers is given in the following section. If the network representation of the system is not very complex, in particular, if there are no splits or mergers of streams, the residual equations can be simplified as illustrated in Table 3. Alternative residual and component equations can be formulated
556
Thermal Systems Optimization
Pa6 6 Ambien Pressure Ambient ha6 •
in m• aCf HaCfin
Const. Voltage Source
Condenser Fan
•
9
WCf
•
out Cf
ma Pa7 ha7
7
•
HaCfout •
in in m• aCd HaCd
•
P3 h3
out out m• Cd H Cd
in in m• Cd H Cd
Condenser
3
2
P2 h2
•
•
in in m• Ex H Ex
Pa8 ha8
out out m• aCd HaCd
8
•
m• Cout H Cout
Ambient Pressure Const. Voltage Source
Expansion Valve
Compressor
•
WC 5 Pa12 12 Ambient Pressure ha12 • m• aEout HaEout
•
out out m• Ex H Ex
P4 h4
•
m• Cin H Cin 1
4 • in E
Evaporator
• in E
m H
Const. Voltage Source
• out E
H
P1 h1
m• aEin HaEin 11 •
m• aEfout HaEfout Evaporator Fan
•
WEf
•
Pa10 ha10
Figure 1
m •
Pa11 ha11 13
• out E
m• a Efin Ha Efin
Ambient Pressure 10 Network representation of vapor compression system with air fans.
and the choice of the appropriate formulation can have a significant influence on the convergence of the simulation. The convergence of numerical equation solvers typically depends on the quality of the initial guess values. Special care must be given to provide good guess values to the equation solver. Other sources of divergence are invalid inputs to thermophysical property functions and invalid inputs to component models. Both instances can occur during the course of the
Table 1 Residual Equations for Vapor Compression System R1 R2 R3 R4
⫽ ⫽ ⫽ ⫽
m ˙ out C out m ˙ Cd m ˙ out Ex m ˙ out E
⫹ ⫹ ⫹ ⫹
m ˙ in Cd m ˙ in Ex m ˙ in E m ˙ in C
R4 ⫽ ⫺MSystem ⫹ MC ⫹ MCd ⫹ MEx ⫹ ME out in R5 ⫽ m ˙ Cf ⫹ ma ˙ Cd out R6 ⫽ m ˙ Ef ⫹ ma ˙ Ein
2
Optimization Toolbox
557
Table 2 Examples for Forms of Component Equations at Steady State 兩m ˙ in ˙ Cout兩 ⫽ m ˙C C 兩 ⫽ 兩m m ˙ C, h2 ⫽ ƒC (P1, h1, P2, compressor parameter) out 兩m ˙ in ˙ Cd 兩⫽m ˙ Cd Cd兩 ⫽ 兩m out 兩ma ˙ in ˙ Cd 兩 ⫽ ma ˙ Cd Cd兩 ⫽ 兩ma m ˙ Cd, h3, ma ˙ Cd, ha8 ⫽ ƒCd (P1, h1, P2, Pa7, ha7, Pa8, condenser parameter) out 兩ma ˙ in ˙ Cf 兩 ⫽ ma ˙ Cf Cf 兩 ⫽ 兩ma ma ˙ Cf, h7 ⫽ ƒCf (Pa6, ha6, Pa7, condenser fan parameter)
....
iteration and may cause the premature termination of the simulation, which would have been successful if the functions handled the instance appropriately. For instance, instead of terminating the simulation if a function call attempts to compute the density of a fluid at negative absolute pressure, the function could return the lower limit of the density and report a warning to the user. The low density may produce a high value of the residual equation and may cause the iteration to step into a more appropriate direction. The success of a simulation can often be greatly enhanced when all functions and subroutines are as robust as possible and return appropriate outputs for all input values. Other system parameters, such as reliability or noise and vibration, can be associated with the system design through specific correlations provided by component manufacturers, though test data which are accessible to the simulation through data bases or other means available to the system designer. The general approach illustrated by the example of a vaporcompression system can be applied to all thermal systems by substituting the junction properties, flow rates, and component models with the appropriate variables and models. Equation Solvers Simultaneous equation solving is frequently an integral part of any mathematical model. The available equation solvers can be classified based on the types of equations they solve, as follows: 1. Simultaneous linear equations. These solve problems of the form AX ⫽ B, where A, X, and B are matrices of order m ⫻ n, m, and n, respectively. Many routines are available for these types of problems, such as LU decomposition and QR decomposition (Press et al., 2002). 2. Simultaneous nonlinear equations. These methods are discussed here. One-dimensional equations (problems with one variable and one unknown) are seldom encountered in thermal systems simulation and optimization. This section discusses gradient-
Table 3 Simplified Residual Equations for System Without Splits and Mergers in 0⫽m ˙ out ˙ Cd C ⫹ m out 0⫽m ˙ Cd ⫹ m ˙ in Ex 0⫽m ˙ out ˙ Ein Ex ⫹ m R1 ⫽ m ˙ out ˙ in E ⫹ m C
R2 ⫽ ⫺MSystem ⫹ MC ⫹ MCd ⫹ MEx ⫹ ME out in R3 ⫽ m ˙ Cf ⫹ ma ˙ Cd out R4 ⫽ m ˙ Ef ⫹ ma ˙ Ein
558
Thermal Systems Optimization
Table 4 Schemes for Calculation of Iteration Step Scheme Name
Step Calculation
Comments
Steepest descent
ssd ⬅ si ⴝ ⴚGfi
Newton-Raphson
sn ⬅ si ⴝ ⴚHifi ␣⫽1
Broyden’s method (unmodified)
si ⫽ ⴚHifi ␣: 兩兩fi⫹1兩兩 ⬍ 兩兩fi兩兩 Hi⫹1 ⫽ Hi ⫺ (Hi yi ⫺ si␣i)siT Hi / siT Hi yi
Levenberg Marquardt method
slm ⬅ Obtained by solving (JT J ⫹ I)slm ⫽ ⫺Gf is damping parameter, updated at each step. Alternates between the Newton step and the steepest descent step. Radius of trust region updated at every iteration (Dennis & Schnabel, 1996).
Powell’s dogleg method
Search is in the direction in which f decreases rapidly. The most basic method of all. Very efficient, with quadratic convergence close to the solution. Involves the calculation of H at every iteration. The inverse of the Jacobian is required only at the first step of iterations. In subsequent steps, it is updated using the strategy shown. Method used for nonlinear least-squares problem and nonlinear equation solving. Can be combined with secant updates to avoid repeated Jacobian calculations. Based on the concept of trust region. Can be combined with secant updates to avoid repeated Jacobian calculations.
based algorithms for solving multiple simultaneous equations. Other methods for onedimensional root finding are available but not discussed here. The reader is referred to Press et al. (2002). The general algorithm for a gradient-based nonlinear equation solver can be summarized as follows: 1. Get an initial guess x0 2. Get an initial value for Jacobian matrix J. J is the transpose of the matrix of the gradients 3. xi ⫽ x0 4. Compute new direction and step s 5. Compute new x and the length ␣ xi⫹1 ⫽ xi ⫹ ␣si
Table 5 Different Stopping Criteria Used in Nonlinear Equation Solving Stopping Criteria
Representation
Function residuals
兩兩f兩兩 ⬍⫽ is some tolerance value rel(xi)j ⬍⫽ 兩(xi)j ⫺ (xi⫺1)j 兩 rel(xi)j ⫽ max{兩(xi)j 兩,typ(xi)j} (xi)j ⫽ jth component of x in the ith iteration. 兩兩f兩兩⬁ ⬍⫽
Change in solution in successive steps
Maximum value of the components of f
Comments Most used criteria Indication that the algorithm has stalled
2
Optimization Toolbox
559
6. Test convergence 7. Compute Ji⫹1 8. Repeat from step 4. Several schemes are available for calculating the new direction s, the step length ␣, and the convergence criteria and for computing the Jacobian for subsequent time steps. A summary of all of these schemes is provided in the following text. This list of equation solving schemes is not meant to be complete. Let G denotes the gradient matrix of f. H denotes the inverse of J, i.e., J⫺1 f denotes the vector of function values and 兩兩 兩兩 denotes the Euclidian norm yi ⫽ fi⫹1 ⫺ fi Other methods based on the second derivatives are also available in the literature. Cost Estimation In contrast to engineering-level system simulation, cost estimation is less challenging on a mathematical level. The challenge in accurately estimating the cost of a thermal system arises from the availability and uncertainty of cost data. Cost data vary with the source that they are obtained from, they vary in time, and they may even be subject to the negotiation skills of the system designer. While most companies can accurately predict the cost of in-house manufacturing, equipment cost estimates from suppliers are less accurate. Fuel and other operating costs probably represent the most volatile cost category. Construction and installation costs can contribute significantly to the total costs, depending on size and type of thermal system. A large number of costs for components in thermal systems are provided in costestimating guides such as Building Construction Cost Data (Waier, 2004), Chemical Engineering, Vol. 6: Chemical Engineering Design (Sinnott, 2003), and Thermal Design and Optimization (Bejan et al., 1996) along with methods for estimating fuel, operating, construction, and installation costs. These references generally provide component cost data associated with characteristic parameters for the equipment, such as heat-transfer area as a parameter for the costs of the heat exchanger. Whenever possible it is recommended to obtain cost data directly from the equipment supplier. Cost data typically present a discontinuous variable in the system simulation, since they often do not scale continuously with equipment size. The optimization procedure either can interpolate over the discontinuities and round the optimization result to the closest available equipment size or can treat the cost as a discontinuous function. In this case, it is necessary to use an optimization driver that is capable of handling discontinuous (discrete) variables, such as genetic algorithms.
2.2
Optimization Drivers An engineering optimization process in its simplest form can be visualized as shown in Fig. 2. There are two basic components, the optimization driver and the simulation tool. The optimization driver is a numerical implementation of some optimization algorithm. This optimization driver needs a function that can, with a given set of inputs (the design variables over which we optimize), provide the output (the objective value that we wish to optimize) and any other relevant simulation information, such as constraints.
560
Thermal Systems Optimization Opt im izat ion Driver
Sim ulat ion Tool
Figure 2 Engineering optimization process.
From an implementation point of view, the above approach also demonstrates the power of component-based computer code development. Here we assume that eventually all of the optimization algorithms and the system and component models are used in the form of computer code. Ideally, we would want to have a library of optimization routines or drives from which we can choose an optimization routine or adopt a hybrid approach wherein the intermediate results of one routine are fed to another more effective routine. Similarly, we would want to have a library of component models, for example, a shell-and-tube heat exchanger model, an air-cooled heat exchanger model, a turbine model, a compressor model, etc. These models can then be put together to assemble a thermal system or can be used individually to evaluate component performance. This component-based approach allows us to achieve system-level objectives with the freedom of changing the lowest level (individual component) variables. The simulation tool is the numerical model of the thermal system or the component, for example, the model of a vapor compression cycle or the model of an air-cooled heat exchanger. Choosing an Optimization Driver After having a library of optimization routines and the required component models to design and optimize a system, the next logical question is which optimization routine to choose. This section elaborates on several criteria that come into play when making such a decision. Objectives. Objectives are performance measures that the designer wants to optimize. A vapor compression cycle system coefficient of performance (COP) is an example of objective. Another example is the cost of the system. The objective function can be linear or nonlinear, continuous or discontinuous in its domain. The designer can have an analytical expression for the objective function or use a result of model simulation. Independent Variables. Independent variables are the variables over which the problem is optimized. These are the variables that are changed / varied during an optimization process to arrive at an optimal solution. The continuity of the independent variables needs to be considered. For example, the tube length of a heat exchanger is a continuous variable, whereas the fan model number in an air-handling system is a discrete variable. Whether an objective function value exists for all possible values of the independent variables or not, this needs to be gracefully conveyed to the optimization driver so that it can proceed in alternative search directions. Derivative / Gradient Information. The gradient or the derivative information is required to improve the estimates of a solution in a nonlinear equation solving or an optimization process. Gradient computation can be simple if an analytical expression for the gradients is
2
Optimization Toolbox
561
available. Alternatively the designer can use a finite difference technique. Using a finite difference technique will involve additional function calls of the objective function, which might not be feasible when the objective function is computationally expensive. Number of Objectives. Many real-world problems are multiobjective. Technically multiobjective optimization is very different from single-objective optimization. In single-objective optimization the driver tries to find a solution that is usually the global minimum or maximum. In a multiobjective optimization problem, there may not exist one solution that is superior to other solutions with respect to all the objectives. As a result the designer has to make a trade-off between such solutions. Many a times, in a multiobjective optimization problem, there exists a set of solutions that is better than the remaining solutions in the design space with respect to all the objectives, but within this set, the solutions are not better than each other with respect to all the objectives. Such a set is called the Pareto set or the trade-off set and the solutions are called Pareto solutions, trade-off solutions, or nondominated solutions. The choice of different optimization routines is covered by A. Ravindran and G.V. Reklaitis in Chapter 24 of the Materials and Mechanical Design volume of this handbook. The case study discussed in Section 3.1 will provide a practical example of the above decisions. In conclusion of this section, a relatively new search and optimization technique termed genetic algorithms is introduced, which is also used in the case study discussed in Section 3.1. Genetic Algorithms Genetic algorithms (GA), first put forth by John Holland (1996) are part of a broader class of evolutionary computation methods. Based on the principal of natural evolution, they mimic on a mathematical level the biological principals of population: generations, inheritance, and selection of individuals based on the survival of the fittest. Genetic algorithms maintain a population of candidate solutions. Each of these candidates is evaluated for their fitness in terms of the corresponding objective function value. Then the candidates with the best fitness are transformed, based on ‘‘crossover’’ and ‘‘mutation’’ into new candidate solutions. Thus, a new population is created and normally the fitness of the best individual increases from generation to generation. Some of the advantages of genetic algorithms are explained below: 1. They need only one scalar value, which is the fitness or the objective value of the function that is being optimized. Thus, the objective function is a black box object as far as the genetic algorith is concerned. 2. They do not need any gradient, i.e., first- or second-derivative information. This can result in significant computation savings, if the objective function is computationally expensive. 3. Genetic algorithms maintain a population of candidate solutions, i.e., they simultaneously search in multiple directions as opposed to deterministic search methods that search in one direction at a time. 4. They can handle discrete and continuous variables at the same time. Representation and Genetic Operators. In a genetic algorithm (binary coded) a given candidate solution is known as a chromosome, and is represented as a series of binary digits (viz. 0’s and 1’s). These chromosomes are manipulated via genetic operators of selection, mutation, and crossover to form new chromosomes or candidate solutions. More information on genetic algorithms can be found in D. E. Goldberg’s book listed in the Bibliography.
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Thermal Systems Optimization Case 2 in the next section demonstrates the use of genetic optimization algorithms for multiobjective optimization problems.
3
METHODOLOGY The optimization of thermal systems requires the repeated evaluation of physics-based system simulations and cost functions. The number of system evaluations ranges from a few dozen for single-objective low-dimensional optimization tasks with deterministic algorithms to tens of thousands of evaluations for multiobjective, high-dimensional tasks with nondeterministic schemes. Especially for optimization tasks with large numbers of system evaluations, it is critical to reduce the computational time for the system evaluation by implementing fast simulation methods or by increasing the computer capacity, possibly by parallel computing. Whenever possible it is recommended to break out separate optimization tasks from the entire system simulation. This method is especially helpful for complex thermal systems. At a stage when the major technology decisions for a system are identified and the optimization focuses on the design of individual component parameters it is often possible to optimize functional groups of the thermal system under the assumption of constant boundary condition between the functional group and the rest of the system. In the design of the vapor compression system from Section 2.1 the optimization task may be to find an evaporator–fan combination that provides a certain cooling capacity at minimum cost. Instead of evaluating the performance of the entire system for each evaporator-fan combination during the optimization, the designer can optimize the functional group evaporator, fan, and fan motor. By assuming reasonable values for the evaporator refrigerant inlet pressure and mass flow rate and minimizing the refrigerant pressure drop and cost of the functional group, the designer can find a design close to the optimum. The refrigerant inlet pressure and mass flow rate can be updated repeatedly by evaluating the system performance with the design found in the previous optimization of the functional group. This method can reduce the number of entire system evaluations. However, it may result in a higher number of component evaluations of the functional group and the designer must base her / his decision for the optimization strategy on experience and sound judgment. The following case studies illustrate optimization strategies for various thermal systems. The case studies are presented as examples to guide the designer in the development of successful optimization applications. While the general scheme for most optimization tasks is identical—find a system with maximum performance and minimum costs—each optimization task has its individual particularities that make a generalized approach very challenging. In the end the design engineer must use his / her sound understanding of the underlying physics, his / her experience, and creativity.
3.1
Case Studies Case 1—Constrained Single-Objective Optimization of an Automotive Air-Conditioning System The objective of this example is to maximize the efficiency measured as coefficient of performance (COP) of an automotive air-conditioning system. The constraint is the cooling capacity and the overall volume of the heat exchangers. Space is a valuable commodity in automotive applications and the space available for the heat exchangers of the airconditioning system usually cannot be changed once the chassis and engine designs are completed. The optimization follows the approach of optimizing functional groups separated from the rest of the system as laid out in the introduction of Section 3. While the overall goal of
3
Methodology
563
the optimization is maximum efficiency of the system, the heat exchangers are optimized separately from the rest of the system. Three performance parameters of the heat exchangers affect the system efficiency: Heat load, refrigerant pressure drop, and air-side pressure drop. The heat load is a constraint determined by the capacity of the system and thus not an optimization objective. The refrigerant and air-side pressure drops are combined in a fitness function f of the form ƒ ⫽ ␣ ⌬ PRefrigerant ⫹  ⌬ PAir where ␣ and  are weight functions, which must be determined by the designer based on experience or other parameters. The component with the smallest fitness value will perform best in the air-conditioning system. The optimization is subject to the constraints of a given heat load and a maximum overall volume of the heat exchangers. The heat load for the evaporator is given by the required cooling capacity. The heat load for the condenser is given by the capacity and an estimated energy efficiency of the system: ˙ Condenser ⫽ Q ˙ Evaporator ⫹ W ˙ Compressor Q ˙ Compressor ⫽ Q ˙ Evaporator / COP W
While the COP of the system depends on the performance of the heat exchangers and is thus not known beforehand, an experienced designer can start with a good guess for the efficiency, perform the optimization, and update the guess to find a solution closer to the global optimum. For this case study only the first step of this iteration is shown. The following geometric parameters are variables in this optimization (microchannel heat exchangers are used for the condenser and evaporator): • Fin spacing
• Number of ports per tube • Port height
• Fin thickness • Number of tubes per pass
Tubes NTubes=8 Tube Cross Section
Port Height
Ports NPorts=5
Fin Thickness Fin Spacing
Figure 3 Geometric parameters of microchannel heat exchanger optimization.
564
Thermal Systems Optimization
Refrigerant
Evaporator 2
Evaporator 3
Condenser Pass 1
Evaporator 1
Evaporator 4
Pass 2 Pass 3
Air Receiver
Pass 4 Condenser Pass 5
Air
Condenser
Refrigerant
Evaporator
Figure 4 Condenser and evaporator geometry.
Figure 3 illustrates the geometric parameters of the microchannel heat exchangers. The heat exchangers are simulated with a detailed, segmented model. The heat exchanger geometry is coded as a binary string and passed to a genetic algorithm. Figure 4 illustrates the condenser and evaporator geometry. Figure 5 shows the network representation of the refrigerant circuit. The starting point for the optimization is the design of commercially available automotive condensers and evaporators. Figure 6 shows the pressure drop and fitness value of the evaporator over the generation number. Note that the goal of the optimization was to minimize the value of the fitness function. Table 6 lists the geometric parameters of the original system and the result of the optimization. Figure 7 shows the efficiency and capacity of the original and optimized systems at various temperature conditions and engine speeds. Figure 8 shows the relative performance improvement of the automotive air-conditioning system due to the optimized heat exchangers. Case 2—Constrained Multiobjective Optimization of a Condensing Unit This study performs a multiobjective optimization of a fan-coil unit with respect to minimizing cost and maximizing heat rejection capacity. The condensing unit in consideration is a traditional tube–fin heat exchanger. The individual components of the condensing unit include tubes, fins, fans, cabinet, etc. There is also a manufacturing cost associated with these components.
Condenser
Sub-Cooler
TXV
Outdoor Air
Evaporator 1
Indoor Air Figure 5
Receiver Outdoor Air Evaporator 2
Evaporator 3
Compressor Evaporator 4
Indoor Indoo Air Network representation of automotive refrigerant circuit.
3
565
Methodology
Refrigerant Pressure Drop Air Side Pressure Drop Fitness Value
14000
Fitness Value
Pressure Drop
12000
10000
8000
6000
4000
2000 0
200
400
600
800
1000
1200
1400
Generation Figure 6 Performance of evaporator over generation number.
The independent variables are Tube diameter (OD). Off-the-shelf four different tube sizes are available and are being used in this study. The tube diameter sizes are not continuous; as a result there are four discrete choices. Fin spacing (FPI). Expressed in terms of fins per inch. The study uses 11 different values ranging from 6 to 16 fins per inch. Tube length. Tubes are cut from a coil of tubes; as a result any tube length is possible. This is a continuous variable. From a heat exchanger point of view, the heat rejection capacity increases with tube length, but so does the cost. Fan models ( fan ID). For the particular coil, 20 different fan models are available, along with the required performance data, such as static pressure drop, noise, power consumption, and frame width. Number of fans (NFan). The baseline coil length is fairly long, and hence multiple fans are required to drive the air flow across the coil. Number of parallel circuits. The coil comprises of several parallel refrigerant circuits, with each circuit having a fixed number of tubes. The number of parallel circuits affects the coil height.
Table 6 Geometric Properties of Original and Optimized Heat Exchangers
Original condenser Optimized condenser Original evaporator Optimized evaporator
Fin Spacing (mm)
Fin Thickness (mm)
Port Height (mm)
No. of Tubes per Pass
No. of Ports per Tube
1.05 0.82 0.81 1.31
0.15 0.11 0.15 0.104
1 2.48 2.9 2.2
18 / 8 / 7 / 4 / 4 7 / 11 / 5 / 3 / 3 9 8
1 4 2 4
566
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
15 A @900 RPM
15 A @1500 RPM
15 A @2500 RPM
45 A @1500 RPM
45 A @2500 RPM
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
15 A @900 RPM
15 A @1500 RPM
Figure 7 COP and capacity of original and optimized systems.
45 A @900 RPM
Original System Optimized System
Capacity [kW]
COP
15 A @2500 RPM
45 A @900 RPM
45 A @1500 RPM
45 A @2500 RPM
Original System Optimized System
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Methodology
567
Performance Improvement of Optimized System [%]
8.0 Improvement COP Improvement Capacity
7.0 6.0 5.0 4.0 3.0 2.0 1.0 0.0 15 A @900 RPM
15 A @1500 RPM
15 A @2500 RPM
45 A @900 RPM
45 A @1500 RPM
45 A @2500 RPM
Figure 8 Relative performance improvement of optimized systems.
The constraints in this problem include manufacturing as well as performance-based constraints. Constraint 1. The total combined width of the fans should be less than the specified cabinet width, so that the fans can fit into the cabinet. Constraint 2. The fan chosen should be able to provide the required static pressure head for the coil, i.e., the fan pressure head should be equal to the air pressure drop through the coil. Constraint 3. The coil height must be less than a specified maximum height. Constraint 4. The refrigerant side pressure drop must be within acceptable lower and upper limits. For this problem a multiobjective genetic algorithm is used. Figure 9 shows the representation of a single condensing unit as seen by the genetic algorithm. As seen from the independent variables and the constraints the problem has two objectives, four constraints, one continuous variable, and four discrete variables. This is an example of constrained multiobjective optimization with mixed variables. A multiobjective genetic algorithm is used for this problem. In Fig. 10, the inputs are generated by the optimization algorithms and are supplied to the condenser model. The condenser model is a very detailed simulation based on a segmented approach. After the condenser model is executed the outputs are transferred to the multiobjective genetic algorithm. The condenser model is coupled with the optimization algorithm, wherein the model is evaluate repeatedly with different input variables until the termination criteria of the genetic algorithm is satisfied. Figure 11 shows the pseudo code for the multiobjective genetic algorithm.
Nt OD Fan I D FPI NFan Tube Length 1 0 0 1 0 1 1 0 0 0 1 1 0 0 0 0 1 1 1 1 0 0 0 0 0 Figure 9 Binary representation of a condensing unit in a genetic algorithm.
568
Thermal Systems Optimization Inputs Tu be Length FPI Fan ID Fan Count No. of Circuits
Fan Routine
Fin and Tube Data Correlations Air Distribution Refrigerant Inlet Condition Refrigerant Flow Rate
Solver Outlet Condition Pressure drops T and P each segment
Cost Routine
Condenser Model
Outputs Heat Load Pressure Drop Tu be Length Total Fan Width Cost
Figure 10 Flow chart of an air-cooled heat exchanger model.
Considering all the choices, the total solutions space consists of over 125,000 different condensing units. The genetic algorithm evaluated only 5500 of them, which is 4.4% of the total solution space. The results of the optimization runs are shown in Figs. 12–14. Figure 12 shows how the number of Pareto solutions increases with respect to algorithm generations. This is consistent with the principal of natural evolution where successive generations are better or fitter than the previous. Figure 13 shows the Pareto curve for the solution. It can be seen that there are several different choices for the designer to pick from and the decision will have to be based on other factors, such as fan power requirements and pressure drops. The baseline case is also shown in Fig. 13. For the same heat rejection capacity it is possible to reduce the coil cost by 16% or, for the same coil cost, the heat rejection capacity can be increased by 12%. Figure 14 shows other condensing units evaluated by the genetic algorithm, which were found infeasible because they violated one or more of the constraints.
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Methodology
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Start Initialize Condenser Model Initialize GA While (GA not done) { g = g + 1 “Update generation count� Evaluate { Decode Variables Run Condenser Model Calculate constraint violation Return (Cost, Heat load) and constraint violation } Perform Non-Dominated Sorting Assign Rank Assign Fitness Perform Selection, Crossover & Mutation } Print Results End Figure 11 Pseudo code for multiobjective genetic algorithm used in case study 1.
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Figure 12 Number of Pareto solutions versus genetic algorithm generations.
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Figure 13 Pareto curve for the results from the multiobjective optimization of a condensing unit.
Infeasible & Pareto Solutions for Condenser Units 1.6 Pareto Solutions Infeasible Solutions 1.4
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Figure 14 Infeasible and Pareto solution evaluated by genetic algorithm.
Bibliography
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For all the given solutions the average reduction in cost is found to be 10%, while the average increase in the heat rejection capacity is found to be 7%. It can be concluded that genetic algorithms are very powerful tools for mixed-variable optimization.
SYMBOLS COP ƒ G H h, ha I J M, m ˙ P, Pa, ⌬ P ˙ Q R W ˙ P W, x ␣,  兩兩 兩兩, 兩兩 兩兩⬁
coefficient of performance function gradient of function vector inverse of Jacobian J Specific enthalpy, specific enthalpy of air Identity matrix Jacobian matrix mass, mass flow rate pressure, air pressure, pressure difference heat flow rate residual work Power vector Step length, weight function Damping parameter Tolerance value Euclidian norm, infinity norm
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