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Finned-tube condenser design optimization using thermoeconomic isolation
Applied Thermal Engineering 30 (2010) 2096e2102
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Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng
Finned-tube condenser design optimization using thermoeconomic isolation Susan W. Stewart a, *, Samuel V. Shelton b a b
The Pennsylvania State University, Applied Research Laboratory, State College, PA 16804, USA Georgia Institute of Technology Strategic Energy Institute, Atlanta, GA 30332, USA
a r t i c l e i n f o
a b s t r a c t
Article history: Received 29 December 2009 Accepted 18 May 2010 Available online 23 May 2010
Using a detailed system model as a comparison, this study shows that isolating the condenser component and optimizing it independently by minimizing the entropy generation in the condenser component alone, also known as thermoeconomic isolation, can be a practical way to design the condenser for optimum air-conditioning system efficiency. This study is accomplished by comparing the optimum design determined by maximizing the entire system’s COP, an undisputed method, with the optimum design determined by minimizing the entropy generation in the isolated condenser component, with consistent constraints used for the two methods. The choice of component junction constraints used in the isolated model is critical and discussed in detail. The resulting optimum designs from the isolated model produced a COP within 0.6%e1.7% of the designs found by optimizing the COP using an entire system model. Additionally, a 65% reduction in computation time was achieved. Ó 2010 Elsevier Ltd. All rights reserved.
Keywords: Finned-tube Heat exchanger Optimization Thermoeconomic isolation
1. Introduction The performance of most energy systems is dominated by heat exchanger design. A primary example is the air-cooled condenser heat exchanger used in single-family air-conditioning systems with a cooling capacity that is typically 8.8 kW (2.5 tons). In the US, these air-conditioning units must meet US DOE minimum efficiency standards based on annual performance weighted with operating hours over a wide ambient temperature range. With the recent ban of R-22 in new equipment, new condenser designs are necessary using environmentally friendlier refrigerants. Additionally, in January 2006 the minimum air conditioner system efficiency requirement was raised by 30% to a Seasonal Energy Efficiency Rating (SEER) of 13. This efficiency improvement can be accomplished with minimal added cost with the new refrigerants via heat exchanger system design optimization. Optimization of finned-tube heat exchangers is a complex and difficult task requiring the determination of over a dozen different finned-tube heat exchanger design interrelated parameters, with appropriate constraints. Additionally, the competing effects of improved heat transfer with amplified air and refrigerant frictional pressure drop make it difficult to determine the relative goodness of a design. Therefore, the appropriate selection of a figure-of-merit is very important. The most common figure-of-merit for an air-
* Corresponding author. Tel.: þ1 814 863 5381. E-mail address: [email protected] (S.W. Stewart). 1359-4311/$ e see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.applthermaleng.2010.05.018
conditioning system is the efficiency, which is expressed as the coefficient of performance (COP). Minimum entropy generation for the entire cycle can also be used because there is a direct correspondence between COP and entropy generation. Both of these methods require the entire system, including fans, to be modeled in order to determine how changes in the condenser design affect the system performance. Alternatively, many researchers have optimized the design of heat exchangers using a component based figure-of-merit. The problem with these techniques is that quite often the figures-ofmerit used are purely heuristic and have no theoretical basis. For instance, the use of minimum heat transfer area, cost, friction factor or a heat transfer parameter like Colburn j-factor as fitness factors give little information about the merits of the heat exchanger designs regarding overall system performance. The concept of thermoeconomic isolation is a method that also falls into the component based category. In this case the figure-ofmerit is minimum entropy generation in the isolated condenser component. The idea that the design obtained from minimizing the entropy generation in an isolated component coincides with the design from maximizing the system performance (COP) stems from the fact that system entropy generation and COP are related through fundamental relations [1]. Since the isolated model is lacking feedback from the rest of the system as the design changes, it is very difficult if not impossible to predict that the resulting component design is the best from the system performance viewpoint by just looking at an individual component. This has been the issue with thermoeconomic isolation, and why its theoretical
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development has been limited. However, von Spakovsky and Evans [2] wrote that although thermoeconomic isolation is an ideal condition that real world systems can only approach, if approached closely enough both detailed and practical component and system optimizations are possible. The current study shows that thermoeconomic isolation can be used as a practical design tool for optimizing finned-tube condenser heat exchangers with the goal of optimizing the COP of the air-conditioning system they are designed for, but only if appropriate constraints are used in the isolated model. 2. Backrgound 2.1. Thermoeconomic isolation The technique of Entropy Generation Minimization (EGM) has been in use since the 1970s and is discussed in detail in Bejan [3]. According to Bejan and Pfister [4], “the ‘energy conservation’ value of a heat transfer augmentation technique can be best measured in terms of the technique’s ability to reduce the rate of entropy generation (irreversibility, exergy destruction) in the heat transfer device in which it is implemented”. The central theme of the EGM method is that by minimizing entropy generation, performance is optimized. Bejan [3e5] gives many examples of situations in which EGM can be used to optimize performance; however, most cases are very basic in nature and have constraints that make their direct application to a real world design problem impractical. The fact that system entropy generation and first law efficiency are directly related stems from the idea of thermoeconomic isolation. This terminology is rooted in a type of analysis known as thermoeconomic optimization, which involves a combination of an exergy/availability analysis of a system with an economical analysis for optimization of thermal systems. A value is placed on the “lost work” from irreversibilities in the system [6]. An early contribution to this field was by Tribus and Evans [7]. Thermoeconomic “isolation”, however, is defined by Muñoz and von Spakovsky [8] as the ability to optimize independently each unit of a system and yet still arrive at the optimum system as a whole. A second law analysis using availability, exergy, or minimum entropy generation analysis is required. Tapia and Moran [9] state that system components can be regarded as isolated (in thermoeconomic isolation) from one another when the proper value is assigned to the availability at the various component junctions. Under these conditions, Tapia and Moran assert that thermoeconomic isolation guarantees that optimizing a component of an overall thermal system by itself coincides with optimization of the system as a whole. In the current study (as many others have found) these component junction values vary with changing component designs. Therefore, varying the design of one component can decrease entropy generation, or irreversibility, in that component while increasing it in another. However, by selecting appropriate variables and component junction values to fix, this ideal condition can be approached. This fact has rarely been studied or discussed in any detail. Many studies inherently assume that thermoeconomic isolation is valid, investigating entropy generation minimization (or other figures-of-merit) in just the heat exchanger component without vigorous regard to the system in which it is placed or considering constraints at the component junctions. McClintock [10] first reports the opportunity for minimizing irreversibility in heat exchangers via design selections in 1951, while Bejan [11], Sekuli c [12], Witte and Shamsundar [13] and others conducted foundational studies on the second law analysis of heat exchangers. Building on these early works, Hesselgreaves [14] addresses the
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inconsistencies between methodologies and defines a nondimensional entropy generation number for optimization purposes, restricted to the analysis of perfect gas flow. Saboya and da Costa [15] compared the minimum irreversibility for a counterflow heat exchanger with other simple heat exchanger configurations. The above analyses are for very simple heat exchanger types, while several other studies have investigated more complicated geometries and conditions. San and Jan [16] apply a second law analysis to a wet cross flow heat exchanger and Saechan and Wongwises [17] investigate the optimum configuration of cross flow plate finned-tube condensers using the second law. Guo et al. [18] develops a field synergy number (Nu/RePr), providing an indicator of the synergy between the velocity field and the heat flow, as a figure-of-merit for optimizing a shell-and-tube heat exchanger. Caputo et al. [19] also investigates the optimum design for shell-and-tube heat exchangers using an economic figure-ofmerit that accounts for capital investment and energy costs related to pumping. Lin and Lee [20] applied the analysis of Bejan and Pfister [4] to a wavy plate finned-tube heat exchanger. Additionally, Schenone et al. [21] performed a second law analysis to optimize the fin geometry of offset stripefin heat exchangers. All of these studies assumed that by optimizing the heat exchanger component the entire system would be optimized, but none attempted to address the validity of this assumption. Rancruel and von Spakovsky [22] applied a decomposition strategy developed by Muñoz and von Spakovsky [23], called Iterative Local-Global Optimization (ILGO), toward an optimal synthesis/design and operation of an advanced tactical aircraft system. They found ILGO to be the first decomposition strategy to successfully closely approach the theoretical condition of ‘‘thermoeconomic isolation’’ when applied to highly complex, highly dynamic, non-linear systems. 2.2. Heat exchanger modeling and optimization Several authors, such as Vardhan and Dhar [24], Bensafi et al. [25] and Corberan and Melon [26], have developed comprehensive simulation models of the individual finned-tube heat exchanger component with reasonable accuracy (w1%e30% error) compared to experimental data. Each of these models uses a nodal analysis approach and requires significant computation time to obtain performance results; therefore, these models were not used for optimization purposes. Jiang et al. [27] also developed a simulation tool for design of finned-tube coils. While the model developed by Jiang et al. has not yet been experimentally validated, it seems to be able to account for several detailed aspects of the coil, such as complex circuitry, various flow configurations (counter-cross, parallel-cross) and nonuniform air distributions. In general there is a focus on the refrigerant side design rather than the air side. The model discretizes the tube into segments and solves the momentum and energy equations for each segment alternatively and repeatedly until convergence is obtained. In addition to the simulation tool, Jiang et al. mention an example optimization case using a genetic algorithm to optimize the number of rows, number of tubes per row, option of parallelcross flow or counter-cross flow, and tube diameter. The objective function was minimum total heat transfer area at a given heat load subject to maximum allowable pressure drops of the refrigerant flow and air flow. This optimization requires detailed constraint information (pressure drops). Additionally, the objective function is subjective. The optimization results in a smaller surface area heat exchanger for the price at whatever allowable pressure drops are specified. But it is not discussed how or why one would choose
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a particular pressure drop and nothing on the air side of the heat exchanger is optimized. Tayal et al. [28] also looked at searching for optimum individual heat exchanger components using a combinatorial (mixed discrete and continuous variable) optimization scheme such as the genetic algorithm, using minimum heat transfer area as well as cost as objective functions in their optimization procedures. Tayal et al. conclude that genetic algorithms are equally effective as compared to simulated annealing in solving black-box optimization model problems. Their study focuses on shell-and-tube heat exchangers for process industries. Caputo et al. [19] and Özçelik [29] also use a genetic algorithm for optimization of a shell-and-tube heat exchanger. Richardson et al. [30] developed a program that simulates a vapor compression system. This program is then used to optimize the system level variables using gradient based and genetic optimization routines. Parameters included in this simulation program are: refrigerant charge, COP, weight, capacity, and cost. Optimization objective functions used were COP, capacity and system weight. This study did not optimize the design of the individual components; rather it used a collection of specific components and found the best combination of them for specified inputs. Additionally, the heat exchanger models used in this simulation program are simplistic, assuming infinite air flow rate and fixed properties at the inlet regime for refrigerant pressure drop calculations. The current study builds on the preceding development work of Wright [31] and Aspelund [32]. A model was developed in Engineering Equation Solver (EES) [33] of an air-conditioning system using R-410a as the working fluid. This model includes a detailed simulation of the components of the air-conditioning system for various designs, including the compressor, finned-tube condenser, evaporator, and expansion valve. The condenser component is the focus of the model incorporating the best available simulations for the air side and refrigerant side pressure drops and heat transfer coefficients based on R-410a as the working fluid. A design optimization search technique (using the Simplex Method by Nelder and Mead [34]) was implemented to optimize 10 controllable, operational, and geometric design parameters of the condenser with a seasonal COP figure-of-merit. The details of the system model can be found in Stewart [35]. The methodology used to define the isolated condenser model is detailed in the next section.
3. Condenser model In order to perform thermoeconomic isolation, the component of interest must be entirely “isolated” from the rest of the system. By doing this, the component ceases to give or receive feedback from the rest of the system. However, in order to produce a properly posed and complete model, additional operating constraints are required. A total of five external junction type constraints are needed. Referencing Fig. 1, the options from which to choose are:
_ Q_ cond ; DTSH ; DTSC ; P1air ; P2air ; P2r ; P3r ; T1air ; T2air ; T2r ; T3r ; mr; and Tsat including the pressures and temperatures at the entrance and exit of the air side (1air, 2air) and refrigerant sides (2r 3r), respectively, the mass flow rate of the refrigerant, the heat transfer rate from the condenser, the degree of superheat entering the condenser (note that this is not the typical definition of superheat in an air-conditioning system, which usually describes the superheat exiting the evaporator), the degrees of sub-cool exiting the condenser (which is a parameter set by the refrigerant charge in the system) and also the saturation temperature in the condenser.
Fig. 1. Detailed condenser flow diagram.
For the condenser of a residential air-conditioning system, two of the needed five constraints are rather straightforward; i.e., the inlet air pressure and temperature are assumed to be ambient. Therefore, the P1air is taken as 1 bar, and T1air was taken as 27.8 C (82 F), which was found by Stewart [35] to give a nearly identical COP as the seasonal COP used by the US Department of Energy in rating residential air conditioners. These values were used in the overall system model as well. The pressure drop and temperature increase on the air side are then calculated by the model and cannot be constrained. Three additional constraints are needed to complete the isolated model. The appropriate selection of these three constraints was found to be extremely important in determining whether the results from the thermoeconomic isolation optimization were close to the results from the system optimization. Therefore, the decision making process is detailed here. Practically, one parameter would give information about the refrigerant entrance conditions (P2r, T2r, Tsat, DTSH), the second parameter would give information about the refrigerant exit conditions (P3r, T3r, Tsat, DTSC), and the third would give information about the heat transfer scaling between the air side and the refrigerant side. While almost any combination of three additional parameters will result in a properly posed model, certain combinations will not give as good results due to their variations with changing condenser design when the condenser is integrated into a system. To find the best combination of parameters the following requirements should be met: (1) the parameters should not require detailed non-condenser system information to arrive at a value, and (2) they should not vary greatly with different heat exchanger designs when operating in the system. Because of these two requirements, explicit temperature and pressure information are not good options; therefore, P2r, P3r, T2r, T3r and even the saturation temperature in the condenser are taken out of consideration as they will all vary dramatically with changing heat exchanger design. The remaining options to specify the refrigerant inlet and outlet conditions are DTSH and DTSC, respectively. With these temperature differences chosen, the only decision left to be made is the size scaling parameter, which could be either the refrigerant mass flow rate or the heat transfer rate between the refrigerant and the air. Both of these situations were investigated in more detail. Prescribing the mass flow rate of the refrigerant in the isolated condenser model does not fix the total heat transfer rate in the heat exchanger. This is due to the fact that as the saturation pressure varies between heat exchanger designs, the heat of condensation varies. Taken with fixed inlet superheat and exit sub-cool, the heat transfer rate increases as the saturation pressure decreases. In the system model, the refrigerant mass flow rate varies to compensate for this, while the total heat transfer rate stays relatively constant. Therefore, fixing the heat transfer rate is a more appropriate size scale constraint.
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As shown by equations (1) (neglecting the work of the fans) and (2), the heat transfer rate is dependent only on the COP and the desired evaporator cooling capacity, e.g., size of the system for which the condenser is being designed:
COP ¼
Q_ evap Q_ evap Desired Output ¼ ¼ _ _ Required Input W comp Q cond Q_ evap
Manifold
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Manifold
1/FP
Xl
(1)
Xt
Rearranging and solving for Q_ cond :
1 Q_ cond ¼ Q_ evap 1 þ COP
Air Inlet (Vac)
(2)
With a refrigerating capacity fixed at 8.8 kW and a typical COP of approximately 4, this gives $Q cond ¼ 11 kW. Note that $Q cond is only weakly dependent on COP; e.g., a 25% change in COP produces only a 4e6% change in $Q cond . In practical application, a small change in operating conditions can result in a liquidevapor mixture exiting the condenser, which, due to limited expansion valve flow area, would back up behind the valve until a high enough pressure is reached to fully condense the vapor. The expansion valve would than be wide open and have no control over the evaporator superheat. While the optimum subcool is actually 0 C, to avoid this situation under the normal range of operating conditions, the sub-cool was specified as 5 C when operating at 35 C ambient temperature. Using this design condition the system model then calculates the sub-cool for the ambient condition of 27.8 C (82 F), at which the system was optimized. The resulting calculated sub-cool at 27.8 C was normally slightly higher than 5 C depending on the condenser design, so for the isolated model the value was fixed at 5 C for simplicity. The value for the condenser inlet superheat was set at 25 C (45 F) in the isolated model. This value was determined by looking at some typical model results from system designs. 4. Optimization parameters Table 1 shows the parameters necessary to define the design of the finned-tube condenser considered in this optimization study. Many are also depicted in Fig. 2. In this heat exchanger design optimization, a single objective function is used in the optimization procedure. Initially, this objective function is seasonal COP, while minimum condenser entropy generation is considered as an alternative to COP in the isolated condenser case (thermoeconomic isolation). Initially, the design parameters were not constrained in
Refrigerant outlet
Fig. 2. Condenser design parameters (shown: 4 rows, 2 circuits, 6 tubes per row).
the optimization procedure except for those that were limited by the range of experimental data used to develop the empirical correlations employed by this study (Xl, Xt, FP, Vac, Lh, Lp) as discussed in Stewart [35] and shown in Table 1. As expected, some variables optimized to their limits. When the limits in these cases were zero or infinity, those design parameters were then constrained to practical limits, as discussed below. Heat exchanger price is typically based on a multiple of material coast assuming mass production; therefore, the cost of the heat exchanger material was used in this study as an indicator for heat exchanger cost. The volume of each type of metal was calculated and multiplied by its density and cost per weight to determine the material cost for each heat exchanger design. Since increasing material cost of a heat exchanger will increase the COP, the cost must be constrained to a maximum value for each optimization. The costs of the materials are based on published values of aluminum (US$1.54/kg) and copper (US$1.76/kg) from the London Metals Exchange [36]. It should be noted that these values have experienced large fluctuations over the last several years and, therefore, may vary from these values in the future. When frontal area is not constrained, the fixed cost design leads to the largest frontal area (Af / N) corresponding to one row. This situation yields the minimum air pressure drop. Therefore, the larger the frontal area the better, if space and cost allow. Because of this, the frontal area was specified, or constrained, for each
Table 1 Design parameters.
28 Design parameters
Upper limit
0 0 0 7.94 0.157 17.8 12.7 1 1 1
Geometric: Frontal area (Af) [m2] Aspect ratio (AR) Fin thickness (tfin) [mm] Tube diameter (D) [mm] Fin pitch (FP) [mm1] Transverse tube spacing (Xt) [mm] Longitudinal tube spacing (Xl) [mm] Number of rows (rows) Tubes per row (tpr) Number of circuits (circ)
N N N 7.94 0.71 30.5 28 N N N
0.79 1.7
Operational: Air velocity (Vac) [m/s] Sub-cool (TSC) [ C] Louvers: Louver height (Lh) [mm] Louver pitch (Lp) [mm]
5.3 N 1.4 3.75
26
Xt [mm]
Lower limit
0.91 0
Refrigerant inlet
24
22 Isolated System
20 10
20
30 40 Condenser Cost [US$]
Fig. 3. Transverse tube spacing vs. condenser cost (0.75 m2 frontal area).
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S.W. Stewart, S.V. Shelton / Applied Thermal Engineering 30 (2010) 2096e2102
30
4
25
3
Air Velocity [m/s]
Xl [mm]
2100
20
15
System 0
10 20
30 40 Condenser Cost [US$]
1 Isolated
Isolated System 10
2
50
10
20
30 40 Condenser Cost [US$]
50
Fig. 4. Longitudinal tube spacing vs. condenser cost (0.75 m2 frontal area).
Fig. 6. Air velocity vs. condenser cost (0.75 m2 frontal area).
optimization while the effects of varying frontal area were investigated by comparing optimum designs using different frontal area constraints, (0.5, 0.75, 1.0, and 1.25 m2) at the same condenser cost ($25). If the overall heat exchanger aspect ratio (width divided by height) were not restricted, the design converges to a single very long finned-tube (AR / N), since tube bends have pressure drop but no heat transfer. Therefore, the aspect ratio was limited by a maximum of three. The outside A/C condensing/compressor unit is typically a cube in shape with the condenser bent around three sides. For all of the optimizations the tube diameter was fixed at 7.94 mm (5/1600 ). This is because Aspelund [32] found that the smaller the tube (D / 0), the better the COP, with little improvement beyond 7.94 mm. Benefits of using smaller diameter tubes include smaller form drag caused by the tube, higher refrigerant side heat transfer coefficients due to smaller hydraulic diameter, and less refrigerant inventory in the system [37]. Also, the fin thickness was ultimately fixed at 0.15 mm (0.00600 ). When left in as a search parameter, the solution always converged to a design with thinner and thinner fins (tfin / 0) while making the fin pitch larger and decreasing the air velocity to adjust for the increased pressure drop. This makes sense theoretically, however, in reality extremely thin fins are not structurally durable, and dirt and dust will clog the fins when the fin spacing is too small. Because of this, the fin thickness was constrained to a practical minimum value of 0.15 mm.
The number of rows multiplied by the longitudinal tube spacing determines the depth of the condenser, while the tubes per row multiplied by the transverse tube spacing determines the height. The number of circuits determines the number of parallel tube passages the refrigerant mass flow rate is divided amongst by the manifold. These circuitry parameters were not constrained. Note that the number of tubes per circuit, tubes per row, and number of rows all must be discrete values in a real design. In the optimization process of this study, this restriction was not considered. Instead hypothetical continuous optimum designs are found for the circuitry parameters so that clear design trends are more evident.
0.8
Fin Pitch [1/mm]
0.7 0.6 0.5 0.4
Isolated System
0.3 10
20
30 40 Condenser Cost [US$]
Fig. 5. Fin pitch vs. condenser cost (0.75 m2 frontal area).
50
5. Results and discussion The resulting system and isolated condenser models were run under the conditions described in the previous section using the optimization search scheme as discussed in Stewart [35]. The resulting optimum designs for the system model and the isolated condenser model are compared in the figures below. For the COP calculation, the optimum design determined by the isolated condenser optimization was then run in the system model to determine the system COP for that design. Figs. 3e5 show the optimum transverse tube spacing, the longitudinal tube spacing, and the fin pitch, respectively, vs. varying condenser material cost for a 0.75-m2 frontal area. It can be seen from these figures that the resulting optimum designs from both the system and the isolated model optimizations are quite similar to each other in both trend and value. Additionally, the optimum number of rows for both cases are identical. The isolated model optimization search scheme was developed to choose an optimum design based on minimizing the entropy generation in the condenser. The components of this entropy generation are heat transfer through a finite temperature difference and both air and refrigerant pressure drop terms. These temperature and pressure effects compete with each other, i.e., as the pressure drop decreases the heat transfer coefficient decreases. Due to the trading off of these effects, the optimum design will converge to the case with a minimum of irreversibility. Therefore, as the isolated optimization scheme reduced the air side velocity (Fig. 6, for 0.75 m2 frontal area), it reduced both the air and refrigerant side pressure drops. This is because in order to maintain the constant condenser heat transfer rate, there was a slight increase in the average saturation temperature and pressure in the condenser and a slight increase in the refrigerant side mass flow rate, but this flow rate was split into more (18.2%e29.3%) parallel circuits (number of circuits) resulting in a lower pressure drop on the refrigerant side
S.W. Stewart, S.V. Shelton / Applied Thermal Engineering 30 (2010) 2096e2102
some of the design parameters showed significant differences in their optimum values (as much as 29% different). With a 65% reduction in computation time using the isolated model, this provides a very practical and effective method for designing finnedtube condenser heat exchangers. Caution, however, should be taken when using thermoeconomic isolation in choosing the appropriate external constraints. The procedure will give results with multiple combinations of external constraints but these optimum results will lack in accuracy with improper choices.
4.4 4.3
COP
2101
4.2 4.1 4
References
Isolated System
3.9 10
20 30 40 Condenser Cost [US$]
50
Fig. 7. COP vs. condenser cost (0.75 m2 frontal area).
compared to the corresponding system optimized design (e.g., at $25, 0.75 m2: 69.84 kPa vs. 119.1 kPa). The net result of these effects reduced the entropy generation in the condenser, which was the figure-of-merit for the isolated model. However, when this isolated optimum design is placed in the context of the overall system, the reduction in pressure drop in the condenser creates the need for a larger irreversibility in the expansion valve. Also, the increase in saturation pressure in the condenser (with a fixed DTSH) and the increased refrigerant mass flow rate requires a larger compressor power, also creating more irreversibility. The net result of these increasing entropy effects in other components offset the lower entropy generation in the condenser. The overall effects of these model differences can be seen in Fig. 7, which shows COP vs. varying condenser material cost for a fixed frontal area of 0.75 m2. It can be seen that the COPs from the system model optimizations are slightly higher than the isolated condenser optimized designs. The variation ranges from 0.6% at $45 to 1.7% at $15. So, even though some of the design parameters differed by as much as 29% between the two methods, the isolated condenser optimization did produce designs very close in COP to those produced by using an entire system model to optimize the condenser design. The 65% reduction in computation time for the isolated model makes this a very attractive and practical option for the design optimization of finned-tube condenser heat exchangers, with a caution that a prudent choice of constraints must be considered. 6. Conclusions Using a detailed system model as a comparison, this study shows that isolating the condenser component and optimizing it independently by minimizing the entropy generation in the condenser component alone (also known as thermoeconomic isolation) can be a practical way to design the condenser for optimum air-conditioning system efficiency. It was found that the most important aspect required for a successful isolated model optimization is a proper set of external or junction constraints. In the current study, constraining the condenser heat transfer rate, condenser exit sub-cool, and the condenser entering superheat were found to give the best results. These parameters vary little in the system model with changing condenser designs and they do not require detailed system information in order to specify them within reasonable accuracy. The resulting optimum designs from the isolated model produced a COP within 0.6%e1.7% of the designs found by optimizing the COP using an entire system model, despite the fact that
[1] Alefeld G., What are thermodynamic losses and how to measure them? A future for energy. In: Proceedings of the World Energy Symposium, Italy, May 1990. [2] M.R. von Spakovsky, R.B. Evans, The Optimal Design and Performance of Components in Thermal Systems, Second-Law Analysis on Heat/Mass Transfer and Energy Conversion, vol. 97, ASME HTD, 1988. [3] A. Bejan, Entropy Generation Through Heat and Fluid Flow. Wiley, New York, 1982. [4] A. Bejan, P.A. Pfister, Evaluation and heat transfer augmentation techniques based on their impact on entropy generation. Letters in Heat and Mass Transfer 7 (1980) 97e106. [5] A. Bejan, Entropy Generation Minimization, the Method of Thermodynamic Optimization of Finite-Size Systems and Finite-time Processes. CRC Press, New York, 1996. [6] M.J. Moran, H.N. Shapiro, Fundamentals of Engineering Thermodynamics, fourth ed. John Wiley & Sons, New York, 2003. [7] Tribus M., Evans R.B., A contribution to the theory of thermoeconomics. UCLA Engineering Department 1962; Report no. 62e36. [8] J.R. Muñoz, M.R. von Spakovsky, Decomposition in energy system synthesis/ design optimization for stationary and aerospace applications. Journal of Aircraft 40 (1) (2003) (JaneFeb). [9] C.F. Tapia, M.J. Moran, Computer-aided design and optimization of heat exchangers. ASME AES 2 (1) (1986) 93e103. [10] McClintock F.A., The design of heat exchangers for minimum irreversibility. Presented at the ASME Annual Meeting 1951, 51-A-108. [11] A. Bejan, The concept of irreversibilityin heat exchanger design: counterflow heat exchanger for gas-to-gas applications. Transactions ASME Journal of Heat Transfer 99 (1977) 374e380. [12] D.P. Sekuli c, Entropy generation in a heat exchanger. Heat Transfer Engineering 7 (1e2) (1986) 83e88. [13] L.C. Witte, N. Shamsundar, A thermodynamic efficiency concept for heat exchange devices. Journal of Engineering Power 105 (1983) 199e203. [14] J.E. Hesselgreaves, Rationalisation of second law analysis of heat exchangers. International Journal of Heat and Mass Transfer 43 (2000) 4189e4204. [15] F.E.M. Saboya, C.E.S.M. da Costa, Minimum irreversibility criteria for heat exchanger configurations. Journal of Energy Resources Technology 121 (1999) 241e246. [16] J.-Y. San, C.-L. Jan, Second-law analysis of a wet crossflow heat exchanger. Energy 25 (2000) 939e955. [17] P. Saechan, S. Wongwises, Optimal configuration of cross flow plate finned tube condenser based on the second law of thermodynamics. International Journal of Thermal Sciences 47 (2008) 1473e1481. [18] J. Guo, M. Xu, L. Cheng, The application of field synergy number in shell-and-tube heat exchanger optimization design. Applied Energy 86 (10) (2009) 2079e2087. [19] A.C. Caputo, P.M. Pelagagge, P. Salini, Heat exchanger design based on economic optimization. Applied Thermal Engineering 28 (2008) 1151e1159. [20] W.W. Lin, D.J. Lee, Second law analysis on a pinefin array under cross flow. International Journal of Heat and Mass Transfer 40 (8) (1997) 1937e1945. [21] C. Schenone, L. Tagliafico, G. Tanda, Second law analysis for offset strip-fin heat exchangers. Heat Transfer Engineering 12 (1) (1991) 19e27. [22] D.F. Rancruel, M.R. von Spakovsky, Decomposition with thermoeconomic isolation applied to the optimal synthesis/design and operation of an advanced tactical aircraft system. Energy 31 (2006) 3327e3341. [23] J.R. Muñoz, M.R. von Spakovsky, Decomposition in energy system synthesis/ design optimization for stationary and aerospace applications. AIAA Aircraft 39 (6) (2003) (special issue). [24] A. Vardhan, P.L. Dhar, A new procedure for performance prediction of air conditioning coils. International Journal of Refrigeration 21 (1) (1998) 77e83. [25] A. Bensafi, S. Borg, D. Parent, CYRANO: a computational model for the detailed design of plate-fin-and-tube heat exchangers using pure and mixed refrigerants. International Journal of Refrigeration 20 (3) (1997) 218e228. [26] J.M. Corberan, M.G. Melon, Modeling of plate finned tube evaporators and condensers working with R134a. International Journal of Refrigeration 21 (4) (1998) 273e284. [27] Jiang H., Aute V., Radermacher R., A user friendly simulation and optimization tool for design of coils. Ninth International Refrigeration and Air Conditioning Conference at Purdue 2002; Paper # R5e1. [28] M. Tayal, Y. Fu, U. Diwekar, Optimal design of heat exchangers: a genetic algorithm framework. Industrial and Engineering Chemistry Research 38 (1999) 456e467.
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[29] Y. Özçelik, Exergetic optimization of shell and tube heat exchangers using a genetic based algorithm. Applied Thermal Engineering 27 (2008) 1849e1856. [30] Richardson D., Jiang H., Lindsay D., Radermcaher R., Optimization of vapor compression systems via simulation, Ninth International Refrigeration and Air Conditioning Conference at Purdue, 2002; Paper # R1e3. [31] Wright M.F., Plate-fin-and-tube condenser performance and design for refrigerant R-410a air-conditioner. M.S. Thesis, Georgia Institute of Technology, 2000. [32] Aspelund K.A., Optimization of plate-fin-and-tube condenser performance and design for refrigerant R-410A air-conditioner. M.S. Thesis, Georgia Institute of Technology, 2001.
[33] Klein S.A., Alvarado F.L., Engineering equation solver, F-Chart Software, 2009; Ver. 6.5.6.3. [34] J.A. Nelder, R. Mead, A Simplex Method for Function Minimization. Computer Journal 7 (1965) 308e313. [35] Stewart S.W., Enhanced finned-tube condenser design and optimization. Ph.D. Dissertation, Georgia Institute of Technology. 2003. [36] London Metals Exchange, October 2003. Available at:
Contents lists available at ScienceDirect
Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng
Finned-tube condenser design optimization using thermoeconomic isolation Susan W. Stewart a, *, Samuel V. Shelton b a b
The Pennsylvania State University, Applied Research Laboratory, State College, PA 16804, USA Georgia Institute of Technology Strategic Energy Institute, Atlanta, GA 30332, USA
a r t i c l e i n f o
a b s t r a c t
Article history: Received 29 December 2009 Accepted 18 May 2010 Available online 23 May 2010
Using a detailed system model as a comparison, this study shows that isolating the condenser component and optimizing it independently by minimizing the entropy generation in the condenser component alone, also known as thermoeconomic isolation, can be a practical way to design the condenser for optimum air-conditioning system efficiency. This study is accomplished by comparing the optimum design determined by maximizing the entire system’s COP, an undisputed method, with the optimum design determined by minimizing the entropy generation in the isolated condenser component, with consistent constraints used for the two methods. The choice of component junction constraints used in the isolated model is critical and discussed in detail. The resulting optimum designs from the isolated model produced a COP within 0.6%e1.7% of the designs found by optimizing the COP using an entire system model. Additionally, a 65% reduction in computation time was achieved. Ó 2010 Elsevier Ltd. All rights reserved.
Keywords: Finned-tube Heat exchanger Optimization Thermoeconomic isolation
1. Introduction The performance of most energy systems is dominated by heat exchanger design. A primary example is the air-cooled condenser heat exchanger used in single-family air-conditioning systems with a cooling capacity that is typically 8.8 kW (2.5 tons). In the US, these air-conditioning units must meet US DOE minimum efficiency standards based on annual performance weighted with operating hours over a wide ambient temperature range. With the recent ban of R-22 in new equipment, new condenser designs are necessary using environmentally friendlier refrigerants. Additionally, in January 2006 the minimum air conditioner system efficiency requirement was raised by 30% to a Seasonal Energy Efficiency Rating (SEER) of 13. This efficiency improvement can be accomplished with minimal added cost with the new refrigerants via heat exchanger system design optimization. Optimization of finned-tube heat exchangers is a complex and difficult task requiring the determination of over a dozen different finned-tube heat exchanger design interrelated parameters, with appropriate constraints. Additionally, the competing effects of improved heat transfer with amplified air and refrigerant frictional pressure drop make it difficult to determine the relative goodness of a design. Therefore, the appropriate selection of a figure-of-merit is very important. The most common figure-of-merit for an air-
* Corresponding author. Tel.: þ1 814 863 5381. E-mail address: [email protected] (S.W. Stewart). 1359-4311/$ e see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.applthermaleng.2010.05.018
conditioning system is the efficiency, which is expressed as the coefficient of performance (COP). Minimum entropy generation for the entire cycle can also be used because there is a direct correspondence between COP and entropy generation. Both of these methods require the entire system, including fans, to be modeled in order to determine how changes in the condenser design affect the system performance. Alternatively, many researchers have optimized the design of heat exchangers using a component based figure-of-merit. The problem with these techniques is that quite often the figures-ofmerit used are purely heuristic and have no theoretical basis. For instance, the use of minimum heat transfer area, cost, friction factor or a heat transfer parameter like Colburn j-factor as fitness factors give little information about the merits of the heat exchanger designs regarding overall system performance. The concept of thermoeconomic isolation is a method that also falls into the component based category. In this case the figure-ofmerit is minimum entropy generation in the isolated condenser component. The idea that the design obtained from minimizing the entropy generation in an isolated component coincides with the design from maximizing the system performance (COP) stems from the fact that system entropy generation and COP are related through fundamental relations [1]. Since the isolated model is lacking feedback from the rest of the system as the design changes, it is very difficult if not impossible to predict that the resulting component design is the best from the system performance viewpoint by just looking at an individual component. This has been the issue with thermoeconomic isolation, and why its theoretical
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development has been limited. However, von Spakovsky and Evans [2] wrote that although thermoeconomic isolation is an ideal condition that real world systems can only approach, if approached closely enough both detailed and practical component and system optimizations are possible. The current study shows that thermoeconomic isolation can be used as a practical design tool for optimizing finned-tube condenser heat exchangers with the goal of optimizing the COP of the air-conditioning system they are designed for, but only if appropriate constraints are used in the isolated model. 2. Backrgound 2.1. Thermoeconomic isolation The technique of Entropy Generation Minimization (EGM) has been in use since the 1970s and is discussed in detail in Bejan [3]. According to Bejan and Pfister [4], “the ‘energy conservation’ value of a heat transfer augmentation technique can be best measured in terms of the technique’s ability to reduce the rate of entropy generation (irreversibility, exergy destruction) in the heat transfer device in which it is implemented”. The central theme of the EGM method is that by minimizing entropy generation, performance is optimized. Bejan [3e5] gives many examples of situations in which EGM can be used to optimize performance; however, most cases are very basic in nature and have constraints that make their direct application to a real world design problem impractical. The fact that system entropy generation and first law efficiency are directly related stems from the idea of thermoeconomic isolation. This terminology is rooted in a type of analysis known as thermoeconomic optimization, which involves a combination of an exergy/availability analysis of a system with an economical analysis for optimization of thermal systems. A value is placed on the “lost work” from irreversibilities in the system [6]. An early contribution to this field was by Tribus and Evans [7]. Thermoeconomic “isolation”, however, is defined by Muñoz and von Spakovsky [8] as the ability to optimize independently each unit of a system and yet still arrive at the optimum system as a whole. A second law analysis using availability, exergy, or minimum entropy generation analysis is required. Tapia and Moran [9] state that system components can be regarded as isolated (in thermoeconomic isolation) from one another when the proper value is assigned to the availability at the various component junctions. Under these conditions, Tapia and Moran assert that thermoeconomic isolation guarantees that optimizing a component of an overall thermal system by itself coincides with optimization of the system as a whole. In the current study (as many others have found) these component junction values vary with changing component designs. Therefore, varying the design of one component can decrease entropy generation, or irreversibility, in that component while increasing it in another. However, by selecting appropriate variables and component junction values to fix, this ideal condition can be approached. This fact has rarely been studied or discussed in any detail. Many studies inherently assume that thermoeconomic isolation is valid, investigating entropy generation minimization (or other figures-of-merit) in just the heat exchanger component without vigorous regard to the system in which it is placed or considering constraints at the component junctions. McClintock [10] first reports the opportunity for minimizing irreversibility in heat exchangers via design selections in 1951, while Bejan [11], Sekuli c [12], Witte and Shamsundar [13] and others conducted foundational studies on the second law analysis of heat exchangers. Building on these early works, Hesselgreaves [14] addresses the
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inconsistencies between methodologies and defines a nondimensional entropy generation number for optimization purposes, restricted to the analysis of perfect gas flow. Saboya and da Costa [15] compared the minimum irreversibility for a counterflow heat exchanger with other simple heat exchanger configurations. The above analyses are for very simple heat exchanger types, while several other studies have investigated more complicated geometries and conditions. San and Jan [16] apply a second law analysis to a wet cross flow heat exchanger and Saechan and Wongwises [17] investigate the optimum configuration of cross flow plate finned-tube condensers using the second law. Guo et al. [18] develops a field synergy number (Nu/RePr), providing an indicator of the synergy between the velocity field and the heat flow, as a figure-of-merit for optimizing a shell-and-tube heat exchanger. Caputo et al. [19] also investigates the optimum design for shell-and-tube heat exchangers using an economic figure-ofmerit that accounts for capital investment and energy costs related to pumping. Lin and Lee [20] applied the analysis of Bejan and Pfister [4] to a wavy plate finned-tube heat exchanger. Additionally, Schenone et al. [21] performed a second law analysis to optimize the fin geometry of offset stripefin heat exchangers. All of these studies assumed that by optimizing the heat exchanger component the entire system would be optimized, but none attempted to address the validity of this assumption. Rancruel and von Spakovsky [22] applied a decomposition strategy developed by Muñoz and von Spakovsky [23], called Iterative Local-Global Optimization (ILGO), toward an optimal synthesis/design and operation of an advanced tactical aircraft system. They found ILGO to be the first decomposition strategy to successfully closely approach the theoretical condition of ‘‘thermoeconomic isolation’’ when applied to highly complex, highly dynamic, non-linear systems. 2.2. Heat exchanger modeling and optimization Several authors, such as Vardhan and Dhar [24], Bensafi et al. [25] and Corberan and Melon [26], have developed comprehensive simulation models of the individual finned-tube heat exchanger component with reasonable accuracy (w1%e30% error) compared to experimental data. Each of these models uses a nodal analysis approach and requires significant computation time to obtain performance results; therefore, these models were not used for optimization purposes. Jiang et al. [27] also developed a simulation tool for design of finned-tube coils. While the model developed by Jiang et al. has not yet been experimentally validated, it seems to be able to account for several detailed aspects of the coil, such as complex circuitry, various flow configurations (counter-cross, parallel-cross) and nonuniform air distributions. In general there is a focus on the refrigerant side design rather than the air side. The model discretizes the tube into segments and solves the momentum and energy equations for each segment alternatively and repeatedly until convergence is obtained. In addition to the simulation tool, Jiang et al. mention an example optimization case using a genetic algorithm to optimize the number of rows, number of tubes per row, option of parallelcross flow or counter-cross flow, and tube diameter. The objective function was minimum total heat transfer area at a given heat load subject to maximum allowable pressure drops of the refrigerant flow and air flow. This optimization requires detailed constraint information (pressure drops). Additionally, the objective function is subjective. The optimization results in a smaller surface area heat exchanger for the price at whatever allowable pressure drops are specified. But it is not discussed how or why one would choose
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a particular pressure drop and nothing on the air side of the heat exchanger is optimized. Tayal et al. [28] also looked at searching for optimum individual heat exchanger components using a combinatorial (mixed discrete and continuous variable) optimization scheme such as the genetic algorithm, using minimum heat transfer area as well as cost as objective functions in their optimization procedures. Tayal et al. conclude that genetic algorithms are equally effective as compared to simulated annealing in solving black-box optimization model problems. Their study focuses on shell-and-tube heat exchangers for process industries. Caputo et al. [19] and Özçelik [29] also use a genetic algorithm for optimization of a shell-and-tube heat exchanger. Richardson et al. [30] developed a program that simulates a vapor compression system. This program is then used to optimize the system level variables using gradient based and genetic optimization routines. Parameters included in this simulation program are: refrigerant charge, COP, weight, capacity, and cost. Optimization objective functions used were COP, capacity and system weight. This study did not optimize the design of the individual components; rather it used a collection of specific components and found the best combination of them for specified inputs. Additionally, the heat exchanger models used in this simulation program are simplistic, assuming infinite air flow rate and fixed properties at the inlet regime for refrigerant pressure drop calculations. The current study builds on the preceding development work of Wright [31] and Aspelund [32]. A model was developed in Engineering Equation Solver (EES) [33] of an air-conditioning system using R-410a as the working fluid. This model includes a detailed simulation of the components of the air-conditioning system for various designs, including the compressor, finned-tube condenser, evaporator, and expansion valve. The condenser component is the focus of the model incorporating the best available simulations for the air side and refrigerant side pressure drops and heat transfer coefficients based on R-410a as the working fluid. A design optimization search technique (using the Simplex Method by Nelder and Mead [34]) was implemented to optimize 10 controllable, operational, and geometric design parameters of the condenser with a seasonal COP figure-of-merit. The details of the system model can be found in Stewart [35]. The methodology used to define the isolated condenser model is detailed in the next section.
3. Condenser model In order to perform thermoeconomic isolation, the component of interest must be entirely “isolated” from the rest of the system. By doing this, the component ceases to give or receive feedback from the rest of the system. However, in order to produce a properly posed and complete model, additional operating constraints are required. A total of five external junction type constraints are needed. Referencing Fig. 1, the options from which to choose are:
_ Q_ cond ; DTSH ; DTSC ; P1air ; P2air ; P2r ; P3r ; T1air ; T2air ; T2r ; T3r ; mr; and Tsat including the pressures and temperatures at the entrance and exit of the air side (1air, 2air) and refrigerant sides (2r 3r), respectively, the mass flow rate of the refrigerant, the heat transfer rate from the condenser, the degree of superheat entering the condenser (note that this is not the typical definition of superheat in an air-conditioning system, which usually describes the superheat exiting the evaporator), the degrees of sub-cool exiting the condenser (which is a parameter set by the refrigerant charge in the system) and also the saturation temperature in the condenser.
Fig. 1. Detailed condenser flow diagram.
For the condenser of a residential air-conditioning system, two of the needed five constraints are rather straightforward; i.e., the inlet air pressure and temperature are assumed to be ambient. Therefore, the P1air is taken as 1 bar, and T1air was taken as 27.8 C (82 F), which was found by Stewart [35] to give a nearly identical COP as the seasonal COP used by the US Department of Energy in rating residential air conditioners. These values were used in the overall system model as well. The pressure drop and temperature increase on the air side are then calculated by the model and cannot be constrained. Three additional constraints are needed to complete the isolated model. The appropriate selection of these three constraints was found to be extremely important in determining whether the results from the thermoeconomic isolation optimization were close to the results from the system optimization. Therefore, the decision making process is detailed here. Practically, one parameter would give information about the refrigerant entrance conditions (P2r, T2r, Tsat, DTSH), the second parameter would give information about the refrigerant exit conditions (P3r, T3r, Tsat, DTSC), and the third would give information about the heat transfer scaling between the air side and the refrigerant side. While almost any combination of three additional parameters will result in a properly posed model, certain combinations will not give as good results due to their variations with changing condenser design when the condenser is integrated into a system. To find the best combination of parameters the following requirements should be met: (1) the parameters should not require detailed non-condenser system information to arrive at a value, and (2) they should not vary greatly with different heat exchanger designs when operating in the system. Because of these two requirements, explicit temperature and pressure information are not good options; therefore, P2r, P3r, T2r, T3r and even the saturation temperature in the condenser are taken out of consideration as they will all vary dramatically with changing heat exchanger design. The remaining options to specify the refrigerant inlet and outlet conditions are DTSH and DTSC, respectively. With these temperature differences chosen, the only decision left to be made is the size scaling parameter, which could be either the refrigerant mass flow rate or the heat transfer rate between the refrigerant and the air. Both of these situations were investigated in more detail. Prescribing the mass flow rate of the refrigerant in the isolated condenser model does not fix the total heat transfer rate in the heat exchanger. This is due to the fact that as the saturation pressure varies between heat exchanger designs, the heat of condensation varies. Taken with fixed inlet superheat and exit sub-cool, the heat transfer rate increases as the saturation pressure decreases. In the system model, the refrigerant mass flow rate varies to compensate for this, while the total heat transfer rate stays relatively constant. Therefore, fixing the heat transfer rate is a more appropriate size scale constraint.
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As shown by equations (1) (neglecting the work of the fans) and (2), the heat transfer rate is dependent only on the COP and the desired evaporator cooling capacity, e.g., size of the system for which the condenser is being designed:
COP ¼
Q_ evap Q_ evap Desired Output ¼ ¼ _ _ Required Input W comp Q cond Q_ evap
Manifold
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Manifold
1/FP
Xl
(1)
Xt
Rearranging and solving for Q_ cond :
1 Q_ cond ¼ Q_ evap 1 þ COP
Air Inlet (Vac)
(2)
With a refrigerating capacity fixed at 8.8 kW and a typical COP of approximately 4, this gives $Q cond ¼ 11 kW. Note that $Q cond is only weakly dependent on COP; e.g., a 25% change in COP produces only a 4e6% change in $Q cond . In practical application, a small change in operating conditions can result in a liquidevapor mixture exiting the condenser, which, due to limited expansion valve flow area, would back up behind the valve until a high enough pressure is reached to fully condense the vapor. The expansion valve would than be wide open and have no control over the evaporator superheat. While the optimum subcool is actually 0 C, to avoid this situation under the normal range of operating conditions, the sub-cool was specified as 5 C when operating at 35 C ambient temperature. Using this design condition the system model then calculates the sub-cool for the ambient condition of 27.8 C (82 F), at which the system was optimized. The resulting calculated sub-cool at 27.8 C was normally slightly higher than 5 C depending on the condenser design, so for the isolated model the value was fixed at 5 C for simplicity. The value for the condenser inlet superheat was set at 25 C (45 F) in the isolated model. This value was determined by looking at some typical model results from system designs. 4. Optimization parameters Table 1 shows the parameters necessary to define the design of the finned-tube condenser considered in this optimization study. Many are also depicted in Fig. 2. In this heat exchanger design optimization, a single objective function is used in the optimization procedure. Initially, this objective function is seasonal COP, while minimum condenser entropy generation is considered as an alternative to COP in the isolated condenser case (thermoeconomic isolation). Initially, the design parameters were not constrained in
Refrigerant outlet
Fig. 2. Condenser design parameters (shown: 4 rows, 2 circuits, 6 tubes per row).
the optimization procedure except for those that were limited by the range of experimental data used to develop the empirical correlations employed by this study (Xl, Xt, FP, Vac, Lh, Lp) as discussed in Stewart [35] and shown in Table 1. As expected, some variables optimized to their limits. When the limits in these cases were zero or infinity, those design parameters were then constrained to practical limits, as discussed below. Heat exchanger price is typically based on a multiple of material coast assuming mass production; therefore, the cost of the heat exchanger material was used in this study as an indicator for heat exchanger cost. The volume of each type of metal was calculated and multiplied by its density and cost per weight to determine the material cost for each heat exchanger design. Since increasing material cost of a heat exchanger will increase the COP, the cost must be constrained to a maximum value for each optimization. The costs of the materials are based on published values of aluminum (US$1.54/kg) and copper (US$1.76/kg) from the London Metals Exchange [36]. It should be noted that these values have experienced large fluctuations over the last several years and, therefore, may vary from these values in the future. When frontal area is not constrained, the fixed cost design leads to the largest frontal area (Af / N) corresponding to one row. This situation yields the minimum air pressure drop. Therefore, the larger the frontal area the better, if space and cost allow. Because of this, the frontal area was specified, or constrained, for each
Table 1 Design parameters.
28 Design parameters
Upper limit
0 0 0 7.94 0.157 17.8 12.7 1 1 1
Geometric: Frontal area (Af) [m2] Aspect ratio (AR) Fin thickness (tfin) [mm] Tube diameter (D) [mm] Fin pitch (FP) [mm1] Transverse tube spacing (Xt) [mm] Longitudinal tube spacing (Xl) [mm] Number of rows (rows) Tubes per row (tpr) Number of circuits (circ)
N N N 7.94 0.71 30.5 28 N N N
0.79 1.7
Operational: Air velocity (Vac) [m/s] Sub-cool (TSC) [ C] Louvers: Louver height (Lh) [mm] Louver pitch (Lp) [mm]
5.3 N 1.4 3.75
26
Xt [mm]
Lower limit
0.91 0
Refrigerant inlet
24
22 Isolated System
20 10
20
30 40 Condenser Cost [US$]
Fig. 3. Transverse tube spacing vs. condenser cost (0.75 m2 frontal area).
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S.W. Stewart, S.V. Shelton / Applied Thermal Engineering 30 (2010) 2096e2102
30
4
25
3
Air Velocity [m/s]
Xl [mm]
2100
20
15
System 0
10 20
30 40 Condenser Cost [US$]
1 Isolated
Isolated System 10
2
50
10
20
30 40 Condenser Cost [US$]
50
Fig. 4. Longitudinal tube spacing vs. condenser cost (0.75 m2 frontal area).
Fig. 6. Air velocity vs. condenser cost (0.75 m2 frontal area).
optimization while the effects of varying frontal area were investigated by comparing optimum designs using different frontal area constraints, (0.5, 0.75, 1.0, and 1.25 m2) at the same condenser cost ($25). If the overall heat exchanger aspect ratio (width divided by height) were not restricted, the design converges to a single very long finned-tube (AR / N), since tube bends have pressure drop but no heat transfer. Therefore, the aspect ratio was limited by a maximum of three. The outside A/C condensing/compressor unit is typically a cube in shape with the condenser bent around three sides. For all of the optimizations the tube diameter was fixed at 7.94 mm (5/1600 ). This is because Aspelund [32] found that the smaller the tube (D / 0), the better the COP, with little improvement beyond 7.94 mm. Benefits of using smaller diameter tubes include smaller form drag caused by the tube, higher refrigerant side heat transfer coefficients due to smaller hydraulic diameter, and less refrigerant inventory in the system [37]. Also, the fin thickness was ultimately fixed at 0.15 mm (0.00600 ). When left in as a search parameter, the solution always converged to a design with thinner and thinner fins (tfin / 0) while making the fin pitch larger and decreasing the air velocity to adjust for the increased pressure drop. This makes sense theoretically, however, in reality extremely thin fins are not structurally durable, and dirt and dust will clog the fins when the fin spacing is too small. Because of this, the fin thickness was constrained to a practical minimum value of 0.15 mm.
The number of rows multiplied by the longitudinal tube spacing determines the depth of the condenser, while the tubes per row multiplied by the transverse tube spacing determines the height. The number of circuits determines the number of parallel tube passages the refrigerant mass flow rate is divided amongst by the manifold. These circuitry parameters were not constrained. Note that the number of tubes per circuit, tubes per row, and number of rows all must be discrete values in a real design. In the optimization process of this study, this restriction was not considered. Instead hypothetical continuous optimum designs are found for the circuitry parameters so that clear design trends are more evident.
0.8
Fin Pitch [1/mm]
0.7 0.6 0.5 0.4
Isolated System
0.3 10
20
30 40 Condenser Cost [US$]
Fig. 5. Fin pitch vs. condenser cost (0.75 m2 frontal area).
50
5. Results and discussion The resulting system and isolated condenser models were run under the conditions described in the previous section using the optimization search scheme as discussed in Stewart [35]. The resulting optimum designs for the system model and the isolated condenser model are compared in the figures below. For the COP calculation, the optimum design determined by the isolated condenser optimization was then run in the system model to determine the system COP for that design. Figs. 3e5 show the optimum transverse tube spacing, the longitudinal tube spacing, and the fin pitch, respectively, vs. varying condenser material cost for a 0.75-m2 frontal area. It can be seen from these figures that the resulting optimum designs from both the system and the isolated model optimizations are quite similar to each other in both trend and value. Additionally, the optimum number of rows for both cases are identical. The isolated model optimization search scheme was developed to choose an optimum design based on minimizing the entropy generation in the condenser. The components of this entropy generation are heat transfer through a finite temperature difference and both air and refrigerant pressure drop terms. These temperature and pressure effects compete with each other, i.e., as the pressure drop decreases the heat transfer coefficient decreases. Due to the trading off of these effects, the optimum design will converge to the case with a minimum of irreversibility. Therefore, as the isolated optimization scheme reduced the air side velocity (Fig. 6, for 0.75 m2 frontal area), it reduced both the air and refrigerant side pressure drops. This is because in order to maintain the constant condenser heat transfer rate, there was a slight increase in the average saturation temperature and pressure in the condenser and a slight increase in the refrigerant side mass flow rate, but this flow rate was split into more (18.2%e29.3%) parallel circuits (number of circuits) resulting in a lower pressure drop on the refrigerant side
S.W. Stewart, S.V. Shelton / Applied Thermal Engineering 30 (2010) 2096e2102
some of the design parameters showed significant differences in their optimum values (as much as 29% different). With a 65% reduction in computation time using the isolated model, this provides a very practical and effective method for designing finnedtube condenser heat exchangers. Caution, however, should be taken when using thermoeconomic isolation in choosing the appropriate external constraints. The procedure will give results with multiple combinations of external constraints but these optimum results will lack in accuracy with improper choices.
4.4 4.3
COP
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4.2 4.1 4
References
Isolated System
3.9 10
20 30 40 Condenser Cost [US$]
50
Fig. 7. COP vs. condenser cost (0.75 m2 frontal area).
compared to the corresponding system optimized design (e.g., at $25, 0.75 m2: 69.84 kPa vs. 119.1 kPa). The net result of these effects reduced the entropy generation in the condenser, which was the figure-of-merit for the isolated model. However, when this isolated optimum design is placed in the context of the overall system, the reduction in pressure drop in the condenser creates the need for a larger irreversibility in the expansion valve. Also, the increase in saturation pressure in the condenser (with a fixed DTSH) and the increased refrigerant mass flow rate requires a larger compressor power, also creating more irreversibility. The net result of these increasing entropy effects in other components offset the lower entropy generation in the condenser. The overall effects of these model differences can be seen in Fig. 7, which shows COP vs. varying condenser material cost for a fixed frontal area of 0.75 m2. It can be seen that the COPs from the system model optimizations are slightly higher than the isolated condenser optimized designs. The variation ranges from 0.6% at $45 to 1.7% at $15. So, even though some of the design parameters differed by as much as 29% between the two methods, the isolated condenser optimization did produce designs very close in COP to those produced by using an entire system model to optimize the condenser design. The 65% reduction in computation time for the isolated model makes this a very attractive and practical option for the design optimization of finned-tube condenser heat exchangers, with a caution that a prudent choice of constraints must be considered. 6. Conclusions Using a detailed system model as a comparison, this study shows that isolating the condenser component and optimizing it independently by minimizing the entropy generation in the condenser component alone (also known as thermoeconomic isolation) can be a practical way to design the condenser for optimum air-conditioning system efficiency. It was found that the most important aspect required for a successful isolated model optimization is a proper set of external or junction constraints. In the current study, constraining the condenser heat transfer rate, condenser exit sub-cool, and the condenser entering superheat were found to give the best results. These parameters vary little in the system model with changing condenser designs and they do not require detailed system information in order to specify them within reasonable accuracy. The resulting optimum designs from the isolated model produced a COP within 0.6%e1.7% of the designs found by optimizing the COP using an entire system model, despite the fact that
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