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Thermal efficiency maximization for H- and X-shaped heat exchangers based on constructal theory
Applied Thermal Engineering 91 (2015) 456e462
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Research paper
Thermal efficiency maximization for H- and X-shaped heat exchangers based on constructal theory Lingen Chen a, b, c, *, Huijun Feng a, b, c, Zhihui Xie a, b, c, Fengrui Sun a, b, c a
Institute of Thermal Science and Power Engineering, Naval University of Engineering, Wuhan, 430033, PR China Military Key Laboratory for Naval Ship Power Engineering, Naval University of Engineering, Wuhan, 430033, PR China c College of Power Engineering, Naval University of Engineering, Wuhan, 430033, PR China b
h i g h l i g h t s Constructal optimizations of H- and X-shaped heat exchangers are carried out. Maximum thermal efficiency is taken as optimization objective. Thermal efficiency is defined as ratio of heat transfer rate to total pumping power. Optimal constructs of H- and X-shaped heat exchangers are obtained. Thermal efficiency of X-shaped heat exchanger is larger than that of H-shaped.
a r t i c l e i n f o
a b s t r a c t
Article history: Received 29 May 2015 Accepted 13 August 2015 Available online 21 August 2015
Constructal optimizations of H- and X-shaped heat exchangers are carried out by taking the maximum thermal efficiency (the ratio of the dimensionless heat transfer rate to the dimensionless total pumping power) as optimization objective. The constraints of total tube volumes and spaces occupied by heat exchangers are considered in the optimizations. For the H-shaped heat exchanger, the thermal efficiency decreases when the dimensionless mass flow rate increases. For the higher order of the X-shaped heat exchanger, when the order number is 3, the thermal efficiency of the heat exchanger with Murry law is increased by 68.54% than that with equal flow velocity in the tubes, and by 435.46% than that with equal cross section area of the tubes. © 2015 Elsevier Ltd. All rights reserved.
Keywords: Constructal theory Maximum thermal efficiency H-shaped heat exchanger X-shaped heat exchanger Generalized thermodynamic optimization
1. Introduction Heat exchanger is an important device used in various industries, such as metallurgy industry, chemical industry, spaceflight, etc. To improve the energy utilization of the industry, one of the important tasks is to reduce the energy losses of the heat exchangers [1]. To satisfy different energy utilization requirements and solve the entropy generation paradox [2,3], many scholars studied the performances of various heat exchangers based on the optimization objectives of entropy generation number [4e6], modified entropy generation number [7,8], exergy [9] and thermaleconomic [10], etc.
* Corresponding author. Institute of Thermal Science and Power Engineering, Naval University of Engineering, Wuhan, 430033, PR China. Tel.: þ86 27 83615046; fax: þ86 27 83638709. E-mail addresses: [email protected], [email protected] (L. Chen). http://dx.doi.org/10.1016/j.applthermaleng.2015.08.029 1359-4311/© 2015 Elsevier Ltd. All rights reserved.
Constructal theory [11e20] is a powerful theory in illustrating and solving various design problems, and it also has exhibited its power in the optimal designs of heat exchangers [21e31]. Ordonez and Bejan [21] optimized the parallel-plate counterflow heat exchanger by taking minimum entropy generation as optimization objective, and obtained the optimal channel spacing ratio, total contact area and the capacity rate ratio of the heat exchanger. Vargas and Bejan [22] studied the counterflow heat exchanger of an aircraft environmental control system based on entropy generation minimization method, and the result showed that the optimal construct of the core with finned surfaces was nearly the same as that with smooth plates. Bejan [23] optimized the channel spacings of a dendritic heat exchanger based on maximum volumetric heat transfer rate, and obtained a better performance of the vascularized heat exchanger over the parallel one in Hagen-poiseuille regime. da Silva et al. [24] built three kinds of multi-scale models for treeshaped heat exchangers, and investigated the characteristics of
L. Chen et al. / Applied Thermal Engineering 91 (2015) 456e462
Nomenclature A cp D h i k L M m_ Nu n ni q T U V W
area, m2 specific heat, J/kg/K diameter, m heat transfer coefficient, W/m2/K current order of the heat exchanger thermal conductivity, W/m/K length, m dimensionless mass flow rate flow rate, kg/s Nusselt number order number of the heat exchanger number of the tubes heat transfer rate, W temperature, K overall heat transfer coefficient, W/m2/K volume, m3 pumping power, W
the global thermal resistances versus the pumping powers. They also discussed the applications of tree-shaped heat exchangers into actual devices, and further carried out experiments for the diskshaped counterflow heat exchanger [25]. Zimparov et al. [26] compared the performances of two tree-shaped heat exchangers. The results showed that the counterflow heat exchanger was superior to the parallel one for a higher value of pumping power, and the conclusion was reversed for a lower value of pumping power. Based on the multi-scale models for tree-shaped heat exchangers in Ref. [24], Manjunath and Kaushik [27,28] further investigated their entropy generation and thermo-economic performances, and concluded that their performances were improved compared with those of the convectional heat exchangers. Moreover, some scholars also implemented constructal optimizations of the shelltube [29] and underground [30,31] heat exchangers. In the studies of the dendritic channel networks, Chen and his groups [32e34] investigated the network's thermal efficiency performances, and found that these networks had the lower pressure drop and pumping power requirement, more uniform temperature and better thermal efficiency compared with the serpentine channels. The results also showed that the thermal efficiency performance was a perfect objective to simultaneously evaluate the thermal and fluid flow performances (TFFPs) of the network. The TFFPs of the H-shaped heat exchangers were analyzed in Refs. [24,26], respectively, and the experiments for the disk-shaped heat exchanger were carried out in Ref. [25]. A vasculature with X-shaped tube structure was optimized by Feng et al. [35], and the TFFPs were evidently improved. Considering these situations, an X-shaped heat exchanger will be investigated in this paper, and the thermal efficiency will be taken as the optimization objective by using the constructal theory. Compared with the heat exchanger's structure adopted in Refs. [24,26], the structure adopted in this paper is modified, and the TFFPs are simultaneously considered by choosing the maximum thermal efficiency as optimization objective. The performance of the X-shaped heat exchanger will be compared with that of the corresponding Hshaped heat exchanger.
Greek symbols tube angle, rad temperature difference, K thermal efficiency kinematic viscosity, m2/s density, kg/m3
a DT hqW n r
Subscripts c cold fluid e crossed tube h hot fluid in inlet opt optimal 0,1,2,3,… index of the tube Superscript ~ dimensionless
composed of many pairs of tubes with different lengths and diameters. In the upper part of the heat exchanger, the single-phase _ temperature Th,in, density r, hot fluid (mass flow rate m_ n ¼ m, thermal conductivity k and specific heat cp) enters the root of the heat exchanger, and releases heat to the cold fluid when it flows through each tube (diameter Di, length Li and tube number ni, i ¼ 0,1,2,3,…,n). Each branching stream (flow rate m_ 0 ) flows out of the heat exchanger from its canopy located at the center of the elemental area. The hot fluid is then collected by pressure equalizer, and finally flows into the hot fluid circulator unit. In the lower part of the heat exchanger, the structures and the mass flow rates are identical to the corresponding ones in the upper part. The same
2. Constructal optimization for H-shaped heat exchanger An H-shaped counterflow heat exchanger in the rectangular area is shown in Fig. 1 [24]. The H-shaped heat exchanger is
457
Fig. 1. H-shaped heat exchanger in a rectangular area [24].
458
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kind of single-phase cold fluid (inlet mass flow rate m_ 0 and inlet temperature Tc,in) flows through the cold side of the tube network from its canopy to the root. The tube walls of the cold and hot sides are adiabatic from the surroundings. The flows in cold and hot side tubes are both in Hagenepoiseuille regime. The relationships of the sizes and mass flow rates of the Hshaped heat exchanger are
_ ni Li ¼ 2i=2 L0 ; Di ¼ 2i=3 D0 ; m_ i ¼ 2in m; ¼ 2ni ; ði ¼ 0; 1; 2; …; nÞ
(1)
where the tube-diameter ratio of the heat exchanger obeys the Murray law [36]. Both the total volume of the tubes and space occupied by the Hshaped heat exchanger are, respectively, given by n X
V ¼ 2$
ni Li
i¼0
pD2i 4
(2)
A ¼ 2n ð2L0 $2L1 Þ
(3)
From Eqs. (1)e(3), D0 and L0 can be, respectively, rewritten as
31=2 1=6 V 2 1 25 2n 5 D0 ¼ 4 9 n pA1=2 212þ 3 24þ2
n X pkNu 1 DT$ ni L i $ 1=2 T 2 pkNuA h;in Tc;in i¼0 1þn 21=4 2 2 1 M1 1þn ¼ 21=4 2 2 1 þ 4M1 2 21=2
~¼ q
n _ X L V2 ~ ¼ 2$128mn m_ i i4 $ W pr i¼0 Di 4p kNu cp 2 ðn=rÞA5=2 1 n 3 1 n p2 M12 242 26þ6 1 ¼ 3 21=6 1
(10)
(11)
The definition of the heat exchanger's thermal efficiency was given as the ratio of the heat transfer quantity to the total pumping work in Refs. [32e34]. According to this definition, the thermal efficiency of the heat exchanger in Fig. 1 can be given as the ratio of the dimensionless heat transfer rate to the dimensionless total pumping power. From Eqs. (10) and (11), the thermal efficiency of the H-shaped heat exchanger is
2
L0 ¼ 2
(4) hqW
nþ2þ12
2
A1=2
n X
3 1þn 2 2 1 2n=2 21=6 1 1þn 1þn 3 h i 21=4 2 2 1 þ 4M1 2 21=2 p2 M1 2 6 1 (12)
(5)
For the same heat capacity rate of the two fluids, the temperature difference DT between the two fluids along the length direction is a constant [24]. In each tube, the heat exchanger's heat transfer rate is equal to that released by the hot fluid, which is the energy balance equation of the coupled heat transfer. For the specified inlet temperatures Tc,in and Th,in of the cold and hot fluids, summing the energy balance equations in each tube, the heat exchanger's heat transfer rate becomes:
DT
~ q ¼ ¼ ~ W
_ p Th;in Tc;in DT ni pUi Di Li ¼ mc
(6)
i¼0
From Eq. (12), the thermal efficiency hqW is a function of the DMFR M1 and the order number n of the heat exchanger, and one can carry out performance analysis and optimization of the Hshaped heat exchanger by varying M1 and n. Fig. 2 shows the effect of n on the characteristic of the thermal efficiency hqW versus the DMFR M1. From Fig. 2, one can see that for the fixed n, hqW decreases when M increases, and the heat exchanger's performance becomes weak. For a lower and fixed M1, the thermal efficiency of the heat exchanger decreases when n increases. For a higher and fixed M1, with the increase in n, the efficiency decreases first and then increases, and it reaches its minimum when n ¼ 4.
where Ui(i ¼ 0,1,…,n) is the overall heat transfer coefficient (HTC) without considering the thermal resistances of the tube walls:
1 1 1 ¼ þ Ui hi hi
(7)
where hi(i ¼ 0,1,…,n) is the HTC of the fluid in one tube:
hi ¼
kNu Di
(8)
where Nu is the Nusselt number, which is assumed to be identical in each tube [24]. Substituting Eqs. (1), (4) and (5) into Eqs. (6)e(8) yields
DT ¼
4M1 2 21=2 Th;in Tc;in 3 n 24þ2 21=4 þ 8 25=2 M1
(9) _ mc
where the dimensionless mass flow rate (DMFR) is M1 ¼ pkNuAp 1=2 . From Eqs. (1), (4), (5) and (9), the dimensionless total heat transfer rate and total pumping power of the H-shaped heat exchanger can be, respectively, given by
Fig. 2. Effect of n on the characteristic of hqW versus M1.
L. Chen et al. / Applied Thermal Engineering 91 (2015) 456e462
In the performance analysis of the heat exchanger, the local pressure losses are ignored. From Eqs. (1) and (3), the svelteness number of the H-shaped heat exchanger is [30].
¼
external flow length scale LTotal Li ¼ 1=3 ¼ i¼0 internal flow length scale V V 1=3 A1=2 1 2ðnþ1Þ=2 $ V 1=3 1 21=2 $2n=2þ5=4
(13) ðnþ1Þ=2
(15)
. 1=4 2 1=8 4~ L 2$ ~L A De ¼ h . i1=2 ~ 2~ p 4þD L V
1=2
12 A In Eq. (13), the magnitude of ð12 1=2 Þ$2n=2þ5=4 is 1. If V 1=3 is large enough, Sv will be large, and the neglect of the local pressure losses is validated [13].
3. Constructal optimization for X-shaped heat exchanger 3.1. Element volume of X-shaped heat exchanger To further improve the performance of the H-shaped heat exchanger, one can try to make a structure improvement based on the second order assembly of H-shaped heat exchanger as shown in Fig. 3, that is, the elemental volume of X-shaped heat exchanger. In the upper part, the single-phase hot fluid (mass flow rate m_ 0 ) enters the root of the heat exchanger, flows through one round tube (diameter D0 and length L0) and four crossed tubes (diameter De and length Le), and flows out of the heat exchanger (mass flow rate me) from its canopy located at the center of the quarter of the elemental volume. The cold side structure of the heat exchanger is the same as that at hot side. The total tube volume and the space occupied by the elemental volume of X-shaped heat exchanger are
p V ¼ 2$ 4D2e Le þ D20 L0 4
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4L2e L20
For the specified V and A, substituting Eq. (15) into Eq. (14) yields
Pn
Sv ¼
A ¼ 4L0
459
Le ¼
A1=2 1=2 2 1=4 4 ~L 2~L
(16)
(17)
~ ¼ D0 and ~L ¼ L0 . where D De Le For the specified inlet temperatures Tc,in and Th,in of the cold and hot fluids, the energy balance equation and the temperature difference of the cold and hot fluids can be, respectively, given by
DT ðL0 þ 4Le Þ ¼ M2 Th;in Tc;in DT 1=2 2A DT ¼
2A1=2 M2 Th;in Tc;in L0 þ 4Le þ 2A1=2 M2
(18)
(19)
_ mc
where the DMFR is M2 ¼ pkNuAp 1=2 . From Eqs. 16e19, the dimensionless total heat transfer rate and total pumping power of the elemental volume of the X-shaped heat exchanger can be, respectively, given by
pkNu 1 DT$ðL0 þ 4Le Þ$ 2 pkNuA1=2 Th;in Tc;in 4 þ ~L M2 ¼ 1=2 2 1=4 4 ~L 4þ~ L þ 4M2 ~L
~¼ q (14)
! _ L Le V2 ~ ¼ 2$128mn m_ 04 þ m_ e 4 $ W 2 pr De 4p kNu cp ðn=rÞA5=2 D0 4 2 ~ 2 ~L ~ þ 4~L 4 þ D p2 M22 D ¼ 3=2 2 3=4 ~ 4~ 4~ L 2D L
(20)
(21)
~ can be From Eq. (21), the dimensionless total pumping power W ~ minimized by varying D. The optimal diameter ratio of the tubes is ~ opt ¼ 22=3 , and the corresponding dimensionless total pumping D power can be rewritten as
2 21=3 p2 M22 22=3 þ ~ L 2 þ 21=3 ~ L ~ ¼ W 3=4 ~L3=2 4 ~L2
Fig. 3. Elemental volume of X-shaped heat exchanger.
(22)
One can see that the optimal diameter ratio of the tubes above is equal to that of the vascular network with X-shaped structure obtained in Ref. [35]. This is because the fluid flow performances of the elemental volumes with X-shaped structure are optimized in this paper and Ref. [35], and the same optimal diameter ratio of the tubes can be obtained due to the same X-shaped fluid flow structure. From Eqs. (20) and (22), the thermal efficiency of the elemental volume is
460
hqW
L. Chen et al. / Applied Thermal Engineering 91 (2015) 456e462
~ q ¼ ¼ ~ W
3=2 2 3=4 ~ 4þ~ L 4~ L L 2 1=2 2 1=4 4~ L 4 þ ~L þ 4M2 ~ 21=3 p2 M2 22=3 þ ~ L 2 þ 21=3 ~L L
From Eq. (23), for the fixed DMFR M2, hqW is only the function of ~ L, and hqW can be maximized by varying ~ L. The total tube volume V and the area A occupied by the elemental volume in Eqs. (14) and (15) are the constraints in the model. However, the tube diameter De and length Le can be obtained by solving Eqs. (14) and (15), therefore, the constraints of the thermal efficiency maximization are disappeared. The optimization of hqW can be implemented by varying ~ L in its reasonable range. Because there is only one optimization variable (~ L) in hqW, the available range of ~ L is divided into many equidistant segments in the optimization of hqW. When the approximate location of the extreme point is found in a small range, the range is further divided into many refined equidistant segments. By using this method, the maximum value of hqW and the corresponding optimal value of ~L (~ Lopt ) can be obtained. Fig. 4 shows the effects of the DMFR M2 on the optimal tube 0 length ratio ~ Lopt ¼ ðL0 =Le Þopt and optimal shape ~ Lopt ¼ 2L0;opt = qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ½2$ ð2Le;opt Þ2 L20;opt ¼ ~ Lopt Þ1=2 of the elemental volLopt $ð4 ~ ume. From Fig. 4, one can see that when M2 << 1, the optimal tube length ratio is ~ Lopt / 0:918, and the corresponding optimal shape is
0 ~ Lopt /0:517. When M2 >> 1, the optimal tube length ratio is 0 ~ Lopt / 0:776, and the corresponding optimal shape is ~Lopt /0:421. One can see that the optimal tube length ratios above are different from those of the vascular network with X-shaped structure obtained in Ref. [35]. This is because the thermal performances of the elemental volumes with X-shaped structure are optimized both in this paper and Ref. [35], and the different optimal tube length ratios can be obtained due to the different heat transfer boundary conditions. For the same shape, the total tube volume V and the space A occupied by the elemental volume, one can compare the thermal efficiency performances of the elemental volume of X-shaped heat exchanger and second order assembly of H-shaped heat exchanger. This requires the tube length ratio of X-shaped heat exchanger satisfies L0/Le ¼ 2/31/2. Numerical calculations show that the thermal efficiency of the elemental volume of X-shaped heat exchanger is increased by
0 Fig. 4. Characteristics of ~Lopt and ~Lopt versus M2.
(23)
44.48% than that of the second order assembly of H-shaped heat exchanger when M2 << 1, and this increment is 46.11% when M2 >> 1. Therefore, due to the structure improvement of the heat exchanger, the performance of the X-shaped heat exchanger is superior to that of the H-shaped one.
3.2. Higher order assembly of X-shaped heat exchanger As shown in Fig. 5, a higher order assembly (order number n) is composed of a number (2n) of elemental volumes. The elemental volumes are collected by several multi-scale tubes (Di ; Li ; m_ i ; ni ði ¼ 1; 2; :::; nÞ). The tube numbers and flow rates of the X-shaped heat exchanger are given as follows
. _ in ; i ¼ 1;2;…;n ne ¼ 2nþ2 ; ni ¼ 2ni ; m_ e ¼ m_ 2nþ2 ; m_ i ¼ m2
(24)
The total tube volume and space occupied by the X-shaped heat exchanger are n X p V ¼ 2$ D2i Li 4D2e Le þ 4 i¼0
A ¼ 2n $4L0
!
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4L2e L20
(25)
(26)
From Eqs. (25) and (26), the tube lengths and diameters of the Xshaped heat exchanger can be, respectively, given by
Fig. 5. Higher order assembly of X-shaped heat exchanger.
L. Chen et al. / Applied Thermal Engineering 91 (2015) 456e462
. pffiffiffi . De ¼ D0 22=3 ; Di ¼ 2i=3 D0 ; Le ¼ 3L0 2; Li ¼ 2i=2 L0
L0 ¼ 2
A1=2 nþ2þ12
(27)
(28)
2
2 D0 ¼ 4
461
VA1=2 p1
31=2
25 2n 1 5 31 n 9 n 212þ2 $31=2 þ 212þ 3 24þ2 21=6 1
(29)
Similar to Eq. (19), the temperature difference of the heat exchanger's two fluids is
DT ¼
29=4 M2 2þ21=2 Th;in Tc;in 21=2 21þ2 þ22þ2 $31=2 22þ2 $31=2 8$21=4 M2 þ4$23=4 M2 (30) n
3
n
n
From Eqs. 27e30, the dimensionless total heat transfer rate, dimensionless total pumping power, and thermal efficiency of the X-shaped heat exchanger are, respectively, given by
! n X pkNu 1 ~¼ DT$ 2nþ2 Le þ ni Li $ q 1=2 T 2 pkNuA h;in Tc;in i¼0
(31)
! n _ Le X L V2 ~ ¼ 2$128mn m_ e 4 þ mi i4 $ W 2 pr De i¼0 Di 4p kNu cp ðn=rÞA5=2
(32)
hqW ¼
Fig. 6. Effect of n on the characteristic of hqW versus M2.
equal cross section area of the tubes. These increments will continue to increase when the order number n increases. Therefore, compared the X-shaped multi-scale heat exchanger with the Xshaped single-scale heat exchanger, the performance of the multiscale heat exchanger is greatly improved. For the shell and tube or plate-fin heat exchangers, the flow channel is single-scale one, which may become one of the weak point in comparison with the multi-scale heat exchanger in this paper.
n 3 pffiffiffi pffiffiffi ~ 3þn pffiffiffi 3þn 3þn 4þn 3 n n n pffiffiffi q 2 21þ2 22þ2 3 þ 2 2 3 M2 2 9$21=3 þ 18 2 9$22=3 þ 9$2 6 þ 2 2 3$2 3 ¼ 24þ2 21=6 1 ~ W pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi 5þn 7þn 10þn n pffiffiffi n pffiffiffi 7 n pffiffiffi þ9$2 6 þ 3$2 6 9$2 6 3 3 þ 9$21=6 3 9$21=3 3 3$22=3 3 þ 3$25=6 3 3$22þ6 3 3$21þ3 3 þ 3$26þ3 3 pffiffiffiio1 pffiffiffihpffiffiffi 11þn pffiffiffi 3þn pffiffiffi n n pffiffiffi 2 21þ2 22þ2 3 þ 2 2 3 þ 4$21=4 p2 M2 2 þ 2 þ 3$2 6 3 þ 3 6
From Eq. (33), one can carry out constructal optimization of the X-shaped heat exchanger by varying the DMFR M2 and order number n. Fig. 6 shows the effect of n on the characteristic of the thermal efficiency hqW versus the DMFR M2. From Fig. 6, one can see that the thermal efficiency versus the DMFR characteristic of the X-shaped heat exchanger is similar to that of the H-shaped heat exchanger in Fig. 2. For a higher mass flow rate, the thermal efficiency of the Xshaped heat exchanger reaches its minimum when n ¼ 3, which is different from the corresponding characteristic shown in Fig. 2. Fig. 7 shows the effect of the tube-diameter ratio on the characteristic of the thermal efficiency hqW versus the DMFR M2. When the ratio is Di ¼ 2i/3D0 (i ¼ 1,2,…,n), the ratio obeys Murry law [36]; when the ratio is Di ¼ 2i/2D0(i ¼ 1,2,…,n), the flow velocities in the tubes of the X-shaped heat exchanger are equal to each other; when the ratio is Di ¼ D0(i ¼ 1,2,…,n), all the tube cross section areas of the X-shaped heat exchanger are equal to each other. From Fig. 7, one can see that when n ¼ 3, the thermal efficiency of the X-shaped heat exchanger with Murry law is increased by 68.54% than that with equal flow velocity in the tubes, and by 435.46% than that with
(33)
Fig. 7. Effect of the tube-diameter ratio on the characteristic of hqW versus M2.
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L. Chen et al. / Applied Thermal Engineering 91 (2015) 456e462
4. Conclusions Constructal optimizations of the H- and X-shaped heat exchangers are carried out in this paper. The thermal efficiencies of the heat exchangers are maximized, and their optimal constructs are obtained. The results show that: for the same shape, total tube volume and the occupying space, the thermal efficiency of the elemental X-shaped heat exchanger is increased by 44.48% than that of the second order assembly of H-shaped heat exchanger when M2 << 1, and this increment is 46.11% when M2 >> 1. Therefore, the performance of the heat exchanger with X-shaped structure is superior to that with H-shaped structure due to its structure improvement. The performance of the heat exchanger is greatly improved by adopting multi-scale structure compared with the single-scale one. The optimization results obtained in this paper can have some applications in electronics cooling, bioengineering, bioreactors, and selective membranes, etc. Acknowledgements This work is supported by the National Natural Science Foundation of China (Grant Nos. 51176203 and 51206184) and the Natural Science Foundation for Youngsters of Naval University of Engineering (Grant No. HGDQNJJ15007). The authors wish to thank the reviewers for their careful, unbiased and constructive suggestions, which led to this revised manuscript. References [1] Q.W. Wang, M. Zeng, T. Ma, X.P. Du, J.F. Yang, Recent development and application of several high-efficiency surface heat exchangers for energy conversion and utilization, Appl. Energy 135 (2014) 748e777. [2] A. Bejan, The concept of irreversibility in heat exchanger design: counterflow heat exchangers for gas-to-gas applications. Trans, ASME, J. Heat. Transf. 99 (3) (1977) 374e380. [3] A. Bejan, Second law analysis in heat transfer, Energy 5 (8e9) (1980) 720e732. [4] A. Bejan, Entropy Generation Through Heat and Fluid Flow, Wiley, New York, 1982. [5] A. Bejan, Entropy Generation Minimization, CRC Press, Boca Raton FL, 1996. [6] M. Babaelahi, S. Sadri, H. Sayyaadi, Multi-objective optimization of a crossflow plate heat exchanger using entropy generation minimization, Chem. Engng. Tech. 37 (1) (2014) 87e94. [7] J.F. Guo, X.L. Huai, Optimization design of heat exchanger in an irreversible regenerative Brayton cycle system, Appl. Therm. Engng. 58 (1e2) (2013) 77e84. [8] T. Wenterodt, C. Redecker, H. Herwig, Second analysis for sustainable heat and energy transfer: the entropic potential concept, Appl. Energy 139 (2015) 376e383. [9] Y. Lara, P. Lisbona, A. Martinez, L.M. Romeo, Design and analysis of heat exchanger networks for integrated Ca-looping systems, Appl. Energy 111 (2013) 690e700. € [10] C. Ezgi, N. Ozbalta, I. Girgin, Thermohydraulic and thermoeconomic performance of a marine heat exchanger on a naval surface ship, Appl. Therm. Engng. 64 (1e2) (2014) 413e421.
[11] A. Bejan, Shape and Structure, from Engineering to Nature, Cambridge University Press, UK, Cambridge, 2000. [12] A.K. da Silva, Constructal Multi-scale Heat Exchanger, Duke University, USA, 2005. Ph. D. Dissertation. [13] S. Lorente, A. Bejan, Svelteness, freedom to morph, and constructal multi-scale flow structures, Int. J. Therm. Sci. 44 (12) (2005) 1123e1130. [14] A. Bejan, S. Lorente, Design with Constructal Theory, Wiley, New Jersey, 2008. [15] L. Rocha, A O. Convection in Channels and Porous Media: Analysis, Optimization, and Constructal Design, VDM Verlag Dr Mueller Aktiengesellschaft & Co. KG, Deutschland, 2009. [16] G. Lorenzini, S. Moretti, A. Conti, Fin Shape Thermal Optimization Using Bejan's Constructal Theory, Morgan & Claypool Publishers, San Francisco, USA, 2011. [17] L.G. Chen, Progress in study on constructal theory and its application, Sci. China Tech. Sci. 55 (3) (2012) 802e820. [18] A. Bejan, S. Lorente, Constructal law of design and evolution: physics, biology, technology, and society, J. Appl. Phys. 113 (15) (2013) 151301. [19] K. Manjunath, S.C. Kaushik, Second law thermodynamic study of heat exchangers: a reviews, Renew. Sustain. Energy Rev. 40 (2014) 348e374. [20] A. Bejan, Constructal law: optimization as design evolution, Trans. ASME J. Heat. Transf. 137 (6) (2015) 061003. [21] J.C. Ordonez, A. Bejan, Entropy generation minimization in parallel-plates counterflow heat exchangers, Int. J. Energy Res. 24 (10) (2000) 843e864. [22] J.V.C. Vargas, A. Bejan, Thermodynamic optimization of finned crossflow heat exchangers for aircraft environmental control systems, Int. J. Heat. Fluid Flow. 22 (6) (2001) 657e665. [23] A. Bejan, Dendritic constructal heat exchanger with small-scale crossflows and larger-scales counterflows. Int, J. Heat. Mass Transf. 45 (23) (2002) 4607e4620. [24] A.K. da Silva, S. Lorente, A. Bejan, Constructal multi-scale tree-shaped heat exchanger, J. Appl. Phys. 96 (3) (2004) 1709e1718. [25] A.K. da Silva, A. Bejan, Dendritic counterflow heat exchanger experiments, Int. J. Therm. Sci. 45 (9) (2006) 860e869. [26] V.D. Zimparov, A.K. da Silva, A. Bejan, Constructal tree-shaped parallel flow heat exchangers, Int. J. Heat. Mass Transf. 49 (23e24) (2006) 4558e4566. [27] K. Manjunath, S.C. Kaushik, Entropy generation and thermo-economic analysis of constructal heat exchanger, Heat. Trans. Res. 43 (1) (2014) 39e60. [28] K. Manjunath, S.C. Kaushik, Second law analysis of unbalanced constructal heat exchanger, Int. J. Green Energy 11 (2) (2014) 173e192. [29] J. Yang, A.W. Fan, W. Liu, A.M. Jacobi, Optimization of shell-and-tube heat exchangers conforming to TEMA standards with designs motivated by constructal theory, Energy Convers. manage. 78 (2014) 468e476. [30] A. Bejan, S. Lorente, R. Anderson, Constructal underground designs for ground-coupled heat pumps, Trans. ASME J. Sol. Energy Engng 136 (1) (2014) 011019. [31] M.K. Rodrigues, R. da Silva Brum, J. Vaz, L.A.O. Rocha, E.D. dos Santos, L.A. Isoldi, Numerical investigation about the improvement of the thermal potential of an earth-air heat exchanger (EAHE) employing the constructal design method, Renew. Energy 80 (2015) 538e551. [32] Y.P. Chen, P. Cheng, An experimental investigation on the thermal efficiency of fractal tree-like microchannel nets, Int. Comm. Heat. Mass Transf. 32 (7) (2005) 931e938. [33] Y.P. Chen, C.B. Zhang, M.H. Shi, Y.C. Yang, Thermal and hydrodynamic characteristics of constructal tree-shaped minichannel heat sink, AIChE J. 56 (8) (2010) 2018e2029. [34] C.B. Zhang, Y.P. Chen, R. Wu, M.H. Shi, Flow boiling in constructal tree-shaped minichannel network, Int. J. Heat. Mass Transf. 54 (1e3) (2011) 202e209. [35] H.J. Feng, L.G. Chen, Z.H. Xie, F.R. Sun, Constructal entropy generation rate minimization for X-shaped vascular networks, Int. J. Therm. Sci. 92 (2015) 129e137. [36] C.D. Murray, The physiological principle of minimal work, in the vascular system, and the cost of blood-volume, Acad. Nat. Sci. 12 (3) (1926) 207e214.
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Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng
Research paper
Thermal efficiency maximization for H- and X-shaped heat exchangers based on constructal theory Lingen Chen a, b, c, *, Huijun Feng a, b, c, Zhihui Xie a, b, c, Fengrui Sun a, b, c a
Institute of Thermal Science and Power Engineering, Naval University of Engineering, Wuhan, 430033, PR China Military Key Laboratory for Naval Ship Power Engineering, Naval University of Engineering, Wuhan, 430033, PR China c College of Power Engineering, Naval University of Engineering, Wuhan, 430033, PR China b
h i g h l i g h t s Constructal optimizations of H- and X-shaped heat exchangers are carried out. Maximum thermal efficiency is taken as optimization objective. Thermal efficiency is defined as ratio of heat transfer rate to total pumping power. Optimal constructs of H- and X-shaped heat exchangers are obtained. Thermal efficiency of X-shaped heat exchanger is larger than that of H-shaped.
a r t i c l e i n f o
a b s t r a c t
Article history: Received 29 May 2015 Accepted 13 August 2015 Available online 21 August 2015
Constructal optimizations of H- and X-shaped heat exchangers are carried out by taking the maximum thermal efficiency (the ratio of the dimensionless heat transfer rate to the dimensionless total pumping power) as optimization objective. The constraints of total tube volumes and spaces occupied by heat exchangers are considered in the optimizations. For the H-shaped heat exchanger, the thermal efficiency decreases when the dimensionless mass flow rate increases. For the higher order of the X-shaped heat exchanger, when the order number is 3, the thermal efficiency of the heat exchanger with Murry law is increased by 68.54% than that with equal flow velocity in the tubes, and by 435.46% than that with equal cross section area of the tubes. © 2015 Elsevier Ltd. All rights reserved.
Keywords: Constructal theory Maximum thermal efficiency H-shaped heat exchanger X-shaped heat exchanger Generalized thermodynamic optimization
1. Introduction Heat exchanger is an important device used in various industries, such as metallurgy industry, chemical industry, spaceflight, etc. To improve the energy utilization of the industry, one of the important tasks is to reduce the energy losses of the heat exchangers [1]. To satisfy different energy utilization requirements and solve the entropy generation paradox [2,3], many scholars studied the performances of various heat exchangers based on the optimization objectives of entropy generation number [4e6], modified entropy generation number [7,8], exergy [9] and thermaleconomic [10], etc.
* Corresponding author. Institute of Thermal Science and Power Engineering, Naval University of Engineering, Wuhan, 430033, PR China. Tel.: þ86 27 83615046; fax: þ86 27 83638709. E-mail addresses: [email protected], [email protected] (L. Chen). http://dx.doi.org/10.1016/j.applthermaleng.2015.08.029 1359-4311/© 2015 Elsevier Ltd. All rights reserved.
Constructal theory [11e20] is a powerful theory in illustrating and solving various design problems, and it also has exhibited its power in the optimal designs of heat exchangers [21e31]. Ordonez and Bejan [21] optimized the parallel-plate counterflow heat exchanger by taking minimum entropy generation as optimization objective, and obtained the optimal channel spacing ratio, total contact area and the capacity rate ratio of the heat exchanger. Vargas and Bejan [22] studied the counterflow heat exchanger of an aircraft environmental control system based on entropy generation minimization method, and the result showed that the optimal construct of the core with finned surfaces was nearly the same as that with smooth plates. Bejan [23] optimized the channel spacings of a dendritic heat exchanger based on maximum volumetric heat transfer rate, and obtained a better performance of the vascularized heat exchanger over the parallel one in Hagen-poiseuille regime. da Silva et al. [24] built three kinds of multi-scale models for treeshaped heat exchangers, and investigated the characteristics of
L. Chen et al. / Applied Thermal Engineering 91 (2015) 456e462
Nomenclature A cp D h i k L M m_ Nu n ni q T U V W
area, m2 specific heat, J/kg/K diameter, m heat transfer coefficient, W/m2/K current order of the heat exchanger thermal conductivity, W/m/K length, m dimensionless mass flow rate flow rate, kg/s Nusselt number order number of the heat exchanger number of the tubes heat transfer rate, W temperature, K overall heat transfer coefficient, W/m2/K volume, m3 pumping power, W
the global thermal resistances versus the pumping powers. They also discussed the applications of tree-shaped heat exchangers into actual devices, and further carried out experiments for the diskshaped counterflow heat exchanger [25]. Zimparov et al. [26] compared the performances of two tree-shaped heat exchangers. The results showed that the counterflow heat exchanger was superior to the parallel one for a higher value of pumping power, and the conclusion was reversed for a lower value of pumping power. Based on the multi-scale models for tree-shaped heat exchangers in Ref. [24], Manjunath and Kaushik [27,28] further investigated their entropy generation and thermo-economic performances, and concluded that their performances were improved compared with those of the convectional heat exchangers. Moreover, some scholars also implemented constructal optimizations of the shelltube [29] and underground [30,31] heat exchangers. In the studies of the dendritic channel networks, Chen and his groups [32e34] investigated the network's thermal efficiency performances, and found that these networks had the lower pressure drop and pumping power requirement, more uniform temperature and better thermal efficiency compared with the serpentine channels. The results also showed that the thermal efficiency performance was a perfect objective to simultaneously evaluate the thermal and fluid flow performances (TFFPs) of the network. The TFFPs of the H-shaped heat exchangers were analyzed in Refs. [24,26], respectively, and the experiments for the disk-shaped heat exchanger were carried out in Ref. [25]. A vasculature with X-shaped tube structure was optimized by Feng et al. [35], and the TFFPs were evidently improved. Considering these situations, an X-shaped heat exchanger will be investigated in this paper, and the thermal efficiency will be taken as the optimization objective by using the constructal theory. Compared with the heat exchanger's structure adopted in Refs. [24,26], the structure adopted in this paper is modified, and the TFFPs are simultaneously considered by choosing the maximum thermal efficiency as optimization objective. The performance of the X-shaped heat exchanger will be compared with that of the corresponding Hshaped heat exchanger.
Greek symbols tube angle, rad temperature difference, K thermal efficiency kinematic viscosity, m2/s density, kg/m3
a DT hqW n r
Subscripts c cold fluid e crossed tube h hot fluid in inlet opt optimal 0,1,2,3,… index of the tube Superscript ~ dimensionless
composed of many pairs of tubes with different lengths and diameters. In the upper part of the heat exchanger, the single-phase _ temperature Th,in, density r, hot fluid (mass flow rate m_ n ¼ m, thermal conductivity k and specific heat cp) enters the root of the heat exchanger, and releases heat to the cold fluid when it flows through each tube (diameter Di, length Li and tube number ni, i ¼ 0,1,2,3,…,n). Each branching stream (flow rate m_ 0 ) flows out of the heat exchanger from its canopy located at the center of the elemental area. The hot fluid is then collected by pressure equalizer, and finally flows into the hot fluid circulator unit. In the lower part of the heat exchanger, the structures and the mass flow rates are identical to the corresponding ones in the upper part. The same
2. Constructal optimization for H-shaped heat exchanger An H-shaped counterflow heat exchanger in the rectangular area is shown in Fig. 1 [24]. The H-shaped heat exchanger is
457
Fig. 1. H-shaped heat exchanger in a rectangular area [24].
458
L. Chen et al. / Applied Thermal Engineering 91 (2015) 456e462
kind of single-phase cold fluid (inlet mass flow rate m_ 0 and inlet temperature Tc,in) flows through the cold side of the tube network from its canopy to the root. The tube walls of the cold and hot sides are adiabatic from the surroundings. The flows in cold and hot side tubes are both in Hagenepoiseuille regime. The relationships of the sizes and mass flow rates of the Hshaped heat exchanger are
_ ni Li ¼ 2i=2 L0 ; Di ¼ 2i=3 D0 ; m_ i ¼ 2in m; ¼ 2ni ; ði ¼ 0; 1; 2; …; nÞ
(1)
where the tube-diameter ratio of the heat exchanger obeys the Murray law [36]. Both the total volume of the tubes and space occupied by the Hshaped heat exchanger are, respectively, given by n X
V ¼ 2$
ni Li
i¼0
pD2i 4
(2)
A ¼ 2n ð2L0 $2L1 Þ
(3)
From Eqs. (1)e(3), D0 and L0 can be, respectively, rewritten as
31=2 1=6 V 2 1 25 2n 5 D0 ¼ 4 9 n pA1=2 212þ 3 24þ2
n X pkNu 1 DT$ ni L i $ 1=2 T 2 pkNuA h;in Tc;in i¼0 1þn 21=4 2 2 1 M1 1þn ¼ 21=4 2 2 1 þ 4M1 2 21=2
~¼ q
n _ X L V2 ~ ¼ 2$128mn m_ i i4 $ W pr i¼0 Di 4p kNu cp 2 ðn=rÞA5=2 1 n 3 1 n p2 M12 242 26þ6 1 ¼ 3 21=6 1
(10)
(11)
The definition of the heat exchanger's thermal efficiency was given as the ratio of the heat transfer quantity to the total pumping work in Refs. [32e34]. According to this definition, the thermal efficiency of the heat exchanger in Fig. 1 can be given as the ratio of the dimensionless heat transfer rate to the dimensionless total pumping power. From Eqs. (10) and (11), the thermal efficiency of the H-shaped heat exchanger is
2
L0 ¼ 2
(4) hqW
nþ2þ12
2
A1=2
n X
3 1þn 2 2 1 2n=2 21=6 1 1þn 1þn 3 h i 21=4 2 2 1 þ 4M1 2 21=2 p2 M1 2 6 1 (12)
(5)
For the same heat capacity rate of the two fluids, the temperature difference DT between the two fluids along the length direction is a constant [24]. In each tube, the heat exchanger's heat transfer rate is equal to that released by the hot fluid, which is the energy balance equation of the coupled heat transfer. For the specified inlet temperatures Tc,in and Th,in of the cold and hot fluids, summing the energy balance equations in each tube, the heat exchanger's heat transfer rate becomes:
DT
~ q ¼ ¼ ~ W
_ p Th;in Tc;in DT ni pUi Di Li ¼ mc
(6)
i¼0
From Eq. (12), the thermal efficiency hqW is a function of the DMFR M1 and the order number n of the heat exchanger, and one can carry out performance analysis and optimization of the Hshaped heat exchanger by varying M1 and n. Fig. 2 shows the effect of n on the characteristic of the thermal efficiency hqW versus the DMFR M1. From Fig. 2, one can see that for the fixed n, hqW decreases when M increases, and the heat exchanger's performance becomes weak. For a lower and fixed M1, the thermal efficiency of the heat exchanger decreases when n increases. For a higher and fixed M1, with the increase in n, the efficiency decreases first and then increases, and it reaches its minimum when n ¼ 4.
where Ui(i ¼ 0,1,…,n) is the overall heat transfer coefficient (HTC) without considering the thermal resistances of the tube walls:
1 1 1 ¼ þ Ui hi hi
(7)
where hi(i ¼ 0,1,…,n) is the HTC of the fluid in one tube:
hi ¼
kNu Di
(8)
where Nu is the Nusselt number, which is assumed to be identical in each tube [24]. Substituting Eqs. (1), (4) and (5) into Eqs. (6)e(8) yields
DT ¼
4M1 2 21=2 Th;in Tc;in 3 n 24þ2 21=4 þ 8 25=2 M1
(9) _ mc
where the dimensionless mass flow rate (DMFR) is M1 ¼ pkNuAp 1=2 . From Eqs. (1), (4), (5) and (9), the dimensionless total heat transfer rate and total pumping power of the H-shaped heat exchanger can be, respectively, given by
Fig. 2. Effect of n on the characteristic of hqW versus M1.
L. Chen et al. / Applied Thermal Engineering 91 (2015) 456e462
In the performance analysis of the heat exchanger, the local pressure losses are ignored. From Eqs. (1) and (3), the svelteness number of the H-shaped heat exchanger is [30].
¼
external flow length scale LTotal Li ¼ 1=3 ¼ i¼0 internal flow length scale V V 1=3 A1=2 1 2ðnþ1Þ=2 $ V 1=3 1 21=2 $2n=2þ5=4
(13) ðnþ1Þ=2
(15)
. 1=4 2 1=8 4~ L 2$ ~L A De ¼ h . i1=2 ~ 2~ p 4þD L V
1=2
12 A In Eq. (13), the magnitude of ð12 1=2 Þ$2n=2þ5=4 is 1. If V 1=3 is large enough, Sv will be large, and the neglect of the local pressure losses is validated [13].
3. Constructal optimization for X-shaped heat exchanger 3.1. Element volume of X-shaped heat exchanger To further improve the performance of the H-shaped heat exchanger, one can try to make a structure improvement based on the second order assembly of H-shaped heat exchanger as shown in Fig. 3, that is, the elemental volume of X-shaped heat exchanger. In the upper part, the single-phase hot fluid (mass flow rate m_ 0 ) enters the root of the heat exchanger, flows through one round tube (diameter D0 and length L0) and four crossed tubes (diameter De and length Le), and flows out of the heat exchanger (mass flow rate me) from its canopy located at the center of the quarter of the elemental volume. The cold side structure of the heat exchanger is the same as that at hot side. The total tube volume and the space occupied by the elemental volume of X-shaped heat exchanger are
p V ¼ 2$ 4D2e Le þ D20 L0 4
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4L2e L20
For the specified V and A, substituting Eq. (15) into Eq. (14) yields
Pn
Sv ¼
A ¼ 4L0
459
Le ¼
A1=2 1=2 2 1=4 4 ~L 2~L
(16)
(17)
~ ¼ D0 and ~L ¼ L0 . where D De Le For the specified inlet temperatures Tc,in and Th,in of the cold and hot fluids, the energy balance equation and the temperature difference of the cold and hot fluids can be, respectively, given by
DT ðL0 þ 4Le Þ ¼ M2 Th;in Tc;in DT 1=2 2A DT ¼
2A1=2 M2 Th;in Tc;in L0 þ 4Le þ 2A1=2 M2
(18)
(19)
_ mc
where the DMFR is M2 ¼ pkNuAp 1=2 . From Eqs. 16e19, the dimensionless total heat transfer rate and total pumping power of the elemental volume of the X-shaped heat exchanger can be, respectively, given by
pkNu 1 DT$ðL0 þ 4Le Þ$ 2 pkNuA1=2 Th;in Tc;in 4 þ ~L M2 ¼ 1=2 2 1=4 4 ~L 4þ~ L þ 4M2 ~L
~¼ q (14)
! _ L Le V2 ~ ¼ 2$128mn m_ 04 þ m_ e 4 $ W 2 pr De 4p kNu cp ðn=rÞA5=2 D0 4 2 ~ 2 ~L ~ þ 4~L 4 þ D p2 M22 D ¼ 3=2 2 3=4 ~ 4~ 4~ L 2D L
(20)
(21)
~ can be From Eq. (21), the dimensionless total pumping power W ~ minimized by varying D. The optimal diameter ratio of the tubes is ~ opt ¼ 22=3 , and the corresponding dimensionless total pumping D power can be rewritten as
2 21=3 p2 M22 22=3 þ ~ L 2 þ 21=3 ~ L ~ ¼ W 3=4 ~L3=2 4 ~L2
Fig. 3. Elemental volume of X-shaped heat exchanger.
(22)
One can see that the optimal diameter ratio of the tubes above is equal to that of the vascular network with X-shaped structure obtained in Ref. [35]. This is because the fluid flow performances of the elemental volumes with X-shaped structure are optimized in this paper and Ref. [35], and the same optimal diameter ratio of the tubes can be obtained due to the same X-shaped fluid flow structure. From Eqs. (20) and (22), the thermal efficiency of the elemental volume is
460
hqW
L. Chen et al. / Applied Thermal Engineering 91 (2015) 456e462
~ q ¼ ¼ ~ W
3=2 2 3=4 ~ 4þ~ L 4~ L L 2 1=2 2 1=4 4~ L 4 þ ~L þ 4M2 ~ 21=3 p2 M2 22=3 þ ~ L 2 þ 21=3 ~L L
From Eq. (23), for the fixed DMFR M2, hqW is only the function of ~ L, and hqW can be maximized by varying ~ L. The total tube volume V and the area A occupied by the elemental volume in Eqs. (14) and (15) are the constraints in the model. However, the tube diameter De and length Le can be obtained by solving Eqs. (14) and (15), therefore, the constraints of the thermal efficiency maximization are disappeared. The optimization of hqW can be implemented by varying ~ L in its reasonable range. Because there is only one optimization variable (~ L) in hqW, the available range of ~ L is divided into many equidistant segments in the optimization of hqW. When the approximate location of the extreme point is found in a small range, the range is further divided into many refined equidistant segments. By using this method, the maximum value of hqW and the corresponding optimal value of ~L (~ Lopt ) can be obtained. Fig. 4 shows the effects of the DMFR M2 on the optimal tube 0 length ratio ~ Lopt ¼ ðL0 =Le Þopt and optimal shape ~ Lopt ¼ 2L0;opt = qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ½2$ ð2Le;opt Þ2 L20;opt ¼ ~ Lopt Þ1=2 of the elemental volLopt $ð4 ~ ume. From Fig. 4, one can see that when M2 << 1, the optimal tube length ratio is ~ Lopt / 0:918, and the corresponding optimal shape is
0 ~ Lopt /0:517. When M2 >> 1, the optimal tube length ratio is 0 ~ Lopt / 0:776, and the corresponding optimal shape is ~Lopt /0:421. One can see that the optimal tube length ratios above are different from those of the vascular network with X-shaped structure obtained in Ref. [35]. This is because the thermal performances of the elemental volumes with X-shaped structure are optimized both in this paper and Ref. [35], and the different optimal tube length ratios can be obtained due to the different heat transfer boundary conditions. For the same shape, the total tube volume V and the space A occupied by the elemental volume, one can compare the thermal efficiency performances of the elemental volume of X-shaped heat exchanger and second order assembly of H-shaped heat exchanger. This requires the tube length ratio of X-shaped heat exchanger satisfies L0/Le ¼ 2/31/2. Numerical calculations show that the thermal efficiency of the elemental volume of X-shaped heat exchanger is increased by
0 Fig. 4. Characteristics of ~Lopt and ~Lopt versus M2.
(23)
44.48% than that of the second order assembly of H-shaped heat exchanger when M2 << 1, and this increment is 46.11% when M2 >> 1. Therefore, due to the structure improvement of the heat exchanger, the performance of the X-shaped heat exchanger is superior to that of the H-shaped one.
3.2. Higher order assembly of X-shaped heat exchanger As shown in Fig. 5, a higher order assembly (order number n) is composed of a number (2n) of elemental volumes. The elemental volumes are collected by several multi-scale tubes (Di ; Li ; m_ i ; ni ði ¼ 1; 2; :::; nÞ). The tube numbers and flow rates of the X-shaped heat exchanger are given as follows
. _ in ; i ¼ 1;2;…;n ne ¼ 2nþ2 ; ni ¼ 2ni ; m_ e ¼ m_ 2nþ2 ; m_ i ¼ m2
(24)
The total tube volume and space occupied by the X-shaped heat exchanger are n X p V ¼ 2$ D2i Li 4D2e Le þ 4 i¼0
A ¼ 2n $4L0
!
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4L2e L20
(25)
(26)
From Eqs. (25) and (26), the tube lengths and diameters of the Xshaped heat exchanger can be, respectively, given by
Fig. 5. Higher order assembly of X-shaped heat exchanger.
L. Chen et al. / Applied Thermal Engineering 91 (2015) 456e462
. pffiffiffi . De ¼ D0 22=3 ; Di ¼ 2i=3 D0 ; Le ¼ 3L0 2; Li ¼ 2i=2 L0
L0 ¼ 2
A1=2 nþ2þ12
(27)
(28)
2
2 D0 ¼ 4
461
VA1=2 p1
31=2
25 2n 1 5 31 n 9 n 212þ2 $31=2 þ 212þ 3 24þ2 21=6 1
(29)
Similar to Eq. (19), the temperature difference of the heat exchanger's two fluids is
DT ¼
29=4 M2 2þ21=2 Th;in Tc;in 21=2 21þ2 þ22þ2 $31=2 22þ2 $31=2 8$21=4 M2 þ4$23=4 M2 (30) n
3
n
n
From Eqs. 27e30, the dimensionless total heat transfer rate, dimensionless total pumping power, and thermal efficiency of the X-shaped heat exchanger are, respectively, given by
! n X pkNu 1 ~¼ DT$ 2nþ2 Le þ ni Li $ q 1=2 T 2 pkNuA h;in Tc;in i¼0
(31)
! n _ Le X L V2 ~ ¼ 2$128mn m_ e 4 þ mi i4 $ W 2 pr De i¼0 Di 4p kNu cp ðn=rÞA5=2
(32)
hqW ¼
Fig. 6. Effect of n on the characteristic of hqW versus M2.
equal cross section area of the tubes. These increments will continue to increase when the order number n increases. Therefore, compared the X-shaped multi-scale heat exchanger with the Xshaped single-scale heat exchanger, the performance of the multiscale heat exchanger is greatly improved. For the shell and tube or plate-fin heat exchangers, the flow channel is single-scale one, which may become one of the weak point in comparison with the multi-scale heat exchanger in this paper.
n 3 pffiffiffi pffiffiffi ~ 3þn pffiffiffi 3þn 3þn 4þn 3 n n n pffiffiffi q 2 21þ2 22þ2 3 þ 2 2 3 M2 2 9$21=3 þ 18 2 9$22=3 þ 9$2 6 þ 2 2 3$2 3 ¼ 24þ2 21=6 1 ~ W pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi 5þn 7þn 10þn n pffiffiffi n pffiffiffi 7 n pffiffiffi þ9$2 6 þ 3$2 6 9$2 6 3 3 þ 9$21=6 3 9$21=3 3 3$22=3 3 þ 3$25=6 3 3$22þ6 3 3$21þ3 3 þ 3$26þ3 3 pffiffiffiio1 pffiffiffihpffiffiffi 11þn pffiffiffi 3þn pffiffiffi n n pffiffiffi 2 21þ2 22þ2 3 þ 2 2 3 þ 4$21=4 p2 M2 2 þ 2 þ 3$2 6 3 þ 3 6
From Eq. (33), one can carry out constructal optimization of the X-shaped heat exchanger by varying the DMFR M2 and order number n. Fig. 6 shows the effect of n on the characteristic of the thermal efficiency hqW versus the DMFR M2. From Fig. 6, one can see that the thermal efficiency versus the DMFR characteristic of the X-shaped heat exchanger is similar to that of the H-shaped heat exchanger in Fig. 2. For a higher mass flow rate, the thermal efficiency of the Xshaped heat exchanger reaches its minimum when n ¼ 3, which is different from the corresponding characteristic shown in Fig. 2. Fig. 7 shows the effect of the tube-diameter ratio on the characteristic of the thermal efficiency hqW versus the DMFR M2. When the ratio is Di ¼ 2i/3D0 (i ¼ 1,2,…,n), the ratio obeys Murry law [36]; when the ratio is Di ¼ 2i/2D0(i ¼ 1,2,…,n), the flow velocities in the tubes of the X-shaped heat exchanger are equal to each other; when the ratio is Di ¼ D0(i ¼ 1,2,…,n), all the tube cross section areas of the X-shaped heat exchanger are equal to each other. From Fig. 7, one can see that when n ¼ 3, the thermal efficiency of the X-shaped heat exchanger with Murry law is increased by 68.54% than that with equal flow velocity in the tubes, and by 435.46% than that with
(33)
Fig. 7. Effect of the tube-diameter ratio on the characteristic of hqW versus M2.
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4. Conclusions Constructal optimizations of the H- and X-shaped heat exchangers are carried out in this paper. The thermal efficiencies of the heat exchangers are maximized, and their optimal constructs are obtained. The results show that: for the same shape, total tube volume and the occupying space, the thermal efficiency of the elemental X-shaped heat exchanger is increased by 44.48% than that of the second order assembly of H-shaped heat exchanger when M2 << 1, and this increment is 46.11% when M2 >> 1. Therefore, the performance of the heat exchanger with X-shaped structure is superior to that with H-shaped structure due to its structure improvement. The performance of the heat exchanger is greatly improved by adopting multi-scale structure compared with the single-scale one. The optimization results obtained in this paper can have some applications in electronics cooling, bioengineering, bioreactors, and selective membranes, etc. Acknowledgements This work is supported by the National Natural Science Foundation of China (Grant Nos. 51176203 and 51206184) and the Natural Science Foundation for Youngsters of Naval University of Engineering (Grant No. HGDQNJJ15007). The authors wish to thank the reviewers for their careful, unbiased and constructive suggestions, which led to this revised manuscript. References [1] Q.W. Wang, M. Zeng, T. Ma, X.P. Du, J.F. Yang, Recent development and application of several high-efficiency surface heat exchangers for energy conversion and utilization, Appl. Energy 135 (2014) 748e777. [2] A. Bejan, The concept of irreversibility in heat exchanger design: counterflow heat exchangers for gas-to-gas applications. Trans, ASME, J. Heat. Transf. 99 (3) (1977) 374e380. [3] A. Bejan, Second law analysis in heat transfer, Energy 5 (8e9) (1980) 720e732. [4] A. Bejan, Entropy Generation Through Heat and Fluid Flow, Wiley, New York, 1982. [5] A. Bejan, Entropy Generation Minimization, CRC Press, Boca Raton FL, 1996. [6] M. Babaelahi, S. Sadri, H. Sayyaadi, Multi-objective optimization of a crossflow plate heat exchanger using entropy generation minimization, Chem. Engng. Tech. 37 (1) (2014) 87e94. [7] J.F. Guo, X.L. Huai, Optimization design of heat exchanger in an irreversible regenerative Brayton cycle system, Appl. Therm. Engng. 58 (1e2) (2013) 77e84. [8] T. Wenterodt, C. Redecker, H. Herwig, Second analysis for sustainable heat and energy transfer: the entropic potential concept, Appl. Energy 139 (2015) 376e383. [9] Y. Lara, P. Lisbona, A. Martinez, L.M. Romeo, Design and analysis of heat exchanger networks for integrated Ca-looping systems, Appl. Energy 111 (2013) 690e700. € [10] C. Ezgi, N. Ozbalta, I. Girgin, Thermohydraulic and thermoeconomic performance of a marine heat exchanger on a naval surface ship, Appl. Therm. Engng. 64 (1e2) (2014) 413e421.
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