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Arabian Journal for Science andEngineering ISSN 1319-8025Volume 38Number 2 Arab J Sci Eng (2013) 38:365-372DOI 10.1007/s13369-012-0447-3

“Volume-Point” Mass Transfer ConstructalOptimization Based on Triangular Element

Huijun Feng, Lingen Chen, Zhihui Xie &Fengrui Sun

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Arab J Sci Eng (2013) 38:365–372DOI 10.1007/s13369-012-0447-3

RESEARCH ARTICLE - SPECIAL ISSUE - MECHANICAL ENGINEERING

“Volume-Point” Mass Transfer Constructal OptimizationBased on Triangular Element

Huijun Feng · Lingen Chen · Zhihui Xie ·Fengrui Sun

Received: 19 February 2012 / Accepted: 3 May 2012 / Published online: 5 December 2012© King Fahd University of Petroleum and Minerals 2012

Abstract Based on constructal theory, the constructs ofa “volume-point” mass transfer model based on triangularelement with Darcy flow and Hagen–Poiseuille flow in thechannel are optimized by taking dimensionless maximumpressure drop minimization as optimization objective, andthe optimal constructs of the “volume-point” mass transfermodel are obtained. It is shown that with the increase ofconstruct order, some rules are presented in the optimiza-tion results. Compared the constructal optimization resultsbased on triangular elements with those based on rectangu-lar element, the dimensionless pressure drop of the triangularelement with Darcy flow in the channel is smaller than that ofthe rectangular element for the fixed C0, but the conclusion isreversed for the subsequent assemblies; the optimal constructof each order assembly based on triangular element withHagen–Poiseuille flow in the channel are different from thatbased on rectangular element with Hagen–Poiseuille flow inthe channel. For the different requirements of the mass trans-fer systems, the optimization results obtained based on trian-gular element can provide some guidelines for the constructaldesigns of the “volume-point” mass transfer problems withporous medium.

Keywords Constructal theory · Porous medium ·Maximum pressure drop minimization · Volume-point masstransfer · Triangular element · Generalized thermodynamicoptimization

H. Feng · L. Chen (B) · Z. Xie · F. SunCollege of Power Engineering,Naval University of Engineering, Wuhan 430033,People’s Republic of Chinae-mail: lgchenna@yahoo.com; lingenchen@hotmail.com

List of Symbols

A Area of the construct, m2

C Product of dimensionless permeabilityand volume fraction

D Width of the channel, mH Width of the construct, mK Permeability, m2

L Length of the construct, mm Mass flow rate, kg/sm

′′′Mass flow rate per unit volume, kg/s/m3

n Number of the constituents�P Pressure drop, Pa

Greek Symbols

φ Volume fractionμ Viscosity, kg/s/m

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ρ Density, kg/m3

ν Kinematic viscosity, m2/s

Subscripts

m Minimumopt Optimum0 Elemental volume1 First order assembly construct2 Second order assembly construct

1 Introduction

Constructal theory [1–6] is a powerful theory in illustratingand solving various design problems, such as the engineer-ing design problems of heat transfer systems [7,8], fluid flowsystems [9–17] as well as steam equipments [18,19], the evo-lution problems of design in nature [12,20–23] and socialdynamics problems of social systems [24–26], etc. In theseproblems, “volume-point” heat conduction problem is oneof hotspots in the constructal theory. Bejan [27], firstly car-ried out constructal optimization of a “volume-point” heatconduction model with rectangular element, and obtainedthe optimal constructs of the model. Many scholars furtheroptimized various “volume-point” heat conduction modelsbased on various elements by using constructal theory, suchas rectangular elements with different constraint conditionsand calculation methods [28–35] triangular elements [36,37]tapered conducting paths and elements [38–42] as well asthree-dimensional elements [43–45]

Constructal theory can also be applied to the optimiza-tion of fluid flow systems analogy to heat conduction sys-tems, such as mass transfer problems with porous mediums[46–55] as well as heat and mass transfer problems [56–59].In analyses of “volume-point” mass transfer problems withporous mediums, Bejan and Errera [47] considered a “vol-ume-point” mass transfer model with rectangular elementand high permeability material in the channel analogy toheat conduction model [27] and obtained the optimal distri-bution of high permeability material and the optimal shapesof the constructs; the open space channels instead of the highpermeability channels were also considered, and the opti-mal shapes of the constructs with Hagen–Poiseuille flow inthe channel were derived. Errera and Bejan [48] further car-ried out constructal optimization of the Bejan and Errera’s[47] model based on complete numerical simulations of theDarcy flow through the multi-component porous medium andby releasing the constraint that the next order assembly wasassembled by the optimized last one, and compared the opti-mization results with those obtained in Ref. [47]. Moreover,Bejan [49,50] built the three-dimensional “volume-point”mass transfer model based on rectangular elemental vol-ume and round tube channel with Hagen–Poiseuille flow,

and carried out constructal optimization to obtain the optimalconstructs of the three-dimensional “volume-point” model.Through the analyses of many “volume-point” flow prob-lems, Bejan discovered that any finite-size portion of thecomposite could have its shape optimized such that its overallresistance to flow was minimal [50].

Based on Ref. [47], the “volume-point” mass transfermodel with triangular element and high-permeability mate-rial in the channel analogy to heat conduction model basedon triangular element [35] will be considered in this paper.The Darcy flow in the channel filled with high-permeabilitymaterial and Hagen–Poiseuille flow in the open space chan-nel will be considered, respectively. The exact method willbe adopted to calculate the pressure drop in the channel inan exact way. Constructal optimization will be carried out bytaking minimum dimensionless maximum pressure drop ofthe construct as optimization objective, and the optimal con-structs obtained will compare with those obtained in Ref. [47]with rectangular element.

2 Optimization of Triangular Elemental Volume

A triangular elemental volume (H0 × L0 × 1/2) generatesmass flow at a constant rate m volumetrically is shown inFig. 1. The mass flow rate per unit volume is a constant m

′′′ =m/(H0 × L0 ×1/2). For simplification, it is assumed that thetriangular elemental volume is two dimensional. In this case,the elemental area size A0 = (H0×L0)/2 is constant, but theratio H0/L0 is free to vary. The single-phase flow generatedin triangular elemental isotropic porous medium with a lowpermeability K is in the Darcy regime [60]. The mass flow,driven by the pressure field �P0(x, y), is first collected bya high permeability channel (K0, D0, L0) located on thelonger axes of the triangular elemental area, and then flowsto the origin located at point M0 along the high-permeabilitychannel. The boundary of the triangular elemental area isimpermeable except for the origin point M0. Similar to the“volume-point” heat conduction problem based on triangu-lar element [35], it is assumed that the permeability of K0

Fig. 1 Triangular elemental volume

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material is much higher than that of the K material (K0 �K ), the area occupied by K0 material is much smaller thanthat occupied by K material (φ0 = 2D0/H0 � 1), and thetriangular elemental area is slender enough (H0 � L0). Withthese assumptions, the mass flow direction is approximatelyparallel to y-direction in the K material, and is approximatelyparallel to x-direction in the K0 material.

According to Ref. [47], combining the local mass conti-nuity equation with Darcy law in K material, one can obtain

K

μ

∂2 P

∂y2 + m′′′

ρ= 0 (1)

where μ and ρ are the viscosity and density of the flow,respectively. The boundary conditions are

∂ P

∂y= 0, y = H0

2

(1 − x

L0

)(2)

P = P(x, 0), y = 0 (3)

From Eqs. (1)–(3), the pressure drop distribution of the tri-angular element in Fig. 1 is

�P0(x, y)= m′′′ν

2K

[H0 y

(1− x

L0

)− y2

]+�P0(x, 0) (4)

where ν = μ/ρ is kinematic viscosity.According to Ref. [47], combining the local mass conti-

nuity eqnarray with Darcy law in K0 material, one can obtain

K0

μ

∂2 P

∂x2 + m′′′

H0

ρD0= 0 (5)

The boundary conditions are

∂ P

∂x= 0, x = L0 (6)

P = P(0, 0), x = 0 (7)

From Eqs. (5)–(7), the pressure drop distribution along the xaxis is

�P0(x, 0) = m′′′νH0

K0 D0

(L0x

2− x2

2+ x3

6L0

)(8)

From Eqs. (4) and (8), the pressure drop distribution of thetriangular element can be determined as

�P0(x, y) = m′′′ν

2K

[H0 y

(1 − x

L0

)− y2

]

+ m′′′νH0

K0 D0

(L0x

2− x2

2+ x3

6L0

)(9)

For the case y < 0, the pressure drop can be obtained byreplacing H0 to −H0 in Eq. (9). The position of the pointswhere the maximum pressure locates is found by solving

∂[�P0(x, y)]/∂x = 0 and ∂[�P0(x, y)]/∂y = 0, simulta-neously. The solutions are

x1 = L0, y1 = 0 (10)

x2 = L0 − K0φ0 H20

4L0, y2 = K0φ0 H3

0

8L20

(11)

where K0 = K0/K . Substituting Eqs. (10) or (11) into (9),the dimensionless maximum pressure drop of the triangularelement becomes

�P0 = �P0

m ′′′ν A0/K

= 2

3K0φ0· L0

H0+ K 2

0 φ20

192·(

H0

L0

)5

(12)

�P0 is minimized with respect to the aspect ratio H0/L0 ofthe triangular element, and the corresponding optimizationresults are

(H0/L0)opt =(

2

5

)1/6

· 2√K0φ0

(13)

�P0,m =(

2

5

)5/6 1√K0φ0

(14)

When K0φ0 >> 1, Eq. (13) agrees with H0 � L0, and theassumption that the triangular element is slender enough isreasonable.

The optimal construct in Ref. [47] based on rectangu-lar element and the corresponding dimensionless maximumpressure drop can be, respectively, given by

(H0/L0)opt = 2√K0φ0

(15)

�P0,m = 1

2√

K0φ0

(16)

Compared the optimal construct of triangular element withthat of rectangular element, the optimal construct based ontriangular element is more slender, and the dimensionlessmaximum pressure drop is decreased by 6.80 %. This meansthat the optimal construct based on triangular element canhelp to improve its mass transfer performance than that basedon rectangular element.

3 Optimization of the First Order Assembly

One way to assemble the optimized triangular elemental vol-umes is shown in Fig. 2. A large number (n1) of triangularelemental volumes (A0 ×1) are distributed on the both sidesof a high permeability channel (K1, D1, L1). The elementalmass currents are collected by the K1 material, and the outer

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Fig. 2 First order assembly construct

boundary of A1 is impermeable except for the D1 patch overthe origin M1, through which the collected mass current isled to the outside. The height of the first order assembly isH1 = 2L0, and its length is given by L1 = n1 H0/2. In thiscase, the number (n1) of triangular element is free to vary.The volume fraction of K1 material in the first order assemblyis defined as φ1 = D1L1/A1, one has D1 = H1φ1/2.

In the following, the exact method, different from themethod in Ref. [47], is applied to calculate the pressuredrop distribution along the K1 channel. The pressure dropdistribution along the K1 channel with mass input at pointsM11, M12, . . ., M1,n1/2 and no mass input between two fol-lowing points M1, j−1 and M1, j is analyzed as one-dimen-sional mass transfer problem similar to the heat conductionproblem in Ref. [35], and it can be given by

d2 P

dx2 = 0 (17)

According to the Darcy law, the boundary conditions alongthe K1 channel are

P = PM1, j−1 at x = (2 j − 3)H0

2(18)

K1 D1dP

dx= [n1−2( j −1)] m

′′′ν A0 at x =(2 j −1)

H0

2(19)

where 2 ≤ j ≤ n1/2. From Eqs. (17)–(19), the pressure dropbetween M1, j−1 and M1, j becomes

PM1, j −PM1, j−1 = [n1−2( j −1)] m′′′ν A0 H0

K1 D12 ≤ j ≤ n1

2

(20)

A similar procedure can be applied to the interval M1 andM11

PM11 − PM1 = n1m′′′ν A0 H0

2K1 D1(21)

The pressure drop along the K1 channel can be derived byaccumulating the pressure drops of all intervals on the K1

channel

�PM1,n1/2 M1 = (PM11 − PM1) +n1/2∑j=2

(PM1, j − PM1, j−1)

= m′′′ν A0 H0n2

1

4K1 D1(22)

From Eqs. (14) and (20), the maximum pressure drop of thefirst order assembly becomes

�P1 = �P0,m + �PM1,n1/2 M1 =(

2

5

)5/6 m′′′ν A0

K

1√K0φ0

+ m′′′ν A0 H0n2

1

4K1 D1(23)

The corresponding dimensionless maximum pressure dropof the first order assembly is

�P1 = �P1

m ′′′ν A1/K

= 21/3 · 52/3n21 + 4K1φ1

27/6 · 55/6n1 K1φ1

√K0φ0

(24)

where K1 = K1/K . �P1 can be optimized with respect tothe number (n1) of constituents. The optimization results aregiven by

n1,opt = 25/6

51/3

√K1φ1 (25)

�P1,m = 2√5K0φ0 K1φ1

(26)

(H1

L1

)opt

=√

5K0φ0

K1φ1(27)

(D1

D0

)opt

=(

5

2

)1/6φ1

φ0

√K0φ0 (28)

The optimal constructs of the first order assembly based onrectangular element in Ref. [47] can be, respectively, givenby

n1,opt =√

2K1φ1 (29)

�P1,m = 1√2K0φ0 K1φ1

(30)

(H1

L1

)opt

=√

2K0φ0

K1φ1(31)

(D1

D0

)opt

= φ1

φ0

√K0φ0 (32)

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Fig. 3 Second order assemblyconstruct

Compared the optimal construct based on triangular ele-ment with that based on rectangular element in Ref. [47],the optimal number (n1,opt) of triangular elements in the firstorder construct is smaller than that of rectangular element inthe first order construct, the shape of the triangular first orderconstruct is tubbier than that of the rectangular first order con-struct, the ratio of the first order and elemental high perme-ability channels of the triangular first order construct is largerthan that of the rectangular first order construct. However, thedimensionless maximum pressure drop of the triangular firstorder construct is larger than that of the rectangular first orderconstruct. For the aesthetic and compatibility requirementsof the mass transfer systems, these systems may not alwaysbe manufactured based on rectangular element, and the opti-mal constructs based on triangular element remains one ofthe optimal designs of the mass transfer systems.

4 Optimization of the Second and Higher-OrderAssemblies

The new second order assembly construct shown in Fig. 3follows the same steps as in the first order assembly con-structs. The outer boundary of A2 is impermeable except forthe D2 patch over the origin M2, through which the collectedmass current is led to the outside. The aspect ratio (H2/L2)

of the new construct or the number (n2) of constituentsinside the new construct, is free to vary. The volume fractionof the K2 material in the second order assembly is defined asφ2 = D2L2/A2, one has D2 = H2φ2/2.

Similar to the analysis of the first order assembly, fromEqs. (20) and (24), the maximum pressure drop of the sec-ond order assembly becomes

�P2 = m′′′ν A1

K

2√5K0φ0 K1φ1

+ m′′′ν A1 H1n2

2

4K2 D2(33)

The corresponding dimensionless maximum pressure dropof the second order assembly is

�P2 = �P2

m ′′′ν A2/K

= (5n22 K0φ0 + 8K2φ2)

√K1φ1

4√

5√

K0φ0 K1φ1 K2φ2n2

(34)

where K2 = K2/K . �P2 can be optimized with respect tothe number (n2) of constituents, and similar steps can be fol-lowed as Eqs. (17)–(28) in the optimization of the higher-order assemblies. The optimization results for each orderassembly are listed in Table 1.

From Table 1, one can see that when the conductanceratio Ci/Ci−1(i ≥ 1) is large, the optimal number (ni,opt)

of constituents becomes large, and the optimal shapes ofthe constructs become slender; when the conductance ratioCi/Ci−1 → 1, ni,opt becomes small, and the analysis shouldbe refined as shown in Ref. [27]. When ni,opt is large, the masscurrent in the Ki channel can be considered to be linear withthe length; when ni,opt is small, this assumption is unrea-sonable, and the exact method without any approximationsadopted in this paper is still valid, which is the major differ-ence between the exact method in this paper and the approx-imate method in Ref. [47]. The minimum dimensionless

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Table 1 Constructaloptimization results for eachorder assembly based ontriangular element with Darcyflow in the channel

where Ci = Ki φi

i ni,opt

(Di

Di−1

)opt

(HiLi

)opt

�Pi,m

0 – –27/6C−1/2

051/6 ( 2

5 )5/6C−1/20

1 25/6

51/3 C1/21 ( 5

2 )1/6φ−10 φ1C1/2

0 (5C0C−11 )1/2 2 · (5C0C1)

−1/2

2 23/2

51/2 (C2C−10 )1/2 2

51/2 φ−11 φ2(C

−10 C1)

1/2 (2C1C−12 )1/2 (2−1C1C2)

−1/2

3 2(C3C−11 )1/2 φ−1

2 φ3(2C−11 C2)

1/2 (2C2C−13 )1/2 (2−1C2C3)

−1/2

≥ 3 2(Ci C−1i−2)

1/2 φ−1i−1φi (2C−1

i−2Ci−1)1/2 (2Ci−1C−1

i )1/2 (2−1Ci Ci−1)−1/2

Table 2 Constructaloptimization results for eachorder assembly based ontriangular element withHagen–Poiseuille flow in thechannel

where K = K/A0

i ni,opt

(Di

Di−1

)opt

(HiLi

)opt

�Pi,m

0 – – 217/9·31/3 K 1/3

71/9φ0

22/9·34/3 K 1/3

77/9φ0

1φ

1/20 φ

3/21

2·31/6·145/18 K 2/371/9φ1

28/9·31/3 K 1/3

2·147/18 K 1/3φ1/20

31/6φ3/21

12·61/2

71/2 K (φ0φ1)−3/2

2 31/3(φ1φ2)3/2

25/18·77/9 K 2/3φ0

31/6φ2(φ−10 φ1)1/2

147/18 K 1/3

28/9·77/18 K 1/3φ1/20

31/6φ3/22

24·27/9·31/6·75/18 K 5/3

φ1/20 φ3

1φ3/22

3 213/18·31/3(φ2φ3)3/2

77/9 K 2/3φ0

21/9·31/6(φ−10 φ2)1/2φ3

77/18 K 1/3

147/18 K 1/3φ1/20

31/6φ3/23

112·35/6·141/18 K 7/3φ1/20

(φ1φ2)3φ3/23

≥ 3 2i−3·213/18·31/3(φi−1φi )3/2

77/9 K 2/3φ0

2(i−3)/2·21/9·31/6(φ−10 φi−1)1/2φi

77/18 K 1/3

77/18 K 1/3φ1/20

2(i−4)/2·21/9·31/6φ3/2i

24+50i−9i2

18 ·3 116 − i

3 ·7− 2318 + 7i

9 K2i+1

3 φ2i−5

20

(∏i−1

j=1 φ3j )φ

3/2i

maximum pressure drop of the triangular element is smallerthan that of the rectangular element obtained in Ref. [47] forthe fixed C0, but the conclusion is reversed for the subsequentassemblies. With the given Ki and φi , ni,opt, (Di/Di−1)opt

and (Hi/Li )opt are different for the element, first and sec-ond order assemblies based on rectangular and triangularelements, but these optimization results are the same for thethird and higher-order assemblies.

5 Channels with Hagen–Poiseuille Flow

The optimal constructs of the “volume-point” mass transfermodel based on triangular element bathed by Darcy flow inthe high permeability channel are obtained Sects. 2–4. Whenthe channels are not filled with permeable materials but sim-ply open spaces, the optimization results will be different.Assuming that the spacing Di is sufficiently small, the per-meability Ki in the channel with Hagen–Poiseuille flow canbe expressed as following [61]

Ki = D2i /12 (i = 0, 1, 2, . . .) (35)

Similar to Eq. (9), the pressure drop distribution of the trian-gular element bathed by Hagen–Poiseuille flow in the chan-nel can be determined as

�P0(x, y) = m′′′ν

2K

[H0 y

(1 − x

L0

)− y2

]

+12m′′′νH0

D30

(L0x

2− x2

2+ x3

6L0

)(36)

For the case y < 0, the pressure drop can be obtained byreplacing H0 to −H0 in Eq. (34). Following the steps shownin Sects. 2–4, one can obtain the optimal constructs of eachorder assembly with Hagen–Poiseuille flow in the channel.The optimization results are listed in Table 2.

From Table 2, some rules can be obtained in the expres-sions of the optimization results ni,opt, (Di/Di−1)opt,

(Hi/Li )opt and �Pi,m with the increase of the construct order(i ≥ 3). The constructal optimization results for each orderassembly with Hagen–Poiseuille flow in the channel are obvi-ously different from those shown in Table 1 with Darcy flow.Compared the constructal optimization results based on tri-angular element and Hagen–Poiseuille flow in the channelwith those based on rectangular element in Ref. [47], theoptimal construct of the triangular element is more slenderthan that of the rectangular element, but the conclusion isreversed for the subsequent assemblies; the minimum dimen-sionless maximum pressure drop for each triangular assem-bly is always larger than that of the rectangular assembly forthe fixed φi and K , however, the optimal constructs based ontriangular element remains one of the optimal designs of themass transfer systems due to the aesthetic and compatibilityrequirements of the mass transfer systems.

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6 Conclusions

The construct of a “volume-point” mass transfer model basedon triangular element is considered in this paper. The chan-nel collected the mass flow from the low permeability porousmedium is filled with high permeability porous medium withDarcy flow. The optimal constructs of the “volume-point”mass transfer model are obtained by taking dimensionlessmaximum pressure drop minimization as optimization objec-tive. The optimization steps are also extends to applicationswhere the high permeability channels are replaced by theempty spaces with Hagen–Poiseuille flow in the channel. It isshown that with the increase of construct order, some rules arepresented in the optimization results. When the conductanceratio Ci/Ci−1(i ≥ 1) is large, the optimal number (ni,opt) ofconstituents of the triangular element with Darcy flow in thechannel becomes large, the mass current in the Ki channelcan be considered to be linear with the length, and the optimalshapes of the constructs become slender; when ni,opt is small,the exact method without any approximations adopted in thispaper is still valid, which is the major difference between theexact method in this paper and the approximate method inRef. [47]. For the given Ki and φi , ni,opt, (Di/Di−1)opt and(Hi/Li )opt are different for the elemental, first and secondorder assemblies based on rectangular and triangular ele-ments with Darcy flow in the channel, but these optimizationresults are the same for the third and higher-order assem-blies. For the given K and φi , and the optimal shapes of eachorder assembly based on rectangular and triangular elementswith Hagen–Poiseuille flow in the channels are different fromeach other. The minimum dimensionless maximum pressuredrop of the triangular element with Darcy flow in the chan-nel is smaller than that of the rectangular element obtainedin Ref. [47] for the fixed C0, but the conclusion is reversedfor the subsequent assemblies. The minimum dimensionlessmaximum pressure drop of each triangular assembly withHagen–Poiseuille flow in the channel is always larger thanthat of the rectangular assembly for the fixed φi and K . Forthe different requirements of the mass transfer systems, thesesystems may not always be manufactured based on rectangu-lar element, and the optimal constructs based on triangularelement remains one of the optimal designs of the mass trans-fer systems.

The optimization results obtained in this paper illustratethat the “volume-point” flow systems can have their optimalstructures which lead to the minimal overall flow resistancesin the flow systems [50]. Moreover, one can release moreconstraints similar to the “volume-point” heat conductionproblems [36,38–43] to further reduce the overall flow resis-tances of the constructs.

Acknowledgments This work is supported by the National NaturalScience Foundation of China (Grant No. 51176203) and the Natural

Science Foundation for Youngsters of Naval University of Engineering(Grant No. HGDQNJJ11008). The authors wish to thank the reviewersfor their careful, unbiased and constructive suggestions, which led tothis revised manuscript.

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14. Ordonez, J.C.; Chen, S.; Vargas, J.V.C.; Dias, F.G.; Gardolinski,J.E.F.C.; Vlassov, D.: Constructal flow structure for a single SOFC.Int. J. Energy Res. 31(14), 1337–1357 (2007)

15. Bejan, A.: Science and technology as evolving flow architectures.Int. J. Energy Res. 32(15), 1399–1417 (2008)

16. Lorente, S.: Vascularized materials as designed porous media. Int.J. Energy Res. 33(2), 211–220 (2009)

17. Xia, L.; Lorente, S.; Bejan, A.: Constructal design of distributedcooling on the landscape. Int. J. Energy Res. 35(9), 805–812 (2011)

18. Kim, Y.S.; Lorente, S.; Bejan, A.: Distribution of size in steamturbine power plants. Int. J. Energy Res. 33(11), 989–998 (2009)

19. Kim, Y.; Lorente, S.; Bejan, A.: Steam generator structure. Con-tinuous model and constructal design. Int. J. Energy Res. 35(4),336–345 (2011)

20. Bejan, A.: How nature takes shape: extensions of constuctal the-ory to ducts, river, turbulence, cracks, dendritic crystals and spatialeconomics. Int. J. Therm. Sci. 38(8), 653–663 (1999)

21. Bejan, A.; Marden, J.H.: Unifying constructal theory for scaleeffects in running, swimming and flying. J. Exp. Biol. 209(2), 238–248 (2006)

22. Bejan, A.; Lorente, S.: The constructal law and the evolution ofdesign in nature. Phys. Life Rev. 8(3), 209–240 (2011)

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23. Bejan, A.; Lorente, S.: The constructal law and the design of thebiosphere: nature and globalization. Trans. ASME J. Heat Transf.133(1), 011001 (2011)

24. Bejan, A.: Street network theory of organization in nature. J. Adv.Transp. 30(2), 85–107 (1996)

25. Bejan, A.; Merkx G.W.: Constructal Theory of Social Dynamics.Springer, New York (2007)

26. Bejan A.; Lorente, S.; Miguel, A.F.; Reis, A.H.: Constructal HumanDynamics, Security and Sustainability. IOS Press, Amsterdam(2009)

27. Bejan, A.: Constructal-theory network of conducting paths forcooling a heat generating volume. Int. J. Heat Mass Transf. 40(4),799–816 (1997)

28. Ledezma, G.; Bejan, A.; Errera, M.: Constructal tree networks forheat transfer. J. Appl. Phys. 82(1), 89–100 (1997)

29. Almogbel, M.; Bejan, A.: Constructal optimization of nonuniform-ly distributed tree-shaped flow structures for conduction. Int. J.Heat Mass Transf. 44(22), 4185–4194 (2001)

30. Ghodoossi, L.; Egrican, N.: Exact solution for cooling of elec-tronics using constructal theory. J. Appl. Phys. 93(8), 4922–4929(2003)

31. Wu, W.; Chen, L.; Sun, F.: On the “area to point” flow problembased on constructal theory. Energy Convers. Manage. 48(1), 101–105 (2007)

32. Chen, L.; Wei, S.; Sun, F.: Constructal entransy dissipation min-imization for “volume-point” heat conduction. J. Phys. D Appl.Phys. 41(19), 195506 (2008)

33. Karakas, A.; Camdali, U.; Tunc, M.: Constructal optimization ofheat generating volumes. Int. J. Exergy 6(5), 637–654 (2009)

34. Wu, W.; Chen, L.; Sun, F.: Heat-conduction optimization based onconstructal theory. Appl. Energy 84(1), 39–47 (2007)

35. Wei, S.; Chen, L.; Sun, F.: Constructal entransy dissipation minimi-zation for “volume-point” heat conduction without the premise ofoptimized last-order construct. Int. J. Exergy 7(5), 627–639 (2010)

36. Ghodoossi, S.; Egrican, N.: Conductive cooling of triangularshaped electronics using constructal theory. Energy Convers. Man-age. 45(6), 811–828 (2004)

37. Wei, S.; Chen, L.; Sun, F.: “Volume-point” heat conduction con-structal optimization with entransy dissipation minimization objec-tive based on triangular element. Therm. Sci. 14(4), 1075-1088(2010)

38. Neagu, M.; Bejan, A.: Constructal-theory tree networks of ‘con-stant’ thermal resistance. J. Appl. Phys. 86(2), 1136–1144 (1999)

39. Zhou, S.; Chen, L.; Sun, F.: Optimization of constructal vol-ume-point conduction with variable cross-section conducting path.Energy Convers. Manage. 48(1), 106–111 (2007)

40. Wei, S.; Chen, L.; Sun, F.: The volume-point constructal optimi-zation for discrete variable cross-section conducting path. Appl.Energy 86(7/8), 1111–1118 (2009)

41. Wei, S.; Chen, L.; Sun, F.: Constructal optimization of discrete andcontinuous variable cross-section conducting path based on en-transy dissipation rate minimization. Sci. China Tech. Sci. 53(6),1666–1677 (2010)

42. Xiao, Q.; Chen, L.; Sun, F.: Constructal entransy dissipation rateminimization for heat conduction based on a tapered element. Chin.Sci. Bull. 56(22), 2400–2410 (2011)

43. Ledezma, G.A.; Bejan, A.: Constructal three-dimensional trees forconduction between a volume and one point. Trans. ASME J. HeatTransf. 120(4), 977–984 (1998)

44. Neagu, M.; Bejan, A.: Three-dimensional tree constructs of ‘con-stant’ thermal resistance. J. Appl. Phys. 86(12), 7107–7115 (1999)

45. Alebrahim, A.; Bejan, A.: Constructal trees of circular fins forconductive and convective heat transfer. Int. J. Heat Mass Transf.42(19), 3585–3597 (1999)

46. Bejan, A.: Advanced Engineering Thermodynamics, 2nd edn.Wiley, New York (1997)

47. Bejan, A.; Errera M.R.: Deterministic tree networks for fluid flow:geometry for minimal flow resistance between a volume and onepoint. Fractals 5(4), 685–695 (1997)

48. Errera, M.R.; Bejan, A.: Tree networks for flows in compositeporous media. J. Porous Media 2(1), 1–18 (1999)

49. Bejan, A.: Constructal tree network for fluid flow between a finite-size volume and one source or sink. Rev. Gen. Therm. 36(8), 592–604 (1997)

50. Bejan, A.: Constructal theory: from thermodynamic and geometricoptimization to predicting shape in nature. Energy Convers. Man-age. 39(18), 1705–1718 (1998)

51. Ordonez, J.C.; Bejan, A.; Cherry, R.S.: Designed porous media:Optimally nonuniform flow structures connecting one point withmore points. Int. J. Therm. Sci. 42(9), 857–870 (2003)

52. Bejan, A.; Dincer, I.; Lorente, S.; Miguel, A.F.; Reis, A.H.: Porousand Complex Flow Structures in Modern Technologies. Springer,New York (2004)

53. Ingham, D.B.; Bejan, A.; Mamut, E.; Pop, I.: Emerging Technolo-gies and Techniques in Porous Media. Kluwer, Dordecht (2004)

54. Lorente, S.: Constructal view of electrokinetic transfer throughporous media. J. Phys. D Appl. Phys. 40(9), 2941–2947 (2007)

55. Lorente, S.; Bejan, A.: Constructal design of vascular porous mate-rials and electrokinetic mass transfer. Transp. Porous Media 77(2),305–322 (2009)

56. Azoumah, Y.; Mazet, N.; Neveu, P.: Constructal network for heatand mass transfer in a solid–gas reactive porous medium. Int. J.Heat Mass Transf. 47(14), 2961–2970 (2004)

57. Azoumah, Y.; Neveu, P.; Mazet, N.: Constructal design combinedwith entropy generation minimization for solid–gas reactors. Int.J. Therm. Sci. 45(7), 716–728 (2006)

58. Azoumah, Y.; Neveu, P.; Mazet, N.: Optimal design of thermo-chemical reactors based on constructal approach. AIChE J. 53(5),1257–1266 (2007)

59. Zhou, S.; Chen, L.; Sun, F.: Constructal entropy generation min-imization for heat and mass transfer in a solid–gas reactor basedon triangular element. J. Phys. D Appl. Phys 40(11), 3545–3550(2007)

60. Nield, D.A; Bejan, A.: Convection in Porous Media. 2nd edn.Springer, New York (1999)

61. Bejan, A.: Convection Heat Transfer, 2nd edn. Wiley, New York(1995)

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