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Conductive cooling of triangular shaped electronics using constructal theory
Energy Conversion and Management 45 (2004) 811–828 www.elsevier.com/locate/enconman
Conductive cooling of triangular shaped electronics using constructal theory Lotfollah Ghodoossi a
a,*
, Nil€ ufer E grican
b
Department of Mechanical Engineering, Istanbul Technical University, 80191 Gumussuyu, Istanbul, Turkey b Department of Mechanical Engineering, School of Engineering and Architecture, Yeditepe University, 26 Agustos Yerlesimi, Kayisdagi Caddesi 81120 Kayisdagi/Istanbul, Turkey Received 15 March 2003; accepted 16 July 2003
Abstract Conductive cooling of electronics falls in the category of the more general ‘‘area to point’’ flow problem. Heat generated in a fixed area is to be discharged to a heat sink located on the border of the heat generating area through relatively high conductive link(s). This will maintain a limited temperature difference between the hot spot inside the heat generating area and the heat sink. The solution procedure starts with heat transfer analysis and geometric optimization of the smallest heat generating area. Assembly of optimized smallest areas in a fixed but larger heat generating area by introducing a new high conductive link and geometric optimization of the new area leads to achieving the goal of conductive cooling of the larger area. The sequence of assembly of optimized areas in a relatively larger area and geometric optimization of this area is continued until the required area size to be cooled is obtained. The process of assembly and optimization steps leads to formation of a tree network of high conductive links inside the heat generating area. Along with geometric optimization of the heat generating area in each step, the tree network of high conductive links is optimized with respect to high conductive material allocation in the heat generating area as well. Ó 2003 Elsevier Ltd. All rights reserved. Keywords: Constructal design; Optimization; Tree networks; Electronics cooling; Heat transfer; Conduction
*
Corresponding author. Tel.: +90-212-2931-1300x2452; fax: +90-212-2450795. E-mail addresses: [email protected] (L. Ghodoossi), [email protected] (N. E grican).
0196-8904/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0196-8904(03)00190-0
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Nomenclature A AP D H k0 kP k^ L n q q000 T x, y
heat generating area area of high conductive links width of high conductive link height of heat generating area conductivity of electronic material conductivity of high conductive material conductivity ratio length of heat generating area number of constituents volumetric heat generation rate heat generation rate per unit volume temperature Cartesian coordinates
Greek symbol / volume fraction of high conductive material Subscripts i order of constructs min minimum opt optimum
1. Introduction Conductive cooling of electronics falls in the category of the more general ‘‘area to point’’ flow problem, which has applications in a wide range of science branches, such as biology [1], economics [2–4], urban transportation [5], heat transfer [6–11] etc [12]. Area to point flow studies in heat transfer problems [6–11] have mainly concentrated on conductive cooling of electronics based on constructal theory. Conductive cooling of electronics using constructal-theory was exposed and solved first by Bejan [13,14]. The solution of Bejan [13,14] to the conductive cooling of electronics using constructal theory was improved by Ghodoossi and Egrican [15]. The specific problem in the works of Bejan [13,14] and Ghodoossi and Egrican [15] is to cool a uniform heat generating fixed size area by distributing links of fixed amounts of high conductive material in the area. The links of high conductive material discharge the generated heat to a heat sink located on the border of the heat generating area. The objective is to minimize the thermal resistance of the heat generating area to maintain a limited maximum temperature in the area for the purpose of safe operation of the electronic device. Solution to the so defined problem starts with heat transfer analysis and geometric optimization of the smallest heat generating area. Assembly of optimized smallest areas in a fixed but larger heat generating area by introducing a new high conductive link and performing a double optimization process leads to achieving the goal of conductive cooling of the larger area. The new larger area is optimized with respect to both the geometry of the area and
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high conductive material allocation in the area. Assembly of this newly optimized larger area in a relatively much larger area by introducing another high conductive link and applying a double optimization process will provide conductive cooling of the much larger area. The sequence of assembly of optimized areas in relatively larger areas and double optimization of the areas is continued until the required area size to be cooled is obtained. The works done in Refs. [13–15] are solutions for the conductive cooling of electronics, all formed by assembly of ‘‘rectangular’’ heat generating areas. However, small and smart electronic devices may be manufactured not in ‘‘rectangular’’ form. Some other forms may be used because of aesthetic and compatibility requirements of the electronic devices. In this paper, the constructal solution method is applied to the problem of cooling of electronics formed of ‘‘triangular’’ heat generating areas.
2. Triangular elemental area Assume a triangular area of base H0 and height L0 , shown in Fig. 1, that generates heat at a constant rate q volumetrically. Heat generation is uniform such that the heat generation per unit volume is constant ðq000 ¼ q=ððH0 L0 =2Þx1Þ ¼ constantÞ. The area size A0 is constant, but the ratio, H0 =L0 is free to vary. The heat generated over the area is first conducted to a relatively high conductive link of width D0 , which is located on the axis of symmetry of the triangular elemental area, from which it is then channeled to a heat sink located at point M0 . It is assumed that the triangular elemental area is slender enough to have one dimensional (y-direction) heat conduction on the heat generating area. The boundary of the triangular elemental heat generating area is adiabatic except for the heat sink point located at point M0 . It is also assumed that the thermal conductivity of the high conductive link (kP ) is much higher than the thermal conductivity of the electronic material (k0 ) and the area occupied by the high conductive material is much smaller than the area of electronic material. The objective is to minimize the maximum thermal resistance between a point in the heat generating area and the heat sink point M0 . The maximum thermal resistance is defined in the form of DT =ðq000 AÞ where DT represents the temperature difference between the hot spot and the heat sink point. The governing differential equation for steady state one dimensional heat conduction with uniform heat generation in the electronic material is
Fig. 1. Triangular elemental area.
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o2 T q000 þ ¼0 oy 2 k0
ð1Þ
The boundary conditions for the electronic material are oT ¼0 oy
at y ¼ yb
T ¼ T0 ðxÞ
at y ¼ 0
ð2Þ ð3Þ
where yb denotes the ordinate of boundary points located on both sides of triangular elemental area, H0 x ð4Þ 1 yb ¼ L0 2 Integrating Eq. (1) subject to boundary conditions (2) and (3) will result in q000 2 q000 H0 x y þ T0 ðxÞ y þ 1 T ðx; yÞ ¼ L0 2k0 2k0
ð5Þ
Heat conduction along the high conductive link can be assumed as conduction in a fin. The amount of heat input from the electronic material to the high conductive link is a function of x. Performing an energy balance for a differential dx length at point x will result in the governing differential equation for steady-state one-dimensional heat conduction in the high conductive link as d2 T0 q000 H0 x ¼ 1 ð6Þ dx2 kP D0 L0 The boundary conditions for the high conductive link are dT0 ¼0 dx
at x ¼ L0
ð7Þ
T0 ¼ TM0
at x ¼ 0
ð8Þ
Integrating Eq. (6) subject to boundary conditions (7) and (8) will give the temperature distribution along the high conductive link as q000 H0 x3 x2 L0 ð9Þ T0 ðxÞ ¼ þ x þ TM0 kP D0 6L0 2 2 Substituting Eq. (9) in Eq. (5) will give the temperature distribution over the triangular elemental heat generating area. q000 2 q000 H0 x q000 H0 x3 x2 L0 ð10Þ T ðx; yÞ TM0 ¼ y þ 1 yþ þ x L0 2k0 2k0 kP D0 6L0 2 2 Eq. (10) is valid for y > 0. There is no need to derive the temperature distribution for y < 0 because the temperatures are equal for symmetrical points with respect to x axis.
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The position of the points involving maximum temperature is found by solving oT ðx; yÞ=ox ¼ 0 and oT ðx; yÞ=oy ¼ 0, simultaneously. Two points are found. The coordinates of these points are x1 ¼ L0 ;
y1 ¼ 0
ð11Þ
x2 ¼ L0
D0 H0 kP ; 2L0 k0
y2 ¼
D0 H02 kP 4L20 k0
ð12Þ
The first point shows the position of maximum temperature on the high conductive link that locates naturally farthest from the heat sink on the right edge of the high conductive link (point P0 ). The second point characterizes the position of the hot spot on the electronic material. The coordinates of the hot spot found in Eq. (12) show that this point lies exactly on the upper border of the triangular elemental heat generating area. Another geometric feature of hot spot is that it is very close to the first point as the triangular elemental area is slender ðH0 =L0 1Þ. The first and second points are not the same points mathematically, but it can be said that they are the same points physically. The maximum temperature difference between the hot spot and the heat sink ðDT0 Þ is derived by substituting the coordinates of the hot spot in Eq. (10). " # 2 q000 A0 D20 A30 kP 2L0 k0 þ ð13Þ DT0 ¼ 6k0 k0 D0 kP L80 Eq. (13) is nondimensionalized to obtain the maximum thermal resistance. Note that q000 , A0 and k0 are constants. " # 5 DT0 1 1 ^ 2 H0 2 L0 þ ð14Þ ¼ k /0 q000 A0 =k0 3 64 L0 k^/0 H0 where /0 ¼
AP0 D0 ¼2 A0 H0
ð15Þ
kP ð16Þ k^ ¼ k0 /0 and k^ are volume fraction of the high conductive material and conductivity ratio, respectively. An important feature of the heat transfer problem analyzed above is investigated now. The thermal resistance can be minimized geometrically. The thermal resistance in Eq. (14) is minimized with respect to the aspect ratio ðH0 =L0 Þ of the triangular elemental heat generating area. The results are !1=2 H0 27=6 1 ¼ ð17Þ L0 opt 51=6 k^/0 DT0;min 25=6 ¼ q000 A0 =k0 55=6
1 ^ k /0
!1=2 ð18Þ
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Using Eqs. (17) and (18), the minimized maximum temperature difference of the triangular elemental heat generating area is found as DT0;min ¼
1 q000 H02 24=3 52=3 k0
ð19Þ
The minimized thermal resistance of the rectangular elemental heat generating area in Refs. [13,14] is found as !1=2 DT0;min 1 1 ¼ ð20Þ q000 A0 =k0 RECT: 2 k^/0 By comparing Eq. (18) with Eq. (20), it is seen that the minimized thermal resistance of the triangular elemental area is approximately 7% lower than that of the rectangular elemental area.
3. First order assembly construct If the electronic material area to be cooled is larger than the triangular elemental area that is just optimized, what can be done? One way to overcome such a problem is to enlarge A0 subject to Eq. (17), but Eq. (19) shows that enlarging the size of the elemental area increases the maximum temperature difference, which puts us distant from the goal of maintaining the lowest maximum temperature difference in the electronic material. In contrast, Eq. (19) advises to design the elemental area in the smallest size that manufacturing restrictions allow. A more efficient way to enlarge the cooled area is shown in Fig. 2. A number of optimized triangular elemental areas are assembled on the upper and lower sides of a new high conductive link of width D1 . This new and larger area is called the first order assembly construct. Heat generated at any elemental electronic material area is first conducted to the associated D0 link, and then, the heat collected in the D0 link, is directed to the D1 link. Finally, the heat collected in the D1 link is conducted toward the heat sink located at point M1 . Once again, the boundaries of the heat generating areas are adiabatic except for the heat sink point M1 . The aspect ratio ðH1 =L1 Þ of the construct or, in other words, the number of constituents
Fig. 2. First order assembly construct.
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n1 inside the construct is free to vary. We will look for an optimal aspect ratio or an optimal number of constituents that minimize the maximum thermal resistance between any point in the new construct and the heat sink point M1 . Clearly, the maximum temperature occurs on the farthest elements of the new construct relative to the M1 heat sink point. The maximum temperature difference between hot spot P1 and the heat sink, DT1 , may be divided into two parts, the temperature difference between the points P1 and M1n01 and the temperature difference between the points M1n01 and M1 . DT1 ¼ DTM1n0 P1 þ DTM1 M1n0 1
ð21Þ
1
where n01 ¼ n1 =2
ð22Þ
DTM1n0 P1 is, in fact, the minimized maximum temperature difference of the elemental area calculated 1in Eq. (19). DTM1n0 P1 ¼ 1
1 q000 H02 24=3 52=3 k0
ð23Þ
Now, we will calculate DTM1 M1n0 the temperature difference along the D1 link. The temperature distribution1 along the D1 link is analyzed as a fin problem with heat input at points M11 ; M12 ; . . . ; M1n01 . The temperature distribution for steady-state one-dimensional heat conduction with no heat input between two following points M1;j1 and M1;j is governed by d2 T ¼0 dx2 The boundary conditions are T ¼ TM1;j1
at x ¼ ð2j 3Þ
ð24Þ
H0 ðpoint M1;j1 Þ 2
ð25Þ
dT H0 ð26Þ ¼ ½n1 2ðj 1Þq000 A0 at x ¼ ð2j 1Þ ðpoint M1;j Þ dx 2 Integrating Eq. (24) subject to the boundary conditions gives the temperature distribution for the M1;j1 M1;j interval. ½n1 2ðj 1Þq000 A0 H0 x ð2j 3Þ T TM1;j1 ¼ ð27Þ k P D1 2 kP D1
The temperature difference between the points M1;j1 and M1;j is found by substituting ð2j 1Þ H20 in place of x in Eq. (27), TM1;j TM1;j1 ¼
½n1 2ðj 1Þq000 A0 H0 kP D1
ð28Þ
Eq. (28) is valid for j P 2. Applying a similar procedure to the M1 M11 interval will result in TM11 TM1 ¼
n1 q000 A0 H0 kP D1 2
ð29Þ
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The temperature difference along the D1 link is equal to the sum of the temperature differences of all intervals on the D1 link, DTM1 M1n0 ¼ ðTM11 TM1 Þ þ 1
n1 =2 X ðTM1;j TM1;j1 Þ
ð30Þ
j¼2
By substituting Eqs. (28) and (29) in Eq. (30) and after some algebra, the temperature difference along the D1 link is found as DTM1 M1n0 ¼ 1
q000 A0 H0 2 n 4kP D1 1
ð31Þ
So far, the temperature difference along the D1 link is found. The maximum temperature difference of the first order assembly construct can be calculated by substituting Eqs. (23) and (31) in Eq. (21). DT1 ¼
1 q000 H02 q000 A0 H0 2 þ n 24=3 52=3 k0 4kP D1 1
ð32Þ
Similar to the case of the elemental area, the maximum temperature difference of the first order assembly construct is nondimensionalized to obtain the maximum thermal resistance. Note that q000 , A1 ðA1 ¼ n1 A0 Þ and k0 are constants. DT1 1 H02 1 H0 ¼ þ n1 000 4=3 2=3 q A1 =k0 2 5 A0 n1 4k^D1
ð33Þ
Eq. (33) shows that the maximum thermal resistance of the construct can be minimized with respect to the number of constituents n1 . The minimization results in 1=2 25=6 ^ D1 ð34Þ n1;opt ¼ 1=3 k 5 L0 1=2 DT1;min 1 1 ¼ H0 q000 A1 =k0 21=6 51=3 k^L0 D1
ð35Þ
Finally, the problem could be optimized with respect to the number of constituents inside the construct, but Eq. (35) shows that the problem can be optimized with respect to D1 , the width of the new high conductive link in the construct, as well. This is done below. The volume fraction of high conductive material in the construct is defined as /1 ¼
AP1 A1
ð36Þ
where AP1 is the total area of the high conductive material in the construct, AP1 ¼ D1 L1 þ n1;opt D0 L0
ð37Þ
Combining Eqs. (15) and (37) with Eq. (36) and taking into account that A1 ¼ n1;opt A0 will result in D1 ¼ L0 ð/1 /0 Þ
ð38Þ
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Placing Eq. (38) in Eq. (35) and using Eq. (17) will read DT1;min 2 1 ¼ 1=2 000 ^ q A1 =k0 5 k ½/0 ð/1 /0 Þ1=2
ð39Þ
The condition that minimizes the thermal resistance in Eq. (39) is /1 ¼ 2/0
ð40Þ
Inserting Eq. (40) into Eqs. (38) and (39) will give the optimal width of the new high conductive link and twice minimized thermal resistance of the construct, respectively. D1 51=6 ¼ 2=3 ðk^/1 Þ1=2 ð41Þ D0 opt 2 DT1;min;min 4 1 ¼ 1=2 5 k^/1 q000 A1 =k0
ð42Þ
The twice minimized maximum temperature difference of the construct is deduced from Eq. (42), 1 q000 H02 DT1;min;min ¼ 1=3 2=3 ð43Þ 2 5 k0 The optimal number of constituents inside the construct is calculated by importing Eq. (41) into Eq. (34) 21=3 1=2 n1;opt ¼ 1=3 ðk^/1 Þ ð44Þ 5 The optimal height and length of the construct are H1;opt ¼ 2L0 ¼ 21=2 51=4 A1=2 1
ð45Þ
n1;opt 21=2 1=2 H0 ¼ 1=4 A1 2 5 The optimal aspect ratio of the construct is obtained by dividing Eq. (45) by Eq. (46) H1 ¼ 51=2 L1 opt L1;opt ¼
ð46Þ
ð47Þ
The optimization process of the first order assembly construct is completed now. All optimal geometric features, minimized maximum thermal resistance and minimized maximum temperature difference of the construct are derived and known through Eqs. (41)–(47). Comparing Eq. (43) with Eq. (19) shows that the maximum temperature difference in the first order assembly construct is exactly doubled relative to the maximum temperature difference of the elemental area. This result was actually predictable because optimization of the problem in a completely mathematical sense without importing any approximations to the physical nature of the problem dictates the existence of an ‘‘equipartition principle’’ in the maximum temperature difference of the construct. This means that half of the total temperature drop in the construct must occur in the electronic material ðk0 Þ and the other half in the high conductive link D1 (kP material). The temperature drop for the electronic material ðDTM1n0 P1 Þ is known at the start of the 1
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optimization process. It is equal to the minimized maximum temperature difference of the elemental area calculated in Eq. (19). The optimization arranges the geometric parameters of the construct in such a way that the temperature drop along the D1 link is exactly equal to the temperature drop in the electronic material, which is known through Eq. (19). This means that as a result of the optimization, the temperature drop of the first order assembly construct will be doubled relative to the temperature drop of the elemental area. It is worth mentioning that the first order assembly construct, in this paper, is optimized mathematically and no approximation is made. Therefore, the total temperature drop, namely, ðq000 H02 =k0 Þ=ð21=3 52=3 Þ is divided into two equal parts as a result of the optimization. Half of the total temperature difference, ðq000 H02 =k0 Þ=ð24=3 52=3 Þ, is dropped along the M1n01 P1 path in the electronic material ðk0 Þ and the other half along the D1 link in the high conductive material ðkP Þ.
4. Second order assembly construct The analysis done at the beginning of Section 3 is valid here once again. In order to enlarge the area to be cooled, one may enlarge size of the elemental areas making the first order assembly construct, but Eqs. (19) and (43) show that this will raise the maximum temperature difference of the construct, which is an unwanted situation in this paper. Therefore, the procedure followed in Section 3 to make a large area will be used again. The new second order assembly construct, shown in Fig. 3, is composed of n2 number of optimized first order assembly constructs. A2 , the area size of the heat generating volumes in the new construct is fixed, but H2 =L2 , the aspect ratio of the construct or n2 , the number of constituents inside the construct are free to vary. The boundaries of the heat generating areas are adiabatic except for the heat sink point located at M2 . We will look for an optimal aspect ratio or an optimal number of constituents that minimize the maximum thermal resistance between any point inside the construct and the heat sink point M2 .
Fig. 3. Second order assembly construct.
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Similar steps will be followed as in the optimization of the first order assembly construct. The maximum temperature difference ðDT2 Þ between the point P2 , which involves the highest temperature in the construct, and the heat sink point M2 is expressed as DT2 ¼ DTM2n0 P2 þ DTM2 M2n0 2
ð48Þ
2
where n02 ¼ n2 =2
ð49Þ
DTM2n0 P2 is equal to the minimized maximum temperature difference of the first order assembly 2 construct calculated in Eq. (43). DTM2n0 P2 ¼ 2
1 q000 H02 21=3 52=3 k0
ð50Þ
By applying similar calculation to that outlined in Eqs. (24)–(31), the temperature difference along the D2 link is found as DTM2 M2n0 ¼ 2
n1 q000 A0 H1 2 n 4kP D2 2
ð51Þ
The maximum temperature difference of the second order assembly construct is found by substituting Eqs. (50) and (51) in Eq. (48). DT2 ¼
1 q000 H02 n1 q000 A0 H1 2 þ n 21=3 52=3 k0 4kP D2 2
ð52Þ
Eq. (52) is nondimensionalized using q000 , A2 ðA2 ¼ n2 A1 ¼ n2 n1 A0 Þ and k0 , that are constants, to obtain the maximum thermal resistance of the construct. DT2 2 1 1 L0 ¼ 1=2 þ n2 ^ ^ 5 k /0 n2 2k D2 2 =k0
q000 A
ð53Þ
The maximum thermal resistance is minimized with respect to n2 , the number of constituents inside the construct. The results are 1=2 2 D2 ð54Þ n2;opt ¼ 1=4 5 /0 L0 DT2;min 2 1 ¼ 1=4 000 q A2 =k0 5 k^
L0 /0 D2
1=2 ð55Þ
Optimization with respect to the number of constituents is followed by optimization with respect to D2 , the width of the new high conductive link in the second order assembly construct. The volume fraction of high conductive material in the construct is defined as /2 ¼
AP2 A2
where AP2 is the total area of high conductive material in the construct.
ð56Þ
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AP2 ¼ n2 AP1 þ D2 L2
ð57Þ
Combining Eqs. (15), (36) and (57) with Eq. (56) results in 1 H0 ð/2 2/0 Þðk^/0 Þ1=2 51=3 Placing Eq. (58) in Eq. (55) and rearranging will read D2 ¼
21=6
DT2;min 1 ¼ 21=2 000 q A2 =k0 k^½/0 ð/2 2/0 Þ1=2
ð58Þ
ð59Þ
The condition that minimizes the thermal resistance in Eq. (59) is /2 ¼ 4/0
ð60Þ
The optimal number of constituents in the construct is found by combining Eqs. (60) and (58) with Eq. (54). n2;opt ¼
4 51=2
ffi 1:8
ð61Þ
This result is very interesting. It says that the optimal number of constituentspinside the second ffiffiffi order assembly construct is not a large number, but it is exactly equal to 4= 5 and it is independent of the physical properties and geometric values. It is clear that from an engineering point of view, the number of constituents cannot be a fractional figure. Therefore, the optimal number of optimized first order assembly constructs making the second order assembly construct should be 2. Among the two degrees of freedom, namely, the number of constituents n2 and the width of the high conductive link D2 , that exist in the problem, one of them, that is, the number of constituents in the second order assembly construct is known now. So, the problem should be optimized with respect to D2 only. The optimization steps in Eqs. (56)–(60) with respect to D2 are repeated subject to n2 ¼ 2. The minimized thermal resistance of the construct is found as pffiffiffi pffiffiffi 2 DT2;min;n2 ¼2 ð2 2 þ 5Þ 1 1 pffiffiffi ¼ ffi 2:8677 ð62Þ ^ ^ q000 A2 =k0 4 5 k /2 k /2 Note that the length of the D2 link is not equal to L2 , see Fig. 4. Because of the shortage of high conductive material, it is cut at point M21 . The same is applied for all constructs with 2 or 4 constituents analyzed in this paper. Optimal geometric features for n2;opt ¼ 2 are derived as H2;opt ¼ 2L1;opt ¼
2
1=2
A2
ð63Þ
L2;opt ¼ H1;opt ¼ 51=4 A1=2 2
ð64Þ
H2 L2
¼ opt
2 51=2
51=4
ð65Þ
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Fig. 4. Second order assembly construct with 2 number of constituents.
The optimal width of the high conductive link for n2;opt ¼ 2 is D2 ¼ 23=2 D1 opt
ð66Þ
The minimized maximum temperature difference of the construct for n2;opt ¼ 2 is found as DT2;min ¼
23=2 þ 51=2 q000 H02 211=6 52=3 k0
ð67Þ
The minimized maximum temperature difference of the second order assembly construct in Eq. (67) is 23=2 þ 51=2 =23=2 ffi 1:79 times higher than the minimized maximum temperature difference of the first order assembly construct in Eq. (43). This ratio was expected to be equal to 2 (equipartition principle) if the construct could be optimized mathematically, but the problem could not be optimized in a completely mathematical sense as the number of constituents was intentionally changed from the optimal value of 1.8 to the best possible value of 2.
5. Third order assembly construct To see what happens next, a third order assembly construct is analyzed. Optimization of the third order assembly construct requires following the same steps as in the first and second order assembly constructs. The third order assembly construct is optimized with respect to both the number of constituents n3 and the width of the high conductive link D3 . The optimization leads to ð68Þ n3;opt ¼ 23=2 þ 51=2 =21=2 ffi 3:58
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It is understood that the best third order assembly construct is the one with the number of constituents equal to either 2 or 4. Optimization of the third order assembly construct with n3 ¼ 2 and 4 results in 3=2 2 2 þ 51=2 þ 2 DT3;min;n3 ¼2 1 1 ¼ ffi 2:7899 ð69Þ 000 3 1=2 ^ ^ q A3 =k0 2 5 k /3 k /3 3=2 2 2 þ 51=2 þ 213=12 31=2 DT3;min;n3 ¼4 1 1 ¼ ffi 3:7996 000 19=6 1=2 ^ ^ q A3 =k0 2 5 k /3 k /3
ð70Þ
Comparing Eq. (69) with Eq. (70) shows that the thermal resistance for n3 ¼ 2 is lower than that for n3 ¼ 4. So, the best third order assembly construct is the one composed of 2 optimized second order assembly constructs. The optimal geometric features, optimal width of the high conductive link and minimized maximum temperature difference of the third order assembly construct for n3 ¼ 2 are recorded below. 1=2
H3;opt ¼ 2L2;opt ¼ 21=2 51=4 A3 L3;opt ¼ H2;opt ¼
H3 L3 D3 D2
21=2 1=2 A 51=4 3
ð71Þ ð72Þ
¼ 51=2
ð73Þ
¼2
ð74Þ
23=2 þ 51=2 þ 2 q000 H02 211=6 52=3 k0
ð75Þ
opt
opt
DT3;min ¼
minimized maximum temperature difference of the third order assembly construct has risen The 23=2 þ 51=2 þ 2 = 23=2 þ 51=2 ffi 1:39 times relative to the minimized maximum temperature difference of the second order assembly construct.
6. Fourth order assembly construct The fourth order assembly is the last construct studied in this paper. A double optimization with respect to the number of constituents n4 and the width of the high conductive link D4 leads to n4;opt ¼
23=2 þ 51=2 þ 2 ffi 2:23 21=2 51=2
ð76Þ
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Optimization of the fourth order construct with n4 ¼ 2 and 4 results in 3=2 2 2 þ 3:51=2 þ 2 DT4;min;n4 ¼2 1 1 ¼ ffi 3:7201 000 4 1=2 q A4 =k0 2 5 k^/4 k^/4
825
ð77Þ
3=2 2 2 þ 51=2 þ 2 þ 23=2 31=2 51=2 DT4;min;n4 ¼4 1 1 ¼ ffi 4:5376 ð78Þ 000 5 1=2 ^ ^ q A4 =k0 2 5 k /4 k /4 The best fourth order assembly construct is the one with n4 ¼ 2 for which the thermal resistance is lower. The optimal geometric features, optimal width of the high conductive link and minimized maximum temperature difference of the fourth order assembly construct for n4 ¼ 2 are recorded below. 2 1=2 ð79Þ H4;opt ¼ 2L3;opt ¼ 1=4 A4 5 1=2
L4;opt ¼ H3;opt ¼ 51=4 A4 H4 2 ¼ 1=2 L4 opt 5
D4 D3
ð80Þ ð81Þ
¼2
ð82Þ
opt
23=2 þ 3:51=2 þ 2 q000 H02 ð83Þ 211=6 52=3 k0 minimized maximum temperature difference of the fourth order assembly construct has risen The 23=2 þ 3:51=2 þ 2 = 23=2 þ 51=2 þ 2 ffi 1:63 times relative to the minimized maximum temperature difference of the third order assembly construct. Proceeding in assembly orders will be continued until the required area size to be cooled is achieved. On the other hand, the maximum temperature difference rises as the assembly order goes up. So, the temperature limit of the electronic material is another criterion to stop proceeding in assembly orders. DT4;min ¼
7. Discussion and conclusion An important and distinctive specification of constructal theory is that every detail of the constructs can be anticipated in a completely deterministic manner. This specification of constructal theory is investigated in this paper. All internal and external geometric details of the constructs are derived and known. None of the geometric details is assumed in advance. All are derived as a result of optimization. Accordingly, the thermal resistance and temperature drop of the constructs are calculated at optimum conditions. The optimization process determines the best number of constituents in each construct as well. The interesting result is that the optimization leads to a large number of constituents in the first
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Thermal Resistance
order construct, but this number remains equal to 2 for all higher order constructs. That is, a bifurcation appears for the second and higher order constructs. Bifurcation has also appeared in other constructal works [13,15] related to conductive cooling of electronics. This coincidence pushes a search for the generality of bifurcation in constructal solutions for area to point flow problems. The equipartition principle is another important feature of optimization applied to the constructal solution in this paper. Mathematical optimization of the problem requires the existence of equipartition for the temperature drop of the constructs. That is, half of the total temperature drop of an optimal construct must be realized in the electronic material and the other half in the high conductive material. This is the case in the first order assembly construct that is optimized in a completely mathematical sense, but the equipartition principle no longer holds in the higher order constructs because changing the optimal fractional number of constituents to a nonoptimal, but the best, even integer disturbed the mathematical optimization of these constructs. If the optimal numbers of the constituents were not changed, the temperature drop of the constructs would be doubled in going from an assembly order to the next. In order to compare the thermal resistance in the various constructs, the results must be introduced on the same basis and equal conditions. It is supposed that the volume fraction of high conductive material is the same for all constructs. Therefore, in the expressions for minimal thermal resistance /i in the ith order construct is replaced with a unique /. This enables us to perform the following discussion. The variation of thermal resistance with respect to the value of ðk^/Þ for the elemental area and first order construct is shown in Fig. 5. For ðk^/Þ < 14:74, the thermal resistance of the elemental area is lower than that of the first order construct. For ðk^/Þ > 14:74, the situation is reversed, and the first order construct produces the lower thermal resistance. Therefore, depending on the amount of high conductive material and the conductivity ratio, one can decide upon the best construct for maintaining optimal operation conditions of electronics. The figure 14.74 is a criterion to transition from the elemental area to the first order construct. Analysis shows that when the product of the conductivity ratio and the volume fraction of high conductive material to be used is higher than 14.74, then it makes sense to design electronics in accordance with the first order construct. If this product is lower than 14.74 then increasing the internal complexity of the electronics by transition from the elemental area to the first order construct will cause a negative effect on the performance of the electronics.
10
Elemental Area First Order Construct
12
14
16
18
20
kΦ Fig. 5. Variation of thermal resistance with respect to the value of ðk^/Þ for the elemental area and first order construct.
Thermal Resistance
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1
2 3 Assembly Order
827
4
Temperature Drop
Fig. 6. Variation of thermal resistance with respect to the order of the constructs.
0
1 2 3 Assembly Order
4
Fig. 7. Variation of temperature drop with respect to the order of the constructs.
The variations of thermal resistance and temperature drop with respect to the order of the constructs are shown in Figs. 6 and 7, respectively. The thermal resistance of the constructs shows an increasing trend, but with fluctuation, as the order of the constructs rises. Consequently, cooling of a larger area by increasing the internal complexity of the tree network of high conductive links will have a cost. In order to cool larger areas using constructal theory, one should be satisfied to pay for the increase in thermal resistance. Contrary to expectation, increasing the internal complexity of the tree network of high conductive links does not decrease the thermal resistance. This is an important conclusion of this paper. Fig. 7 shows that the temperature drop in the constructs increases as well if the internal complexity of the constructs increases. This is an expected situation because the temperature drop is proportional to the thermal resistance. The increase in thermal resistance causes the increase in temperature drop as the order of the constructs rises. References [1] Bejan A. The tree of convective heat streams: its thermal insulation function and the predicted 3/4-power relation between body heat loss and body size. Int J Heat Mass Transfer 2001;44:699–704. [2] Bejan A, Badescu V, De Vos A. Constructal theory of economics structure generation in space and time. Energy Convers Mgmt 2000;41:1429–51. [3] Ghodoossi L, Egrican N. Flow area optimization in point to area or area to point flows. Proceedings of ESDA2002, 6th Biennial Conference on Engineering Systems Design and Analysis. Istanbul, Turkey: 8–11 July 2002.
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[4] Ghodoossi L, E grican N. Flow area structure generation in point to area or area to point flows. Proceedings of ECOS2002, 15th International Conference on Efficiency, Costs, Optimization, Simulation and Environmental Impact of Energy Systems. Berlin, Germany: 3–5 July 2002. [5] Bejan A, Ledezma GA. Streets tree network and urban growth: Optimal geometry for quickest access between a finite-size volume and one point. Physica A 1998;255:211–7. [6] Rocha LAO, Lorente S, Bejan A. Constructal design for cooling a disc-shaped area by conduction. Int J Heat Mass Transfer 2002;45(8):1643–52. [7] Almogbel M, Bejan A. Constructal optimization of nonuniformly distributed tree-shaped flow structures for conduction. Int J Heat Mass Transfer 2001;44:4185–95. [8] Neagu M, Bejan A. Three-dimensional tree constructs of ÔconstantÕ thermal resistance. J Appl Phys 1999;86(12):7107–15. [9] Neagu M, Bejan A. Constructal-theory tree networks of ÔconstantÕ thermal resistance. J Appl Phys 1999;86(2):1136–44. [10] Almogbel M, Bejan A. Conduction trees with spacings at the tips. Int J Heat Mass Transfer 1999;42:3739–56. [11] Ledezma GA, Bejan A. Constructal three-dimensional trees for conduction between a volume and one point. J Heat Transfer 1998;120:977–84. [12] Bejan A. Constructal theory: From thermodynamic and geometric optimization to predicting shape in nature. Energy Convers Mgmt 1998;39(16–18):1705–18. [13] Bejan A. Constructal-theory network of conducting paths for cooling a heat generating volume. Int J Heat Mass Transfer 1997;40(4):799–816. [14] Bejan A. Shape and structure, from engineering to nature. Cambridge, UK: Cambridge University Press; 2000. p. 52–61. [15] Ghodoossi L, Egrican N. An exact solution for cooling of electronics using constructal-theory. J Appl Phys 2003;93(8):4922–9.
Conductive cooling of triangular shaped electronics using constructal theory Lotfollah Ghodoossi a
a,*
, Nil€ ufer E grican
b
Department of Mechanical Engineering, Istanbul Technical University, 80191 Gumussuyu, Istanbul, Turkey b Department of Mechanical Engineering, School of Engineering and Architecture, Yeditepe University, 26 Agustos Yerlesimi, Kayisdagi Caddesi 81120 Kayisdagi/Istanbul, Turkey Received 15 March 2003; accepted 16 July 2003
Abstract Conductive cooling of electronics falls in the category of the more general ‘‘area to point’’ flow problem. Heat generated in a fixed area is to be discharged to a heat sink located on the border of the heat generating area through relatively high conductive link(s). This will maintain a limited temperature difference between the hot spot inside the heat generating area and the heat sink. The solution procedure starts with heat transfer analysis and geometric optimization of the smallest heat generating area. Assembly of optimized smallest areas in a fixed but larger heat generating area by introducing a new high conductive link and geometric optimization of the new area leads to achieving the goal of conductive cooling of the larger area. The sequence of assembly of optimized areas in a relatively larger area and geometric optimization of this area is continued until the required area size to be cooled is obtained. The process of assembly and optimization steps leads to formation of a tree network of high conductive links inside the heat generating area. Along with geometric optimization of the heat generating area in each step, the tree network of high conductive links is optimized with respect to high conductive material allocation in the heat generating area as well. Ó 2003 Elsevier Ltd. All rights reserved. Keywords: Constructal design; Optimization; Tree networks; Electronics cooling; Heat transfer; Conduction
*
Corresponding author. Tel.: +90-212-2931-1300x2452; fax: +90-212-2450795. E-mail addresses: [email protected] (L. Ghodoossi), [email protected] (N. E grican).
0196-8904/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0196-8904(03)00190-0
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Nomenclature A AP D H k0 kP k^ L n q q000 T x, y
heat generating area area of high conductive links width of high conductive link height of heat generating area conductivity of electronic material conductivity of high conductive material conductivity ratio length of heat generating area number of constituents volumetric heat generation rate heat generation rate per unit volume temperature Cartesian coordinates
Greek symbol / volume fraction of high conductive material Subscripts i order of constructs min minimum opt optimum
1. Introduction Conductive cooling of electronics falls in the category of the more general ‘‘area to point’’ flow problem, which has applications in a wide range of science branches, such as biology [1], economics [2–4], urban transportation [5], heat transfer [6–11] etc [12]. Area to point flow studies in heat transfer problems [6–11] have mainly concentrated on conductive cooling of electronics based on constructal theory. Conductive cooling of electronics using constructal-theory was exposed and solved first by Bejan [13,14]. The solution of Bejan [13,14] to the conductive cooling of electronics using constructal theory was improved by Ghodoossi and Egrican [15]. The specific problem in the works of Bejan [13,14] and Ghodoossi and Egrican [15] is to cool a uniform heat generating fixed size area by distributing links of fixed amounts of high conductive material in the area. The links of high conductive material discharge the generated heat to a heat sink located on the border of the heat generating area. The objective is to minimize the thermal resistance of the heat generating area to maintain a limited maximum temperature in the area for the purpose of safe operation of the electronic device. Solution to the so defined problem starts with heat transfer analysis and geometric optimization of the smallest heat generating area. Assembly of optimized smallest areas in a fixed but larger heat generating area by introducing a new high conductive link and performing a double optimization process leads to achieving the goal of conductive cooling of the larger area. The new larger area is optimized with respect to both the geometry of the area and
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high conductive material allocation in the area. Assembly of this newly optimized larger area in a relatively much larger area by introducing another high conductive link and applying a double optimization process will provide conductive cooling of the much larger area. The sequence of assembly of optimized areas in relatively larger areas and double optimization of the areas is continued until the required area size to be cooled is obtained. The works done in Refs. [13–15] are solutions for the conductive cooling of electronics, all formed by assembly of ‘‘rectangular’’ heat generating areas. However, small and smart electronic devices may be manufactured not in ‘‘rectangular’’ form. Some other forms may be used because of aesthetic and compatibility requirements of the electronic devices. In this paper, the constructal solution method is applied to the problem of cooling of electronics formed of ‘‘triangular’’ heat generating areas.
2. Triangular elemental area Assume a triangular area of base H0 and height L0 , shown in Fig. 1, that generates heat at a constant rate q volumetrically. Heat generation is uniform such that the heat generation per unit volume is constant ðq000 ¼ q=ððH0 L0 =2Þx1Þ ¼ constantÞ. The area size A0 is constant, but the ratio, H0 =L0 is free to vary. The heat generated over the area is first conducted to a relatively high conductive link of width D0 , which is located on the axis of symmetry of the triangular elemental area, from which it is then channeled to a heat sink located at point M0 . It is assumed that the triangular elemental area is slender enough to have one dimensional (y-direction) heat conduction on the heat generating area. The boundary of the triangular elemental heat generating area is adiabatic except for the heat sink point located at point M0 . It is also assumed that the thermal conductivity of the high conductive link (kP ) is much higher than the thermal conductivity of the electronic material (k0 ) and the area occupied by the high conductive material is much smaller than the area of electronic material. The objective is to minimize the maximum thermal resistance between a point in the heat generating area and the heat sink point M0 . The maximum thermal resistance is defined in the form of DT =ðq000 AÞ where DT represents the temperature difference between the hot spot and the heat sink point. The governing differential equation for steady state one dimensional heat conduction with uniform heat generation in the electronic material is
Fig. 1. Triangular elemental area.
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o2 T q000 þ ¼0 oy 2 k0
ð1Þ
The boundary conditions for the electronic material are oT ¼0 oy
at y ¼ yb
T ¼ T0 ðxÞ
at y ¼ 0
ð2Þ ð3Þ
where yb denotes the ordinate of boundary points located on both sides of triangular elemental area, H0 x ð4Þ 1 yb ¼ L0 2 Integrating Eq. (1) subject to boundary conditions (2) and (3) will result in q000 2 q000 H0 x y þ T0 ðxÞ y þ 1 T ðx; yÞ ¼ L0 2k0 2k0
ð5Þ
Heat conduction along the high conductive link can be assumed as conduction in a fin. The amount of heat input from the electronic material to the high conductive link is a function of x. Performing an energy balance for a differential dx length at point x will result in the governing differential equation for steady-state one-dimensional heat conduction in the high conductive link as d2 T0 q000 H0 x ¼ 1 ð6Þ dx2 kP D0 L0 The boundary conditions for the high conductive link are dT0 ¼0 dx
at x ¼ L0
ð7Þ
T0 ¼ TM0
at x ¼ 0
ð8Þ
Integrating Eq. (6) subject to boundary conditions (7) and (8) will give the temperature distribution along the high conductive link as q000 H0 x3 x2 L0 ð9Þ T0 ðxÞ ¼ þ x þ TM0 kP D0 6L0 2 2 Substituting Eq. (9) in Eq. (5) will give the temperature distribution over the triangular elemental heat generating area. q000 2 q000 H0 x q000 H0 x3 x2 L0 ð10Þ T ðx; yÞ TM0 ¼ y þ 1 yþ þ x L0 2k0 2k0 kP D0 6L0 2 2 Eq. (10) is valid for y > 0. There is no need to derive the temperature distribution for y < 0 because the temperatures are equal for symmetrical points with respect to x axis.
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The position of the points involving maximum temperature is found by solving oT ðx; yÞ=ox ¼ 0 and oT ðx; yÞ=oy ¼ 0, simultaneously. Two points are found. The coordinates of these points are x1 ¼ L0 ;
y1 ¼ 0
ð11Þ
x2 ¼ L0
D0 H0 kP ; 2L0 k0
y2 ¼
D0 H02 kP 4L20 k0
ð12Þ
The first point shows the position of maximum temperature on the high conductive link that locates naturally farthest from the heat sink on the right edge of the high conductive link (point P0 ). The second point characterizes the position of the hot spot on the electronic material. The coordinates of the hot spot found in Eq. (12) show that this point lies exactly on the upper border of the triangular elemental heat generating area. Another geometric feature of hot spot is that it is very close to the first point as the triangular elemental area is slender ðH0 =L0 1Þ. The first and second points are not the same points mathematically, but it can be said that they are the same points physically. The maximum temperature difference between the hot spot and the heat sink ðDT0 Þ is derived by substituting the coordinates of the hot spot in Eq. (10). " # 2 q000 A0 D20 A30 kP 2L0 k0 þ ð13Þ DT0 ¼ 6k0 k0 D0 kP L80 Eq. (13) is nondimensionalized to obtain the maximum thermal resistance. Note that q000 , A0 and k0 are constants. " # 5 DT0 1 1 ^ 2 H0 2 L0 þ ð14Þ ¼ k /0 q000 A0 =k0 3 64 L0 k^/0 H0 where /0 ¼
AP0 D0 ¼2 A0 H0
ð15Þ
kP ð16Þ k^ ¼ k0 /0 and k^ are volume fraction of the high conductive material and conductivity ratio, respectively. An important feature of the heat transfer problem analyzed above is investigated now. The thermal resistance can be minimized geometrically. The thermal resistance in Eq. (14) is minimized with respect to the aspect ratio ðH0 =L0 Þ of the triangular elemental heat generating area. The results are !1=2 H0 27=6 1 ¼ ð17Þ L0 opt 51=6 k^/0 DT0;min 25=6 ¼ q000 A0 =k0 55=6
1 ^ k /0
!1=2 ð18Þ
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Using Eqs. (17) and (18), the minimized maximum temperature difference of the triangular elemental heat generating area is found as DT0;min ¼
1 q000 H02 24=3 52=3 k0
ð19Þ
The minimized thermal resistance of the rectangular elemental heat generating area in Refs. [13,14] is found as !1=2 DT0;min 1 1 ¼ ð20Þ q000 A0 =k0 RECT: 2 k^/0 By comparing Eq. (18) with Eq. (20), it is seen that the minimized thermal resistance of the triangular elemental area is approximately 7% lower than that of the rectangular elemental area.
3. First order assembly construct If the electronic material area to be cooled is larger than the triangular elemental area that is just optimized, what can be done? One way to overcome such a problem is to enlarge A0 subject to Eq. (17), but Eq. (19) shows that enlarging the size of the elemental area increases the maximum temperature difference, which puts us distant from the goal of maintaining the lowest maximum temperature difference in the electronic material. In contrast, Eq. (19) advises to design the elemental area in the smallest size that manufacturing restrictions allow. A more efficient way to enlarge the cooled area is shown in Fig. 2. A number of optimized triangular elemental areas are assembled on the upper and lower sides of a new high conductive link of width D1 . This new and larger area is called the first order assembly construct. Heat generated at any elemental electronic material area is first conducted to the associated D0 link, and then, the heat collected in the D0 link, is directed to the D1 link. Finally, the heat collected in the D1 link is conducted toward the heat sink located at point M1 . Once again, the boundaries of the heat generating areas are adiabatic except for the heat sink point M1 . The aspect ratio ðH1 =L1 Þ of the construct or, in other words, the number of constituents
Fig. 2. First order assembly construct.
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n1 inside the construct is free to vary. We will look for an optimal aspect ratio or an optimal number of constituents that minimize the maximum thermal resistance between any point in the new construct and the heat sink point M1 . Clearly, the maximum temperature occurs on the farthest elements of the new construct relative to the M1 heat sink point. The maximum temperature difference between hot spot P1 and the heat sink, DT1 , may be divided into two parts, the temperature difference between the points P1 and M1n01 and the temperature difference between the points M1n01 and M1 . DT1 ¼ DTM1n0 P1 þ DTM1 M1n0 1
ð21Þ
1
where n01 ¼ n1 =2
ð22Þ
DTM1n0 P1 is, in fact, the minimized maximum temperature difference of the elemental area calculated 1in Eq. (19). DTM1n0 P1 ¼ 1
1 q000 H02 24=3 52=3 k0
ð23Þ
Now, we will calculate DTM1 M1n0 the temperature difference along the D1 link. The temperature distribution1 along the D1 link is analyzed as a fin problem with heat input at points M11 ; M12 ; . . . ; M1n01 . The temperature distribution for steady-state one-dimensional heat conduction with no heat input between two following points M1;j1 and M1;j is governed by d2 T ¼0 dx2 The boundary conditions are T ¼ TM1;j1
at x ¼ ð2j 3Þ
ð24Þ
H0 ðpoint M1;j1 Þ 2
ð25Þ
dT H0 ð26Þ ¼ ½n1 2ðj 1Þq000 A0 at x ¼ ð2j 1Þ ðpoint M1;j Þ dx 2 Integrating Eq. (24) subject to the boundary conditions gives the temperature distribution for the M1;j1 M1;j interval. ½n1 2ðj 1Þq000 A0 H0 x ð2j 3Þ T TM1;j1 ¼ ð27Þ k P D1 2 kP D1
The temperature difference between the points M1;j1 and M1;j is found by substituting ð2j 1Þ H20 in place of x in Eq. (27), TM1;j TM1;j1 ¼
½n1 2ðj 1Þq000 A0 H0 kP D1
ð28Þ
Eq. (28) is valid for j P 2. Applying a similar procedure to the M1 M11 interval will result in TM11 TM1 ¼
n1 q000 A0 H0 kP D1 2
ð29Þ
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The temperature difference along the D1 link is equal to the sum of the temperature differences of all intervals on the D1 link, DTM1 M1n0 ¼ ðTM11 TM1 Þ þ 1
n1 =2 X ðTM1;j TM1;j1 Þ
ð30Þ
j¼2
By substituting Eqs. (28) and (29) in Eq. (30) and after some algebra, the temperature difference along the D1 link is found as DTM1 M1n0 ¼ 1
q000 A0 H0 2 n 4kP D1 1
ð31Þ
So far, the temperature difference along the D1 link is found. The maximum temperature difference of the first order assembly construct can be calculated by substituting Eqs. (23) and (31) in Eq. (21). DT1 ¼
1 q000 H02 q000 A0 H0 2 þ n 24=3 52=3 k0 4kP D1 1
ð32Þ
Similar to the case of the elemental area, the maximum temperature difference of the first order assembly construct is nondimensionalized to obtain the maximum thermal resistance. Note that q000 , A1 ðA1 ¼ n1 A0 Þ and k0 are constants. DT1 1 H02 1 H0 ¼ þ n1 000 4=3 2=3 q A1 =k0 2 5 A0 n1 4k^D1
ð33Þ
Eq. (33) shows that the maximum thermal resistance of the construct can be minimized with respect to the number of constituents n1 . The minimization results in 1=2 25=6 ^ D1 ð34Þ n1;opt ¼ 1=3 k 5 L0 1=2 DT1;min 1 1 ¼ H0 q000 A1 =k0 21=6 51=3 k^L0 D1
ð35Þ
Finally, the problem could be optimized with respect to the number of constituents inside the construct, but Eq. (35) shows that the problem can be optimized with respect to D1 , the width of the new high conductive link in the construct, as well. This is done below. The volume fraction of high conductive material in the construct is defined as /1 ¼
AP1 A1
ð36Þ
where AP1 is the total area of the high conductive material in the construct, AP1 ¼ D1 L1 þ n1;opt D0 L0
ð37Þ
Combining Eqs. (15) and (37) with Eq. (36) and taking into account that A1 ¼ n1;opt A0 will result in D1 ¼ L0 ð/1 /0 Þ
ð38Þ
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Placing Eq. (38) in Eq. (35) and using Eq. (17) will read DT1;min 2 1 ¼ 1=2 000 ^ q A1 =k0 5 k ½/0 ð/1 /0 Þ1=2
ð39Þ
The condition that minimizes the thermal resistance in Eq. (39) is /1 ¼ 2/0
ð40Þ
Inserting Eq. (40) into Eqs. (38) and (39) will give the optimal width of the new high conductive link and twice minimized thermal resistance of the construct, respectively. D1 51=6 ¼ 2=3 ðk^/1 Þ1=2 ð41Þ D0 opt 2 DT1;min;min 4 1 ¼ 1=2 5 k^/1 q000 A1 =k0
ð42Þ
The twice minimized maximum temperature difference of the construct is deduced from Eq. (42), 1 q000 H02 DT1;min;min ¼ 1=3 2=3 ð43Þ 2 5 k0 The optimal number of constituents inside the construct is calculated by importing Eq. (41) into Eq. (34) 21=3 1=2 n1;opt ¼ 1=3 ðk^/1 Þ ð44Þ 5 The optimal height and length of the construct are H1;opt ¼ 2L0 ¼ 21=2 51=4 A1=2 1
ð45Þ
n1;opt 21=2 1=2 H0 ¼ 1=4 A1 2 5 The optimal aspect ratio of the construct is obtained by dividing Eq. (45) by Eq. (46) H1 ¼ 51=2 L1 opt L1;opt ¼
ð46Þ
ð47Þ
The optimization process of the first order assembly construct is completed now. All optimal geometric features, minimized maximum thermal resistance and minimized maximum temperature difference of the construct are derived and known through Eqs. (41)–(47). Comparing Eq. (43) with Eq. (19) shows that the maximum temperature difference in the first order assembly construct is exactly doubled relative to the maximum temperature difference of the elemental area. This result was actually predictable because optimization of the problem in a completely mathematical sense without importing any approximations to the physical nature of the problem dictates the existence of an ‘‘equipartition principle’’ in the maximum temperature difference of the construct. This means that half of the total temperature drop in the construct must occur in the electronic material ðk0 Þ and the other half in the high conductive link D1 (kP material). The temperature drop for the electronic material ðDTM1n0 P1 Þ is known at the start of the 1
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optimization process. It is equal to the minimized maximum temperature difference of the elemental area calculated in Eq. (19). The optimization arranges the geometric parameters of the construct in such a way that the temperature drop along the D1 link is exactly equal to the temperature drop in the electronic material, which is known through Eq. (19). This means that as a result of the optimization, the temperature drop of the first order assembly construct will be doubled relative to the temperature drop of the elemental area. It is worth mentioning that the first order assembly construct, in this paper, is optimized mathematically and no approximation is made. Therefore, the total temperature drop, namely, ðq000 H02 =k0 Þ=ð21=3 52=3 Þ is divided into two equal parts as a result of the optimization. Half of the total temperature difference, ðq000 H02 =k0 Þ=ð24=3 52=3 Þ, is dropped along the M1n01 P1 path in the electronic material ðk0 Þ and the other half along the D1 link in the high conductive material ðkP Þ.
4. Second order assembly construct The analysis done at the beginning of Section 3 is valid here once again. In order to enlarge the area to be cooled, one may enlarge size of the elemental areas making the first order assembly construct, but Eqs. (19) and (43) show that this will raise the maximum temperature difference of the construct, which is an unwanted situation in this paper. Therefore, the procedure followed in Section 3 to make a large area will be used again. The new second order assembly construct, shown in Fig. 3, is composed of n2 number of optimized first order assembly constructs. A2 , the area size of the heat generating volumes in the new construct is fixed, but H2 =L2 , the aspect ratio of the construct or n2 , the number of constituents inside the construct are free to vary. The boundaries of the heat generating areas are adiabatic except for the heat sink point located at M2 . We will look for an optimal aspect ratio or an optimal number of constituents that minimize the maximum thermal resistance between any point inside the construct and the heat sink point M2 .
Fig. 3. Second order assembly construct.
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Similar steps will be followed as in the optimization of the first order assembly construct. The maximum temperature difference ðDT2 Þ between the point P2 , which involves the highest temperature in the construct, and the heat sink point M2 is expressed as DT2 ¼ DTM2n0 P2 þ DTM2 M2n0 2
ð48Þ
2
where n02 ¼ n2 =2
ð49Þ
DTM2n0 P2 is equal to the minimized maximum temperature difference of the first order assembly 2 construct calculated in Eq. (43). DTM2n0 P2 ¼ 2
1 q000 H02 21=3 52=3 k0
ð50Þ
By applying similar calculation to that outlined in Eqs. (24)–(31), the temperature difference along the D2 link is found as DTM2 M2n0 ¼ 2
n1 q000 A0 H1 2 n 4kP D2 2
ð51Þ
The maximum temperature difference of the second order assembly construct is found by substituting Eqs. (50) and (51) in Eq. (48). DT2 ¼
1 q000 H02 n1 q000 A0 H1 2 þ n 21=3 52=3 k0 4kP D2 2
ð52Þ
Eq. (52) is nondimensionalized using q000 , A2 ðA2 ¼ n2 A1 ¼ n2 n1 A0 Þ and k0 , that are constants, to obtain the maximum thermal resistance of the construct. DT2 2 1 1 L0 ¼ 1=2 þ n2 ^ ^ 5 k /0 n2 2k D2 2 =k0
q000 A
ð53Þ
The maximum thermal resistance is minimized with respect to n2 , the number of constituents inside the construct. The results are 1=2 2 D2 ð54Þ n2;opt ¼ 1=4 5 /0 L0 DT2;min 2 1 ¼ 1=4 000 q A2 =k0 5 k^
L0 /0 D2
1=2 ð55Þ
Optimization with respect to the number of constituents is followed by optimization with respect to D2 , the width of the new high conductive link in the second order assembly construct. The volume fraction of high conductive material in the construct is defined as /2 ¼
AP2 A2
where AP2 is the total area of high conductive material in the construct.
ð56Þ
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AP2 ¼ n2 AP1 þ D2 L2
ð57Þ
Combining Eqs. (15), (36) and (57) with Eq. (56) results in 1 H0 ð/2 2/0 Þðk^/0 Þ1=2 51=3 Placing Eq. (58) in Eq. (55) and rearranging will read D2 ¼
21=6
DT2;min 1 ¼ 21=2 000 q A2 =k0 k^½/0 ð/2 2/0 Þ1=2
ð58Þ
ð59Þ
The condition that minimizes the thermal resistance in Eq. (59) is /2 ¼ 4/0
ð60Þ
The optimal number of constituents in the construct is found by combining Eqs. (60) and (58) with Eq. (54). n2;opt ¼
4 51=2
ffi 1:8
ð61Þ
This result is very interesting. It says that the optimal number of constituentspinside the second ffiffiffi order assembly construct is not a large number, but it is exactly equal to 4= 5 and it is independent of the physical properties and geometric values. It is clear that from an engineering point of view, the number of constituents cannot be a fractional figure. Therefore, the optimal number of optimized first order assembly constructs making the second order assembly construct should be 2. Among the two degrees of freedom, namely, the number of constituents n2 and the width of the high conductive link D2 , that exist in the problem, one of them, that is, the number of constituents in the second order assembly construct is known now. So, the problem should be optimized with respect to D2 only. The optimization steps in Eqs. (56)–(60) with respect to D2 are repeated subject to n2 ¼ 2. The minimized thermal resistance of the construct is found as pffiffiffi pffiffiffi 2 DT2;min;n2 ¼2 ð2 2 þ 5Þ 1 1 pffiffiffi ¼ ffi 2:8677 ð62Þ ^ ^ q000 A2 =k0 4 5 k /2 k /2 Note that the length of the D2 link is not equal to L2 , see Fig. 4. Because of the shortage of high conductive material, it is cut at point M21 . The same is applied for all constructs with 2 or 4 constituents analyzed in this paper. Optimal geometric features for n2;opt ¼ 2 are derived as H2;opt ¼ 2L1;opt ¼
2
1=2
A2
ð63Þ
L2;opt ¼ H1;opt ¼ 51=4 A1=2 2
ð64Þ
H2 L2
¼ opt
2 51=2
51=4
ð65Þ
L. Ghodoossi, N. E grican / Energy Conversion and Management 45 (2004) 811–828
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Fig. 4. Second order assembly construct with 2 number of constituents.
The optimal width of the high conductive link for n2;opt ¼ 2 is D2 ¼ 23=2 D1 opt
ð66Þ
The minimized maximum temperature difference of the construct for n2;opt ¼ 2 is found as DT2;min ¼
23=2 þ 51=2 q000 H02 211=6 52=3 k0
ð67Þ
The minimized maximum temperature difference of the second order assembly construct in Eq. (67) is 23=2 þ 51=2 =23=2 ffi 1:79 times higher than the minimized maximum temperature difference of the first order assembly construct in Eq. (43). This ratio was expected to be equal to 2 (equipartition principle) if the construct could be optimized mathematically, but the problem could not be optimized in a completely mathematical sense as the number of constituents was intentionally changed from the optimal value of 1.8 to the best possible value of 2.
5. Third order assembly construct To see what happens next, a third order assembly construct is analyzed. Optimization of the third order assembly construct requires following the same steps as in the first and second order assembly constructs. The third order assembly construct is optimized with respect to both the number of constituents n3 and the width of the high conductive link D3 . The optimization leads to ð68Þ n3;opt ¼ 23=2 þ 51=2 =21=2 ffi 3:58
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It is understood that the best third order assembly construct is the one with the number of constituents equal to either 2 or 4. Optimization of the third order assembly construct with n3 ¼ 2 and 4 results in 3=2 2 2 þ 51=2 þ 2 DT3;min;n3 ¼2 1 1 ¼ ffi 2:7899 ð69Þ 000 3 1=2 ^ ^ q A3 =k0 2 5 k /3 k /3 3=2 2 2 þ 51=2 þ 213=12 31=2 DT3;min;n3 ¼4 1 1 ¼ ffi 3:7996 000 19=6 1=2 ^ ^ q A3 =k0 2 5 k /3 k /3
ð70Þ
Comparing Eq. (69) with Eq. (70) shows that the thermal resistance for n3 ¼ 2 is lower than that for n3 ¼ 4. So, the best third order assembly construct is the one composed of 2 optimized second order assembly constructs. The optimal geometric features, optimal width of the high conductive link and minimized maximum temperature difference of the third order assembly construct for n3 ¼ 2 are recorded below. 1=2
H3;opt ¼ 2L2;opt ¼ 21=2 51=4 A3 L3;opt ¼ H2;opt ¼
H3 L3 D3 D2
21=2 1=2 A 51=4 3
ð71Þ ð72Þ
¼ 51=2
ð73Þ
¼2
ð74Þ
23=2 þ 51=2 þ 2 q000 H02 211=6 52=3 k0
ð75Þ
opt
opt
DT3;min ¼
minimized maximum temperature difference of the third order assembly construct has risen The 23=2 þ 51=2 þ 2 = 23=2 þ 51=2 ffi 1:39 times relative to the minimized maximum temperature difference of the second order assembly construct.
6. Fourth order assembly construct The fourth order assembly is the last construct studied in this paper. A double optimization with respect to the number of constituents n4 and the width of the high conductive link D4 leads to n4;opt ¼
23=2 þ 51=2 þ 2 ffi 2:23 21=2 51=2
ð76Þ
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Optimization of the fourth order construct with n4 ¼ 2 and 4 results in 3=2 2 2 þ 3:51=2 þ 2 DT4;min;n4 ¼2 1 1 ¼ ffi 3:7201 000 4 1=2 q A4 =k0 2 5 k^/4 k^/4
825
ð77Þ
3=2 2 2 þ 51=2 þ 2 þ 23=2 31=2 51=2 DT4;min;n4 ¼4 1 1 ¼ ffi 4:5376 ð78Þ 000 5 1=2 ^ ^ q A4 =k0 2 5 k /4 k /4 The best fourth order assembly construct is the one with n4 ¼ 2 for which the thermal resistance is lower. The optimal geometric features, optimal width of the high conductive link and minimized maximum temperature difference of the fourth order assembly construct for n4 ¼ 2 are recorded below. 2 1=2 ð79Þ H4;opt ¼ 2L3;opt ¼ 1=4 A4 5 1=2
L4;opt ¼ H3;opt ¼ 51=4 A4 H4 2 ¼ 1=2 L4 opt 5
D4 D3
ð80Þ ð81Þ
¼2
ð82Þ
opt
23=2 þ 3:51=2 þ 2 q000 H02 ð83Þ 211=6 52=3 k0 minimized maximum temperature difference of the fourth order assembly construct has risen The 23=2 þ 3:51=2 þ 2 = 23=2 þ 51=2 þ 2 ffi 1:63 times relative to the minimized maximum temperature difference of the third order assembly construct. Proceeding in assembly orders will be continued until the required area size to be cooled is achieved. On the other hand, the maximum temperature difference rises as the assembly order goes up. So, the temperature limit of the electronic material is another criterion to stop proceeding in assembly orders. DT4;min ¼
7. Discussion and conclusion An important and distinctive specification of constructal theory is that every detail of the constructs can be anticipated in a completely deterministic manner. This specification of constructal theory is investigated in this paper. All internal and external geometric details of the constructs are derived and known. None of the geometric details is assumed in advance. All are derived as a result of optimization. Accordingly, the thermal resistance and temperature drop of the constructs are calculated at optimum conditions. The optimization process determines the best number of constituents in each construct as well. The interesting result is that the optimization leads to a large number of constituents in the first
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Thermal Resistance
order construct, but this number remains equal to 2 for all higher order constructs. That is, a bifurcation appears for the second and higher order constructs. Bifurcation has also appeared in other constructal works [13,15] related to conductive cooling of electronics. This coincidence pushes a search for the generality of bifurcation in constructal solutions for area to point flow problems. The equipartition principle is another important feature of optimization applied to the constructal solution in this paper. Mathematical optimization of the problem requires the existence of equipartition for the temperature drop of the constructs. That is, half of the total temperature drop of an optimal construct must be realized in the electronic material and the other half in the high conductive material. This is the case in the first order assembly construct that is optimized in a completely mathematical sense, but the equipartition principle no longer holds in the higher order constructs because changing the optimal fractional number of constituents to a nonoptimal, but the best, even integer disturbed the mathematical optimization of these constructs. If the optimal numbers of the constituents were not changed, the temperature drop of the constructs would be doubled in going from an assembly order to the next. In order to compare the thermal resistance in the various constructs, the results must be introduced on the same basis and equal conditions. It is supposed that the volume fraction of high conductive material is the same for all constructs. Therefore, in the expressions for minimal thermal resistance /i in the ith order construct is replaced with a unique /. This enables us to perform the following discussion. The variation of thermal resistance with respect to the value of ðk^/Þ for the elemental area and first order construct is shown in Fig. 5. For ðk^/Þ < 14:74, the thermal resistance of the elemental area is lower than that of the first order construct. For ðk^/Þ > 14:74, the situation is reversed, and the first order construct produces the lower thermal resistance. Therefore, depending on the amount of high conductive material and the conductivity ratio, one can decide upon the best construct for maintaining optimal operation conditions of electronics. The figure 14.74 is a criterion to transition from the elemental area to the first order construct. Analysis shows that when the product of the conductivity ratio and the volume fraction of high conductive material to be used is higher than 14.74, then it makes sense to design electronics in accordance with the first order construct. If this product is lower than 14.74 then increasing the internal complexity of the electronics by transition from the elemental area to the first order construct will cause a negative effect on the performance of the electronics.
10
Elemental Area First Order Construct
12
14
16
18
20
kΦ Fig. 5. Variation of thermal resistance with respect to the value of ðk^/Þ for the elemental area and first order construct.
Thermal Resistance
L. Ghodoossi, N. E grican / Energy Conversion and Management 45 (2004) 811–828
1
2 3 Assembly Order
827
4
Temperature Drop
Fig. 6. Variation of thermal resistance with respect to the order of the constructs.
0
1 2 3 Assembly Order
4
Fig. 7. Variation of temperature drop with respect to the order of the constructs.
The variations of thermal resistance and temperature drop with respect to the order of the constructs are shown in Figs. 6 and 7, respectively. The thermal resistance of the constructs shows an increasing trend, but with fluctuation, as the order of the constructs rises. Consequently, cooling of a larger area by increasing the internal complexity of the tree network of high conductive links will have a cost. In order to cool larger areas using constructal theory, one should be satisfied to pay for the increase in thermal resistance. Contrary to expectation, increasing the internal complexity of the tree network of high conductive links does not decrease the thermal resistance. This is an important conclusion of this paper. Fig. 7 shows that the temperature drop in the constructs increases as well if the internal complexity of the constructs increases. This is an expected situation because the temperature drop is proportional to the thermal resistance. The increase in thermal resistance causes the increase in temperature drop as the order of the constructs rises. References [1] Bejan A. The tree of convective heat streams: its thermal insulation function and the predicted 3/4-power relation between body heat loss and body size. Int J Heat Mass Transfer 2001;44:699–704. [2] Bejan A, Badescu V, De Vos A. Constructal theory of economics structure generation in space and time. Energy Convers Mgmt 2000;41:1429–51. [3] Ghodoossi L, Egrican N. Flow area optimization in point to area or area to point flows. Proceedings of ESDA2002, 6th Biennial Conference on Engineering Systems Design and Analysis. Istanbul, Turkey: 8–11 July 2002.
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[4] Ghodoossi L, E grican N. Flow area structure generation in point to area or area to point flows. Proceedings of ECOS2002, 15th International Conference on Efficiency, Costs, Optimization, Simulation and Environmental Impact of Energy Systems. Berlin, Germany: 3–5 July 2002. [5] Bejan A, Ledezma GA. Streets tree network and urban growth: Optimal geometry for quickest access between a finite-size volume and one point. Physica A 1998;255:211–7. [6] Rocha LAO, Lorente S, Bejan A. Constructal design for cooling a disc-shaped area by conduction. Int J Heat Mass Transfer 2002;45(8):1643–52. [7] Almogbel M, Bejan A. Constructal optimization of nonuniformly distributed tree-shaped flow structures for conduction. Int J Heat Mass Transfer 2001;44:4185–95. [8] Neagu M, Bejan A. Three-dimensional tree constructs of ÔconstantÕ thermal resistance. J Appl Phys 1999;86(12):7107–15. [9] Neagu M, Bejan A. Constructal-theory tree networks of ÔconstantÕ thermal resistance. J Appl Phys 1999;86(2):1136–44. [10] Almogbel M, Bejan A. Conduction trees with spacings at the tips. Int J Heat Mass Transfer 1999;42:3739–56. [11] Ledezma GA, Bejan A. Constructal three-dimensional trees for conduction between a volume and one point. J Heat Transfer 1998;120:977–84. [12] Bejan A. Constructal theory: From thermodynamic and geometric optimization to predicting shape in nature. Energy Convers Mgmt 1998;39(16–18):1705–18. [13] Bejan A. Constructal-theory network of conducting paths for cooling a heat generating volume. Int J Heat Mass Transfer 1997;40(4):799–816. [14] Bejan A. Shape and structure, from engineering to nature. Cambridge, UK: Cambridge University Press; 2000. p. 52–61. [15] Ghodoossi L, Egrican N. An exact solution for cooling of electronics using constructal-theory. J Appl Phys 2003;93(8):4922–9.