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Optimum Profiles for AsymmetricalLongitudinal Fins in Annular DuctsGiampietro FabbriPublished online: 29 Oct 2010.
To cite this article: Giampietro Fabbri (2000) Optimum Profiles for Asymmetrical Longitudinal Fins in AnnularDucts, Heat Transfer Engineering, 21:2, 40-52, DOI: 10.1080/014576300271013
To link to this article: http://dx.doi.org/10.1080/014576300271013
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Heat Transfer Engineering , 21:40–52, 2000Copyright C°° 2000 Taylor & Francis0145–7632/00 $12.00 + .00
Optimum Profilesfor AsymmetricalLongitudinal Finsin Annular Ducts
GIAMPIETRO FABBRIDipartimento di Ingegneria Energetica, Nucleare e del Controllo Ambientale,Universita degli studi di Bologna, Italy
In the present work the geometry of annular ducts with asymmetrical longitudinal � ns is optimizedin order to enhance the heat transfer under laminar coolant � ow conditions. The heat transferred isalso maximized for a given amount of material or hydraulic resistance. Polynomial pro� les areassigned to the two lateral � n surfaces. Velocity and temperature distributions on the annular ductcross section are determined with the help of a � nite-element model. A global heat transfercoef� cient and an equivalent Nusselt number are then calculated. Lastly, optimum asymmetrical � nsobtained by means of a genetic algorithm are shown for different situations and their performance iscompared with those of optimum symmetrical � ns.
To enhance heat transfer, � nned surfaces are com-monly used in various engineering sectors. The elec-tronic industry [1] or the compact heat exchanger sec-tor [2], for example, employ � nned surfaces to removehigher heat � uxes from more and more miniaturizedcomponents. Therefore, in the last years, the need to re-duce the volume and the weight of � nned dissipators hasbecome more and more important. Many researchershave studied the problem of optimizing the shape of the� nned surfaces in order to increase the heat transfer ef-fectiveness and reduce the dimensions and the weightof thermal dissipator systems.
To this aim, a variety of � n pro� les has been stud-ied. Since the 1920s, parabolic, triangular, and undulatepro� les have been proposed for longitudina l � ns [3–7].Some of them [8–14] have been demonstrated as hav-
Address correspondenc e to Dott. Ing. Giampietro Fabbri, D. I. E. N.C. A., Via Zannoni 452, 40134 Bologna, Italy. E-mail: [email protected]
ing a noticeably improved effectiveness under particularconditions , but a de� nitive solution to the problem ofoptimizing the pro� le of the � n has not yet been foundfor many situations .
Different factors in� uence the heat transfer effec-tiveness of � nned surfaces. The most important factorsare the � n conductance, the extent of the heat transfersurface between the solid and the convection � uid, andthe local heat transfer coef� cient. In many cases, eachof these factors depends on the others. To enhance theconductance of the � ns for a given value of thermalconductivity, it is necessary to increase the thicknessand reduce the height. On the other hand, the extent ofthe � n must be augmented to increase the heat trans-fer surface. Therefore, if the material or weight whichis available for the � n is constrained, the � n thicknessmust be reduced. Lastly, a complex relationship existsbetween the local heat transfer coef� cient and the shapeor the spacing of the � ns.
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The dependence of the local convective heat trans-fer coef� cient on the spacing and height of longitudi-nal � ns cooled by a laminar � ow has been studied bymany authors [15–17]. Under these conditions , it hasbeen observed that the local heat transfer coef� cientchanges greatly along the lateral pro� les of the � ns, de-pending on the � n height and the distance between the� ns. In the case of a shrouded array of rectangular � nsthe local heat transfer coef� cient is less variable whenthe height or the distance is shorter. Moreover, the lo-cal convection coef� cient at the � n base is reduced bydecreasing the � n height or spacing. Similar relation-ships between the laminar convection coef� cient andthe � n height or spacing exist for longitudina l � ns withstraight lateral pro� les located in cylindrical and annu-lar ducts.
For longitudina l � ns located in plain, cylindrical, andannular ducts, the effect of changes in the lateral pro� leon the local laminar convective heat transfer coef� cienthas been considered in recent works of ours [18–20].In these works, aimed at solving the problem of opti-mizing the lateral pro� le of the � ns for given volume orhydraulic resistance of the duct, the local heat transfercoef� cient on the � nned surface has been demonstratedas being very sensitive to the � uid dynamic conditionsdetermined by the � n pro� le. As a consequence, theimprovements in the � nned duct heat transfer effec-tiveness, which can be obtained by optimizing the � npro� le, depend much more on the heat transfer coef� -cient increment than on the � n surface extension or theconductance enhancement.
As in our previous works [18–20], in most of thestudies performed on the optimization of the � n shape,longitudina l � ns which have symmetrical lateral pro-� les have been investigated . The symmetry of the lateralpro� les simpli� es the treatment of the problem with re-gard to the boundary conditions assignment and to theextent of the domain studied. Symmetry conditions al-low zero momentum and heat � uxes to be imposed anda less extended domain to be considered. This is par-ticularly convenient when the solution is obtained in anumerical way, to reduce the number of nodes or ele-ments. Nevertheless, in many situations the adoption ofsymmetrical � ns does not provide the best solution interms of heat transfer effectiveness. This is particularlytrue when the � n volume is constrained and the localheat transfer coef� cient is very sensitive to the shape ofthe � ns. For a given volume, asymmetrical � ns provide,in fact, a more extended heat transfer surface and allowmore appropriate shapes for the channels between the� ns. In the present work, we then continue the study[20] of the optimization of the heat dissipation througha � nned annular duct into a � uid in laminar � ow by in-vestigating the improvements which can be obtained in
different situations by assigning an asymmetrical crosssection to the � ns.
Longitudinal � ns with asymmetrical cross sectionsare not particularly onerous to produce. In fact, � nnedwalls with polynomial lateral pro� le � ns as consideredin [20] can be obtained by extrusion of melted metalthrough an appropriate die. An external cursor can besubsequently employed to rectify the wall and give the� nal shape to the � n. The same technology can be uti-lized to realize asymmetrical polynomial lateral pro-� les. In this case, to make the manufacturing easier, anexcessive displacement in the angular direction of the� n middle line is to be avoided. The effective conve-nience of asymmetrical � ns with respect to symmet-rical ones must be evaluated by considering the costsof preparing, installing, and substituting the dies andcursors.
THE MATHEMATICAL MODEL
Let us consider an annular duct composed of an inter-nal � nned wall and an external adiabatic smooth surfaceas in Figure 1a. All � ns are identical and have an asym-metrical cross section. The � nned wall is crossed by aheat � ux q 0 0 which is uniformly imposed on its internalsurface. A coolant passes through the duct in laminar� ow.
Figure 1 Finned tube geometry: (a) cross section; (b) subdivisionof a portion of the cross section in � nite elements.
heat transfer engineering vol. 21 no. 2 2000 41
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Let us choose a cylindrical coordinate system whosez axis is coincident with the duct axis and is directed asthe coolant � ow. Since all � ns are identical, the thermalstate of the system is a periodic function of the angu-lar coordinate q . The heat transfer performances of thesystem can then be determined by studying a portion ofit delimited by the lines \ 1 and \ 2 which pass throughthe middle points of two adjacent � ns as shown in Fig-ure 1b.
Let Ri be the inner radius of the internal wall, Re
the radius of the external surface, a the � n height in theradial direction, s the un� nned wall thickness, and 2 b
the angle between \ 1 and \ 2. Moreover, let the angularcoordinate q be equal to f1(r) on the lateral pro� le ofthe lower half-� n of Figure 1b and to 2 b ¡ f2(r) on thelateral pro� le of upper half-� n, f1(r ) and f2(r) beingarbitrary functions of the radial coordinate r .
Let us suppose that the system is in steady state, nat-ural convection is negligible compared to the forcedone, the � uid properties are uniform, the viscous dissi-pation within the coolant is negligible, and velocity andtemperature pro� les are fully developed. Under suchconditions the velocity vector v is parallel to the z axis.The coolant � ow is then described by the followingequation:
1
r
¶¶ r (r ¶ v
¶ r)+1
r 2
¶ 2v
¶ q 2 =1
l
dp
dz(1)
where l is the dynamic viscosity and p is a general-ized pressure, which includes the gravitation potential.Equation (1) must be integrated by imposing as bound-ary conditions that velocity is zero on the contact surfacebetween � uid and solid and
v[r, x 1(r )] = v [r, x 2(r)]
8 Ri + s + a · r · Re (2)
[¶ v¶ N][r, x 1(r )]
= [¶ v¶ N][r, x 2(r)]
8 Ri + s + a · r · Re (3)
x 1(r) and x 2(r) being functions which provide thevalue of the angular coordinate in \ 1 and \ 2, respec-tively, and N the coordinate which is normal to the twolines.
Since the temperature pro� le is fully developed andthe conductive heat � ux in the z direction is constant andcan be neglected in thermal balances, the temperaturedistribution in the coolant is described by the following
equation:
1
r
¶¶ r (r ¶ Tc
¶ r )+1
r 2
¶ 2Tc
¶ q 2 =q cp
kcv
¶ Tc
¶ z(4)
where kc is the thermal conductivity, q is the density andcp is the speci� c heat of the coolant. The temperaturedistribution in the � nned wall is instead described byLaplace’s equation:
1
r
¶¶ r (r ¶ T f
¶ r )+1
r 2
¶ 2T f
¶ q 2 = 0 (5)
Equations (4) and (5) must be integrated by imposingas boundary conditions that temperature and heat � uxesin normal to the surface direction are identical in � uidand solid on the contact surface, heat � ux per unit ofsurface in radial direction is equal to q 0 0 on the smoothsurface of the � nned wall and is zero on the insulatedexternal surface, and
Tc[r, x 1(r )] = Tc[r, x 2(r)]
8 Ri + s + a · r · Re (6)
T f [r, x 1(r )] = T f [r, x 2(r)]
8 Ri · r · Ri + s + a (7)
[¶ Tc
¶ N ][r, x 1(r)]= [¶ Tc
¶ N][r, x 2(r )]
8 Ri + s + a · r · Re (8)
[¶ T f
¶ N ][r, x 1(r)]= [¶ T f
¶ N ][r, x 2(r )]
8 Ri · r · Ri + s + a (9)
The temperature in one point of the section must alsobe imposed.
Velocity and temperature distribution s can be conve-niently determined, as in [20], by using a � nite-elementmethod. The portion of the cross section of the � nnedannular duct can be subdivided in an array of elementsdelimited by two concentric arches and two segments(Figure 1b). By joining the middle points of the oppositesides, each element can be subdivided into four subele-ments. Moreover, in each element the velocity and thetemperature can be approximated by an interpolation ofthe values (vi and ti , respectively) which they assume
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in the four nodes at the corners of the element:
v(r, q ) = ^i
Fi (r, q ) vi t(r, q ) = ^i
Fi (r, q ) ti
(10)
Fi being form factors.Let us suppose that, in the � rst instance, conditions
(2), (6), and (7) are not necessarily veri� ed and momen-tum and heat � uxes through \ 1 and \ 2 are zero. In thesame way as in [20], from a balance of the pressure andthe viscous forces which act on the subelements whichsurround each knot and from a balance of the conduc-tive and convective heat � uxes, the following systemscan be derived:
A ¤ V =1
l
dp
dzB (11)
C ¤ T = D (12)
A and C being momentum and heat transfer matrices,respectively, V, B, and T vectors containing the veloc-ity, the total transversal surface, and the temperatureassociated to each node, and
D =q 0 0
kc(Ri 2 b
w tE ¤ V ¡ L) (13)
where w t is the total volume � ow rate through the con-duit section, E is a surface-averaging matrix, and L isa vector containing the total perimeter crossed by q 0 0
which is associated to each node [20].Conditions (2)–(3) and (6)–(9) can now be imposed
by modifying matrices A and C and vectors V, T, B,and D in the following way:
A = (A11 + A12 + A21 A13 + A23
A31 + A32 A33)
(14)
C = (C11 + C12 + C21 C13 + C23
C31 + C32 C33)
V = (V1
V3) T = (T1
T3) (15)
B = (B1 + B2
B3) D = (D1 + D2
D3) (16)
where indices 1 and 2 refer to the nodes on the lines\ 1 and \ 2, respectively, and index 3 refers to all othernodes. In this way velocities and surfaces of the nodeson \ 2 are eliminated from system (11) and their contri-butions to viscous and pressure forces are attributed tovelocities and surfaces of the nodes on \ 1. Similarly,
temperatures, surfaces, and perimeters of the nodes on\ 2 are eliminated from system (12) and their contri-butions to convective and conductive heat � uxes areattributed to the nodes on \ 1.
By distinguishin g nodes on the contact surface be-tween � uid and solid, in which velocity is known, thefollowing system is then obtained:
Auu ¤ Vu =1
l
dp
dzBu (17)
where index u refers to the nodes in which the velocityis unknown. Since known velocities are zero, their con-tribution has been omitted. By solving system (17) thevelocity distribution in the portion of the � nned annularduct section is determined.
Let us assign an arbitrary temperature to a node. Bydistinguishin g this node from the others, whose temper-ature is unknown, the following system is obtained:
Cuu ¤ Tu = Du ¡ Cun ¤ Tn (18)
index n referring to the node whose temperature isknown. By solving system (18) the temperature distri-bution in the portion of the cross section is determinedas a function of Tn . The value of this temperature doesnot in� uence the heat transfer characteristics, becausethe system is linear.
At this stage, bulk temperature, global heat transfercoef� cient, equivalent Nusselt number, and normalizedhydraulic resistance can be de� ned as in [20] and cal-culated as follows:
Tb =1
w t^i
^k
eikvk tk (19)
h =q 0 0
Tmax ¡ Tb(20)
Nue =h2(Re ¡ Ri )
kc(21)
z =( ¡ dp / dz)
p w t / b /8 l
p R4i ( d 4 ¡ 1) f 1 ¡ [1/ (ln d )][( d 2 ¡ 1)/ ( d 2 + 1)]g
(22)
where the summation index i is extended to all nodesof the coolant and Tmax is the maximum temperature,which obviously occurs on the internal surface of the� nned wall. Parameter z indicates how many times thehydraulic resistance of the annular duct increases dueto the presence of the � ns.
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GEOMETRY OPTIMIZATION
Theproblem of optimizing the geometry of the � nnedannular duct independently from its size can be stud-ied by referring the geometric parameters to the innerradius:
d =Re
Ria =
a
Rir =
s
Rig =
r
Ri
(23)u 1( g ) = f1( g Ri ) u 2( g ) = f2( g Ri )
To the pro� le functions u 1 and u 2 a polynomial formcan be assigned as in [20]. If n1 and n2 are the poly-nomial orders, the two functions are univocally deter-mined by the values u 1i and u 2i which they assume inn1 + 1 and n2 + 1 points equally spaced in the radialdirection between g = 1 + r and g = 1 + r + a . Ob-viously, the rotation of the coordinate system does notaffect the thermal behavior of the � nned annular duct.Therefore, the origin of q coordinate can be chosen inorder to let the two pro� le functions be equal at the be-ginning of the � n ( g = r ). In this way u 10 results inbeing equal to u 20 and the number of describing pa-rameters can be reduced by one. Moreover, two � nnedannular ducts whose cross sections are specular presentthe same thermal performance. The number of possi-ble � n pro� les can then be reduced by imposing, forexample, that the derivative of u 1 at the � n base is notnegative.
The problem now consists of � nding the combina-tions of parameters d , a , b , r , u 1i , and u 2i which allowthe maximum Nue to be obtained respecting some con-straining conditions . A genetic algorithm [11, 14, 21]similar to that of [20] can be used to solve such a prob-lem. The following changes must be made.
After parameter reproduction the � n can be too thinor too thick. If this occurs, parameters u 1i or u 2i mustbe resized. Let u 3( g ) be the sum of u 1( g ) and u 2( g ),u 3min and u 3max the minimum and the maximum val-ues which u 3( g ) assumes for g between 1 + r and1 + r + a . In order to let u 3( g ) be no less and no greaterthan limit functions q min( g ) and q max( g ), respectively,u 1i can be changed in the following way:
u 3i =
ì ïïïíïïïî
u 3max ¡u 3max ¡ q min( g min)
u 3max ¡ u 3( g min)[u 3max ¡ u 3(1 ¡ a i / n1)] if u 3( g min) < q min( g min)
8 i = 0, 1, . . . , n1
u 3min +q max( g max) ¡ u 3min
u 3( g max) ¡ u 3min[u 3(1 ¡ a i / n1) ¡ u min] if u 3( g max) > q max( g max)
(24)
ˆu 1i = u 1i + u 3i ¡ u 3(1 ¡a i
n1) (25)
where g min and g max are the values of g at the pointswhere the maximum of q min ¡ u 3 and u 3 ¡ q max, re-spectively, occur and ˆu 1i are the new parameter values.
RESULTS
Optimized annular duct geometries with asymmetri-cal � ns have been found for different situations . In somecases, constraints have been imposed on the hydraulicresistance z and on the normalized average thicknessof the � nned wall ¯r , which can be de� ned as the nor-malized thickness of an un� nned cylindrical wall of thesame cross sectional area as the � nned one. To com-pare the performance of the optimized asymmetrical� ns with those of optimized symmetrical � ns, some op-timizations have been also carried out by assigning thesame pro� le to the two � n lateral surfaces. As in [20],the solution of the optimization problem has been lim-ited to the case of d equal to 2.
For the � nite-element model, a grid of 32 £ 52 el-ements (33 £ 53 nodes) was employed. To investigatethe grid dependence of the model predictions, the sametests as in [20] have been carried out, obtaining the sameresults again.
In the genetic algorithm, populations of 20 samplesand a selection percentage equal to 20 were established.Samples were reproduced by copying their parameterswith random errors which were uniformly distributedbetween ¡ 10% and +10% of the parameter values.The genetic algorithm was stopped after 100 genera-tions from the time in which an improvement was nolonger observed. As a prototype, an annular duct hasbeen employed whose � ns had � rst-order polynomialpro� les. The sum of u 1( g ) and u 2( g ) was equal to 0.5 b
or to a value depending on the available solid volumeand x 2(Ri g ) varied linearly between 0 at the base and0.25 b at the tip of the � ns. An asymmetrical prototypewas employed to let the algorithm escape from sym-metrical solutions which often provided local maximain Nue. Moreover, by maintaining n1 + n2 + 2 � n pro-� le describing parameters and imposing no constrainton the derivatives of the pro� le functions at the � n base,it has been possible to make the algorithm faster and to
avoid its stopping in correspondence with asymmetricalsolutions providing local maxima in Nue.
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Figure 2 Finned tube geometries which maximize Nue when r + a is 0.8, b and k are p / 16, and c is 500.
To ensure the structural integrity of the � nned wall,the following constraints have been imposed in the ge-netic algorithm:
r ¸ g s (26)
u 1( g ) + u 2( g ) ¸ q min( g ) = k s / g (27)
Moreover, to ensure a uniformdistribution of the coolantin the duct, the following constraints have been im-posed:
a + r = g d (28)
u 1( g ) + u 2( g ) · q max( g ) = b ¡k d
g(29)
The latter constraint has been imposed since, as ob-served in [20], the algorithm tried to extend a or r inorder to create separated narrow channels. Lastly, toavoid an excessive displacement in angular direction ofthe � n middle line, which could create problems in pro-duction, the following constraint has been introduced:
maxg
[x 1(Ri g )] ¡ ming
[x 1(Ri g )] · k (30)
In order to compare performances of symmetricaland asymmetrical � ns, parameters g s , k s , g d , and k d
have been chosen in accordance with [20]:
g s = 0.05 k s = 0.05 g d = 0.8 k d = 0.05
(31)
Moreover, k has been chosen equal to b or 2 b .
Some � nned annular duct geometries which maxi-mize Nue are shown in Figure 2 for n1 equal to n2 andranging from 1 to 4, b and s equal to p / 16, and c equalto 500. The value chosen for c corresponds to the caseof a � nned wall made of copper and cooled by water.No constraint has been imposed on the available solidvolume nor on the hydraulic resistance. The describingparameters of these optimum geometries are reportedin Table 1 together with the average � nned tube thick-ness, the equivalent Nusselt number, and the normal-ized hydraulic resistance. The table also contains thecharacteristic parameters of the optimum geometriesof annular ducts with asymmetrical � ns, which will bediscussed below. Geometries obtained with the geneticalgorithm have been rotated in order to let u 10 be equalto u 20.
The optimized asymmetrical � ns of Figure 2 pro-vide greater equivalent Nusselt numbers than optimizedsymmetrical � ns with polynomial pro� les of the sameorder under the same conditions [20]. Relative differ-ences are not very large, particularly when the polyno-mial order is low. However, equivalent Nusselt num-bers of second- and third-order asymmetrical � ns arehigher than those of third- and fourth-order symmet-rical � ns, respectively. The greatest improvement be-tween the heat transferred through asymmetrical andsymmetrical � ns of the same order is obtained in cor-respondence with a polynomial order equal to 3. Inthis case, for asymmetrical � ns the equivalent Nus-selt number is nearly 7% higher than for symmetrical� ns. Referring to the heat � ux which can be dissipatedthrough an annular duct with optimum zero-order � ns,asymmetrical third-order � ns provide a 62% increment,while symmetrical third-order � ns provide a 52% incre-ment. Therefore, the adoption of an asymmetrical cross
heat transfer engineering vol. 21 no. 2 2000 45
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Tabl
e1
Cha
ract
eris
tic
para
met
ers
ofso
me
opti
mum
geom
etri
esof
annu
lar
duct
sw
ithas
ymm
etri
cal�
ns(c
ases
are
list
edin
the
sam
eor
der
they
are
disc
usse
din
the
text
)
kb
cn 1
n 2¯r
ra
u10
u11
u12
u13
u14
u21
u22
u23
u24
Nu e
z
p/1
6p
/16
500
11
0.49
70.
050.
750.
0471
0.34
46—
——
¡0.
0388
——
—13
4.2
35.8
1p
/16
p/1
650
02
20.
506
0.05
10.
749
0.10
250.
1839
0.35
77—
—¡
0.00
27¡
0.03
36—
—13
8.7
36.2
3p
/16
p/1
650
03
30.
486
0.05
30.
747
0.07
570.
1071
0.23
630.
3359
—0.
0648
¡0.
0474
0.00
78—
147.
336
.1p
/16
p/1
650
04
40.
479
0.05
10.
749
0.11
590.
0857
0.17
40.
2475
0.35
870.
0606
0.03
36¡
0.04
92¡
0.01
9915
1.3
34.9
5p
/8p
/850
04
40.
605
0.07
90.
721
0.20
580.
2729
0.49
950.
6317
0.73
920.
1539
0.05
56¡
0.05
14¡
0.01
2810
9.4
45.3
p/2
4p
/24
500
44
0.37
20.
050.
750.
0738
0.03
460.
0687
0.12
320.
235
0.03
720.
0225
¡0.
0457
¡0.
0155
178.
232
.81
p/1
6p
/16
500
22
0.2
0.05
10.
749
0.04
06¡
0.05
450.
1705
——
0.09
58¡
0.05
54—
—77
.74
9.22
p/1
6p
/16
500
44
0.2
0.05
0.75
0.06
52¡
0.06
74¡
0.12
81¡
0.03
350.
1304
0.10
960.
1707
0.06
750.
0936
81.8
10.9
3p
/16
p/1
650
02
20.
30.
050.
750.
0603
¡0.
0734
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8
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Figure 3 Velocity distributions in the cross section in the casesof Figure 2. Curves are drawn every 10% of the maximum velocityfrom 10% to 90%.
section allows nearly a 20% enlargement in the im-provement in the heat transfer of third-order polynomialpro� le � ns.
In the four cases of Figure 2, the increments whichcan be obtained in Nue by increasing the polynomialorder depend on two main factors. Higher-order asym-metrical � ns, as well as symmetrical ones, allow bettervelocity distribution so that higher velocity and temper-ature gradients are induced near the � nned wall surface(Figures 3 and 4). Moreover, they present a more ex-tended heat transfer surface. In particular, third- andfourth-order asymmetrical lateral pro� les create curvesin the channels between the � ns, extending their lengthwithout excessively reducing their width. On the otherhand, the extension of the channel height also reducesthe � n cross-sectional area. Therefore, when the heattransfer surface is already suf� ciently extended, anincrement can be less effective. For this reason thedifference between the equivalent Nusselt numbers ofthird- and second-order � ns is considerably higher thanbetween the equivalent Nusselt numbers of fourth- andthird-order � ns.
Referring to the performances of symmetrical � ns,the improvements of asymmetrical � ns are larger whenthe number of � ns is lower. Figure 5 shows the ge-ometries of annular ducts with 8 and 24 asymmetri-cal fourth-order � ns which maximize Nue under thesame conditions of Figure 2 and k equal to b . In thesame cases the equivalent Nusselt numbers of the opti-mum symmetrical � n geometries are equal to 100.3 and
Figure 4 Temperature distributions in the cross section in thecases of Figure 2. Curves are drawn every 10% of the differencebetween maximum and minimum temperature from 10% to 90%.
175.2, respectively. Therefore, the adoption of asym-metrical lateral pro� les provide a 9% improvement inthe � rst case and only a 1.7% improvement in the sec-ond. Moreover, for a number of � ns equal to 32, ithas been found that symmetrical � ns perform the best.This result can be explained by considering that, whenthe number of � ns is large, � ns are thin. Therefore,the extension of the length of the channels in the an-gular direction negligibly increases the heat transfer
Figure 5 Finned tube geometries which maximize Nue whenr + a is 0.8, b and k are p / 8 and p / 24, and c is 500.
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Figure 6 Finned tube geometries which maximize Nue when r + a is 0.8, b and k are p / 16, c is 500 and ¯r is constrained to differentvalues.
surface and noticeably reduces the conductance ofthe � n.
In the above-considered cases, optimum asymmetri-cal � ns require less material than optimum symmetri-cal � ns of the same order. When the � nned wall solidvolume is constrained, the improvements in the heattransfer of asymmetrical � ns increase. For n1 and n2
both equal to 2 and 4, b and k equal to p / 16, c equalto 500, and ¯r constrained to three different values, the� nned annular duct geometries which maximize Nue
are shown in Figure 6. In the six cases, referring tothe equivalent Nusselt numbers of the optimum sym-metrical � ns of the same order [20], asymmetrical � nsprovide increments ranging from 10.4% (n1 = n2 = 2,¯r = 0.4) to 26.6% (n1 = n2 = 2, ¯r = 0.2). Whenthe normalized average � nned wall thickness is con-strained to 0.2, referring to the heat � ux which canbe dissipated through an annular duct with optimumzero-order-pro� le � ns under the same conditions [20],asymmetrical second-order � ns provide a 41.8% incre-ment, while symmetrical third-order � ns provide a 12%increment. The adoption of an asymmetrical cross sec-tion in this case allows a nearly 250% enlargement ofthe improvement in the heat transfer of second-orderpolynomial pro� le � ns. Moreover, it is interesting toobserve that, when the normalized average � nned wallthickness is constrained to 0.4, fourth-order asymmet-rical � ns provide a 83.2% increment in the heat dissi-pated, referring to that removed by zero-order-pro� le� ns [20].
Performance of symmetrical and asymmetrical opti-mum � ns are compared together in Figures 7a and 7bfor unconstrained and constrained solid volume. More-over, maximum equivalent Nusselt numbers obtainedfor n1 and n2 ranging independently from 1 to 4 arereported in Figure 7c.
When the available solid volume is constrained, largerimprovements in the heat transfer can be obtained with
asymmetrical � ns by allowing a greater angular dis-placement of the � n middle line. Figure 8 shows the� nned annular duct geometries which maximize Nue
when both n1 and n2 are equal to 2 and 4, c is equalto 500, ¯r is constrained to 0.4, b is equal to p / 16, andk is equal to p / 8. Referring to the equivalent Nusseltnumber of symmetrical � ns of the same order, in thiscase, second- and fourth-order asymmetrical � ns pro-vide 23% and 22% increments, respectively. These in-crements are more than twice those obtained by lettingk be equal to b under the same conditions (10.4% and11%, respectively). Moreover, it must be noticed that,referring to the equivalent Nusselt number of optimumzero-order-pro� le � ns under the same conditions (75.88[20]), the fourth-order geometry of Figure 8 provides a100% increment.
When the ratio between thermal conductivity of solidand � uid is a magnitude order lower, improvements inthe heat transfer of asymmetrical � ns are more limited.Figure 9 shows the geometries which maximize Nue
when both n1 and n2 are equal to 2 and 4, b and k areequal to p / 16, ¯r is constrained to 0.4, and c is equal to50. Referring to the equivalent Nusselt number of opti-mum symmetrical � ns of the same order under the sameconditions, in this case, second- and fourth-order asym-metrical � ns provide only 5.2% and 5.6% increments,respectively. However, second-order asymmetrical � nsperform almost as well as fourth-order symmetrical � ns.
Optimum annular duct geometries with symmetricaland asymmetrical � ns have been found by constrain-ing the normalized hydraulic resistance to be no greaterthan 10, 15, 20, 25, 30 for both n1 and n2 equal to 4,b and k equal to p / 16, and c equal to 500. In all casesthe value of z in the geometry obtained with the ge-netic algorithm is equal to the constraining value. Threeof the optimum geometries with asymmetrical � ns areshown in Figure 10. In the � ve cases, the equivalentNusselt numbers of the optimum symmetrical � ns have
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Figure 7 Comparison between performances of optimum geometries with symmetrical (dashed line) and asymmetrical (continuous line)� ns when r + a is 0.8, b and k are p / 16, c is 500, and ¯r is unconstrained (±) and constrained to 0.4 (*), 0.3 ( £ ), and 0.2 (+). EquivalentNusselt numbers (a, c) and percental increments (b) refer to the equivalent Nusselt number of the zero-order � n pro� le geometry of the samevolume class.
been found as being equal to 80.15, 100.6, 117.4, 132.8,and 141.8, respectively. Referring to these values, op-timum asymmetrical � ns provide 5.0%, 8.15%, 8.18%,5.6%, and 5.0% increments. When z is constrained tobe no greater than 20, referring to the equivalent Nusselt
Figure 8 Finned tube geometries which maximize Nue when ¯r isconstrained to 0.4, r + a is 0.8, c is 500, b is p / 16, and k is p / 8.
number of the optimum zero-order pro� le � ns (82.49),optimum symmetrical and asymmetrical � ns provide42.3% and 54% increments, respectively. Under suchconditions , the adoption of an asymmetrical cross sec-tion then allows a 28% enlargement of the improvement
Figure 9 Finned tube geometries which maximize Nue whenr + a is 0.8, b and k are p / 16, c is 50 and ¯r is constrained to 0.4.
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Figure 10 Finned tube geometries which maximize Nue whenr + a is 0.8, b and k are p / 16, c is 500, and z is constrained todifferent values.
in the heat transfer of fourth-order polynomial pro� le� ns.
REMARKS
The results obtained demonstrate that, in � nned an-nular ducts cooled by a � uid in laminar � ow, it is pos-sible to noticeably improve the heat transfer perfor-mances of polynomial lateral pro� le � ns [20] in somesituations by assigning to them an asymmetrical crosssection. The improvement depends on the maximum al-lowed displacement between the points of the � n middleline in the angular direction. For example, in the case ofan annular duct internal � nned wall made of copper andcooled by water with an average thickness equal to 0.4times the inner radius and with 16 � ns, it was demon-strated [20] that the optimum fourth-order polynomialpro� le provides a 65% increment in the dissipated heat� ux with respect to that removed by assigning to the� ns a zero-order pro� le and the same amount of mate-rial. Under the same conditions , it has been found thatoptimum asymmetrical � ns provide more than 80% and100% increments when the maximum � n middle linedisplacement in the angular direction is equal to half ofthe angle between the � n ( b ) and to the whole angle(2 b ), respectively.
Referring to the thermal performances of symmetri-cal polynomial � ns, the improvements of asymmetrical� ns are larger when the number of � ns is low, and tend to
vanish when this parameter increases. In particular, un-der the considered conditions , for a 32-� n annular duct,no improvement has been obtained. Moreover, improve-ments of asymmetrical � ns increase when the amountof available material is constrained and decreases whenthe ratio between the thermal conductivity in solid andin � uid is reduced.
Referring to the thermal performances of zero-order-pro� le � ns, the improvements of asymmetrical � ns de-pend on the available solid volume, the ratio between thethermal conductivity in solid and � uid, and the numberof � ns in the same way as the improvements of higher-order symmetrical polynomial � ns.
In general, optimum asymmetrical � ns require lessmaterial and present a lower hydraulic resistance thansymmetrical polynomial � ns of the same order do, whenthe available solid volume is not constrained. On thecontrary, when the available solid volume is constrained,the hydraulic resistance of symmetrical � ns is lower.
The improvements in the heat transfer which havebeen observed by increasing the polynomial order ofasymmetrical � ns are due to two main factors. A betterdistribution of velocity and temperature is induced in thechannels between the � ns. Moreover, a more extendedheat transfer surface is created between solid and � uid.In particular, in third- and fourth-order � ns, one or bothpro� les are wavy, depending on the available material.The shape of these � ns allows curves to be created in thechannels and the heat transfer surface to be extended.
CONCLUSIONS
The problem of maximizing the heat dissipationthrough a � nned annular duct into a � uid in laminar� ow has been studied by assigning asymmetrical poly-nomial lateral pro� les to the longitudina l � ns. A geneticalgorithm has been used to optimize the � nned annularduct geometry for given volume of the wall or hydraulicresistance. Optimum pro� les have been found for dif-ferent polynomial orders, number of � ns, and ratio be-tween the thermal conductivity of the � nned wall andthe coolant � uid.
NOMENCLATURE
a height of the � ns, m (ft)A momentum transfer matrixB transversal surface vector, m2 (ft2)cp coolant speci� c heat, J/kg K (Btu/lb ±F)C heat transfer matrixD heat source vector, K (±R)ei k elements of matrix E, m2 (ft2)E surface averaging matrix, m2 (ft2)
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f1, f2 angular coordinate values on the � n pro� lesas a function of r , rad
Fi form factorsh global heat transfer coef� cient, W/m2 K
(Btu/hr ft2 ±F)kc thermal conductivity of the coolant, W/m K
(Btu/hr ft ±F)L vector containing the element perimeters
crossed by q 0 0 , m (ft)n1, n2 � n pro� le polynomial orderN coordinate normal to the contour linesNue equivalent Nusselt numberp generalized pressure, N/m2 (lbf/in.2)q 0 0 heat � ux per unit of surface, W/m2
(Btu/hr ft2)r radial coordinate, m (ft)Ri � nned wall inner radius, m (ft)Re external surface radius, m (ft)s un� nned wall thickness, m (ft)t temperature, K (±R)ti temperature of the i th node, K (±R)T temperature vector, K (±R)Tb bulk temperature of the coolant, K (±R)Tc temperature of the coolant, K (±R)T f temperature of the � nned wall, K (±R)Tmax maximum temperature on the � nned wall in-
ternal surface, K (±R)v coolant velocity, m/s (ft/s)V velocity vector, m/s (ft/s)vi coolant velocity of the i th node, m/s (ft/s)w t total coolant volume � ow rate, m3 / s (ft3 / s)z longitudina l coordinate, m (ft)a normalized height of the � nsb half-angle between middle lines of two ad-
jacent � ns, radc ratio between � nned wall and coolant ther-
mal conductivityd ratio between Re and Ri
g normalized radial coordinateg d constraint value for a + r imposed to ensure
structural integrityg s lower limit value for r imposed to ensure
uniform coolant distributionz normalized hydraulic resistanceq angular coordinate (rad)k d lower limit value for the normalized arch be-
tween the pro� les of two adjacent � ns, im-posed to ensure uniform coolant distribution
k s lower limit value for the normalized arch be-tween the pro� les of a � n, imposed to ensurestructural integrity
l dynamic viscosity, kg/m s (lb/ft s)q coolant density, kg/m3 (lb/ft3)r normalized un� nned wall thickness
¯r normalized average wall thicknessk maximum allowed displacement in angular
direction of the � n middle line, radu 1, u 2 angular coordinate values on the � n pro� les
as a function of g , radu 1i , u 2i � n pro� le describing parametersx 1, x 2 angular coordinate values in \ 1 and \ 2, re-
spectively, as a function of r , rad\ 1, \ 2 contours of the domain studied
Superscripts and Subscripts for Matrices and Vectors
ˆ modi� ed to impose periodicity conditions1 refers to the nodes on the contour line \ 1
2 refers to the nodes on the contour line \ 2
3 refers to the nodes which are not on the con-tour lines \ 1 and \ 2
n indicates known elementsu indicates unknown elements
REFERENCES
[1] Bar-Cohen, A., and Kraus, A. D., Advances in Thermal Mod-eling of Electronic Components and Systems, chap. 3, ASMEPress Series, New York, 1990.
[2] Kays, W. M., and London, A. L., Compact Heat Exchangers,3d ed., chap. 1, McGraw-Hill, New York, 1984.
[3] Schmidt, E., Die Warmeuebertragung durch Rippen, Ver. Dt.Ing., vol. 70, pp. 885–951, 1926.
[4] Duf� n, R. J., A Variational Problem Relating to Cooling Fins,J. Math. Mech., vol. 8, pp. 47–56, 1959.
[5] Maday, C. J., The Minimum Weight One-Dimensional StraightFin, ASME J. Eng. Ind., vol. 96, pp. 161–165, 1974.
[6] Snider, A. D., and Kraus, A. D., The Quest for the OptimumLongitudinal Fin Pro� le, Heat Transfer Eng., vol. 8, no. 2, pp.19–25, 1987.
[7] Tsukamoto, Y., and Seguchi, Y., Shape Optimization Problemfor Minimum Volume Fin, Heat Transfer Jpn. Res., vol. 13,pp. 1–19, 1984.
[8] Snider, A. D., Kraus, A. D., Graff, S., Rodriguez, M., andKusmierczyk, A. G., Optimal Fin Pro� les. Classical and Mod-ern, Proc. 9th Int. Heat Transfer Conf., Jerusalem, vol. 4, pp.15–19, 1990.
[9] Aziz, A., Optimum Dimensions of Extended Surfaces Oper-ating in a Convective Environment, Appl. Mech. Rev., vol. 45,no. 5, pp. 155–173, 1992.
[10] Spiga, M., and Fabbri, G., Ef� cienza di dissipatori a pro-� lo sinusoidale, Proc. 12th UIT Natl. Congr., L’Aquila, Italy,pp. 197–204, 1994.
[11] Lorenzini, E., Spiga, M., and Fabbri, G., A Polynomial Fin Pro-� le Optimization, Int. J. Heat and Technol., vol. 12, no. 1–2,pp. 137–144, 1994.
[12] Fabbri, G., and Lorenzini, G.,Analisi numerica bidimensionaledi dissipatori a pro� lo sinusoidale,Proc. 13th UIT Natl.Congr.,Bologna, Italy, pp. 491–499, 1995.
[13] Aziz, A., and Kraus, A. D., Optimum Design of Radiating andConvecting-Radiating Fins, Heat Transfer Eng., vol. 17, no. 3,pp. 44–78, 1996.
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[14] Fabbri, G., A Genetic Algorithm for Fin Pro� le Optimization,Int. J. Heat Mass Transfer, vol. 40, no. 9, pp. 2165–2172, 1997.
[15] Sparrow, E. M., Baliga, B. R., and Patankar, S. V., ForcedConvection Heat Transfer from a Shrouded Fin Array withand without Tip Clearance, ASME J. Heat Transfer, vol. 100,pp. 572–579, 1978.
[16] Masliyah, J. H., and Nandakumar, K., Heat Transfer in In-ternally Finned Tubes, ASME J. Heat Transfer, vol. 98, pp.119–128, 1976.
[17] Soliman, H. M., Chau, T. S., and Trupp, A. C., Analysis ofLaminar Heat Transfer in Internally Finned Tubes with Uni-form Outside Wall Temperature, ASME J. Heat Transfer, vol.102, pp. 598–604, 1980.
[18] Fabbri, G., Optimization of the Heat Transfer through FinnedDissipator Cooled by Laminar Flow, Int. J. Heat Fluid Flow,vol. 19, no. 6, pp. 644–654, 1998.
[19] Fabbri, G., Heat Transfer Optimization in Internally FinnedTubes under Laminar Flow Conditions, Int. J. Heat MassTransfer, vol. 41, no. 10, pp. 1243–1253, 1998.
[20] Fabbri, G., Heat Transfer Optimization in Finned AnnularDucts under Laminar Flow Conditions, Heat Transfer Eng.,vol. 19, no. 4, pp. 42–54, 1998.
[21] Queipo, N., Devarakonda, R., and Humphrey, J. A. C., GeneticAlgorithms for Thermosciences Research: Application to theOptimized Cooling of Electronic Components, Int. J. HeatMass Transfer, vol. 37, no. 6, pp. 893–908, 1994.
Giampietro Fabbri performs theoretical and ex-perimental research on heat transfer and � uid dy-namics at the Department of Energetic, Nuclearand Environment Control Engineering of the Uni-versity of Bologna, Italy. He earned the B.S.-M.S.in Electronic Engineering in 1988 from the Uni-versity of Bologna and the Ph.D. in Bioegineeringin 1993 from the same university. Since 1992 hehas been a University Researcher in Nuclear En-
gineering. Since 1996 he has been teaching � uid dynamics at the Universityof Bologna.
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