Constructal entransy dissipation rate minimization for a heat generating volume cooled by forced convection

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1 Article Engineering Termopysics September 011 Vol56 No7: doi: /s Constructal entransy dissipation rate minimization or a eat generating volume cooled by orced convection XIAO QingHua, CHEN ingen * & SUN FengRui College o Naval Arcitecture and Power, Naval University o Engineering, Wuan 30033, Cina Received Marc 1, 011; accepted July 5, 011 Te geometry o a eat generating volume cooled by orced convection is imized by applying te entransy dissipation extremum principle and constructal teory, wile te imal spacing between te adjacent tubes and te imal diameter o eac tube are obtained based on entransy dissipation rate minimization Te results o tis wor sow tat te imal constructs based on entransy dissipation rate minimization and maximum temperature dierence minimization, respectively, are clearly dierent For te ormer, te porosity o te volume o cannels allocated to te eat generating volume is 1/; wile or te latter, te larger te porosity is, te better te perormance will be Te imal construct o te ormer greatly decreases te mean termal resistance and improves te global eat transer perormance o te system compared wit te imal construct o te latter Tis is identical to te essential requirement o te entransy dissipation extremum principle tat te required eat transer temperature dierence is minimal wit te same eat transer rate (te given amount o eat generated in te eat generating volume based on te entransy dissipation extremum principle entransy dissipation extremum principle, constructal teory, orced convection, entransy dissipation rate, generalized termodynamic imization Citation: Xiao Q H, Cen G, Sun F R Constructal entransy dissipation rate minimization or a eat generating volume cooled by orced convection Cinese Sci Bull, 011, 56: , doi: /s *Corresponding autor ( lgcenna@yaoocom; lingencen@otmailcom Te necessity to cool electronic devices escalates wit te continued increase in cip integration level, pacaging density and operating requency, all leading to a steep increase in te generated eat density Nowadays, te eat transer imization teory involves constructal teory [1 16], entropy generation minimization teory [11,1,17,18] and te ield synergy principle [19 6] Since Bejan studied te development o te street networ [1], put orward te constructal teory and applied it to imization problems involving eat conduction [3], constructal teory as been developing rapidly in simply low systems [7 37] Ordonez [7] considered a inite volume tat consisted o many eat generating components wit uniorm rate o internal eat generation and imized te equidistant internal low cannels tat were cooled by natural and orced convection, by using te minimization o maximum temperature dierence as te objective Bello-Ocende and Bejan [8] enanced te eat transer rate by canging te sape o te cannels in te eat generating volume By taing te maximum eat transer rate as te imization objective and air as te eat transer medium, Matos et al [9,30] aded staggered inned circular and elliptic tubes or orced convection and obtained te imal spacing between te tubes Wals and Grimes [31] studied te structures o te low cannels to obtain te minimum requirements or orced convection cooling solutions Muzyca et al [3] discussed te eat transer perormances o dierent sapes o micro cannels involving parallel plates, rectangles, ellipses, triangles and polygons including teir squareness and roundness Robbe and Sciubba [33] studied te imal internal cooling geometry o a prismatic slab, and considered our inds o distribution layout o te cannels and teir comparative diameters Kim et al [3] studied dendritic vascularization or countering intense Te Autor(s 011 Tis article is publised wit open access at Springerlincom csbscicinacom wwwspringercom/scp

2 Xiao Q H, et al Cinese Sci Bull September (011 Vol56 No7 967 eating rom te side to control te maximum temperature o te system, and urter considered te transient beavior o vascularized walls exposed to sudden eating [35] Roca et al [36] put orward a constructal imization metod or designing tree-saped vascular walls in multilayer slabs Wang et al [37] designed tree-saped vascularization and developed a metod to control te maximum temperature o te eat generating volume using noting more tan natural convection Res [7 37] only relected te temperature limitation o te ins wit eat transer rate maximization (maximum termal resistance minimization Guo et al [6,38] proposed a new pysical quantity called entransy tat describes te eat transer ability (Re [39] reerred to it as eat transer potential capacity and te entransy dissipation extremum principle to relect global eat transer perormance Te pysical meaning o entransy was urter expounded troug researc into, or example, pysical mecanisms o eat conduction and electro-termal simulation experiments [0,1] Many scolars [ 59] ave sown great interest in and ave also studied all inds o eat transer imizations based on entransy dissipation Wei et al [60 65], Xie et al [66 69] and Xiao et al [70 7] combined te entransy dissipation extremum principle wit constructal teory and conducted constructal imizations or a series o eat transer problems Re [7] only required tat te temperature o te electrical devices encapsulation sould not exceed te temperature limitation, and did not consider ow to most eectively remove te eat generated Consequently, based on Re [7], tis paper will combine constructal teory wit te entransy dissipation extremum principle and re-imize te uniormly eated volume cooled by orced convection, to obtain te imal construct corresponding to te minimization o entransy dissipation rate (ie best eat transer perormance, and compare it wit te construct corresponding to te minimization o maximum temperature dierence Te entransy dissipation extremum principle proposed by Guo et al [38] is stated as ollows: or a ixed boundary eat lux, te eat transer process is imized wen te entransy dissipation is minimized (minimum temperature dierence; wile or a ixed boundary temperature, te eat transer process is imized wen te entransy dissipation is maximized (maximum eat lux Te eat transer problem belongs to te entransy dissipation extremum principle wit a ixed boundary eat lux Te smaller te entransy dissipation rate, te better te eat transer eect Tat is, wen te entransy dissipation rate is minimized, te mean termal resistance in te eat transer process will be minimized and te perormance o te system will be imal Structure o te eat generating volume As sown in Figure 1, a inite volume tat consists o many eat generating components can be regarded as a cuboid control-volume wit a uniorm rate o internal eat generation Te cuboid volume as a rontal area A, lengt and eat generation rate q (W/m 3 in a solid material o termal conductivity s Te eat generated is removed rom te volume by te coolant in multiple cannels cooled by orced convection Te coolant is a single pase luid wit constant speciic eat c p and termal conductivity Te spacing between tube axes is S, and eac tube as a lengt and a diameter d One assumes tat te arrangement o tube centers is square as sown in Figure 1 Te eat generated inside te saded square wit side S is conducted entirely into te low cannel situated at its center Tis saded square element is based on te area A as constructed Te square perimeter 1 Deinition o entransy dissipation rate Entransy is a new pysical quantity relecting te eat transer ability o an object as deined in [38] Te entransy dissipation o a wole volume is deduced as [38] E E dv q Td, v (1 vφ v φ were q is te termal current density vector, and T is te temperature gradient Te equivalent termal resistance or multi-dimensional eat conduction problems wit a speciied eat lux boundary condition is given as ollows [38]: R E Q, ( v vφ were Q is te termal current Figure 1 Model o eat generating volume [7]

3 968 Xiao Q H, et al Cinese Sci Bull September (011 Vol56 No7 is adiabatic For ease o analysis [7], te square element can be substituted by a cylindrical element o equivalent diameter D as in Figure 1, and tus, πd /S and te number o cylindrical elements is A n (3 π D Te cylindrical perimeter is adiabatic wit te same temperature Te total eat generated is π q nq ( D d ( Anoter simpliying assumption is tat eac cylindrical element is suiciently slender (D < so tat te axial conduction is negligible Tat is, te eat generated in eac annular cross-section (q π(d d / is conducted radially in a solid sell o ticness (D d/ Te eat lux q troug te tube wall into te luid can be determined by te ollowing equation: q nq πd (5 1 arge-diameter limit Te cylindrical element or tis extreme condition is sown in Figure Te tube diameter d is suiciently greater tan te boundary layer ticness o te coolant so tat te temperature dierence between te luid and te tube wall is negligible Te core region o te luid is still as cold as at te inlet Te conduction dierential equation in te solid sell is 1 d + + 0, r dr T d T q dr s wit te boundary conditions: dt s q and d r T T ( x, (7 r d/ w r d/ were T w (x is te temperature o te tube wall By solving te equation one can obtain: q D r r d T( x, r Tw ( x ln + (8 s d 8 (6 Because te eat lux in te tube wall can be regarded as uniorm, te temperature dierence o te tube wall is [8] 08qx 5 Tw( x Tmin ( Pr 05, Re 5 10, 1/3 1/ x (9 Pr Re x were Pr v/α, Re x Ux/v, and U serves as ree stream velocity to te boundary layer [7]: /3 1/ 1/ dδp U 656ρν Te temperature dierence o te cylindrical element is T( x, r, θ T min r 1/3 1/ d + D ln r qd Be x 3058 d +, /3 1/6 A s d ( D d (10 (11 Δ were Be P is te dimensionless group pressure drop μα [7] Te total entransy dissipation rate o te wole volume is vφ π 0 D 0 0 E n q qd A r d + D ln r ( D d 1/3 1/ Be x d /3 1/6 d s qd AD d d /3 d + 8 s ( ( /3 1/3 3 ln 039 D d Be 1/3 D d D + D d rddd r x θ (1 vφ vφ Its corresponding mean termal resistance R and mean eat transer temperature dierence ΔT are, respectively, presented as E / q and E / q Te mean termal resistance and mean eat transer temperature dierence are nondimensionalized as R R ( A 1 + φ( 1 φ /3 1/3 039 Be K, (13 Figure Model o cylindrical element wit large-diameter limit [7] ΔT Δ T R, q ( A (1

4 Xiao Q H, et al Cinese Sci Bull September (011 Vol56 No7 969 were φ d /D is te porosity o te volume o cannels allocated to te total cuboid volume, and K s ( 1 φ 1/ lnφ 1/ 1 (15 Te actor in square bracets in eq (15 is a 1st order number In most applications / s << 1 [7], so K can be expected to be considerably smaller tan 1 Eq (13 conirms te expectation tat te dimensionless mean termal resistance eeps increasing wit increasingly larger diameter d (or D Small-diameter limit Te cylindrical element or tis extreme condition is sown in Figure 3 Te tube is suiciently slender, so tat its lengt is covered mainly by ully developed low and te Nu number result [8] (ie Nu q x d/ 36 in te tube wit uniorm eat lux is obtained According to te Newton cooling equation, one can obtain te temperature dierence between te tube wall and te core o te luid at any axial position q q d Tw( x T ( x (16 36 Te irst law o termodynamics yields te dierential equation: π duρcpdt πdq d x, (17 wit te boundary conditions: x T, 0 Tmin x (18 In eq (17 te mean velocity U is [7] U Δ 3μ d P Solving eq (17 yields te temperature o te luid: 3 18q T x x+ T ( 3 dbe min (19 (0 Te temperature o te solid material o te cylindrical element as te same orm as in eq (8 According to eqs (8, (16 and (0 one can obtain te temperature dierence o te cylindrical element D q T ( x, r,θ A r d + D log r 1 18 d + x + 36 Bed s ( D d E as + Tmin (1 Te total entransy dissipation rate o te wole volume is vφ D π Dq q A n r d + D log 1 18 r d + x + 36 Bed s ( D d q 6 3 ln Aφ( 1 φ Be + 8( 1 φ s rddd rxθ φ φ d ( Its corresponding mean termal resistance R is presented E vφ q Te mean termal resistance is nondimensionalized as R q E vφ ( A φ( φ K 36 Be (3 From te above equation, i te ollowing inequality is valid or a muc smaller d/: or 6 1 >> K +, 36 ( Be 1/ d 6 < < + 03 K Be 1/ in te small-diameter limit eq (3 can be reduced as, (5 ( 1 φ From [7] one can obtain 16 R d φ Be (6 Figure 3 Model o cylindrical element wit small-diameter limit [7] d 18 << 0 1/ Be 1/ (7

5 970 Xiao Q H, et al Cinese Sci Bull September (011 Vol56 No7 According to te comparison between eqs (5 and (7, eq (5 is valid because o te inequality K << 1 Eq (6 conirms te expectation tat or te given φ te dimensionless mean termal resistance eeps decreasing wit increasing diameter d (or D 3 Optimal tube diameter and minimum mean termal resistance From te above analyses, in te large-diameter limit, te dimensionless mean termal resistance eeps increasing wit increasing diameter, wile in te small-diameter limit, te dimensionless mean termal resistance eeps decreasing wit increasing diameter Based on te intersection o asymptotes metod [7], tere exists an imal tube diameter, d, leading to minimum mean termal resistance Intersecting te two asymptotes (ie eqs (13 and (6 yields /3 8/3 Be d KBe d Solving eq (8 yields B (8 1/ e F( K, (9 were F is te dimensionless unction o K as sown in Figure Te imal tube diameter and its corresponding dimensionless mean termal resistance are F Be 1/, (30 and 16Be 1 / R,m φ φ F ( 1 (31 Eq (31 sows tat R,m irst decreases, and ten increases wit te increase o φ In oter words, tere exists an imal φ or D/d leading to a minimum R, By imizing m eq (31 again one can obtain φ or (D/d and its corresponding twice-minimized dimensionless mean termal resistance as φ 1/, ( Dd, (3 1/ 6Be and R, mm (33 F In eqs (31 and (33, te subscripts, m and mm, denote once-imized and twice-imized dimensionless mean termal resistances, respectively 3 Comparison between dierent imization objectives Te imal tube diameter, d, based on te minimization o maximum temperature dierence is obtained by te ollowing equation [7]: /3 8/3 3057Be K + Be d 1, (3 were 1/ lnφ 1 K Solving eq (3 yields s 1 φ B 1/ e ( K, (35 F were F is te dimensionless unction o K as sown in Figure 5 Te imal tube diameter and its corresponding dimensionless mean termal resistance are F Be 1/, (36 and 1 / 16Be R t, φ F (37 ( 1 φ Table 1 lists te imal constructs based on te minimi- Figure Optimal tube diameter (d/ or tis paper Figure 5 Optimal tube diameter (d/ rom [7]

6 Xiao Q H, et al Cinese Sci Bull September (011 Vol56 No7 971 Table 1 Optimal constructs or dierent imization objectives Optimization objectives (d/ (D/d R Minimization o entransy dissipation rate (tis paper FBe 1/ 6Be 1/ /F 6Be 1/ /F Minimization o maximum temperature dierence ([7] F Be 1/ Better perormance wit a Tese are derived rom [7] 1/ 16Be F even smaller value φ( 1 φ a ΔT 1/ 16Be φ F ( 1 φ a zation o entransy dissipation rates and maximum temperature dierences Te results o [7] sow tat te perormance improves as te porosity increases, or as D/d decreases Tat is, wen te minimization o maximum temperature dierence is taen as te imization objective, te better designs ave relatively tin sells o eat generating material, and relatively iger volumetric eat generating rates (q or te total eat generation (q Tis paper demonstrates te imal constructal design, ie φ 1/ or (D/d For visual comparison, / s 1/5 and φ 1/5 are set, and tus, K 00 and K 005 Figures and 5 sow tat F 3571 and F 391 are obtained Te calculations sow tat te imal constructs based on entransy dissipation rate minimization and maximum temperature dierence minimization are clearly dierent Te proiles o te imized cylindrical elements based on entransy dissipation rate minimization and maximum temperature dierence minimization are sown in Figure 6 Te number o cylindrical elements or te ormer is muc larger tan tat or te latter Calculations sow tat te number o cylindrical elements or te ormer is 18 times tat or te latter Based on entransy dissipation rate minimization, te mean termal resistance is decreased by 31%, and te eat transer temperature dierence is even smaller Because te result based on te entransy dissipation extremum principle sould be better, te eat transer perormance o te eat generating volume is improved Conclusions Te geometry o te eat generating volume cooled by orced convection is imized by combining te entransy dissipation extremum principle wit constructal teory Te imal spacing between adjacent tubes and te imal diameter o eac tube are obtained based on te intersection o asymptotes metod or minimizing te entransy dissipation rate Te imized results and imal constructs based on entransy dissipation rate minimization and maximum temperature dierence minimization, respectively, are clearly dierent For te ormer, te imal design means φ 1/ or (D/d ; wile or te latter, te better design means an increasing φ or a decreasing D/d Te results sow tat or te ormer te imal construct greatly decreases te mean termal resistance, and clearly improves Figure 6 Optimal constructs (proiles o cylindrical elements based on te two imization objectives (a Minimization o entransy dissipation rate; (b minimization o maximum temperature dierence te global eat transer perormance o te eat generating volume Te mean termal resistance as deined based on entransy dissipation rate relects te global eat transer perormance A lower mean termal resistance means better eat transer perormance, lower mean temperature dierence and a iger eiciency o eat transer Te imal construct corresponding to te minimization o maximum temperature dierence sows te temperature limitation o te encapsulation o te electrical devices cooled by orced convection, does not relect te global eat transer perormance In tis paper, te imal construct corresponding to te minimization o entransy dissipation rate relects te global eat transer perormance, and improves te global eat transer perormance o te system more eiciently Te constructal design corresponding to te minimization o entransy dissipation rate sould be aded or designing eat generating volumes cooled by orced convection wen engineering or sae conditions Te wor and results provided in tis paper enric te constructal teory and entransy dissipation extremum principle, and urter develop te entransy dissipation extremum principle wit some signiicance Tis wor was supported by te National Natural Science Foundation o Cina ( , te Program or New Century Excellent Talents in University o Cina (NCET and te Foundation or Autors o National Excellent Doctoral Dissertations o Cina ( Bejan A Street networ teory o organization in nature J Adv Trans, 1996, 30: Bejan A Sape and Structure, rom Engineering to Nature Cambridge: Cambridge University Press, 000

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