Constructal entransy dissipation rate minimization for disc-to-point heat conduction

Transcription

1 Article Engineering Thermohysics January Vol.6 No.: doi:.7/s--8-8 SPECIAL TOPICS: Constructal entransy dissiation rate imization for disc-to-oint heat conduction XIAO Qingua, CEN LinGen * & SUN Fengui Postgraduate School of Naval University of Engineering, Wuhan, China eceived March, ; acceted May 7, Based on constructal theory, disc-to-oint heat conduction is otimized by imizing the entransy dissiation rate whereby a critical oint is detered that distributes the high-conductivity material according to otimized radial or branch atterns. The results show that the critical oint is detered by the roduct of the thermal conductivity ratio of the two materials and the volume fraction of the high-conductivity material allocated to the entire volume. The notion of otimal heat transfer erformance can be attributed to the disc based on the entransy dissiation extremum rincile. Comaring the results based on ED imization entransy dissiation rate imization with those based on MTD imization maximum temerature difference imization, one finds that the erformance derived from the two otimization rocedures are different. When the roduct of the thermal conductivity ratio and volume fraction is, the critical oint of the former rocedure is that for which the nondimensional radius of the disc equals.7, while that of the latter rocedure is that for which this radius of the disc equals.8. Comaring heat transfer erformances from the two rocedures, the mean heat transfer temerature difference is decreased more for the former rocedure thereby receiving an imroved erformance quota. constructal theory, entransy dissiation rate, disc-to-oint heat conduction, generalized thermodynamic otimization Citation: Xiao Q, Chen L G, Sun F. Constructal entransy dissiation rate imization for disc-to-oint heat conduction. Chinese Sci Bull,, 6:, doi:.7/s--8-8 The necessity to cool electronic devices grows ever higher as electronic devices become ever smaller. Convective and radiative means are not feasible because of the finite saces involved with small-scaled electronic devices. eat conduction aears as the best otion. Since the constructal theory was ut forward by Bejan and alied to otimization roblems involving heat conduction [], constructal theory has been develoing raidly [ ] and has rovided new research imetus into heat transfer roblems [6 8]. Cooling electronic devices can be described as a disc-to-oint heat conduction roblem; essentially, how to detere the otimal distribution of a high-conductivity material through a given disc such that the heat generated in the disc is conveyed most effectively to a single oint. ocha et al. [9] detered the otimal structure and distribution of the *Corresonding author lgchenna@yahoo.com; lingenchen@hotmail.com high-conductivity material by decreasing the thermal resistance via imizing maximum temerature differences as the otimization criterion for heat conduction roblems within a disc. da Silva et al. [] studied a disc uniformly-slotted with radially-distributed rectangular blades and otimized the blade structures made of high-conductivity material by following the same otimization rocedure. ocha et al. [] further validated their analytic solution of [9] with numerical simulations, otimized the tree channels with loos and comared heat conduction erformances of radial-atterned and branch-atterned discs. owever, constructal otimizations of disc-oint heat conduction roblems do not reflect global heat conduction erformances by imizing maximum thermal resistances maximum temerature differences as in [9 ]. From definitions of heat transfer otential caacity and heat transfer otential caacity dissiation function obtained from The Authors. This article is ublished with oen access at Sringerlin.com csb.scichina.com

2 Xiao Q, et al. Chinese Sci Bull January Vol.6 No. heat transfer theory, Guo et al. [] ointed out that their hysical meanings can be reinterreted as a heat transfer ability measure and its dissiation rate in heat transfer rocesses resectively. Guo et al. [] introduced definitively a new hysical quantity called entransy that describes the heat transfer ability and roosed the entransy dissiation extremum rincile stated as follows: for a fixed boundary heat flux, the conduction rocess is otimized when the entransy dissiation is imized imum temerature difference, while for a fixed boundary temerature, the conduction rocess is otimized when the entransy dissiation is maximized maximum heat flux. An equivalent thermal resistance for multi-dimensional heat conduction roblems is defined based on entransy dissiation. An equivalent thermal resistance reflects an average heat transfer effect. The hysical meaning of entransy was further exounded with research into, for examle, hysical mechanisms of heat conduction and electro-thermal simulation exeriments [ 6]. Many scholars [7 9] have shown great interest and have also studied heat transfer otimizations based on imizing entransy dissiation rate. Wei et al. combined for the first time the entransy dissiation extremum rincile with constructal theory and otimized rectangular [6] and triangular [6] elements. esults of otimization showed that constructs based on ED imization could decrease the mean temerature difference more effectively than those based on MTD imization. The otimal constructs without the remise of otimized last-order construct based on ED imization were also obtained [6]. The reason for the difference between constant and discrete-variable cross-sectional conducting ath elements based on ED imization was analyzed and a universal constructal otimization method through self-similar organization based on ED imization was obtained [6]. Wei et al. [6,6] further derived the mean temerature difference of an electromagnet and made a multi-discilinary constructal otimization of the electromagnet by combining magnetic induction and heat transfer roblems. Xie et al. otimized the heating unit with either a rectangular cavity [66] or a T-shaed cavity [67] based on constructal ED imization. ef. [9] illustrated a hierarchical strategy, in otimizing the structure and distribution of high-conductivity material of the heat generating disc based on constructal MTD imization, and detered a critical oint in which the high-conductivity material is distributed according to the otimized radial attern or branch attern that decreases the global thermal resistance of the entire disc. Based on [9], the structure and distribution of high-conductivity material of the disc will be otimized based on constructal ED imization, and the obtained results will be comared with those obtained in [9]. Definition of entransy dissiation rate Entransy which is a new hysical quantity reflecting heat transfer ability of an object was defined in [] as E vh QvhT, where Q vh is the thermal caacity of an object with constant volume, T reresents the thermal otential. The entransy dissiation function which reresents the entransy dissiation er unit time and er unit volume is deduced [] as q T, hφ where q is thermal current density vector, and T is the temerature gradient. In steady-state heat conduction, h φ can be calculated as the difference between the entransy inut and the entransy outut of the object, i.e.. hφ h,in h,out The entransy dissiation rate of the whole volume in the volume-to-oint conduction is E vhφ E hφd. v v The equivalent thermal resistance for multi-dimensional heat conduction roblems with secified heat flux boundary condition is given as follows []: Q h vhφ h, where Q h is the thermal current. The mean temerature difference for multi-dimensional heat conduction can be exressed as adial-atterned disc Δ T Q 6 h h. The cooling model of radial-atterned disc is shown in Figure [9]. The radius of disc is and the thermal conductivity is. The heat current q π, where is the heat generation rate which is generated uniformly in the disc flows through the high-conductivity aths towards the center T of the disc. The thermal conductivity of the material is and the thicness is D. The number of the high-conductivity aths which are distributed uniformly over the disc is N. The temerature of the disc is above T, and the rim is adiabatic. According to the distribution of high-conductivity aths, the disc can be divided into a number of equal sectors, Nπ /. The radial boundary dotted line of each

3 Xiao Q, et al. Chinese Sci Bull January Vol.6 No. sector is adiabatic. Each sector as deicted in Figure is a fundamental element and N sectors fit in a comlete disc arrangement. One assumes N so that each sector is sufficiently slender to be aroximated by an isosceles triangle of base and height. The rofile of the element is fixed A, 7 where and vary. The fraction of high-conductivity material allocated to the disc is also fixed, D D φ. 8 The asect ratio of the element, /, and the number of the elements is unnown. The ratio of the thermal conductivities is assumed to be large, /. The direction of the heat conduction in the element is considered to be arallel to the y-axis and that in the blade arallel to the r-axis. The element has r-axial symmetry, so one only needs to consider the uer-half sector. The equation for heat conduction through the area is [] with the boundary conditions T +, y /,, 9 T y y r T T, r y, where Tr is the temerature of the central blade. Solving eq. 9 yields y T T r r y [ ]. The equation for heat conduction through the blade is with boundary conditions T D + r, r r T / r, r, Solving eq. yields T T, r. r Tr T r. 6 D 6 Eliating Tr from eqs. and 6 yields the distribution of temerature differences of the element for y> Figure adial-atterned disc [9]. y T T [ r y ] r r. 7 φ 6 The entransy dissiation rate of the element is r vhφ [ ] q T r, y T drdy 8 A +. 6 φ 8 By otimizing eq. 8 with resect to /, the otimal /, the corresonding imal entransy dissiation rate, the mean temerature difference and maximum temerature difference are, resectively, Figure Elemental sector [9]. / φ ot, 9

4 Xiao Q, et al. Chinese Sci Bull January Vol.6 No. q A / vhφ, m φ, φ φ / / Δ T Aq.96 Aq, φ φ 7 / / Δ T,max Aq.67 Aq. 6 owever, the otimal asect ratio of the element obtained in [9] based on MTD imization was / φ ot. Moreover, the corresonding mean temerature difference and the maximum temerature difference are, resectively, φ φ 7 / / Δ T Aq.6 Aq, φ φ / / Δ T,max Aq. Aq. From eqs. 9 and, one can see that the otimal construct of the element based on ED imization is different from that based on MTD imization. The imal values of the mean temerature difference of heat transfer and of the maximum temerature difference based on the former solid line and the latter dotted line imizations are shown in Figure. One can see that the otimal construct based on the former can decrease the mean temerature difference to a great extent comared with the otimal construct based on the latter, with a clear imrovement in the heat transfer erformance. The asect ratio eq. 9 deteres the number of elements which fit in a comlete disc arrangement / / Not π φ.7 6 φ, 6 so eq. 9 shows that the otimal asect ratio agrees with the above assumtion. The area of the entire disc is detered by eq. 6, π / A π NA φ A. 7 And thus, the radius of the disc is / A / / / φ.8 7 A φ. 8 The corresonding dimensionless mean thermal resistance and dimensionless maximum thermal resistance of the entire disc are, resectively, T 6 Δ h, ''' q A/ π φ 9 max th, ''' q A/ π φ T Δ. The radius of the disc is not detered but results from the otimization, and deends on A, φ and. The roduct φ is an imortant quantity. Large φ signifies more high-conductivity material, a more slender element and a disc with larger radius. These roerties will be discussed further in association with Figure. The number of disc elements in the disc and the radius of the disc obtained in [9] based on MTD imization were, resectively, π / / N ot φ.7 φ, / / / / φ φ A /.9 A. The corresonding dimensionless mean thermal resistance and dimensionless maximum thermal resistance are, resectively, T Δ, max t qa / π φ ht, ΔT. A/ π φ Figure The mean temerature difference and maximum temerature difference of element. eqs. and are derived from the results in [9] by the author of this aer. Comaring eq. 6 with, and 8 with one can see that both the number of elements and the radius of the disc based on ED imization are smaller than those based on MTD imization. That is, the A in eq. or is not equal to the A in eq. 9 or. Therefore, the mean thermal resistance and the maximum thermal resistance of the entire radial-atterned disc based on ED imization are not comared with those based on MTD

5 6 Xiao Q, et al. Chinese Sci Bull January Vol.6 No. imization in this aer. Branch-atterned disc This aer considers the branch-atterned disc with first order assembly [9]. ere, only the erimeter of the branchatterned disc is assembled by many elemental sectors of the radial-atterned disc, as shown in Figure.,, q,, T, and all have the same meaning as discussed in the revious section. The difference from the revious is the distribution of high conductivity material: one blade thicness is D stretches radially to a distance away from the disc center, continues with a number n of tributaries with thicness D and length that terate on the rim. The A sector with one stem,d and n tributaries,d, as shown in Figure, is analyzed in this section. T c is the confluence of the stem and tributaries. To aly the results of the last section, the A sector is considered to be a combination of n small A sectors of asect ratio / and a central A sector of asect ratio / [9]. One assumes that each eriheral sector of radius is slender enough such that eq. 9 is correct: / φ, D φ, A. The T c end of the high-conductivity blade of the central sector of radius is not adiabatic, so eq. 9 for the central sector is incorrect. For this reason, the asect ratio / of the central sector is free to vary. α,, A 6 where the ti angle α of the central sector is a function of n and. The number of eriheral A elements of the disc is Nπ/, and the number of central A sectors of the disc is N/n. Then, angle α is πn n na α N / / φ And thus, the area of a central sector is. φ / α na A A / φ / /, 7 8 where. Not all of the A elements have the radius, so eq. 8 is aroximate. For the central A sector the equation for heat conduction through the blade is with boundary conditions T D r r r +, 9 Figure First order assembly of disc [9]. Solving eq. 9 yields T r D + na, r, r T T, r. T r T A + na D q D r [ na r + r ]. 6 Invoing the boundary condition T T c at r yields T T A + na c. D Figure First order assembly of sector [9]. The distribution of the temerature in the area is the same as eq., so the distribution of temerature difference of central A sector for y> is

6 Xiao Q, et al. Chinese Sci Bull January Vol.6 No. 7 T T r y y + D r na r + r. D 6 A+ na The entransy dissiation rate of central sector A is r [ ] T r, y T drdy vhφc A A D. + 6 A + na D The entransy dissiation rate of A element is comosed of the entransy dissiation rate introduced by the heat current qa flowing through A element and the entransy dissiation rate introduced by the heat current qa flowing through high-conductivity blade of central sector A. In fact, it can be derived from the definition of the entransy dissiation rate: vhφ T T drdy A T T drd y+ T T drd y. 6 c A A The entransy dissiation rate introduced by the heat current qa flowing through A element the first term on the right-hand side of eq. 6 can be given by eq.. The entransy dissiation rate introduced by the heat current qa flowing through high-conductivity blade of central sector the second term on the right-hand side of eq. 6 can be given by A Tc T. Thus, the entransy dissiation rate of A element is q A / vhφ φ + A Tc T q A φ + A A + na. 7 / D The entransy dissiation rate of A sector is + n vhφ vhφc vhφ A D A D + 6 A A / naq φ + + A+ na. D After some algebra, eq. 8 becomes c 8 nq A vhφ n 7 D φ n + n D φ φ φ φ φ φ n nd φ n + nd D n φ n + n φ + nd φ, 9 where D D/ D is the ratio of thicnesses of high- / conductivity blades, and / A is the dimensionless radius of the disc. The corresonding mean temerature difference of heat transfer is qa Δ T 6 D φ 6 φ φ φ φ φ φ φ φ n + 6 n + n D + n nd n + nd D n φ n +. n + nd φ The fraction of high-conductivity material allocated to the disc is N A ND + D φ n π π / / φ φ D φ φ +. n Eq. deteres the relationshi among n, D,, and φ Because φ and are fixed e.g. φ. and, the relation between D and φ is obtained when n and are given. Figure 6 shows the dimensionless mean temerature difference T ΔT / A / based on ED imization, the dimensionless maximum temerature difference T ΔTmax / A / based on MTD imization, and D versus φ characteristis with n and. From

7 8 Xiao Q, et al. Chinese Sci Bull January Vol.6 No. Figure 6 T, T and D versus φ characteristics. the figure, one can see that the otimal constructs corresonding to T and T resectively, are different from each other: the values of φ,ot are. and.9 resectively, the values of D ot are. and.76 resectively, and the values of T corresonded to T and T are.78 and.7 resectively. The calculations show that the mean temerature difference of the A sector corresonding to the former decreases by 6.% comared with that corresonding to the latter. The otimal construct based on ED imization can imrove the heat transfer efficiency. Figure 7 shows the effects of the dimensionless radius on the otimal constructs φ,ot, D ot, T or T based on ED imization solid line and MTD imization dotted lines resectively. From the figure, one can see that when increases φ,ot increases, D ot decreases and T first decreases then increases based on ED imization, T increases, φ,ot decreases and D ot increases slowly based on MTD imization. Evidently, the otimal constructs corresonding to T and T are different from each other. One can see also that when is small the values of D ot and φ,ot corresonding to T and T resectively, tend to be equal. and all. These henomena Note that all the solid lines vanish below.7 the dotted lines vanish below.8 will be exlained in the following text. Figure 7 shows the otimal number of eriheral elements: N φ ot / π,ot /. N ot increases as increases solid line in Figure 7, and there exists a similar law based on MTD imization [9] dotted line. Because φ,ot based on ED imization increases with n, one sees from eq. that N ot increases with n. The behavior is different from the relation based on MTD imization where N ot is indeendent of n. For the same, N ot corresonding to T is smaller than N ot corresonding to T. They tend to be equal as increases. Figure 7 also shows the otimal dimensionless radius of the central sector of the disc: φ /,ot,ot,ot. / A When,ot shrins to zero hence φ,ot φ,.7. The branch-atterned disc reduces to a radial-atterned disc. For this reason, the solid lines in Figure 7 vanish below.7. Similarly, the dotted lines in Figure 7 vanish below.8 based on MTD imization [9].,ot based on ED imization and,ot based on MTD imization both increase and tend to be equal as increases. For the same, the former is larger than the latter. The former increases as n increases, but the latter is indeendent of n. Figure 8 shows the effects of the number n of tributaries on the otimal constructs based on ED imization solid line and MTD imization dotted line resectively, with. From the figure, one can see that the otimal constructs φ,ot, D ot, T of the former increase with n. For the latter, D ot also increases with n, but φ,ot and T are both indeendent of n and equal.9 and.7, resectively. Comaring the solid lines with the dotted lines in the figure, one sees that φ corresonding to the former is larger, but the Figure 7,ot D ot corresonding the former Effect of dimensionless radius on the otimal construct.

8 Xiao Q, et al. Chinese Sci Bull January Vol.6 No. 9 sionless radius of the radial-atterned disc is cf. eq. 8 / φ /. Figure 8 Effect of the number of elemental branches on the otimal construct. is smaller. A larger D ot means that an elemental insert D is thinner relative to the stem D. In each of the cases otimized in Figures 6 8 the elemental area and dimensionless radius of the disc were fixed. This means that the imization of dimensionless mean thermal resistance of the entire disc is equivalent to the imization of T : ΔT T h π / π 6 6πD φ 6 φ φ n nd φ φ φ + φ φ φ n + 6 n + n D n 7 nd D n φ + n n + nd φ. Because T is almost roortional to, the imized dimensionless mean thermal resistance h, T /π is almost roortional to. h, deends on,, φ and n. owever, the imized dimen- sionless maximum thermal resistance /π t, T [9] is indeendent of n as shown in Figure 9. Based on the same elemental area and the same amounts and roerties of conductive materials φ,, one comares h, with h for different n as shown in Figure 9 solid line. The dimen- This dimensionless radius is fixed when and φ are fixed. The dimensionless radius of a branch-at- terned disc in Figure can be free to vary. This is why in Figure 9 h is a oint and h is a curve. From the solid, lines in the figure, one can see that, increases with n. When exceeds.7, the global resistance is smaller when the high-conductivity material is distributed according to the otimized branched attern. That is,.7 is a critical oint that whether the high-conductivity material is distributed according to the otimized radial attern or branched attern. Tables and list the ey results of arameters of some oints. Based on ED imization, radial-atterned disc has elemental sectors for.7, and the branch-atterned disc has eriheral elements for and 6 eriheral elements for. Table Otimal construct based on ED imization this aer n h N h, ot,ot D φ ot,ot Figure 9 Dimensionless mean thermal resistances of the radial-atterned and branch-atterned discs and the results in [9]. Table Otimal construct based on MTD imization ref. [9] n t t, ot D φ ot,ot h N,ot

9 Xiao Q, et al. Chinese Sci Bull January Vol.6 No. Based on Figure 9 solid line, Figure solid line shows the effects of and φ on the dimensionless mean thermal resistance with n, and. φ.. The dotted lines reroduce the results in [9]. The oint h and the curve h in Figure 9 with n, is a secial case of Figure. Figure includes all these results with n. From the solid lines in the figure, one can see that when the dimensionless radius exceeds that of the otimized branch-atterned disc, lower mean thermal resistance is achieved when the high- conductivity material is distributed according to the branch attern. The mean thermal resistance can also be decreased and the heat transfer erformance is imroved by increasing φ. Figure shows the comarison between the mean thermal resistances based on both ED and MTD imizations, with different n. The dotted lines which are derived from the results of [9] vanish below.8. From the figure, one can see that the difference between the two mean thermal resistances decreases as decreases. The former is smaller than the latter and clearly imroves the heat transfer erformance of the disc with the larger value of. Figure Effects of and φ on dimensionless mean thermal resistances of the radial- and branch-atterned discs and the results of [9]. Figure Dimensionless mean thermal resistances based on h and t. Conclusions The entransy dissiation extremum rincile rovides new academic justification and criterion for heat transfer otimizations. This aer imoses imum temerature differences for given heat flux that satisfies the imum entransy dissiation rincile. When the entransy dissiation rate is imal, the mean temerature difference of heat transfer is imal and the heat transfer erformance is otimal. The constructal roblem of cooling a disc was analyzed and discussed based on ED imization. The results showed that the construct of otimal heat transfer erformance of the system can be designed based on this entransy dissiation extremum rincile. From Figures, 6, one can see that the difference between the otimal constructs for the two objectives is evident. φ,ot corresonding to the former is larger than φ,ot corresonding to the latter, and D ot corresonding to the former is smaller than D ot corresonding to the latter for a given. When increases, φ,ot increases, D ot decreases, and T first decreases and then increases based on ED imization; T increases, φ,ot decreases and D ot increases slowly based on MTD imization. When φ. and, for the former.7 is the critical oint detering whether the high-conductivity material is distributed according to radial or branch atterns, and for the latter,.8 is the critical oint. These are all the differences for the two otimization objectives, one based on ED imization and the other on MTD imization. Additionally, comared with otimal construct based on the latter, the otimal construct based on the former can decrease the mean temerature difference of heat transfer to a greater extent, and clearly imroves the heat transfer erformance. The constructal otimization based on MTD imization amounted to limiting maximum temeratures, and the constructal otimization based on ED imization amounted to decreasing the mean temerature difference of heat transfer of the disc and imroving the heat transfer erformance. Therefore, the otimal constructal design corresonding to the imal mean thermal resistance should be adoted by limiting maximum temerature. When cooling ever larger heat-generating discs, the second order assembly of the branch-atterned disc can be considered based on the wor of this aer. The sector with ti angle β in Figure is comosed of a number of otimized A sectors as discussed in section. Deductions rocess of entransy dissiation rate of these second-order assemblies are similar as in section. The curve of the mean thermal resistance of the second order assembly will intersect the existing curve Figure 9 of the mean thermal resistance of the first order assembly, and the intersecting oint will be the critical oint that the high-conductivity material

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