International Journal of Heat and Mass Transfer

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1 International Journal of Heat an Mass Transfer 54 (211) Contents lists available at ScienceDirect International Journal of Heat an Mass Transfer journal homepage: Cooling augmentation using microchannels with rotatable separating plates A.-R.A. Khale a, K. Vafai b, a Department of Thermal Engineering an Desalination Technology, King Abulaziz University, P.O. Box 824, Jeah 21589, Saui Arabia b Department of Mechanical Engineering, University of California, Riversie, Riversie, CA 92521, USA article info abstract Article history: Receive 9 January 211 Receive in revise form 9 February 211 Accepte 9 February 211 Available online 21 April 211 Keywors: Flexible Microheat exchanger Microchannel Convection Seals Cooling augmentation using ouble layere (DL) microchannels separate by rotatable plates is investigate in this work. The analyze evices of the propose configuration are (A) the flexible microheat exchanger, an (B) the DL-flexible microchannel evice. The moment of the pressure forces on the separating plate is relate to its rotational angle through its flexible supports. Energy equations for the flowing fluis are solve numerically using an iterative finite-ifference metho. Comparisons with obtaine close-form solutions uner fully evelope conitions are performe an excellent agreement is obtaine. It is foun that the effectiveness an the heat transfer rate per unit pumping power for the flexible microheat exchanger are always higher than that for the rigi one. Moreover, DL-flexible microchannels evices are foun to provie more cooling effects per unit pumping power than rigi ones at flow Reynols numbers below specific values, an at stiffness number an aspect ratio above certain values. These specific values are foun to vary with the magnitue of the heating loa. DL-microchannels with rotatable separating plates can be utilize in several applications such as electronic cooling. Ó 211 Elsevier Lt. All rights reserve. 1. Introuction Many electronic evices incorporate large integration ensity of chips in a small area such as VLSI components. These evices consume large amount of electrical energy which is issipate as heat. The amount of issipate heat is usually very large an can be more than 1 W/cm 2 [1]. As such, conventional cooling technologies may not work. Recent evelopments in microfabrication technologies have enable reaily accessible fabrication of microchannel heat sinks primarily for cooling of electronic components [2 8]. These evices can be arrange in a single layere (SL) micro-passages such as those escribe in the works of Lee an Vafai [9] an Feorov an Viskanta [1]. In aition, they can be arrange in ouble layere (DL) microchannel passages which was invente an reporte by Vafai an Zhu [11]. DL-microchannels are foun to provie aitional cooling capacity an they can ecrease the coolant temperature graients along the microchannel length. It shoul be mentione that SL-microchannel heat sinks can be either single microchannel system [12] or multiple microchannel system [9]. Microchannels can have a wavy shape [13] in orer to enhance heat transfer. The successive evelopments in microchannels technologies have reveale certain esigns that are capable to increase flui mixing within the flui volume hence increasing the heat transfer passively [14]. Corresponing author. Tel.: fax: aress: vafai@engr.ucr.eu (K. Vafai). The main isavantage of conventional microchannel cooling evices is the increase coolant temperature as very large heating loas are issipate by a relatively small coolant flow rates. As such, Vafai an Zhu [9] propose the DL-microchannels heat sink evices. Another solution is to utilize flexible microchannels. These types of microchannels which were evelope in the works of Khale an Vafai [15 17] an Vafai an Khale [18] reuce the coolant temperature because volumes of both the flow passage an the supporting seals are expanable. This effect causes an increase in the coolant flow rate. The volume expansion in flexible microchannels is ue to pressure forces. These forces can be ue to an increase in the pressure rop across the microchannel [18]. Also, it can be ue to gas pressure if the supporting seals contain close gas-cavities in contact with the heate surface. As such, any increase in the gas pressure ue to excessive heating prouces extra expansion in the volume [15,17]. As the expansion in the flow passage volume may result in slight reuction in the convection heat transfer coefficient [18], a new configuration of DL-flexible microchannels evice is consiere here. In this work, a DL-microchannel cooling evice with rotatable separating plate is propose an analyze for possible augmenting in the cooling effects. The separating plate is assume to be supporte via anti-leaking flexible seals. The only allowable motion for that plate is the rotational motion about a pivot ro. This ro is taken to be aligne along the microchannel center line normal to its sies bounaries. Two ifferent evices base on the above configuration are consiere. They are (A) the flexible micro heat exchanger an (B) the heate DL-flexible microchannel evice /$ - see front matter Ó 211 Elsevier Lt. All rights reserve. oi:1.11/j.ijheatmasstransfer

2 A.-R.A. Khale, K. Vafai / International Journal of Heat an Mass Transfer 54 (211) Nomenclature A maximum relative isplacement of the separating plate, /H o A k col to hot fluis thermal conuctivities ratio, k c /k h Al col to hot fluis ynamic viscosities ratio, l c /l h Aq col to hot fluis ensities ratio, q c /q h C c,c h col an hot fluis thermal capacities per unit with [W m 1 K 1 ] maximum isplacement of the separating plate [m] E o imensionless elastic parameter efine in Eq. (11) H microchannel height [m] H microchannel imensionless height efine in Eq. (1) H o half main microchannel height [m] h convection heat transfer coefficient [W m 2 K 1 ] K stiffness of the supporting seals per unit separating plate with [N] Ka stiffness number efine in Eq. (3) k thermal conuctivity [W m 2 K 1 ] L microchannel length [m] Nu local Nusselt number efine Eqs. (47), (48) an (2) P t total ieal pumping power requirement [W m 1 ] p mean pressure [N m 2 ] q heat transfer rate per unit with [W m 1 ] Pr Prantl number, lc p /k Re Reynols number, qu h H o /l T temperature fiel [K] T m mean bulk temperature [K] U local overall heat transfer coefficient [W m 2 K 1 ] U e equivalent overall heat transfer coefficient [W m 2 K 1 ] u velocity fiel [m s 1 ] u imensionless velocity fiel efine in Eqs. (19) an (2) x x y y axial micro-passage coorinate [m] imensionless axial micro-passage coorinate efine in Eqs. 3(c, ) transverse micro-passage coorinate [m] imensionless transverse coorinates efine in Eqs. 18(a, b) Greek symbols e effectiveness of the heat exchanger efine in Eq. (43) e c, e h effectiveness of col an hot fluis flows efine in Eqs. (35) an (3) c 1 c 2 first an secon performance inicators efine in Eqs. (44) an (4) k s imensionless maximum heate plate temperature efine in Eq. (3) l flui ynamic viscosity [kg m 1 s 1 ] h imensionless temperature efine in Eqs. 18(c, ) h m mean bulk imensionless temperature efine in Eqs. (41) an(42) P imensionless ieal pumping power requirement efine in Eq. (33) q flui ensity [kg m 3 ] Subscripts 1 inlet 2 exit c col flui f thermally fully evelope h hot flui m mean value The linear elasticity theory [19] is applie to flexible seals supporting the separating plate to relate the moment of the pressure forces on that plate to its rotational angle. The energy equations for both fluis are solve numerically for general conitions an analytically uner special conitions. As such, the effectiveness of the flexible micro heat exchanger (evice A) an other performance inicators for both evices (A) an (B) are calculate. The avantages of the propose evice in cooling attributes over the performance of the DL-rigi microchannel evice are examine. a Inlet col flui port Exit col flui port 2. Problem formulation 2.1. Moeling of the maximum relative isplacement of the moving plate Consier a wie microchannel of height 2H o which is much smaller than its length L. Consier that this microchannel is ivie into two ientical microchannels by a highly conuctive an inflexible separating plate mounte about a pivot axis (O). The pivot axis is taken to be a ro aligne along the normal centerline axis of the evice as seen in (Fig. 1). Accoringly, the allowable motion of the separating plate is the rotational motion about that pivot axis. Two fluis at ifferent temperatures are allowe to flow in counter-irection insie the lower an upper microchannels. The hot flui which is at temperature T h1 enters the lower microchannel an flows towars the left irection along the x h -axis as shown in Fig. 1. However, the col flui which is at temperature T c1 (T c1 < T h1 ) is allowe to flow in the upper microchannel towars right irection along an the x c -axis as shown in Fig. 1. The heights Exit hot flui port b Pivot ro Separating plate (Seale against leakage from all sies using flexible seals) P c1 T c1 P h2 Y c x c Col Flow Case A: Insulate plate Case B: uniformly heate plate H c(x c) H o H o Inlet hot flui port L/2 L/ 2 H h(x h) Insulate plate Hot Flow x h Separating plate P c2 P h1 T h1 y h Fig. 1. (a) 3D view of the DL-microchannel evice with rotatable separating plate, an (b) sie view schematic of the evice an the coorinate system.

3 3734 A.-R.A. Khale, K. Vafai / International Journal of Heat an Mass Transfer 54 (211) of the microchannels enote by H h,c (H h an H c ) vary with the axial coorinates accoring to the following relationships [2]: H hc ðx hc Þ H hcðx hc Þ H o 1 þ A f1 2A r x hc g: ð1 2Þ The subscript h stans for the hot flui while the subscript c stans for the col flui. The parameters A, A r an the imensionless variable x are the maximum relative isplacement of separating plate, the aspect ratio an the imensionless axial coorinate, respectively. They are given by A H o A r H o L x hc x hc L ð3a-þ where is the maximum isplacement of the separating plate The with of the microchannel is taken to be much larger than its height in orer to have enough pressure forces to cause rotation of the separating plate. As such, the two-imensional flow moel [18] is aopte here an the steay state Reynols equations for the microchannels are given by p hc ð4 5Þ x hc H 3 hc x hc where p h an p c are the mean pressures insie the lower an upper microchannels, respectively. The solution to Eqs. (4) an (5) can be arrange in the following form: 2 p hc ðx hc Þ p h1c1 p h2c2 p h1c1 1 Hðx hc Þ 2 ð 7Þ 1 where p h1 an p h2 are the inlet an exit pressures of the hot flui, respectively. Moreover, p c1 an p c2 are the inlet an exit pressures of the col flui, respectively. We allow the motion of the separating plate to be restraine by the stiffness effect of the elastic seals supporting it [18]. As such, the net moment of pressure forces applie on that plate about the pivot axis per unit with, P M o, is equal to X M K tan 1 ð2=lþ ð8þ where K is the stiffness of the supporting seals per unit with of the separating plate. When A r <.5 as in microchannel applications, the rotational angle which is equal to tan 1 (2/L) can be approximate by the value (2/L). Also it shoul be note that P M o is equal to the following expression when A r <.5: X Z L M p h fl=2 x h gx h Z L p c fx c L=2gx c : Substituting Eq. (9) into Eq. (8) while utilizing Eqs. () an (7) an solving Eq. (8) yiels the following relationship: 2 n o 1 A 2 E o ð1þ 2A ln 1 where E o is the imensionless elastic parameter which is equal to E o ðp h1 p h2 Þþðp c1 p c2 Þ 4KA r =L 2 : ð11þ It is ifficult to obtain the inverse of Eq. (11) analytically. However utilizing statistical softwares, A is correlate to E o by the following functional form: A a 1 þ a 2 E o þ a 3 E 2 o þ a 4E 3 o þ a 5E 4 o 1 þ a E 4 o þ a 7E 4 o þ a 8E 4 o þ a : ð12þ 9E 4 o ð9þ The following a 1 a 9 coefficients were foun to prouce the correlation coefficient R 2 = 1. for 999 equally space A -values (maximum eviation of A from exact is 1.48% at A =.1): a 1 1: a 2 :1851 a 3 :179 a 4 :11819 a 5 :15295 a : a 7 :11314 a 8 :19251 a 9 :15283 ð13a-iþ 2.2. Moeling of flow an heat transfer insie the two microchannels Both fluis are consiere to be Newtonian fluis having constant average properties though compressible fluis can be use in microchannel applications [21]. The momentum an energy equations insie each microchannel are given by [18]: p þ l 2 u hc x hc ð14 15Þ 2 q 2 T hc hc ðc p Þ hc u hc k hc ð1 2 hc where l, u, T, q, c p an k are the ynamic viscosity of the flui, velocity fiel, temperature fiel, flui ensity, flui specific heat an the flui thermal conuctivity, respectively. Define the following imensionless variables: y hc y hc Hðx hc Þ h hc T hcðx hc y hc Þ T h1c1 T c1h1 T h1c1 : ð18a-þ The imensionless velocity fiels u hc ðy hc Þ are obtaine by solving Eqs. (14) an (15). This will result u hc ðy hc Þ u hcðx hc y hc Þ u hmcm ðx hc Þ y hcð1 y hc Þ ð19 2Þ The mean velocities u hm ðx h Þ an u cm ðx c Þ are relate to pressure graients insie the microchannels accoring to the following relationships: u hmcm ðx hc Þ H2 p o hc 2 u hmocmo H 12l hc ðx hc Þ hc x hc H hc ðx hc Þ CF ð21 22Þ where u hmo an u cmo are the mean velocities of the hot an col fluis when =, respectively. The quantities u hmo an u cmo an the mean velocities correction factor CF are given by u hmocmo 1 12l hc Dp hc L H 2 o ð23 24Þ CF 4A ð1 þ A Þ 2 2 ð25þ 1 where Dp h = p h1 p h2 an Dp c = p c1 p c2. Imposing the variables given in Eqs. (3) an (18), on Eqs. (1) an (17) results in the following imensionless hc Re hc Pr hc ðcfþu hc H h hc ð2 2 hc where Re an Pr are the Reynols number an the Prantl number, respectively. The Reynols numbers are efine base on the following expressions: Re hc q hc u hmocmoh o l hc ð28a bþ The imensionless parameter E o can be expresse in terms of the flow Reynols numbers as follows:

4 (! ) E o 3 A q A 4 r Ka Re A 2 h þ Re c ð29þ l where Ka, Aq an Al which are given below are the imensionless stiffness number, col to hot flui ensity ratio an col to hot flui ynamic viscosity ratio, respectively: Ka K l 2 c =q c A q q c q k A l l c l k : A.-R.A. Khale, K. Vafai / International Journal of Heat an Mass Transfer 54 (211) ð3-32þ power requirement to that of the rigi micro heat exchanger (case with E o = ). This can be expresse by e c 1 CF P tj Eo e : ð44þ e Eo P t e Eo The local hot an col flows convection heat transfer coefficients h h an h c, respectively, are efine hc T hm:cm ðx hc Þ T h:c ðx hc y hc H hc Þ khc h hc hc x hc y hc H hc 2.3. The ieal total pumping power requirement The total ieal pumping power requirements, P t, can be expresse in terms of the imensionless parameters as follows: P t P l 3 c =ðq2 c H2 o Þ Dp _m ( h h=q h þ Dp c _m c =q c l 3 c =ðq2 c H2 o Þ 12 Re 2 c A þ A! ) 2 q Re 2 r A 3 h CF l ð33þ where _m h an _m c are the mass flow rate of hot an col fluis, respectively Case (A): insulate DL-microchannels with rotatable separating plate (flexible micro heat exchanger case) The bounary conitions for this case are given by h h ðx h y h Þ h c ðx c y c Þ ð34a bþ h c ðx c 1=A r x h y c 1Þ 1 h h ðx h y c h xh y h 1: xh y h ð34cþ ( ) H h ðx h c A k H c ðx c 1=A r x h ð34þ c xc1=a r x h y ð34e fþ c xcy c where A k = k c /k h. For this case, the effectivenesses of the hot an col fluis, e h an e c are efine as follows: e hc q _m hc ðc p Þ hc ðt h1 T c1 Þ q ðqc p Þ hc u hmcm H hc ðt h1 T c1 Þ ð35 3Þ where q is the rate of heat transfer between the two fluis per unit with. Recall that q is given by q ðqc p Þ h u hm H h ft h1 T hm2 gðqc p Þ c u cm H c ft cm2 T c1 g ð37 38Þ where T hm2 an T cm2 are the mean bulk temperature at the exit of the hot an col fluis, respectively. The mean bulk temperatures are given by: Z Hhc u hc T hmcm T hc y hc : ð39 4Þ u hmcm In terms of the imensionless parameters, the parameters e h an e c are given by e hc h hmcm ðx hc 1=A r Þ Z 1 u hc ðy hc Þh h ðx hc 1=A r y hc Þy hc : ð41 42Þ Accoringly, the effectiveness of the micro heat exchanger, e, which is the maximum value of e h an e c [22] can be expresse by ð43þ e e h if e h > e c e c if e c > e h Lets efine the first performance inicator, c 1, as the ratio of the rate of heat transfer between the two fluis per its ieal total pumping ð45 4Þ Thus, the local hot an col section Nusselt numbers Nu h an Nu c are equal to Nu hc h! hch hc 1 k hc fh hmcm ðx hc Þ h h ðx hc y hc hc xhc y hc 1 ð47 48Þ Approximate analytical solution for fully evelope conitions. Uner fully evelope conitions, axial graient of the temperature ifference across any section of the flexible micro heat exchanger can be consiere to be very small. Accoringly, Nu h ffi Nu c ffi Nu f = 2.95 [23]. The latter number is for the case of laminar flow between an insulate plate an a plate subject to a constant heat flux. The heat transfer rate across a ifferential element within the flexible micro heat exchanger containing both fluis is given by: q C h T hm C c T cm UfT hm T cm gx h T hm T cm x h ð1=h h Þþð1=h c Þ ð49a-þ where C h = q h (c p ) h u hm H h an C c = q c (c p ) c u cm H c. Note that U is the overall heat transfer coefficient. In terms of imensionless parameters, Eq. 49(c) can be expresse as q ( ) Nu f H h fa k 1gþ2 T hm T cm H o =k c x h : ð5þ The ifferential of the mean bulk temperatures ifference across any section can be obtaine using Eqs. 49(a c) as ðt hm T cm Þ 1 1 ( ) Nu f k c =H o x h : ð51þ T hm T cm C h C c H h fa k 1gþ2 Integrating both sies of Eq. (51) over the micro heat exchanger length results in the following expression: n o 1 T hm2 T c1 A k þ n oa : ð52þ T h1 T cm2 1 þ A k The power U is given by U A k 1 Pe h Pe c Nu f 2A r A ða k 1ÞCF : ð53þ Using Eqs. (35) an (52) e h can be represente as n n o. n oo U 1 A k þ 1 þ A 1 A k e h n n o. n oo U : ð54þ 1 Pe h A k Pe c A k þ 1 þ A 1 A k Similarly, using Eqs. (3) an (52), e c can be represente as n n o. n oo U 1 A k þ 1 þ A 1 A k e c n n o. n oo U : ð55þ A k Pe c A Pe k þ 1 þ A h 1 A k

5 373 A.-R.A. Khale, K. Vafai / International Journal of Heat an Mass Transfer 54 (211) Lets introuce e = MAX(e h, e c ), as the nominal efinition of the counter flow micro heat exchanger effectiveness [22], given by e 1 expð NTU½1 C ŠÞ 1 C expð NTU½1 C ŠÞ : ð5þ Accoringly, NTU will be the equivalent number of transfer units which is given by U e L NTU qc p u mo min H ocf ð57þ where U e is the equivalent overall heat transfer coefficient. Note that C = C min /C max where C min an C max are the minimum an maximum values of C h an C c values, respectively. Utilizing Eqs. (54) an (55), the following relationship can be obtaine: 1 þ Ak U e ðu e Þ Ka!1 1 2A ðcfþ 1 A k ln A k þð1 þ A Þ=ð1 A Þ 1 þ A k ð1 þ A Þ=ð1 A Þ : ð58þ Case (B): DL-microchannels with rotatable separating plates subject to uniform heat flux, q s, from below The bounary conition given by Eq. 34(e) is change for this case to the A q H h ðx h Þ ð59þ h xh y h where A q is the imensionless heat flux parameter which can be presente as A q q s H o k h ðt h1 T c1 Þ : ðþ The local convection heat transfer coefficient for the heate plate sie, h s, is efine as h s ft h ðx h y h Þ T hm ðx h Þg q s : ð1þ Thus, the local Nusselt number for the heate plate sie, Nu s, can be presente as Nu s h sh h ðx h Þ k h A q H h ðx h Þ fh h ðx h y h Þ h hm ðx h Þg : ð2þ An important inicator for this case is the ratio of the ifference between the maximum heate surface temperature an inlet col flui temperature to the ifference between the hot an col inlet temperatures. This ratio, which is represente by k s,is k s ½T hðx h y h ÞŠ max T c1 T h1 T c1 1 ½h h ðx h y h ÞŠ min : ð3þ Defining the secon performance inicator, c 2, as the ratio of k s -value for the present heate DL-flexible micorchannel to that for the conventional rigi one (case with Ka? 1) when both operate uner the same ieal pumping power, it can be presente by k c s 2 ð4þ k s j K!1 P t : The relation between the hot flui Reynols number for the present case, Re h, an that for the conventional rigi one, Re hr, that satisfies the constraint of Eq. (4) is given by vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1! u A 3 Re h t CF 1 l Re 2 A 2 c þ Re2 hr ð5þ CF q when both microchannels have the same col flow Reynols number. Equation (5) can be solve by an iteration proceure, as CF is a function of both Re c an Re h. 3. Numerical methoology Eqs. (2) an (27) which are couple through their bounary conitions given by Eqs. 34(c,), are solve numerically using an iterative metho. The implicit-finite-ifference metho given by Blottner [24] is utilize for the present problem. These equations were iscretize using three-points central ifferencing quotients for the first an secon erivative terms with respect to y h an y c irections an an attempt was mae to create a stable implicit finite ifference scheme [25]. For the first erivative with respect to x h an x c irections, two points backwar ifferencing quotients was use. The finite ifference equations for Eqs. (2) an (27) along with their bounary conitions for the flexible microheat exchanger case are given by ðh hc Þ ij ðh hc Þ i 1j Re hc Pr hc CFðH hc Þ i ðu hc Þ j Dx hc ðh hcþ ij 1 2ðh hc Þ ij þðh hc Þ ijþ1 ð 7Þ Dy 2 hc ðh h Þ i1j ðh c Þ ii1jj ðh h Þ ijn 1 ðh c Þ iim iþ1jjn ( ) ðh c Þ iim iþ1jjn ðh c Þ iim iþ1jjn 1 ðh h Þ ijn A k ðh h Þ i þðh h Þ ijn 1 ðh c Þ iim iþ1 ðh h Þ ij1 ðh h Þ ij2 ðh c Þ iijj1 ðh c Þ iijj2 ð8a-fþ where (i, j) an (ii, jj) are the location of the iscretize points in the numerical gris for the lower an upper microchannels, respectively, an Dx h an Dx c are the istances between the two consecutive i an ii vertical lines in the numerical gri, respectively, an Dy h an Dy c are the istances between the two consecutive j an jj horizontal lines in the numerical gri, respectively. The resulting tri-iagonal systems of algebraic equations obtaine from iscretizing Eqs. () an (7) at a given i an ii-sections, respectively, were solve using the Thomas algorithm [24]. The solution was base on an initial estimate of the temperature at the separating plate, (h h ) i,j=n. The same proceure was repeate for the consecutive i an ii values while the temperatures at the separating plate were correcte using Eq. 8(). The numerical results were compare with the analytical solution given earlier. An excellent agreement was foun between the numerical an analytical results. In what follows, both hot an col fluis are taken to be liqui water. The maximum col flow Reynols number was taken as Re c = 33. This correspons to a col flow mean velocity of u cm =.11 m/s when the average height of the microchannel is H o = 3 lm. The maximum stiffness number was taken as Ka = This correspons to supporting seals stiffness per unit separating plate with of K =.12 N. 4. Results an iscussion 4.1. Effect of E o on the maximum relative isplacement an CFcorrection factor The variations of the maximum isplacement of the separating plate (A ) an the mean velocity correction factor (CF) with the imensionless elastic parameter, E o, are shown in Fig. 2. The elastic parameter increases as the stiffness of the separating plate ecreases. This effect increases the separating plate rotation causing an increase in A. As can be seen, the CF-mean velocity correction factor ecreases as E o increases. This effect is clearly seen in Fig. 2.

6 A.-R.A. Khale, K. Vafai / International Journal of Heat an Mass Transfer 54 (211) A.9 A.9 CF CF E O Fig. 2. Effects of the elastic parameter E o on A an CF..9.7 ε..5.4 Ka = Re h=15 Re h=25 using Eqs.(54, 55) Ka A k=.915, A μ=2.3, A ρ=1.1, 1/A r=4, Pr c=8.25, Pr h= Re c Fig. 4. Effects of Re c on e for the flexible micro heat exchanger case..9.7 ε Flexible micro heat exchanger case Nusselt number istributions an the valiity of the analytical solution Effects of the imensionless axial istance (x) an the aspect ratio (A r ) on the hot an col flui local Nusselt numbers (Nu h, Nu c ), respectively, are illustrate in Fig. 3. This figure is presente for the case when both fluis have the same thermal capacitance (RehPr h = A k Re c Pr c ). For long microchannels such as when 1/A r =, both Nu h an Nu c converge to a constant value for most of x h -values. This value is very close to Nu f = This correspons to the case of fully evelope channel flow between an insulate plate an a plate subject to constant heat flux. Thermal entry region effect is clearly seen in Fig. 3 for the case with 1/A r = 1, corresponing to short flexible micro heat exchangers. For this case, Both Nu h an Nu c ecrease as istances from the inlet sections increase as shown in Fig Variation of effectiveness an first performance inicator with Re c The variation of the flexible microheat exchanger effectiveness (e) with the col flui Reynols number (Re c ) is shown in Fig. 4. Also, shown in Fig. 4 is a comparison between analytical an numerical results. As it can be seen, there is an excellent agreement between these results. The increase in Re c causes an increase in the col flui thermal capacitance (C c ). As Re c approaches Re c = 7.25 (when Re h = 15) or Re c = (when Re h = 25), the value of C c approaches C h. This results in e h = e c an prouces the minimum effectiveness (e = e min ) as shown in Fig. 4. IfRe c < 7.25 (when Re h = 15) or if Re c < (when Re h = 25), the temperature ifference for the col flui is larger than that for the hot flui thus, e = e c > e min. However, e = e h > e min if Re c > 7.25 (when Nu A k=.915, A μ=2.3, A ρ=1.1, Pr c=8.25, Pr h=3.5, Re c=7.2, Re h=15 Nu h Nu c Nu f 1/A r=1 1/A r = 5.Nu x h Ka = Fig. 3. Effects of x h an A r on Nu h an Nu c for the flexible micro heat exchanger case Re h = 15) orifre c > (when Re h = 25). Therefore, e ecreases as Re c increases until it reaches e = e min after which it starts to increase as Re c increases as epicte in Fig. 5. For rigi micro heat exchangers (K? 1), the effectiveness is smaller than that for the flexible micro heat exchangers as shown in Fig. 4. It is notice from Fig. 5 that the first performance inicator (c 1 ) is always larger than one, inicating that the propose flexible micro heat exchanger transfers more heat than the rigi one uner constant pumping power. As such, the superiority of flexible micro heat exchangers over rigi ones is establishe Variation of effectiveness an the first performance inicator with Ka The increase in the imensionless stiffness number (Ka) is accomplishe by an increase in the supporting seals stiffness per unit with of the separating plate (K). As Ka ecreases, the separating plate supporting seals softness increases. As such, the separating plate maximum relative isplacement (A ) increases as Ka ecreases. As a consequence, velocities near the separating plate in regions very close to the fluis exits increases. Also, expansions of the microchannels near the inlet regions increase, which allow the thermal bounary layers to evelop further ownstream. Thus, convection heat transfer coefficients are expecte to increase as Ka ecreases. As such, e increases as Ka ecreases as can be seen in Fig.. Similarly, it is notice from this figure that c 1 is always larger than one an it increases as Ka ecreases. Again, this estabilishes the superiority of flexible micro heat exchangers over rigi ones Variation of the first performance inicator with aspect ratio The increase in the microchannel length causes a ecrease in the aspect ratio. As such, the moment of the pressure forces across γ 1 γ A k=.915, A μ=2.3, A ρ=1.1, 1/A r=4, Pr c=8.25, Pr h=3.5 Re h=15 Re h=25 Ka = Re c Fig. 5. Effects of Re c on c 1 for the flexible micro heat exchanger case

7 3738 A.-R.A. Khale, K. Vafai / International Journal of Heat an Mass Transfer 54 (211) ε γ A k=.915, A μ=2.3, A ρ=1.1, 1/A r=4, Pr c=8.25, Pr h= Ka x1 the separating plate increases. Thus, the maximum relative isplacement A increases which augments the first performance inicator as shown in Fig DL-flexible microchannel evice ε Re c=7.25, Re h=15 γ 1 Re c=12.9, Re h=25 Fig.. Effects of P c on e an c 1 for the flexible micro heat exchanger case. Fig. 8 shows that the secon performance inicator (c 2 ) is smaller than one for the small col flow Reynols numbers. This is expecte because large pressure rops which inuce large Reynols numbers prouce large moments of forces that can cause closing of flows passages. This effect increases c 2 significantly above one. On the other han, it is notice from Fig. 9 that the secon performance inicator is smaller than one for large stiffness numbers. Similar effects can be conclue for the role of the aspect ratio on the secon performance inicator as shown in Fig. 1. It is seen from Figs. 8 1 that c 2 has local minimum near the switch points. At these points, the increase in Re c or the ecrease in both Ka an A r values cause the values of c 2 to be larger than one. Therefore, the superiority of the heate microchannels with rotatable separating plates can be implie for moerate aspect ratios an moerate Reynols an stiffness numbers. Typical heate plate temperature an Nusselt number istributions are shown in Fig. 11 for ifferent col flow Reynols numbers Feasibility of the microchannels with rotatable separating plates evices The inlet an outlet ports of the passages of the propose evice can be rille on the upper an lower plates as shown in Fig. 1(a). 1.5 γ A q = 3 A q = 5 A q = 5 A q = 3 Re c=7.25 Re c= A k=.915, A μ=2.3, A ρ=1.1, 1/A r=4, Pr c=8.25, Pr h=3.5, Re hr= Ka x1 Fig. 9. Effects of Ka on c 2 for the heate DL-flexible microchannel case Ka = Re c=11 Re c= γ γ γ Ar Fig. 7. Effects of 1/A r on on c 1 for the flexible micro heat exchanger case λ s A k=.915, A μ=2.3, A ρ=1.1, 1/A r=4, Pr c=8.25, Pr h=3.5 Ka Ka = Re λ hr=15 s γ 2 Re hr= A q = Re c γ 2 Fig. 8. Effects of Re c on k s an c 2 for the DL-flexible microchannel case. ( x, y ) 1 θ h =.5.5 A k=.915, A q = 5, A μ=2.3, A ρ=1.1, Pr c=8.25, Pr h=3.5, Re hr= /A r Fig. 1. Effects of Rec on k s an c 2 for the DL-flexible microchannel case Ka = θh ( x, y = ) Nu s Re c=13 Re c=21 1. Re c=1. Re c= A k=.915, A q = 5, A μ=2.3, Pr c=8.25,re A ρ=1.1, 1/A r=4, hr= x Re c=13 Re c= Nu s Fig. 11. Effects of Re c an x on Nu s an the surface temperature for the DL-flexible microchannel case. 5.

8 A.-R.A. Khale, K. Vafai / International Journal of Heat an Mass Transfer 54 (211) For this configuration, changes in inlet an exit heights with operating conitions have no influence on inlet an exit port imensions. Moreover, the expecte leakage of flui flow between the passages of the propose evice can be controlle by having well seale separating plates using carefully esigne soft seals. The manufacturing of these kins of seals which shoul exhibit high softness, high strength an high anti-leaking sealant attributes is possible consiering the current avance fabrication methos. Finally, the possibility of having eflecte separating plate other than that ue to its rotation can be eliminate by having multiple micro-passages system within each layer [9]. For this case, each micro-passage is isolate against the other one by using sies mae of anti-leaking soft seals which provie an aitional support to the separating plate. 5. Conclusions Heat transfer insie DL-microchannels evices with rotatable separating plates was consiere in this work. Two ifferent evices having ifferent bounary conitions are analyze. These are (A) the flexible micro heat exchanger an (B) the heate DLflexible microchannels evice. The rotational angle of the separating plate is relate to the moment of pressure forces acting on it through the flexible supports. Appropriate forms of the couple energy equations for both flows were solve using an iterative implicit-finite-ifference metho. The numerical results for case (A) were valiate against obtaine close-form solutions base on fully evelope thermal conitions. It was foun out that flexible micro heat exchangers always have a higher effectiveness than the rigi ones. Moreover, flexible micro heat exchangers were foun to transfer more heat than the rigi ones operating uner the same pumping power. Further, the DL-flexible microchannels evice (evice B) was foun to provie more cooling effects per unit pumping power than the rigi evices for flow Reynols numbers below specific values. In aition, the cooling attributes for evice B were foun to be better than those for the heate DL-rigi microchannels evice at stiffness numbers an aspect ratios above certain values. These specific values were foun to vary with the heat loa. DL-microchannels evices with rotatable separating plates can be use in several applications such as electronic cooling. References [1] J.-Y. Jung, H.-S. Oh, H.-Y. Kwak, Force convective heat transfer of nanofluis in microchannels, Int. J. Heat Mass Transfer 52 (29) [2] D.B. Tuckerman, D.B. Pease, High-performance heat sinking for VLSI, IEEE Electron. Dev. Lett. EDL-2 (1981) [3] L.J. Missaggia, J.N. Walpole, Z.L. Liau, R.J. Philips, Microchannel heat sinks for two imensional high-power-ensity ioe laser arrays, IEEE J. Quant. Electron. 25 (1989) [4] M.B. 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