A Study of two phases heat transport capacity in a Micro Heat Pipe

Transcription

1 WSEAS TRANSACTIONS n INFORMATION SCIENCE APPLICATIONS A Study f tw phases heat transprt capacity in a Micr Heat Pipe Cheng-Hsing Hsu, Kuang-Yuan Kung,*, Shu-Yu Hu, Ching-Chuan Chang Department f Mechanical Engineering Chung-Yuan Christian Uniersity.,* Department f Mechanical Engineering Nanya Institute f Technlgy. kykung@nanya.edu.tw Abstract: Present study mdifies Ctter s mdel by using the dimensinless liquid flw shape factr, K, t predict the maximum heat transprt capacity t discus the effects f cntact angle. The results indicated that as the dimensinless liquid flw shape factr, K, decreases, the frictin effects n the apr-liquid interface flw, L, increases, the liquid flw influenced by the apr flw als increases. The predicted maximum heat transprt capacity agrees well with Babin s experimental data f a cpper-water micr heat pipe under, L = cntact angle, α = 0. In a micr heat pipe, the results indicated that bth the maximum heat transprt capacity K increases with a increasing cntact angleα. Key-Wrds: Ctter s mdel, Cntact angle, Heat pipe, Heat transprt capacity,shape factr, Dimensinless Intrductin Heat pipes are widely used in related heat transfer applicatin. Analyses f heat pipes, bth numerical experimental, hae been perfrmed by many inestigatrs [-5]. Due t the cmplexity f getting as analytical slutin fr heat pipe peratin, sme mdels withut the effects f shearing stress n the liquid-apr interface has been intensiely study by [-4]. The first analytical mdel fr micr-heat pipe was presented by Ctter []. The capillary pressure gradient due t the ariatin f radius curature was assumed t be the driing frce f the liquid flw, zer liquid-apr interface elcity, laminar incmpressible apr flw regin. Ha Petersn [] applied Xu [3] gerning equatin t mdify Ctter mdel. They defined dimensinless shape factr reprted the effects f the cntact angle f liquid apr interface. Ayyaswamy [4] analyzed a steady flw in a gre by Galerkin bundary methd. He cncluded that the effects f the cntact angle f slid-liquid n the aerage elcity are minus under small gre tip angle. Babin et al [5] reprted the experimental results f a mm heat pipe f Ctter s mdel by using water as wrking fluid in cpper r siler. Because it is imprtant t knw the effects f shearing stress n the liquid-apr interface, many researches [6-] hae been depicted it. Ma et al. [6] inestigated the effects f shearing frce n the liquid-apr interface inducing the pressure gradient in the fluid flw in a triangular gre. They cncluded that the frictinal factr Reynlds number is a functin f gre tip angle, the cntact angle f slid-apr, the liquid-apr interface. The results are presented theretical experimental cnsistency. One f the mst imprtant characteristics f micr heat pipe is the maximum heat transprt capacity, which has been discussed by [7-]. Faghri [0] theretically experimentally inestigated bth axially gred heat pipes flat miniature heat pipes. The maximum heat transfer capability in axially gred heat pipes is subjected t the gemetry f the heat pipes the cntact angle f liquid-slid interface. The present paper deals with the numerical predictin f Micr heat pipes. It shws the lume flw rate, maximum heat transprt capacity input pwer f micr capillary gres under the influence f the shearing stress f interface, cntact angle length. The numerical results are cmpared t the experimental data f Babin [] Ma et al. [6] mdel. Theretical Analysis It is clearly that the apr flw directin strngly impacts the flw elcity behair f liquid surface[5]. But the shearing stress n the liquid-apr interface was neglect in Ctter s mdel [], which results in a larger heat transfer than real situatin. The fundamental assumptins are tw dimensinal Newtnian incmpressible flw, fully deelped laminar apr flw n slip cnditin n liquid-slid interface [6]. The mmentum equatin can be written as u u dp r + = r r r r θ μ dz () where u is axial elcity, z the crrespnding axial crdinate, P the static pressure, μ the dynamic ISSN: Issue 7, lume 6, July 009

2 WSEAS TRANSACTIONS n INFORMATION SCIENCE APPLICATIONS iscsity θ gre angle. N slip cnditins at the wall are assumed. θ = 0, u = 0 () θ = φ, u = 0 (3) In rder t btain a general slutin, tw bundary cnditins in r directin are assumed. A trapezid shape in liquid regin is frmed when r = r,the liquid surface elcity is defined as u r zer liquid elcity at r = r. As r 0, the slutin relates t that btained fr a triangle gre, clsed n these three sides. Based n Ma et al. [6] dimensinless parameters as r u+ ( dp dz) * 4μ u = r ( dp dz) 4μ (4) x = ln r r (5) φ is half-channel angle. The gemetry n the interface f liquid-apr depends majr n the shape f the liquid flw passage with lw Bnd number cnditins [6].They als describe radius as: csφ r = r w cs( φ θ ) () csαcs( φ θ) sin φ cs αsin ( φ θ) r= r w cs ( α+ φ) () Furier transfrmatin methd superpsitin is applied t the Laplace gerning equatin (6), where the exact slutin is subjected t fit the bundary cnditins gien in equatins (7)-(0). The analytical elcity u can be shwn as [6] r u = ( dp dz) 4μ ( a x) Hp sinh x + sinh n φ φ θ ( ) sin π n n n π = sinh a φ φ n r ( ) θ ( φ θ) sinh sinh r n + a a + sin x π n a φ = sinh a + a r r Where (3) The dimensinless gerning equatin bundary cnditins are * * u u + = 0 x θ (6) * r x = = θ = 0, r (7) u e at r * x u = = e at θ = φ, r (8) * u at x = = 0, r ur + ( dp dz) * 4μ r u = at x= ln r r ( dp dz) 4μ (9) (0) a = ln r r (4) r ur + ( dp dz) 4μ Hp = r ( dp dz) 4μ (5) u r is the aerage elcity f liquid surface at r = r. This alue depends n bth the liquid apr flw characteristics prperties. Due t an irregular crss-sectin is shwn due t the fractinal interactin f liquid flw apr flw. It is quite difficult t get a general slutin f the aerage elcity f liquid surface u r. T aid the rising difficulty f getting analytical slutin, assume the liquid surface is free t the influence f frictinal influences. The mmentum equatin can be written under these assumptins duf dp μ rdθ = dz (6) where u f is the free liquid surface elcity withut the effects f apr shearing stress the crrespnding bundary cnditins are: uf = 0 at θ = 0 (7) uf = 0 at θ = φ. (8) Integrating the mmentum equatin, the slutin f equatin (6) with these bundary cnditins is: ISSN: Issue 7, lume 6, July 009

3 WSEAS TRANSACTIONS n INFORMATION SCIENCE APPLICATIONS u f r dp = 4μ dz { csαcs( φ θ) sin φ cs αsin ( φ θ) } cs( α+ φ) w ( 4φθ θ ) (9) The aerage elcity f the liquid flw is preferred in the actual situatin, it is written as: φ ud f θ 0 u f φ dθ = 0 (0) Then the ne dimensinal aerage elcity f the free liquid surface can be btained as: φ dp u f = φ 0 4μ dz rw { csαcs( φ θ) sin φ cs αsin ( φ θ) } cs( α + φ) φθ θ dθ ( 4 ) () Since the ne dimensinal assumptin is insufficient fr the actual case, a crrectin factr is necessary t mdify it. This cefficient, K, can be determined experimentally. Utilizing the experimental data f Ayyaswamy et al.[4], Ma et al. [] presented that the cefficient K, withut ntificatin, is simplified t be the alue f fr a 60 channel angle. But the cefficient K shuld ary with the channel apex angle θ, by neglecting this effect Ma et al. [] assumed the cefficient K t be a cnstant. u f = Ku f () where u f is defined in equatin (). Ma et al. [3]defined the actual elcity f the free liquid surface a dimensinless liquid-apr interface flw number L as: Δu L = u f (3) Δ u = u f ur (4) / u f is the aerage elcity f the free liquid surface. Ma et al [] demnstrates L including the crrectin factr K which is frm Ayyaswamy [4] experimental data. But crrectin factr K cannt shw bth f the effects n the liquid-apr interface liquid-slid interface. Ma et al. [] define a dimensinless liquid-apr interface flw number L subjected t the frictinal impacts n the liquid-apr interface. The effects f shear stresses at the liquid surface are taken int cnsideratin, which is due t frictinal liquid-apr interactin n the liquid flw. This paper is based upn Ma et al. [] cnclusin by adjusting L t study the impacts f the liquid-apr interface n rhelgy. The aerage elcity at r is: ur = u ( ) f L (5) cmbine equatins () (5), u r can be shwn as: u r K( L) φ dp = φ 0 4μdz { csαcs( φ θ) sin φ cs αsin ( φ θ) } cs ( α+ φ) r w φθ θ dθ ( 4 ) (6) Frm the equatin (6), the aerage elcity f the liquid surface is related tα, φ, θ, r w, dp / dz, K L. The flw directin f the liquid the apr elcity is alues f the dimensinless apr flw number, L, which is well explained by Ma et al. []. In real situatin, the alue f L can be btained by the methd f Schlichting. The elcity distributin f the liquid fluid in a gre at any psitin ( r, θ ) can be ealuated in equatin (3), while the alue f u r is btained. Furthermre, The lume flw rate, Q, thrugh any crssectin can be expressed as: ISSN: Issue 7, lume 6, July 009

4 WSEAS TRANSACTIONS n INFORMATION SCIENCE APPLICATIONS Q = φ r 0 r = 4 urdrdθ ( dp dz) μ 0 φ r 4 n ( ) r sinh a + csh a n= φ 8φ r φ sinh a φ 4φ r r θ + Hp sinh a csh a sin r φ 8φ r φ φ n r ( ) θ ( φ θ) sinh sinh n r + π a a + dθ 8a a φ + a sinh + a 4 4 φ r r ( dp dz) 4μ dθ 0 4 r (7) The parameters r, r, α Hp in equatin (7) are determined frm equatins (), (), (4) (5), respectiely. Cmparing Ma et al. [] analysis, Ctter [], by a dimensinless shape factr, K, presented the liquid mass flw rate as: K l ρlal dp m l = 8πμ dz (8) Where A is the liquid flw crss sectin area, ρ the l liquid density m = Q ρ. t l Q t is the liquid lume flw rate in heat pipe. Rearrange the abe equatin (8), Kmay be btained as 8πμ lnq Kl = dp Al dz (9) n is the gre numbers f heat pipe, Q the lume flw rate. The analytical elcity is gained by Furier transfrm methd linear superpsitin t analysis the gerning equatin fr a -shape gre. By the mmentum equatin near the liquid-apr interface t demnstrate the elcity near the interface, als define a dimensinless liquid-apr interface flw number L t shw the effect f shearing stress n the liquid elcity t calculate the liquid lume flw rate. Substitute the elcity distributin int Ctter s mdel define a dimensinless liquid flw factr K which shws the frictin effects n liquid-apr interface. These mdify the maximum heat transfer in Ctter,s equatin[] makes the numerical results mre clse t the Babin s experimental data[5]. l 3 Results Discussin The influence f shearing stress n liquid-apr interface in a micr heat pipe is predicted. T erify the mdel predictins, results are cmpared with the experimental data reprted by [] the numerical results by [9]. The frictinal frces n the liquid-apr interface culd reduce the elcity f liquid flwing frm cndenser area tward eaprating area. Calculate the lume flw rate f liquid-wick in -shape gre by analytical methd induce dimensinless parameter f flw factr L t inspect the frictinal frce effects n rhelgy. Frm equatin (9), K is prprtinal t lume flw rate Q, Q is related t the elcity scale crss sectin area A l, meanwhile the frictinal frce n the liquid interface induced by the reerse directin between apr flw liquid flw, thereafter reduced the liquid elcity. Operating temperature lies in 40 c 70 C in Babin et al[5], the cntact angle n the slid-liquid interface are in0 c 50 C. Pertersn Ma [8] presents the dimensinless apr-liquid interface flw number as L =0; ; ;.5. Accrding t Babin[5] experimental data fr a square heat pipe, the calculated result f Ha Pertersn [9] reprts that the alue f K =0.94, β =.343 L e =.7mm, L a =3.6mm, L c =.7mm. This research tries t inestigate the effects f shearing stress n liquid-apr interface. Fig. presents the relatinship between a dimensinless liquid flw factr K the cntact angle f slid-liquid interface L, the K changes alng with the L alue. It shws that the reerse steam flwing d hae the frictinal influences. And it als demnstrated that K increases alng with the increasing cntact angle f slid - fluid. Babin[5] biusly neglects the influence f cntact angle by fixed alue f K =0.6 Ctter[] simply put K =. Use the analytical methd in present study 0 under the cnditin L =0 with cntact angle f 0, the alue f K is 0.3, which is less than Ctter s predictin. Fig. shws the ariatin f K with the slid liquid cntact angle α in a square gre fr different alues f L. It is bsered that the lwer alue f K will shwn as increasing alue f L under same cntact angle α. Since the flw directin ISSN: Issue 7, lume 6, July 009

5 WSEAS TRANSACTIONS n INFORMATION SCIENCE APPLICATIONS f apr is reersed t the flw directin f liquid, it tends t induce a frictinal frce n the liquid-apr interface, the effects f apr flw reducing the liquid flw by frictin n interface is shwn. It is als shws that the alue f K increase with the increasing alue f cntact angle α.the physical dimensins f the triangular heat pipe are chsen as: Le = La = Lc = 5mm, β =. 433, K =0.86. By using the analytical methd in present study, ne can get the alues f K. The K inles a crrectin factr cmparing t the K in Ma et al. [0] researches, which is directly deried frm Xu[3]. Ma K et al. [0] defines l as 4πc Kl = 4 κβ (30) where 3 c= 4c tan φ, π c = + φ, tanφ β = nc (3) The dimensinless liquid flw shape factr, K,is calculated t be K = by Ma et al.[6, 9], using the parameter K induced by the influence n the slid-liquid cntact angle withut the impacts f the shearing stress n liquid-apr interface. Under the cnditin f α = 0 ; L = 0, (i.e. withut the impacts f the shearing stress n liquid-apr interface),we can get the alue K = 0. 9, which is lwer than Ctter s predicted K alue. The maximum heat transprt capacity is impacted by the alues f K. The maximum heat transprt capacity presents a similar tendency between Ctter s predictin [] Babin s[5] experimental data, which is nly half the alues f Ctter s predictin. Fig.3 t Fig.6 shw the relatinship between the peratinal temperature the maximum heat transfer rate when the rund heat pipe is ψ= 30. The cntact angles (α) are respectiely 0, 0, With changing f the peratinal temperature (T), the impacts f different L n the maximum heat transfer rate are cmpared. Frm these daigrams we learn that when L becmes greater, the maximum heat transfer rate will decrease. Thereby we find that the frictin shear stress f air flw fluid flw wil lwer the maximum heat transfer rate. Fig.7 t Fig.0 shw the relatinship between the peratinal temperature the maximum heat transfer rate when the rund heat pipe is ψ= 30. The number f flw f the dimensinless liquid-air cntact surface is respectiely L =0, L =, L = L =.5. With changing f the peratinal temperature (T), the alues f the maximum heat transfer rate at different cntact angles (α) are cmpared. Frm bsering these daigrams we learn that the higher the cntact angle, the greater the maximum heat transfer rate. Fig. t Fig.6 shw the relatinship between the peratinal temperature the maximum heat transfer rate when the rund heat pipe is ψ= 30. Respectiely the peratinal temperatures are 40, 45, 50, 55, With changing f the cntact angles (α), the alues f the maximum heat transfer rate at different peratinal temperature (T) are cmpared. Frm these daigrams we can tell that the higher the peratinal temperature, the greater the maximum heat transfer rate. Cmparing the utcmes abe with Babin experiment Ctter predictin, we can see that as the cntact angles (α) increases, the thermal perfrmance is enhanced; that as number f flw f the dimensinless liquid-air cntact surface L increases, the thermal perfrmance will be cmprmised; that as the peratinal temperature (T) rises, the thermal perfrmance will be impred. Fig.7 presents the relatinship between the peratin temperature the maximum heat transprt capacity fr a square micr heat pipe under different alues f L. The cntact angle f slid-liquid interface is 0, The results presents cmparisn f the maximum heat capacity with the numerical results gien by Ctter the experimental results gien by Babin[5]. Fig.7 shws the maximum heat capacity increase with increasing alue f the cntact angle f the slid-liquid interface, but it decreased with higher alues f L. The shearing stress induced by the frictin frce n the liquid-apr interface will reduce the maximum heat transprt capacity. In ther wrd, the reerse directin between the flw directin f liquid apr will hinder the maximum heat transprt capacity f the heat pipe. Fig.8 t Fig.0 shw the relatinship between the included angle (ψ) f the trugh pening the maximum heat transfer rate when the peratinal temperature f the rund heat pipe is T= 40. The cntact angles (α) are respectiely 0, ISSN: Issue 7, lume 6, July 009

6 WSEAS TRANSACTIONS n INFORMATION SCIENCE APPLICATIONS Fig. t Fig.6 shw the relatinship between the included angle (ψ) f the trugh pening the maximum heat transfer rate when the cntact angles f the rund heat pipe is α= 0. Respectiely the peratinal temperatures are 40, 45, 50, 55, Fig.7 t Fig.9 presents the relatinship between the peratin temperature the maximum heat transprt capacity fr a square micr heat pipe under different alues f the cntact angle n slid-liquid interface with L =0; ; ;.5. It shws the maximum heat capacity increase with increasing alue f the cntact angle f the slid-liquid interface. The maximum heat transprt capacity is cincide with the experimental results f Babin [5] under L =. 5 α = 0, the same result f the nset f dryut is presented as the experimental data f Babin[5] under α = 0. Fr circular heat pipe when ψ=3 (gre angle 6 ) may btain the biggest thermal prperties. 4 Cnclusin A mathematical mdel in square gres with apr flw crssing the liquid-apr interface in micr-heat pipes is deelped [0]. Under the same cntact angle f liquid-slid interface gre angle, the maximum heat transfer rate K will decrease with increasing alues f L. The fractinal frce n the liquid-pr interface will slw dwn the liquid elcity lwer the maximum heat transfer rate the alues f K. Under the same dimensinless parameter f liquid-slid interface L gre angle, the maximum heat transfer rate K will increase with increasing alues f cntact angleα. Acknwledgement: The present inestigatin was supprted by Natinal Science Cuncil f Taiwan under granted NSC-97--E References: [] Ctter, T. P., Prceedings Prspects f the Micr Heat Pipe, Prceedings the 5th Internatinal Heat Pipe Cnference, Tsukuba, Japan. 984, pp [] Ha, J. M. Petersn, G. P., The Heat Transprt Capacity f Micr Heat Pipes, Transactin f ASME Jurnal f Heat Transfer, l. 0, 998, pp [3] Xu, X., Carey,. P., Film Eapratin Frm a Micr-Gred Surface-An Apprximate Heat Transfer Mdel Its Cmparisn with Experimental Data, AIAA J. f Thermphysics Heat Transfer, l.4, N.4, 990, pp [4] Ayyaswamy, P. S., Cattn, I. Edwards, D.K., Capillary flw in triangular gres, ASME J, l.4, 974, pp [5] Babin, B.R., Petersn, G. P., Wu, D., Steady-Sate Mdeling Testing f a Micr Heat Pipe, ASME Jurnal f Heat Transfer,l., 990, pp [6] Ma, H. B., Petersn, G. P. Lu, X. J., The Influence f apr-liquid Interactins n the Liquid Pressure Drp in Triangular Micrgres, Int. J. Heat mass Transfer, l. 37, N.5, 994, pp. -9. [7] Khrustale, D., Faghri, A., Thermal analysis f a micr heat pipe, ASME Jurnal f heat transfer, l.6, 994, pp [8] Ma, H. B., Petersn, G. P., Theretical Analysis f the Maximum Heat Transprt in Triangular Gres - a Study f Idealized Micr Heat Pipe, ASME Jurnal f Heat Transfer, l.8, 996, pp [9] Ma, H. B., Petersn, G. P., Experiment Inestigatin f the Maximum Heat Transprt in Triangular Gre, ASME Jurnal f Transactins, l.8, 996, pp [0] Khrustale. D., Faghri. A., Thermal Characteristics f Cnentinal Flat Miniature Axially Gred Heat Pipe, ASME Jurnal f Transactin, l.7, 995, pp [] J. an Beal E. Peeters, Ealuatin f the first micr cmbined heat pwer fr scial husing Belgium, Prceedings f the 6th WSEAS Internatinal Cnference n Pwer Systems, 006. [] Kung, K.Y., Hsu, Cheng Hsing Ku, G.C., A study f liquid-apr interface interactin effects in a micr heat pipe WSEAS Transactins n Mathematics, l.5, N.8, 006, pp [3] Nerg, Janne, Thermal analysis f a high-speed slid-rtr inductin mtr with a slitted slid-rtr, WSEAS Transactins n Circuits Systems, l.5, N.3, 006, pp [4] Attar, S. Sharifian, Yagub, M.C.E., Mhammadi, F., Simulatin f electr-thermal effects in deice circuit, WSEAS Transactins n Circuits Systems, l.5, N.7, 006, pp ISSN: Issue 7, lume 6, July 009

7 WSEAS TRANSACTIONS n INFORMATION SCIENCE APPLICATIONS [5] Fraguela, Andrés, Infante, Juan-Antni, Rams, Ángel Manuel, Rey, Jsé María, Identificatin f a heat transfer cefficient when it is a functin depending n temperature, WSEAS Transactins n Mathematics, l.7, N.4, 008, pp L=0 L= L= L=.5 K l α cntact angle(degree) Fig. 3 The maximum heat transprt capacity ersus peratin temperature fr arius L α= 0. Fig. Dependence f K n the cntact angle α fr arius L with Square shape heat pipe. 0.6 L=0 L= L= L= K l α cntact angle(degree) Fig. 4 The maximum heat transprt capacity ersus peratin temperature fr arius L α= 0. Fig. Dependence f K n the cntact angle α fr arius L with equilateral triangle heat pipe. Fig. 5 The maximum heat transprt capacity ersus peratin temperature fr arius L α= 30. ISSN: Issue 7, lume 6, July 009

8 WSEAS TRANSACTIONS n INFORMATION SCIENCE APPLICATIONS Fig. 6 The maximum heat transprt capacity ersus peratin temperature fr arius L α= 40. Fig.9 The maximum heat transprt capacity ersus peratin temperature fr arius α L =. Fig. 7 The maximum heat transprt capacity ersus peratin temperature fr arius α L =0. Fig. 0 The maximum heat transprt capacity ersus peratin temperature fr arius α L =.5. Fig. 8 The maximum heat transprt capacity ersus peratin temperature fr arius α L =. Fig. The maximum heat transprt capacity ersus cntact angle fr arius L T= 40. ISSN: Issue 7, lume 6, July 009

9 WSEAS TRANSACTIONS n INFORMATION SCIENCE APPLICATIONS Fig. The maximum heat transprt capacity ersus cntact angle fr arius L T= 45. Fig. 5 The maximum heat transprt capacity ersus cntact angle fr arius L T= 60. Fig. 3 The maximum heat transprt capacity ersus cntact angle fr arius L T= 50. Fig. 6 The maximum heat transprt capacity ersus cntact angle fr arius L T= 65. Fig. 4 The maximum heat transprt capacity ersus cntact angle fr arius L T= 55. Fig. 7 The maximum heat transprt capacity ersus channel angle fr arius L α = 0. ISSN: Issue 7, lume 6, July 009

10 WSEAS TRANSACTIONS n INFORMATION SCIENCE APPLICATIONS Fig. 8 The maximum heat transprt capacity ersus channel angle fr arius L α= 0. Fig. The maximum heat transprt capacity ersus channel angle fr arius L T= 40. Fig. 9 The maximum heat transprt capacity ersus channel angle fr arius L α= 30. Fig. The maximum heat transprt capacity ersus channel angle fr arius L T= 45. Fig. 0 The maximum heat transprt capacity ersus channel angle fr arius L α= 40. Fig. 3 The maximum heat transprt capacity ersus channel angle fr arius L T= 50. ISSN: Issue 7, lume 6, July 009

11 WSEAS TRANSACTIONS n INFORMATION SCIENCE APPLICATIONS Fig. 4 The maximum heat transprt capacity ersus channel angle fr arius L T= 55. Fig. 7 The maximum heat transprt capacity ersus channel angle fr arius T L =0. Fig. 5 The maximum heat transprt capacity ersus channel angle fr arius L T= 60. Fig.8 The maximum heat transprt capacity ersus channel angle fr arius peratin temperature T L =. Fig. 6 The maximum heat transprt capacity ersus channel angle fr arius L T= 65. Fig. 9 The maximum heat transprt capacity ersus channel angle fr arius T L =. ISSN: Issue 7, lume 6, July 009