Journal of Alied Fluid Mechanic, Vol. 9, No.,. 1569-1577, 216. Available online at www.jafmonline.net, ISSN 1735-3572, EISSN 1735-365. DOI: 1.18869/acadub.jafm.68.235.21 Numerical Invetigation of Heat Tranfer and Fluid Flow around the Rectangular Flat Plane Confined by a Cylinder under Pulating Flow G. Li, Y. Zheng, H. Yang and Y. Xu Deartment of Energy and Environment Sytem Engineering, Zhejiang Univerity of Science and Technology, Hangzhou, 3123, P. R. China Correonding Author Email: gnlene@163.com (Received October 8, 21; acceted October 1, 215) ABSTRACT Fluid flow around and heat tranfer from a rectangular flat lane with contant uniform heat flux in laminar ulating flow i tudied, and comared with our exerimental data. Quantitatively accurate, econd-order cheme for time, ace, momentum and energy are emloyed, and fine mehe are adoted. The numerical reult agree well with exerimental data. Reult found that the heat tranfer enhancement i caued by the relative low temerature gradient in the thermal boundary layer, and by the lower urface temerature in ulating flow. In addition, the heat tranfer reitance i much lower during revere flow eriod than that during forward flow eriod. The flow reveral eriod i about 18 degree behind the ulating reure wave. Beide, ectrum reult of the imulated averaged urface temerature howed that the temerature fluctuate in multile-eaked mode when the amlitude of the imoed ulation i larger, wherea the temerature fluctuate in a ingle-eaked mode when the amlitude of the imoed ulation i mall. Keyword: Pulating flow; Heat tranfer enhancement; Comutational fluid dynamic. NOMENCLATURE A D c f h H L Nu=hL/λ Nur=Nu/Nu q Re Rh,d Rh,t Rh,ave S h St T T Tw t ulating velocity comonent diameter of the airflow tunnel heat caacity of air ulating frequency convection coefficient height of the airflow tunnel length of the late heater Nuelt number heat tranfer enhancement factor reure amlitude Inut heat flux Reynold number time averaged heat reitance ace averaged heat reitance total averaged heat reitance area of the late heater heat ource Strouhal number tatic temerature inlet temerature averaged urface temerature time u u w y y + zb α λ ρ ε ν σb σ θ Subcrit rad rm intant velocity vector average inlet velocity axial velocity ditance of the firt grid centroid dimenionle ditance ditance along the flat lane abortivity of the heater thermal conductivity of air denity of air emiivity of the heater kinematic vicoity Stefan-Boltzmann cont thermal boundary layer thickne hae angle ulation radiation root mean quare teady
G. Li et al. / JAFM, Vol. 9, No.,. 1569-1577, 216. 1. INTRODUCTION The fluid flow and convective heat tranfer in ulating flow i encountered in many engineering alication, uch a the comact heat exchanger, cooling ytem of nuclear reactor, ule combutor, ule tube dry cooler, electronic cooling ytem and arterial flow inide human body. Therefore, great interet i being attracted to find out in what degree doe the ulation of airflow enhance the heat tranfer roce. Many analytical, exerimental and numerical tudie exit. However, due to it comlicated nature, i.e. many arameter of ulating flow affect the heat tranfer roce, reviou literature continue a to whether the ulation enhance the heat tranfer roce or not. In thoe work concluded in oitive effect by ulation, the exactly hyical mechanim of in what way doe the ulation enhance the heat tranfer roce i till unclear, and need further invetigation. Detail CFD (Comutational Fluid Dynamic) tudie are needed to find out thi mechanim. However, even CFD tudie of heat tranfer in ulating flow are difficult until the large comuter reource could be available in recent year. CFD tudie of heat tranfer need fine comutational grid, the length of which hould be much maller than the thickne of thermal boundary layer. Among thoe imortant ulating arameter affecting the heat tranfer roce, the ulating frequency and reure amlitude are two mot imortant one. However, reviou work how a variety of contradictory reult. In the exerimental tudie, (Habib et al. 22) invetigated the heat tranfer enhancement of a tube with uniform heat flux in laminar ulating flow at variou condition (f=1-29.5 Hz, Re=78-1987). A heat tranfer model Nur = Nur (f, Re) wa develoed with the deviation of about 1% in different range of Reynold number. (Dec et al.1992) tudied the heat tranfer rate in ule combution tail ie in a erie of condition (f=67-11 Hz, Re=375). It concluded that the reure amlitude mut be trong enough to create flow reveral in order to obtain oitive heat tranfer enhancement factor. (Kearney et al. 21) invetigated the time reolved tructure of a thermal boundary layer in laminar ulating channel flow, concluding that the flow reveral i not neceary for heat tranfer enhancement. A few year later, (Moon et al. 25) confirmed Kearney concluion (Kearney et al. 21). Moon work reorted heat tranfer enhancement from a rectangular heated block array in a ulating channel flow at variou condition ( f=1-1 Hz, A=.2-.3). Reult howed that the heat tranfer enhancement factor can be larger than 1.2. Many other valuable exerimental tudie reorted an enhancement in heat tranfer due to the ulation (Zohir et al. 26, Ji et al. 28, Baffigi and Bartoli 21). However, ome tudie uch a Elayed et al. work (Elhafei et al. 28), reorted the ulation would downgrade the heat tranfer comared to that in teady flow. In analytical tudie, (Faghri et al. 1979) reorted that larger heat tranfer enhancement factor could be obtained at higher ulating frequency. However, (Hemida et al. 22) found that the heat tranfer enhancement factor increae with reure amlitude, but decreae with ulating frequency. Some tudie (Chang and Tucker, 2, Chattoadhyay et al. 26) reorted that the ulation ha no effect on the heat tranfer roce. Other work reented valuable analyi about the influence of Prandtl number (Moon et al. 25), hae lag and urface temerature ditribution (Nika et al. 27) on the heat tranfer roce. In numerical tudie, (Thyagewaran 2) develoed a near-wall turbulence model for heat tranfer in ulating flow. (Wang and Zhang 25) found that there i an otimum Womerley number at which heat tranfer i maximally enhanced, which i confirmed by Akdag work (Akdag 21). (Selimefendigil et al. 212) invetigated the nonlinearity of the unteady heat tranfer of a cylinder in ulating cro flow. They develoed an accurate, low-order model for the heat ource dynamic in a Rijke combutor, which i tudied carefully in one of our reviou work (Li et al. 28). The mechanim controlling the heat tranfer enhancement in ulating flow i unclear, and need to be further exlored. Exerimental data i limited, and it i difficult to meaure the velocity/temerature rofile inide the flow/thermal boundary layer. Few CFD tudie exit (Thyagewaran 2, Wang and Zhang 25, Akdag 21, Selimefendigil et al. 212), and more detail analyi i obviou needed, uch a the ulating mode of the temerature field, time evolution of thermal boundary layer and ace ditribution of heat tranfer reitance. In thi work, a detail CFD tudy wa carried out to exlore the mechanim controlling the heat tranfer in ulating flow. Numerical reult were comared with our exerimental data, and detail dicuion were made. 2. NUMERICAL MODEL Thi art reent the detail of the governing equation and the boundary condition. A brief introduction on the exerimental etu and exerimental rocedure i alo reented in order to make the reent tudied roblem more ecified. 2.1 Governing Equation Prediction of the fluid flow and heat tranfer in ulating flow require CFD comutation with a comreible flow model. However, uch comutation demand huge comutational reource, ince the heat tranfer in ulating flow involve variou hyical henomena covering a wide range of length and time cale. Conidering that the ulation wave length are much larger than the characteritic length of the lane heater, the unteady, incomreible Navier-Stoke equation along with the energy equation i olved with ANSYS-FLUENT 13. egregated olver, a follow, 157
G. Li et al. / JAFM, Vol. 9, No.,. 1569-1577, 216. Continuum equation ui x i Momentum conervation equation 2 u u i iu j 1 u i t x 2 j x i x j Energy conervation equation 2 T u jt k T 2 h t x j c x j (1) (2) (3) where i, j=1,2,3, ρ i the air kinetic vicoity, c i the heat caacity, u i air flow velocity vector, x denote the atial coordinate, tand for tatic reure, T i the tatic temerature. Combining with very fine mehe, the laminar flow method give direct numerical imulation reult. In thi work, relative fine mehe, econd-order cheme for time, ace, momentum and energy, are ued. Such comutation allow week turbulence to be roerly olved under the reent Reynold number of 219. Radiation i taken into conideration, and the P1 model i emloyed. The lane heater i uoed to be a grey obtacle, and the emiivity ε i et to be.68 (Jone et al. 1977). The heat ource in Eq. (3) i a follow (ANSYS Inc. 21), b h T T () c c Where σb i the Stefan-Boltzmann contant, ε and α i the emiivity and abortivity of heater, reectively. 2.2 Boundary Condition The comutational zone and the comutational mehe in the croing ection of the lane heater are hown in Fig. 1, which i conitent with the exerimental tet ection (Li et al. 21). The comutational zone i a circle channel with inner diameter D=68 mm and height H=21 mm. A rectangular lane heater with dimenion of 7 mm 2 mm 1.5 mm i fixed in the middle of the channel. The 7 mm ide of the heater i aligned with channel axi. The Reynold number baed on the averaged channel velocity and the channel diameter i 219. Conidering that the comutation hould cover detail information inide the boundary layer, the length cale of the firt grid around the heater fulfill the following demand (Breuer et al. 23) y<y=.283 mm y+=1 (5) y (u/νzb) -.5 (6) Where y i the ditance to the wall from the adjacent grid centroid, u i the averaged velocity inide the circle channel, and zb denote the ditance along the wall from the tarting oint of the boundary layer. Block artition grid method i emloyed to ave comuter reource and comutation time. The number of comutational mehe i about 2,5, with emhaize around the lane heater. The effect of the y / y + on the numerical reult when rm=75 Pa i hown in Table 1. A hown in Table 1, y / y + =.35 i fine enough ince the temerature difference i le than.5 K for the cae of y / y + =.28. Table 1 Effect of y / y + on the numerical reult y / y + Grow Factor Number of Grow Traniton attern Tw (K).28 1.5 16 :2 333.1.35 1.5 16 :2 333..63 1.5 16 :2 33.7.78 1.5 16 :2 335.6 1.17 1.5 16 :2 337. Fig. 1. Comutational zone and the meh ditribution in the croing ection. Temerature deended hyical roertie of air (Yang and Tao 1998) including denity, ecific heat caacity, thermal conductivity and vicoity, are ued with everal UDF (Uer Define Function) rogram. At the inlet and outlet, velocity inlet and reure outlet boundary condition are imoed, reectively. The ulation i imoed to the inlet uing a UDF rogram to create reure ocillation (Roux et al. 211), a follow, u(=u(1+ain2πf (7) where A denote the ulating comonent of velocity at the inlet urface. All UDF rogram are written in C language, and interret into the main rogram. The heat flux wall boundary condition i emloyed to decribe the heated lane, in which the wall temerature i calculated with the following equation (ANSYS Inc. 21), T w q q h rad T (8) where Tw i the wall temerature, q i total heat flux from the heated lane, which equal to. W in thi work. qrad i the radiation heat flux, which equal to h in Eq. (). h denote the convection heat tranfer coefficient. In thi work, the ulating frequency i maintained at 15 Hz, and A i modified to roduce different reure amlitude (2-175 1571
G. Li et al. / JAFM, Vol. 9, No.,. 1569-1577, 216. Pa) according to exerimental condition. Time te i et to be 1-3. The global convergence for continuity, momentum and energy reidual are et to be 1 -, 1 -, and 1-7, reectively. Detail of the exerimental etu and the exerimental rocedure can be found in our reviou tudie (Li et al. 21, Li et al. 213). In brevity, the air flow rate i et to be 1 666.7 ml/ by a ma flow controller. Two-tage erforated late are intalled to inure the uniformity of inlet airflow. A High Temerature Co-fired Ceramic (HTCC) rectangular lane heater with ame dimenion a Fig. 1 i fixed in the middle of the channel. A loudeaker i intalled utream the inlet to roduce ulating wave. A corrugated vibration-abortive tube i intalled to avoid vibration tranmiion from the loudeaker to the HTCC heater. A reure tranducer i intalled to meaure the reure wave. Then ectrum analyi i carried out to find out the ulating frequency and reure amlitude. The ower of the HTCC heater i ket to be.w, and the ulating frequency i maintained at f=15 Hz, and the reure amlitude, rm, i varied in the range of 25-175 Pa. All meaured exerimental data have an inaccuracy le than.1% (Li et al. 21). 3. RESULTS AND DISCUSSIONS Verification of the CFD model i reented at firt with the hel of exerimental data. Then the temerature ditribution and the temerature ulating mode are dicued. Finally, the mechanim of heat tranfer enhancement in ulating flow i exlored. 3.1 Verification of CFD Model The imulated averaged urface temerature and the exerimental data i hown in Fig. 2. The heat tranfer enhancement factor Nur can be calculated with above temerature data, which i hown in Fig. 3. The calculation follow the following formula,. q S T T ) h( T T ) (9) b ( hl Nu (1) Nu Nur Nu q q b b A( T A( T T ) ( T T ) ( T T ) T ) (11) where q and T are the conumed ower of the lane heater and the averaged urface temerature of the HTCC heater in ulating airflow, reectively, wherea the q and T are the correonding value in teady airflow, reectively. λ and λ are the heat conductivity of the air in ulating airflow and in teady airflow, reectively. T i the temerature of inlet airflow, and it i maintained at 296.5 K. A hown in Fig. 2, the imulated temerature agree well with the exerimental data. However, CFD comutation over-redict (about 3. K, le than 1.%) the averaged urface temerature in ulating condition with large reure amlitude (rm 1 Pa). A a reult, the imulated heat tranfer enhancement factor i under-redicted (about 1.%) in thoe cae, hown in Fig. 3. Thi finding reveal that the accurate rediction of temerature field i critical to redict the heat tranfer enhancement factor. Fig. 2. Comarion between imulated averaged urface temerature and exerimental data. Fig. 3. Comarion between imulated heat tranfer enhancement factor and exerimental data. Small dicreancy of temerature field could caue very large difference in rediction of heat tranfer enhancement factor. Therefore, fine comutational mehe, eecially around the heater, hould be adoted. It i exected that the averaged urface temerature i alo time deended. In other word, Fig. 2 reent the averaged value of the averaged urface temerature. The firt average refer to time averaged value, and the econd average refer to atial averaged value. The time evolution of the imulated averaged urface temerature (atial averaged) i hown in Fig.. Single-eaked fluctuation (15 Hz) could be found in ulating condition with low reure amlitude ( rm 5 Pa). However, econd harmonic fluctuation are obviouly excited when the reure amlitude i 1572
G. Li et al. / JAFM, Vol. 9, No.,. 1569-1577, 216. large ( rm 1Pa). The eak-eak amlitude (about 3. K to 7. K) of the averaged urface temerature decreae firtly, and then increae with the reure amlitude. The correonding ectrum analyzed reult are hown in Fig. 5. The temerature amlitude at 15 Hz decreae firtly, and then increae with reure amlitude. However, the temerature amlitude at 3 Hz i larger than that at 15 Hz when the reure amlitude lie between 1 Pa and 175 Pa. Thi henomenon i accomanied by the dicreancy between imulated averaged urface temerature and the exerimental data. However, there are no evidence to correlate thee two henomena, and thi need to be further tudied. frequency of the CFD comutation i 1-3 and 15 Hz, reectively. Obviou difference can be found in Fig. 6. The temerature level of the averaged temerature field in ulating condition i much lower than that in teady flow condition. Therefore, it worth to robe into the ulation mode of the temerature field in ulating condition. Firtly, a ul-ation cycle wa elected when the reure amlitude equal 75 Pa, which i hown in Fig. 7. Fig. 7 alo reent the time evolution of z-velocity. Secondly, eight intant are elected, which are hown in Fig. 7, marked with mall circle. (a) Steady (b) Pulation Fig.. Time hitory of the imulated averaged urface temerature. Fig. 6. Temerature field of the teady tate and the averaged temerature field of the ulating tate when rm=75 Pa. Fig. 5. Sectrum reult of the imulated averaged urface temerature ulation. 3.2 Temerature Field The averaged temerature field in the lane of x= m i reented in Fig. 6. In the teady flow condition, the temerature increae in the z direction near the wall urface, and the temerature ditribution hae i conitent with thoe in textbook. In the ulating condition, the temerature field i an averaged temerature field from 66 intant temerature field within a ulation cycle, ince the time te and ulating Fig. 7. One ulation cycle for the tudy of temerature field when rm=75 Pa The correonding 8 temerature field are hown in Fig. 8. The local temerature near the wall along the z direction could reach 38 K ( θ=, θ=.96π), however, the averaged local temerature are le than 33 K hown in Fig. 6. The temerature field fluctuate with time, and aear in a ecial mode. In the firt half cycle of reure wave, heated area locate in the utream zone when the flow i in forward, wherea thee area change to locate in 1573
G. Li et al. / JAFM, Vol. 9, No.,. 1569-1577, 216. the downtream zone in the econd half cycle when the flow i in reveral. much larger than that in teady flow condition, which i hown in Fig. 1. Conidering that the heat tranfer reitance i direct roortional with the temerature difference, and i invere roortional with boundary layer thickne. Therefore, the heat tranfer reitance in ulating condition i much le than that in teady flow condition. Fig. 9. Ditribution of the imulated temerature in the line (y=-.75 m, x= m,.7 m z.1 m) in the teady tate and in the ulation tate when rm=75 Pa. Fig. 8. Ocillating temerature field during one cycle in the lane of x= m when rm=75 Pa Thi imlie that the flow reveral occurred in the econd half cycle, in other word, the flow reveral i inconitent with the oitive reure wave. One oible mechanim for the dilacement between the ocillating temerature and the flow reveral may caued by the inconitent between the reure-induced ulating velocity and the ulating reure during one cycle. The highet reure occur when the higher temerature fluid i in the utream zone, acting a a iton to the incoming fluid. Thi i confirmed in Fig. 7-8. According the reviou tudie (Pedley, 1976), the actual ulating velocity amlitude deart from the reure-baed ulating velocity amlitude excited by reure gradient when the Strouhal number St >.3, and lag behind the ulating reure wave. In the reent work, the etimated Strouhal number i larger than 2., therefore, the flow reveral lag behind the ulating reure wave, and cauing the above ocillating temerature mode. The ditribution of imulated temerature near the left ide of lane heater i hown in Fig. 9. In teady flow condition, the temerature increae harly along the heater, and then increaed in a tender way. However, the temerature ditribution i very flat in the ulating condition. In order to find out the mechanim controlling the heat tranfer enhancement in ulating flow, thermal boundary need to be tudied. The averaged thickne of thermal boundary layer in ulating condition i Fig. 1. Ditribution of the imulated thermal boundary layer thickne in the area (y< m, x= m,.7 m z.1 m) in the teady tate and in the ulation tate when rm=75 Pa. 3.3 Heat Tranfer Enhancement Mechanim A hown in Fig. 3, both the exeriment and CFD comutation how that the heat tranfer could be enhanced greatly in ulating flow. However, the mechanim controlling thi enhancement i till unclear. In order to make ome dicuion, everal ecified heat tranfer reitance are defined a follow, R h, d Tw ( z, 299.5 dt ( z, (12) 157
G. Li et al. / JAFM, Vol. 9, No.,. 1569-1577, 216. R R h, t h, ave Tw ( z, 299.5 dz ( z, Tw ( z, 299.5 dzdt ( z, (13) (1) where the temerature of 299.5 K i the definition of the location of thermal boundary, and equal to 11% of the inlet temerature. Rh,d, Rh,t and Rh,ave are the time averaged heat tranfer reitance, ace averaged heat tranfer reitance and total averaged heat tranfer reitance, reectively. Tw i the urface temerature, and δ i the thickne of thermal boundary layer. Time averaged heat tranfer reitance and ace averaged heat tranfer reitance are hown in Fig. 11 and Fig. 12, reectively. In teady flow condition, the curve in Fig. 11-12 are calculated from Fig. 6(a). In ulating condition, the curve in Fig. 11-12 are calculated from 66 intant cae within a comlete cycle hown in Fig. 7. It hould be reminded that Fig.11 how the atial ditribution of the heat tranfer reitance and Fig.12 how the time evolution of the heat tranfer reitance. time. However, the heat tranfer reitance in anytime i lower than that in teady flow condition. Reult found that the heat tranfer reitance i much lower during revere flow eriod than that during forward flow eriod. The flow reveral eriod i about 18 degree behind the ulating reure wave. Thi finding exand the concluion of reviou work (Dec et al. 1992, Thyagewaran 2, Wang and Zhang 25, Akdag 21, Selimefendigil et al. 212), i.e. the finding in thi work reveal that the heat tranfer i more enhanced in flow reveral eriod during one ulation cycle. Fig. 12. Evolution of the imulated heat reitance in ulation cycle when rm=75 Pa.. CONCLUSION Fig. 11. Ditribution of the imulated heat reitance in the teady tate and in the ulation tate when rm=75 Pa. The heat tranfer reitance hown in Fig. 11 decreae along the lane heater in both teady and ulating flow condition. A hown in Fig. 9-1, the temerature increae along the heater, while the boundary thickne alo increae along the heater, which i invere roortional to the heat tranfer reitance. In other word, the temerature gradient inide the thermal boundary layer i the dominant arameter. The final reult reveal that the temerature gradient in ulating flow i much le than that in teady flow condition. The total averaged heat tranfer reitance i 155 K/m 2 in teady flow condition, while it equal to 552 K/m 2 in ulating flow condition. Thi imlie that the heat tranfer enhancement in ulating flow i generated by larger maller temerature gradient in the thermal boundary layer, and by lower urface temerature. It i worth to find out which time create the lowet heat tranfer reitance during one ulation cycle. A hown in Fig. 12, ace averaged heat tranfer reitance fluctuate with The heat tranfer from a rectangular lane heater in teady and ulating flow i exlored numerically and exerimentally. Fine comutational mehe, econd-order cheme for time, ace, momentum and energy, are emloyed. The numerical reult agree well with exerimental data. Several concluion are drawn a follow, (1) Heat tranfer enhancement in ulating flow i generated by relative low temerature gradient in the thermal boundary layer, and by lower urface temerature. (2) Heat tranfer reitance i much lower during the revere flow eriod than that during forward flow eriod. In addition, the flow reveral eriod i about 18 degree behind the ulating reure wave. (3) The imulated ace averaged urface temerature of the lane heater i time variant, and fluctuate in multile-eaked mode when the amlitude of the imoed ulation wa larger enough. ACKNOWLEDGEMENTS Thi work ha been uorted by the National Natural Science Foundation of China ( 517615, 517616, 51161), the Zhejiang Provincial Natural Science Foundation of China (Z111695), 1575
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