Friction resistance for gas flow in smooth microtubes

中国科学(英文版)E0027中国科学E旧万数据资源系统WANFANG DATA ( CHINAINFO)SCIENCE IN CHINA E数字化期刊2000Vol43No2P.171-17Friction resistance for gas flow insmooth microtubesDU Dongxing(杜东兴)Department of Engineering Mechanics, Tsinghua University, Beijing 100084, ChinaLI Zhixin(李志信)Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China)GUO ZEn(过增元)Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China)Abstract: A new tube-cutting method was used to measure the pressure and Mach number distributionalong a microtube of 108.3 H m. Experiments were also performed concerning the average Fanning frictionfactors of five kinds of microtubes whose diameters range from 80.0 to 166.6 H m. It is found that thepressure distribution in a microtube becomes nonlinear at a high Mach number and the product of measuredaverage Fanning friction factors Cf and Reynolds number Re is higher than 16. Numerical results show thatthe gas compressibility leads to a variation of the velocity profile from parabolic, and results in a largevelocity gradient at the tube inner wall surface. The transition from laminar to turbulence in microtubes alsooccurs at Re=2 300, and the phenomenon of early transition is not observed in the experimentsKeywords: microtubes, gas flow, friction resistance. 4Micro-channels of diameter ranging from 100 to 0.1 u have found applications in such areas asintegrated cooling of the electronic components, micro heat exchangers, biochemical applications, andfuture.Therefore, the study of characteristics of microscale heat and momentum transfer is becoming an nearmicroelectromechanical systems(MEMS). It is more than likely that new applications will be found in the limportant subject of recent investigationHowever, the experimental data reported by different authors are not in accord with each other The resultsof Wuand Little[ 1] showed higher friction factors than those expected from classical Moody-Chart Pfahler[2, 3] observed that the measured friction coefficients were 15% lower than those predicted by conventionaltheory. Choi 4] reported that the product of friction factor f and Reynolds number Refor gas flow inmicrotubes whose inside diameters range from 3.0 to 812u mis 50.253.3, Which islower than 64 forincompressible laminar flow. The experimental results of H arley et al. [5] was in good agreement with thetheoretical predictions assuming isothermal, locally fully developed gas flow.Compressibility is an important influential factor to the characteristics of gas flow and heat transfer inmicrochannels. Some researchers concluded that compressibility can bring on extra pressure drop due to gasacceleration 6, 7], but have little effect on the characteristics of frictional resistance for gas flow in microtubes.The heat transfer group of T singhua University conducted some theoretical and experimental investigations onthe effect of compressibility. The analytical results and numerical simulations presented by Guo et al. [89]showed that the local fanning friction factor is a function of local Mach number in microtubes, and the frictionfactor can be up to 30% higher than classical value when the local Mach number is 0. 5. the experimental resultsof liet al. 10 also presented ahigher friction factor for gas flow in microtubes中国煤化工Since no instruments with fine spatial resolution were available, most experCNMHGmicrotubes were limited to the measurement of global quantities such as the mass flow rate, the inlet and outletpressures of the microchannel. Few researchers got the pressure distribution although it is important for thefe/EVqk/zgkx-egx200000000hmn(第1/8页)2010323155546中国科学(英文版)E0027study on the characteristics of fluid flow in microchannels. Pong et al. 11] measured the pressure distributionalong a micro-sized channel which is 3 000 u min length, 1.2 u m in height and 40 u m in width and observedthat the pressure distribution is not a linear function of the streamwise distance and the effect of Knudsonnumber is obvious.In this paper, a new"tube- cutting method is used to measure the pressure distribution along aperformed to discuss the effect of compressibility for gas filow in microtube d Numerical simulation is also1 Experimental principleThe basic principle of the tube- cutting method is based on the following fact: for steady adiabatic subsonicgas flow in microtubes, the mass flow rate G and the back pressure pb is one-to-one correspondence when theinlet stagnation pressure po and the stagnation temperature To are kept constant. In other words, thecharacteristics of fluid flow in a certain microtube can be determined by three parameters po to pb or po, to gThe experimental principle can be schemed by fig. 1, in which tube 1 and tube 2 are microtubes with thesame diameter and material Point M is on tube 1 and the distance between point a and point M is equal to thelength of tube 2. Keeping the inlet gas parameters constant, po, to of two microtubes equal each other and adjustthe back pressure of tube 2 to equate the mass flow rate of tube 2 to that of tube 1. From the analysis above, thecharacteristics of fluid flow in the part of AM of tube 1 is the same with tube 2, and the outlet pressure of tube 2,pg, is equal to the static pressure pM of tube 1. For subsonic tube flow, pB is equal to the back pressure pbWhich can be measured conveniently and accurately by conventional method. In fact, the essential of the tube.cutting method is to let the flow condition in section AM of tube 1 reappear using the shorter tube 2. Thepressure distribution along microtube 1 then can be obtained indirectly by employing microtubes of differerlength in this way the average characteristics of frictional resistance can be obtained by the method presented inref.[10]Tube lTFig. 1. Experimental principle.2 Experimental apparatusThe test microtubes were prepared using glass- drawing technology and precise knowedge of themicrotubes dimension is extremely important for the accurate evaluation of experimental results. The diameterof the microtubes was measured with a calibrated optical microscope and a scanning electron microscope(SEM), and the results obtained using the two methods agree within 2.5% of each other. Fig. 2 shows an image ofone of the test microtubes under SEM. It can be seen from the image that thetest中国煤化工 ed as asmooth microtube. the dimension of the microtube for pressure distributionCNMHGle 1While the dimensions of microtubes employed for the measurement of the average characteristics of frictionalresistance are shown in table 2.fle/EVqk/zgkx-egx200000000hmn(第2/8页)2010323155546中国科学(英文版)E0027Fig. 2. Cross section of a microtube under SEMTable 1 D imension of the microtube for pressure distribution measurementd;/Hm Different length l /mm219312390477108355765373.581.0Table 2 Microtubes employed for the measurement of the average characteristics of frictional resistanceNo d; /H ml/mm0031.22108.342903132.14804149366251663613The experimental apparatus is depicted schematically in fig 3. Nitrogen gas flows from a regulated highpressure cylinder into a buffer chamber, goes through the test tube into the back pressure chamber, then flowspast the pressure regulating valve and stationary flow valve, finally passes through a flow rate measurement unitinto the atmosphere. The pressure regulation valve and stationary flow valve make it possible to regulate the backpressure of microtubes continually and steadily. In the flow measurement unit, the volumetric flow rate of the gasflow is determined by timing the exit gas replacing the water in avessel whose volume was precisely measuredbeforehand∠=1112凵中国煤化工CNMHGFig. 3. Experimental apparatus. 1, High-pressure nitrogen cylinder; 2, pressure regulator; 3, ourTer chamber; 4,thermocouple: 5, potentiometer; 6, pressure transmitter; 7, precise millivoltmeter; 8, connecting piece: 9,testmicrotube: 10, back pressure chamber; 11, pressure regulation valve; 12, stationary flow valve; 13, flow ratefle//EVqk/zgkx-egx200000000hmn(第3/8页)2010323155546中国科学(英文版)E0027measurement unit.3 Results and discussion3.1 Pressure distribution along the microtubeThe tube-cutting method was used to measure the axial pressure distribution in the microtube whosedimensions are listed in table 1. Table 3 lists three different experimental conditions and fig 4 and fig. 5 show theexperimental results.t.0kp. cond. No. 1Exp, cond. No 2Exp cond. No. 308Numerical results060.4000204060.81.02.214fAvrEFig. 4. Pressure distribution along the microtube.a E xp cond. ND. IExp cond. No.2▲Exp. cond N03050402(rid/ReFig. 5. Local Mach number variation along the microtube.T able 3 Experimental conditions for the pressure distribution measurementNo. Stagnation pressure po/Pa Stagnation temperature To/K Inlet Mach number Min284130299.500087383491299.100.12477861299.100.14中国煤化工CNMHGfe/EVqk/zgkx-egx200000000hmn(第4/8页)2010323155546中国科学(英文版)E0027It is noted from fig, 4 and fig. 5 that at the forepart of the microtube, the Mach number is small and thepressure distribution is approximately linear; while at the rear part the mach number increases rapidly and thepressure distribution manifests different trend under different experimental conditions. The outlet Mach numberof experimental condition No. 1 is 0. 243, and the pressure drop still remains approximately linear due to lowercompressibility effect Under the experimental condition No. 2, the Mach number at the tube exit is higher(0.439)and the pressure drop deviates from linear distribution. In experimental condition No. 3, the outletMach number is up to 0.643 and the effect of compressibility is large enough, which leads to a higher pressuregradient at the exit of the microtube.Corresponding to three experimental conditions, numerical results were also presented by solid lines in fig4 and fig. 5. It is obvious that the calculation results agree well with experimental ones and then it can beconcluded that the tube-cutting method is feasible3.2 Average characteristics of frictional resistance in microtubeFig 6 shows the product of the measured average Fanning friction factor crand Reynolds number Reversus average Mach number Mave in microtubes. It is noted that at lower average Mach number( Mae<0.3)Which corresponds to lower gas compressibility effect, the cfRe in microtubes is almost the same as that ofincompressible laminar tube flow, while at higher Mave region(Mave>0. 3), the measured CrReis consistentlyhigher than the classical value and increases with the increasing average Mach number. For example, the value ofCrReisup to 18 at Mave=0.5, which is 12% higher than 16. It can be concluded from experimental results thatthe compressibility leads to a higher frictional resistance for gas flow in microtubes.d=108 3um20+166md=149.JmCrRes……”18000.1003506Fig. 6. Fricitional resistance for gas flow in microtubes.The frictional resistance of internal flow is dominated by the velocity gradient at the wall surface of the tube,but the velocity gradient cannot be determined until the velocity profile is known clearly Two dimensionalmodel for compressible gas flow in a smooth circular tube was used to analyze the experimental results. Fig. 7shows the dimensionless velocity profile at different local Mach number, from which it can be seen that thevelocity profile in microtubes is no longer parabolic, but is a function of local Mach number. the higher the localMach number, the more the velocity profile deviates from the parabolic one. It can be concluded that the gascompressibility in microtubes brings about an increase of velocity gradient at wall surface and therefore leads to ahigher frictional factor. The experimental results are clearly interpreted by numerical simulation中国煤化工CNMHGfle//EVqk/zgkx-egx200000000hmn(第5/8页)2010323155546中国科学(英文版)E00270004Fig. 7. Velocity profile at different local Mach number 1, Parabolic; 2, Mloc=0. 2 3, Mloc=0.3; 4, Mloc=0.5; 5,M1=063,3 Discussion on flow transition in microtubesTransition from laminar to turbulent flow in tubes is accompanied by a noticeable change in the law ofresistance For internal microflow systems, different researchers got different and even conflicting results. Wuetal. [1 and Peng et al. 12] reported the early transition from laminar to turbulent flow in microchannels, butChoi et al. 4] did not find the early transition phenomenonTodetermine the critical Reynolds number for gas flow in microtubes, the experimental results of Fanningfriction factors versus Reynolds number are plotted in fig 8. It can be clearly seen that when Reynolds number issmaller than 1 800, Cr is almost the same as the theoretical value for laminar flow in macrotubes, if 1 800Re2300, then cf is a little bit higher than classical values due to higher compressibility. While for Re> 2300, therelation between Cf and Re changes remarkably which cannot be totally attributed to the effect ofcompressibility and it indicates that the internal microflow undergoes a transition from laminar to turbulentregime. Then it can be concluded that the critical Reynolds number for gas flow in microtubes is about 2 300,which is in substantial agreement with the results in macro tubes=800um1083um0.05d-132.1pmd-1493pm+4=166.3m00lFig. 8. Fanning frictional factor versus reynolds number3. 4 Uncertainty analysisAn approximate analysis of uncertainty was carried out, using Darcy equation for inromnrescihlenipe flow.中国煤化工CNMHG△p=4CrReRe d 2fle/EVqk/zgkx-egx200000000hmn(第6/8页)2010323155546中国科学(英文版)E00207Eq (1)can also be arranged asCoRe(2)u The independent variables in the experiments are mass flow rate G, pressure drop A p, tube diameter d andlength l. The experimental uncertainty of Cf Re can be derived usinga(c Re)ol(CRe)DdTake the experiment for microtube No. 1 listed in table 2 as an example the relative error of g, a p, d, l is1.5%, 2%,0.1% and 2% respectively and the relative error of measured CfRepredicted by eg. (3)is.4%Toverify the veracity of the above approximate method, the experimental uncertainty of tube No. 1 was alsoanalyzed numerically by means of the method provided in ref. 13]. It can be calculated that the uncertaintiesof the experimental value of Cf Re 8.6%-9.5%, Which agrees well with the results of the approximateanalysis. The exactness of the approximate method was proven positively by calculation results.Using eq (3), the experimental uncertainties of Cf Re for microtubes No1-No.5 can be estimated to be8.4%, 6.5%, 5.6%, 5.8% and 5.7% respectively It can also be deduced from eg (3)that the error of diametermeasurement is the primary influential factor to the precision of experimental results and the experimentaluncertainties are controlled below 10%4 Conclusion1)A tube- cutting experimental method can be used to measure the pressure and Mach number distributionin a microtube. The experimental results for the microtube with diameter of 108.3u m indicate that the pressuredrop remains approximately linear due to lower Mach number at the forepart of the microtube, while at therearpart, the pressure drop increases and deviates from linear distribution as a result of higher compressibility effect.2)The product of the measured average Fanning friction faco C, f and the Reynolds number Reinmicrotubes is also the same as that predicted by Moody chart when the average Mach number isless than 0.3,whereas the value of ceRe became larger and increases with increasing average Mach number when Mave>0.33)The numerical analysisindicates that the higher frictional resistance for gas fiow in microtubes wasattributed to the gas compressibility: the gas acceleration in microtubes leads to a shape variation of the velocityprofile, resulting in an increasing velocity gradient at the wall surface of the tube and the consequent increase ofthe Fanning friction factor4) It is deduced from the present experimental results that the transition from laminar to turbulence inmicrotubes occurs at Re=2 300, which is consistent with the result in macro tubes. The phenomenon of earlytransition in microtubes reported by some researchers is not observed in the experiments.5)Uncertainty analysis shows that the inaccuracy of diameter measurement of microtubes dominates theoverall error of experiments. Extreme care was taken on the measurement of the diameter of microtubes and theexperimental uncertainty was controlled below 10%.中国煤化工CNMHGReferencesfle//EVqk/zgkx-egx200000000hmn(第7/8页)2010323155546中国科学(英文版)E0027I Wu, P. Y, Little, W.A., Measurement of friction factors for the flow of gases in very fine channelsused for microminiature Joule-Thomson refrigerators, Cryogenics, 1983, 23(5):2732 Pfahler, J, Harley, J, Bau, H, Liquid and gas transport in small channels, ASME DSC, 1990, 19: 149[3] Pfahler, J, Harley, J, Bau, H, Gas and liquid flow in small channels, ASME DSC, 1991, 32: 494 Choi, S.B., Barron, R.F., Warrington, R.Q., Fluid flow and heat transfer in microtubes, ASMEDSC,1991,32:123[5]Harley, J. C, Huang, Y. F, Bau, H. 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P, Wang, B X, Friction flow characteristics of water flowing throughrectangular microtubes, Experimental Heat Transfer, 1994, 7: 249[13 Moffat, R.J., Contributions to the theory of single-sample uncertainty analysis, J. Fluid Engng1982,104:250.R中国煤化工CNMHGfle/EVqk/zgkx-egx200000000hmn(第8/8页)2010323155546