Mechanics of Low-Temperature Phase Transition in Liquid-Filled Elastic Capillary Tube

TAO Ze; LI Moxiao; TI Fei; LIU Yonggang; LIU Shaobao; LU Tianjian

doi:10.21656/1000-0887.420301

Volume 42 Issue 10

Oct. 2021

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Article Navigation > Applied Mathematics and Mechanics > 2021 > 42(10): 1045-1061

TAO Ze, LI Moxiao, TI Fei, LIU Yonggang, LIU Shaobao, LU Tianjian. Mechanics of Low-Temperature Phase Transition in Liquid-Filled Elastic Capillary Tube[J]. Applied Mathematics and Mechanics, 2021, 42(10): 1045-1061. doi: 10.21656/1000-0887.420301

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Mechanics of Low-Temperature Phase Transition in Liquid-Filled Elastic Capillary Tube

doi: 10.21656/1000-0887.420301

TAO Ze^1,2,
LI Moxiao^1,2,
TI Fei^1,2,
LIU Yonggang^1,2,
LIU Shaobao^{1,2
,
,},
LU Tianjian^1,2

1. State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, P.R.China;
2. MIIT Key Laboratory of Multifunctional Lightweight Materials and Structures, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, P.R.China

Funds:

11902155)

The National Natural Science Foundation of China(12032010

Received Date: 2021-09-30
Rev Recd Date: 2021-10-14

Abstract

Abstract

Liquid-filled elastic capillaries are a kind of standard component in life body (e.g., capillary blood vessel and plant vessel) and engineering fields (e.g., MEMS microchannel resonators and heat pipes). Under sufficiently low-temperature, the liquid in a capillary tube will undergo a phase transition and exhibit a frozen-heave effect, which may cause deformation, damage, and even fracture of the tube wall. In this study, we established the governing equation of an elastic capillary tube, with temperature gradient, interfacial tension, and frozen-heave effect accounted for, and solved the equation for stresses developed in the tube wall during freezing. It is demonstrated that stress distribution in tube wall is influenced by the thermoelastocapillary number and the freezeoelastocapillary number, both dependent upon wall thickness. Results obtained in this study are not only helpful for understanding the prevention of frozen-heave failure, but also provide theoretical guidance for tailoring the freezing resistance of microfluidic devices used in MEMS.
- phase transition,
- interfacial tension,
- frost heaving,
- thermoelastocapillary number,
- freezeoelastocapillary number

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References(68)

References

BOWDEN R. Neuroanatomy: an illustrated colour text[J]. Clinical Anatomy,1996,9(1): 349.

[2]YANG S, GONG X, QI Y, et al. Comparative study of variations in mechanical stress and strain of human blood vessels: mechanical reference for vascular cell mechano-biology[J]. Biomechanics and Modeling in Mechanobiology,2020,19(2): 519-531.

[3]BABU A M, MENON A. Distribution of gum and gum-resin ducts in plant body: certain familiar features and their significance[J]. Flora,1990,184(4): 257-261.

[4]DARHUBER A A, TROIAN S M. 表面应力调制的微流控技术的驱动原理[J]. 力学进展, 2007,37(1): 113-129.(DARHUBER A A, TROIAN S M. Principles of microfluidic a ctuation by modulation of surface stresses[J]. Advances in Mechanics,2007,37(1): 113-129.(in Chinese))

[5]黄永光, 刘世炳, 陈涛, 等. 基于微通道构型的微流体流动控制研究[J]. 力学进展, 2009,39(1): 69-78.(HUANG Yongguang, LIU Shibing, CHEN Tao, et al. Studies on microfluid flow controls based on the configuration of microchannel[J]. Advances in Mechanics,2009,39(1): 69-78.(in Chinese))

[6]YAO S C, TANG X, HSIEH C C, et al. Micro-electro-mechanical systems (MEMS)-based micro-scale direct methanol fuel cell development[J]. Energy,2006,31(5): 636-649.

[7]XUAN X, SINTON D, LI D. Thermal end effects on electroosmotic flow in a capillary[J]. International Journal of Heat and Mass Transfer,2004,47(14/16): 3145-3157.

[8]PETTIGREW K, KIRSHBERG J, YERKES K, et al. Performance of a MEMS based micro capillary pumped loop for chip-level temperature control[C]//IEEE International Conference on Micro Electro Mechanical Systems. 2001: 427-430.

[9]闫寒, 张文明. 微纳通道谐振器检测与表征中的动力学问题[J]. 力学进展, 2019,49(1): 274-311.(YAN Han, ZHANG Wenming. Dynamics problems of micro/nano channel resonators for detection and characterization[J]. Advances in Mechanics,2019,49(1): 274-311.(in Chinese))

[10]LANDERS J P. Handbook of Capillary Electrophoresis[M]. 2nd ed. Crc Press, 1996.

[11]ISSAQ H J, CHAN K C, LI Q, et al. Multidimensional high performance liquid chromatography-capillary electrophoresis separation of a protein digest: an update[J]. Electrophoresis,2001,22(6): 1133-1135.

[12]MIRZAJANZADEH M, DESHPANDE V S, FLECK N A. The swelling of cellulose foams due to liquid transport[J]. Journal of the Mechanics and Physics of Solids,2020,136(1): 103707.

[13]CAMPANA D, DI P J, SAITA F A. A 2-D model of Rayleigh instability in capillary tubes: surfactant effects[J]. International Journal of Multiphase Flow,2004,30(5): 431-454.

[14]BICO J, REYSSAT E, ROMAN B. Elastocapillarity: when surface tension deforms elastic solids[J]. Annual Review of Fluid Mechanics,2018,50(1): 629-659.

[15]HOMSCHOTEN J W V, ESCALANTE M, TAS N R, et al. Elastocapillary filling of deformable nanochannels[J]. Journal of Applied Physics,2007,101(9): 094310.

[16]SINGH K, LISTER J R, VELLA D. A fluid-mechanical model of elastocapillary coalescence[J]. Journal of Fluid Mechanics,2014,745(2): 621-646.

[17]DAS S, MARCHAND A, ANDREOTTI B, et al. Elastic deformation due to tangential capillary forces[J]. Physics of Fluids,2011,23(7): 827.

[18]HADJITTOFIS A, LISTER J R, SINGH K, et al. Evaporation effects in elastocapillary aggregation[R/OL]. 2016. [2021-10-14]. https://arxiv.org/pdf/1506.07241.pdf.

[19]SMALYUKH I I, CHERNYSHUK S, LEV B I, et al. Ordered droplet structures at the liquid crystal surface and elastic-capillary colloidal interactions[J]. Physical Review Letters,2004,93(11): 117801.

[20]HOBERG T B, VERNEUIL E, HOSOI A E. Elastocapillary flows in flexible tubes[J]. Physics of Fluids,2014,26(12): 122103.

[21]XUAN C, BIGGINS J. Finite wavelength surface-tension driven instabilities in soft solids, including instability in a cylindrical channel through an elastic solid[J]. Physical Review E,2016,94(2): 023107.

[22]LIU S, WU Y, YANG F, et al. Vibration of a liquid-filled capillary tube[J]. Journal of the Mechanical Behavior of Biomedical Materials,2020,106: 103745.

[23]WU Y, LI M, YIN J, et al. Hydrostatic pressure and interfacial tension induce mode instability in wave propagation along a liquid-filled microtubule[J]. Physics of Fluids,2020,32(3): 031901.

[24]MAZOUCHI A, HOMSY G M. Thermocapillary migration of long bubbles in cylindrical capillary tubes[J]. Physics of Fluids,2000,12(3): 542-549.

[25]RAO P, LI T, WU Z, et al. Ductile “ice”: frozen hydrogels with high ductility and compressive yielding strength[J]. Extreme Mechanics Letters,2019,28: 43-49.

[26]MORELLE X P, ILLEPERUMA W R, TIAN K, et al. Highly stretchable and tough hydrogels below water freezing temperature[J]. Advanced Materials,2018,30(35): 1801541.

[27]COUSSY O. Poromechanics of freezing materials[J]. Journal of the Mechanics and Physics of Solids,2005,53(8): 1689-1718.

[28]RABIN Y, OLSON P, TAYLOR M J, et al. Gross damage accumulation on frozen rabbit liver due to mechanical stress at cryogenic temperatures[J]. Cryobiology,1997,34(4): 394-405.

[29]RABIN Y, STEIF P S. Thermal stresses in a freezing sphere and its application to cryobiology[J]. Journal of Applied Mechanics,1998,65(2): 328-333.

[30]GILPIN R R. The effects of dendritic ice formation in water pipes[J]. International Journal of Heat and Mass Transfer,1977,20(6): 693-699.

[31]GILPIN R R. Modes of ice formation and flow blockage that occur while filling a cold pipe[J]. Cold Regions Science and Technology, 1981,5(2): 163-171.

[32]ALEXIADES V. Mathematical Modeling of Melting And Freezing Processes[M]. CRC Press, 1992.

[33]CONDE R, PARRA M T, CASTRO F, et al. Numerical model for two-phase solidification problem in a pipe[J]. Applied Thermal Engineering,2004,24(17/18): 2501-2509.

[34]LIU Z, MULDREW K, WAN R G, et al. Measurement of freezing point depression of water in glass capillaries and the associated ice front shape[J]. Physical Review E,2003,67(6): 061602.

[35]LIU Z, MULDREW K, WAN R G, et al. Retardation of ice growth in glass capillaries: measurement of the critical capillary radius[J]. Physical Review E,2004,69(2): 021611.

[36]JAIN A, MIGLANI A, HUANG Y, et al. Ice formation modes during flow freezing in a small cylindrical channel[J]. International Journal of Heat and Mass Transfer,2018,128: 836-848.

[37]张驰, 张涛, 韩涛, 等. 管壁温度非恒定条件下单管冻结温度场解析计算[J]. 煤炭科学技术, 2012,40(3): 20-23, 82.(ZHANG Chi, ZHANG Tao, HAN Tao, et al. Analysis calculation on single pipeline freezing temperature field under non-constant condition of pipe wall temperature[J]. Coal Science and Technology, 2012,40(3): 20-23, 82.(in Chinese))

[38]胡向东, 陈锦, 汪洋, 等. 环形单圈管冻结稳态温度场解析解[J]. 岩土力学, 2013,34(3): 874-880.(HU Xiangdong, CHEN Jin, WANG Yang, et al. Analytical solution to steady-state temperature field of single-circle-pipe freezing[J]. Rock and Soil Mechanics,2013,34(3): 874-880.(in Chinese))

[39]MYERS T G, LOW J. An approximate mathematical model for solidification of a flowing liquid in a microchannel[J]. Microfluidics and Nanofluidics,2011,11(4): 417-428

[40]LELEA D, NISHIO S, TAKANO K. The experimental research on microtube heat transfer and fluid flow of distilled water[J]. International Journal of Heat And Mass Transfer,2004,47(12/13): 2817-2830.

[41]JAIN A. Characterization of flow freezing in small channels for ice valve applications[D]. PhD Thesis. West Lafayette: Purdue University, 2019.

[42]尹超男, 沈财敏. PVC自来水管结冻物理特性的分析研究[J]. 水利科技与经济, 2016,22(12): 22-27.(YIN Chaonan, SHEN Caimin. Aanlysis on the frozen physical characteristics of PVC tap water pipe[J]. Water Conservancy Science and Technology and Economy,2016,22(12): 22-27.(in Chinese))

[43]BRAGA S L, MILON J J. Visualization of dendritic ice growth in supercooled water inside cylindrical capsules[J]. International Journal of Heat and Mass Transfer,2012,55(13/14): 3694-3703.

[44]AKYURT M, ZAKI G, HABEEBULLAH B. Freezing phenomena in ice-water systems[J]. Energy Conversion and Management,2002,43(14): 1773-1789.

[45]申彪, 廖振强, 李洪强, 等. 厚壁圆筒热-结构耦合应力分析[J]. 弹箭与制导学报, 2019,39(3): 49-52.(SHEN Biao, LIAO Zhenqiang, LI Hongqiang, et al. Calculating for thermo-mechanical coupling stress in thick wall cylinders[J]. Journal of Projectiles, Rockets, Missiles and Guidance,2019,39(3): 49-52.(in Chinese))

[46]中原一郎, 涩谷寿一, 土田荣一郎, 等. 弹性力学手册[M]. 关正西, 李跃明, 译. 西安: 西安交通大学出版社, 2014.(NAKAHARA Ichiro, SHIBUYA Juichi, TSUCHIDA Eiichiro. Handbook of Elasticity[M]. GUAN Zhengxi, LI Yueming, transl. Xi’an: Xi’an Jiaotong University Press, 2014.(Chinese version))

[47]徐芝纶. 弹性力学[M]. 4版. 北京: 高等教育出版社, 2006.(XU Zhilun. Elasticity Mechanics[M]. 4th ed. Beijing: Higher Education Press, 2006.(in Chinese))

[48]JAIN A, MIGLANI A, WEIBEL J A, et al. The effect of channel diameter on flow freezing in microchannels[J]. International Journal of Heat and Mass Transfer,2020,157(11): 119718.

[49]高世桥, 刘海鹏. 毛细力学[M]. 北京: 科学出版社, 2010.(GAO Shiqiao, LIU Haipeng. Capillary Mechanics[M]. Beijing: Science Press, 2010.(in Chinese))

[50]欧阳芳, 鲁世强, 方军, 等. 塑性应变对21-6-9高强不锈钢管瞬时弹性模量的影响[J]. 塑性工程学报, 2019,26(3): 209-217.(OUYANG Fang, LU Shiqiang, FANG Jun, et al. Effect of plastic strain on instantaneous elastic modulus of 21-6-9 high strength stainless steel tube[J]. Journal of Plasticity Engineering,2019,26(3): 209-217.(in Chinese))

[51]施季华. 有机玻璃动力弹性模量的测定[J]. 苏州教育学院学报, 1996(1): 83-84.(SHI Jihua. Plexiglass-determination of dynamic elastic modulus[J]. Journal of Suzhou Institute of Education,1996(1): 83-84.(in Chinese))

[52]马洪顺, 邢国平. 聚乙烯合成塑料管弹性模量等指标测定的试验研究[J]. 试验技术与试验机, 1989(6): 10-13.(MA Hongshun, XING Guoping. Experimental study on elastic modulus of polyethylene synthetic plastic tube[J]. Test Technology and Testing Machine, 1989(6): 10-13.(in Chinese))

[53]JIANG H, SUN Y, ROGERS J A, et al. Post-buckling analysis for the precisely controlled buckling of thin film encapsulated by elastomeric substrates[J]. International Journal of Solids and Structures,2008,45(7/8): 2014-2023.

[54]CAI S, CHEN D, SUO Z, et al. Creasing instability of elastomer films[J]. Soft Matter, 2012,8: 1301-1304.

[55]姜宗来, 何光篪. 人冠状动脉的生物力学特性, Ⅲ: 静态弹性模量与血管结构[J]. 第三军医大学学报, 1989,11(4): 252-256.(JIANG Zonglai, HE Guangchi. Biomechanical properties of human coronary arteries, Ⅲ: static elastic modulus and vascular structure[J]. Journal of Third Military Medical University,1989,11(4): 252-256.(in Chinese))

[56]MULLER A, WAPLER M C, WALLRABE U. A quick and accurate method to determine the Poisson's ratio and the coefficient of thermal expansion of PDMS[J]. Soft Matter,2018,15(4): 779-784.

[57]DUGAS J, MICHEL P, MARTIN L, et al. Behavior of the refractive index and of the coefficient of thermal expansion of silicone with temperature[J]. Applied Optics,1986,25(21): 3807-3808.

[58]郭耀, 李刚, 贾成艳, 等. 冰力学参数的超声波测试研究[J]. 极地研究, 2016,28(1): 152-157.(GUO Yao, LI Gang, JIA Chengyan, et al. Study of ultrasonic test in the measurements of mechanical properties of ice[J]. Chinese Journal of Polar Research,2016,28(1): 152-157.(in Chinese))

[59]KAEWKULCHAI G. Dynamic progressive collapse of frame structures[D]. PhD Thesis. Austin: The University of Texas at Austin, 2003.

[60]ZWALLY H J, GIOVINETTO M B, LI J, et al. Mass changes of the Greenland and Antarctic ice sheets and shelves and contributions to sea-level rise: 1992—2002[J]. Journal of Glaciology,2005,51(175): 509-527.

[61]KELL G S, WHALLEY E. Reanalysis of the density of liquid water in the range 0~150 ℃ and 0~1 kbar[J]. Journal of Chemical Physics,1975,62(9): 3496-3503.

[62]刘传安, 罗小凤. 用恒流量热器测定水的比热容[J]. 大学物理, 2006,25(4): 40-42.(LIU Chuan’an, LUO Xiaofeng. Determination of the specific heat capacity of water using a constant flow calorimeter[J]. College Physics,2006,25(4): 40-42.(in Chinese))

[63]CHAUDHARY G, LI R. Freezing of water droplets on solid surfaces: an experimental and numerical study[J]. Experimental Thermal and Fluid Science,2014,57: 86-93.

[64]GRUNDKE K, MICHEL S, KNISPEL G, et al. Wettability of silicone and polyether impression materials: characterization by surface tension and contact angle measurements[J]. Colloids and Surfaces A: Physicochemical and Engineering Aspects,2008,317(1/3): 598-609.

[65]TAN J, FENG S, et al. Effect of counterions on micellization of pyrrolidinium based silicone ionic liquids in aqueous solutions[J]. Journal of Chemical and Engineering Data,2014,59(6): 1830-1834.

[66]ADAMSON A W, SHIRLEY F P, KUNICHIKA K T. Contact angles on molecular solids,Ⅰ: ice[J]. Journal of Colloid and Interface Science,1970,34(3): 461-468.

[67]EVERETT D H. Thermodynamics of frost damage to porous solids[J]. Transactions of the Faraday Society,1961,57(5): 1541-1551.

[68]OSS C J V, GIESE R F, WENTZEK R, et al. Surface tension parameters of ice obtained from contact angle data and from positive and negative particle adhesion to advancing freezing fronts[J]. Journal of Adhesion Science and Technology,1992,6(4): 503-516.

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