浆氮的制备及其在水平管内的流动与相变换热特性研究
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摘要
高温超导电缆具有在低温下零电阻的特性，用于远距离电力输送中可显著降低功率损耗，因此引起了各国研究者的广泛关注和研究。浆氮中含有的固氮颗粒在融化时吸收相变潜热，从而提高了其换热性能，同时浆氮还具有温度低且稳定、密度大、输运性能好等优点，因此被认为是高温超导电缆理想的制冷剂。目前低温浆体的研究尚处于起步阶段，对其各方面性质及相关机理的认识不深入。本文针对高温超导电缆冷却等应用中涉及的相关问题，从实验和理论两方面入手对浆氮的制备、测量、管内的流动与换热特性进行了深入的研究。主要结论如下：
开展了浆氮冻结-融化制备技术的可视化研究工作，分析固氮形成的热力学过程，获得不同条件下气液界面上生成的固氮层图片。通过对固氮层形成过程的深入研究，分析浆氮制备过程中真空泵抽速以及冻结和融化段时间等对固氮层状态、结构以及所生成固氮颗粒性质的影响。搅拌作用加速了颗粒的老化过程，使得制备过程中新鲜固氮颗粒的结构随时间发生明显的变化。基于可视化研究结果对冻结-融化制备技术进行优化，得到平均直径约1.0mm且尺寸形状较均匀的固氮颗粒，所制备的浆氮质量较高。
采用电容式密度计测量浆氮的密度，并利用整体屏蔽法进行改进以克服其寄生电容大、抗电磁干扰能力弱的缺点。理论和实验分析表明，采用整体屏蔽法可显著降低测量线路的寄生电容，测量结果接近探测电容器本身的电容值；密度计的抗电磁干扰能力提高了一个数量级，电容测量噪声减小至±0.001pF之内，其稳定性和分辨力得到显著提高。分析测量电压的频率对分辨率及灵敏度的影响从而得到电容式密度计的最佳测量频率为1.0MHz。非差动式和差动式密度计的有效灵敏度分别为0.415pF和3.717pF，与理论值接近。优化后电容式密度计的测量精度达到±0.25%。
建立基于颗粒动力学的欧拉-欧拉两相流模型。采用颗粒动力学模型模拟颗粒间的碰撞及颗粒与液相湍流的相互作用，采用各相k-ε紊流模型以充分考虑液相与固相颗粒间的作用力，以Gunn模型模拟液固两相间传热过程，考虑浆氮的融化以及由相变引起的相间动量和能量的传递。将模拟结果分别与文献中多组常规浆体的实验结果进行对比，从而验证了两相流模型的可靠性。在浆氮的模拟方面，对关键性模型参数（如颗粒-颗粒恢复碰撞系数、颗粒-壁面碰撞恢复系数和镜面系数）进行敏感性分析，获得适用于浆氮颗粒特性的模型参数群。浆氮的模型结果包括流动压降、流速分布和局部换热系数均与文献中实验结果相吻合。
对浆氮在10.0mm的水平圆管内的流动特性进行研究。实验测量了浆氮在管内流动的压降及流速，在粘性摩擦与机械摩擦的共同作用下浆氮的流动压降明显高于过冷液氮，且随固相体积分数的增加而增大。采用欧拉-欧拉两相流模型模拟得到了不同条件下浆氮压降、固相体积分数分布、流速分布等详细信息。浆氮在水平管内流动可划分为三种流型：拟均质流、非均质流及推移床流，分析不同流型下颗粒分布、流速分布等及对浆氮流动的影响，并获得相应的流型转变速度，即悬浮速度及沉降速度。以Einstein公式计算浆氮表观粘度并进而得到浆氮雷诺数，在实验结果的基础上得到适用于浆氮水平管流动的经验关联式。
实验测量了浆氮在水平长直圆管内加热流动时进出口流体温度、壁面温度和流体流速，在此基础上得到浆氮管内强制对流换热特性。由于固氮颗粒融化吸收相变潜热，浆氮的局部换热系数高于同条件下的过冷液氮，且随固相体积分数的增加而增大。采用欧拉-欧拉两相流模型对浆氮的对流换热过程进行模拟，计算结果与实验结果相吻合，并得到浆氮管内加热流动时相间传热传质过程的细节。分析浆氮对流换热过程中的沿程变化规律，发现流动充分发展后由于流速较高固氮融化量少，浆氮的换热性能基本保持稳定。由于相间换热系数受固相含量的影响，颗粒在截面上分布的不同造成管内浆氮流体温度分布随流型而变化。在拟均质流中流体温度关于管轴线近似对称分布，而在非均质流中，颗粒不均匀分布使得管下部的浆氮温度明显低于上部。参照一般换热经验关联式，在实验数据的基础上总结得出浆氮水平管内对流换热经验关联式。
High-temperature superconductive power cable which is capable of reducing powertransmission loss has attracted a great deal of attention as the energy crisis occurred allover the world in recent years. Slush nitrogen contains small solid nitrogen particlessuspending in liquid nitrogen, leading to higher density and lower temperature than liquidnitrogen. More importantly, solid nitrogen can absorb the latent heat, improving the heatcapacity of slush nitrogen and keeping the temperature change of fluid very small, andtherefore slush nitrogen is considered as the excellent coolant for the high-temperaturesuperconductive power cable. However, the researches so far are not sufficient to supportits application. The present article mainly experimentally and theoretically investigatedseveral key issues of slush nitrogen, including the production technology, the densitydetermination, the flow and heat transfer in the horizontal pipe. The main conclusions areshown below:
The production technology of slush nitrogen with the freeze-thaw method was firstlyinvestigated by visualization. The thermodynamic process during the solidification ofliquid nitrogen was analyzed and the photographs of solid nitrogen layers created on thevapor/liquid interface under different conditions were taken in the experiments. On thebasis of the study on the formation of solid nitrogen layers, the effect of the pumping speedof vacuum pump and the time of freeze and thaw cycles on the properties of solid layers,furthermore the created solid nitrogen particles was achieved. Additionally, due to theagitation effect, the aging of solid particles was observed in the experiment, and the freshflocculent particles were gradually changed into the dense near-spherical particles with a smooth surface. According to the visualization results the freeze-thaw productiontechnology was optimized and the created solid particles have the average diameter of1.0mm, and slush nitrogen with high quality was produced in the experiment.
The density of slush nitrogen was determined by the capacitance-type densimeter, andthe bulk shield method was employed to reduce the influence of parasitic capacitance andto improve the anti-interference ability of the densimeter. Theoretical and experimentalresults indicated that the parasitic capacitance of the measuring circuit was significantlydecreased and the measured capacitance was very close to the value of the densimeter itself.The anti-interference ability was highly increased, and the noise of the capacitancemeasurement was suppressed within±0.001pF, indicating the high stability and theresolution of the densimeter. The optimal frequency was determined as1.0MHz, and theeffective sensitivities of the non-differential and differential type densimeter were0.415pFand3.717pF, respectively, which were in good agreement with the theoretical values. Theimproved capacitance-type densimeter had the accuracy of±0.25%for the densitymeasurement.
The Eulerian-Eulerian multiphase model incorporated with the kinetic theory ofgranular flow was developed, where the dispersed phase was considered as a continuousfluid, interpenetrating and interacting with the liquid phase. The motion of solid particleswas modeled by the kinetic theory of granular flow and the per-phase k-ε turbulence modelwas utilized in order to comprehensively consider interaction of two phases. The Gunnmodel was used to calculate the interphase heat transfer coefficient, and the melting ofsolid nitrogen and the momentum and energy transfer between two phases caused by thephase change was taken into account. The developed multiphase model was firstlyvalidated by comparing the numerical results with the experimental results of generalslurry flows from literature. The proper parameters for slush nitrogen in the model,including the particle-particle and particle-wall restitution coefficients and the specularitycoefficient, were determined by the sensitivity analysis. The numerical results of pressuredrop, velocity distribution and heat transfer coefficient of slush nitrogen obtained by theEulerian-Eulerian model agreed well with the experimental results from literature.
The flow characteristics of slush nitrogen in the horizontal pipe with the diameter of10.0mm was investigated. It was found that pressure drop of slush nitrogen was alwayshigher than that of subcooled liquid nitrogen due to the viscous and the mechanicalfrictions, and increased with the solid volume fraction. The detailed distributions of solidparticles and the velocity over the cross section were obtained in the modeling. The flowpatterns of slush nitrogen in the horizontal pipe of10.0mm in diameter were determined as:pseudo-homogenous flow, heterogeneous flow and bedload flow, and the flowcharacteristics and the associated mechanisms in various flow patterns were investigated.Based on the experimental results the transition velocities, namely the suspending velocityand the limit deposit velocity, were obtained. The effective viscosity of slush nitrogen wascalculated by the Einstein equation and the correlation of friction factor and slushReynolds number for slush nitrogen with various solid volume fraction was obtained.
The fluid temperature at the inlet and outlet, the wall temperature and the meanvelocity of slush nitrogen flowing through the heated pipe were measured and the heattransfer characteristics were obtained. Solid nitrogen particles were capable of absorbingheat latent, leading to the local heat transfer coefficient of slush nitrogen higher than that ofsubcooled liquid nitrogen and increasing with the solid volume fraction. Numericalsimulation was conducted with the Eulerian-Eulerian multiphase model to clarify the heatand mass transfer process in slush nitrogen flow. The local heat transfer coefficient of slushnitrogen was found nearly constant in the fully developed flow because the melt of solidnitrogen was small at the high velocity. The effect of non-uniform distribution of solidparticles on the heat transfer characteristics of slush nitrogen in various flow patterns wasinvestigated. According to the experimental results the heat transfer correlation of slushnitrogen was obtained.
开展了浆氮冻结-融化制备技术的可视化研究工作，分析固氮形成的热力学过程，获得不同条件下气液界面上生成的固氮层图片。通过对固氮层形成过程的深入研究，分析浆氮制备过程中真空泵抽速以及冻结和融化段时间等对固氮层状态、结构以及所生成固氮颗粒性质的影响。搅拌作用加速了颗粒的老化过程，使得制备过程中新鲜固氮颗粒的结构随时间发生明显的变化。基于可视化研究结果对冻结-融化制备技术进行优化，得到平均直径约1.0mm且尺寸形状较均匀的固氮颗粒，所制备的浆氮质量较高。
采用电容式密度计测量浆氮的密度，并利用整体屏蔽法进行改进以克服其寄生电容大、抗电磁干扰能力弱的缺点。理论和实验分析表明，采用整体屏蔽法可显著降低测量线路的寄生电容，测量结果接近探测电容器本身的电容值；密度计的抗电磁干扰能力提高了一个数量级，电容测量噪声减小至±0.001pF之内，其稳定性和分辨力得到显著提高。分析测量电压的频率对分辨率及灵敏度的影响从而得到电容式密度计的最佳测量频率为1.0MHz。非差动式和差动式密度计的有效灵敏度分别为0.415pF和3.717pF，与理论值接近。优化后电容式密度计的测量精度达到±0.25%。
建立基于颗粒动力学的欧拉-欧拉两相流模型。采用颗粒动力学模型模拟颗粒间的碰撞及颗粒与液相湍流的相互作用，采用各相k-ε紊流模型以充分考虑液相与固相颗粒间的作用力，以Gunn模型模拟液固两相间传热过程，考虑浆氮的融化以及由相变引起的相间动量和能量的传递。将模拟结果分别与文献中多组常规浆体的实验结果进行对比，从而验证了两相流模型的可靠性。在浆氮的模拟方面，对关键性模型参数（如颗粒-颗粒恢复碰撞系数、颗粒-壁面碰撞恢复系数和镜面系数）进行敏感性分析，获得适用于浆氮颗粒特性的模型参数群。浆氮的模型结果包括流动压降、流速分布和局部换热系数均与文献中实验结果相吻合。
对浆氮在10.0mm的水平圆管内的流动特性进行研究。实验测量了浆氮在管内流动的压降及流速，在粘性摩擦与机械摩擦的共同作用下浆氮的流动压降明显高于过冷液氮，且随固相体积分数的增加而增大。采用欧拉-欧拉两相流模型模拟得到了不同条件下浆氮压降、固相体积分数分布、流速分布等详细信息。浆氮在水平管内流动可划分为三种流型：拟均质流、非均质流及推移床流，分析不同流型下颗粒分布、流速分布等及对浆氮流动的影响，并获得相应的流型转变速度，即悬浮速度及沉降速度。以Einstein公式计算浆氮表观粘度并进而得到浆氮雷诺数，在实验结果的基础上得到适用于浆氮水平管流动的经验关联式。
实验测量了浆氮在水平长直圆管内加热流动时进出口流体温度、壁面温度和流体流速，在此基础上得到浆氮管内强制对流换热特性。由于固氮颗粒融化吸收相变潜热，浆氮的局部换热系数高于同条件下的过冷液氮，且随固相体积分数的增加而增大。采用欧拉-欧拉两相流模型对浆氮的对流换热过程进行模拟，计算结果与实验结果相吻合，并得到浆氮管内加热流动时相间传热传质过程的细节。分析浆氮对流换热过程中的沿程变化规律，发现流动充分发展后由于流速较高固氮融化量少，浆氮的换热性能基本保持稳定。由于相间换热系数受固相含量的影响，颗粒在截面上分布的不同造成管内浆氮流体温度分布随流型而变化。在拟均质流中流体温度关于管轴线近似对称分布，而在非均质流中，颗粒不均匀分布使得管下部的浆氮温度明显低于上部。参照一般换热经验关联式，在实验数据的基础上总结得出浆氮水平管内对流换热经验关联式。
High-temperature superconductive power cable which is capable of reducing powertransmission loss has attracted a great deal of attention as the energy crisis occurred allover the world in recent years. Slush nitrogen contains small solid nitrogen particlessuspending in liquid nitrogen, leading to higher density and lower temperature than liquidnitrogen. More importantly, solid nitrogen can absorb the latent heat, improving the heatcapacity of slush nitrogen and keeping the temperature change of fluid very small, andtherefore slush nitrogen is considered as the excellent coolant for the high-temperaturesuperconductive power cable. However, the researches so far are not sufficient to supportits application. The present article mainly experimentally and theoretically investigatedseveral key issues of slush nitrogen, including the production technology, the densitydetermination, the flow and heat transfer in the horizontal pipe. The main conclusions areshown below:
The production technology of slush nitrogen with the freeze-thaw method was firstlyinvestigated by visualization. The thermodynamic process during the solidification ofliquid nitrogen was analyzed and the photographs of solid nitrogen layers created on thevapor/liquid interface under different conditions were taken in the experiments. On thebasis of the study on the formation of solid nitrogen layers, the effect of the pumping speedof vacuum pump and the time of freeze and thaw cycles on the properties of solid layers,furthermore the created solid nitrogen particles was achieved. Additionally, due to theagitation effect, the aging of solid particles was observed in the experiment, and the freshflocculent particles were gradually changed into the dense near-spherical particles with a smooth surface. According to the visualization results the freeze-thaw productiontechnology was optimized and the created solid particles have the average diameter of1.0mm, and slush nitrogen with high quality was produced in the experiment.
The density of slush nitrogen was determined by the capacitance-type densimeter, andthe bulk shield method was employed to reduce the influence of parasitic capacitance andto improve the anti-interference ability of the densimeter. Theoretical and experimentalresults indicated that the parasitic capacitance of the measuring circuit was significantlydecreased and the measured capacitance was very close to the value of the densimeter itself.The anti-interference ability was highly increased, and the noise of the capacitancemeasurement was suppressed within±0.001pF, indicating the high stability and theresolution of the densimeter. The optimal frequency was determined as1.0MHz, and theeffective sensitivities of the non-differential and differential type densimeter were0.415pFand3.717pF, respectively, which were in good agreement with the theoretical values. Theimproved capacitance-type densimeter had the accuracy of±0.25%for the densitymeasurement.
The Eulerian-Eulerian multiphase model incorporated with the kinetic theory ofgranular flow was developed, where the dispersed phase was considered as a continuousfluid, interpenetrating and interacting with the liquid phase. The motion of solid particleswas modeled by the kinetic theory of granular flow and the per-phase k-ε turbulence modelwas utilized in order to comprehensively consider interaction of two phases. The Gunnmodel was used to calculate the interphase heat transfer coefficient, and the melting ofsolid nitrogen and the momentum and energy transfer between two phases caused by thephase change was taken into account. The developed multiphase model was firstlyvalidated by comparing the numerical results with the experimental results of generalslurry flows from literature. The proper parameters for slush nitrogen in the model,including the particle-particle and particle-wall restitution coefficients and the specularitycoefficient, were determined by the sensitivity analysis. The numerical results of pressuredrop, velocity distribution and heat transfer coefficient of slush nitrogen obtained by theEulerian-Eulerian model agreed well with the experimental results from literature.
The flow characteristics of slush nitrogen in the horizontal pipe with the diameter of10.0mm was investigated. It was found that pressure drop of slush nitrogen was alwayshigher than that of subcooled liquid nitrogen due to the viscous and the mechanicalfrictions, and increased with the solid volume fraction. The detailed distributions of solidparticles and the velocity over the cross section were obtained in the modeling. The flowpatterns of slush nitrogen in the horizontal pipe of10.0mm in diameter were determined as:pseudo-homogenous flow, heterogeneous flow and bedload flow, and the flowcharacteristics and the associated mechanisms in various flow patterns were investigated.Based on the experimental results the transition velocities, namely the suspending velocityand the limit deposit velocity, were obtained. The effective viscosity of slush nitrogen wascalculated by the Einstein equation and the correlation of friction factor and slushReynolds number for slush nitrogen with various solid volume fraction was obtained.
The fluid temperature at the inlet and outlet, the wall temperature and the meanvelocity of slush nitrogen flowing through the heated pipe were measured and the heattransfer characteristics were obtained. Solid nitrogen particles were capable of absorbingheat latent, leading to the local heat transfer coefficient of slush nitrogen higher than that ofsubcooled liquid nitrogen and increasing with the solid volume fraction. Numericalsimulation was conducted with the Eulerian-Eulerian multiphase model to clarify the heatand mass transfer process in slush nitrogen flow. The local heat transfer coefficient of slushnitrogen was found nearly constant in the fully developed flow because the melt of solidnitrogen was small at the high velocity. The effect of non-uniform distribution of solidparticles on the heat transfer characteristics of slush nitrogen in various flow patterns wasinvestigated. According to the experimental results the heat transfer correlation of slushnitrogen was obtained.
引文
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[39] K. Matsuo, M. Ikeuchi, A. Machida, K. Yasuda, Fundamental study of pipe flowand heat transfer characteristics of slush nitrogen, Advances in CryogenicEngineering: Transactions of the Cryogenic Engineering Conference-CEC Vol.51(2005)1025-1032.
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[42] A. Levy, Two-fluid approach for plug flow simulations in horizontal pneumaticconveying, Powder Technology112(2000)263-272.
[43] A. Ramadan, P. Skalle, A. Saasen, Application of a three-layer modeling approachfor solids transport in horizontal and inclined channels, Chemical EngineeringScience60(10)(2005)2557-2570.
[44]车得福,李会雄,多相流及其应用,西安交通大学出版社,2007.
[45] P. Frawley, A.P. O'Mahony, M. Geron, Comparison of Lagrangian and EulerianSimulations of Slurry Flows in a Sudden Expansion, Journal of Fluids Engineering132(9)(2010)1-12.
[46] J. Ishimoto, R. Ono, Numerical study of the two-phase flow characteristics of slushnitrogen, Cryogenics45(4)(2005)304-316.
[47] F. Dobran, Liquid and gas-phase distributions in a jet with phase change, JournalName: J. Heat Transfer;(United States); Journal Volume:110:4(1988) Medium:X; Size: Pages:955-960.
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[50] K. Zhu, C.K. Wong, S.M. Rao, C.-H. Wang, Pneumatic conveying of granularsolids in horizontal and inclined pipes, AIChE Journal50(8)(2004)1729-1745.
[51] K. Ohira, A. Ota, Y. Mukai, T. Hosono, Numerical study of flow and heat-transfercharacteristics of cryogenic slush fluid in a horizontal circular pipe (SLUSH-3D),Cryogenics52(7–9)(2012)428-440.
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[53] S.B. Savage, D.J. Jeffrey, The stress tensor in a granular flow at high shear rates,Journal of Fluid Mechanics110(1981)255-272.
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[55] M. Shahinpour, G. Ahmadi, A kinetic theory for the rapid flow of rough identicalspherical particles and the evolution of fluctuation: Advances in the mechanicasand the flow of granlar materials II, Switzerland: Andermannsforf,1983.
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[57] P.C. Johnson, R. Jackson, Frictional-collisional constitutive relations for granularmaterials, with application to plane shearing, Journal of Fluid Mechanics176(1987)67-93.
[58] V. Jiradilok, D. Gidaspow, S. Damronglerd, W.J. Koves, R. Mostofi, Kinetic theorybased CFD simulation of turbulent fluidization of FCC particles in a riser,Chemical Engineering Science61(17)(2006)5544-5559.
[59] W. Duangkhamchan, F. Ronsse, F. Depypere, K. Dewettinck, J.G. Pieters,Comparison and evaluation of interphase momentum exchange models forsimulation of the solids volume fraction in tapered fluidised beds, ChemicalEngineering Science65(10)(2010)3100-3112.
[60] L. Chen, Y. Duan, W. Pu, C. Zhao, CFD simulation of coal-water slurry flowing inhorizontal pipelines, Korean Journal of Chemical Engineering26(4)(2009)1144-1154.
[61] R. Span, E.W. Lemmon, R.T. Jacobsen, W. Wagner, A. Yokozeki, A ReferenceEquation of State for the Thermodynamic Properties of Nitrogen for Temperaturesfrom63.151to1000K and Pressures to2200MPa, Journal of Physical andChemical Reference Data29(6)(2000)1361-1433.
[62] B.A. Younglove, Thermophysical Properties of Fluids. I. Argon, Ethylene,Parahydrogen, Nitrogen, Nitrogen Trifluoride, and Oxygen, Journal of Physical andChemical Reference Data11(1982)1-353.
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[65] C.F. Sindt, A summary of the characterization study of slush hydrogen, Cryogenics10(1970)372-380.
[66] J.D. Jackson, Classical Electrodynamics,3rd ed., John Wiley&Sons, Inc,1999.
[67] J.W. Stewart, Dielectric Polarizability of Fluid Para-Hydrogen, Journal of ChemicalPhysics40(1964)3297-3306.
[68] L.K. Baxter, Capacitive Sensors: Design and Applications, Wiley-IEEE Press, NewYork,1996.
[69] D. Gidaspow, R. Bezburuah, J. Ding, Hydrodynamics of circulating fluidized beds:Kinetic theory approach, in Fluidization VII, Proceedings of the7th EngineeringFoundation Conference on Fluidization, Gold Coast, Australia,1992, pp.75-82.
[70] M. Syamlal, T.J. O'Brien, Computer simulation of bubbles in a fluidized bed,Washington, DC, USA,1989, pp.22-31.
[71] C. Wen, Y. Yu, Mechanics of fluidization, Chem. Eng. Prog. Symp. Ser.62(62)(1966)100.
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[73] M.R. Maxey, J.J. Riley, Equation of motion for a small rigid sphere in anonuniform flow, Physics of Fluids26(1983)883-889.
[74] S. Elghobashi, On predicting particle-laden turbulent flows, Applied ScientificResearch52(4)(1994)309-329.
[75] S. Ogawa, A. Umemura, N. Oshima, On the equations of fully fluidized granularmaterials, Zeitschrift für Angewandte Mathematik und Physik (ZAMP)31(4)(1980)483-493.
[76] D.J. Gunn, Transfer of heat or mass to particles in fixed and fluidised beds,International Journal of Heat and Mass Transfer21(4)(1978)467-476.
[77] S.V. Patankar, Numerical Heat Transfer and Fluid Flow, Hemisphere, WashingtonDC,1980.
[78] D.R. Kaushal, K. Sato, T. Toyota, K. Funatsu, Y. Tomita, Effect of particle sizedistribution on pressure drop and concentration profile in pipeline flow of highlyconcentrated slurry, International Journal of Multiphase Flow31(7)(2005)809-823.
[79] D.R. Kaushal, Y. Tomita, Experimental investigation for near-wall lift of coarserparticles in slurry pipeline using γ-ray densitometer, Powder Technology172(3)(2007)177-187.
[80] C.S. Campbell, F. Avila-Segura, Z. Liu, Preliminary observations of a particle liftforce in horizontal slurry flow, International Journal of Multiphase Flow30(2)(2004)199-216.
[81] R. Skartlien, D. Drazen, D.C. Swailes, A. Jensen, Suspensions in turbulent liquidpipe fow: Kinetic modelling and added mass effects, International Journal ofMultiphase Flow35(2009)1017-1035.
[82] S. Benyahia, M. Syamlal, T.J. O'Brien, Study of the ability of multiphasecontinuum models to predict core-annulus flow, AIChE Journal53(10)(2007)2549-2568.
[83] C.C. Pain, S. Mansoorzadeh, C.R.E. de Oliveira, A study of bubbling and sluggingfluidised beds using the two-fluid granular temperature model, International Journalof Multiphase Flow27(3)(2001)527-551.
[84] M.Y. Louge, E. Mastorakos, J.T. Jenkins, Role of particle collisions in pneumatictransport, Journal of Fluid Mechanics231(1991)345-359.
[85] E.J. Bolio, J.A. Yasuna, J.L. Sinclair, Dilute turbulent gas-solid flow in risers withparticle-particle interactions, AIChE Journal41(6)(1995)1375-1388.
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