Ferrofluid Convection in a Hydrodynamic Loop: Analysis of Temperature Field

Purpose of research. To construct a method for the analysis of temperature measurement results in an experimental investigation of thermomagnetic convection in a closed hydrodynamic circuit.

Methods. The experiment was carried out using a long closed hydrodynamic loop made of a thin tube of circular cross section. A short vertical segment of the loop, located in a gradient magnetic field with an intensity amplitude of 24 kA/m, was heated. The heat was removed from the entire surface of the loop tubes by blowing them with thermostatic air. The experiments were carried out with colloidal solution of magnetite in paraffin stabilized with oleic acid. Control measurements in zero magnetic field were carried out using pure illuminating paraffin. An exponential temperature distribution was established in the steady-state flow regime along the circuit. The exponent was measured. The results of the temperature measurements were analysed using approximate solution of the convective heat transfer equation in the cooled section of the circuit. The solution used a parabolic approximation of the velocity profile and the smallness of the molecular axial heat flux compared with the convective axial heat flux.

Results. It is shown that the exponent measured in the experiment is sufficient to obtain information about the intensity of the axial heat flux. The calculation formula for Nusselt number taking into account geometric parameters of the circuit, fluid properties and heat exchange conditions has been proposed. Dependence of Nusselt number on Rayleigh heat number is plotted for all series of measurements. It has been found that the heat transfer along the loop has increased by a factor of 3 to 3.5 due to the thermomagnetic convection mechanism in comparison with the results of the control experiments.

Conclusion. The method of analysis of temperature measurements during experimental investigation of thermomagnetic convection in a closed long hydrodynamic loop has been proposed. The method has been experimentally tested.

1. Bejan A. Convection Heat Transfer. Fourth ed. Hoboken, John Willey & Sons, Inc, 2013. 685 р.

2. Lienhard J.H. Heat Transfer Textbook. 5th ed. Cambridge, MA, Phlogiston Press, 2019. 784 р.

3. Kolchanov N. V., Putin G. V. Gravitational convection of magnetic colloid in a horizontal layer. International Journal of Heat and Mass Transfer, 2015, vol. 89, pp. 90–101. https://doi.org/10.1016/j.ijheatmasstransfer.2015.05.050

4. Belyaev A. V., Smorodin B. L. Convection of a ferrofluid in an alternating magnetic field. Journal of Applied Mechanics and Theoretical Physics., 2009, vol. 50, pp. 558–565. https://doi.org/10.1007/s10808-009-0075-1

5. Jitender S. Energy relaxation for transient convection in ferrofluids. Physical Review E, 2010, vol. 82(2), pp. 026311. https://doi.org/10.1103/PhysRevE.82.026311

6. Matrura P., Lücke M. Thermomaghetic convection in a ferrofluid layer exposed to time-periodic magnetic field. Physical Review E, 2009, vol. 80(2), pp. 026314. https://doi.org/10.1103/ PhysRevE.80.026314

7. Lange A., Odenbach S. Patterns of thermomagnetic convection in magnetic fluids subjected to spatially modulated magnetic fields. Physical Review E, 2011, vol. 83(6), pp. 066305. https://doi.org/10.1103/PhysRevE.83.066305

8. Rahman H., Suslov S. Thermomagnetic convection in a layer of ferrofluid placed in a uniform oblique external magnetic field. Journal of Fluid Mechanics, 2015, vol. 764, pp. 316–348. https://doi.org/10.1017/jfm/2014.709

9. Bozhko A. A., Putin G. F. Osobennosti konvektivnogo teploperenosa v magnitnyh nanozhidkostyah [Features of convective heat transfer in magnetic nanofluids]. Vestnik Permskogo universiteta. Matematika. Mekhanika. Informatika = Bulletin of Perm University. Mathematics. Mechanics. Informatics, 2012, vol. 4(12), pp. 25–30.

10. Krauzina M. T., Bozhko A. A., Krauzin P. V., Suslov S. A. The use of ferrofluids for heat removal: Advantage or Disadvantage? Journal of Magnetism and Magnetic Materials, 2018, vol. 54(1), pp. 241–244. https://doi.org/10.1016/j.jmmm.2016.08.085

11. Yamaguchi H., Niu X. D., Zhang X. R., Yoshikava K. Experimental and numerical investigation of natural convection of magnetic fluids in cubic cavity Journal of Magnetism and Magnetic Materials, 2009, vol. 321(2), pp. 3665–3670. https://doi.org/10.1016/j.jmmm.2009.07.013

12. Krakov M. S., Nikiforov I. V., Reks A. G. Influence of the uniform magnetic field on natural convection in cubic enclosure: Experiment and Numerical simulation. Journal of Magnetism and Magnetic Materials, 2005, vol. 289, pp. 272–274. https://doi.org/10.1016/j.jmmm.2004.11.077

13. Yanhong C., Decai Li. Experimental investigation on convection heat transfer characteristics of ferrofluid in a horizontal channel under a non-uniform magnetic field. Applied Thermal Engineering, 2019, vol. 163, pp. 114306. https://doi.org/10.1016/j.applthermaleng.2019.114306.

14. Soltanipour H. Two-phase simulation of magnetic field effect on the ferrofluid forced convection in a pipe considering Brownian diffusion, thermophoresis, and magnetophoresis. The European Physical Journal Plus, 2020, vol. 135(9), pp. 702. https://doi.org/10.1140/epjp/s13360-020-00725w.

15. Shojaeizadeh E., Veysi F., Goudarzi K. Heat transfer and thermal efficiency of a labfabricated ferrofluid-based single-ended tube solar collector under the effect of magnetic field: An experimental study. Applied Thermal Engineering, 2020, vol. 164, pp. 114510. https://doi.org/10.1016/j.applthermaleng.2019.114510

16. Toshio A., Joo-Lyun K., Okuyama K., Lasek A. Controllability of convective heat transfer of magnetic fluid in a circular tube. Journal of Magnetism and Magnetic Materials, 1993, vol. 122, pp. 297–300. / ; 12(2):

17. Xuan Y., Lian W. Electronic cooling using automatic energy transport device based on thermomagnetic effect. Applied Thermal Engineering, 2011, vol. 31(8), pp. 1487–1494. https://doi.org/10.1016/j.applthermaleng.2011.01.033.

18. Mehdi Bahirael, Marteza Handi. Automatic cooling by means of thermomagnetic phenomenon of magnetic nanoflud in a torroidal loop. Applied Thermal Engineering, 2016, vol. 107, pp. 700–708. https://doi.org/10.1016/j.applthermaleng.2016.07.021

19. Fumoto K., Yamagishi H., Ikegava M. A mini heat transport device based on thermosensitive magnetic fluid. Nanoscale and Microscale Thermophysical Engineering, 2007, vol. 11, pp. 201–210. https://doi.org/10.1080/15567260701333869

20. Koskov M. A., Pshenichnikov A. F. Konvekciya magnitnoj zhidkosti v zamknutom gidrodinamicheskom konture [Convection of Magnetic Fluid in a Closed Hydrodynamic Loop]. Vestnik Permskogo universiteta. Fizika = Bulletin of Perm University. Physics, 2021, vol. 2, pp. 14-22. https://doi.org/10.17072/1994-3598-2021-2-14-22

21. Termodat-25 M6. Rukovodstvo pol'zovatelya [User’s guide]. Available at: https://termodat.ru/catalog/termodat-25m6/termodat_25m6. (accessed 23.03.2022)

22. Berkovsky B. M., Medvedev V. F., Krakov M. S. Magnitnye zhidkosti [Magnetic Fluids]. Moscow, Khimiya Publ., 1985. 240 р.

23. Basu D. N. Dynamic frequency response of a single-phase natural circulation loop under an imposed sinusoidal excitation. Annals of Nuclear Energy, 2019, vol. 132, pp. 603-614. https://doi.org/doi.org/10.1016/j.anucene.2019.06.050

24. Drozdov S. M. Laminarnaya konvektsiya vyazkoi i teploprovodnoi zhidkosti v zamknutom kanale [Simulation of the onset on nonstationary and chaos in a hydrodynamic system governed by a small number of degrees of freedom]. Uchenye zapiski TsAGI = Scientific notes of TsAGI, 2001, vol. 36, pp. 40–53. https://doi.org/10.1023/A:1018863206798.

25. Didital Library of Mathematical Functions. Kummer Functions. Available at: https://dlmf.nist.gov/13.2. (accessed 15.03.2022)

26. Isachenko V. P., Osipova V. A., Sukomel A. S. Teploperedacha [Heat Transfer]. Moscow, Energiya Publ., 1957. 487 p.