Generalization of Vapor Bubble Size during Unsteady Boiling With the Use of Two Stage Optimization Method

This study aims to apply a novel technique devised by the authors to process the results of thermal physics experiments. The paper describes a two-stage technique for identifying coefficients of mathematical models from observed experimental data. The technique is based on the maximum likelihood method and is informed by the errors of all sensors used to obtain parameter measurements. Stage 1 of the technique minimizes the maximum relative error over all measured parameters, which allows gross measurement errors to be identified in qualitative terms and reduces the maximum relative error down to acceptable values. At Stage 2, we propose to use the method of weighted least absolute values to minimize the sum of absolute values of relative errors of all measured parameters. The technique was applied to process the results of thermal physics experiments aimed at generalizing the size of vapor bubbles of various types during unsteady heating of a vertical steel cylindrical heater surrounded by an upward flow of water subcooled to the saturation temperature. The numerical simulations reported in this study attest to the high quality of the proposed two-stage technique for identifying coefficients of mathematical models. The study also presents a comparative analysis of the results obtained by the classical least squares method and the novel two-stage technique.

Keywords

thermal physics experiments; coefficient identification; mathematical model; maximum likelihood criterion; weighted least absolute values method; least squares method.

References

1. Wentzel E.S. Probability Theory. Moskow, Mir, 1982.
2. Eadie W.T, Dryard D., James F.E., Roos M., Sadoulet B. Statistical Methods in Experimental Physics. Amsterdam, North-Holland Puplishing Company, 1971.
3. Kler A.M., Alekseiuk V.E., Levin A.A., Khan P.V. [A Method for Processing the Results of Thermophysical Experiments Based on Solving Two Types of Problems of Nonlinear Mathematical Programming]. Informacionnye i matematicheskie tehnologii v nauke i upravlenii [Information and Mathematical Technologies in Science and Management], 2022, no. 4 (28), pp. 32-49. (in Russian)
4. Gilman L., Baglietto E. A Self-Consistent, Physics-Based Boiling Heat Transfer Modeling Framework for Use in Computational Fluid Dynamics. International Journal of Multiphase Flow, 2017, vol. 95, pp. 35-53. DOI: 10.1016/j.ijmultiphaseflow.2017.04.018
5. Hoang Nhan Hien, Song Chul H., Chu In Cheol, Euh Dong Jin. A Bubble Dynamics-Based Model for Wall Heat Flux Partitioning During Nucleate Flow Boiling. International Journal of Heat and Mass Transfer, 2017, no. 112, pp. 454-464. DOI: 10.1016/j.ijheatmasstransfer.2017.04.128
6. Levin A.A., Khan P.V. Characteristics of Nucleate Boiling Under Conditions of Pulsed Heat Release at the Heater Surface. Applied Thermal Engineering, 2019, vol. 149, pp. 1215-1222. DOI: 10.1016/j.applthermaleng.2018.12.126
7. Unal H.C. Maximum Bubble Diameter, Maximum Bubble-Growth Time and Bubble-Growth Rate During the Subcooled Nucleate Flow Boiling of Water up to 17.7 MN/m^2. International Journal of Heat and Mass Transfer, 1976, vol. 19, no. 6, pp. 643-649. DOI: 10.1016/0017-9310(76)90047-8
8. Levin A.A. Choosing Average Values When Determining Characteristics of the Unsteady Boiling of Liquid. Bulletin of the South Ural State University, Series: Mathematical Modelling, Programming and Computer Software, 2021, vol. 14, no. 3, pp. 99-105. DOI: 10.14529/mmp210308
9. Chu In-Cheol, No Hee Cheon, Song Chul-Hwa. Bubble Lift-Off Diameter and Nucleation Frequency in Vertical Subcooled Boiling Flow. Journal of Nuclear Science and Technology, 2011, vol. 48, no. 6, pp. 936-949. DOI: 10.1080/18811248.2011.9711780
10. Prodanovic V., Fraser D., Salcudean M. Bubble Behavior in Subcooled Flow Boiling of Water at Low Pressures and Low Flow Rates. International Journal of Multiphase Flow, 2002, vol. 28, no. 1, pp. 1-19. DOI: 10.1016/S0301-9322(01)00058-1
11. Levin A.A., Khan P.V. Experimental Observation of the Maximum Bubble Diameter in Non-Stationary Temperature Field of Subcooled Boiling Water Flow. International Journal of Heat and Mass Transfer, 2018, vol. 124, pp. 876-883. DOI: 10.1016/j.ijheatmasstransfer.2018.03.078
12. Levin A.A., Khan P.V. Intensification of Non-Stationary Nucleate Boiling at Increasing Flow Velocity. Heat Transfer Engineering, 2022, vol. 43, no. 3-5, pp. 388-396. DOI: 10.1080/01457632.2021.1874682
13. Levin A.A., Khan P.V. Effect of Micro-Sized Vapor Bubbles on Heat Transfer at Different Heater Temperature Rise Rate. Technical Physics Letters, 2024, vol. 50, no. 4, pp. 19-22. DOI: 10.21883/0000000000
14. Kler A.M., Zharkov P.V., Epishkin N.O. Parametric Optimization of Supercritical Power Plants using Gradient Methods. Energy, 2019, article ID: 116230, 42 p. DOI: 10.1016/j.energy.2019.116230
15. Alekseiuk V. Improving the Efficiency of the Three-Stage Technique of Mathematical Model Identification of Complex Thermal Power Equipment. ENERGY-21 - Sustainable Development and Smart Management, Irkutsk, 2020, vol. 209, article ID: 03002, 9 p. DOI: 10.1051/e3sconf/202020903002
16. Zabuga F.V., Alekseiuk V.E. Research Based on Mathematical Modelling of CHP-10 Power unit No 5 "Baikal Energy Company" LLC to Assess the Efficiency of Its Modernisation. International Conference of Young Scientists "Energy Systems Research 2021", Irkutsk, 2021, vol. 289, article ID: 02002, 9 p. DOI: 10.1051/e3sconf/202128902002