Abstract: The aim of this study is to determine the heat transfer coefficient between the heated surface
and the boiling fluid flowing in a vertical minichannel on the basis of experimental data. The
calculation model is based on Beck’s method coupled with the FEM and Trefftz functions.
The Trefftz functions used in the Hermite interpolation are employed to construct the shape
functions in the FEM. The unknown local values of the heat transfer coefficient at the foil-
fluid contact surface are calculated from Newton’s law. The temperature of the heated foil
and the heat flux on the foil surface are determined by solving a two-dimensional inverse
heat conduction problem. The study is focused on the identification of the heat transfer
coefficients in the subcooled boiling region and the saturated nucleate boiling region. The
results are compared with the data obtained through the one-dimensional method. The
investigations also reveal how the smoothing of measurement data affects calculation results.
B I B L I O G R A F I A1. Beck J.V., Blackwell B., Clair C.R.St., 1985, Inverse Heat Conduction. Ill-Posed Problems,
Wiley-Interscience Publ., New York
2. Beck J.V., 1970, Nonlinear estimation applied to the nonlinear inverse heat conduction problem,
International Journal of Heat and Mass Transfer, 13, 703-716
3. Ciałkowski M.J., 2002, New type of basic functions of FEM in application to solution of inverse
heat conduction problem, Journal of Thermal Science, 11, 163-171
4. Ciałkowski M.J., Grysa K., 2010, Trefftz method in solving the inverse problems, Journal
Inverse Ill-Posed Problems, 18, 595-616
5. Ciałkowski M.J., Frąckowiak A., 2002, Solution of the stationary 2D inverse heat conduction
problem by Treffetz method, Journal of Thermal Science, 11, 148-162
6. Duda P., Taler J., 2009, A new method for identification of thermal boundary conditions
in water-wall tubes of boiler furnaces, International Journal of Heat and Mass Transfer, 52,
1517-1524
7. Grysa K., Hożejowska S., Maciejewska B., 2012, Compensatory calculus and Trefftz functions
applied to local heat transfer coefficient determination in a minichannel, Journal of Theoretical
and Applied Mechanics, 50, 087-1096
8. Grysa K., Maciejewska B., 2013, Trefftz functions for the non-stationary problems, Journal of
Theoretical and Applied Mechanics, 51, 251-264
9. Herrera I., 2000, Trefftz method: a general theory, Numerical Methods for Partial Differential
Equations, 16, 561-580
10. Hożejowska S., Maciejewska B., Poniewski M.E., 2015, Numerical analysis of boiling twophase
flow in mini- and microchannels, [In:] Encyclopedia of Two-Phase Heat Transfer and Flow.
I. Fundamentals and Method, Thome J.R. (Edit.), World Scientific Publishing Co Ltd., New Jersey
11. Hożejowska S., Piasecka M., 2014, Equalizing calculus in Trefftz method for solving twodimensional
temperature field of FC-72 flowing along the minichannel, Heat and Mass Transfer,
50, 1053-1063
12. Hożejowska S., Piasecka M., Poniewski M.E., 2009, Boiling heat transfer in vertical minichannels.
Liquid crystal experiments and numerical investigations, International Journal of Thermal
Sciences, 48, 1049-1059
13. Kincaid D., Cheney W., 2002, Numerical Analysis: Mathematics of Scientific Computing, 3rd
ed., Brooks/Cole Publishing Company, Belmont, California
14. Kompis V., Konkol F., Vasko M., 2001, Trefftz-polynomial reciprocity based FE formulations,
Computer Assisted Mechanics and Engineering Sciences, 8, 385-395
15. Kruk B., Sokała M., 1999, Sensitivity coefficients and heat polynomials in the inverse heat
conduction problems, Journal of Applied Mathematics and Mechanics, ZAMM, 3, 693-694
16. Kruk B., Sokała M., 2000, Sensitivity coefficients applied to two-dimensional transient inverse
heat conduction problems, Journal of Applied Mathematics and Mechanics, ZAMM, 81, 945-946
17. Kurpisz K., Nowak A.J., 1992, BEM approach to inverse heat conduction problem, Engineering
Analysis with Boundary Elements, 10, 291-297
18. Le Niliot C., Lefevre F., 2004, A parameter estimation approach to solve the inverse problem
of point heat sources identification, International Journal of Heat and Mass Transfer, 47, 827-841
19. Li Z.-C., Lu T.-T., Huang H.-T., Cheng A.H.-D., 2006, Trefftz, collocation, and other boundary
methods – a comparison, Numerical Methods for Partial Differential Equations, 23, 1-52
20. Lin D.T.W., Yan W.-M., Li H.-Y., 2008, Inverse problem of unsteady conjugated forced convection
in parallel plate channels, International Journal of Heat and Mass Transfer, 51, 993-1002
21. Maciejewska B., 2004, Application of the modified method of finite elements for identification
of temperature of a body heated with a moving heat source, Journal of Theoretical and Applied
Mechanics, 42, 771-787
22. Maciąg A., 2011, The usage of wave polynomials in solving direct and inverse problems for
two-dimensional wave equation, International Journal for Numerical Methods in Biomedical Engineering,
27, 1107-1125
23. Ozer B.A., Oncel A.F., Hollingsworth D.H., Witte L.C., 2011, A method of concurrent
thermographic-photographic visualization of flow boiling in a minichannel, Experimental Thermal
and Fluid Science, 35, 1522-1529
24. Piasecka M., 2013, Determination of the temperature field using liquid crystal thermography
and analysis of two-phase flow structures in research on boiling heat transfer in a minichannel,
Metrology and Measurement Systems, XX, 205-216
25. Piasecka M., 2014a, Flow boiling heat transfer in a minichannel with enhanced heating surface,
Heat Transfer Engineering, 35, 903-912
26. Piasecka M., 2014b, Laser texturing, spark erosion and sanding of the surfaces and their practical
applications in heat exchange devices, Advanced Material Research, 874, 95-100
27. Piasecka M., 2014c, The use of enhanced surface in flow boiling heat transfer in a rectangular
minichannels, Experimental Heat Transfer, 27, 231-255
28. Piasecka M., 2015, Impact of selected parameters on boiling heat transfer and pressure drop in
minichannels, International Journal of Refrigeration, 56, 198-212
29. Piasecka M., Maciejewska B., 2012, The study of boiling heat transfer in vertically and horizontally
oriented rectangular minichannels and the solution to the inverse heat transfer problem
with the use of the Beck method and Trefftz functions, Experimental Thermal and Fluid Science,
38, 19-32
30. Piasecka M., Maciejewska B., 2013, Enhanced heating surface application in a minichannel
flow and the use of the FEM and Trefftz functions for the solution of inverse heat transfer problem,
Experimental Thermal and Fluid Science, 44, 23-33
31. Piasecka M., Maciejewska B., 2015, Heat transfer coefficient during flow boiling in a minichannel
at variable spatial orientation, Experimental Thermal and Fluid Science, 68, 459-467
32. Piasecka M., Strąk K., Maciejewska B., 2016, Calculations of flow boiling heat transfer in
a minichannel using data from LCT and IRT, Heat Transfer Engineering, 5, 14, 1-51
33. Shi J., Wang J., 2009, Inverse problem of transpiration cooling for estimating wall heat flux by
LTNE model and CGM method, International Journal of Heat and Mass Transfer, 52, 2714-2720
34. Tikhonov A.N., Arsenin V.Y., 1977, Solution of Ill-Posed Problems, Wiley, New York
35. Trefftz E., 1926, Ein Gegenst¨uck zum Ritzschen Verfahren, 2 Int. Kongress f¨ur Technische
Mechanik, Z¨urich, 1926, 131-137
36. 36 Tseng A.A., Chen T.C., Zhao F.Z., 1995, Direct sensitivity coefficient method for solving
two-dimensional inverse heat conduction problems by finite-element scheme, Numerical Heat
Transfer, Part B: Fundamentals, 27, 291-307
37. Tseng A.A., Chen T.C., Zhao F.Z., 1996, Multidimensional inverse transient heat conduction
problems by direct sensitivity coefficent method using a finite-element scheme, Numerical Heat
Transfer, Part B: Fundamentals, 29, 365-38 
DOI