Skip to main content
SpringerLink
Log in

Uncertainty Quantification Through use of the Monte Carlo Method in a One-Dimensional Heat Conduction Experiment

  • Published:
International Journal of Thermophysics Aims and scope Submit manuscript

Abstract

The correct expression of temperature measurement results is of the utmost importance for experimental research in thermal sciences. Temperature measurements are used in heat transfer models to estimate various parameters, either in direct or inverse problems. The reliability of these parameter values depends mainly on the uncertainty associated with the measured temperature. This paper deals with the application of Monte Carlo method for uncertainty quantification, in an experimental model of heat transfer that describes the behavior of a homogeneous, isotropic and linear solid. Temperature measurements were carried out using a type K thermocouple, considering a nominal measuring range from − 5 °C to 110 °C, at a given point in an AISI 304 stainless steel sample, specifically a massive cylindrical billet. The sample was placed in an experimental setup, and it was submitted to a one-dimensional steady-state thermal field, with boundary conditions of prescribed temperature and prescribed heat flux. The uncertainty associated with temperature was assessed using the Monte Carlo method, and the obtained results were compared with those calculated by the Guide to the Expression of Uncertainty in Measurement (GUM). Noteworthy in this study was that the temperature simulated values follow a Gaussian probability distribution function. The expanded uncertainty (k =2.00) associated with temperature (in Kelvin) was 0.42 % about the measured average temperature. The results presented herein can be useful for those cases when the mechanical component is not fully accessible physically. Therefore, using the temperature measured in a particular region and since the heat conduction problem is unidimensional in a steady state, it is possible to estimate the temperature in any section.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

Abbreviations

AISI:

American Iron and Steel Institute

AWG:

American wire gauge

CIPM:

International Committee for Weights and Measures

GUM:

Guide to the expression of uncertainty in measurement

IEC:

International Electrotechnical Commission

ISO:

International organization for standardization

JCGM:

Joint Committee for Guides in Metrology

MC:

Monte Carlo

PDF:

Probability density function

Y :

Output variable

W :

Input variable

w :

Uncertainty sources

u :

Standard uncertainty

u c :

Combined standard uncertainty

U :

Expanded uncertainty

N :

Number of variables

r :

Correlation coefficient between the uncertainty sources

k :

Coverage factor

v eff :

Degree of freedom

u i :

Standard uncertainty associated with each input variable

v i :

Degrees of freedom of each input variable

M :

Number of MC trials

L :

Sample height, mm

x :

Spatial coordinate, mm

T :

Temperature, °C

T 0 :

Boundary temperature, °C

q″ :

Heat flux, W·m−2

λ :

Thermal conductivity, W·m−1·K−1

Ttc :

Temperature values indicated by the thermocouple, °C

Tg :

Temperature values indicated by the glass thermometer,  °C

E :

Systematic error, °C

Tc :

Measured temperature indicated by the thermocouple during calibration, °C

\(\bar{T}tc\) :

Average for the values indicated by the thermocouple, °C

ΔCTg :

Correction associated with the glass thermometer calibration, °C

V :

Electrical voltage, V

I :

Electrical current, A

\(\bar{T}\) :

Average temperature, °C

ΔR :

Correction associated with the thermocouple resolution, °C

ΔC :

Correction due to the measurement system calibration, °C

n :

Number of temperature readings

References

  1. C. Soize, Uncertainty Quantification: An accelerated course with advanced applications in computational engineering (Springer International Publishing, Switzerland, 2017), pp. 1–15

    Book  Google Scholar 

  2. S. Salicona, Measurement uncertainty: an approach via the mathematical theory of evidence (Springer, US, New York, 2007), pp. 1–14

    Google Scholar 

  3. JCGM 100, Evaluation of measurement data—guide to the expression of uncertainty in measurement (GUM), GUM 1995 with minor corrections (Joint Committee for Guides in Metrology, BIPM, Paris, 1995), p. 2008

    Google Scholar 

  4. R.V. Arencibia, C.C. Souza, H.L. Costa, A. Pitarelli-Filho, J. Braz. Soc. Mech. Sci. Eng. (2015). https://doi.org/10.1007/s40430-014-0157-8

    Article  Google Scholar 

  5. R. Lima, R. Sampaio, J. Braz. Soc. Mech. Sci. Eng. (2018). https://doi.org/10.1007/s40430-018-1079-7

    Article  Google Scholar 

  6. K. Rost, K. Wendt, F. Härtig, Precis. Eng. (2016). https://doi.org/10.1016/j.precisioneng.2016.01.001

    Article  Google Scholar 

  7. M. Désenfant, M. Priel, Measurement (2017). https://doi.org/10.1016/j.measurement.2016.10.022

    Article  Google Scholar 

  8. M.V. Rezende, J.E.S. Leal, R.R. Pires, M.T. Piza Paes, F.F. Ramos Neto, S.D. Franco, R.V. Arencibia, J. Braz. Soc. Mech. Sci. Eng. (2019). https://doi.org/10.1007/s40430-019-1817-5

    Article  Google Scholar 

  9. JCGM 200:2012—International vocabulary of metrology—basic and general concepts and associated terms (VIM) (Joint Committee for Guides in Metrology, BIPM, Paris, 2012)

  10. H. Schwenke, B.R.L. Siebert, F. Wäldele, H. Kunzmann, CIRP Ann. (2000). https://doi.org/10.1016/S0007-8506(07)62973-4

    Article  Google Scholar 

  11. JCGM 101—Evaluation of Measurement Data—Supplement 1 to the guide to the expression of uncertainty in measurement—propagation of distributions using a Monte Carlo Method, 1st edn. (Joint Committee for Guides in Metrology, BIPM, Paris, 2008)

  12. D. Theodorou, L. Meligotsidou, S. Karavoltsos, A. Burnetas, M. Dassenakis, M. Scoullos, Talanta (2011). https://doi.org/10.1016/j.talanta.2010.11.059

    Article  Google Scholar 

  13. C. Wang, Z. Qiu, Y. Yang, Int. J. Heat Mass Transfer (2016). https://doi.org/10.1016/j.ijheatmasstransfer.2016.03.094

    Article  Google Scholar 

  14. G. Guimarães, P.C. Philippi, P. Thery, Rev. Sci. Instrum. (1995). https://doi.org/10.1063/1.1145592

    Article  Google Scholar 

  15. J.M. Owen, P.J. Newton, G.D. Lock, Int. J. Heat Fluid Fl. (2003). https://doi.org/10.1016/S0142-727X(02)00207-2

    Article  Google Scholar 

  16. V. L. Borges, S. M. M. Lima e Silva, G. Guimarães, Inverse Probl. Sci. Eng. (2006) https://doi.org/10.1080/17415970600573700

  17. P. Rosenkranz, Int. J. Thermophys. (2011). https://doi.org/10.1007/s10765-010-0879-5

    Article  Google Scholar 

  18. P.R. Muniz, R.A. Kalid, S.P.N. Cani, R.S. Magalhães, Opt. Eng. (2014). https://doi.org/10.1117/1.OE.53.7.074101

    Article  Google Scholar 

  19. A.P. Fernandes, M.B. dos Santos, G. Guimarães, Appl. Math. Model. (2015). https://doi.org/10.1016/j.apm.2015.02.012

    Article  Google Scholar 

  20. H. Jin, G. Lin, L. Bai, M. Amjad, E.P. BandarraFilho, D. Wen, Sol. Energy (2016). https://doi.org/10.1016/j.solener.2016.09.021

    Article  Google Scholar 

  21. Z. Xusheng, Z. Lianyu, T. Xiaojun, Proc. CIRP (2016). https://doi.org/10.1016/j.procir.2016.10.078

    Article  Google Scholar 

  22. T.S. Bording, S.B. Nielsen, N. Balling, Int. J. Heat Mass Transfer (2019). https://doi.org/10.1016/j.ijheatmasstransfer.2018.12.104

    Article  Google Scholar 

  23. D.F.M. Pico, L.R.R. da Silva, P.S. Schneider, E.P. Bandarra Filho, Int. J. Refrig (2019). https://doi.org/10.1016/j.ijrefrig.2018.12.009

    Article  Google Scholar 

  24. L.R. de Oliveira, S.R.F.L. Ribeiro, M.H.M. Reis, V.L. Cardoso, E.P. Bandarra Filho, Diam. Relat. Mater. (2019). https://doi.org/10.1016/j.diamond.2019.05.004

    Article  Google Scholar 

  25. D.F.M. Pico, L.R.R. da Silva, O.S.H. Mendoza, E.P. Bandarra Filho, Int. J. Heat Mass Transfer 1, 1 (2020). https://doi.org/10.1016/j.ijheatmasstransfer.2020.119493

    Article  Google Scholar 

  26. B. Hay, J.R. Flitz, J. Hameury, L. Rongione, Int. J. Thermophys. (2005). https://doi.org/10.1007/s10765-005-8603-6

    Article  Google Scholar 

  27. M.J. Assael, K.D. Antoniadis, I.N. Metaxa, S.K. Mylona, J.A.M. Assael, J. Wu, M. Hu, Int. J. Thermophys. (2015). https://doi.org/10.1007/s10765-015-1964-6

    Article  Google Scholar 

  28. K.D. Antoniadis, G.J. Tertsinidou, M.J. Assael, W.A. Wakeham, Int. J. Thermophys. (2016). https://doi.org/10.1007/s10765-016-2083-8

    Article  Google Scholar 

  29. A. Benisek, E. Dachs, Cryogenics (2018). https://doi.org/10.1016/j.cryogenics.2008.08.005

    Article  Google Scholar 

  30. A. Araújo, S. Silvano, N. Martins, Infrared Phys. Technol. (2014). https://doi.org/10.1016/j.infrared.2014.07.018

    Article  Google Scholar 

  31. J. Mazo, A.T. El Badry, J. Carreras, M. Delgado, D. Boer, B. Zalba, Appl. Therm. Eng. (2015). https://doi.org/10.1016/j.applthermaleng.2015.07.047

    Article  Google Scholar 

  32. A. Araújo, Measurement (2016). https://doi.org/10.1016/j.measurement.2016.08.007

    Article  Google Scholar 

  33. T.A. Moreira, A.R.A. Colmanetti, C.B. Tibiriçá, J. Braz. Soc. Mech. Sci. Eng. (2019). https://doi.org/10.1007/s40430-019-1763-2

    Article  Google Scholar 

  34. M.B.M. Taib, J.P.M. Trusler, Int. J. Thermophys. (2020). https://doi.org/10.1007/s10765-020-02700-0

    Article  Google Scholar 

  35. J. Caro-Corrales, K. Cronin, X. Gao, V. Cregan, J. Food Eng. (2010). https://doi.org/10.1016/j.jfoodeng.2010.02.014

    Article  Google Scholar 

  36. K. Shahanaghi, P. Nakhjiri, Measurement (2010). https://doi.org/10.1016/j.measurement.2010.03.008

    Article  Google Scholar 

  37. L.L. Martins, A.S. Ribeiro, J. Alves e Sousa, A.B. Forbes, Int. J. Thermophys. (2012). https://doi.org/10.1007/s10765-011-1005-z

    Article  Google Scholar 

  38. D.K. Banerjee, Fire Saf. J. (2013). https://doi.org/10.1016/j.firesaf.2013.08.012

    Article  Google Scholar 

  39. M. Noorkami, J.B. Robinson, Q. Meyer, O.A. Obeisun, E.S. Fraga, T. Reisch, P.R. Shearing, D.J.L. Brett, Int. J. Hydrogen Energy (2014). https://doi.org/10.1016/j.ijhydene.2013.10.156

    Article  Google Scholar 

  40. B.W. Weber, C.J. Sung, M.W. Renfro, Combust. Flame (2015). https://doi.org/10.1016/j.combustflame.2015.03.001

    Article  Google Scholar 

  41. D.K. Banerjee, Fire Saf. J. (2016). https://doi.org/10.1016/j.firesaf.2016.03.005

    Article  Google Scholar 

  42. R. Palencar, P. Sopkuliak, J. Palencar, S. Duris, E. Suroviak, M. Halaj, Meas. Sci. Rev. (2017). https://doi.org/10.1515/msr-2017-0014

    Article  Google Scholar 

  43. A.C.G.C. Diniz, M.E.K. de Almeida, J.N.S. Vianna, A.B.S. Oliveira, A.T. Fabro, J. Braz. Soc. Mech. Sci. Eng. (2017). https://doi.org/10.1007/s40430-016-0619-2

    Article  Google Scholar 

  44. N.L. Bhirud, R.R. Gawande, J. Braz. Soc. Mech. Sci. Eng. (2017). https://doi.org/10.1007/s40430-017-0869-7

    Article  Google Scholar 

  45. R.A.M. Ferreira, B.P.A. Silva, G.G.D. Teixeira, R.M. Andrade, M.P. Porto, Measurement (2019). https://doi.org/10.1016/j.measurement.2018.09.036

    Article  Google Scholar 

  46. P.R.N. Childs, J.R. Greenwood, C.A. Long, Rev. Sci. Instrum. (2000). https://doi.org/10.1063/1.1305516

    Article  Google Scholar 

  47. J.V. Nicholas, D.R. White, Traceable temperatures: an introduction to temperature measurement and calibration, 2nd edn. (Wiley, New Jersey, 2001), pp. 125–158

    Book  Google Scholar 

  48. A.K. Kaminise, G. Guimarães, M.B. da Silva, Int. J. Adv. Manuf. Technol. (2014). https://doi.org/10.1007/s00170-014-5898-0

    Article  Google Scholar 

  49. L.R.R. da Silva, A. Favero Filho, E.S. Costa, D.F.M. Pico, W.F. Sales, W.L. Guesser, A.R. Machado, Proc. Manuf. (2018). https://doi.org/10.1016/j.promfg.2018.07.056

    Article  Google Scholar 

  50. Y.T. Iç, E.S. Güler, Z.E. Çakir, Int. J. Thermophys. (2019). https://doi.org/10.1007/s10765-019-2516-2

    Article  Google Scholar 

  51. D.C. Ripple, K.M. Garrity, NCSLI Meas. (2006). https://doi.org/10.1080/19315775.2006.11721305

    Article  Google Scholar 

  52. ISO/IEC 17025, General Requirements for the Competence of Testing and Calibration Laboratories (ISO, 2017)

  53. L.F.S. Carollo, A.L.F. Lima e Silva, S.M.M. Lima e Silva, Meas. Sci. Technol. (2012). https://doi.org/10.1088/0957-0233/23/6/065601

    Article  Google Scholar 

  54. J. R. FerreiraOliveira, L. R. R. Lucena, R. P. B. Reis, C. J. Araújo, C. R. BezerraFilho, Proc. Braz. Cong. Therm. Sci. Eng. (2018). https://doi.org/10.26678/ABCM.ENCIT2018.CIT18-0444

  55. J.V. Pearce, V. Montag, D. Lowe, W. Dong, Int. J. Thermophys. (2011). https://doi.org/10.1007/s10765-010-0892-8

    Article  Google Scholar 

  56. M. Noriega, R. Ramírez, R. López, M. Vaca, J. Morales, H. Terres, A. Lizardi., S. Chávez, J. Phys. (2015) https://doi.org/10.1088/1742-6596/582/1/012029

  57. Omega®, Temperature Measurement: Thermocouple sensors, connectors, wire, surface probes and accessories, Vol. 2 (Omega®, United Kingdom, 2017), pp. 5–9

Download references

Acknowledgments

The authors would like to thank the Brazilian financing agencies Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES), Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) and Fundação de Amparo à Pesquisa de Minas Gerais (FAPEMIG) for supporting the development of this research.

Author information

Authors and Affiliations

  1. School of Mechanical Engineering, Federal University of Uberlândia, Uberlândia, MG, Brazil

    José Ricardo Ferreira-Oliveira & Rosenda Valdés Arencibia

  2. Department of Mechanical Engineering, Federal University of Campina Grande, Campina Grande, PB, Brazil

    Luiz Roberto Rocha de Lucena, Carlos José de Araújo & Celso Rosendo Bezerra-Filho

  3. Department of Engineering and Technology, Federal University of the Semi-Arid, Mossoró, RN, Brazil

    Rômulo Pierre Batista dos Reis

Authors
  1. José Ricardo Ferreira-Oliveira
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Luiz Roberto Rocha de Lucena
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Rômulo Pierre Batista dos Reis
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Carlos José de Araújo
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Celso Rosendo Bezerra-Filho
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Rosenda Valdés Arencibia
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to José Ricardo Ferreira-Oliveira.

Ethics declarations

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ferreira-Oliveira, J.R., de Lucena, L.R.R., dos Reis, R.P.B. et al. Uncertainty Quantification Through use of the Monte Carlo Method in a One-Dimensional Heat Conduction Experiment. Int J Thermophys 41, 140 (2020). https://doi.org/10.1007/s10765-020-02724-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10765-020-02724-6

Keywords

Search

Navigation