Skip to main content
SpringerLink
Log in

Estimation of slip flow parameters in microscale conjugated heat transfer problems

  • Technical Paper
  • Published:
Journal of the Brazilian Society of Mechanical Sciences and Engineering Aims and scope Submit manuscript

Abstract

In this work, it is proposed the direct and inverse analyses of the forced convection of an incompressible gas flow within rectangular channels in the range of the slip flow regime by taking into account the wall conjugation and the axial conduction effects. The Generalized Integral Transform Technique (GITT) combined with the single-domain reformulation strategy is employed in the direct problem solution of the three-dimensional steady forced convection formulation. A non-classical eigenvalue problem that automatically accounts for the longitudinal diffusion operator is here proposed. The Bayesian framework implemented with the maximum a posteriori objective function is used in the formulation of the inverse problem, whose main objective is to estimate the temperature jump coefficient, the velocity slip coefficient, and the Biot number, using only external temperature measurements, as obtained, for instance, with an infrared measurement system. A comprehensive numerical investigation of possible experimental setups is performed in order to verify the influence of the Biot number, wall thickness, and Knudsen number on the precision of the unknown parameters estimation.

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

Similar content being viewed by others

Abbreviations

\({\mathrm{Bi}}\) :

Biot number

\(CI_{i}\) :

Relative measure of the confidence interval of the estimated value \(\hat{P}_i\)

\(D_h\) :

Hydraulic diameter

\(h_e\) :

Convective heat transfer coefficient

\(\mathbf {J}\) :

Jacobian matrix

\(\mathcal {J}\) :

Scaled sensitivity coefficients

k :

Thermal conductivity

K :

Dimensionless thermal conductivity

\({\mathrm{Kn}}\) :

Knudsen number

\(L_x\) :

Distance from the channel centerline to the external face of the channel wall (x direction)

\(L_y\) :

Distance from the channel centerline to the external face of the channel wall (y direction)

M :

Truncation order of the eigenfunction expansion (eigenvalue problem solution)

N :

Truncation order of the temperature eigenfunction expansion

n :

Norm of the eigenfunction \(\psi (X,Y)\)

\(N_p\) :

Dimension of the vector \(\mathbf {P}\)

\(N_d\) :

Dimension of the vector \(\mathbf {Y}\)

\(\mathbf {n}\) :

Outward-drawn normal vector

\(\mathbf {P}\) :

Vector of parameters

\(\mathbf {P}_{\mathrm{exact}}\) :

Vector with the exact values of the sought parameters

\({\mathrm{Pe}}\) :

Péclet number

\({\mathrm{Pr}}\) :

Prandtl number

\({\mathrm{Re}}\) :

Reynolds number

S :

Objective function

T :

Temperature

\(T_\infty \) :

Ambient temperature

u :

Fully developed flow velocity

U :

Dimensionless flow velocity

\(\mathbf {V}\) :

Covariance matrix of the prior information

\(\mathbf {W}\) :

Covariance matrix of the experimental errors

y :

Transversal coordinate

X, Y :

Dimensionless transversal coordinates

\(\mathbf {Y}\) :

Vector of temperature measurements

z :

Longitudinal coordinate

Z :

Dimensionless longitudinal coordinate

\(Z_f\) :

Dimensionless channel length

\(\alpha _f\) :

Thermal diffusivity of the fluid

\(\alpha _m\) :

Tangential momentum accommodation coefficient

\(\alpha _t\) :

Thermal accommodation coefficient

\(\beta \) :

General temperature jump coefficient in the 3D formulation

\(\beta _t\) :

Wall temperature jump coefficient

\(\beta _v\) :

Wall velocity slip coefficient

\(\epsilon _{\mathrm{fic}}\) :

Dimensionless thickness of the fictitious layer

\(\gamma \) :

Specific heat ratio

\(\lambda \) :

Molecular mean free path

\(\Omega \) :

Auxiliary eigenfunctions

\(\omega \) :

Eigenvalue corresponding to the eigenfunction \(\Omega \)

\(\psi \) :

Temperature eigenfunctions

\(\eta \) :

Eigenvalue corresponding to the eigenfunction \(\psi \)

\({\varvec{\mu }}\) :

Mean vector of the prior density

\(\pi \) :

Probability density function

\(\sigma _e\) :

Standard deviation of the experimental errors

\(\sigma _{P_{i}}\) :

Standard deviation of the estimated parameter \(P_i\)

\(\theta \) :

Dimensionless temperature

\(\nu \) :

Kinematic viscosity

\({\mathrm{ac}}\) :

Quantity corresponding to the axial conduction term

\({\mathrm{av}}\) :

Average

\({\mathrm{f}}\) :

Fluid flow region

\({\mathrm{fic}}\) :

Quantity corresponding to the fictitious layer

\({\mathrm{in}}\) :

Quantity corresponding to the entrance of the channel

\({\mathrm{int}}\) :

Interface position

i, j, m, n :

Indices

\({\mathrm{s}}\) :

Solid region (channel walls)

\({\mathrm{w}}\) :

Quantity corresponding to the external face of the channel wall

X :

Quantity corresponding to the X direction

Y :

Quantity corresponding to the Y direction

\(^{*}\) :

Domain including the fictitious layer

\(\hat{}\) :

Estimated value

\(+\) :

Upper bound of the confidence interval

−:

Lower bound of the confidence interval

References

  1. Sobhan CB, Peterson GP (2008) Microscale and nanoscale heat transfer: fundamentals and engineering applications. CRC Press, Boca Raton

    Book  Google Scholar 

  2. Morini GL (2004) Single-phase convective heat transfer in microchannels: a review of experimental results. Int J Therm Sci 43(7):631–651

    Article  Google Scholar 

  3. Rosa P, Karayiannis TG, Collins MW (2009) Single-phase heat transfer in microchannels: the importance of scaling effects. Appl Therm Eng 29(17–18):3447–3468

    Article  Google Scholar 

  4. Zhang W-M, Meng G, Wei X (2012) A review on slip models for gas microflows. Microfluid Nanofluid 13(6):845–882

    Article  Google Scholar 

  5. Ameel TA, Wang X, Barron RF, Warrington RO (1997) Laminar forced convection in a circular tube with constant heat flux and slip flow. Microscale Thermophys Eng 1(4):303–320

    Article  Google Scholar 

  6. Larrodé FE, Housiadas C, Drossinos Y (2000) Slip-flow heat transfer in circular tubes. Int J Heat Mass Transf 43(15):2669–2680

    Article  MATH  Google Scholar 

  7. Tunc G, Bayazitoglu Y (2001) Heat transfer in microtubes with viscous dissipation. Int J Heat Mass Transf 44(13):2395–2403

    Article  MATH  Google Scholar 

  8. Hooman K, Ejlali A (2010) Effects of viscous heating, fluid property variation, velocity slip, and temperature jump on convection through parallel plate and circular microchannels. Int Commun Heat Mass Transf 37(1):34–38

    Article  Google Scholar 

  9. Xiao N, Elsnab J, Ameel T (2009) Microtube gas flows with second-order slip flow and temperature jump boundary conditions. Int J Therm Sci 48(2):243–251

    Article  Google Scholar 

  10. Norouzi M, Rezaie MR (2018) An exact analysis on heat convection of nonlinear viscoelastic flows in isothermal microtubes under slip boundary condition. J Braz Soc Mech Sci Eng 40(9):472

    Article  Google Scholar 

  11. Yu S, Ameel TA (2001) Slip-flow heat transfer in rectangular microchannels. Int J Heat Mass Transf 44(22):4225–4234

    Article  MATH  Google Scholar 

  12. Tunc G, Bayazitoglu Y (2002) Heat transfer in rectangular microchannels. Int J Heat Mass Transf 45(4):765–773

    Article  MATH  Google Scholar 

  13. Mikhailov MD, Cotta RM (2005) Mixed symbolic-numerical computation of convective heat transfer with slip flow in microchannels. Int Commun Heat Mass Transf 32(3–4):341–348

    Article  Google Scholar 

  14. Cotta RM, Knupp DC, Naveira-Cotta CP (2016) Analytical heat and fluid flow in microchannels and microsystems. Springer, Berlin

    Book  MATH  Google Scholar 

  15. Renksizbulut M, Niazmand H, Tercan G (2006) Slip-flow and heat transfer in rectangular microchannels with constant wall temperature. Int J Therm Sci 45(9):870–881

    Article  Google Scholar 

  16. Kuddusi L (2007) Prediction of temperature distribution and nusselt number in rectangular microchannels at wall slip condition for all versions of constant wall temperature. Int J Therm Sci 46(10):998–1010

    Article  Google Scholar 

  17. Siginer DA, Akyildiz FT, Boutaous M (2019) Unsteady gaseous poiseuille slip flow in rectangular microchannels. J Braz Soc Mech Sci Eng 41(7):286

    Article  Google Scholar 

  18. Maranzana G, Perry I, Maillet D (2004) Mini-and micro-channels: influence of axial conduction in the walls. Int J Heat Mass Transf 47(17–18):3993–4004

    Article  MATH  Google Scholar 

  19. Nunes JS, Cotta RM, Avelino MR, Kakaç S (2010) Conjugated heat transfer in microchannels. In: Kakaç S, Kosoy B, Li D, Pramuanjaroenkij A (eds) Microfluidics based microsystems. NATO science for peace and security series A: chemistry and biology. Springer, Dordrecht

  20. Maranzana G, Perry I, Maillet D (2004) Modeling of conjugate heat transfer between parallel plates separated by a hydrodynamically developed laminar flow by the quadrupole method. Numer Heat Transf Part A 46(2):147–165

    Article  Google Scholar 

  21. Knupp DC, Cotta RM, Naveira-Cotta CP, Kakaç S (2015) Transient conjugated heat transfer in microchannels: integral transforms with single domain formulation. Int J Therm Sci 88:248–257

    Article  Google Scholar 

  22. Knupp DC, Naveira-Cotta CP, Renfer A, Tiwari MK, Cotta RM, Poulikakos D (2015) Analysis of conjugated heat transfer in micro-heat exchangers via integral transforms and non-intrusive optical techniques. Int J Numer Methods Heat Fluid Flow 25(6):1444–1462. https://doi.org/10.1108/HFF-08-2014-0259

    Article  MATH  Google Scholar 

  23. Knupp DC, Naveira-Cotta CP, Cotta RM (2014) Theoretical-experimental analysis of conjugated heat transfer in nanocomposite heat spreaders with multiple microchannels. Int J Heat Mass Transf 74:306–318

    Article  Google Scholar 

  24. Knupp DC, Cotta RM, Naveira-Cotta CP (2015) Fluid flow and conjugated heat transfer in arbitrarily shaped channels via single domain formulation and integral transforms. Int J Heat Mass Transf 82:479–489

    Article  Google Scholar 

  25. Souza J R B, Lisboa K M, Allahyarzadeh A B, de Andrade G J A, Loureiro J B R, Naveira-Cotta C P, Silva Freire A P, Orlande H R B, Silva G A L, Cotta R M (2016) Thermal analysis of anti-icing systems in aeronautical velocity sensors and structures. J Braz Soc Mech Sci Eng 38(5):1489–1509

    Article  Google Scholar 

  26. Knupp DC, Mascouto FS, Abreu LAS, Naveira-Cotta CP, Cotta RM (2018) Conjugated heat transfer in circular microchannels with slip flow and axial diffusion effects. Int Commun Heat Mass Transf 91:225–233

    Article  Google Scholar 

  27. Agrawal A, Prabhu SV (2008) Survey on measurement of tangential momentum accommodation coefficient. J Vac Sci Technol A Vac Surf Films 26(4):634–645

    Article  Google Scholar 

  28. Rader DJ, Castaneda JN, Torczynski JR, Grasser TW, Trott WM (2005) Measurements of thermal accommodation coefficients. Technical report, Sandia National Laboratories

  29. Sharipov F (2011) Data on the velocity slip and temperature jump on a gas-solid interface. J Phys Chem Ref Data 40(2):023101

    Article  Google Scholar 

  30. McCormick NJ (2005) Gas-surface accommodation coefficients from viscous slip and temperature jump coefficients. Phys Fluids 17(10):107104

    Article  MATH  Google Scholar 

  31. Naveira-Cotta CP, Cotta RM, Orlande HRB (2010) Inverse analysis of forced convection in micro-channels with slip flow via integral transforms and bayesian inference. Int J Therm Sci 49(6):879–888

    Article  Google Scholar 

  32. Naveira-Cotta CP (2016) Direct-inverse problem analysis in the thermal characterization of microsystems. In: Rebay M, Kakaç S, Cotta RM (eds) Microscale and nanoscale heat transfer: analysis, design, and application. CRC Press, Boca Raton

    Google Scholar 

  33. Colin S (2012) Gas microflows in the slip flow regime: a critical review on convective heat transfer. J Heat Transf 134(2):020908

    Article  Google Scholar 

  34. Knupp DC, Naveira-Cotta CP, Cotta RM (2013) Conjugated convection-conduction analysis in microchannels with axial diffusion effects and a single domain formulation. J Heat Transf 135(9):091401

    Article  Google Scholar 

  35. Wolfram S et al (1996) Mathematica. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  36. Kaipio J, Somersalo E (2006) Statistical and computational inverse problems, vol 160. Springer, Berlin

    MATH  Google Scholar 

  37. Orlande HRB, Fudym O, Maillet D, Cotta RM (2011) Thermal measurements and inverse techniques. CRC Press, Boca Raton

    Book  Google Scholar 

  38. Özişik MN, Orlande HRB (2000) Inverse heat transfer: fundamentals and applications. CRC Press, Boca Raton

    Google Scholar 

  39. Beck JV, Arnold KJ (1977) Parameter estimation in engineering and science. James Beck, Denver

    MATH  Google Scholar 

  40. Knupp DC, Naveira-Cotta CP, Ayres JVC, Cotta RM, Orlande HRB (2012) Theoretical-experimental analysis of heat transfer in nonhomogeneous solids via improved lumped formulation, integral transforms and infrared thermography. Int J Therm Sci 62:71–84

    Article  MATH  Google Scholar 

  41. Figueira da Silva E, Cotta RM (1996) Benchmark results for internal forced convection through integral transformation. Int Commun Heat Mass Transf 23(7):1019–1029

    Article  Google Scholar 

  42. Knupp DC, Naveira-Cotta CP, Ayres JVC, Orlande HRB, Cotta RM (2012) Space-variable thermophysical properties identification in nanocomposites via integral transforms, bayesian inference and infrared thermography. Inverse Probl Sci Eng 20(5):609–637

    Article  MathSciNet  MATH  Google Scholar 

  43. Knupp DC, Naveira-Cotta CP, Orlande HRB, Cotta RM (2013) Experimental identification of thermophysical properties in heterogeneous materials with integral transformation of temperature measurements from infrared thermography. Exp Heat Transf 26(1):1–25

    Article  Google Scholar 

Download references

Acknowledgements

This study was financed in part by CAPES—Coordenação de Aperfeiçoamento de Pessoal de Nível Superior—Brasil, Finance Code 001. The authors would also like to thank the other sponsoring agencies, CNPq—Conselho Nacional de Desenvolvimento Científico e Tecnológico, and FAPERJ—Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro.

Author information

Author notes

  1. Renato M. Cotta

    Present address: General Directorate of Nuclear and Technological Development, DGDNTM, Brazilian Navy, Ministry of Defense, Brasilia, Brazil

Authors and Affiliations

  1. LEMA - Laboratory of Experimentation and Numerical Simulation in Heat and Mass Transfer, Department of Mechanical Engineering and Energy, Polytechnic Institute, Rio de Janeiro State University, IPRJ/UERJ, Rua Bonfim 25, Vila Amelia, Nova Friburgo, RJ, 28625-570, Brazil

    Géssica R. Silva, Diego C. Knupp & Antônio J. Silva Neto

  2. LabMEMS - Laboratory of Nano and Microfluidics and Microsystems, Mechanical Engineering Department - PEM, POLI/COPPE, Federal University of Rio de Janeiro, UFRJ, Cx. Postal 68503, Rio de Janeiro, RJ, 21945-970, Brazil

    Carolina P. Naveira-Cotta & Renato M. Cotta

Authors
  1. Géssica R. Silva
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Diego C. Knupp
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Carolina P. Naveira-Cotta
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Renato M. Cotta
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Antônio J. Silva Neto
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Diego C. Knupp.

Additional information

Technical Editor: Francis HR Franca, Ph.D.

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

Silva, G.R., Knupp, D.C., Naveira-Cotta, C.P. et al. Estimation of slip flow parameters in microscale conjugated heat transfer problems. J Braz. Soc. Mech. Sci. Eng. 42, 263 (2020). https://doi.org/10.1007/s40430-020-02328-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40430-020-02328-z

Keywords

Search

Navigation