Skip to main content
SpringerLink
Log in

Turbulent heat and mass transfer over a rotating disk for the Prandtl or Schmidt numbers much larger than unity: an integral method

  • Original
  • Published:
Heat and Mass Transfer Aims and scope Submit manuscript

Abstract

Turbulent heat and mass transfer of a rotating disk for Prandtl and Schmidt numbers much larger than unity was modeled using an integral method validated against empirical equations of different authors for Sherwood numbers. As shown, decrease in relative thickness of thermal/diffusion boundary layers with increasing local radii entails additional increase of the exponent at the Reynolds number in expressions for Nusselt and Sherwood numbers in comparison with air flows.

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

a :

Thermal diffusivity (m2/s)

b :

Outer radius of disk (m)

c f/2:

Friction coefficient \( \left( { =\tau_{\text{w}} /\left( {\rho V_{*}^{2} } \right)} \right) \)

C :

Concentration (mol/m3)

c p :

Isobaric specific heat (J/(kg K))

D m :

Diffusion coefficient (m2/s)

h m :

Mass transfer coefficient (m/s)

h m,av :

Average mass transfer coefficient (m/s) \( ( { ={\frac{2}{{b^{2} }}}\int\nolimits_{0}^{b} {h_{\text{m}} rdr} }) \)

k :

Thermal conductivity (W/(m K))

K 1, K 2 :

Constants in Eqs. 1 and 2

n R :

Exponent in Eqs. 1 and 2

n * :

Exponent in Eq. 3

n, n T :

Exponents in Eqs. 19 and 21

Nu :

Nusselt number \( \left( { =q_{\text{w}} r/[k\Updelta T]} \right) \)

Nu av :

Average Nusselt number \( \left( { =q_{{{\text{w}},{\text{av}}}} b/[k\Updelta T_{\text{av}} ]} \right) \)

Pr :

Prandtl number (=ν/a)

q w :

Local heat flux at the wall (W/m2)

q w,av :

Average heat flux at the wall (W/m2) \(( {={{\int\nolimits_{0}^{b} {q_{\text{w}} rdr} } \mathord{\left/ {\vphantom {{\int\nolimits_{0}^{b} {q_{\text{w}} rdr} } {\int\nolimits_{0}^{b} {rdr} }}} \right. \kern-\nulldelimiterspace} {\int\nolimits_{0}^{b} {rdr} }}}) \)

r, φ, z :

Radial, tangential and axial coordinate (m)

Re ω :

Local Reynolds number (=ωr 2/ν)

Re φ :

Reynolds number (=ωb 2/ν)

Re V* :

Reynolds number \( \left( { ={{V_{*} \delta } \mathord{\left/ {\vphantom {{V_{*} \delta } \nu }} \right. \kern-\nulldelimiterspace} \nu }} \right) \)

Sc :

Schmidt number (=ν/D m)

Sh :

Sherwood number \( \left( { =h_{\text{m}} r/D_{\text{m}} } \right) \)

Sh av :

Average Sherwood number \( \left( { =h_{\text{m,av}} b/D_{\text{m}} } \right) \)

St :

Stanton number \( \left( { =q_{\text{w}} /\rho c_{p} V_{*} \Updelta T} \right) \)

T :

Temperature (K)

T + :

Temperature in wall coordinates (K) \( \left( { =(T_{\text{w}} - T){{\uprho}}c_{\rm p} V_{\tau } /q_{\text{w}} } \right) \)

tanφ:

Tangent of the flow swirl angle \( \left( { =v_{\rm r} /(\omega \, r - v_{\varphi } )} \right) \)

v r :

Radial, tangential and axial

v φ, v z :

Velocity components (m/s)

V :

Total velocity (m/s) \(( { =\left[ {{\text{v}}_{\rm r}^{2} + ({\text{v}}_{\varphi } - \omega \, r)^{2} } \right]^{1/2} }) \)

V + :

Velocity in wall coordinates (m/s) \( \left( { =V/V_{\tau } } \right) \)

V τ :

Friction velocity (m/s) (=(τw/ρ)1/2)

V * :

Characteristic velocity (m/s) \( \left( { =\omega \, r(1 + {{\upalpha}}^{2} )^{1/2} } \right) \)

z + :

Wall coordinate \( \left( { =zV_{\tau } /{{\upnu}}} \right) \)

α:

Value of tanφ at z = 0

δ:

Boundary layer thickness (m)

δT :

Thermal/diffusion boundary layer thickness (m)

Δ:

Dimensionless ratio (= δ T /δ)

\( \Updelta T \) :

Temperature difference (K) \(( { = T_{w} - T_{\infty } }) \)

\( \Updelta T_{\text{av}} \) :

Average temperature difference (K) \( ( { = {{\int\nolimits_{0}^{b} {(T{}_{w} - T_{\infty } )rdr} } \mathord{\left/ {\vphantom {{\int\nolimits_{0}^{b} {(T{}_{w} - T_{\infty } )rdr} } {\int\nolimits_{0}^{b} {rdr} }}} \right. \kern-\nulldelimiterspace} {\int\nolimits_{0}^{b} {rdr} }}}) \)

Θ:

Dimensionless temperature \( \left( { = (T - T_{\text{w}} )/(T_{\infty } - T_{\text{w}} )} \right) \)

μ:

Dynamic viscosity (Pa s)

ν:

Kinematic viscosity (m2/s)

ξ:

Dimensionless coordinate (= z/δ)

ξ T :

Dimensionless coordinate (= z/δ T )

ρ:

Density (kg/m3)

τwr :

Radial shear stress at the wall (Pa) \( ( = \mu (d{\text{v}}_{\rm r} /dz)_{z = 0} ) \)

τwφ :

Tangential shear stress at the wall (Pa) \( \left( { = \mu (d{\text{v}}_{\varphi } /dz)_{z = 0} } \right) \)

τw :

Total wall shear stress (Pa) \( ( { = (\tau_{wr}^{2} + \tau_{w\varphi }^{2} )^{1/2} }) \)

ω:

Angular speed of the disk (1/s)

av:

Average value

w:

Wall (z = 0)

T:

Thermal boundary layer

1:

Boundary of a sub-layer

∞:

Infinity

References

  1. Daguenet M (1968) Etude du transport de matière en solution, a l’aide des électrodes a disque et a anneau tournants. Int J Heat Mass Transf 11(11):1581–1596

    Article  Google Scholar 

  2. Deslouis C, Tribollet B, Viet L (1980) Local and overall mass transfer rates to a rotating disk in turbulent and transition flows. Electrochim Acta 25(8):1027–1032

    Article  Google Scholar 

  3. Dossenbach O (1976) Simultaneous laminar and turbulent mass transfer at a rotating disk electrode. Berichte der Bunsen Gesellschaft Phys Chem Chem Phys 80(4):341–343

    Google Scholar 

  4. Ellison BT, Cornet I (1971) Mass transfer to a rotating disk. J Electrochem Soc 118(1):68–72

    Article  Google Scholar 

  5. Mohr CM, Newman J (1976) Mass transfer to a rotating disk in transitional flow. J Electrochem Soc 123(11):1687–1691

    Article  Google Scholar 

  6. Levich VG (1962) Physicochemical Hydrodynamics. Prentice-Hall, Englewood Cliffs

    Google Scholar 

  7. Newman JS (1991) Electrochemical systems, 2nd edn. Prentice-Hall, Englewood Cliffs

    Google Scholar 

  8. Chen Y-M, Lee W-T, Wu S-J (1998) Heat (mass) transfer between an impinging jet and a rotating disk. Heat Mass Transf 34(2–3):101–108

    Google Scholar 

  9. He Y, Ma LX, Huang S (2005) Convection heat and mass transfer from a disk. Heat Mass Transf 41(8):766–772

    Article  Google Scholar 

  10. Kreith F, Taylor JH, Chong JP (1969) Heat and mass transfer from a rotating disk. Trans ASME J Heat Transf 81:95–105

    Google Scholar 

  11. Tien CL, Campbell DT (1963) Heat and mass transfer from rotating cones. J Fluid Mech 17:105–112

    Article  MATH  Google Scholar 

  12. Janotková E, Pavelek M (1986) A naphthalene sublimation method for predicting heat transfer from a rotating surface. Strojnícky časopis 37(3):381–393 (in Czech)

    Google Scholar 

  13. Shevchuk IV (2008) A new evaluation method for Nusselt numbers in naphthalene sublimation experiments in rotating-disk systems. Heat Mass Transf 44(11):1409–1415

    Article  Google Scholar 

  14. Dorfman LA (1963) Hydrodynamic resistance and the heat loss of rotating solids. Oliver and Boyd, Edinburgh

    Google Scholar 

  15. Owen JM, Rogers RH (1989) Flow and heat transfer in rotating-disc systems. Vol 1: Rotor–stator Systems. Research Studies Press, Taunton

    Google Scholar 

  16. Sparrow EM, Gregg JL (1959) Heat transfer from a rotating disc to fluids at any Prandtl number. Trans ASME J Heat Transf 81:249–251

    Google Scholar 

  17. Shevchuk IV (2000) Turbulent heat transfer of rotating disk at constant temperature or density of heat flux to the wall. High Temp 38(3):499–501

    Article  Google Scholar 

  18. Shevchuk IV, Buschmann MH (2005) Rotating disk heat transfer in a fluid swirling as a forced vortex. Heat Mass Transf 41(12):1112–1121

    Article  Google Scholar 

  19. Popiel CzO, Boguslawski L (1975) Local heat-transfer coefficients on the rotating disk in still air. International J Heat Mass Transf 18(1):167–170

    Article  Google Scholar 

  20. Elkins CJ, Eaton JK (1997) Heat transfer in the rotating disk boundary layer. Department of Mechanical Engineering, Thermosciences Division Report TSD–103, Stanford University, USA

  21. Cardone G, Astarita T, Carlomagno GM (1997) Heat transfer measurements on a rotating disk. Int J of Rotating Mach 3(1):1–9

    Article  Google Scholar 

  22. Kawase Y, De A (1982) Turbulent mass transfer from a rotating disk. Electrochim Acta 27(10):1469–1473

    Article  Google Scholar 

  23. Wasan DT, Tien CL, Wilke CR (1963) Theoretical correlation of velocity and eddy viscosity for flow close to a pipe wall. AIChE J 9(4):567–569

    Article  Google Scholar 

  24. Law CG Jr, Pierini P, Newman J (1981) Mass transfer to rotating disks and rotating rings in laminar, transition and fully-developed turbulent flow. Int J Heat Mass Transf 24(5):909–918

    Article  Google Scholar 

  25. Paterson JA, Greif R (1973) Transport to a rotating disk in turbulent flow at high Prandtl or Schmidt number. Trans ASME J Heat Transf 95(4):566–568

    Google Scholar 

  26. Shevchuk IV (2005) A new type of the boundary condition allowing analytical solution of the thermal boundary layer equation. Int J Therm Sci 44(4):374–381

    Article  Google Scholar 

  27. Shevchuk IV, Khalatov AA (1997) The integral method of calculation of heat transfer in the turbulent boundary layer on a rotating disk: quadratic approximation of the tangent of the flow-swirl angle. Promyshlennaya Teplotekhnika 19(4–5):145–150 (in Russian)

    Google Scholar 

  28. Kabkov VIa (1976) Characteristics of turbulent boundary-layer on a smooth disk rotating in a large volume. Teplofizika i Teplotekhnika 28:119–124

    Google Scholar 

  29. Karman Th von (1921) Über laminare und turbulente Reibung. Z angew Math Mech 1 (4):233–252

    Google Scholar 

  30. Littel HS, Eaton JK (1994) Turbulence characteristics of the boundary layer on a rotating disk. J Fluid Mech 266:175–207

    Article  Google Scholar 

  31. Itoh M, Hasegawa I (1994) Turbulent boundary layer on a rotating disk in infinite quiescent fluid. JSME Int J Ser B 37(3):449–456

    Google Scholar 

  32. Suga K (2007) Computation of high Prandtl number turbulent thermal fields by the analytical wall-function. Int J Heat Mass Transf 50(25–26):4967–4974

    Article  MATH  Google Scholar 

  33. Kader BA (1981) Temperature and concentration profiles in fully turbulent boundary layers. Int J Heat Mass Transf 24(9):1541–1544

    Article  Google Scholar 

Download references

Acknowledgments

The research results presented in this work were obtained in part due to the gratefully acknowledged support of the Research Fellowship of the Alexander von Humboldt Foundation taken by the author at TU Dresden, Germany.

Author information

Authors and Affiliations

  1. MBtech Powertrain GmbH, Salierstr. 38, 70736, Fellbach-Schmiden, Germany

    Igor V. Shevchuk

Authors
  1. Igor V. Shevchuk
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Igor V. Shevchuk.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Shevchuk, I.V. Turbulent heat and mass transfer over a rotating disk for the Prandtl or Schmidt numbers much larger than unity: an integral method. Heat Mass Transfer 45, 1313–1321 (2009). https://doi.org/10.1007/s00231-009-0505-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00231-009-0505-x

Keywords

Search

Navigation