A theoretical investigation is carried out to study natural convection around a vertical thin cylinder, or a needle, heated at uniform and constant wall heat flux in order to compare the analytical solutions of the present work with previous experimental results. The local non-similarity solution with the first level of truncation, proposed by Minkowycz and Sparrow, is used. The temperature and velocity distributions are calculated for fluids with several Prandtl numbers. The analytical solutions of this work are compared to experimental results carried out with needles of diameters ranging from 0.6 to 1.5 mm and fluids with Prandtl numbers in the range Pr = 0.7–730. The agreement is reasonable good.
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Abbreviations
- Symbols :
-
- D:
-
Diameter of the needle (m)
- f :
-
Dimensionless stream function
- \(f^{\prime}=\frac{\partial f}{\partial\eta}\) :
-
- g :
-
Acceleration of gravity (m·s−2)
- \(Gr_{0}^{\ast} =g \beta q_{w}r_{0}^{4}/(k \nu)\) :
-
Modified Grashof number
- h :
-
Heat transfer coefficient (W·m−2·K−1)
- h m :
-
Mean heat transfer coefficient (W·m−2·K−1)
- k :
-
Thermal conductivity (W·m−1·K−1)
- L :
-
Needle length (m)
- Nu = h x / k :
-
Nusselt number
- Nu m = h m x / k :
-
mean Nusselt number
- Pr = ν / α :
-
Prandtl number
- q w :
-
Wall heat flux (W·m−2)
- r :
-
Radial coordinate (m)
- \(Ra = Gr_{0}^{\ast}Pr\) :
-
Rayleigh number
- r 0 :
-
Needle radius (m)
- T :
-
Temperature (K)
- T ∞ :
-
Stagnant temperature (K)
- u :
-
Axial velocity (m·s−1)
- w :
-
Radial velocity (m·s−1)
- x :
-
Axial coordinate (m)
- Greek Symbols :
-
- α :
-
Thermal diffusivity (m2·s−1)
- β :
-
Coefficient of thermal expansion (K−1)
- η :
-
Pseudo-similarity variable, Eq. (6)
- ν :
-
Kinematics viscosity (m·s−1)
- θ :
-
Dimensionless temperature, Eq. (9)
- \(\theta^{\prime}=\frac{\partial\;\theta}{\partial\;\eta}\) :
-
- θ 0 :
-
Dimensionless wall temperature
- ρ :
-
Density (kg·m−3)
- ξ :
-
Stretched x coordinate, Eq. (7)
- ψ :
-
Stream function, Eq. (8), (m3·s−1)
- Subscript :
-
- c:
-
Critical
- m:
-
Mean
- max:
-
Maximum
References
Elenbaas W. (1948) . J. Appl. Phys. 19:1148
E. M. Sparrow and J. L. Gregg, Trans. ASME 435 (1956).
E. M. Sparrow and J. L. Gregg, Trans. ASME 1823 (1956).
Minkowycz W.J., Sparrow E.M. (1974) . Trans. ASME, J. Heat Transfer 96:178
Kuiken H.K. (1968) . Int. J. Heat Mass Transfer 11:1141
Fujii T., Uehara H. (1970) . Int. J. Heat Mass Transfer 13:607
F. Gori and M. Pietrafesa, in Fundamental Experimental Measurements in Heat Transfer, ASME-WAM, Paper HT-7A-5, HTD Vol. 179, D. E. Beasley and J. L. S. Chen, eds., Book N. H00663–1991, Atlanta (1991), pp. 83–98.
F. Gori and M. Pietrafesa, Proc. 10th Int. Heat Transfer Conf., Vol. 5, Brighton, United Kingdom (1994), pp. 349–354.
F. Gori, P. Coppa, and M. Pietrafesa, Adv. Eng. Heat Transfer, Proc. Second Baltic Heat Transfer Conf., Southampton (1995), pp. 101–111.
Gori F., Marino C., Pietrafesa M. (2001) . Int. Commun. Heat Mass Transfer 28:1091
F. Gori and S. Corasaniti, 5th World Conf. Exptal. Heat Transfer, Fluid Mechanics and Thermodyn., Vol. 2 (2001), pp. 1257–1262.
F. Gori and S. Corasaniti, HTD-24152, Int. Mech. Eng. Cong. Expo. (IMECE), ASME (2001), pp. 1–8.
Gori F., Corasaniti S. (2001) . Microgravity and Space Station Utilization 2:23
Gori F., Corasaniti S. (2002) . J. Heat Transfer 126:1001
F. Gori and P. Coppa, Proc. ESDA 2002:6th Biennial Conf. Eng. Systems Design Anal., Instanbul, Turkey, July 8–11(2002).
E. J. Le Fevre and A. J. Ede, Proc. IX Congress for Appl. Mech., Vol. 4, Brussels (1956), pp. 175–183.
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Gori, F., Serranò, M.G. & Wang, Y. Natural Convection along a Vertical Thin Cylinder with Uniform and Constant Wall Heat Flux. Int J Thermophys 27, 1527–1538 (2006). https://doi.org/10.1007/s10765-006-0130-6
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DOI: https://doi.org/10.1007/s10765-006-0130-6