Correlations for Nusselt Number in Free Convection from an Isothermal Inclined Square Plate by a Numerical Simulation

Jasim Abdulateef; Ayad Hassan

doi:doi:10.11648/j.ajma.20150302.11

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Correlations for Nusselt Number in Free Convection from an Isothermal Inclined Square Plate by a Numerical Simulation

Jasim Abdulateef, Ayad Hassan

Published in American Journal of Mechanics and Applications (Volume 3, Issue 2)

Received: 4 April 2015 Accepted: 12 April 2015 Published: 7 May 2015

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Abstract

This paper presents a numerical study on three-dimensional transient natural convection from an inclined isothermal square plate. The finite difference approach is used to solve the governing equations, in which buoyancy is modeled via the Boussinesq approximation. The complete Navier-Stokes equations are transformed and expressed in term of vorticity and vector potential. The transformed equations are solved using alternating direction implicit (ADI) method for parabolic portion of the problem and successive over relaxation (SOR) for the elliptic portion. Solutions for laminar case are obtained up to Grashof number of 5x104 as well as the inclination angles were varied from 0o to 180o with 30o intervals, and the Prandtl number of 0.7 is considered. The results are shown in terms of isothermal plots, and the local and average Nusselt numbers are also presented. The simulation results show that the main process of heat transfer is conduction for Grashof number less than 103 and convection for Grashof number larger than 103. It is also found that, the values of Nusselt number show fairly large dependence on inclination angle and there is a significant difference in heat transfer rates between the upward and downward orientation. The average Nusselt number increases to 20% at the vertical position compared to horizontal position then decreases with increasing inclination of plate at downward orientation. Based on the results obtained, correlations have been proposed to evaluate the Nusselt numbers of both upward and downward orientation. Validations of the present results are made through comparison with available numerical and experimental data, and a good agreement was obtained.

Published in	American Journal of Mechanics and Applications (Volume 3, Issue 2)
DOI	10.11648/j.ajma.20150302.11
Page(s)	8-18
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2015. Published by Science Publishing Group

Keywords

Natural Convection, Laminar Flow, Vorticity and Vector Potential, Inclined Isothermal Plate

References

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[2]	M. Al-Arabi, M. K. El-Riedy, Natural convection heat transfer from isothermal horizontal plates of different shapes, Int. J. Heat Mass Transfer 19 (1976):1399-1404.
[3]	Goldstein, R. J., and Kei-Shun Lau, Laminar natural convection from a horizontal plate and the influence of plate-edge extensions, Journal of Fluid Mechanics 129 (1983): 55-75.
[4]	Lewandowski, W. M., and P. Kubski, Effect of the use of the balance and gradient methods as a result of experimental investigations of natural convection action with regard to the conception and construction of measuring apparatus, Heat and Mass Transfer 18 (1984): 247-256.‏
[5]	Kitamura, Kenzo, and Fumiyoshi Kimura, Heat transfer and fluid flow of natural convection adjacent to upward-facing horizontal plates, International journal of heat and mass transfer 38 (1995): 3149-3159.‏
[6]	Lewandowski, Witold M., et al., Free convection heat transfer and fluid flow above horizontal rectangular plates, Applied energy 66 (2000): 177-197.‏
[7]	Pretot, S., B. Zeghmati, and G. Le Palec, Theoretical and experimental study of natural convection on a horizontal plate, Applied thermal engineering 20 (2000): 873-891.‏
[8]	Rafah Aziz, Instruction system to study free convection heat transfer from isothermal square flat surface, 2000, M.Sc. Thesis, University of Technology, Baghdad.
[9]	Martorell, Ingrid, Joan Herrero, and Francesc X. Grau, Natural convection from narrow horizontal plates at moderate Rayleigh numbers, International Journal of heat and mass transfer 46 (2003): 2389-2402.
[10]	Kozanoglu, Bulent, and Jorge Lopez, Thermal boundary layer and the characteristic length on natural convection over a horizontal plate, Heat and mass transfer 43 (2007): 333-339.
[11]	Chen, T. S., Hwa-Chong Tien, and Bassem F. Armaly. "Natural convection on horizontal, inclined, and vertical plates with variable surface temperature or heat flux." International journal of heat and mass transfer 29 (1986): 1465-1478.
[12]	Lewandowski, Witold M. "Natural convection heat transfer from plates of finite dimensions." International journal of heat and mass transfer 34 (1991): 875-885.
[13]	Wei, J. J., B. Yu, and Y. Kawaguchi, Simultaneous natural-convection heat transfer above and below an isothermal horizontal thin plate, Numerical Heat Transfer 44 (2003): 39-58.
[14]	Ayad K. Hassan, Prediction of three dimensional natural convection from heated disk and rings at constant temperature, Engineering and Technology 5 (2003):229-248.
[15]	Ayad K. Hassan, Prediction of three dimensional natural convection from a horizontal isothermal square plate, Journal of Engineering 13 (2006):742-755.
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[18]	Gerald, Curtis F., Patrick O. Wheatley, and Fengshan Bai. Applied numerical analysis. New York: Addison-Wesley, 1989.‏

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APA Style

Jasim Abdulateef, Ayad Hassan. (2015). Correlations for Nusselt Number in Free Convection from an Isothermal Inclined Square Plate by a Numerical Simulation. American Journal of Mechanics and Applications, 3(2), 8-18. https://doi.org/10.11648/j.ajma.20150302.11

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ACS Style

Jasim Abdulateef; Ayad Hassan. Correlations for Nusselt Number in Free Convection from an Isothermal Inclined Square Plate by a Numerical Simulation. Am. J. Mech. Appl. 2015, 3(2), 8-18. doi: 10.11648/j.ajma.20150302.11

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AMA Style

Jasim Abdulateef, Ayad Hassan. Correlations for Nusselt Number in Free Convection from an Isothermal Inclined Square Plate by a Numerical Simulation. Am J Mech Appl. 2015;3(2):8-18. doi: 10.11648/j.ajma.20150302.11

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@article{10.11648/j.ajma.20150302.11,
  author = {Jasim Abdulateef and Ayad Hassan},
  title = {Correlations for Nusselt Number in Free Convection from an Isothermal Inclined Square Plate by a Numerical Simulation},
  journal = {American Journal of Mechanics and Applications},
  volume = {3},
  number = {2},
  pages = {8-18},
  doi = {10.11648/j.ajma.20150302.11},
  url = {https://doi.org/10.11648/j.ajma.20150302.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajma.20150302.11},
  abstract = {This paper presents a numerical study on three-dimensional transient natural convection from an inclined isothermal square plate. The finite difference approach is used to solve the governing equations, in which buoyancy is modeled via the Boussinesq approximation. The complete Navier-Stokes equations are transformed and expressed in term of vorticity and vector potential. The transformed equations are solved using alternating direction implicit (ADI) method for parabolic portion of the problem and successive over relaxation (SOR) for the elliptic portion. Solutions for laminar case are obtained up to Grashof number of 5x104 as well as the inclination angles were varied from 0o to 180o with 30o intervals, and the Prandtl number of 0.7 is considered. The results are shown in terms of isothermal plots, and the local and average Nusselt numbers are also presented. The simulation results show that the main process of heat transfer is conduction for Grashof number less than 103 and convection for Grashof number larger than 103. It is also found that, the values of Nusselt number show fairly large dependence on inclination angle and there is a significant difference in heat transfer rates between the upward and downward orientation. The average Nusselt number increases to 20% at the vertical position compared to horizontal position then decreases with increasing inclination of plate at downward orientation. Based on the results obtained, correlations have been proposed to evaluate the Nusselt numbers of both upward and downward orientation. Validations of the present results are made through comparison with available numerical and experimental data, and a good agreement was obtained.},
 year = {2015}
}

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TY  - JOUR
T1  - Correlations for Nusselt Number in Free Convection from an Isothermal Inclined Square Plate by a Numerical Simulation
AU  - Jasim Abdulateef
AU  - Ayad Hassan
Y1  - 2015/05/07
PY  - 2015
N1  - https://doi.org/10.11648/j.ajma.20150302.11
DO  - 10.11648/j.ajma.20150302.11
T2  - American Journal of Mechanics and Applications
JF  - American Journal of Mechanics and Applications
JO  - American Journal of Mechanics and Applications
SP  - 8
EP  - 18
PB  - Science Publishing Group
SN  - 2376-6131
UR  - https://doi.org/10.11648/j.ajma.20150302.11
AB  - This paper presents a numerical study on three-dimensional transient natural convection from an inclined isothermal square plate. The finite difference approach is used to solve the governing equations, in which buoyancy is modeled via the Boussinesq approximation. The complete Navier-Stokes equations are transformed and expressed in term of vorticity and vector potential. The transformed equations are solved using alternating direction implicit (ADI) method for parabolic portion of the problem and successive over relaxation (SOR) for the elliptic portion. Solutions for laminar case are obtained up to Grashof number of 5x104 as well as the inclination angles were varied from 0o to 180o with 30o intervals, and the Prandtl number of 0.7 is considered. The results are shown in terms of isothermal plots, and the local and average Nusselt numbers are also presented. The simulation results show that the main process of heat transfer is conduction for Grashof number less than 103 and convection for Grashof number larger than 103. It is also found that, the values of Nusselt number show fairly large dependence on inclination angle and there is a significant difference in heat transfer rates between the upward and downward orientation. The average Nusselt number increases to 20% at the vertical position compared to horizontal position then decreases with increasing inclination of plate at downward orientation. Based on the results obtained, correlations have been proposed to evaluate the Nusselt numbers of both upward and downward orientation. Validations of the present results are made through comparison with available numerical and experimental data, and a good agreement was obtained.
VL  - 3
IS  - 2
ER  -

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Author Information

Jasim Abdulateef

Mechanical Engineering Department, Diyala University, Diyala, Iraq
Ayad Hassan

Material Engineering Department, University of Technology, Baghdad, Iraq

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APA Style

Jasim Abdulateef, Ayad Hassan. (2015). Correlations for Nusselt Number in Free Convection from an Isothermal Inclined Square Plate by a Numerical Simulation. American Journal of Mechanics and Applications, 3(2), 8-18. https://doi.org/10.11648/j.ajma.20150302.11

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ACS Style

Jasim Abdulateef; Ayad Hassan. Correlations for Nusselt Number in Free Convection from an Isothermal Inclined Square Plate by a Numerical Simulation. Am. J. Mech. Appl. 2015, 3(2), 8-18. doi: 10.11648/j.ajma.20150302.11

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AMA Style

Jasim Abdulateef, Ayad Hassan. Correlations for Nusselt Number in Free Convection from an Isothermal Inclined Square Plate by a Numerical Simulation. Am J Mech Appl. 2015;3(2):8-18. doi: 10.11648/j.ajma.20150302.11

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@article{10.11648/j.ajma.20150302.11,
  author = {Jasim Abdulateef and Ayad Hassan},
  title = {Correlations for Nusselt Number in Free Convection from an Isothermal Inclined Square Plate by a Numerical Simulation},
  journal = {American Journal of Mechanics and Applications},
  volume = {3},
  number = {2},
  pages = {8-18},
  doi = {10.11648/j.ajma.20150302.11},
  url = {https://doi.org/10.11648/j.ajma.20150302.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajma.20150302.11},
  abstract = {This paper presents a numerical study on three-dimensional transient natural convection from an inclined isothermal square plate. The finite difference approach is used to solve the governing equations, in which buoyancy is modeled via the Boussinesq approximation. The complete Navier-Stokes equations are transformed and expressed in term of vorticity and vector potential. The transformed equations are solved using alternating direction implicit (ADI) method for parabolic portion of the problem and successive over relaxation (SOR) for the elliptic portion. Solutions for laminar case are obtained up to Grashof number of 5x104 as well as the inclination angles were varied from 0o to 180o with 30o intervals, and the Prandtl number of 0.7 is considered. The results are shown in terms of isothermal plots, and the local and average Nusselt numbers are also presented. The simulation results show that the main process of heat transfer is conduction for Grashof number less than 103 and convection for Grashof number larger than 103. It is also found that, the values of Nusselt number show fairly large dependence on inclination angle and there is a significant difference in heat transfer rates between the upward and downward orientation. The average Nusselt number increases to 20% at the vertical position compared to horizontal position then decreases with increasing inclination of plate at downward orientation. Based on the results obtained, correlations have been proposed to evaluate the Nusselt numbers of both upward and downward orientation. Validations of the present results are made through comparison with available numerical and experimental data, and a good agreement was obtained.},
 year = {2015}
}

Copy | Download

TY  - JOUR
T1  - Correlations for Nusselt Number in Free Convection from an Isothermal Inclined Square Plate by a Numerical Simulation
AU  - Jasim Abdulateef
AU  - Ayad Hassan
Y1  - 2015/05/07
PY  - 2015
N1  - https://doi.org/10.11648/j.ajma.20150302.11
DO  - 10.11648/j.ajma.20150302.11
T2  - American Journal of Mechanics and Applications
JF  - American Journal of Mechanics and Applications
JO  - American Journal of Mechanics and Applications
SP  - 8
EP  - 18
PB  - Science Publishing Group
SN  - 2376-6131
UR  - https://doi.org/10.11648/j.ajma.20150302.11
AB  - This paper presents a numerical study on three-dimensional transient natural convection from an inclined isothermal square plate. The finite difference approach is used to solve the governing equations, in which buoyancy is modeled via the Boussinesq approximation. The complete Navier-Stokes equations are transformed and expressed in term of vorticity and vector potential. The transformed equations are solved using alternating direction implicit (ADI) method for parabolic portion of the problem and successive over relaxation (SOR) for the elliptic portion. Solutions for laminar case are obtained up to Grashof number of 5x104 as well as the inclination angles were varied from 0o to 180o with 30o intervals, and the Prandtl number of 0.7 is considered. The results are shown in terms of isothermal plots, and the local and average Nusselt numbers are also presented. The simulation results show that the main process of heat transfer is conduction for Grashof number less than 103 and convection for Grashof number larger than 103. It is also found that, the values of Nusselt number show fairly large dependence on inclination angle and there is a significant difference in heat transfer rates between the upward and downward orientation. The average Nusselt number increases to 20% at the vertical position compared to horizontal position then decreases with increasing inclination of plate at downward orientation. Based on the results obtained, correlations have been proposed to evaluate the Nusselt numbers of both upward and downward orientation. Validations of the present results are made through comparison with available numerical and experimental data, and a good agreement was obtained.
VL  - 3
IS  - 2
ER  -

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