{"id":20351,"date":"2018-11-19T18:57:41","date_gmt":"2018-11-19T18:57:41","guid":{"rendered":"http:\/\/sitepourvtc.com\/?page_id=20351"},"modified":"2023-02-14T13:15:49","modified_gmt":"2023-02-14T13:15:49","slug":"what-is-rayleigh-number","status":"publish","type":"page","link":"https:\/\/sitepourvtc.com\/nuclear-engineering\/heat-transfer\/introduction-to-heat-transfer\/characteristic-numbers\/what-is-rayleigh-number\/","title":{"rendered":"What is Rayleigh Number"},"content":{"rendered":"
The Rayleigh\u00a0number<\/strong> is a dimensionless number named after\u00a0 Lord Rayleigh. The Rayleigh\u00a0number<\/strong> is closely related to the Grashof number, and both numbers are used to describe natural convection (Gr) and heat transfer by natural convection (Ra).<\/div><\/div>\n

The Rayleigh\u00a0number<\/strong>\u00a0is simply defined as the product of the Grashof number<\/a>, which describes the relationship between buoyancy and viscosity within a fluid, and the Prandtl number<\/a>, which describes the relationship between momentum diffusivity and thermal diffusivity.<\/p>\n

Rax<\/sub> = Grx<\/sub> . Pr<\/span><\/strong><\/p>\n

The\u00a0Grashof number<\/strong> is defined as the ratio of the buoyant to a viscous force acting on a fluid in the velocity boundary layer. Its role in natural convection is much like that of the Reynolds number<\/a> in forced convection.\u00a0Natural convection\u00a0occurs if this motion and mixing are caused by density<\/a> variations resulting from temperature differences within the fluid. Usually, the density decreases due to increased temperature<\/a> and causes the fluid to rise. This motion is caused by the buoyant force. The major force that resists the motion is the viscous force. The Grashof number is a way to quantify the opposing forces.<\/p>\n

The Rayleigh number<\/strong> is used to express heat transfer in natural convection. The magnitude of the Rayleigh number is a good indication of whether the natural convection boundary layer is laminar or turbulent. The simple empirical correlations for the average Nusselt number, Nu, in natural convection are of the form:<\/p>\n

Nux<\/sub>\u00a0= C. Rax<\/sub>n<\/sup><\/span><\/strong><\/p>\n

The values of the constants C<\/strong> and n<\/strong> depend on the geometry of the surface and the flow regime<\/a>, which is characterized by the range of the Rayleigh number<\/strong>. The value of n is usually n = 1\/4 for laminar flow and n = 1\/3 for turbulent flow.<\/p>\n

The\u00a0Rayleigh\u00a0number<\/strong> is defined as:<\/p>\n

\"rayleigh<\/a><\/p>\n

where:<\/p>\n

g is the acceleration due to Earth’s gravity<\/p>\n

\u03b2 is the coefficient of thermal expansion<\/p>\n

Twall<\/sub> is the wall temperature<\/p>\n

T\u221e<\/sub> is the bulk temperature<\/p>\n

L is the vertical length<\/p>\n

\u03b1 is the\u00a0thermal diffusivity<\/a><\/p>\n

\u03bd is the kinematic viscosity.<\/p>\n

For gases \u03b2 = 1\/T where the temperature is in K. For liquids, \u03b2 can be calculated if the variation of density with the temperature at constant pressure is known. For a vertical flat plate, the flow turns turbulent<\/a> for the value of:<\/p>\n

Rax<\/sub>\u00a0= Grx<\/sub>\u00a0. Pr >\u00a0109<\/sup>\u00a0<\/span><\/strong><\/p>\n

As in forced convection, the microscopic nature of flow and convection correlations are distinctly different in the laminar<\/a> and turbulent<\/a> regions.<\/p>\n

Example:\u00a0Rayleigh Number <\/strong><\/h2>\n

\"Rayleigh<\/a>A vertical plate is maintained at 50\u00b0C in 20\u00b0C air. Determine the height at which the boundary layer will turn turbulent if turbulence sets in at Ra = Gr.Pr = 109<\/sup>.<\/p>\n

Solution:<\/p>\n

The property values required for this example are:<\/p>\n

\u03bd = 1.48 x 10-5<\/sup> m2<\/sup>\/s<\/p>\n

\u03c1 = 1.17 kg\/m3<\/sup><\/p>\n

Pr = 0.700<\/p>\n

\u03b2 = 1\/ (273 + 20) = 1\/293<\/p>\n

We know the natural circulation becomes turbulent at approximately Ra = Gr.Pr > 109<\/sup>, which is fulfilled at the following height:<\/p>\n

\"grashof<\/a><\/p>\n

\u00a0
<\/span>References:<\/div>
Heat Transfer:<\/strong>\n
    \n
  1. Fundamentals of Heat and Mass Transfer, 7th Edition. Theodore L. Bergman, Adrienne S. Lavine, Frank P. Incropera. John Wiley & Sons, Incorporated, 2011. ISBN: 9781118137253.<\/li>\n
  2. Heat and Mass Transfer. Yunus A. Cengel. McGraw-Hill Education, 2011. ISBN: 9780071077866.<\/li>\n
  3. Fundamentals of Heat and Mass Transfer.\u00a0C. P. Kothandaraman.\u00a0New Age International, 2006, ISBN:\u00a09788122417722.<\/li>\n
  4. U.S. Department of Energy, Thermodynamics, Heat Transfer and Fluid Flow.\u00a0DOE Fundamentals Handbook,\u00a0Volume\u00a02\u00a0of\u00a03.\u00a0May\u00a02016.<\/li>\n<\/ol>\n

    Nuclear and Reactor Physics:<\/strong><\/p>\n

      \n
    1. J. R. Lamarsh, Introduction to Nuclear Reactor Theory, 2nd ed., Addison-Wesley, Reading,\u00a0MA (1983).<\/li>\n
    2. J. R. Lamarsh, A. J. Baratta, Introduction to Nuclear Engineering, 3d ed., Prentice-Hall, 2001, ISBN: 0-201-82498-1.<\/li>\n
    3. W. M. Stacey, Nuclear Reactor Physics, John Wiley & Sons, 2001, ISBN: 0- 471-39127-1.<\/li>\n
    4. Glasstone, Sesonske. Nuclear Reactor Engineering: Reactor Systems Engineering,\u00a0Springer; 4th edition, 1994, ISBN:\u00a0978-0412985317<\/li>\n
    5. W.S.C. Williams. Nuclear and Particle Physics.\u00a0Clarendon Press; 1 edition, 1991, ISBN:\u00a0978-0198520467<\/li>\n
    6. G.R.Keepin. Physics of Nuclear Kinetics.\u00a0Addison-Wesley Pub. Co; 1st edition, 1965<\/li>\n
    7. Robert Reed Burn, Introduction to Nuclear Reactor Operation, 1988.<\/li>\n
    8. U.S. Department of Energy, Nuclear Physics and Reactor Theory.\u00a0DOE Fundamentals Handbook,\u00a0Volume 1 and 2.\u00a0January\u00a01993.<\/li>\n
    9. Paul Reuss, Neutron Physics.\u00a0EDP Sciences, 2008.\u00a0ISBN: 978-2759800414.<\/li>\n<\/ol>\n

      <\/strong>Advanced Reactor Physics:<\/strong><\/p>\n

        \n
      1. K. O. Ott, W. A. Bezella, Introductory Nuclear Reactor Statics, American Nuclear Society, Revised edition (1989), 1989, ISBN: 0-894-48033-2.<\/li>\n
      2. K. O. Ott, R. J. Neuhold, Introductory Nuclear Reactor Dynamics, American Nuclear Society, 1985, ISBN: 0-894-48029-4.<\/li>\n
      3. D. L. Hetrick, Dynamics of Nuclear Reactors, American Nuclear Society, 1993, ISBN: 0-894-48453-2.\u00a0<\/span><\/li>\n
      4. E. E. Lewis, W. F. Miller, Computational Methods of Neutron Transport, American Nuclear Society, 1993, ISBN: 0-894-48452-4.<\/li>\n<\/ol>\n<\/div><\/div><\/div>
        <\/div><\/div><\/div>
        <\/div>
        <\/div><\/div>
        \n

        See above:<\/h2>\n

        Characteristic Numbers<\/i> <\/span><\/a><\/p><\/div><\/div>

        <\/div><\/div>\n","protected":false},"excerpt":{"rendered":"

        The Rayleigh\u00a0number\u00a0is simply defined as the product of the Grashof number, which describes the relationship between buoyancy and viscosity within a fluid, and the Prandtl number, which describes the relationship between momentum diffusivity and thermal diffusivity. Rax = Grx . Pr The\u00a0Grashof number is defined as the ratio of the buoyant to a viscous force … Read more<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":20296,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"generate_page_header":""},"_links":{"self":[{"href":"https:\/\/sitepourvtc.com\/wp-json\/wp\/v2\/pages\/20351"}],"collection":[{"href":"https:\/\/sitepourvtc.com\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/sitepourvtc.com\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/sitepourvtc.com\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/sitepourvtc.com\/wp-json\/wp\/v2\/comments?post=20351"}],"version-history":[{"count":3,"href":"https:\/\/sitepourvtc.com\/wp-json\/wp\/v2\/pages\/20351\/revisions"}],"predecessor-version":[{"id":37254,"href":"https:\/\/sitepourvtc.com\/wp-json\/wp\/v2\/pages\/20351\/revisions\/37254"}],"up":[{"embeddable":true,"href":"https:\/\/sitepourvtc.com\/wp-json\/wp\/v2\/pages\/20296"}],"wp:attachment":[{"href":"https:\/\/sitepourvtc.com\/wp-json\/wp\/v2\/media?parent=20351"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}