In fluid dynamics, the general equation of heat transfer is a nonlinear partial differential equation describing specific entropy production in a Newtonian fluid subject to thermal conduction and viscous forces:[1][2]
where is the specific entropy, is the fluid's density, is the fluid's temperature, is the material derivative, is the thermal conductivity, is the dynamic viscosity, is the second Lamé parameter, is the flow velocity, is the del operator used to characterize the gradient and divergence, and is the Kronecker delta.
If the flow velocity is negligible, the general equation of heat transfer reduces to the standard heat equation. It may also be extended to rotating, stratified flows, such as those encountered in geophysical fluid dynamics.[3]
Derivation[edit]
Extension of the ideal fluid energy equation[edit]
For a viscous, Newtonian fluid, the governing equations for mass conservation and momentum conservation are the continuity equation and the Navier-Stokes equations:
Equation for entropy production[edit]
Note that the thermodynamic relations for the internal energy and enthalpy are given by:
Application[edit]
This equation is derived in Section 49, at the opening of the chapter on "Thermal Conduction in Fluids" in the sixth volume of L.D. Landau and E.M. Lifshitz's Course of Theoretical Physics.[1] It might be used to measure the heat transfer and air flow in a domestic refrigerator,[4] to do a harmonic analysis of regenerators,[5] or to understand the physics of glaciers.[6]
See also[edit]
References[edit]
- ^ a b Landau, L.D.; Lifshitz, E.M. (1987). Fluid Mechanics (PDF). Course of Theoretical Physics. Vol. 6 (2nd ed.). Butterworth-Heinemann. pp. 192–194. ISBN 978-0-7506-2767-2. OCLC 936858705.
- ^ Kundu, P.K.; Cohen, I.M.; Dowling, D.R. (2012). Fluid Mechanics (5th ed.). Academic Press. pp. 123–125. ISBN 978-0-12-382100-3.
- ^ Pedlosky, J. (2003). Waves in the Ocean and Atmosphere: Introduction to Wave Dynamics. Springer. p. 19. ISBN 978-3540003403.
- ^ Laguerre, Onrawee (2010-05-21), Farid, Mohammed M. (ed.), "Heat Transfer and Air Flow in a Domestic Refrigerator", Mathematical Modeling of Food Processing (1 ed.), CRC Press, pp. 453–482, doi:10.1201/9781420053548-20, ISBN 978-0-429-14217-8, retrieved 2023-05-07
- ^ Swift, G. W.; Wardt, W. C. (October–December 1996). "Simple Harmonic Analysis of Regenerators". Journal of Thermophysics and Heat Transfer. 10 (4): 652–662. doi:10.2514/3.842.
- ^ Cuffey, K. M. (2010). The physics of glaciers. W. S. B. Paterson (4th ed.). Burlington, MA. ISBN 978-0-12-369461-4. OCLC 488732494.
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Further reading[edit]
- Vallis, G.K. (2006). Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-Scale Circulation. Cambridge University Press. ISBN 978-0-521-84969-2.
- Yilmaz, T.; Cihan, E. (September 1993). "General equation for heat transfer for laminar flow in ducts of arbitrary cross-sections". International Journal of Heat and Mass Transfer. 36 (13): 3265–3270. doi:10.1016/0017-9310(93)90009-U. ISSN 0017-9310.