# On the Energy Conservation Formulation for Flows in Porous Media Including Viscous Dissipation Effects

## Abstract

Energy conservation formulation needs to be carefully conducted when dealing with natural or mixed convection problems if viscous dissipation is considered. A violation of the Energy Conservation Principle exists if only the viscous dissipation term is considered, and if its counterpart, the work of pressure forces is not also considered in the internal energy conservation equation. This is true both for flows in clear fluid domains and for flows in fluid-saturated porous domains. In this chapter general detailed energy conservation formulations are conducted, first for flows in clear fluid domains and then for flows in fluid-saturated porous domains. Main conclusions obtained from the model for flows in clear fluid domains are equally relevant for flows in fluid-saturated porous domains. Considering an enclosure of rigid walls where steady natural convection takes place, it is shown that, when integrating over the overall domain, the viscous dissipation (energy source) has its symmetric on the work of pressure forces (energy sink). Globally heat entering the enclosure equals the heat leaving it, no matter the thermal boundary conditions considered at the enclosure walls. It is shown in an exact way that this result applies both for domains filled with a clear fluid or filled with a fluid-saturated porous medium. Main conclusions concerning verification of the Energy Conservation Principle are extracted from what happens in steady natural convection in enclosures filled with fluid-saturated porous domains and then to what happens in steady natural convection in open fluid-saturated porous domains, in unsteady natural convection problems in fluid-saturated porous domains, and also in steady or unsteady mixed convection problems in fluid-saturated porous domains.

## Keywords

Energy Conservation Viscous Dissipation Mixed Convection Pressure Force Energy Balance Equation## References

- 1.Costa, V.A.F.: Thermodynamics of natural convection in enclosures with viscous dissipation. Int. J. Heat Mass Transf.
**48**, 2333–2341 (2005)MATHCrossRefGoogle Scholar - 2.Costa, V.A.F.: On natural convection in enclosures filled with fluid-saturated porous media including viscous dissipation. Int. J. Heat Mass Transf.
**49**, 2215–2226 (2006)MATHCrossRefGoogle Scholar - 3.Nield, D.A.: Resolution of a paradox involving viscous dissipation and nonlinear drag in a porous medium. Transport Porous Med.
**41**, 349–357 (2000)MathSciNetCrossRefGoogle Scholar - 4.Nield, D.A.: The modeling of viscous dissipation in a saturated porous medium. ASME J. Heat Transf.
**129**1459–1463 (2007)CrossRefGoogle Scholar - 5.Costa, V.A.F.: Discussion: “The modeling of viscous dissipation in a saturated porous medium” (Nield, D.A., ASME J. Heat Transfer
**129**, 1459–1463 (2007)). ASME J. Heat Transfer**131**, 025502-1-2 (2009)Google Scholar - 6.Nield, D.A.: Closure to “Discussion of ‘The modeling of viscous dissipation in a saturated porous medium” (ASME J. Heat Transfer
**131**, p. 025501 (2009)). ASME J. Heat Transfer**131**, 025502-1 (2009)CrossRefGoogle Scholar - 7.Costa, V.A.F.: Comment on the paper “I.A. Badruddin, Z.A. Zainal, Z.A. Khan, Z. Mallick, Effect of viscous dissipation and radiation on natural convection in a porous medium embedded within vertical annulus. Int. J. Thermal Sci.
**46**(3) (2007) 221–227”. Int. J. Thermal Sci.**49**, 1874–1875 (2010)Google Scholar - 8.Nield, D.A., Barletta, A.: Comment on the comment by V.A.F. Costa, IJTS 49 (9) (2010) 1874-1875, on the paper by I.A. Badruddin, Z.A. Zainal, Z.A. Khan, Z. Mallick “Effect of viscous dissipation and radiation on natural convection in a porous medium embedded within vertical annulus” IJTS
**46**(3), 221–227 (2007). Int. J. Thermal Sci.**49**, 2491–2492 (2010)Google Scholar - 9.Bird, R.B., Stewart, W.E., Lightfoot, E.N.: Transport Phenomena, 2nd edn. Wiley, New York (2002)Google Scholar
- 10.Bejan, A.: Adv. Eng. Thermodyn, 3rd edn. Wiley, Hoboken (2006)Google Scholar
- 11.Nield, D.A., Bejan, A.: Convection in Porous Media, 3rd edn. Springer, New York (2006)Google Scholar