General equation of heat transfer

Short description: Entropy production in Newtonian fluids

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]

ρTDsDt⏟Heat Gain=∇⋅(κ∇T)⏟Thermal Conduction+μ2(∂vi∂xj+∂vj∂xi−23δij∇⋅𝐯)2+ζ(∇⋅𝐯)2⏟Viscous Dissipation

where

is the specific entropy,

is the fluid's density,

is the fluid's temperature,

D/Dt

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

δij

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]

## Contents

* 1 Derivation
* 1.1 Extension of the ideal fluid energy equation
* 1.2 Equation for entropy production
* 2 Application
* 3 See also
* 4 References
* 5 Further reading

## Derivation

### Extension of the ideal fluid energy equation

For a viscous, Newtonian fluid, the governing equations for mass conservation and momentum conservation are the continuity equation and the Navier-Stokes equations:∂ρ∂t=−∇⋅(ρ𝐯)ρD𝐯Dt=−∇p+∇⋅σwhere p is the pressure and σ is the viscous stress tensor, with the components of the viscous stress tensor given by:σij=μ(∂vi∂xj+∂vj∂xi−23δij∇⋅𝐯)+ζδij∇⋅𝐯The energy of a unit volume of the fluid is the sum of the kinetic energy ρv2/2≡ρk and the internal energy ρε, where ε is the specific internal energy. In an ideal fluid, as described by the Euler equations, the conservation of energy is defined by the equation:∂∂t[ρ(k+ε)]+∇⋅[ρ𝐯(k+h)]=0where h is the specific enthalpy. However, for conservation of energy to hold in a viscous fluid subject to thermal conduction, the energy flux due to advection ρ𝐯(k+h) must be supplemented by a heat flux given by Fourier's law 𝐪=−κ∇T and a flux due to internal friction −σ⋅𝐯. Then the general equation for conservation of energy is:∂∂t[ρ(k+ε)]+∇⋅[ρ𝐯(k+h)−κ∇T−σ⋅𝐯]=0

### Equation for entropy production

Note that the thermodynamic relations for the internal energy and enthalpy are given by:ρdε=ρTds+pρdρρdh=ρTds+dpWe may also obtain an equation for the kinetic energy by taking the dot product of the Navier-Stokes equation with the flow velocity 𝐯 to yield:ρDkDt=−𝐯⋅∇p+vi∂σij∂xjThe second term on the righthand side may be expanded to read:vi∂σij∂xj=∂∂xj(σijvi)−σij∂vi∂xj≡∇⋅(σ⋅𝐯)−σij∂vi∂xjWith the aid of the thermodynamic relation for enthalpy and the last result, we may then put the kinetic energy equation into the form:ρDkDt=−ρ𝐯⋅∇h+ρT𝐯⋅∇s+∇⋅(σ⋅𝐯)−σij∂vi∂xjNow expanding the time derivative of the total energy, we have:∂∂t[ρ(k+ε)]=ρ∂k∂t+ρ∂ε∂t+(k+ε)∂ρ∂tThen by expanding each of these terms, we find that:ρ∂k∂t=−ρ𝐯⋅∇k−ρ𝐯⋅∇h+ρT𝐯⋅∇s+∇⋅(σ⋅𝐯)−σij∂vi∂xjρ∂ε∂t=ρT∂s∂t−pρ∇⋅(ρ𝐯)(k+ε)∂ρ∂t=−(k+ε)∇⋅(ρ𝐯)And collecting terms, we are left with:∂∂t[ρ(k+ε)]+∇⋅[ρ𝐯(k+h)−σ⋅𝐯]=ρTDsDt−σij∂vi∂xjNow adding the divergence of the heat flux due to thermal conduction to each side, we have that:∂∂t[ρ(k+ε)]+∇⋅[ρ𝐯(k+h)−κ∇T−σ⋅𝐯]=ρTDsDt−∇⋅(κ∇T)−σij∂vi∂xjHowever, we know that by the conservation of energy on the lefthand side is equal to zero, leaving us with:ρTDsDt=∇⋅(κ∇T)+σij∂vi∂xjThe product of the viscous stress tensor and the velocity gradient can be expanded as:σij∂vi∂xj=μ(∂vi∂xj+∂vj∂xi−23δij∇⋅𝐯)∂vi∂xj+ζδij∂vi∂xj∇⋅𝐯=μ2(∂vi∂xj+∂vj∂xi−23δij∇⋅𝐯)2+ζ(∇⋅𝐯)2Thus leading to the final form of the equation for specific entropy production:ρTDsDt=∇⋅(κ∇T)+μ2(∂vi∂xj+∂vj∂xi−23δij∇⋅𝐯)2+ζ(∇⋅𝐯)2In the case where thermal conduction and viscous forces are absent, the equation for entropy production collapses to Ds/Dt=0 \- showing that ideal fluid flow is isentropic.

## Application

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

* Dissipation
* Heat transfer
* Turbulent kinetic energy

## References

1. ↑ 1.0 1.1 Landau, L.D.; Lifshitz, E.M. (1987) (in en). Fluid Mechanics. Course of Theoretical Physics. 6 (2nd ed.). Butterworth-Heinemann. pp. 192–194. ISBN 978-0-7506-2767-2. OCLC 936858705. https://phys.au.dk/~srf/hydro/Landau+Lifschitz.pdf.
2. ↑ 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.
3. ↑ Pedlosky, J. (2003). Waves in the Ocean and Atmosphere: Introduction to Wave Dynamics. Springer. pp. 19. ISBN 978-3540003403.
4. ↑ Laguerre, Onrawee (2010-05-21), Farid, Mohammed M., ed., "Heat Transfer and Air Flow in a Domestic Refrigerator" (in en), Mathematical Modeling of Food Processing (CRC Press): pp. 453–482, doi:10.1201/9781420053548-20, ISBN 978-0-429-14217-8, https://www.taylorfrancis.com/books/9781420053548/chapters/10.1201/9781420053548-20, retrieved 2023-05-07
5. ↑ 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. https://dokumen.tips/amp/documents/simple-harmonic-analysis-of-regenerators.html.
6. ↑ Cuffey, K. M. (2010). The physics of glaciers. W. S. B. Paterson (4th ed.). Burlington, MA. ISBN 978-0-12-369461-4. OCLC 488732494.

## Further reading

* 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. Bibcode: 1993IJHMT..36.3265Y. https://www.sciencedirect.com/science/article/abs/pii/001793109390009U.

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