7.3 Weak Form of the Energy Equation

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To obtain the weak form of Equation (7.16), we proceed as explained in Section 1.12.

Multiplying by an infinitesimal perturbation of the temperature δT and integrating over V

yields

_

V

ρc ˙ T δT dv = −_

V

qk

,kδT dv + _

V

_ρh + σ : ˙_p + qi : ˙α − β : ˙_eθref_ δT dv. (7.18)

Integrating the first term on the right-hand side by parts, one obtains

−_

V

qk

,kδT dv = −_

A

qkδT dak + _

V

qkδT,k dv (7.19)

leading to

_

V

ρc ˙ T δT dv − _

V

qkδT,k dv

= −_

A

qkδT dak + _

V

_ρh + σ : ˙_ p + qi : ˙α − β : ˙_ eθref_ δT dv. (7.20)

The entropy inequality, Equation (5.17), requires that

q ・ ∇T ≥ 0 (7.21)

308 HEAT TRANSFER

which implies that q must be at least a linear function of ∇T :

qk = −κkl(T )T , l (7.22)

that is, for a zero-temperature gradient, there is no heat flux. In Equation (7.22), the coefficients

κkl are generally a function of the temperature. It is a nonlinear equation of the

temperature.

The flux in the first term on the right-hand side of Equation (7.20) is the heat flux

entering the body through its surface. It consists of three parts:

1. a convective part, which is more or less linear in T:

qk

conv

= h(T )(T − Te)nk (7.23)

where Te is the environmental temperature, h(T ) is the convective coefficient and n

is the normal to the surface.

2. a radiation part, which is highly nonlinear (Incropera and DeWitt 2002)

qk

rad

= A(T )(θ 4 − θ4

e )nk (7.24)

where θe is the absolute environmental temperature (in Kelvin) and A(T ) is the

product of the Stefan–Boltzmann constant σ with the emissivity _(T ):

A(T ) = σ _(T ). (7.25)

The emissivity is a property of the surface and takes values between zero and one.

It is a measure of how well the surface emits radiation. For a perfect black body,

_ = 1.

3. any other known flux

qk = qnk. (7.26)

Summarizing, the heat equation for small strains and small temperature deviations from a

reference temperature yields

_

V

ρc(T ) ˙ T δT dv + _

V

κklT,lδT,k dv

= −_

A

h(T )(T − Te)δT nk dak − _

A

A(T )(θ 4 − θ4

e )δT nk dak

− _

A

qδT nk dak + _

V

_ρh + σ : ˙_ p + qi : ˙α − β : ˙_ eθ_ δT dv. (7.27)

In the last term, θref was replaced by θ, which also corresponds to a second-order correction.

Indeed,

β : ˙_eθ = β : ˙_ eθref + β : ˙_ eT (7.28)

where β : ˙_eT = O(___T ). Equation (7.27) is highly nonlinear because of the radiation

term. Furthermore, the temperature dependence of the materials constants in Equation (7.27)

cannot be neglected and must be taken into account through an iterative procedure.

HEAT TRANSFER 309

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