Entropy inequality for the whole system

The use of the entropy inequality to obtain governing equations is demon-strated in the following. It is shown how Fick's law can be derived from the classical denition of chemical potentialµ, with a specic choice of Helmholtz free energyψ. Consider the entropy inequality for the case with a single tem-perature for the whole system, as

ρDη

Dt ≥ −div (h/θ) +ρr/θ (A.1.171) where the entropy inux vectorhcan be expressed as (A.1.168) and (A.1.107) to yield the form

ρDη

Dt ≥ −div q/θ− XN

j=1

ρ_j

θ −T^T_j/ρ_j +ψ_j+ ¹₂u²_jI u_j

!

+ρr/θ (A.1.172) Using the denition of the chemical potentialK_jin (A.1.165) reduce (A.1.172) to

ρDη

Dt ≥ −div q/θ− XN

j=1

ρ_j

θ K_j +¹₂u²_jI u_j

!

+ρr/θ (A.1.173) The chemical potential K_j is assumed simplied as K_j = µ_jI, using this in (A.1.173) and rewrite (A.1.173) in terms of the temperature θ, for further use, yields

θρDη

Dt ≥ −θdiv 1 θ q−

XN

j=1

ρ_j µ_j+ ¹₂u²_j u_j

!!

+ρr (A.1.174) The last term in (A.1.174) is found in the energy balance for the system (A.1.113) and inserting this yields

θρDη

Dt −ρDe

Dt ≥ −θdiv 1 θ q−

XN

j=1

ρ_j µ_j+ ¹₂u²_j u_j

!!

−trTL

+ divq− XN

j=1

ρ_ju_j·b_j (A.1.175) The terms divq and θdiv ^q_θ

in (A.1.175) can be rewritten as θdivq

θ

=−q

θ ·grad (θ) + divq (A.1.176)

Inserting (A.1.176) in (A.1.175), yields θρDη

Dt −ρDe

Dt ≥ θdiv 1 θ

XN

j=1

ρ_j µ_j+ ¹₂u²_j u_j

!!

−trTL

+q

θ ·grad (θ)− XN

j=1

ρ_ju_j·b_j (A.1.177) For the further rewriting of (A.1.177) consider the following relation θdiv 1

θ XN

j=1

ρ_j µ_j +¹₂u²_j u_j

!!

=θ XN

j=1

ρ_ju_j ·grad 1

θ µ_j +¹₂u²_j +

XN

j=1

µ_j +¹₂u²_j

div (ρ_ju_j) (A.1.178) and the mass balance (A.1.47) multiplied by µ_j+ ¹₂u²_j

, which is ρ

XN

j=1

Dc_j

Dt µ_j +¹₂u²_j

= − XN

j=1

µ_j +¹₂u²_j

div(ρ_ju_j)

+ XN

j=1

µ_j+ ¹₂u²_j ˆ

c_j (A.1.179) Substituting the rst term on the right-hand side of (A.1.177) in (A.1.178) and substituting the last term on the right-hand side of (A.1.178) in (A.1.179), yields the inequality

θρDη

Dt −ρDe Dt +ρ

XN

j=1

Dc_j

Dt µ_j +¹₂u²_j

≥ XN

j=1

ρ_ju_j ·

θgrad 1

θ µ_j +¹₂u²_j

−b_j

+ XN

j=1

µ_j +¹₂u²_j ˆ

c_j −trTL+q

θ ·grad (θ) (A.1.180) For future use consider the relation between the internal energy e and the inner internal energy e_I, as shown in (A.1.104), with the expansion u²_j = u_j·u_j. The material derivative of the internal energy ^De_Dt in (A.1.180) is

De

Dt = De_I Dt + 1

2 XN

j=1

Dc_j Dt u²_j +

XN

j=1

c_jDu_j

Dt ·u_j (A.1.181)

Substituting (A.1.181) into (A.1.180) and using the derivative of (A.1.163) for the inner internal energy e_I yields the sought entropy inequality form

0≤ρ −ηDθ

Dt − Dψ_I Dt +

XN

j=1

Dc_j Dt µ_j

!

− XN

j=1

ρ_ju_j ·

θgrad 1

θ µ_j +¹₂u²_j

−b_j +Du_j Dt

− XN

j=1

µ_j+ ¹₂u²_j ˆ

c_j+ trTL−q

θ ·grad (θ) (A.1.182)

A.2 Mixture theory for multispecies in multi phases

The following section will show a detailed review of the derivation of multi-phase and multi-constituent mixture theory also referred to as hybrid mixture theory (HMT). Consider a mixture of R continuous bodies B1, ...BR in a three dimensional physical space, with the possibility of the dierent bodies to occupy common physical space. As a result of thisX_aj andX_bjare allowed to occupy the the same spatial position in the space.

The notation of the derivatives is extended compared to the single phase mixture in order to include multiple phases. The velocity of the constituents, the phases and the whole mixture are dened as

x⁰_αj = ∂χ_αj X_αj, t

∂t ; x⁰_α = ∂χ_α(X_α, t)

∂t ; x⁰ = ∂χ(X, t)

∂t (A.2.1) The material time derivatives are also extended to include multiple phases whereΓ is an arbitrary property. The material time derivative following the j constituent in the α phase is

D_αj(Γ) Dt = ∂Γ

∂t +grad(Γ)x⁰_αj (A.2.2) The material time derivative following the α phase is

D_α(Γ) Dt = ∂Γ

∂t +grad(Γ)x⁰_α (A.2.3) The material time derivative for the whole mixture is given as

D(Γ) Dt = ∂Γ

∂t +grad(Γ)x⁰ (A.2.4)

The diusion velocity is dened in two levels, one for the constituents with respect to a phase and one for the phases with respect to the whole mixture, these are dened as

u_αj =x⁰_αj−x⁰_α; u_α =x⁰_α−x⁰ (A.2.5) The denitions of the material derivatives and the diusion velocities, leads to the following relations between the material derivatives

(a) : D_αj(Γ)

Dt −D_α(Γ)

Dt =grad(Γ)u_αj (b) : D_α(Γ)

Dt − D(Γ)

Dt =grad(Γ)u_α

(A.2.6)

which is used in the the following derivations.

The dierence between describing multiple-phases and multiple-species is that a clear boundary exist between phases, whereas the species do not have clear physical boundaries. In HMT a volume ratio ε_α of the phases α in a bodyB is dened as

ε_α = dv_α

dv (A.2.7)

where v_a is the volume of the phase α and v is the volume of the whole mixture. The denition (A.2.7) implies that

XN

α=1

ε_α = 1 (A.2.8)

must be satised. The density of the whole mixture determined from the summation of the phase masses dm = PN

α=1ρ_αdv_α is the mass of the total mixture, determines as

dm=ρdv= XM

α=1

ρ_αdv_α (A.2.9)

which yields the density of the whole mixture as ρ=

XM

α=1

ε_αρ_α (A.2.10)

The density of the whole mixture is described by summation of the den-sities of the phases, recall from Sec. A.1.1 that the density of a phase is described by the summation of densities of the species, see equation (A.1.19)

and repeated here with the phase indexα andj = 1...N as the total number of species in the phase

ρ_α = XN

j=1

ρ_αj (A.2.11)

The equations (A.2.10) and (A.2.11) relates the phases and species to the mixture.

A.2.1 Mass balance

The mass balance postulate for an M - phase and N - species mixture is given as

∂

∂t ˆ

Ω

ε_αρ_αjdv=−

˛

∂Ω

ε_αρ_αjx⁰_αj ·ds+ ˆ

Ω

ˆ

r_αj+ ˆc_αj

dv (A.2.12) where rˆ_αj is the mass exchange term between the phases. Using the diver-gence theorem for the rst term on the right hand side of (A.2.12), yields the local form of mass balance as

∂ε_αρ_αj

∂t +div

ε_αρ_αjx⁰_αj

= ˆr_αj+ ˆc_αj α = 1, ..., M j = 1, .., N (A.2.13) The local form of the mass balance for the phase α given as

∂ε_αρ_α

∂t +div(ε_αρ_αx⁰_α) = ˆr_α (A.2.14) where the summation of the mass exchange termrˆ_αj yields the left-hand side of (A.2.14)

ˆ r_α =

XN

j=1

ˆ

r_αj (A.2.15)

Summation of the constituents in (A.2.13), together with (A.1.19) and (A.1.22) enables the comparison of (A.2.13) and (A.2.14), Using (A.2.15) yeilds the summation of the chemical interactions in the α phase as

XN

j=1

ˆ

c_αj = 0 (A.2.16)

which is equal to the result obtained in (A.1.43) for the single phase system.

The further derivation to include multiple phases in the mixture is similar to

the constituent derivation. The postulate for the mass balance of the mixture

is ∂ρ

∂t +div(ρx⁰) = 0 (A.2.17) summation of (A.2.14), over the phases and use

x⁰ = 1 ρ

XR α=1

ρ_αx⁰_α(x, t) (A.2.18) with (A.2.8), (A.2.10) and nally compare with (A.2.17), yields

XN

α=1

ˆ

r_α = 0 (A.2.19)

With the results in (A.2.16) and (A.2.19), it is shown that a summation over the constituents and the phases yields the result for the mixture as a whole.

As described earlier, it is convenient to work with the concentrations of the species due to the fact no direct boundary exist between the species. The mass balance for a single phase written in terms of species concentrations is given in (A.1.47) and extended in the following to include multiple-phases.

The equation (A.2.13) written in terms of the diusion velocity (A.1.23), yields

∂ε_αρ_αj

∂t +div ε_αρ_αju_αj

+div ε_αρ_αjx⁰_α

= ˆr_αj + ˆc_αj (A.2.20) The rst term on the left-hand side of (A.2.20) written in terms of the species concentration,c_αj, is

∂ε_αρ_αj

∂t = ∂ε_αc_αjρ_α

∂t =c_αj∂ε_αρ_α

∂t +ε_αρ_α∂c_αj

∂t (A.2.21)

The divergence product rule used on the third term in (A.2.20) together with (A.1.20) gives the result

div c_αjε_αρ_αx⁰_α

=c_αjdiv(ε_αρ_αx⁰_α) +ε_αρ_αx⁰_αgradc_αj (A.2.22) Inserting (A.2.21) and (A.2.22) into (A.2.20) and collect the terms including c_αj, yields

ˆ

r_αj+ ˆc_αj = c_αj

∂ε_αρ_α

∂t +div(ε_αρ_αx˙_α)

+ε_αρ_α∂c_αj

∂t +div ε_αρ_αju_αj

+ε_αρ_αx˙_αgradc_αj (A.2.23)

where the mass balance for the phase (A.2.14) is identied in the square bracket. Using (A.1.33) with Γ =c_αj simplies (A.2.24) to

ε_αρ_αD_αc_αj

Dt +div ε_αρ_αju_αj

= ˆr_αj + ˆc_αj −c_αjrˆ_α (A.2.24) which is the material derivative of the mass balance for the species, expressed in terms of the diusion velocity and the concentration of the species.

The mass balance for the phase can be rewritten in terms of the diusion velocity of the phase, using Γ =ε_αρ_α in (A.2.3) to obtain

D_α(ε_αρ_α)

Dt = ∂ε_αρ_α

∂t +grad(ε_αρ_α)x⁰_α (A.2.25) and the identity

div(ε_αρ_αx⁰_α) = ε_αρ_αdiv(x⁰_α) +grad(ε_αρ_α)x⁰_α (A.2.26) Combining (A.2.25) and (A.2.26) with (A.2.14), yields

D_α(ε_αρ_α)

Dt +ε_αρ_αdiv(x⁰_α) = ˆr_α (A.2.27) The equation (A.2.27) written in terms of the diusion velocity, with respect to the whole mixture is

D(ε_αρ_α)

Dt +div(ε_αρ_αu_α) +ε_αρ_αdiv(x⁰) = ˆr_α (A.2.28) The mass balance for the whole mixture is obtained by using ρin (A.2.4) to get

D(ρ) Dt = ∂ρ

∂t +grad(ρ)x⁰ (A.2.29)

and using the identity

div(ρx⁰) = ρdiv(x⁰) +grad(ρ)x⁰ (A.2.30) Combining (A.2.29) and (A.2.30) yields the mass balance of the whole mix-ture, that is

D(ρ)

Dt +ρdiv(x⁰) = 0 (A.2.31)

A.2.2 Momentum balance

The momentum balance for a mixture, withαphases andj species is derived.

Consider a system ofα phases and j species which expands the postulate of momentum balance (A.1.48) to include momentum gain for the j species from other phases in the mixture by ˆt_αj. The local form of the momentum balance is

∂

ε_αjρ_αjx⁰_αj

∂t = −div

ε_αjρ_αjx⁰_αj⊗x⁰_αj

+div ε_αjT_αj

+ε_αjρ_αjb_αj +ˆt_αj + ˆp_αj + ˆc_αjx⁰_αj + ˆr_αjx⁰_αj (A.2.32) where the mass balance (A.2.13) can be identied by similar mathematical steps as shown from (A.1.53) to (A.1.55) and therefore (A.2.25) reduces to

ε_αρ_αjD_αjx⁰_αj

Dt =div ε_αT_αj

+ε_αρ_αjb_αj + ˆt_αj+ ˆp_αj (A.2.33) The local postulate for the momentum balance of the phases is dened as

ε_αρ_αD_αx⁰_α

Dt =div(ε_αT_α) +ε_αρ_αb_α+ ˆt_α (A.2.34) Summation over the species of (A.2.33) yields

XN

j=1

ε_αρ_αjD_αjx⁰_αj Dt =

XN

j=1

div ε_αT_αj

+ε_αρ_αjb_αj + ˆt_αj+ ˆp_αj

(A.2.35) where the inner stress tensorT_I,α of the α phase is dened as

T_I,α= XN

j=1

T_αj (A.2.36)

and the body force for the phase is dened as b_α = 1

ρ_α XN

j=1

ρ_αjb_αj (A.2.37)

Inserting (A.2.36) and (A.2.37) into (A.2.35), yields XN

j=1

ε_αρ_αjD_αjx⁰_αj

Dt =div(ε_αT_I,α) +ε_αρ_αb_α+ XN

j=1

ˆt_αj + XN

j=1

pˆ_αj (A.2.38)

General relation 2:

A general relation for a arbitrary property Γ(x, t) is shown in order to com-pare the summation of the species (A.2.38) to the postulate of the phase (A.2.34). Consider the property Γ(x, t) and assume that the relation be-tween the species level and the phase level is described by

Γ_α = 1 ρ_α

XN

j=1

ρ_αjΓ_αj = XN

j=1

c_αjΓ_αj (A.2.39) The material time derivative of the general property is

D_αΓ_α Dt =

XN

j=1

D_α c_αjΓ_αj

Dt =

XN

j=1

"

Γ_αjD_α c_αj

Dt +c_αjD_α Γ_αj Dt

#

(A.2.40) For the further derivation, consider (A.2.6)-(a) with Γ = Γ_αj which yields the time derivative of the last term in the square brackets in (A.2.40). Mul-tiplying (A.2.40) by ε_αρ_α, shows that a part of the rst term in the square brackets is equal to the mass balance shown in (A.2.24). Inserting these relation into (A.2.40) yields the form

ε_αρ_αD_αΓ_α

Dt =ε_αρ_α XN

j=1

"

Γ_αjD_α c_αj

Dt +c_αjD_α Γ_αj Dt

#

= XN

j=1

"

Γ_αjε_αρ_αD_α c_αj

Dt +ε_αρ_αc_αj

D_αj(Γ_αj)

Dt −grad(Γ_αj)u_αj

#

= XN

j=1

ε_αρ_αc_αjD_αj(Γ_αj)

Dt −ε_αρ_αc_αjgrad(Γ_αj)u_αj

−Γ_αjdiv ε_αρ_αju_αj

+ Γ_αj ˆr_αj+ ˆc_αj−c_αjrˆ_α

(A.2.41) The divergence term in (A.2.41) is rewritten by the identity

div ε_αρ_αjΓ_αju_αj

=ε_αρ_αjgrad Γ_αj

u_αj+ Γ_αjdiv ε_αρ_αju_αj

(A.2.42) Combining (A.2.41) and (A.2.42) yields the desired nal relation as

ε_αρ_αD_αΓ_α

Dt =

XN

j=1

ε_αρ_αc_αjD_αj(Γ_αj)

Dt −div ε_αρ_αjΓ_αju_αj +Γ_αj rˆ_αj+ ˆc_αj−c_αjˆr_α

(A.2.43)

Using the general relation (A.2.43), with Γ =x⁰, yields ε_αρ_αD_αx⁰_α

Dt =

XN

j=1

ε_αρ_αc_αjD_αj(x⁰_αj)

Dt −div ε_αρ_αjx⁰_αj ⊗u_αj +x⁰_αj rˆ_αj + ˆc_αj −c_αjrˆ_α

(A.2.44) Applying the diusion velocity for the j'th species, x⁰_αj = u_αj +x⁰_α in the divergence term of (A.2.44) yields

ε_αρ_αD_αx⁰_α

Dt =

XN

j=1

ε_αρ_αc_αjD_αj(x⁰_αj) Dt −div

XN

j=1

ε_αρ_αju⁰_αj⊗u_αj

−div XN

j=1

ε_αρ_αju_αj

⊗x⁰_α+ XN

j=1

ˆ

r_αj + ˆc_αj −c_αjrˆ_α x⁰_α

+ XN

j=1

ˆ

r_αj+ ˆc_αj −c_αjrˆ_α

u_αj (A.2.45)

The fact that PN

j=1(ρ_αju_αj) = 0 is used to reduce (A.2.45) further, which eliminates the second divergence term in (A.2.45). Furthermore, it is noted that PN

j=1 rˆ_αj + ˆc_αj −c_αjrˆ_α

= 0 is obtained for the mass balance and the fact thatPN

j=1c_αj = 1, which yields a reduced form of (A.2.45), as ε_αρ_αD_αx⁰_α

Dt =

XN

j=1

ε_αρ_αc_αjD_αj(x⁰_αj) Dt −div

XN

j=1

ε_αρ_αju_αj ⊗u_αj

+ XN

j=1

ˆ

r_αj + ˆc_αj −c_αjrˆ_α

u_αj (A.2.46)

by substituting the rst term on right-hand side of (A.2.46) with (A.2.38), to get

ε_αρ_αD_αx⁰_α

Dt = div ε_αT_I,α− XN

j=1

ε_αρ_αju⁰_αj⊗u_αj!

+ε_αρ_αb_α

+ XN

j=1

ˆt_αj+ XN

j=1

ˆ p_αj +

XN

j=1

ˆ

r_αju⁰_αj + XN

j=1

ˆ c_αju⁰_αj

−rˆ_α ρ_α

XN

j=1

ρ_αju⁰_αj (A.2.47)

where the last term is eliminated by PN

j=1(ρ_αju_αj) = 0 with the use of the expansion ofc_αj.

The momentum balance for the species (A.2.47) and the phase (A.2.34) are compatible which yields the criteria

ε_αT_α =ε_αT_I,α− XN

j=1

ε_αρ_αju_αj ⊗u_αj

(A.2.48)

ˆt_α = XN

j=1

ˆt_αj + XN

j=1

ˆ

r_αju_αj (A.2.49)

and

XN

j=1

ˆ p_αj+

XN

j=1

ˆ

c_αju_αj = 0 (A.2.50) where the second order tensor ρ_αju⁰_αj⊗u⁰_αj is the so-called Reinholds stress tensor.

A similar approach is used to show that the summation of the phases is compatible with the postulate for the whole mixture. The local postulate for the momentum balance of the whole mixture is

ρDx⁰

Dt =divT+ρb (A.2.51)

Direct denition for summations over the phases are given for the stress tensor and the body force as

T_I = XN

α=1

ε_αT_α (A.2.52)

and

b= 1 ρ

XN

α=1

ε_αρ_αb_α (A.2.53)

where (A.2.8) is used. By use of the denitions (A.2.52) and (A.2.53) in (A.2.34) yields

XN

α=1

ε_αρ_αD_αx⁰_α

Dt =divT_I+ρb+ XN

α=1

ˆt_α (A.2.54)

General relation 3:

A relation similar to (A.2.43), is established between the phase description an the whole mixture description. First consider the arbitrary variable Γ, with the denition

ρΓ = XN

α=1

ε_αρ_αΓ_α (A.2.55)

The material derivative of (A.2.55) with respect to the mixture is D(ρΓ)

Dt = XN

α=1

Γ_αD(ε_αρ_α)

Dt +ε_αρ_αD(Γ_α) Dt

(A.2.56) The rst term on the left-hand side of (A.2.56) is (A.2.28), substituting this and using (A.2.6)-(b) with Γ = Γ_α, yields

D(ρΓ)

Dt =

XN

α=1

Γ_α(−div(ε_αρ_αu_α)−ε_αρ_αdiv(x⁰) + ˆr_α) +ε_αρ_α

D_α(Γ_α)

Dt −grad(Γ_α)u_α

(A.2.57) The rst divergence term and the gradient term on the left-hand side, is equivalent to the divergence of the product of the variables

div(ε_αρ_αΓ_αu_α) =ε_αρ_αgrad(Γ_α)u_α+ Γ_αdiv(ε_αρ_αu_α) (A.2.58) Substituting (A.2.58) into (A.2.57), yields

D(ρΓ)

Dt =

XN

α=1

−div(ε_αρ_αΓ_αu_α)−ε_αρ_αΓ_αdiv(x⁰) +Γ_αrˆ_α+ε_αρ_αD_α(Γ_α)

Dt

(A.2.59) Applying the product rule to the right-hand side of (A.2.59) and by using PN

α=1ε_αρ_αΓ_αdiv(x⁰) = ρΓdiv(x⁰), show that the mass balance for the whole mixture (A.2.31) is identied in the below square brackets, that is

Γ

D(ρ)

Dt +ρdiv(x⁰)

+ρD(Γ) Dt =

XN

α=1

−div(ε_αρ_αΓ_αu_α)

+ Γ_αrˆ_α+ε_αρ_αD_α(Γ_α) Dt

(A.2.60)

which reduce (A.2.60) to ρD(Γ)

Dt =−div XN

α=1

(ε_αρ_αΓ_αu_α) + XN

α=1

Γ_αrˆ_α+ XN

α=1

ε_αρ_αD_α(Γ_α)

Dt (A.2.61) Using the general relation (A.2.61) with Γ =x⁰and the denition of the diusion velocity for the phase x⁰_α =u_α+x⁰, yields

ρD(x⁰)

Dt = −div XN

α=1

(ε_αρ_αu_α⊗u_α)−div XN

α=1

(ε_αρ_αu_α)⊗x⁰

+ XN

α=1

ˆ r_αu_α+

XN

α=1

ˆ r_αx⁰+

XN

α=1

ε_αρ_αD_α(x⁰_α)

Dt (A.2.62)

The relation between the momentum for the whole mixture and the summa-tion of the momentum of the phases is established in (A.2.62). The second di-vergence term in (A.2.62) is eliminated by the denitionPN

α=1(ε_αρ_αu_α) = 0 and it is shown by the compatible mass balance terms that PN

α=1ˆr_α = 0. Furthermore, combining (A.2.62) and (A.2.54) reduces (A.2.62) to

ρD(x⁰) Dt =div

XN

α=1

(T_α−ε_αρ_αu_α⊗u_α) +ρb+ XN

α=1

ˆt_α+ XN

α=1

ˆ

r_αu_α (A.2.63) Comparing (A.2.63) with (A.2.51) show that

T= XN

α=1

(T_α−ε_αρ_αu_α⊗u_α) (A.2.64)

and XN

α=1

ˆt_α+ XN

α=1

ˆ

r_αu_α = 0 (A.2.65)

A.2.3 Angular momentum balance

The angular momentum is derived for the whole mixture in Sec. A.1.3, where it is shown that the stress tensor must be symmetric T=TT and also that the inner stress tensor must be symmetric T_I = T_I^T. Since the symmetry condition is decduced from the whole mixture, then it is directly transferable to the multi-phase and multi-species approach. The summation over the species yields

XN

j=1

M_αj = XN

j=1

T_αj −TT

αj

= 0 (A.2.66)

and with summation over the phases yields XN

α=1

XN

j=1

M_αj = XN

α=1

XN

j=1

T_αj−TT

αj

= 0 (A.2.67)

A.2.4 Energy balance

Energy balance postulates are given in Sec. A.2.4.2 where compatible ver-sions of the postulates between the whole mixture, the phases and the species are obtained. The compatible versions is deduced with help from, among other things, relations given in Sec. A.2.4.1.

A.2.4.1 Denitions for energy balances

The relation for summation of the external heat r_α and the inner internal energy e_I,α for the α phase is

r_α = 1 ρ_α

XN

j=1

ρ_αjr_αj; e_I,α= 1 ρ_α

XN

j=1

ρ_αje_αj (A.2.68) The internal energy of the phase is related to the energy density e_α for the phase, as

e_α =e_I,α− 1 2ρ_α

XN

j=1

ρ_αj u_αj2

(A.2.69)

where u_αj2

=u_αj·u_αj.

The inner heat ux q_I,α for the α phase is dened as

q_I,α= XN

j=1

q_αj −T^T_αju_αj +ρ_αje_αju_αj

(A.2.70)

which is related to the heat uxq_α for the phase as

q_α =q_I,α+1 2

XN

j=1

ρ_αj u_αj2

u_αj (A.2.71)

A quantity k_α related to the heat ux is introduced as k_α =

XN

j=1

q_αj+ρ_αje_αju_αj

= q_I,α+ XN

j=1

T^T_αju_αj

= q_α− XN

j=1

ρ_αj

− 1

ρ_αjT^T_αj+ 1

2 u_αj2

I

u_αj (A.2.72) Similar denitions is given for the whole mixture, The denition for the external heat r_α and the inner internal energy e_I,α for the whole mixture is given as

r = 1 ρ

XN

α=1

ε_αρ_αr_α; e_I = 1 ρ

XN

α=1

ε_αρ_αe_α (A.2.73) The internal energy e_I is related to the energy density e the whole mixture as

e=eI− 1 2ρ

XN

α=1

ε_αρ_α(u_α)² (A.2.74) where (u_α)² =u_α·u_α.

The inner heat ux for the mixture is dened as q_I =

XN

α=1

q_α−T^T_αu_α+ε_αρ_αe_αu_α

(A.2.75) which is related to the heat ux qfor the whole mixture as

q=q_I +1 2

XN

α=1

ε_αρ_α(u_α)²u_α (A.2.76) A quantity k related to the heat ux qis introduced as

k = XN

α=1

(ε_αq_α+ε_αρ_αe_αu_α)

= qI+ XN

α=1

T^T_αu_α

= q− XN

j=1

−T^T_α+ε_αρ_α1

2(u_α)²I

u_α (A.2.77)

A.2.4.2 Energy balance for constituents

The energy balance postulate for the j'th specie in the α phase is

∂

∂t ˆ

Ω

ε_αρ_αj

e_αj+ ¹₂ x⁰_αj

2 dv=

−

˛

∂Ω

ε_αρ_αj

e_αj +¹₂

x⁰_αj 2

x⁰_αj ·ds +

˛

∂Ω

ε_α

T_αjx⁰_αj −q_αj

·ds +

ˆ

Ω

h

ε_αρ_αjr_αj +ε_αρ_αjx⁰_αj ·b_αj +x⁰_αj · ˆp_αj+ˆt_αj

+ ˆe_αj+ ˆQ_αj i

dv +

ˆ

Ω

ˆ c_αj

e_αj+ ¹₂

x⁰_αj 2

dv +

ˆ

Ω

ˆ r_αj

e_αj +¹₂ x⁰_αj

2

dv (A.2.78) Using the mathematical operations similar to (A.1.86)-(A.1.98) yields the local form of the energy balance for the species, that is

ε_αρ_αjD_αje_αj

Dt = tr ε_αT_αjL_αj

−div ε_αq_αj + +x⁰_αj ·

div ε_jT_αj

+ε_αρ_αjb_αj + ˆp_αj+ˆt_αj

−ε_αρ_αjD_αjx⁰_αj Dt

+ε_αρ_αjr_αj + ˆe_αj+ ˆQ_αj (A.2.79) The momentum balance for the species is identied in the square brackets in (A.2.79) and cancels out. The reduced form of (A.2.79) is

ε_αρ_αjD_αje_αj

Dt = tr ε_αT_αjL_αj

−div ε_αq_αj

+ε_αρ_αjr_αj

+ˆe_αj+ ˆQ_αj (A.2.80)

The summation over the species of (A.2.80) where the denition (A.2.68)-(b) is used, is

ε_α XN

j=1

ρ_αjD_αje_αj

Dt = tr XN

j=1

ε_αT_αjL_αj

−div XN

j=1

ε_αq_αj

+ε_αρ_αr_α

+ XN

j=1

ˆ e_αj+

XN

j=1

Qˆ_αj (A.2.81)

The left-hand side of (A.2.81) is the rst term of the right-hand side of (A.2.43) with Γ = e. The summation of e_αj over the species is (A.2.68) which is the internal energy of the phase e_I,α, which change the notation of the left-hand side of (A.2.39) to Γ_I,α with Γ = e and further into (A.2.43), which is

ε_α XN

j=1

ρ_αjD_αje_αj

Dt = ε_αρ_αD_αe_I,α Dt +div

XN

j=1

ε_αρ_αje_αju_αj

− XN

j=1

ˆ

r_αj + ˆc_αj −c_αjrˆ_α

e_αj (A.2.82)

Combining (A.2.81) and (A.2.82) yields

ε_αρ_αD_αe_I,α

Dt = tr XN

j=1

ε_αT_αjL_αj

−div XN

j=1

ε_αq_αj

+ε_αρ_αr_α

+ XN

j=1

ˆ e_αj+

XN

j=1

Qˆ_αj−div XN

j=1

ε_αρ_αje_αju_αj

+ XN

j=1

ˆ

r_αj+ ˆc_αj−c_αjrˆ_α

e_αj (A.2.83)

wherek_αis identied by combining the divergence terms and (A.2.83) reduces to

ε_αρ_αD_αe_I,α

Dt = tr XN

j=1

ε_αT_αjL_αj

−div(ε_αk_α) +ε_αρ_αr_α+ XN

j=1

ˆ e_αj +

XN

j=1

Qˆ_αj

+ XN

j=1

ˆ

r_αj+ ˆc_αj −c_αjrˆ_α

e_αj (A.2.84)

In the following, the energy balance postulate for the α phase is brought to a form, which is compatible with (A.2.84). The postulate for the energy

balance for theα phase is

∂

∂t ˆ

Ω

ε_αρ_α e_α+¹₂(x⁰_α)²

dv = −

˛

∂Ω

ε_αρ_α e_α+ ¹₂(x⁰_α)²

x⁰_α·ds+ +

˛

∂Ω

ε_α(T_αx⁰_α−q_α)·ds+ +

ˆ

Ω

[ε_αρ_αr_α+ε_αρ_αx⁰_α·b_α+ +x⁰_α·ˆt_α+ ˆQ_α

i dv+ +

ˆ

Ω

ˆ

r_α e_α+ ¹₂(x⁰_α)²

dv (A.2.85) Using the divergence theorem on the rst and second term on the right-hand side and rewrite the left-right-hand side by mathematical steps similar to (A.1.86)-(A.1.96), yields a reduced local form of (A.2.81) as

ε_αρ_αD_α

Dt e_α+¹₂ (x⁰_α)²

= div(ε_α(T_αx⁰_α−q_α)) +ε_αρ_αr_α

+ε_αρ_αx⁰_α·b_α+x⁰_α·ˆt_α+ ˆQ_α (A.2.86) Note that the momentum balance for theα phase has not been identied in (A.2.86). The left-hand side of (A.2.86), where the velocity term is written by its denition (x⁰_α)² =x⁰_α·x⁰_α and expanded by the product rule is

ε_αρ_αD_αe_α

Dt +ε_αρ_αD_α Dt

1

2x⁰_α·x⁰_α

=ε_αρ_αD_α

Dte_α+ε_αρ_αx⁰_α·D_αx⁰_α

Dt (A.2.87) The rst part of divergence term on the right-hand side of (A.2.86), expanded by the product rule and using the denition of the velocity gradient, L_α = grad(x⁰_α), is

div(ε_αT_αx⁰_α) =x⁰_α·div(ε_αT_α) +ε_αtr(T_αL_α) (A.2.88) The third term in (A.2.86), expressed in terms of the phase velocity and the diusion velocity (A.2.5)-(a), is

XN

j=1

ε_αρ_αjx⁰_αj·b_αj = XN

j=1

ε_αρ_αj u_αj +x⁰_α

·b_αj (A.2.89) It is shown in (A.2.37) that summation over the species of the body force b_αj yields the body force of the phaseb_α. Using this condition with (A.2.89) results in

XN

j=1

ε_αρ_αj u_αj +x⁰_α

·b_αj =ε_αρ_αb_α·x⁰_α+ε_α XN

j=1

ρ_αju_αj ·b_αj (A.2.90)

In document A Coupled Transport and Chemical Model for Durability Predictions of Cement Based Materials (Sider 182-200)