Compositional Flow in Fractured Porous Media: Mathematical Background and Basic Physics - Selected Topics of Computational and Experimental Fluid Mechanics - page 22

C k ʱ ∂

T
∂

ˆ S

ʱ ˁ ʱ

c k ʱ

t +

c k ʱ C k ʱ (ʻ ʱ ˁ v + J

ʱ ) ·∇

T + C k ʱ ( I k ʱ −∇· d k ʱ ) T

=−∇· p k ʱ v ʱ +

T − ʵ˃ SB T 4 H k ʱ d k ʱ − ˆ k T , k ʱ ∇

+

Q k ʱ ,
(90)

where we have used the definitions

U k ʱ =

,

C k ʱ T (91)

p k ʱ
ˁ k ʱ ,

H k ʱ = U k ʱ + (92)

for the specific internal energy and enthalpy of constituent ( k , ʱ ). Summing up over

all species, making use of the definitions

n

U ʱ = c k ʱ U k ʱ , (93)

k = 1

n

C

ʱ =

c k ʱ C k ʱ , (94)

k = 1

n

p ʱ = p k ʱ , (95)

k = 1

for the specific internal energy, heat capacity, and pressure of phase , together with

relation ( 34 ) for the correction factor for energy advection, and then summing up
over all phases, we obtain the temperature equation for the compositional flow

ʱ

C ∂

T
∂

ˆˁ

t +

ʳ C ˁ C v + C ʱ J ʱ ·∇ T

ʱ

T

n

=−∇·

p ʱ v ʱ − H k ʱ D k ʱ ·∇ c k ʱ − ˆ k T ∇

ʱ ʱ

k = 1

n

− ʵ˃ SB T 4 − T C k ʱ [ I k ʱ +∇· ( D k ʱ ) ] + Q . (96)

ʱ

k = 1

Using relations ( 70 ) and ( 71 ) into Eq. ( 96 ) we recover the form of the temperature
equation for the composite system (i.e., the solid and the fluid phases). For a reactive
solid phase, a term of the form

n

− T C kR I kR (97)

k = 1

must be added on the right-hand side of Eq. ( 96 ).