Handbook of Solvents - George Wypych - ChemTech - Ventech!

7.2 Bubbles dynamics and boiling 359
1(
)
−
p = p + G ρ ρ − ρ + G θ [7.2.16]
0 20 0
0 30
Thermodynamic equation of state for non-relaxing liquid at small deviations from
equilibrium can be written as follows
p p
p = p + ( )
T
⎛ ⎞
⎜ ⎟ − +
⎝ ⎠
⎛
∂
∂ ⎞
0 ρ ρ0
⎜ ⎟
∂ρ
⎝∂
⎠
T = T0
ρ= ρ0
θ
[7.2.17]
The thermal expansion coefficient, α, and the isothermal bulk modulus, K is, are defined
as 4
∂ρ
∂
α =−ρ ρ
∂ ρ ρ
∂ρ
⎛ ⎞
⎛ p⎞
0⎜⎟ , Kis
= 0⎜⎟
[7.2.18]
⎝ T ⎠
⎝ ⎠
= 0 T = T0
Therefore, equation [7.2.17] can be rewritten in the form
( )
−1
p = p + K ρ ρ − ρ + αK θ [7.2.19]
0 is 0
0
is
From [7.2.16] and [7.2.18] it follows that G 20 =K is, G 30 = αK is.
In rheology of polymers complex dynamic modulus, G k*, is of special importance. It
is introduced to describe periodic deformations with frequency, ω, and defined according to:
G
∞
*
k = ∫
0
()( )( + )
2
1+
( ωλ)
F λ ωλ ωλ i dλ
k
[7.2.20]
Equations of motion of the liquid follow from momentum and mass conservation
laws. In the absence of volume forces they mean:
�
dv
ρ =∇ ⋅ σ
[7.2.21]
dt
dρ
�
+ ρ∇
⋅ v = 0 [7.2.22]
dt
For polymeric solution the stress tensor, σ, is defined according to [7.2.10], [7.2.11].
To close the system, it is necessary to add the energy conservation law to equations [7.2.21],
[7.2.22]. In the case of liquid with memory it has the form 2
t t
2 ∂
∂
k∇ θ−T
− ′ ′ ′ = − ′
0 3 0 4
∂t
∂t
∫G ( t t) ekk( t) dt T ∫G
( t t)
−∞ −∞
t d
∂θ
t′
∂ ′
∞
t t 1
G4( t − t′ ⎛ − ′ ⎞
−
) = G40 −∫F4( λ)
exp ⎜−
⎟ dλ, G40 = ρ0cvT0[7.2.23]
⎝ λ ⎠
0