Universidad de Carabobo Autoridades

240 Damelys Zabala and Aurélie Wender / Revista Ingeniería UC , Vol. 23, No. 3, Diciembre 2016, 237-246
7. Determine the ∆ρ s for the step change in the
density of the reference component ρ s . ( N p =
500, [13]), with equation (11).
∆ρ s = ρe s,L − ρe s,V
N p
(11)
8. Solving the set of equations (7):
F (i) = √ c i
(
µs
(
ρ1 , ρ 2 , ..ρ nc
) − µ
e
s
)
− √ c s
(
µi
(
ρ1 , ρ 2 , ..ρ nc
) − µ
e
i
) = 0 (12)
For the solution of the n c − 1 algebraic
equations, (equation (12)), the initialization
of the ρ i (is) is required at each k step
(sequence from 8a to 8c).
a) The matrix equation (14) is used for
finding the first estimate of the n c − 1
derivatives dρ i
dρ s
n∑
c −1 n∑
c −1 {[ √ √ ] }
ci ∂µ s cs ∂µ i dρ j
−
∂ρ
i=1 j=1 j ∂ρ j dρ s
√ √
cs ∂µ i ci ∂µ s
= −
∂ρ s ∂ρ s
(13)
[ √ √ ]{ }
ci ∂µ s cs ∂µ i dρ j
−
=
∂ρ j ∂ρ j dρ s
{ √ √ }
cs ∂µ i ci ∂µ s
−
∂ρ s ∂ρ s
(14)
b) The n c derivatives dµ i /dρ j required in
equation 14 are evaluated numerically
with δρ j = 1 × 10 −6 using the equation
15 for the chemical potential of the
component i in the mixture:
µ i = µ res
i
Where, µ res
i
[ ( )]
Pre f
+ RT 1 − ln (15)
ρ i RT
= residual contribution of
the chemical potential of i in the mixture,
[kJ/kmol]
c) The first estimation for the k step concentrations
will be done with equations
16 and 17:
ρ k s = ρ k−1
s + ∆ρ s (16)
ρ k i = ρ k−1
i +
( ) dρi
∆ρ s (17)
dρ s
d) With this first estimate for concentrations
ρ i , the system of algebraic
equations is solved by Newton-Raphson
method, with N iterations (until in
equation 19 the biggest of the |∆ρ i | ≤
1 × 10 −7 ). (The definitive concentrations
for the k step are then achieved).
⎛
⎜⎝
∂F(1) ∂F(1)
∂ρ 1
∂F(n c −1)
∂ρ nc−1
∂F(n c −1)
∂ρ 1 ∂ρ nc−1
⎛
⎞⎟⎠ ⎜⎝
∂F(1)
∂ρ nc−1
∂F(n c −1)
∂ρ nc−1
ρ k 1
.
ρ k n c −1
⎞
⎟⎠ =
− ( F(1) . . . F(n c − 1) ) t
⎛
⎛
∂F(1)
∂ρ
+ 1
. . .
ρ
⎜⎝ ∂F(n c −1)
∂ρ 1
. . .
⎞⎟⎠
k−1 ⎞
1
ρ
⎜⎝
k−1
2
⎛ ∂F(1)
∂ρ 1
· · ·
.
. . .
⎜⎝ ∂F(n c −1)
∂ρ 1
· · ·
∂F(1)
∂ρ nc−1
.
∂F(n c −1)
∂ρ nc−1
⎛ ⎞⎟⎠
⎜⎝
ρ k−1
n c −1
∆ρ 1
..
∆ρ nc −1
= − ( F(1) ... F(n c − 1) ) t
⎟⎠ (18)
⎞
⎟⎠
(19)
For the next k, the procedure from step
8a) is repeated.
9. Once the iterative process for getting “ρ i ” is
finished, the density profile for the reference
component “ρ s (z)” must be verified and
for doing that, the grand thermodynamic
potential, Φ(ρ) is needed and it can be
calculated by equation 20.
∆Φ (ρ 1 , ρ 2 . . .ρ nc ) = Φ(ρ) − Φ e
= A ∑nc
V − (
ρi µ e )
i + P
e
i=1
nc
∑ (
= a 0 − ρi µ e )
i + P
e
i=1
(20)
Where Φ e = −P e , the equilibrium pressure,
[Pa]
A = Helmholtz energy [J]
V = volume [m 3 ]
a 0 = Helmholtz energy density [J/m 3 ].
The Helmholtz energy density a 0
calculated with equation (21).
can be
a 0 (ρ) = ρµ(ρ) − P(ρ) (21)
Revista Ingeniería UC, ISSN: 1316–6832, Facultad de Ingeniería, Universidad de Carabobo.