Page 371 - Characterization and Properties of Petroleum Fractions - M.R. Riazi

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8. APPLICATIONS: ESTIMATION OF TRANSPORT PROPERTIES 351
L 14:25 L

a b

τ= 2L = 1 = 1 = 1 = 2 ≈ 1.4
2LCosθ Cosθ Cos45 ° 2
2
FIG. 8.9—Approximate calculation of tortuosity (τ).

size and shape of particles are all different, value of τ varies [50] suggest that for calculation of diffusion coefficients of
from 3 to 5. gases in porous solids (i.e., catalytic reactors) effective diffu-
In a porous media τ is related to the formation resistivity sion coefficients can be calculated from the following equa-
factor and porosity as tion:

(8.72) τ = (Fφ) n 1 −1.5
(8.75) D eff = φ D
where F is the resistivity and φ is the porosity, both are di-
mensionless parameters. φ is the fraction of connected empty This equation can be obtained from Eq. (8.70) by assuming
n
space in a porous media and F is an indication of electrical re- τ = φ 1.5 .
sistance of materials that form the porous media and is always
greater than unity. n 1 is a dimensionless empirical parameter
--`,```,`,``````,`,````,```,,-`-`,,`,,`,`,,`---
that depends on the type of porous media. Theoretically, value 8.4 INTERRELATIONSHIP AMONG
of n 1 in Eq. (8.72) is one; however, in practice n 1 is taken as TRANSPORT PROPERTIES
1.2. Various relations between F and φ are given by Amyx
et al. [48] and Langness et al. [49]. One general relation is In previous sections three transport properties of μ, k, and D
given as follows [48]:
were introduced. In the predictive methods for these molecu-
lar properties, there exist some similarities among these prop-
−m
(8.73) F = aφ
erties. Most of the predictive methods for transport properties
of dense fluids are developed through reduced density, ρ r .In
where parameters a and m are specific of a porous media. addition, diffusion coefficients of dense fluids and liquids are
Parameter m is called cementation factor and it is specifi- related to viscosity. Riazi and Daubert developed several re-
cally a characteristic of a porous media and it usually varies lationships between μ, k, and D based on the principle of
from 1.3 to 2.5. Some researchers have attempted to correlate dimensional analysis [37]. For example, they found that for
parameter m with porosity and resistivity. For some reser- liquids ln (μ 2/3 D/T) versus ln (T/T b ) is linear and obtained the
voirs a = 0.62 and m = 2.15, while for some other reservoirs, following relations:
when φ> 0.15, a = 0.75 and m = 2 and for φ< 0.15, a = 1
and m = 2. By combining Eqs. (8.72) and (8.73) with n 1 = 1.2
and a = 1: μ 2/3 D = 6.3 × 10 −8 T 0.7805 for liquids except water
T T b
(8.74) τ = φ 1.2−1.2m μ 2/3 −8 T 1.0245
T D = 10.03 × 10 T b for liquid water
Equation (8.74) can be combined with Eq. (8.70) to estimate
effective diffusion coefficients in a porous media. Parameter (8.76)
min Eq. (8.74) can be taken as an adjustable parameter, while where μ is liquid viscosity in cp (mPa · s), T is temperature
2
for simplicity, parameter n in Eq. (8.70) can be taken as unity. in kelvin, D is liquid self-diffusivity in cm /s, and T b is nor-
In practical applications, engineers use simpler relations mal boiling point in kelvin. For example, for n-C 5 in which T b
between tortuosity and porosity. For example, Fontes et al. is 309 K the viscosity and self-diffusion coefficient at 25 C
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