Page 366 - Characterization and Properties of Petroleum Fractions - M.R. Riazi

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346 CHARACTERIZATION AND PROPERTIES OF PETROLEUM FRACTIONS
TABLE 8.8—Order of magnitude of binary diffusion coefficient and its concentration dependency
for various systems [35].
Order of magnitude Activation energy (E), Concentration
2
Type of system of D,cm /s kcal/mol dependence
Gas–gas (vapor–gas) 0.1–1.0 E < 5 Very weak
Gas–liquid ∼10 −5 E ≤ 5 Weak
Normal liquids 10 −5 –10 −6 5–10 ±100%
Polymer solutions 10 −5 –10 −8 10–20 ±1000%
Gas or liquid in polymer or solids ∼10 −12 –10 −15 E ≥ 40 Factor of 1000%
respectively. The order of magnitude of N Sc in liquids such as relation is developed for gas–gas diffusivities [1, 3]:
3
water is 10 .
Diffusion coefficient like any other thermodynamic prop- 3 × π × 10 −1 k 3 B 1/2 1 1 1/2 T 3/2
erty is a function of the state of a system and depends on D AB = 8 × π 3 2m A + 2m B P d A +d B 2
T, P, and concentration (i.e., x i ). One theory that describes (8.56) 2
molecular diffusion is based on the assumption that molecu- 2
lar diffusion requires a jump in their energy level. This energy where D AB is in cm /s, k B is the Boltzman’s constant (1.381 ×
−23
is called activation energy and is shown by E A . This activation 10 J/K), T is temperature in kelvin, P is the pressure in
energy, although not the same, is very similar to the activa- bar, m is the molecular mass in kg [M/N A , i.e., m A = M A ×
−3
23
tion energy required for a chemical reaction to occur. Heavier 10 /(6.022 × 10 )], and d is the hard sphere molecular diam-
−9
molecules have higher activation energy and as a result lower eter in m (1 nm = 1 × 10 m). Values of d may be determined
diffusion coefficients. Based on this theory, dependency of D from measured viscosity or thermal conductivity data by Eqs.
with T can be expressed by Arrhenius-type equation in the (8.2) and (8.30), respectively. For example, for CH 4 value of d
following form: from viscosity is 0.414 nm while from thermal conductivity is
0.405 nm. For O 2 ,H 2 and CO 2 , values of d are 0.36, 0.272, and
0.464 nm, respectively [3]. As an example, the self-diffusion
coefficient of CH 4 at 1 bar and 298 K from the kinetic the-
E A
(8.55) D = D o exp − ory is calculated as m A = m B = m = 2.66 × 10 −26 kg, d A = d B =
RT
2
d = 0.414 × 10 −9 m and from Eq. (8.56) D AB = 0.194 cm /s.
Thus one can calculate diffusion coefficient from viscosity
where D o is a constant (with respect to T), E A is the activation data through calculation of molecular diameter. For gases
energy, R is the gas constant, and T is the absolute tempera- at low pressures D varies inversely with pressure, while it is
−1/2
ture. The order of magnitude of D and E A in various systems proportional to T 3/2 . Furthermore, D A varies with M A , that
and concentration dependency of D are shown in Table 8.8. is, heavier molecules have lower diffusivity under the same
Diffusion coefficients depend on the ability of molecules to conditions of T and P. In practical cases molecular diame-
move. Therefore, larger molecules have more difficulty to ters can be estimated from liquid molar volumes in which
move and consequently their diffusivity is lower. Similarly in actual data are available, as will be seen in Eq. (8.59). A more
liquids where the space between molecules is small, diffusion accurate equation for estimation of diffusivity of ideal gases
coefficients are lower than in the gases. Increase in T would was derived independently by Chapman and Enskog from the
increase diffusion coefficients, while increase in P decreases kinetic theory and is known as Chapman–Enskog equation,
diffusivity. The effect of P on diffusivity of liquids is less than which may be written as [1, 9]
its effect on the diffusivity of gases. At very high pressures, 0.5

1 1 --`,```,`,``````,`,````,```,,-`-`,,`,,`,`,,`---
0.5
−5
values of diffusion coefficients of liquids approach their val- 2.2648 × 10 T +
◦ M A M B
ues for the gas phase. At the critical point, both liquid and (ρD AB) = 2
σ AB
AB
gas phases have the same diffusion coefficient called critical
diffusion coefficient and it is represented by D C . σ AB = σ A + σ B
In this section, methods of estimation of diffusion coef- 2
ficients in gases and liquids as well as in multicomponent σ i = 0.1866V ci 1/3 Z −6/5
ci
systems and the effect of porous media on diffusivity are pre- 1.06036
∗
sented. In the last part a new method different from conven- AB = ∗ 0.1561 + 0.193 exp −0.47635T AB + 1.76474
tional methods for experimental measurement of diffusion T AB
coefficients in dense hydrocarbon fluids (both gases and liq- × exp −3.89411T ∗ + 1.03587 exp −1.52996T ∗

uids) is presented. AB AB
T ∗
AB = T/ε AB
ε AB = (ε A ε B) 1/2
ε i = 65.3T ci Z 18/5
8.3.1 Diffusivity of Gases at Low Pressures ci
(8.57)
Similar to viscosity and thermal conductivity, kinetic theory
provides a relatively accurate relation for diffusivity of rigid where (ρD AB) represents the product of density–diffusivity
◦
(hard) molecules with different size. Based on this theory, for of ideal gas at low-pressure conditions according to the
gases at low pressures (ideal gas conditions) the following Chapman–Enskog theory and is in mol/cm · s. ε and σ are the

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