2.6 Diffusion Coefficient or Diffusivity of Gases
2.6.2 Determination of Gas Phase Diffusivity
Values of diffusivities of gases may be obtained from literature, predicted from available equations or by conducting suitable experiments.
Data available in literature
Experimental values of gas phase diffusivities at different temperatures and pressures for a number of systems are available in the International Critical Tables (ICT 1929). Some selected values are also available in several books (Perry et al. 1997 and Treybal 1985). These values are quite reliable and can be used wherever available.
A list of some gas phase diffusivities for some common systems is given in Table 2.1.
Table 2.1 Diffusivities of Some Common Gas Pairs at 1 atm Pressure Gas pair
A number of equations, mostly empirical have been proposed for the prediction of gas phase diffusivities. Some of them are quite reliable.
Based on the kinetic theory of gases, Jeans and co-workers (Bird et al. 1960) developed an expression for diffusivities of monoatomic gases. Gilliland (1934) determined the constants of their expression from published experimental data and proposed the following simple empirical equation for prediction of diffusivities of binary gas mixtures:
DAB = 435.7 (2.42)
where,
DAB = diffusivity of components A in B, cm2/s T = temperature, K
P = total pressure, N/m2
VA and VB = molar volumes of A and B respectively, m3/kmol MA and MB = molecular weights of A and B, respectively.
Molar volumes of compounds may be estimated from Kopp’s law of additive volumes.
EXAMPLE 2.8 (Estimation of diffusivity in gases using Gilliland equation): Estimate the diffusivity of ammonia in air at standard atmospheric pressure and 0°C by using Gilliland equation.
Given: Molecular volume of ammonia and air are 25.8 and 29.9, respectively.
Solution: T = 273 K, P = 1.013 # 105 N/m2 For ammonia: MA = 17, VA = 25.8
For air: MB = 29, VB = 29.9
Substituting these values in Eq. (2.42), we obtain
DAB = 435.7 # 10-4
= 1.615 # 10-5 m2/s
Fuller et al. (1966) had proposed the following empirical equation for predicting binary gas phase diffusivities up to moderate pressures:
DAB = 0.00100 (2.43)
where,
DAB = diffusivity of components A and B, cm2/s T = temperature, K
P = total pressure, atm
∑yA and ∑yB = sum of atomic diffusion volumes of A and B respectively MA and MB = molecular weights of A and B, respectively.
Equation (2.43) is simple but quite reliable. In most cases, the error is within ±5%.
Atomic diffusion volumes for some common molecules and structural groups have been presented in Table 2.2.
Table 2.2 Atomic diffusion volumes for use in Eq. (2.43) Atomic and Structural diffusion volume increments, u
Carbon 16.5 Chlorine 19.5 Fuller et al., estimate the diffusivity of naphthalene vapour in air at a temperature of 0°C and a total pressure of 1 atm.
Solution: Let us call naphthalene (C10H8) A and air B. Hence, MA = 128.16, MB = 29.
From Table 2.2, atomic diffusion volume increments are as follows:
For naphthalene (A): (∑u)A = (10 # 16.5) + (8 # 1.98) - (2 # 20.2) = 140.44 For air (B): (∑u)B = 20.1
Substituting the values in Eq. (2.43), we get
DAB = 0.00100
= 0.0605 cm2/s.
Chapman and Enskog independently proposed an equation for prediction of binary gas-phase diffusivities. This equation is a theoretical equation based on the kinetic theory of gases and gives fairly reliable values of DAB. The diffusivity strongly depends upon binary interaction parameters of the two components. Chapman and Enskog used the Lennard-Jones potential function (Chapman and Cowling 1970) to calculate the interaction parameters.
The equation proposed by Chapman and Enskog is as follows:
DAB = (2.44)
where,
DAB = diffusivity of components A and B, m2/s T = temperature, K
P = total pressure, atm.
MA and MB = molecular weights of A and B, respectively vAB = molecular separation at collision = (vA + vB)/2 vA and vB = molecular diameters of A and B, respectively, Å
= collision integral, function of (kT/fAB), where k is the Boltzmann’s constant fAB = energy of molecular attraction between molecules of A and B = Ö .
Values of v and f for some common substances are given in Table 2.3, and values of collision integral are given in Table 2.4.
If values of v and f are not available, the same may be estimated from the properties of the fluid at the critical point (c), the liquid at the normal boiling point (b), or the solid at the melting point (m), by following very approximate relations (Bird et al. 2006):
f/k = 0.77 Tc and v = 0.841 Vc1/3 or, v = 2.44 (Tc/Pc)1/3 f/k = 1.15 Tb and v = 1.166 Vb1/3
f/k = 1.92 Tm and v = 1.222 Vm1/3
where, f/k and T are in K, v is in Å, V is in cm3/gmol, and Pc is in atm.
Table 2.3 Lennard-Jones potential for some common substances Compound v, Å f/k B, K
Carbon dioxide 3.941 195.2 Table 2.4 Values of collision integral, Ω
k BT/fAB D,AB k BT/fAB D,AB 0.30 2.662 2.00 1.075 0.35 2.476 2.10 1.057 0.40 2.318 2.20 1.041 0.45 2.184 2.30 1.026
0.50 2.066 2.40 1.012 0.55 1.966 2.50 0.9996 0.60 1.877 2.60 0.9878 0.65 1.798 2.70 0.9770 0.70 1.729 2.80 0.9672 0.75 1.667 2.90 0.9576 0.80 1.612 3.00 0.9490 0.85 1.562 3.20 0.9328 0.90 1.517 3.40 0.9186 0.95 1.476 3.60 0.9058 1.00 1.439 3.80 0.8942 1.05 1.406 4.00 0.8836 1.10 1.375 5.00 0.8422 1.15 1.346 6.00 0.8124 1.20 1.320 7.00 0.7896 1.25 1.296 8.00 0.7712 1.30 1.273 9.00 0.7556 1.35 1.253 10.00 0.7424 1.40 1.233 20.00 0.6640 1.45 1.215 30.00 0.6232 1.50 1.198 40.00 0.5960 1.55 1.182 50.00 0.5756 1.60 1.167 60.00 0.5596 1.65 1.153 70.00 0.5464 1.70 1.140 80.00
0.5352
1.75 1.128 90.00 0.5256 1.80 1.116 100.00 0.5130 1.85 1.105 200.00 0.4644 1.90 1.094 400.00 0.4170 1.95 1.084
EXAMPLE 2.10 (Estimation of binary gas diffusivity using Chapman-Enskog equation): Estimate the diffusivity of sulphur dioxide in air at standard atmospheric pressure and 400°C by using Equation of Chapman-Enskog.
Solution: Let SO2 = A, and Air = B.
From Table 2.3, For air:
fA/kB = 78.6 K, vA = 3.711 Å = 3.711 # 10-10 m = 0.3711 nm
For SO2: fB/kB = 335.4 K, vB = 4.112 Å = 4.112 # 10-10 m = 0.4112 nm and for the system:
vAB = (vA + vB) = (3.711 + 4.112) = 3.9115 Å = 0.39115 nm fAB = kB(78.6 335.4)1/2
=
From Table 2.4, Ω = 0.88
Using Chapman-Enskog Eq. (2.44 )
DSO2-air =
= 5.40 # 10-5 m2/s = 0.54 cm2/s
EXAMPLE 2.11 (Rate of diffusion of a gas through the pores of a catalyst): A nickel catalyst for the hydrogenation of ethylene has a mean pore diameter of 100 Å. Calculate the diffusivities of hydrogen for this catalyst at (i) temperature 100°C, pressure 1 atm and (ii) temperatrue 100 °C, pressure 10 atm in a hydrogen-ethane mixture.
Solution: Let hydrogen = A and ethane = B.
From Table 2.3,
For H2 (A): fA/kB = 59.7 K; vA = 2.827 Å = 2.827 # 10-10 m
For C2H6 (B): fB/kB = 215.7 K; vB = 4.443 Å = 4.443 # 10-10 m and for the system:
vAB = (vA + vB) = (2.827 + 4.443) = 3.635 Å fAB = kB (59.7 # 215.7)]1/2
= = 3.287.
From Table 2.4, = 0.92
Given, T = 100°C = 373 K, MA = 2.016, MB = 30.05.
Substituting the values in Chapman-Enskog Eq. (2.44), we get
DH2-C2H6 =
(i) when P = 1 atm, DH2-C2H6 = 8.01 # 10-5 m2/s = 0.801 cm2/s (ii) when P = 10 atm, DH2-C2H6 = 8.01 # 10-6 m2/s = 0.0801 cm2/s
EXAMPLE 2.12 (From the value of gas phase diffusivity at a given temperature, prediction of diffusivity for the system at another temperature): The diffusion coefficient of H2-N2 system is 7.84 # 10-5 m2/s at 1 atm and 298 K.
(i) Find out the diffusivity at 350 K and 500 K using Chapman-Enskog equation.
(ii) Using equation D = D0 (P0/P)(T/T0)n, find the diffusion coefficients at the above temperatures assuming n = 1.75 and 1.5.
Solution:
(i) For H2(A): fA/kB = 59.7 K, vA = 2.827 Å = 2.827 # 10-10 m, MA = 2.016 (ii) For N2(B): fB/kB = 71.4 K, vB = 3.798 Å = 3.798 # 10-10 m, MB = 28.0134 and for the system:
vAB = (vA + vB) = (2.827 + 3.798) = 3.3125 Å fAB = kB(fAfB)1/2 = kB(59.7 # 71.4)1/2
= = 5.36.
From Table 2.4, AB = 0.8314.
Using Chapman-Enskog Eq. (2.44), we have
DH2-N2 at 350 K =
= 9.724 # 10-5 m2/s = 0.972 cm2/s Similarly,
DH2-N2 at 500 K (when kBT/fAB = 7.66, AB = 0.7775) = 17.76 10-5 m2/s.
(ii) D350 = D298 = 10.388 # 10-5 m2/s and D500 = D298 = 19.392 # 10-5 m2/s
Proceeding in the same way, the values of diffusivity at 350 K and 500 K with n = 1.5 are found to be D350 = 9.979 # 10-5 m2/s, and D500 = 17.039 # 10-5 m2/s, respectively.
Comparison of the computed values using different methods as above, shows that the diffusivity values obtained assuming n = 1.5 are in good agreement with those obtained from Chapman-Enskog Equation.
Prediction of the diffusivity using Eq. (2.44) gives rise to values within 6% for nonpolar gas pairs at low density, if the collision constants are available from viscosity data. However, the error increases to about 10% if the Lennard-Jones parameters are estimated from thermodynamic data (Hirschfelder et. al. 1949; Tee et al. 1966). The Chapman-Enskog theory fails to predict a variation in the diffusion coefficient with concentration. For example, diffusivity in some binary gas mixtures such as chloroform-air may vary as much as 9% with concentration, whereas in other systems like methanol-air, the diffusivity is independent of the concentration.
Figure 2.6 Determination of diffusivity by Stefan tube.
The Chapman-Enskog Equation for the transport coefficients is valid for the range 200-1000 K.
Below 200 K, quantum effects become important (Hirschfelder et al. 1954), and above 1000 K the Lennard-Jones potential function is no longer applicable. If the force constants are derived from diffusion data instead of the usual viscosity data, the equations may be extended to 1200 K. Above 1200 K, the force constants should be evaluated from molecular beam scattering experiments.
The equations were developed for dilute gases composed of nonpolar, spherical, monoatomic molecules. Empirical functions and correlations, i.e. the Lennard-Jones potential must be used, with
the net result that the equations are remarkable for their ability to predict diffusivity for many gas mixtures. Agreement in nonpolar gas mixtures is excellent, even for polyatomic molecules. For polar-nonpolar gas mixtures the same equations are used with different combining laws (Hirschfelder et al. 1954).
Experimental determination of gas phase diffusivity
Several methods are available for experimental determination of gas phase diffusivity. Two of them use discussed here.
Stefan tube method: This method is suitable for measuring gas phase diffusivity of binary mixtures where one of the components (A) is available as a liquid at the conditions of the experiment. The other component (B) is a gas not soluble in A.
As shown in Figure 2.6, the experimental set-up consists of a narrow glass tube 5-6 mm id, sealed at the bottom, held vertically and joined at the top with a larger diameter horizontal tube. The horizontal tube is sometimes provided with straightners to avoid turbulence. The vertical tube is filled with the experimental liquid A up to a certain distance from the top while the gas B flows through the horizontal tube. The entire system is maintained at a constant temperature. The molecules of evaporated A diffuses through a mixture of A and B, reaches the top and is swept away by the gas B.
Being insoluble in liquid A, B is not diffusing. Hence, this is a case of diffusion of one component through the stagnant layer of another component, i.e. gas. Assuming pseudo-steady state operation, the liquid level in the vertical tube varies very slowly. The experiment should be continued for a long time to reduce experimental error. The distances of the liquid level from the top open end at start and end of the experiment, and also the duration of the experiment along with the ambient temperature and barometric pressure are to be measured. The drop in the liquid level should be accurately measured, preferably with a travelling microscope.
Let z be the distance of the liquid level from top open end at any instant i, then from Eq. (2.25), the diffusional flux may be expressed as
NA =
Equating the above flux with the drop dz in liquid level over time di,
(2.45) Integrating and rearranging, we get
(2.46)
where, z1 and z2 are the distances of the liquid surface from the top at the beginning and at the end of the experiment.
The partial pressure (p A1) of component A at the liquid surface is the vapour pressure of A at the temperature of experiment. At the top open end of the tube, p A2 may be assumed zero since vapours
of A are swept away by B.
EXAMPLE 2.13 (Experimental determination of diffusivity of a gas by Stefan-tube method): A narrow vertical glass tube is filled with liquid toluene up to a depth of 2 cm from the top open end placed in a gentle current of air. After 150 hr, the liquid level has dropped to 4.4 cm from the top. The temperature is 25°C and the total pressure 1.013 # 105 N/m2.
At 25°C, the vapour pressure of toluene is 29.2 mm Hg and the density of liquid toluene is 850 kg/m3.
Neglecting the effects of convective currents and counter diffusion of air, calculate the diffusivity of toluene in air at 25°C and 1.013 # 105 N/m2 pressure.
Solution: From Eq. (2.46), we get
DAB =
T = 25°C = 298 K, P = 1.013 105 N/m2, q = 150 hr = 5.4 # 105 s. z1 = 0.02 m, z2 = 0.044 m, R = 8314 Nm/(kmol)(K).
Molecular weight of toluene = 92 For toluene at 25°C:
Vapour pressure = 29.2 mm Hg = 3.89 # 103 N/m2,
Density = 850 kg/m3, p A1 = 29.2 mm Hg = 3.89 # 103 N/m2, p B1 = 101.3 # 103 - 3.89 # 103 = 97.41 # 103 N/m2
p A2 = 0, p B2 = 101.3 # 103 N/m2.
Substituting the values in Eq. (2.46):
DAB =
= 0.82 # 10-5
Diffusivity of toluene in air = 0.82 # 10-5 m2/s.
Twin Bulb Method: This method is suitable for measuring the diffusivities of permanent gases or of components that are gases at least at the conditions of the experiment.
As shown in Figure 2.7, the experimental set-up consists of two large bulbs of volumes V1 and V2, connected by a narrow tube fitted with a stop cock or a plug type valve. Let the cross sectional area and the length of the connecting tube be a and z respectively.
Figure 2.7 The twin bulb set-up.
Before starting the experiment the valve in the connecting tube is closed, the two bulbs are evacuated, washed several times with the respective gases to be filled in that bulb. Then one bulb is filled with pure A while the other with pure B, both being at the same pressure.
The experiment is then started by opening the valve in the connecting tube. During the experiment the contents of the two bulbs should be constantly stirred to maintain uniform concentration and the entire system should be kept at constant temperature and pressure. Since the two bulbs are large and are at equal and constant pressure, equimolal counter-diffusion takes place through the connecting tube.
Concentrations (or partial pressures) of these two components will change only very slowly so that pseudo-steady state conditions may be assumed. The total pressure (P) in the bulbs, the duration of the experiment (q ), and the partial pressures of component A in the two bulbs ( p A1 and p A2) are to be noted.
If p A1 and p A2 (p A1 > p A2) be the partial pressures of component A in bulb 1 and bulb 2 at any instant q, the steady state rate of transport of A from bulb 1 to bulb 2 can be expressed as follows in terms of Eq. (2.27):
aNA = = -aNB (2.47)
Since p A1 and p A2 are the instantaneous partial pressures of component A in the two bulbs, the rates of change of partial pressures are given by
aNA = - (2.48)
aNA = (2.49)
Combining Eqs. (2.48) and (2.49), we obtain - (p A1 - p A2) = aRT NA
=
or, - (2.50)
Integrating Eq. (2.50) with the following limits:
q = 0, (p A1 - p A2) = (P - 0) = P; and q = q , (p A1 - p A2) = (p A1 - p A2) yields
ln (2.51)
On the basis of the measurements made, DAB can be estimated from Eq. (2.51).