Interfacial geometry and its effect on the estimation of binary gas diffusivities in an isothermal Stefan column

Abstract

The Stefan column was developed in the 19th Century to estimate binary gas diffusivities, D_AB’s, and this goal has been achieved by many research groups. In the classical device, volatile liquid A is placed at the bottom and overlaid with stagnant gas B. A slow gas B sweep is provided at the top to remove the diffused gas A. However, column “end effects” have been largely ignored or neglected during data analysis. The present study addressed the effect of interfacial curvature on diffusivity determination. Curvature affects the interfacial transport area and the diffusion path length of gas A. Cylindrical aluminum punches with flat or curved tips were used to imprint solid naphthalene surfaces. The naphthalene-containing tubes were subjected to standard Stefan column sublimation-diffusion experiments at ∼60 °C and atmospheric pressure. With the mass loss from each tube obtained gravimetrically and the punch/tube dimensions, the experimental diffusivities, D_AB,exp, were determined using the solution to a one-dimensional mass transport model for gas A. Tubes with flat interfaces had the smallest D_AB,exp errors relative to the literature, while the curved surfaces (spherical and elliptical) had the largest when assumed to be flat in the calculations. When curvature was accounted for mathematically, the D_AB,exp errors changed sign and magnitude depending on the actual punch shape and system dimensions. This is the first study to show that neglect of interfacial curvature can lead to significant errors in D_AB,exp obtained using the isothermal Stefan column method. Therefore, curvature corrections should be considered whenever accurate D_AB,exp’s are sought.

Keywords:

• Cylindrical punch with specific tip geometry
• Diffusion path length corrections
• End effects in an isothermal Stefan column;
• Experimental binary gas diffusivities
• Interfacial curvature and gas diffusivity determination
• Naphthalene/air sublimation-diffusion process
• One-dimensional gas transport modeling
• Simulated meniscus profile

Acknowledgements

The following personnel at the University of Puerto Rico, Mayagüez, are acknowledged for facilitating the completion of this study:

1. Prof. I. Y. Padilla, Director, and P. M. Torres, technician, Environmental Engineering Laboratory, Department of Civil Engineering and Surveying, for making available their experimental facilities;

2. Technician U. Almodóvar and students C. Hernández and R. P. Quiñones, Center for Advanced Aerospace Engineering and Manufacturing Technologies, Department of Mechanical Engineering, for preparing the cylindrical aluminum punches; and

3. The Interlibrary Loans Section of the General Library for tracking the archival references.

A special recognition is extended to University of Wisconsin Professors R. B. Bird, W. E. Stewart, and E. N. Lightfoot for being a source of academic inspiration and for their timeless Transport Phenomena text. In addition, student authors M. Moreno, I. Moreno, M. Jaime, and S. G. Maisonet received undergraduate research credit for carrying out and analyzing the naphthalene/air sublimation-diffusion experiments. Finally, the anonymous reviewers’ insightful comments and suggestions are acknowledged.

Nomenclature

Note: The letter codes assigned to the various naphthalene groups are given in the Theoretical Background section. The entries below include unsubscripted, subscripted, and superscripted letters.

Latin letters

1D, 3D	=	dimensionality of the transport system under consideration
a	=	aluminum punch dimension (Figure 3), m
A	=	species A
A0, …, A3	=	constants in the Ambrose et al. (1975) correlation for the vapor pressure of solid naphthalene
AR	=	gas phase height to column inside diameter ratio (initial value in the descending interface case; a constant in the stationary interface case), -
b	=	aluminum punch dimension (Figure 3), m
B	=	species B
c	=	molar density of an ideal gas mixture, mol/m3
d	=	inside diameter of the horizontal glass tubing during the sublimation-diffusion experiments (Figure 4), m
D	=	Fickian diffusivity of a specific gas in a binary mixture (subscripted), m2/s
error	=	DAB,exp deviation from a literature value calculated according to Eq. (20)(16a, b) $$3\mathrm{D}\mathrm{ }\text{differential}\mathrm{ }\text{interfacial}\mathrm{ }\text{area}\mathrm{ }=\text{ r x dθ dϕ }=\text{ r x }\frac{\text{dθ}}{\text{dx}}\mathrm{ }\text{dx}\text{ dϕ}$$(16a, b) , %
E1, E2, E3	=	dimensionless temperature functions in the Ambrose et al. (1975) correlation for the vapor pressure of solid naphthalene, -
$\mathcal{\ell }$	=	vertical distance from the top of the diffusion column to a specified location on the solid-gas interface, also known as the diffusion path length of gas A, m
m	=	mass (subscripted), g
MW	=	molecular mass of a pure gas or solid species (subscripted), g/mol
n	=	sample size of the naphthalene groups used in the sublimation-diffusion experiments, -
N	=	1D (axial) component of the total molar flux vector for gas A (includes convective and diffusive transport; subscripted), mol/(m2·s)
p	=	unsubscripted: absolute pressure of a gas mixture; subscripted: vapor pressure of a pure solid at a specified temperature; subscripted: critical pressure of a given species, atm (1 atm = 1.01325E + 05 N/m2)
r	=	unsubscripted: radial distance from the ellipse center (0, 0) to an arbitrary interfacial point (x, -y) in the fourth quadrant as illustrated in Figure 2; subscripted: naphthalene-containing glass tube radius, m
R	=	universal gas constant, atm·m3/(mol·K)
Re	=	dimensionless Reynolds number giving the ratio of inertial to viscous forces acting on a fluid at a specified location in a flow system, -
S	=	cross-sectional area of the naphthalene-containing tube used in the sublimation-diffusion experiments, m2
T	=	unsubscripted: temperature; subscripted: gas temperature at a specific axial location within the diffusion column; subscripted: upper and lower bounds of a temperature range, K; superscripted: dimensionless operating temperature required to calculate ΩD,AB, -
V	=	cross-sectional-average speed of the sweeping gas stream measured at the outlet of the horizontal glass tubing during the sublimation-diffusion experiments, m/s
x	=	horizontal coordinate or distance in Figure 2, m
y	=	unsubscripted: vertical coordinate or distance in Figure 2, m; subscripted: mole fraction of gas A at any axial location within the diffusion column, -
z	=	unsubscripted: axial coordinate along the diffusion column, where z = 0 corresponds to the bottom of the liquid A phase (Figure 1); subscripted: a specific location within the column such as the solid-gas interface or the top, m

Greek letters

α	=	clockwise-sweeping angle from the x-axis in Figure 2
Δt	=	elapsed time interval after starting a sublimation-diffusion experiment, s
ε	=	Lennard-Jones (12-6) potential parameter representing the maximum energy of attraction between two gas molecules (subscripted), J
ϕ	=	circumferential angle in Figure 2
κ	=	Boltzmann’s constant, J/K
ν	=	kinematic viscosity of atmospheric air (assumed to be a single, bone-dry species B) at the ambient dry-bulb temperature, m2/s
θ	=	zenith angle in Figure 2
σ	=	Lennard-Jones (12-6) potential parameter representing the characteristic molecular diameter of a pure gas or of a binary gas mixture (subscripted), Å
Ω	=	collision integral for diffusion of gases A and B (subscripted), -

Subscripts

1, 2	=	axial positions from the bottom of the column: 1, liquid-gas or solid-gas interface; 2, top
A	=	gas A
AB	=	A-B gas pair
AB,CE	=	pertaining to the Fickian binary gas diffusivity of A in B estimated using the Chapman-Enskog kinetic theory for low-density gases (Chapman and Cowling 1953; Bird et al. 2007)
AB,exp	=	pertaining to the Fickian binary gas diffusivity of A in B estimated with a 1D mass transport model for gas A using experimental data
AB,lit	=	pertaining to the Fickian binary gas diffusivity of A in B obtained from the literature
A,lost	=	total mass of solid A lost from each tube during the sublimation-diffusion experiment
A,lost,annular	=	mass of solid A lost from the annular portion of the interface during the sublimation-diffusion experiment (Figure 2)
A,lost,curved	=	mass of solid A lost from the curved portion of the interface during the sublimation-diffusion experiment (Figure 2)
Az	=	species A at axial position z
Az1	=	species A at axial position z1
Az2	=	species A at axial position z2
B	=	gas B
c	=	critical property of a species
D,AB	=	diffusion of gas A in B
max	=	maximum
min	=	minimum
tube	=	naphthalene-containing tube used in the sublimation-diffusion experiments
vap,A	=	vapor pressure of solid A at a specified temperature
z1	=	axial coordinate of the solid-gas interface

Superscripts

*

=

dimensionless absolute temperature