Research Article Detailed Mathematical Analysis of Solute Transport in Armfield’s Liquid Diffusion Apparatus A solute’s diffusivity is a key property in the design and analysis of mass transfer systems. A simple method to determine such coefficients in aqueous media is Armfield’s diffusion apparatus, in which reservoir (donor) and bath (receiver) compartments are separated by a honeycomb array of liquid-filled cylindrical pores. The solute of interest, i.e., NaCl, diffuses from the reservoir and pores into the bath. The electrical conductivity of the bath solution, which is proportional to the NaCl molarity, is tracked as a function of time at constant temperature. A comprehensive mathematical model is presented which combines simultaneous solute transport in the three compartments, allowing estimation of the aqueous NaCl diffusivity from experimental data. The model improves existing theoretical analyses of such data and can be adapted to study other challenging systems. Keywords: Diffusivity, Mass transfer system, Solute transport Received: April 20, 2013; revised: September 04, 2013; accepted: October 29, 2013 DOI: 10.1002/ceat.201300261 1 Introduction Solute transport is an important subject for chemical engineers since diffusion and convection affect the performance of most chemical, biological, and environmental systems. This report addresses solute diffusion in liquids, specifically NaCl in water. This study was motivated by the author’s recent experience in a laboratory where students carry out mass transfer experi- ments leading to estimates of a solute’s diffusion coefficient D. Many ingenious methods have been devised to determine D in various media: diaphragm cells, particle uptake by thin films, Taylor dispersion, and interferometry, to name a few [1]. The method employed in the Unit Operations Laboratory, University of Puerto Rico, uses Armfield’s liquid diffusion ap- paratus (product code CERB) operated along the lines of the diaphragm cells. The CERB, available since 1982, is depicted in Figs. 1 a and 1 b. It consists of a glass U-shaped reservoir com- partment containing the donor NaCl solution, a polyvinyl chloride disc with a drilled honeycomb array of parallel cylin- drical pores separating the external compartments, and a cylin- drical acrylic housing or bath compartment acting as the NaCl receiver. Experimentally, the reservoir and pores are initially filled with an aqueous NaCl solution (range: 0.2–3.0 M; solu- bility: 5.4 M at 20 °C [2]), and the U-tube positioned vertically in the bath solution, leaving the pore openings just under the liquid surface. A dip-in conductivity electrode immersed in the bath solution is used to track its electrical conductivity (proportional to NaCl molarity) with time. The most common approach for analyzing diaphragm cell data is to write solute mass balances for one or more compart- ments, and to simplify the equations through assumptions such as: (i) perfectly mixed reservoir and bath solutions; (ii) neg- ligible volume available to the solute within the diaphragm; and (iii) pseudo-steady-state conditions in the diaphragm throughout the experiment [3–7]. The last assumption implies the instantaneous establishment of a linear concentration pro- file across the diaphragm, leading to simple algebraic expres- sions for estimating D from concentration-time data collected from the external compartments. Experimentally, perfect mix- ing may be attained by stirring at increasing intensities, in which case vorticity effects may become important, while the diaphragm’s relative volume may be adjusted by judicious selection of its porosity and overall system dimensions. On the other hand, the applicability of pseudo-steady-state conditions in the diaphragm may be assessed from the solute’s dimen- sionless diffusion time s given by: s  tD L 2 (1) where t represents the experimental diffusion time, L the dia- phragm thickness or diffusion path length, and D an effective Chem. Eng. Technol. 2014, 37, No. 1, 49–58 © 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.cet-journal.com Carlos A. Ramírez University of Puerto Rico, Department of Chemical Engineering, Mayagüez, Puerto Rico. – Correspondence: Prof. Carlos A. Ramírez (carlos.ramirez9@upr.edu), University of Puerto Rico, Department of Chemical Engineering, Mayagüez, Puerto Rico 00681. Mass transfer system 49