###### Abstract  
A liquid-phase adsorption technology has been widely applied to address environmental problems related to the removal of pollutants from aqueous streams. Simple and effective methods for determining mass transfer parameters, including intra-particle and fluid-to-solid film resistances, are crucial for designing adsorption processes. The efficient simplified diffusion technique (ES technique), based on a completely mixed batch reactor (CMBR), is proposed in this study to address these needs. In this study, we compare three diffusivity (Ds) determination methods: the rigorous diffusion technique (R technique), the simplified diffusion technique (S technique), and the ES technique. Although the simulation results from the R technique are excellent, it is a very complicated and time-consuming approach that is not convenient for practical use. The S technique provides a much simpler approach, but its results are only valid in cases where the contribution of fluid film resistance is negligible (Biot number > 40). The ES technique proposed in this study can overcome those limitations. The estimation errors of the ES technique are significantly smaller than that of the S technique when compared with the R technique. The proposed ES technique would be very useful for field applications to determine diffusivity for aqueous adsorption systems. adsorption; diffusivity; CMBR; efficient simplified diffusion technique +
Footnote †: journal: Applied Physics Letters
Another approach to reduce the fluid-to-solid film resistance is using the differential reactor (shallow bed) technique [10; 11]. When intra-particle diffusion is controlled by surface diffusion, concentration decay curves can be estimated using an analytical solution. The intra-particle diffusivity can be determined by comparing the experimental data and theoretical concentration decay curve [12]. However, when the fluid-to-solid film resistance cannot be neglected in the experiment, the separation of intra-particle resistance and fluid-to-solid film resistance would be required to obtain accurate diffusivity values [8; 13; 14], and the differential reactor technique may not be valid. In addition, one of the main drawbacks of that technique is that a large volume of fluid is required for the test. The completely mixed batch reactor (CMBR) method provides another choice for estimating the diffusivity in liquid systems. The test procedure is relatively simple and only requires a few liters of liquid, providing a much more environmentally friendly approach relative to the long column or differential reactor methods [2]. Unfortunately, the fluid-to-solid film mass transfer resistances cannot be ignored in these experiments, and an empirical formula for estimating the value of that resistance is not available. For example, [PERSON] and [PERSON] (1974) investigated the adsorption rates of 2-dodecylbenzene sulfonate on four different activated carbons and proposed a simple technique based on a Linear Driving Force Diffusion (L.D.F.) Model [15]. Unfortunately, the simplicity of the model and computation technique applied in that study led to significant errors in their estimates of diffusivity. Therefore, a better diffusion model is desirable. For example, an intraparticle diffusion method has been proposed by [PERSON] et al. (1996) for the investigation of two phenolic compounds (p-chlorophenol (PCP) and p-nitrophenol (PNP)) onto granular activated carbon (GAC-CAL) [16]. One of the main limitations of that method is that it does not consider the resistance within the fluid-to-solid film in the CMBR, but that resistance can be important for CMBR processes. A method that considers both intraparticle diffusion and fluid-to-solid film resistances was proposed by [PERSON] et al. (2010), where the intraparticle diffusivity was determined by matching the numerical concentration decay curves with the experimentally measured values [2]. Although that simulation result performs fairly well, the method is very complicated and time-consuming.