## 1 Introduction  
Membrane contactors are widely used for separating and recovering pollutant species, especially in trace amounts [1; 2; 3]. In recent years, polymer inclusion membranes (PIMs) have attracted a lot of interest in a wide range of separation process, and they have grown in popularity, especially for small organic compounds [4] and metal ion separation from aqueous solutions [5; 6], such as chromium [7], aluminum [8], Au(III) [9], copper (II) [10], zinc [11], cadmium [12], cobalt [13], and lead [14]. Moreover, PIMs have reached the agricultural sector for the extraction of pesticides that, due to their toxicity, pose a great risk to human health [15]. The first use of PIMs has been suggested more than 50 years ago and considered both liquid and dense membranes [16; 17; 18]. In the literature, in fact, scientists classified them in the category of liquid membranes while others hesitate between liquid and solid. PIMs are essentially composed of a polymer dissolved in a suitable solvent; the carrier (liquid phase) is added and physically trapped in the membrane matrix providing significant stability compared to other liquid membranes [19]. Most PIMs contain a plasticizer or modifier for an appropriate separation or specific application. These membranes became attractive in separation processes and presented an alternative technique for solvent extraction. They are considered environmentally friendly and economic because of the simple procedure for their preparation and the very low amount of solvent and extractant used [18; 20; 21; 22; 23; 24]. In general, polyvinyl chloride (PVC) and cellulose triacetate (CTA) were used as basic materials [25; 26; 27; 28; 29; 30]. The removal of chromium(VI) from industrial effluents and contaminated waters is very important because of its high toxicity and impact on both human health and the environment. Chromium (VI) extraction has been widely investigated using PIM, and the most used devices have been double compartment or microdevices [18; 25; 31; 32]. All systems investigated by researchers have been double cell systems or microsystem channels. [PERSON] et al. [33] reported the extraction of Cr(VI) by polymeric inclusion membranes based on CTA and Aliquat-336; a high permeability combined with stability, comparable to supported liquid membrane, was observed. Similar results were obtained by [PERSON] et al. [34] in their comparative study between supported liquid membranes SLM and PIMs using CTA and Aliquat-336 as carriers. [PERSON] et al. [35] reported a high stability of PIMs based on CTA/Aliquat-336 for Cr(VI) transport in a double cell system, with a constant permeability coefficient after 12 cycles of use. A comprehensive review on PIM applications and the systems used is reported by [PERSON] [18].  
In the present work, for the first time, a multiframe flat sheet membrane module was designed and used for the extraction of Cr(VI) containing a series of chambers alternating feed and receiving phases. With respect to the systems reported in the literature, the multiframe module is an example of a compact design that could be adopted to scale up the technique. To optimize the efficiency of the system, design of experiment (DOE) has been used to evaluate all parameters of the system and their interactions. In particular, for the first time, DOE was used for investigating parameters of polymeric inclusion membranes in a multiframe plates module. This methodology allows to reduce the number of experiments, which avoids product waste and loss of time [36; 37; 38].