## 1 Introduction  
Power generation using pressure-retarded osmosis (PRO) has garnered significant due to its potential for sustainable and continuous electricity generation. However, several technical challenges related to its practical implementation remain unresolved [1]. Among these challenges is internal concentration polarization (ICP), which is one of the most critical issues that need to be addressed [2]. ICP is a phenomenon that depends on the feed and draw solution concentrations as well as on the structural parameters of the membrane. Its effects could be minimized from the design of the membrane itself or by maintaining operating conditions that help mitigate its effects [3]. In PRO, as well as in various other membrane processes, a high shear rate induced by turbulence near the membrane surface is desirable. This turbulence enhances water flux flow through the membrane and mitigates the effects of concentration polarization (CP) and reduces membrane fouling [4, 5, 6, 7, 8]. The use of high-velocity flow rates generates this turbulence, but the required working pressures of PRO and the use of high-velocity flow rates requires more energy from the pumps, diminishing the net energy output. There is some research on other membrane processes that have employed rotational kinetic energy to produce high shear rates in the area near the membrane. The work of [PERSON] et al. in [9] defines the fundamental principles and implements two prototypes of centrifugal reverse osmosis (CRO). [PERSON] et al., in [10], studies the favorable effects of centrifugal and Coriolis acceleration for a centrifugal membrane separation (CMS) process are evaluated with a numerical model. [PERSON] et al., in [11], experiments to evaluate the use of CMS to enhance water flux in rotating reverse osmosis (RO) are presented. [PERSON] and [PERSON], in [12], present a dynamic model for rotating reverse osmosis with the Navier-Stokes equation for cylindrical Taylor-Couette flow. The same authors, in [13], evaluate the control of scale formation in rotating reverse osmosis. More recently, [PERSON] and [PERSON], in [14], present a CRO process that can reduce the specific energy consumption relative to a single-stage RO process by progressively adjusting the trans-membrane pressure with a set radius and a variable speed. Some studies have explored the technical challenges and energy consumption associated with applying rotational kinetic energy to membrane processes. Experimental, mathematical, and numerical evidence suggests that rotational kinetic energy can help mitigate the negative effects of CP and membrane fouling in these processes. Furthermore, these benefits can be achieved with lower energy consumption. Due to the similarities between membrane-based processes, many of the technologies developed for RO have been adapted for use in PRO. It could be of interest to evaluate the effects of rotational kinetic energy applied to PRO. A literature review focused on the mathematical models used specifically for PRO revealed that three fundamental theories have been used to develop models: the solution-diffusion model, irreversible thermodynamics, and mass interchange (See Table 1). Among the three fundamental theories, the solution-diffusion model is the most used for PRO. In broad terms, permeation through a membrane is defined by driving forces like pressure, temperature, and concentration. These forces are interrelated between them, and the resulting driving force is the gradient of the chemical potential \(\mu\)[15]. Regarding the process for obtaining the solution-diffusion model, it is highlighted that to calculate the chemical potential \(\mu\), usually, only the concentration, temperature, and pressure gradients are considered. Let us consider that other driving forces, like electromagnetic fields, gravity, or inertia, are present and may directly or indirectly influence the distribution of species and thus the chemical potential of solutions. With this, it is possible to develop new expressions to explain osmotic processes.