## 2 Considering Other Impulse Forces  
Generally speaking, in osmotic processes, the most significant impulse forces are the concentration potential and the convective potential as expressed by Equation (A1). Nonetheless, other kinds of forces could affect these processes. For example, the author [PERSON], in [16], presents an equation for the chemical potential of solutions with ionic flows where the electric field is considered an important impulse force. \[\mu_{i}=\mu_{i}^{0}+R\ T\text{ln}(a_{i})+\hat{v}_{i}(P)+F_{z}\ E \tag{1}\]  
where \(a\) is the activity of component \(i\), \(F_{z}\) is the Faraday constant  
The term that corresponds to kinetic energy has two components, a linear velocity \(v_{fi}\) that corresponds to the flow velocity produced by the applied pressure, and a component due to the angular velocity \(\omega_{i}\). The kinetic energy will be determined by the following equation:  
\[E_{k}=\left[\frac{m_{i}\left(\overrightarrow{v_{fi}}+\overrightarrow{\omega_{ i}}\times\overrightarrow{r}\right)^{2}}{2}\right] \tag{4}\]  
If Equation (4) is expanded and simplified using some properties of scalar and vector product, the kinetic energy can be expressed as follows:  
\[E_{k}=\left[\left(\frac{m_{i}\overrightarrow{v_{fi}}^{2}}{2}\right)+m_{i} \left(\overrightarrow{v_{fi}}\times\overrightarrow{\omega_{i}}\right) \overrightarrow{r_{i}}+\left(\frac{\overrightarrow{\omega_{i}}^{2}\times \left(m_{i}\overrightarrow{r_{i}}^{2}\right)}{2}\right)\right] \tag{5}\]  
In Equation (5), the left-side term corresponds to the kinetic energy due to the fluid velocity generated by the applied hydraulic pressure, the central term corresponds to the kinetic energy due to Coriolis acceleration produced by the cross-product between the velocity of the fluid and the angular velocity, and the right side term corresponds to the rotational kinetic energy that is determined by the half of the cross-product of the square of the angular velocity with the moment of inertia of the rotational body. According to what has been previously described, Equation (3) can be written as follows:  
\[ \begin{split}\mu_{i}=\left[\mu_{i}^{0}+R\;T\;ln\left(\frac{ \gamma_{i}\Omega_{i}}{\rho_{i}m_{i}}\right)\right]_{Ec}+\left[\vartheta_{i}( P_{i}-P_{sat})\right]_{Eff}+\left[\left(\frac{m_{i}\overrightarrow{v_{fi}}^{2}}{2} \right)+m_{i}(\overrightarrow{v}_{fi}\right.\\ \left.\times\overrightarrow{\omega_{i}}).\overrightarrow{r_{i}} +\left(\frac{\overrightarrow{\omega_{i}}^{2}\times\left(\overrightarrow{ l_{i}}\right)}{2}\right)\right]_{Ek}\end{split}  \tag{6}\]  
where \(\overrightarrow{l_{i}}=m_{i}.\overrightarrow{r_{i}}^{2}\) is the total moment of inertia. In the case of the system of Figure 1, the rotational axis does not go through the center of mass of the PRO module, and according to the Huygens-Steiner theorem, also known as the parallel axis theorem, we can describe the moment of inertia of the system as \(I_{pa}=I_{cm}+m\;r^{2}\), where \(I_{cm}\) is the moment of inertia of an object when the rotational axis goes through the center of mass, plus the product of the total mass by the square of the perpendicular distance between the parallel axis. Considering the  
Figure 1: Schematic of a PRO system with an angular velocity. Considering Figure A1 again, we can propose an energy difference between the chemical potential of the bulk draw solution (\(\mu_{Db}\)) and of the area near the membrane (\(\mu_{Dm}\)). Even though there can be differences between the velocities, concentrations, or pressures between the bulk of the solutions and the area near the membrane, by being a lossless closed system, the internal energies should remain equal. We propose the equation \(\mu_{Db}=\mu_{Dm}\). \[