## 2 Considering Other Impulse Forces
#### 2.2.1 Feed Solution Energy Balance
After simplifying the expression, we obtain the following:  
\[J_{w}=\frac{D_{w}\ C_{Db}\ C_{Db}\ C_{D}}{l\ R\ T}\bigg{(}\overset{\rightarrow}{ \
u}_{F}\Delta\Pi-\overset{\rightarrow}{\
u}_{F}(P_{Db}-P_{Fb})-\frac{1}{2} \Big{[}\omega_{Db}^{2}\times I_{Db}-\omega_{Fb}^{2}\times I_{Fb}-2 Cor_{Fb} \Big{]}\bigg{)} \tag{20}\]  
Now, substituting Equation (7) in Equation (20) and writing the molar mass \(m_{i}\) in terms of the molar density \(\rho_{i}\) and the molar volume \(\
u^{*}_{i}\),  
\[ \begin{array}{l}J_{w}=-\frac{D_{w}\ C_{Db}\ C_{Db}\ C_{Db}}{I\ R\ T}(\vartheta _{F}\ \Delta\Pi-\vartheta_{F}(P_{Db}-P_{Fb})\\ -\vartheta_{F}\frac{1}{2}\bigg{[}\omega_{Db}^{2}\times\rho_{Db}\ \bigg{(}0. \overset{\rightarrow}{\
u}_{F}^{2}\overset{\rightarrow}{\
u}_{,Db}+ \overset{\rightarrow}{\
u}_{M,Db}\bigg{)}\\ -\omega_{Fb}^{2}\times\rho_{Fb}\bigg{(}0.\overset{\rightarrow}{\
u}_{,Fb} \overset{\rightarrow}{\
u}_{,Fb}+\overset{\rightarrow}{\
u}_{,M,Fb}\bigg{)} \bigg{]}\\ +\vartheta_{F}\Big{[}\rho_{Fb}\overset{\rightarrow}{\
u}_{,Fb}\times \overset{\rightarrow}{\omega}_{Fb}.\overset{\rightarrow}{\tau}_{,Mrb}\Big{]} \end{array}  \tag{21}\]  
Finally, considering that the difference between the flow velocity, angular velocity, and the radius of the bulk of the feed and draw solutions is very small, Equation (21) can be written in a compact form as  
\[J_{w}=A\ (\Delta\Pi-\Delta P-\Delta P_{RKE}) \tag{22}\]  
where  
\[ \begin{array}{l}A=-\frac{D_{w}C_{Db}C_{Db}\ \delta_{F}}{IRT}\\ \Delta\Pi=iRT(C_{Db}-C_{Fb})\\ \Delta P=(P_{Db}-P_{Fb})\\ \Delta P_{RKE}=0.5\bigg{(}\omega_{Db}^{2}\times\bigg{(}0.5\overset{ \rightarrow}{\tau}_{,Db}^{2}+\overset{\rightarrow}{\tau}_{,MDb}^{2}\bigg{)} \bigg{)}(\rho_{Db}-\rho_{Fb})-\Big{(}\rho_{Fb}\overset{\rightarrow}{\
u}_{,Fb }\times\overset{\rightarrow}{\omega}_{Fb}.\overset{\rightarrow}{\tau}_{,MFb} \bigg{)}\end{array} \]  
Equation (22) describes a process where diffusion through the semipermeable membrane is driven by the potential differences of three types of energy: chemical energy, represented by the osmotic potential \(\Delta\Pi\); convective energy, arising from the applied hydraulic pressure \(\Delta P\); and rotational kinetic energy, represented by the dynamic pressure \(\Delta P_{RKE}\). The term \(\Delta P_{RKE}\) indicates that the equivalent hydraulic pressure exerted on the membrane by rotational kinetic energy is determined by the angular velocity \(\omega\), the total radius that will be simplified as \(r_{M}\), the fluid velocity of the draw solution \(v_{-\overset{\rightarrow}{f}}\), and the densities of the feed and draw solutions, which depend on their concentrations. While the channel heights for the feed and draw solutions could also influence the resulting pressure, this effect is beyond the scope of this study. Depending on the directions of vectors \(\overset{\rightarrow}{\omega}\), \(\overset{\rightarrow}{\
u}\), and \(\overset{\rightarrow}{r}\), within the spiral membrane module, the resultant dynamic pressure \(\Delta_{RKE}\) can either support or oppose the osmotic pressure \(\Delta\Pi\). However, the local hydrodynamic effects on the membrane surface can help mitigate the negative impacts of concentration polarization CP or membrane fouling in real time. These overall positive effects suggest that further research into this technology is well justified. Although some studies have explored the use of rotational kinetic energy to modify flow through membranes in osmotic processes like RO, to the best of the authors' knowledge, this is the first analytical model to explicitly incorporate rotational kinetic energy in PRO.