## 2 Considering Other Impulse Forces
#### 2.2.1 Feed Solution Energy Balance  
For the feed solution, it is necessary to consider that the main component of the fluid's velocity \(v_{fF}\), due to the spiral-wound module geometry, which varies its direction inside the \(ZY\)-plane (\(0,j,k\)) parallel to the membrane surface. Even though components of this velocity could affect diffusion through the membrane, as the feed solution flow velocity is usually small, this contribution will be neglected. Now, concerning the rotational motion, just as in the draw solution, the vector product of the square of the angular velocity and the radius will be on the \(Z\) axis and will contribute directly to diffusion through the membrane. The Coriolis acceleration results of the vector product of the flow velocity \(v_{fF}\) (\(0,j,k\)) and angular velocity \(\omega_{F}\) (\(i,\,0,\,0\)), because of this, will alternate between zero and a maximum value in the \(Y\) or \(Z\) axis, respectively, and will contribute to diffusion through the membrane. In the vector product for Coriolis energy, the angular velocity \(\omega\) appears in the second position, unlike in the vector product for centrifugal energy, where it occupies the first position. As a result, when \(\overset{\rightarrow}{v}_{fF}\) is aligned with \(\overset{\rightarrow}{r}_{M}\), the Coriolis energy will have the opposite direction to the centrifugal energy. For ease in handling the equations, we will call the Coriolis energy as follows:  
\[Cor_{i}=m_{i}\Bigl{(}\overset{\rightarrow}{v}_{fi}\times\overset{\rightarrow }{\omega}_{i}\Bigr{)}.\overset{\rightarrow}{r}_{i}\]  
With these considerations, the energy balance for the Feed solution will be \(\mu_{Fm}=\mu_{Fb}\). \[ \begin{split}\mu_{F}^{0}+R\ T\ln\Bigl{(}\frac{\gamma_{Em}}{\rho_ {Fm}}\frac{C_{Fm}}{m_{Fm}}\Bigr{)}+\hat{v}_{Fm}(P_{Fm}-P_{sat,Fm})+\left( \overset{\overset{\omega_{Dm}^{2}}{\omega_{Db}\times\overset{\rightarrow}{I} _{Em}}}{2}+Cor_{Fm}\right)=\mu_{F}^{0}+\\ R\ T\ln\Bigl{(}\frac{\gamma_{Eb}}{\rho_{Fb}}\frac{C_{Fb}}{m_{Pb}} \Bigr{)}+\hat{v}_{Fb}(P_{Fb}-P_{sat,Fb})+\left(\overset{\overset{\omega_{Db} ^{2}}{\omega_{Db}\times\overset{\rightarrow}{I}_{Eb}}}{2}+Cor_{Fb}\right) \end{split}  \tag{12}\]In this case, additionally, from the considerations related to the volumes and the incompressibility of the solution, it is considered that \(P_{Fm}=P_{Db}\), and after simplifying Equation (12), we arrive at the following equation:  
\[C_{Fm}=C_{Fb}\ G_{F}\ e^{[\frac{1}{2 RT}\cdot(\omega_{p_{D}}^{2}\times I_{Fb}+2 Cor_{Fb}-\omega_{p_{m}}^{2}\times I_{Fm}-2 Cor_{Fm})-\frac{\overset{\gamma}{ \
u}F}{RT}(P_{Db}-P_{Fb})]} \tag{13}\]  
where  
\[G_{F}=\frac{\gamma_{Fb}\ \rho_{Fm}}{\gamma_{Fm}\ \rho_{Fb}}\]  
If we consider a semipermeable membrane, as shown in Figure 11, and assume a solvent flux primarily driven by diffusion, applying [PERSON]'s law leads to the following expression:  
\[J_{w}=\frac{D_{w}}{l}(C_{Dm}-C_{Fm}) \tag{14}\]  
When we replace Equations (11) and (13) in Equation (14),  
\[\begin{array}{l}J_{w}\\ =\frac{D_{m}}{l}\left(C_{Db}\ G_{D}\ e^{[\frac{1}{2 RT  
If, in Equation (18), we consider the exponential function definition (\(\text{e}^{x}\approx 1+\text{x}\) for \(-1<\text{x}<1\)),  
\[ \begin{array}{l}J_{w}=\frac{D_{w}\ C_{Db}\ C_{Db}\ C_{Db}}{IRT}\Big{(}\Big{[}1 +\frac{1}{2}\big{(}\omega_{Db}^{2}\times I_{Db}-\omega_{Dm}^{2}\times I_{Dm} \big{]}\Big{)}-\Big{[}1+\frac{1}{2}\big{(}\omega_{Fb}^{2}\times I_{Fb}+\\ 2 Cor_{Fb}-\omega_{Fm}^{2}\times I_{Fm}-2 Cor_{Fm}\big{)}-\overset{\rightarrow}{ \
u}_{F}(P_{Db}-P_{Fb})+\overset{\rightarrow}{\
u}_{F}\Delta\Pi\Big{]}\Big{)} \end{array}  \tag{19}\]  
In Equation (19), we must consider that, due to the flux through the membrane, the kinetic energy in the area near the membrane will be equal for the feed and draw solutions. Due to this, \(\omega^{2}\times I_{Dm}=\omega^{2}\times I_{Fm}+2\ Cor_{Fm}\).