# Mathematical Modeling of Screw Press Configuration for Processing Safflower Oil
## 3 Results and Discussion
### Mathematical Modeling of the Separation and Output Dependence during the Pressing Process  
For the calculation, it is necessary, first of all, to replace the existing press with its calculation (model) scheme. All features of the existing press that are not essential must be excluded from the model scheme, and only those that are principal to the calculation of the process and the unit must be retained, giving a comparatively simple model of the press machine. Figure 5: Safflower processing products: (**a**) safflower oil; (**b**) oilcake. Considering the motion of a viscous medium, the Navier-Stokes equation [20] is written as (Equation (5)):  
\[ \begin{array}{l}\frac{\partial v_{x}}{\partial x}+v_{x}\frac{\partial v_{x}}{ \partial x}+v_{y}\frac{\partial v_{x}}{\partial y}+v_{z}\frac{\partial v_{x}}{ \partial z}=F_{x}-\frac{1}{\rho}\times\frac{\partial p}{\partial x}+v\left( \frac{\partial^{2}v_{x}}{\partial x^{2}}+\frac{\partial^{2}v_{x}}{\partial y^{2 }}+\frac{\partial^{2}v_{y}}{\partial z^{2}}\right),\\ \frac{\partial v_{y}}{\partial x}+v_{x}\frac{\partial v_{y}}{\partial x}+v_{y} \frac{\partial v_{y}}{\partial y}+v_{z}\frac{\partial v_{y}}{\partial z}=F_{y}- \frac{1}{\rho}\times\frac{\partial p}{\partial y}+v\left(\frac{\partial^{2}v _{x}}{\partial x^{2}}+\frac{\partial^{2}v_{y}}{\partial y^{2}}+\frac{\partial ^{2}v_{y}}{\partial z^{2}}\right),\\ \frac{\partial v_{z}}{\partial t}+v_{x}\frac{\partial v_{z}}{\partial x}+v_{y} \frac{\partial v_{z}}{\partial y}+v_{z}\frac{\partial v_{z}}{\partial z}=F_{z} -\frac{1}{\rho}\times\frac{\partial p}{\partial x}+v\left(\frac{\partial^{2} v_{z}}{\partial x^{2}}+\frac{\partial^{2}v_{z}}{\partial y^{2}}+\frac{\partial ^{2}v_{z}}{\partial z^{2}}\right),\end{array}  \tag{5}\]  
where \(v\) is the kinematic viscosity, m\({}^{2}\)/s; and \(F_{x}\),\(F_{y}\), and \(F_{z}\) are the mass force projections on the co-ordinate axes. To solve the problem of continuum mechanics, the Poisson equation [21] can be written as follows (Equation (6)):  
\[\frac{\partial^{2}v_{Z}}{\partial x^{2}}+\frac{\partial^{2}v_{Z}}{\partial y^{ 2}}=-\frac{\Delta p}{\eta_{\ast\phi}l}, \tag{6}\]  
where \(\eta_{\ast\phi}\) is the effective viscosity, Pa-s; \(\Delta p\) is the pressure difference, Pa; and \(l\) is the channel length, m.  
To determine the volumetric capacity, let us calculate the double integral:  
\[Q=\int\limits_{0}^{H}\int\limits_{0}^{W}\theta_{z}dxdy,\]  
where \(Q\) is the press capacity, m\({}^{3}\)/s; and \(v\) is the velocity along the channel axis, m/s. Using [PERSON]'s equation, let us determine the productivity of the screw channel at idle running without the impact of an extraneous mechanism, taking into account the geometric features of the screw channel (Figure 6, Equation (7)):  
\[Q_{\rm IIK}=\frac{K_{\rm IIK}}{\eta_{\ast\phi}}\Delta p, \tag{7}\]  
where \(K_{\rm IIK}\) is the pressing channel coefficient; \(\Delta\rho\) is the pressure drop in the matrix forming device, Pa; and \(\eta_{\ast\phi}\) is the effective viscosity, Pa-s.  
Figure 6: Dependence of pressing channel coefficient (**a**) on capacity and pressure at free running of the pressing screw (**b**). This case in the presence of \(\Delta p\) has already been considered in the calculation of the pressing channel. Solving Equations (1) and (2) together, we obtain the following solution (Equation (8)):  
\[ \begin{split} v_{Z}=\frac{4\times v}{\pi}\sum\limits_{m=1,3,5}^{ \infty}\frac{1}{m}\times\frac{5\text{H}(\frac{\pi mw}{w})}{\text{H}(\frac{\pi mw }{w})}\times\sin(\frac{\pi mw}{w})-\frac{1}{\eta}\times\frac{dp}{dz}\times\\ \times\left\{\frac{v^{2}}{2}-\frac{vH}{2}+\frac{4\times H^{2}}{ \pi^{3}}\times\sum\limits_{m=1,3,5}^{\infty}\frac{1}{m^{3}}\times\frac{c\text {H}\left[\frac{\pi mw}{2H}\right]}{c\text{H}(\frac{\pi mw}{2})}\times\sin( \frac{\pi mw}{H})\right\}\end{split}