## 3 Results and Discussion
### Physical Modeling of the Process of Centrifugation of Meat-Bone Cracklings
#### 3.2.1 Derivation of the 1 st Criterion Equation  
The main task of optimization was to obtain the maximum productivity in terms of meat-bone cracklings yield (\(G\)) at the optimal speed (\(n\)) of the centrifuge rotor. The method of dimensionality analysis was applied [27]. Consider the desired productivity \(G\) as a function of a number of arguments--physical parameters. \[G=f\left(\mu,\,\rho,\,n,\,D\right) \tag{3}\]  
where  
\(G\)--specific mass productivity of the centrifuge for meat-bone cracklings, kg/s m\({}^{2}\). \(\mu\)--dynamic viscosity coefficient of liquid fat raw material, Pa-s.  
\(\rho\)--density of fat, kg/m\({}^{3}\). \(n\)--centrifuge drum speed, s\({}^{-1}\). \(D\)--inner diameter of the centrifuge drum, m.  
According to the second similarity theorem [28], the functional dependence of physical processes always has the form of an exponential equation. Here:  
\[G=B\cdot\mu^{x}\cdot\rho^{y}\cdot\mu^{z}\cdot D^{q} \tag{4}\]  
\(B\)--numerical coefficient;  
\(x,y,z,q\)--degree indices. We have five variables. Therefore, according to the \(\pi\)-theorem, the number of criteria will be determined:  
\[\Theta=5-3=2.\]  
Make a dimensional base of the quantities in Equation (4):  
\([G]=\) kg/s\(\cdot\)m\({}^{2}=\) kg\(\cdot\)s\({}^{-1}\cdot\)m\({}^{2}\)  
\([\mu]=\) Pa\(\cdot\)s = N\(\cdot\)s/m\({}^{2}=\) kg\(\cdot\)m\(\cdot\)s/s\({}^{2}\cdot\)m\({}^{2}=\) kg/s\({}^{-1}\) m\({}^{-1}\)  
\([\rho]=\) kg/m\({}^{3}=\) kg\(\cdot\)m\({}^{-3}\)  
\([n]=\) s\({}^{-1}\)  
\([D]=\) m  
Substitute the dimensions of the quantities into Equation (4):  
\[\text{kg}\cdot\text{s}^{-1}\cdot\text{m}^{2}=\text{B}\cdot(\text{kg}/\text{s }^{-1}\cdot\text{m}^{-1})^{x}\;(\text{kg}\cdot\text{m}^{-3})^{y}\cdot(\text{ s}^{-1})^{z}\cdot(\text{m})^{q} \tag{5}\]  
Convert (5):  
\[\text{kg}\cdot\text{s}^{-1}\cdot\text{m}^{-2}=\text{B}\cdot(\text{kg})^{x+y }\cdot(\text{s})^{-x-z}\cdot(\text{m})^{-x-3y+q}\]  
\(k\)g: \(1=x+y\)  
\(s\): \(-1=-x-z\)  
\(m\): \(-2=-x-3y+q\)  
Express all unknown indices through \"\(x\)\". \(y=1-x\)  
\(z=1-x\)\[-2=-x-3y+q\] \[q=x+3y-2=x+3\cdot(1-x)-2=x+3-3x-2=1-2x.\]  
Substitute the indices into Equation (4). We obtain:  
\[G_{\text{un}}=\text{B}\cdot\mu^{x}.\frac{\rho}{\rho^{x}}.\frac{n}{n^{x}}.\frac{D }{D^{2x}}\]  
We obtain Equation (6):  
\[G=\text{B}\cdot\left(\frac{\mu}{\rho\cdot n\cdot D^{2}}\right)^{x}\cdot\rho \cdot n\cdot D \tag{6}\]  
Check the dimensionality of the individual complexes included in Equation (6). \[\left(\frac{\mu}{\rho\cdot n\cdot D^{2}}\right)=\frac{\text{Pa}\cdot\text{s}}{ \frac{\text{kg}\cdot\text{1}\cdot\text{m}^{2}}{\text{m}^{3}\cdot\text{s}}}= \frac{\frac{\text{N}\cdot\text{s}}{\text{m}^{2}}}{\text{kg}\cdot\text{m}\cdot \text{s}}=\frac{\frac{\text{kg}\cdot\text{m}}{\text{kg}}}{\text{m}\cdot\text{s }}=1\]  
Consequently, the inverse of the modified Reynolds criterion for centrifugation \(Re_{\text{\emph{ef}}}\) is obtained:  
\[G=B\cdot\mu^{x}\cdot\rho^{y}\cdot\mu^{z}\cdot D^{q} \tag{4}\]0  
\[G=B\cdot\mu^{x}\cdot\rho^{y}\cdot\mu^{z}\cdot D^{q} \tag{4}\]1  
\(Re_{\text{\emph{ef}}}\) characterizes the flow regime of the liquid fluid at centrifugation along with the Froude criterion \(Fr=n^{2}\cdot D/1800\). \[G=B\cdot\mu^{x}\cdot\rho^{y}\cdot\mu^{z}\cdot D^{q} \tag{4}\]2  
The left-hand side of Equation (6) is divided by the complex \(\rho\cdot n\cdot D\):  
\[G=B\cdot\mu^{x}\cdot\rho^{y}\cdot\mu^{z}\cdot D^{q} \tag{4}\]3  
The obtained fractional expression (8) is a dimensionless complex--the criterion of productivity \(P\) (in this case by squash). Here, the value \(\rho\cdot n\cdot D\), the inverse of the specific productivity, shows the resistance \(R_{p}\) to the value of productivity, which is proportional to the density of the medium (\(\rho\)), the drum speed (\(n\)), and the determining geometric dimension--the inner diameter of the drum (\(D\)) (Table 1).