## 2 Materials and Methods
### Methods
#### 2.2.2 Theoretical Equations  
Threshing Force  
Separating seeds from, for example, pods, spikes or heads requires a detachment force F, which can be expressed as a function of the cylinder's linear speed, the friction coefficient between crop parts and the friction coefficient between the crops and the thresher parts [24]:  
\[F=F_{c}+F_{r} \tag{1}\]  
where \(F\) is the total threshing force (N), \(F_{c}\) is the cylinder's impact force (N) and \(F_{r}\) represents the friction forces.  
Figure 3: (**a**) The loop threshing bar and (**b**) a threshing tooth.  
Figure 2: Designed model (**a**) and prototype (**b**) of the threshing roller and the screw conveyor, showing the loop threshing bar (1), the screw conveyor (2) and the main threshing shaft (3).  
The cylinder impact force can be calculated as a function of the feeding rate and the crop linear speed inside the threshing unit with the following equation [8]:  
\[F_{c}=q(v_{2}-v_{1}) \tag{2}\]  
where \(q\) is the crop feeding rate in kg/s, \(v_{2}\) is the crop speed at the end of the cylinder in m/s and \(v_{1}\) is the crop speed as it enters the cylinder in m/s.  
Usually, \(v_{2}\) depends upon the cylinder's linear speed, and it can be predicted from the following equation:  
\[v_{2}=a\ v \tag{3}\]  
where \(a\) is an empirical figure related to the cylinder length, straw moisture content, threshing bar shape, feeding rate and physical properties of the thresher. For a thresher 0.8 m in length with a moisture content of 15-25% and a feeding rate of 12,600 kg/h, the coefficient \(a\) was determined equal to 0.70-0.85 [25], and \(v\) is the linear speed of the cylinder:  
\[F_{r}=fF \tag{4}\]  
where \(f\) is a coefficient for the rasp-bar thresher type, which is equal to (0.65-0.75) [8], and \(F\) is the force needed for threshing crops (i.e., the total threshing force) [26].  
By inserting Equations (2)-(4) into Equation (1), the resulting equation is as follows:  
\[F=q\ (a\ v-v_{1})+fF \tag{5}\]  
The required threshing force to overcome the seed's attachment force can be calculated from Equation (6). According to this equation, the diameter of the threshing roller and the critical roller rotational speed are estimated, as well as the range of the feeding rate. It was taken into account that the directed force of the seeds should not exceed 8.5 N (the lowest average breaking force for under-studying seed at the same moisture content, which was determined by using a TMS-PRO texture analyzer device (Spectronic CamSpec Ltd., TMS-PRO, Chicago, IL, USA)):  
\[\text{F}=\left[\frac{q(av-v_{1})}{1-f}\right] \tag{6}\]  
The following equation is used to determine the suitable diameter of the steel shafts with almost no axial load [26], and it was designed with a safety factor of 3.81:  
\[d^{3}=\frac{16\ n}{\pi\tau_{max}}\sqrt{\left(K_{t}\ M_{t}\right)^{2}+\left(K_{ b}\ M_{b}\right)^{2}} \tag{7}\]  
where \(n\) is a safety factor, \(\tau_{max}\) is the allowable stress (40-55 MPa), \(K_{t}\) and \(K_{b}\) are the combined shock and fatigue factors applied to bending and torsional moment, respectively, \(M_{t}\) is the torsional moment (N.mm) and \(M_{b}\) is the bending moment (N.mm). \(K_{b}\) = 1.2-2.0, and \(K_{t}\) = 1.0-1.5 [26].  
The seed damage percentage was measured from the following equation:  
\[\text{Seed damage percentage}\ =\frac{d}{t}\times 100 \tag{8}\]  
where \(d\) is the mass of the damaged seed (g) and \(t\) is the total mass of the sample (g).  
The percentage of unthreshed seed was calculated with the following equation:  
\[\text{Ut}_{t}=\frac{\text{Ut}_{m}}{\text{Ut}_{m}+\text{S}_{outlet}+\text{ damaged seed}}\times 100 \tag{9}\]where Ut is the percentage of unthreshed seed, Ut_m_ is the mass of the unthreshed seed (g) and S_outlet_ is the mass of the threshed seeds at the main seed outlet.