2.2.1. Theoretical Discharge Capacity of Inclined Spiral Fertilizer Discharger
The basic structural parameters established for the inclined spiral fertilizer discharger, focusing on its core component (double-head spiral fertilizer discharging wheel) in this study, are as follows: spiral blade outer diameter
D1 = 50 mm, spiral blade inner diameter
D2 = 26 mm, pitch
S = 36 mm, and average blade thickness
k = 2 mm. These parameters ensure compliance with the maximum fertilizer-discharge capacity of 375 kg/hm² as specified in GB/T35487-2017. To determine the maximum rotational speed of the fertilizer-discharge wheel, the fertilizer composite coefficient
C was identified based on reference [
16], and the maximum speed of the spiral fertilizer-discharge wheel was calculated using Equation (1).
where
C is the fertilizer composite factor,
C = 50;
D1 is the spiral blade outer diameter, mm;
Nmax is the maximum speed of the spiral fertilizer wheel, r/min.
Both the inclined spiral fertilizer discharger and the traditional spiral fertilizer discharger operate based on the spiral conveyor mechanism [
17]. However, the primary distinction between these two types of theoretical fertilizer dischargers lies in the fertilizer filling rate within the discharger shell cavity. This can be calculated using Formulas (4) and (5) to determine the theoretical fertilizer-discharge amount for the inclined spiral discharger, where the filling coefficient is assumed to be 1.
where
q is the fertilizer-discharge per cycle, g;
D2 is the spiral blade inner diameter, mm;
S is the pitch of the spiral, mm;
j is the number of spiral blades;
k is the average blade thickness, mm;
ρ is the fertilizer density, g/mm
3;
φ is the filling factor of fertilizer-discharge spirals, %.
where
Q is the fertilizer discharged quantity, g;
t is the rotation time, s;
ω is the fertilizer wheel speed, rad/s.
2.2.2. Fertilizer-Discharge Performance Parameter Analysis
During the operation of the inclined spiral fertilizer discharger, fertilizer particles come into contact with the cavity wall. To determine the minimum inclination angle of the discharger and its impact on performance, a force analysis was conducted at the point of maximum friction on the fertilizer, assuming ideal ambient conditions with no air humidity. This analysis included an examination of the fertilizer’s motion state, as illustrated in
Figure 2.
where
Wf is the energy loss due to friction between shell and fertilizer, J;
Ff is the friction on fertilizer, N;
μ is the coefficient of friction;
θ is the tilting angle of the fertilizer-discharge unit, °;
R is the overall fertilizer center of mass radius, m;
L is the fertilizer overall actual distance traveled, m.
where
vp is the overall fertilizer moving speed along
θ inclination angle, m/s;
γ is helix inclination, °.
where
Wv is the kinetic energy of fertilizer slanting along
θ inclination angle, J;
vr is the overall fertilizer circumferential velocity, m/s.
The above discussion reveals that the tilt angle
θ of the inclined spiral fertilizer discharger significantly affects its fertilizer-discharge performance.
θ should exceed the angle of repose of the fertilizer, which, according to relevant literature [
18,
19,
20], typically ranges from 20 to 30 degrees for commonly used fertilizers. For the inclined spiral discharge mechanism to ensure proper cavity filling, the fertilizer needs to undergo a backflow process along the inner wall of the cavity. Without this backflow condition, the fertilizer would simply be pushed outward along the cavity’s inner wall during discharge, preventing proper accumulation and full filling of the fertilizer discharger, thus deviating from its design concept. Therefore, considering the practical contact characteristics of the material, the inclined angle
θ of the inclined spiral fertilizer discharger is set to not be less than 30° to ensure effective and consistent fertilizer discharge.
As depicted in
Figure 3, the distance from the terminating spiral blade of the fertilizer-discharge wheel to the fertilizer outlet (referred to as
l) impacts the fertilizer filling rate and delivery performance. This additional distance is crucially considered, with its primary influence on the filling space typically extending approximately three-quarters of the blade’s pitch.
where
l is the fertilizer-discharge wheel termination spiral blade to fertilizer outlet distance, mm;
Wl is the kinetic energy of fertilizer on
l during the discharging of fertilizer, J.
Inclined spiral fertilizer discharger in the process of transporting fertilizer can be based on the principle of following the law of conservation of energy, expressed as the by the following formula:
where
WT is the inclined spiral fertilizer discharger total fertilizer delivery capacity, J;
Wo is other energy loss, J.
Previous analysis found that the size of the inclined spiral fertilizer discharger
l directly impacts the fertilizer filling rate. For this reason, the tilting angle of 35° is an example of the theoretical analysis of the filling process concerning the fertilizer discharger fertilizer simulation state, with the use of SolidWorks 2022 software (Dassault Systemes, Waltham, MA, USA) for the slanting spiral fertilizer discharger
z-distance of the fertilizer filling condition in three-dimensional modeling [
21], as shown in
Figure 4.
The evaluation function of the software was used to measure the volumetric attributes and the three-dimensional model of the fertilizer capacity space, and the three-dimensional stacking state of the spiral discharge wheel was selected for volumetric identification, as shown in
Figure 5.
To explore the relationship between different values of
l and the filling rate, volumetric analysis was conducted to examine how variations in
l affect the filling of the fertilizer exhauster cavity. Formula (9) was employed for parametric modeling and volumetric assessment, revealing trends in cavity
z-distance filling rates. The findings are presented in
Figure 6. From 0 to 20 mm, the cavity filling rate increased significantly; when the value for reaching the full filling state was greater than 20 mm, the filling rate remained unchanged, and it can be seen that
l has a direct impact on the cavity filling condition. When
l is longer, the filling condition is without fluctuations and exhibits a smooth phenomenon;
l was comprehensively determined up to 40 mm.
where
e is the fertilizer filling rate of cavity
z-distance, %;
Vm is the volume of space occupied by fertilizer of cavity
z-distance, mm
3;
V is the overall spatial volume of cavity z-distance, mm
3.
A higher fertilizer filling rate enhances the uniformity of fertilizer discharge. Based on the analysis above, it is evident that adjusting the tilting angle θ and size l of the fertilizer discharger influences the speed of fertilizer transport by the inclined spiral mechanism. Therefore, optimizing both θ and l of the fertilizer discharger is essential to achieve optimal performance.