2.1.1. Theory Consideration
The relatively mature corn online moisture content sensor is generally a parallel plate capacitor designed based on the dielectric properties of materials, and its structure is modeled in
Figure 1a. The measuring cavity is composed of insulating material, the metal plate placed on the outer wall of the cavity is an electrode plate, and the cavity is filled with corn to be measured. The structure shown in
Figure 1a is equivalent to a series capacitance model when the edge effect of the capacitance sensor is ignored, as shown in
Figure 1b.
When the corn reaches the state to be measured in the measurement cavity, the output capacitance (
C) of sensor is:
where
ε0 is the vacuum dielectric constant, which is 8.85 pF/m [
22];
εm is the relative dielectric constant of the corn;
d is the distance between the two parallel plates; and
S is the area of plate. The medium between two plates included dry matter, water, and air. The total volume (
V) and the total relative dielectric constant (
εm) of above three can be expressed, respectively, as:
where
V1,
V2, and
V3 are the volume of dry matter, water, and air, respectively;
ε1,
ε2, and
ε3 the relative dielectric constant of these three, respectively. Substituting Equation (3) into Equation (1), the Equation (4) can be obtained:
The porosity of the filling layer in the measurement cavity is calculated using Equation (5):
The moisture content of corn feeding into the measurement cavity is calculated as:
where
Md is the moisture content, dry basis, and
ρ1 and
ρ2 are the density of water and dry matter, respectively.
Defining
K1, and
K2 as follows, and substituting Equations (5) and (6) into Equation (4), the Equation (9) can be obtained:
where
G is the total mass of corn filled into the measurement cavity, which generally changed little and can be regarded as a constant value under the constraint of the determined installation position of the level sensor. According to Equation (9), the output capacitance of sensor is mainly related to
Md,
p,
ε1,
ε2,
ε3,
ρ1,
ρ2,
V, and
G. Among the above parameters,
V,
ρ2, and
G were the determined constants;
ρ1 is a function depending on
Md [
22]; and
ε1,
ε2, and
ε3 are also related to temperature
T. Therefore, the feedback capacitance value of the sensor in practical application is a function depending on
Md,
p, and
t. However, the porosity in Equation (9), which is not only affected by the air volume among grains but also by the air volume inside grains [
23], is one of the important factors affecting the dielectric properties of materials, especially in online moisture content measurement devices installed to monitor the extremely complex process of industrial corn drying.
2.1.2. Mathematical Modeling
Based on the series capacitance calculation method, Equations (10) and (11) were obtained for calculating the corn capacitance and the capacitance of the cavity outer wall, respectively:
where
Cm and
Cc are the capacitance of the corn and outer wall, respectively;
εm and
εc are the dielectric constant of the corn and outer wall, respectively; and
ym and
yc are the thickness of corn and outer wall, respectively. Therefore, the total capacitance to be measured is the sum of the series capacitance, which can be calculated using Equation (12):
where
Ctot is the total capacitance value of the series capacitance to be measured.
Equation (12) shows that the measured capacitance is only related to the relative dielectric constant, which corresponds to the moisture content one-by-one under the determined electric field frequency, corn porosity, and temperature. Thus, Equation (13) can be obtained:
where
M is the moisture content of the corn to be measured, in wet basis. Equation (14) can be obtained by differentiating Equation (13):
A relationship exists between the resolution of the capacitance measurement circuit and the resolution of the moisture sensor, as shown in Equation (15):
where
RC and
Rm are the resolution of the capacitance measurement circuit and moisture sensor, respectively. Assuming the change rate of the corn dielectric constant with moisture content was
K, the full-range resolution of the moisture sensor, as shown in Equation (16), can be obtained from Equation (14):
From Equation (16), when the corn dielectric constant changes little with the moisture content, the resolution of moisture sensor decreases with decreasing moisture content. Therefore, in the process of designing a sensor, it is necessary to ensure that the measurement near the safe moisture content meets the minimum resolution requirements.
On the other side, due to the internal friction of corn, the corn gradually forms a cone in the process of flowing into measurement cavity. Thus, the chamber will not be filled with corn with an unreasonable sensor structure design, as shown in
Figure 2a.
However, the level sensor still judges that the chamber is full of corn at this time, which causes measurement error. Increasing the height of the cavity can cover the measurement area with corn. When the grain filling method is determined, grain with same moisture content has the same repose angle. Therefore, when the material’s falling point is at the top of measurement cavity, as shown in
Figure 2b, the required increase in height has the largest value, which should satisfy Equation (17):
where ∆
z is the required increase in height,
xm is the width of parallel plate, and
α is the repose angle. Based on the mathematical relationship, when
xm is equal to
ym, the minimum value of ∆
z is obtained, and the structure of the designed sensor is the most compact. Thus, the height and the required increase in height of cavity can be calculated using Equations (18) and (19), respectively:
To summarize, the constraint conditions of the moisture sensor’s structural parameters are as follows:
where
Cdy is the measurable dynamic capacitance range of the measurement circuit, [
Cmax,
Cmax] is the measurable statistic capacitance range,
L is the corn bridge size, and
δ is the minimum safe thickness of the measurement cavity.