Creation of a Mathematical Model for Bedding Manure Grinding and Homogenizing

Mathematical description and methods for calculating coarse grinding of cattle waste in screw and knife grinders.

Screw grinding and calculation method.

Studies of the granulometric composition of grinded particles of manure straw obtained in a screw chopper were performed by scientists [4], who established that the percentage of straw particles in the dose depends on humidity and rotor rotation speed, i.e., at n=200, it is 55% and straw splitting is 18.9%, and at n = 600 it is 86.7% and straw splitting is 45%.

In a one-factor experiment program, various aspects of the chopper-spreader were studied, such as productivity, degree of grinding, and power consumption. The factors examined were rotor speed, number of blades, angle of shear plate installation, and moisture content of the manure. From the group of control factors for the experiment, the following were selected: $x_1$ rotor rotation speed rpm, $x_2$ number of blades, pcs, $x_3$ installation angle of the shear plate, $x_4$ and manure moisture (Table 1).

The response function is the estimated indicator Eoc. It was chosen as an optimization criterion.

$E o c=\frac{N_{\text {spec }}}{Q}\left[1+a_{c f} \frac{l c f}{L}\right]$       (1)

where, N_spec is the power; Q is the performance; L is the average length of the stems of unchopped straw; a_cf is the share of the coarse fraction in the grinded product; l_cf is average length of coarse particles.

Determination of specific energy consumption for grinding bedding manure with a screw grinder.

Table 1. Levels of factor variation

The resulting regression equation looks like:

$\begin{aligned} Y_N=18.439- 0.011 x_1+0.00001 x_1^2-1.269 x_2 -0.096 x_3+0.0003 x_3^2-0.272 x_4 +0.001 x_4^2+0.0005 x_1 x_2 +0.000003 x_1 x_4+0.003 x_2 x_3 --0.005 x_2 x_4-0.0005 x_3 x_4\end{aligned}$                (2)

According to the research results, it was found that with an optimal speed value of 549.85 rpm, an optimal installation angle of the shear plate of 79.3, and an optimal number of blades of 6, sufficient productivity and crushing of bedding manure up to 50-100 mm are ensured. The moisture content of the manure can be anything within the range of 18-76%, since the chopper will provide the required dimensions within the range of 50-100 mm.

Analysis of the received dependencies shows that with an increase in the rotor speed from 200 to 1000 revolutions, the average length of grinded particles decreased from 20.1 to 4.1 cm, and the grinding degree increased from 1.21 to 5.92. The throughput of the unit increases from 0.109 to 1,240 t/h. It is also noted that along the length (50-100 mm) grinding of straw particles in manure is achieved at n=550 rpm, the number of blades is 6 and the installation angle of the shear plate is 79°.

Also, the reduction ratio depends on the angle of installing a shear plate, for example, when grinding bedding manure with a moisture content of 76% at 600 drum revolutions, at φ=17° and z=6, the grinding coefficient is λ=5.92, and when the angle of the shear plate is φ=105°, λ=3.11.

As a result, it was established that λ the grinding coefficient increases with a larger number of blades. For example, with two blades λ=3.47; with six blades λ=4.1. When grinding manure with a moisture content of W=76%, the maximum grinding ratio is achieved at 500-700 rpm of the shredder rotor. However, the grinding degree received in this case is not sufficient for advanced processing of manure in the bioreactor, so the resulting dose is subsequently sent to knife grinding.

Knife grinding and calculation method

The mathematical calculation of knife grinding of bedding manure can be divided into the following stages [4]:

1). Calculation of the grinding time. The grinding time is determined by the formula:

$t=V / Q$      (3)

where, t is grinding time (h), V is volume of manure, m³, Q is crusher productivity (m³/h);

The particle size is determined by the formula:

$d=2 \pi v /(P q)$      (4)

where, d is particle size (mm), π is mathematical constant equal to 3.14, v is the blade rotation speed (rpm), P is the crusher power (kW), q is the manure density (kg/m³).

2). Calculation of the knife grinding of bedding manure in the Python program with the following input data:

Python capacity=10 # m³/h, power =100 # kW, speed =1000 # rpm.

feed_rate =5 # m³/h.

The program will produce the following results: grinding time 0.5 h, productivity 20 m³/h, specific energy intensity of grinding 5 kW/m³, particle size 2.5 mm. Thus, when particle size is 2.5 mm the dose is supplied to cavitation destruction to obtain a homogeneous environment.

Cavitation destruction and calculation method

To describe the necessary biotechnological processes for producing biogas, we use a tree structure graph [10, 13] (Figure 2).

When constructing a dynamic model of grinding and dispersing bedding manure, we accept the above assumptions. Taking them into account, a dynamic model of the biotechnological processing of bedding manure will look like (Figure 3).

Using fuzzy logical equations that connect the membership functions of input and output variables at the technological and raw material level with the biogas release Q:

$Q=f Q(A, B)$      (5)

where,

A=fa(a1,a2,a3,a4,a5).

B=fb(b1,b2,b3,b4,b5). b2=f(w1,w2).

A is the quality of the feedstock; B is the quality of the technological process; a1-origin of raw materials; a2-moisture content of raw materials; a3-presence of impurities; a4-raw material viscosity; a5-density of raw materials; b1-reactor loading volume; b2-temperature quality; b3-mixing quality; b4-raw material processing time; b5-optimal grinding; t1-temperature regime; t2-temperature stability.

Thus, knowing the values of the theoretical parameters of the desired function (Table 2), and using approximation tools, we will perform the analysis. First, let us make a table of auxiliary quantities:

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Figure 2. Graphics of biotechnological processes for obtaining biogas

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Figure 3. Dynamic model of the bedding manure processing

Table 2. Auxiliary quantities

Let us calculate the coefficients a and b for the hyperbolic regression equation $\widehat{\mathrm{y}}=\alpha+\frac{b}{x}$ using the known formulas:

$\begin{gathered}\underline{b=\frac{n \sum \frac{y_i}{x_i}-\sum \frac{1}{x_i} \sum y_i}{n \sum \frac{1}{x_i^2}-\left(\sum \frac{1}{x_i}\right)^2}} \\ =\frac{6 \times 234.721-2.45 \times 295.71}{6 \times 1.49138889-2.45^2} \approx 232.1369\end{gathered}$             (6)

$\begin{gathered}a=\frac{1}{n} \sum y_i-\frac{b}{n} \sum \frac{1}{x_i} \\ =\frac{1}{6} \times 295.71-\frac{232.1369}{6} \times 2.45 \approx-45.5042\end{gathered}$                   (7)

Thus, the required regression equation has the form:

$\begin{gathered}f(x):=161.11 \cdot x^{-1.705} x:=(123456) \hat{y}= \\ -45.5042+\frac{232.1369}{x}\end{gathered}$       (8)

x is the number of passes, and a is the particle size of the grinded mixture after x-th pass.

Let us analyze the graph of the regression equation using the Mathcad program (Figure 4).

$\begin{gathered}x:=(123456) . \\ f(x):=-45.5042+\frac{232,1369}{x} . \\ {[(x>0) \text { and }(y>0)] .}\end{gathered}$

Correlation index:

$\begin{gathered}R=\sqrt{1-\frac{\sum\left(y_i-\hat{y}_i\right)^2}{\sum\left(y_i-\bar{y}_i\right)^2}} \\ =\sqrt{1-\frac{15.194042}{27976.6806}} \approx 0.9725\end{gathered}$              (9)

Determination index: $R^2=0.9725^2 \approx 0.9457$.

Average approximation error:

$\begin{aligned} \bar{A}=\frac{1}{n} \sum\left|\frac{y_i-\hat{y}_i}{y_i}\right| & \times 100 \%=\frac{4.033}{6} \times 100 \% \\ & \approx 67.2172 \%\end{aligned}$         (10)

Analyzing the graph, we can conclude that the function $\hat{y}=-45.5042+\frac{232,1369}{x}$ describes the grinding process only up to $x<=4$ or up to a grinding of 14 microns. With large refinement, the approximation error increases, and the data becomes incorrect.

To find a function describing the grinding process after approximately 14 μn, we will use the “Trend Line” function in Excel (Figure 5).

The average approximation error is 17.89%, which shows a good “fit” of the resulting function to the parameters under study. Let us model this function in Mathcad (Figure 6).

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Figure 4. Regression equation graph 

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Figure 5. Trend line function

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Figure 6. Implementation of a power function in Mathcad

Thus, processing cattle excrement (grinding) is described by a system of equations.

$\left\{\begin{array}{c}f(x)=45.5042+\frac{232.1369}{x} \\ f(x)=161.11 x^{-1.705}\end{array}\right.$        (11)

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7b.png

Figure 7. Processing cattle excrement

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Figure 8. System limit definitions

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Figure 9. Linear regression

Let us model this function in Mathcad (Figure 7).

To ascertain the system's conditions, the point where the graphs intersect is identified (see Figure 8).

The system of equations based on the obtained data accepts the following restrictions

$\left\{\begin{array}{c}f(x)=45.5042+\frac{232.1369}{x}, x<4(f(x)>=18) \\ f(x)=161.11 x^{-1.705}, x>3(f(x)<18)\end{array}\right.$        (12)

We need to find the relationship between the productivity of the disperser and the number of passes performed. The relationship of the parameters in the table is described by a linear equation, so we will use the linear regression analysis tool in Excel, the “ЛИНЕЙН” (LINEIN) function (Figure 9).

The required inequality takes the form where x is the number of passes Q=0.79x+7.436.

To illustrate the general formula for processing raw materials using selected parameters such as processing duration and loaded volume, we establish the processing time equation, determining the required number of cycles per pass as N=V/4.69. Here, V represents the volume of raw materials loaded (in liters), and 4.69 refers to the system pipes' total volume.

The processing time for a single pass is calculated as Tob=N*9.61, where 9.61 corresponds to the total duration of one cycle.

Next, we express Tob as Tob=V*9.61/4.69*x=2.049*V*x, representing the full duration required to complete the raw material processing task. The equation for X is X=Tob/(2.049*V)=0.49 Tob/V. This derived value of X is then substituted back into the system of equations.

$\begin{aligned} & f(x)=45.5042+\frac{232.1369 * V}{0.49 * T_{0б}}, x \leq 4(x \geq 14 \mu n) \\ & f(x)=161.11\left(\frac{0.49 * T_{0б}}{V}\right)^{-1.705}, x>4(x<14 \mu n)\end{aligned}$            (13)

Simplifying we get

$\begin{aligned} & \left\{f(x)=45.5042+437.7 \frac{V}{T_{0б}}, x \leq 4(x \geq 14 \mu n)\right. \\ & f(x)=161.11\left(\frac{0.49 * T_{0б}}{V}\right)^{-1.705}, x>4(x<14 \mu n)\end{aligned}$          (14)

The time needed to grind a given value (μn) can be calculated using this function:

$T_{\omega б}=\frac{437.7 \mathrm{~V}}{L+45.5042}$        (15)

In this context, V represents the volume of raw materials being processed (liters), L denotes the target output dimension (microns), and Tob signifies the necessary duration to finish the process (seconds).

To display the general mathematical model of the biotechnological process for processing cattle excrement, it is necessary to add performance parameters to the function. Considering Q=0.79x+7.436, then x=(Q-7.436)/0.79, we get a system of equations:

$\begin{gathered}f(x)=-45.5042+\frac{183.39}{Q-7.436} \\ =-45.5042+\frac{183.39}{V p \omega^2 t-7.436} \\ \frac{437.7 V}{T_{o б}}=\frac{183.39}{(Q * 1000 / 3600)-7.436} \\ 437.7 V *\left(\left(Q * \frac{1000}{3600}\right)-7.436\right)-183.39 T_{o б}=0\end{gathered}$                             (16)

We receive from the equations an inversely proportional relationship between grinding sizes and productivity.

Figure 10 shows the results of solving the resulting system of equations in the program.

$\begin{gathered}\mathrm{f}^1=-45.5042+\frac{232.1369}{\mathrm{x}} \\ \mathrm{f}^2=161.11 \times \mathrm{x}^{-1.705} \\ \mathrm{Q}=0.79 * \mathrm{x}+7.436\end{gathered}$

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Figure 10. System of equations for processing raw materials

Modeling the grinding degree of the substrate and the biogas release from the bioreactor

For the mathematical description and modeling of the technological process of substrate methane fermentation in bioreactors, the author chose the Conto model [14], which considers the hydrodynamics and heat exchange of the fermentable substrate and uses the K₆ coefficient, which describes the effect of bubbling mixing on the efficiency of the process.

$\begin{gathered}\frac{d B}{d \tau}=V \cdot v ; \\ V=\frac{B_0 \cdot S \cdot K_6}{\tau_{\text {бр }}} \cdot\left(1-\frac{K}{\mu_T \cdot \tau_{\text {бр }}-1+K}\right) \\ \frac{d Q}{q \tau}=-u \frac{d Q}{d \tau}-k \cdot \Delta F_{\text {зм }} \cdot\left(T_T-T\right) \\ Q d \tau=k \cdot \Delta F \mathbf{3 м} \cdot T 1-T\end{gathered}$                        (17)

where, V-volume of biogas release (m³); V-speed; K_b-bubbling mixing coefficient; T_br-fermentation time; K-kinetic parameter; Q is the amount of heat required to heat the bioreactor (J); u is the speed of movement of the hot flow through the pipe (m/s); k-heat transfer coefficient (W/(m²*K)); $\Delta F_{\text {зм }}$-heat transfer area of the bioreactor (m²); Т_т-hot flow temperature (K); T is the temperature of the fermentable substrate in the bioreactor (K);

$\mu_m$ depends on the temperature of the fermentation process:

$\mu_m=0.013 \cdot T-0.129$      (18)

where, T-temperature (℃).

Kinetic parameter for bedding manure:

$K=0.8+0.016 \cdot e^{0.06 \cdot S_0}$      (19)

where, S₀-concentration of organic matter.

The concentration of organic matter is determined by the formula:

$S_0=\frac{G_{d r y} \cdot S_1}{W_б}$      (20)

$G_{d r y}=G_б * C_{d r y}$      (21)

determined by the formula:

$W_б=G_б/ \rho_{\bar{б}}$       (22)

where, ρ_b-substrate density (kg/m³).

Biomass density:

$p_{\text {б }}=1000+2.4 \cdot(100-b)$     (23)

where, b-humidity of the fermented substrate (%).

The daily biogas release from the reactor depends on the daily methane release:

$L_{Б \mathrm{\Gamma}}=\frac{L_{C H 4}}{C_{C H 4}^{\%}} * 100$       (24)

where, L_CH4-daily methane release into biogas (m³/day).

$L_{C H 4}=\frac{L_{0 C H 4 * V_{\text {peakT }}}^{\min }}{k_3} * 100$        (25)

where, L_0CH4-specific daily methane release (m³/day), V_min-90%.

The minimum reactor volume is determined by the formula:

$V_{\min }=W_б\cdot T_{б \mathrm{p}}$      (26)

The specific daily methane yield is determined by the formula:

$L_{0 C H 4}=\frac{B_0 \cdot S_0}{T_{6 \mathrm{p}}} \cdot \frac{1-K}{T_{6 \mathrm{p}} \cdot \mu_{\mathrm{M}}-1+K}$       (27)

The use of this mathematical model allows to determine operating parameters: raw material loading temperature; humidity; process time; volume of biogas produced; stirring frequencies and effective structural and geometric dimensions of the bioreactor; mixing device; heating systems [11, 15].

The results of calculating the biogas release on grinded and non- grinded substrate from manure-free manure of anaerobic digestion using the Conto method are presented in Figure 11.

Analyzing the curves obtained in Figure 11, a significant effect from grinding bedding manure is visible, increasing the biogas release by 2.7 times. In the calculations, the following values were taken: the density of the grinded substrate, Pmeas=1001 kg/m³, and the density of the non-grinded Pconstant=1020 kg/m³ of the substrate.

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Figure 11. Biogas release from the substrate with grinded and with non-grinded and solids