Numerical Simulation of Deep Bed Cooling Drying Process of Pellet Feed Based on Non-Equilibrium Model

Abstract

In this study, a deep bed cooling drying model based on non-equilibrium model was established for pellet feed. The modified Verma model was used to describe the thin-layer drying rate, and the air temperature coefficient was introduced to optimize the convection heat transfer coefficient. The model was verified by the enterprise production data and laboratory-scale cooling and drying test. The results show that the improved model can accurately predict the changes in feed temperature and moisture and has good applicability to the cooling and drying process under different wind speeds, air temperatures, and humidity. The model lays a foundation for the development of an intelligent control system for a pellet feed cooler and has important engineering value for achieving real-time control of cooling process parameters, improving feed quality stability and energy savings, and reducing energy consumption.

1. Introduction

In the production process of pellet feed, the role of the cooling process is to control the moisture content and temperature of finished pellet feed within a reasonable range [1]. This is crucial to the quality of the feed. The moisture of feed should be kept under safe moisture to avoid mildew during storage and transportation [2,3,4]. After the feed is cooled, the temperature shall not exceed the ambient temperature by 5 °C to prevent the moisture in the feed from evaporating and condensing during storage, resulting in moisture accumulation in the local part of the bagged feed and causing local mildew. At present, many feed production enterprises in China can only rely on manual experience to control feed moisture and temperature, which is inefficient, and the feed moisture is generally too low or too high [5]. In addition, in order to reduce the feed temperature to safety requirements, the widespread problem of excessive cooling has led to serious energy waste [5]. Therefore, it is necessary to build a deep bed cooling and drying model for pellet feed to predict the feed temperature and moisture and further achieve the real-time control of feed temperature and moisture.

Considering the diversity of materials involved in heat and mass transfer and the effects observed in each material, the study of drying kinetics is combined with the mathematical modeling and evaluation of thermodynamic properties, which is worth researchers’ attention to various products [6,7,8]. Therefore, although many basic works have been completed long ago, the parametric model based on solving equations is still an important research field, and the further development of this field is still continuing [9,10,11,12,13,14].

Xin Li et al. [15] established a dual diffusion heat and moisture transfer model based on the principle of thermal non-equilibrium. The drying process of soybean beds was simulated using a model, and the effects of drying air temperature and inlet velocity on drying were analyzed. Dhiraj Kumar et al. [16] developed a non-equilibrium drying model to analyze the coupled heat and mass transfer phenomena in the steady-state countercurrent deep bed drying process of rapeseed. The control equations include moisture mass balance, enthalpy balance, heat transfer rate, and drying rate equations. Z. Naghavi et al. [17] proposed a non-equilibrium model for fixed deep bed drying of brown rice grains. A computer program was developed and implemented in MATLAB 7.6 environment based on a discrete equation system to simulate the grain drying process and predict the characteristics of a deep bed dryer.

However, the simulation model for the cooling and drying process of pellet feed has not been deeply studied. Therefore, studying the heat and mass transfer mechanism of pellet feed cooling and drying process, establishing a deep bed cooling and drying heat and mass transfer model, and predicting the moisture content and temperature changes of finished pellet feed during the cooling process can fill the gap in this area. Further establishing an intelligent control system for pellet feed coolers based on deep bed cooling and drying models and adjusting cooling process parameters in real time according to predicted results are of great significance for improving feed product quality stability, increasing production efficiency and energy conservation, and reducing energy consumption [18,19,20,21,22].

In this study, a deep bed cooling drying model based on a non-equilibrium model was established for pellet feed. The equations of the model include moisture balance, feed enthalpy balance, air enthalpy balance, and drying rate equations in the process of feed cooling. For the accuracy of the data, the modified Verma model [23] developed by our team was selected as the thin layer drying model (drying rate equation) used to describe the drying rate of a thin layer. The relevant important parameters of the thin layer model have a clear corresponding relationship with wind temperature and wind speed and can be calculated through correlation analysis and experimental measurement [24]. This makes the deep bed cooling and drying model established in this study have wide applicability and extrapolation ability [25]. In addition, the heat transfer and balance in the process of cooling and drying with ambient air are different from that of hot air drying with air as a heat source. Therefore, taking pellet feed as the heat source, the feed enthalpy balance and air enthalpy balance equations for cooling and drying were established.

The actual production data of pellet feed cooling and drying link on the production line were collected in a feed factory in the suburb of Shenyang, Liaoning Province, China. Based on the collected data, the deep bed cooling drying model of pellet feed was evaluated. According to the rule that the predicted value of feed temperature deviated, the air temperature coefficient h_T was added to the calculation formula of convective heat transfer coefficient h [26], and the accuracy of the model was verified again.

A laboratory-scale cooler was built. Taking pig pellet feed as the experimental object, the deep bed cooling and drying experiment was carried out, and sufficient detailed cooling and drying data were collected. Through the comparison between the changes in feed temperature and moisture and the predicted values during the cooling and drying process, the scientificity and rationality of the model were proved.

The purpose of this study is to establish a scientific and reasonable deep bed cooling and drying model of pellet feed with wide applicability and high prediction accuracy and lay the foundation for further establishing a set of intelligent control systems for a feed cooler with high practical value based on the model.

2. Materials and Methods

2.1. Development of Model Equation

In the development of the pellet feed cooling and drying model, the following assumptions were made [17,27,28,29,30]:

• Dry air is an ideal gas mixture;
• The air flow is of the plug type;
• During the drying process, the shrinkage of particles can be ignored with the change in moisture and temperature;
• The temperature gradient in a single particle can be ignored;
• The heat conduction between particles can be ignored;
• The specific heat of wet air and pellet feed is constant in a short time;
• The wall of the cooler is insulated, and the heat capacity can be ignored.

The feed bed is divided into several thin layers. Taking a single thin layer (Figure 1) as the object, the equations of water balance, feed enthalpy balance, air enthalpy balance, and drying rate were established [5].

2.1.1. Moisture Balance Equation

The mass change of moisture carried by the air passing through the thin layer in unit time is

$${\rho }_{wa}{v}_{a}S{\displaystyle \frac{\partial H_w}{\partial x}}dxdt$$

(1)

where ρ_wa is the density of wet air (kg/m³), v_a is the average flow rate of air in the cooling bin (m/s), S is the cross-sectional area of the material layer (m²), and H_w is the moisture content of air wet basis (%).

The mass change of moisture content in pellet feed in the thin layer per unit time was as follows:

$$(1-\epsilon ){\rho }_p({\displaystyle \frac{1}{1-M}})S{\displaystyle \frac{\partial M}{\partial t}}dxdt$$

(2)

where ԑ is the porosity inside the feed bed, ρ_p is the real density of dry feed (kg/m³), and M is the real-time wet basis moisture of feed (%, wb).

Ignoring the change in air humidity in the pores of the thin layer, the mass change of moisture carried by the air passing through the thin layer in unit time is equal to the mass change of moisture contained in pellet feed in the thin layer [5].

$${\rho }_{wa}{v}_{a}S{\displaystyle \frac{\partial H_w}{\partial x}}dxdt=-(1-\epsilon ){\rho }_p({\displaystyle \frac{1}{1-M}})S{\displaystyle \frac{\partial M}{\partial t}}dxdt$$

(3)

By simplifying the moisture balance equation (Equation (3)), Equation (4) can be obtained.

$${\displaystyle \frac{\partial H_w}{\partial x}}=-{\displaystyle \frac{(1-\epsilon ){\rho }_p(\frac{1}{1-M})}{{\rho }_{wa}{v}_a}}·{\displaystyle \frac{\partial M}{\partial t}}$$

(4)

2.1.2. Enthalpy Balance Equation of Pellet Feed

The heat transfer and balance during the cooling and drying process of feed using ambient air are different from hot air-drying using air as the heat source. It is necessary to establish feed enthalpy balance and air enthalpy balance equations for the cooling and drying process using pellet feed as the heat source.

Heat released by temperature drop of feed:

$$\left(1-\epsilon \right){(\rho }_p{C}_p+{{\rho }_{p}C}_{w}M({\displaystyle \frac{1}{1-M}})){\displaystyle \frac{\partial T_p}{\partial t}}Sdxdt$$

(5)

where C_p is the specific heat of dry feed (kJ/kg·°C), C_w is the specific heat of liquid water (kJ/kg·°C), and T_p is the real-time temperature of pellet feed (°C).

The heat absorbed by the evaporation of moisture in feed:

$${h_v\rho }_{wa}{v}_a{\displaystyle \frac{\partial H_w}{\partial x}}Sdxdt$$

(6)

where h_v is the evaporation heat of water (kJ/kg).

Sensible heat transferred to air by convection:

$$ha\left(T_p-T_a\right)Sdxdt$$

(7)

where h is the convective heat transfer coefficient (kJ/m²·s·°C), a is the specific surface area (m²/m³), and T_a is the real-time air temperature (°C).

The heat released by the temperature drop of the feed is equal to the heat absorbed by the evaporation of moisture in the feed plus the sensible heat transferred to the air by convection [5].

$$\left(1-\epsilon \right){(\rho }_p{C}_p+{{\rho }_{p}C}_{w}M({\displaystyle \frac{1}{1-M}})){\displaystyle \frac{\partial T_p}{\partial t}}Sdxdt={{h_v\rho }_{wa}v}_a{\displaystyle \frac{\partial H_w}{\partial x}}Sdxdt+ha\left(T_p-T_a\right)Sdxdt$$

(8)

By simplifying the enthalpy balance equation of pellet feed (Equation (8)), Equation (9) can be obtained.

$${\displaystyle \frac{\partial T_p}{\partial t}}={\displaystyle \frac{{h_v\rho }_{wa}{v}_a\frac{\partial H_w}{\partial x}+ha\left(T_p-T_a\right)}{-\left(1-\epsilon \right){(\rho }_p{C}_p+{{\rho }_{p}C}_{w}M(\frac{1}{1-M}))}}$$

(9)

2.1.3. Enthalpy Balance Equation of Air

The sensible heat obtained by air passing through a thin layer of feed per unit time:

$$(1-H_w){{\rho }_{wa}v}_{a}S{C}_{da}{\displaystyle \frac{\partial T_a}{\partial x}}dxdt+H_w{{\rho }_{wa}v}_{a}S{C}_v{\displaystyle \frac{\partial T_a}{\partial x}}dxdt$$

(10)

where C_da is the specific heat of dry air (kJ/kg·°C) and C_v is the specific heat of water vapor (kJ/kg·°C).

Heat released by temperature drop of water vapor:

$${{C_v(T_p-T_a)\rho }_{wa}v}_{a}S{\displaystyle \frac{\partial H_w}{\partial x}}dxdt$$

(11)

The sensible heat obtained by the air passing through the feed thin layer in unit time is equal to the sensible heat transferred to the air by convection plus the heat released by the temperature drop of water vapor [5].

$$\left(1-H_w\right){{\rho }_{wa}v}_{a}S{C}_{da}{\displaystyle \frac{\partial T_a}{\partial x}}dxdt+H_w{{\rho }_{wa}v}_{a}S{C}_v{\displaystyle \frac{\partial T_a}{\partial x}}dxdt=ha\left(T_p-T_a\right)Sdxdt{{{+C}_v(T_p-T_a)\rho }_{wa}v}_{a}S{\displaystyle \frac{\partial H_w}{\partial x}}dxdt$$

(12)

By simplifying the enthalpy balance equation of air (Equation (12)), Equation (13) can be obtained.

$${\displaystyle \frac{\partial T_a}{\partial x}}={\displaystyle \frac{ha\left(T_p-T_a\right)}{\left[\left(1-H_w\right)C_{da}+H_w{C}_v\right]{\rho }_{wa}{v}_a}}+{\displaystyle \frac{C_v\left(T_p-T_a\right)}{\left(1-H_w\right)C_{da}+H_w{C}_v}}·{\displaystyle \frac{\partial H_w}{\partial x}}$$

(13)

2.1.4. Thin Layer Drying Model

The thin layer drying model describes the change rate of moisture in the material thin layer. In this study, the modified Verma model (Equation (14)) developed by our team was selected. The parameters of this semi theoretical thin-layer drying model have a clear corresponding relationship with wind temperature and wind speed, which can be calculated through correlation analysis and experimental measurement. This makes the deep bed cooling and drying model established in this study have wide applicability and practical value.

$$MR={\displaystyle \frac{M-M_e}{M_0-M_e}}=A\exp \left(-k_0{t}^n\right)+\left(1-A\right)\exp \left(-k_1{t}^n\right)$$

(14)

where MR is the moisture ratio; M₀ is the initial feed moisture (%, wb); M_e is the feed equilibrium moisture (%, wb); t is the cumulative cooling and drying time (min); and a, k₀, k₁, and n are the model parameters.

When the diameter d of feed particles is in the range of 0.003~0.0045 m, the value of A is 0.85, the value of k₁ is 0.025, and the values of k₀ and n are calculated from Equations (15) and (16) [23].

$$k_0=\left(1.9557-305.34d\right)\left(0.001v_a+0.0005\right)T_a+\left(0.0136-2.6794\right)v_a+4.458d+0.0033$$

(15)

$$n=0.006T_a+0.63$$

(16)

Finding the derivative of Equation (14) with respect to time t yields Equation (17).

$${\displaystyle \frac{\partial M}{\partial t}}=(M_0-M_e)\left[-k_{0}n{t}^{n-1}·Aexp\left(-k_0{t}^n\right)-k_{1}n{t}^{n-1}·(1-A)exp\left(-k_1{t}^n\right)\right]$$

(17)

2.1.5. Equilibrium Moisture Content of Feed

As shown in Equation (18), the Generalized D’Arcy and Watt model (GDW model) [31,32] developed by Furmaniak [31] was used to calibrate the equilibrium moisture content of pellet feed. This model can introduce air temperature variables and maintain good fitting accuracy. This ensures that the GDW model will not affect the applicability of the thin-layer drying model in practical applications [23].

$$M_e={\displaystyle \frac{M_m\left(C/T_a\right)RH}{1+\left(C/T_a\right)RH}}·{\displaystyle \frac{1-b\left(1-w\right)RH}{1-bRH}}$$

(18)

where RH is the relative humidity of air; M_m is the moisture content of a single layer; and C, b, and w are the model parameters.

For pig pellet feed, the value of M_m is 0.07682, the value of C is 357.6073, the value of b is 1.00287, and the value of w is 0.26751 [23].

2.2. Parameters of the Model

2.2.1. Physical Parameters Related to Pellet Feed

The bulk density ρ_b of absolute dry pellet feed was determined by a bulk density tester. The bulk density ρ_b of pig pellet feed used in this study was 600.01 kg/m³. The porosity ԑ has a strong influence on the heat and mass transfer efficiency between the drying medium and particles [33]. The porosity ԑ of stacked feed pellets was determined by the oil immersion method. The porosity ԑ of pig pellet feed was 0.4 [5]. The true density ρ_p of absolute dry pellet feed is calculated by Equation (19). The true density ρ_p of pig pellet feed was 1000.02 kg/m³ [5].

$${\rho }_p={\displaystyle \frac{{\rho }_b}{1-\epsilon }}$$

(19)

The diameter of the ring mold for the production of the feed is 0.0035 m, the measured average diameter d of the pellet feed is 0.0039 m, and the average length l is 0.0106 m [5]. The specific surface area a can be calculated by Equation (20).

$$a={\displaystyle \frac{2(1-\epsilon )(d+2l)}{dl}}$$

(20)

The specific heat C_p of absolute dry pellet feed varies with temperature, which can be calculated by Equation (21) [5].

$$C_p=1.12834+0.00449T_p$$

(21)

The heat of evaporation h_v of water can be calculated by Equation (22). The specific heat of water C_w is 4.1874 kJ/kg·°C.

$$h_v=-2.3414T_p+2493.5$$

(22)

2.2.2. Physical Parameters Related to Air

The density of humid air, ρ_wa, can be calculated using Equation (24). The air dry basis moisture content H_d and air wet basis moisture content H_w can be calculated using Equations (25) and (26) [5].

$$P_s=133.322\times {10}^{\left(7.96681-{\displaystyle \frac{1668.21}{T_a+228}}\right)}$$

(23)

$${\rho }_{wa}={\displaystyle \frac{P-0.3774RH{P}_s}{287(T_a+273.15)}}$$

(24)

$$H_d={\displaystyle \frac{0.6219RH{P}_s}{P-RH{P}_s}}$$

(25)

$$H_w={\displaystyle \frac{0.6219RH{P}_s}{P-0.3781RH{P}_s}}$$

(26)

where P_s is the saturated water vapor pressure (Pa); P is the air pressure, taking 101,325 Pa (standard atmospheric pressure); and RH is the relative humidity of the air.

The specific heat of dry air C_da is 1.013 kJ/kg·°C, the specific heat of steam C_v is 1.8469 kJ/kg·°C, and the specific heat of wet air C_wa (kJ/kg·°C) can be calculated by Equation (27) [5].

$$C_{wa}=\left(1-H_w\right)C_{da}+H_w{C}_v$$

(27)

The dynamic viscosity μ_wa (Pa·s) of wet air can be calculated by Equation (28) [5].

$${\mu }_{wa}={\displaystyle \frac{{\mu }_{da}+1.268{\mu }_v{H}_d}{1+1.268H_d}}$$

(28)

$${\mu }_{da}=\left(17.4945+4.779\times {10}^{-2}{T}_a-3.5256\times {10}^{-5}{T}_a^2\right)\times {10}^{-6}$$

(29)

$${\mu }_v=\left(8.1804+4.011\times {10}^{-2}{T}_a-1.7858\times {10}^{-5}{T}_a^2\right)\times {10}^{-6}$$

(30)

where μ_da is the dynamic viscosity of dry air (Pa·s) and μ_v is the dynamic viscosity of water vapor (Pa·s).

The convective heat transfer coefficient h (kJ/m²·s·°C) can be calculated by Equation (31) [34].

$$h=5.633C_{wa}{\displaystyle \frac{{\rho }_{wa}{v}_a}{\epsilon }}{\left({\displaystyle \frac{d{\rho }_{wa}{v}_a}{\epsilon {\mu }_{wa}}}\right)}^{-0.34}\times {10}^{-3}$$

(31)

2.3. Numerical Solution

To solve the equations of the deep bed drying model, the initial and boundary conditions must be known [17]. The initial condition is the initial state of pellet feed and air at each position in the feed bed; the boundary condition is the state of ambient air.

The initial conditions:

$$\begin{gathered} RH\left(x,0\right)={RH}_0 \\ {T}_a\left(x,0\right)=T_{a,0} \\ {T}_p\left(x,0\right)=T_{p,0} \\ M\left(x,0\right)=M_0 \end{gathered}$$

The boundary conditions:

$$\begin{gathered} RH\left(0,t\right)={RH}_{input} \\ {T}_a\left(0,t\right)=T_{a,input} \end{gathered}$$

To solve the equations by numerical method, it is necessary to discretize the feed bed in space (Figure 2) and the drying process in time [17]. Based on the comprehensive consideration of model accuracy and calculation amount, the feed thin layer thickness ∆x is set to 0.02 m, and the time step ∆t is set to 5 s [5]. The time step will fluctuate slightly with the actual application. The position x of each thin layer can be expressed as x_i = iΔx [5]. I is the thin layer number, and the number sequence is from bottom to top. The cooling time of each thin layer can be expressed as t_i,j = jΔt, and j is the cumulative number of time steps of each thin layer [5].

Based on the consideration of simplifying the computational complexity and reducing the amount of calculation, the solving process of feed and air state was simplified. In a single time step, the state of the feed thin layer and air is calculated after a single feed thin layer completes the cooling of a time step, and then this air state is used to calculate the state of the feed thin layer and air after the cooling of the next feed thin layer in the time step until the state of all the thin layers is calculated.

By discretizing the partial derivatives of Equations (4), (9), (13) and (17), Equations (32)–(35) can be obtained to form an algebraic equation group [17]. Equation (35) can be solved directly by relying on known information, and then, Equations (32)–(34) can be solved in turn to complete the solution of the equation group.

$${\displaystyle \frac{{H_w}_{i+1}^{j+1}-{H_w}_i^j}{∆x}}=-{\displaystyle \frac{(1-\epsilon ){\rho }_p(\frac{1}{1-M})}{{\rho }_{wa}{v}_a}}·{\displaystyle \frac{M_i^{j+1}-M_i^j}{∆t}}$$

(32)

$${\displaystyle \frac{{T_p}_i^{j+1}-{T_p}_i^j}{∆t}}={\displaystyle \frac{ha\left(T_p-T_a\right)}{-\left(1-\epsilon \right){(\rho }_p{C}_p+{{\rho }_{p}C}_{w}M(\frac{1}{1-M}))}}+{\displaystyle \frac{{h_v\rho }_{wa}{v}_a}{-\left(1-\epsilon \right){(\rho }_p{C}_p+{{\rho }_{p}C}_{w}M(\frac{1}{1-M}))}}·{\displaystyle \frac{{H_w}_{i+1}^{j+1}-{H_w}_i^j}{∆x}}$$

(33)

$${\displaystyle \frac{{T_a}_{i+1}^{j+1}-{T_a}_i^j}{∆x}}={\displaystyle \frac{ha\left(T_p-T_a\right)}{\left[\left(1-H_w\right)C_{da}+H_w{C}_v\right]{\rho }_{wa}{v}_a}}+{\displaystyle \frac{C_v\left(T_p-T_a\right)}{\left(1-H_w\right)C_{da}+H_w{C}_v}}·{\displaystyle \frac{{H_w}_{i+1}^{j+1}-{H_w}_i^j}{∆x}}$$

(34)

$${\displaystyle \frac{M_i^{j+1}-M_i^j}{∆t}}=(M_0-M_e)\left[-k_{0}n{t}^{n-1}·Aexp\left(-k_0{t}^n\right)-k_{1}n{t}^{n-1}·(1-A)exp\left(-k_1{t}^n\right)\right]$$

(35)

In order to obtain the state of feed and air at any time point at any position in the cooler, it is necessary to solve the equation group for each thin layer in order according to the time step. To solve the state of feed and air at any position in the first time step (j = 1), initial conditions and boundary conditions are required. The solution of the subsequent time step depends on the calculation results and boundary conditions of the previous time step. In a single time step, the solution of the state of the first thin layer depends on the boundary conditions, and the solution of the state of the remaining thin layers depends on the calculation results of the previous thin layer [5].

During the cooling and drying process, the height of the feed bed gradually accumulates to the set value and is maintained at that height during the feeding and discharging processes. Therefore, the cooling time of each thin layer is independently accumulated.

According to the above calculation steps, the model operation program is written in Python language. The calculation and data summary of the model are carried out by the program. The version of Python used in this study is 3.13.1.

2.4. Experimental Apparatus and Procedure

2.4.1. Preliminary Verification of the Model

The actual production data of cooling and drying of pig pellet feed were collected on the production line of a feed factory in the suburb of Shenyang, Liaoning Province, China. The actual production data was input into the model operation program, and the simulation results were compared with the measured data. The rule of deviation in the predicted values of feed temperature and moisture was analyzed, and the model parameters were adjusted according to the results. And, the accuracy of the adjusted model was verified for the prediction of feed temperature and moisture after cooling.

2.4.2. Model Validation Experiment

As shown in Figure 3, a laboratory-scale cooler was built. The main components are the cooling bin, frequency conversion fan, feed air shutoff, blanking device, control box, and industrial control computer. The control system is deployed in the industrial computer, and the control box is used to regulate the line and supply power for the cooler. The size of the cooling bin is 0.35 × 0.35 × 0.5 m³ (length × width × height). The full bin is filled with 30 kg, and the effective charge is 15 kg. The fan speed is controlled by the frequency converter to obtain the required wind speed.

The improved model is deployed in the control system of the laboratory-scale cooler. The control system is also written in Python. The cooler was used for the deep bed cooling and drying test of pellet feed, and sufficient detailed cooling and drying data were collected (including the real-time monitoring of feed temperature in each thin layer).

At the same time of the feed deep bed cooling and drying test, the model will predict the real-time temperature and moisture content of the pellet feed at different positions in the material layer according to the information collected by the control system, such as the initial temperature and moisture content of the pellet feed, the initial temperature and initial relative humidity of the air, and the real-time wind speed on the cross section in the cooling bin. The accuracy and scientificity of the model were verified by comparing the differences between the changes in feed temperature and moisture, and the predicted values during the cooling and drying process.

Microwave moisture sensors (ACO, MMS, Beijing, China) are installed at the inlet and outlet of the cooling bin to monitor the initial and final moisture content of pellet feed. Several temperature sensors (ChuangJiMei, PT-100, Hengshui, China) are equipped on the right vertical direction of the cooling bin to monitor the feed temperature of each feed layer. Air temperature and humidity sensors (Renke, cos-c5-4, Jinan, China) are installed at the air inlet and outlet of the cooling bin to monitor the initial and final temperature and humidity of the air. The air outlet is equipped with a wind speed sensor (Renke, FS-V05, Jinan, China) to monitor the air flow through the material layer.

The deep bed cooling and drying test of pellet feed was carried out according to the actual production of feed. The total time for feed cooling and drying (the time from the pellet feed entering the cooling bin to leaving the cooling bin) is set at 850 s. The total height of the feed bed is set at 0.2 m, which is divided into 10 thin layers. As shown in

Table 1. Ambient temperature and humidity conditions.

Ambient Air Temperature (°C)	Ambient Air Humidity
14.30	0.291
19.50	0.285
24.60	0.297

, the test was conducted under temperature and humidity conditions at three different levels. The average wind speed in the cooling bin is set at three levels: 0.5 m/s, 0.6 m/s, and 0.7 m/s.

As shown in Figure 3, a small portion of pellet feed is intercepted from the pipe under the granulator of the feed production line and enters the laboratory scale cooler. The pellet feed enters the cooling bin at a constant speed through the feed air shutoff. Under the action of negative pressure, the ambient air enters the cooling bin from the blanking port and passes through the material layer from bottom to top to take away the heat and moisture of pellet feed. When the cooling time reaches 850 s, the height of the feed bed exceeds 0.2 m, the feed at the bottom of the feed bed leaves the cooling bin from the blanking port of the cooler, and the height of the feed bed decreases by 0.02 m. When the height of the feed bed exceeds 0.2 m again, the feed at the bottom leaves the cooling bin from the blanking port again, and so on. The finished feed product after completing the cooling process is shown in Figure 4.

The real-time monitoring data during the test and the prediction data of the model are recorded by the industrial computer.

2.4.3. Experimental Materials

In order to verify the value of popularization and application of the model, the validation experiment was not conducted in the feed factory in Shenyang, but in a feed factory in the suburbs of Beijing, China. The feed selected in the validation experiment is the same as the feed selected in the feed factory in Shenyang when collecting data, which is pig pellet feed. The formula, preparation temperature, and actual diameter of the feed selected in the two feed factories are consistent. The relevant data of pig pellet feed are detailed in Section 2.2.1. As described in Section 2.4.2 of this article, the pig pellet feed required for the experiment is directly intercepted from the feed production line through the pipeline. About 30 kg of pellet feed was used in each experiment, and about 270 kg of pellet feed was used in 9 experiments.

3. Results and Discussion

3.1. Preliminary Verification and Improvement of the Model

Table 2. Cooling and drying data of pellet feed collected in feed plant.

Batch	Wind Speed (m/s)	Initial Air Temperature (°C)	Initial Air Humidity	Initial Feed Temperature (°C)	Initial Feed Moisture	Height of Feed Bed (m)	Drying Time (s)
1	0.688	8.00	0.394	79.0	0.1580	0.70	868
2	0.688	9.00	0.432	78.0	0.1520	0.70	889
3	0.688	14.00	0.353	78.0	0.1560	0.70	868
4	0.592	14.00	0.353	78.0	0.1500	1.00	1200
5	0.688	17.00	0.359	78.0	0.1560	0.70	875
6	0.688	19.00	0.392	78.0	0.1550	0.70	847
7	0.688	23.00	0.348	79.0	0.1560	0.70	910
8	0.592	23.00	0.348	79.0	0.1520	1.00	1200
9	0.688	29.71	0.514	78.0	0.1481	0.70	826
10	0.688	30.84	0.493	75.0	0.1513	0.70	770
11	0.688	30.84	0.493	75.0	0.1455	0.70	833
12	0.688	33.38	0.537	79.9	0.1481	0.70	854
13	0.688	33.76	0.566	82.1	0.1538	0.70	868
14	0.688	34.82	0.524	81.2	0.1522	0.70	847
15	0.688	35.00	0.477	81.0	0.1498	0.70	840

shows the cooling and drying data of pellet feed collected in the feed factory. In the 15 groups of data, the height of the feed bed has two types, 0.7 and 1.0 m, and the corresponding wind speed is 0.6875 and 0.5919 m/s, respectively. Other cooling and drying conditions include different air temperatures and humidity, drying times, and the initial temperatures and moisture of feed, which also have sufficient dispersion ranges. Therefore, the production conditions covered by the 15 groups of feed cooling and drying production data are sufficient to complete the preliminary verification of the model.

The predicted and measured results of the feed deep bed cooling and drying model on the temperature and moisture of pellet feed are shown in Figure 5 and Figure 6, respectively. As shown in Figure 5, the predicted temperature of each batch of feed is much higher than the measured value, with large error. As shown in Figure 6, the predicted value of moisture in each batch of feed is generally slightly higher than the measured value, and the error is random and has no obvious rule.

There is a big error in the prediction of feed temperature. Therefore, the problem appears in the enthalpy balance equation (Equation (9)) of pellet feed. The parameters involved in the enthalpy balance equation of pellet feed, such as the specific surface area a of feed and the specific heat C_w of water, are fixed values. The calculation formulas of evaporation heat of water h_v and wet air density ρ_wa have been verified for a long time. The calculation formula of convective heat transfer coefficient h (Equation (31)) will vary according to different application scenarios and objects. As shown in Figure 7, the error between the predicted value and the measured value of feed temperature generally increases with the decrease in initial air temperature T_a. The reason for the error is that the calculation formula of convective heat transfer coefficient h is insufficient. Therefore, as shown in Equation (36), the air temperature coefficient h_T is added to Equation (31).

$$h=h_{T}5.633C_{wa}{\displaystyle \frac{{\rho }_{wa}{v}_a}{\epsilon }}{\left({\displaystyle \frac{d{\rho }_{wa}{v}_a}{\epsilon {\mu }_{wa}}}\right)}^{-0.34}\times {10}^{-3}$$

(36)

The calculation steps and logic of the calculation program for the pellet feed deep bed cooling drying model changed. The cooling and drying conditions, and the temperature and moisture of the feed after the cooling and drying were taken as the inputs of the model calculation program. The air temperature coefficient h_T was used as the output of the calculation program. Then, 15 batches of cooling and drying data were input, and the value of air temperature coefficient h_T corresponding to each batch of data was inversely calculated. As shown in Figure 8, the value of h_T was linearly and negatively correlated with the initial air temperature. The relationship between initial air temperature and air temperature coefficient h_T was analyzed by linear regression. The relationship between h_T value and initial air temperature is shown in Equation (37). The R² value was 0.9832.

$$h_T=-0.03689T_a+2.2845$$

(37)

The calculation formula of convective heat transfer coefficient h is calculated, the program is re-run, the prediction results of the model are obtained, and the prediction accuracy of the model is verified for feed temperature and moisture after cooling.

Table 3. Root mean square error between predicted and measured values of feed temperature and moisture.

Model Version	Ambient Air Humidity
	Feed Temperature	Feed Moisture
Original model	6.6827	0.001779
Improved model	0.3570	0.000756

shows the root mean square error between the predicted and measured values of feed temperature and moisture of 15 batches before and after the improvement of the model. After the improvement of the model, the root mean square error between the predicted value and the measured value is much lower than that before the improvement. It can be seen from Figure 9 and Figure 10, and Table 3 that the improved model can accurately predict the temperature and moisture of pellet feed after deep bed cooling and drying. The improved calculation formula of convective heat transfer coefficient h not only greatly improves the prediction accuracy of the model for feed temperature but also improves the prediction accuracy for feed moisture. This shows that the prediction results of feed temperature and moisture are interactive. Specifically, in the calculation logic of the deep bed drying model, the calculation of the thin layer state is affected by the calculation results of the previous thin layer and the calculation of each thin layer state in a time step is affected by the calculation results of the previous time step. Therefore, the calculation results of the heat balance of pellet feed will affect the calculation results of moisture balance with the progression of the thin layer and the accumulation of the time step.

3.2. Results and Analysis of Verification Experiments

Table 4. Initial feed state and air state of validation experiment.

Batch	Wind Speed (m/s)	Ambient Air Temperature (°C)	Ambient Air Humidity	Initial Feed Temperature (°C)	Initial Feed Moisture
1	0.5	14.30	0.291	69.9	0.1454
2	0.6	14.30	0.291	69.9	0.1455
3	0.7	14.30	0.291	69.9	0.1454
4	0.5	19.50	0.285	69.9	0.1455
5	0.6	19.50	0.285	69.9	0.1454
6	0.7	19.50	0.285	69.9	0.1454
7	0.5	24.60	0.297	70.0	0.1456
8	0.6	24.60	0.297	70.0	0.1454
9	0.7	24.60	0.297	70.0	0.1455

shows the initial temperature, initial moisture of feed, and initial temperature and humidity of air in the validation test. Figure 11, Figure 12, Figure 13 and Figure 14 show the predicted and measured values of the state of feed and air at different positions when the drying time of the lowest feed thin layer (thin layer No. 1) reaches 850 s (when the cooling drying is completed). Figure 11 shows the measured and predicted values of the temperature of each feed thin layer. Figure 12 shows the predicted value of moisture of each feed thin layer, the measured value of initial moisture of feed, and the measured value of moisture of the lowest feed thin layer. Figure 13 and Figure 14 show the predicted value of temperature and humidity of air after passing through each feed thin layer, the measured value of ambient air temperature and humidity, and the measured value of air temperature and humidity after passing through the last thin layer. There are 10 thin layers in the feed bed. The feed state of the 11th thin layer in Figure 11 and Figure 12 represents the initial state of the feed before entering the cooler, and the air state of the 11th thin layer in Figure 13 and Figure 14 represents the state of the air after passing through the last thin layer (thin layer No. 10).

As shown in Figure 11 and Figure 12, the temperature and moisture of pellet feed increased with the decrease in feed thin layer position. As shown in Figure 13, the temperature of the air increases with the increase in the position of the thin layer, and the lower the wind speed, the greater the temperature rise. As shown in Figure 14, with the increase in the position of the thin layer, the relative humidity of the air may rise first and then fall. For example, when the initial temperature of the air is 14.3 °C, the relative humidity begins to decrease as the air passes through the seventh thin layer. It can be seen from Figure 13 that this is caused by the rapid rise in air temperature. It can be seen that with the increase in the position of the thin layer, the increase in the absolute humidity of the air is inevitable, but the relative humidity may rise first and then fall. Especially when the initial temperature of the air is low, the absolute humidity of the air will be relatively low and this phenomenon is most likely to occur.

As shown in Figure 11, the cooling efficiency of pellet feed increases with the decrease in initial air temperature and the increase in wind speed. As shown in Figure 12, the drying efficiency of pellet feed increases with the increase in initial air temperature and wind speed [35,36,37]. The change in moisture content of pellet feed is analyzed with the position of the thin layer under different initial air temperatures and wind speeds in Figure 12. It can be seen that under the condition of low initial air temperature, the influence of wind speed change on feed drying efficiency is limited. When the initial temperature of the air is low, the cooling efficiency is high and the drying efficiency is low, which is consistent with the actual production problems of the feed factory in winter. It can be seen from the above analysis that the problem can be solved by increasing the height of the feed bed and appropriately reducing the wind speed. Increasing the height of the feed bed can increase the cooling and drying time to reduce feed moisture. Air passing through the higher feed bed or at a lower speed through the feed bed can obtain a higher temperature, which can improve the drying efficiency of the high feed thin layer.

Table 5. Root mean square error between predicted and measured values of feed state and air state.

Data	Feed Temperature	Feed Moisture	Air Temperature	Air Humidity
RMSE	0.134536	0.000141	0.137558	0.001161

shows the root mean square error between the predicted value and the measured value of the state of the lowest feed thin layer (thin layer No. 1) and the state of the air after passing through the last thin layer (thin layer No. 10) in the validation experiment. The root mean square error values of the four kinds of data are at a low level. Comparing the measured and predicted values of the state (temperature and moisture) of feed thin layer 1 (the lowest thin layer) (Figure 11 and Figure 12) and the measured and predicted values of the state (temperature and moisture) of the air at the position of feed thin layer 11 (the air after passing through the feed bed) (Figure 13 and Figure 14), it can be seen that the error is very small. The above results clearly show that the deep bed cooling and drying model constructed in this study has enough accuracy and can accurately describe the deep bed cooling and drying characteristics of pellet feed. In addition, as shown in Figure 11, the model has good accuracy in predicting the temperature of a feed thin layer at different positions in the feed bed, indicating the rationality and scientificity of the model.

4. Conclusions

The experimental results and model simulation results showed that the cooling efficiency of pellet feed increased with the decrease in initial air temperature and the increase in wind speed; the drying efficiency increased with the increase in initial air temperature and wind speed; and the influence of wind speed on drying efficiency is limited under low-temperature conditions. The drying effect can be optimized by increasing the height of the feed bed and reducing the wind speed appropriately.

In this study, a non-equilibrium model for deep bed cooling drying of pellet feed was established by integrating the water balance, feed enthalpy balance, air enthalpy balance and drying rate equations. The equations of the model are solved by discretization and numerical methods. The air temperature coefficient is introduced to optimize the calculation formula of convective heat transfer coefficient, so that the model has good prediction accuracy for the temperature and moisture of the finished pellet feed under different cooling and drying conditions and can accurately predict the temperature changes in the thin layer at different positions in the feed bed. The model can accurately describe the deep bed cooling and drying process of pellet feed and provides a theoretical basis for the development of an intelligent control system for a pellet feed cooler. It has important engineering application value for achieving the real-time control of process parameters, improving the stability of feed quality and energy saving.