# Generalized Mathematical Model of the Grain Drying Process

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Generalized Mathematical Model of a Grain Dryer

#### 2.1. Decomposition of the Drying Process

#### 2.2. Loading and Unloading Operations

#### 2.3. Drying Chamber

#### 2.3.1. Classification of the Models

#### 2.3.2. Generalized Drying Model

- Input variables describing impurities and moist grain streams at the drying chamber inlet $\left({S}_{g\left(in\ast \right)},{S}_{i\left(in\ast \right)},{S}_{w\left(in\ast \right)}\right)$;
- Process parameters that are influenced by grain type, such as kernel geometry, bulk density ${\rho}_{\left(dr\right)}$, equilibrium moisture content ${u}_{r}$, and effective water diffusion coefficient $D$;
- Control variables represented by the initial parameters of drying air (mass flow rate ${S}_{a\left(in\ast \right)}$, humidity ${\phi}_{a\left(in\right)}$, temperature ${T}_{a\left(in\right)}$, average velocity ${v}_{a\left(in\right)}$);
- Dryer type and structural parameters;
- Average drying time in the drying chamber ${t}_{dr}$.

_{0}and T

_{g0}is supplied in successive iterations. Therefore, the initial parameters of the i-th layer are average values. The above implies that the model will not provide information about moisture content distribution, but it will estimate the average moisture content at the dryer outlet. The drying algorithm for the cross-flow mode can be expressed as follows:

#### 2.4. Grain Cooling

- (a)
- The parameters of the drying air stream ${S}_{a\left(in\ast \right)}$ are replaced with the parameters of the cooling air stream ${S}_{a\left(in\right)}^{\prime}$ (Figure 3) with the use of the below Formula (26):$$\{\begin{array}{l}{S}_{a\left(in\ast \right)}\to {S}_{a\left(in\right)}^{\prime}\hfill \\ {\phi}_{a}\to {\phi}_{a}^{\prime}\hfill \\ {T}_{a}\to {T}_{a}^{\prime}\hfill \end{array}$$
- (b)
- An empirical formula for calculating the drying coefficient and equilibrium moisture content [46] was selected for the cooling process based on the thin-layer drying equation (Equation (6)).
- (c)
- The heat transfer coefficient was calculated with the use of Equation (10) by considering the parameters of cooling air and the direction of heat flow during grain cooling;
- (d)
- The initial conditions described by Equation (13) were modified as follows:
- in a batch dryer:$$\{\begin{array}{l}\left({t}^{\prime}=0\wedge {j}^{\prime}=1\right)\Rightarrow {\forall}_{{i}^{\prime}\in <1,{I}^{\prime}>}\left[\left({u}_{0{i}^{\prime}{j}^{\prime}}^{\prime}={u}_{iJ}\right)\wedge \left({T}_{g0{i}^{\prime}{j}^{\prime}}^{\prime}={T}_{giJ}\right)\right]\hfill \\ \left({I}^{\prime}=I\right)\wedge \left({J}^{\prime}=J\xb7\frac{{t}_{co}}{{t}_{dr}}\right)\hfill \end{array}$$
- in a continuous-flow dryer:$$\{\begin{array}{l}\left({t}^{\prime}=0\wedge {j}^{\prime}=1\right)\Rightarrow {\forall}_{{i}^{\prime}\in <1,{I}^{\prime}>}\left[\left({u}_{0{i}^{\prime}{j}^{\prime}}^{\prime}={u}_{IJ}\right)\wedge \left({T}_{g0{i}^{\prime}{j}^{\prime}}^{\prime}={T}_{gIJ}\right)\right]\hfill \\ \left({I}^{\prime}=I\xb7\frac{{V}_{co}}{{V}_{dr}}\right)\wedge \left({t}_{co}={t}_{dr}\right)\wedge \left({J}^{\prime}=J\right)\hfill \end{array}$$

- (e)
- The boundary conditions described by Equation (14) were modified as follows:$$\begin{array}{c}\left({t}^{\prime}\in <0,{t}_{ch\u0142}>\wedge {i}^{\prime}=1\right)\Rightarrow {\forall}_{{j}^{\prime}\in <1,{J}^{\prime}>}\left[\left({T}_{a0{i}^{\prime}{j}^{\prime}}^{\prime}={T}_{a0}^{\prime}\right)\wedge \left({\phi}_{0{i}^{\prime}{j}^{\prime}}^{\prime}={\phi}_{0}^{\prime}\right)\right]\end{array}$$
- (f)
- In “iterative” algorithms for the first layer of cooled grain, the initial parameters ${u}_{0}$ and ${T}_{g0}$ were replaced with parameters ${u}_{0}^{\prime}$ and ${T}_{g0}^{\prime}$ of the grain layer evacuated from the drying chamber in successive time steps (Equations (18)–(24)):$$\{\begin{array}{l}{u}_{0}\to {u}_{0}^{\prime}={u}_{IJ}\hfill \\ {T}_{g0}\to {T}_{g0}^{\prime}={T}_{gIJ}\hfill \end{array}$$

#### 2.5. Drying Air Source

## 3. Results and Discussion

#### 3.1. Butch Dryer Model

^{3}s

^{−1}to 1.9 m

^{3}s

^{−1}in steps of 0.1 m

^{3}s

^{−1}. The results and the relationships between energy consumption and drying time vs. air temperature at the drying chamber inlet and air stream volume are presented in Figure 7.

^{3}s

^{−1}and temperature was 55–65 °C, when air stream volume was below 0.6 m

^{3}s

^{−1}and temperature was 70–75 °C, and when air stream volume was below 0.7 m

^{3}s

^{−1}and temperature was 80–90 °C. The above results account only for heat consumption, but not the energy consumed by the supply fan; therefore, optimal solutions can be expected in an area where both parameters interact.

#### 3.2. Continuous-flow dryer Models

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

Mathematical Symbol | Description |

$\left(a\alpha \right)$ | heat transfer coefficient in W·m^{−3}·K^{−1} |

${a}_{1}$$,{a}_{2}$ | grain damage coefficients |

$c$ | specific heat in J·kg^{−1}·K |

$h$ | height of the grain layer in m |

$k$ | drying coefficient in s^{−1} |

$r$ | heat of water vaporization from grain in J·kg^{−1} |

$t$ | time in s |

$u$ | moisture content of grain dry basis in kg·kg^{−1} |

$x$ | moisture content of grain wet basis in kg·kg^{−1} |

$D$ | effective water diffusion coefficient in m^{2}·s^{−1} |

$E$ | nominal capacity in kg·s^{−1} |

$S$ | mass flow stream in kg·s^{−1} |

$V$ | volume of a dryer’s functional unit in m^{3} |

$T$ | air temperature in K |

$Q$ | heat stream in W·m^{−2}, |

$\mathsf{\Delta}{S}_{g}$ | grain loss stream in kg·s^{−1} |

$\mathsf{\Delta}{S}_{w}$ | stream of evacuated water in kg·s^{−1} |

$\mathsf{\Delta}U$ | mass of evacuated water in kg·kg^{−1}·s^{−1} |

$\rho $ | density in kg·m^{−3} |

$\phi $ | relative humidity in % |

$v$ | velocity in m·s^{−1} |

Subscripts | |

$a$ | Air |

$dr$ | drying chamber |

$g$ | grain dry matter |

$i$ | impurities (material other than grain, MOG) |

$in$ | dryer inlet |

$in\ast $ | inlet of a dryer’s functional unit |

$lo\ast $ | loading unit |

ou | dryer outlet |

$ou\ast $ | outlet of a dryer’s functional unit |

$un\ast $ | unloading unit |

w | Water |

Superscripts | |

‘ | stream mixing in a continuous flow dryer |

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**Figure 3.**Decomposition of the grain dryer into functional units. 1. Loading unit; 2. drying chamber; 3. cooling unit; 4. unloading unit; 5. source of drying air.

**Figure 4.**Classification of drying models. (

**a**) Immobile grain layer; (

**b**) cross-flow; (

**c**) counter-flow; (

**d**) parallel-flow.

**Figure 6.**The influence of the number of layers on the drying time of wheat, canola, and corn grain.

**Figure 7.**The influence of air temperature and air stream volume on energy consumption and drying time.

**Figure 8.**A comparison of the simulated daily capacity of selected batch dryers with their actual capacity. (

**a**) Wheat; (

**b**) canola; (

**c**) corn.

**Figure 10.**Simulated changes in the moisture content of successive grain layers dried in the counter-flow mode.

**Figure 11.**Simulation of changes in the moisture content of successive grain layers dried in the parallel-flow mode. (

**a**) Cyclic drying of grain in a very thick layer; (

**b**) overdrying in the initial drying period; (

**c**) simulation of a drying process conducted in 20% in the mixed-flow mode.

**Figure 12.**Simulated changes in the moisture content of successive grain layers dried in the cross-flow mode.

**Figure 13.**A comparison of the simulated daily capacities of selected continuous-flow dryers and the capacities specified by the manufacturers.

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Myhan, R.; Markowski, M.
Generalized Mathematical Model of the Grain Drying Process. *Processes* **2022**, *10*, 2749.
https://doi.org/10.3390/pr10122749

**AMA Style**

Myhan R, Markowski M.
Generalized Mathematical Model of the Grain Drying Process. *Processes*. 2022; 10(12):2749.
https://doi.org/10.3390/pr10122749

**Chicago/Turabian Style**

Myhan, Ryszard, and Marek Markowski.
2022. "Generalized Mathematical Model of the Grain Drying Process" *Processes* 10, no. 12: 2749.
https://doi.org/10.3390/pr10122749