# Mathematical Modelling of Rotary Drum Dryers for Alfalfa Drying Process Control

^{1}

^{2}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. The Complete Mathematical Model

_{A}and the product T

_{P}inside the dryer is also shown. Two zones can be identified in the dryer: in the first, long L

_{I}

_{−}

_{C}, the product reduces its moisture content from the initial value X

_{I}to the critical value X

_{C}; in the second, long L

_{C}

_{−}

_{E}, the product reaches the final moisture content X

_{E}.

#### 2.1.1. First Zone of the Dryer (L_{I−C})

_{I−C}, the moisture content X of the product is higher than the critical content X

_{C}; therefore, the temperature T

_{P}is constant and equal to the wet-bulb temperature T

_{WB}[17,18].

_{A}is the air temperature meeting the area dA; and T

_{WB}is the product temperature, which is assumed to be equal to the wet-bulb temperature of the air.

_{A}:

_{DAI}is the mass flow rate of dry air coinciding with the mass flow rate of hot air entering the dryer, as the air humidity at the inlet is very low (only 0.01 kg/kg

_{D.A.}); c

_{A}and c

_{V}are the specific heat of dry air and of vapor, respectively; dT

_{A}is the infinitesimal variation of the air temperature when it touches the area dA; and G

_{EV}is the mass flow rate of the vapor that originates from the product, which increases along the first zone of the dryer.

_{EV}from the elemental area dA of the product, it must receive the infinitesimal heat transfer rate dq from Equations (1) and (2). The relationship between the elemental mass flow rate and infinitesimal heat transfer rate is

_{I−C}is the thermal energy, in the first zone I − C of the dryer (Figure 1), required to produce 1 kg of superheated steam at the air temperature T

_{A}. It is considered to be a constant.

_{AI}is the air temperature at the inlet of the dryer; T

_{AC}is the air temperature at the end of the first zone, where z = L

_{I−C}corresponds to the critical moisture content of the product (Figure 1); and G

_{EV(I−C)}is the total mass flow rate produced by the product along the first zone I − C.

_{A}= T

_{AC}at the point C where the length of the drum is L

_{I−C}, becomes

_{I−C}is the length of the I − C zone of the dryer (see Figure 1).

_{I}is the moisture content of the product at the inlet, and X

_{C}is the critical moisture content of the product at the end of the I − C zone (Figure 1).

_{WB}is the temperature of the air at the wet-bulb condition; ${x}_{AI}$ is the absolute humidity of air at the dryer inlet, with its value of 0.01 kg/kg

_{D.A.}considered constant, due to the external air temperature of 25 °C and at 50% relative humidity also being considered constant; x

_{WB}is the absolute humidity of the saturated air (i.e., in the wet-bulb condition); p

_{SWB}is the vapor pressure in the wet-bulb condition (saturated air); and p

_{atm}is the atmospheric pressure, as the dryer operates at this pressure.

_{WB}with respect to the air inlet temperature T

_{AI}.

#### 2.1.2. Second Zone of the Dryer (L_{C−E})

_{E}, which is the product moisture content at the exit of dryer; ${L}_{C-E}$, which is the length of the second zone of the drum dryer; T

_{PE}, the product temperature at the exit of the dryer, which is greater than the wet-bulb temperature T

_{WB}[20]; and ${r}_{C-E}$, which is the thermal energy necessary to produce 1 kg of superheated vapor at the air temperature T

_{A}(Figure 1), and is equal to the difference in enthalpy [21] between the superheated vapor at T

_{A}and the water contained in the product to be dried at the temperature T

_{P}. The value of r

_{C−E}will be greater than that of r

_{I−C}for the first zone with X > X

_{C}as, below a certain value of the moisture content X (i.e., lower than the critical one X

_{C}), evaporation of the bound water requires thermal energy greater than that for free-form water.

_{C}(Figure 1).

#### 2.2. The Simplified Mathematical Model

_{A}) coming from the product does not participate in the production of dq, due to the lowering of the temperature equal to dT

_{A}—then the ODE can be simplified, and its solution provides the following equation:

_{I−C}in the I − C zone as

_{mL(I−C)}in the I − C zone.

_{AI}—often over 500 °C [1]—for which the error with Equation (18)—and, therefore, Equation (20)—is not negligible, as demonstrated in Section 3.5. The same Equation (20) has been used in [17,19,20], precisely because the conveyor belt dryers presented and studied used air inlet temperatures T

_{AI}below 150 °C.

_{WB}depends on the air inlet temperature, according to Equations (11)–(13), and as the product exit temperature T

_{PE}must remain limited to avoid fire risks (particularly, within 5–10 °C above the T

_{WB}) in Equation (22), only two unknowns remain: the air temperature T

_{AI}and T

_{AE}.

_{EV(I−C)}and G

_{EV(C−E)}depend on the moisture content of the product at the inlet X

_{I}and exit X

_{E}of the drum, according to Equations (10) and (17), Equation (24) provides the value of the air inlet temperature which ensures the achievement of the required moisture content X

_{E}, based on simplified mathematical modelling.

#### 2.3. The Rotary Drum Dryer and the Product

#### 2.4. The Mass Flow Rate of the Product G_{PI}

_{PI}as equal to 90% of that measured at the moment of flooding.

^{3}/revolution) were taken and weighed, in order to measure the unit mass m

_{T}(kg/revolution) from the loader for each of the five moisture values (X

_{I}). Considering the relationship between the dry mass m

_{D}and the moisture content and wet mass m

_{T}of alfalfa, ${m}_{D}=\frac{{m}_{T}}{1+{X}_{I}}$, it was found that m

_{D}is equal to 3.8 kg/revolution and is invariant with respect to moisture content. Therefore, the mass flow rate of alfalfa at the dryer inlet G

_{PI}was calculated using the following equation:

_{C}denotes the loader’s rotation (rps).

#### 2.5. The Mass Flow Rate of the Dry Air G_{DAI}

_{A}was measured at the intake of the heater (Figure 1) by means of orifice plates—one on the primary air for combustion and one on the secondary air. Therefore, the air was at the external condition, with an average temperature T

_{ext}= 25 °C, an average absolute humidity ${x}_{ext}={x}_{AI}=$ 0.01 kg/kg

_{D.A.}, and with density ρ

_{ext}= 1.18 kg/m

^{3}assumed to be constant. Considering the vapor component (x

_{ext}only 0.01) of this external humid air mixture to be negligible, and considering the mass conservation law, the mass flow rate of the hot dry air G

_{DAI}at the inlet of the drum is given by

#### 2.6. Experimental Assessment of the Thermal Energy r_{I−C} and of the Convective Heat Transfer Coefficient α·f

_{PI}, an experiment was also carried out to determine the values of the thermal energy r

_{I−C}of the first zone, where X > X

_{C}, and of the convective heat transfer coefficient multiplied by the transverse dimension α·f.

_{AI}were chosen, such that the reduced heat transfer rate supplied by the air to the product did not allow the exit moisture content X

_{E}to be lower than the critical moisture content X

_{C}. Equation (5) was used to calculate the thermal energy r

_{I−C}, while Equation (9) was used for the convective heat transfer coefficient α·f, both with the foresight to replace the length L

_{I−C}with that of the drum L

_{TOT}. Using the two equations to obtain these two quantities, r

_{I−C}and α·f, involves measuring the inlet and exit temperatures, the alfalfa and air mass flow rates, and the alfalfa moisture content at inlet X

_{I}and exit X

_{E}. PT100 resistance thermometers and data loggers were used to measure and register the T

_{AI}and T

_{AE}air temperatures, and the alfalfa temperature at the exit of the dryer, T

_{PE}, was measured using an infrared thermometer. The mass flow rates were obtained according to the measurements and calculations described in Section 2.4 and Section 2.5. Finally, the moisture content of the alfalfa at the inlet X

_{I}and exit X

_{E}was measured using a thermobalance.

#### 2.7. Experimental Assessment of the Thermal Energy r_{C−E} and Evaluation of the Accuracy of Mathematical Modelling

_{AI}, increased by 170 °C compared to the previous experiment. Thus, they turned out to be similar to those suggested in [1] and similar to those adopted by the operator of the dryer, by virtue of his personal experience. In this way, there was certainty that the exit moisture content of the alfalfa would be lower than the critical one and a high probability that the commercial moisture content expected for the storage of the alfalfa (X

_{E}= 0.111) would be reached.

_{I}, the following were measured: the air temperatures at the inlet T

_{AI}and exit T

_{AE}; the temperature of the alfalfa at the exit T

_{PE}; the moisture content of the alfalfa at the exit X

_{E}. The mass flow rates for the alfalfa G

_{PI}were those obtained from the tests on the flooding risks (see Section 2.4) and mass flow rates of the air G

_{DAI}were those calculated using the method indicated in Section 2.5.

_{I−C}and of the convective heat transfer coefficient α·f determined with the method indicated in Section 2.6.

_{AC}, where the product assumes the critical moisture content X

_{C}. With Equation (8), the distance of point C with respect to the inlet was calculated, following which the length of the drum of the second zone, L

_{C−E}= L

_{TOT}− L

_{I−C}, was calculated. With Equation (14), the value of the thermal energy r

_{C−E}corresponding to the second zone C − E was calculated (Figure 1).

_{PE}of the alfalfa at the exit of the dryer were calculated. These values were compared with the measured ones, allowing us to verify the accuracy of the proposed mathematical model.

## 3. Results

#### 3.1. Product Mass Flow Rate G_{PI}

_{PI}to avoid product blockage in the drum (flooding), as indicated in Section 2.4, are provided in Table 2, together with the five moisture values of the products X

_{I}introduced into the drum dryer. By applying non-linear regression (R

^{2}= 0.994), the following relationship between the optimal mass flow rate of the product G

_{PI}and the moisture content X

_{I}was obtained:

#### 3.2. Thermal Energy r_{I−C} and Convective Heat Transfer Coefficient α·f

_{I}, Table 2 shows the hot air inlet temperatures T

_{AI}chosen with reduced values, according to the criterion indicated in Section 2.6 (i.e., to have an exit moisture content of the product X

_{E}higher than the critical one; also shown in Table 2). Clearly, the temperatures T

_{AI}increased with the moisture content of the product at the inlet X

_{I}, given that, as the amount of water to be evaporated increases, it is necessary to increase the heat transfer rate from the air to the product. As the (logarithmic) mean temperature difference between the air and the product must also increase, there is also an increase in the air exit temperature T

_{AE}. Table 2 shows these measured temperatures T

_{AE}and the wet-bulb temperatures T

_{WB}calculated using Equations (11)–(13). These were found to be superimposable on the measured product exit temperatures T

_{PE}, confirming that, with X > X

_{C}, the presence of water on the surface of the product allows it to remain at T

_{WB}.

_{DAext}introduced into the heater, measured as indicated in Section 2.5. As already mentioned, for the law of conservation of mass, neglecting the contribution of that of the fuel, it is equal to the mass flow rate of the dry air entering the drum G

_{DAI}. Note that G

_{DAI}decreased with an increase in the air inlet temperature T

_{AI}and, therefore, in the air exit temperature T

_{AE}. This fact can be explained by the decrease in the density of the air at the intake of the centrifugal fan (Figure 1). In fact, this turbomachine should keep the volumetric flow rate constant, as it depends on the diameter of the impeller, the inclination of the blades, and the rotation speed, which are fixed. Therefore, if the fan intake air is hotter, a reduction in density and, therefore, in the mass flow rate of the air at the drum exit follows. This is reflected in a reduction in the mass flow rate G

_{DAI}at the drum inlet. However, it appears that with warmer and less dense air, the pressure drop in the drum is reduced. Therefore, the fan shifts the operating point towards a partial increase in the volumetric flow rate, according to the characteristic curve, but which is not sufficient to keep the mass flow rate G

_{DAI}constant.

_{I−C}obtained by Equation (5), introducing into it: the measured temperatures T

_{AI}and T

_{AE}, the latter in place of T

_{AC}; c

_{A}and c

_{V}, as shown in Table 1; and the air and vapor mass flow rates G

_{DAI}and G

_{EV}, presented in Table 2. G

_{EV}was calculated using Equation (10).

_{I−C}turned out to be higher than that obtained in previous experience on a conveyor belt dryer [17]. In fact, it should be noted that thermal energy includes the latent heat, the heat to overheat the steam generated up to the temperature T

_{A}, and the heat losses from the walls of the drum. In the rotary drum dryer, these latter two contributions are clearly higher than that in a conveyor belt dryer, due to the much higher T

_{A}temperature and the rotation, which increases the convective heat transfer coefficient. By applying non-linear regression (R

^{2}= 0.935), the following relationship was found between the thermal energy in the first zone, where X > X

_{C}, r

_{I−C}, and the product inlet moisture content X

_{I}:

^{2}= 0.945), the following relationship between the convective heat transfer coefficient multiplied by the transverse dimension α·f and the moisture content X

_{I}was determined:

#### 3.3. Thermal Energy r_{C−E} and Evaluation of the Accuracy of the Mathematical Model

_{C−E}in the second zone C − E of the drum (Figure 1) where X < X

_{C}, are shown in Table 3. The r

_{C−E}was calculated by applying Equation (14), which required the knowledge of other quantities, such as the following:

- -
- -
- The air exit temperatures T
_{AE}, which were measured and shown in Table 3; - -
- The product exit moisture contents X
_{E}, which were measured and shown in Table 3; - -
- The wet-bulb temperatures T
_{WB}, calculated using Equations (11)–(13); - -
- The air temperatures T
_{AC}at point C, where the alfalfa was at critical moisture content, calculated by means of Equation (5); - -
- The air inlet mass flow rates G
_{DAI}, measured as indicated in Section 2.5; - -
- The vapor mass flow rates of the first zone I − C (Figure 1) G
_{EV(I−C)}, calculated by means of Equation (10); - -
- The vapor mass flow rates of the second zone C − E (Figure 1) G
_{EV(C−E)}, calculated by means of Equation (17).

_{C−E}of the second zone (Figure 1) were inserted into Equation (15), in order to calculate the values of the product exit temperature T

_{PE}, which are shown in Table 3 together with those measured during the tests. The maximum relative error was 2.4% and, in all cases, the differences were not statistically significant. Therefore, we found the complete mathematical modelling results from these first experimental tests to be accurate.

^{2}= 0.164), the following relationship was found between the thermal energy in the second zone, where X < X

_{C}, r

_{C−E}, and the product inlet moisture content X

_{I}:

^{2}= 0.996), the following relationship was found between the air inlet mass flow rate G

_{DAI}and the air inlet temperature T

_{AI}:

#### 3.4. Drying Control Using Complete Mathematical Modelling

_{I−C}and r

_{C−E}, and the product inlet mass flow rate G

_{PI}, all as a function of the product inlet moisture content X

_{I}, based on the relationships presented in Equations (29), (28), (30) and (27), respectively.

_{E}as a constant, equal to the commercial value of 0.111 (10% w.b.). It is important that X

_{E}remains constant and equal to the commercial value as if X

_{E}< 0.111, then the energy consumption and the risk of fire are both increased. Meanwhile, if X

_{E}> 0.111, then the shelf-life and the quality of the alfalfa may be reduced.

_{E}to be constant and equal to 0.111, it is necessary to act on the air inlet temperature; that is, it is necessary to calculate T

_{AI}using the modelling equations after setting the desired value of X

_{E}. The equations are the following:

- -
- Equation (10), to obtain the mass flow rate of evaporated water G
_{EV(I−C)}from the first zone I − C (Figure 1), knowing the product moisture content X_{I}and X_{C}; - -
- Equation (17), to obtain the mass flow rate of evaporated water G
_{EV(C−E)}from the second zone C − E (Figure 1), knowing the product moisture content X_{I}, X_{C}, and X_{E}(as it is forced to be equal to 0.111); - -
- Equations (5), (8), (14), and (15), to obtain the four unknowns (i.e., the three temperatures T
_{AI}, T_{AE}, and T_{AC}, as well as the length of the first zone of the drum L_{I−C}). The wet-bulb temperature T_{WB}and the air mass flow rate G_{DAI}also appear in the system of equations. However, T_{WB}can be determined through Equations (11)–(13), in which the unknown T_{WB}is a function of the temperature T_{AI}. The air mass flow rate G_{DAI}is a function of T_{AI}through Equation (31). The system of Equations (5), (8), (14), (15), (11), (12), (13), and (31) must be solved with a recursive procedure, which is easily conducted in spreadsheets, as the equations implicitly contain the unknowns.

_{PE}and the wet-bulb T

_{WB}were between 5 and 8.7 °C, thus reducing the risk of fire.

#### 3.5. Drying Control Using Simplified Mathematical Model

_{AI}is higher than that obtained by the complete mathematical model, which is also presented in Table 5 for a direct comparison. The simplified model matches the results of the complete model for T

_{AI}below about 150 °C, while it becomes less accurate at the higher temperatures essential for drying products with higher inlet moisture X

_{I}. For example, in test no. 5, the simplified model suggested a T

_{AI}of 604 °C, compared to the 558 °C derived by the complete model, presenting a difference of +8.2%.

_{AI}higher than those required to achieve the expected product exit content X

_{E}. This means that the product will exit with a lower content than expected. To estimate this, the T

_{AI}of 604 °C was implemented in the complete mathematical model in the case of test no. 5, and it was found that X

_{E}was 0.076, compared to the desired 0.111. This excessive product drying leads to a waste of energy (+8.2%) and, above all, a higher product exit temperature T

_{PE}which, in test no. 5, was found to be 17 °C higher than the T

_{WB}(i.e., about 82 °C), thus increasing the risk of fire.

## 4. Discussion and Conclusions

_{PE}, which resulted in a maximum relative error of only 2.4%, compared to the experimental values.

_{I}of alfalfa (from a minimum of about 0.4 to a maximum of about 2). In all cases, the mathematical models were required to maintain the product exit moisture content X

_{E}constant and equal to 0.111, which is precisely the drying control task. The simplified model provided higher air inlet temperatures T

_{AI}than the complete model. The T

_{AI}differences were found to be negligible for reduced X

_{I}values (i.e., when the air entered with a T

_{AI}lower than 150 °C), but they gradually became more marked at higher X

_{I}. For X

_{I}equal to 2 (in the fifth simulation test), the two T

_{AI}temperatures were 604 °C and 558 °C, respectively, with a difference of +8.2% for the simplified model, notably reflected in an equal increase in energy consumption.

_{E}. To determine the actual value of the moisture content X

_{E}with the T

_{AI}temperatures proposed by simplified mathematical model, these temperatures were implemented in the complete mathematical model, thus obtaining, in the extreme case of the fifth test, an X

_{E}of 0.076 and, above all, a product exit temperature of 82 °C (17 °C higher than the wet-bulb temperature)—a very high value, increasing the risk of fire and reducing the quality of the product. This fact indicates the good opportunity provided by the proposed model, in order to better realize drying control.

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Rotary drum dryer mass flows and diagram of the temperature of the air (red line) and the product (blue line). The product has final moisture content X

_{E}< X

_{C}.

**Figure 2.**Diagram of the temperature of the air T

_{A}(red line) and product T

_{P}(blue line) vs. the exchange surface A. The elemental area dA is equal to the product of elemental length dz and the quantity f (see Figure 3).

**Figure 3.**Generic section of the drum containing N product elements. In the foreground, the i-th element with perimeter f

_{i}is shown. The average value of the sum of the perimeters f

_{i}of all elements in the generic drum section constitutes the “transverse dimension,” $f={\sum}_{i=1}^{N}{f}_{i},$ which is multiplied by the infinitesimal axial length dz to give the elemental area dA = f·dz.

Quantity | Symbol | Value |
---|---|---|

Drum diameter | D (m) | 2.1 |

Total drum length | L_{T} (m) | 12.2 |

Drum rotation | N (R.P.M.) | 6 |

Specific heat of dry air | c_{A} (J K^{−1} kg^{−1}) | 1005 |

Specific heat of vapor | c_{V} (J K^{−1} kg^{−1}) | 1926 |

Quantity | Symbol | Test n. 1 | Test n. 2 | Test n. 3 | Test n. 4 | Test n. 5 |
---|---|---|---|---|---|---|

Inlet moisture content | X_{I} | 0.410 ± 0.029 | 0.580 ± 0.027 | 0.761 ± 0.034 | 1.237 ± 0.031 | 2.030 ± 0.049 |

Exit moisture content | X_{E} | 0.305 ± 0.013 | 0.353 ± 0.009 | 0.368 ± 0.012 | 0.362 ± 0.011 | 0.387 ± 0.016 |

Critical moisture content [20] | X_{C} | 0.290 | 0.290 | 0.290 | 0.290 | 0.290 |

Air inlet temperature | T_{AI} (°C) | 70.3 ± 0.6 | 120.5± 0.9 | 179.2 ± 0.8 | 300.4 ± 1.0 | 400.6 ± 0.8 |

Air exit temperature | T_{AE} (°C) | 36.2 ± 0.7 | 51.1 ± 1.0 | 68.4 ± 0.6 | 99.2 ± 0.8 | 117.6 ± 0.9 |

Wet-bulb temperature | T_{WB} (°C) | 29.4 | 37.8 | 44.7 | 53.9 | 58.9 |

Product exit temperature | T_{PE} (°C) | 28.9 ± 0.5 | 37.2 ± 0.4 | 44.0 ± 0.6 | 53.0 ± 0.6 | 58.2 ± 0.7 |

Product inlet mass flow rate | G_{PI} (kg s^{−1}) | 1.061 ± 0.032 | 1.028 ± 0.031 | 0.996 ± 0.033 | 0.928 ± 0.029 | 0.851 ± 0.031 |

Air inlet mass flow rate | G_{DAext} = G_{DAI}(kg s ^{−1}) | 8.982 | 8.646 | 8.236 | 7.413 | 6.687 |

Vapor mass flow rate | G_{EV} (kg s^{−1}) | 0.0792 | 0.1475 | 0.2221 | 0.3627 | 0.4614 |

Thermal energy (X > X_{C}) | r_{I−C} (kJ kg^{−1}) | 3536 | 4124 | 4246 | 4313 | 4380 |

Convect. heat transf. coef. x transverse dimension | α·f (W m^{−1}K^{−1}) | 1334 | 1322 | 1218 | 1092 | 1047 |

**Table 3.**Experimental and calculated data from tests described in Section 2.7.

Quantity | Symbol | Test n. 1 | Test n. 2 | Test n. 3 | Test n. 4 | Test n. 5 |
---|---|---|---|---|---|---|

Inlet moisture content | X_{I} (d.b.) | 0.410 ± 0.029 | 0.580 ± 0.027 | 0.761± 0.034 | 1.237 ± 0.031 | 2.030 ± 0.049 |

Exit moisture content | X_{E} (d.b.) | 0.136 ± 0.012 | 0.145 ± 0.010 | 0.147 ± 0.009 | 0.126 ± 0.011 | 0.086 ± 0.009 |

Critical moisture content | X_{C} (d.b.) | 0.290 | 0.290 | 0.290 | 0.290 | 0.290 |

Air inlet temperature | T_{AI} (°C) | 240.2 ± 1.0 | 290.4 ± 0.9 | 350.1 ± 0.9 | 469.3 ± 0.8 | 570.7 ± 1.1 |

Air temperature at point C | T_{AC} (°C) | 200.0 | 188.8 | 193.2 | 209.9 | 215.4 |

Air exit temperature | T_{AE} (°C) | 84.4 ± 0.7 | 91.1 ± 0.8 | 101.2 ± 0.8 | 122.8 ± 1.2 | 134.6 ± 0.9 |

Wet-bulb temperature | T_{WB} (°C) | 49.9 | 53.3 | 56.7 | 62.0 | 65.4 |

Product inlet mass flow rate | G_{PI} (kg s^{−1}) | 1.061 ± 0.032 | 1.028 ± 0.031 | 0.996 ± 0.033 | 0.928 ± 0.029 | 0.851 ± 0.031 |

Air inlet mass flow rate | G_{DAI} (kg s^{−1}) | 7.968 | 7.462 | 6.941 | 6.110 | 5.549 |

Vapor mass flow rate (first zone I − C) | G_{EV(I−C)}(kg s ^{−1}) | 0.0904 | 0.1884 | 0.2662 | 0.3928 | 0.4885 |

Vapor mass flow rate (second zone C − E) | G_{EV(C−E)}(kg s ^{−1}) | 0.1158 | 0.0943 | 0.0812 | 0.0681 | 0.0574 |

Convect. heat transf. coef. x transverse dimension | α·f (W m^{−1}K^{−1}) | 1334 | 1322 | 1218 | 1092 | 1047 |

Thermal energy (X < X_{C}) | r_{C−E} (kJ kg^{−1}) | 8102 | 7863 | 7980 | 7930 | 7922 |

Product exit temperature measured | T_{PE} (°C) | 54.3 ± 0.6 | 60.7 ± 0.7 | 64.1 ± 0.8 | 68.0 ± 0.6 | 72.9 ± 0.7 |

Product exit temperature calculated | T_{PE} (°C) | 55.1 | 59.6 | 62.6 | 68.8 | 74.3 |

**Table 4.**Data simulated from the complete mathematical model used for drying control to maintain the product exit moisture content X

_{E}at a constant level (=0.111).

Quantity | Symbol | Test n. 1 | Test n. 2 | Test n. 3 | Test n. 4 | Test n. 5 |
---|---|---|---|---|---|---|

Inlet moisture content | X_{I} (d.b.) | 0.410 | 0.580 | 0.761 | 1.237 | 2.030 |

Imposed exit moisture content | X_{E} (d.b.) | 0.111 | 0.111 | 0.111 | 0.111 | 0.111 |

Critical moisture content | X_{C} (d.b.) | 0.290 | 0.290 | 0.290 | 0.290 | 0.290 |

Air inlet temperature | T_{AI} (°C) | 270.0 | 322.0 | 383.0 | 479.0 | 558.0 |

Air temperature at point C | T_{AC} (°C) | 228.5 | 218.0 | 223.8 | 219.0 | 203.1 |

Air exit temperature | T_{AE} (°C) | 88.4 | 94.0 | 107.2 | 124.2 | 132.1 |

Wet-bulb temperature | T_{WB} (°C) | 52.0 | 55.2 | 58.1 | 62.3 | 65.0 |

Product inlet mass flow rate | G_{PI} (kg s^{−1}) | 1.061 | 1.028 | 0.996 | 0.928 | 0.851 |

Air inlet mass flow rate | G_{DAI} (kg s^{−1}) | 7.631 | 7.254 | 6.811 | 6.115 | 5.543 |

Vapor mass flow rate (first zone I − C) | G_{EV(I−C)}(kg s ^{−1}) | 0.0904 | 0.1884 | 0.2662 | 0.3928 | 0.4885 |

Vapor mass flow rate (second zone C − E) | G_{EV(C−E)}(kg s ^{−1}) | 0.1348 | 0.1167 | 0.1014 | 0.0743 | 0.0503 |

Convect. heat transf. coef. x transverse dimension | α·f (W m^{−1}K^{−1}) | 1334 | 1322 | 1218 | 1092 | 1047 |

Thermal energy (X < X_{C}) | r_{C−E} (kJ kg^{−1}) | 8102 | 7863 | 7980 | 7930 | 7922 |

Product exit temperature measured | T_{PE} (°C) | 57.0 | 60.4 | 64.8 | 69.6 | 73.7 |

Product exit temperature calculated | T_{PE} (°C) | 55.1 | 59.6 | 62.6 | 68.8 | 74.3 |

**Table 5.**Comparison between the air and product temperatures simulated with the complete mathematical model and the simplified one. The goal is to control the dryer, through the air inlet temperature value, to keep the product exit moisture content X

_{E}constant (at 0.111). The last line shows the exit moisture content calculated with the complete model after having implemented the air inlet temperatures simulated using the simplified model.

Quantity | Symbol | Test n. 1 | Test n. 2 | Test n. 3 | Test n. 4 | Test n. 5 |
---|---|---|---|---|---|---|

Inlet moisture content | X_{I} (d.b.) | 0.410 | 0.580 | 0.761 | 1.237 | 2.030 |

Imposed exit moisture content | X_{E} (d.b.) | 0.111 | 0.111 | 0.111 | 0.111 | 0.111 |

Air inlet temperature from complete model | T_{AI-compl} (°C) | 270.0 | 322.0 | 383.0 | 479.0 | 558.0 |

Air inlet temperature from simplified model | T_{AI-simpl} (°C) | 278.0 | 334.0 | 401.0 | 511.0 | 604.0 |

Air exit temperature from complete model | T_{AE-compl} (°C) | 88.4 | 94.0 | 107.2 | 124.2 | 132.1 |

Air exit temperature from simplified model | T_{AE-simpl} (°C) | 90.1 | 96.1 | 109.0 | 125.8 | 138.4 |

Product exit temperature from complete model | T_{PE-compl} (°C) | 57.0 | 60.4 | 64.8 | 69.6 | 73.7 |

Product exit temperature from simplified model | T_{PE-simpl} (°C) | 57.5 | 61.1 | 65.3 | 70.7 | 81.9 |

Exit moisture content from complete model and T_{AI-simpl} | X_{E} (d.b.) | 0.105 | 0.100 | 0.094 | 0.080 | 0.076 |

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**MDPI and ACS Style**

Friso, D. Mathematical Modelling of Rotary Drum Dryers for Alfalfa Drying Process Control. *Inventions* **2023**, *8*, 11.
https://doi.org/10.3390/inventions8010011

**AMA Style**

Friso D. Mathematical Modelling of Rotary Drum Dryers for Alfalfa Drying Process Control. *Inventions*. 2023; 8(1):11.
https://doi.org/10.3390/inventions8010011

**Chicago/Turabian Style**

Friso, Dario. 2023. "Mathematical Modelling of Rotary Drum Dryers for Alfalfa Drying Process Control" *Inventions* 8, no. 1: 11.
https://doi.org/10.3390/inventions8010011