# Mathematical Modelling of Conveyor-Belt Dryers with Tangential Flow for Food Drying up to Final Moisture Content below the Critical Value

## Abstract

**:**

_{F}lower than the critical one X

_{C}(X

_{F}< X

_{C}). In fact, this work follows a precedent in which a mathematical model was developed through the differentiation of the drying rate equation along the dryer belt with the hypothesis that the final moisture content X

_{F}of the product was higher than the critical one X

_{C}. The relationships between the extensive quantities (air flow rate and product flow rate), the intensive quantities (temperatures, moisture content and enthalpies) and the dimensional ones (length and width of the belt) were then obtained. Finally, based on these relationships, the rules for an optimized design for X

_{F}< X

_{C}were obtained.

## 1. Introduction

_{F}higher than the critical one X

_{C}(X

_{F}> X

_{C}) and the air that continuously changes its temperature inside the dryer in order to offer a series of design guidelines. In the present work, with the aim of completing the study related to the case of final moisture content of the product lower than the critical one (X

_{F}< X

_{C}), broader mathematical modelling will be performed for these dryers. Finally, after the experimental validation of the mathematical modelling, the design guidelines will be proposed.

## 2. Materials and Methods

#### 2.1. Mathematical Modelling

_{A}and the product T

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

_{I-C}, where the product maintains the moisture content X higher than the critical moisture content X

_{C}and, last, long L

_{C-E}where the moisture content of the product X is lower than the critical one X

_{C}.

#### 2.1.1. First Zone of the Dryer L_{I-C}

_{I-C}, the product is characterized by a moisture content X higher than the critical one X

_{C}, and therefore it maintains its temperature T

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

_{WB}[1]. In this first zone, the equations proposed in the previous work [1] are valid. The first equation, called (12) in [1], concerns the heat transfer rate q

_{I-C}that the warm and dry air of initial mass flow rate G

_{AI}releases when cooling from the input temperature T

_{AI}to temperature T

_{AC}corresponding to the achievement of critical moisture content (Figure 1):

_{A}is the specific heat of dry air; η is the corrective coefficient introduced in [2] and related to the heat losses. For the pilot dryer used in the experiments performed in this work, η is equal to 0.965.

_{I-C}exchanged between the air and the product:

_{mL(I-C)}is the logarithmic mean temperature difference in the I-C zone; A

_{I-C}is the total area of the product inside the I-C zone of the dryer.

_{I-C}is the length of the I-C zone of the dryer (Figure 1). Therefore, the equation that gives the heat transfer rate exchanged between air and product is:

_{EV(I-C)}is the mass flow rate of the evaporated water in the I-C zone; H

_{I}is the input height and B

_{I}is the input width of the bulk product on the belt; v

_{Belt}is the belt speed; ρ

_{BulkI}is the bulk density of the input product; X

_{I}is the input moisture content; X

_{C}is the critical moisture content; r

_{I-C}is the thermal energy, in the I-C zone, required to produce 1 kg of superheated steam at the air temperature T

_{A}; r

_{I-C}is equal to the difference in enthalpy [3] between the superheated steam at T

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

_{P}. As indicated in [1] r

_{I-C}is considered constant and has an average value of 2617 kJ kg

^{−1}.

#### 2.1.2. Second Zone of the Dryer L_{C-E}

_{C}), the first equation proposed concerns the heat transfer rate q

_{C-E}that the warm air of initial flow rate G

_{AI}releases when cooling from the temperature T

_{AC}to the exit temperature T

_{AE}. This equation is similar to (1):

_{AI}is the mass flow rate of drying air; c

_{A}is its specific heat; η is the corrective coefficient introduced in [2], related to the heat losses and equal to 0.965 for the pilot dryer used in the experiments. The steam mass flow rate coming from the previous L

_{I-C}dryer area and mixed with the air, has been neglected because analyzing the data indicated in [1] it is about 2% of the dry air flow rate G

_{AI}and therefore two orders of magnitude less than G

_{AI}.

_{C-E}exchanged between air and product, is similar to (2):

_{mL(C-E)}is logarithmic mean temperature difference in the C-E zone; f is the transverse dimension shown in Figure 5 of [1]; L

_{C-E}is the length of the C-E zone of the dryer (Figure 1).

_{C-E}required by the water to evaporate and, similarly to (3), it is:

_{EV(C-E)}is the mass flow rate of the evaporated water in the C-E zone; H

_{I}is the input height and B

_{I}is the input width of the bulk product on the belt; v

_{Belt}is the belt speed; ρ

_{BulkC}is the bulk density in the point where the product has the critical moisture content; X

_{C}is the critical moisture content; X

_{F}is the final moisture content; r

_{C-E}is the thermal energy, in the C-E zone, required to produce 1 kg of superheated steam at the air temperature T

_{A}and is equal to the difference in enthalpy [3] between the superheated steam at T

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

_{P}. The thermal energy r

_{C-E}is currently unknown.

_{BulkC}of the product at point C to the input bulk density ρ

_{BulkI}.

_{D}is the dry mass; m

_{W}is the mass of water; V is the bulk volume; we obtain:

_{C-E}zone with X < X

_{C}is characterized by a drying constrained to the process of internal diffusion of water towards the surface of the product. The diffusion of water inside the product is described by Fick’s second law, i.e., by a partial differential equation (PDE) that in the one-dimensional case is: $\frac{\partial X}{\partial t}=D\frac{{\partial}^{2}\mathrm{X}}{\partial {y}^{2}}$.

_{C}is the critical moisture content of the product and corresponds to the initial one of the drying in the L

_{C-E}zone; X

_{eq}is the equilibrium moisture content; D is the mass diffusivity of water inside the product; δ is half the thickness of the plate.

_{EV}is the instantaneous mass flow rate of evaporated water; m

_{D}is the dry mass present in the L

_{C-E}section of the dryer and is [2]: ${m}_{D}=\frac{{H}_{I}\xb7{B}_{I}\xb7{\rho}_{BulkC}\xb7{L}_{C-E}}{1+{X}_{C}}$, that is, based on (7): ${m}_{D}=\frac{{H}_{I}\xb7{B}_{I}\xb7{\rho}_{BulkI}\xb7{L}_{C-E}}{1+{X}_{I}}$. Furthermore, since the belt speed v

_{Belt}is constant and: ${v}_{Belt}=\frac{z}{t}$, the previous (9) becomes:

_{EV}varies along the horizontal coordinate z within the L

_{C-E}zone of the dryer (Figure 1) according to the exponential function as shown in Figure 2.

_{EV(C-E)}can be calculated as follows:

_{1}and C

_{2}are constants to be determined by experiments.

_{C-E}zone where the moisture content is X < X

_{C}.

#### 2.2. Design Guidelines

_{I-C}, q

_{C-E}, G

_{AI}, ρ

_{BulkI}, T

_{AC}, L

_{I-C}, L

_{C-E}and T

_{PE}. Therefore, this system of equations has a solution, which will be found in the next sub-paragraphs, where the design guidelines will be also proposed.

#### 2.2.1. Air Temperature T_{AC} at Critical Moisture Content of the Product

_{AI}and exit T

_{AE}of the dryer can be imposed a priori [1], the combination of Equations (1) and (3) gives:

_{AC}temperature (Figure 1):

#### 2.2.2. Length of Dryer L_{I-C} to Reach Critical Moisture Content X_{C}

_{I-C}corresponding to the zone with X > X

_{C}:

#### 2.2.3. Mass Flow Rate of Drying Air G_{AI}

_{AI}:

#### 2.2.4. Length of Dryer L_{C-E} to Reduce Moisture Content from Critical Value X_{C} to Final One X_{F}

_{C-E}where the product dries from X

_{C}to X

_{F}:

#### 2.2.5. Temperature Difference Δt_{b} at the Dryer Exit and Product Exit Temperature T_{PE}

_{AE}is the air exit temperature and T

_{PE}is the product exit temperature:

_{b}can be obtained by an iterative method easily through a spreadsheet. As seen in Section 2.2.1, the T

_{AE}temperature can be set a priori and, therefore, the product exit temperature ${T}_{PE}={T}_{AE}-\Delta {T}_{b}$ can be obtained.

#### 2.2.6. Known Quantities and Experimental Quantities

- The input and exit temperatures of the drying air T
_{AI}and T_{AE}can be defined using the guidelines 3.1.1 proposed in [1]. - The compound quantity F·α is the product of the convective heat transfer coefficient α and the form factor $F=\frac{f}{{H}_{I}\xb7{B}_{I}}$, where f is the transverse dimension (Figure 5 in [1]) and H
_{I}and B_{I}are the height and width of the bulk product placed above the belt, respectively. The quantity F·α is measured as indicated in the guidelines 3.1.9 of [1]. - The bulk density ρ
_{BulkI}is measured as indicated in the next point I) of Section 2.3. - The thermal energy r
_{I-C}has an average value of 2617 kJ kg^{−1}, as indicated in guidelines 3.1.6 of [1]. - The constants C
_{1}and C_{2}, where the first one can be connected to the diffusivity of the water inside the product and to the size and the second concerns the initial delay, must be determined using the experimental method that will be described in the next Section 2.3. The same experimental method will also allow us to determine the critical moisture content X_{C}and the equilibrium moisture content X_{eq}. - To determine the lengths L
_{I-C}and L_{C-E}of the dryer, it is necessary to impose the belt speed v_{Belt}and the height H_{I}and the width B_{I}of the bulk product above the belt. These three quantities must be chosen a priori in order to reach the total mass flow rate of evaporated water G_{EV(I-E)}foreseen for the dryer. In fact, it is known that the quantity characterizing a dryer, both technically and commercially, is the G_{EV(I-E)}quantity. Therefore, the designer must start from: a known value of the mass flow rate G_{EV(I-E)}; the input and final moisture content and from bulk density of the product; an equation obtained by adding the mass flow rate of evaporated water G_{EV(I-C)}in the L_{I-C}zone, and the one G_{EV(C-E)}of the L_{C-E}zone:$${G}_{EV\left(I-E\right)}={G}_{EV\left(I-C\right)}+{G}_{EV\left(C-E\right)}={H}_{I}\xb7{B}_{I}\xb7{v}_{Belt}\xb7{\rho}_{BulkI}\frac{{X}_{I}-{X}_{F}}{1+{X}_{I}}$$

_{I}, B

_{I}and hence can obtain the v

_{Belt}.

- g.
- Finally, the thermal energy r
_{C-E}in the L_{C-E}length zone with X < X_{C}must be measured. The r_{C-E}will be greater than the r_{I-C}of the zone with X > X_{C}, since below a certain value of the moisture content X, lower than the critical one X_{C}, the evaporation of the bound water requires thermal energy greater than that for free-form water. In Section 2.3 the iterative method for determining the experimental values of r_{C-E}will be described.

#### 2.3. Experimental Procedure and Equipment

- (I)
- to quantify the constants C
_{1}, C_{2}, the critical moisture content X_{C}and the equilibrium moisture content X_{eq}of the alfalfa were as foreseen in the previous point**e.**of Section 2.2.6. The four quantities, C_{1}, C_{2}, X_{C}and X_{eq}can be obtained from the experimental drying curve plotted in a diagram (time, moisture content), by interpolating the moisture content data measured each minute on a sample of the product inserted in the pilot dryer. The sample of alfalfa of initial moisture content X_{I}was placed with a height H_{I}equal to 0.05 m on a thin aluminum plate 1 m long and 0.3 m wide, i.e. like the dryer belt width. In turn, the plate was placed on a precision balance placed on the locked belt of the dryer. Only the drying air at the temperature T_{AI}equal to 60 °C was forced by the fan to lick the product sample. The total mass of the sample m_{T}= m_{W}+ m_{D}, dry mass m_{D}plus water mass m_{W}, was measured each minute. Knowing [2] that ${m}_{D}=\frac{{m}_{TI}}{1+{X}_{I}}$, where m_{TI}is the initial mass of the sample, the moisture content X of the product at instant t in which the mass of the alfalfa sample m_{T}is measured, can be calculated as follows:$$X=\frac{{m}_{T}}{{m}_{D}}-1=\frac{{m}_{T}}{{m}_{TI}}\left(1+{X}_{I}\right)-1$$_{I}is measured on a separate sample of the same alfalfa, by weighing the mass before and after drying in an oven at 135 ° C for two hours. Therefore, the drying curve plotted on the t-X diagram allows obtaining C_{1}, C_{2}, X_{C}and X_{eq}(see also the next Section 3); - (II)
- to quantify the thermal energy r
_{C-E}during drying with X<X_{C}by means of an iterative procedure as provided in point**g.**of the previous 2.2.6. The iterative procedure consists of three steps. In the first step, r_{C-E}is assumed equal to r_{I-C}that is 2617 kJ kg^{-1}and a preliminary design of the dryer is performed according to guidelines 2.2. However, some guidelines are overturned because the pilot dryer already has a predetermined length L_{T}=L_{I-C}+L_{C-E}=6 m. Therefore, in this case the sequence of calculations is seen to impose L_{T}= 6 m to derive the belt speed v_{Belt}. For the rest of the quantities, the guidelines are the same as in Section 2.2, thus determining the air temperature T_{AC}and T_{AE}, the mass flow rate of the drying air G_{AI}and the product exit temperature T_{PE}, using equations in the Section 2.2.1, Section 2.2.3 and Section 2.2.5. The second step consists in carrying out a test on the pilot dryer functioning as required by the preliminary design. During the test, the actual temperatures T’_{PE}and T’_{AE}are measured, which will be different from those of the preliminary design T_{PE}and T_{AE}. The third step consists in looking for the r_{C-E}value which, inserted in the equations of guidelines 2.2, allows to restore the initial values of the T_{PE}and T_{AE}temperatures of the preliminary project, to the experimental values T’_{PE}and T’_{AE}. Since the equations of guidelines 2.2 are implementable in a spreadsheet, it is very easy to perform this third step; - (III)
- to validate the mathematical model described in 2.1 and the design guidelines described in 2.2.

## 3. Results

_{C}, X

_{eq}, C

_{1}and C

_{2}, the series of tests was conducted as described in

**purpose (I)**of the previous Section 2.3. Figure 3 shows the time-moisture content diagram (t-X) called drying curve. The value of the critical moisture content X

_{C}was measured at the point where the straight section, which represents the drying phase at constant rate (dX/dt = R

_{C}), begins to flex, exactly when the derivative dX/dt changes by 1% with respect to the value of the slope of the line equal to R

_{C}.

_{eq}.

_{1}and C

_{2}, the diagram of Figure 4 was made using data of Figure 3 relating to the phase with decreasing drying rate, i.e., for X ≤ X

_{C}. In fact, the time on the abscissa is t’ = t − t

_{C}, where t

_{C}is the time to reach the critical moisture content X

_{C}. In ordinate there is the quantity: $\mathrm{ln}\left(\frac{X-{X}_{eq}}{{X}_{C}-{X}_{eq}}\right).$

_{2}and intercept equal to ln(C

_{1}), therefore it is sufficient to look for the regression line in the diagram of Figure 4, to obtain ln(C

_{1}) = 0.1385 and C

_{2}= 0.0026. Table 2 summarizes the results obtained.

_{C-E}required to evaporate 1 kg of water during drying with X < X

_{C}, the procedure adopted is that described in purpose II) of the previous Section 2.3. The results of the three steps are indicated in Table 3.

_{C}, r

_{C-E}is 2617 kJ/kg, i.e., the same as in the I-C section, where X > X

_{C}. In addition, a total length L

_{T}equal to 6 m was set for this preliminary design, which is the length of the pilot dryer (Table 1), in order to obtain the belt speed v

_{Belt}, through the design guidelines of Section 2.2. Instead, the belt speed v

_{Belt}is normally chosen, so that the mass flow rate of evaporated water G

_{EV(I-E)}foreseen for the dryer is satisfied through Equation (21), and consequently, through the design guidelines 2.2, the length L

_{T}is calculated.

_{PE}(red font) and the air exit temperature T

_{AE}(red font). As reported in the second column (2nd step) of Table 3, the values of these quantities are lower than those expected from the preliminary design: an experimental T’

_{AE}= 52.7 °C instead of design T

_{AE}= 57 °C and an experimental T’

_{PE}= 46.5 °C instead of design T

_{PE}= 55.6 °C. It is clear the experimental T’

_{AE}lower than the design T

_{AE}indicates that the air has more cooled to transmit a greater heat flow rate towards the product to be dried. This is an indication that the product requires a thermal energy r

_{C-E}greater than that assumed in the preliminary design, which was 2617 kJ kg

^{−1}. The decrease in the experimental T’

_{PE}compared to the design T

_{PE}can also be explained by the need to transmit a greater heat flow rate to the product due to r

_{C-E}greater than 2617 kJ kg

^{−1}. In fact, the experimental T’

_{PE}settled on a value that produced a logarithmic mean temperature difference ΔT’

_{mL(C-E)}of Equation (5) equal to 13.9 °C against the 8.5 °C of the ΔT

_{mL(C-E)}calculated during the design of the first step. Therefore, the increase in ΔT

_{mL(C-E)}is 63.2% and implies that the heat flow rate q

_{C-E}has to increase by the same amount based on Equation (5), therefore based on Equation (6) also the thermal energy r

_{C-E}has to increase by 63.2% i.e., from 2617 to 4271 kJ kg

^{−1}. This last value coincides with that obtained in the third step (Table 3) according to the procedure of purpose (II) of Section 2.3, confirming that the calculations are correct.

_{C-E}equal to 4271 kJ kg

^{−1}as just determined. For a broader validation, the experiments were doubled with the air input temperature T

_{AI}set on two values: 120 °C and 100 °C.

_{AI}equal to 120 °C. The values of the second column are those detected during the test with T

_{AI}equal to 120 °C and concern: the product moisture content at the input X

_{I}and at the exit X

_{F}; the input bulk density ρ

_{BulkI}; the air exit temperature T

_{AE}; the air temperature T

_{AC}at point C (Figure 1); the product exit temperature T

_{PE}. All the other quantities of the first column (design), particularly the length of the dryer L

_{T}, the mass air flow rate G

_{AI}and the speed of the belt v

_{Belt}resulted in the same value also during the test (second column) due to the adjustments imposed on the pilot dryer.

_{AI}equal to 100 °C.

_{AI}of 120 °C and 100 °C, there were no significant differences between the experimental values and the expected ones from the design on: the air exit temperatures T

_{AE}; the product exit temperatures T

_{PE}; the air temperatures T

_{AC}in point C of Figure 1. Even the product exit moisture content X

_{F}did not show significant differences between the experimental values and those expected from the design.

_{F}to the air exit temperature T

_{AE}, valid for final moisture content greater than the critical one (X

_{F}> X

_{C}). In this way, it was possible to indicate the indirect measurement of the final moisture content X

_{F}, useful for optimized adjustment of the drying operation, through the simpler, faster and cheaper measurement of the air exit temperature T

_{AE}. Therefore, with the validated equations of the mathematical modelling of Section 2.1 proposed for drying with final moisture content lower than the critical one (X

_{F}< X

_{C}), the comparison between (1), (3), (4), (6) and (7) gives the following equation in place of (12) of [2]:

_{C}, its input moisture content X

_{I}, its bulk density ρ

_{BulkI}, its bulk dimensions on the belt B

_{I}and H

_{I}, the thermal energy values r

_{I-C}and r

_{C-E}, the mass flow rate of air G

_{AI}, its input temperature T

_{AI}, the belt speed v

_{Belt}and by measuring the air exit temperature T

_{AE}, Equation (24) shows that the final moisture content X

_{F}is promptly obtained. The same equation, appropriately implemented in a PLC, suggests how to correct the X

_{F}, if this is not the expected value, for example by changing the belt speed v

_{Belt}, or the air input temperature T

_{AI}.

_{AI}on the following three variables was analyzed: the length of the dryer L

_{T}; the product temperature at the exit T

_{PE}; the mass flow rate of drying air G

_{AI}. Since this air temperature at the input T

_{AI}is chosen by the designer within a range that depends on the nature of the product [1], the simulation has highlighted the influence of this choice on the three variables defined above. The series of diagrams on the left of Figure 5a–c represents the trend of the three variables as a function of T

_{AI}which has values between 100 °C and 150 °C. During the simulation, the following quantities were kept constant: the belt speed v

_{Belt}equal to 0.004 m s

^{−1}; the air temperature at the exit T

_{AE}equal to 60 °C; the mass flow rate of evaporated water in the dryer G

_{EV}equal to 23 kg h

^{−1}. It is noted that the dryer length decreases significantly. This positive effect is reflected in a more compact system, but there is a slight increase in the product temperature at the exit T

_{PE}, by just a couple of degrees, which is therefore acceptable, and a reduction in the mass flow rate of drying air G

_{AI}to keep the mass flow rate of evaporated water G

_{EV}constant given the increase by T

_{AI}. Therefore, the space problem that afflicts these dryers can be mitigated with an increase in T

_{AI}.

_{T}, temperature T

_{PE}, mass flow rate G

_{AI}—vs. the air temperature at the exit T

_{AE}which is reduced from 60 °C to 50 °C, keeping constant: the belt speed v

_{Belt}equal to 0.004 m s

^{−1}; the air temperature at the input T

_{AI}equal to 120 °C; the mass flow rate of evaporated water in the dryer G

_{EV}equal to 23 kg h

^{−1}. The figure shows a slight increase in the length of dryer L

_{T}of about 0.5 m (+8%), an important decrease of the product temperature at the exit T

_{PE}which decreases from 55.7 °C to 42.8 °C improving the organoleptic and nutritional quality. It should be remembered that the product temperature T

_{P}remains at the wet bulb temperature T

_{WB}up to point C (Figure 1) which is located at about 4 m from the input of the dryer. Only in the last 2.75 m, where the moisture content of the product is below the critical one, does the temperature slowly rise up to T

_{PE}.

## 4. Conclusions

_{F}> X

_{C}). In this work, the mathematical modelling and design guidelines have been extended to the more general situation of a final moisture content lower than the critical one (X

_{F}< X

_{C}).

_{C-E}, necessary to evaporate the water when the moisture content is lower than the critical one (X < X

_{C}). In fact, it is always greater than that for X > X

_{C}since starting from a certain moisture content below the critical one X

_{C}, there is bound water which requires higher energy to evaporate compared to latent heat.

_{C}. The mathematical modelling and the equations for the design guidelines, including the method for calculating the thermal energy r

_{C-E}, have been experimentally confirmed.

_{F}> X

_{C}. It was an equation that correlated the air exit temperature T

_{AE}with the final moisture content of the product X

_{F}which needs to be known promptly and continuously during the operation of the dryer for the optimization and reduction of energy consumption [73] and for the control of thermal damage to the dried product.

_{F}< X

_{C}, able to link the final moisture content X

_{F}to the air exit temperature T

_{AE}. If the equation is implemented in a dryer adjustment PLC, it will be able to keep the X

_{F}value constant and optimized, controlling the T

_{AE}which is much easier, faster and less expensive to detect than direct methods for measuring the X

_{F}.

_{AI}and, respectively, at the exit T

_{AE}. This last analysis showed that by reducing the mass flow rate of drying air G

_{AI}, with the same T

_{AI}, the temperature T

_{AE}and consequently the exit temperature of the product T

_{PE}can be reduced with, consequently, less thermal damage to food.

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**The conveyor-belt dryer with tangential flow and diagram of the temperature of the air and product with final moisture content X

_{F}lower than the critical one X

_{C}(X

_{F}< X

_{C}).

**Figure 2.**Mass flow rate of evaporated water G

_{EV}vs. z-coordinate within the L

_{C-E}zone, corresponding to the direction of the belt in the C-E zone where the moisture content is X < X

_{C}.

**Figure 3.**Experimental drying curve of alfalfa. From it, the critical moisture content X

_{C}and the equilibrium moisture content X

_{eq}can be obtained.

**Figure 4.**Alfalfa drying curve modified to obtain the constants C

_{1}and C

_{2}, the first can be linked to the diffusivity of the water inside the product and to the size, the second to the initial delay.

**Figure 5.**Simulation using the equations of the design guidelines. Left, histograms vs. air temperature at the dryer input T

_{AI}(with v

_{Belt}, T

_{AE}and G

_{EV}kept constant), of the: (

**a**) dryer length L

_{T}; (

**b**) alfalfa temperature at the exit of the dryer T

_{PE}; (

**c**) mass flow rate of the drying air G

_{AI}. Right, histograms of the same variables, but vs. air temperature at the dryer exit T

_{AE}with v

_{Belt}, T

_{AI}and G

_{EV}kept constant: (

**d**) dryer length L

_{T}; (

**e**) alfalfa temperature at the exit of the dryer T

_{PE}; (

**f**) mass flow rate of the air G

_{AI}.

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

Belt width | B_{I} (m) | 0.3 |

Total Belt length | L_{T} (m) | 6.0 |

Alfalfa bulk height | H_{I} (m) | 0.05 |

Flow section of the drying air | A_{A} (m^{2}) | 0.15 |

Form factor·Convective heat transfer coefficient [1] | F·α (W·m^{−3}·K^{−1}) | 5144 |

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

Alfalfa input moisture content (D.B.) | X_{I} | 1.688 ± 0.105 |

Alfalfa input bulk density | ρ_{BulkI} (kg·m^{−3}) | 183 ± 7.6 |

Alfalfa critical moisture content (D.B.) | X_{C} | 0.290 |

Alfalfa equilibrium moisture content (D.B.) | X_{eq} | 0.041 |

Coefficient related to delay | C_{1} | 1.149 |

Coefficient related to diffusivity | C_{2} | 0.0026 |

**Table 3.**Results of the three steps to obtain the thermal energy r

_{C-E}concerning the humidity X < X

_{C}. The first step refers to the results of the preliminary design with r

_{C-E}equal to r

_{I-C}= 2617 kJ/kg. The second step concerns the experimental measurement of moisture content and temperatures of the air and of the product (red font). The third step concerns the r

_{C-E}value found (red font) through the equations of the design guidelines 2.3 by imposing the experimental temperatures (red font).

Quantity | Symbol | 1st Step Preliminary Design | 2nd Step Exper. Value | 3rd Step Search for r _{C-E} |
---|---|---|---|---|

Thermal energy | r_{C-E} (kJ kg^{−1}) | 2617 | 4271 | |

Input moisture content | X_{I} | 1.688 | 1.688 ± 0.105 | 1.688 |

Final moisture content | X_{F} | 0.122 | 0.121 ± 0.01 | 0.122 |

Input bulk density | ρ_{BulkI} (kg m^{−3}) | 183 | 183 ± 7.6 | 183 |

Critical moisture content | X_{C} | 0.290 | = | 0.290 |

Equilibrium moisture content | X_{eq} | 0.041 | = | 0.041 |

Air input temperature | T_{AI} (°C) | 120 | 119.7 ± 1.2 | 120 |

Air exit temperature | T_{AE} (°C) | 57 | 52.7 ± 1.1 | 52.7 |

Belt velocity | v_{Belt} (m s^{−1}) | 0.0036 | = | 0.0036 |

Air temperature in C | T_{AC} (°C) | 63.8 | = | 63.8 |

Dryer length I-C (Figure 1) | L_{I-C} (m) | 3.55 | = | 3.55 |

Dryer length C-E (Figure 1) | L_{C-E} (m) | 2.45 | = | 2.45 |

Total dryer length | L_{T} (m) | 6.00 | = | 6.00 |

Product exit temperature | T_{PE} (°C) | 55.6 | 46.5 ± 0.7 | 46.5 |

Air input mass flow rate | G_{AI} (kg s^{−1}) | 0.246 | 0.246 ± 0.006 | 0.246 |

Quantity | Symbol | Design | Exper. Value | Design | Exper. Value |
---|---|---|---|---|---|

Air input temperature | T_{AI} (°C) | 120 | 119.7 ± 1.2 | 100 | 100.9 ± 1.1 |

Thermal energy (X > X_{C}) | r_{I-C} (kJ kg^{−1}) | 2617 | = | 2617 | = |

Thermal energy (X < X_{C}) | r_{C-E} (kJ kg^{−1}) | 4271 | = | 4271 | = |

Input moisture content | X_{I} | 1.688 | 1.688 ± 0.105 | 1.688 | 1.688 ± 0.105 |

Final moisture content | X_{F} | 0.122 | 0.120 ± 0.01 | 0.122 | 0.124 ± 0.009 |

Input bulk density | ρ_{BulkI} (kg m^{−3}) | 183 | 183 ± 7.6 | 183 | 183 ± 7.6 |

Critical moisture content | X_{C} | 0.290 | = | 0.290 | = |

Equilibrium moisture content | X_{eq} | 0.041 | = | 0.041 | = |

Air exit temperature | T_{AE} (°C) | 57 | 56.7 ± 1.0 | 57 | 56.8 ± 0.9 |

Belt velocity | v_{Belt} (m s^{−1}) | 0.00369 | = | 0.00344 | = |

Air temperature in C | T_{AC} (°C) | 67.3 | 66.9 ± 0.9 | 64.1 | 64.3 ± 0.8 |

Dryer length I-C (Figure 1) | L_{I-C} (m) | 3.47 | = | 3.64 | = |

Dryer length C-E (Figure 1) | L_{C-E} (m) | 2.53 | = | 2.36 | = |

Total dryer length | L_{T} (m) | 6.00 | = | 6.00 | = |

Product exit temperature | T_{PE} (°C) | 52 | 52.4 ± 0.6 | 52.1 | 52.2 ± 0.7 |

Air input mass flow rate | G_{AI} (kg s^{−1}) | 0.270 | 0.269 ± 0.006 | 0.368 | 0.369 ± 0.005 |

Evaporated water flow rate | G_{EV} (kg s^{−1}) | 0.00591 | = | 0.00550 | = |

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**MDPI and ACS Style**

Friso, D.
Mathematical Modelling of Conveyor-Belt Dryers with Tangential Flow for Food Drying up to Final Moisture Content below the Critical Value. *Inventions* **2021**, *6*, 43.
https://doi.org/10.3390/inventions6020043

**AMA Style**

Friso D.
Mathematical Modelling of Conveyor-Belt Dryers with Tangential Flow for Food Drying up to Final Moisture Content below the Critical Value. *Inventions*. 2021; 6(2):43.
https://doi.org/10.3390/inventions6020043

**Chicago/Turabian Style**

Friso, Dario.
2021. "Mathematical Modelling of Conveyor-Belt Dryers with Tangential Flow for Food Drying up to Final Moisture Content below the Critical Value" *Inventions* 6, no. 2: 43.
https://doi.org/10.3390/inventions6020043