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Article

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

by
Dario Friso
Department of Land, Environment, Agriculture and Forestry, University of Padova, Agripolis, Viale dell’Università 16, 35020 Legnaro, Italy
Inventions 2021, 6(2), 43; https://doi.org/10.3390/inventions6020043
Submission received: 29 April 2021 / Revised: 9 June 2021 / Accepted: 10 June 2021 / Published: 13 June 2021

## Abstract

:
This work presents the mathematical modeling of the conveyor-belt dryer with tangential flow operating in co-current, which has the advantage of improving the preservation of the organoleptic and nutritional qualities of the dried food. On the one hand, it is a more cumbersome dryer than the perforated cross flow belt dryer but, on the other hand, it has a low air temperature in the final section where the product has a low moisture content and, therefore, it is more heat sensitive. The results of the mathematical modeling allowed a series of guidelines to be developed for a rational design of the conveyor-belt dryer with tangential flow for the specific case of the moisture content of the final product XF lower than the critical one XC (XF < XC). 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 XF of the product was higher than the critical one XC. 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 XF < XC were obtained.

## 1. Introduction

In previous papers [1,2], the conveyor-belt dryer with tangential flow operating in co-current had been indicated as the dryer that shows the advantage of possible lower thermal damage, especially for heat-sensitive food. However, its use is infrequent with respect to the through-circulation conveyor-belt dryer (perforated belt) [1,3]. In fact, having the hot air temperature approximately constant throughout the length of the dryer when the air enters the product, this dryer is more compact and easier to design and, given its high diffusion, it has been the subject of theoretical and experimental studies also to give indications for its design [4,5,6,7,8,9,10].
Furthermore, the thermo-hygrometric exchanges during drying are described by differential equations [11] that can be solved in closed form or with numerical methods. The scientific literature is rich in mathematical models of heat and mass exchanges developed with numerical or empirical solutions to describe the drying phenomenon when the temperature of the air entering or touching the product remains constant. Among the food products that can be mentioned are: apple [12,13,14,15,16,17], apricot [18], banana [19,20,21,22,23], carrot [24], cassava [25], coconut [26], coroba [27], cumbeba [28], fisher [29], ginger [30], jujuba [31], kiwi [32], mandarin [33], mango [34,35,36,37], mussel [38], quince [39], papaya [40], pear [41], potato [42], rice [43], sultanas [44], taraxacum [45], tomato [46,47], turnip [48], generic fruits [49,50,51,52,53,54] and generic foods [55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71].
Trying to fill the gap in the study of conveyor-belt dryers with tangential flow, in a previous work [1] mathematical modelling was developed of the thermo-hygrometric exchanges between the product with final moisture content XF higher than the critical one XC (XF > XC) 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 (XF < XC), 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

Figure 1 shows the schematic of a conveyor-belt dryer with tangential flow. The typical diagram of the temperatures of the air TA and the product TP inside the dryer is also shown. Inside, two zones can be identified: first long LI-C, where the product maintains the moisture content X higher than the critical moisture content XC and, last, long LC-E where the moisture content of the product X is lower than the critical one XC.

#### 2.1.1. First Zone of the Dryer LI-C

In the first zone, long LI-C, the product is characterized by a moisture content X higher than the critical one XC, and therefore it maintains its temperature TP constant and equal to the wet bulb temperature TWB [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 qI-C that the warm and dry air of initial mass flow rate GAI releases when cooling from the input temperature TAI to temperature TAC corresponding to the achievement of critical moisture content (Figure 1):
$q I − C = G A I · c A ( T A I − T A C ) · η$
where: cA 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.
The second equation, called (13) in [1], concerns the heat transfer rate qI-C exchanged between the air and the product:
$q I − C = α · A I − C · Δ T m L ( I − C )$
where: α is the convective heat transfer coefficient; ΔTmL(I-C) is the logarithmic mean temperature difference in the I-C zone; AI-C is the total area of the product inside the I-C zone of the dryer.
From Equation (18) of [1]: $A I − C = f · L I − C$; where: f is the transverse dimension shown in Figure 5 of [1]; LI-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:
$q I − C = α · f · L I − C · Δ T m L ( I − C )$
The third equation refers to the heat transfer rate required by the water in the product to evaporate. It was indicated with the number (20) in [1]:
$q I − C = G E V ( I − C ) · r I − C = H I · B I · v B e l t · ρ B u l k I · X I − X C 1 + X I · r I − C$
where: GEV(I-C) is the mass flow rate of the evaporated water in the I-C zone; HI is the input height and BI is the input width of the bulk product on the belt; vBelt is the belt speed; ρBulkI is the bulk density of the input product; XI is the input moisture content; XC is the critical moisture content; rI-C is the thermal energy, in the I-C zone, required to produce 1 kg of superheated steam at the air temperature TA; rI-C is equal to the difference in enthalpy [3] between the superheated steam at TA and the water contained in the product to be dried at the temperature TP. As indicated in [1] rI-C is considered constant and has an average value of 2617 kJ kg−1.

#### 2.1.2. Second Zone of the Dryer LC-E

In order to write the mathematical modelling of the C-E drying zone in which the product moisture content is lower than the critical one (X < XC), the first equation proposed concerns the heat transfer rate qC-E that the warm air of initial flow rate GAI releases when cooling from the temperature TAC to the exit temperature TAE. This equation is similar to (1):
$q c − E = G A I · c A ( T A C − T A E ) · η$
where: GAI is the mass flow rate of drying air; cA 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 LI-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 GAI and therefore two orders of magnitude less than GAI.
The second equation, concerning the heat transfer rate qC-E exchanged between air and product, is similar to (2):
$q C − E = α · f · L C − E · Δ T m L ( C − E )$
where: α is the convective heat transfer coefficient; ΔTmL(C-E) is logarithmic mean temperature difference in the C-E zone; f is the transverse dimension shown in Figure 5 of [1]; LC-E is the length of the C-E zone of the dryer (Figure 1).
The third equation refers to the heat transfer rate qC-E required by the water to evaporate and, similarly to (3), it is:
$q C − E = G E V ( C − E ) · r C − E = H I · B I · v B e l t · ρ B u l k C · X C − X F 1 + X C · r C − E$
where: GEV(C-E) is the mass flow rate of the evaporated water in the C-E zone; HI is the input height and BI is the input width of the bulk product on the belt; vBelt is the belt speed; ρBulkC is the bulk density in the point where the product has the critical moisture content; XC is the critical moisture content; XF is the final moisture content; rC-E is the thermal energy, in the C-E zone, required to produce 1 kg of superheated steam at the air temperature TA and is equal to the difference in enthalpy [3] between the superheated steam at TA and the water contained in the product to be dried at the temperature TP. The thermal energy rC-E is currently unknown.
A fourth equation must be added to correlate the bulk density ρBulkC of the product at point C to the input bulk density ρBulkI.
Considering: $ρ B u l k I = m D + m W I V$; and: $ρ B u l k C = m D + m W C V$, where: mD is the dry mass; mW is the mass of water; V is the bulk volume; we obtain:
$ρ B u l k C ρ B u l k I = 1 + X C 1 + X I$
Finally, the LC-E zone with X < XC 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: $∂ X ∂ t = D ∂ 2 X ∂ y 2$.
In the case of a simple geometry such as the thin plate to which many food products are similar and excluding the initial period corresponding to the phenomenon of delay, Perry [72] suggests a solution of the PDE, such as:
$X − X e q X C − X e q = 8 π 2 e − D · π 2 4 · δ 2 t$
where: X is the moisture content as a function of the drying time t; XC is the critical moisture content of the product and corresponds to the initial one of the drying in the LC-E zone; Xeq is the equilibrium moisture content; D is the mass diffusivity of water inside the product; δ is half the thickness of the plate.
By differentiating we obtain:
$d X d t = − ( X C − X e q ) · 2 D δ 2 · e − D · π 2 4 · δ 2 t$
Since [1]: $d X d t = − G E V m D$, where: GEV is the instantaneous mass flow rate of evaporated water; mD is the dry mass present in the LC-E section of the dryer and is [2]: $m D = H I · B I · ρ B u l k C · L C − E 1 + X C$, that is, based on (7): $m D = H I · B I · ρ B u l k I · L C − E 1 + X I$. Furthermore, since the belt speed vBelt is constant and: $v B e l t = z t$, the previous (9) becomes:
$G E V = H I · B I · ρ B u l k I · L C − E 1 + X I · 2 D δ 2 · ( X C − X e q ) · e − D · π 2 4 · δ 2 · v B e l t z$
This equation indicates that the mass flow rate of water evaporated GEV varies along the horizontal coordinate z within the LC-E zone of the dryer (Figure 1) according to the exponential function as shown in Figure 2.
Therefore, the integral average evaporated water flow rate GEV(C-E) can be calculated as follows:
$G E V ( C − E ) = 1 L C − E ∫ 0 L C − E H I · B I · ρ B u l k I · L C − E 1 + X I · 2 D δ 2 · ( X C − X e q ) · e − D · π 2 4 · δ 2 · v B e l t z d z$
Then:
$G E V ( C − E ) = H I · B I · ρ B u l k I · v B e l t 1 + X I · 8 π 2 · ( X C − X e q ) · [ 1 − e − D · π 2 4 · δ 2 · v B e l t L C − E ]$
However, since food products do not always have the plate shape, the previous equation must be rewritten to make the mathematical model more general:
$G E V ( C − E ) = H I · B I · ρ B u l k I · v B e l t 1 + X I · C 1 · ( X C − X e q ) · [ 1 − e − C 2 v B e l t L C − E ]$
where: C1 and C2 are constants to be determined by experiments.
The heat flow rate required for this mass flow rate of evaporated water is:
$q C − E = H I · B I · ρ B u l k I · v B e l t 1 + X I · C 1 · ( X C − X e q ) · ⌈ 1 − e − C 2 v B e l t L C − E ⌉ · r C − E$
Equation (13) is added to the other four Equations (4)–(7) to perform mathematical modelling of the drying in the LC-E zone where the moisture content is X < XC.

#### 2.2. Design Guidelines

In the previous paragraph, eight equations were found, namely (1), (2), (3), (4), (5), (6), (7) and (13), in which there are eight unknowns: qI-C, qC-E, GAI, ρBulkI, TAC, LI-C, LC-E and TPE. 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 TAC at Critical Moisture Content of the Product

Since the air temperature at the input TAI and exit TAE of the dryer can be imposed a priori [1], the combination of Equations (1) and (3) gives:
$G A I · c A · ( T A I − T A C ) · η = H I · B I · v B e l t · ρ B u l k I · X I − X C 1 + X I · r I − C$
The combination of Equation (4) with (6) and (7) gives:
$G A I · c A · ( T A C − T A E ) · η = H I · B I · v B e l t · ρ B u l k I · X C − X F 1 + X I · r C − E$
The division of (14) with (15), after a few steps, gives the TAC temperature (Figure 1):
$T A C = T A E · ( X I − X C ) · r I − C + T A I · ( X C − X F ) · r C − E ( X I − X C ) · r I − C + ( X C − X F ) · r C − E$
The TAC value allows us to calculate ΔTc (Figure 1): $Δ T c = ( T A C − T W B )$, where TWB is the temperature of the product equal to the wet bulb temperature [1].

#### 2.2.2. Length of Dryer LI-C to Reach Critical Moisture Content XC

The combination of Equation (2) with (3) and the observation [1] that $f · α H I · B I = F · α$, where F is the form factor of the product, gives the length of the dryer LI-C corresponding to the zone with X > XC:
$L I − C = v B e l t · ρ B u l k I · r I − C F · α · Δ T m L ( I − C ) · X I − X C 1 + X I$
Equation (17) is similar to (23) of [1].

#### 2.2.3. Mass Flow Rate of Drying Air GAI

The combination of Equation (1) with (3) gives the mass flow rate of drying air GAI:
$G A I = B I · H I · v B e l t · ρ B u l k I c A · ( T A I − T A C ) · η · X I − X C 1 + X I · r I − C$

#### 2.2.4. Length of Dryer LC-E to Reduce Moisture Content from Critical Value XC to Final One XF

The combination of Equation (4) with (13), gives the length of the dryer LC-E where the product dries from XC to XF:
$L C − E = − v B e l t C 2 · ln [ 1 − G A I · c A · ( T A C − T A E ) · η C 1 · H I · B I · ρ B u l k I · v B e l t · r C − E · 1 + X I X C − X e q ]$

#### 2.2.5. Temperature Difference Δtb at the Dryer Exit and Product Exit Temperature TPE

The combination of Equation (4) with (5) gives the temperature difference between the air and the product at the exit of the dryer (Figure 1): $Δ T b = T A E − T P E$, where TAE is the air exit temperature and TPE is the product exit temperature:
The temperature difference ΔTb can be obtained by an iterative method easily through a spreadsheet. As seen in Section 2.2.1, the TAE temperature can be set a priori and, therefore, the product exit temperature $T P E = T A E − Δ T b$ can be obtained.

#### 2.2.6. Known Quantities and Experimental Quantities

• The input and exit temperatures of the drying air TAI and TAE can be defined using the guidelines 3.1.1 proposed in [1].
• The compound quantity α is the product of the convective heat transfer coefficient α and the form factor $F = f H I · B I$, where f is the transverse dimension (Figure 5 in [1]) and HI and BI are the height and width of the bulk product placed above the belt, respectively. The quantity α is measured as indicated in the guidelines 3.1.9 of [1].
• The bulk density ρBulkI is measured as indicated in the next point I) of Section 2.3.
• The thermal energy rI-C has an average value of 2617 kJ kg−1, as indicated in guidelines 3.1.6 of [1].
• The constants C1 and C2, 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 XC and the equilibrium moisture content Xeq.
• To determine the lengths LI-C and LC-E of the dryer, it is necessary to impose the belt speed vBelt and the height HI and the width BI 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 GEV(I-E) foreseen for the dryer. In fact, it is known that the quantity characterizing a dryer, both technically and commercially, is the GEV(I-E) quantity. Therefore, the designer must start from: a known value of the mass flow rate GEV(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 GEV(I-C) in the LI-C zone, and the one GEV(C-E) of the LC-E zone:
$G E V ( I − E ) = G E V ( I − C ) + G E V ( C − E ) = H I · B I · v B e l t · ρ B u l k I X I − X F 1 + X I$
In this equation, the designer can choose the values of HI, BI and hence can obtain the vBelt.
g.
Finally, the thermal energy rC-E in the LC-E length zone with X < XC must be measured. The rC-E will be greater than the rI-C of the zone with X > XC, since below a certain value of the moisture content X, lower than the critical one XC, 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 rC-E will be described.

#### 2.3. Experimental Procedure and Equipment

The experimental activity was carried out using a pilot dryer. The pilot dryer was the same as in previous works [1,2] and Table 1 summarizes its geometrical data. All the tests were carried out by drying alfalfa consisting of leaves attached to the stems cut in pieces 5 cm long. In the table the value of the quantity F·α relative of alfalfa is also reported. This quantity F·α was experimentally evaluated in the previous work [1].
The measuring instruments used were: infrared thermometer for the alfalfa temperature; PT100 resistance thermometers for the input and exit temperature of the dryer; precision balance for weighing the sample before and after dehydration in an oven for two hours at 135 °C to measure the moisture content of the product at the input and exit; a pitot anemometer; data logger. The bulk density was calculated after measuring the mass and volume of the samples. Five replicates were made for each test.
The three purposes of the experiments were:
(I)
to quantify the constants C1, C2, the critical moisture content XC and the equilibrium moisture content Xeq of the alfalfa were as foreseen in the previous point e. of Section 2.2.6. The four quantities, C1, C2, XC and Xeq 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 XI was placed with a height HI 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 TAI equal to 60 °C was forced by the fan to lick the product sample. The total mass of the sample mT = mW + mD, dry mass mD plus water mass mW, was measured each minute. Knowing [2] that $m D = m T I 1 + X I$, where mTI 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 mT is measured, can be calculated as follows:
$X = m T m D − 1 = m T m T I ( 1 + X I ) − 1$
where the initial moisture content XI 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 C1, C2, XC and Xeq (see also the next Section 3);
(II)
to quantify the thermal energy rC-E during drying with X<XC 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, rC-E is assumed equal to rI-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 LT=LI-C+LC-E=6 m. Therefore, in this case the sequence of calculations is seen to impose LT = 6 m to derive the belt speed vBelt. For the rest of the quantities, the guidelines are the same as in Section 2.2, thus determining the air temperature TAC and TAE, the mass flow rate of the drying air GAI and the product exit temperature TPE, 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 TPE and TAE. The third step consists in looking for the rC-E value which, inserted in the equations of guidelines 2.2, allows to restore the initial values of the TPE and TAE 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

To determine the quantities XC, Xeq, C1 and C2, 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 XC was measured at the point where the straight section, which represents the drying phase at constant rate (dX/dt = RC), 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 RC.
When the measured moisture content value remained constant for five consecutive readings (5 min), it was defined as the equilibrium moisture content Xeq.
To determine the constants C1 and C2, the diagram of Figure 4 was made using data of Figure 3 relating to the phase with decreasing drying rate, i.e., for X ≤ XC. In fact, the time on the abscissa is t’ = t − tC, where tC is the time to reach the critical moisture content XC. In ordinate there is the quantity: $ln ( X − X e q X C − X e q ) .$
Starting from Equation (13) and returning to having time as an independent variable, it becomes:
$ln X − X e q X C − X e q = ln C 1 − C 2 · t$
This is the equation of a line that has a negative slope equal to −C2 and intercept equal to ln(C1), therefore it is sufficient to look for the regression line in the diagram of Figure 4, to obtain ln(C1) = 0.1385 and C2 = 0.0026. Table 2 summarizes the results obtained.
To determine the average thermal energy rC-E required to evaporate 1 kg of water during drying with X < XC, 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.
During the first step, the dryer was preliminarily designed by imposing a thermal energy in the C-E section, where X < XC, rC-E is 2617 kJ/kg, i.e., the same as in the I-C section, where X > XC. In addition, a total length LT 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 vBelt, through the design guidelines of Section 2.2. Instead, the belt speed vBelt is normally chosen, so that the mass flow rate of evaporated water GEV(I-E) foreseen for the dryer is satisfied through Equation (21), and consequently, through the design guidelines 2.2, the length LT is calculated.
Therefore, the data of the first column (first step) are related to this preliminary design (blue font). In the second step, it was planned to carry out the experimental survey on the pilot dryer, running with the data obtained in the first step. Among the experimental measurements detected, the most important were the product exit temperature TPE (red font) and the air exit temperature TAE (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 TAE = 57 °C and an experimental T’PE = 46.5 °C instead of design TPE = 55.6 °C. It is clear the experimental T’AE lower than the design TAE 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 rC-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 TPE can also be explained by the need to transmit a greater heat flow rate to the product due to rC-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 ΔTmL(C-E) calculated during the design of the first step. Therefore, the increase in ΔTmL(C-E) is 63.2% and implies that the heat flow rate qC-E has to increase by the same amount based on Equation (5), therefore based on Equation (6) also the thermal energy rC-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.
The purpose (III) of the experiments, described in Section 2.3, concerned the validation of the mathematical modelling 2.1 and of the design guidelines 2.2, using the value of the thermal energy rC-E equal to 4271 kJ kg−1 as just determined. For a broader validation, the experiments were doubled with the air input temperature TAI set on two values: 120 °C and 100 °C.
Table 4 shows the results obtained. The values of the quantities in the first column are those of the design carried out according to guidelines 2.3 with TAI equal to 120 °C. The values of the second column are those detected during the test with TAI equal to 120 °C and concern: the product moisture content at the input XI and at the exit XF; the input bulk density ρBulkI; the air exit temperature TAE; the air temperature TAC at point C (Figure 1); the product exit temperature TPE. All the other quantities of the first column (design), particularly the length of the dryer LT, the mass air flow rate GAI and the speed of the belt vBelt resulted in the same value also during the test (second column) due to the adjustments imposed on the pilot dryer.
The third and fourth columns are similar to the first and second respectively, but with air input temperature TAI equal to 100 °C.
For both TAI 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 TAE; the product exit temperatures TPE; the air temperatures TAC in point C of Figure 1. Even the product exit moisture content XF did not show significant differences between the experimental values and those expected from the design.
As consequence of these positive results validating mathematical modelling 2.1 and design guidelines 2.2, a part of the equations of the model can be used to extend a result that was obtained in the previous work [2]. In [2], an equation was obtained, numbered (12), which correlated the final moisture content XF to the air exit temperature TAE, valid for final moisture content greater than the critical one (XF > XC). In this way, it was possible to indicate the indirect measurement of the final moisture content XF, useful for optimized adjustment of the drying operation, through the simpler, faster and cheaper measurement of the air exit temperature TAE. 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 (XF < XC), the comparison between (1), (3), (4), (6) and (7) gives the following equation in place of (12) of [2]:
$X F = X C + ( X I − X C ) r I − C r C − E + G A I · c A · ( 1 + X I ) B I · H I · v B e l t · ρ B u l k I · r I − C · ( T A E − T A I ) · η$
Knowing the critical moisture content of the product XC, its input moisture content XI, its bulk density ρBulkI, its bulk dimensions on the belt BI and HI, the thermal energy values rI-C and rC-E, the mass flow rate of air GAI, its input temperature TAI, the belt speed vBelt and by measuring the air exit temperature TAE, Equation (24) shows that the final moisture content XF is promptly obtained. The same equation, appropriately implemented in a PLC, suggests how to correct the XF, if this is not the expected value, for example by changing the belt speed vBelt, or the air input temperature TAI.
The last result concerns a simulation using the equations of the design guidelines. The influence of the air temperature at the input TAI on the following three variables was analyzed: the length of the dryer LT; the product temperature at the exit TPE; the mass flow rate of drying air GAI. Since this air temperature at the input TAI 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 TAI which has values between 100 °C and 150 °C. During the simulation, the following quantities were kept constant: the belt speed vBelt equal to 0.004 m s−1; the air temperature at the exit TAE equal to 60 °C; the mass flow rate of evaporated water in the dryer GEV 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 TPE, by just a couple of degrees, which is therefore acceptable, and a reduction in the mass flow rate of drying air GAI to keep the mass flow rate of evaporated water GEV constant given the increase by TAI. Therefore, the space problem that afflicts these dryers can be mitigated with an increase in TAI.
The series of diagrams on the right of Figure 5d–f provides the three variables—length LT, temperature TPE, mass flow rate GAI—vs. the air temperature at the exit TAE which is reduced from 60 °C to 50 °C, keeping constant: the belt speed vBelt equal to 0.004 m s−1; the air temperature at the input TAI equal to 120 °C; the mass flow rate of evaporated water in the dryer GEV equal to 23 kg h−1. The figure shows a slight increase in the length of dryer LT of about 0.5 m (+8%), an important decrease of the product temperature at the exit TPE which decreases from 55.7 °C to 42.8 °C improving the organoleptic and nutritional quality. It should be remembered that the product temperature TP remains at the wet bulb temperature TWB 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 TPE.

## 4. Conclusions

In the food industry, the drying operation is still very widespread with the use of various types of system. In this work, the focus was on the conveyor-belt dryer with tangential flow that is underused because it is more cumbersome and more difficult to design, but it has the advantage of greater respect for heat-sensitive food products and, therefore, greater diffusion in the future it may be able to improve food quality. This is the dryer already studied in two previous works [1,2] for the case of final moisture content higher than critical moisture content (XF > XC). 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 (XF < XC).
First, all the equations of the thermo-hygrometric exchange have been set up. The associated mathematical modelling allowed us to define design guidelines and also a method to calculate the thermal energy rC-E, necessary to evaporate the water when the moisture content is lower than the critical one (X < XC). In fact, it is always greater than that for X > XC since starting from a certain moisture content below the critical one XC, there is bound water which requires higher energy to evaporate compared to latent heat.
Among the eight equations of the mathematical modelling there is one describing the diffusion phenomenon of water inside the product when the moisture content of the product X is lower than the critical one XC. The mathematical modelling and the equations for the design guidelines, including the method for calculating the thermal energy rC-E, have been experimentally confirmed.
A part of the equations of the mathematical modelling presented here in Section 2.1 was also used to extend the result obtained in a previous work [2] dedicated to drying with the final moisture content of the product exceeding the critical one, XF > XC. It was an equation that correlated the air exit temperature TAE with the final moisture content of the product XF 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.
Therefore, among the results of this work there is also the new equation extended to the case XF < XC, able to link the final moisture content XF to the air exit temperature TAE. If the equation is implemented in a dryer adjustment PLC, it will be able to keep the XF value constant and optimized, controlling the TAE which is much easier, faster and less expensive to detect than direct methods for measuring the XF.
Finally, the mathematical modelling and equations of the design guidelines were used to simulate the influence of air temperatures at the input TAI and, respectively, at the exit TAE. This last analysis showed that by reducing the mass flow rate of drying air GAI, with the same TAI, the temperature TAE and consequently the exit temperature of the product TPE can be reduced with, consequently, less thermal damage to food.

## Funding

This research received no external funding.

Not applicable.

Not applicable.

## Data Availability Statement

The data presented in this study is contained within the article.

## Conflicts of Interest

The author declares no conflict 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 XF lower than the critical one XC (XF < XC).
Figure 1. The conveyor-belt dryer with tangential flow and diagram of the temperature of the air and product with final moisture content XF lower than the critical one XC (XF < XC).
Figure 2. Mass flow rate of evaporated water GEV vs. z-coordinate within the LC-E zone, corresponding to the direction of the belt in the C-E zone where the moisture content is X < XC.
Figure 2. Mass flow rate of evaporated water GEV vs. z-coordinate within the LC-E zone, corresponding to the direction of the belt in the C-E zone where the moisture content is X < XC.
Figure 3. Experimental drying curve of alfalfa. From it, the critical moisture content XC and the equilibrium moisture content Xeq can be obtained.
Figure 3. Experimental drying curve of alfalfa. From it, the critical moisture content XC and the equilibrium moisture content Xeq can be obtained.
Figure 4. Alfalfa drying curve modified to obtain the constants C1 and C2, 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 4. Alfalfa drying curve modified to obtain the constants C1 and C2, 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 TAI (with vBelt, TAE and GEV kept constant), of the: (a) dryer length LT; (b) alfalfa temperature at the exit of the dryer TPE; (c) mass flow rate of the drying air GAI. Right, histograms of the same variables, but vs. air temperature at the dryer exit TAE with vBelt, TAI and GEV kept constant: (d) dryer length LT; (e) alfalfa temperature at the exit of the dryer TPE; (f) mass flow rate of the air GAI.
Figure 5. Simulation using the equations of the design guidelines. Left, histograms vs. air temperature at the dryer input TAI (with vBelt, TAE and GEV kept constant), of the: (a) dryer length LT; (b) alfalfa temperature at the exit of the dryer TPE; (c) mass flow rate of the drying air GAI. Right, histograms of the same variables, but vs. air temperature at the dryer exit TAE with vBelt, TAI and GEV kept constant: (d) dryer length LT; (e) alfalfa temperature at the exit of the dryer TPE; (f) mass flow rate of the air GAI.
Table 1. Geometrical and operational data of the pilot dryer.
Table 1. Geometrical and operational data of the pilot dryer.
QuantitySymbolValue
Belt widthBI (m)0.3
Total Belt lengthLT (m)6.0
Alfalfa bulk heightHI (m)0.05
Flow section of the drying airAA (m2)0.15
Form factor·Convective heat transfer coefficient [1]F·α (W·m−3·K−1)5144
Table 2. Results from experimental drying curve.
Table 2. Results from experimental drying curve.
QuantitySymbolValue
Alfalfa input moisture content (D.B.)XI1.688 ± 0.105
Alfalfa input bulk densityρBulkI (kg·m−3)183 ± 7.6
Alfalfa critical moisture content (D.B.)XC0.290
Alfalfa equilibrium moisture content (D.B.)Xeq0.041
Coefficient related to delayC11.149
Coefficient related to diffusivity C20.0026
Table 3. Results of the three steps to obtain the thermal energy rC-E concerning the humidity X < XC. The first step refers to the results of the preliminary design with rC-E equal to rI-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 rC-E value found (red font) through the equations of the design guidelines 2.3 by imposing the experimental temperatures (red font).
Table 3. Results of the three steps to obtain the thermal energy rC-E concerning the humidity X < XC. The first step refers to the results of the preliminary design with rC-E equal to rI-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 rC-E value found (red font) through the equations of the design guidelines 2.3 by imposing the experimental temperatures (red font).
QuantitySymbol1st Step
Preliminary Design
2nd Step
Exper. Value
3rd Step
Search for rC-E
Thermal energyrC-E (kJ kg−1)2617 4271
Input moisture content XI1.6881.688 ± 0.1051.688
Final moisture contentXF0.1220.121 ± 0.010.122
Input bulk densityρBulkI (kg m−3)183183 ± 7.6183
Critical moisture content XC0.290=0.290
Equilibrium moisture content Xeq0.041=0.041
Air input temperatureTAI (°C)120119.7 ± 1.2120
Air exit temperatureTAE (°C)5752.7 ± 1.152.7
Belt velocityvBelt (m s−1)0.0036=0.0036
Air temperature in CTAC (°C)63.8=63.8
Dryer length I-C (Figure 1)LI-C (m)3.55=3.55
Dryer length C-E (Figure 1)LC-E (m)2.45=2.45
Total dryer lengthLT (m)6.00=6.00
Product exit temperatureTPE (°C)55.646.5 ± 0.746.5
Air input mass flow rateGAI (kg s−1)0.2460.246 ± 0.0060.246
Table 4. Results of the validation of the mathematical modeling and design guidelines.
Table 4. Results of the validation of the mathematical modeling and design guidelines.
QuantitySymbolDesignExper. ValueDesignExper. Value
Air input temperatureTAI (°C)120119.7 ± 1.2100100.9 ± 1.1
Thermal energy (X > XC)rI-C (kJ kg−1)2617=2617=
Thermal energy (X < XC)rC-E (kJ kg−1)4271=4271=
Input moisture content XI1.6881.688 ± 0.1051.6881.688 ± 0.105
Final moisture contentXF0.1220.120 ± 0.010.1220.124 ± 0.009
Input bulk densityρBulkI (kg m−3)183183 ± 7.6183183 ± 7.6
Critical moisture content XC0.290=0.290=
Equilibrium moisture content Xeq0.041=0.041=
Air exit temperatureTAE (°C)5756.7 ± 1.05756.8 ± 0.9
Belt velocityvBelt (m s−1)0.00369=0.00344=
Air temperature in CTAC (°C)67.366.9 ± 0.964.164.3 ± 0.8
Dryer length I-C (Figure 1)LI-C (m)3.47=3.64=
Dryer length C-E (Figure 1)LC-E (m)2.53=2.36=
Total dryer lengthLT (m)6.00=6.00=
Product exit temperatureTPE (°C)5252.4 ± 0.652.152.2 ± 0.7
Air input mass flow rateGAI (kg s−1)0.2700.269 ± 0.0060.3680.369 ± 0.005
Evaporated water flow rateGEV (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