# Conveyor-Belt Dryers with Tangential Flow for Food Drying: Mathematical Modeling and Design Guidelines for Final Moisture Content Higher Than the Critical Value

## Abstract

**:**

_{F}of the product is higher than the critical moisture content X

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

_{F}> X

_{C}were obtained and experimentally evaluated.

## 1. Introduction

_{A}and its specific humidity x.

_{A}and specific humidity x, in the length of the dryer. Therefore, the mathematical description of the thermo-hygrometric exchange between the air and the product is complicated precisely because of these variabilities. Below, a mathematical model will be developed to determine the average flow rate of evaporated water G

_{EV}of the dryer and then to design it when the output product is still with a moisture content higher than the critical moisture content (X

_{F}> X

_{C}).

## 2. Materials and Methods

#### 2.1. Preliminaries

^{−1}) is the drying rate; X = m

_{W}/m

_{D}(kg/kg) is the moisture content (dry basis) of the product without dimensions; m

_{W}is the mass of water and m

_{D}is the dry mass; G

_{EV}= dm

_{W}/dt is the flow rate of evaporated water (kg/s).

_{A}(K) is the air temperature; T

_{WB}(K) is the wet bulb temperature of the air. It is also the temperature of the food product if it has a moisture content X greater than the critical one X

_{C}. In other words, if the product is moist enough to behave like pure water; α (Wm

^{−1}K

^{−1}) is the convection coefficient; A (m

^{2}) is the area of the product lapped by the air; r (J/kg) should be the latent heat of the water vapor at the wet bulb temperature T

_{WB}, but it assumes a greater value as indicated by [55].

_{C}, T

_{WB}is constant and if the air temperature T

_{A}also remains constant, then the drying rate R becomes constant and is called R

_{C}.

_{C}, the (1) can be rewritten:

_{I}is the initial moisture content and X

_{F}è is the final moisture content of the period at constant speed. Therefore, the drying time t

_{C}at a constant rate is:

_{A}and with X

_{F}> X

_{C}.

_{C}, the water runs out on the surface of the product. This does not prevent drying since water from the inside comes by diffusion through the internal mass. However, the process is conditioned precisely by the reduced speed of water diffusion towards the surface. Therefore, the balance between the heat transfer rate q, from the hot air to the product, and the heat transfer rate q that accompanies the flow rate of evaporated water G

_{EV}from the product to the air [55], occurs with a product temperature T

_{P}higher than that of the wet bulb T

_{WB}. Equation (2) is still valid provided that the temperature T

_{P}replaces the T

_{WB}, which confirms the reduction of G

_{EV}and therefore also of the drying rate R.

#### 2.2. Mathematcal Modeling of the Conveyor-Belt Dryer with Tangential Flow

_{EV}in the entire dryer, it is necessary to proceed with the integration of (6). This requires knowledge of the mathematical relationship between the air temperature T

_{A}and the coordinate along which the dryer belt develops.

_{F}> X

_{C}, that is the output product is still with a moisture content higher than the critical one. This means that the product maintains its T

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

_{WB}air throughout the dryer.

_{A}air and the T

_{P}product.

_{A}is the air temperature when it comes into contact with the dA area; T

_{P}= T

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

_{A}of the air (in its dry component). In fact, it undergoes a decrease by flowing over the elementary area dA, as it warms to the same area dA (Figure 3):

_{A}is the mass flow rate of dry air; c

_{A}is the specific heat of dry air; dA is the elementary area; dT

_{A}is the elementary decrease in air temperature when it laps the dA area. The negative sign appears to make the heat transfer rate dq positive since dT

_{A}is negative.

_{PI}, was considered equal to the wet bulb T

_{WB}. This is an acceptable approximation because, in the enthalpy balance of the dryer [55], the thermal energy required to heat the dry component of the product, from T

_{PI}to T

_{WB}, is less than 1% of the total thermal energy supplied by the hot air. While the thermal energy necessary to heat the water contained in the product from T

_{PI}to T

_{WB}is approximately 3% and will be taken into consideration during the development of the mathematical model.

_{A}and moving on to integration, we have:

_{TOT}is the total area of the product surface inside the dryer; and T

_{AI}and T

_{AE}are the dryer input and exit air temperatures (Figure 3). The result of the integration is:

_{A}·c

_{A}to the right, and multiplying both members by ${T}_{AI}-{T}_{AE}$, we have:

_{AI}to T

_{AE}. Having admitted that it goes completely to the product without dispersion to the outside, then the left term of (11) becomes the heat exchange equation between the air and the product. Just add and subtract the T

_{WB}temperature inside the parenthesis, to have $\left({T}_{AI}-{T}_{AE}\right)=\left(\left({T}_{AI}-{T}_{WB}\right)-\left({T}_{AE}-{T}_{WB}\right)\right)=\Delta {T}_{a}-\Delta {T}_{b}$. Definitely:

_{EV}from the product bed within the tunnel of the dryer is:

_{A}air as we will see in Section 3.1.6.

#### 2.3. Experimental Equipment

_{I}= 0.3 m wide and L

_{TOT}= 6 m long, was used (Figure 4). The scheme of its operation was the same as in Figure 3. The dryer was fed with alfalfa distributed on the belt for a height (H

_{I}) of 0.05 m. The geometric and operational characteristics are shown in Table 1.

## 3. Results

#### 3.1. Design Guideline Conveyor-Belt Dryer with Tangential Flow for Food with X_{F} > X_{C}

#### 3.1.1. Input and Exit Temperatures of the Drying Air

_{AI}and at the exit T

_{AE}of the dryer, must be defined in order to calculate the logarithmic mean temperature difference in ΔT

_{mL}. The input temperature must be established according to the characteristics of the product to be dried. It may be greater, the smaller and more uniform the size of the product, and the more permeable the surface of the product and the more humid the entering product are. All these facts allow to keep the product sufficiently humid on the surface and therefore its temperature equal to the wet bulb. Furthermore, it is necessary to avoid denaturing some components of the product. For example, and with reference to the grain, the maximum recommended T

_{AI}for maize is 115–120 °C, while oilseeds, such as soy, can be subjected to a T

_{AI}temperature of no more than 90 °C. For rice, it must be below 70 °C. For herbaceous products, due to the size of the leaves and the stems, as well as to the high permeability of the surfaces, the T

_{AI}can reach 500 °C [64] and, if very humid, even 800 °C [65].

_{AE}, it must be adequately higher than that of the wet bulb T

_{WB}. In fact, even if the product were pure water, this temperature T

_{WB}would be reached by the air at the end of a continuous dryer only if it had infinite length (Figure 3).

_{mL}, between the air and the product, high enough. As is known, it is assumed here that the product is throughout the dryer with a moisture content higher than the critical value, and therefore, at a temperature equal to that of the wet bulb T

_{WB}. In order to have good values of the logarithmic mean temperature difference: $\Delta {T}_{mL}\frac{\left(\Delta {T}_{a}-\Delta {T}_{b}\right)}{\mathrm{ln}\left(\frac{\Delta {T}_{a}}{\Delta {T}_{b}}\right)}$, both ΔT

_{a}and ΔT

_{b}must be high; as shown in Figure 3, the latter is high if the temperature of the air at the exit of the dryer T

_{AE}is also high.

_{AE}is set a few tens of degrees higher than that of the exit product. For example, with a T

_{AI}of 120 °C, the T

_{WB}, provided by the psychrometric diagram, is 34 °C and therefore the final temperature of the T

_{AE}air can be set between 50 °C and 60 °C.

#### 3.1.2. Flow Rate of Evaporated Water and Final Moisture Content

_{EV}must be considered a datum of the problem. It is related to the flow rate of the product to be dried G

_{I}, the input moisture content X

_{I}and the final moisture content X

_{F}to be reached during the process. Indeed, the manufacturers identify the performance of the dryers, in their production range, precisely through the flow rate of evaporated water expressed in kg/h.

_{WI}of the product and the final one m

_{WF}, divided by the time interval Δt required for this evaporation: ${G}_{EV}=\frac{{m}_{WI}-{m}_{WF}}{\Delta t}$. The total mass of wet product (dry mass m

_{D}plus water mass m

_{WI}) that enters the dryer in this time interval Δt is the flow rate of the wet product, i.e., at the conditions of input into the dryer: ${G}_{I}=\frac{{m}_{D}+{m}_{WI}}{\Delta t}$.

_{D}and defining the moisture content on a dry basis $X=\frac{{m}_{W}}{{m}_{D}}$, we have: $\frac{{G}_{EV}}{{G}_{I}}=\frac{{X}_{I}-{X}_{F}}{1+{X}_{I}}$.

_{I}, X

_{F}and G

_{I}, relating to the wet product and its drying process, the determination of G

_{EV}, as the initial datum of the problem, is immediate:

#### 3.1.3. Wet Product Flow Rate

_{I}can be defined as the volumetric flow rate of the bulk wet product Q

_{BulkI}multiplied by the bulk density of the wet product at the input of the dryer, ρ

_{BulkI}: ${G}_{I}={\rho}_{BulkI}\xb7{Q}_{BulkI}$.

_{BulkI}since it is equal to the initial section multiplied by the feed speed of the product, equal to the belt speed v

_{Belt}: ${Q}_{BulkI}={B}_{I}\cdot {H}_{I}\cdot {v}_{Belt}$. It is therefore very easy to control Q

_{BulkI}both by adjusting the initial height of the bulk product H

_{I}(B

_{I}is the initial width) with a toothed roller and by adjusting the speed of advancement of the belt v

_{Belt}. Therefore:

#### 3.1.4. Area of the Product Lapped by the Air

_{TOT}, exposed to the drying air, can be imagined as the length, L

_{TOT}, of the belt of the dryer multiplied by a transverse dimension f (Figure 5):

#### 3.1.5. Convection Coefficient

_{A}, in addition to other parameters. We can proceed to the calculation of α through the formulas between the dimensionless numbers of Nusselt, Reynolds and Prandtl Nu = f (Re, Pr) [55,66].

#### 3.1.6. Thermal Energy r

_{A}is equal to the difference in enthalpy [55] of the superheated steam at T

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

_{PI}:

_{W}is the water specific heat capacity equal to 4.187 kJ kg

^{−1}K

^{−1}; λ

_{WB}is the latent heat at the wet bulb temperature T

_{WB}; c

_{SV}is the superheated steam specific heat capacity equal to 1.92 kJ kg

^{−1}K

^{−1}.

_{A}between 343 K (70 °C) and 423 K (150 °C) and considering an initial product temperature T

_{PI}of 293 K (20 °C), the thermal energy r assumes a value between 2542 and 2694 kJ kg

^{−1}. If an arithmetic mean value r of 2617 kJ kg

^{−1}is assumed, a standard error of 1.6% and a maximum error of 2.9% are verified. These errors must be accepted as a consequence of the imposition r = constant. Otherwise, the mathematical modeling developed in the previous Section 2.2 would have been much more complicated and consequently the design guidelines would become impractical.

#### 3.1.7. Flow Rate of Drying Air

_{A}the flow rate of the water G

_{EV}contained in the product, so that the moisture content drops from X

_{I}to X

_{F}. For this reason, this heat transfer rate q will be related to the G

_{EV}given by Equation (17) and to the thermal heat r provided by Equation (19):

_{A}, i.e., by Equation (12). By equating (12) with (20), we can finally derive the flow rate of drying air G

_{A}:

#### 3.1.8. Length of the Dryer

_{I}, H

_{I}and f in a single parameter that could be called a form factor, $F=\frac{f}{{B}_{I}\xb7{H}_{I}}$:

_{TOT}, after the values of the dryer belt speed, v

_{Belt}, and the initial moisture content X

_{I}and the final moisture content X

_{F}, were chosen.

_{mL}as in Section 3.1.1; (2) the value of thermal energy r as in Section 3.1.6; the value of the bulk density of the wet product at the beginning of the dryer ρ

_{BulkI}, with the experimental method as in the next Section 3.1.9; the value of the product, F·α, of the form factor F and of the convection coefficient α, with experimental method as in the next Section 3.1.9.

#### 3.1.9. Experimental Evaluation of F·α

_{I}· H

_{I}product section identical to that of the belt dryer to be designed, but with a short length, L

_{D}. Above the product, high H

_{I}, flows the drying air with the same characteristics and the same flow rate G

_{A}(and speed v

_{A}) as the dryer.

_{D}is short, the exponential decrease in T

_{A}can be approximated with a segment from T

_{AI}to T

_{AD}. Therefore, the average air temperature T

_{Aa}above the product is: ${T}_{Aa}=\frac{\left({T}_{AI}+{T}_{AD}\right)}{2}$. The heat transfer rate associated with the decrease in air temperature is: $q={c}_{A}\xb7{G}_{A}\left({T}_{AI}-{T}_{AD}\right)$. This heat transfer rate must equal that exchanged between the air and the product: $q=\alpha \xb7A\left({T}_{Aa}-{T}_{WB}\right)$. Since the area of product A lapped by the air is (18) $A=f\xb7{L}_{D}$ and remembering that: $F=\frac{f}{{B}_{I}\xb7{H}_{I}}$, in the absence of thermal loss through the walls and measuring the temperatures T

_{AI}, T

_{AE}and T

_{WB}, we have:

#### 3.1.10. Adjustment of Parameters of the Dryer

_{I}constant, the dryer, sized with the L

_{TOT}length via Equation (23), will ensure that the product will be dried at the final moisture content X

_{F}expected according to the programmed G

_{PI}flow rate of the wet product.

_{F}:

_{I}increases, the final X

_{F}increases as well, failing the food safety objective due to the consequent increase in the water activity of the product subsequently stored. Conversely, a decrease in the final moisture content X

_{F}can also occur when X

_{I}decreases, with an unnecessary energy consumption and an increased risk of fire/explosion in the final section of the dryer if the product is dusty. At certain concentrations, the dry dust raised and floating in the drying air can produce explosions from any accidental spark.

_{F}, and if this is not the desired one, intervenes accordingly on one of the parameters in Equation (25). For example, the feed speed of the product, through v

_{Belt}, can be varied, decreasing it for example if the X

_{F}is too high, thus leaving the product in the dryer for a longer time.

_{F}is still too high, then one can proceed by increasing the inlet temperature of the drying air T

_{AI}. It also follows a partial increase in the exit temperature of the air T

_{AE}. Since the product remains at the temperature of the wet bulb, there is an increase in the logarithmic mean temperature difference between the air and the product, ΔT

_{mL}, which leads to an increase in the heat transfer rate q from the air to the product, intensifying the evaporated water flow rate G

_{EV}. The Equation (25) confirms it.

#### 3.2. Experimental Results

_{A}and alfalfa T

_{P}ones, together with the relative standard deviation (S.D.), at the input and the exit of the dryer. In addition, the table shows the mean value and the S.D. of the alfalfa moisture content at the input and exit of the dryer. There was no significant difference between the experimental values of the alfalfa temperature T

_{P}and the wet bulb temperature T

_{WB}, quantified with the psychrometric chart.

_{AI}, T

_{AE}e T

_{P}temperatures were used to calculate the logarithmic mean temperature difference ΔT

_{mL}showed in Table 2.

_{AD}, at distance from input z = 1.5 m, was measured together with the T

_{AI}temperature value, the alfalfa temperature value T

_{P}= T

_{WB}and the length L

_{D}= 1.5 m. This allowed to use Equation (24) which gave an approximate value of F·α equal to:

_{A}, vs. the distance z from the input of the dryer.

## 4. Conclusions

_{F}< X

_{C}) in further future work.

## Funding

## Conflicts of Interest

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**Figure 1.**Through-circulation conveyor-belt dryer (perforated belt) in co-current: (

**left**) side view; (

**right**) front view.

**Figure 3.**The conveyor-belt dryer with tangential flow and a diagram of the temperatures of the air and the product with a moisture content X

_{F}> X

_{C}.

**Figure 5.**Conveyor-belt dryer with tangential flow in co-current: transverse dimension f; velocity of the drying air v

_{A}; velocity of the belt v

_{Belt}; length of the belt L

_{TO}

_{T}; height of the product bed H; width of the product bed B; elementary area of the product exposed to the air dA; elementary length dz.

**Figure 6.**Experimental curve of the air temperature decay, T

_{A}, vs. the horizontal distance, z, from the input of the dryer.

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

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

Belt length | L_{TOT} (m) | 6.0 |

Belt speed | v_{Belt} (m/s) | 0.005 |

Air input velocity | v_{AI} (m/s) | 2.6 |

Air section | A_{A} (m^{2}) | 0.15 |

Air input volumetric flow rate | Q_{AI} (m^{3}/s) | 0.395 |

Air input temperature | T_{AI} (K) | 393 |

Air input density | ρ_{AI} (kg/m^{3}) | 0.896 |

Air mass flow rate | G_{AI} (kg/s) | 0.354 |

Alfalfa input moisture content (D.B.) | X_{l} | 1.892 ± 0.110 |

Alfalfa input moisture content (W.B.) | Y_{I} (%) | 65.4 ± 1.3 |

Alfalfa input bulk density | ρ_{BulkI} (kg/m^{3}) | 197 ± 7.5 |

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

Air input temperature | T_{AI} ± S.D. (K) | 392.2 ± 1.3 |

Air exit temperature | T_{AE} ± S.D. (K) | 331.4 ± 1.2 |

Air temperature at z = 1.5 m as batch dryer for F·α assessment | T_{AD} ± S.D. (K) | 368.8 ± 1.2 |

Alfalfa input temperature | T_{PI} (K) | 310.7 ± 0.6 |

Alfalfa exit temperature | T_{PE} (K) | 311.6 ± 0.9 |

Log. mean temperature difference | ΔT_{mL} (K) | 45.2 |

Simulated batch dryer length | L_{D} (m) | 1.5 |

Alfalfa final moisture content (D.B.) | X_{F} ± S.D. | 0.332 ± 0.016 |

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**MDPI and ACS Style**

Friso, D.
Conveyor-Belt Dryers with Tangential Flow for Food Drying: Mathematical Modeling and Design Guidelines for Final Moisture Content Higher Than the Critical Value. *Inventions* **2020**, *5*, 22.
https://doi.org/10.3390/inventions5020022

**AMA Style**

Friso D.
Conveyor-Belt Dryers with Tangential Flow for Food Drying: Mathematical Modeling and Design Guidelines for Final Moisture Content Higher Than the Critical Value. *Inventions*. 2020; 5(2):22.
https://doi.org/10.3390/inventions5020022

**Chicago/Turabian Style**

Friso, Dario.
2020. "Conveyor-Belt Dryers with Tangential Flow for Food Drying: Mathematical Modeling and Design Guidelines for Final Moisture Content Higher Than the Critical Value" *Inventions* 5, no. 2: 22.
https://doi.org/10.3390/inventions5020022