# Optimization Method for the Evaluation of Convective Heat and Mass Transfer Effective Coefficients and Energy Sources in Drying Processes

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{T}and mass h

_{m}transfer, as well as the energy source due to ultrasound absorption ΔQ and the critical moisture content X

_{cr}. The main aim of this work is to present a numerical algorithm for the evaluation of these parameters. In modeling and simulations of the heat and mass transfer processes in the available literature, the Runge–Kutta method and ready-made solvers (e.g., implemented in MatLab) are often used [24,25,26]. This method of the ordinary differential equations solution is not stable for long-term simulation. Therefore, the Adams–Bashforth method was used in this work [27]. The model parameters were estimated on the basis of the ultrasound-assisted drying kinetics realized experimentally. The estimation was based on the inverse problem solution, obtained by the use of optimization techniques. The non-gradient Rosenbrock optimization method was chosen because of its efficiency [28].

## 2. Mathematical Model

_{s}is the dry mass; A

_{m}and A

_{T}denote the surfaces of mass and heat exchange, respectively; ${\phi}_{a}$ is the relative air humidity in ambient air; ${\phi |}_{\partial B}$ is the air humidity close to the dried sample surface; p

_{vs}is the temperature-dependent saturated vapor pressure; c

_{s}and c

_{l}denote the specific heat of dry solid and moisture, respectively; l is the latent heat of evaporation; h

_{m}and h

_{T}denote the mass and heat transfer coefficients, respectively; and ΔQ is the heat in the material sample due to the absorption of ultrasonic waves. Both equations are determined as a result of the mass and energy balances. Equation (1) describes the mass exchange: The mass accumulation of moisture (left hand side) is equal to the convective moisture mass flux (right hand side). Equation (2) describes the heat transfer between the dried material and the surrounding air: The heat accumulation (left hand side) is equal to the heat flux delivered by convection (the first term to the right hand side) minus the heat consumed by evaporation (the second term to the right hand side) plus the heat of ultrasonic wave absorption (the third term to the right hand side).

_{V}denotes the volumetric shrinkage coefficient, X

_{0}is the initial moisture content, and X represents the actual moisture content. The surfaces A

_{m}and A

_{T}(Equations (1) and (2)) denote the surfaces of mass and heat exchange, respectively. If the dried material is placed on a moisture-impermeable support, the surfaces A

_{m}and A

_{T}are different. If, on the other hand, the material experiences constant motion during drying, the evaporation area is not limited. In this case, the heat and mass exchange occurs through the whole material surface. Then, the surfaces A

_{m}and A

_{T}are equal. In the rotary dryer used in our tests (Figure 1), the material is not supported and experiences constant motion, and both heating and drying take place over the whole surface. Hence, a simple transformation of Equation (3) gives the area of both mass and heat exchange as a function of humidity, described by the following equation:

_{cr}, the free moisture film is reduced and the drying rate decreases. The falling drying rate period (FDRP) starts. During this period, the temperature of the dried material rises to the equilibrium temperature.

_{0}, X

_{cr}, and X

_{eq}are the initial, critical, and equilibrium values of the moisture contents, respectively and ${\phi}_{a}$ is the relative humidity of the drying medium. The critical moisture point X

_{cr}describes the transition between the constant drying rate period (CDRP) and the falling drying rate period (FDRP).

_{vs}is a function of temperature:

## 3. Solution Method

## 4. Model Parameter Determination

_{T}and mass h

_{m}transfer coefficients; the heat source ΔQ describing the absorption of ultrasonic waves; and the critical relative humidity φ

_{cr}

_{2}.

## 5. Materials and Methods

_{s}and the volumetric shrinkage coefficient α

_{V}. Their values should be determined before starting the optimization procedure. The value of the material specific heat c

_{s}as a function of temperature was calculated based on formulas described in the literature [33]. The procedure employed for calculating the specific heat of the moist material was as follows. First, the mass fractions of the components contained in the apples and carrots were determined on the basis on the data from the National Food Institute of the Technical University of Denmark [34]. The individual specific heats of all constituents as temperature functions (polynomials) were determined on the basis of [33]. Then, the specific heat of dry matter was obtained as the sum of the products of the individual specific heats and the mass fractions of the components. The specific heat of the wet material is the sum of the specific heat of dry matter and the specific heat of water (temperature-dependent) multiplied by the moisture content X (see Equation (2)). The volumetric shrinkage coefficient, α

_{V}, was determined on the basis of an additional set of experiments. During the experiments, the materials were subject to slow convection drying. The volumes of fresh material samples were measured before drying and in given time intervals during the process. The measurement was carried out with the use of the gravimetrical method based on Archimedes’ law. In this method, the volume is calculated on the basis of weight measurement in air and water. Due to the amount of water removed by the sample, its weight in water is less than in air. The weight difference enables the sample volume measurement.

_{wb}, whereas for apples, it was 0.83 ± 0.01kg/kg

_{wb}. The drying processes were performed until the final moisture content of the carrots reached 0.05 kg/kg

_{wb}and that of the apples reached 0.1 kg/kg

_{wb}. During drying tests, the controller maintained the set process parameters, and collected all the data in its internal memory at constant time intervals. Only the material’s temperature was measured independently using a Dwyer HTDL-30 wireless temperature data logger, which allowed the collection of this parameter for material placed in a rotating drum. The wireless Dwyer sensor is a small electronic tube which ends with an elastic thermocouple. The setup was placed in a drier drum where the thermocouple was placed inside the samples in its center. The sample was secured against slipping from thermocouple by a very thin string.

_{a}, was about 40, 50, and 70 °C. When ultrasound was applied in drying schemes CVUD_404200, CVUD_502200, and CVUD_702200, the material temperature T increased above the drying medium temperature T

_{a}due to the absorption of ultrasonic waves. This phenomenon follows from the additional heat source ΔQ.

## 6. Numerical Results

_{cr}for all drying schemes were assumed to be equal to the initial one X

_{0}.

_{cr}

_{2}were estimated on the basis of the drying kinetics (experimental data), and are given in Table 2.

_{m}, h

_{T}, φ

_{cr}

_{,}and ΔQ). Examples of optimization procedure paths are shown in Figure 4. The chart shows three optimization paths (red, blue, and green). Each of these paths starts at a different point in space (points 1, 2, and 3 in Figure 4). For clarity, the chart was made with the standardized variable ranging from zero to the maximum value occurring during the calculation. Each of these paths was obtained as a result of the optimization procedure starting from a different point in the optimization variable space. It was found, regardless of the starting point of the performed calculations, that the same optimal point was obtained (see Figure 4). This means that the objective function, f, had only one minimum, and that the proposed algorithm was convergent to this solution.

_{T}and mass h

_{m}convective transfer, the additional heat source ΔQ, and the critical relative humidity φ

_{cr2}, were determined using the optimization procedure described previously (in the section on the estimation of model parameters). It was found that ultrasound application improves all parameters (h

_{m}, h

_{T}, and φ

_{cr}

_{2}) describing the convective heat and mass transfer between the samples and the drying medium. It was also found that the heating effect of ultrasound application caused by the ultrasound’s absorption ΔQ was very small. The power of the energy absorbed was less than 1% of the ultrasound generator’s power.

## 7. Discussion

^{−5}, 1.73 × 10

^{−4}, and 2.80 × 10

^{−4}kg/m

^{2}s, for the strawberries, raspberries, and green pepper, respectively. The application of ultrasound resulted in an increase of the coefficient by 96% for the strawberries, 45% for the raspberries, and 26% for the green pepper. These results indicate that the mass transfer coefficient depends on the material to be dried. The values of the coefficient of the mass transfer obtained in the present work presented in Table 2 and Table 3 are an order of greater value. This results from the differences in the movement of the material in the dryer. The pouring of the material in rotating cylinder construction improves the heat and mass transfer. The obtained results show that both estimated transfer coefficients depend on the material to be dried, the dryer construction, and the drying conditions.

## 8. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

Symbol | Designation | Unit |

${c}_{s}$ | specific heat of dry solid | J/kgK |

${c}_{l}$ | specific heat of moisture (water) | J/kgK |

$f$ | objective function | 1 |

${h}_{m}$ | mass transfer coefficient | kg/m^{2}s |

${h}_{T}$ | heat transfer coefficient | J/m^{2}sK |

$l$ | latent heat of evaporation | J/kg |

${m}_{s}$ | dry mass | kg |

${p}_{vs}$ | saturated vapor pressure | Pa |

$t$ | time | s |

$A$ | surface of sample | m^{2} |

${A}_{0}$ | initial surface of sample | m^{2} |

${A}_{m}$ | surface of mass exchange | m^{2} |

${A}_{T}$ | surface of heat exchange | m^{2} |

MC | moisture content dry basis | kg/kg_{db} |

$T$ | absolute temperature | K |

${T}_{0}$ | initial temperature | K |

${T}_{a}$ | ambient air temperature | K |

${T}_{\mathrm{exp}}$ | experimental value of temperature | K |

${T}_{\mathrm{max}}$ | maximal value of temperature | K |

${T}_{\mathrm{min}}$ | minimal value of temperature | K |

${T}_{\mathrm{num}}$ | numerical value of temperature | K |

$V$ | volume of sample | m^{3} |

${V}_{0}$ | initial volume of sample | m^{3} |

$X$ | moisture content dry basis | kg/kg_{db} |

${X}_{0}$ | initial moisture content dry basis | kg/kg_{db} |

${X}_{cr}$ | critical moisture content dry basis | kg/kg_{db} |

${X}_{cr2}$ | second critical moisture content dry basis | kg/kg_{db} |

${X}_{eq}$ | equilibrium moisture content dry basis | kg/kg_{db} |

${X}_{\mathrm{exp}}$ | experimental value of moisture content | kg/kg_{db} |

${X}_{\mathrm{max}}$ | maximal value of moisture content | kg/kg_{db} |

${X}_{\mathrm{min}}$ | minimal value of moisture content | kg/kg_{db} |

${X}_{\mathrm{num}}$ | numerical value of moisture content | kg/kg_{db} |

${\alpha}_{V}$ | volumetric shrinkage coefficient | 1 |

$\phi $ | relative air humidity | 1 |

${\phi}_{a}$ | relative air humidity in ambient air | 1 |

${\phi |}_{\partial B}$ | air humidity close to the dried sample surface | 1 |

${\phi}_{cr2}$ | second critical air relative humidity | 1 |

$\Delta Q$ | heat due to absorption of ultrasonic waves | W |

$\Phi \left(t,X,t\right)$ | function—simplifying designation | kg/s |

$\Psi \left(t,X,t\right)$ | function—simplifying designation | K/s |

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**Figure 1.**The scheme of the hybrid dryer: 1—blower (fan); 2—AUS controller; 3—AUS preamplifier; 4—microwave feeders; 5—heater; 6—pneumatic valve; 7—air outlet; 8—pyrometer; 9—drum drive; 10—microwave generators; 11—balance; 12—rotatable drum; 13—AUS ultrasound transducer; and 14—control unit.

**Figure 2.**Schematic presentation of the drying rate function: (

**a**) Existence of the constant drying period, and (

**b**) existence of the two stages of the falling drying rate period.

**Figure 3.**Drying kinetics of the ultrasound-assisted convective drying of carrot samples: (

**a**) Drying curves and (

**b**) temperature evolution curves. Drying kinetics of the ultrasound-assisted convective drying of apple samples: (

**c**) Drying curves and (

**d**) temperature evolution curves.

**Figure 4.**Optimization paths of the mass and heat transfer coefficients and the critical relative humidity (h

_{m}, h

_{T}, and φ

_{cr}) (drying scheme CVUD_502200).

**Figure 5.**Drying kinetics of ultrasound-assisted convective drying: (

**a**) Drying of apple—drying scheme CVUD_404200, and (

**b**) drying of carrot—drying scheme CVUD_502200.

Scheme No. and Abbreviation | Convective Drying Parameters | Ultrasound Power If Used |
---|---|---|

Apples | ||

1—CV_404 | 40 °C, air flow 4 m/s (effective 0.8 m/s) | – |

2—CVUD_404200 | 40 °C, air flow 4 m/s (effective 0.8 m/s) | 200 W |

Carrots | ||

1—CV_502 | 50 °C, air flow 2 m/s (effective 0.4 m/s) | – |

2—CV_702 | 70 °C, air flow 2 m/s (effective 0.4 m/s) | – |

3—CVUD_502200 | 50 °C, air flow 2 m/s (effective 0.4 m/s) | 200 W |

4—CVUD_702200 | 70 °C, air flow 2 m/s (effective 0.4 m/s) | 200 W |

Scheme No. and Abbreviation | Calculated Parameters | Assumed Value | Value of the Objective Function $\mathit{f}({\mathit{h}}_{\mathit{m}},{\mathit{h}}_{\mathit{T}},\mathbf{\Delta}\mathit{Q},{\mathit{\phi}}_{\mathit{c}\mathit{r}2})$ | |||
---|---|---|---|---|---|---|

h_{m} (kg/m^{2}s) | h_{T} (J/m^{2}sK) | φ_{cr}_{2} (1) | ΔQ (W) | X_{cr}_{2} (kg/kg) | ||

1—CV_502 | 0.000416 | 24.3 | 0.191 | – | 0.3 | 0.6589 |

2—CV_702 | 0.00116 | 20.8 | 0.194 | – | 1.0 | 0.2682 |

3—CVUD_502200 | 0.000773 | 44.5 | 0.323 | 0.584 | 0.3 | 0.1048 |

4—CVUD_702200 | 0.00140 | 40.5 | 0.275 | 0.887 | 0.8 | 0.1219 |

Scheme No. and Abbreviation | Calculated Parameters | Value of the Objective Function $\mathit{f}({\mathit{h}}_{\mathit{m}},{\mathit{h}}_{\mathit{T}},\mathbf{\Delta}\mathit{Q},{\mathit{X}}_{\mathit{c}\mathit{r}})$ | ||
---|---|---|---|---|

h_{m} (kg/m^{2}s) | h_{T} (J/m^{2}sK) | ΔQ (W) | ||

1—CV_404 | 0.000501 | 22.5 | – | 0.1235 |

2—CVUD_404200 | 0.000843 | 29.4 | 0.158 | 0.1321 |

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**MDPI and ACS Style**

Stasiak, M.; Musielak, G.; Mierzwa, D.
Optimization Method for the Evaluation of Convective Heat and Mass Transfer Effective Coefficients and Energy Sources in Drying Processes. *Energies* **2020**, *13*, 6577.
https://doi.org/10.3390/en13246577

**AMA Style**

Stasiak M, Musielak G, Mierzwa D.
Optimization Method for the Evaluation of Convective Heat and Mass Transfer Effective Coefficients and Energy Sources in Drying Processes. *Energies*. 2020; 13(24):6577.
https://doi.org/10.3390/en13246577

**Chicago/Turabian Style**

Stasiak, Marcin, Grzegorz Musielak, and Dominik Mierzwa.
2020. "Optimization Method for the Evaluation of Convective Heat and Mass Transfer Effective Coefficients and Energy Sources in Drying Processes" *Energies* 13, no. 24: 6577.
https://doi.org/10.3390/en13246577