# Study on Sublimation Drying of Carrot and Simulation by Using Cellular Automata

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{2}laser perforation [13], along with the integration of multiple drying technologies [14]. Nonetheless, these measures often come at the expense of compromising the quality of the final dried product. Hence, the search for appropriate external control of fundamental variables such as product size, vacuum chamber pressure, and temperature remains one of the most effective approaches [15]. In contrast to the traditional trial-and-error approach, numerical models have emerged as powerful tools for investigating the transfer phenomena during drying. These models serve as guides for analyzing and interpreting experimental data, thus reducing development time and associated costs. Curcio et al. [16] employed a simplified finite element model to gain insights into the various transfer phenomena occurring during convective drying of cylindrical carrot samples. This model, representative of actual drying behavior, exhibited a favorable agreement between predicted and experimental results. El-Maghlany et al. [17] developed a comprehensive multiphase porous media transport model to numerically investigate the freeze-drying process across multiple dimensions and different heating methods. This study extensively explored the effects of key parameters involved, providing valuable insights. Capozzi et al. [18] employed computational fluid dynamics (CFD) to determine the porosity, pore size, curvature, and permeability of particle accumulation structures formed through spray freezing. Subsequently, these parameters were utilized to describe mass transfer during the freeze-drying process.

## 2. Materials and Methods

#### 2.1. Materials

#### 2.2. Experimental Procedure

#### 2.3. Model of Sublimation Drying

_{1}) into the frozen region. Conversely, the superior plate remains separated from the material. Given the low-pressure environment, heat transfer primarily occurs through radiation (q

_{2}). Although the heat effects from radiation and convection (q

_{3}) on the material’s flanks depend on its arrangement and quantity [23], under low-pressure conditions, their contribution is generally minimal and often considered trifling [24].

- The material itself is isotropic, featuring uniform heat and mass transfer in the frozen region.
- The change in the volume of the material during sublimation is ignored.
- The sublimation interface exists between the dried region and the frozen region, which is continuous and infinitesimal in thickness.
- The concentration of water vapor is in equilibrium with ice at the sublimation interface.
- The porous matrix formed by ice sublimation is rigid in structure, and the matrix is permeable, which enables the vapor flux to circulate.

#### 2.3.1. Governing Equations

_{1}and k

_{3}are the bulk diffusivity constants, and k

_{2}and k

_{4}are the self-diffusivity constants. Their expressions and required parameters for calculation are listed in Table 1.

#### 2.3.2. The Initial and Boundary Conditions

#### 2.4. Cellular Automata Model

## 3. Results and Discussion

#### 3.1. Simulation Results

#### 3.2. Comparison of Moisture Content Curve

^{2}) reaching 99.4%. Both sets of curves displayed a similar exponential trend, with the moisture content decreasing rapidly in the early stage and gradually slowing down over time. This nonlinear decrease aligns with the drying behavior typically observed in biological materials [38,39,40]. The high initial drying rate can be attributed to the shorter diffusion length and larger sublimation interface area. However, as the drying process progresses, the interface area decreases, resulting in a smaller sublimation region and a larger dried region. Consequently, internal heat and mass transfer resistance increases, leading to a decline in the drying rate during the later stages. Notably, the measured moisture content was invariably lower than the simulated data. This discrepancy may arise from the loss of carrot mass and deformation during sublimation drying. In the non-shrinking process, the effective diffusion coefficient is generally higher compared to situations involving shrinkage [41], which was not considered in our simulations. Additionally, a slightly augmented measured dehydration rate, compared to the simulated rate, might stem from the model neglecting the desorption of partially bound water during sublimation drying [42].

#### 3.3. Comparison of Temperature Curve

^{2}) of 97.6%. This validates the model’s proficiency in capturing temperature variation during sublimation drying. As observed in the figure, in the early stage, the temperature rise at the center of the cell is relatively gradual and stable. This is because the heat received at the center of the material is solely derived from internal heat transfer. Nevertheless, post roughly 210 min of drying, a pronounced inflection point materializes on the temperature graph. This indicates that the sublimation interface traverses through the central cross-section of the carrot at this particular stage. Subsequently, the geometric center of the carrot enters the dried region, where external heat diffuses through the porous medium into the material with higher thermal conductivity, leading to a more rapid increase in temperature [33].

## 4. Conclusions

^{2}reached 99.4% and 97.6%, respectively. This study confirms the capability of cellular automata in simulating the sublimation drying behavior of carrots, allowing for the exploration of various external condition combinations to derive optimal parameters. With proper modeling and parameter adjustments, it can be tailored for simulating materials of similar structures.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

${\mathrm{C}}_{01}$ | constant dependent only upon the structure of the porous medium and giving relative D’Arcy flow permeability |

${\mathrm{C}}_{1}$ | constant dependent only upon the structure of the porous medium and giving relative Knudsen flow permeability |

${\mathrm{C}}_{2}$ | constant dependent only upon the structure of the porous medium and giving the ratio of bulk diffusivity within the porous medium to the free gas bulk diffusivity |

${\mathrm{c}}_{\mathrm{p}}$ | specific heat capacity at constant pressure |

${\mathrm{D}}_{\mathrm{w},\mathrm{i}\mathrm{n}}$ | free gas mutual diffusivity in a binary mixture of water vapor and inert gas |

F | view factor for radiative heat transfer |

$\mathrm{H}(\mathrm{t},\mathrm{r})$ | geometric shape of the moving interface, a function of time and radial distance |

${\mathrm{h}}_{\mathrm{v}}$ | convective heat transfer coefficient |

k | thermal conductivity |

${\mathrm{k}}_{1},{\mathrm{k}}_{3}$ | bulk diffusivity constant |

${\mathrm{k}}_{2},{\mathrm{k}}_{4}$ | self-diffusivity constant |

${\mathrm{K}}_{\mathrm{i}\mathrm{n}}$ | knudsen diffusivity for inert gas |

${\mathrm{K}}_{\mathrm{m}\mathrm{x}}$ | mean Knudsen diffusivity for binary gas mixture |

${\mathrm{K}}_{\mathrm{w}}$ | knudsen diffusivity for water vapor |

L | thickness of sample |

M | molecular weight |

N | mass flux |

P | partial pressure in the dried layer |

Q | heat flux |

r | space coordinate of radial distance |

R | radius of sample |

${\mathrm{R}}_{\mathrm{g}}$ | ideal gas constant |

t | drying time |

T | temperature |

v | velocity of moving interface |

z | space coordinate of distance along the thickness of the sample |

Z | value of z at the moving interface |

Greek letters | |

$\mathsf{\alpha}$ | thermal diffusivity |

$\u2206{\mathrm{H}}_{\mathrm{s}}$ | heat of sublimation of ice (J/kg) |

$\mathsf{\delta}$ | emissivity of the material surface. |

$\mathsf{\epsilon}$ | porosity of sample |

${\mathsf{\mu}}_{\mathrm{m}\mathrm{x}}$ | viscosity of vapor phase in pores of the dried layer |

$\mathsf{\rho}$ | density |

$\mathsf{\sigma}$ | Stefan–Boltzmann constant |

Subscripts | |

e | effective |

I | frozen region |

II | dried region |

in | inert gas |

lp | lower heating plate |

r | r direction |

up | upper heating plate |

w | water vapor |

z | z direction |

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**Figure 1.**Schematic of the experimental freeze-dryer: (1) cold trap, (2) drying chamber, (3) heating plates, (4) micro-computer system, (5) freezer, (6) vacuum pump.

**Figure 2.**The classical sublimation drying process for carrot slices. R and L represent the radius of and the thickness of the carrot, respectively.

**Figure 3.**The simulation results obtained from the cellular automata model at different times during the sublimation drying process. (

**a**) t = 0; (

**b**) t = 30 min; (

**c**) t = 60 min; (

**d**) t = 300 min.

Parameters | Value | Unit |
---|---|---|

${\mathrm{C}}_{01}$ | 7.219 × 10^{−15} | m^{2} |

${\mathrm{C}}_{1}$ | 3.85583 × 10^{−4} | m |

${\mathrm{C}}_{2}$ | 0.921 | – |

${\mathrm{k}}_{1}$ | ${\mathrm{k}}_{1}={\mathrm{C}}_{2}{\mathrm{D}}_{\mathrm{w},\mathrm{i}\mathrm{n}}^{0}{\mathrm{K}}_{\mathrm{w}}/{(\mathrm{C}}_{2}{\mathrm{D}}_{\mathrm{w},\mathrm{i}\mathrm{n}}^{0}+{\mathrm{K}}_{\mathrm{m}\mathrm{x}}\mathrm{p})$ | m^{2}/s |

${\mathrm{k}}_{2},{\mathrm{k}}_{4}$ | ${\mathrm{k}}_{2}={\mathrm{k}}_{4}=({\mathrm{K}}_{\mathrm{w}}{\mathrm{K}}_{\mathrm{i}\mathrm{n}}/{(\mathrm{C}}_{2}{\mathrm{D}}_{\mathrm{w},\mathrm{i}\mathrm{n}}^{0}+{\mathrm{K}}_{\mathrm{m}\mathrm{x}}\mathrm{p}))+({\mathrm{C}}_{01}/{\mathsf{\mu}}_{\mathrm{m}\mathrm{x}}$) | m^{4}/N·s |

${\mathrm{k}}_{3}$ | ${\mathrm{k}}_{3}={\mathrm{C}}_{2}{\mathrm{D}}_{\mathrm{w},\mathrm{i}\mathrm{n}}^{0}{\mathrm{K}}_{\mathrm{i}\mathrm{n}}/{(\mathrm{C}}_{2}{\mathrm{D}}_{\mathrm{w},\mathrm{i}\mathrm{n}}^{0}+{\mathrm{K}}_{\mathrm{m}\mathrm{x}}\mathrm{p})$ | m^{2}/s |

${\mathrm{D}}_{\mathrm{w},\mathrm{i}\mathrm{n}}^{0}$ | 8.729 × 10^{−7}(T_{0} + T_{int})^{2.334} | kg/ms^{3} |

${\mathrm{K}}_{\mathrm{m}\mathrm{x}}$ | ${\mathrm{K}}_{\mathrm{m}\mathrm{x}}=({\mathrm{P}}_{\mathrm{w}}/\mathrm{p}){\mathrm{K}}_{\mathrm{w}}+({\mathrm{P}}_{\mathrm{i}\mathrm{n}}/\mathrm{p}){\mathrm{K}}_{\mathrm{i}\mathrm{n}}$ | m^{2}/s |

${\mathrm{K}}_{\mathrm{w}}$ | ${\mathrm{K}}_{\mathrm{w}}={\mathrm{C}}_{1}{(\mathrm{R}\mathrm{T}/{\mathrm{M}}_{\mathrm{w}})}^{0.5}$ | m^{2}/s |

${\mathrm{K}}_{\mathrm{i}\mathrm{n}}$ | ${\mathrm{K}}_{\mathrm{i}\mathrm{n}}={\mathrm{C}}_{1}{(\mathrm{R}\mathrm{T}/{\mathrm{M}}_{\mathrm{i}\mathrm{n}})}^{0.5}$ | m^{2}/s |

${\mathsf{\mu}}_{\mathrm{m}\mathrm{x}}$ | [18.4858(T^{1.5}/(T + 650))] | kg/ms |

L | 10 | mm |

R | 40 | mm |

${\mathrm{T}}^{0}$ | 243.15 | K |

${\mathrm{T}}_{\mathrm{u}\mathrm{p}}$ | 303.15 | K |

${\mathrm{T}}_{\mathrm{l}\mathrm{p}}$ | 263.15 | K |

${\mathrm{T}}_{\mathrm{c}}$ | 303.15 | K |

${\mathsf{\rho}}_{\mathrm{I}}$ | 1000 | kg/m^{3} |

${\mathsf{\rho}}_{\mathrm{II}}$ | 236 | kg/m^{3} |

${\mathsf{\rho}}_{\mathrm{II},\mathrm{e}}$ | 388 | kg/m^{3} |

${\mathrm{c}}_{\mathrm{p},\mathrm{g}}$ | 1616.6 | J/(kg∙K) |

${\mathrm{c}}_{\mathrm{p},\mathrm{II},\mathrm{e}}$ | 2590 | J/(kg∙K) |

${\mathrm{c}}_{\mathrm{p},\mathrm{I}}$ | 1930 | J/(kg∙K) |

${\mathrm{k}}_{\mathrm{I}}$ | 2.68 | W/(m∙K) |

${\mathrm{k}}_{\mathrm{II}}$ | 0.18 | W/(m∙K) |

${\mathrm{k}}_{\mathrm{II},\mathrm{e}}$ | 680[12.98 × 10^{−8}P + 39.806 × 10^{−6}] | W/(m∙K) |

$\mathsf{\epsilon}$ | 0.581 | – |

${\mathrm{M}}_{\mathrm{w}}$ | 18 | g/mol |

${\mathrm{M}}_{\mathrm{i}\mathrm{n}}$ | 28 | g/mol |

${\mathrm{P}}_{\mathrm{w}}^{0}$ | 1.07 | Pa |

${\mathrm{P}}_{\mathrm{i}\mathrm{n}}^{0}$ | 24 | Pa |

$\u2206{\mathrm{H}}_{\mathrm{s}}$ | 2.7912 × 10^{3} | kJ/kg |

${\mathrm{R}}_{\mathrm{g}}$ | 8.314 | J/(mol∙K) |

$\mathsf{\sigma}$ | 5.67 × 10^{−8} | W/(m^{2}∙K^{4}) |

$\mathsf{\delta}$ | 0.85 | – |

${\mathrm{F}}_{\mathrm{c}}$ | 0.75 | – |

${\mathrm{F}}_{\mathrm{u}\mathrm{p}}$ | 0.795 | – |

${\mathrm{F}}_{\mathrm{l}\mathrm{p}}$ | 0.00809 | – |

${\mathrm{h}}_{\mathrm{v}}$ | 26 | W/(m^{2}∙K) |

Cell Status | Symbols | Description |
---|---|---|

Plate | P (gray) | For fixed cells, heat exchange occurs only with the bottom layer of the carrot. |

Frozen State | FS (yellow) | The initial state of the carrot cells, with the level of water content indicated by the number of blue dots. |

Dried State | DS (red) | As the moisture content decreases below a certain threshold, the cell transitions into a dried state, and at this point, the number of blue dots in the cell is 0. |

Air | Air (white) | Referring to the air within the drying chamber, heat exchange transpires between the air and the material through radiation and convective mechanisms. |

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## Share and Cite

**MDPI and ACS Style**

Shao, J.; Jiao, F.; Nie, L.; Wang, Y.; Du, Y.; Liu, Z.
Study on Sublimation Drying of Carrot and Simulation by Using Cellular Automata. *Processes* **2023**, *11*, 2507.
https://doi.org/10.3390/pr11082507

**AMA Style**

Shao J, Jiao F, Nie L, Wang Y, Du Y, Liu Z.
Study on Sublimation Drying of Carrot and Simulation by Using Cellular Automata. *Processes*. 2023; 11(8):2507.
https://doi.org/10.3390/pr11082507

**Chicago/Turabian Style**

Shao, Jiayuan, Fan Jiao, Lili Nie, Ying Wang, Yihan Du, and Zhenyu Liu.
2023. "Study on Sublimation Drying of Carrot and Simulation by Using Cellular Automata" *Processes* 11, no. 8: 2507.
https://doi.org/10.3390/pr11082507