# Crystallisation Degree Analysis during Cryopreservation of Biological Tissue Applying Interval Arithmetic

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Governing Equations

^{−1}·K

^{−1}).

_{f}, W

_{w}, H

_{f}are the microchannel dimensions (see Figure 2), T

_{f}is the temperature of the working fluid, α is the external heat transfer coefficient, and $\overline{\mathsf{\eta}}$ is the interval fin efficiency. The interval heat flux is defined as [40]:

**n**is the normal vector.

#### 2.2. Numerical Model

^{2}= 0.989 for the thermal conductivity and R

^{2}= 0.999 for the specific heat):

## 3. Results

^{−6}m, where the number of nodes is m = 100 (the number of elements is l = 99). The initial parameters are as follows: T

_{0}= 22 °C and χ

_{0}= 0. The calculations have been performed using the finite difference method supplemented by interval arithmetic rules.

_{w}+ H

_{s}). The width of the interval is relatively small; hence, an approximation is prepared for 0–0.02 s and 0.40–0.43 s, which confirms that the resulting temperature course is in the form of intervals. Note that the red line and the blue line represent the upper and lower limits of the interval, respectively. It can be seen that stabilisation of the temperature to the assumed temperature by working fluid is achieved within a few seconds. In the whole sample, the minimum temperature is reached after 14.1 s.

^{−7}while for warming it is χ = 0.999. It can be said that the lower risk of cell damage caused by crystallisation is during the cooling process. However, for warming, the degree of crystallisation increases as a result of recrystallisation and then decreases to 0 after exceeding the DTR; while for cooling, the degree of crystallization remains constant after passing DTR.

^{−11}. Whereas for heating, the passing through the DTR lasts 0.057 s and the crystallisation rate is less than 2.4 × 10

^{−3}. These results are significantly lower than indicated in our simulation. However, it is important to consider a criterion that defines a certain maximum value of the degree of crystallisation, above which crystallisation can damage the sample. It is assumed that χ < 10

^{−6}, thus, both the results from Zhou et al. [8] and from our model fulfill this criterion for cooling and heating after recrystallization disappears.

_{w}+ H

_{s}) at selected moments of the simulation time.

## 4. Discussion

^{−6}. It can be observed that for cooling, this condition is achieved and the phenomenon of ice crystallisation can be neglected. For warming, the degree of crystallisation is higher than the given maximum value. However, after the recrystallisation time, the rate rapidly falls to zero after passing through the DTR as a result of heating the sample structure. Thus, it can be assumed that the crystallisation is not damaging the sample.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 4.**Interval thermal conductivity as a function of time during cooling for (

**a**) chip wall; (

**b**) sample layer.

**Figure 5.**Specific heat coefficient as a function of time during cooling for (

**a**) chip wall; (

**b**) sample layer.

**Figure 10.**Interval temperature in the cross-section of the sample at simulation time t = 0.04 s: (

**a**) for cooling; (

**b**) for warming.

Parameter | Value | |
---|---|---|

Dimensions [8,39] | ||

W_{f} | [m] | 5 × 10^{−5} |

W_{w} | [m] | 2.5 × 10^{−5} |

H_{f} | [m] | 3.5 × 10^{−4} |

H_{w} | [m] | 1 × 10^{−4} |

H_{s} | [m] | 1 × 10^{−4} |

Working fluid parameters [8,44,50] | ||

T_{f} | [°C] | −196 (for cooling)/40 (for warming) |

α | [W·m^{−2}·K^{−1}] | 1.048 × 10^{4} (for cooling)/4.74 × 10^{4} (for warming) |

k_{a} | [s^{−1}·K^{−1}] | 3.933 × 10^{7} (for cooling)/1.287 (for warming) |

Crystallisation properties [42,51] | ||

T_{m} | [K] | 243.5 |

Q | [J·mol^{−1}] | 4.187 × 10^{3} |

Other [52] | ||

L_{h} | [J·kg^{−1}] | 334 × 10^{3} |

${\mathsf{\rho}}_{h}$ | [kg·m^{−3}] | 1000 |

${\mathsf{\rho}}_{w}$ | [kg·m^{−3}] | 2330 |

Time, t [s] | Interval Thermal Conductivity, $\overline{\mathbf{\lambda}}$ [W·m ^{−1}·K^{−1}]
| Interval Specific Heat Coefficient, $\overline{\mathit{c}}$ × 10 ^{3} [J·kg^{−1}·K^{−1}]
| Interval Temperature, $\overline{\mathit{T}}$ [°C] | Interval Degree of Crystallisation, $\overline{\mathit{\chi}}$ × 10 ^{−8} |
---|---|---|---|---|

During cooling | ||||

0.0 | [0.987; 1.091] | [2.651; 2.930] | [22.000; 22.000] | [0.000; 0.000] |

0.1 | [2.125; 2.120] | [2.611; 2.612] | [−41.007; −40.677] | [0.3976; 0.3789] |

0.2 | [2.582; 2.578] | [2.522; 2.523] | [−72.112; −71.873] | [6.108; 6.182] |

0.4 | [3.195; 3.193] | [2.383; 2.383] | [−121.158; −120.973] | [6.832; 6.932] |

0.6 | [3.516; 3.515] | [2.295; 2.295] | [−152.115; −152.025] | [6.832; 6.932] |

0.8 | [3.651; 3.650] | [2.253; 2.253 | [−166.894; −166.848] | [6.832; 6.932] |

1.0 | [3.720; 3.720] | [2.229; 2.229] | [−175.088; −175.040 | [6.832; 6.932] |

During warming | ||||

0.0 | [3.6872; 4.0753] | [2.061; 2.278] | [−196.000; −196.000] | [6.832; 6.932] |

0.1 | [2.485; 2.496] | [2.542; 2.540] | [−65.211; −65.984] | [1.6 × 10^{5}; 1.3 × 10^{5}] |

0.2 | [2.442; 2.443] | [2.551; 2.550] | [−62.254; −62.270] | [5.352 × 10^{7}; 5.301 × 10^{7}] |

0.4 | [1.303; 1.310] | [2.750; 2.749] | [7.811; 7.427] | [7.811 × 10^{8}; 0.000] |

0.6 | [0.879; 0.879] | [2.814; 2.814] | [30.397; 30.364] | [0.000; 0.000] |

0.8 | [0.794; 0.794] | [2.827; 2.827] | [34.752; 34.739] | [0.000; 0.000] |

1.0 | [0.754; 0.754] | [2.832; 2.832] | [36.765; 36.758] | [0.000; 0.000] |

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**MDPI and ACS Style**

Piasecka-Belkhayat, A.; Skorupa, A.
Crystallisation Degree Analysis during Cryopreservation of Biological Tissue Applying Interval Arithmetic. *Materials* **2023**, *16*, 2186.
https://doi.org/10.3390/ma16062186

**AMA Style**

Piasecka-Belkhayat A, Skorupa A.
Crystallisation Degree Analysis during Cryopreservation of Biological Tissue Applying Interval Arithmetic. *Materials*. 2023; 16(6):2186.
https://doi.org/10.3390/ma16062186

**Chicago/Turabian Style**

Piasecka-Belkhayat, Alicja, and Anna Skorupa.
2023. "Crystallisation Degree Analysis during Cryopreservation of Biological Tissue Applying Interval Arithmetic" *Materials* 16, no. 6: 2186.
https://doi.org/10.3390/ma16062186