Abstract

Tissue freezing has significant applications in cryopreservation and cryosurgery processes. The freezing rate is an important factor during the cryopreservation process. To improve and make an accurate estimation of the freezing rate, radiation heat transfer plays a major role. To analyze in details, a two-dimensional coupled conduction–radiation model is developed. The tissue is frozen from the left side while the other sides are at the initial temperature. Finite volume method (FVM) is used to discretize both the radiative transfer equation (RTE) and energy equation. The algebraic equation after discretization is solved by the tri-diagonal matrix algorithm. The radiative heat flux is calculated by solving the RTE. The energy equation provides the temperature field. The enthalpy-porosity method is used to update the liquid volume fraction and thus the freezing front is captured. A linearization technique is proposed to linearize the radiative source term in the energy equation to avoid chances of divergence of the solution. The present model is first validated with the results of the existing literature and a good agreement is found. The effects of different parameters such as conduction–radiation parameter, scattering albedo, extinction coefficient and Stefan number on the prediction of temperature field, and the position of the freezing front are studied in details. It is found that a decrease in the values of conduction–radiation parameter, scattering albedo, and increase in the values of extinction coefficient and Stefan number cause more radiative heat loss. Hence, the freezing rate is improved and more frozen region is observed.

References

References
1.
Wang
,
Z.
,
Wu
,
H.
,
Zhao
,
G.
,
Liao
,
X.
,
Chen
,
F.
,
Wu
,
J.
, and
Hu
,
X.
,
2007
, “
One-Dimensional Finite-Difference Modeling on Temperature History and Freezing Time of Individual Food
,”
J. Food Eng.
,
79
(
2
), pp.
502
510
. 10.1016/j.jfoodeng.2006.02.012
2.
Whittingham
,
D. G.
,
Leibo
,
S. P.
, and
Mazur
,
P.
,
1972
, “
Survival of Mouse Embryos Frozen to −196°C and −269°C
,”
Science
,
178
(
4059
), pp.
411
414
. 10.1126/science.178.4059.411
3.
Singh
,
S.
, and
Kumar
,
S.
,
2015
, “
Freezing of Biological Tissues During Cryosurgery Using Hyperbolic Heat Conduction Model
,”
Math. Model. Anal.
,
20
(
4
), pp.
443
456
. 10.3846/13926292.2015.1064486
4.
Rall
,
W. F.
, and
Fahy
,
G. M.
,
1985
, “
Ice-Free Cryopreservation of Mouse Embryos at −196°C by Vitrification
,”
Nature
,
313
(
14
), pp.
573
575
. 10.1038/313573a0
5.
Fahy
,
G. M.
,
MacFarlane
,
D. R.
,
Angell
,
C. A.
, and
Meryman
,
H. T.
,
1984
, “
Vitrification as an Approach to Cryopreservation
,”
Cryobiology
,
21
(
4
), pp.
407
426
. 10.1016/0011-2240(84)90079-8
6.
Zhang
,
Y.
,
Zhao
,
G.
,
Chapal Hossain
,
S. M.
, and
He
,
X.
,
2017
, “
Modeling and Experimental Studies of Enhanced Cooling by Medical Gauze for Cell Cryopreservation by Vitrification
,”
Int. J. Heat Mass Transf.
,
114
, pp.
1
7
. 10.1016/j.ijheatmasstransfer.2017.06.036
7.
Kandra
,
D.
,
2004
, “
Tissue Interactions With Lasers and Liquid Nitrogen: A Novel Cryopreservation Method
,” LSU Masters’ thesis.
8.
Zheng
,
Y.
,
Zhao
,
G.
,
Zhang
,
Y.
, and
Gao
,
R.
,
2018
, “
On-Chip Loading and Unloading of Cryoprotectants Facilitate Cell Cryopreservation by Rapid Freezing
,”
Sens. Actuators B
,
255
, pp.
647
656
. 10.1016/j.snb.2017.08.084
9.
Müller-Schweinitzer
,
E.
,
2009
, “
Cryopreservation of Vascular Tissues Cryopreservation of Vascular Tissues
,”
Organogenesis
,
5
(
3
), pp.
97
104
. 10.4161/org.5.3.9495
10.
Su
,
F.
,
Zhao
,
N.
,
Deng
,
Y.
, and
Ma
,
H.
,
2017
, “
An Ultrafast Vitrification Method for Cell Cryopreservation
,”
ASME J. Heat Transfer
,
140
(
1
), p.
012001
. 10.1115/1.4037327
11.
Loken
,
S. D.
, and
Demetrick
,
D. J.
,
2005
, “
A Novel Method for Freezing and Storing Research Tissue Bank Specimens
,”
Hum. Pathol.
,
36
(
9
), pp.
977
980
. 10.1016/j.humpath.2005.06.016
12.
Alhamdan
,
A.
,
Hassan
,
B.
,
Alkahtani
,
H.
, and
Abdelkarim
,
D.
,
2015
, “
Cryogenic Freezing of Fresh Date Fruits for Quality Preservation During Frozen Storage
,”
J. Saudi Soc. Agric. Sci.
,
17
(
1
), pp.
9
16
. 10.1016/j.jssas.2015.12.001
13.
Ahmadikia
,
H.
, and
Moradi
,
A.
,
2012
, “
Non-Fourier Phase Change Heat Transfer in Biological Tissues During Solidification
,”
Heat Mass Transf.
,
48
(
9
), pp.
1559
1568
. 10.1007/s00231-012-1002-1
14.
Moradi
,
A.
, and
Ahmadikia
,
H.
,
2012
, “
Numerical Study of the Solidification Process in Biological Tissue With Blood Flow and Metabolism Effects by the Dual Phase Lag Model
,”
Proc. Inst. Mech. Eng. Part H J. Eng. Med.
,
226
(
5
), pp.
406
416
. 10.1177/0954411912441305
15.
Kumar
,
D.
,
Singh
,
S.
,
Sharma
,
N.
, and
Rai
,
K. N.
,
2018
, “
Verified Non-Linear DPL Model With Experimental Data for Analysing Heat Transfer in Tissue During Thermal Therapy
,”
Int. J. Therm. Sci.
,
133
, pp.
320
329
. 10.1016/j.ijthermalsci.2018.07.031
16.
Zhao
,
G.
,
Zhang
,
H. F.
,
Guo
,
X. J.
,
Luo
,
D. W.
, and
Gao
,
D. Y.
,
2007
, “
Effect of Blood Flow and Metabolism on Multidimensional Heat Transfer During Cryosurgery
,”
Med. Eng. Phys.
,
29
(
2
), pp.
205
215
. 10.1016/j.medengphy.2006.03.005
17.
Singh
,
S.
, and
Kumar
,
S.
,
2014
, “
Numerical Study on Triple Layer Skin Tissue Freezing Using Dual Phase Lag Bio-Heat Model
,”
Int. J. Therm. Sci.
,
86
, pp.
12
20
. 10.1016/j.ijthermalsci.2014.06.027
18.
Kumar
,
S.
, and
Katiyar
,
V. K.
,
2010
, “
Mathematical Modeling of Freezing and Thawing Process in Tissues: A Porous Media Approach
,”
Int. J. Appl. Mech.
,
02
(
3
), pp.
617
633
. 10.1142/S1758825110000688
19.
Deng
,
Z. S.
, and
Liu
,
J.
,
2004
, “
Numerical Simulation of 3-D Freezing and Heating Problems for Combined Cryosurgery and Hyperthermia Therapy
,”
Numer. Heat Transf. Part A Appl.
,
46
(
6
), pp.
587
611
. 10.1080/10407780490487740
20.
Jaunich
,
M.
,
Raje
,
S.
,
Kim
,
K.
,
Mitra
,
K.
, and
Guo
,
Z.
,
2008
, “
Bio-Heat Transfer Analysis During Short Pulse Laser Irradiation of Tissues
,”
Int. J. Heat Mass Transf.
,
51
(
23–24
), pp.
5511
5521
. 10.1016/j.ijheatmasstransfer.2008.04.033
21.
Kim
,
K.
, and
Guo
,
Z.
,
2007
, “
Multi-Time-Scale Heat Transfer Modeling of Turbid Tissues Exposed to Short-Pulsed Irradiations
,”
Comput. Methods Programs Biomed.
,
86
(
2
), pp.
112
123
. 10.1016/j.cmpb.2007.01.009
22.
Guo
,
Z.
, and
Kumar
,
S.
,
2007
, “
Discrete-Ordinates Solution of Short-Pulsed Laser Transport in Two-Dimensional Turbid Media
,”
Appl. Opt.
,
40
(
19
), pp.
3156
3163
. 10.1364/AO.40.003156
23.
Mukherjee
,
A.
,
Mishra
,
S. C.
, and
Mondal
,
P. K.
,
2017
, “
Numerical Analysis of Combined-Mode Dual-Phase-Lag Heat Conduction and Radiation in an Absorbing, Emitting, and Scattering Cylindrical Medium
,”
Numer. Heat Transf. Part A Appl.
,
71
(
7
), pp.
769
788
. 10.1080/10407782.2017.1308708
24.
Zhou
,
J.
,
Zhang
,
Y.
, and
Chen
,
J. K.
,
2008
, “
Non-Fourier Heat Conduction Effect on Laser-Induced Thermal Damage in Biological Tissues
,”
Numer. Heat Transf. Part A Appl.
,
54
(
1
), pp.
1
19
. 10.1080/10407780802025911
25.
Singh
,
R.
,
Das
,
K.
,
Okajima
,
J.
,
Maruyama
,
S.
, and
Mishra
,
S. C.
,
2015
, “
Modeling Skin Cooling Using Optical Windows and Cryogens During Laser Induced Hyperthermia in a Multilayer Vascularized Tissue
,”
Appl. Therm. Eng.
,
89
, pp.
28
35
. 10.1016/j.applthermaleng.2015.06.006
26.
Abrams
,
M.
, and
Viskanta
,
R.
,
2010
, “
The Effects of Radiative Heat Transfer Upon the Melting and Solidification of Semi-Transparent Crystals
,”
ASME J. Heat Transfer
,
96
(
2
), pp.
184
190
. 10.1115/1.3450162
27.
Mishra
,
S. C.
, and
Stephen
,
A.
,
2011
, “
Combined Mode Conduction and Radiation Heat Transfer in a Spherical Geometry With Non-Fourier Effect
,”
Int. J. Heat Mass Transf.
,
54
(
13–14
), pp.
2975
2989
. 10.1016/j.ijheatmasstransfer.2011.02.053
28.
Yi
,
H. L.
,
Tan
,
H. P.
, and
Zhou
,
Y.
,
2011
, “
Coupled Radiation and Solidification Heat Transfer Inside a Graded Index Medium by Finite Element Method
,”
Int. J. Heat Mass Transf.
,
54
(
13–14
), pp.
3090
3095
. 10.1016/j.ijheatmasstransfer.2011.03.006
29.
Anteby
,
I.
,
Shai
,
I.
, and
Arbel
,
A.
,
2000
, “
Numerical Calculations for Combined Conduction and Radiation Transient Heat Transfer in a Semi-Transparent Medium
,”
Numer. Heat Transf. Part A Appl.
,
37
(
4
), pp.
359
371
. 10.1080/104077800274226
30.
Wang
,
J. Y.
,
Gao
,
Z. X.
, and
Lee
,
C. H.
,
2015
, “
An Iterative Technique for Coupled Conduction-Radiation Heat Transfer in Semi-Transparent Media
,”
Numer. Heat Transf. Part A Appl.
,
67
(
11
), pp.
1208
1231
. 10.1080/10407782.2014.955370
31.
Raj
,
R.
,
Prasad
,
A.
,
Parida
,
P. R.
, and
Mishra
,
S. C.
,
2006
, “
Analysis of Solidification of a Semi-Transparent Planar Layer Using the Lattice Boltzmann Method and the Discrete Transfer Method
,”
Numer. Heat Transf. Part A Appl.
,
49
(
3
), pp.
279
299
. 10.1080/10407780500359828
32.
Lapka
,
P.
, and
Furmański
,
P.
,
2010
, “
Fixed Grid Simulation of Radiation-Conduction Dominated Solidification Process
,”
ASME J. Heat Transfer
,
132
(
2
), p.
023504
. 10.1115/1.4000188
33.
Mukherjee
,
A.
, and
Mondal
,
P. K.
,
2018
, “
Analysis of Heat Transfer Through Optically Participating Medium in a Concentric Spherical Enclosure: The Role of Dual-Phase-Lag Conduction and Radiation
,”
ASME J. Therm. Sci. Eng. Appl.
,
10
(
4
), p.
041022
. 10.1115/1.4040283
34.
Mishra
,
S. C.
,
Krishna
,
N. A.
,
Gupta
,
N.
, and
Chaitanya
,
G. R.
,
2008
, “
Combined Conduction and Radiation Heat Transfer With Variable Thermal Conductivity and Variable Refractive Index
,”
Int. J. Heat Mass Transf.
,
51
(
1–2
), pp.
83
90
. 10.1016/j.ijheatmasstransfer.2007.04.018
35.
Mishra
,
S. C.
,
Behera
,
N. C.
,
Garg
,
A. K.
, and
Mishra
,
A.
,
2008
, “
Solidification of a 2-D Semi-Transparent Medium Using the Lattice Boltzmann Method and the Finite Volume Method
,”
Int. J. Heat Mass Transf.
,
51
(
17–18
), pp.
4447
4460
. 10.1016/j.ijheatmasstransfer.2008.02.003
36.
Kim
,
K.
, and
Guo
,
Z.
,
2004
, “
Ultrafast Radiation Heat Transfer in Laser Tissue Welding and Soldering
,”
Numer. Heat Transf. Part A Appl.
,
46
(
1
), pp.
23
40
. 10.1080/10407780490457365
37.
Guo
,
Z.
,
2001
, “
Ultrashort Laser Transport in Turbid Media
,” Ph.D. thesis.
38.
Modest
,
M. F.
,
2013
,
Radiative Heat Transfer
,
Academic Press
,
New York, NY
.
39.
Chai
,
J. C.
,
Lee
,
H. S.
, and
Patankar
,
S. V.
,
1994
, “
Finite Volume Method for Radiation Heat Transfer
,”
J. Thermophys. Heat Transf.
,
8
(
3
), pp.
419
425
. 10.2514/3.559
40.
Patankar
,
S.
,
1980
,
Numerical Heat Transfer and Fluid Flow
,
Hemisphere
,
New York, NY
.
41.
Brent
,
A. D.
,
Voller
,
V. R.
, and
Reid
,
K. J.
,
1988
, “
Enthalpy-Porosity Technique for Modeling Convection-Diffusion Phase Change: Application to the Melting of a Pure Metal
,”
Numer. Heat Transf.
,
13
(
3
), pp.
297
318
. 10.1080/10407788808913615
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