Numerical Modeling of Heat and Mass Transfer during Cryopreservation Using Interval Analysis

Abstract

In the paper, the numerical analysis of heat and mass transfer proceeding in an axially symmetrical articular cartilage sample subjected to the cryopreservation process is presented. In particular, a two-dimensional (axially symmetrical) model with imprecisely defined parameters is considered. The base of the heat transfer model is given by the interval Fourier equation and supplemented by initial boundary conditions. The phenomenon of cryoprotectant transport (Me2SO) through the extracellular matrix is described by the interval mass transfer equation. The liquidus-tracking (LT) method is used to control the temperature, which avoids the formation of ice regardless of the cooling and warming rates. In the LT process, the temperature decreases/increases gradually during addition/removal of the cryoprotectant, and the articular cartilage remains on or above the liquidus line so that no ice forms, independent of the cooling/warming rate. The discussed problem is solved using the interval finite difference method with the rules of directed interval arithmetic. Examples of numerical computations are presented in the final part of the paper. The obtained results of the numerical simulation are compared with the experimental results, realized for deterministically defined parameters.

Numerical Study of Heat and Mass Transfer during Cryopreservation Process with Application of Directed Interval Arithmetic

“…It is also important to know the interval extracellular and intracellular concentrations, which are described by the following interval equations [ 13 , 14 , 15 , 18 , 22 , 27 , 40 , 41 ]: where is the interval diffusion coefficient, is the cell volume (the superscript 0 represents the initial moment of simulation), is the osmotically inactive volume of cells (for chondrocytes it is equal to 0.41 ), where is the isotonic volume of cells. The interval normalized cell volume is defined: …”

“…The interval diffusion coefficient of CPA in extracellular matrix can be estimated by the Einstein–Stokes equation [ 41 , 42 , 43 ]: where k B is the Boltzmann constant, equal to 1.38 × 10 −23 J·K −1 , r s is the radius of the spherical particle molecule, and η is the dynamic viscosity.…”

“…The interval temperature coupled to the interval mass transfer model (compare with Equations (1), (2) and (15)) is determined using the interval Fourier equation [ 13 , 41 ]: where is the interval thermal conductivity and is the interval volumetric specific heat.…”

“…The boundary nodes are located at the distance 0.5 h or 0.5 k with respect to the real boundary (h , k are the steps of regular mesh in directions r and z ), respectively (see Figure 2 ). This approach gives a better approximation of the Neumann and Robin boundary conditions [ 38 , 41 , 43 ].…”

“…The approximate form of the interval energy equation (Equation (16)) for the internal nodes ( i , j ) and transition t s− 1 → t s is the following [ 41 , 43 ]: where are the shape functions of differential mesh, while the thermal resistances are defined by the following formulas: …”