Heat and Mass Transfer Models and Measurements for Low-Temperature Storage of Biological Systems

  • Shahensha M. Shaik
  • Ram DevireddyEmail author
Reference work entry


Living systems are routinely exposed to low temperatures, and their corresponding response has been of scientific and medical interest for centuries. The basic physicochemical phenomena that govern the response of the living systems to subzero temperatures are complex and interactively coupled so that the prediction and regulation of the associated events are often difficult. However, during the past several decades, significant progress has been made in devising useful techniques to further our understanding of biological systems at ultralow temperatures. This chapter presents some of the heat and mass transfer models and associated measurements in biological systems exposed to low temperatures specifically as applied to cryopreservation protocols.



Cross-sectional area of the extracellular space


Effective membrane surface area available for water transport


Isotonic (initial) cell surface area


Chemical potential of the intracellular fluid


Chemical potential of extracellular fluid


Constant cooling rate


Biophysically determined optimum cooling rate


Specific heat capacity


Cryoprotective agent

\( {C}_{CPA}^O \)

Concentration of the CPA outside the cell (extracellular space)

\( {C}_{CPA}^I \)

Concentration of the CPA inside the cell (intracellular space)


Diffusivity coefficient


Extracellular matrix


Apparent activation energy for Lpg


Apparent activation energy for Lpg in the presence of CPAs


Fractional mass of extracellular water in the medium


Fractional mass of intracellular water in the medium


Intracellular ice formation


Solute (CPA) flux across the cell membrane


Total flux across the cell membrane


Thermal conductivity


Latent heat of fusion per unit mass of solution


Length of the Krogh cylinder


Cell membrane hydraulic conductivity which measures the volume flow induced by a hydrostatic pressure difference


Cell membrane hydraulic conductivity at TR


Cell membrane hydraulic conductivity TR in the presence of CPAs


Number of moles of solutes in the cell as calculated from initial cell osmolarity


Number of water molecules


Initial (isotonic) number of water molecules


Probability of intracellular ice formation


Spatial coordinate


Sinusoid (vascular) radius in the Krogh cylinder model


Initial sinusoid radius in the Krogh cylinder model


Universal gas constant


Goodness of fit parameter


Surface-catalyzed ice nucleation




Reference temperature (273.15 K)




Cell volume


Volume-catalyzed ice nucleation


Initial isotonic cell volume


Osmotically inactive cell volume


Mole fraction of the CPA


Distance between adjacent sinusoid centers in the Krogh cylinder model


Mole fraction of the intracellular water


Experimental data points


Model predicted data points

\( \overline{y} \)

Normal mean of the experimental data points


Concentration difference between the intracellular and extracellular space for solute (CPA)


The “log mean” osmolality of the permeable solute


Concentration difference between the intracellular and extracellular space for water


Hydrostatic pressure difference

\( \frac{\partial {C}_I}{\partial t} \)

Local unsteady CPA concentration term

\( \frac{\partial \left({vC}_I\right)}{\partial x} \)

Convection transport term

\( D\frac{\partial^2{C}_I}{\partial {x}^2} \)

Diffusion term, assuming constant D

Greek Symbols


Local convective velocity


Partial molar volume of water


Cell membrane permeability to cryoprotective agent (m3/N-s)


Reflection coefficient or the “relative permeability” of the membrane to the solute


Uncertainties in the experimental data points




Disassociation constant for sodium chloride


Residual variance

\( {\overline{\chi}}^2 \)

Total variance


Mass fraction of the medium which has not yet released latent heat during freezing


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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringLouisiana State UniversityBaton RougeUSA

Section editors and affiliations

  • Ram Devireddy
    • 1
  1. 1.Department of Mechanical and Industrial EngineeringLouisiana State UniversityBaton RougeUSA

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