Devashish Shrivastava

An Analytical Study of Heat Transfer in Finite Tissue With Two Blood Vessels and Uniform Dirichlet Boundary Conditions

Counter-current (vessel-vessel) heat transfer has been postulated as one of the most important heat transfer mechanisms in living systems. Surprisingly, however the accurate quantification of the vessel-vessel, and vessel-tissue, heat transfer rates has never been performed in the most general and important case of a finite, unheated/heated tissue domain with noninsulated boundary conditions. To quantify these heat transfer rates, an exact analytical expression for the temperature field is derived by solving the 2-D Poisson equation with uniform Dirichlet boundary conditions

An analytical study of ‘Poisson conduction shape factors’ for two thermally significant vessels in a finite, heated tissue

To conveniently and properly account for the vessel to vessel and vessel to tissue heat transfer rates to predict in vivo tissue temperature distributions, this paper analyses two different types of Poisson conduction shape factors (PCSFs) for unheated and/or uniformly heated, non-insulated, finite tissue domains. One is related to the heat transfer rate from one vessel to another (vessel-vessel PCSF (VVPCSF)) and the other is related to the vessel to tissue heat transfer rates (vessel-tissue PCSF (VTPCSF)). Two alternative formulations for the VTPCSFs are studied; one is based on the difference between the vessel wall and tissue boundary temperatures, and the other on the difference between the vessel wall and the average tissue temperatures. The effects of a uniform source term and of the diameters and locations of the two vessels on the PCSFs are studied for two different cases: one, when the vessel wall temperatures are lower than the tissue boundary temperature, i.e., the vessels cool the tissue, and vice versa. Results show that, first, the VVPCSFs are only geometry dependent and they do not depend on the applied source term and the vessel wall and tissue boundary temperatures. Conversely, the VTPCSFs are strong functions of the source term and of the temperatures of the vessel walls and tissue boundary. These results suggest that to account for the vessel to vessel heat transfer rates, the VVPCSFs can be evaluated solely based on the vessel network geometry. However, to account for the vessel to tissue heat transfer rates, the VTPCSFs should be used iteratively while solving for the tissue temperature distributions. Second, unlike the tissue boundary temperature-based VTPCSFs which may become singular only in heated tissues, the average tissue temperature-based VTPCSFs have the potential to become singular in both unheated and heated tissues. These results suggest that caution should be exercised in the use of the VTPCSFs since they may approach singularity by virtue of their definition and thus may introduce large errors in the evaluation of tissue temperature distribution. Presented results are new and complementary to the previous shape factor results since these include the effect of (1) source term and (2) unequal vessel-tissue heat transfer rates from the two vessels to the tissue.

A General Analytical Approach for Defining Effectiveness in Ideal Two Fluid Heat Exchangers

A general analytical development, based on the first and second laws of thermodynamics, is used to define the maximum possible heat transfer in an ideal two-fluid exchanger as well as the maximum possible temperature differences for both fluid streams. It is shown that the conventional expression for the maximum possible heat transfer in ideal two-fluid heat exchangers is a special case of the general expressions. The application of both the first and second laws of thermodynamics in defining the maximum possible heat transfer and maximum possible temperature difference provides only one expression (instead of two different expressions) for either stream which is a measure of both thermal and temperature effectiveness of the particular stream. Differences between the conventional and proposed effectiveness values are presented as functions of the capacity ratio and NTU. These data are used to demonstrate the advantages of the new definitions for maximum heat transfer and maximum temperature difference in ideal two-fluid heat exchangers.Copyright

A Comparison Between 2-D and 3-D Conduction Shape Factors

Conduction shape factors are frequently used in a variety of heat transfer applications to evaluate heat transfer from one three-dimensional body to another three-dimensional body. Previous investigators have used conduction shape factors derived using the 2-D cross-section of the 3-D geometries for non-heating conditions as approximations to 3-D conduction shape factors with heating and no-heating present. This paper investigates the suitability of neglecting the axial conduction and power deposition in deriving expressions for conduction shape factors for the case of a single, cylindrical vessel imbedded concentrically in a cylindrical, uniformly heated tissue matrix. It is shown that 1) conduction shape factors are functions of the deposited power and the temperature distribution and 2) the magnitudes of conduction shape factors are affected significantly by axial conduction.Copyright

Poisson Conduction Shape Factors for “Mixed Case” Counter-Current Heat Transfer Applications

The effects of a source term and geometry on vessel-vessel and vessel-tissue Poisson conduction shape factors (VVPCSFs and VTPCSFs) are studied for uniformly heated, finite, non-insulated tissues for the ‘mixed case’ i.e., when the tissue boundary temperature lies in between the two vessel wall temperatures. In addition, two alternative formulations for the VTPCSFs are compared; while both formulations use the vessel wall temperature, one uses the tissue boundary temperature, and the other the area averaged tissue temperature. Results show that the VVPCSFs are only geometry dependent and do not depend on the applied power or the two vessel wall and tissue boundary temperatures. Conversely, the VTPCSFs are strong functions of these variables.Copyright

An Analytical Derivation of 2-D Conduction Shape Factors

New, improved formulations to evaluate two commonly used 2-D conduction shape factors, standard and average, are presented for a single, circular vessel eccentrically imbedded in a uniformly heated circular tissue matrix in terms of three non-dimensional parameters. The standard conduction shape factor is defined based on the vessel wall temperature and the outer wall temperature. The average conduction shape factor is defined based on the average tissue temperature and the vessel wall temperature. It is shown that both types of 2-D conduction shape factors are functions of the deposited power. The need to use proper expressions to evaluate conduction shape factors to accurately estimate the heat transfer from/to a region of interest when heating is present is stressed.Copyright

Theoretical Evaluation of Convective Heat Transfer Coefficients in Conjugated Problems Using the Second Law of Thermodynamics

The Tissue Convective Energy Balance Equation (TCEBE) is a recently derived, general bio-heat transfer equation from which a derivation of Pennes BHTE can be obtained. To accurately implement the TCEBE it is necessary to obtain expressions for the values of the overall heat transfer coefficients between the blood vessels and the tissue. One of the requisite steps in evaluating the overall heat transfer coefficients is to estimate convective heat transfer coefficients between the blood vessels and the tissue. To achieve this goal, a new technique is presented using the second law of thermodynamics to estimate convective heat transfer coefficients in conjugated problems at steady state. One dimensional results obtained are qualitatively similar to two dimensional results from the analytical solution for conjugated problems.© 2002 ASME