An analytic solution of one-dimensional steady-state Pennes’ bioheat transfer equation in cylindrical coordinates

Based on the Pennes’ bioheat transfer equation, a simplified one-dimensional bioheat transfer model of the cylindrical living tissues in the steady state has been set up for application in limb and whole body heat transfer studies, and by using the Bessel’s equation, its corresponding analytic solution has been derived in this paper. With the obtained analytic solution, the effects of the thermal conductivity, the blood perfusion, the metabolic heat generation, and the coefficient of heat transfer on the temperature distribution in living tissues are analyzed. The results show that the derived analytic solution is useful to easily and accurately study the thermal behavior of the biological system, and can be extended to such applications as parameter measurement, temperature field reconstruction and clinical treatment.     

A closed-form solution of DPL bioheat transfer problem with time-periodic boundary conditions

A closed-form (long-time) solution of one-dimensional dual-phase lag bioheat transfer problem with consistent time-periodic boundary conditions (BCs) is presented in this paper for planar, cylindrical, and spherical skin tissue for a newly developed solution methodology. The steady-periodic solution is composed of a steady-state part and an oscillating part; corresponding to the constant and oscillating parts of BCs, respectively. Using the superposition principle, these two parts are split into two problems, which are solved separately. The steady-state part is fairly straightforward to obtain, while for the oscillating part, an alternate Laplace transform (LT) approach is proposed in this work. It is demonstrated that for sinusoidal BCs, a closed-form solution in the time domain can be obtained by evaluating an approximate convolution integral, which emulates the effect of the inverse LT. The obtained closed-form solution is free of any series summation or numerical inversion, thereby, making it computationally very efficient compared with conventional LT and eigenfunctions-based approaches. The current methodology is verified with the established eigenfunctions expansion-based methodology. It can be seen that the long-time solutions obtained by these two approaches are almost identical. The verified methodology is further extended for the time-periodic nonsinusoidal BCs. The ease of implementation and simplicity of the new methodology for both sinusoidal and nonsinusoidal BCs is demonstrated using a few test cases. It is evident from the results that the developed methodology leads to an efficient and accurate solution.     

Estimation of temperature distribution in biological tissue by using solutions of bioheat transfer equation

A living body has a system for maintaining its temperature. We have investigated the heat transfer characteristics common to each organ and therapy using heat transfer. The one‐dimensional bioheat transfer equation with bioheat generation was converted into a dimensionless form and solved by Laplace transformation on the assumption that biological tissue is homogeneous. A dimensionless steady‐state solution and transient solution were derived analytically. These solutions can represent the characteristics of the temperature distribution common to each organ. Comparison with numerical solutions has confirmed that these solutions can be applied to estimate the temperature distribution of inhomogeneous biological tissue. It is proved that the size of the region where temperature change occurs, the steady‐state thermal penetration depth, is decided by biological properties. Furthermore, the time needed to reach a steady state, or the time it takes for biological tissue to reach a steady state, is calculated by using these solutions. Additionally, a temperature chart was proposed for each organ or tissue. This chart can serve as a guideline for medical doctors in formulating thermal therapy. © 2008 Wiley Periodicals, Inc. Heat Trans Asian Res, 37(6): 374– 386, 2008; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/htj.20210     

A general bioheat transfer model based on the theory of porous media

A volume averaging theory (VAT) established in the field of fluid-saturated porous media has been successfully exploited to derive a general set of bioheat transfer equations for blood flows and its surrounding biological tissue. A closed set of macroscopic governing equations for both velocity and temperature fields in intra- and extravascular phases has been established, for the first time, using the theory of anisotropic porous media. Firstly, two individual macroscopic energy equations are derived for the blood flow and its surrounding tissue under the thermal non-equilibrium condition. The blood perfusion term is identified and modeled in consideration of the transvascular flow in the extravascular region, while the dispersion and interfacial heat transfer terms are modeled according to conventional porous media treatments. It is shown that the resulting two-energy equation model reduces to Pennes model, Wulff model and their modifications, under appropriate conditions. Subsequently, the two-energy equation model has been extended to the three-energy equation version, in order to account for the countercurrent heat transfer between closely spaced arteries and veins in the circulatory system and its effect on the peripheral heat transfer. This general form of three-energy equation model naturally reduces to the energy equations for the tissue, proposed by Chato, Keller and Seiler. Controversial issues on blood perfusion, dispersion and interfacial heat transfer coefficient are discussed in a rigorous mathematical manner.     

A unified heat transfer model in a pressurized volumetric solar receivers

In the present work, we developed an overall mathematical model adequately describing the main heat transfer processes in a pressurized volumetric receiver. The key components, a windowed cavity, incorporating with the irradiated surface of the absorber, were theoretically modeled as a closed diffuse-gray surfaces system. Accordingly, a boundary condition for the absorber concerning its porous structure surface was developed using net radiation method (NRM) under local thermal non-equilibrium (LTNE) condition. The same method is also applied to the back cavity. Then a modified P1 approximation with collimated irradiation was introduced to incorporate the radiation transfer penetrating in the absorber. The major characteristic of the heat transfer behavior combining radiation, thermal conduction, and convection in the windowed cavity, absorber and the back cavity, are detailedly presented. Also, the key design parameters, such as those relating to pore structure (φ and d_p), the volumetric heat transfer coefficient h_v, the emissivity ε for window and absorber, and their thickness L_a and L_g were systematically analyzed. Optimization design can be carried out for both of the solar thermal system and the receiver itself in the future work based on our model.     

A numerical study on nonlinear DPL model for analyzing heat transfer in tissue during thermal therapy

In the present paper, therapeutic treatment of infected tumorous cells has been studied through mathematical modeling and simulation of heat transfer in tissues by using a nonlinear dual-phase lag bioheat transfer model with Dirichlet boundary condition. The components of volumetric heat source in this model such as blood perfusion and metabolism are assumed experimentally validated temperature-dependent function, which gives more accurate temperature distribution in tissues through this model. We have used the finite difference and RK (4, 5) techniques of numerical methods to solve the proposed problem and obtained the exact solution in a particular case. After comparison, we got a good agreement between them. We have used dimensionless quantities throughout this paper. The effect of relaxation and thermalization time with respect to dimensionless temperature distribution has been analyzed in the treatment process.     

Numerical analysis of phase-change heat transfer in axisymmetric biological tissue during nano-cryosurgery

The current work is concerned with the numerical investigation of phase-change heat transfer in a cylindrical-shaped biological tissue during nano-cryosurgery. So, the two-dimensional axisymmetric biological tissue with a single cryoprobe is considered to understand the freezing process inside it during nano-cryosurgery. This problem has been numerically simulated using the commercial software ANSYS Fluent 2020 R2. The results obtained are verified with the data available in the literature, and good agreement was found between them. After that, the grid-independent study is carried out to select the optimum element size. Then, the effect of various parameters, such as the radius and insertion depth of the cryoprobe and internal heat source, on the temperature distribution inside the biological tissue is studied. The effect of different nanoparticles, such as Au, Fe₃O_4, and MgO, on the temperature field are investigated, and it was found that the MgO nanoparticles reach the threshold temperature limit early and reduce cryoinjury to the surrounding healthy tissue. This study may help understand the freezing process inside the biological tissue during nano-cryosurgery.     

A bioheat transfer model: Forced convection in a channel occupied by a porous medium with counterflow

An illustrative model for bioheat transfer is developed. An analytical solution is obtained for forced convection in a parallel plate channel occupied by a layered saturated porous medium with counterflow, the dominant feature that distinguishes bioheat transfer from other forms of heat transfer. The case of asymmetrical constant heat-flux boundary conditions is considered and the Brinkman model is employed for the porous medium. It is found that the Nusselt number Nu is zero when the mean velocity is zero, and negative values can be attained.     

Generalized dual-phase lag bioheat equations based on nonequilibrium heat transfer in living biological tissues

Based on a nonequilibrium heat transfer model in the living tissue obtained by performing volume average to the local instantaneous energy equations for blood and tissues, the dual-phase lag bioheat equations with blood or tissue temperature as sole unknown temperature are obtained by eliminating the tissue or blood temperature from the nonequilibrium model. The present dual-phase model successfully overcame the drawbacks of the existing dual-phase lag bioheat equation obtained by simply modifying the classical Pennes bioheat equation. Under the dual-phase model developed in this work, the phase lag times are expressed in terms of the properties of blood and tissue and the interphase convective heat transfer coefficient and blood perfusion rate. The phase lag times for heat flux and temperature gradient for the living tissue are estimated using the available properties from the literature. It is found that the phase lag times for heat flux and temperature gradient for the living tissue are very close to each other.     

Nonlinear analysis of dual-phase lag bio-heat model in living tissues induced by laser irradiation

Abstract

This paper provides a method for determining a numerical solution of the thermal damage of living tissues using a nonlinear dual phase lag model. Due to the nonlinearity of the basic equations, the finite element approach is adopted to solve such problems. The numerical outcomes obtained by the finite element technique are also compared with the existing experimental study to verify the accuracy of the numerical calculations. Based on the formulation of Arrhenius, the thermal damages to the tissues are estimated by the denatured protein range. Numerical results for temperatures are presented graphically. Also, the comparisons between the numerical outcomes and the existing experimental data show that the present mathematical models are effective tools to evaluate the bioheat transfer in a spherical living tissue.     

Exergy transfer characteristics of forced convective heat transfer through a duct with constant wall temperature

The exergy transfer characteristics of fluid flow and heat transfer inside a circular duct under fully developed laminar and turbulent forced convection are presented. Temperature is kept constant at the duct wall. The exergy transfer Nusselt number is put forward and the analytical expressions for exergy transfer Nusselt number are obtained as functions of heat transfer Nusselt number, Reynolds number, Prandtl number, etc. The variations of the local and mean convective exergy transfer coefficient, non-dimensional exergy flux, exergy transfer rate, etc. with operating parameters are presented graphically. By reference to a smooth duct and taking air as working fluid, a numerical analysis of the influence of the Reynolds number and non-dimensional cross-sectional position on exergy transfer characteristics has been conducted. The results show that the process parameters and configuration in the fluid flow and heat transfer inside a duct should be properly selected so that the forced convection process could have the best exergy utilization. In addition, the results corresponding to the exergy transfer and energy transfer are compared.     

Exergy transfer characteristics of forced convective heat transfer through a duct with constant wall heat flux

The examination of exergy transfer characteristics caused by forced convective heat transfer through a duct with constant wall heat flux for thermally and hydrodynamic fully developed laminar and turbulent flows has been presented. The exergy transfer Nusselt number is put forward and the dependence relationships of the exergy transfer Nusselt number on the heat transfer Nusselt number, Reynolds number and Prandtl number are obtained. Expressions involving relevant variables for the local and mean convective exergy transfer coefficient, non-dimensional exergy flux and exergy transfer rate, etc. have been derived. By reference to a smooth duct, the numerical results of exergy transfer characteristics for fluids with different Prandtl number are obtained and the effect of the Reynolds number and non-dimensional cross-sectional position on exergy transfer characteristics is analyzed. In addition, the results corresponding to the exergy transfer and energy transfer are compared.     

CFD analysis on heat transfer through different extended surfaces

The present study includes computational fluid dynamics analysis and comparison of heat enhancement through different extended surfaces, especially in rectangular and square conductive and nonconductive fins. Computational and numerical analysis of heat transfer from a rectangular extended surface and a pin-finned plate studied to calculate the average Nusselt number in parallel, vertical direction placed along the sidewall. The total rise of the mean Nusselt number is noticed around 36% in pin-finned plate with respect to a plain plate. This is examined with optimal fin spacing of S_v with L ratio equals to 0.2 and S_h with W ratio equals to 0.25, height of extended surfaces 24 mm with 45° angle of inclination. The mean Nusselt number reduces with a rise in the angle of inclination and also increases with a rise in aspect ratio. The present study reveals that inline and staggered arrangements do not yield appreciably different results. The maximum average Nusselt number difference between conductive and nonconductive fins is around 5% for S_h per W ratio 0.33 and S_v per L ratio 0.2 at an angle of inclination 45°, fin height of 6 mm (height to thickness ratio 2).     

Radiation-induced mass transfer through membranes

The influence of excitation of molecules on mass transfer in membranes is investigated theoretically. It is shown that excitation of the molecules in the gas phase at one side of the membrane can lead to the occurrence of the resulting mass flux in an initially equilibrium system.     

An Analytic Solution of One-dimensional Steady-state Pennes'''' Bioheat Transfer Equation in Cylindrical Coordinates

Based on the Pennes' bioheat transfer equation, a simplified one-dimensional bioheat transfer model of the cylindrical living tissues in the steady state has been set up for application in limb and whole body heat transfer studies, and by using the Bessel's equation, its corresponding analytic solution has been derived in this paper. With the obtained analytic solution, the effects of the thermal conductivity, the blood perfusion, the metabolic heat generation, and the coefficient of heat transfer on the temperature distribution in living tissues are analyzed. The results show that the derived analytic solution is useful to easily and accurately study the thermal behavior of the biological system, and can be extended to such applications as parameter measurement, temperature field reconstruction and clinical treatment.     

Development of a unified heat transfer model with special emphasis on thermal conduction

It is postulated that energy fields in space and matter are quantized by an entity having coincident properties of wave and virtual matter. A unified link between thermal radiation and conduction was established as confirmed by Planck's quantum theory, and a unified heat transfer equation was developed. One-dimensional, steady-state thermal conduction was solved by using approximate scatter and mobility models in this equation. Coefficient of thermal conductivity of various materials were calculated and compared with the literature. Even with approximate scatter and mobility models, a good agreement was observed.     

Augmentation of heat transfer through passive techniques

The thermal performance of energy preservation systems is greatly improved by increasing miniaturization and boosting. These are imaginative (or Promethean) techniques to enhance heat transfer. Enhancement methods of heat transfer draw great attention in front of the industrial sector because of their ability to provide energy savings and raise the economic efficiency of thermal systems. Three techniques these methods are categorized; those are active, passive, and compound. Different types of components are used in passive methods because of the transfer/working fluid flow path to the enhancement of the heat transfer rate. In this article, the subject of the review was the passive heat transfer enhancement methods including inserts (conical strips, winglets, twisted tapes, baffles), porous materials, coil/helical/spiral tubes, rough surfaces (corrugated/ribbed surfaces), extended surfaces (fins) and nanofluids (mono and hybrid nanofluid). Recent passive heat transfer enhancement techniques are studied in this article as they are cost-effective and reliable, and also comparably passive methods do not need any extra power to promote the energy conversion systems' thermal efficiency than active methods. In the passive approaches, various components are applied to the heat transfer/working fluid flow path to improve the heat transfer rate. The passive heat transfer enhancement methods studied in this article include inserts (twisted tapes, conical strips, baffles, winglets), extended surfaces (fins), porous materials, coil/helical/spiral tubes, rough surfaces (corrugated/ribbed surfaces), and nanofluids (mono and hybrid nanofluid). From the pioneers' research work, it is clear that a lower twist ratio and lower pitch, lesser winglet angles can provide more heat transfer rate and a little bit more friction factor. In the case of nanofluids, a little bit of pumping power is enhanced. Finally, heat transfer enhancement is compared with the thermal performance factor, which is more than unity.     

Calculating heat transfer through windows

To calculate the energy performance of buildings, one must know the heat-transfer characteristics of the windows as functions of environmental variables, such as temperature and wind speed. Window designs are becoming more complex in response to the need for energy conservation. In this paper, we develop a general procedure for calculating the net energy flux through the glazed area of a window composed of an arbitrary number of solid layers. These layers, which may have thin-film coatings, can have any specified solar and thermal radiation properties and enclosed spaces between solid layers can contain either air or other gases. We verified our results by comparing them with experimental measurements of heat flow using a calibrated hot-box.     

A lattice Boltzmann model for axisymmetric thermal flows through porous media

In this paper, a new lattice Boltzmann model at representative elementary volume (REV) scale is proposed for axisymmetric thermal flows in porous media. In this model, a new equilibrium distribution function including porosity is proposed to describe the cases of variable porosity, and simple force term is adopted to achieve a better numerical stability. The numerical experiments, including five benchmark problems, demonstrate that the proposed LBM can be served as a feasible and efficient method for axisymmetric thermal flows through porous media.