The document describes a numerical simulation comparing parabolic and hyperbolic models of bioheat transfer during radiotherapy treatment of cancer. Temperature distributions are estimated using the Pennes, thermal wave, and dual phase lag models. Results show parabolic and hyperbolic models produce similar results for small phase lag times, but differ significantly for large lag times, suggesting hyperbolic models better represent complex biological systems. Thermal damage is also calculated, with the thermal wave model predicting higher damage than the dual phase lag or Pennes models. Time required for tumor necrosis is longer for higher phase lag values.
A Review on Nanofluids Thermal Properties Determination Using Intelligent Tec...IJSRD
Nanofluids are the dispersion of nano-sized particles into base fluids. Nanofluids have wide scope for applying as coolant in many of the engineering fields because of its higher thermal conductivity and more desirable thermal properties. Numerous mathematical models and experimental models have been proposed to predict the thermo physical properties for the past two decades. It has been noticed that many discrepancies between the mathematical and experimental results of thermo physical properties of nanofluids, in particular, thermal conductivity and viscosity. To mitigate those discrepancies, Intelligent Techniques with flexible mathematical structure that is capable of identifying complex non-linear relationships between input and output data were utilized to accurately predict the thermal properties of nanofluids. The data mining model based on genetic neural network has been widely applied to the procedure of data mining on thermal physical properties of nanofluids to acquire the pattern knowledge. This paper is to review the thermal conductivity of nanofluids research publications which are inter linked with soft computing tools. The outcome of this review shall lead to optimize the nanofluids properties while applying heat transfer nanofluids and to reduce the experimental test runs and number of hypothesis posed by different investigators.
An Inverse Heat Conduction Problem With Heat Flux MeasurementsJoaquin Hamad
This document describes a study that aims to develop a methodology for estimating time-dependent blood perfusion from heat flux measurements. A probe is designed with a thin heat flux sensor that records heat flux when placed in contact with tissue during a controlled thermal event, like air flow cooling. A mathematical model simulates the thermal event at the contact region. An inverse method is developed to simultaneously estimate time-dependent blood perfusion and thermal contact conductance by minimizing differences between measured and simulated heat fluxes, without prior knowledge of their functional forms. The method is tested on simulated data and shown to feasibly solve the inverse problem and estimate the two unknowns.
In this paper, the breast cyst diagnosis based on the surface temperature profile is proposed.
The temperature profiles of normal breasts and breasts with cyst have been investigated. Heat transfer inside
the breast is modeled by the Penne’s bio-heat equationand solved by the finite element method using COMSOL
Multiphysics software. A two dimensional (2-D) cross-section of the breast is modeled by a semicircle, whereas
the cyst is modeled by a small circle. The temperature distribution on the breast surface for the normal breast
and the breast with a cyst of various sizes were determined. The results show the difference in the surface
temperature profiles of the normal breast and of the breast with a cyst of different sizes.
Thermal Energy on Water and Oil placed Squeezed Carreau Nanofluids FlowMOHAMMED FAYYADH
this research work is focused on the numerical study regarding Carreau nanofluids’ squeezed flow via a permeable sensor surface. The nanofluids’ thermal conductivity is considered to be dependent on temperature. A convenient transformation is employed to reorganize governing equations into ordinary differential equations. The Runge–Kutta method and shooting technique are employed to accurately solve the boundary layer momentum as well as heat equations. Graphical and tabular aids are used to evaluate the solutions of applicable parameter with regards to temperature as well as the rate of heat transfer. In this work, a comparison is done from three nanofluids, i.e. copper, oxide aluminum and SWCNTs (nanoparticles) based fluids (water, crude oil and ethylene glycol) to improve heat transfer. It is found that the temperature dimensionless was dropped and dominated with the squeezed flow parameter and nanoparticle volume fraction parameter. That is for all nanomaterials. When compared with water and ethylene glycol, crude oil is cooler and a thinner thermal boundary layer is presented. For the rate of heat transfer (Nusselt number) was higher in: Ethylene glycol- SWCNT with high permeable velocity parameter 0.2, Ethylene glycol- SWCNT with low squeeze flow parameter 0.1 and Ethylene glycol- oxide aluminum with low nanoparticle volume fraction 0.05
This document summarizes a study that used artificial neural networks to model and identify dynamic indoor thermal comfort based on the PMV index. The study developed equations to model thermal comfort based on factors like air temperature, humidity, clothing insulation, and metabolism. An artificial neural network was then trained using these equations to approximate the nonlinear relationship between inputs like temperature and outputs like predicted mean vote. Simulation results showed the neural network model could accurately track desired thermal sensations and matched existing fuzzy logic models of human thermal comfort. The neural network approach provides a practical method for real-time identification of thermal comfort that is better than traditional manual calculations.
This document provides an overview of heat transfer and haptic sensing. It discusses thermal haptic sensing and how it involves two phases of sensation - when touching a real object and a virtual object. It also describes common thermal haptic systems, human thermal sensor characteristics, and several models for modeling haptic heat transfer including those based on electrical analogies and finite difference calculations. Finally, it notes some drawbacks of thermal haptics such as the long response time of thermal sensors and lack of understanding of human thermal perception.
Study on Thermal and Hydrodynamic Indexes of a Nanofluid Flow in a Micro Heat...A Behzadmehr
The paper numerically presents laminar forced convection of a nanofluid flowing in a duct at microscale.
Results were compared with both analytical and experimental data and observed good concordance with
previous studies available in the literature. Influences of Brinkman and Reynolds number on thermal and
hydrodynamic indexes have been investigated. For a given nanofluid, no change in efficiency (heat dissipation
to pumping power) was observed with an increasing in Reynolds number. It was shown that the pressure was
decrease with an increase in Brinkman number. Dependency of Nu increment changes with substrate material.
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology.
A Review on Nanofluids Thermal Properties Determination Using Intelligent Tec...IJSRD
Nanofluids are the dispersion of nano-sized particles into base fluids. Nanofluids have wide scope for applying as coolant in many of the engineering fields because of its higher thermal conductivity and more desirable thermal properties. Numerous mathematical models and experimental models have been proposed to predict the thermo physical properties for the past two decades. It has been noticed that many discrepancies between the mathematical and experimental results of thermo physical properties of nanofluids, in particular, thermal conductivity and viscosity. To mitigate those discrepancies, Intelligent Techniques with flexible mathematical structure that is capable of identifying complex non-linear relationships between input and output data were utilized to accurately predict the thermal properties of nanofluids. The data mining model based on genetic neural network has been widely applied to the procedure of data mining on thermal physical properties of nanofluids to acquire the pattern knowledge. This paper is to review the thermal conductivity of nanofluids research publications which are inter linked with soft computing tools. The outcome of this review shall lead to optimize the nanofluids properties while applying heat transfer nanofluids and to reduce the experimental test runs and number of hypothesis posed by different investigators.
An Inverse Heat Conduction Problem With Heat Flux MeasurementsJoaquin Hamad
This document describes a study that aims to develop a methodology for estimating time-dependent blood perfusion from heat flux measurements. A probe is designed with a thin heat flux sensor that records heat flux when placed in contact with tissue during a controlled thermal event, like air flow cooling. A mathematical model simulates the thermal event at the contact region. An inverse method is developed to simultaneously estimate time-dependent blood perfusion and thermal contact conductance by minimizing differences between measured and simulated heat fluxes, without prior knowledge of their functional forms. The method is tested on simulated data and shown to feasibly solve the inverse problem and estimate the two unknowns.
In this paper, the breast cyst diagnosis based on the surface temperature profile is proposed.
The temperature profiles of normal breasts and breasts with cyst have been investigated. Heat transfer inside
the breast is modeled by the Penne’s bio-heat equationand solved by the finite element method using COMSOL
Multiphysics software. A two dimensional (2-D) cross-section of the breast is modeled by a semicircle, whereas
the cyst is modeled by a small circle. The temperature distribution on the breast surface for the normal breast
and the breast with a cyst of various sizes were determined. The results show the difference in the surface
temperature profiles of the normal breast and of the breast with a cyst of different sizes.
Thermal Energy on Water and Oil placed Squeezed Carreau Nanofluids FlowMOHAMMED FAYYADH
this research work is focused on the numerical study regarding Carreau nanofluids’ squeezed flow via a permeable sensor surface. The nanofluids’ thermal conductivity is considered to be dependent on temperature. A convenient transformation is employed to reorganize governing equations into ordinary differential equations. The Runge–Kutta method and shooting technique are employed to accurately solve the boundary layer momentum as well as heat equations. Graphical and tabular aids are used to evaluate the solutions of applicable parameter with regards to temperature as well as the rate of heat transfer. In this work, a comparison is done from three nanofluids, i.e. copper, oxide aluminum and SWCNTs (nanoparticles) based fluids (water, crude oil and ethylene glycol) to improve heat transfer. It is found that the temperature dimensionless was dropped and dominated with the squeezed flow parameter and nanoparticle volume fraction parameter. That is for all nanomaterials. When compared with water and ethylene glycol, crude oil is cooler and a thinner thermal boundary layer is presented. For the rate of heat transfer (Nusselt number) was higher in: Ethylene glycol- SWCNT with high permeable velocity parameter 0.2, Ethylene glycol- SWCNT with low squeeze flow parameter 0.1 and Ethylene glycol- oxide aluminum with low nanoparticle volume fraction 0.05
This document summarizes a study that used artificial neural networks to model and identify dynamic indoor thermal comfort based on the PMV index. The study developed equations to model thermal comfort based on factors like air temperature, humidity, clothing insulation, and metabolism. An artificial neural network was then trained using these equations to approximate the nonlinear relationship between inputs like temperature and outputs like predicted mean vote. Simulation results showed the neural network model could accurately track desired thermal sensations and matched existing fuzzy logic models of human thermal comfort. The neural network approach provides a practical method for real-time identification of thermal comfort that is better than traditional manual calculations.
This document provides an overview of heat transfer and haptic sensing. It discusses thermal haptic sensing and how it involves two phases of sensation - when touching a real object and a virtual object. It also describes common thermal haptic systems, human thermal sensor characteristics, and several models for modeling haptic heat transfer including those based on electrical analogies and finite difference calculations. Finally, it notes some drawbacks of thermal haptics such as the long response time of thermal sensors and lack of understanding of human thermal perception.
Study on Thermal and Hydrodynamic Indexes of a Nanofluid Flow in a Micro Heat...A Behzadmehr
The paper numerically presents laminar forced convection of a nanofluid flowing in a duct at microscale.
Results were compared with both analytical and experimental data and observed good concordance with
previous studies available in the literature. Influences of Brinkman and Reynolds number on thermal and
hydrodynamic indexes have been investigated. For a given nanofluid, no change in efficiency (heat dissipation
to pumping power) was observed with an increasing in Reynolds number. It was shown that the pressure was
decrease with an increase in Brinkman number. Dependency of Nu increment changes with substrate material.
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology.
Study of Magnetohydrodynamic Fluid Flows and Their Applicationsijtsrd
In this paper a detailed study is presented on magnetohydrodynamic fluid flows with their basic governing equations. Basic concept of magnetohydrodynamic is discussed in detail. The results of various problems done by researchers are presented and addressed properly. The various applications of magnetohydrodynamic fluid flows have been presented. Dr. Vishal Saxena ""Study of Magnetohydrodynamic Fluid Flows and Their Applications"" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-4 | Issue-2 , February 2020, URL: https://www.ijtsrd.com/papers/ijtsrd29931.pdf
Paper Url : https://www.ijtsrd.com/mathemetics/applied-mathematics/29931/study-of-magnetohydrodynamic-fluid-flows-and-their-applications/dr-vishal-saxena
Investigation of the Effect of Nanoparticles Mean Diameter on Turbulent Mixed...A Behzadmehr
Abstract
Turbulent mixed convection of a nanofluid (water/Al2O3, Φ=.02) has been studied numerically. Two-phase
mixture model has been used to investigate the effects of nanoparticles mean diameter on the flow parameters. Nanoparticles distribution at the tube cross section shows that the particles are uniformly dispersed. The non-uniformity of the particles distribution occurs in the case of large nanoparticles and/or high value of the Grashof numbers. The study of particle size effect showed that the effective Nusselt number and turbulent intensity increases with the decreased of particle size.
EXPERIMENTAL INVESTIGATION ON BOILING HEAT TRANSFER USING R134AJournal For Research
The document experimentally investigates the heat transfer characteristics of R134a during boiling in horizontal mini channels. Several correlations for predicting boiling heat transfer are evaluated by comparing their predicted heat transfer coefficients to experimental data for R134a. The Liu and Wu correlation and Tran correlation provided the best predictions, with mean absolute deviations below 15% and 40% respectively, depending on the channel diameter. It is concluded that boiling heat transfer is strongly influenced by heat flux and Lazarek and Black correlation is most suitable for mini channels, while Tran correlation also performs well for some channel diameters.
1. article in mathematical problems in engineering 2020MohamedSANNAD2
This document summarizes a numerical study that simulated natural convection heat transfer in a cubical cavity filled with nanofluids. The cavity is heated by a partition maintained at a hot temperature, while the right and left walls are kept at a cold temperature and the rest are adiabatic. Results show that increasing the Rayleigh number, volume fraction of nanoparticles, and size of the heating partition all improve heat transfer. Copper-based nanofluids provided the best thermal transfer. The study analyzes temperature, velocity and heat transfer to understand how nanofluids affect natural convection in 3D enclosures.
This document summarizes a book on the epidemiology and diffusion of viruses with a focus on the role of latitude, air pollutants, and humidity. It discusses several viruses including SARS, MERS, influenza, and COVID-19. It reviews literature finding associations between increased air pollutants like PM2.5 and higher risk of influenza-like illness. Some studies found temperature could impact COVID-19 transmission, with an optimal temperature range. Experiments with influenza in guinea pigs found that cold, dry conditions favor airborne transmission. The role of atmospheric conditions in the seasonality and spread of influenza over large geographic areas is discussed.
MHD Chemically Reacting and Radiating Nanofluid Flow over a Vertical Cone Emb...IJLT EMAS
In this study, we examine the combined effects of
thermal radiation, chemical reaction on MHD hydromagnetic
boundary layer flow over a vertical cone filled with nanofluid
saturated porous medium under variable properties. The
governing flow, heat and mass transfer equations are
transformed into ordinary differential equations using similarity
variables and are solved numerically by a Galerkin Finite
element method. Numerical results are obtained for
dimensionless velocity, temperature, nanoparticle volume
fraction, as well as the skin friction, local Nusselt and Sherwood
number for the different values of the pertinent parameters
entered into the problem. The effects of various controlling
parameters on these quantities are investigated. Pertinent
results are presented graphically and discussed quantitatively.
The present results are compared with existing results and found
to be good agreement. It is found that the temperature of the
fluid remarkably enhances with the rising values of Brownian
motion parameter (Nb).
- The document presents a study of relaxation processes and ultrasonic attenuation in KDP-type ferroelectrics.
- It uses a four-particle cluster model Hamiltonian considering proton-lattice interactions and anharmonicity up to fourth order. The proton and phonon Green's functions are evaluated using this Hamiltonian.
- Collective mode frequencies and widths are calculated, relating them to the relaxational behavior and ultrasonic attenuation. Temperature dependence of these properties is discussed in terms of a relaxational soft mode.
- Relaxation times calculated from attenuation data, dielectric data, and spectral line widths are compared, showing similar temperature dependence in the paraelectric phase. The results suggest relaxational behavior of dielectric and attenuation properties in
Effects of Thermal Radiation and Chemical Reaction on MHD Free Convection Flo...IJERA Editor
This paper analyzes the radiation and chemical reaction effects on MHD steady two-dimensional laminar
viscous incompressible radiating boundary layer flow over a flat plate in the presence of internal heat generation
and convective boundary condition. It is assumed that lower surface of the plate is in contact with a hot fluid
while a stream of cold fluid flows steadily over the upper surface with a heat source that decays exponentially.
The Rosseland approximation is used to describe radiative heat transfer as we consider optically thick fluids.
The governing boundary layer equations are transformed into a system of ordinary differential equations using
similarity transformations, which are then solved numerically by employing fourth order Runge-Kutta method
along with shooting technique. The effects of various material parameters on the velocity, temperature and
concentration as well as the skin friction coefficient, the Nusselt number, the Sherwood number and the plate
surface temperature are illustrated and interpreted in physical terms. A comparison of present results with
previously published results shows an excellent agreement.
Radiation and Mass Transfer Effects on MHD Natural Convection Flow over an In...IJMER
A numerical solution for the unsteady, natural convective flow of heat and mass transfer along an inclined plate is presented. The dimensionless unsteady, coupled, and non-linear partial differential conservation equations for the boundary layer regime are solved by an efficient, accurate and unconditionally stable finite difference scheme of the Crank-Nicolson type. The velocity, temperature, and concentration fields have been studied for the effect of Magnetic parameter, buoyancy ratio parameter, Prandtl number, radiation parameter and Schmidt number. The local skin-friction, Nusselt number and Sherwood number are also presented and analyzed graphically.
Effect of Radiation on Mixed Convection Flow of a Non-Newtonian Nan fluid ove...IJMER
International Journal of Modern Engineering Research (IJMER) is Peer reviewed, online Journal. It serves as an international archival forum of scholarly research related to engineering and science education.
This document reviews research on the heat transfer of nanofluids when an electric or magnetic field is applied. It discusses how applied fields can affect the heat transfer performance and mechanisms of nanofluids. While studies show fields can significantly impact nanofluid heat transfer, there are differing opinions on their exact effects and mechanisms. The document aims to analyze the mechanism of thermal conductivity enhancement in nanofluids and how applied fields induce chaotic convection and heat transfer enhancement.
Study of Forced Convection Heat Transfer with Single phase and mixture phase ...IOSRJMCE
In this study, forced convection heat transfer of nanoliquids is done using both single-phase and mixture-phase models and the results are compared with experimental results. The governing equations of the study here are discretized using the finite volume method. Hybrid differencing scheme is used to calculate the face values of the control volumes. A code is written using SIMPLER algorithm and then solved using the MATLAB engine. The mixture-phase model studied here, considers two slip mechanisms between nanoparticle and base-fluid, namely Brownian diffusion and thermophoresis. Al2O3-water nanofluid is used for the study of nanofluid and the study shows significant increase in convective heat transfer coefficient while the mixturephase model demonstrates slightly lower values than the single-phase model. The study is done with various nanoparticle concentrations and Reynolds numbers. With increasing particle concentration and Reynolds number, the convective heat transfer coefficient increases and as well as the shear stress. For low concentrations of the nanoparticle, Nusselt number is slightly lower than the base fluid and as the concentration increases, the Nusselt number also rises higher than the base fluid
Adaptive gender-based thermal control system IJECEIAES
A closed loop adaptive gender-based thermal control system (AG-TCS) is designed, modelled, analysed and tested. The system has the unique feature of adapting to the surrounding environment as a function of the number of humans present and the gender ratio. The operation of the system depends on a unique interface between a radio frequency identification (RFID) device and an imaging device, both of which are correlated and interfaced to a controller. Testing of the system resulted in smooth transition and shape conversion of the response curve, which proved its adaptability. Three mathematical equations describing the internal mechanisms of the AG-TCS are presented and have been proven to optimally reflect the original statistical data covering both genders.
Numerical Study of Heat Transfer in Ternary Nanofluid Over a Stretching Sheet...Atif75347
The new method of enhancing heat transfer through tri- hybrid nanofluid is discussed in the current study and represented in differential equation system.
Provides up to date information on DSC, recent developments and applicability. Recommended for those seeking up-to-date information on thermal analysis instruments.
Variation of dose distribution with depth and incident energy using EGSnrc Mo...iosrjce
IOSR Journal of Applied Physics (IOSR-JAP) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of physics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in applied physics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
The document discusses statistical analysis of wavelet coefficients of thermographs for characterizing breast cancer. It presents a wavelet-based technique to detect breast cancer in thermographs. Haar, biorthogonal, and reverse biorthogonal wavelets are analyzed and it is found that Haar wavelets provide better results in representing the temperature variations in cancer-affected regions. The methodology involves applying discrete wavelet transforms to segmented thermographs and calculating statistical measures like mean and standard deviation of the approximation and detail coefficients. The absolute difference between corresponding left and right segments is used to detect the presence of cancer.
The document discusses statistical analysis of wavelet coefficients of thermographs for characterizing breast cancer. It presents a wavelet-based technique to detect breast cancer in thermographs. Haar, biorthogonal, and reverse biorthogonal wavelets are analyzed and it is found that Haar wavelets provide better results in representing the temperature variations in cancer-affected regions. The methodology involves applying discrete wavelet transforms to segmented thermographs and calculating statistical measures like mean and standard deviation of the approximation and detail coefficients. The absolute difference between corresponding left and right segments is used to detect abnormal regions indicative of cancer.
THERMAL CONDUCTIVITY OF NANOFLUIDS PREPARED FROM BIOBASED NANOMATERIALS DISPE...IAEME Publication
In the present study, experimental investigation on thermal conductivity of green
nanofluids prepared from coconut fibre-based nanoparticles and suspended in 60:40
ethylene glycol (EG) water (W) mixture was carried out. The measurement of thermal
conductivity was conducted at 15 °C to 60 °C at mass fractions of 0.04 wt%, 0.08
wt%, 0.5 wt% and 1 wt%. The results show deterioration in thermal conductivity with
an increasing temperature. Also the deterioration increased as the mass fraction
increased.
This document discusses determining the thermal conductivity of liquids using the transient hot disk method. It analyzes the effects of natural convection numerically and experimentally to determine when natural convection begins during testing for different fluids. A correlation is developed to predict the onset of natural convection based on the fluid's properties, Rayleigh number, and sensor power level. This correlation will help establish an accurate method for measuring liquid thermal conductivity using transient techniques by determining the maximum test time before natural convection occurs.
Design and optimization of ion propulsion dronebjmsejournal
Electric propulsion technology is widely used in many kinds of vehicles in recent years, and aircrafts are no exception. Technically, UAVs are electrically propelled but tend to produce a significant amount of noise and vibrations. Ion propulsion technology for drones is a potential solution to this problem. Ion propulsion technology is proven to be feasible in the earth’s atmosphere. The study presented in this article shows the design of EHD thrusters and power supply for ion propulsion drones along with performance optimization of high-voltage power supply for endurance in earth’s atmosphere.
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Study of Magnetohydrodynamic Fluid Flows and Their Applicationsijtsrd
In this paper a detailed study is presented on magnetohydrodynamic fluid flows with their basic governing equations. Basic concept of magnetohydrodynamic is discussed in detail. The results of various problems done by researchers are presented and addressed properly. The various applications of magnetohydrodynamic fluid flows have been presented. Dr. Vishal Saxena ""Study of Magnetohydrodynamic Fluid Flows and Their Applications"" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-4 | Issue-2 , February 2020, URL: https://www.ijtsrd.com/papers/ijtsrd29931.pdf
Paper Url : https://www.ijtsrd.com/mathemetics/applied-mathematics/29931/study-of-magnetohydrodynamic-fluid-flows-and-their-applications/dr-vishal-saxena
Investigation of the Effect of Nanoparticles Mean Diameter on Turbulent Mixed...A Behzadmehr
Abstract
Turbulent mixed convection of a nanofluid (water/Al2O3, Φ=.02) has been studied numerically. Two-phase
mixture model has been used to investigate the effects of nanoparticles mean diameter on the flow parameters. Nanoparticles distribution at the tube cross section shows that the particles are uniformly dispersed. The non-uniformity of the particles distribution occurs in the case of large nanoparticles and/or high value of the Grashof numbers. The study of particle size effect showed that the effective Nusselt number and turbulent intensity increases with the decreased of particle size.
EXPERIMENTAL INVESTIGATION ON BOILING HEAT TRANSFER USING R134AJournal For Research
The document experimentally investigates the heat transfer characteristics of R134a during boiling in horizontal mini channels. Several correlations for predicting boiling heat transfer are evaluated by comparing their predicted heat transfer coefficients to experimental data for R134a. The Liu and Wu correlation and Tran correlation provided the best predictions, with mean absolute deviations below 15% and 40% respectively, depending on the channel diameter. It is concluded that boiling heat transfer is strongly influenced by heat flux and Lazarek and Black correlation is most suitable for mini channels, while Tran correlation also performs well for some channel diameters.
1. article in mathematical problems in engineering 2020MohamedSANNAD2
This document summarizes a numerical study that simulated natural convection heat transfer in a cubical cavity filled with nanofluids. The cavity is heated by a partition maintained at a hot temperature, while the right and left walls are kept at a cold temperature and the rest are adiabatic. Results show that increasing the Rayleigh number, volume fraction of nanoparticles, and size of the heating partition all improve heat transfer. Copper-based nanofluids provided the best thermal transfer. The study analyzes temperature, velocity and heat transfer to understand how nanofluids affect natural convection in 3D enclosures.
This document summarizes a book on the epidemiology and diffusion of viruses with a focus on the role of latitude, air pollutants, and humidity. It discusses several viruses including SARS, MERS, influenza, and COVID-19. It reviews literature finding associations between increased air pollutants like PM2.5 and higher risk of influenza-like illness. Some studies found temperature could impact COVID-19 transmission, with an optimal temperature range. Experiments with influenza in guinea pigs found that cold, dry conditions favor airborne transmission. The role of atmospheric conditions in the seasonality and spread of influenza over large geographic areas is discussed.
MHD Chemically Reacting and Radiating Nanofluid Flow over a Vertical Cone Emb...IJLT EMAS
In this study, we examine the combined effects of
thermal radiation, chemical reaction on MHD hydromagnetic
boundary layer flow over a vertical cone filled with nanofluid
saturated porous medium under variable properties. The
governing flow, heat and mass transfer equations are
transformed into ordinary differential equations using similarity
variables and are solved numerically by a Galerkin Finite
element method. Numerical results are obtained for
dimensionless velocity, temperature, nanoparticle volume
fraction, as well as the skin friction, local Nusselt and Sherwood
number for the different values of the pertinent parameters
entered into the problem. The effects of various controlling
parameters on these quantities are investigated. Pertinent
results are presented graphically and discussed quantitatively.
The present results are compared with existing results and found
to be good agreement. It is found that the temperature of the
fluid remarkably enhances with the rising values of Brownian
motion parameter (Nb).
- The document presents a study of relaxation processes and ultrasonic attenuation in KDP-type ferroelectrics.
- It uses a four-particle cluster model Hamiltonian considering proton-lattice interactions and anharmonicity up to fourth order. The proton and phonon Green's functions are evaluated using this Hamiltonian.
- Collective mode frequencies and widths are calculated, relating them to the relaxational behavior and ultrasonic attenuation. Temperature dependence of these properties is discussed in terms of a relaxational soft mode.
- Relaxation times calculated from attenuation data, dielectric data, and spectral line widths are compared, showing similar temperature dependence in the paraelectric phase. The results suggest relaxational behavior of dielectric and attenuation properties in
Effects of Thermal Radiation and Chemical Reaction on MHD Free Convection Flo...IJERA Editor
This paper analyzes the radiation and chemical reaction effects on MHD steady two-dimensional laminar
viscous incompressible radiating boundary layer flow over a flat plate in the presence of internal heat generation
and convective boundary condition. It is assumed that lower surface of the plate is in contact with a hot fluid
while a stream of cold fluid flows steadily over the upper surface with a heat source that decays exponentially.
The Rosseland approximation is used to describe radiative heat transfer as we consider optically thick fluids.
The governing boundary layer equations are transformed into a system of ordinary differential equations using
similarity transformations, which are then solved numerically by employing fourth order Runge-Kutta method
along with shooting technique. The effects of various material parameters on the velocity, temperature and
concentration as well as the skin friction coefficient, the Nusselt number, the Sherwood number and the plate
surface temperature are illustrated and interpreted in physical terms. A comparison of present results with
previously published results shows an excellent agreement.
Radiation and Mass Transfer Effects on MHD Natural Convection Flow over an In...IJMER
A numerical solution for the unsteady, natural convective flow of heat and mass transfer along an inclined plate is presented. The dimensionless unsteady, coupled, and non-linear partial differential conservation equations for the boundary layer regime are solved by an efficient, accurate and unconditionally stable finite difference scheme of the Crank-Nicolson type. The velocity, temperature, and concentration fields have been studied for the effect of Magnetic parameter, buoyancy ratio parameter, Prandtl number, radiation parameter and Schmidt number. The local skin-friction, Nusselt number and Sherwood number are also presented and analyzed graphically.
Effect of Radiation on Mixed Convection Flow of a Non-Newtonian Nan fluid ove...IJMER
International Journal of Modern Engineering Research (IJMER) is Peer reviewed, online Journal. It serves as an international archival forum of scholarly research related to engineering and science education.
This document reviews research on the heat transfer of nanofluids when an electric or magnetic field is applied. It discusses how applied fields can affect the heat transfer performance and mechanisms of nanofluids. While studies show fields can significantly impact nanofluid heat transfer, there are differing opinions on their exact effects and mechanisms. The document aims to analyze the mechanism of thermal conductivity enhancement in nanofluids and how applied fields induce chaotic convection and heat transfer enhancement.
Study of Forced Convection Heat Transfer with Single phase and mixture phase ...IOSRJMCE
In this study, forced convection heat transfer of nanoliquids is done using both single-phase and mixture-phase models and the results are compared with experimental results. The governing equations of the study here are discretized using the finite volume method. Hybrid differencing scheme is used to calculate the face values of the control volumes. A code is written using SIMPLER algorithm and then solved using the MATLAB engine. The mixture-phase model studied here, considers two slip mechanisms between nanoparticle and base-fluid, namely Brownian diffusion and thermophoresis. Al2O3-water nanofluid is used for the study of nanofluid and the study shows significant increase in convective heat transfer coefficient while the mixturephase model demonstrates slightly lower values than the single-phase model. The study is done with various nanoparticle concentrations and Reynolds numbers. With increasing particle concentration and Reynolds number, the convective heat transfer coefficient increases and as well as the shear stress. For low concentrations of the nanoparticle, Nusselt number is slightly lower than the base fluid and as the concentration increases, the Nusselt number also rises higher than the base fluid
Adaptive gender-based thermal control system IJECEIAES
A closed loop adaptive gender-based thermal control system (AG-TCS) is designed, modelled, analysed and tested. The system has the unique feature of adapting to the surrounding environment as a function of the number of humans present and the gender ratio. The operation of the system depends on a unique interface between a radio frequency identification (RFID) device and an imaging device, both of which are correlated and interfaced to a controller. Testing of the system resulted in smooth transition and shape conversion of the response curve, which proved its adaptability. Three mathematical equations describing the internal mechanisms of the AG-TCS are presented and have been proven to optimally reflect the original statistical data covering both genders.
Numerical Study of Heat Transfer in Ternary Nanofluid Over a Stretching Sheet...Atif75347
The new method of enhancing heat transfer through tri- hybrid nanofluid is discussed in the current study and represented in differential equation system.
Provides up to date information on DSC, recent developments and applicability. Recommended for those seeking up-to-date information on thermal analysis instruments.
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2. Journal of the Brazilian Society of Mechanical Sciences and Engineering (2020) 42:62
1 3
62 Page 2 of 13
Pennes [5] has proposed one of the earliest mathematical
models for governing qualitative relationship between tissue
and arterial temperature. According to this, arterial blood is
assumed to be constant at the core temperature and venous
temperature to be equal to that of the tissue temperature.
The model assumes Fourier heat transfer to be applicable
to tissue and that the effect of blood perfusion is uniformly
distributed in space thus not accounting for the effect of
directional blood flow. This model was quite successful
in interpreting temperatures in various cases. However, it
is not very convincing as the model permits infinite speed
of thermal disturbance propagation. Further, no means is
provided in accounting for non-homogeneity present in
biological systems. Catteneo [6] has introduced a new heat
transfer model which eliminates this paradox of instantane-
ous propagation by introducing wave-like relation between
heat flux and temperature gradient. This is a hyperbolic heat
transfer equation. Mitra et al. [7] have shown experimental
evidence of hyperbolic heat transfer in processed bologna
meat. They have used meats at different temperatures and
brought them into contact suddenly and used thermocouples
to measure the instantaneous temperature distributions. They
have estimated a thermal phase lag time value to be around
16 s. Kaminski [8] has also conducted some experiments
with material with non-homogeneous inner structure and has
found a phase lag time value of 20 s. Herwig and Beckert
[9] and Graumann and Peters [10] have conducted similar
experiments as above but found that Fourier heat transfer
model was sufficient to represent their observations, thus
asserting that Mitra et al. [7] and Kaminski [8] works to be
misinterpreted. However, nothing can be ruled out as there
is no means to know and compare the actual constitution of
the materials used by different researchers.
Further in Thermal wave model, there are abrupt jumps
in temperatures which are again unphysical. This was over-
come by dual phase lag (DPL) model consisting of two
phase lag time values τq and τT. Minkowycz et al. [11]
have considered local thermal non-equilibrium. In their
theoretical study, they have shown that local thermal equi-
librium conditions depend on mean pore size, interstitial
heat transfer coefficient and other thermo-physical proper-
ties. Hooshmand et al. [12] have considered similar non-
equilibrium model and have solved them using separation
of variables and Duhamel’s integral methods. Zhou et al.
[13] have used finite volume method to solve DPL model
for an axisymmetric domain. All three of them have shown
that DPL model reduces to Pennes’ model only when both
τq and τT are zero, unlike Liu and Wang [14] who have sug-
gested that DPL reduces to Pennes’ model even when they
are equal, not necessarily to zero. Antaki [15] interpreted
τq to be a measure of delay in conduction and τT to be a
measure of conduction along microscopic paths. Museux
et al. [16] have experimentally observed skin burns when
a porcine tissue was exposed to two different lasers of
different wavelengths. They have compared their results
with those predicted by Pennes’ model using finite element
method. Degree of burns was estimated using different
thermal damage models to find which matches with the
experimental results the best.
Jiang et al. [17] have used finite difference method to
solve Pennes’ bioheat transfer equation and Arrhenius
equations to evaluate thermal damage in a one-dimensional
multi-layer model and have studied the effects of thermo-
physical properties and physical dimensions of the domain
on the temperature and thermal damage function distribu-
tions. Bedin and Bazan [18] have considered a two-dimen-
sional bioheat model and obtained an explicit Fourier-
based solution. Further, they have used a pseudospectral
collocation method to construct highly accurate numerical
solutions for their problem. The model is subjected to con-
vective boundary conditions and space-dependent perfu-
sion coefficient. Tung et al. [19] have compared Pennes’
model and thermal wave model for their physical differ-
ences. Analytical results have been illustrated for radiofre-
quency heating and laser heating used for refractive error
correction in the cornea. Ahmadikia et al. [20] have used
Laplace transform method to obtain analytical solutions to
Pennes’ and Thermal wave models applied to a one-dimen-
sional skin domain. Transient temperature responses have
been studied with the skin surface subjected to a constant,
cosine and pulse train heat fluxes. Lakhssassi et al. [21]
have developed analytical solutions to Modified Pennes’
model which accounts for blood perfusion variation with
temperature on a one-dimensional domain. The obtained
analytical solutions were used to conduct various paramet-
ric studies like the effect on temperature distributions due
to variation in thermal diffusivity, temperature depend-
ent and independent blood perfusion. Deng and Liu [22]
have solved transient three-dimensional bioheat transfer
equations subject to convective, radiative and evaporative
boundary conditions with the Monte Carlo method. Using
statistical principles, they have proposed various thermal
criteria for disease diagnostic.
Motivated by the previous studies and noting down the
limitations like physically relevant values of phase lag
were not considered, we aim to analyze temperature distri-
butions using a developed computational model by incor-
porating them. This enables us to understand as to which
model is better at predicting temperatures in real-life situ-
ations and at a faster time and lesser computational cost.
Finite difference method is used for the analysis. Effect of
applying these models to a two-dimensional [2D] domain
is considered. Further, the effect of phase lag time is also
studied. Time required for temperature to reach 42.5 °C in
entire tumor region is estimated.
3. Journal of the Brazilian Society of Mechanical Sciences and Engineering (2020) 42:62
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Page 3 of 13 62
2 Mathematical models
2.1 Heat transfer
In the present work, we will analyze three bioheat transfer mod-
els—Pennes’ model of bioheat transfer (PMBT), thermal wave
model of bioheat transfer (TWMBT) and dual phase lag (DPL)
model. Constitutive equation for heat transfer used in PMBT is
For TWMBT it is
And for DPL model it is
Here q is heat flux (W/m2
), k is the thermal conductiv-
ity (W/m2
) of the material under discussion, and T (°C) is
temperature of the material. τq (s) and τT (s) are the thermal
lag times for heat flux and temperature gradient, respectively.
Considering metabolic heat generation and blood perfusion,
bioheat transfer equation becomes [5]
where ρ is density (kg/m3
), c is mass specific heat (J/kgK) of
tissue, Qm is volumetric heat generation, wb is blood perfu-
sion (m3
/(m3
s)), ρb is density(kg/m3
), and Tb is temperature
(°C) of blood. When we use the aforementioned different
constitutive equations, we get different Bioheat transfer
models. It is assumed here that Tb represents arterial tem-
perature of blood and to be at some constant, generally equal
to body core temperature of around 37 °C and that the blood
leaves the tissue at a venous temperature equal to that of the
tissue. Using Eqs. (3) and (4), we get DPL model as
If τT is zero in the above equation, we get TWMBT and
when τq is also zero, we get PMBT. The above equation
considers all the physical properties to be constant.
(1)
q = −k∇T
(2)
q
(
t + 𝜏q
)
= −k∇T
(3)
q
(
t + 𝜏q
)
= −k∇T
(
t + 𝜏T
)
(4)
𝜌c(𝜕T∕𝜕t) = −∇q + Qm + wb𝜌bcb
(
Tb − T
)
(5)
𝜌c𝜏q
(
𝜕2
T∕𝜕t2
)
+ 𝜌c
(
1 +
(
wb𝜌bcb𝜏q
)
∕𝜌c
)
(𝜕T∕𝜕t)
= k∇2
T + k𝜏T
(
𝜕∇2
T∕𝜕t
)
+ Qm + wb𝜌bcb
(
Tb − T
)
2.2 Thermal damage model
Widely studied Arrhenius protein denaturation burn integral
equation, proposed by Henriques and Mortitz [23], will be
used here to estimate the thermal damage in the skin when
exposed to radiation. The equation is as follows
where Ω is the dimensionless burn parameter. Here, A is the
frequency factor of Arrhenius equation for rate of chemical
reaction. The value of the material parameter equivalent to
it is taken to be 3.1 × 1098
s−1 and the ratio of activation
energy of necrosis, Ea, to the universal gas constant, R, is
taken to be 75,000. In basal layer, thermal damage can be
estimated by performing the integration in Eq. (6). A first-
degree burn is considered to occur when 0.53Ω1 and
T44 °C. Similarly, a second-degree burn is considered to
occur when Ω = 1 and T 44 °C and a third-degree burn
when Ω = 10,000 and T 44 °C. French Society for Burn
Study and Treatment (SFETB) has classified burns and the
corresponding physical effects as in Table 1 [16].
3
Physical problem description
We have adopted the skin as the physical domain from
Verma et al. [24] for the present study as it is more realistic
in approach. It is a two-dimensional domain of size 2L ×L
with three layers. Figure 1 represents the line diagram show-
ing the skin layers with tumor.
Tumor is placed in the top of Dermis, representing the
common place where cells are generally located and hence
a probable site for melanoma occurrence. We will be ana-
lyzing only right half of the above full domain. This is to
reduce computational effort. Initial condition and boundary
conditions are shown in Table 2. Steady-state temperature,
Tsteady, can be solved from Eq. (5) with boundary conditions
(iii), (iv) and (v). Later, when t0, boundary condition (vi)
is applied at the tumor location until the entire tumor region
attains the temperature which ensures killing of all tumor
cells.
(6)
𝛺 =
t
∫
0
Ae−
Ea
RT dt
Table 1 Degree of Burns and their Physical effects [16]
Thermal damage factor Degree of burn Physical effects
0.53 𝛺 1 1st degree Superficial epidermal involvement
1 𝛺 10, 000 2nd degree Whole thickness epidermal involvement; Basal membrane disruption; Papillary dermis involvement
10,000Ω 3rd degree Full-thickness epidermal necrosis, including hair follicles; complete basal membrane necrosis; deep
dermis/hypodermis involvement
4. Journal of the Brazilian Society of Mechanical Sciences and Engineering (2020) 42:62
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62 Page 4 of 13
4 Numerical method
Finite difference method (FDM) is used, with central differ-
ence scheme and implicit scheme for spatial and temporal
discretization of second-order derivatives. These expressions
are second-order accurate. Discretizing the second derivative
with time as that present in the first term of Eq. (5) is done
as in Eq. (7).
The second term in the right side of Eq. (5) can be dis-
cretized as follows
Spatial discretization is according to the standard central
difference schemes and hence not shown explicitly. After
(7)
𝜌c𝜏q𝜕2
T
𝜕t2
≈ 𝜌c𝜏q
(
Tt+Δt
P
− 2Tt
P
+ Tt−Δt
P
)
Δt2
(8)
k𝜏T𝜕∇2
T
𝜕t
≈ k𝜏T
(
∇2
Tt+Δt
− ∇2
Tt
)
Δt
spatial and temporal discretization is done, all the terms are
arranged into the standard form as below
Uniform grid is adopted, and hence the coefficients
become as follows
where
Successive over relaxation (SOR) method is used to solve
the discretized Eq. (9). Once the temperatures are obtained in
each time step, boundary temperatures are calculated using the
boundary conditions (iii), (iv) and (v). Then these temperature
(9)
aPTP = aETE + aWTW + aNTN + aSTS + b
(10)
aE = aW = aN = aS =
(
k
𝜌cΔx2
)
×
(
1 +
𝜏T
Δt
)
=
(
k
𝜌cΔy2
)
×
(
1 +
𝜏T
Δt
)
(11)
b = at
P
Tt
P
− at
E
Tt
E
− at
W
Tt
W
− at
N
Tt
N
− at
S
Tt
S
− at−Δt
P
Tt−Δt
P
+ SC
(12)
at
E
= at
N
= at
N
= at
S
=
k𝜏T
𝜌cΔtΔx2
(13)
at
P
=
2𝜏q
Δt2
+
(
1 +
wb𝜌bcb𝜏q
𝜌c
)
×
1
Δt
+ at
E
+ at
W
+ at
N
+ at
S
(14)
at−Δt
P =
𝜏q
Δt2
(15)
SC = −SPTb +
Qm
𝜌c
(16)
SP = −
wb𝜌bcb
𝜌c
(17)
aP = aE + aW + aN + aS +
𝜏q
Δt2
+
(
1 +
wb𝜌bcb𝜏q
𝜌c
)
×
1
Δt
− SP
Fig. 1 Schematic of the domain used in the study
Table 2 Initial Conditions and Boundary Conditions
Sl. no. Boundary condition Location and time of application
1. T = Tsteady At t=0
2. 𝜕T∕𝜕t = 0 At t=0
3. 𝜕T∕𝜕x = 0 At left and right boundaries of the domain and t≥0
4. T=37 °C At the bottom boundary and t≥0 as many of the large arteries are at this location.
5. 𝜕T∕𝜕y = h ×
(
T∞ − T
)
At entire top boundary when t=0. And at top boundary except the part straight
above the tumor when t0
6. 𝜕T∕𝜕y = q At the part of top boundary straight above the tumor when t0
5. Journal of the Brazilian Society of Mechanical Sciences and Engineering (2020) 42:62
1 3
Page 5 of 13 62
values are used to solve temperatures in the next time step. It
is to be noted with care that any random selection of time step
value Δt may result in unphysical results. To avoid this, time
step Δt and grid sizes Δx and Δy should be chosen in such a
way that [25]
For the case when thermal conductivity, k, changes with
space, the discretization is as follows. For the discussion, kX
means thermal conductivity at the node X and ̄
kX means aver-
age of thermal conductivities at node P and node X, i.e.,
Using the standard form and uniform grid system, the coef-
ficients change as follows
(18)
aE = aW = aN = aS at−Δt
P
(19)
̄
kX =
(
kX + kP
)
∕2
(20)
aE =
( ̄
kE
𝜌cΔx2
+
̄
kN
2𝜌cΔxΔy
−
̄
kS
2𝜌cΔxΔy
)
×
(
1 +
𝜏T
Δt
)
(21)
aW =
( ̄
kW
𝜌cΔx2
−
̄
kN
2𝜌cΔxΔy
+
̄
kS
2𝜌cΔxΔy
)
×
(
1 +
𝜏T
Δt
)
(22)
aN =
( ̄
kE
2𝜌cΔxΔy
−
̄
kW
2𝜌cΔxΔy
+
̄
kN
𝜌cΔy2
)
×
(
1 +
𝜏T
Δt
)
(23)
aS =
(
−
̄
kE
2𝜌cΔxΔy
+
̄
kW
2𝜌cΔxΔy
+
̄
kS
𝜌cΔy2
)
×
(
1 +
𝜏T
Δt
)
(24)
b = at
P
Tt
P
− at
E
Tt
E
− at
W
Tt
W
− at
N
Tt
N
− at
S
Tt
S
− at−Δt
P
Tt−Δt
P
+ SC
(25)
at
E
=
( ̄
kE
Δx
+
̄
kN
2Δy
−
̄
kS
2Δy
)
×
(
𝜏T
𝜌cΔtΔx
)
(26)
at
W
=
( ̄
kW
Δx
−
̄
kN
2Δy
+
̄
kS
2Δy
)
×
(
𝜏T
𝜌cΔtΔx
)
(27)
at
N
=
( ̄
kE
2Δx
−
̄
kW
2Δx
+
̄
kN
Δy
)
×
(
𝜏T
𝜌cΔtΔy
)
(28)
at
S
=
(
−
̄
kE
2Δx
+
̄
kW
2Δx
+
̄
kS
Δy
)
×
(
𝜏T
𝜌cΔtΔy
)
(29)
at
P
=
2𝜏q
Δt2
+
(
1 +
wb𝜌bcb𝜏q
𝜌c
)
×
1
Δt
+ at
E
+ at
W
+ at
N
+ at
S
It can be observed that when thermal conductivity is uni-
form, all the equations from Eq. (20) to (33) get reduced to
Eq. (10) to (17).
5 Results and discussion
5.1 Validation
For the current study, the thermo-physical properties and
the geometry used are presented in Table 3 [24]. Blood
is considered to have a temperature Tb = 37 °C, density
ρb = 1052 kg/m3
, mass specific heat = 3800 J/kgK, heat
transfer coefficient of the ambient atmosphere h = 20 W/
m2
K, and the ambient temperature T∞ =20 °C. A length of
L=6 cm is considered as shown in Fig. 1. Numerical models
developed for the heterogeneous medium are validated with
analytical results for both PMBT [26] and TWMBT [14] as
shown in Figs. 2 and 3, respectively. It may be noted that the
analytical results considered for validation were developed
for homogeneous medium. However, there is no loss of gen-
erality as the codes developed for heterogeneous medium
can be used for homogeneous medium by just assigning
same thermo-physical parameters to various regions in the
computational domain considered for the current study.
Temperature distributions along the depth of the skin at
various instances are shown in Fig. 2. Figure 2a shows tem-
perature distribution variation with time when skin is irradi-
ated with constant heat flux of 250 W/m2
[26], and Fig. 2b
shows temperature distribution variation with time when a
(30)
at−Δt
P =
𝜏q
Δt2
(31)
SC = −SPTb +
Qm
𝜌c
(32)
SP = −
wb𝜌bcb
𝜌c
(33)
aP = aE + aW + aN + aS +
𝜏q
Δt2
+
(
1 +
wb𝜌bcb𝜏q
𝜌c
)
×
1
Δt
− SP
Table 3 Parameters of different tissues [24]
Type of tissue Epidermis Dermis Subcutaneous Tumor
Thickness (m) 0.0001 0.0015 0.0044 0.001
𝜌cp
(
J/m3
K
)
4.2 × 106
4.2 × 106
4.2 × 106
4.2 × 106
k (W/mK) 0.21 0.30 0.21 0.59
wb (m3
/m3
s) 0 1.63 × 10−3
1.0 × 10−3
5.0 × 10−3
Qm
(
W/m3
)
400 400 400 4000
6. Journal of the Brazilian Society of Mechanical Sciences and Engineering (2020) 42:62
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62 Page 6 of 13
sinusoidal heat flux of 250+200 cos(0.02t) W/m2
is used.
In these figures, lines represent the temperature distributions
obtained using the present code and the discrete symbols are
those obtained from the analytical study of Deng and Liu
[26]. It can be seen that there is a slight difference between
present results with that of [26]. In Deng and Liu [26] work,
the analytical solution is derived based on Green’s function
method. Also, it can be noticed in their work that they have
omitted zero-order terms in the Green function series in the
analytical solution. We believe that the difference between
the two results may be due to the above reason. But there is
a qualitative agreement between both results.
For validation with [26], the thermo-physical prop-
erties used are density of blood and skin as 1000 kg/m3
,
mass specific heat of blood and skin as 4200 J/kg °C, body
core temperature as 37 °C, thermal conductivity of skin as
0.5 W/m °C, blood perfusion rate as 0.0005 ml/s/ml and
metabolic heat generation rate as 33,800 W/m3
. Tempera-
ture of the surrounding fluid is taken as 25 °C and natural
convective heat transfer coefficient as 10 W/m2
°C.
Figure 3a shows the comparison between present results
with that of [14], and Fig. 3b represents comparison between
present results with that of [14] for the PMBT, TWMBT and
DPL models. For this validation case, the heat flux is 2 W/
m2
; phase lag times used are τq =16 s and τT =0.05 s. From
Fig. 2 Validation with [26]
Fig. 3 Validation with [14]
7. Journal of the Brazilian Society of Mechanical Sciences and Engineering (2020) 42:62
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both the figures, it is clear that a good matching is obtained
between our results with that of [14] which confirms the
validity of the developed model. Further, it can be observed
that in Fig. 3a at the time of 5 s, there are unphysical tem-
perature oscillations. This is due to the condition explained
in Eq. (18) [25]. The condition in Eq. (18) can be met either
by reducing the spatial grid size or by increasing time step
size. It is recommended to reduce the grid size as the effect
of τT which is generally in the order of 0.01–10 s [27] will
not be captured if time step size is large.
5.2 Temperature distributions
For the present study, τq =16 s and τT =0.5 s are used. Fig-
ure 4 shows the temperature distributions predicted by three
different bioheat transfer models at the end of 7 s. It can be
observed that at this time, temperature estimated by PMBT
at the lower right corner of the tumor region is higher than
that of those predicted by other models. On the other hand,
temperatures at the surface are predicted to be very high
by TWMBT and DPL models when compared to PMBT.
This is because energy easily diffuses through the domain
according to PMBT, while according to the other two, it
takes more time to travel to deeper parts of the skin. Very
high temperatures at the surface predicted by the hyper-
bolic models can be explained as follows. Same amount of
energy, 11,000 W/m2
, is being applied to the domain for all
the models. However, according to PMBT, energy diffuses
to a larger domain and hence temperature rise is small over
a larger domain, while hyperbolic models suggest energy
distribution in smaller domain and hence large temperature
rise over small domain. This is in agreement with energy
conservation.
In Fig. 4b, it can be observed that TWMBT predicts sharp
temperature gradients typical to wave type of propagation.
The effect of τT can be observed from the third figure in
Fig. 4c that it has a diffusing effect at the wave front pre-
dicted by the TWMBT. This also explains the slight lower
surface temperature predicted by DPL model. Such behavior
was observed in experiment I in Mitra et al. [7]. Figure 5
shows the temperature distributions predicted by three dif-
ferent bioheat transfer models at the end of 15 s of expo-
sure. In Fig. 5a PMBT predicts that the effect of applied
heat flux is observed in the entire part of the domain shown.
Temperature at the corner of the tumor is between 38 and
40 °C already. Figure 5b, c shows the temperature distribu-
tion according to TWMBT and DPL models, respectively.
It is found that maximum temperature rises by about 10 °C
according to PMBT, while it is only about 5 °C according
to the hyperbolic models for same duration of exposure to
radiation. This is due to the wave type of propagation that
the temperature rise by hyperbolic models is not as high as
that suggested by PMBT.
Figure 6 shows temperature distributions at 7 s of expo-
sure to radiation. However, this time, τq and τT are taken to
be 22 s and 7 s, respectively, indicative of more complex
tissues like those with greater vasculature [27]. Compar-
ing Figs. 4b and 6b, we can observe that the increase in τq
from 15 to 22 s retards the energy penetration further, thus
modeling the effect of contact resistances due to greater
complexity. Here again, there is greater temperature of
90 °C predicted at the surface due to greater retardation
and hence greater energy concentration while it is only
85 °C in the case of Fig. 4b. Higher value of τT has a
greater diffusing effect on the thermal wave of TWMBT
as seen in Fig. 6c. It can be observed in Fig. 7 that for the
case of more complex tissues, for a given time of radiation
exposure, the order of temperature rise from 7 s of expo-
sure to 15 s of exposure is same for all the models. This is
because the thermal wave is not yet completely formed by
this time for the considered values of thermal phase lag
time. As thermal phase lag time for temperature gradient
Fig. 4 Temperature distributions at 7 s of exposure with 11,000 W/m2
heat flux with τq =16 s and τT =0.5 s
8. Journal of the Brazilian Society of Mechanical Sciences and Engineering (2020) 42:62
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62 Page 8 of 13
τT increases, the heat transfer mechanism becomes more
diffusive, as seen in Fig. 7c. Though τT has only diffusing
effect at the wave front only, as its magnitude increases,
its effect spreads into a larger domain. The higher the τq,
the steeper the temperature gradient at the wave front, and
the higher the τT, the smoother the wave front becomes.
Figure 8 shows temperatures along a line passing through
the tumor at a quarter width line, that is x = 0.0005 m,
Fig. 5 Temperature distributions at 15 s of exposure with 11,000 W/m2
heat flux with τq =16 s and τT =0.5 s
Fig. 6 Temperature distributions at 7 s of exposure with 11,000 W/m2
heat flux with τq =22 s and τT =7 s
Fig. 7 Temperature distributions at 15 s of exposure with 11,000 W/m2
heat flux with τq =22 s and τT =7 s
9. Journal of the Brazilian Society of Mechanical Sciences and Engineering (2020) 42:62
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parallel to y-axis at 7 s of radiation exposure. The thermal
wave propagation is clearly found in the figure. It may be
observed that PMBT overestimates temperatures in deeper
parts of the domain; however, the temperature variation is
very smooth. On the other hand, the temperatures predicted
by hyperbolic models exhibit abrupt changes. From Fig. 8b,
we see that as τT increases, the temperature distribution pre-
dicted by DPL model moves away from that predicted by
TWMBT toward the one predicted by PMBT. Effect of the
applied heat flux is experienced at a depth of y=0.0057 m
when τq =16 s, while it is experienced only till a depth of
y=0.0058 m when τq =22 s at same 7 s of exposure. From
Fig. 9a, it may be seen that temperature predicted by all the
heat transfer models is about to reach the tumor killing value
of 42.5 °C throughout the tumor region at a close time inter-
val. Though temperature according to PMBT is supposed
to be higher than that according to hyperbolic models, it
is still below the threshold tumor killing temperature. Fur-
ther, for the present case, the thermal wave happens to have
reached entire depth of tumor, which is y=0.0049 m from
the surface. Hence, for the case of τq =16 s and τT =0.5 s,
the predicted time of exposure to ensure killing of entire
tumor is almost same by all the models as presented in
Table 4. In Fig. 9b, such behavior is not observed. There
Fig. 8 Temperature distributions at 7 s of exposure with 11,000 W/m2
heat flux with a τq =16 s and τT =0.5 s and b τq =22 s and τT =7 s near
quarter tumor along a line parallel to y-axis
Fig. 9 Temperature distributions at 15 s of exposure with 11,000 W/m2
heat flux with a τq =16 s and τT =0.5 s and b τq =22 s and τT =7 s near
quarter tumor along a line parallel to y-axis
10. Journal of the Brazilian Society of Mechanical Sciences and Engineering (2020) 42:62
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is a considerable difference in the depth and temperatures
predicted by the three models. Time taken for necrosis of
cancer cells to occur for this case of τq = 22 s and τT = 7 s
is presented in Table 4. It can be seen that it takes around
21.4 s for cancer killing according to PMBT, while it takes
about 50.2 s and 56.0 s according to TWMBT and DPL
models, respectively.
5.3 Thermal damage
Figure 10 shows the thermal damage at 7 s of radiation
exposure for the case of τq =16 s and τT =0.5 s. It may be
observed that PMBT predicts much lower level of thermal
damage than the hyperbolic models. At this instant, only a
second-degree burn is seen as per PMBT and over a small
area of the tissue. As can be seen in the case of TWMBT in
Fig. 10b, a third-degree burn is seen over a large part of the
tissue above the tumor. The diffusing effect of phase lag time
for temperature gradient τT is evident from a slight decrease
in the area of burn as seen in Fig. 10c. Thermal damage in
the tumor region is set to zero as thermal damage of skin is
only studied. Figure 11 shows the thermal damage at 15 s
of radiation exposure for the case of τq =16 s and τT =0.5 s.
In the case of PMBT, a small region of third-degree burn is
seen in Fig. 11a. In the case of TWMBT and DPL models,
the region of third-degree burn is almost unchanged when
compared to that for 7 s of exposure, while the region of
second and first-degree burns has increased to similar extent
in both the models. It can be seen that the effect of having
small τT is not very significant. Further, PMBT does not
predict thermal damage beyond the region of radiation expo-
sure, i.e., beyond the region immediately above the tumor.
However, the hyperbolic models do predict such behavior
because the concentration of energy has been explained ear-
lier in Sect. 5.2.
Thermal damage at 7 s of exposure to radiation with
τq = 22 s and τT = 7 s is shown in Fig. 12. From Fig. 12b,
it can be observed that the extent of third-degree damage
is more in this case as compared to that present in the case
of Fig. 10b. Thermal damage has reduced in this case for
DPL model than that present in Fig. 10c. For 15 s of radia-
tion exposure, thermal damage is shown in Fig. 13. From
Fig. 13b, the damage is seen not to have increased a lot
from that in Fig. 11b. However, thermal damage is decreased
considerably in the case of DPL model as is observed when
Figs. 11c and 13c are compared. Increasing the value of
τq increases the thermal damage while increasing the value
of τT reduces the thermal damage. From the heat transfer
models, a DPL model would predict same results as that of
PMBT only when both τq and τT are zeroes. In [14] it was
Table 4 Time taken for killing of tumor cells
Values of phase lag time Model Estimated time for necrosis
of cancer cells to occur (s)
τq =16 s and τT =0.5 s PMBT 21.4
TWMBT 21
DPL 21.4
τq =22 s and τT =7 PMBT 21.4
TWMBT 50.2
DPL 56.0
Fig. 10 Thermal damage at 7 s of exposure with11, 000 W
m2
heat flux with 𝜏q = 16s and 𝜏T = 0.5 s
Fig. 11 Thermal damage at 15 s of exposure with11, 000 W
m2
heat flux with 𝜏q = 16s and 𝜏T = 0.5 s
11. Journal of the Brazilian Society of Mechanical Sciences and Engineering (2020) 42:62
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proposed that even when they are equal to each other, not
necessarily to zero, DPL will predict same results as that
of PMBT. However, in [12] and [13] it can be found that
when both τq and τT are equal but not to zero, there are dis-
crepancies between parabolic, i.e., PMBT, and hyperbolic,
i.e., TWMBT and DPL, models. In our study, we have ana-
lyzed this case of both phase lag time values being equal to
each other, for a particular value of τq =7 s and τT =7 s and
found that there is not much difference in the temperature
fields predicted by both PMBT and DPL at 15 s of exposure
with 11,000 W/m2
heat flux near quarter tumor line paral-
lel to y-axis as shown in Fig. 14. The small oscillations at
y=0.0016 m can be attributed to a very small deviation from
condition depicted in Eq. (18).
5.4 Effect of realistic heat flux distribution
In all the previous studies, a uniform heat flux was consid-
ered over the tumor. However, this is not realistic and the
fact that the intensity of the projected radiation usually fades
out toward the end of the laser irradiation area can reduce
the temperature attained in the domain and thereby compel
one to increase the time of duration of radiation exposure to
ensure necrosis of the cancer cells. For studying this condi-
tion, we assumed Gaussian distribution shown in Eq. (34) of
heat flux to be a good approximation of the actual heat flux
distribution, as shown in Fig. 15. Here, qmax is the maximum
intensity applied, and xt is the half of the width of the tumor,
shown in Fig. 1 along x-axis.
(34)
𝐪 = 𝐪𝐦𝐚𝐱
(
𝐞
− 𝐱2
𝐱2
𝐭
)
We observed significant changes in the time required for
the cancer cell necrosis as shown in Table 5, only for the
cases of PMBT and DPL but not in the case of TWMBT
when τq and τT are considered to be 22 s and 7 s, respec-
tively. For the case of PMBT, it has increased by about 8 s,
while it has increased by about 5 s for the case of DPL model
when we go from uniform to Gaussian distribution. This is
a substantial change and, hence, must be taken care when
simulations are carried out to estimate the time required for
the treatment.
Fig. 12 Thermal damage at 7 s of exposure with110, 00 W/m2
heat flux with 𝜏q = 22 s and 𝜏T = 7 s
Fig. 13 Thermal damage at 15 s of exposure with11, 000 W/m2
heat flux with 𝜏q = 22 s and 𝜏T = 7 s
Fig. 14 Effect of thermal phase lag time being equal to each other
and non-zero value
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6 Conclusion
A two-dimensional computational model based on finite dif-
ference method is developed to understand temperature dis-
tributions in biological tissue. Three bioheat transfer models
are compared and analyzed. Finite difference method is used
for discretization. The developed computational model is
validated with analytical results of previous researchers.
Later, temperature distribution in the domain is determined
using different bioheat transfer models. It is observed that
heat reaches quickly to deeper layers of skin according to
PMBT. However, higher temperatures and hence suggest-
ing energy concentration at surface layers are predicted by
TWMBT and DPL models. The effect of τT is understood to
have a diffusing effect on heat at the wave front predicted by
TWMBT model. Time of exposure of heat source for killing
tumor cells is estimated according to different models. It is
observed that time of treatment is not always increased due
to the phase lag times as the thermal wave will have reached
the corner of the tumor before the temperature predicted by
PMBT is already reached the threshold cancer cell killing
temperature. The higher the value of τq, the greater is the
resistance to the penetration of heat; hence, only a small
part of the domain is affected by the applied heat flux. For
complex tissues which generally have very large values of
phase lag times, the time required for cancer cell necro-
sis is highest according to DPL followed by TWMBT and
PMBT. TWMBT predicts highest thermal damage among
all the three models. When a realistic heat flux distribution,
like that of a Gaussian distribution, is considered, a signifi-
cant rise in the estimated time for cancer cell necrosis is
observed. It is recommended that time for necrosis to occur
should be determined by all the three models and whatever
is higher should be used for designing the treatment protocol
so as not to leave a chance for some cancer cells being left
behind, which may lead to resurrection of tumor.
Acknowledgement This research was supported by National Institute
of Technology Karnataka, Surathkal. We extend our thanks to Prof.
Prasenjith Rath, IIT Bhuvaneshwar, for his valuable comments.
Compliance with ethical standards
Conflict of interest We declare that we do not have any commercial or
associative interest that represents a conflict of interest in connection
with the work submitted.
References
1. GLOBOCAN (2008) Cancer incidence and mortality world-
wide. https://www.iarc.fr/en/media-centre/iarcnews/2010/globo
can2008.php. Accessed 09 July 2018
2. Delaney G, Jacob S, Featherstone C, Barton M (2005) The role
of radiotherapy in cancer treatment: estimating optimal utiliza-
tion from a review of evidence-based clinical guidelines. Cancer
104(6):1129–1137
3. Barnett GC et al (2009) Normal tissue reactions to radiotherapy:
towards tailoring treatment dose by genotype. Nat Rev Cancer
9(2):134–142
4. Bleehen NM (1982) Hyperthermia in the treatment of cancer.
Br J Cancer 45:96
5. Pennes HH (1948) Analysis of tissue and arterial blood tempera-
tures in the resting human forearm. Appl Physiol 1(2):93–122
6. Cattaneo C (1958) A form of heat-conduction equations which
eliminates the paradox of instantaneous propagation. Comptes
Rendus 247:431–433
7. Mitra K, Kumar S, Vedevarz A, Moallemi MK (1995) Experi-
mental evidence of hyperbolic heat conduction in processed
meat. J Heat Transf 117(3):568
8. Kaminski W (1990) Hyperbolic heat conduction equation for
materials with a nonhomogeneous inner structure. J Heat Transf
112(3):555
9. Herwig H, Beckert K (2000) Experimental evidence about the
controversy concerning Fourier or non-Fourier heat conduction
Fig. 15 Representation of Gaussian distribution type of applied heat
flux
Table 5 Effect of Gaussian heat flux
Values of phase lag time Model Estimated time for necrosis
of cancer cells to occur (s)
τq =22 s and τT =7 s
Uniform heat flux
Pennes’ 21.4
TWM 50.2
DPL 56.0
τq =22 s and τT =7
Gaussian heat flux distribution
Pennes’ 29.2
TWM 50.4
DPL 61.4
13. Journal of the Brazilian Society of Mechanical Sciences and Engineering (2020) 42:62
1 3
Page 13 of 13 62
in materials with a nonhomogeneous inner structure. Heat mass
Transf 36:387–392
10. Graûmann A, Peters F (1999) Experimental investigation of heat
conduction in wet sand. Heat Mass Transf 35:1–6
11. Minkowycz WJ, Haji-Sheikh A, Vafai K (1999) On departure
from local thermal equilibrium in porous media due to a rapidly
changing heat source: the Sparrow number. Int J Heat Mass
Transf 42:3373–3385
12. Hooshmand P, Moradi A, Khezry B (2015) Bioheat transfer analy-
sis of biological tissues induced by laser irradiation. Int J Therm
Sci 90:214–223
13. Zhou J, Zhang Y, Chen JK (2009) An axisymmetric dual-phase-
lag bioheat model for laser heating of living tissues. Int J Therm
Sci 48(8):1477–1485
14. Liu K, Wang J (2014) Analysis of thermal damage to laser irradi-
ated tissue based on the dual-phase-lag model. Int J Heat Mass
Transf 70:621–628
15. Antaki PJ (2005) New interpretation of non-Fourier heat conduc-
tion in processed meat. J Heat Transf 127(2):189
16. Museux N, Perez L, Autrique L, Agay D (2012) Skin burns after
laser exposure: histological analysis and predictive simulation.
Burns 38(5):658–667
17. Jiang SC, Ma N, Li HJ, Zhang XX (2002) Effects of thermal prop-
erties and geometrical dimensions on skin burn injuries. Burns
28(8):713–717
18. Bedin L, Bazán FSV (2014) On the 2D bioheat equation with
convective boundary conditions and its numerical realization via
a highly accurate approach. Appl Math Comput 236:422–436
19. Tung MM, Trujillo M, LópezMolina JA, Rivera MJ, Berjano
EJ (2009) Modeling the heating of biological tissue based on
the hyperbolic heat transfer equation. Math Comput Model
50(5–6):665–672
20. Ahmadikia H, Fazlali R, Moradi A (2012) Analytical solution of
the parabolic and hyperbolic heat transfer equations with constant
and transient heat flux conditions on skin tissue. Int Commun Heat
Mass Transf 39(1):121–130
21. Lakhssassi EK, Semmaoui H (2010) Investigation of nonlinear
temperature distribution in biological tissues by using bioheat
transfer equation of Pennes’ type. Nat Sci 2(3):131–138
22. Deng ZS, Liu J (2004) Mathematical modeling of temperature
mapping over skin surface and its implementation in thermal dis-
ease diagnostics. Comput Biol Med 34(6):495–521
23. Henriques R Jr, Moritz FC (1947) Studies of thermal injury: I.
The conduction of heat to and through skin and the temperatures
attained therein. A theoretical and an experimental investigation.
Am J Pathol 23(4):530
24. Verma K, Rath P, Mahapatra SK (2017) Assessment of thermal
damage during skin tumor treatment using thermal wave model:
a realistic approach. J Heat Transf 139(5):051102
25. Li J, Cheng P, Peterson GP, Xu JZ (2005) Rapid transient heat
conduction in multilayer materials with pulsed heating boundary.
Numer Heat Transf Part A Appl 47(7):633–652
26. Deng Z-S, Liu J (2002) Analytical study on bioheat transfer prob-
lems with spatial or transient heating on skin surface or inside
biological bodies. J Biomech Eng 124(6):638
27. Zhang Y (2009) Generalized dual-phase lag bioheat equations
based on nonequilibrium heat transfer in living biological tissues.
Int J Heat Mass Transf 52(21–22):4829–4834
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