scholarly journals Solving bio-heat transfer multi-layer equation using Green’s Functions method

2021 ◽  
Vol 2090 (1) ◽  
pp. 012150
Author(s):  
de Oliveira Eduardo Peixoto ◽  
Gilmar Guimaräes
Keyword(s):  
Heat Transfer ◽  
Green's Functions ◽  
Green’S Functions ◽  
Transfer Problem ◽  
Blood Perfusion ◽  
The Body ◽  
Mathematical Background ◽  
Number Of Layers ◽  
Pennes Equation ◽  
Transfer Problems

Abstract An analytical method using Green’s Functions for obtaining solutions in bio-heat transfer problems, modeled by Pennes’ Equation, is presented. Mathematical background on how treating Pennes’ equation and its μ2T term is shown, and two contributions to the classical numbering system in heat conduction are proposed: inclusion of terms to specify the presence of the fin term, μ2T, and identify the biological heat transfer problem. The presentation of the solution is made for a general multi-layer domain, deriving and showing general approaches and Green’s Functions for such n number of layers. Numerical examples are presented to simplify human skin as a two-layer domain: dermis and epidermis, accounting metabolism as a heat source, and blood perfusion only at the dermis. Time-independent summations in the series-solution are written in closed forms, leading to better convergence along the boundaries. Details on obtaining the two-layer solution and its eigenvalues are presented for boundary conditions of prescribed temperature inside the body and convection at the surface, such as its intrinsic verification.

A New Model Towards Thermal Mixing and Stratification in Passive Containment

2013 ◽  
Author(s):  
He Zhang ◽  
Fenglei Niu ◽  
Yu Yu ◽  
Peipei Chen
Keyword(s):  
Heat Transfer ◽  
System Analysis ◽  
Cooling System ◽  
Transfer Problem ◽  
Ambient Fluid ◽  
Simulation Code ◽  
Thermal Mixing ◽  
Lumped Parameter ◽  
Passive Safety System ◽  
Transfer Problems

Thermal mixing and stratification often appears in passive containment cooling system (PCCS), which is an important part of passive safety system. So, it is important to accurately predict the temperature and density distributions both for design optimization and accident analysis. However, current major reactor system analysis codes only provide lumped parameter models which can only get very approximate results. The traditional 2-D or 3-D CFD methods require very long simulation time, and it’s not easy to get result. This paper adopts a new simulation code, which can be used to calculate heat transfer problems in large enclosures. The new code simulates the ambient fluid and jets with different models. For the ambient fluid, it uses a one-dimensional model, which is based on the thermal stratification and derived from three conservation equations. While for different jets, the new code contains several jet models to fully simulate the different break types in containment. Now, the new code can only simulate rectangular enclosures, not the cylinder enclosure. So it is meaningful for us to modify the code to simulate the actual containment, then it can be applied to solve the heat transfer problem in PCCS accurately.

Scalable Heat Transfer Characterization on Film Cooled Geometries Based on Discrete Green’s Functions

2020 ◽  
Author(s):  
Jorge Saavedra ◽  
Venkat Athmanathan ◽  
Guillermo Paniagua ◽  
Terrence Meyer ◽  
Doug Straub ◽  
...  
Keyword(s):  
Heat Transfer ◽  
Heat Flux ◽  
Flat Plate ◽  
Convective Heat Transfer ◽  
Convective Heat ◽  
Green's Functions ◽  
Green’S Functions ◽  
Numerical Procedure ◽  
Local Heat Transfer ◽  
Sensitivity Matrix

Abstract The aerothermal characterization of film cooled geometries is traditionally performed at reduced temperature conditions, which then requires a debatable procedure to scale the convective heat transfer performance to engine conditions. This paper describes an alternative engine-scalable approach, based on Discrete Green’s Functions (DGF) to evaluate the convective heat flux along film cooled geometries. The DGF method relies on the determination of a sensitivity matrix that accounts for the convective heat transfer propagation across the different elements in the domain. To characterize a given test article, the surface is discretized in multiple elements that are independently exposed to perturbations in heat flux to retrieve the sensitivity of adjacent elements, exploiting the linearized superposition. The local heat transfer augmentation on each segment of the domain is normalized by the exposed thermal conditions and the given heat input. The resulting DGF matrix becomes independent from the thermal boundary conditions, and the heat flux measurements can be scaled to any conditions given that Reynolds number, Mach number, and temperature ratios are maintained. The procedure is applied to two different geometries, a cantilever flat plate and a film cooled flat plate with a 30 degree 0.125” cylindrical injection orifice with length-to-diameter ratio of 6. First, a numerical procedure is applied based on conjugate 3D Unsteady Reynolds Averaged Navier Stokes simulations to assess the applicability and accuracy of this approach. Finally, experiments performed on a flat plate geometry are described to validate the method and its applicability. Wall-mounted thermocouples are used to monitor the surface temperature evolution, while a 10 kHz burst-mode laser is used to generate heat flux addition on each of the discretized elements of the DGF sensitivity matrix.

Meshless Local Petrov-Galerkin Method for Heat Transfer Analysis

2013 ◽  
Author(s):  
Singiresu S. Rao
Keyword(s):  
Heat Transfer ◽  
Heat Transfer Coefficient ◽  
Boundary Conditions ◽  
Transfer Coefficient ◽  
Transfer Problem ◽  
Radiation Heat Transfer ◽  
Heat Transfer Problem ◽  
Radiation Heat ◽  
Mlpg Method ◽  
Transfer Problems

A meshless local Petrov-Galerkin (MLPG) method is proposed to obtain the numerical solution of nonlinear heat transfer problems. The moving least squares scheme is generalized, to construct the field variable and its derivative continuously over the entire domain. The essential boundary conditions are enforced by the direct scheme. The radiation heat transfer coefficient is defined, and the nonlinear boundary value problem is solved as a sequence of linear problems each time updating the radiation heat transfer coefficient. The matrix formulation is used to drive the equations for a 3 dimensional nonlinear coupled radiation heat transfer problem. By using the MPLG method, along with the linearization of the nonlinear radiation problem, a new numerical approach is proposed to find the solution of the coupled heat transfer problem. A numerical study of the dimensionless size parameters for the quadrature and support domains is conducted to find the most appropriate values to ensure convergence of the nodal temperatures to the correct values quickly. Numerical examples are presented to illustrate the applicability and effectiveness of the proposed methodology for the solution of heat transfer problems involving radiation with different types of boundary conditions. In each case, the results obtained using the MLPG method are compared with those given by the FEM method for validation of the results.

Scalable Heat Transfer Characterization on Film Cooled Geometries Based on Discrete Green’s Functions

2021 ◽  
Vol 143 (2) ◽  
Author(s):  
Jorge Saavedra ◽  
Venkat Athmanathan ◽  
Guillermo Paniagua ◽  
Terrence Meyer ◽  
Doug Straub ◽  
...  
Keyword(s):  
Heat Transfer ◽  
Heat Flux ◽  
Flat Plate ◽  
Convective Heat Transfer ◽  
Convective Heat ◽  
Green's Functions ◽  
Green’S Functions ◽  
Numerical Procedure ◽  
Local Heat Transfer ◽  
Sensitivity Matrix

Abstract The aerothermal characterization of film-cooled geometries is traditionally performed at reduced temperature conditions, which then requires a debatable procedure to scale the convective heat transfer performance to engine conditions. This paper describes an alternative engine-scalable approach, based on Discrete Green’s Functions (DGF) to evaluate the convective heat flux along film-cooled geometries. The DGF method relies on the determination of a sensitivity matrix that accounts for the convective heat transfer propagation across the different elements in the domain. To characterize a given test article, the surface is discretized in multiple elements that are independently exposed to perturbations in heat flux to retrieve the sensitivity of adjacent elements, exploiting the linearized superposition. The local heat transfer augmentation on each segment of the domain is normalized by the exposed thermal conditions and the given heat input. The resulting DGF matrix becomes independent from the thermal boundary conditions, and the heat flux measurements can be scaled to any conditions given that Reynolds number, Mach number, and temperature ratios are maintained. The procedure is applied to two different geometries, a cantilever flat plate and a film-cooled flat plate with a 30 degree 0.125 in. cylindrical injection orifice with length-to-diameter ratio of 6. First, a numerical procedure is applied based on conjugate 3D unsteady Reynolds-averaged Navier–Stokes (URANS) simulations to assess the applicability and accuracy of this approach. Finally, experiments performed on a flat plate geometry are described to validate the method and its applicability. Wall-mounted thermocouples are used to monitor the surface temperature evolution, while a 10 kHz burst-mode laser is used to generate heat flux addition on each of the discretized elements of the DGF sensitivity matrix.

THREE‐DIMENSIONAL INDUCED POLARIZATION AND ELECTROMAGNETIC MODELING

Geophysics ◽  
1975 ◽  
Vol 40 (2) ◽  
pp. 309-324 ◽  
Author(s):  
Gerald W. Hohmann
Keyword(s):  
Integral Equation ◽  
Green's Functions ◽  
Green’S Functions ◽  
Three Dimensional ◽  
Induced Polarization ◽  
The Body ◽  
Scale Model ◽  
Two Dimensional ◽  
Electromagnetic Modeling ◽  
The Earth

The induced polarization (IP) and electromagnetic (EM) responses of a three‐dimensional body in the earth can be calculated using an integral equation solution. The problem is formulated by replacing the body by a volume of polarization or scattering current. The integral equation is reduced to a matrix equation, which is solved numerically for the electric field in the body. Then the electric and magnetic fields outside the inhomogeneity can be found by integrating the appropriate dyadic Green’s functions over the scattering current. Because half‐space Green’s functions are used, it is only necessary to solve for scattering currents in the body—not throughout the earth. Numerical results for a number of practical cases show, for example, that for moderate conductivity contrasts the dipole‐dipole IP response of a body five units in strike length approximates that of a two‐dimensional body. Moving an IP line off the center of a body produces an effect similar to that of increasing the depth. IP response varies significantly with conductivity contrast; the peak response occurs at higher contrasts for two‐dimensional bodies than for bodies of limited length. Very conductive bodies can produce negative IP response due to EM induction. An electrically polarizable body produces a small magnetic field, so that it is possible to measure IP with a sensitive magnetometer. Calculations show that horizontal loop EM response is enhanced when the background resistivity in the earth is reduced, thus confirming scale model results.

Numerical Simulation of Enhanced Skin Thermal Signature of Female Breast Tumor Subjected to Forced Convection

2003 ◽  
Author(s):  
A. Gupta ◽  
L. Hu ◽  
J. P. Gore ◽  
L. X. Xu
Keyword(s):  
Heat Transfer ◽  
Flow Field ◽  
Skin Temperature ◽  
Forced Convection ◽  
Infrared Imaging ◽  
Transfer Problem ◽  
Blood Perfusion ◽  
Skin Surface ◽  
Temperature Distributions ◽  
Thermal Signature

Early detection is considered to be the best defense against breast cancer and imaging plays a very important role in screening and in the diagnosis of symptomatic women. Infrared thermal imaging of skin temperature changes caused by a malignant tumor in breast is a rapidly developing detection modality with potential for functional detection. Knowledge and control of environmental factors which affect the skin temperature can reduce misinterpretations and false diagnosis associated with infrared imaging. A bio heat transfer based numerical model was utilized to study the energy balance in healthy and malignant breasts subjected to low velocity forced convection in a wind tunnel. Existing estimates of metabolic heating rates and previous measurements of temperature distributions along the radial direction in a region intersecting a known tumor and a comparable region in the healthy breast of the same patient were used to estimate the blood perfusion rates for the tumor. A simplified structural and thermal model was used for representing the changes within and around the tumor. Steady state temperature distributions on the skin surface of the breasts were obtained by numerically solving the conjugate heat transfer problem. Parametric studies on the influences of the airflow on the skin thermal expression of tumors were performed. It was found that the presence of tumor may not be clearly shown due to the irregularity of the skin temperature distribution induced by the flow field. Image processing techniques could be employed to eliminate the effects of the flow field and thermal noise and significantly improve the thermal signature of the tumor on the skin surface.

Dynamic Observer Method Based on Modified Green’s Functions for Robust and More Stable Inverse Algorithms

2008 ◽  
Author(s):  
Priscila F. B. Sousa ◽  
Ana P. Fernandes ◽  
Vale´rio Luiz Borges ◽  
George S. Dulikravich ◽  
Gilmar Guimara˜es
Keyword(s):  
Heat Transfer ◽  
Transfer Function ◽  
Green's Functions ◽  
Transfer Functions ◽  
Green’S Functions ◽  
Three Dimensional ◽  
Original Method ◽  
Identification Algorithm ◽  
Transfer Function Model ◽  
Function Model

This work presents a modified procedure to use the concept of dynamic observers based on Green’s functions to solve inverse problems. The original method can be divided in two distinct steps: i) obtaining a transfer function model GH and; ii) obtaining heat transfer functions GQ and GN and building an identification algorithm. The transfer function model, GH, is obtained from the equivalent dynamic systems theory using Green’s functions. The modification presented here proposes two different improvements in the original technique: i) A different method of obtaining the transfer function model, GH, using analytical functions instead of numerical procedures, and ii) Definition of a new concept of GH to allow the use of more than one response temperature. Obtaining the heat transfer functions represents an important role in the observer method and is crucial to allow the technique to be directly applied to two or three-dimensional heat conduction problems. The idea of defining the new GH function is to improve the robustness and stability of the algorithm. A new dynamic equivalent system for the thermal model is then defined in order to allow the use of two or more temperature measurements. Heat transfer function, GH can be obtained numerically or analytically using Green’s function method. The great advantage of deriving GH analytically is to simplify the procedure and minimize the estimative errors.

Inverse Heat Transfer Analysis of Micro Heater Strength and Locations for Hyperthermia Treatment of Brain Tumors

2007 ◽  
Author(s):  
Maral Biniazan ◽  
Kamran Mohseni
Keyword(s):  
Heat Transfer ◽  
Brain Tumors ◽  
Transfer Problem ◽  
The Body ◽  
Least Square ◽  
Bioheat Transfer ◽  
Heat Transfer Analysis ◽  
Whole Body ◽  
Transient Temperature ◽  
Inverse Heat Transfer

Hyperthermia, also called thermal therapy or thermotherapy, is a type of cancer treatment in which the aim is to maintain the surrounding healthy tissue at physiologically normal temperatures and expose the cancerous region to high temperatures between 43°C–45°C. Several methods of hyperthermia are currently under study, including local, regional, and whole-body hyperthermia. In local hyperthermia, Interstitial techniques are used to treat tumors deep within the body, such as brain tumors. heat is applied to the tumor, usually by probes or needles which are inserted into the tumor. The heat source is then inserted into the probe. Invasive interstitial heating technique offer a number of advantages over external heating approaches for localizing heat into small tumors at depth. e. g interstitial technique allows the tumor to be heated to higher temperatures than external techniques. This is why an innovative internal hyperthermia research is being conducted in the design of an implantable microheater [1]. To proceed with this research we need complete and accurate data of the strength, number and location of the micro heaters, which is the objective of this paper. The location, strength, and number of implantable micro heaters for a given tumor size is calculated by solving an Inverse Heat Transfer Problem (IHTP). First we model the direct problem by calculating the transient temperature field via Pennies bioheat transfer equation. A nonlinear least-square method, modified by addition of a regularization term, Levenberg Marquardt method is used to determine the inverse problem [2].

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