Analysis of hyperbolic Pennes bioheat equation in perfused homogeneous biological tissue subject to the instantaneous moving heat source

AbstractThe one-dimensional hyperbolic Pennes bioheat equation under instantaneous moving heat source is solved analytically based on the Eigenvalue method. Comparison with results of in vivo experiments performed earlier by other authors shows the excellent prediction of the presented closed-form solution. We present three examples for calculating the Arrhenius equation to predict the tissue thermal damage analysis with our solution, i.e., characteristics of skin, liver, and kidney are modeled by using their thermophysical properties. Furthermore, the effects of moving velocity and perfusion rate on temperature profiles and thermal tissue damage are investigated. Results illustrate that the perfusion rate plays the cooling role in the heating source moving path. Also, increasing the moving velocity leads to a decrease in absorbed heat and temperature profiles. The closed-form analytical solution could be applied to verify the numerical heating model and optimize surgery planning parameters.

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Green’s functions for soft materials containing a hard line inhomogeneity

Mathematics and Mechanics of Solids ◽

10.1177/1081286519850225 ◽

2019 ◽

Vol 24 (11) ◽

pp. 3614-3631 ◽

Cited By ~ 2

Author(s):

Pengyu Pei ◽

Guang Yang ◽

Cun-Fa Gao

Keyword(s):

Thermal Expansion ◽

Heat Source ◽

Closed Form ◽

Hilbert Problem ◽

Closed Form Solution ◽

Soft Materials ◽

Form Solution ◽

Soft Material ◽

Elastic Plane ◽

Rigid Line

The linear elastic plane deformation of a soft material containing a rigid line inhomogeneity subjected to a concentrated force, a concentrated moment, and a point heat source was studied. Distinct from the existing rigid line inhomogeneity model which neglects the deformation of the inhomogeneity induced by both the mechanical stresses and thermal expansion, the current model allows for the thermal expansion-induced stretch and rotation of the inhomogeneity. In this context, we derive the closed-form solution for the full stress field in the soft material by solving the corresponding Riemann–Hilbert problem. In particular, our solution can serve as the Green’s function to establish other analytical solutions for more practical and complicated problems in this area. Several numerical examples are presented to illustrate our closed-form solution corresponding to the thermal loading. It is found that the presence of the heat source contributes significantly to the rigid rotation of inhomogeneity, and the thermal expansion-induced stretch of the inhomogeneity has a great impact on the stress intensity factors at the inhomogeneity tips.

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Closed-form and numerical solutions to the laser heating process

Proceedings of the Institution of Mechanical Engineers Part C Journal of Mechanical Engineering Science ◽

10.1243/0954406981521105 ◽

1998 ◽

Vol 212 (2) ◽

pp. 141-151 ◽

Cited By ~ 28

Author(s):

B S Yilbas ◽

M Sami ◽

A Al-Farayedhi

Keyword(s):

Closed Form ◽

Laser Heating ◽

Numerical Solutions ◽

Closed Form Solution ◽

Form Solution ◽

Heat Of Fusion ◽

Heating Process ◽

Temperature Profiles ◽

Depth Analysis ◽

Analytical Approaches

The laser processing of engineering materials requires an in-depth analysis of the applicable heating mechanism. The modelling of the laser heating process offers improved understanding of the machining mechanism. In the present study, a closed-form solution for a step input laser heating pulse is obtained and a numerical scheme solving a three-dimensional heat transfer equation is introduced. The numerical solution provides a comparison of temperature profiles with those obtained from the analytical approach. To validate the analytical and numerical solutions, an experiment is conducted to measure the surface temperature and evaporating front velocity during the Nd—YAG laser heating process. It is found that the temperature profiles resulting from both theory and experiment are in a good agreement. However, a small discrepancy in temperatures at the upper end of the profiles occurs. This may be due to the assumptions made in both the numerical and the analytical approaches. In addition, the equilibrium time, based on the energy balance among the internal energy gain, conduction losses and latent heat of fusion, is introduced.

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Pipeline Walking due to Temperature Transient: Enhanced Analytical Calculations

Volume 4: Pipelines, Risers, and Subsea Systems ◽

10.1115/omae2020-18360 ◽

2020 ◽

Author(s):

Daniel Carneiro ◽

Renata Carvalhal

Keyword(s):

Closed Form ◽

Closed Form Solution ◽

Form Solution ◽

Transient Temperature ◽

Temperature Profiles ◽

Linear Temperature ◽

Temperature Front ◽

Non Linear ◽

Closed Form Analytical Solution ◽

Remarkable Agreement

Abstract Pipeline walking induced by transient temperature profiles as the pipeline heats up is assessed. Firstly, an integrated, closed-form solution is given for a problem for which only an ‘incremental’ solution was available [1]. This involves a short pipeline unrestrained at both ends subject to a linearly ramping temperature front. Secondly, the range of validity of this solution is significantly extended, whilst still presenting it in closed-form. Results are compared with previously published FEA results, presenting remarkable agreement. The closed-form analytical solution is then compared with FEA of more realistic transients (non-linear temperature front). Results show that the linear simplification can introduce excessive conservatism. The FEA results are then examined, and the reason for the excessive conservatism is found to be associated to early expansion of the cold end, which is not observed with the simplified linear front. A simplified incremental solution for non-linear transients is proposed. This is shown to be simple and effective in improving prediction for the range over which where the closed form solution is most conservative.

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A Closed-Form Solution for Temperature Profiles in Selective Laser Melting of Metal Additive Manufacturing

Materials Science Forum ◽

10.4028/www.scientific.net/msf.982.98 ◽

2020 ◽

Vol 982 ◽

pp. 98-105

Author(s):

Steven Y. Liang ◽

Jin Qiang Ning ◽

Elham Mirkoohi

Keyword(s):

Finite Element ◽

Heat Source ◽

Closed Form ◽

Heat Loss ◽

Heat Sink ◽

Prediction Accuracy ◽

Closed Form Solution ◽

Form Solution ◽

Laser Melting ◽

Temperature Prediction

This paper presents a closed-form solution for the temperature prediction in selective laser melting (SLM). This solution is developed for the three-dimensional temperature prediction with consideration of heat input from a moving laser heat source, and heat loss from convection and radiation on the part top boundary. The consideration of heat transfer boundary condition and latent heat in the closed-form solution leads to an improvement on the understanding of thermal development and prediction accuracy in SLM, and thus the usefulness of the analytical model in the temperature prediction in real applications. A moving point heat source solution is used to calculate the temperature rise due to the heat input. A heat sink solution is used to calculate the temperature drop due to heat loss from convection and radiation on the part boundary. The heat sink solution is modified from a heat source solution with equivalent power due to heat loss from convection and radiation, and zero-moving velocity. The temperature solution is then constructed from the superposition of the linear heat source solution and linear heat sink solution. Latent heat is considered using a heat integration method. Ti-6Al-4V is chosen to test the presented model with the assumption of isotropic and homogeneous material. The predicted molten pool dimensions are compared to the documented values from the finite element method and experiments in the literature. The presented model has improved prediction accuracy and significantly higher computational efficiency compared to the finite element model.

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EXACT ANALYTICAL SOLUTION OF BIOHEAT EQUATION SUBJECTED TO INTENSIVE MOVING HEAT SOURCE

Journal of Mechanics in Medicine and Biology ◽

10.1142/s0219519417500816 ◽

2017 ◽

Vol 17 (05) ◽

pp. 1750081 ◽

Cited By ~ 6

Author(s):

MOHAMMAD REZA TALAEE ◽

ALI KABIRI

Keyword(s):

Analytical Solution ◽

Heat Source ◽

Numerical Solutions ◽

Exact Analytical Solution ◽

Blood Perfusion ◽

Temperature Profiles ◽

Moving Heat Source ◽

Bioheat Equation ◽

Line Heat Source ◽

Moving Line

Presented is the analytical solution of Pennes bio-heat equation, under localized moving heat source. The thermal behavior of one-dimensional (1D) nonhomogeneous layer of biological tissue is considered with blood perfusion term and modeled under the effect of concentric moving line heat source. The procedure of the solution is Eigen function expansion. The temperature profiles are calculated for three tissues of liver, kidney, and skin. Behavior of temperature profiles are studied parametrically due to the different moving speeds. The analytical solution can be used as a verification branch for studying the practical operations such as scanning laser treatment and other numerical solutions.

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