Transfer Equation Research Articles

Temperature and water content estimation in soils of the semi‐arid region of Brazil using finite difference and CFD

AbstractDetermining the temperature and water content of soil, at a given instant or along time, is fundamental to understand several soil‐related phenomena and processes. Evaporation, aeration, chemical‐reaction rates and types, biological processes such as germination and growth of seeds, root development, nutrient and water uptake by roots, and decomposition of organic matter by microbes, are all strongly influenced by soil temperature. On the other hand, infiltration of water through the soil surface allows soil to temporarily store water, making it available for uptake by plants and organisms living in soil. Furthermore, soil water content is closely related to physical and chemical properties of soil, such as oxygen content and demand, which impacts root breathing, microbial activity and soil chemical balance. The accurate evaluation of these two parameters and their interconnection is even relevant in semi‐arid regions, where climate conditions are particularly difficult, such as the north‐eastern zone of Brazil. Thus, the use of computational models and coupled approaches are imperative for rigorous descriptions. This work presents a contribution to estimate soil temperature and water content, by solving the heat transfer equation and the Richards equation, respectively, through finite differences. As input, the model uses the experimental material composition of the soil, the time‐dependent temperature profile at the surface and information about the regional rain regime. Three different numerical approaches were implemented: explicit, simple implicit and the Crank–Nicolson method. The calculations for temperature and water content of the soil obtained with these computational models were compared with the results from Computational Fluid Dynamics (CFD). The relative differences between the numerical methods were less than 0.006% by solving the heat transfer equation and less than 2.75% using the Richards equation. The maximum relative differences within the model, including both a constant and a variable water‐content profile, were 3.28%. The results from the computational model using the CFX tool have maximum relative differences of 0.6%, which contributes to verifying the accuracy of the implemented methods.

Investigation of time-dependent mechanical loading on the start-stop process of functionally graded rotating disks under thermal conditions

Rotating disks are used in industries, such as aerospace, automotive, and power plants. These disks are under time-dependent mechanical loading when the machine starts or stops working conditions. Thermal stresses caused by temperature changes with this time-dependent mechanical load create dangerous conditions. Therefore, the estimation of the elastic limit angular velocity and acceleration as a criterion for the initiation of plastic deformations has special importance in the start-stop process. An analytical Homotopy perturbation method is used to solve the heat transfer equation and the governing Navier equations in both radial and tangential directions of functionally graded rotating disks with non-uniform thickness. The Tamura-Tomoto-Ozawa model is implemented to calculate the yield stress at a different radius of the disk. The obtained results will be verified with the higher-order finite difference method. The effect of thickness parameter, type of thermal and boundary conditions on the angular velocity and acceleration, and also the starting radius of the plastic deformations are investigated by numerical examples. It is shown that by defining the appropriate temperature gradient on the outer surface of the disk and the parameters in the time-dependent angular velocity function, the level of stresses can be controlled and optimized in all working processes.

A numerical study on MHD Casson fluid flow in a non-uniform rough channel with temperature-dependent properties using OHAM

This study explores the complex dynamics of magnetohydrodynamic (MHD) Casson fluid in a non-uniform rough channel, focusing on the effects of temperature-dependent viscosity and variable thermal conductivity under no-slip boundary conditions. The study employs an innovative approach by utilising a rough surface with irregular textures to analyse flow patterns and assess drag forces on channel objects. A novel mathematical model, governed by continuity, momentum, and heat transfer equations, is developed and transformed into dimensionless, nonlinear Ordinary Differential Equations (ODEs) using non-dimensional quantities and fundamental assumptions. The Optimal Homotopy Analysis Method (OHAM) is applied to solve these equations to enhance convergence speed and accuracy. The research explores the impact of surface roughness on velocity profiles and temperature distributions under various physical constraints. Numerical simulations are conducted to determine skin friction coefficients and Nusselt numbers. Furthermore, the study examines the influence of confined boluses on fluid flow in diverse physiological conditions. A comprehensive analysis is performed to elucidate the combined effects of surface roughness on fluid passage, including flow separation, pattern alterations, pressure distribution and drop, heat transfer characteristics, and flow resistance. The intricate interplay between temperature-dependent viscosity, varying thermal conductivity, and surface roughness is thoroughly investigated to explain the complex dynamics of MHD Casson fluid movement in non-uniform channels. Implementing a magnetic field over the rough, non-uniform channel is found to provide stability and prevent fluid overflow. This research has significant real-world applications, including soil erosion prevention, blood flow regulation in arteries, and optimisation of hydropower channels and penstocks. By enhancing our understanding of flow dynamics through rough and non-uniform channels, this study contributes valuable insights into both theoretical fluid mechanics and practical engineering applications.

Exact Solutions for Radiative Transfer with Partial Frequency Redistribution

The construction of exact solutions for radiative transfer in a plane-parallel medium has been addressed by Hemsch and Ferziger in 1972 for a partial frequency redistribution model of the formation of spectral lines consisting in a linear combination of frequency coherent and fully incoherent scattering. The method of solution is based on an eigenfunction expansion of the radiation field, leading to two singular integral equations with Cauchy or Cauchy-type kernels, that have to be solved one after the other. We reconsider this problem, using as starting point the integral formulation of the radiative transfer equation, where the terms involving the coupling between the two scattering mechanisms are clearly displayed, as well as the primary source of photons. With an inverse Laplace transform, we recover the singular integral equations previously established and, with Hilbert transforms as in the previous work, recast them as boundary value problems in the complex plane. Their solutions are presented in detail for an infinite and a semi-infinite medium. The coupling terms are carefully analyzed and consistency with either the coherent or the incoherent limit is systematically checked. We recover the important result of the previous work that an exact solution exists for an infinite medium, whereas for a semi-infinite medium, which requires the introduction of half-space auxiliary functions, the solution is given by a Fredholm integral equation to be solved numerically. The solutions of the singular integral equations are used to construct explicit expressions providing the radiation field for an arbitrary primary source and for the Green’s function. An explicit expression is given for the radiation field emerging from a semi-infinite medium.

New dimensionless equation for mass transfer at acetone evaporation under forced convection

Equations of evaporation rate were collected from literature for different liquids. These relationships directly result in the evaporation rate or the Sherwood number. Experiments were made for vaporization of acetone from a heated vessel, at different air velocities and temperatures of air and liquid surface. The effect of the different setting parameters, and the different directions of the heat flow on the evaporation rate were studied. As expected, when the air velocity and the temperature of the liquid were higher, the evaporation was faster, but when the temperature of the air was higher, the evaporation rate was lower. Based on the measurement results, a new dimensionless equation was created to determine the evaporation rate of acetone. The new equation and the correlations from the literature were compared with the results determined from the measurement data. Two of the ten correlations did not prove to be applicable to the measurement results, three correlations showed serious deviations, while five correlations gave fairly good results. Among these five correlations, the new dimensionless equation proved to be the most accurate for determining the evaporation rate of acetone in the examined range.

A TDOA sequence estimation method of underwater sound source based on hidden Markov model

To address the time difference of arrival (TDOA) estimation problem in the passive positioning system with wideband underwater motion sound sources and distributed hydrophones, a hidden Markov model-based (HMM) TDOA sequence estimation method is proposed in this paper. The method estimates the TDOA with multi-frame output of cross-correlated signals received on hydrophones. The transfer equation of the TDOA is established as a first-order hidden Markov process by analyzing the motion characteristics of the moving sound source and delays obtained from different hydrophones. Dynamic assignment of the HMM parameters is proposed to address the inconsistent change rate of the TDOA. We then achieve an HMM expression of the TDOA sequence by fitting the transfer equation and dynamic assignment of parameters into the HMM. Then, the Viterbi algorithm (VA) is applied to distinguish the optimal sequence of the TDOA among ambiguous estimations. To deal with the problem of data loss or unreliable issues caused by interferences, a data prediction algorithm which could produce possible time delays is added to VA to avoid the impact of outliers on the estimation results. By utilizing multi-frame processing, the proposed method reduces the signal-to-noise ratio (SNR) requirement of single frames since it does not require accurate estimations of TDOA for each frame. Moreover, the method adapts to a lower SNR, which has significant advantages in terms of whole sequence estimation compared with common methods. The results from the simulations and lake experiments validated the proposed TDOA sequence estimation method.

Thickness evaluation of organic coating using active long-pulse transmission thermography

The thickness of the optically translucent coating was evaluated using the transmission thermography. The transmitted temperature of a two-layer structure was theoretically analysed based on the equation of 1D heat transfer in the depth direction, and accordingly, the method for the measurement of coating thickness in a two-layer structure was established based on a long-pulse transmission thermography. The coating thickness was determined based on the characteristic time at the maximum half-rise temperature. Then, the proposed method was experimentally validated and calibrated by coating specimen with a different coating thickness. The thickness measurement method was further applied to measure an uneven coating specimen fabricated by mechanical grinding. The measurements were compared with a 3D digital image correlation method and the averaged relative error was less than 4%. Finally, thermal excitation, sampling rate of thermography and translucence of organic coating were discussed for accurate measurement.

Modeling and optimization for the continuous catalytic reforming process based on the hybrid surrogate optimization model

To address the modeling and optimization challenges of the complex reaction system in the continuous catalytic reforming process, a new integrated simulation and optimization framework is presented. First, a detailed mechanism model is established based on a reaction network involving 32 components and 50 reactions, coupled with mass transfer, heat transfer, pressure drop, and catalyst deactivation equations. Then, to solve the differential-algebraic equations in the mechanism model, a multi-objective hybrid optimization method with the adaptive infill strategy is introduced. GAMS and MATLAB are integrated to perform a joint iterative solution. Finally, two cases are conducted with the proposed algorithm. Results show that the mechanism model calculation deviations are below 4 % of reactor temperature, pressure, and composition distribution, and the Pareto front of various production plans is obtained. The accurate simulation and rapid trade-off optimization among the key goals can be achieved to provide scientific decision support for enterprise production.

Thermal modeling and simulation applied to novel microwave hybrid heating process

Owing to the unique encouraging characteristics of microwave hybrid heating, primarily volumetric heating, and additional potentials such as being repeatable, quick, economical, and green; it has been utilized in various processing techniques. The efficient joining of mild steel pipes through microwave hybrid heating in a multimode applicator at 2.45 GHz and 900 W for an operational time of 480 s has already been performed. The modeling and simulation of the process have been performed in this research paper as the numerical analysis of the working environment is crucial for evaluating various aspects of a technique, decreases process-design cycle time, and found to be more economical than experimental trials. The numerical analysis provides in-depth insight-taking into consideration of the electromagnetic field distribution, its interaction with the materials, heat generation and transfer, along with the thermal analysis of the experimental assembly, in addition to the comprehensive parametric analysis. The numerical model of the assembled set-up was developed in order to simulate a real-life heating environment by solving electromagnetic and heat transfer equations and providing analytically predicted results with an accuracy of 3.75% against the experimental results. The analytical modeling and simulation have been strategically fragmented into three phases which are pre-processing, processing, and post-processing phase and elucidated extensively, providing a systematic working of the analytical model. This research will be utilized further in optimizing the microwave hybrid heating process in order to make it time-efficient and inexpensive for its applications to industrial environments.

Model development using hybrid method for prediction of drug release from biomaterial matrix

A comprehensive multi-scale computational strategy was developed in this study based on mass transfer and machine learning for simulation of drug concentration distribution in a biomaterial matrix. The controlled release was modeled and validated via the hybrid model. Mass transfer equations along with kinetics models were solved numerically and the results were then used for machine learning models. We investigated the performance of three regression models, namely Decision Tree (DT), Random Forest (RF), and Extra Tree (ET) in predicting medicine concentration (C) based on r and z data. Hyper-parameter optimization is conducted using Glowworm Swarm Optimization (GSO). Results revealed high predictive accuracy across all models, with ET demonstrating superior performance, achieving a coefficient of determination value (R2) of 0.99854, an RMSE of 1.1446E-05, and a maximum error of 6.49087E-05. DT and RF also exhibit notable performance, with coefficients of determination equal to 0.99571 and 0.99655, respectively. These results highlight the effectiveness of ensemble tree-based methods in accurately predicting chemical concentrations, with Extra Tree (ET) Regression emerging as the most promising model for this specific dataset.

Thermal response function method: A method for predicting the transient surface temperature of black-box objects

In engineering practice, thermal analysis of objects with unknown internal structure and/or thermophysical properties, and uncertainties in contact thermal resistances, is very challenging and even impossible using the traditional approach of direct solving the heat transfer equation. In this work, a thermal response function method (TRFM) is proposed for predicting the transient surface temperature of ‘black-box’ objects (i.e., unknown internal structure and thermophysical properties). The method relies on an introduced measurable quantity called thermal response function, which characterizes the thermal response characteristics of an object. Using the measured thermal response functions as input parameters, the transient temperature distribution on the surface of a black-box object under arbitrary external heat flux boundary condition can be predicted through linear superposition. Proof-of-concept simulations and experiments are conducted to demonstrate the feasibility and effectiveness of the TRFM method. The predicted surface temperature distribution under various external heat fluxes using TRFM agree well with the reference results. The results show that the TRFM is very promising as a solution of the challenging problem of predicting the transient surface temperature of black-box objects, with potential application for thermal imaging modeling of complex objects.

Multi-physics analysis of squeezing fluid flow with rotational, magnetic field, and heat transfer interactions

This research paper presents a comprehensive study on the intricate interplay of rotational, magnetic field and heat transfer effects in squeezing fluid flow, utilizing the foundational framework of Navier–Stokes equations. Fluid flow in confined geometries has significant applications in various industrial processes, including lubrication systems, microfluidics, and heat exchangers. The integration of rotational motion, magnetic fields, and heat transfer phenomena into this context holds immense potential for optimizing performance and efficiency. The study commences with the derivation and analysis of the coupled Navier–Stokes equations, incorporating the Coriolis force due to rotational motion, the Lorentz force arising from magnetic fields, and the energy equation for heat transfer. This comprehensive model accounts for the complex interactions between these physical phenomena, offering a holistic perspective on squeezing fluid flow dynamics. Numerical analysis are conducted to investigate the behavior of fluids under the influence of rotational, magnetic field, and heat transfer effects. The results reveal intricate flow patterns, including vortex formation and heat transfer enhancement. Furthermore, the impact of key parameters such as rotational speed, magnetic field strength, and temperature gradients on flow characteristics is systematically analyzed. The Dufour effect pertains to the transfer of energy resulting from a gradient in mass concentration, occurring as a correlated consequence of irreversible processes. It functions as the inverse of the Soret effect. The temperature changes due to the concentration gradient, amplifying as the Dufour number increases. The most significant temperature rise is observed at the midpoint of the fluid domain. Conversely, the Soret number behaves in a similar but opposite manner. Tables 4–10 has been created to offer a comprehensive analysis of the optimal alignment between Homotopy Analysis Method (HAM) and BVP4c. Specifically, it demonstrates a continuous decrease in skin friction for ξSo, ξRd, and ξR2, while showing an increase for ξSc, ξDo, and ξRem. Meanwhile, the skin friction remains constant for ξPr. The research centers on analyzing heat and mass transfer properties within an incompressible fluid confined between two rotating and stretching disks amidst an unpredictable flow pattern. The study employs a blend of analytical and numerical approaches to solve the governing equations, delving into the intricate fluid dynamics and the intricate processes of heat and mass transfer within this intricate system.

A physics-guided automated machine learning approach for obtaining surface radiometric temperatures on sunny days based on UAV-derived images

Urban surface radiometric temperatures, approximate to the surface kinetic temperatures, are predominantly retrieved using satellites or unmanned aerial vehicles (UAVs) and exhibit pronounced spatiotemporal variations. Despite numerous methods ranging from empirical to physical models for obtaining urban microscale surface radiometric temperatures via UAVs, challenges remain given the limited physical significance and substantial professional barriers to method application. Against this background, this study introduces a novel and straightforward approach for acquiring spatially distributed radiometric temperatures on sunny days without understanding the complex radiative transfer process as well as acquiring low-altitude atmospheric parameters. An automated machine learning was employed to train a model capable of efficiently estimating radiometric temperatures. Training and testing datasets were created based on the urban radiative transfer equation, incorporating three independent variables: UAV-measured surface brightness temperature, broadband emissivity, and sky view factor, which collectively represent the diverse thermal environments across different surface characteristics and urban layouts during sunny transitional and summer seasons. The model's accuracy was subsequently confirmed through direct comparisons with radiometric temperatures retrieved from UAV-collected multimodal images and kinetic temperatures synchronously collected on the ground across four periods. The results indicate that AutoGluon achieved high accuracy (MAE: 0.04 K; RMSE: 0.06 K; R2: 0.99). Additional ground measurement validations further demonstrated the model's reliability, with absolute biases on sunlit surfaces maintained within 1.25 K. Given its capability for real-time, high-spatial-resolution mapping of radiometric temperatures (April test: 8.70 cm, July test: 6.89 cm) in urban microscales with considerable heterogeneity, such a method is envisioned to be an effective tool for the dynamic monitoring and management of thermal environments at the microscale level in urban settings.

Simultaneous heat, moisture, and salt transfer in porous building materials considering osmosis flow: Part 1: Theoretical modeling based on nonequilibrium thermodynamics

Salt weathering is a common deterioration phenomenon that affects outdoor cultural properties, and it is important to precisely predict the heat, moisture, and salt transfer in porous materials to suppress salt weathering. Osmosis and osmotic pressure were considered in the field of soil research, especially in clay research, but not in the field of outdoor cultural properties and building materials, which are the main target of salt weathering. Osmosis in clay is supposed to be caused by its surface charge. However, it has been suggested that sandstones and bricks that constitute cultural properties and buildings also have surface charge as clay. Thus, osmosis and osmotic pressure can occur in building materials, which may lead to materials degradation. In this study, we derive basic equations, based on nonequilibrium thermodynamics, for the simultaneous heat, dry air, water vapor, liquid water, cation, and anion transfer in building materials by considering osmosis. This equation was compared with existing model for heat and moisture transfer equations as well as models that considered the salt transfer. Based on the previous research for osmosis in clay, we summarized conditions under which osmosis occurs in building materials and presented an outlook for modeling the physical properties of materials related to osmosis.

Statistical investigation of heat transfer efficiency on a ternary nanoflow over a stretching surface

Ternary hybrid nanofluids offer significant advantages in heat transfer due to their enhanced thermal conductivity compared to traditional fluids. Investigating their flow behavior over expanding surfaces can contribute to the development of more efficient heat exchangers and cooling systems in the electronics, energy production, and transportation industries. This paper presents a detailed analysis of the flow and thermal transfer characteristics of a ternary hybrid nanofluid (SiO2–Cu–Al2O3/H2O) over a radially stretching surface, utilizing both numerical techniques (Keller Box method) and statistical techniques. Furthermore, the model includes factors like convective heat transport, radiation, internal fluid friction, and suction. Similarity transformations transform the governing equations for fluid flow and heat transfer into a system of nonlinear ordinary differential equations. Then the equations are solved numerically using the Keller Box method with the aid of MATLAB software. Additionally, we have employed statistical techniques such as correlation analysis, probable error estimation, and multivariate regression to validate and ensure the accuracy of the numerical outcomes. Parameter ranges for Biot number, radiation, unsteadiness, Eckert number, and energy generation parameter are 0.1 ≤ Mp ≤ 0.7, 0.1 ≤ Ks ≤ 1.5, 0.2 ≤ Ps ≤ 0.8, 0.1 ≤ EN ≤ 0.4, 0.5 ≤ RN ≤ 2, 1≤ BT ≤ 2.5, 0.1 ≤ Hsc ≤ 0.7 and found to significantly impact heat transfer behavior within these bounds. This study utilizes 3D surface plots to visualize the influence of key parameters on engineering quantities. R squared values show that the data strongly match the regression model. The findings demonstrate excellent concordance between the predicted values and the actual Nusselt number and skin friction measurements. Biot number has a very strong correlation with heat transfer, with minimal chance of error. And the same is evidenced by the correlation matrix and multiple regression analysis. Eckert number also exhibits a negative correlation with the Nusselt number. This translates to a situation where increasing Eckert number directly leads to a decrease in the rate of heat transfer, with no margin for error. Energy generation parameter demonstrates a negative correlation with heat transfer. However, there's a slight possibility of error, with an estimated error percentage of around 0.0017.

Reconstruction of infrared absorption and scattering coefficient distributions of semitransparent medium using Adam algorithm combined with adjoint model

Precise reconstruction of infrared optical coefficient distributions within semitransparent medium holds great significance in engineering field. This study introduces the Adam algorithm to reconstruct the infrared absorption and scattering coefficient distributions of a 2D semitransparent slab. The discrete ordinate method (DOM) is utilized to solve the time-domain radiative transfer equation (TD-RTE), and the radiative intensity at boundary is employed as input signal. The gradient calculation of the objective function employs the adjoint equation method (AEM). Results demonstrate that the adaptive moment estimation (Adam) algorithm can simultaneously estimate the infrared absorption and scattering coefficient distributions with high accuracy. Compared to traditional optimization methods, the Adam algorithm demonstrates significant superiority in terms of reconstruction accuracy, computational efficiency, and tolerance to measurement errors. Furthermore, it is also appropriate for determining the infrared optical parameters of different internal structures. Balancing both calculation accuracy and computation time, a recommended learning rate of η = 50 is suggested for the reconstruction task. The Adam algorithm presents limitations for reconstructing the distributions in scattering-dominant medium. The normalized root-mean-square errors of the absorption and scattering coefficients respectively increase to 0.2311 and 0.0655 when the standard measurement error is increased to 20 %.

Two-phase flow simulation algorithm for numerical estimation of relative phase permeability curves of porous materials

Abstract The paper presents an algorithm for three-dimensional modelling of two-phase flows on the scale of pore size order for numerical evaluation of relative phase permeability curves of porous materials. Such an evaluation is performed based on the results of numerical simulation of primary drainage with subsequent waterflooding. In this case, models of porous materials based on three-dimensional tomographic images of rocks are used. The simulation of the flow considers the Stokes equation and the Cahn–Hilliard equation for modelling phase transfer, which allows us to determine phases using the concentration function. The combination of the phase field method and finite difference method makes it possible to correctly take into account the contact angle and stably calculate surface tension forces in domains with complex topology.

Impact of Multi-Scattered LiDAR Returns in Fog.

In the context of autonomous driving, the augmentation of existing data through simulations provides an elegant solution to the challenge of capturing the full range of adverse weather conditions in training datasets. However, existing physics-based augmentation models typically rely on single scattering approximations to predict light propagation under unfavorable conditions, such as fog. This can prevent the reproduction of important signal characteristics encountered in a real-world environment. Consequently, in this work, Monte Carlo simulations are employed to assess the relevance of multiple-scattered light to the detected LiDAR signal in different types of fog, with scattering phase functions calculated from Mie theory considering real particle size distributions. Bidirectional path tracing is used within the self-developed GPU-accelerated Monte Carlo software to compensate for the unfavorable photon statistics associated with the limited detection aperture of the LiDAR geometry. To validate the Monte Carlo software, an analytical solution of the radiative transfer equation for the time-resolved radiance in terms of scattering orders is derived, thereby providing an explicit representation of the double-scattered contributions. The results of the simulations demonstrate that the shape of the detected signal can be significantly impacted by multiple-scattered light, depending on LiDAR geometry and visibility. In particular, double-scattered light can dominate the overall signal at low visibilities. This indicates that considering higher scattering orders is essential for improving AI-based perception models.

Online Portfolio Selection of Fuzzy Mean Regression Strategy Considering Investor Sentiment Based on Text Data

Investors are often affected by emotion, cognition, and other psychological factors in stock trading when making decisions. At present, people can use machine learning and other technologies to obtain a massive amount of text data from the Internet to mine information related to investor behavior and sentiment. Building intelligent online portfolio trading strategies that consider investor sentiment has become an important topic and key challenge in the financial field. Therefore, this paper explores how to use text data to depict investor sentiment, fuzzifies historical stock price data, designs a new weight transfer equation, and finally obtains a novel fuzzy mean regression strategy that considers investor sentiment based on text data. We conduct empirical tests on this strategy by using the stock price data selected from CSI300 constituent stocks, as well as the text data of investors’ opinions on the internet. The results show that the strategy proposed in this study has a higher Calmar ratio than other mean regression strategies previously studied.

Analytical solution of three-dimensional temperature field for skin tissue considering blood perfusion rates under laser irradiation and thermal damage analysis

An analytical solution of the three-dimensional transient temperature distribution for skin tissue when heated by a laser heat source is presented in this work. An equation for the three-dimensional bioheat transfer of skin tissue is obtained firstly, and the transient temperature value at any point inside the skin tissue can be solved using the separation variable method and the Newton-Cotes method. Previous research primarily focused on semi-infinite domains, while this work presents an analytical solution for finite domains. A comparison of present analytical solution with simulation software results and numerical solution is given to demonstrate the feasibility of the analytical solution. The three-dimensional temperature distribution of skin tissue irradiated by a laser is theoretically studied. The extent of thermal damage to skin tissue is assessed by substituting the analytical solution of the temperature field for laser-irradiated skin tissue into the thermal damage equation. The calculations in this work reveal that the variation of temperature field in biological tissues is dependent not only on the laser incident light intensity and the optical parameters of the tissues but also on the heat transfer properties of the biological tissues, such as the blood perfusion rates.