在本文中經由建立適當的統御方程式、邊界條件及初始條件，以拉氏轉換法(Laplace transform method)，求得一半解析—半數值解（analytical-numerical solution），再配合黎曼和近似法(Riemann-sum approximation method)，探討各種拋物線型、雙曲線型及修正拋物線型擴散問題，諸如無限熱傳速度之傅立葉(Fourier)熱傳導問題、有限熱傳速度的非傅立葉(non-Fourier)及修正的非傅立葉(modified non-Fourier)熱傳導問題。文中所使用之半解析—半數值法，其解題步驟乃是先以拉氏轉換法，將統御方程式、邊界條件及初始條件中的時間項移除後，求得以拉氏轉換變數 為參數的半解析解，再配合黎曼和近似法撰寫程式，將半解析解轉換成以物理時間 為參數的數值解。由於一般以數值方法解有限熱傳速度的問題時，在波前(wavefront)附近常會發生劇烈數值振盪(numerical oscillation)的現象，若採用本文所使用之方法，並取適當的收斂條件(相對誤差設為 )，則可將數值振盪的情形消除。
　　
　　在本文中首先針對一微小球體材質，兩側受到雷射光加熱時，依材質之熱傳導係數或吸收率（absorptivity）大小，採表面均勻加熱模式（spatially uniform surface-heating model）或表面非均勻加熱模式(spatially non-uniform surface-heating model)，對Biot數、入射雷射光源強度、球體半徑及極角(polar angle)做參數探討，求得微小球體材質溫度場分佈之解析—數值解，並與實驗值加以比較，結果發現非常吻合。
　　
　　在傳統的熱擴散理論中，傅立葉定律假設熱量傳遞速度為無窮大，然而當具有很大的溫度梯度、溫度趨近於絕對零度時、極短時間的熱變化或極高的熱通量等情況出現時，此一假設必須加以修正。目前所使用的修正模式主要有雙曲線型熱傳方程式(hyperbolic heat conduction equation)、雙相延遲模式(dual-phase-lag)及修正拋物線型熱波方程式(modified parabolic thermal wave equation)。
　　
　　由於有限熱傳速度的非傅立葉效應僅發生於極短時間內，該效應會隋時間增加而遞減，因此在本文中分別以傅立葉定律、雙曲線型熱傳方程式及修正拋物線型熱波方程式對球體及平板分析所得之結果，其差異將會隨時間的增長而消失。當材質之鬆弛時間（relaxation time）愈小、熱擴散係數（thermal diffusivity）愈大，則熱波傳遞速度愈快，且達到穩定狀態所需的時間愈短，熱傳導係數（thermal conductivity）對熱波傳遞速度大小則沒有影響。若以雙曲線型熱傳方程式分析問題，當熱波到達邊界或不同材質交界面（interface）時，熱波將一分為二：傳遞波（transmitted waves）及反射波（reflected waves），因此其溫度場分佈變化極為劇烈，導致可能會發生材質內部溫度高於邊界作用溫度之情形；而以修正拋物線型熱波方程式分析問題時，熱波不會反射及互相干擾，其溫度場分佈變化較為平緩，且不會發生材質內部溫度高於邊界溫度之情形。
　　
　　隨著時代的進步、科技的發展，短時間的脈衝雷射光（pulsed laser beams）可應用在焊接（welding）、切削、表面處理、表面清潔及加熱生物組織（heating biological tissues）等領域。雷射光源可為連續或脈衝的、固定或移動的、及點、線或面的加熱源。當以一移動雷射光束加熱薄棒時，分別採用具有有限熱波傳遞速度之雙曲線型及修正拋物線型熱波方程式加以分析，結果發現影響其溫度分佈的主要因素為雷射光源強度、移動速度、熱波傳遞速度及材質鬆弛時間，而對流係數僅在長時間才有效果，且當雷射光源移動速度及熱波傳遞速度大小相近時，將會在薄棒內產生較高的溫度場。以雙曲線型熱傳方程式分析所得之溫度場，將高於傅立葉定律或修正拋物線型熱波方程式所得之結果，且溫度場在波前附近變化更為劇烈。

　The analytical-numerical technique, which is based on the Laplace transformation and the Riemann-sum approximation, is employed to predict the temperature and heat flux histories in the materials of Fourier, non-Fourier, and modified non-Fourier heat conduction problems. The whole solution processes for solving temperature or heat flux histories in the materials are set and nondimensionalized the appropriate governing equations, initial conditions, and boundary conditions for the analyzed problems first of all. Then the Laplace transform technique is employed to deal with the dimensionless time-derivative terms in the governing equations, initial conditions, and boundary conditions. When using numerical method to solve hyperbolic heat conduction equation, how to suppress the numerical oscillation in the vicinity of the wavefront is the major difficulty. In the present study, the convergence criterion is set as to avoid the numerical oscillation.

　The dynamic thermal behavior of a micro-spherical particle due to pulsed laser heating source is investigated and compared with the experimental data. According to the values of thermal diffusivity and absorptivity of the micro-spherical particle, the spatially uniform heating and non-uniform heating of the surface model are used to predict the surface temperature histories of the micro-spherical particles, respectively. The effects of different parameters such as the Biot number, the intensities of the incident laser beams, the radii of micro-spherical particles, and the polar angles are studied and presented. Good agreements are found between the analytical-numerical solutions and experimental data.

　In the classical theory of diffusion, Fourier law of heat conduction is used to describe the relation between the heat flux vector and the temperature gradient and assumed that the heat propagation speeds are infinite. When the heat transfer situations include extremely high temperature gradients, temperatures near absolute zero, extremely large heat fluxes, and extremely short transient duration, the heat propagation speeds are finite, and the Fourier law should be modified, the modified models include the thermal wave model (or hyperbolic heat conduction equation, HHCE), the phase-lag concept, the dual-phase-lag (DPL), and modified parabolic thermal wave equation (MPTWE). The results obtained using the hyperbolic heat conduction equation, and modified parabolic thermal wave equation are compared with the classical Fourier law, which is the parabolic heat conduction equation. Although the thermal waves can be split into two waves, transmitted and reflected waves, respectively, and travel in the opposing direction under hyperbolic heat conduction equation, but the modified parabolic thermal wave equation rejects interference and reflection of thermal wave. The speeds of heat propagation are finite, as revealed in the temperature and heat flux calculated by using the hyperbolic heat conduction equation and modified parabolic thermal wave equation. The smaller the dimensionless relaxation time is or the larger the thermal diffusivity is, the larger the thermal wave propagation speed is. The effect of thermal conductivity on thermal wave propagation speed is negligible.

　The results obtained using modified parabolic thermal wave equation and hyperbolic heat conduction equation, the heat propagation speeds are finite, while using Fourier law the heat propagation speeds are infinite. The solutions obtained by modified parabolic thermal wave equation and Fourier law are always consistent with the second law of thermodynamics, while by hyperbolic heat conduction equation violate the second law of thermodynamics sometimes. When the relaxation time is close to zero or the system reaches steady state, all the solutions obtained under modified parabolic thermal wave equation, hyperbolic heat conduction equation, and Fourier law are the same.

　The lasers are widely used as a welding, cutting, surface treatment, surface cleaning, or heating biological tissues tool. The lasers can be considered as continuous or pulsed, stationary or moving, and point, line or surface heat sources. The affections of heat conduction in the rod with finite thermal wave propagation speed on the dimensionless temperatures distribution predicted using hyperbolic heat conduction equation, and modified parabolic thermal wave equation are investigated and presented in the present paper. The effects of different parameters such as the dimensionless relaxation time , moving heat source speeds , dimensionless heat convective losses , and the dimensionless strength of the heating source term are also analyzed and presented. The temperature profiles in the rod predicted by modified parabolic thermal wave equation and Fourier law are smooth than by hyperbolic heat conduction equation. The temperature profiles obtained using hyperbolic heat conduction equation are significantly steep in the vicinity of the thermal wave front. The temperature profiles reveal the wave nature of heat propagation under hyperbolic heat conduction equation and modified parabolic thermal wave equation, and the thermal wave propagation speeds are finite.

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