應用熱波理論探討二維非等向性介質之暫態熱傳現象

Abstract

傳統傅立葉熱傳導理論，假設熱通量與溫度梯度之間直接成比例，這意味著熱的傳播速率為無窮大，亦表示局部的溫度改變將引起介質內任一點同時發生溫度的變化，即使距離為無限遠處。雖然熱的傳播速率為無窮大是不合乎實際的物理現象，然而對大部份的工程應用，傅立葉熱傳導理論仍然可合理的解釋絕大多數熱傳現象。但是在許多工程應用中，當溫度非常接近絕對零度、或產生極大的溫度梯度與極短的時間等情況，熱是以波的形式來傳遞，其速度為有限值，且對整個系統的熱傳具有重大的影響，因此傅立葉熱傳導理論必須做適度的修正。 為了探討熱波傳播速度為有限值，熱傳導修正方程式 q ( r, t+tau ) = - k defT( r, t) 此意指熱通量及溫度梯度間允許時間延遲。事實上，延遲時間常數 與質子碰撞熱流發生時間有關，同時也是介質本身熱慣性的測量值。 本論文的主題是藉著應用熱波理論來討論二維暫態的熱傳導現象。本論文分別討論兩種邊界條件：等溫邊界及絕熱邊界。此外也探討物質的非等向性對熱傳之影響，討論的例子包括傳導係數比 (K= 1, 4 )及時間延遲常數比 (tau = 1, 2, 4 )。本文使用格林函數 (Green function)方法解答上述邊界值問題。由研究結果顯示二維熱波傳播時熱波波前溫度將會急驟變化及於尾端產生相反方向之波，同時波受到有限區之熱及邊界情況等交互作用及反射現象將更為複雜。

In classical Fourier heat conduction theory, heat flux is postulated to be directly proportional to temperature gradient. That implies an infinite speed of propagation of the thermal wave, indicting that a local change in temperature cause an instantaneous perturbation in the temperature at each point in the medium, even if the intervening distances are infinitely large. Although an infinite speed of heat propagation is nonphysical, for most engineering applications, this approximation is quite acceptable. However, in situations dealings with very low temperature near absolute zero, an extremely short transient duration, and an extremely high rate change of temperature or heat flux. The Fourier equation breaks down because the heat propagation velocity in such situations becomes finite and dominant. To consider the finite speed of wave propagation, a damped-wave model has been proposed that uses a variety of reasoning and derivations. Researcher suggested a modified heat flux model of the form q ( r, t+tau ) = - k defT ( r, t) This means that the heat wave model allows a time lag between the heat flux and the temperature gradient. In fact, the relaxation time is associated with the communication ‘time’ between phonons (phonon-phonon collisions) necessary for commencement of heat flow and is a measurement of the thermal inertia of a medium. The objective of this work is to discuss the two-dimensional heat conduction phenomenon by applying the thermal wave theory. Two kinds of boundary conditions; either a constant wall temperature or adiabatic boundary condition, were considered. In addition, the influence of thermal conductivity ratio K and the relaxation time ratio(tau) on the heat transfer phenomenon were examined in detail. In this work, the Green’s function technique is adopted to solve the above boundary value problem. The results show that the disturbance induces a severe thermal wave front that traverses the medium with a sharp peak at the leading edge and generated a negative trailer that follows behind the wave front. Moreover, the reflection and interaction of thermal waves are complicated by two factors, the finite area of the thermal disturbance and the boundary conditions.