本文應用線性最小均方根誤差法(linear least-squares error method)來求解有限加熱/冷卻區段之層流與紊流管流的穩態共軛熱傳逆向問題(Inverse Heat Transfer Problem, IHTP)，分析過程並不忽略管壁的熱傳導作用。藉由流體內的少數量測點來估測外管壁上之溫度、熱通量或者隔熱層的不規則幾何邊界形狀。並且，結合函數指定法(Function Specification Method, FSM)於管流的暫態共軛熱傳逆算問題，估測隨時間而變的入口流溫度與外管壁熱通量。
　　首先，使用有限差分法(finite-difference method)將欲求解之管流穩態熱傳逆問題的統御方程式離散化(discretization)，以建構一線性矩陣方程式。藉由排列逆問題之矩陣方程式，使未知狀態(如邊界條件、初始條件、熱傳係數等)可用一行矩陣(column matrix)明確地表示出來；再以有限點之溫度量測資料代入此線性模型中，利用線性最小均方根誤差法將問題最佳化以求解此逆模型。為增進管流暫態熱傳逆問題解析運算之穩定與準確性，在排列出模型的矩陣方程式後，合併導入未來時間原理，並指定待估測值的未來時間函數，最後再以線性最小均方根誤差法求得穩定、準確之估測值。
　　本文中利用直接問題(direct problem)所求得的溫度值來模擬實際的溫度量測，並考慮量測誤差大小對於估測結果的影響；也將探討量測位置與量測點數目左右估測結果之準確度的程度。結果顯示量測點位置愈靠近待測標的，其準確度愈高；也確認本逆算法於量測誤差為3%時，仍可求得令人滿意的估測結果。
　　本方法在分析物理問題時，可以避開以往傳統方法所須利用的迭代程序，也因此不需初始猜測值；僅利用一次的運算程序，即可直接解出未知條件，所以計算上較一般傳統方法快速與精確。
　　文中分別使用循次函數指定法(Sequential Function Specification Method, SFSM)與本文所使用的全域函數指定法(Whole Domain Function Specification Method, WDFSM)來分析層流管流的暫態共軛熱傳逆算問題，以比較其優缺點；並且於全域法中分別比較兩種未來時間之未知狀態元素的典型假設(常數與線性)對於估測結果的影響。最後，本文並提出一個簡單函數修正法來增進估測值的準確度。本文所提的逆向矩陣法合併函數指定法可應用於一維、二維甚至三維之問題，故此方法可成為研究逆問題之一有效方法。

This study addresses the inverse problem of a finite cooled/heated length on the heat transfer characteristics of laminar or turbulent flows through thick-walled circular tubes. Using temperature measurements taken at several different locations within the fluid, the linear least-squares-error method is used to estimate the unknown heat flux on the external surface of the circular pipe and the unknown shape of the thermal insulation. Furthermore, the function specification method is used to estimate the time-varying inlet temperature and the outer-wall heat flux simultaneously on the basis of temperature measurements taken at two different locations within the pipe flow.
While determining the steady unknown boundary conditions of the pipe flow, the present approach in this study rearranges the matrix forms of the governing differential equations, and then combines the reverse matrix method and the linear least-squares-error method. Another study about a transient inverse heat transfer problem is solved using a whole domain estimated technique with the function specification method and the linear least-squares-error method to determine the unknown boundary conditions of the pipe flow.
The temperature data obtained from the direct problem are used to simulate the temperature measurement, and the influence of errors in these measurements upon the precision of the estimated results is considered. This study also considers the influence of the locations and numbers of sensors used upon the accuracy of the estimated results. The results indicate that the accuracy of the estimated results is improved by taking temperature measurements in locations close to the unknowns. The results confirm that the proposed methods are capable of yielding accurate results even when errors in the temperature measurements are present.
The proposed methods provide several advantages compared to traditional methods: (1) it yields a solution within a single computational iteration, (2) no prior information is required regarding the functional form of the quantities of interest, (3) no initial guesses of the unknown parameter values are required, and (4) the inverse problem can be solved in a linear domain.
This study also compares the application of the whole domain function specification method (WDFSM) and the sequential function specification method (SFSM) to the inverse problem of transient conjugate heat transfer of laminar forced convection in a circular pipe. These two inverse methods are used to estimate the time-varying inlet temperature and the outer-wall heat flux simultaneously on the basis of temperature measurements taken at two different locations within the pipe flow. The numerical results reveal that the estimations obtained from the WDFSM method are marginally better than those obtained from the SFSM approach.
Finally, the performance of two classical algorithms (e.g. uniform and linear function) used in the whole domain function specification method (WDFSM) to obtain simultaneous estimates of the time-varying inlet temperature and outer-wall heat flux are compared. Additionally, this study proposes a modification to the linear assumption employed in the conventional WDFSM method to improve its estimation performance. The numerical results confirm that the proposed algorithm yields slightly more accurate estimates of the unknowns than the two classic algorithms.

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