課程名稱︰熱傳學 課程性質︰必修 課程教師︰黃美嬌 開課學院：工學院 開課系所︰機械系 考試日期（年月日）︰101/10/19 考試時限（分鐘）：180 是否需發放獎勵金：Yes! （如未明確表示，則不予發放） 試題 : Heat Transfer First Exam 19/10/101 Problem 1 (20%): Answer the following questions in brief but clearly in CHINESE: (a) It is claimed that the temperature profile in a medium must be perpendicular to an insulated surfave. Is this a valid claim? Explain. (b) What is the thermal contact resistance? (c) How should the convection and the radiation resistances at a surface be connected, in parallel or in series? Explain. (d) Consider a cylindrical pipe of radius r1 which is insulated with some material. Sketch and explain the variation of the rate of heat transfer from the insulated pipe to the surrounding air against the radius of the insulated pipe r2. Problem 2 (40%): The diagram shows a conical section fabricated from pure aluminum. It is of circular cross section having diameter D = a x^1/2. The small end is located at x1 and the large end at x2. The end temperatures are T1 and T2, while the lateral surface is well insulated. (a) From the principle of energy conservation, derive the governing equation for the temperature distribution, assuming 1D steady and without any additional heat generation. (b) Solve (a) to find an expression for the temperature distribution T(x). Calculate the total heat transfer rate Q-dot. (c) In class, we have shown that the thermal conduction resistance for a slab of wall having a thickness of L, a cross-sectional area A, and a conductivity k is Rcond = L / kA. Consider the conical section as a combination of many pieces of slabs, each of which has a cross-sectional area A(x) and a thickness dx, its thermal resistance is thus dRcond = dx / kA(x). The total thermal resistance is thus the effective thermal resistance, Rcond, of these pieces of slabs. Using the limiting case Σdx = ∫dx as dx→0, find the effective thermal resistance Rcond. (d) Find the total heat transfer rate Q-dot using the result of (c) and compare it with your answer in (b). Are they the same? ＿＿＿＿ ／ ＿＿ ＼dr T1▁▃▅▆▇█ T2 ／ ／ ＼↗＼ A(r) = 4πr^2 ｜ ／ r↗＼ ｜ ─‧─‧─‧─‧─‧─ ｜｜ ／ ｜｜ dV = A(r)dr ｜｜ ｜｜ │x1 ▇▅▃▂▁ ｜ ＼ ／ ｜ ├─→│ x2 │ ＼ ＼ ／ ／ ├───────→│ ＼ ￣￣ ／ ￣￣￣￣ Problem 3 (40%): Steel balls of diameter D are annealed by heating to TH and then slowly cooling to a surface temperature of TL in an air environment for which has a temperature T∞ and a convectoin heat transfer coefficient h. Assume the properties of the steel, namely the thermal conductivity k, density ρ, and specific heat c, are all constant. (a) Write down the definition of Biot number and explain its physical meaning. What is the requirement for assuming a "lumped" system? (b) Assuming a lumped system, find the time required for the cooling process. (c) Now consider the spatial effect. From the principle of energy conservation, taking a spherical shell as your control volume as shown above, derive the proper governing equation for the evolution of the dimensionless temperature Θ(r*,t*) ≡ (T(r,t) - T∞)/(TH - T∞) of the steel balls T(r,t), where r* ≡ r/D and t* ≡ αt/D^2. Also write down the initial condition and boundary conditions required for solving this problem. (d) Using the method of separation of variables, i.e. assuming Θ(r*,t*) = F(r*)G(t*), find the associated ordinary differential equations for F(r*) and G(t*). Also write down the boundary conditions required for solving F(r*) in the present problem. (Do not solve.) --

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