[試題] 101上 黃美嬌 熱傳學 第一次期中考
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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