推 hmonkey:感謝! 02/17 18:44
課程名稱︰熱傳學
課程性質︰機械系大三上必修
課程教師︰陳希立
開課學院：工學院
開課系所︰機械系
考試日期（年月日）︰2006.11.17
考試時限（分鐘）：
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
1. (10％) Steady state temperature measurements made by an array of
thermocouples inserted in a one-dimensional homogeneous solid slad in the
d dT
direction of heat flow (x) indicate that ─（──）＜0. There is no heat
dx dx
generation or sink in the medium. What conclusion can be drawn from these
experimental observations?
2. (15％) A long homogenerous resistance wire of radius ｒ0 with thermal
conductivity ｋ is being used to boil water at a uniform temperature Ｔ∞,
as shown in (Fig-1). Heat is generated in the wire uniformly at a rate of
‧ 3
ｇ(ｗ/ｃｍ ). If the outer convective boiling heat transfer coefficient ｈ
is very large. Obtion a relation for the temperature distribution, and
determine the temperture at the centerline of the wire when steady
operating conditions are reached.
１ δ δＴ １ δ δＴ δ δＴ ‧ δＴ
Note: ─ ──(ｋｒ──)＋── ──(ｋｒ──)＋──(ｋ──)＋ｇ＝ρＣ──
ｒ δｒ δｒ ｒ^2 δφ δφ δｚ δｚ δｔ
│
│～．～│
Water │ │ │←Ｔ∞
│ │ │
│ ． │
│ 0├─┼──→ｒ
│．│ │ｒ0
│ｇ． │
│ │ │
│～│～│ Fig-1
│
3. (20％) A very thin, square plate (side = L, thickness = t, Fig-2) protudes
from a wall. The temperture of the wall is Ｔw, and the temperture of the
fluid is Ｔa. The convective heat transfer coefficient at the upper surface
of the plate is ｈ1(ｘ,ｙ), while the coefficient at the lower surface of
the plate is ｈ2(ｘ,ｙ). The heat loss from the edges of the plate can be
neglected. Assume steady two-dimensional conditions in the plate and write
the differential equation which governs the plate temperture distribution.
State the boundary conditions.
◢
◢█
◢██
◢███
◢██◤ｙ██████◤
██◤↗██████◤
█◤ ─→ｘ███◤
█████
████◤
███◤
██◤
█◤ Fig-2
◤
4. (10％) Consider a plane wall of thickness ２Ｌ, thermal conductivity ｋ,
initially at a uniform temperture Ｔi. At time t=0, it is placed a large
medium that is at a constant temperture Ｔ∞ and kept in that medium for
t＞0. Heat transfer takes place by convection with a constant heat transfer
coefficient ｈ. Sketch the temperture profiles of plane wall for any time
ｔ at limiting conditions of ｈＬ/ｋ＞＞0 and ｈＬ/ｋ＜＜1.
5. (20％) Consider the semi-infinite solid maintained at some initial
temperture Ｔi. The surface temperture is suddenly lower and maintained at
a temperture Ｔo for t＞0.
(a) Write the energy equation and associated initial and boundary conditions.
Ｔ －Ｔi ｘ
(b) Let θ = ──── and ξ = ────. Transform the equation at part (a)
Ｔo－Ｔi ２√αｔ
to ordinary differential equation.
(c) Show that the temperture profile θ can be expressed as θ = erfc(ξ),
where erfc(ξ) is the complementary error function, defined as
２ ξ -ｕ^2 ∞ -ｕ^2 √π
erfc(ξ)＝1－──∫ e ｄｕ, note that ∫ e ｄｕ＝──.
√π 0 0 ２
ｋＡ(Ｔ0－Ｔi)
(d) Prove that at the surface (x=0) the heat flow is ｑ0 = ───────.
√παｔ
(e) Under what conditions can plane wall be treated as a semi-infinite
medium?
2 。 -6 2
6. (25％) A short brass (ｋ= 50 Ｗ/ｍ．Ｃ, α= 33 .9×10 ｍ/ｓ) cylinder of
diameter Ｄ＝10 cm and hight Ｈ＝12 cm is initially at a uniform temperture
。 。
Ｔi = 120 Ｃ. The cylinder is now placed in atmospheric air at 25 Ｃ, where
heat transfer takes place by convection, with a heat transfer coefficient
2 。
of ｈ= 1200 Ｗ/ｍ．Ｃ. Calculate the temperture at
(a) the center of the cylinder and
(b) the center of the top surface of the cylinder 2.5 min after the start of
the cooling.
│
┌──┼──┐──┬
│ │ │ ↑
│ ‧ │
。 │ │ │ Ｌ
Ｔ∞＝25Ｃ │ ｘ↑ │
│ │ │ ↓
│ 0├─→┤──┼
│ │ ｒ │ｒ0 ↑
│ ‧ │
│ │ 。 │ Ｌ
│Ｔi＝120Ｃ│
│ │ │ ↓
└──┼──┘──┴ Fig-3
│
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