Chapter 10. Heat Transfer by Conduction
Three heat flow mechanisms:
Conduction (전도)
Convection (대류) natural convection (자연대류) Radiation(복사) forced convection (강제대류)
Basic Law of Conduction
* Fourier’s law
dx k dT dA
dq =−
q : rate of heat flow A: surface area T: temperature x : distance normal to surface k : thermal conductivity
--- Eq. (10.1)
General expressions of Fourier’s law in all three directions:
T z k
T y
T x
k T dA
dq ⎟⎟= − ∇
⎠
⎜⎜ ⎞
⎝
⎛
∂ + ∂
∂ + ∂
∂
− ∂
=
* Thermal conductivity (열전도도) k
dy µ du τ = − Fourier’s law에서의 비례상수
Å Newton’s law에서의 점도에 해당
Rate of heat flow (열흐름속도) q 의 단위: W or Btu/h dT/dx 의 단위: oC/m or oF/ft
∴ 열전도도 k 의 단위: W/m·oC or Btu/ft·h·oF For small ranges of T, k = constant
For larger T ranges, k = a + bT
solids: 17 W/m·oC for stainless steel 415 W/m·oC for silver
k values 0.35 W/m·oC for glass liquids: 0.5 W/m·oC for water gases: 0.024 W/m·oC for air
Solid having low k Æ “insulators” ex) polystyrene foam dx
k dT dA
dq =−
Steady-State Conduction ( 정상상태 열전도 )
Å neither accumulation nor depletion of heat within the slab q is constant along the path of heat flow
Heat flow into the tank Heat flow from the tank
B k T x x
T kT
dx k dT A
q = ∆
−
= −
−
=
1 2
1
2 --- Eq. (10.7) x: distance from hot side B: thickness of slab
R T A
q = ∆
thermal resistance (B/k) 1/R : conductance (k/B) cf.)
Ex. 10.1) A layer of pulverized cork (insulator)
0.152 m
T2 = 4.4 oC T1 = 82.2 oC
q = ?
k : 0.036 W/moC at 0 oC 0.055 W/moC at 93.3 oC A = 2.32 m2
q (the rate of heat flow) ?
W 3 . 53
152 . 0
8 . 77 0
3 . 93
036 . 0 055 . 3 0 . 43 036 . 0 32 . 2
=
⎟×
⎠
⎜ ⎞
⎝
⎛
−
× − +
×
∆ =
= B
k T A q
* Compound resistances in series
In steady heat flow,
C B
A q q
q = =
C B
A T T
T
T = ∆ +∆ +∆
∆
A k q B
A A A
A k q B
B
B B k A
q B
C C C
R T R
R R
T k
B k
B k
B
T A
q
C B
A C
C B
B A
A
= ∆ +
+
= ∆ +
+
= ∆
∴ / / /
&
C C B
B A
A
R T R
T R
T R
T = ∆ = ∆ = ∆
∆ A
q A q A q A
qA = B = C = ⇒
Ex. 10.2) A flat furnace wall constructed of a layer of Sil-o-cel brick backed by a common brick
BB = 0.229 m
T2 = 76.6 oC T1 = 760 oC
BA = 0.114 m Sil-o-cel
brick
common brick
Tx
kA = 0.138 W/m oC kB= 1.38 W/m oC kA kB
(a) Heat loss through the wall, q = ? Assume A = 1 m2
W 81 . 693 693.81W/m
5 683.4/0.98
C 683.4 C/W
m 985 . 0
159 . 0 826
. 0
2
o o
2
=
=
=
∴
=
∆
= +
=
=
=
=
=
q q/A
T R
R R
k R B
k R B
B A
B B B
A A A
(b) Temperature of the interface between the two bricks
C 186.9 C
08 . 573 826
. 0 / 985
. 0 / 4 . 683 /
/ = ∆ = ∆ ∆ = o ∴ = 1 ∆ = o
∆T R TA RA TA TA Tx T- TA
(c) In case that the contact between the two bricks is poor and the contact resistance is 0.088 m2 oC/W, the heat loss q = ?
W 9 . 636 /
C/W m
073 . 1 088 . 0 985 .
0 + = 2 o ∴ = ∆ =
= q T R
R
* Heat flow through a cylinder
Logarithmic mean radius vs. arithmetic mean radius cylinder length: L
dr
ro ri
) 2 ( rL dr
kdT
q = − π
i o
o i L
i o
o T i
T r
r
r r
T T A k
r r
T T L q k
q dT Lk r
dr o
i o
i
−
= −
= −
∫ ⇒
= −
∫
) (
) / ( ln
) )(
2 (
2π π
L r AL = 2π L
) / ( ln o i
i L o
r r
r r = r −
: logarithmic mean radius (로그평균반지름)
rL ra
Ex. 10.3) A tube of 60 mm OD insulated with a 50 mm silica foam layer and a 40 mm cork layer Calculate the heat loss q of pipe in W/m ?
For silica layer, kA = 0.055 W/m oC
kB= 0.05 W/m oC 50 mm
30 mm
40 mm
silica cork 150 oC
30 oC layer B
layer A
) / ( ln o i
i L o
r r
r r = r −
) 30 / 80 ( ln
30 80−
L = r
For cork layer,
) 80 / 120 ( ln
80 120−
L = r
) (
7748 . 0
) (
3522 . 0
2 ) 2
( )
(
o x B
x i A
L B
L A
B o x B B B
A x i A A A
T T L q
T T L q
L r A
L r x A
T T A q k
x T T A q k
−
=
−
=
⇒
=
− =
− =
= π π
At steady state, q = qA = qB ∴ q/L = 29.1W
Unsteady-State Conduction ( 비정상상태 열전도 )
Heat input – Heat output = Accumulation of heat
2 2
x T t
T
∂
= ∂
∂
⇒ ∂
α
thermal diffusivity (열확산계수) 단위: [m2/s]
cf.) D : mass diffusivity : kinematic viscosity
Solutions are available for certain simple shapes, such as infinite slab (Eq. 10.20), infinitely long cylinder (Eq. 10.21) and sphere (Eq. 10.22).
Æ Fig. 10.5
Cp
k
= ρ
ν
Unsteady-state conduction in solid slab
Fig. 10.5. Average temperature during unsteady-state hearing or cooling of a large slab, an infinitely long cylinder, or a sphere.
