heat transfer with internal heat...
TRANSCRIPT
Heat Transfer with Internal Heat Generation
Dr. Md. Zahurul Haq
ProfessorDepartment of Mechanical Engineering
Bangladesh University of Engineering & Technology (BUET)Dhaka-1000, Bangladesh
[email protected]://teacher.buet.ac.bd/zahurul/
ME 307: Heat Transfer Equipment Design
http://teacher.buet.ac.bd/zahurul/ME307/
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Dr. Md. Zahurul Haq (BUET) HT with Internal Heat Generation ME 307 (2017) 1 / 15
Conduction Equation:
∇2T +
_qg
k=
1
α
∂T
∂t
∇2 =
∂2
∂x 2 + ∂2
∂y2 + ∂2
∂z 2 (x , y, z ) coordinate1r
∂∂r
(
r ∂∂r
)
+ 1r2
∂2
∂θ2 + ∂2
∂z 2 (r , θ, z ) coordinate1r2
∂∂r
(
r2 ∂∂r
)
+ 1r2 sin θ
∂∂θ
(
sinθ ∂∂θ
)
+ 1r2 sin2 θ
∂2
∂φ2 (r , θ, φ) coordinate
Conduction Equation: 1D & steady-state:
1
rn
∂
∂r
(
rn ∂T
∂r
)
+
_qg
k= 0
n =
0 (x , y , z ) coordinate (rectangular )
1 (r , θ, z ) coordinate (cylindrical)
2 (r , θ, φ) coordinate (sphecial)
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Dr. Md. Zahurul Haq (BUET) HT with Internal Heat Generation ME 307 (2017) 2 / 15
Thermal Conductivity, k
T737
Temperature dependence of the thermal conductivity of selected solids.
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Dr. Md. Zahurul Haq (BUET) HT with Internal Heat Generation ME 307 (2017) 3 / 15
T738
Temperature dependence of k of
selected gases at normal pressures.
T739
Temperature dependence of k of
selected non-metallic liquids under
saturated conditions.
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Dr. Md. Zahurul Haq (BUET) HT with Internal Heat Generation ME 307 (2017) 4 / 15
Heat Conduction & Convection
T740
Heat transfer through a plane wall. (a) Temperature distribution. (b)
Equivalent thermal circuit.
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Dr. Md. Zahurul Haq (BUET) HT with Internal Heat Generation ME 307 (2017) 5 / 15
T741
Hollow cylinder with convective surface conditions.
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Dr. Md. Zahurul Haq (BUET) HT with Internal Heat Generation ME 307 (2017) 6 / 15
System with Internal Heat Generation
System with Internal Heat Generation
Nuclear reactors
Electric heater & conductors
Exothermic chemical reactions
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Dr. Md. Zahurul Haq (BUET) HT with Internal Heat Generation ME 307 (2017) 7 / 15
System with Internal Heat Generation
Plane Wall with Uniform Heat Generation
T674
BC: T = Tw at x ± L
DE: d2Tdx 2 +
_qg
k = 0
⇒ T (x ) = −
_qg
2k x 2 + C1x + C2
=⇒
T (x ) = Tw +
_qg
2k L2[
1 −(
xL
)2]
if T (x = 0) = T0 :
⇒T0 = Tw +
_qg
2k L2
=⇒
T−TwT0−Tw
= 1 −(
xL
)2
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Dr. Md. Zahurul Haq (BUET) HT with Internal Heat Generation ME 307 (2017) 8 / 15
System with Internal Heat Generation
T675
Conduction in a plane wall with uniform heat generation. (a)
Asymmetrical boundary conditions. (b) Symmetrical boundary
conditions. (c) Adiabatic surface at midplane.
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Dr. Md. Zahurul Haq (BUET) HT with Internal Heat Generation ME 307 (2017) 9 / 15
System with Internal Heat Generation
Radial System with Internal Heat Generation
T676
BCs:
T (r0) = Ts
dTdr |r=0 = 0
DE: 1r
ddr
(
r dTdr
)
+
_qg
k = 0
⇒ T (r) = −
_qg
4k r2 + C1 ln r + C2
=⇒
T (r) = Ts +
_qg
4k r20
[
1 −(
rr0
)2]
if T (r = 0) = T0 :
⇒T0 = Ts +
_qg
4k r20
=⇒
T (r )−Ts
T0−Ts= 1 −
(
rr0
)2
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Dr. Md. Zahurul Haq (BUET) HT with Internal Heat Generation ME 307 (2017) 10 / 15
System with Internal Heat Generation
contd. . . .
T676
Applying an overall energy balance:
_qg(πr20 L) = h(2πr0L)(Ts − T∞)
=⇒Ts = T∞ +
_qg r02h
Tmax = T0 = Ts +
_qg
4k r20 = T∞ +
_qg r04hc
(
2 + hcrok
)
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Dr. Md. Zahurul Haq (BUET) HT with Internal Heat Generation ME 307 (2017) 11 / 15
System with Internal Heat Generation
Holman Ex. 2-6 ⊲ A current of 200 A is passed through a stainless-steel
wire [k =19 W/moC] 3 mm in diameter. The resistivity of the steel may be
taken as 70 µΩcm, and the length of the wire is 1 m. The wire is submerged
in a liquid at 110oC and experiences a convection heat-transfer coefficient of 4
kW/m2oC. Calculate the center temperature of the wire.
_qg = 560.2 MW/m3
To = 231.6oC
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Dr. Md. Zahurul Haq (BUET) HT with Internal Heat Generation ME 307 (2017) 12 / 15
System with Internal Heat Generation
Kreith Ex. 2-5 ⊲ Heat is generated at 7.5× 107W/m3 rate in nuclear reactor
uranium rods of 0.05 m diameter. These rods are jacketed by an annulus in
which water at 120oC is circulated. If hav = 55000 W/m2 K, and for uranium,
k = 29.5 W/m K, determine the center temperature of the uranium fuel rods.
T677
Ts = 137.50C
T0 = 534.30C
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Dr. Md. Zahurul Haq (BUET) HT with Internal Heat Generation ME 307 (2017) 13 / 15
System with Internal Heat Generation
Critical Thickness of Insulation
T700
T701
Rtotal = Rcond + Rconv =ln(r/ri )2πkL + 1
2πrLh∞
For maximum heat loss, dRtotaldr = 1
2πkrL − 12πr2Lh∞
= 0 → r = kh∞
Critical thickness of insulation , rcr = kh∞
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Dr. Md. Zahurul Haq (BUET) HT with Internal Heat Generation ME 307 (2017) 14 / 15
System with Internal Heat Generation
Dewitt Ex. 3-4 ⊲ Critical thickness of insulation, k = 0.055 W/mK, ri =
5mm, Ti = 100oC, T∞ = 25oC.
T696
0 5 10 15 20 25 30
t [mm]
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
6.5
7.0
R[m
K/W
]
Critical Thickness of Insulation
Rcond
Rconv
Rtot
Q
10.0
10.5
11.0
11.5
12.0
12.5
13.0
13.5
14.0
14.5
15.0
Q[W
]
T697
rcr = k/h = 11 mm, t = rcr − ri = 6 mm
Q = 14.5 W
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Dr. Md. Zahurul Haq (BUET) HT with Internal Heat Generation ME 307 (2017) 15 / 15