ME 150 – Heat and Mass Transfer
Chap. 15.2: Emp. Correlations - Natural Convection
Emperical Correlations for Natural Convection Summary of dimensionless numbers relevant for natural convection Boussinesq number Bo for inertia-dominated natural convection Bo H = Ra H ⋅ Pr =
g ⋅ β ⋅ (Tw − T∞ )⋅ H 3
α
2
Buoyancy/Heat transfer
Grasshof number Gr for friction-dominated natural convection Ra H g ⋅ β ⋅ (Tw − T∞ )⋅ H 3 GrH = = Pr ν2
Buoyancy/Viscosity
Rayleigh number is the geometric mean: Prof. Nico Hotz
Ra H = (Bo H ⋅ GrH )
1
2
1
ME 150 – Heat and Mass Transfer
Chap. 15.3: Natural Convection – Real Cases
Semi-empirical correlations for Nu for real cases: Laminar natural convection over vertical plate: 14
Valid for all Pr:
h ⋅H ⎛ Pr ⎞ 14 Nu = = ⎜ ⋅ Ra ⎟ H 12 k ⎝ Pr + 0.986 ⋅ Pr + 0.492 ⎠
Turbulent natural convection over vertical plate: 9 12 Valid for: 10 < Ra H < 10
Criterion for turbulent flow:
Nu =
h ⋅H = 0.13 ⋅ Ra1H3 k
RaH ,crit ≈ 109
Prof. Nico Hotz
2
ME 150 – Heat and Mass Transfer
Chap. 15.3: Natural Convection – Real Cases
Vertical plate: Equation of Churchill and Chu (1975) for laminar and turbulent natural convection (all RaH): 16 h ⋅H ⎛⎜ 0.387 ⋅ Ra H Nu = = 0.825 + 9 16 ⎜ k ( ) 1 + 0 . 492 / Pr ⎝
[
⎞ ⎟ 8 27 ⎟ ⎠
2
]
Horizontal plate: Characteristic length L instead of height H L=
A Area = P Perimeter
Prof. Nico Hotz
3
ME 150 – Heat and Mass Transfer
Chap. 15.3: Natural Convection – Real Cases
To be distinguished: stabil and instabil flow stabil: Fig. (a) and (d) Nu L =
Cold plate
h⋅ L 14 = 0.27 ⋅ RaL k
105 ≤ RaL ≤ 1010
instabil: Fig. (b) and (c) Nu L = 0.54 ⋅ RaL1 4 104 ≤ RaL ≤ 107 Nu L = 0.15 ⋅ RaL1 3
Hot plate
107 ≤ RaL ≤ 1011 Prof. Nico Hotz
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ME 150 – Heat and Mass Transfer
Chap. 15.3: Natural Convection – Real Cases
Cylinder in cross-flow (Churchill and Chu):
16 ⎛ ⎞ 0 . 387 ⋅ Ra D ⎜ ⎟ Nu D = 0.6 + 8 27 9 16 ⎜ ⎟ 1 + (0.559 / Pr ) ⎝ ⎠
[
]
2
for RaD ≤ 1012
Spheres:
Nu D = 2 +
0.589 ⋅ Ra D1 4
[1 + (0.469 / Pr ) ]
9 16 4 9
Prof. Nico Hotz
for Pr ≥ 0.7 and RaD ≤ 1011
5
ME 150 – Heat and Mass Transfer
Chap. 15.3: Natural Convection – Real Cases
Vertical cavities (e.g. windows):
Nu = 0.42 ⋅ Ra L
1/ 4
⎛ H ⎞ ⋅ Pr 0.012 ⋅ ⎜ ⎟ ⎝ L ⎠
−0.3
H < 40 L 1 < Pr < 2 ⋅ 10 4 10 <
10 4 < Ra L < 10 7
Essential:
aspect ratio H/L tight gap reduces circulation circulation roll exists for Ra > 1000 (air: gap > 1 cm)
Prof. Nico Hotz
6
ME 150 – Heat and Mass Transfer
Chap. 15.4: Natural Convection – Example
Example: Convective heat transfer at window of fireplace characteristic value: Rayleigh number k = 33.8 x 10-3 W/mK ν = 26.4 x 10-6 m2/s α = 38.3 x 10-6 m2/s Pr = 0.69 β = 0.0025 K-1 W = 1.02 m H = 0.71 m
g ⋅ β ⋅ (T0 − T∞ )⋅ H 3 RaH = = 1.813 ⋅109 α ⋅ν
RaH ,crit ≈ 109
Transition region laminar-turbulent, using equation by Churchill und Chu for Nusselt number Prof. Nico Hotz
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ME 150 – Heat and Mass Transfer
Chap. 15.4: Natural Convection – Example
Calculation of Nusselt number: 2
16 ⎞ h ⋅H ⎛⎜ 0.387 ⋅ RaH ⎟ = 147 Nu H = = 0.825 + 8 27 9 16 ⎜ ⎟ k 1 + (0.492 / Pr ) ⎝ ⎠
[
]
Calculation of heat transfer coefficient: k ⋅ Nu H 33.8 ⋅10 −3 ⋅147 h = = = 7.0 W/(m2 ⋅ K) H 0.71
Calculation of heat flux: q = h ⋅W ⋅ H ⋅ ΔT = 7 ⋅1.02 ⋅ 0.71 ⋅ (232 − 23) =1060 W
Note: We did not consider thermal radiation through the window, which can be an order of magnitude higher than convective heat transfer (in this case). Prof. Nico Hotz
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ME 150 â&#x20AC;&#x201C; Heat and Mass Transfer
Prof. Nico Hotz
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