Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

1

Emperical Correlations for Natural Convection

Chap. 15.2: Emp. Correlations - Natural Convection

( )2

3

PrRaBoα

β HTTg wHH

⋅−⋅⋅=⋅= ∞

Summary of dimensionless numbers relevant for natural convection

( )2

3

PrRaGr

νHTTg wH

H⋅−⋅⋅

== ∞β

Boussinesq number Bo for inertia-dominated natural convection

Grasshof number Gr for friction-dominated natural convection

Buoyancy/Heat transfer

Buoyancy/Viscosity

Rayleigh number is the geometric mean: ( ) 21GrBoRa HHH ⋅=

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

2

4141

21 492.0Pr986.0PrPr

HRakHhNu ⋅⎟

⎠

⎞⎜⎝

⎛+⋅+

=⋅

=

Semi-empirical correlations for Nu for real cases:

Laminar natural convection over vertical plate:

Valid for all Pr:

3113.0 HRakHhNu ⋅=⋅

=

Turbulent natural convection over vertical plate:

129 1010 << HRaValid for:

9, 10≈critHRaCriterion for turbulent flow:

Chap. 15.3: Natural Convection – Real Cases

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

3

( )[ ]

2

278169

61

Pr/492.01

Ra387.0825.0Nu⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

+

⋅+=

⋅= H

kHh

Vertical plate: Equation of Churchill and Chu (1975) for laminar and turbulent natural convection (all RaH):

Horizontal plate:

Characteristic length L instead of height H

PerimeterArea

PAL ==

Chap. 15.3: Natural Convection – Real Cases

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

4

To be distinguished: stabil and instabil flow

105

41

1010

27.0

≤≤

⋅=⋅

=

L

LL

Ra

RakLhNu

stabil: Fig. (a) and (d)

instabil: Fig. (b) and (c)

1 4

4 7

1 3

7 11

0.54

10 10

0.15

10 10

L L

L

L L

L

Nu RaRa

Nu RaRa

= ⋅

≤ ≤

= ⋅

≤ ≤

Cold plate

Hot plate

Chap. 15.3: Natural Convection – Real Cases

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

5

( )[ ]

2

278169

61

Pr/559.01

387.06.0⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

+

⋅+= D

DRaNu

Cylinder in cross-flow (Churchill and Chu):

for RaD ≤ 1012

( )[ ] 94169

41

Pr/469.01

589.02

+

⋅+= D

DRaNu

Spheres:

for Pr ≥ 0.7 and RaD ≤ 1011

Chap. 15.3: Natural Convection – Real Cases

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

6

74

4

3.0012.04/1

1010102Pr1

4010

Pr42.0

<<

⋅<<

<<

⎟⎠

⎞⎜⎝

⎛⋅⋅⋅=

−

L

L

Ra

LH

LHRaNu

Vertical cavities (e.g. windows):

Essential: aspect ratio H/L tight gap reduces circulation

circulation roll exists for Ra > 1000 (air: gap > 1 cm)

Chap. 15.3: Natural Convection – Real Cases

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

7

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

( ) 93

0 10813.1 ⋅=⋅

⋅−⋅⋅= ∞

ναβ HTTgRaH

9, 10≈critHRa

Example:

Convective heat transfer at window of fireplace

characteristic value: Rayleigh number

Transition region laminar-turbulent, using equation by Churchill und Chu for Nusselt number

Chap. 15.4: Natural Convection – Example

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

8

( )[ ]147

Pr/492.01

387.0825.02

278169

61

=⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

+

⋅+=

⋅= H

HRa

kHhNu

Calculation of Nusselt number:

Calculation of heat transfer coefficient:

K)W/(m0.771.0147108.33 2

3

⋅=⋅⋅

=⋅

=−

HNukh H

Calculation of heat flux:

( ) W10602323271.002.17 =−⋅⋅⋅=Δ⋅⋅⋅= THWhq

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).

Chap. 15.4: Natural Convection – Example

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

9