Convection:
General 4/6/2012
Free Considerations
and Results for
Vertical and Horizontal Plates
Chapter 9
Sections 9.1 through 9.6.2, 9.9
General Considerations
Lecture 16-2
General Considerations
Free convection refers to fluid motion induced by buoyancy forces.
Buoyancy forces may arise in a fluid for which there are density gradients
and a body force that is proportional to density.
In heat transfer, density gradients are due to temperature gradients and the
body force is gravitational.
Stable and Unstable Temperature Gradients
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General Considerations (cont)
Lecture 16-3
Free Boundary Flows
Occur in an extensive (in principle, infinite), quiescent (motionless
at locations far from the source of buoyancy) fluid.
Plumes and Buoyant Jets:
Free Convection Boundary Layers
Boundary layer flow on a heated or cooled surface Ts T induced
by buoyancy forces.
General Considerations (cont)
Lecture 16-4
Pertinent Dimensionless Parameters
Grashof Number:
GrL
g Ts T L3
Buoyancy Force
Viscous Force
2
L characteristic length of surface
thermal expansion coefficient (a thermodynamic property of the fluid)
1
T p
Liquids: Tables A.5, A.6
Perfect Gas: =1/T K
Rayleigh Number:
RaL GrL Pr
g Ts T L3
2
4/6/2012
Lecture 16-5
"Heat and Mass Transfer: A Practical
Approach, 3/e" By Yunus A. engel
The flow regime on a vertical plate becomes
turbulent at Grashof numbers greater than 109
IF
Natural convection is small
volume expansion coefficient
Lecture 16-6
"Heat and Mass Transfer: A Practical
Approach, 3/e" By Yunus A. engel
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General Considerations (cont)
Lecture 16-7
Mixed Convection
A condition for which forced and free convection effects are comparable.
Exists if
Gr
L
/ Re 2 o 1
L
- Free convection GrL / Re 2 1
L
- Forced convection GrL / Re2 1
L
Heat Transfer Correlations for Mixed Convection:
n
n
Nu n Nu FC Nu NC
assisting and transverse flows
- opposing flows
n3
Vertical Plates
Lecture 16-8
Vertical Plates
Free Convection Boundary Layer Development on a Heated Plate:
Ascending flow with the maximum velocity occurring in the boundary layer
and zero velocity at both the surface and outer edge.
How do conditions differ from those associated with forced convection?
How do conditions differ for a cooled plate Ts T ?
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Vertical Plates (cont)
Lecture 16-9
Form of the x-Momentum Equation for Laminar Flow
2
u u u g T T u
x
y
y 2
Net Momentum Fluxes Buoyancy Force
( Inertia Forces)
Viscous Force
Temperature dependence requires that solution for u (x,y) be obtained
concurrently with solution of the boundary layer energy equation for T (x,y).
2
u T T T
x
y
y 2
The solutions are said to be coupled.
Vertical Plates (cont)
Lecture 16-10
Nusselt Numbers Nu x and Nu L :
1/ 4
Gr
Nu x hx x
k
4
g Pr
dT
d
1/ 4
0
Gr
x
4
g Pr
0.75 Pr1/ 0.609 2
1.221 Pr
1/ 2
1.238 Pr
1/ 4
0 Pr
L
h 1 o h dx Nu L 4 NuL
L
3
Transition to Turbulence
Amplification of disturbances
depends on relative magnitudes
of buoyancy and viscous forces.
Transition occurs at a critical
Rayleigh Number.
Rax , c Grx , c Pr
g Ts T x3
109
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Lecture 16-11
Horizontal Plates
Horizontal Plates
Buoyancy force is normal, instead of parallel, to the plate.
Flow and heat transfer depend on whether the plate is heated or cooled and
whether it is facing upward or downward.
Hot Surface Facing Upward or Cold Surface Facing Downward
Ts T
Ts T
Nu L 0.54 Ra1/ 4
L
Nu L 0.15 Ra1/ 3
L
10
10
4
RaL 107
7
RaL 1011
How does h depend on L when Nu L Ra1 / 3 ?
L
RaL GrL Pr
g Ts T L3
Lecture 16-12
Hot Surface Facing Downward or Cold Surface Facing Upward
Horizontal Plates (cont)
Ts T
Nu L 0.27 Ra1/ 4
L
Ts T
10
5
RaL 1010
Why do these flow conditions yield smaller heat transfer rates than those
for a heated upper surface or cooled lower surface?
RaL GrL Pr
g Ts T L3
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RaL GrL Pr
g Ts T L3
Lecture 16-13
Lecture 16-14
Problem 9.18
Problem 9.18
Why is the frost at the bottom ?
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Lecture 16-15
"Heat and Mass Transfer: A Practical Approach, 3/e" By Yunus A.
engel
Lecture 16-16
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Lecture 16-17
Lecture 16-18
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Lecture 16-19
Cylinders
Lecture 16-20
The Long Horizontal Cylinder
Boundary Layer Development and Variation of the Local Nusselt Number
for a Heated Cylinder:
The Average Nusselt Number:
0.387 Ra1/ 6
D
Nu D 0.60
9 /16 8 / 27
1 0.559 / Pr
2
RaD 1012
How do conditions change for a cooled cylinder?
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Lecture 16-21
Spheres
Spheres
The Average Nusselt Number:
Nu D 2
0.589 Ra1/ 4
D
1 0.469 / Pr 9 /16
4/9
Enclosures
Lecture 16-22
Enclosures
Rectangular Cavities
Characterized by opposing walls of different temperatures, with the
remaining walls well insulated.
RaL
g T1 T2 L3
q h T1 T2
Horizontal Cavity 0, 180deg
Vertical Cavity 90 deg
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Enclosures (cont)
Lecture 16-23
Horizontal Cavities
Heating from Below 0
RaL RaL , c 1708 :
Fluid layer is thermally stable.
Nu L hL 1
k
1708 Ra L 5 104 :
Thermal instability yields a regular convection pattern in the form of roll cells.
3 105 RaL 7 109 :
Buoyancy driven flow is turbulent
Nu L 0.069 Ra1/ 3 Pr 0.074
L
Enclosures (cont)
Lecture 16-24
Heating from Above 180 deg
Fluid layer is unconditionally stable.
Nu L 1
Vertical Cavities
Nu L hL 1
k
Heated from below:
For Ra > 1708
For Ra <1708,
Nu L 0.069 Ra1/ 3 Pr 0.074
L
3
RaL 10 :
Nu L 1
RaL 103 :
A primary cellular flow is established, as the core
becomes progressively more quiescent, and
secondary (corner) cells develop with increasing
RaL .
Correlations for Nu L Eqs. (9.50) - (9.53).
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