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Free Convection: General Considerations and Results for Vertical

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Free Convection:
General Considerations
and Results for
Vertical and Horizontal Plates
Chapter 9
Sections 9.1 through 9.6.2, 9.9
General Considerations
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
General Considerations (cont)
• Free Boundary Flows
пѓ� Occur in an extensive (in principle, infinite), quiescent (motionless
at locations far from the source of buoyancy) fluid.
пѓ� Plumes and Buoyant Jets:
• Free Convection Boundary Layers
пѓ� Boundary layer flow on a heated or cooled surface пЂЁ T s п‚№ Tп‚Ґ пЂ© induced
by buoyancy forces.
General Considerations (cont)
• Pertinent Dimensionless Parameters
пѓ� Grashof Number:
g пЃў пЂЁ T s пЂ­ Tп‚Ґ пЂ© L
3
G rL пЂЅ
пЃ®
B u o yan cy F o rce
2
V isco u s F o rce
L п‚® characteristic length of surface
пЃў п‚® thermal expansion coeffi cien t (a thermo dynamic property of the flui d)
  
пЃў пЂЅпЂ­1пѓ§
пѓ·
  T  p
Liquids: пЃў п‚® T ables A.5, A.6
P erfect G as: пЃў =1/ T пЂЁ K пЂ©
пѓ� Rayleigh Number:
g пЃў пЂЁ T s пЂ­ Tп‚Ґ пЂ© L
3
R a L пЂЅ G rL P r пЂЅ
пЃ®пЃЎ
General Considerations (cont)
• Mixed Convection
пѓ� A condition for which forced and free convection effects are comparable.
пѓ� Exists if
пЂЁGr
L
/ ReL пЂ©
2
0 пЂЁ1 пЂ©
- Free convection п‚® пЂЁ G rL / R e L пЂ© 1
2
- Forced convection п‚® пЂЁ G rL / R e L пЂ© 1
2
пѓ� Heat Transfer Correlations for Mixed Convection:
N u п‚» N u FC п‚± N u NC
n
n
n
пЂ« п‚® assisting and transverse flow s
- п‚® opposing flow s
nп‚»3
Vertical Plates
Vertical Plates
• Free Convection Boundary Layer Development on a Heated Plate:
пѓ� Ascending flow with the maximum velocity occurring in the boundary layer
and zero velocity at both the surface and outer edge.
пѓ� How do conditions differ from those associated with forced convection?
пѓ� How do conditions differ for a cooled plate пЂЁ T s пЂј Tп‚Ґ пЂ© ?
Vertical Plates (cont)
• Form of the x-Momentum Equation for Laminar Flow
u
u
 u
пЂ«пЃµ
пЂЅ g пЃў пЂЁ T пЂ­ Tп‚Ґ пЂ© пЂ« пЃ®
2
x
y
y
2
u
Net Momentum Fluxes Buoyancy Force
( Inertia Forces)
Viscous Force
пѓ� Temperature dependence requires that solution for u (x,y) be obtained
concurrently with solution of the boundary layer energy equation for T (x,y).
T
T
 T
u
пЂ«пЃµ
пЂЅпЃЎ
2
x
y
y
2
– The solutions are said to be coupled.
Vertical Plates (cont)
• Similarity Solution
пѓ� Based on existence of a similarity variable, пЃЁ through which the x-momentum
equation may be transformed from a partial differential equation with twoindependent variables ( x and y) to an ordinary differential equation expressed
exclusively in terms of пЃЁ .
y пѓ¦ Gr пѓ¶
пЃЁ п‚є пѓ§ xпѓ·
xпѓЁ 4 пѓё
1/ 4
пѓ� Transformed momentum and energy equations:
пЂЄ
f п‚ўп‚ўп‚ў пЂ« 3 ff п‚ўп‚ў пЂ­ 2 пЂЁ f п‚ў пЂ© пЂ« T пЂЅ 0
2
T
пЂЄ п‚ўп‚ў
пЂ« 3 P r fT
f п‚ў пЂЁпЃЁ пЂ© п‚є
df
dпЃЁ
пЂЄп‚ў
пЂЅ
пЂЅ0
пЂ­1 / 2
x
G rx
пЂЁ
пЂ©u
2пЃ®
T
пЂЄ
п‚є
T пЂ­ Tп‚Ґ
T s пЂ­ Tп‚Ґ
Vertical Plates (cont)
пѓ� Numerical integration of the equations yields the following results for
f п‚ў пЂЁпЃЁ пЂ© and T :
пЂЄ
пѓ� Velocity boundary layer thickness пЂЁ пЃ¤ пЂ© п‚® пЃЁ п‚» 5 for P r пЂѕ 0.6
пѓ�
пѓ¦ G rx пѓ¶
P r пЂѕ 0.6 : пЃ¤ пЂЅ 5 x пѓ§
пѓ·
пѓЁ 4 пѓё
пЂ­1 / 4
пЂЅ 7.07
x
пЂЁ G rx пЂ©
1/ 4
п‚µ x
1/ 4
Vertical Plates (cont)
пѓ� Nusselt Numbers пЂЁ N u x an d N u L пЂ© :
пѓ¦ G rx пѓ¶
hx
Nux пЂЅ
пЂЅ пЂ­пѓ§
пѓ·
k
пѓЁ 4 пѓё
g пЂЁ Pr пЂ© пЂЅ
hпЂЅ
1/ 4
пЂЄ
dT
dпЃЁ
пЃЁ пЂЅ0
0.75 P r
пЂЁ 0.609 пЂ« 1.221 P r
пѓ¦ G rx пѓ¶
пЂЅпѓ§
пѓ·
пѓЁ 4 пѓё
1/ 2
пЂ« 1.238 P r пЂ©
• Transition to Turbulence
пѓ� Amplification of disturbances
depends on relative magnitudes
of buoyancy and viscous forces.
пѓ� Transition occurs at a critical
Rayleigh Number.
g пЃў пЂЁ T s пЂ­ Tп‚Ґ пЂ© x
пЃ®пЃЎ
g пЂЁ Pr пЂ©
1/ 2
1 L
4
пѓІo h dx п‚® N u L пЂЅ N u L
L
3
R a x , c пЂЅ G rx , c P r пЂЅ
1/ 4
3
п‚» 10
9
1/ 4
пЂЁ0 пЂј
Pr пЂј п‚Ґ пЂ©
Vertical Plates (cont)
• Empirical Heat Transfer Correlations
пѓ� Laminar Flow пЂЁ R a L пЂј 10 9 пЂ© :
1/ 4
N u L пЂЅ 0.68 пЂ«
0.670 R a L
пѓ©1 пЂ« пЂЁ 0.492 / P r пЂ© 9 / 16 пѓ№
пѓ«
пѓ»
4/9
пѓ� All Conditions:
Nu L
пѓ¬
пѓј
1/ 6
0 .3 8 7 R a L
пѓЇ
пѓЇ
пЂЅ пѓ­ 0 .8 2 5 пЂ«
4/9 пѓЅ
9 / 16
пѓ©1 пЂ« пЂЁ 0 .4 9 2 / P r пЂ©
пѓ№
пѓЇ
пѓЇ
пѓ«
пѓ»
пѓѕ
пѓ®
2
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.
• Heated Surface Facing Upward or Cooled Surface Facing Downward
T s пЂѕ Tп‚Ґ
T s пЂј Tп‚Ґ
N u L пЂЅ 0.54 R a L
пЂЁ10
4
пЂј R a L пЂј 10
7
N u L пЂЅ 0.15 R a L
пЂЁ10
7
пЂј R a L пЂј 10
11
1/ 4
1/ 3
How does h depend on L when N u L п‚µ R a 1L / 3 ?
пЂ©
пЂ©
Horizontal Plates (cont)
• Heated Surface Facing Downward or Cooled Surface Facing Upward
T s пЂѕ Tп‚Ґ
T s пЂј Tп‚Ґ
N u L пЂЅ 0.27 R a L
1/ 4
пЂЁ10
5
пЂј R a L пЂј 10
10
пЂ©
пѓ� Why do these flow conditions yield smaller heat transfer rates than those
for a heated upper surface or cooled lower surface?
Problem: Solar Receiver
Problem 9.31: Convection and radiation losses from the surface of a
central solar receiver.
q conv
Too = 300 K
q S = 10 5 W /m 2
q rad
KNOWN: Dimensions and emissivity of cylindrical solar receiver. Incident solar flux.
Temperature of ambient air.
FIND: (a) Heat loss and collection efficiency for a prescribed receiver temperature, (b)
Effect of receiver temperature on heat losses and collector efficiency.
ASSUMPTIONS: (1) Steady-state, (2) Ambient air is quiescent, (3) Incident solar flux is
uniformly distributed over receiver surface, (4) All of the incident solar flux is absorbed by
the receiver, (5) Negligible irradiation from the surroundings, (6) Uniform receiver surface
temperature, (7) Curvature of cylinder has a negligible effect on boundary layer development,
(8) Constant properties
Problem: Solar Receiver (cont)
PROPERTIES: Table A-4, air (Tf = 550 K): k = 0.0439 W/mпѓ—K, пЃ® = 45.6 п‚ґ 10-6 m2/s, пЃЎ =
66.7 п‚ґ 10-6 m2/s, Pr = 0.683, пЃў = 1.82 п‚ґ 10-3 K-1.
ANALYSIS: (a) The total heat loss is
q пЂЅ q rad пЂ« q co n v пЂЅ A s пЃҐпЃі Ts пЂ« h A s пЂЁ Ts пЂ­ Tп‚Ґ
4
пЂ©
With RaL = gпЃў (Ts - Tп‚Ґ)L3/пЃ®пЃЎ = 9.8 m/s2 (1.82 п‚ґ 10-3 K-1) 500K (12m)3/(45.6 п‚ґ 66.7 п‚ґ 10-12
m4/s2) = 5.07 п‚ґ 1012, the Churchill and Chu correlation yields
пѓ¬
k пѓЇ
пѓЇ
h пЂЅ
пѓ­ 0.825 пЂ«
L пѓЇ
пѓЇпѓ®
2
пѓј
1/ 6
пѓЇпѓЇ
0.387 R a L
0.0439 W / m пѓ— K
2
пЂЅ
пЃ» 0.825 пЂ« 42.4 пЃЅ пЂЅ 6.83 W / m 2 пѓ— K
пѓЅ
8 / 27
12 m
пѓЇ
пѓ©1 пЂ« пЂЁ 0.492 / P r пЂ© 9 / 16 пѓ№
пѓЄпѓ«
пѓєпѓ»
пѓЇпѓѕ
Hence, with As = пЃ°DL = 264 m2
2
q пЂЅ 264 m п‚ґ 0.2 п‚ґ 5.67 п‚ґ 10
пЂ­8
6
2
W / m пѓ—K
4
пЂЁ 800 K пЂ©
5
4
пЂ« 264 m п‚ґ 6.83 W / m пѓ— K пЂЁ 500 K пЂ©
2
6
q пЂЅ q rad пЂ« q conv пЂЅ 1.23 п‚ґ 10 W пЂ« 9.01 п‚ґ 10 W пЂЅ 2.13 п‚ґ 10 W
2
Problem: Solar Receiver (cont)
With
7
A s q п‚ўп‚ўs пЂЅ 2.64 п‚ґ 10 W ,
пѓ¦ A q п‚ўп‚ў пЂ­ q
пЃЁ пЂЅпѓ§ s s
пѓЁ A s q п‚ўп‚ўs
the collector efficiency is
пЂЁ
7
2.64 п‚ґ 10 пЂ­ 2.13 п‚ґ 10
пѓ¶
пѓ· 100 пЂЅ
7
2.64 п‚ґ 10 W
пѓё
6
пЂ© W пЂЁ100 пЂ© пЂЅ 91.9%
5E6
100
4E6
95
Collector efficiency, %
Heat rate, W
(b) As shown below, because of its dependence on temperature to the fourth power, q rad
increases more significantly with increasing T s than does qconv, and the effect on the efficiency
is pronounced
3E6
2E6
1E6
0
600
700
800
900
Receiver temperature, K
Convection
Radiation
Total
90
85
80
1000
75
600
700
800
900
1000
Receiver temperature, K
COMMENTS: The collector efficiency is also reduced by the inability to have a perfectly
absorbing receiver. Partial reflection of the incident solar flux will reduce the efficiency by at
least several percent.
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