Free Convection:General Considerations

and Results forVertical 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

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 induced by buoyancy forces.

sT T

General Considerations (cont)

• Pertinent Dimensionless Parameters

Grashof Number:

3

2

Buoyancy ForceViscous Force

sL

g T T LGr

Rayleigh Number:

3

Pr sL L

g T T LRa Gr

characteristic length of surfaceL

(a thermothermal dynamic expansion coeffi property of the cien fl d)t ui

1

pT

Liquids: Tables A.5, A.6

Perfect Gas: =1/ KT

General Considerations (cont)

• Mixed Convection

A condition for which forced and free convection effects are comparable.

Exists if

2/ Re 0 1L LGr

2- Free convection / Re 1L LGr 2- Forced convection / Re 1L LGr

Heat Transfer Correlations for Mixed Convection:

n n nFC NCNu Nu Nu

assisting and transverse flows- opposing flows

3n

Vertical Plates

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 ?sT T

Vertical Plates (cont)

• Form of the x-Momentum Equation for Laminar Flow

2

2u u uu g T Tx y y

Net Momentum Fluxes( Inertia Forces)

Buoyancy Force 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

2T T Tux y y

– The solutions are said to be coupled.

Vertical Plates (cont)

• Similarity Solution

Based on existence of a similarity variable, through which the x-momentum equation may be transformed from a partial differential equation with two- independent variables ( x and y) to an ordinary differential equation expressed exclusively in terms of .

1/ 4

4xGry

x

Transformed momentum and energy equations:

23 2 0f ff f T

3Pr 0T fT

1/ 2

2 xs

T Tdf xf Gr u Td T T

Vertical Plates (cont)

Numerical integration of the equations yields the following results for and :f T

Velocity boundary layer thickness 5 for Pr 0.6

1/ 41/ 4

1/ 4Pr 0.6 : 5 7.074

x

x

Gr xx xGr

Vertical Plates (cont)

Nusselt Numbers and :LxNu Nu

1/ 4 1/ 4

0

Pr4 4

x xx

Gr Grhx dTNu gk d

1/ 2

1/ 41/ 2

0.75 PrPr 0 Pr

0.609 1.221 Pr 1.238 Prg

1 43

LL Loh hdx Nu Nu

L

• Transition to Turbulence

Amplification of disturbances depends on relative magnitudes of buoyancy and viscous forces.

Transition occurs at a critical Rayleigh Number.

39

, , Pr 10sx c x c

g T T xRa Gr

Vertical Plates (cont)

• Empirical Heat Transfer Correlations

Laminar Flow 910 :LRa

1/ 4

4 / 99 /16

0.6700.68

1 0.492 / Pr

LL

RaNu

All Conditions:

2

1/ 6

4 / 99 /16

0.3870.825

1 0.492 / Pr

LL

RaNu

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

sT T sT T

1/ 4 4 70.54 10 10L L LNu Ra Ra

1/ 3 7 110.15 10 10L L LNu Ra Ra

How does depend on L when 1/ 3 ?L LNu Rah

Horizontal Plates (cont)

• Heated Surface Facing Downward or Cooled Surface Facing Upward

sT T sT T

1/ 4 5 100.27 10 10L L LNu Ra Ra

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.

T = 300 Koo

q conv

q rad

q = 10 S5 W /m 2

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/mK, = 45.6 10-6 m2/s, = 66.7 10-6 m2/s, Pr = 0.683, = 1.82 10-3 K-1.

ANALYSIS: (a) The total heat loss is

4rad conv s s s sq q q A T hA T T

With RaL = g (Ts - T)L3/ = 9.8 m/s2 (1.82 10-3 K-1) 500K (12m)3/(45.6 66.7 10-12 m4/s2) = 5.07 1012, the Churchill and Chu correlation yields

2

1/ 62 2L

8 / 279 /16

k 0.387 Ra 0.0439 W / m Kh 0.825 0.825 42.4 6.83 W / m K

L 12m1 0.492 / Pr

Hence, with As = DL = 264 m2

42 8 2 4 2 2q 264m 0.2 5.67 10 W / m K 800K 264m 6.83W / m K 500K

6 5 6rad convq q q 1.23 10 W 9.01 10 W 2.13 10 W

Problem: Solar Receiver (cont)

With 7s sA q 2.64 10 W, the collector efficiency is

7 6s s

7s s

2.64 10 2.13 10 WA q q100 100 91.9%

A q 2.64 10 W

(b) As shown below, because of its dependence on temperature to the fourth power, qrad increases more significantly with increasing Ts than does qconv, and the effect on the efficiency is pronounced

600 700 800 900 1000

Receiver temperature, K

0

1E6

2E6

3E6

4E6

5E6

Hea

t rat

e, W

ConvectionRadiationTotal

600 700 800 900 1000

Receiver temperature, K

75

80

85

90

95

100

Co

llect

or

effi

cie

ncy

, %

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.