Transcript Free Convection: General Considerations and Results for Vertical

Free Convection:
General Considerations
and Results for
Vertical and Horizontal Plates
Ts  T
1
Ts  T
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
2
• 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.
3
4
• Pertinent Dimensionless Parameters
 Grashof Number:
GrL 
g  Ts  T  L3
2

Buoyancy Force
Viscous Force
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 
Pr 
 Rayleigh Number :
RaL  GrL Pr 
g  Ts  T  L3

 Molecular diffusion momentum


Molecular diffusion heat
• Mixed Convection
 A condition for which forced and free convection effects are comparable.
 Exists if
 Gr
2
 0 1
/
Re
L
L
- Free convection   GrL / Re 2L   1
- Forced convection   GrL / Re 2L   1
 Heat Transfer Correlations for Mixed Convection:
n
n
Nu n  NuFC
 Nu NC
  assisting and transverse flows
-  opposing flows
n3
5
Vertical Plates
6
• Free Convection Boundary Layer Development on a Heated Plate:
STEADY STATE
 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  ?
• Form of the Continuity Equation (=cte)
u v

0
x y
• Form of the x-Momentum Equation for Laminar Flow (constant properties)
2

u

u

u 
 g  T  T    u2
x
y
y
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

T

T

u

  T2
x
y
y
– The solutions are said to be coupled.
7
7C
• Non-dimensional equations
u v

0
x y
 u *  v*

0
*
*
x y
u u   u  g  T  T     u2
x
y
y
2
GrL 
g  TS  T  L3
2
2

T

T

u

  T2
x
y
y
;
*
GrL *
 u*
1  2u*
*  u
u
v

T 
*
*
2
Re L  y * 2
x
y
Re L
*
Re L 
U0  L

*
 T*
1
 2T*
*  T
u
v

*
*
x
y
Pr Re L  y* 2
*
8
• Similarity Solution
 Based on existence of a similarity variable,  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  .

1/ 4
y  Grx 


x 4 
f ( ) 
 ( x , y)
 4 Grx
4

4
 2  1/ 2
u

Grx f '  
y x 
v
 2 y  1/ 2

Grx f '  
x x 2 
 Transformed momentum and energy equations:
f   3 ff   2  f    T   0
2
T   3Pr fT   0
f    
df
 x  Grx1/ 2  u
d 2
T 
T  T
Ts  T
 Numerical integration of the equations yields the following results for
f    and T  :
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 Velocity boundary layer thickness      5 for Pr  0.6
1/ 4
 Gr 
 Pr  0.6 :   5 x  x 
 4 
1/ 4
 7.07
1/ 4
x

x
1/ 4
 Grx 
y Gr
   x 
x 4 
10
 Nusselt Numbers  Nux and Nu L  :
1/ 4
 Gr 
Nu x  hx    x 
k
 4 
g  Pr  
1/ 4

dT
d
 0
 Gr 
 x
 4 
0.75 Pr1/ 2
 0.609  1.221 Pr
1/ 2
h  1  h dx  Nu L  4 NuL
L
3
1/ 4
1 L
k  g Ts  _ T 
h   h dx  

L 0
L
4 2

4  Gr 
  L
3 4 
• Transition to Turbulence
 Amplification of disturbances
depends on relative magnitudes
of buoyancy and viscous forces.
 Transition occurs at a critical
Rayleigh Number.
g  Ts  T  x3

 0  Pr   
 1.238 Pr 
L
o
Rax , c  Grx ,c Pr 
g  Pr 
 109
1/ 4
gPr 
1/ 4
gPr  
L
0
dx

1/4
x
11
• Empirical Heat Transfer Correlations
 Laminar Flow  RaL  109  :
Nu L  0.68 
0.670 Ra1/L 4
1   0.492 / Pr 9 /16 


4/9
 All Conditions:


1/ 6
0.387 RaL


Nu L  0.825 

9 /16 4 / 9
1   0.492 / Pr   


 

2
CHURCHILL E CHU
 What if the vertical plate is subjected to a constant heat flux (temperature is varying)?
In this case, the same correlations are applied, but the non-dimensional parameters (Nusselt
and Rayleigh numbers) are defined in terms of the temperature difference at the midpoint of
the plate:
q s''
h
T1 / 2
T1 / 2  Ts
x L / 2
 T
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
Ts  T
Ts  T
4
Nu L  0.54 Ra1/
L
3
Nu L  0.15 Ra1/
L
10
10
4
 RaL  107 
7
 RaL  1011 
3
How does h depend on L when Nu L  Ra1/
L ?
IT IS INDEPENDENT OF L !
12
• Heated Surface Facing Downward or Cooled Surface Facing Upward
Ts  T
4
Nu L  0.27 Ra1/
L
13
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?
BECAUSE THE PLATE IMPEDES THE ASCENDING/DESCENDING NATURAL
CONVECTION FLOW THAT HAS TO MOVE HORIZONTALLY. THIS MAKES
CONVECTION HEAT TRANSFER INEFFECTIVE.
7A
•Continuity Equation
  u    v 

0
 ( = constant)
x
y
u v

0
x y
•x- and y-Momentum Equations for Laminar Flow (constant properties)
u
u
 ,
y
x
u
v
,
x
v
y
u  v
u
u
1 p
v


x
y
 x
1    u 2  u v   1    u v 
     g
 2      

 x   x 3  x y    y   y x 
u
v
v
1 p
v


x
y
 y
1    v 2  u v   1    u v 
   
 2      

 y   y 3  x y    x   y x 

p
0
y
•Energy Equation for Laminar Flow (constant properties)
u
u
 ,
y
x
v
,
x
v
y
T
T

y
x
 T
 T    T    T 
 
 C p  u
v
 k
  q 
k  
 y  x  x  y  y 
 x
 u v  2
 u  2  v  2  2  u v  2 
     2           
 y x 
 x   y   3  x y  
2
T
T
 2 T    u  
u
v

    
2
x
y
 y  C p  y  
T
T
 2T
u
v

x
y
 y2
7B
8A
•BOUNDARY CONDITIONS
y  0:
uv0
y*  0 :
u *  v*  0
y:
u0
y*   :
 0:
:
T  Ts
T*  1
T  T
u*  0
f f' 0
T*  1
f' 0
T*  0
T*  0
f    
df
 x  Grx1/ 2  u
d 2
T 
T  T
Ts  T
9A
•Nusselt number
1/ 4
 Gr 
Nu x  hx    x 
k
 4 
1/ 4
 Gr 
dT 
 x
d   0  4 
10A
g  Pr 
hx

q '' x

k
  Nu x 
k Ts  T 
q ''  hTs  T 

Nu x 
q ''   k
T
y
y0
T
y
y0
 T*
 Ts  T 

 0




1/ 4
*

1  Gr   T

 Ts  T   x 
y
x  4    0 
k  Gr 
q  Ts  T   x 
x 4 
''
1/ 4
k  Gr 
Nu x  Ts  T   x 
x 4 
 T*

1/ 4
 T*

 0
1/ 4
 0
x
 Gr 
  x 
k Ts  T 
 4 
 T*

 0