Transcript Transient Heat Conduction in Large Biot Number Systems

Convection in Flat Plate Boundary Layers
P M V Subbarao
Associate Professor
Mechanical Engineering Department
IIT Delhi
A Universal Similarity Law ……
Hyper sonic Plane
Boundary Layer Equations
Consider the flow over a parallel flat plate.
Assume two-dimensional, incompressible, steady flow
with constant properties.
Neglect body forces and viscous dissipation.
The flow is nonreacting and there is no energy
generation.
The governing equations for steady two dimensional
incompressible fluid flow with negligible viscous
dissipation:
Boundary Conditions
0
u
0
0
Twall
T
Scale Analysis
Define characteristic parameters:
L : length
u ∞ : free stream velocity
T ∞ : free stream temperature
General parameters:
x, y : positions (independent variables)
u, v : velocities (dependent variables)
T : temperature (dependent variable)
also, recall that momentum requires a pressure gradient
for the movement of a fluid:
p : pressure (dependent variable)
Define dimensionless variables:
x
x 
L
*
v
v 
u
*
y
y 
L
*
p
p  2
u 
*
u
u 
u
*
T  Ts

T  Ts
Similarity Parameters:
Re 
u L


Pr 

Pe  Re Pr
u v
 * 0
*
x y
*
*
u
p
1 u
* v
u
v
 * 
*
*
*2
x
y
x
Re L y
*
*
*
2 *
*
p
0
*
y
*

1 
* 
u
v

*
*
*2
x
y
Re L Pr y
2
*
Boundary Layer Parameters
• Three main parameters (described below) that are used to
characterize the size and shape of a boundary layer are:
• The boundary layer thickness,
• The displacement thickness, and
• The momentum thickness.
• Ratios of these thickness parameters describe the shape of the
boundary layer.
Boundary Layer Thickness
• The boundary layer thickness: the thickness of
the viscous boundary layer region.
• The main effect of viscosity is to slow the fluid
near a wall.
• The edge of the viscous region is found at the
point where the fluid velocity is essentially
equal to the free-stream velocity.
• In a boundary layer, the fluid asymptotically
approaches the free-stream velocity as one
moves away from the wall, so it never actually
equals the free-stream velocity.
• Conventionally (and arbitrarily), the edge of the
boundary layer is defined to be the point at
which the fluid velocity equals 99% of the freestream velocity:
u y   0.99u
u   0.99u
• Because the boundary layer thickness is defined in terms of the
velocity distribution, it is sometimes called the velocity
thickness or the velocity boundary layer thickness.
• Figure illustrates the boundary layer thickness. There are no
general equations for boundary layer thickness.
• Specific equations exist for certain types of boundary layer.
• For a general boundary layer satisfying minimum boundary
conditions:
u (0)  0;
u ( )  u ;
u
y
The velocity profile that satisfies above conditions:
 2 y y2 
u  u 
 2 
  
0
y 
Similarity Solution for Flat Plate Boundary Layer
u
1 u
* u
u
v

*
*
*2
x
y
Re L y
*
*
2 *
*


*
u  *
&
v  *
y
x
2
2
3
  
  
1 
 * *2 
*
*
*
*3
y x y x y
Re L y
*
Similarity variables :

f   
u
x *
u
&
u
y
x *
  
  
1 
 * *2 
*
*
*
*3
y x y x y
Re L y
2
2
3
Substitute similarity variables:
3
2
d f
d f
2 3f
0
2
d
d
Boundary conditions:
df
 f 0  0
d  0
and
df
1
d  
Blasius Similarity Solution
•Conclusions from the Blasius solution:
 
x
,
  
and
 
1
u
Further analysis shows that:

5.5

x
Re x
Where:
u x
Re x 

Variation of Reynolds numbers
All Engineering Applications
Laminar Velocity Boundary Layer
The velocity boundary layer thickness for laminar flow over a
flat plate:
5.5
 x
Re x
as u∞ increases, δ decreases (thinner boundary layer)
The local friction coefficient:
and the average friction coefficient over some distance x:
Methods to evaluate convection heat transfer
• Empirical (experimental) analysis
– Use experimental measurements in a controlled lab setting to
correlate heat and/or mass transfer in terms of the appropriate nondimensional parameters
• Theoretical or Analytical approach
– Solving of the boundary layer equations for a particular geometry.
– Example:
• Solve for 
• Use evaluate the local Nusselt number, Nux
• Compute local convection coefficient, hx
• Use these (integrate) to determine the average convection
coefficient over the entire surface
– Exact solutions possible for simple cases.
– Approximate solutions also possible using an integral method
Empirical method
T , U 
Twing surface
• How to set up an experimental test?
• Let’s say you want to know the heat transfer rate of an airplane
wing (with fuel inside) flying at steady conditions………….
• What are the parameters involved?
– Velocity,
–wing length,
– Prandtl number,
–viscosity,
– Nusselt number,
• Which of these can we control easily?
• Looking for the relation:
Nu  C Re mL Pr n
Experience has shown the following relation works well:
Experimental test setup
 Power input
T , U 
L
insulation
T , U 
•Measure current (hence heat transfer) with various fluids and test conditions for
•Fluid properties are typically evaluated at the mean film temperature
Similarity Variables
Laminar Thermal Boundary Layer: Blasius Similarity Solution
2




1


*
*
u
v

*
*
*2
x
y
Re L Pr y
Boundary conditions:
 0  0
    1
Ts  T
Similarity Direction

Direction of similarity
d  Pr d

f
0
2
d
2 d
2
u
y
x
This differential equation can be solved by numerical integration.
One important consequence of this solution is that, for pr >0.6:

1/ 3
 0.332 pr
  0
Local convection heat transfer coefficient:
 T  Ts  
hTs  T   k fluid 
 *
 L  y
k fluid 
hx 
*
L y
y* 0
y * 0
hx  k fluid
 u  
 
 x    0
Local Nusselt number:
hx x

 u  
 u x  
Nux 
x  
 
 Re x

k fluid
  0
 x    0
     0
hx x
Nux 
 0.332 Re x pr1/ 3
k fluid
Average heat transfer coefficient:
L
havg
L
1
1 k fluid
  hx dx  
0.332 Re x pr1/ 3dx
L0
L0 x
L
havg
1 k fluid
u 1/ 3 dx

0.332
pr 
L x

x
0
havg  2hx
Nuavg 
havg L
k fluid
 0.664 Re L pr 1/ 3
pr  0.6
For large Pr (oils):
For small Pr (liquid metals):

 th
y
 th

y
x
x
Pr > 1000
Pr < 0.1
Fluid viscosity greater
than thermal diffusivity

Fluid viscosity less than
thermal diffusivity

A single correlation, which applies for all Prandtl numbers,
Has been developed by Churchill and Ozoe..
Nu x 
0.338 Re x pr 1/ 3
  0.0468  2 3 
1  
 
  pr  


Nuavg  2 Nu x
1
4
Pex  100
Transition to Turbulence
• When the boundary layer changes from a laminar flow to a turbulent
flow it is referred to as transition.
• At a certain distance away from the leading edge, the flow begins to
swirl and various layers of flow mix violently with each other.
• This violent mixing of the various layers, it signals that a transition
from the smooth laminar flow near the edge to the turbulent flow
away from the edge has occurred.
Flat Plate Boundary Layer Trasition
Important point:
–Typically a turbulent boundary
layer is preceded by a laminar
boundary layer first upstream
– need to consider case with
mixed boundary layer
conditions!
L
1  xc

hx    hlam dx   hturb dx 
L0

xc
Turbulent Flow Regime
• For a flat place boundary layer becomes turbulent at Rex ~
5 X 105.
• The local friction coefficient is well correlated by an
expression of the form
1
C f , x  0.059 Re x
Re x  107
5
4
Local Nusselt number:
Nux  0.029 Re x 5 pr1/ 3
0.6  pr  60
Mixed Boundary Layer
• In a flow past a long flat plate initially, the boundary layer
will be laminar and then it will become turbulent.
• The distance at which this transitions starts is called critical
distance (Xc) measured from edge and corresponding
Reynolds number is called as Critical Reynolds number.
• If the length of the plate (L) is such that 0.95  Xc/L  1,
the entire flow is approximated as laminar.
• When the transition occurs sufficiently upstream of the
trailing edge, Xc/L  0.95, the surface average coefficients
will be influenced by both laminar and turbulent boundary
layers.
Leading
Edge
havg , L
havg , L
Xc
L
Trailing
Edge
xc
L


1


  hlam, x dx   hturb, x dx 
L

xc
0

1 x
4 L

2 c
k
dx
 13
 u 
 u  5 dx 
 0.332   1  0.0296   1 dx pr
L
  0x 2
   xc x 5 


On integration:
Nuavg , L
4
4
1
1




 0.664 Re xc2  0.037 Re L 5  Re xc5  pr 3



Nuavg , L  0.037 Re L 5  A pr


4
1
3
For a smooth flat plate: Rexc = 5 X 105
Nuavg , L
4
1


 0.037 Re L 5  871 pr 3


For very large flat plates: L >> Xc, in general for ReL > 108
4
Nuavg , L  0.037 Re L 5 pr
1
3
Cylinder in Cross Flow
Cylinder in Cross Flow
Smooth circular cylinder
where
Valid over the ranges 10 < Rel < 107 and 0.6 < Pr < 1000
Array of Cylinders in Cross Flow
• The equivalent diameter is calculated as four
times the net flow area as layout on the tube
bank (for any pitch layout) divided by the wetted
perimeter.
For square pitch:
For triangular pitch:
Number of tube centre lines in a Shell:
Ds is the inner diameter of the shell.
Flow area associated with each tube bundle between baffles is:
where A s is the bundle cross flow area, Ds is the inner diameter of
the shell, C is the clearance between adjacent tubes, and B is the
baffle spacing.
the tube clearance C is expressed as:
Then the shell-side mass velocity is found with
Gshell
m shell

As
Shell side Reynolds Number:
Shell-Side Heat Transfer Coefficient