convection in flat plate boundary layers
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Convection in Flat Plate Boundary Layers](https://reader036.vdocument.in/reader036/viewer/2022081506/568145a4550346895db2987b/html5/thumbnails/1.jpg)
Convection in Flat Plate Boundary Layers
P M V Subbarao
Associate Professor
Mechanical Engineering Department
IIT Delhi
A Universal Similarity Law ……
![Page 2: Convection in Flat Plate Boundary Layers](https://reader036.vdocument.in/reader036/viewer/2022081506/568145a4550346895db2987b/html5/thumbnails/2.jpg)
Hyper sonic Plane
![Page 3: Convection in Flat Plate Boundary Layers](https://reader036.vdocument.in/reader036/viewer/2022081506/568145a4550346895db2987b/html5/thumbnails/3.jpg)
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.
![Page 4: Convection in Flat Plate Boundary Layers](https://reader036.vdocument.in/reader036/viewer/2022081506/568145a4550346895db2987b/html5/thumbnails/4.jpg)
The governing equations for steady two dimensional incompressible fluid flow with negligible viscous dissipation:
![Page 5: Convection in Flat Plate Boundary Layers](https://reader036.vdocument.in/reader036/viewer/2022081506/568145a4550346895db2987b/html5/thumbnails/5.jpg)
Boundary Conditions
0
0
Twall
u
0
T
![Page 6: Convection in Flat Plate Boundary Layers](https://reader036.vdocument.in/reader036/viewer/2022081506/568145a4550346895db2987b/html5/thumbnails/6.jpg)
Scale Analysis
Define characteristic parameters:
L : length
u ∞ : free stream velocity
T ∞ : free stream temperature
![Page 7: Convection in Flat Plate Boundary Layers](https://reader036.vdocument.in/reader036/viewer/2022081506/568145a4550346895db2987b/html5/thumbnails/7.jpg)
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)
![Page 8: Convection in Flat Plate Boundary Layers](https://reader036.vdocument.in/reader036/viewer/2022081506/568145a4550346895db2987b/html5/thumbnails/8.jpg)
Define dimensionless variables:
L
xx *
L
yy *
u
uu*
u
vv*
s
s
TT
TT
2*
u
pp
Lu
Re
Similarity Parameters:
Pr PrRePe
![Page 9: Convection in Flat Plate Boundary Layers](https://reader036.vdocument.in/reader036/viewer/2022081506/568145a4550346895db2987b/html5/thumbnails/9.jpg)
0*
*
*
*
y
v
x
u
2*
*2
*
*
*
**
*
**
Re
1
y
u
x
p
y
vv
x
uu
L
2*
2
**
**
PrRe
1
yyv
xu
L
0*
*
y
p
![Page 10: Convection in Flat Plate Boundary Layers](https://reader036.vdocument.in/reader036/viewer/2022081506/568145a4550346895db2987b/html5/thumbnails/10.jpg)
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.
![Page 11: Convection in Flat Plate Boundary Layers](https://reader036.vdocument.in/reader036/viewer/2022081506/568145a4550346895db2987b/html5/thumbnails/11.jpg)
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 free-stream velocity:
uu
y99.0
uu 99.0
![Page 12: Convection in Flat Plate Boundary Layers](https://reader036.vdocument.in/reader036/viewer/2022081506/568145a4550346895db2987b/html5/thumbnails/12.jpg)
• 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:
0 ;)( ;0)0(
y
y
uuuu
The velocity profile that satisfies above conditions:
2
22
yy
uu
![Page 13: Convection in Flat Plate Boundary Layers](https://reader036.vdocument.in/reader036/viewer/2022081506/568145a4550346895db2987b/html5/thumbnails/13.jpg)
Similarity Solution for Flat Plate Boundary Layer
2*
*2
*
**
*
**
Re
1
y
u
y
uv
x
uu
L
**
** &
xv
yu
Similarity variables :
**
& x
uy
ux
u
f
3*
3
2*
2
***
2
* Re
1
yyxyxy L
![Page 14: Convection in Flat Plate Boundary Layers](https://reader036.vdocument.in/reader036/viewer/2022081506/568145a4550346895db2987b/html5/thumbnails/14.jpg)
022
2
3
3
d
fdf
d
fd
Substitute similarity variables:
Boundary conditions:
1 and 000
d
dff
d
df
3*
3
2*
2
***
2
* Re
1
yyxyxy L
![Page 15: Convection in Flat Plate Boundary Layers](https://reader036.vdocument.in/reader036/viewer/2022081506/568145a4550346895db2987b/html5/thumbnails/15.jpg)
Blasius Similarity Solution
u
1 and , x
•Conclusions from the Blasius solution:
![Page 16: Convection in Flat Plate Boundary Layers](https://reader036.vdocument.in/reader036/viewer/2022081506/568145a4550346895db2987b/html5/thumbnails/16.jpg)
Further analysis shows that:
xx Re
5.5
Where:
xu
xRe
![Page 17: Convection in Flat Plate Boundary Layers](https://reader036.vdocument.in/reader036/viewer/2022081506/568145a4550346895db2987b/html5/thumbnails/17.jpg)
Variation of Reynolds numbers
All Engineering Applications
![Page 18: Convection in Flat Plate Boundary Layers](https://reader036.vdocument.in/reader036/viewer/2022081506/568145a4550346895db2987b/html5/thumbnails/18.jpg)
Laminar Velocity Boundary Layer
The velocity boundary layer thickness for laminar flow over a flat plate:
as u∞ increases, δ decreases (thinner boundary layer)
The local friction coefficient:
and the average friction coefficient over some distance x:
x
xRe
5.5
![Page 19: Convection in Flat Plate Boundary Layers](https://reader036.vdocument.in/reader036/viewer/2022081506/568145a4550346895db2987b/html5/thumbnails/19.jpg)
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 non-dimensional 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
![Page 20: Convection in Flat Plate Boundary Layers](https://reader036.vdocument.in/reader036/viewer/2022081506/568145a4550346895db2987b/html5/thumbnails/20.jpg)
Empirical method
• 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:
Experience has shown the following relation works well:
UT ,
surface wingT
nmLCNu PrRe
![Page 21: Convection in Flat Plate Boundary Layers](https://reader036.vdocument.in/reader036/viewer/2022081506/568145a4550346895db2987b/html5/thumbnails/21.jpg)
Experimental test setup
UT ,inputPower
insulation
L
UT ,
•Measure current (hence heat transfer) with various fluids and test conditions for
•Fluid properties are typically evaluated at the mean film temperature
![Page 22: Convection in Flat Plate Boundary Layers](https://reader036.vdocument.in/reader036/viewer/2022081506/568145a4550346895db2987b/html5/thumbnails/22.jpg)
![Page 23: Convection in Flat Plate Boundary Layers](https://reader036.vdocument.in/reader036/viewer/2022081506/568145a4550346895db2987b/html5/thumbnails/23.jpg)
Similarity Variables
![Page 24: Convection in Flat Plate Boundary Layers](https://reader036.vdocument.in/reader036/viewer/2022081506/568145a4550346895db2987b/html5/thumbnails/24.jpg)
Laminar Thermal Boundary Layer: Blasius Similarity Solution
Boundary conditions: 1 00
2*
2
**
**
PrRe
1
yyv
xu
L
TTs
![Page 25: Convection in Flat Plate Boundary Layers](https://reader036.vdocument.in/reader036/viewer/2022081506/568145a4550346895db2987b/html5/thumbnails/25.jpg)
Similarity Direction
Direction of similarity
x
uy
0 2
Pr2
2
d
df
d
d
![Page 26: Convection in Flat Plate Boundary Layers](https://reader036.vdocument.in/reader036/viewer/2022081506/568145a4550346895db2987b/html5/thumbnails/26.jpg)
This differential equation can be solved by numerical integration.
One important consequence of this solution is that, for pr >0.6:
3/1
0
332.0 pr
Local convection heat transfer coefficient:
0
**
y
fluidx yL
kh
0
**
y
sfluids yL
TTkTTh
![Page 27: Convection in Flat Plate Boundary Layers](https://reader036.vdocument.in/reader036/viewer/2022081506/568145a4550346895db2987b/html5/thumbnails/27.jpg)
Local Nusselt number:
0
x
ukh fluidx
000
Re
xfluid
xx
xu
x
ux
k
xhNu
3/1Re332.0 prk
xhNu x
fluid
xx
![Page 28: Convection in Flat Plate Boundary Layers](https://reader036.vdocument.in/reader036/viewer/2022081506/568145a4550346895db2987b/html5/thumbnails/28.jpg)
Average heat transfer coefficient:
L
xfluid
L
xavg dxprx
k
Ldxh
Lh
0
3/1
0
Re332.011
L
fluidavg
x
dxpr
u
x
k
Lh
0
3/1332.01
xavg hh 2
6.0 Re664.0 3/1 prprk
LhNu L
fluid
avgavg
![Page 29: Convection in Flat Plate Boundary Layers](https://reader036.vdocument.in/reader036/viewer/2022081506/568145a4550346895db2987b/html5/thumbnails/29.jpg)
y
x
th
For large Pr (oils):
Pr > 1000
y
x
th
For small Pr (liquid metals):
Pr < 0.1
Fluid viscosity greater than thermal diffusivity
Fluid viscosity less than thermal diffusivity
![Page 30: Convection in Flat Plate Boundary Layers](https://reader036.vdocument.in/reader036/viewer/2022081506/568145a4550346895db2987b/html5/thumbnails/30.jpg)
A single correlation, which applies for all Prandtl numbers,Has been developed by Churchill and Ozoe..
100
0468.01
Re338.0
41
32
3/1
xx
x Pe
pr
prNu
xavg NuNu 2
![Page 31: Convection in Flat Plate Boundary Layers](https://reader036.vdocument.in/reader036/viewer/2022081506/568145a4550346895db2987b/html5/thumbnails/31.jpg)
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.
![Page 32: Convection in Flat Plate Boundary Layers](https://reader036.vdocument.in/reader036/viewer/2022081506/568145a4550346895db2987b/html5/thumbnails/32.jpg)
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
xcturb
xc
lamx dxhdxhL
h 1
0
![Page 33: Convection in Flat Plate Boundary Layers](https://reader036.vdocument.in/reader036/viewer/2022081506/568145a4550346895db2987b/html5/thumbnails/33.jpg)
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
7x
51
, 10Re Re059.0
xxfC
Local Nusselt number: 60 0.6 Re029.0 3/154
prprNu xx
![Page 34: Convection in Flat Plate Boundary Layers](https://reader036.vdocument.in/reader036/viewer/2022081506/568145a4550346895db2987b/html5/thumbnails/34.jpg)
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.
![Page 35: Convection in Flat Plate Boundary Layers](https://reader036.vdocument.in/reader036/viewer/2022081506/568145a4550346895db2987b/html5/thumbnails/35.jpg)
Xc
L
LeadingEdge Trailing
Edge
L
x
xturb
x
xlamLavg
c
c
dxhdxhL
h ,
0
,,
1
31
51
54
0 21
21
, 0296.0332.0 prdxx
dxu
x
dxu
L
kh
L
x
x
Lavg
c
c
![Page 36: Convection in Flat Plate Boundary Layers](https://reader036.vdocument.in/reader036/viewer/2022081506/568145a4550346895db2987b/html5/thumbnails/36.jpg)
On integration:
31
54
54
21
, ReRe037.0Re664.0 prNucxLcxLavg
31
54
, Re037.0 prANuLLavg
For a smooth flat plate: Rexc = 5 X 105
31
54
, 871Re037.0 prNuLLavg
![Page 37: Convection in Flat Plate Boundary Layers](https://reader036.vdocument.in/reader036/viewer/2022081506/568145a4550346895db2987b/html5/thumbnails/37.jpg)
For very large flat plates: L >> Xc, in general for ReL > 108
31
54
, Re037.0 prNuLLavg
![Page 38: Convection in Flat Plate Boundary Layers](https://reader036.vdocument.in/reader036/viewer/2022081506/568145a4550346895db2987b/html5/thumbnails/38.jpg)
Cylinder in Cross Flow
![Page 39: Convection in Flat Plate Boundary Layers](https://reader036.vdocument.in/reader036/viewer/2022081506/568145a4550346895db2987b/html5/thumbnails/39.jpg)
![Page 40: Convection in Flat Plate Boundary Layers](https://reader036.vdocument.in/reader036/viewer/2022081506/568145a4550346895db2987b/html5/thumbnails/40.jpg)
![Page 41: Convection in Flat Plate Boundary Layers](https://reader036.vdocument.in/reader036/viewer/2022081506/568145a4550346895db2987b/html5/thumbnails/41.jpg)
![Page 42: Convection in Flat Plate Boundary Layers](https://reader036.vdocument.in/reader036/viewer/2022081506/568145a4550346895db2987b/html5/thumbnails/42.jpg)
Cylinder in Cross Flow
![Page 43: Convection in Flat Plate Boundary Layers](https://reader036.vdocument.in/reader036/viewer/2022081506/568145a4550346895db2987b/html5/thumbnails/43.jpg)
Smooth circular cylinder
where
Valid over the ranges 10 < Rel < 107 and 0.6 < Pr < 1000
![Page 44: Convection in Flat Plate Boundary Layers](https://reader036.vdocument.in/reader036/viewer/2022081506/568145a4550346895db2987b/html5/thumbnails/44.jpg)
![Page 45: Convection in Flat Plate Boundary Layers](https://reader036.vdocument.in/reader036/viewer/2022081506/568145a4550346895db2987b/html5/thumbnails/45.jpg)
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.
![Page 46: Convection in Flat Plate Boundary Layers](https://reader036.vdocument.in/reader036/viewer/2022081506/568145a4550346895db2987b/html5/thumbnails/46.jpg)
For square pitch:
For triangular pitch:
![Page 47: Convection in Flat Plate Boundary Layers](https://reader036.vdocument.in/reader036/viewer/2022081506/568145a4550346895db2987b/html5/thumbnails/47.jpg)
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.
![Page 48: Convection in Flat Plate Boundary Layers](https://reader036.vdocument.in/reader036/viewer/2022081506/568145a4550346895db2987b/html5/thumbnails/48.jpg)
the tube clearance C is expressed as:
Then the shell-side mass velocity is found with
s
shellshell A
mG
Shell side Reynolds Number:
![Page 49: Convection in Flat Plate Boundary Layers](https://reader036.vdocument.in/reader036/viewer/2022081506/568145a4550346895db2987b/html5/thumbnails/49.jpg)
Shell-Side Heat Transfer Coefficient