Convection in Boundary Layers

1
Convection in Boundary Layers

• P M V Subbarao
• Associate Professor
• Mechanical Engineering Department
• IIT Delhi

A tiny layer but very significant..
2
Momentum Vs Thermal Effects
n Potential for diffusion of momentum change
(Deficit or excess) created by a solid
boundary. a Potential for Diffusion of thermal
changes created by a solid boundary.
Prandtl Number The ratio of momentum diffusion
to heat diffusion.
Other scales of reference
Length of plate L Free stream velocity uoo
3
(No Transcript)
4
This dimensionless temperature gradient at the
wall is named as Nusselt Number
Local Nusselt Number
5
Average Nusselt Number
6
Computation of Dimensionless Temperature Profile

• First Law of Thermodynamics for A CV
• Energy Equation for a CV
• How to select A CV for External Flows ?

Relative sizes of Momentum Thermal Boundary
Layers
7
Liquid Metals Pr ltltlt 1
y
1.0
8
y
1.0
Gases Pr 1.0
9
y
1.0
Water 2.0 lt Pr lt 7.0
10
y
1.0
OilsPr gtgt 1
11
The Boundary Layer A Control Volume
For pr lt 1
12
Reynolds Transport Theorem

• Total rate of change of any extensive property B
  of a system(C.M.) occupying a control volume C.V.
  at time t is equal to the sum of
• a) the temporal rate of change of B within the
  C.V.
• b) the net flux of B through the control surface
  C.S. that surrounds the C.V.

The relation between A CM and CV for conservation
of any extensive property B.
13
Conservation of Mass

• Let b1, the B mass of the system, m.

The rate of change of mass in a control mass
should be zero.
Above integral is true for any shape and size of
the control volume, which implies that the
integrand is zero.
14
Conservation of Momentum

• Let bV, the B momentum of the system, mV.

The rate of change of momentum for a control mass
should be equal to resultant external force.
Momentum equation of per unit volume
15
For a boundary layer
For an incompressible flow
16
Conservation of Energy

• Let be, the B Energy of the system, me.

The rate of change of energy of a control mass
should be equal to difference of work and heat
transfers.
Energy equation per unit volume
17
Using the law of conduction heat transfer
The net Rate of work done on the element is
From Momentum equation N S Equations
18
Then
19
For an Incompressible fluid
Substitute the work done by shear stress
This is called the first law of thermodynamics
for fluid motion.
20
Invoking conservation of mass
First law for a fluid motion
21
F is called as viscous dissipation.
22
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.
23
The governing equations for steady two
dimensional incompressible fluid flow with
negligible viscous dissipation
24
Boundary Conditions
25
Scale Analysis
Define characteristic parameters L length u 8
free stream velocity T 8 free stream
temperature
26
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)
27
Define dimensionless variables
28
Similarity parameters can be derived that relate
one set of flow conditions to geometrically
similar surfaces for a different set of flow
conditions
29
(No Transcript)
30
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 thicknesses describe the shape of
  the boundary layer.

31
Boundary Layer Thickness

• The boundary layer thickness, signified by , is
  simply the thickness of the viscous boundary
  layer region.
• Because 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), we define the
  edge of the boundary layer to be the point at
  which the fluid velocity equals 99 of the
  free-stream velocity

32

• 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

The velocity profile that satisfies above
conditions
33
Further analysis shows that
Where
34
Variation of Reynolds numbers
35
Laminar Velocity Boundary Layer
The velocity boundary layer thickness for laminar
flow over a flat plate
as u8 increases, d decreases (thinner boundary
layer)
The local friction coefficient
and the average friction coefficient over some
distance x
36
Laminar Thermal Boundary Layer
Boundary conditions
37
This differential equation can be solved by
numerical integration. One important consequence
of this solution is that, for pr gt0.6
Local convection heat transfer coefficient
38
Local Nusselt number
39
Average heat transfer coefficient
40
A single correlation, which applies for all
Prandtl numbers, Has been developed by Churchill
and Ozoe..
41
Turbulent Flow

• 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

Local Nusselt number
Local Sherwood number