Industrial Utilization of Theory of Thermal Boundary Layers
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Transcript Industrial Utilization of Theory of Thermal Boundary Layers
Enhancement of Heat Transfer
P M V Subbarao
Associate Professor
Mechanical Engineering Department
IIT Delhi
Invention of Compact Heat Transfer Devices
Heat transfer enhancement
• Enhancement
• Increase the convection coefficient
Introduce surface roughness to enhance turbulence.
Induce swirl.
• Increase the convection surface area
Longitudinal fins, spiral fins or ribs.
Heat Transfer Enhancement using Inserts
Heat Transfer Enhancement using Inserts
Heat transfer enhancement :Coiling
• Helically coiled tube
• Without inducing turbulence or additional heat transfer
surface area.
• Secondary flow
FREE CONVECTION
P M V Subbarao
Associate Professor
Mechanical Engineering Department
IIT Delhi
Its free, No operating cost!……..
Its Natural …..
Natural Convection
Where we’ve been ……
• Up to now, have considered forced convection, that is
an external driving force causes the flow.
Where we’re going:
• Consider the case where fluid movement is by
buoyancy effects caused by temperature differential
Events due to natural convection
•
•
•
•
Weather events such as a thunderstorm
Glider planes
Radiator heaters
Hot air balloon
• Heat flow through and on outside of a double pane
window
• Oceanic and atmospheric motions
• Coffee cup example ….
Small velocity
Natural Convection
• New terms
– Volumetric thermal expansion coefficient
– Grashof number
– Rayleigh number
• Buoyancy is the driving force
– Stable versus unstable conditions
• Nusselt number relationship for laminar free
convection on hot or cold surface
• Boundary layer impacts: laminar turbulent
Buoyancy is the driving force
• Buoyancy is due to combination of
– Differences in fluid density
– Body force proportional to density
– Body forces namely, gravity, also Coriolis force in
atmosphere and oceans
• Convection flow is driven by buoyancy in unstable conditions
• Fluid motion may be
(no constraining surface) or along a surface
Buoyancy is the driving force
• Free boundary layer flow
Heated wire or hot pipe
A heated vertical plate
• We focus on free convection flows bounded by a surface.
• The classic example is
Ts T
Extensive, quiescent
fluid
u(x,y)
Ts
T
g
x
y
u
v
Governing Equations
• The difference between the two flows (forced flow and free
flow) is that, in free convection, a major role is played by
buoyancy forces.
X g
Very important
•Consider the x-momentum equation.
u
u
1 P
2u
u v
g 2
x
y
x
y
•As we know, p / y 0 , hence the x-pressure gradient in the
boundary layer must equal that in the quiescent region outside the
boundary layer.
Pascal Law :
P
- g
x
u
u
1
2u
u v g g 2
x
y
y
u
u
u
2
u
v
g
x
y
y
2
Buoyancy force
Governing Equations
• Define , the volumetric thermal
expansion coefficient.
1
T P
For an ideal gas : P
RT
1
Thus :
T
P
RT
Not for liquids and non-ideal gases
1
1
T
T T
(T T )
Density gradient is due to the temperature gradient
Governing Equations (cont’d)
• Now, we can see buoyancy effects replace pressure gradient in the
momentum equation.
u
u
2u
u v g (T T ) v 2
x
y
y
•The buoyancy effects are confined to the momentum equation, so
the mass and energy equations are the same.
u v
0
x y
T
T
T u
v
2
x
y
y
c p y
2
u
2
Strongly coupled and must be solved simultaneously
Dimensionless Similarity Parameter
x
x
L
u
u
u0
and
and
y
y
L
v
v
u0
T T
T
Ts T
*
where L is a characteri stic length, and
u 0 is an arbitrary reference velocity
• The x-momentum and energy equations are
*
*
2 *
g
(
T
T
)
L
u
u
1
u
*
*
*
s
u
v
T
*
*
2
x
y
u0
Re L y *2
*
*
2 *
T
T
1
T
u * * v* *
x
y
Re L Pr y *2
Dimensionless Similarity Parameter (cont’d)
• Define new dimensionless parameter,
g (Ts T ) L u0 L
g (Ts T ) L
GrL
2
2
u0
2
3
•Grashof number in natural convection is analogous to the Reynolds
number in forced convection.
•Grashof number indicates the ratio of the buoyancy force to the
viscous force.
•Higher Gr number means increased natural convection flow
GrL
1 forced
2
Re L
GrL
1
2
Re L
natural
Laminar Free Convection on Vertical Surface
• As y : u = 0, T = T
• As y 0 : u = 0, T = Ts
Ts T
u(x,y)
Ts
• With little or no external driving flow,
Re 0 and forced convection effects
can be safely neglects
T
g
x
y
u
v
GrL
1
2
Re L
Nu L f (GrL , Pr)
Analytical similarity solution for the local Nusselt number in
laminar free convection
1/ 4
hx GrL
Nu x
k 4
f (Pr)
Where
f Pr
0.75 Pr
0.609 1.221
Average Nusselt # =
Pr 1.238 Pr
1/ 4
1/ 4
h L 4 GrL
NuL
k
3 4
f (Pr)
Effects of Turbulence
• Just like in forced convection flow, hydrodynamic instabilities
may result in the flow.
• For example, illustrated for a heated vertical surface:
• Define the Rayleigh number for relativemagnitude of
buoyancy and viscous forces
Ra x ,c Grx ,c Pr
g (Ts T ) x
3
Ts T
Effects of Turbulence
• Transition to turbulent flow greatly effects heat transfer rate.
Empirical Correlations
Typical correlations for heat transfer coefficient developed from
experimental data are expressed as:
hL
NuL
CRa Ln
k
Ra L GrL Pr
g Ts T L3
n 1 / 4
n 1 / 3
For Turbulent
For Laminar
Vertical Plate at constant Ts
Log10 Nu L
Log10 RaL
•Alternative applicable to entire Rayleigh number range (for constant Ts)
Nu L 0.825
1 (0.492 / Pr) 9 /16
0.387 Ra1L/ 6
8 / 27
Vertical Cylinders
•Use same correlations for vertical flat plate if:
D ~ 35
1/ 4
L GrL
2
Inclined Plate
Horizontal Plate
Cold Plate (Ts < T)
Hot Plate (Ts > T)
Empirical Correlations : Horizontal Plate
•Define the characteristic length, L as
As
L
P
•Upper surface of heated plate, or Lower surface of cooled plate :
Nu L 0.54 Ra1L/ 4
Nu L 0.15 Ra1L/ 3
104 RaL 107
107 RaL 1011
•Lower surface of heated plate, or Upper surface of cooled plate :
1/ 4
Nu L 0.27 RaL
10
5
RaL 10
Note: Use fluid properties at the film temperature
10
Ts T
Tf
2
Empirical Correlations : Long Horizontal Cylinder
•Very common geometry (pipes, wires)
•For isothermal cylinder surface, use general form equation
for computing Nusselt #
hD
NuD
CRa Dn
k
Constants for general Nusselt number Equation
RaD
C
n
1010 - 10 2
0.675
0.058
10 2 - 10 2
1.02
0.148
102 - 104
0.850
0.188
104 - 107
0.480
0.250
107 - 1012
0.125
0.333