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Transcript
Convection
Prepared by: Nimesh Gajjar
CONVECTIVE HEAT TRANSFER
20°C
5 m/s
AIR
.
Q
20°C
a) Forced Convection
AIR
.
Q
Warmer air
rising
b) Free Convection
No convection
current
AIR
c) Conduction
Convection heat transfer involves
• fluid motion
• heat conduction
The fluid motion enhances the heat
transfer, since it brings hotter and cooler
chunks of fluid into contact, initiating
higher rates of conduction at a greater
number of sites in fluid. Therefore, the rate
of heat transfer through a fluid is much
higher by convection than it is by
conduction. In fact, the higher the fluid
velocity, the higher the rate of heat
transfer.
Convection heat transfer strongly depends on
• fluid properties dynamic viscosity , thermal conductivity k, density
and specific heat
• fluid velocity V
• Geometry and the roughness of the solid surface
• Type of fluid flow (such as being laminar or turbulent).
NEWTON’S LAW OF COOLING
Qconv  hAs  Ts  T 
h = Convection heat transfer coefficient
As = Heat transfer surface area
Ts = Temperature of the surface
T= Temperature of the fluid sufficiently far from the surface
LOCAL HEAT FLUX

qconv
  hl Ts  T 
qconv
hl is the local convection coefficient
TOTAL HEAT TRANSFER RATE Qconv
Qconv 
 q
conv
As
Qconv  Ts  T 
dAs
1
h
As

 h dA
l
s
As
hl dAs
As
Local and total convection transfer (a) Surface of arbitrary shape. (b) Flat plate.
q”
V ,T
U  ,T
dAs
q”
As, Ts
As, Ts
x
dx
L
A fluid flowing over a stationary surface comes to a complete stop at
the surface because of the no-slip condition.
Uniform
approach
velocity, V
Relative velocity
of fluid layers
Zero velocity
at the surface
Solid Block
A similar phenomenon occurs for the temperature. When two bodies at
different temperatures are brought into contact, heat transfer occurs until
both bodies assume the same temperature at the point of contact.
Therefore, a fluid and a solid surface will have the same temperature at the
point of contact. This is known as NO-TEMPERATURE-JUMP CONDITION.
An implication of the no-slip and the no-temperature jump conditions
is that heat transfer from the solid surface to the fluid layer adjacent
to the surface is by pure conduction, since the fluid layer is motionless,
qconv  qcond  k fluid
T
dy
y 0
T represents the temperature distribution in the fluid  T y  y 0 is the
temperature gradient at the surface.
  hl Ts  T 
qconv
h
 k fluid  T y  y  0
Ts  T
NUSSELT NUMBER
hLc
Nu 
k
k is the thermal conductivity of the fluid
Lc is the characteristic length
Heat transfer through a fluid layer of
thickness L and temperature difference
T2
Heat transfer through the fluid layer
will be by convection when the fluid
involves some motion and by
conduction when the fluid layer is
motionless.
.
Fluid
layer
qconv  hT
Q
T1
ΔT = T2 – T1
qcond  k
T
L
qconv
hL
hT


 Nu
qcond k T L
k
Nusselt number - enhancement of heat transfer through a fluid layer as a result of
convection relative to conduction across the same fluid layer.
Larger the Nusselt number, the more effective the convection.
Nu = 1 for a fluid layer - heat transfer across the layer by pure conduction
Internal and external flows
EXTERNAL FLOW - The flow of an unbounded fluid over a surface such
as a plate, a wire, or a pipe
INTERNAL FLOW - flow in a pipe or duct, if the fluid is completely
bounded by solid surfaces
External flow
Air
Internal flow
Water
Laminar versus Turbulent Flow
Some flows are smooth and orderly while others are rather chaotic. The highly
ordered fluid motion characterized by smooth streamlines is called laminar. The flow
of high-viscosity fluids such as oils at low velocities is typically laminar.
The highly disordered fluid motion that typically occurs at high velocities
characterized by velocities fluctuations is called turbulent. The flow of low-viscosity
fluids such as air at high velocities is typically turbulent. The flow regime greatly
influences the heat transfer rates and the required power for pumping
Laminar
Dye
Pipe
Transitional
Q = VA
Dye Streak
Turbulent
Smooth well rounded
Entrance
Osborne Reynolds in 1880’s, discovered that the flow regime depends mainly on the
ratio of the inertia forces to viscous forces in the fluid.
The Reynolds number can be viewed as the ratio of the inertia forces to
viscous forces acting on a fluid volume element.
Inertia forces VLc  VLc
Re 


Viscous


One, Two and Three Dimensional Flows
V(x, y, z) in cartesian or V(r, , z) in cylindrical coordinates
velocity profile
(remains unchanged)
V(r)
r
z
One-dimensional flow in a circular pipe
VELOCITY BOUNDARY LAYER
Laminar boundary-layer
V
Transition
Turbulent boundary-layer
region
U
U
Turbulent layer
y
Buffer layer
Laminar layer
x
xcr
V
Boundary-layer thickness, δ
Relative velocity of
fluid layers
U  V
INVISCID FLOW REGION
0.99V
δ
Zero velocity at the
surface
(No slip condition)
BOUNDARY LAYER
REGION
Development of a boundary layer on a surface is due to the no-slip condition
Surface Shear Stress
Skin friction coefficient
u
s  
y
s  Cf
y 0
V 2
Friction force over the entire surface
2
Ff  C f As
V 2
2
THERMAL BOUNDARY LAYER
Free Stream
T
Uniform
Temperature T
t
Ts
Thermal
boundary layer
Ts  0.99  (T  Ts )
Thermal boundary layer on a flat plate (the fluid is hotter than the plate surface)
The thickness of the thermal boundary layer, at any location along the
surface is define as the distance from the surface at which the
temperature difference T – Ts equals 0.99(T– Ts).
For the special case of Ts = 0, we have T = 0.99 at the outer edge of the
thermal boundary layer, which is analogous to u = 0.99 for the velocity
boundary layer.
 Shape of the temperature profile in the thermal boundary layer dictates
the convection heat transfer between a solid surface and the fluid flowing
over it.
 In flow over a heated (or cooled) surface, both velocity and thermal
boundary layers will develop simultaneously.
 Noting that the fluid velocity will have a strong influence on the
temperature profile, the development of the velocity boundary layer
relative to the thermal boundary layer will have a strong effect on the
convection heat transfer.
PRANDTL NUMBER
The relative thickness of the velocity and the thermal boundary layers is
described by the dimensionless parameter Prandtl number, defined as
Molecular diffusivity of momentum  C p
Pr 
 
Molecular diffusivity of heat

k
TYPICAL RANGES OF PRANDTL NUMBERS FOR COMMON FLUIDS
Fluid
Pr
Liquid metals
0.004-0.030
Gases
0.7-1.0
Water
1.7-13.7
Light organic fluids
5-50
Oils
50-100,000
Glycerin
2000-100,000
 v,x 
La min ar :
 v,x 
Turbulent :
5x
Re1x 2
0.382 x
C f ,x 
and
and
C f ,x 
Re1x 5
AVERAGE SKIN FRICTION COEFFICIENT
0.664
Re1x 2
0.0592
Re1x 5
, Rex  5  105
, 5  105  Rex  107
L
1
C f   C f ,x dx
L0
La min ar :
Cf 
L

1 0.664
dx
L 0 Re1x 2
L
0.664  Vx 

 
L 0   
0.664  V 

 
L  
1 2
1 2
Turbulent :
dx
L
x1 2
2  0.664  VL 

 
L
 
1.328
 12
ReL
1
2
1 2
0
Cf 
1.328
Re1L 2
0.074
Re1L 5
Re  5  105
5  105  ReL  107
Problem: Engine oil at 60° C flows over the upper surface of a 5-m long flat plate
whose temperature is 20° C with a velocity of 2 m/s (Fig 2.12). Determine the total
drag force and the rate of heat transfer per unit width of the entire plate.
T  60 C
V2m s
Oil
Ts  20 C
A
L= 5 m
Known: Engine oil flows over a flat plate.
Find: The total drag force and the rate of heat transfer per unit width of the plate are
to be determined.
Assumptions:
The flow is steady and incompressible.
The critical Reynolds number is . Recr  5 105
T f  Ts  T  2   20  60  2  40o C
  876 kg m3
Pr  2870
k  0.144 W m   C
  242  106 m 2 s
ReL 
VL


 2 m s  5 m 
242 10-6 m 2 s
C f  1.328 ReL0.5  1.328   41322
FD  C f As
Nu 
V 2
2

 6.533 103  5 1m2

which is less than the
critical Reynolds number
 41322
0.5
 6.533103
876 kg m   2 m s 
3
2
2
 1N

= 57.23 N

2 

 1kg  m s 
hL
0.5
 0.664 ReL0.5 Pr1 3  0.664   41322   28701 3  1918
k
0.144 W m   C
k
2 o
h  Nu =
1918  55.2 W m  C
L
5m



Q  hAs T  Ts   55.2 W m2  C 5 1m2  60  20  C =11, 049 W
o
o
Note that, heat transfer is always from the higher-temperature medium to the lowertemperature one. In this case, it is from the oil to the plate. The heat transfer rate is per
m width of the plate. The heat transfer for the entire plate can be obtained by
multiplying the value obtained by the actual width of the plate.
Problem: A long 10-cm diameter steam pipe whose external surface
temperature is 110oC passes through some open area that is not
protected against the winds (Fig. 2.23). Determine the rate of heat
loss from the pipe per unit length of its length when the air is at 1
atm pressure and 10oC and the wind is blowing across the pipe at a
velocity of 8 m/s.
Schematic:
Wind
Known: A steam pipe is exposed to windy air.
Find: The rate of heat loss from the steam is to be determined.
Ts = 110o C
Assumptions:
•Steady operating conditions exist.
•Radiation effects are negligible.
•Air is an ideal gas.
The properties of air at the average film temperature of
T f  Ts  T  2
k  0.02808 W m  C
= (110 + 10)/2 =60
Pr = 0.7202
 = 1.896  10-5 m2 s
Re 
Nu 
VD


8m s  0.1m 
 4.219  104
1.896  105 m2 s
12
13
0.62 Re Pr
hD
 0.3 
14
k
1   0.4 Pr 2 3 


 0.3 

0.62 4.219  104
= 124.44

12
  Re 5 8 
1  
 
282
,
000

 

 0.7202 1 3 
1   0.4 0.7202 

2 3 1 4

45
45
4 5 8 

1   4.219  10 
  282,000 




h
k
0.02808 W m C
Nu =
124.44   34.94 W m 2 C
D
0.1m
As  pL   DL    0.1m 1m   0.314 m 2



Q  hAs Ts  T   34.94 W m 2 C 0.314 m 2 110  10  C = 1097.3 W
Important numbers
Nusselt number
Reynolds number
Prandtl number
Grashof number
Biot number
hL
Nu 
k
Re L 
Pr 
Gr 
Convection
Conduction
UoL
Inertial force

Viscous force

Momentum diffusivity
a
Thermal diffusivity
g Ts  T L3
hL
Bi 
k solid
2
Buoyancy forces
Viscous forces
thermal internal resistance
surface film resistance
Reference book: Fundamentals of Heat and Mass Transfer, Incropera & DeWitt