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This led to the introduction of Newton’s equation.
) ( T T
fh
A
Q
s [W] (3.1)T
s: temperature of the wall surfaceTf: temperature of the fluid away from the wall
A
: heat transfer areaTs, Tf and A are measurable quantities. Evaluation of the convective coefficient h then
completes the parameters necessary for heat transfer calculations. The convective coefficient h is evaluated in some limited cases by mathematical analytical methods and in most cases by experiments.
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3.2 Flow equations and boundary layer
In this study, we are going to assume that the student is familiar with the flow governing equations. The interested reader can refer to basic fluid dynamics textbooks for their derivation.
However, for clarity, we are going to state the flow equations to provide a more clear picture for the boundary layer simplifications mentioned below.
Fluid flow is covered by laws of conservation of mass, momentum and energy. The conservation of mass is known as the continuity equation. The conservation of momentum is described by the Navier-Stokes equations while the conservation of energy is described by the energy equation.
The flow equations for two dimensional steady incompressible flow in a Cartesian coordinate system are:
w 0
w w w
y v x u
(Continuity) (3.2a)
Fx
y u x
u x
p y
v u x
u u ¸¸
¹
¨¨ ·
©
§
w
w w
w w
w
¸¸¹
¨¨ ·
©
§
w
w w w
2 2 2
P
2U
(x-momentum) (3.2b)
Fy
y v x
v y
p y
v v x
u v ¸¸¹
¨¨ ·
©
§
w
w w
w w
w
¸¸¹
¨¨ ·
©
§
w
w w w
2 2 2
P
2U
(y-momentum) (3.2c)
)
¸¸
¹
¨¨ ·
©
§
w
w w
¸¸ w
¹
¨¨ ·
©
§
w
w w
w
P
U
22 22y T x
k T y v T x u T cp
(Energy) (3.2d)
where uand vare the flow velocities in the x and y directions,
T
is the temperature,p
is the pressure,U
,P
andc
pare the fluid density, viscosity and specific heat at constant pressure,
F
xandF
yare the body forces in the x and y directions and
)
is the dissipation function.3.2.1 The velocity boundary layer
Figure 3.1 shows fluid at uniform velocity Ufapproaching a plate and the resulting development of the velocity boundary layer. When the fluid particles make contact with the surface they assume zero velocity. These particles then tend to retard the motion of particles in the fluid layer above them which in term retard the motion of particles above them and so on until at a
distance
y G
from the plate this effect becomes negligible and the velocity of the fluid particles is again almost equal to the original free stream velocityUf.Download free ebooks at BookBooN.com 61
The retardation of the fluid motion which results in the boundary layer at the fluid-solid interface is a result of shear stresses (
W
) acting in planes that are parallel to the fluid velocity. The shear stress is proportional to the velocity gradient and is given bydy P du W
(3.3)
The fluid flow as described above is characterized by two distinct regions: a thin fluid layer – the boundary layer – in which velocity gradient and shear stresses are large and an outer region – the free stream – where velocity gradients and shear stresses are negligible.
The quantity
G
seen in the above figure is called the boundary layer thickness. It is formally defined as the value of y at whichUf
U 0.99 (3.4)
The boundary layer velocity profile refers to the way in which the velocity u varies with distance y from the wall through the boundary layer.
3.2.2 Laminar and turbulent boundary layer
In convection problems it is essential to determine whether the boundary layer is laminar or turbulent. The convective coefficient h will depend strongly on which of these conditions exists.
U
Free stream U
Velocity boundary layer
X x Y
y T
U
Free stream U
Velocity boundary layer
X x Y
y T
x
y T
W į W y
x
U
Free stream U
Velocity boundary layer
X x Y
y T
U
Free stream U
Velocity boundary layer
X x Y
y T
x
y T
W į W y
x
Figure 0-1The velocity boundary layer on a flat plate
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There are sharp differences between laminar and turbulent flow conditions. In laminar boundary layers the fluid motion is highly ordered. Fluid particles move along streamlines. In contrast, fluid motion in the turbulent boundary layer is highly irregular. The velocity fluctuations that exist in this regular form of fluid flow result in mixing of the flow and as a consequence enhance the convective coefficient significantly.
Figure 3.2 shows the flow over a flat plate where the boundary layer is initially laminar. At some distance from the leading edge fluid fluctuations begin to develop. This is the transition region.
Eventually with increasing distance from the leading edge complete transition to turbulence occurs. This is followed by a significant increase in the boundary layer thickness and the convective coefficient, see Figures 3.2 and 3.4. Three different regions can be seen in the
turbulent boundary layer. The laminar sublayer, the buffer layer and turbulent zone where mixing dominates. The location where transition to turbulence exists is determined by the value of theReynolds’s number which is a dimensionless grouping of variables.
P U L Re U
f(3.5a)
Where L is the appropriate length scale.
In the case of the flat plate the length is the distance x from the leading edge of the plate.
Therefore,
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v u streamline
y, v x, u
Laminar
Transition
Turbulent
x u•
u•
u•
Turbulent region
Buffer layer Laminar sublayer v
u streamline
y, v x, u
Laminar
Transition
Turbulent
x u•
u•
u•
Turbulent region
Buffer layer Laminar sublayer