Download Principles of Convection

ERT 216
HEAT & MASS TRANSFER
Sem 2/ 2011-2012
Prepared by;
Miss Mismisuraya Meor Ahmad
School of Bioprocess Engineering
University Malaysia Perlis
HEAT TRANSFER
(2) CONVECTION
PRINCIPLE OF
CONVECTION
Point
• Conduction & convection are similar  both require the
presence of a material medium. But convection requires
the presence of fluid motion.
Heat transfer through the:
 Solid is always by conduction
 Liquid & gas can be by conduction or convection
(depending on the presence of any bulk fluid motion)
Point
Convection heat transfer is complicated  involved fluid motion as well as heat
conduction. The fluid motion enhances heat transfer (fluid velocity ↑, the rate
of heat transfer ↑).
Example of forced convection in daily life
.
We resort to forced convection whenever we want to increased the
rate of heat transfer from a hot object:
1) Turn on the fan on hot summer days to help our body cool more effectively
(the higher the fan speed the better we feel)
2) Stir our soup & blow a hot slice of pizza  to make them cool faster
Physical Mechanism
Example  Consider steady heat transfer through a fluid contained
between two parallel plates at different temp.
Assume no fluid motion, the energy of the hotter fluid
molecules near the hot plate is transferred to the adjacent
cooler fluid molecules. This energy is then transfer to the
next layer of the cooler fluid molecules. This energy is
then transferred to the next layer of the cooler fluid, and
so on until it is transferred to the other plate.
This is what
happen during
conduction
through a fluid
If we used a syringe to draw some fluid near the hot plate & inject it next to the cold plate
repeatedly. So that, it will speed up the heat transfer process (energy is carried to the
other side as a result of fluid motion)  convection
Consider, the cooling of a hot block
with a fan blowing air over its top
surface
We know that heat is transferred from
the hot block to the surrounding cooler
air, & the block eventually cools. The
block cools faster if:
1) The fan is switched to higher
speed
2) Replacing air by water
The convection heat transfer strongly depends on:
1) Fluid properties  dynamic viscosity (μ), thermal conductivity
(k), density (ρ), specific heat (cp) & fluid velocity (v)
2) The geometry & the roughness of the solid surface
3) The type of fluid flow (being streamlined or turbulent)
Thus, the convection heat transfer relations to be rather
complex because of the dependence on so many variables.
The convection is the most complex mechanism of heat
transfer.
Physical Mechanism
Although the complexity of convection, the rate of convection is proportional to
the temp. difference and expressed by Newton’s Law of cooling as:
or
Where:
The convection heat transfer coefficient, h defined as the
rate of heat transfer between a solid surface & a fluid per unit
surface area per unit temp. difference. The h depend on several
of the mentioned variable and thus is difficult to determine.
Physical Mechanism
Fluid flow is often restricted by solid surfaces. Important to understand how
the presence of solid surface affects fluid flow.
Consider  the flow of fluid over a solid surface that is nonporous
Observations indicate that a fluid in motion comes to complete stop at the
surface and assume a zero velocity relative to the surface. That is, a fluid in
direct contact with a solid  stick to the surface due to viscous effects &
there is no slip. This is known as the
no-slip condition.
The layer that sticks to the surface slows the adjacent fluid layer because of viscous
forces between the fluid layers, which slow the next layer. The flow region adjacent
to the wall in which the viscous effects & thus velocity gradients are significant is called
 the
boundary layer
A consequence of the no-slip condition is:
1) All velocity profile must have zero value with respect to the surface at the
point of contact between a fluid & solid surface
2) The surface drag which is the force a fluid exerts on a surface in the flow
direction
Physical Mechanism
An implication of the no-slip condition 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 & can be expressed as:
Heat is then convected away from a surface as a result of fluid motion.
Note that convection heat transfer from a solid surface to a fluid is merely the
conduction heat transfer from the solid surface to the fluid layer adjacent to
the surface. So that, connect the equation for the heat flux to obtain:
Used to get h when
the temp. distribution
within the fluid is
known
Nusselt Number
In convection studies, it is common practice to nondimensionalize
the governing equations & combine the variables, which group
together into dimensionless no.  to reduce the no. of total
variables. It is also common practice to nondimensionalize the
heat transfer coefficient, h with Nusselt no., defined as:
K: thermal conductivity of the fluid
Lc: Characteristic length
Nusselt Number
The physical significance of the Nusselt no.
Consider  a fluid layer of thickness L & temp. difference
Heat transfer through the fluid layer is by :
1) Convection (when fluid involved some motion) 
2) Conduction (when the fluid layer is motionless) 
The ratio gives 
Nusselt Number
Represent the enhancement of
heat transfer through a fluid
layer as a result of convection
relative to conduction across
the same fluid layer.
Nu large  the more effective the convection
Nu = 1  heat transfer across the layer by pure conduction
Classification of Fluid Flows
1) Viscous versus Inviscid Regions of Flow
# When 2 fluid layers move relatively to each other, a friction force develops between
them & the slower layer tries to slow down the faster layer. This internal resistance to
flow is quantified by the fluid property  viscosity.
# Viscosity is caused by cohesive forces between the molecules in liquids & by molecular
collisions in gases. No fluid with zero viscosity.
#Flow in which the frictional effects are significant are called  viscous flows
#Typically regions not close to solid surface, where viscous forces are negligible small
compared to pressure forces. inviscid flow regions
The development of viscous &
inviscid regions of flow as a
result of inserting a flat plate
parallel into a fluid stream of
uniform velocity
Classification of Fluid Flows
2) Internal versus External Flow
Depending on whether the fluid if forced to flow:
1) In a pipe or duct (Internal Flow)  fluid completely bounded by solid surface.
Dominated by the influence of viscosity throughout the flow field.
2) Over a surface such as plate or pipe (External Flow)  unbounded fluid. The
viscous effects are limited to boundary layers near solid surfaces.
3) Compressible versus Incompressible Flow
Depending on the level variation of density during flow.
Incompressible  if the flow density remains nearly constant & the volume of
energy portion of fluids remain unchanged over the course of its motion. Gas
flows as incompressible depends on the Mach number 
where c is
the speed of sound, c=346 m/s
Classification of Fluid Flows
4) Laminar versus Turbulent Flow
# Fluid motion characterized by smooth layer of the fluid  Laminar layer
# Disordered fluid motion typically occur at high velocities & is characterized by
velocity fluctuations  Turbulent Layer
# The flow that alternates between being laminar & turbulent is called  Transitional
5) Natural versus Force Flow
# Depending on how the fluid motion is initiated
1) Forced flow  fluid is forced to flow over a surface or in pipe by
external means such as pump or fan
2) Natural flow  fluid motion is due to natural means such as the
buoyancy effect
Classification of Fluid Flows
6) Steady versus Unsteady Flow
# Steady  no change at a point with time
# Unsteady  change at a point with time
# Uniform  no change with location over a
specified region.
1D, 2D and 3D Flows
A flow field is best characterized by the velocity distribution & thus a flow is said to be
1D, 2D or 3D if the flow velocity varies in 1, 2 or 3 primary dimensions.
Example  a typical fluid flow involves a 3D
geometry & the velocity may vary in all 3 dimensions,
V(x,y,z) in rectangular. However, the variation of
velocity in certain direction can be small relative to
the variation on other directions & can be ignored.
1D, 2D and 3D Flows
Consider  steady flow of a fluid through a circular pipe attached to a large tank.
The fluid velocity everywhere on the pipe surface is zero (because no-slip condition).
The development of the velocity profile in a circular pipe, V=V(r,z)
# The flow is 2D in the entrance region of the pipe since the velocity change
in both r and z directions. The velocity profile develops fully & remains
unchanged after some distance from the inlet and the flow in this region is
said to be fully developed (1D since velocity just varies in the radial rdirection).
Velocity Boundary Layer
Consider  the parallel flow of a fluid over a flat plate. The x-coordinate measured
along the plate surface from the leading edge of the plate in the direction of the flow. The
y-coordinates measured from the surface in the normal direction.
The development of the boundary layer for flow over a flat
plate & the different flow regimes.
# the fluid approaches the plate in the x-direction with a uniform velocity, V which is
practically identical to the free stream velocity over the plate away from the surface.
Concept:
Velocity Boundary Layer
# the velocity of the particles in the first fluid layer adjacent to the plate become  zero
(because of the no-slip condition). This motionless layer slows down the particle of the
neighboring fluid layers as a result of friction between the particles of these two
adjoining fluid layers at different velocity. This fluid layer then slows down the
molecules of the next layer and so on. Thus, the presence of the plate is felt up to
some normal distance from the plate beyond which the free stream velocity remain
essentially unchanged.
As a result, the x-component of the fluid velocity, u varies from 0 at y=0 to 
V at y= . The region of the flow above the plate bounded by
called
“velocity boundary layer “. The boundary layer thickness, define as the
distance y from the surface at which u=0.99V.
The development of a
boundary layer on a surface
is due to the no-slip condition
& friction
Wall shear Stress
From the velocity boundary layer concept (the fluid layer in contact with the surface
tries to drag the plate along via friction), apply a friction force on it.
# Friction force per unit area is called  Shear Stress,
The shear stress at the wall surface is expressed as:
Where:
Cf dimensionless friction coefficient. The value in most
cases is determined experimentally.
ρ  density of the fluid.
Note: Generally, the friction coefficient , Cf varies with location along the surface.
Once the average Cf over a given surface is available, the friction force over the
entire surface is determine 
The friction coefficient , Cf  important parameter in Heat Transfer studies since it
is directly related to the Heat Transfer Coefficient
Thermal Boundary Layer
A thermal boundary layer develops  when a fluid at a specified temp. flows over a
surface that is at different temp.
Consider  the flow of a fluid at a uniform temp. of T∞ over an isothermal flat plate at
temp. Ts. These fluid particle then exchange energy with the particles in the adjoining
fluid layer. The flow region over the surface in which the temp. variation in the direction
normal to the surface is thermal boundary layer.
Thermal boundary layer on a flat plate
(the fluid is hotter than the plate surface)
The thickness of thermal boundary layer,
at any location along the surface is
defined as the distance from the surface at which the temp. difference 
.
Note: the fluid velocity has a strong influence on the temp. 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 & the thermal boundary layer is best described
by the Prandtl no. (dimensionless parameter).
The Prandtl numbers of fluids range from less than 0.01 for liquid metals to more
than 100,000 for heavy oil.
Laminar & Turbulent Flow
The flow regimes:
1) Laminar  characterized by smooth stream-lines & highly-ordered motion
2) Transition  Flow from laminar to turbulent occur over some region in which
the flow fluctuates between laminar & turbulent
3) Turbulent  characterized by velocity fluctuations & highly-disordered motion
Most flows encountered in practice are turbulent. Laminar flows is
encountered when highly viscous fluids such as oils flow in small
pipes.
The turbulent boundary layer can be considered consists of 4 regions
(characterized by the distance from the wall.
1) Viscous sublayer  the very thin layer next
to the wall where viscous effects are dominant
. The velocity profile nearly linear & the flow is
streamlined
2) Buffer layer  the turbulent effects are
becoming significant but the flow still
dominated by viscous effects
3) Overlap layer  the turbulent effects are
much more significant but still not domain
4) Turbulent layer  The turbulent effects
dominate over viscous effects.
# The intense mixing of the fluid in turbulent flow as result of rapid fluctuations
enhances  heat and momentum transfer between fluid particles, which increase
the friction force on the surface & the convection heat transfer rate.
# Both the friction & heat transfer coefficients reach maximum
values when the flow become  fully turbulent
Reynolds Number
The flow regime depends on the ratio of the inertia forces to viscous forces in the
fluid. This ratio is called  Reynolds no.
Where.
V = upstream velocity (equivalent to the free-stream velocity for a flat plate)
Lc = characteristic length of the geometries
= kinematic viscosity of the fluid 
(units: m2/s)
# The critical Reynolds no.  The Reynolds no. at which the flow become turbulent
# The value of the critical Reynolds no. is different for different geometries and flow
conditions. For flow over a flat plate, the general value of the critical Reynolds no is
Where
is the distance from the
leading edge of the plate at which
transition from laminar to turbulent flow
occurs
Reynolds Number
Heat & Momentum Transfer in Turbulent Flow
Concept:
Most flows encountered in eng. practice are turbulence  thus it is important to
understand how turbulence affects wall shear stress & heat transfer. However,
turbulence flow is a complex mechanism dominated by fluctuation.
Turbulent flow  characterized by disorderly & rapid fluctuations of swirling region
of fluid called  eddies. These fluctuation provide an additional mechanism for
momentum & energy transfer.
In laminar flow, fluid particles flow in an orderly manner along pathlines and
momentum & energy are transferred across streamlines by molecular diffusion.
In turbulent flow, the swirling eddies transport mass, momentum & energy to other
regions of flow much more rapidly than molecular diffusion  enhancing mass,
momentum & heat transfer. As a result, turbulent flow is associated with much
higher values of friction, heat transfer & mass transfer coefficient
Derivation of Differential Convection Equations
Consider  the parallel flow of a fluid over a surface. For analysis, take the flow
direction along the surface to be x & the direction normal to the surface to be y and
a differential volume element of length dx, height dy and unit depth in z-direction.
The continuity, momentum and energy equations for steady 2D incompressible
flow with constant properties are determined from mass. Momentum and energy
balance to be:
Where the viscous dissipation function,

Using the boundary layer approximations and a similarity variable, all the
equation mention can be solved for parallel steady incompressible flow over a flat
plate, with the following result
The average friction coefficient and Nusselt no. are expressed in functional form as:
and
The Nusselt no. can be expressed by a simple power law relation of the form:
m & n are constant exponents and the value
of the constant, C depends on geometry.
The Reynolds analogy relates the convection
coefficient to the friction coefficient for fluids with
and is expressed as:
Where;
Stanton Number