# Convection Heat Transfer Cross Rate

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```					                          Convection Heat Transfer

Heat transfer:
Heat transfer (or heat) is thermal energy in transit due to a temperature difference. According
to the 2nd law of thermodynamics heat is transferred from a higher temperature body to a
lower temperature body.

Modes of heat transfer:
(i)    Conduction: Mechanism of heat transfer through a solid or fluid in the absence of any
fluid motion.
(ii)   Convection: Mechanism of heat transfer through a fluid in the presence of bulk fluid
motion.
(iii) Radiation: The energy of the radiation field is transported by electromagnetic waves
9or alternatively, photons0. Radiation heat transfer does not require material medium.

Types of convection heat transfer:
Convection heat transfer depends on how the fluid motion is initiated.
(i)   Natural or free convection;
(ii)  Forced convection;

Natural or free convection:
In natural convection, any fluid motion is caused by natural means such as the buoyancy
effect, which manifests itself as the rise of warmer fluid and the fall of cooler fluid.

Forced convection:
In forced convection, the fluid is forced to flow over a surface or in a tube by external means
such as a pump, blower, or a fan.

Newton’s law of cooling:
It states that the rate of heat flow/ heat transfer from a solid surface of area A, at a
temperature Tw to a fluid at a temperature Ts is
Qconvection = hA (Ts-T).
Convection heat transfer coefficient (h): The rate of heat transfer between a solid surface and
a fluid per unit surface area and per unit temperature difference is called convection heat
transfer coefficient (h).
Qconvection = hA (Ts-T)
Where, h = convection heat transfer coefficient, W/m2.0C

Convection heat transfer coefficient strongly depends on the following fluid properties:
(i)   By decreasing dynamic viscosity,  convection heat transfer coefficient can be
increased.
(ii)  By increasing thermal conduction, K convection heat transfer coefficient can be
increased.
(iii) By increasing specific heat, Cp convection heat transfer coefficient can be increased.
(iv)  By increasing fluid velocity, V convection heat transfer coefficient can be increased.
Convection heat transfer coefficient also depends on:
(v)    Surface geometry;
(vi)   Surface roughness;
(vii)  Type of fluid flow.

Laminar and Turbulent flows:
An essential first step in the treatment of any convection problem is to determine whether the
boundary layer is laminar or turbulent. Surface friction and the convection transfer rates
depend strongly on which of these conditions exists.

Laminar flow:
In the laminar flow, fluid motion is highly ordered and it is possible to identify streamlines
along which particles move.
Fluid motion along a streamline is characterized by velocity components in both the x and y
directions.

Turbulent flow: Fluid motion in the turbulent flow is highly irregular and is characterized by
velocity fluctuations. These fluctuations enhance the transfer of momentum, energy and
species, and hence increase surface friction as well as convection transfer rates.
Fluid mixing resulting from the fluctuations makes turbulent boundary layer thickness larger
and boundary layer profiles (velocity, temperature) flatter than in laminar flow.

Transition flow: Transition flow occurs between laminar and turbulent flow. The transition
from laminar to turbulent flow does not occur suddenly; rather, it occurs over some region in
which the flow hesitates between laminar and turbulent flows before it becomes fully
turbulent.

Transition from laminar to turbulent depends on:
(i)    Surface geometry
(ii)   Surface roughness
(iii) Free stream velocity
(iv)   Surface temperature
(v)    Type of fluid.
The velocity profile is approximately parabolic in laminar flow and becomes flatter in the
turbulent flow with a sharp drop near the surface.

Effect of turbulence:
(i)     Intense mixing of the fluid.
(ii)    Enhance heat and momentum transfer between fluid particles.
(iii)   Increase conduction heat transfer rate.

Types of flow:
(i)     Internal flow: The fluid is completely confined by the inner surfaces of the tube
and there is limit on how much the boundary layer grows.
(ii)    External flow: The fluid has a free surface and thus the boundary layer over the
surface is free to grow indefinitely.
Velocity Boundary layer:

The region of flow that develops from the leading edge of the plate in which the effects of
viscosity are observed is called the boundary layer. Some arbitrary point is used to designate
the y position where the boundary layer ends; this point is usually chosen as the y coordinate
where the velocity becomes 99 percent of the free stream velocity.

Fluid velocity at the surface of the plate is zero (because of no-slip condition), and gradually
increases with distance from the plate. At a sufficiently large distance from the plate, the fluid
velocity becomes equal to the ‘free stream velocity’ V. The region above the plate surface
within which this change of velocity from zero to the free stream value occurs is called the
boundary layer (velocity boundary layer) also called the hydrodynamic boundary layer. The
thickness of this region is called the boundary layer thickness and is denoted by . The
boundary layer thickness increases with the distance x from the leading edge of the plate, i.e.
 = (x).
Initially, the boundary-layer development is laminar, but at some critical distance from the
leading edge, depending on the flow field and fluid properties, small disturbances in the flow
begin to become amplified, and a transition process takes place until the flow becomes
turbulent. The turbulent-flow region may be pictured as a random churning action with
chunks of fluid moving to and fro in all directions.

The transition from laminar to turbulent flow occurs when
u x  u x
          5 10 5 Where, u   free stream velocity, m / sec
        

x = distance from leading edge, m

       Kinematic viscosity, m2/sec

Property variation with time in a turbulent boundary layer:
The thermal boundary layer:

A velocity boundary layer develops when there is fluid flow over a surface; a thermal
boundary layer must develop if the fluid free stream and surface temperatures differ.
Consider flow over an isothermal flat plate. At the leading edge the temperature profile is
uniform, with T(y) = T∞. However, fluid particles that come into contact with the plate
achieve thermal equilibrium at the plate’s surface temperature. In turn, these particles
exchange energy with those in the adjoining fluid layer, and temperature gradients develop in
the fluid. The region of the fluid in which these temperature gradients exist is the thermal
boundary layer, and its thickness is defined as  t . With increasing distance from the leading
edge, the effects of heat transfer penetrate further into the free stream and the thermal
boundary layer grows.
Significance of the boundary layers:

The velocity boundary layer is of extent  x  and is characterized by the presence of velocity
gradient and shear stresses. The thermal boundary is of extent  t (x) and is characterized by
temperature gradients and heat transfer. The principle manifestations of the two boundary
layers are, respectively, surface friction, convection heat transfer. The key boundary layer
parameters are then the friction coefficient Cf and the heat transfer convection coefficient h,
respectively.
For flow over any surface, there will always exist a velocity boundary layer, and hence
surface friction. However, a thermal boundary, and hence convection heat transfer, exists
only if the surface and free stream temperatures differ.

The mean velocity:
The velocity various over the cross section and there is no well-defined free stream, it is
necessary to work with a mean velocity u m when dealing with internal flows. This velocity is
defined such that, when multiplied by the fluid density  and the cross sectional area of the
tube Ac, it provides the rate of mass flow through the tube. Hence m   u m Ac .


Flow in tubes:
The fluid velocity in a tube changes from zero at the surface to a maximum velocity at the
tube centre. A boundary layer develops at the entrance. Eventually the boundary layer fills
the entire tube, and the flow is said to be fully developed. If the flow is laminar, a parabolic
velocity profile is experienced. When the flow is turbulent, a blunter profile is observed. In a
tube, the Reynolds number is again used as a criterion for laminar and turbulent flow.
For
um d
Re d        2300. The flow is usually observed to be turbulent and where d is the tube

diameter.
Again, a range of Reynolds numbers for transition may be observed, depending on the
surface roughness and smoothness of the flow. The generally accepted range for transition is
2000  Re d  4000 .

Energy balance for flow in a tube:

The flow in a tube is completely enclosed, an energy balance may be applied to determine
how the mean temperature Tm  x  various with position along the tube and how the total
convection heat transfer q conv is related to the difference in temperature at the tube inlet and
outlet.
The rate of convection heat transfer to the fluid must equal the rate at which the fluid thermal
energy increases plus the net rate at which work is done in moving the fluid through the
control volume.

Pressure gradient and Friction factor in fully developed flow:
The pressure drop needed to sustain an internal flow because this parameter determines the
pump, blower or fan power requirements. To determine the pressure drop, it is convenient to
work with the Moody (or Darcy) friction factor, which is a dimensionless parameter defined
 dp 
 D
as f    . Where f is the friction factor.
dx
 um 2
2

Resistance to fluid flow, the pressure drop in the flow direction:

m
50kPa
35 kPa

The fanning friction factor:
s
The fanning friction factor is called the friction coefficient, which is defined as C f 
 um 2
2
Where       C f is the friction coefficient or drag coefficient, whose value in most cases is
determined experimentally, and  is the density of the fluid. The friction coefficient, in
general, will vary with location along the surface.

The mean or Bulk Fluid temperature:
The mean or bulk temperature of the fluid at a given cross section is defined in terms of the
thermal energy transported by the fluid as it moves past the cross section. The rate at which
this transport occurs, Et , may be obtained by integrating the product of the mass flux  u 

and the internal energy per unit mass            c T 
p      over the cross section. That is,
Et    u c p c T d Ac

Ac

Hence if a mean temperature is defined such that Et  m c p Tm . Where C p is the specific heat

                                    
of the fluid and m is the mass flow rate. The product m C p Tm AT any cross section along the
tube represents the energy flow with the fluid at that cross section. In the absence of any work
interactions (such as electric resistance heating) , The conservation of energy equation for the
steady flow of a fluid in a tube can be expressed as Q  m C p Te  Ti  .Where T i and Te are
 

the mean fluid temperature at the inlet and exit of the tube, respectively, and Q is the rate of
heat transfer to or from the fluid.
After all, the bulk temperature is the representative of the total energy of the flow at any
particular location. The bulk temperature is used for overall energy balances on systems.

The thermal conditions:
The thermal conditions at the surface of a tube can usually be approximated with reasonable
accuracy to be constant surface temperature (Ts = constant) or constant surface heat flux
q s  cons tan t . For example, the constant surface temperature condition is realized when a

phase change process such as boiling or condensation occurs at the outer surface of a tube.
The constant surface heat flux condition is realized when the tube is subjected to radiation or
electric resistance heating uniformly from all directions. The convection heat flux at any
location on the tube can be expressed as q  hTs  Tm  , where h is the local heat transfer

coefficient and T s and Tm are the surface and the mean fluid temperatures at that location.
Therefore, where h = constant, the surface temperature Ts must change when q s  constant,


and the surface heat flux q s must change when T s = constant. Thus we may have either T s =
constant or q s  constant at the surface of a tube, but not both.


Constant surface heat flux ( q s  constant):

In the case of q s  constant, the rate of heat transfer can also be expressed as

Q  qs A  m C p Te  Ti 
        

qs A
Then the mean fluid temperature at the tube exit becomes Te  Ti     .

mCp
Variation of the tube surface and the mean fluid temperature along the tube for the case of
constant surface heat flux:

Fully developed region

Entry region
Ts

Te

Tm
∆T=Ts-Tm=qs/h
Ti

Constant surface temperature (Ts = constant):

In the case of Ts = constant, the rate of heat transfer is expressed as
                               T  Ti       Te   Ti
Q  hA  Tln , where  Tln  e                            is the logarithmic mean temperature
Ts  Te          Te
Ln              Ln
Ts  Ti          Ti
difference. Here,  Ti  Ts  Ti
and  Te  Ts  Te are the temperature differences between the surface and the fluid at the
inlet and the exit of the tube, respectively. Then the mean fluid temperature at the tube exit in
hA

this case can be determined from Te  Ts  Ts  Ti  e

mCp
.

The variation of the mean fluid temperature along the tube for the case of constant surface
Temperature:

Ts=costant
Ts

∆T=Ts-Tm

Ti

L

Ti

Ts=costant
Hydrodynamic entry region, length and hydrodynamically developed region:

The region from the tube inlet to the point at which the boundary layer merges at the
centerline is called the hydrodynamic entry region, and the length of this region is called the
hydrodynamic entry length. The region beyond the hydrodynamic entry region in which the
velocity profile is fully developed and remains unchanged is called the hydrodynamically
developed region.

Hydrodynamic entry lengths
Lh  0.05 Red  Laminar flow

Lh  10 d  Turbulent flow
Velocity profile in the fully developed region:
Thermal entry region, length and thermally developed region:

The region of flow over which the thermal boundary layer develops and reaches the tube
centre is called the thermal entry region and the length of this region is called the thermal
entry length. The region beyond the thermal entry region in which the dimensionless
temperature profile remains unchanged is called thermally developed region.
The region in which the flow is both hydrodynamically and thermally developed is called the
fully developed flow.

Thermal entry lengths
Lt  0.05 re. Pr d  laminar flow
Lt  10 d  Turbulent flow
For Pr >> 1 and Lh < Lt  Laminar flow.
Axial Variation of the convection heat transfer coefficient for flow in a
tube:

Viscous-energy dissipation function:
The energy equation in the rectangular co-ordinate system for a elemental control volume for
steady, two dimensional (x, y) flow of an incompressible, constant-property fluid when
consider for convection energy, conduction energy and viscous energy is determined as
 T    T       2T  2T 
 C p u
 x v      K 2  2    
       y 

 x
       y  
Where  is the viscous-energy dissipation function and is defined as

 u  2  v  2   v u  2
  2         
                
 x   y    x y 
                  

The left hand side represents the net energy transfer due to mass transfer; on the right hand
side the first term represent the conductive heat transfer, and the last term on the right hand
side is the viscous-energy dissipation in the fluid due to internal fluid friction.

Physical significance of the dimensionless parameters:
The dimensionless parameters such as the Reynolds number, Nusselt number and Prandtl
numbers are introduced and the physical significance of these dimensionless parameters in
the interpretation of the conditions associated with fluid flow or heat transfer is discussed.

The Reynolds number:
The Reynolds number represents the ratio of the inertia to viscous force. This result implies
that viscous forces are dominant for small Reynolds numbers and inertia forces are dominant
for large Reynolds numbers. The Reynolds number is used as the criterion to determine
whether the flow is laminar or turbulent. As the Reynolds number is increased, the inertia
forces become dominant and small disturbances in the fluid may be amplified to cause the
transition from laminar to turbulent.

where:

     vs - mean fluid velocity,
     L - characteristic length (equal to diameter (2r) if a cross-section is circular),
     μ - (absolute) dynamic fluid viscosity,
     ν - kinematic fluid viscosity: ν = μ / ρ,
     ρ - fluid density.

Nusselt number:
The Nusselt number is a dimensionless number that measures the enhancement of heat
transfer from a surface that occurs in a real situation, compared to the heat transferred if just
conduction occurred. Typically it is used to measure the enhancement of heat transfer when
convection takes place.

where

     L = characteristic length, which is simply Volume of the body divided by the Area of
the body (useful for more complex shapes)
     kf = thermal conductivity of the "fluid"
      h = convection heat transfer coefficient

Thus the Nusselt number may be interpreted as the ratio of heat transfer by convection to
conduction across the fluid layer of thickness L. Based on this interpretation, the value of the
Nusselt number equal to unity implies that there is no convection-the heat transfer is by pure
conduction. A large value of the Nusselt number implies enhanced heat transfer by
convection.

The Prandtl number:
The Prandtl number is a dimensionless number approximating the ratio of momentum
diffusivity and thermal diffusivity. The Prandtl number provides a measure of the relative
effectiveness of momentum and energy transport by diffusion in the velocity and thermal
boundary layers, respectively.
where:

     ν is the kinematic viscosity, ν = μ / ρ.
     α is the thermal diffusivity, α = k / (ρ cp).

In heat transfer problems, the Prandtl number controls the relative thickness of the
momentum and thermal boundary layers.

The Stanton number:
The Stanton number is a dimensionless number which measures the ratio of heat transferred
into a fluid to the thermal capacity of fluid. It is used to characterize heat transfer in forced
convection flows.

where;

     h = convection heat transfer coefficient
     ρ = density of the fluid
     cp = specific heat of the fluid
     V = velocity of the fluid

It can also be represented in terms of the fluid's Nusselt, Reynolds, and Prandtl numbers:

where;

     Nu is the Nusselt number
     Re is the Reynolds number
     Pr is the Prandtl number

Heat transfer enhancement:
Several options are available for enhancing heat transfer associated with internal flows.
Enhancement may be achieved by increasing the convection coefficient and/or by increasing
the convection surface area. For example, h may be increased by introducing surface
roughness to enhance turbulence, as, for example, through machining or insertion of a coil-
spring wire. The wire insert provides a helical roughness element in contact with the tube
inner surface. Alternatively, the convection coefficient may be increased by inducing swirl
through insertion of a twisted tape. The insert consists of a thin strip that is periodically
twisted through 3600. Introduction of a tangential velocity component increases the speed of
the flow, particularly near the tube wall. The heat transfer area may be increased by attaching
longitudinal fins to the inner surface, while both the convection coefficient and area may be
increased by using spiral fins or ribs. In evaluating any heat transfer enhancement scheme,
attention must also be given to the attendant increase in pressure drop and hence fan or pump
power requirements.

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Description: Convection Heat Transfer Cross Rate