Chapter 4 - Convection Heat Transfer Updated
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Transcript of Chapter 4 - Convection Heat Transfer Updated
CHAPTER 4
Convection Heat Transfer
Outlines
To examine the methods of calculating convection
heat transfer (particularly, the ways of predicting the
value of convection heat transfer coefficient, h)
Convection heat transfer requires an energy balance
along with an analysis of the fluid dynamics of the
problem concern. So that. The discussion will
consider;
simple relations of fluid dynamics
Boundary layer analysis
Important for basic
understanding of
convection heat
transfer
The region of flow that develops from the leading edge of the plate in which
the effects of viscosity boundary layer (The y position where the
boundary layer ends and the velocity become 0.99 of the free-stream value)
Initially the boundary layer development is laminar but at some
critical distance from the leading edge (depending on the flow field & fluid
properties), small disturbance in the flow begin to become amplified, and
a transition process takes place until the flow become turbulent.
occurs when where
Flat
plate
Tube Laminar flow
Turbulence flow
For transition between
laminar & turbulence
d = Tube diameter
For transition between
laminar & turbulence
Re. no. (Tube)
Other Form:
Define the mass velocity as
So that, mass flow rate
The Reynolds no. may also written
Classification of Fluid Flows
Inviscid Flow
Although no real fluid is inviscid, in some instance the fluid may be treated
as such, and useful to present some of the equation that apply in these
circumstances (the flow at a sufficiently large distance from the plate will
behave as a nonviscous flow system).
or
Where,
Fundamentally a Dynamic Equation
(The Bernoulli equation)
To solve convection heat transfer
coefficient, h we have to:
1) Identify the type of fluid involve to
get the fluid properties
2) State the process
1) Type of Fluid
# Specify the equation of state of fluid to calculate pressure drop in compressible
flow.
An ideal gas
e = Internal Energy
i = Entalpy
Gas
Where:
Where:
Air
2) State the Process
Example: Reversible Adiabatic Flow through a nozzle
The relation involved which is relating the
properties at some points in the flow to the
Mach no. & stagnation properties
Where:
a: Local velocity of sound
For an ideal gas: For air behaving as an ideal gas:
Laminar Boundary Layer on Flat Plate
From the analytical analysis by making a force and momentum balance on the
element yield the momentum equation for the laminar boundary layer with
constant properties.
Can be solved for many
boundary conditions.
For development in this chapter, we
shall satisfied with an approximate
analysis that furnishes an easier
solution without a loss in physical
understanding of process involved.
Laminar Boundary Layer on Flat Plate Consider the boundary layer flow system as shown:
The free-stream velocity outside the boundary
layer is u∞ and the boundary layer thickness is .
We wish to make a momentum-and-force balance
on the control volume bounded by the plane 1, 2,
A-A and the solid wall.
The boundary layer thickness,
Mass flow rate Where:
So that
Example 1: Mass Flow & Boundary-Layer
Thickness
Air at 27 °C and 1 atm flows over a flat plate at a speed of 2 m/s. Calculate the boundary-layer thickness at distances of 20 cm and 40 cm from the leading edge of the plate. Calculate the mass flow that enters the boundary layer between x=20 cm and x= 40 cm. The viscosity of air at 27 ° C is 1.85 x 10-5 kg/ m . s. Assume unit depth in the z direction.
J.P Holman
Solution:
J.P Holman
Dimensionless Numbers in Heat transfer
Dimensionless numbers are tools of heat transfer. They are not actually necessary to solve convection problems but the make it easier.
Key dimensionless numbers used in heat transfer are:
1. Reynolds number (Re)
2. Nusselt number (Nu)
3. Prandtl number (Pr)
4. Grashof number (Gr)
5. Rayleigh number (Ra)
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
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
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.
(Can be found in tables)
Grashof Number
where
This dimensionless number is mainly for natural convection.
It is the ratio of buoyancy forces to the viscous forces. The Grashof
number plays the same role that the Reynolds number plays for forced
convection.
When buoyancy forces overcome viscous forces, the flow starts to become
turbulent.
Rayleigh Number
dimensionless number associated with buoyancy driven flow (also known as free convection or natural convection). When the Rayleigh number is below the critical value for that fluid, heat transfer is primarily in the form of conduction; when it exceeds the critical value, heat transfer is primarily in the form of convection.
This number is just the product of the Grashof Number and the Prandtl number
In general,
Nu = f(Re, Pr, Gr)
The Thermal Boundary layer - Flat Plate
Exist when temperature gradient are present in the flow.
If the fluid properties were constant throughout the flow, an appreciable
variation between the wall and free stream condition which is film temp. Tf define as:
Used Tf to get the fluid properties
from properties fluids table
The Thermal Boundary layer - Flat Plate Consider the system shown
Tw: The temp. of the wall
T∞ : The temp. of the fluid outside the thermal boundary layer
: Thickness of the thermal boundary layer
For the plate heated
over its entire length
Basic: convection/conduction
hx: Heat transfer coefficient in term of the
distance from the leading edge of the plate
0.6 < Pr < 50
Used the average heat transfer coefficient
Case 1
Rex Pr > 100
The Thermal Boundary layer - Flat Plate
For the plate heated
starts at
Case 2
or
Where:
The Thermal Boundary layer - Flat Plate
Constant Heat Flux
To find the distribution of the plate surface temp.
and
From these equation, can
be produce equation
So that
The energy Equation of the boundary layer (heat flow, q)
& the thermal boundary layer (heat flux, qw)
To determine heat flow, q and heat flux, q w
From energy equation of the
boundary layer
From energy equation of the
thermal boundary layer
Heat flow, q
1) Film temp.
2) The properties of fluid at Tf such as kinematic viscosity, thermal conduction
coefficient, heat capacity, Prandtle no.
There is an appreciable variation between wall & free
stream condition, so that, it is recommended that the
properties be evaluated at film temp.
Determine:
3) Rex at x=xL
4) Nusselt No. 0.6 < Pr < 50
Rex Pr > 100
Then
5) Heat transfer coefficient, h
6) The average heat transfer coefficient,
7) Heat Flow,
Heat flux, qw
Example 2: Isothermal Flat Plate Heated
Over Entire Length
Air at 27 °C and 1 atm flows over a flat plate at a speed of 2 m/s. Calculate the boundary-layer thickness at distances of 20 cm and 40 cm from the leading edge of the plate. Calculate the mass flow that enters the boundary layer between x=20 cm and x= 40 cm. The viscosity of air at 27 ° C is 1.85 x 10-5 kg/ m . s. Assume unit depth in the z direction and the plate is heated over its entire length to a temp. of 60 ° C. Calculate the heat transferred in
(a) The first 20 cm of the plate and
(b) The first 40 cm of the plate.
J.P Holman
Solution:
J.P Holman
J.P Holman
Example 3: Flat Plate with Constant Heat
Flux
A 1.0 KW heater is constructed of a glass plate with an electrically
conducting film that produces a constant heat flux. The plate is 60 cm
by 60 cm and placed in an airstream at 27 °C, 1 atm with u∞ = 5 m/s.
Calculate
(a) The average temp. different along the plate and
(b) The temperature difference at the trailing edge.
Example 4: Oil Flow Over Heated Flat Plate
Engine oil at 20 °C is forced over a 20-cm-square plate at a velocity of
1.2 m/s. The plate is heated to a uniform temp. of 60 °C. Calculate
the heat lost by the plate.
J.P Holman
Solution:
J.P Holman
The relation between fluid friction & heat
transfer
The shear stress at the wall be
expressed in term of a friction
coefficient, Cf
Where:
The relation between fluid friction
and heat transfer for Laminar flow
on a flat plate
This is important relation between
friction & heat transfer is the drag force
(D) which is depends on the average
shear stress. The average of shear
stress is a friction coeffiecient Cfx
Drag Force, D = (shear stress) (Area)
Example 5: Drag Force on a Flat Plate
Air at 27 °C and 1 atm flows over a flat plate at a
speed of 2 m/s. Calculate the boundary-layer
thickness at distances of 20 cm and 40 cm from the
leading edge of the plate. Calculate the mass flow
that enters the boundary layer between x=20 cm
and x= 40 cm. The viscosity of air at 27 ° C is 1.85 x
10-5 kg/ m . s. Assume unit depth in the z direction
and the plate is heated over its entire length to a
temp. of 60 ° C. Compute the drag forced exerted
on the first 40 cm of the plate using the analogy
between fluid friction and heat transfer
Turbulent-boundary-layer heat transfer (q)- Flat Plate
1) Determine either the flow is turbulent
region or not. Check based on Re. no.
Turbulent Region
2) Heat transfer (q) from the plate is:
Where:
Turbulent-boundary-layer thickness ( )- Flat Plate
The boundary layer thickness measured when 500000>Re<10000000
The boundary layer is fully turbulent
from the ledge of the plate
The boundary layer follows a laminar
growth pattern up to Rcrit= 5 x 105 and
turbulent growth thereafter
Air at 20◦C and 1 atm flows over a flat plate at 35 m/s. The plate is 75 cm
long and is maintained at 60◦C. Assuming unit depth in the z direction,
i. Calculate the heat transfer from the plate
ii. Calculate the turbulent-boundary-layer thickness at the end of the plate
if it is develop.
(a) from the leading edge of the plate
(b) from the transition point at Recrit.
Example 6: Turbulent Heat Transfer &
boundary Layer Thickness – Flat Plate
i.
ii.
Heat Transfer in Laminar Tube Flow Consider the tube flow system
Aim to calculate the heat transfer
under developed flow condition
(Laminar Flow)
Consider the fluid element derive to get the
velocity and temp. distribution
Heat Transfer in Laminar Tube Flow From the analysis, the analytical solution give;
The velocity at the center of the tube
The velocity
distribution
The temperature
distribution
The Bulk Temperature In tube flow, convection heat transfer coefficient, h
defined by:
Where, Tw : The wall temp. Tb : Bulk temp.
From the analysis, the analytical solution give;
The bulk temp The wall temp
Heat transfer coefficient Heat transfer coefficient in term of the Nusselt No.
Turbulent Flow in a Tube
Velocity profile for turbulent flow in a tube
To determine heat transfer analytically should know the temp.
distribution in the flow
# to obtain temp. distribution, the analysis must take into consideration
the effect of the turbulent eddies in the transfer of heat and momentum)
Turbulent Flow in a Tube
From the analysis, the analytical solution give;
Relates the heat transfer rate to the friction loss in tube flow
Where,
Heat transfer coefficient in term of the Nusselt No.
or
Pr ≈ 1.0
Relation for turbulent heat transfer in smooth tube
# from this analytical solution, shows that h higher than those observed in experiment Pr 2/3
SUMMARY OF
EQUATIONS