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LECTURE 2 Fundamentals of Convection
Flow Over Flat Plates
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conv = hAs (Ts T) Watt
Convection Heat Transfer
By Newtons Law of Cooling, convection heat transfer is
proportional to temperature different :
2
Q
h = convection heat transfer coefficient (W/m2 C) As = heat transfer surface area (m
2)
Ts = temperature of surface (C)
T = temperature of media far from surface (C)
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Heat transfer by conduction,
cond = kAs (T1T2) / L = kAs T/ L
Heat transfer by convection,
conv = hAs (T1T2) = hAs T
By taking ratio heat transfer of convection to conduction
gives :
conv / cond =
3
Q
Q
Nusselt Number
TS = T1 T = T2
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or
Reynolds Number
Reynolds Number (Re) gives relation between inertial force and viscous force of fluid flow.
Re = inertial force / viscous force
VL / VD /
V = free stream velocity (m/s)
L = characteristic length (m) for flat plate
D = diameter (m) for tube/cylinder
= kinematic viscosity (m2/s)
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Prandtl Number
The thickness of thermal boundary layer increases in the flow direction of liquid.
The development of velocity boundary layer relative to
thermal boundary layer will have strong effect on
convection heat transfer
The relative thickness of the velocity and thermal boundary
layers are described by the dimensionless parameter called :
Prandtl Number (Pr)
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Prandtl Number
Heat diffuses very quick in liquid metals and gaseous
(Pr =1 and less) and very slow in water/oil (Pr > 1).
Liquid metals 0.004 0.03
Gaseous 0.7 1.0
Water 1.7 13.7
Oil 50 100,000
Typical values Pr Number :
7
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Properties of Air
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Friction Force over Flat Plate
Because of the friction between the layers, exerting a friction force on the plate.
The friction force over the entire surface is determined as
below, where Cf is friction coefficient
Friction Force, Ff = Cf As V 2 / 2 (N)
The friction force is important parameter in heat transfer
studies since its is directly related to the heat transfer
coefficient.
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Laminar :
Turbulent :
21Re
328.1fC
51Re
074.0fC
Re
1742
Re
074.051fC
Laminar & Turbulent
Friction Coefficient over Flat Plate
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Friction and Pressure Drag
When the flat plate is placed normal to the flow direction,
however, the drag force depends on the pressure only and is
independent of the wall shear since the shear stress in this case
acts in the direction normal to flow.
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Friction and Pressure Drag
The drag force FD depends on the density of the fluid, the
upstream velocity, and the size, shape, and orientation of the
body, among other things.
The drag characteristics of a body is represented by the
dimensionless drag coefficient CD defined as
where A is the frontal area that tends to block the flow. The
frontal area of a cylinder of diameter D and length L, for
example, is A = LD.
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Flow Across Different Bodies
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Plate [ AN = L x w ] Sphere [ AN = D ] Cylinder [ AN = L x D ]
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Limitation of Re over Flat Plate
The transition from laminar to turbulent flow depends on
the surface geometry, surface roughness, surface
temperature, velocity and type of fluid used.
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Limitation of Re over Flat Plate
For fluid flow over the flat surface, the limit of laminar and turbulent flows is given as :
Laminar Flow : Re < 5 x 105
Turbulent Flow : Re 5 x 105
Re = 5 x 105
Transition ?
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Nu, Pr and Re Numbers
Generally, heat transfer analysis on forced convection involves three major numbers : Nusselt number (Nu), Prantdl number (Pr) and Reynolds number (Re).
To simplify the numbers :
C, m and n are constant value that depends on geometry
and flow type of the body (range of Re). The value of
m and n are between 0 to 1.
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Nu = C Rem Prn
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Nusselt Number over Flat Plates
Laminar : Re < 5 x 105 315.0 Pr Re 664.0 Nu
k
hL
Turbulent : Re 5 x 105 318.0 Pr Re 037.0 Nu
k
hL
Laminar & Turbulent : Re 5 x 105
318.0 Pr )871 Re 037.0( Nu k
hL
Nu = C Rem Prn
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The flat plate is assumed to be at uniform temperature during the entire flow.
However, when a flat plate is subjected to uniform heat
flux instead of uniform temperature, the Nu is given as :
Laminar :
Turbulent :
315.0 Pr Re 453.0 Nu k
hL
318.0 Pr Re 0308.0 Nu k
hL
Uniform Temperature/Heat Flux
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q
Uniform temperature
Uniform heat flux
Ts
=W/m2
Uniform Temperature/Heat Flux
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Fluid Properties
The fluid temperature in the thermal boundary layer varies from Ts at the surface to T at the outer edge of boundary.
Therefore, by taking the average, the fluid properties are
usually evaluated at film temperature (Tf ) :
The fluid properties are assumed to remain constant
during the entire flow.
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Tf = (Ts + T) / 2
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Methodology for Solving Problem
Find the fluid properties based on Tf
Calculate the Reynolds Number (Re)
Identify the type of flow, based on Re obtained
Determine h from equation of Nu
Use the right equations
= hAs(TsT)
Cf = Laminar or Turbulent
Ff = Cf A V2 / 2
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Q
Ts = 110 C T= 30 C V = 6 m/s
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5 m
1.5 m
Q
Example 1
Air, at 30C flows with a velocity of 6 m/s over a 1.5 m x 5m
flat plate whose temperature is 110 C. Determine the rate of
heat transfer from the plate if the flows parallel to :
a) 5 m long side b) 1.5 m side
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Properties of Air
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Assume air at 1 atm pressure Tf = (Ts+T)/2 = (110 + 30)/2 = 70C Properties of air at 70C k = 0.02881 W/m.C =1.995 x 10-5 m2/s Pr = 0.7177 Surface are of the flat plate As = W x L = 1.5 x 5 = 7.5 m
2
Example 1
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(a) 5 m long side (L = 5 m) Re = 1503759, Re 5 x 105 - Turbulent Assume the entire flow is turbulent
Nu = 2899 h = 16.7 W/m2 C = 10020 W b) 1.5 m long side (L = 1.5 m) Re = 451128, Re < 5 x 105 - Laminar Nu = 399 h = 7.67 W/m2 C = 4602 W
Q
Q
Example 1
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Forced Convection
EXERCISE 1
Consider a hot automotive engine, which can be approximated
as a 0.5-m-high, 0.40-m-wide, and 0.8-m-long rectangular
block. The bottom surface of the block is at a temperature of
100C and the ambient air is at 20C.
Determine the rate of heat transfer from the bottom surface of
the engine block by convection as the car travels at a velocity
of 80 km/h.
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Properties of Air
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EXERCISE 2
A 15-cm x 15-cm circuit board dissipating 15 W of power
uniformly is cooled by air, which approaches the circuit
board at 20C with a velocity of 6 m/s. Disregarding any heat
transfer from the back surface of the board, determine the
surface temperature of the electronic component.
Forced Convection
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Properties of Air
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EXERCISE 3
Consider a case of air flows over hot plates with 20 cm x 50 cm
surface area. The velocity of air is 5 m/s, the temperature of
each plate is 120C and the surrounding air temperature is 60C.
Determine the rate of heat transfer.
Forced Convection
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Properties of Air