Download Chapters_9,10, 11 Lecture notes

Lumped System Analysis
T (t ) − T∞
= e −bt
Ti − T∞
b=
Spring 2003
hA
ρ V CP
ECE309: Chapters 9,10,11
1
Criteria for Lumped System Analysis
Biot Number = Bi =
Convection at the surface of the body
h ∆T
=
conduction within the body
k ∆T / L
L = Characteristic Length = V / A
Bi =
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hV
≤ 0. 1
kA
ECE309: Chapters 9,10,11
2
1
Convection Heat Transfer
Convection heat transfer strongly
depends on the:
„
„
„
„
„
„
„
„
fluid properties,
dynamic viscosity µ,
thermal conductivity k,
density ρ,
specific heat Cp,
fluid velocity V,
geometry of the solid surface,
type of fluid flow
q& Conv = q&Cond = − k fluid
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∂T
∂y
y =0
h=
ECE309: Chapters 9,10,11
− k fluid ( ∂T / ∂y ) y =0
Ts − T∞
3
Nusselt Number, Nu
q& conv = h ∆T
∆T
q& cond = k
δ
q& conv hδ
=
= Nu
q& cond
k
„
The larger the Nusselt number, the more effective is
the convection. A Nusselt number Nu = 1 for a fluid
layer represents heat transfer by pure conduction.
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ECE309: Chapters 9,10,11
4
2
Velocity Boundary Layer
„
„
The region of the flow above the plate bounded by δV in which
the effects of the viscous shearing forces caused by fluid
viscosity are felt is called the velocity boundary layer.
The thickness of the boundary layer, δv, is arbitrarily defined as
the distance from the surface at which V = 0.99 V∞.
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ECE309: Chapters 9,10,11
5
Laminar & Turbulent Flows
„
„
Laminar flow is characterized by
smooth streamlines and highly
ordered motion,
Turbulent Flow is characterized by
velocity fluctuations and highly
disordered motion.
Re =
‰ Critical
inertia forces
V δ
= ∞
viscouse fources
ν
V∞ Free - stream velocity [m/s]

δ Characteri stic length, [m]
ν Kinematic viscosity ( µ/ρ ), [m2 / s ]

Reynolds number is the Reynolds number at which
the flow becomes turbulent.
Re critical, flat plate ≈ 5 × 10 5
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ECE309: Chapters 9,10,11
6
3
Thermal Boundary Layer
„ A thermal boundary layer develops
when a fluid at a specified temperature
flows over a surface which is at a
different temperature.
„ The thickness of the thermal boundary
layer δt at any location is defined as the
distance from the surface at which the
temperature difference (T-Ts) equals
0.99(T∞-Ts).
„ The thickness of the thermal boundary
layer increases in the flow direction.
„ The relative thickness of the velocity
and the thermal boundary layers is
described by Prandtl Number.
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Pr =
ν µC P
=
α
k
Most Gases,
 Pr ≈ 1

<<
Pr
1
Liquid Metals,

 Pr >> 1 Oils.

ECE309: Chapters 9,10,11
7
Flow Over Flat Plates
„
Laminar:
Nu x =
hx x
= 0.332 Re1x/ 2 Pr 1 / 3
k
(Pr ≥ 0.6)
hL
Nu =
= 0.664 Re1L/ 2 Pr 1/ 3
k
ƒ Turbulent:
Nux =
Nu =
hx x
= 0.0296 Re4x / 5 Pr 1/ 3
k
hL
= 0.037 Re 4L/ 5 Pr 1/ 3
k
 0.6 ≤ Pr ≤ 60



 5 × 105 ≤ Re ≤ 107 
x


 0.6 ≤ Pr ≤ 60



 5 × 105 ≤ Re ≤ 107 
L


ƒ Combined:
h=
x
L

1  cr
hx ,la min ar dx + ∫ hx ,turbulentdx 

L  ∫0
x cr

ƒ All properties are evaluated at film temperature as:
Spring 2003
ECE309: Chapters 9,10,11
Tf =
Ts + T∞
2
8
4
Natural Convection, Physical Mechanism
The magnitude of the natural
convection heat transfer between a
surface and a fluid is directly
related to the mass flow rate of
the fluid.
„
The higher the mass flow rate, the
higher is the heat transfer rate.
„
The mass flow is established by the
dynamic balance of buoyancy and
„
friction.
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ECE309: Chapters 9,10,11
9
Grashof Number, Gr
„
Grashof Number, Gr, is a measure of the
relative magnitudes of the buoyancy force and
the opposing friction force acting on the fluid.
Gr =
buoyancy forces g ∆ ρ V g β ∆ T V gβ (Ts − T∞ )δ 3
=
=
=
viscous forces
ρν 2
ρν
ν2
g

β

 ∆T
δ

ν
Gravitatio nal accelerati on [m/s 2 ]
Coeff. of volume expansion, 1/T , [1/K]
Ts − T∞ , [°C]
Characteri stic length, [m]
kinematic viscosity , µ / ρ , [m 2 /s]
ƒ The Grashof number provides the main criteria in determining whether the
fluid flow is laminar or turbulent in natural convection.
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Gr ≤ 109
Laminar
Gr > 10
Turbulent
9
ECE309: Chapters 9,10,11
10
5
Natural Convection Over Surfaces
Nu =
hδ
= C(Gr Pr) n = C Ra n
k
Where Rayleigh number, Ra, is
Ra = Gr Pr =
„
„
„
gβ (Ts − T∞ )δ 3
ν2
Pr
The values of the constants C and n
depend on the geometry of the surface and
the flow regime,
The value of n is usually ¼ for laminar flow
and 1/3 for turbulent flow.
The value of the constant C is normally less
than 1. (Table 11-1, P 587)
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ECE309: Chapters 9,10,11
11
Natural Convection Inside Enclosures
Ra =
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gβ (T1 − T2 ) δ 3
ν2
Pr
Distance between the hot & cold plates [m]
δ

T1 , T2 Temperatur es of hot & cold surfaces [°C]
ECE309: Chapters 9,10,11
12
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