BASIC CONCEPTS IN CONVECTION
Introduction
• Characteristic of convection: Fluid motion
• Focal point: Determination of heat transfer coefficient
α - součinitel přestupu tepla
• Determination of α:
Temperature distribution in the fluid → α
1
General Observations
TT
ws•
The Role of Fluid Motion
For the electric bulb:
q& w = surface flux
Tw = surface temperature
T∞ = free stream temperature
u∞ = free stream velocity
q&qw′s′
Vu∞∞
T∞
−
+
Fig. 6.1
For a fixed input power how to lower surface temperature?
• Raise or lower u∞ ?
• Change the cooling fluid?
2
Increasing u ∞ or changing the fluid from air to
water lowers surface temperature
Conclusion:
Fluid motion and fluid nature
play important roles in convection
Newton’s law:
q& w = α(Tw − T∞ )
Solve for Tw
q& w
Tw = T∞ +
α
[W/m2]
(1)
fixed
Surface temperature depends on α
3
In general, α is not uniform over a surface
• Local heat transfer coefficient, α = f(S), f(x)
q& w = α(Tw − T∞ ) [W/m2]
q& w
y
x
• Average heat transfer coefficient, α
For α=f(S)
1
α = ∫ αdS
SS
For α=f(x)
1 L
α = ∫ α ( x )dx
L0
(w
)
∞
Q& = α S T − T
w
[W] (2)
(3)
4
Similarly for Mass Transfer (as for Heat Transfer)
Consider a lake or a pond and
its surface from which water
evaporates (or an evaporating
droplet)
Two substances, one labeled A (water vapor)
is transferred into B (dry or humid air).
ρ A,w
mass concentration of substance A – density [kg/m3] –
at surface temperature and assumed in saturated state
ρ A,∞ mass concentration of substance A – at free stream
conditions (temperature, humidity, pressure)
5
q& w = α(Tw − T∞ )
Heat transfer is proportional to temperature difference
Mass transfer is proportional to concentration difference
If
ρ A,w ≠ ρ A,∞
then m A = β ( ρA,w − ρA,∞ ) [kg/s.m2]
(6)
β is mass transfer coefficient [m/s] – souč. přestupu hmoty
For the entire surface S: m& A = β S ( ρ A,w − ρ A,∞ ) [kg/s]
β is an average mass transfer
coefficient [m/s]
1
β = ∫ βdS
SS
(7)
6
• Conclusion:
For a given geometry,
heat transfer coefficient depends on
fluid motion and fluid nature
What is the objective of this chapter?
Examine thermal interaction between a surface and a
moving fluid and determine:
(1) The heat transfer coefficient α
(2) Surface heat flux q& w
(3) Surface temperature Tw
7
Heat transfer coefficient
(1) How is α determined analytically?
(2) Why is α introduced ?
8
Apply Fourier's law to the fluid at surface
∂T
q& w = − λ f
∂y
y =0
(8)
Heat conducted across a thin sticky
layer on the surface
Balance between conducted heat
and heat convected downstream
the surface - consider Newton's Vu∞
∞
law
T∞
(9)
q& w = α(Tw − T∞ )
y
T ( x, y)
x
•
Tsw
Fig. 6.2
Combine Newton’s and Fourier’s laws
∂T
− λf
∂ y y =0
q& w
=
α=
Tw − T∞
Tw − T∞
q& w
(10)
9
• To get
α from (10), we must determine temperature
distribution T ( x , y ) in the fluid and to obtain temperature
gradient in the fluid ∂T / ∂ y y = 0
− λf
∂T
∂y
q& w
y =0
α=
=
Tw − T∞
Tw − T∞
(10)
10
Governing Equations for Convection Heat
Transfer
• Focal point in convection:
Determination of temperature
distribution in a moving fluid
• Basic laws governing temperature distribution:
(1) Conservation of mass
(2) Conservation of momentum
(3) Conservation of energy
11
• Assumptions:
(1) Two-dimensional u(x,y), v(x,y)
(2) Single phase flow (water, air, etc)
Conservation of Mass: The Continuity
Equation
m& y +
y
dy
dx
x
(a)
Fig. 6.4
∂y
dy
m& x +
m& x
dy
∂m& y
∂m& x
dx
∂x
dx
m& y
(b)
12
Apply conservation of mass to an element dxdy:
Rate of mass added - Rate removed
= Rate of mass change within
(a)
Apply (a) Using Fig. 6.4:
+
⎡
−⎢
⎣
∂
⎤ ⎡
+(
)dx ⎥ − ⎢
∂x
⎦ ⎣
∂
⎤ ∂m
+(
)dy ⎥ =
∂y
⎦ ∂t
(b)
= mass flow rate entering element in the x-direction
= mass flow rate entering element in the y-direction
m = mass within element
Express (b) in terms of fluid density and velocity:
13
= ρVA
(c)
A = flow area
V = velocity normal to A
ρ = density
Apply (c) to the element
= ρ udy
= ρ vdx
(d)
(e)
u and v are the velocity components in the x and y
directions
m = mass of element:
m = ρdxdy
(f)
14
(d)–(f) into (b)
∂ ρ ∂ ( ρ u) ∂ ( ρ v )
+
+
=0
∂t
∂x
∂y
(11)
Incompressible fluid: ρ = constant
∂u ∂v
+
=0
∂x ∂ y
(12)
is the continuity equation
15
Conservation of Momentum:
The Navier-Stokes Equations of Motion
Assume: 2-D
Newton's law of motion: Apply to element dxdy in
x-direction
(a)
∑ F x = ma x
a x = acceleration in the x-direction
m = mass of the element
m = ρdxdy
• Acceleration a x :
we need:
(b)
u = u(x,y,t)
16
The total change in u is
∂u
∂u
∂u
du =
dx +
dy +
dt
∂x
∂y
∂t
Divide by dt and note that dx/dt = u and dy/dt = v
∂u
du
∂u ∂u
ax =
=u
+v
+
dt
∂x
∂ y ∂t
(c)
∂u
∂u
u
+v
= convective acceleration
∂x
∂y
∂u
= local acceleration
∂t
• ∑ Fx : Two types of external forces:
17
Body forces, ∑ Fb : Gravity
∑ Fb = ρ g dxdy
(d)
Surface forces, ∑ Fs :
Normal: pressure p and normal stress σ xx
Tangential: shearing stressτ xy
Total external forces:
∑ F x = ∑ Fb + ∑ Fs
(e)
18
Surface forces:
∑
∂ p ∂σ xx ∂τ y x
Fs = ( −
+
)dxdy
+
∂x
∂x
∂y
(τ yx +
∂τ yx
∂y
dy )dx
(σ xx +
σ xx dy
dy
pdy
(f)
dx
τ yx dx
∂σ xx
dx )dy
∂x
∂p
( p + dx )dy
∂x
Fig. 6.5
(e) and (f) into (d)
∑
∂ p ∂σ xx ∂τ yx
Fx = ( ρ g −
+
+
)dxdy
∂x
∂x
∂y
(g)
19
(b), (c) and (g) into (a)
∂u
∂u
∂u
∂ p ∂σ xx ∂τ yx (13)
+
+
ρ( + u + v ) = ρ g −
∂t
∂x
∂y
∂x
∂x
∂y
By analogy: y-direction
∂v
∂v
∂v
∂ p ∂σ yy ∂σ xy
ρ( + u + v ) = − +
+
∂t
∂x
∂y
∂y
∂y
∂x
(14)
Too many unknowns!
• Important assumption: The variables σ xx , σ yy , τ xy ,
and τ yx are eliminated using empirical relations. For
incompressible fluids:
20
σ xx
∂u
= 2μ
∂x
(15)
σ yy
∂v
= 2μ
∂y
(16)
τ xy = τ yx
⎛ ∂u ∂ v ⎞
= μ ⎜⎜
+ ⎟⎟
⎝ ∂y ∂x ⎠
(17)
• Fluids that obey these relations, such as water, air and
oil, are referred to as Newtonian fluids
• Polymers, honey, etc. do not follow these relations
and are known as non-Newtonian fluids
(15)-(17) into (13) and (14), assume constant viscosity
21
∂u
∂u
∂u
∂p
∂ 2u ∂ 2u
ρ ( + u + v )= ρg − + μ ( 2 + 2)
∂t
∂x
∂y
∂x
∂x
∂y
(18)
and
∂v
∂v
∂v
∂p
∂ 2v ∂ 2v
ρ ( + u + v ) = − + μ ( 2 + 2 ) (19)
∂t
∂x
∂y
∂y
∂x
∂y
(18) and (19) are the equations of motion in
rectangular coordinates. They are also known as the
Navier-Stokes equations of motion.
22
Limitations on (18) and (19):
(1) Newtonian fluids
(2) Constant density ρ
(3) Constant viscosity μ
(4) Two-dimensional flow
(5) Gravity pointing in the positive x-direction
23
Conservation of Energy
Assumptions:
(1) Two-dimensional
(2) Negligible changes in kinetic and potential energy
(3) Negligible energy transfer due to normal stresses σ xx
and σ yy , and shearing stress τ xy (viscous dissipation)
(4) Constant properties
24
Apply conservation of energy, to element dxdy :
dE
&
&
&
&
Ein + Eg − Eout = Eak =
dt
[W]
Energy by conduction and convection
25
Energy generation and energy change within the element:
E& g = Q& zdr dxdy
(a)
dE
∂T
&
E ak =
dxdy
= ρc
(b)
dt
∂t
Energy convected into the element (carried with flowing fluid):
Q& conv , x = ρuidy = ρuc pTdy
incompressible fluid
(c)
di = c p dT
Q& conv , y = ρuidx = ρuc pTdx
Energy conducted into the element (carried by molecular
motion - Fourier’s law):
∂T
&
Qcond , x = − λdy
∂x
∂T
&
Qcond , y = − λdx
∂y
(d)
26
(a), (b), (c) and (d) into conservation of energy and using
the continuity equation
Q& zdr
∂T
∂T
∂T
∂ 2T ∂ 2T
+u
+v
= α( 2 + 2 ) +
∂t
∂x
∂y
ρc p
∂x
∂y
(20)
a = thermal diffusivity (součinitel tepelné vodivosti):
λ
a=
ρc p
• Equation (6.20) is the energy equation in rectangular
coordinates for 2-D constant property fluids
27
• Physical significance of each term in (6.20):
Q& zdr
∂T
∂T
∂T
∂ 2T ∂ 2T
+u
+v
= α( 2 + 2 ) +
∂t
∂x
∂y
ρc p
∂x
∂y
(1)
(2)
(3)
(20)
(4)
(1) First term: Local rate of energy change
(2) Second term: Net energy convected with fluid
(3) Third: Net energy conducted in the x and y
directions
(4) Fourth term: Energy generation
28
Summary of the Governing Equations for
Convection Heat Transfer: Mathematical
Implications
Assumptions:
(1) Newtonian fluid
(2) Two-dimensional
(3) Negligible changes in kinetic and potential energy
(4) Constant properties (except in buoyancy)
(5) Gravity is in the positive x-direction
Continuity:
∂u ∂v
+
=0
∂x ∂ y
29
x-momentum:
∂u
∂u
∂u
∂p
∂ 2u ∂ 2u
+u +v
ρ(
) = − + μ( 2 + 2
∂t
∂x
∂y
∂x
∂x
∂y
)
y-momentum:
∂v
∂v
∂v
∂p
∂ 2v ∂ 2v
ρ( +u +v )= − +μ ( 2 + 2)
∂t
∂x
∂y
∂y
∂x
∂y
Energy:
⎛ ∂ 2T ∂ 2T ⎞ Q& zdr
∂T
∂T
∂T
+u
+v
= α⎜⎜ 2 + 2 ⎟⎟ +
∂t
∂x
∂y
∂ y ⎠ ρc p
⎝ ∂x
30
The Boundary Layer Concept:
Simplification of the Governing Equations
Velocity boundary layer:
Under certain conditions the effect of viscosity will be
confined to a thin region near a surface called
the velocity or viscous boundary layer
The edge of this region is defined by the thickness δ, which is
referred to as a distance where u = 0.99 u∞
τw
∂u
Shear stress
Friction
cf = 2
τw = μ
(smykové napětí)
coefficient
ρu∞
∂y y =0
231
Conditions for the existence of the velocity boundary layer:
(1) Streamlined body without flow separation
y
y
x
δ
R
(2) High Reynolds number (Re > 100)
Boundary layer features:
Fig. 6.10
(1) Velocity at the surface vanishes. This is the no-slip
condition
(2) Velocity changes rapidly across the boundary layer
thickness δ . At the edge u ≈ u∞
(3) Viscosity plays no role outside the velocity boundary
layer
32
Thermal boundary layer:
Under certain conditions the
effect of thermal interaction
between a fluid and a surface will
be confined to a thin region near
the surface called the thermal
boundary layer
heating
The edge of this region is defined by the thickness δT
where Tw − T = 0,99 (Tw − T∞ )
∂T
Heat flux transferred at the wall: q& w = − λ f
∂y
y =0
= α (Tw − T∞ )
33
Heat Transfer Coefficient
α=
− λ f ∂T
tangent
∂y y =0
(Tw − T∞ )
heat flux qw
For (Tw-T∞) = const,
what’s the behavior of α?
As the boundary layer increases,
the temperature gradient ∂T ∂x decreases.
Why?
The same temperature difference (Tw-T∞)
∂T
applies to a larger distance –
∂x
decreases and so α does.
∂T
∂y y =0
α
x
34
Conditions for the existence of the thermal boundary layer:
(1) Streamlined body without flow separation
(2) High product of the Reynolds and Prandtl numbers
(Re Pr >100)
ρu∞ L
Peclet Number = Pe = (
μ
)(
cpμ
λ
)=
ρc p u∞ L
λ
(4) Temperature changes rapidly across the thermal
boundary layer thickness δ t . At the edge T ≈ T∞
(5) In general, both velocity and thermal boundary layer
are thin
35
Laminar vs. Turbulent Flow
u
u
turbulent
(a)
t
laminar
(b)
t
Turbulent flow: Random fluctuations in velocity, temperature
Laminar flow: Streamlines are smooth. Fluctuations are absent.
36
• Transition Reynolds number, Re t
• Used to check if the flow is laminar or turbulent
• Ret is determined experimentally
• Its value depends on geometry, surface roughness,
pressure gradient, …
• For uniform flow over a semi-infinite plate:
Re x,trans
u∞ xtrans
=
≅ 500000
ν
y
x
u∞
• For flow through smooth tubes:
Retrans
uD
=
≅ 2300
ν
u
• Magnitude of Retrans can be changed by manipulating
surface roughness, pressure gradient, …
37
Laminar vs. Turbulent Flow
α
Laminar
x
Transition
Turbulent
38
Mathematical Simplifications for Boundary
Layer Flows
• Not always all terms in the momentum equations are
necessary to take into account
•
Often, incompressible flow, ρ=const, often constant
physical quantities λ, µ, negligible mass forces
(gravitational etc.), no internal heat source.
•
Often u>>v,
∂u
∂u ∂v ∂v
>> , ,
∂x ∂y ∂x
∂y
∂T
∂T
>>
∂x
∂y
y
x
u∞
39
Summary of Boundary Layer Equations
for Steady Laminar Flow
Assumptions:
(1) Newtonian fluid
(2) Two-dimensional
(3) Negligible changes in kinetic and potential energy
(4) Constant properties
(5) Streamlined surface
(6) High Reynolds number (Re > 100)
(7) High Peclet number (Pe > 100).
40
(8) Steady state
(9) Laminar flow
(10) No dissipation, Φ = 0
(11) No buoyancy, β = 0
(12) No gravity
(13) No energy generation,
41
∂u ∂v
+
=0
∂x ∂y
Continuity:
1 dp ∞
∂u
∂u
∂ 2u
u +v =−
+ν 2
ρ dx
∂y
∂x
∂y
x-Momentum:
Assuming zero pressure gradient
∂u
∂u
∂ 2u
u +v =ν 2
∂x
∂y
∂y
Energy:
∂T
∂T
∂ 2T
u
+v
=a 2
∂x
∂y
∂y
a=
Both equations identical –
Velocity and temperature
profiles will be similar –
analogy between
momentum and heat transfer
λ
ρcp
thermal diffusivity
souč. teplotní vodivosti
42
Classification of Convection Heat Transfer
1. Forced convection vs. free convection
2. External vs. internal flow
3. Boundary layer flow vs. low Reynolds number flow
4. Compressible vs. incompressible flow
5. Laminar vs. turbulent flow
6. Newtonian vs. non-Newtonian fluid
43
Fluid Properties
Fluid properties needed to solve convection problems:
Specific heat cp
Thermal conductivity λ
Prandtl number Pr
Thermal diffusivity a
Dynamic viscosity μ
Kinematic viscosity ν
Density ρ
44
Heat Transfer Coefficient and Dimensionless
Criteria
From boundary layer
α=
− λ f ∂T
∂y y =0
(Tw − T∞ )
General functional dependence
α = f ( u, L, ρ ,ν , c , λ )
for forced convection
αL Nusselt number
Nu =
λ
uL
Reynolds number
Re =
ν
νρc ν
Prandtl number
Pr =
=
λ a
7 quantities
4 primary dimensions
J/K, kg, m, s
Buckingham π theorem
3 dimensionless similarity
parameters - numbers
45
Formula with dimensionless numbers – correlation equations:
Nu = f (Re, Pr )
forced convection
Actual form of the equation depends on the system:
• Forced convection in a tube – laminar or turbulent (entrance
length L/d, fully developed region)
• Cross flow over a cylinder, tube bundle
• Forced convection for external flow on a flat plate
• Natural convection (another dimensionless number enters
into play – Grasshoff number)
• Flow with viscous dissipation (another dimensionless number
enters into play – Eckert number) etc.
How such equations can be obtained?
Mostly by experiments or by analytical solution
for simple situations or systems (e.g. flow over a flat plate)
46