of 47
7/28/2019 08. Basic Concepts in Convection_Bounday Layers
1/47
1
8. BASIC CONCEPTS IN CONVECTION
Introduction
Characteristic of convection: Fluid motion Focal point: Determination of heat transfer coefficient
- souinitel pestupu tepla Determination of:
Temperature distribution in the fluid
7/28/2019 08. Basic Concepts in Convection_Bounday Layers
2/47
2
General ObservationsThe Role of Fluid Motion
For the electric bulb:
wq& = surface flux
wT = surface temperature
T = free stream temperatureu = free stream velocity
For a fixed input power how to lower surface temperature?
? Raise or lower u Change the cooling fluid?
VT
+
sTsq
1.6Fig.
u
wq&wT
7/28/2019 08. Basic Concepts in Convection_Bounday Layers
3/47
3
water lowers surface temperature toair
from
fluid
the
changingorIncreasing u
Surface temperature depends on
Conclusion:
Fluid motion and fluid nature
play important roles in convection
Solve for wT
qTT ww
&+=
fixed
Newtons law:
( )= TTq ww& (8.1)[W/m2]
7/28/2019 08. Basic Concepts in Convection_Bounday Layers
4/47
4
For =f(x) = L dxxL
0
)(1
(8.3)
Local heat transfer coefficient, = f(S), f(x)
Average heat transfer coefficient,
is not uniform over a surfaceIn general,
( )= TTq ww& [W/m2]
For =f(S) = SdSS1
(8.2)=
TTSQ ww& [W]
wq&
x
y
7/28/2019 08. Basic Concepts in Convection_Bounday Layers
5/47
5
Similarly for Mass Transfer (as for Heat Transfer)
Consider a lake or a pond and
its surface from which water
evaporates (or an evaporatingdroplet)
Two substances, one labeledA (water vapor)
is transferred intoB (dry or humid air).
A,w
mass concentration of substanceA density [kg/m
3
]at surface temperature and assumed in saturated state
A, mass concentration of substanceA at free stream
conditions (temperature, humidity, pressure)
7/28/2019 08. Basic Concepts in Convection_Bounday Layers
6/47
6
For the entire surfaceS: [kg/s]= A,A,wA Sm&
is an average mass transfercoefficient [m/s]
=S
dSS
1
(8.5)
Mass transfer is proportional to concentration difference
( )= TTq ww&
Heat transfer is proportional to temperature difference
is mass transfer coefficient [m/s] sou. pestupu hmoty
(8.4)=A,A,wA
m [kg/s.m2]thenIf A,A,w
7/28/2019 08. Basic Concepts in Convection_Bounday Layers
7/47
7
Conclusion:
naturefluidandmotionfluidondependscoefficienttransferheat
geometry,givenaFor
What is the objective of this chapter?
Examine thermal interaction between a surface and a
moving fluid and determine:
(1) The heat transfer coefficient
(3) Surface temperature Tw(2) Surface heat flux wq
&
7/28/2019 08. Basic Concepts in Convection_Bounday Layers
8/47
8
introduced ?isWhy(2) analytically?determinedisHow)1(
Heat transfer coefficient
7/28/2019 08. Basic Concepts in Convection_Bounday Layers
9/47
9
Combine Newtons and Fouriers laws
(8.8)
=
=
=
TT
yT
TT
q
w
0yf
w
w&
Apply Fourier's law to thefluidat surface
0y
fwy
Tq
=
=& (8.6) Heat conducted across a thin sticky
layer on the surface
Balance between conducted heatand heat convected downstream
the surface - consider Newton's
law( )= TTq ww&
VT x
sT
6.2Fig.
),( yxT
wT
u
wq&(8.7)
7/28/2019 08. Basic Concepts in Convection_Bounday Layers
10/47
10
To get from (8.8), we must determine temperature
distribution in the fluid and to obtain temperature
gradient in the fluid
),(xT
0/ =yyT
(8.8)
=
=
=TT
yT
TT
q
w
0yf
w
w&
7/28/2019 08. Basic Concepts in Convection_Bounday Layers
11/47
11
Governing Equations for Convection Heat
Transfer
Focal point in convection:
a
temperatureofDeterminationfluidmovingindistribution
Basic laws governing temperature distribution:
(3) Conservation of energy
(1) Conservation of mass(2) Conservation of momentum
7/28/2019 08. Basic Concepts in Convection_Bounday Layers
12/47
12
Assumptions:
(1) Two-dimensional u(x,y), v(x,y)(2) Single phase flow (water, air, etc)
Conservation of Mass: The ContinuityEquation
(b)(a) 6.4Fig.
x
y
ym&
dyy
mm
yy
+ &&
xm&dx
x
mm xx + &&
dxdx
dydy
7/28/2019 08. Basic Concepts in Convection_Bounday Layers
13/47
13
Apply conservation of mass to an element dxdy:
removedRate-addedmassofRatewithinchangemassofRate
(a)
Apply (a) Using previous Figure:
= mass flow rate entering element in thex-direction
= mass flow rate entering element in the y-direction
m = mass within element
Express (b) in terms of fluid density and velocity:
(b)
)()(t
mdy
ydx
x =
+
+
7/28/2019 08. Basic Concepts in Convection_Bounday Layers
14/47
14
m = mass of element:
dxdym (f)
= flow area
V= velocity normal to
= density
(c)VA
u and v are the velocity components in thexand y
directions
Apply (c) to the element
(d)udy
(e)vdx
7/28/2019 08. Basic Concepts in Convection_Bounday Layers
15/47
15
(d)(f) into (b)
(8.9)0)()( =++ yx
u
t
v
Incompressible fluid: constant
(8.10)0+
yx
u v
is the continuity equation
7/28/2019 08. Basic Concepts in Convection_Bounday Layers
16/47
16
Conservation of Momentum:
The Navier-Stokes Equations of Motion
Assume: 2-D
Newton's law of motion: Apply to element dxdyx-direction
in
(a)xx maF =xa = acceleration in thex-direction
m = mass of the element
dxdym (b) Acceleration :xa
we need: u = u(x,y,t)
7/28/2019 08. Basic Concepts in Convection_Bounday Layers
17/47
17
The total change in u is
dtt
udy
y
udx
x
udu
++
=Divide by dtand note that dx/dt= u and dy/dt= v
(c)t
u
y
u
x
uu
dt
duax
++
= v
y
u
x
uu
+
v = convective acceleration
tu = local acceleration
xF : Two types of external forces:
7/28/2019 08. Basic Concepts in Convection_Bounday Layers
18/47
18
Surface forces, sF :Normal: pressurep and normal stress xx
Tangential: shearing stress xy
Body forces, bF : GravitydxdygFb (d)
(e)+ sbx FTotal external forces:
7/28/2019 08. Basic Concepts in Convection_Bounday Layers
19/47
19
dxyx
dxdy
y
yxyx )(
+ dyxx dydx
x
xxxx )(
+
pdy dydxx
p
p )(
+dxdy
6.5Fig.
(e) and (f) into (d)
dxdy
yxx
pgF
yxxxx )(
++
(g)
(f)dxdyyxx
pF xyxxs )( ++
Surface forces:
7/28/2019 08. Basic Concepts in Convection_Bounday Layers
20/47
20
By analogy:y-direction
xyy
p
yx
u
t
xyyy +
+
++
)( vvvv (8.12)Too many unknowns!
Important assumption: The variables ,xx
,yy
,xy
yxand are eliminated using empirical relations. For
incompressible fluids:
(b), (c) and (g) into (a)
yxx
pg
y
u
x
uu
t
u yxxx +
+
++
)( v (8.11)
7/28/2019 08. Basic Concepts in Convection_Bounday Layers
21/47
21
Fluids that obey these relations, such as water, air and
oil, are referred to asNewtonian fluids
Polymers, honey, etc. do not follow these relationsand are known as non-Newtonian fluids
(8.13)-(8.15) into (8.11) and (8.12), assume constant
viscosity
(8.13)x
u
xx
= 2
y
yy
= v2 (8.14)
(8.15)
+
=x
v
y
uyxxy
7/28/2019 08. Basic Concepts in Convection_Bounday Layers
22/47
22
(8.16))()( 2
2
2y
u
x
u
x
pgy
u
x
uut
u
+
+
++
2vand
)()(2
2
2
2
yxy
p
yxu
t ++++vvv
vvv (8.17)
(8.16) and (8.17) are the equations of motion inrectangular coordinates. They are also known as the
Navier-Stokes equations of motion.
7/28/2019 08. Basic Concepts in Convection_Bounday Layers
23/47
23
(3) Constant viscosity (4) Two-dimensional flow
(5) Gravity pointing in the positivex-direction
Limitations on (8.16) and (8.17):
(1) Newtonian fluids
(2) Constant density
7/28/2019 08. Basic Concepts in Convection_Bounday Layers
24/47
7/28/2019 08. Basic Concepts in Convection_Bounday Layers
25/47
25
Energy by conduction and convection
Apply conservation of energy, to element dxdy :
dtdEEEE akoutgin ==+ &&&& [W]
7/28/2019 08. Basic Concepts in Convection_Bounday Layers
26/47
7/28/2019 08. Basic Concepts in Convection_Bounday Layers
27/47
27
(a), (b), (c) and (d) into conservation of energy and using
the continuity equation
(8.18)p
zdr
2
2
2
2
c
Q
y
T
x
T
y
Tv
x
Tu
t
T)(
&
+
+
=
+
+
a = thermal diffusivity (souinitel tepeln vodivosti):
pc
a =
Equation (8.18) is the energy equation in rectangularcoordinates for 2-D constant property fluids
7/28/2019 08. Basic Concepts in Convection_Bounday Layers
28/47
28
(1) First term: Local rate of energy change
(2) Second term: Net energy convected with fluid
(3) Third: Net energy conducted in thexandy
directions
(4) Fourth term: Energy generation
Physical significance of each term in (8.18):
(8.18)p
zdr2
2
2
2
c
Q
y
T
x
T
y
Tv
x
Tu
t
T)(
&
+
+
=
+
+
(1) (2) (3) (4)
7/28/2019 08. Basic Concepts in Convection_Bounday Layers
29/47
29
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 positivex-direction
Continuity:
0+
yx
u v
7/28/2019 08. Basic Concepts in Convection_Bounday Layers
30/47
30
x-momentum:
)()(2
2
2
2
y
u
x
ux
p
y
uvx
uut
u+
+
=
+
+
y-momentum:
)()(2
2
2
2
yxy
p
yxu
t +
+
++
vvvv
vv
Energy:
p
zdr2
2
2
2
cQ
yT
xT
yTv
xTu
tT
&
+
+
=
+
+
7/28/2019 08. Basic Concepts in Convection_Bounday Layers
31/47
7/28/2019 08. Basic Concepts in Convection_Bounday Layers
32/47
32
Boundary layer features:
(1) Velocity at the surface vanishes. This is the no-slip
condition
(2) Velocity changes rapidly across the boundary layerthickness . At the edge uu
(3) Viscosity plays no role outside the velocity boundary
layer
Conditions for the existence of the velocity boundary layer:
(2) High Reynolds number (Re > 100)
(1) Streamlined body without flow separation
6.10Fig.
R
y
xy
7/28/2019 08. Basic Concepts in Convection_Bounday Layers
33/47
33
heating
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
The edge of this region is defined by the thickness
whereT
( )= TT0,99TT ww
( )=
=
= TT
y
Tq w
0y
fw&Heat flux transferred at the wall:
7/28/2019 08. Basic Concepts in Convection_Bounday Layers
34/47
34
heat flux qw0yy
T=
tangent
For (Tw-T) = const,whats the behavior of?
x
( )
=
=TT
yT
w
0yf
Heat Transfer Coefficient
As the boundary layer increases,
the temperature gradient decreases.
Why?
The same temperature difference (Tw-T)applies to a larger distance
decreases and so does.
xT
xT
7/28/2019 08. Basic Concepts in Convection_Bounday Layers
35/47
35
(1) Streamlined body without flow separation
(2) High product of the Reynolds and Prandtl numbers
(Re Pr>100)
Luc
c
LuPe
pp ))( === (NumberPeclet
(4) Temperature changes rapidly across the thermal
boundary layer thickness t . At the edge TT(5) In general, both velocity and thermal boundary layer
are thin
Conditions for the existence of the thermal boundary layer:
7/28/2019 08. Basic Concepts in Convection_Bounday Layers
36/47
Transition Reynolds number Re
7/28/2019 08. Basic Concepts in Convection_Bounday Layers
37/47
37
Transition Reynolds number, tRe
Used to check if the flow is laminar or turbulent Ret is determined experimentally
Its value depends on geometry, surface roughness,
pressure gradient,
Magnitude of transRe can be changed by manipulating
surface roughness, pressure gradient,
For uniform flow over a semi-infinite plate:
500000
xu
Retrans
x,trans =
ux
y
For flow through smooth tubes:
2300
DuRetrans = u
L i T b l t Fl
7/28/2019 08. Basic Concepts in Convection_Bounday Layers
38/47
38
Laminar vs. Turbulent Flow
xLaminar Transition Turbulent
7/28/2019 08. Basic Concepts in Convection_Bounday Layers
39/47
39
x-momentum:
)()(2
2
2
2
y
u
x
uxp
yuv
xuu
tu
+
+
=
+
+
y-momentum:
)()(2
2
2
2
yxy
p
yxu
t +
+
++
vvvv
vv
Energy:
p
zdr2
2
2
2
cQ
yT
xT
yTv
xTu
tT
&
+
+
=
+
+
M th ti l Si lifi ti f B d
7/28/2019 08. Basic Concepts in Convection_Bounday Layers
40/47
40
Mathematical Simplifications for Boundary
Layer Flows
Not always all terms in the momentum equations are
necessary to take into account
Often, incompressible flow, =const, often constant
physical quantities , , negligible mass forces
(gravitational etc.), no internal heat source.
Often u>>v,x
v,
y
v,
x
u
y
u
>>
x
T
y
T
>>
ux
y
7/28/2019 08. Basic Concepts in Convection_Bounday Layers
41/47
41
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).
7/28/2019 08. Basic Concepts in Convection_Bounday Layers
42/47
42
(8) Steady state
(9) Laminar flow
(12) No gravity
(10) No dissipation, 0
(11) No buoyancy, 0
(13) No energy generation,
7/28/2019 08. Basic Concepts in Convection_Bounday Layers
43/47
7/28/2019 08. Basic Concepts in Convection_Bounday Layers
44/47
44
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 flow5. Laminar vs. turbulent flow
6. Newtonian vs. non-Newtonian fluid
7/28/2019 08. Basic Concepts in Convection_Bounday Layers
45/47
45
Fluid Properties
Fluid properties needed to solve convection problems:
Specific heat cpThermal conductivity
Prandtl numberPr
Thermal diffusivity a
Dynamic viscosity Kinematic viscosity Density
Heat Transfer Coefficient and Dimensionless
7/28/2019 08. Basic Concepts in Convection_Bounday Layers
46/47
46
Heat Transfer Coefficient and Dimensionless
Criteria
( )
=
=TT
yT
w
0yfFrom boundary layer
7 quantities
4 primary dimensions
J/K, kg, m, s
Buckingham theorem
3 dimensionless similarity
parameters - numbers
General functional dependence
for forced convection),,,,,( cLuf =
LNu = Nusselt number
uL
Re = Reynolds number
a
cPr == Prandtl number
Formula with dimensionless numbers correlation equations:
7/28/2019 08. Basic Concepts in Convection_Bounday Layers
47/47
47
( )PrRe,Nu =
Formula with dimensionless numbers correlation equations:
forced convection
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)
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 entersinto play Grashoff number)
Flow with viscous dissipation (another dimensionless number
enters into play Eckert number) etc.
Actual form of the equation depends on the system: