Forced convection

Convection
• Heat transfer in the presence of a fluid motion on a solid surface
•Various mechanisms at play in the fluid:
- advection → physical transport of the fluid
- diffusion → conduction in the fluid
- generation → due to fluid friction
•But fluid directly in contact with the wall does not move relative to it; hence
direct heat transport to the fluid is by conduction in the fluid only.
T(y)q”
y y
U ∞ T∞
Ts
u(y)
U ∞
( )∞
=
−=


∂
∂
−=′′ TTh
y
T
kq s
y
fconv
0
But depends on the whole fluid motion, and both fluid flow
and heat transfer equations are needed
0=



∂
∂
y
y
T
T(y)
y T∞U ∞
Ts

Convection
Free or natural convection
(induced by buoyancy
forces)
forced convection (driven
externally)
May occur with
phase change
(boiling,
condensation)
Convection
Typical values of h (W/m2
K)
Free convection: gases: 2 - 25
liquid: 50 - 100
Forced convection: gases: 25 - 250
liquid: 50 - 20,000
Boiling/Condensation: 2500 -100,000
Heat transfer rate q = h( Ts-T ∞ )W
h=heat transfer coefficient (W /m2
K)
(h is not a property. It depends on
geometry ,nature of flow,
thermodynamics properties etc.)

T(y)q”
y y
U ∞ T∞
Ts
u(y)
U ∞
Convection rate equation
Main purpose of convective heat
transfer analysis is to determine:
• flow field
• temperature field in fluid
• heat transfer coefficient, h
q’’=heat flux = h(Ts - T∞)
q’’ = -k(∂T/ ∂y)y=0
Hence, h = [-k(∂T/ ∂y)y=0] / (Ts - T∞)
The expression shows that in order to determine h, we
must first determine the temperature distribution in the
thin fluid layer that coats the wall.

• extremely diverse
• several parameters involved (fluid properties, geometry, nature of flow,
phases etc)
• systematic approach required
• classify flows into certain types, based on certain parameters
• identify parameters governing the flow, and group them into meaningful
non-dimensional numbers
• need to understand the physics behind each phenomenon
Classes of convective flows:
Common classifications:
A. Based on geometry:
External flow / Internal flow
B. Based on driving mechanism
Natural convection / forced convection / mixed convection
C. Based on number of phases
Single phase / multiple phase
D. Based on nature of flow
Laminar / turbulent

How to solve a convection problem ?
• Solve governing equations along with boundary conditions
• Governing equations include
1. conservation of mass
2. conservation of momentum
3. conservation of energy
• In Conduction problems, only (3) is needed to be solved.
Hence, only few parameters are involved
• In Convection, all the governing equations need to be
solved.
⇒ large number of parameters can be involved

• Nusselt No. Nu = hx / k = (convection heat transfer strength)/
(conduction heat transfer strength)
• Prandtl No. Pr = ν/α = (momentum diffusivity)/ (thermal diffusivity)
• Reynolds No. Re = U x / ν = (inertia force)/(viscous force)
Viscous force provides the dampening effect for disturbances in the
fluid. If dampening is strong enough ⇒ laminar flow
Otherwise, instability ⇒ turbulent flow ⇒ critical Reynolds number
δ
Laminar Turbulent
δ
Forced convection: Non-dimensional groupings

h=f(Fluid, Vel ,Distance,Temp)
•Fluid particle adjacent to the
solid surface is at rest
•These particles act to retard the
motion of adjoining layers
∀⇒ boundary layer effect
Momentum balance: inertia forces, pressure gradient, viscous forces,
body forces
Energy balance: convective flux, diffusive flux, heat generation, energy
storage
FORCED CONVECTION:
external flow (over flat plate)
An internal flow is surrounded by solid boundaries that can restrict the
development of its boundary layer, for example, a pipe flow. An external flow, on
the other hand, are flows over bodies immersed in an unbounded fluid so that the
flow boundary layer can grow freely in one direction. Examples include the flows
over airfoils, ship hulls, turbine blades, etc.

One of the most important concepts in understanding the external flows is the
boundary layer development. For simplicity, we are going to analyze a
boundary layer flow over a flat plate with no curvature and no external pressure
variation.
laminar turbulent
transition
Dye streakU∞ U∞ U∞
U∞
Hydrodynamic boundary layer
Boundary layer definition
Boundary layer thickness (δ): defined as the distance away from the surface
where the local velocity reaches to 99% of the free-stream velocity, that is
u(y=δ)=0.99U∞. Somewhat an easy to understand but arbitrary definition.
Boundary layer is usually very thin: δ/x usually << 1.

Hydrodynamic and Thermal
boundary layers
As we have seen earlier,the hydrodynamic boundary layer is a region of a
fluid flow, near a solid surface, where the flow patterns are directly
influenced by viscous drag from the surface wall.
0<u<U, 0<y<δ
The Thermal Boundary Layer is a region of a fluid flow, near a solid
surface, where the fluid temperatures are directly influenced by heating or
cooling from the surface wall.
0<t<T, 0<y<δt
The two boundary layers may be expected to have similar characteristics
but do not normally coincide. Liquid metals tend to conduct heat from the
wall easily and temperature changes are observed well outside the dynamic
boundary layer. Other materials tend to show velocity changes well outside
the thermal layer.

Effects of Prandtl number, Pr
δ
δT
Pr >>1
ν >> α
e.g., oils
δ, δT
Pr = 1
ν = α
e.g., air and gases
have Pr ~ 1
(0.7 - 0.9)
W
W
TT
TT
U
u
−
−
∞∞
tosimilar
(Reynold’s analogy)
δT
δ
Pr <<1
ν << α
e.g., liquid metals

Boundary layer equations (laminar flow)
• Simpler than general equations because boundary layer is thin
δ
δT
∞U
WT
∞T
∞U
x
y
• Equations for 2D, laminar, steady boundary layer flow






∂
∂
∂
∂
=
∂
∂
+
∂
∂






∂
∂
∂
∂
+=
∂
∂
+
∂
∂
=
∂
∂
+
∂
∂
∞
∞
y
T
yy
T
v
x
T
u
y
u
ydx
dU
U
y
u
v
x
u
u
y
v
x
u
α
ν
:energyofonConservati
:momentum-xofonConservati
0:massofonConservati
• Note: for a flat plate, 0hence,constantis =∞
∞
dx
dU
U

Exact solutions: Blasius
3
1
2
1
3
1
2
1
PrRe678.0numberNusseltAverage
PrRe339.0numberNusseltLocal
Re
Re
328.11
tcoefficiendragAverage
,Re
Re
664.0
tcoefficienfrictionSkin
Re
99.4
x
knesslayer thicBoundary
0
0
2
2
1
L
xx
L
L
L
fD
y
wx
x
w
f
x
uN
Nu
LU
dxC
L
C
y
uxU
U
C
=
=






===








∂
∂
==
==
=
∞
=
∞
∞
∫ ν
µτ
ν
ρ
τ
δ

Heat transfer coefficient
• Local heat transfer coefficient:
x
k
x
kNu
h xx
x
3
1
2
1
PrRe339.0
==
• Average heat transfer coefficient:
L
k
L
kuN
h L
3
1
2
1
PrRe678.0
==
• Recall: ( ) wallfromrateflowheat,∞−= TTAhq ww• Recall: ( ) wallfromrateflowheat,∞−= TTAhq ww
• Film temperature, Tfilm
For heated or cooled surfaces, the thermophysical properties within
the boundary layer should be selected based on the average
temperature of the wall and the free stream;
( )∞+= TTT wfilm 2
1

Heat transfer coefficient
U∞
x Hydrodynamic
Boundary Layer, δ
Convection
Coefficient, h.
Thermal Boundary
Layer, δt
Laminar Region Turbulent Region
Laminar and turbulent b.l.

Turbulent boundary layer
( )
( )
etc.:wayusualintcoefficienferheat transCalculate*
Re664.0Re036.0PrPrRe036.0
PrRe029.0
Re328.1Re072.0
Re
1
Re072.0
)105(ReRe059.0
:dataalexperimentonbasednscorrelatiousemainlywillWe*
solve.todifficultmore
infinitelybutones,laminarsimilar toareequationsb.l.Turbulent*
)105plate,flatoverflowFor(Re
.turbulenteventuallyandnaltransitiobecomesflowthe),ReRe(
numberReynoldsofvaluecriticalaBeyondwith x.increasesRe*
5.08.08.0
8.0
5.08.0
52.0
5
3
1
3
1
3
1
x
kNu
h
uN
Nu
C
C
x
xU
xcxcL
xx
xcxc
L
LD
xxf
c
c
xc
xcx
x
=
−−=
=
−−=
×>=
×≈=
=
−
∞
ν

• Boundary layer growth: δ ∝ √x
• Initial growth is fast
• Growth rate dδ/dx ∝ 1/√x,
decreasing downstream.
• Wall shear stress: τw ∝ 1/√x
• As the boundary layer grows, the
wall shear stress decreases as the
velocity gradient at the wall becomes
less steep.
Laminar Boundary Layer Development

Determine the boundary layer thickness, the wall shear stress of a laminar water flow
over a flat plate. The freestream velocity is 1 m/s, the kinematic viscosity of the water
is 10-6
m2
/s. The density of the water is 1,000 kg/m3
. The transition Reynolds number
Re=Ux/ν=5×105
. Determine the distance downstream of the leading edge when the
boundary transitions to turbulent. Determine the total frictional drag produced by the
laminar and turbulent portions of the plate which is 1 m long. If the free stream and
plate temperatures are 100 °C and 25 °C, respectively, determine the heat transfer rate
from the plate.
3
2
w
( ) 5 5 10 ( ).
Therefore, for a 1m long plate, the boundary layer grows by 0.005(m),
or 5 mm, a very thin layer.
0.332 0.0105
The wall shear stress, 0.332 ( )
Re
The transition Reyn
x
x
x x m
U
U U
U Pa
x x
ν
δ
ρ ρµ
τ
−
∞
∞ ∞
∞
= = ×
= = =
5U
olds number: Re 5 10 , 0.5( )tr
tr
x
x m
ν
∞
= = × =
Example

2
D
0 0
f
2
The total frictional drag is equal to the integration of the wall shear stress:
0.664
F (1) 0.332 0.939( )
Re
Define skin friction coefficient: C
0.664
for a
1 Re
2
tr tr
tr
x x
w
x
w
f
x
U U
dx U dx N
x
C
U
ρµ ρ
τ
τ
ρ
∞ ∞
∞
∞
= = = =
= =
∫ ∫
laminar boundary layer.
Example (cont..)

µ
ρ DU
D
∞
=Re
Perimeter
Area4×
=hD
D
kN
h
k
Dh
N
AU
C u
u
normal
D ===
∞
;;
forceDrag
2
2
1
ρ
Forced convection over exterior bodies
• Much more complicated.
•Some boundary layer may exist, but it is likely
to be curved and U∞ will not be constant.
• Boundary layer may also separate from the
wall.
• Correlations based on experimental data can
be used for flow and heat transfer calculations
• Reynolds number should now be based on a
characteristic diameter.
• If body is not circular, the equivalent
diameter Dh is used

0.70.0810-102
0.60.26102-10
0.50.5110-40
0.40.75401
mCRe
Pr
Pr
ReuN
65
53
3
D
25.
62.
×
×
−
=
s
m
DC
Flow over circular cylinders
All properties at free stream
temperature, Prs at cylinder
surface temperature

Flow over circular cylinders
Flow patterns for cross
flow over a cylinder at
various Reynolds
numbers

• Thermal conditions
 Laminar or turbulent
 entrance flow and fully developed thermal condition
For laminar flows the thermal entrance length is a function of the
Reynolds number and the Prandtl number: xfd,t/D ≈ 0.05ReDPr,
where the Prandtl number is defined as Pr = ν/α and α is the thermal
diffusitivity.
For turbulent flow, xfd,t ≈ 10D.
FORCED CONVECTION: Internal flow
Thermal entrance region, xfd,t
e.g. pipe flow

Thermal Conditions
• For a fully developed pipe flow, the convection
coefficient is a constant and is not varied along the
pipe length. (as long as all thermal and flow
properties are constant also.)
x
h(x)
xfd,t
constant
• Newton’s law of cooling: q”S = hA(TS-Tm)
Question: since the temperature inside a pipe flow is not constant,
what temperature we should use. A mean temperature Tm is
defined.

Energy Transfer
Consider the total thermal energy carried by the fluid
as (mass flux) (internal energy)v
A
VC TdAρ =∫
Now image this same amount of energy is carried by a body of
fluid with the same mass flow rate but at a uniform mean
temperature Tm. Therefore Tm can be defined as
v
A
m
v
VC TdA
T
mC
ρ
=
∫
&
Consider Tm as the reference temperature of the fluid so that the
total heat transfer between the pipe and the fluid is governed by the
Newton’s cooling law as: qs”=h(Ts-Tm), where h is the local
convection coefficient, and Ts is the local surface temperature.
Note: usually Tm is not a constant and it varies along the pipe
depending on the condition of the heat transfer.

Energy Balance
Example: We would like to design a solar water heater that can heat up the
water temperature from 20° C to 50° C at a water flow rate of 0.15 kg/s. The
water is flowing through a 5 cm diameter pipe and is receiving a net solar
radiation flux of 200 W per unit length (meter). Determine the total pipe length
required to achieve the goal.

Example (cont.)
Questions: (1) How do we determine the heat transfer coefficient, h?
There are a total of six parameters involving in this problem: h, V, D, ν, kf,
cp. The last two variables are thermal conductivity and the specific heat of
the water. The temperature dependence is implicit and is only through the
variation of thermal properties. Density ρ is included in the kinematic
viscosity, ν=µ/ρ. According to the Buckingham theorem, it is possible for
us to reduce the number of parameters by three. Therefore, the convection
coefficient relationship can be reduced to a function of only three
variables:
Nu=hD/kf, Nusselt number, Re=VD/ν, Reynolds number, and
Pr=ν/α, Prandtl number.
This conclusion is consistent with empirical observation, that is
Nu=f(Re, Pr). If we can determine the Reynolds and the Prandtl numbers,
we can find the Nusselt number, hence, the heat transfer coefficient, h.

Convection Correlations
ln(Nu)
ln(Re)
slope m
Fixed Pr
ln(Nu)
ln(Pr)
slope n
Fixed Re
D s
D s
Laminar, fully developed circular pipe flow:
Nu 4.36, when q " constant, (page 543, ch. 10-6, ITHT)
Nu 3.66, when T constant, (page 543, ch. 10-6, ITHT)
Note: t
f
hD
k
⇒
= = =
= =
mhe therma conductivity should be calculated at T .
Fully developed, turbulent pipe flow: Nu f(Re, Pr),
Nu can be related to Re & Pr experimentally, as shown.
⇒ =

Empirical Correlations
4/5
D
s m s m
D
Dittus-Boelter equation: Nu 0.023Re Pr , (eq 10-76, p 546, ITHT)
where n 0.4 for heating (T T ), n 0.3 for cooling (T T ).
The range of validity: 0.7 Pr 160, Re 10,000, / 10.
n
L D
=
= > = <
≤ ≤ ≥ ≥
Note: This equation can be used only for moderate temperature difference with all
the properties evaluated at Tm.
Other more accurate correlation equations can be found in other references.
Caution: The ranges of application for these correlations can be quite different.
D 1/ 2 2/3
D
For example, the Gnielinski correlation is the most accurate
among all these equations:
( / 8)(Re 1000) Pr
Nu (from other reference)
1 12.7( /8) (Pr 1)
It is valid for 0.5 Pr 2000 and 3000 Re 5 10
Df
f
−
=
+ −
< < < < × 6
m
.
All properties are calculated at T .

Example (cont.)
In our example, we need to first calculate the Reynolds number: water at 35°C,
Cp=4.18(kJ/kg.K), µ=7x10-4
(N.s/m2
), kf=0.626 (W/m.K), Pr=4.8.
4
D 1/ 2 2/ 3
4 4(0.15)
Re 5460
(0.05)(7 10 )
Re 4000, it is turbulent pipe flow.
Use the Gnielinski correlation, from the Moody chart, f 0.036, Pr 4.8
( / 8)(Re 1000) Pr (0.
Nu
1 12.7( /8) (Pr 1)
D
m DVD mA
D
f
f
ρ
µ µ π µ π −
= = = = =
×
>
= =
−
= =
+ −
&
&
1/ 2 2/ 3
2
036 / 8)(5460 1000)(4.8)
37.4
1 12.7(0.036 / 8) (4.8 1)
0.626
(37.4) 469( / . )
0.05
f
D
k
h Nu W m K
D
−
=
+ −
= = =

Energy Balance
Question (2): How can we determine the required pipe length?
Use energy balance concept: (energy storage) = (energy in) minus (energy out).
energy in = energy received during a steady state operation (assume no loss)
'( ) ( ),
( ) (0.15)(4180)(50 20)
94( )
' 200
P out in
P in out
q L mC T T
mC T T
L m
q
= −
− −
= = =
&
&
q’=q/L
Tin Tout

Temperature Distribution
s s
s
From local Newton's cooling law:
q hA(T ) ' ( )(T ( ) ( ))
' 200
( ) ( ) 20 0.319 22.7 0.319 ( )
(0.05)(469)
At the end of the pipe, T ( 94) 52.7( )
m m
s m
T q x h D x x T x
q
T x T x x x C
Dh
x C
π
π π
= − ⇒ ∆ = ∆ −
= + = + + = + °
= = °
Question (3): Can we determine the water temperature variation along the pipe?
Recognize the fact that the energy balance equation is valid for
any pipe length x:
'( ) ( ( ) )
' 200
( ) 20 20 0.319
(0.15)(4180)
It is a linear distribution along the pipe
P in
in
P
q x mC T x T
q
T x T x x x
mC
= −
= + = + = +
&
&
Question (4): How about the surface temperature distribution?

Temperature variation for constant heat flux
Note: These distributions are valid only in the fully developed region. In the
entrance region, the convection condition should be different. In general, the
entrance length x/D≈10 for a turbulent pipe flow and is usually negligible as
compared to the total pipe length.
T m x( )
T s x( )
x
0 20 40 60 80 100
20
30
40
50
60
Constant temperature
difference due to the
constant heat flux.

Internal Flow Convection
-constant surface temperature case
Another commonly encountered internal convection condition is when the
surface temperature of the pipe is a constant. The temperature distribution in
this case is drastically different from that of a constant heat flux case. Consider
the following pipe flow configuration:
Tm,i
Tm,o
Constant Ts
Tm Tm+dTm
qs=hA(Ts-Tm)
p
p
s
p s
Energy change mC [( ) ]
mC
Energy in hA(T )
Energy change energy in
mC hA(T )
m m m
m
m
m m
T dT T
dT
T
dT T
= + −
=
= −
=
= −
&
&
&
dx

Temperature distribution
p s
s m
m
mC hA(T ),
Note: q hA(T ) is valid locally only, since T is not a constant
, where A Pdx, and P is the perimeter of the pipe
(T )
Integrate from the inlet to a diatance x downstr
m m
m
m
s P
dT T
T
dT hA
T mC
= −
= −
= − =
−
&
&
,
,
( )
0 0
m
( )
m
0
eam:
(T )
ln(T ) | , where L is the total pipe length
and h is the averaged convection coefficient of the pipe between 0 & x.
1
, or
m
m i
m
m i
T x x x
m
T
s P P
T x
s T
P
x
dT hP P
dx hdx
T mC mC
Ph
T x
mC
h hdx hdx
x
= − = −
−
− = −
=
∫ ∫ ∫
∫
& &
&
0
x
hx=∫

Temperature distribution
,
( )
exp( ), for constant surface temperaturem s
m i s P
T x T Ph
x
T T mC
−
= −
− &
T x( )
x
Tm(x)
Constant surface
temperature
Ts
The difference between the averaged fluid temperature and the surface
temperature decreases exponentially further downstream along the pipe.

Log-Mean Temperature Difference
,
,
, , , ,
For the entire pipe:
( )
exp( ) or
ln( )
( ) (( ) ( ))
( )
ln( )
where is cal
ln( )
m o s o s
P
om i s i P
i
P m o m i P s m i s m o
o i
P i o s s lm
o
i
o i
lm
o
i
T T T hAh PL
mC
TT T T mC
T
q mC T T mC T T T T
T T
mC T T hA hA T
T
T
T T
T
T
T
− ∆
= = − = −
∆− ∆
∆
= − = − − −
∆ − ∆
= ∆ − ∆ = = ∆
∆
∆
∆ − ∆
∆ =
∆
∆
&
&
& &
&
led the log mean temperature difference.
This relation is valid for the entire pipe.

External Heat Transfer
Can we extend the previous analysis to include the situation that some external
heat transfer conditions are given, rather than that the surface temperature is
given. Example: Pipe flow buried underground with insulation. In that case,
the heat transfer is first from the fluid to the pipe wall through convection;
then followed by the conduction through the insulation layer; finally, heat is
transferred to the soil surface by conduction. See the following figure:
Diameter D, insulation thickness t
Soil (ks) temperature Ts
Soil resistance
Resistance of insulator
Convection resistance

Overall Heat Transfer Coefficient
From the previous example, the total thermal resistance can be written as
Rtotal=Rsoil+Rinsulator+Rconvection.
The heat transfer can be expressed as: q=∆Tlm/Rtot=UAs ∆Tlm
by defining the overall heat transfer coefficient UAs=1/Rtot. (Consider U as an
equivalent heat transfer coefficient taking into consideration of all heat transfer
modes between two constant temperature sources.)
We can replace the convection coefficient h by U in the temperature
distribution equation derived earlier:
,
,
1
exp( ) exp( )m o soil o s
m i soil i P P tot
T T T UA
T T T mC mC R
− ∆
= = − = −
− ∆ & &

Free Convection
A free convection flow field is a self-sustained flow driven by the
presence of a temperature gradient. (As opposed to a forced
convection flow where external means are used to provide the
flow.) As a result of the temperature difference, the density field is
not uniform also. Buoyancy will induce a flow current due to the
gravitational field and the variation in the density field. In general,
a free convection heat transfer is usually much smaller compared to
a forced convection heat transfer. It is therefore important only
when there is no external flow exists.
hot
cold
Τ↑ ⇒ ρ↓
T↓ ⇒ ρ↑
Flow is unstable and a circulatory
pattern will be induced.

Basic Definitions
Buoyancy effect:
Warm, ρ
Surrounding fluid, cold, ρ∞
Hot plate
Net force=(ρ∞- ρ)gV
The density difference is due to the temperature difference and it can be
characterized by ther volumetric thermal expansion coefficient, β:
1 1 1
( )P
T T T T
T
ρ ρρ ρ
β
ρ ρ ρ
ρ β
∞
∞
−∂ ∆
= − ≈ − = −
∂ − ∆
∆ ≈ ∆

Grashof Number and Rayleigh Number
Define Grashof number, Gr, as the ratio between the buoyancy force and the
viscous force: 33
2 2
( )Sg T T Lg TL
Gr
ββ
ν ν
∞−∆
= =
• Grashof number replaces the Reynolds number in the convection correlation
equation. In free convection, buoyancy driven flow sometimes dominates the
flow inertia, therefore, the Nusselt number is a function of the Grashof number
and the Prandtle number alone. Nu=f(Gr, Pr). Reynolds number will be
important if there is an external flow. (combined forced and free convection.
• In many instances, it is better to combine the Grashof number and the
Prandtle number to define a new parameter, the Rayleigh number, Ra=GrPr.
The most important use of the Rayleigh number is to characterize the laminar
to turbulence transition of a free convection boundary layer flow. For
example, when Ra>109
, the vertical free convection boundary layer flow over
a flat plate becomes turbulent.

Example
Determine the rate of heat loss from a heated pipe as a result of
natural (free) convection.
Ts=100°C
T∞=0°C D=0.1 m
Film temperature( Tf): averaged boundary layer temperature Tf=1/2(Ts+T ∞)=50 °C.
kf=0.03 W/m.K, Pr=0.7, ν=2×10-5
m2
/s, β=1/Tf=1/(273+50)=0.0031(1/K)
3 3
6
2 5 2
1/ 6
2
9/16 8/ 27
2
( ) (9.8)(0.0031)(100 0)(0.1)
Pr (0.7) 7.6 10 .
(2 10 )
0.387
{0.6 } 26.0 (equation 11.15 in Table 11.1)
[1 (0.559 / Pr) ]
0.03
(26) 7.8( / )
0.1
( ) (7.8)( )(
S
D
f
D
S
g T T L
Ra
Ra
Nu
k
h Nu W m K
D
q hA T T
β
ν
π
∞
−
∞
− −
= = = ×
×
= + =
+
= = =
= − = 0.1)(1)(100 0) 244.9( )
Can be significant if the pipe are long.
W− =

Forced convection

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