CHAPTER 10
CORRELATION EQUATIONS:
FORCED AND FREE CONVECTION
10.1 Introduction
• Correlation equations: Based on experimental data
• Chapter outline: Correlation equations for:
(1) External forced convection over:
Plates
Cylinders
Spheres
(2) Internal forced convection through channels
(3) External free convection over:
Plates
Cylinders
Spheres
1
10.2 Experimental Determination of Heat Transfer Coefficient h
Newton's law of cooling defines h:
q′s′
h=
Ts − T∞
∆V
(10.1)
q ′s′ = surface flux
Ts = surface temperature
T∞ = ambient temperature
Example: Electric heating
Measure: Electric power, Ts , T∞
Use (10.1) to calculate h
q′s′
−•
•
Ts
T∞
•
+
V∞
Fig. 10.1
• Form of correlation equations:
• Dimensionless: Nusselt number Is a dimensionless heat transfer
coefficient.
2
1.Example: Forced convection with no dissipation
hx
Nu x =
= f ( x * ; Re, Pr )
k
(2.52)
Use (2.52) to plan experiments and correlate data
10.3 Limitations and Accuracy of Correlation Equations
All correlation equations have limitations !
• Limitations on:
(1) Geometry
(2) Range of parameters: Reynolds, Prandtl, Grashof, etc.
(3) Surface condition: Uniform flux, uniform temperature, etc.
• Accuracy: Errors as high as 25% are not uncommon!
10.4 Procedure for Selecting and Applying Correlation Equations
(1) Identify the geometry
3
(2) Identify problem classification:
Forced convection
Free convection
External flow
Internal flow
Entrance region
Fully developed region
Boiling
Condensation
Etc.
(3) Define objective: Finding local or average heat transfer coefficient
(4) Check the Reynolds number:
(a) Laminar
(b) Turbulent
(c) Mixed
(5) Identify surface boundary condition:
(a) Uniform temperature
4
(b) Uniform flux
(6) Note limitations on correlation equation
(7) Determine properties at the specified temperature:
(a) External flow: at the film temperature T f
T f = (Ts + T∞ ) / 2
(10.2)
(b) Internal flow: at the mean temperatureTm
(c) However, there are exceptions
(8) Use a consistent set of units
(9) Compare calculated values of h with Table 1.1
10.5 External Forced Convection Correlations
10.5.1 Uniform Flow over a Flat Plate:
Transition to Turbulent Flow
• Boundary layer flow over a semi-infinite flat plate
5
Three regions:
V∞
(1) Laminar
(2) Transition
(3) Turbulent
•
T∞
x
• t
laminar
Re x=t Transition or
x
turbulent
transition
Fig. 10.2
critical Reynolds
number:
Re x t depends on: Geometry, surface finish, pressure gradient, etc.
For flow over a flat plate:
V x
Re xt = ∞ t ≈ 5 × 105
ν
• Examples of correlation equations for plates:
Laminar region, x < xt :
6
Use (4.72a) or (4.72b) for local Nusselt number to obtain local h
Turbulent region, x > xt :
Local h:
hx
Nu x =
= 0.0296( Re x )4 / 5 ( Pr )1 / 3
k
Limitations:
flat plate, constant Ts
5 × 105 < Re x < 107
0.6 < Pr < 60
properties at T f
(10.4a)
(10.4b)
Average h
x
L

1 L
1 t
h=
h( x )dx = 
hL ( x )dx +
ht ( x )dx 
0
0
L
L
xt


∫
∫
∫
(10.5)
7
hL = local laminar heat transfer coefficient
ht = local turbulent heat transfer coefficient
(4.72b) and (10.4a) into (10.5):
1/ 2
k
 V∞ 
h = 0.332 

L 
ν 
0
∫
xt
 V∞ 
+
0
.
0296
ν 
1/ 2


x
dx
dx 
1/ 3
(
)
Pr

x t x1 / 5 

4/5 L
∫
(10.6)
Integrate
h=
{
[
) ] } ( Pr )1 / 3
k
0.664 Re xt 1 / 2 + 0.037 ( Re L )4 / 5 − Re x t 4 / 5
L
(
)
(
Dimensionless form:
NuL =
{
[
) ]}( Pr )1 / 3
1/ 2
4/5
4/5
hL
= 0.664 Re x t
+ 0.037 ( Re L )
− Re x t
k
(
)
(2) Plate at uniform surface temperature
with an insulated leading section
x0=Length of insulated section
(
V∞
T∞
xt
0
•
insulation
(10.7b)
δt
•
xo
x
•
Ts
Fig. 10.3
8
Two cases:
• Laminar flow, x t > x o : Use (5.21) for the local Nusselt number to obtain
local h
•Turbulent flow, x t < x o : The local Nusselt number is
hx 0.0296Re 4x / 5 Pr 1 / 3
Nu x =
=
k
9 / 10 1 / 9
1 − ( xo / x )
[
(10.8)
]
(3) Plate with uniform surface flux
Two regions:
• Laminar flow, 0 < x < xt
Use (5.36) or (5.37) for the local
Nusselt number to obtain local h
V∞
T∞
0
•Turbulent flow, x > xt :
hx
Nu x =
= 0.030Re 4x / 5 Pr 1 / 3
k
xt
x
•
q′s′
Fig. 10.4
(10.9)
9
Properties at T f = (Ts + T∞ ) / 2 and Ts is the average surface temperature
10.5 External Flow Normal to a Cylinder
• For uniform surface temperature or uniform
surface flux
V∞
T∞
θ
Fig. 10.5
Nu L =
5/8
0.62 Re1D/ 2 Pr 1 / 3   Re D  
1 + 
 
1
/
4
  282,000  
Pr 2 / 3
hD
= 0.3 +
k
1 + (4 /
Limitations:
[
)
]

Flow norm al to cylinder
Pe = Re D Pr > 0 . 2
properties at T f
4/5
(10.10a)

(10.10b)
Pe = Peclet number = ReD Pr
10
For Pe < 0.2, use:
hD
1
NuD =
=
k
0.8237 − 0.5 ln Pe
Limitations
(10.11a)
flow normal to cylinder
Pe = Re D Pr < 0.2
properties at T f
10.5.3 External Flow over a Sphere
( )
1/4
hD
0 .4 µ
1/ 2
2/3
Nu D =
= 2 + 0.4 Re D + 0.06 Re D Pr
µs
k
[
Limitations:
]
flow over sphere
3.5 < ReD < 7.6 × 104
0.71 < Pr < 380
1< µ
(10.12a)
(10.12b)
< 3.2
µs
properties at T∞ , µ s at Ts
11
10.6 Internal Forced Convection Correlations
Chapter 7:
Analytic solutions to h for
fully developed laminar flow
Correlation equations for h in the
entrance and fully developed regions
for laminar and turbulent flows
• Transition or critical Reynolds number for smooth tubes:
Re Dt =
uD
ν
≈ 2300
(10.13)
12
10.6.1 Entrance Region: Laminar Flow Through
Tubes at Uniform Surface Temperature
• Two cases:
(1) Fully Developed Velocity, Developing Temperature: Laminar Flow
• Solution: Analytic
• Correlation of analytic
results:
Ts
T
u
FDV
•
developing
0
x
δt
u
temperature
insulation
Fig. 10.6
hD
NuD =
= 3.66 +
k
0.0668 ( D/L ) Re D Pr
2/3
{1 + 0.04 [( D/L) ReD Pr ] }
(10.14a)
13
Limitations:
entrance region of tubes
uniform surface temperature Ts
laminar flow (ReD < 2300)
fully developed velocity
developing temperature
properties at Tm = (Tmi + Tmo ) / 2
(10.14b)
(2) Developing Velocity and Temperature: Laminar flow
hD
1 / 3 µ 
[
]
Nu D =
= 1.86 ( D/L) Re D Pr
µ 
k
 s
0.14
(10.15a)
14
Limitations:
entrance region of tube
uniform surface temperature Ts
laminar flow (ReD < 2300)
developing velocity and temperature
0.48 < Pr < 16700
0.0044 < µ µ < 9.75
s
properties at Tm , µ s at Ts
10.6.2 Fully Developed Velocity and Temperature in Tubes: Turbulent Flow
• Entrance region is short: 10-20 diameters
• Surface B.C. have minor effect on h for Pr > 1
• Several correlation equations for h:
(1) The Colburn Equation: Simple but not very accurate
Nu D =
Limitations:
4/5 1/3
hD
= 0.023Re D Pr
k
(10.16a)
15
fully developed turbulent flow
smooth tubes
ReD > 104
0.7 < Pr < 160
L /D > 60
properties at Tm
(10.16b)
• Accuracy: Errors can be as high as 25%
(2) The Gnielinski Equation: Provides best correlation of experimental
data
Nu D =
2/3
[
]
1
+
(
D
/
L
)
1/2
2/3
8 ) ( Pr − 1) ]
( f 8 )( Re D − 1000) Pr
[1 + 12.7( f
(10.17a)
• Valid for: developing or fully developed turbulent flow
16
Limitations:
2300 < ReD < 5 × 106
0.5 < Pr < 2000
0 < D/L <1
properties at Tm
(10.17b)
• The D/L factor in equation accounts for entrance effects
• For fully developed flow set D/L = 0
The Darcy friction factor f is defined as
∆p D
f =
ρ u2
2 L
(10.18)
For smooth tubes f is approximated by
f = (0.79ln Re D − 1.64) − 2
(10.19)
17
10.6.3 Non-circular Channels: Turbulent Flow
Use equations for tubes. Set D = De (equivalent diameter)
4Af
De =
P
A f = flow area
P = wet perimeter
10.7 Free Convection Correlations
x
10.7.1 External Free Convection Correlations
(1) Vertical plate: Laminar Flow, Uniform Surface
Temperature
u
Ts •
T∞
• Local Nusselt number:
g
y
Fig. 10.7
18
hx 3 
Pr

Nu x =
= 
k 4  2.435 + 4.884 Pr 1 / 2 + 4.953 Pr 
1/ 4
( Ra x )1 / 4
(10.21a)
• Average Nusselt number:

hL 
Pr
Nu L =
= 

1/2
k
 2.435 + 4.884Pr + 4.953Pr 
1/4
( Ra L )1/4
(10.21b)
(10.21a) and (10.21b) are valid for:
Limitations:
vertical plate
uniform surface temperature Ts
laminar, 10 4 < Ra L < 10 9
0 < Pr < ∞
properties at T f
(10.21c)
19
(2) Vertical plates: Laminar and Turbulent, Uniform Surface
Temperature
1/6

h L 
0.387 Ra L

Nu L =
= 0.825 +

8/27
k 
1 + (0.492 /Pr ) 9/16


[
]
2
(10.22a)
Limitations:
vertical plate
uniform surface temperature Ts
laminar, transition, and turbulent
10 −1 < Ra L < 1012
0 < Pr < ∞
properties at T f
(10.22b)
(3) Vertical Plates: Laminar Flow, Uniform Heat Flux
• Local Nusselt number:
20

hx 
Pr 2
*
Nux =
=
Grx 
1/2
k  4 + 9Pr + 10Pr

1/ 5
(10.23)
Determine surface temperature: Apply Newton’s law:
where Grx* is defined as
q′s′
h( x ) =
Ts ( x ) − T∞
Grx*
=
(10.24)
β gq ′s′
4
x
kν 2
(10.25)
(10.24) and (10.25) into (10.23) and solve for Ts ( x ) − T∞
 4 + 9 Pr 1 / 2 + 10 Pr α ν q′s′ 4
Ts ( x ) − T∞ = 
( β g )( k )
Pr


x

1/ 5
(10.26a)
(10.23) and (10.26a) are valid for:
21
vertical plate
laminar, 104 < Grx* Pr < 109
uniform surface flux, q′s′
0 < Pr < ∞
• Properties in (10.26a) depend on surface temperatureTs (x) which is not
known. Solution is by iteration
(4) Inclined plates: Constant surface temperature
• Use equations for vertical plates
θ
• Modify Rayleigh number as:
β gcosθ (Ts − T∞ )
Ra x =
αv
Ts > T∞
Ts < T∞
(10.27)
(a)
g T∞
θ
(b)
Fig. 10.9
22
Limitations:
inclined plate
uniform surface temperatur e Ts
Laminar, Ra L < 109
(10.28)
0 ≤ θ ≤ 60o
(5) Horizontal plates: Uniform surface temperature:
(i) Heated upper surface or cooled lower surface
Nu L = 0.54( Ra L )1 / 4 , 2 × 104 < Ra L < 8 × 106
Nu L = 0.15( Ra L )1 / 3 , 8 × 106 < Ra L < 1.6 × 109
Limitations:
horizontal plate
hot surface up or cold surface down
all properties , except , β , at T f
β at T f for liquids , Ts for gases
(10.29b)
(10.29c)
23
(ii) Heated lower surface or cooled upper surface
Nu L = 0.27( Ra L )1 / 4 , 105 < Ra L < 1010
Limitations:
horizontal plate
hot surface down or cold surface up
all properties, except, β, at Tf
β at Tf for liquids, Ts for gases
Characteristic length L:
L=
(10.30b)
surface area
perimeter
(6) Vertical Cylinders. Use vertical plate correlations for:
D
35
>
for Pr ≥ 1
1
/
4
L (GrL )
(10.32)
(7) Horizontal Cylinders:
24
1/ 6

h D 
0.387( Ra D )
= 0.60 +
Nu D =
8 / 27 
k
9
/
16

1 + (0.559/Pr )

[
Limitations:
(8) Spheres
Limitations:
]
2
(10.33a)
horizontal cylinder
uniform surface temperature or flux
10 − 5 < Ra D < 1012
properties at T f
hD
Nu L =
= 2+
k
0.589( Ra D )1 / 4
[1 + (0.Pr469 ) ]
9 / 16 4 / 9
(10.34a)
sphere
uniform surface temperature or flux
Ra D < 1011
Pr > 0.7
properties at T f
25
10.7.2 Free Convection in Enclosures
Examples:
• Double-glazed windows
• Solar collectors
• Building walls
• Concentric cryogenic tubes
• Electronic packages
Fluid Circulation:
• Driving force: Gravity and unequal surface temperatures
Heat flux:
Newton’s law:
q′′ = h(Th − Tc )
(10.35)
Heat transfer coefficient h:
Nusselt number correlations depend on:
26
• Configuration
• Orientation
• Aspect ratio
• Prandtl number Pr
• Rayleigh numberRa δ
(1) Vertical Rectangular Enclosures
δ
Rayleigh number
β g ( Th − Tc )δ 3
Raδ =
Pr
2
ν
Tc
Tc
(10.36)
L
g
Several equations:
Fig. 10.10
27
hδ
 Pr

Nuδ =
Raδ 
= 0.18 
k
 0.2 + Pr

Valid for
0.29
(10.37a)
vertical rectagular enclosure
L
1<
<2
δ
10 − 3 < Pr < 10 5
Pr
Ra δ > 10 3
0 . 2 + Pr
properties at T = ( T c + T h ) / 2
hδ
 Pr

Nuδ =
Raδ 
= 0.22 
k
 0.2 + Pr

Valid for
0.28
(10.37b)
 L
 δ 
− 0.25
(10.38a)
vertical rectagular enclosure
L
2 < < 10
δ
Pr < 10 5
(10.38b)
10 3 < Raδ < 1010
properties at T = (Tc + Th ) / 2
28
hδ
Nuδ =
= 0.046 [Raδ ]1 / 3
k
Valid for
(10.39a)
vertical rectagular enclosure
L
1 < < 40
δ
(10.39b)
1 < Pr < 20
10 6 < Raδ < 10 9
properties at T = (Tc + Th ) / 2
hδ
0.012
Nuδ =
[Raδ ]
= 0.42 [Pr ]
k
Valid for
0.25
 L
 δ 
−0
0..3
(10.40a)
vertical rectagular enclosure
L
10 < < 40
δ
1 < Pr < 2 × 10
4
(10.40b)
104 < Reδ < 107
properties at T = (Tc + Th ) / 2
29
(2) Horizontal Rectangular Enclosures
• Enclosure heated from below
• Cellular flow pattern develops at critical Rayleigh number Ra δ c = 1708
• Nusselt number:
L
hδ
δ
Nuδ =
= 0.069[Raδ ]1 / 3 [Pr ]0.074
k
Tc
g
Th
(10.41a)
Fig. 10.11
Valid for
horizontal rectangular enclosure
heated from below
3 × 105 < Raδ < 7 × 10 7
properties at T = (Tc + Th ) / 2
(10.41b)
30
δ
(3) Inclined Rectangular Enclosures
• Applications: Solar collectors
• Nusselt number:correlations depend on:
• Inclination angle
• Aspect ratio
•Prandtl number Pr
• Rayleigh numberRa δ
Tc
g
Th
L
θ
For:
Fig. 10.12
0 o < θ < 90 o: heated lower surface, cooled upper surface
90
o
< θ < 180
o
Table 10.1
critical tilt angle
: cooled lower surface,
heated upper surface
• Nusselt number is minimum at
L/δ
θc
1
3
6
12 > 12
25o 53o 60o 67 o 70o
31
a critical angle θ c : Table 10.1

1708 
hδ
Nuδ =
= 1 + 1.441 −

k
 Raδ cosθ 
 ( Raδ cosθ )1 / 3

− 1

18


*
 1708(1.8 sinθ )1.6 
1 −
+
Raδ cosθ


*
(10.42a)
Valid for
inclined rectangular enclosure
L / δ ≤ 12
0 < θ ≤ θc
(10.42b)
∗
set [ ] = 0 when negative
properties at T = (Tc + Th ) / 2
32
hδ
o  Nuδ ( 90 )
0.25 
Nuδ =
= Nuδ (0 )
(sinθ c ) 
o
k
 Nuδ (0 )

o
θ /θ c
(10.43a)
Valid for
inclined rectangular enclosure
L / δ ≤ 12
0 < θ ≤ θc
properties at T = (Tc + Th ) / 2
Nuδ =
Valid for
hδ
= Nuδ (90o ) [sinθ ] 0.25
k
(10.43b)
(10.44a)
inclined rectangular enclosure
all L / δ
o
θ c < θ < 90
properties at T = (Tc + Th ) / 2
(10.44b)
33
[
]
hδ
Nuδ =
= 1 + Nuδ (90o ) − 1 sinθ
k
(10.45a)
Valid for
inclined rectangular enclosure
all L / δ
(10.45b)
90o < θ < 180 o
properties at T = (Tc + Th ) / 2
Do
(4) Horizontal Concentric Cylinders
5
• Flow circulation forT i > T o
Ti
• Flow direction is reversed for T i < T o .
• Circulation enhances thermal conductivity
q′ =
2π keff
ln( Do / Di )
(Ti − To )
To
Di
(10.46)
Fig. 10..13
34
Correlation equation for the effective conductivity keff :
Pr

*
= 0.386 
Ra 
k
 0.861 + Pr

keff
Ra * =
[ln( Do / Di )] 4
[
δ 3 ( Di )− 3 / 5 + ( Do )− 3 / 5
δ=
Valid for
]
5
1/ 4
(10.47a)
Raδ
Do − Di
2
(10.47b)
(10.47c)
concentric cylinders
10 2 < Ra* < 107
(10.47d)
properties at T = (Tc + Th ) / 2
35
10.8 Other Correlations
The above presentation is highly abridged.
There are many other correlation equations for:
• Boiling
• Condensation
• Jet impingement
• High speed flow
• Dissipation
• Liquid metals
• Enhancements
• Finned geometries
• Irregular geometries
• Non-Newtonian fluids
• Etc.
Consult textbooks, handbooks, reports and journals
36