Convection Part1
External Flow
Introduction
Recall:
Convention is the heat transfer mode between a fluid
and a solid or a 2 fluids of different phases
In order to simplify the process we used Newton’s
correlation
q
 hT
A
Where h is the convective heat transfer coefficient also called the
film coefficient.
h is a function of:
Fluid flow
Fluid properties
Geometry of the solid
There are four means to evaluate the heat transfer coefficient
1)
2)
3)
4)
Dimensional analysis
Exact analysis of boundary layer
Approximate integral analysis of the boundary layer
Analogy between energy and momentum transfer
Significant Parameters:
Nusselt Number Nu
v
Ts T
y
x
Ts  T
The heat transfer rate between the surface and the fluid is
qy  hATs  T
At the surface itself


q y   kA  T  Ts 
 y
 y 0
Where k is the thermal conductivity of the fluid. Therefore:


 kA  T  Ts   hATs  T 
 y
 y 0


 y Ts  T 
h 
 y 0

Ts  T 
k


 y Ts  T 
hL 
 y 0
Nu 

Ts  T  L
k
Prandtl Number Pr
Momentum Diffusivity
Thermal Diffusivity



k

C p
The ratio of the momentum diffusivity over the thermal diffusivity is
a combination of fluid properties and is also thougth of as a property
(Named Prandtl Number Pr).
C p
Pr 
k
Dependent on fluid and temperature
Dimensional Analysis of Convective Heat Transfer
Forced Convection:
movement dictated by v
Variable
Symbol
Dimensions
Tube Diameter
D
L
Fluid density
ρ
M L-3
Fluid viscosity
μ
M L-1 t-1
Fluid heat capacity
Cp
Q M –1 T –1
Fluid thermal conductivity
k
Q t –1 L –1 T –1
Velocity
v
L t –1
Heat transfer coefficient
h
Q t –1 L –2 T –1
Using the Buckingham method we group the variables in
dimensionless number:
Dv
a b c d
 1  D k  v   Re 

 2  D k  v Cp  Pr 
e
f
g h
 3  D i k j  k v l h  Nu 
C p
k
hL
k
This dimensional analysis for a forced convection in a circular
conduit indicates the possibility of correlating the variables as
Nu  f1 Re, Pr
Similarly we could have developed the Stanton number instead of the
Nusselt
h
St 
vCp
Free Convection:
movement dictated by buoyancy
Given the coefficient of thermal expansion β:
  0 1  T 
Fbuoyant    0 g  g0T
Variable
Symbol
Dimensions
Significant length
D
L
Fluid density
ρ
M L-3
Fluid viscosity
μ
M L-1 t-1
Fluid heat capacity
Cp
Q M –1 T –1
Fluid thermal conductivity
k
Q t –1 L –1 T –1
Fluid Coef. Therm. Exp.
β
T –1
Gravitational acceleration
G
L t –2
Temperature difference
ΔT
T
Heat transfer coefficient
h
Q t –1 L –2 T –1
Using the Buckingham method we group the variables in
dimensionless number:
Cp
 1  La  b k c  d g eCp  Pr 
k
2  L  k  g  
f
g
h
i
j
L3 g 2
Define the Grashof number as
2
 3  Lk  l k m  n g o T  T
 2 3  Gr
 4  Lp  q k r  s g t h  Nu 
hL
k
This dimensional analysis for a forced convection in a circular
conduit indicates the possibility of correlating the variables as
Nu  f 2 Gr, Pr
Nu vs Re
100
Nu
75
Pr = 2
Pr = 1
Pr = 0.5
50
25
0
100
1000
Re
10000
Nu vs f ( Re,Pr)
Pr = 2
100
Pr = 1
Nu
75
Pr = 0.5
50
25
0
10
30
50
Re0.5 Pr 0.33
70
Selected Dimensionless Groups
Group
Symbol Definition
Grashof Number
Gr
Colburn Factor
jH
Nusselt Number
Nu
hL
k
Prandtl Number
Pr
C p
k
Reynolds
Re
Dv
Stanton Number
St
h
vCp
Modified Nusselt number
Peclet Number
Pe
RePr
Independent heat transfer parameter
L3 g 2 T
2
2
St Pr 3

Interpretation
Ratio buoyancy to viscous forces
Dimensionless heat transfer coefficient
Dimensionless surface temperature gradient
Ratio momentum to thermal diffusivity
Ratio inertia to viscous forces
Laminar Flow
Transition Region
Flat Plate in Parallel Flow
Turbulent Flow
Re x 
Re L 
δ(x)
x
L
Properties of fluid evaluated at the film temperature Tf
Tf 
T  Ts
2
xv

Lv

Forced Convection
Flat Plate in Parallel Flow
Laminar flow: Re<2 x 105
Prandtl number >0.6
The local Nusselt number is
The average Nusselt number
x
1
hx x
1
2
Nu 
 0.332 Re x Pr 3
k
Nu 
1
hL L
1
 0.664Re x2 Pr 3
k
All Prandtl number and Pe >100
1
1
The local Nusselt number is
2
0.3387Re x P r 3
Nu 
1
2
4
3
1  0.0468/ P r
The average Nusselt number
1
1
2
0.6774Re x P r 3
Nu 
1
2
4
3
1  0.0468/ P r




L
Forced Convection
Flat Plate in Parallel Flow
Transition flow: Rec=5 x 105
60>Prandtl number >0.6
3 x 106 >Re > 2 x 105
L
The average Nusselt number
Nu 


4
1
hL L
 0.037Re L5  871 Pr 3
k
Forced Convection
Flat Plate in Parallel Flow
Turbulent flow: Re>3x106
60>Prandtl number >0.6
107 >Re >3 x 106
L
The average Nusselt number
Nu  0.037Re L Pr
4
5
1
3
The local Nusselt number
Nu  0.0296Re x5 Pr 3
4
1
Cylinder in a Cross Flow
Laminar
Transition
Turbulent
v
D
Re D  2 105
Re D 
Dv

Separation
Properties of fluid
evaluated at the film
temperature Tf
v
T  Ts
Tf 
2
D
Re D  2 105
Separation
Forced Convection
Cylinder in a Cross Flow
Nu D C Re Pr
m
D
The average Nusselt number
3
ReD
C
m
0.4-4
0.989
0.330
4-40
0.911
0.385
40-4000
0.683
0.466
4000-40,000
0.193
0.618
40,000-400,000
0.027
0.805
1
If
1
ReDPr>0.2
Nu D  0.3
2
D
0.62Re P r


 0.4
1
Pr
2
1
3
1
3


4
  Re 
D
1

  282000
5
8



4
5
Forced Convection
Various Object in a Cross Flow
The average Nusselt number
Nu D C Re Pr
m
D
1
3
Geometry
ReD
C
m
Square
5x103-105
5x103-105
0.246
0.102
0.588
0.675
0.160
0.0385
0.153
0.638
0.782
0.638
0.228
0.731
Square
D
D
Hexagon
D
Hexagon
D
5x103-1.95x104
1.95x104 -105
5x103-105
D
4x103-1.5x104
Vertical Plate
Sphere in a Cross Flow
Re D 
Dv

All properties of fluid evaluated at
temperature T , except μs at Ts
Restrictions
0.71 < Pr < 380
3.5 < ReD < 7.6x104
 
0.4 

3
2
Nu D 2 0.4Re D 0.06Re D Pr  


 s 
1
2
1
4
Bank of Tubes in a Cross Flow
V
Fluid in cross flow
over tube bank
Aligned Bank of Tubes in a Cross Flow
SL
D
ST
A1
v, T
Properties of fluid evaluated at the film temperature Tf
Re D ,max 
Dvmax 

vmax
ST

v
ST  D
Staggered Bank of Tubes in a Cross Flow
SL
D
A1
ST
v, T
Re D ,max 
Dvmax 

Properties of fluid evaluated at the film temperature Tf
ST




2
S

D

S

D
v

v
If
D
T
max
ST  D
else
vmax 
ST
v
2S D  D 
Number of row (NL) greater or equal to 10
2000 < ReD,max < 40000
Pr > 0.7
Nu D 1.13C 1Re
m
D , max
Pr
1
3
C1 in table 7.5
If number of row is smaller than 10
Nu D
 N L 10 
 C2 Nu D
C2 in table 7.6
 N L 10 
Number of row (NL) greater or equal to 20
1000 < ReD,max < 2x106
500 > Pr > 0.7
m
0.36  P r 

Nu D C Re D ,max P r 
 P rs 
C in table 7.7
1
4
If number of row is smaller than 10
Nu D
 N L  20 
 C2 Nu D
 N L  20 
C2 in table 7.8
Tin  Tout
All properties of fluid evaluated at the average temperature
2
except Pr at T
s
s
In this case the temperature difference in the convective heat
transfer equation is defined as the log-mean temperature
difference ΔTlm
T  T   Ts  To 
Tlm  s i

Ts  Ti 
ln
Ts  To 
Where
Ti is the temperature of the fluid entering the bank
To is the temperature of the fluid leaving the bank
And the outlet temperature can be estimated using
Ts  To   exp  DNh 
 vN S Cp 
Ts  Ti 
T T


Where N is the total number of tube and NT the transverse number
of tube. Finally the heat transfer rate per unit length is
q'  N h DTlm 
Packed Bed
v
Re D 
Dv 

Properties of fluid evaluated at the the average temperature Tin  Tout
2
0.575
 jH  2.06Re D
ε is the porosity or void fraction of the bed (0.3 to 0.5)
Valid for
Pr  0.7
90  Re D  4000
gas flow
Ts  To   exp  Ap ,T h 
 vA Cp 
Ts  Ti 
c ,b


q  h Ap,T Tlm
Ap,T is the total area of the particles and
Ab,c is the bed cross sectional area