CHAPTER 7: EXTERNAL FLOW
Objectives:
1. To examine the expressions for convection heat transfer coefficient for different flow
geometries.
2. To develop the methodology of calculating the heat transfer coefficient.
3. To develop the calculation procedure for flow across a bank of tubes.
7.0 Introduction
 An external flow is a flow in which boundary layers develop freely without constraints
imposed by adjacent surfaces.
 Examples of external flow are flow over a flat plate and flow over curved surfaces such as a
sphere, cylinder, airfoil, or turbine blade.
 The problem of convection is simply that of finding the heat transfer coefficient. There are
two approaches that could be used: theoretical and empirical (i.e. experimental).
7.1 The Empirical Method
 The empirical method involves performing heat transfer measurements under controlled
laboratory conditions.
 The measurements taken usually include the heat transfer rate on the surface and the fluid
velocity and temperature distributions in the velocity and thermal boundary layers,
respectively.
 The data obtained is usually correlated in terms of appropriate dimensionless parameters.
7.2 The Flat Plate in Parallel Flow
 The flow over a flat plate (parallel to the free stream) is shown below.

Laminar boundary layer development begins at the leading edge (x = 0).

Transition to turbulence occurs at the location x = xc which corresponds to the critical
Reynolds number Rex,c.
7.2.1 Laminar Flow over an Isothermal Plate
 The correlations for local Nusselt number and friction coefficient for laminar flow over an
isothermal flat plate is obtained analytically. The results are
Nu x
h x
 x
k
12
0.332 Re x Pr 1 3

 s, x
C f ,x 
 U 2 2
Pr  0.6
1 2
 0.664 Re x
(1)
(2)
 The average heat transfer coefficient for laminar flow is
hx

Nu x

1 x
hx
x 0

h x
 x
k

0.664
k 12 13
Re x Pr
x
12
0.664 Re x Pr 1 3

(3)
Pr  0.6
(4)
 From equations 1 and 4, we see that for laminar flow, h x  2 hx . The same relation holds for
friction coefficient.
7.2.2 Turbulent Flow over an Isothermal Plate
 The correlation for local Nusselt number and friction coefficient are obtained from
experiment. The result for Nusselt number is as follows:
Nu x

45
0.0296 Re x Pr 1 3
0.6  Pr  60
(5)
7.2.3 Mixed Boundary Layer Conditions
 When the plate contains both laminar and turbulent boundary layers, the average heat transfer
coefficient is obtained as follows:
hL 
1  xc

hlam dx 
L  0

L

 xc hturb dx 
where hlam and hturb are obtained from equations 1 and 5, respectively, i.e.
(6)
hL 
12 x
45 L
k 
dx  1 3
u 
u 
c dx
0.332   
 0.0296   
Pr
L
0 x1 2
xc x1 5 
  
  




(7)
Note: A completely turbulent boundary layer (i.e. without a preceding laminar boundary layer)
may be realized by tripping the boundary layer at the leading edge using a fine wire or some
other turbulence promoter.
7.2.4 Unheated Starting Length
 For a situation involving the existence of an unheated starting length (Ts = T) upstream of a
heated section (Ts ≠ T), the development of the boundary layers is as shown below.

There is no heat transfer for 0  x  .

The Nusselt number for this situation is calculated according to equations 7.42 and 7.43
of the textbook.
7.2.5 Flat Plates with Constant Heat Flux Conditions
 For uniform surface heat flux, the Nusselt number correlations are presented in equations 7.45
and 7.46 of the textbook.
 If the surface involves an unheated starting length, the correction of equations 7.42 and 7.43
of the textbook is applied.
7.3 Methodology for a Convection Calculation
 The methodology for calculating the heat transfer coefficient is itemized on page 415 of the
textbook. These rules apply to any flow situation.
 Note that to compute hx, all fluid properties are evaluated at the film temperature, defined as
T  T
Tf  s
2
(8)
7.4 The Cylinder in Cross Flow
 Another common external flow involves fluid motion normal to the axis of a circular cylinder.
 The processes leading to boundary layer separation from the surface and formation of wake in
the downstream have been discussed in the previous course (ME 311).
 The average Nusselt number for flow over a cylinder may be obtained from the following
correlation:
Nu D

hD
k

13
C Re m
D Pr
Pr  0.7
(9)
where C and m are listed in Tables 7.2 and 7.3 of the textbook for circular and noncircular
cylinders, respectively.
Note: All fluid properties are evaluated at the film temperature (defined in equation 8).
7.5 The Sphere
 The flow over a sphere is much like flow over a circular cylinder.
 The Nusselt number for flow over a sphere may be computed from equation 7.56 of the
textbook.
7.6 Flow across Banks of Tubes
 Flow across tube banks is extremely important in heat exchanger applications.
 The geometric arrangement is shown in figure 7.10 of the textbook. One fluid moves over the
tubes. Another fluid at a different temperature passes through the tubes and is treated as an
internal flow (chapter 8).
 The arrangement of tube rows in a bank are either aligned (in-line) or staggered in the
direction of fluid velocity, as shown below.
 The geometric dimensions of a tube bank are defined as follows:
Tube arrangement in a bank. (a) Aligned. (b) Staggered
ST :
transverse pitch
SL :
longitudinal pitch
SD :
diagonal pitch, (staggered arrangement)
NT :
number of tubes in the transverse direction
NL :
number of tubes in the longitudinal direction
N
total number of tubes, N = NTNL
:
 The maximum fluid velocity occurs at the minimum flow area (A1 of aligned arrangement; A1
or A2 for staggered arrangement).
 By mass conservation, the maximum fluid velocity is


Aligned
Staggered
ST
V
ST  D
:
Vmax 
:
ST

 2 S  D  V ,

D
Vmax  
S
 T V,
 ST  D
(10)
if 2 S D  D  ST  D 
(11)
otherwise
 The Reynolds number is based on the maximum velocity, i.e.
Re D, max 
 Vmax D

(12)
 The average Nusselt number for the entire bundle may be obtained from the following
correlation:
Nu D
(for N L 10 ;

13
1.13 C1 Re m
D, max Pr
2000 ReD, max  40,000;
Pr  0.7 )
(13)
where the constants C1 and m are given in Table 7.5 of the textbook.
Note: If NL < 10, a correction factor is applied according to equation 7.61 of the textbook.
 For the purpose of heat transfer and pressure drop calculations, a tube bank is modeled as a
number of uniform and continuous flow channels with uniform surface temperature Ts.

If the fluid enters the tube bank at temperature Ti, the exit temperature To is obtained from
To
Ti


Ts  To
Ts  Ti

 DNh
 exp  
  V N T ST c p





(14)
The heat transfer rate per unit length of the tubes is

q  N h  D Tlm

where ΔTlm is the log mean temperature difference (see the chapter 8 note).

The pressure drop is
 V 2
p  N L X  max

2


f


where the friction factor f and the correction factor X are obtained from figures 7.13 and
7.14 of the textbook.