University of Hail
Faculty of Engineering
DEPARTMENT OF MECHANICAL ENGINEERING
ME 315 – Heat Transfer
Lecture notes
Chapter 7
External flow: Flow over a flat plate, cylinders,
spheres. Tube banks
Prepared by : Dr. N. Ait Messaoudene
Based on:
“Introduction to Heat Transfer”
Incropera, DeWitt, Bergman, and Lavine,
5th Edition, John Willey and Sons, 2007.
2nd semester 2011-2012
Objective:
Our primary objective is to determine convection coefficients for different flow
geometries.
In particular, we wish to obtain specific forms of the functions that represent these
coefficients.
By nondimensionalizing the boundary layer equations in Chapter 6, we found that the
local and average convection coefficients may be correlated (via the Nusselt number)
by equations of the form:
where
and
with
The Empirical Method
Experiment for measuring the
average convection heat
transfer coefficient
electrical power (E.I) = total heat transfer rate (q)
Procedure: Measure , Ts and T∞ for different conditions (I, As=W.L, u ∞, fluid)
the electrical
many different
values of the Nusselt number corresponding to a wide range of
power,
E I, which
is equal
Reynolds and Prandtl
numbers
. to the total heat
transfer rate q
the data may then be represented by an empirical correlation of the form
plot the results on a log–log scale
C, m, and n are often independent of the fluid
The specific values of
the coefficient C and
the exponents m and
n vary with the nature
of the surface
geometry and the
type of flow.
The Flat Plate in Parallel Flow
As with all external flows, the boundary layers develop freely without constraint.
Boundary layer conditions may be entirely laminar, laminar and turbulent, or entirely turbulent.
To determine the conditions, compute Re   u L  u L
L


and compare with the critical Reynolds number for transition to turbulence, Re x , c .
Re L  Re x,c  laminar flow throughout
Re L  Re x,c  transition to turbulent flow at xc / L  Re x,c / Re L
Value of Rex,c depends on free stream turbulence and surface roughness.
Nominally, Re x, c  5  105.
Laminar Flow over an Isothermal Plate: A Similarity Solution
• Based on premise that the dimensionless x-velocity component,
,
and temperature,
, can be represented exclusively in
terms of a dimensionless similarity parameter
• Similarity permits transformation of the partial differential equations associated
with the transfer of x-momentum and thermal energy to ordinary differential
equations of the form
where
B.C’s :
and
The solution to this equation is
termed the Blasius solution
f given by the
Blasius solution
B.C’s :
• Numerical solutions to the momentum and energy equations yield the following results
for important local boundary layer parameters:
And comparing the energy
and momentum equations:
and d 2 f / d 2
 0
and dT * / d
 0.332,
 0
 0.332 Pr1/ 3 for Pr  0.6,
• Average Boundary Layer Parameters:
Average friction factor
where
Blasius
solution:
Average Nusselt number
• The effect of variable properties may be considered
by evaluating all properties at the film temperature.
Tf 
Ts  T
2
Liquid Metals : fluids of small Prandtl number.
A single correlating equation, which applies for all Prandtl numbers, has been recommended
by Churchill and Ozoe
Turbulent Flow over an Isothermal Plate
Local Parameters:
Empirical
Correlations
Average Parameters:
For laminar flow over the entire (or 95%) of the plate, laminar expressions may be used to
compute the average coefficients.
When transition occurs sufficiently upstream of the trailing edge, (xc /L) < 0.95, the surface
average coefficients will be influenced by conditions in both the laminar and turbulent BL.
5
Substituting expressions for the local coefficients and assuming Re x,c  5  10 ,
C f ,L 
0.074 1742

1/ 5
Re L
Re L


Nu L  0.037 Re4L / 5  871 Pr1/ 3
Flat Plates with Constant Heat Flux Conditions
It is also possible to have a uniform surface heat flux, rather than a uniform temperature,
imposed at the plate.
For laminar flow, it may be shown that:
For turbulent flow
If the heat flux is known, the convection coefficient may
be used to determine the local surface temperature
It is not necessary to introduce an average convection coefficient for the purpose of determining
the total heat rate q since
However, an average surface temperature can be determined from:
For laminar flow
In fact, this expression can be used with
obtained for uniform a temperature condition
with only a small error
where
7.3 Methodology for a Convection Calculation
Although we have only discussed correlations for parallel flow over a flat plate, selection
and application of a convection correlation for any flow situation are facilitated by following
a few simple rules.
1. Understand the flow geometry and approximate it properly if necessary.
2. Specify the appropriate reference temperature and evaluate the pertinent fluid properties
at that temperature (some cases might require to include a property ratio to account for the
nonconstant property effect).
3. Calculate the Reynolds number. Determine whether the flow is laminar or turbulent.
4. Decide whether a local or surface average coefficient is required. Recall that for constant
surface temperature, the local coefficient is used to determine the flux at a particular point on
the surface, whereas the average coefficient determines the transfer rate for the entire surface.
5. Select the appropriate correlation.
Problem 7.21: Preferred orientation (corresponding to lower heat loss) and the
corresponding heat rate for a surface with adjoining smooth and
roughened sections.
SCHEMATIC:
Problem 7.24: Convection cooling of steel plates on a conveyor by
air in parallel flow.
The Cylinder in Cross Flow
Flow Considerations
• Conditions depend on special features of boundary layer development, including
onset at a stagnation point and separation, as well as transition to turbulence.
V depends on the
distance x from the
stagnation point.
– Stagnation point: Location of zero velocity  u  0  and maximum pressure.
– Followed by boundary layer development under a favorable pressure gradient
 dp / dx  0 and hence acceleration of the free stream flow  du / dx  0 .
– As the rear of the cylinder is approached, the pressure must begin to increase.
Hence, there is a minimum in the pressure distribution, p(x), after which boundary
layer development occurs under the influence of an adverse pressure gradient
 dp / dx  0, du / dx  0.
– Separation occurs when the velocity gradient
accompanied by flow reversal and a downstream wake.
reduces to zero and is
Location of separation depends on boundary layer transition. Since the momentum of fluid in
a turbulent boundary layer is larger than in the laminar boundary layer, it is reasonable to
expect transition to delay the occurrence of separation.
wake
Define
• Force imposed by the flow is due to the combination of friction and form drag.
A dimensionless drag coefficient is defined :
where Af is the cylinder frontal area of the cylinder (=D.L, L length of the cylinder)
•Convection Heat Transfer
– The Local Nusselt Number:
– The Average Nusselt Number
– Churchill and Bernstein Correlation:
where all properties are evaluated at the film temperature.
– Cylinders of Noncircular Cross Section:
C and m are listed in table 7.3
Each correlation is reasonable over a certain range of conditions, but for most
engineering calculations one should not expect accuracy to much better than 20%.
1. Determine the convection heat transfer coefficient from the experimental observations.
2. Compare the experimental result with the convection coefficient computed from an
appropriate correlation.
2.
Flow Across Tube Banks
• A common geometry for
two-fluid heat exchangers.
• Aligned and Staggered Arrays:
Longitudinal pitch
transverse pitch
Flow around the tubes in the first row of a tube bank is similar to that for a single (isolated)
cylinder in cross flow.
Correspondingly, the heat transfer coefficient for a tube in the first row is approximately equal
to that for a single tube in cross flow.
For downstream rows, flow conditions depend strongly on the tube bank arrangement
• Flow Conditions:
Typically, we wish to know the average heat transfer coefficient for the entire tube
bank.
Nu D  C2 C RemD,max Pr 0.36  Pr/ Prs 

1/ 4


C , m  Table 7.7
aligned
C2  Table 7.8
staggered
All properties are evaluated at Ti  To  / 2 except for Prs.
• Fluid Outlet Temperature (To) :

Ts  To
DNh 
 exp  
 VNT ST c p 
Ts  Ti


N  NT x N L
• Total Heat Rate:
q  hAs T m
As  N  DL 
T  T   Ts  To 
T m  s i
 T T 
n s i 
 Ts  To 
• Pressure Drop:
2
 Vmax

p  N L  
f
 2 
 , f  Figures 7.13 and 7.14
Per unit length
TABLE 7.9 Summary of convection heat transfer correlations for external flow
7.56
7.57
7.11
7.7
7.8
7.54
7.54
7.59
7.58 and 7.60
Problem 7.63: Cooling of extruded copper wire by convection and radiation.
SCHEMATIC:
dqconv
D = 5 mm
dqrad
Tsur = 25oC
Ve = 0.2 m/s
dx
Air
Too = 25oC
V = 5 m/s
ANALYSIS: (a) Applying conservation of energy to a stationary control surface, through which the
wire moves, steady-state conditions exist and Ein  Eout  0.
Hence, with inflow due to advection and outflow due to advection, convection and radiation,
 Ve Ac cp T   Ve Ac cp  T  dT   dqconv  dq rad  0




4 
  Ve  D2 / 4 cp dT   Ddx  h  T  T    T 4  Tsur
  0

dT
4

dx
 Ve D cp


 h  T  T    T 4  T 4 

sur 


(1)
<
Alternatively, if the control surface is fixed to the wire, conditions are transient and the energy balance
is of the form, Eout  Est , or
  D2  dT
4
4


 D dx h  T  T    T  Tsur   
dx  cp


 4

dt


dT
4 
4 

h  T  T    T 4  Tsur

dt
 D cp 




Problem: 7.78 Measurement of combustion gas
temperature with a spherical thermocouple junction.
Use the Whitaker correlation for a sphere
All properties , except μs, are evaluated at T∞.
To be neglected
SCHEMATIC: