Chapter 6: Fundamentals of
Convection
Yoav Peles
Department of Mechanical, Aerospace and Nuclear Engineering
Rensselaer Polytechnic Institute
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Objectives
When you finish studying this chapter, you should be able to:
• Understand the physical mechanism of convection, and its
classification,
• Visualize the development of velocity and thermal boundary layers
during flow over surfaces,
• Gain a working knowledge of the dimensionless Reynolds, Prandtl,
and Nusselt numbers,
• Distinguish between laminar and turbulent flows, and gain an
understanding of the mechanisms of momentum and heat transfer in
turbulent flow,
• Derive the differential equations that govern convection on the
basis of mass, momentum, and energy balances, and solve these
equations for some simple cases such as laminar flow over a flat
plate,
• Nondimensionalize the convection equations and obtain the
functional forms of friction and heat transfer coefficients, and
• Use analogies between momentum and heat transfer, and determine
heat transfer coefficient from knowledge of friction coefficient.
Physical Mechanism of Convection
• Conduction and convection are similar in that both
mechanisms require the presence of a material medium.
• But they are different in that convection requires the presence
of fluid motion.
• Heat transfer through a liquid or gas can be by conduction or
convection, depending on the presence of any bulk fluid
motion.
• The fluid motion enhances heat transfer, since it brings
warmer and cooler chunks of fluid into contact, initiating
higher rates of conduction at a greater number of sites in a
fluid.
• Experience shows that convection heat transfer strongly
depends on the fluid properties:
–
–
–
–
–
dynamic viscosity m,
thermal conductivity k,
density r, and
specific heat cp, as well as the
fluid velocity V.
• It also depends on the geometry and the roughness of the solid
surface.
• The rate of convection heat transfer is observed to be
proportional to the temperature difference and is expressed by
Newton’s law of cooling as
qconv  h Ts  T 
(W/m2 )
(6-1)
• The convection heat transfer coefficient h depends on the
several of the mentioned variables, and thus is difficult to
determine.
• All experimental observations indicate that a fluid in
motion comes to a complete stop at the surface and
assumes a zero velocity relative to the surface (no-slip).
• The no-slip condition is responsible for the development
of the velocity profile.
• The flow region adjacent
to the wall in which the
viscous effects (and thus
the velocity gradients) are
significant is called the boundary layer.
• An implication of the no-slip condition is that heat
transfer from the solid surface to the fluid layer
adjacent to the surface is by pure conduction, and can
be expressed as
qconv  qcond  k fluid
T
y
(W/m2 )
(6-3)
y 0
• Equating Eqs. 6–1 and 6–3 for the heat flux to obtain
h
k fluid  T y  y 0
Ts  T
(W/m2  C)
(6-4)
• The convection heat transfer coefficient, in general,
varies along the flow direction.
The Nusselt Number
• It is common practice to nondimensionalize the heat transfer
coefficient h with the Nusselt number
hLc
Nu 
(6-5)
k
• Heat flux through the fluid layer by convection and by
conduction can be expressed as, respectively:
T
qconv  hT (6-6) qcond  k
(6-7)
L
• Taking their ratio gives
qconv
hT
hL


 Nu (6-8)
qcond k T / L k
• The Nusselt number represents the enhancement of heat transfer
through a fluid layer as a result of convection relative to
conduction across the same fluid layer.
• Nu=1 pure conduction.
Classification of Fluid Flows
•
•
•
•
•
•
•
Viscous versus inviscid regions of flow
Internal versus external flow
Compressible versus incompressible flow
Laminar versus turbulent flow
Natural (or unforced) versus forced flow
Steady versus unsteady flow
One-, two-, and three-dimensional flows
Velocity Boundary Layer
•
•
•
•
•
•
•
•
Consider the parallel flow of a fluid over a flat plate.
x-coordinate: along the plate surface
y-coordinate: from the surface in the normal direction.
The fluid approaches the plate in the x-direction with a uniform
velocity V.
Because of the no-slip condition V(y=0)=0.
The presence of the plate is felt up to d.
Beyond d the free-stream velocity remains essentially unchanged.
The fluid velocity, u, varies from 0 at y=0 to nearly V at y=d.
Velocity Boundary Layer
• The region of the flow above the plate bounded by d
is called the velocity boundary layer.
• d is typically defined as
the distance y from the
surface at which
u=0.99V.
• The hypothetical line of
u=0.99V divides the flow over a plate into two
regions:
– the boundary layer region, and
– the irrotational flow region.
Surface Shear Stress
• Consider the flow of a fluid over the surface of a plate.
• The fluid layer in contact with the surface tries to drag the
plate along via friction, exerting a friction force on it.
• Friction force per unit area is called shear stress, and is
denoted by t.
• Experimental studies indicate that the shear stress for most
fluids is proportional to the velocity gradient.
• The shear stress at the wall surface for these fluids is
expressed as
u
(6-9)
ts  m
(N/m 2 )
y y 0
• The fluids that that obey the linear relationship above are
called Newtonian fluids.
• The viscosity of a fluid is a measure of its resistance to
deformation.
• The viscosities of liquids decrease with temperature, whereas
the viscosities of gases increase with temperature.
• In many cases the flow velocity profile is
unknown and the surface shear stress ts
from Eq. 6–9 can not be obtained.
• A more practical approach in external flow
is to relate ts to the upstream velocity V as
rV 2
(6-10)
ts  Cf
(N/m2 )
2
• Cf is the dimensionless friction coefficient (most cases is
determined experimentally).
• The friction force over the entire surface is determined from
rV 2
(6-11)
Ff  C f As
(N)
2
Thermal Boundary Layer
• Like the velocity a thermal boundary layer develops when a
fluid at a specified temperature flows over a surface that is at
a different temperature.
• Consider the flow of a fluid
at a uniform temperature of
T∞ over an isothermal flat
plate at temperature Ts.
• The fluid particles in the
layer adjacent assume the surface temperature Ts.
• A temperature profile develops that ranges from Ts at the
surface to T∞ sufficiently far from the surface.
• The thermal boundary layer ─ the flow region over the
surface in which the temperature variation in the direction
normal to the surface is significant.
• The thickness of the thermal boundary layer dt at any
location along the surface is defined as the distance
from the surface at which the temperature difference
T(y=dt)-Ts= 0.99(T∞-Ts).
• The thickness of the thermal boundary layer increases
in the flow direction.
• The convection heat transfer rate anywhere along the
surface is directly related to the temperature gradient
at that location.
Prandtl Number
• The relative thickness of the velocity and the
thermal boundary layers is best described by the
dimensionless parameter Prandtl number, defined
as
Molecular diffusivity of momentum  mc p
Pr 
 
(6-12)
Molecular diffusivity of heat

k
• Heat diffuses very quickly in liquid metals (Pr«1)
and very slowly in oils (Pr»1) relative to momentum.
• Consequently the thermal boundary layer is much
thicker for liquid metals and much thinner for oils
relative to the velocity boundary layer.
Laminar and Turbulent Flows
• Laminar flow ─ the flow is characterized by
smooth streamlines and highly-ordered
motion.
• Turbulent flow ─ the flow is
characterized by velocity
fluctuations and
highly-disordered motion.
• The transition from laminar
to turbulent flow does not
occur suddenly.
• The velocity profile in turbulent flow is much fuller than that in
laminar flow, with a sharp drop near the surface.
• The turbulent boundary layer can be considered to consist of
four regions:
–
–
–
–
Viscous sublayer
Buffer layer
Overlap layer
Turbulent layer
• The intense mixing in turbulent flow enhances heat and
momentum transfer, which increases the friction force on the
surface and the convection heat transfer rate.
Reynolds Number
• The transition from laminar to turbulent flow depends on the
surface geometry, surface roughness, flow velocity, surface
temperature, and type of fluid.
• The flow regime depends mainly on the ratio of the inertia forces
to viscous forces in the fluid.
• This ratio is called the Reynolds number, which is expressed for
external flow as
Inertia forces VLc rVLc
Re 


(6-13)
Viscous forces 
m
• At large Reynolds numbers (turbulent flow) the inertia forces are
large relative to the viscous forces.
• At small or moderate Reynolds numbers (laminar flow), the
viscous forces are large enough to suppress these fluctuations and
to keep the fluid “inline.”
• Critical Reynolds number ─ the Reynolds number at which the
flow becomes turbulent.
Heat and Momentum Transfer in
Turbulent Flow
• Turbulent flow is a complex mechanism dominated by
fluctuations, and despite tremendous amounts of research the
theory of turbulent flow remains largely undeveloped.
• Knowledge is based primarily on experiments and the empirical
or semi-empirical correlations developed for various situations.
• Turbulent flow is characterized by random and rapid fluctuations
of swirling regions of fluid, called eddies.
• The velocity can be expressed as the sum
of an average value u and a fluctuating
component u’
u  u u'
(6-14)
• It is convenient to think of the turbulent shear stress as
consisting of two parts:
– the laminar component, and
– the turbulent component.
• The turbulent shear stress can be expressed as
t turb   r u ' v '
• The rate of thermal energy transport by turbulent eddies is
qturb  r c p v ' T '
• The turbulent wall shear stress and turbulent heat transfer
t turb
u
  r u ' v '  mt
y
;
qturb
T
 r c p vT  kt
(6-15)
y
• mt ─ turbulent (or eddy) viscosity.
• kt ─ turbulent (or eddy) thermal conductivity.
• The total shear stress and total heat flux can be
expressed as
u
u
  m  mt 
 r   t 
y
y
(6-16)
T
T
   k  kt 
  r c p    t 
y
y
(6-17)
t turb
and
qturb
• In the core region of a turbulent boundary layer ─
eddy motion (and eddy diffusivities) are much larger
than their molecular counterparts.
• Close to the wall ─ the eddy motion loses its intensity.
• At the wall ─ the eddy motion diminishes because of
the no-slip condition.
In the core region ─ the velocity and temperature profiles
are very moderate.
In the thin layer adjacent to the wall ─ the velocity and
temperature profiles are very steep.
Large velocity and temperature gradients at the
wall surface.
The wall shear stress
and wall heat flux are much larger
in turbulent flow than they
are in laminar
flow.
Derivation of Differential Convection
Equations
• Consider the parallel flow of a fluid over a surface.
• Assumptions:
–
–
–
–
steady two-dimensional flow,
Newtonian fluid,
constant properties, and
laminar flow.
• The fluid flows over the surface with a uniform freestream velocity V, but the velocity within boundary
layer is two-dimensional (u=u(x,y), v=v(x,y)).
• Three fundamental laws:
– conservation of mass  continuity equation
– conservation of momentum  momentum equation
– conservation of energy  energy equation
The Continuity Equation
• Conservation of mass principle ─ the mass can
not be created or destroyed during a process.
• In steady flow:
Rate of mass flow
into the control volume
=
Rate of mass flow
out of the control volume
• The mass flow rate is equal to: ruA
ruA
(6-18)
The fluid leaves the control volume from the left surface at a rate of
ru  dy 1
the fluid leaves the control volume from the right surface at a rate of
u 

r  u  dx   dy 1
x 

Repeating this for the y direction
(6-19)
v+∂v/∂y·dy
and substituting the results into Eq.
u
6–18, we obtain
dy
u+∂u/∂x·dx
r u  dy 1  r v  dx 1 


r u 

u 
v 
x,y
dx   dy 1  r  v  dy   dx 1
x 
y 
(6-20)

dx
v
Simplifying and dividing by dx·dy
u v

0
x y
(6-21) The continuity equation
The Momentum Equation
• The differential forms of the equations of motion in
the velocity boundary layer are obtained by applying
Newton’s second law of motion to a differential
control volume element in the boundary layer.
• Two type of forces:
– body forces,
– surface forces.
• Newton’s second law of motion for the control
volume
Acceleration
Net force (body and surface)
(6-22)
(Mass) X
in a specified direction =
acting in that direction
or
d m  ax  Fsurface, x  Fbody , x
(6-23)
• where the mass of the fluid element within the control
volume is
d m  r  dx  dy 1
(6-24)
• The flow is steady and two-dimensional and thus
u=u(x, y), the total differential of u is
du 
u
u
dx  dy
x
y
(6-25)
• Then the acceleration of the fluid element in the x
direction becomes
du u dx u dy
u
u
ax 


u v
dt x dt y dt
x
y
(6-26)
• The forces acting on a surface are due to pressure and
viscous effects.
• Viscous stress can be resolved into
two perpendicular components:
– normal stress,
– shear stress.
• Normal stress should not be confused with pressure.
• Neglecting the normal stresses the net surface force
acting in the x-direction is
Fsurface , x
 t 
 t P 
 P 
  dy   dx 1  
dx   dy 1  

  dx  dy 1
 x 
 y 
 y x 
  2u P 
 m 2 
  dx  dy 1
x 
 y
(6-27)
•
Substituting Eqs. 6–21, 6–23, and 6–24 into Eq. 6–20 and
dividing by dx·dy·1 gives
 u
u 
 2u P
r u  v   m 2 
y 
y
x
 x
Boundary Layer Approximation
Assumptions:
1) Velocity components:
The x-momentum
equation
(6-28)
u>>v
2) Velocity gradients:
∂v/∂x≈0 and ∂v/∂y≈0
∂u/∂y >> ∂u/∂x
3) Temperature gradients:
∂T/∂y >> ∂T/∂x
•
When gravity effects and other body forces are negligible the
y-momentum equation P  0
(6-29)
y
Conservation of Energy Equation
• The energy balance for any system undergoing any
process is expressed as Ein-Eout=Esystem.
• During a steady-flow process Esystem=0.
• Energy can be transferred by
– heat,
– work, and
– mass.
• The energy balance for a steady-flow control volume
can be written explicitly as
E
in
 Eout 
by heat
  Ein  Eout 
by work
  Ein  Eout 
by mass
0
(6-30)
• Energy is a scalar quantity, and thus energy
interactions in all directions can be combined in one
equation.
Energy Transfer by Mass
• The total energy of a flowing fluid stream per unit
mass is
estream  enthalpy  kinetic  potential
2
2
C pT
gz
V 2 u v 

2
2
• Noting that mass flow rate of the fluid entering the
control volume from the left is ru(dy·1), the rate of
energy transfer to the control volume by mass in the
x-direction is

  mestream  x 
 Ein  Eout by mass,x   mestream x   mestream x  x dx 


(6-31)
  r u  dy 1 c pT 
u 
 T

dx   r c p  u
T
 dxdy
x
x 
 x
• Repeating this for the y-direction and adding
the results, the net rate of energy transfer to the
control volume by mass is determined to be
E
in
 Eout 
by mass

 T
u 
v 
 T
 rcp  u
T
 T  dxdy
 dxdy  r c p  v
x 
y 
 x
 y
 T
T 
 rcp  u
v
 dxdy
y 
 x
(6-32)
• Note that ∂u/∂x+∂v/∂y=0 from the continuity
equation.
Energy Transfer by Heat Conduction
• The net rate of heat conduction to the volume element
in the x-direction is
E
in
 Eout 
by heat , x

Qx 
 
T
 Qx   Qx 
dx     k  dy 1
x
x 
x


 2T
 k 2 dxdy
x

 dx

(6-33)
• Repeating this for the y-direction and adding the
results, the net rate of energy transfer to the control
volume by heat conduction becomes
E
in
 Eout 
by heat
  2T  2T 
 2T
 2T
 k 2 dxdy  k 2 dxdy  k  2  2  dxdy
x
y
y 
 x
(6-34)
Energy Transfer by Work
• The work done by a body force is determined by
multiplying this force by the velocity in the direction
of the force and the volume of the fluid element.
• This work needs to be considered only in the
presence of significant gravitational, electric, or
magnetic effects.
• The work done by pressure (the flow work) is already
accounted for in the analysis above by using enthalpy
for the microscopic energy of the fluid instead of
internal energy.
• The shear stresses that result from viscous effects are
usually very small, and can be neglected in many
cases.
The Energy Equation
• The energy equation is obtained by substituting Eqs.
6–32 and 6–34 into 6–30 to be 2
2
 T  T 
 T
T 
rcp  u
v
k 2  2 
y 
y 
 x
 x
(6-35)
  2T  2T 
 T
T 
rcp  u
v
  k  2  2   m
y 
y 
 x
 x
(6-36)
• When the viscous shear stresses are not negligible,
• where the viscous dissipation function is obtained
after a lengthy analysis to be
 u 2  v 2   u v 2
  2          
 x   y    y x 
(6-37)
• Viscous dissipation may play a dominant role in highspeed flows.
Solution of Convection Equations for a
Flat Plate (Blasius Equation)
• Consider laminar flow of a fluid over
a flat plate.
• Steady, incompressible, laminar flow
of a fluid with constant properties
• Continuity equation
u v

x y
(6-39)
• Momentum equation
u
u
 2u
u v
 2
x
y
y
(6-40)
• Energy equation
T
T
 2T
u
v
 2
x
y
y
(6-41)
Boundary conditions
• At x=0
• At y=0
• As y∞
u  0, y   V ,
u  x,0  0,
T  0, y   T
v  x,0   0, T  x,0   Ts
u  x,    V ,
(6-42)
T  x,    T
• When fluid properties are assumed to be constant,
the first two equations can be solved separately for
the velocity components u and v.
• knowing u and v, the temperature becomes the only
unknown in the last equation, and it can be solved
for temperature distribution.
• The continuity and momentum equations are solved
by transforming the two partial differential equations
into a single ordinary differential equation by
introducing a new independent variable (similarity
variable).
• The argument ─ the nondimensional velocity profile
u/V should remain unchanged when plotted against
the nondimensional distance y/d.
• d is proportional to (x/V)1/2, therefore defining
dimensionless similarity variable as
  y V x
might enable a similarity solution.
(6-43)
• Introducing a stream function y(x, y) as
y
u
y
;
y
v
x
(6-44)
• The continuity equation (Eq. 6–39) is automatically
satisfied and thus eliminated.
• Defining a function f() as the dependent variable as
y
f   
V  x /V
(6-45)
• The velocity components become
y y 
 x df V
df
u

V
V
y  y
V d  x
d
(6-46)

y
 x df V 
1 V  df
v
 V

f 
f

x
V d 2 Vx
2 x  d

(6-47)
• By differentiating these u and v relations, the
derivatives of the velocity components can be shown
to be
u
V d2 f
u
V d2 f
 2u V 2 d 3 f
 
;
V
;

2
2
2
x
2 x d
y
 x d
y
 x d 3
(6-48)
• Substituting these relations into the momentum
equation and simplifying
d3 f
d2 f
2 3f
0
2
d
d
(6-49)
• which is a third-order nonlinear differential equation.
Therefore, the system of two partial differential
equations is transformed into a single ordinary
differential equation by the use of a similarity
variable.
• The boundary conditions in terms of the similarity
variables
f  0   0,
df
 0,
d  0
df
 1 (6-50)
d  
• The transformed equation with its
associated boundary conditions
cannot be solved analytically, and
thus an alternative solution method
is necessary.
• The results shown in Table 6-3 was
obtained using different numerical approach.
• The value of  corresponding to u/V=0.99 is =4.91.
• Substituting =4.91 and y=d into the definition of the
similarity variable (Eq. 6–43) gives 4.91=d(V/x)1/2.
• The velocity boundary layer thickness becomes
d
4.91
4.91x

V x
Re x
(6-51)
• The shear stress on the wall can be determined from its
definition and the ∂u/∂y relation in Eq. 6–48:
u
tw  m
y
y 0
V d2 f
 mV
 x d 2
(6-52)
 0
Substituting the value of the second derivative of f at h=0
from Table 6–3 gives
2
rmV
t w  0.332V
x
0.332 rV

Re x
(6-53)
Then the average local skin friction coefficient becomes
C f ,x 
tw
rV / 2
2
 0.664 Re x 1/ 2
(6-54)
The Energy Equation
• Introducing a dimensionless temperature q as
q  x, y  
T  x, y   Ts
(6-55)
T  Ts
• Noting that both Ts and T are constant, substitution
into the energy equation Eq. 6–41 gives
q
q
 2q
u
v
 2
x
y
y
(6-56)
• Using the chain rule and substituting the u and v
expressions from Eqs. 6–46 and 6–47 into the energy
equation gives
df dq d 1 Vy  df  dq d
d q   
V

f
 2 


d d dx 2 x  d  d dy
d  y 
2
2
(6-57)
• Simplifying and noting that Pr=/ gives
d 2q
dq
2 2  Pr f
0
d
d
Boundary conditions:
(6-58)
q  0  0, q     1
• Obtaining an equation for q as a function of  alone confirms that
the temperature profiles are similar, and thus a similarity solution
exists.
• for Pr=1, this equation reduces to Eq. 6–49 when q is replaced by
df/d.
• Equation 6–58 is solved for numerous values of Prandtl numbers.
• For Pr>0.6, the nondimensional temperature gradient at the surface is
found to be proportional to Pr1/3, and is expressed as
dq
 0.332 Pr1/ 3
d  0
(6-59)
• The temperature gradient at the surface is
dT
dy
y 0
q
 T  Ts 
y
y 0
dq
d
 T  Ts 
d  0 dy
y 0
(6-60)
V
x
• Then the local convection coefficient and Nusselt number become
 0.332 Pr1/ 3 T  Ts 
and
k  T y  y 0
qs
V
1/ 3
hx 

 0.332 Pr k
Ts  T
Ts  T
x
(6-61)
hx x
(6-62)
Nu x 
 0.332 Pr1/ 3 Re1/ 2
Pr>0.6
k
• Solving Eq. 6–58 numerically for the temperature profile for
different Prandtl numbers, and using the definition of the thermal
boundary layer, it is determined that
d d t  Pr1/ 3
Nondimensional Convection Equation
and Similarity
Continuity equation
u v

0
x y
(6-21)
x-momentum equation
 u
u 
 2u P
r u  v   m 2 
y 
y
x
 x
(6-28)
Energy equation
  2T  2T 
 T
T 
rcp  u
v
  k  2  2  (6-35)
y 
y 
 x
 x
• Nondimensionalized variables
T  Ts
x
y
u
v
P
*
*
*
*
*
x 
; y 
; u 
; v 
; P 
; T 
2
L
L
V
V
rV
T  Ts
*
• Introducing these variables into Eqs. 6–21, 6–28, and
6–35 and simplifying give
u* v*
 * 0
*
x y
Continuity equation
(6-64)
x-momentum equation
*
*
2 *
*

u

u
1

u

P
u * *  v* * 
 *
*2
x
y
Re L y
x
(6-65)
Energy equation
*
*
2 *

T

T
1

T
u * *  v* * 
x
y
Re L Pr y*2
(6-66)
with the boundary conditions








u* 0, y*  1 ; u * x* , 0  0 ; u * x* ,   1 ; v* x* , 0  0






T * 0, y*  1 ; T * x* , 0  0 ; T * x* ,   1
(6-67)
• For a given type of geometry, the solutions of
problems with the same Re and Nu numbers are
similar, and thus Re and Nu numbers serve as
similarity parameters.
• A major advantage of nondimensionalizing is the
significant reduction in the number of parameters.
• The original problem involves 6 parameters (L, V, T,
Ts, , ), but the nondimensionalized problem
involves just 2 parameters (ReL and Pr).
6 parameters
L, V, T, Ts,, 
Nondimensionalizing
2 parameters
ReL, Pr
Functional Forms of the Friction and
Convection Coefficient
• From Eqs. 6-64 and 6-65 it can be inferred that

u *  f1 x* , y* , Re L

(6-68)
• Then the shear stress at the surface becomes
u
ts  m
y

y 0
mV u *
L y

*
y*  0
mV
L

f 2 x* , Re L

(6-69)
• Substituting into its definition gives the local
friction coefficient,
C f ,x
ts
mV L
2
*
*
*


f
x
,
Re

f
x
,
Re

f
x
, Re L 





2
L
2
L
3
2
2
rV 2 rV 2
Re L
(6-70)
• Similarly the solution of Eq. 6-66

T *  g1 x* , y* , Re L , Pr

(6-71)
• Using the definition of T*, the convection heat
transfer coefficient becomes
k  T y  y 0 k T  T  T *
k T *

s
(6-72)
h


*
*
Ts  T
L Ts  T  y y 0 L y y 0
*
*
• Substituting this into the Nusselt Number relation
gives
*
hL T
Nu x 
 *
k
y

 g 2 x* , Re L , Pr
y*  0

(6-73)
• It follows that the average Nu and Cf depends on
Nu  g3  Re L , Pr 
;
C f  f 4  Re L 
(6-74)
• These relations are extremely valuable:
– The friction coefficient can be expressed as a function of
Reynolds number alone, and
– The Nusselt number as a function of Reynolds and Prandtl
numbers alone.
• The experiment data for heat transfer is often
represented by a simple power-law relation of the
form:
Nu  C RemL  Pr n
(6-75)
Analogies Between Momentum and
Heat Transfer
• Reynolds Analogy (Chilton─Colburn Analogy) ─ under
some conditions knowledge of the friction coefficient, Cf,
can be used to obtain Nu and vice versa.
• Eqs. 6–65 and 6–66 (the nondimensionalized momentum
and energy equations) for Pr=1 and ∂P*/∂x*=0:
x-momentum equation
*
*
2 *

u

u
1

u
*
*
u
v

*
*
x
y
Re L y*2
(6-76)
Energy equation
*
*
2 *

T

T
1

T
*
*
u
v

*
*
x
y
Re L y*2
(6-77)
• which are exactly of the same form for the dimensionless
velocity u* and temperature T*.
• The boundary conditions for u* and T* are also identical.
• Therefore, the functions u* and T* must be identical.
u *
y*
Reynolds analogy
C f ,x
y*  0
T *
 *
y
Re L
 Nu
2
(6-78)
y*  0
(6-79)
(Pr=1)
• The Reynolds analogy can be extended to a wide range of
Pr by adding a Prandtl number correction.
C f , x  0.664Rex1/ 2 (6-82)
C f ,x
Nu x  0.332 Pr1/ 3 Re1/ 2
Re
 Nu x Pr1 3
2
0.6  Pr  60
Pr>0.6
(6-83)