Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell
5 2 Basic Definitions,
5.2
Definitions Terminology and
Governing Equations
 It is
i important
i
t t to
t define
d fi some basic
b i
definitions, terminologies and criteria that
are often used in internal convective heat
and mass transfer. These include:



Mean velocity,
velocity temperature
temperature, and
concentration
Fullyy developed
p flow,, temperature,
p
, and
concentration profiles
Hydrodynamic, thermal, and concentration
entrance
t
lengths
l
th
Chapter 5: Internal Forced Convective
Heat and Mass Transfer
1
Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell



Figure 5.1a
Fi
5 1 shows
h
the
th development
d
l
t off a velocity
l it
profile inside a duct or tube with uniform inlet
velocity for laminar flow for an incompressible
Newtonian fluid.
The velocity profile at some distance away from
the tube’s inlet no longer changes along the flow
direction, where it is referred to as the fully
developed flow condition.
The fully developed condition is often met at
some distance away from inlet, however there
are also applications in which fully developed
flow is never realized
realized.
Chapter 5: Internal Forced Convective
Heat and Mass Transfer
2
Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell
H d d
Hyrdrodynamic
i entrance
t
length
l th
Fully developed flow
δ
uin
u
r
ro
δ
friction factor
(a) Velocity profile
x
(b) Friction factor
Figure 5.1
5 1 Velocity
V l it profiles
fil andd friction
f i ti factor
f t variation
i ti in
i laminar
l i
flow
fl in
i a circular
i l tube.
t b
Chapter 5: Internal Forced Convective
Heat and Mass Transfer
3
Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell




Momentum,
M
t
thermal,
th
l and
d concentration
t ti
boundary layers form on the inside surface of
the tube. The thickness of the layers
y
increases
in a similar manner as boundary layer flow over
a flat plate (which was presented in detail in
Chapter
p 4).
)
Figure 5.1a shows how the momentum
boundary layer builds up in a pipe along the flow
direction.
direction
At some distance away from the inlet, the
boundary layer fills the flow area.
The flow downstream from this point is referred
to as fully developed flow since the velocity
slope does not change after this point.
Chapter 5: Internal Forced Convective
Heat and Mass Transfer
4
Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell




The distance downstream from the inlet to where
the flow becomes fully developed flow is called
the hydrodynamic entrance
entrance.
If the flow is laminar (Re < 2300 for flow inside
circular tubes),
) the fully
y developed
p velocity
y is a
parabolic shape.
It should be noted that the fluid velocity outside
th b
the
boundary
d
llayer iincreases with
ith x, which
hi h iis
required to satisfy the conservation of mass (or
y) equation.
q
continuity)
The velocity finally reaches a value two times
the inlet velocity uin for fully developed steady
i
incompressible
ibl llaminar
i
flflow iinside
id ttubes.
b
Chapter 5: Internal Forced Convective
Heat and Mass Transfer
5
Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell



It should be noted that the hydrodynamic
entrance length for fully developed flow does not
mean it starts from the point where the friction
coefficient,
w
cf 
(5.3)
u 2 / 2
does not change along the flow.
The friction coefficient variation for laminar flow
inside a circular tube with uniform inlet velocity is
shown in Figure 5.1b.
The friction factor is highest at the entrance and
then decreases smoothly to a constant value,
corresponding
di tto ffully
ll d
developed
l
d flflow.
Chapter 5: Internal Forced Convective
Heat and Mass Transfer
6
Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell

Two factors cause the friction coefficient to be
higher in the entrance region of tubes than in the
f ll de
fully
developed
eloped region
region.


The first factor is due to larger velocity gradient at the
entrance. The gradient decreases along the pipe and
becomes constant before the velocity becomes fully
developed.
Th second
The
d ffactor is
i that
h the
h velocity
l i outside
id the
h
boundary layer must increase to satisfy the
conservation of mass or continuityy equation.
q
Accelerating velocity in the core produces an
additional drag force when its effect is considered in
the friction factor
factor.
Chapter 5: Internal Forced Convective
Heat and Mass Transfer
7
Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell




The turbulent velocity profile and friction
coefficient variation for a circular pipe is shown
in Figure
g
5.2.
Even for very high inlet velocity, there will be
some initial part of entrance over which the
boundary layer is laminar.
This transition from laminar to turbulent is clearly
shown by the sudden increase in momentum
boundary layer thickness as shown in Figure
52
5.2a.
The friction factor variation for turbulent flow in a
pipe entrance is shown in Figure 5
5.2b.
2b
Chapter 5: Internal Forced Convective
Heat and Mass Transfer
8
Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell
Hyrdrodynamic entrance length
Fully developed flow
δ
u
r0
r
δ
(a) Velocity profile
Turbulent boundary
layer
Fully developed
turbulent
frriction factor
Laminar boundary layer
x
(b) Friction factor
Figure 5.2 Velocity profiles and friction factor in turbulent flow in a circular tube.
Chapter 5: Internal Forced Convective
Heat and Mass Transfer
9
Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell


The hydrodynamic entry length required for fully
developed flow should be obtained by a complete solution
of the flow and thermal field in the entrance region.
A rule of thumb to judge whether or not the flow is fully
developed flow for circular pipes is
LH
 0.05 Re
D
for laminar flow (5.4)
LH
 0.625 Re0.25
D
for turbulent flow (5.5)
where LH is the hydrodynamic length and the Reynolds
number is defined by Re  um D

Chapter 5: Internal Forced Convective
Heat and Mass Transfer
10
Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell



A similar behavior is expected for thermal cases with the thermal
boundary layer growth at the entrance of a tube as shown in
Figure 5.2a.
Figure 5.3 corresponds to a case in which there may be an
unheated length in which the velocity is fully developed flow
before heating starts.
One expects that the thermal boundary layer increases in the
thermal entry region before the heat transfer coefficient becomes
constant.
Uniform inlet
temperature
Tin
T-Tw
δT
r
δT
R
Figure 5.3: Temperature development along the flow in a circular tube
Chapter 5: Internal Forced Convective
Heat and Mass Transfer
11
Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell

It should
h ld be
b emphasized
h i d th
thatt th
the requirement
i
t for
f fully
f ll
developed thermal region is when the dimensionless
temperature,
T T
T T



w
Tw  Tm
or
w
Tw  Tc
does not change with distance along the flow direction nor the
absolute temperature, T.
Where Tm and Tc are the mean and centerline temperatures,
respectively.
Similar requirements
q
exist for the fully
y developed
p
concentration profile where θ is replaced with
Cw  C
Cw  C

or
C w  Cm
Cw  Cc
Cm, Cc are mean and centerline concentration (or mass
density) respectively
Chapter 5: Internal Forced Convective
Heat and Mass Transfer
12
Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell

In the subsequent sections we use the following definitions to
define mathematically the fully developed flow and
temperature profiles:
u
uc
or
T T
 w
Tw  Tm
cw  c

cw  cm
or
or
r
u
 f 
um
 ro 
Fully developed flow
(5.6)
r
Tw  T
 g  Fully developed temperature profile (5.7)
Tw  Tc
 ro 
r
cw  c
 h  Fully developed concentration profile (5.8)
cw  cc
 ro 
Chapter 5: Internal Forced Convective
Heat and Mass Transfer
13
Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell

We can now define
f
the local heat and mass transfer
f coefficient
ff
(h
(
and hm) based on mean temperature or concentration.
qw  h  Tw  Tm    k
T
r
(5 9)
(5.9)
r  ro
 w  hm w  m     D
m

r
(5.10)
r  ro
where D is mass diffusivity
 Since we define the fully developed temperature profile as when the
non-dimensional
di
i
l ttemperature
t
profile
fil Tw  T is
i iinvariant
i t
Tw  T
in the flow direction (x-direction), we can write the following
T
equation:
ti

r r ro h
  Tw  T 
 constant 
  constant


r  Tw  Tm  r r
Tw  Tm k
o
(5.11)
Chapter 5: Internal Forced Convective
Heat and Mass Transfer
14
Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell


The above conclusion, that the local heat transfer
coefficient is constant along the flow direction for fully
developed temperature profile, is valid for both constant
wall heat flux and wall temperature conditions.
The requirement for the dimensionless temperature to be
i
invariant
i t ffor fully
f ll d
developed
l
d ttemperature
t
profile
fil can also
l
be presented in the following way:
(5 12)
(5.12)
  Tw  T 

0
x  Tw  Tm 

Differentiating the above equation yields
T dTw  Tw  T  dTw  Tw  T  dTm





x
dx  Tw  Tm  dx  Tw  Tm  dx
(5.13)
Chapter 5: Internal Forced Convective
Heat and Mass Transfer
15
Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell




IIn an external
t
l flow,
fl
the
th heat
h t and
d mass transfer
t
f
coefficients are usually defined by a driving differential
(Tw - T∞) or (ωw - ω∞) where T∞ and ω∞ are the
temperature and mass fraction of the fluid in the free
stream (far away from the wall).
In most cases, T∞ and ω∞ are known and constant for
external flows.
flows However
However, in internal flow configurations
configurations,
there is not usually a well defined temperature or
concentration (mass fraction), except at the inlet and/or
the boundaries.
In internal flow, the temperature and concentrations
probably change both in the axial direction and
perpendicular to the flow direction.
Therefore, there are several choices available for the
driving differential for temperature and concentration in
internal flow.
Chapter 5: Internal Forced Convective
Heat and Mass Transfer
16
Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell

The most common choice for defining the driving temperature
or concentration is based on mean temperature or
concentration (mass fraction or mass density). The mixed
mean fluid temperature or concentration is defined at a given
axial local location based on the convective thermal energy or
mass balance as shown below
1
(5 14)
(5.14)
Tm 
uT  c p dA
d

Aum  m c p , m A
1
(5.15)
 A,m 
u  A dA

A m A
Au
where  A , m is the mean mass density for a given component
A, and m is the mean densityy for the fluid and where mean
velocity is defined as
(5.16)
1
um 
u  dA

A
A m
Chapter 5: Internal Forced Convective
Heat and Mass Transfer
17
Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell


Assuming
A
i constant
t t properties
ti for
f mean velocity,
l it
temperature and mass concentration in the above
equation, we obtain
1
um   udA
(5.17)
A A
Tm 
1
Aum

 A,m 
1
Aum

A
A
uTdA
d
(5 18)
(5.18)
 A udA
((5.19))
We will now focus our attention on two conventional
special boundary conditions; constant wall heat flux or
constant surface temperature.
Chapter 5: Internal Forced Convective
Heat and Mass Transfer
18
Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell



First, consider the constant heat flux or heat rate at the
wall, where it occurs in many applications such as
electronic cooling, electric resistance heating and radiant
heating.
From equation (5.9), since h and qw are constant, we
can conclude
l d th
thatt
Tw  Tm  constant
Differentiating above leads to
dTw dTm

dx
dx

Substituting in equation (5.13) gives us
T dTw dTm



x
dx
d
d
dx
(5.20)
Chapter 5: Internal Forced Convective
Heat and Mass Transfer
19
Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell


Now consider the case of constant surface or
wall temperature, where it also occurs in many
applications including condensers
condensers, evaporators
and any heat exchange surface when the heat
transfer coefficient is extremely high.
Using equation (5.13) and the fact that dTw/dx = 0
for constant surface temperature, we get
T  Tw  T  dTm
(5
(5.21)
21)

x




T
T
m  dx
 w
It should be emphasized that equations (5
(5.20)
20)
and (5.21) are only applicable when the
temperature
p
p
profile is fully
y developed.
p
Chapter 5: Internal Forced Convective
Heat and Mass Transfer
20




Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell
The variations of wall and mean temperature for the fully
d
developed
l
d ttemperature
t
profile
fil along
l
th
the flflow ffor constant
t t
heat rate or surface temperature are shown in Figure
5.4.
Finally to obtain the convective heat and/or mass
Finally,
transfer coefficients, one needs to solve the continuity,
mass, momentum, energy and appropriate species
equations.
equations
It is important to obtain information about the flow by
solving the continuity and momentum equations, in
addition to energy and species equations, in convective
heat and mass transfer problems.
These conservation equations are mostly decoupled,
except
p for circumstances such as a variable p
property
p y or
coupled governing equation or boundary conditions due
to physical circumstances (which happens in
applications such as free convection, absorption,
sublimation,
bli ti
evaporation
ti and
d condensation
d
ti problems).
bl
)
Chapter 5: Internal Forced Convective
Heat and Mass Transfer
21
Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell
Chapter 5: Internal Forced Convective
Heat and Mass Transfer
22
Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell



It is obviously more accurate to solve the complete transport
conservation equations (elliptic form) for internal flow without making
boundary layer assumptions (parabolic form) as discussed in
Ch t 4
Chapter
4.
However, in most cases it is not practical due to complexity of
geometry and/or solution techniques as well as the requirement of
additional
dditi
l boundary
b
d
conditions
diti
in
i both
b th analytical
l ti l or numerical
i l
methods.
For the case of two-dimensional fully developed steady laminar flow
with
ith constant
t t properties,
ti
the
th momentum
t
equation
ti in
i a circular
i l tube,
t b
including boundary conditions, as shown in Chapter 2 are
dp  d  du 

r

dx r dr  dr 
(5.22)
u0
at
r  ro
du
0
dr
at
r0
((5.23))
Chapter 5: Internal Forced Convective
Heat and Mass Transfer
23
Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell


Integrating the above equation twice and using the boundary
conditions yields a parabolic velocity profile.
ro2  dp   r 2 
(5.24))
(5
u
1





4   dx   ro2 
Using the definition of mean velocity um for constant properties
and
d th
the above
b
equation,
ti
we obtain
bt i
um



A
udA
A


0
ro
2 rudr
r
2
o
ro2 dp

8 dx
((5.25))
Equation (5.24) in terms of mean velocity is
 r2 
u  2um 1  2 
 ro 
(5.26)
Chapter 5: Internal Forced Convective
Heat and Mass Transfer
24
Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell

The shear stress at the wall can be calculated
from the velocity gradient at the wall.
4um 
ro dp
u


w  
ro
r r  ro
2 dx

(5.27)
The above result can be presented in terms of
the friction coefficient, cf.
w
8
16
cf 


2
 um / 2 ro  um Re
(5.28)
Chapter 5: Internal Forced Convective
Heat and Mass Transfer
25
Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell

In addition to the above friction coefficient
coefficient, the following
friction factor is also widely used:
(dp / dx) D
f 
 um2 / 2

(5.29)
It follows from eq. (6.27) that
4 w
64
f  2
 4c f 
Re
 um / 2
(5.30)
Chapter 5: Internal Forced Convective
Heat and Mass Transfer
26
Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell


Let’s make an analysis of the energy equation to get a
feeling about the importance of various terms, since
determination of the temperature field in the fluid is
required for the heat transfer coefficients.
To simplify the analysis, let’s consider a two-dimensional
y
g
geometry
y with the following
g assumptions:
p
cylindrical
Steady laminar flow
 Constant properties
 Fully developed flow
 Newtonian incompressible fluid


The energy equation under above assumptions is
2
 1   r T / r   2T 
  u 
T
u
 
 2 
 
r
r
c

x

x

p  r 


(5.31)
Chapter 5: Internal Forced Convective
Heat and Mass Transfer
27
Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell

The above
Th
b
equation
ti iis di
dimensionalized
i
li d using
i th
the ffollowing
ll i
variables to show the effect of axial conduction and viscous
dissipation 
2 x / D
um2
u

u 
, x 
, E
um
R P
Re
Pr
c p T
(5.32)
T  Tr
r
, T  Tin  Tr , r  
, Pe  Re Pr
Tin  Tr
ro
where Tr is a reference temperature and E and Pe are Eckert
and Peclet numbers, respectively.
The resulting dimensionless energy equation is


1    
u 
   r

2 x
r r  r 


 u 
1 

 E Pr   

2
2
 2 Pe x
 r 
2

2
(5.33)
The second term in the right hand side of the above equation is
due to axial heat conduction, and the last term is due to the
viscous dissipation effect
effect.
Chapter 5: Internal Forced Convective
Heat and Mass Transfer
28
Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell



If EPr is small, viscous dissipation can be neglected. This
is true for flow with a low velocity and low Prandtl
number.
The second term on the right hand side (axial heat
conduction) is neglected when the Peclet number, Pe, is
greater
t than
th 100.
100
Axial heat conduction should be accounted for when the
Peclet number is small
small, such as when the Prandtl
number is small corresponding to liquid metals.
Chapter 5: Internal Forced Convective
Heat and Mass Transfer
29
Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell
Example 5.1
 Estimate
the hydrodynamic entry
l
length,
th LH, using
i Bl
Blasius’s
i ’ result
lt ffor
the momentum boundary layer
thickness.
Chapter 5: Internal Forced Convective
Heat and Mass Transfer
30
Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell
Solution
 From Blasius’s solution


5

x Re x1/ 2
F fully
For
f ll d
developed
l
d flflow conditions,
diti
δ = D/2,
/ and
d
therefore
D/2
5

LH
Re x1/ 2
1/ 2
LH Re x

10
D
(5.34)
(5.35)
 0.1Re x1/ 2
Chapter 5: Internal Forced Convective
Heat and Mass Transfer
31