Chapter 4 Fluid Flow, Heat Transfer, and Mass Transfer:
Similarities and Coupling
4.1 Similarities among different types of transport
4.1.1 Basic laws
The transfer of momentum, heat , and species A occurs in the direction of
decreasing vz, T, and wA, as summarized in Fig. 4.1-1. according to Eqs. [1.1-2],
[2.1-1], and [3.1-1]
d z
 yz   
qy  k
dy
dT
j Ay    D A
dy
dw A
dy
Newton’s law of viscosity
Fourier’s law of conduction
Fick’s law of diffution
[4.1-1] also
[1.1-2]
[4.1-2] also
[2.1-2]
[4.1-3] also
[3.1-1] 1
The three basic laws share the same form as follows:
Or
 F lu x o f

tra n sp o rt

 p ro p erty


 g ra d ien t o f
p
ro
p
o
rto
n
a
l




 
   tra n sp o rt

 co n s tan t
 


 p ro p erty
j y    





[4.1-4]
d
dy
[4.1-5]
The three-dimensional forms of these basic laws are summarized in Table 4.1-1.
For constant physical properties, Eqs. [4.1-1] through [4.1-3] can be written
as follows:
d
[4.1-6]
 yz  
(  vz )
dy
2
q y  a
d
dy
j Ay   D A
(  C vT )
d
dy
[4.1-7]
( A )
[4.1-8]
These equations share the same form listed as follows:
 F lu x o f

tr a n s p o r t

 p r o p er ty


 d iffu s ivity


  o f tr a n s p o r t



 p r o p er ty








 g r a d ien t o f

tr a n s p o r t p r o p er ty

 co n cen tr a tio n






[4.1-9]
In other words, , a, and DA are the diffusivities of momentum, heat, and mass,
respectively, and vZ, CvT, and A are the concentration of z momentum, thermal
energy, and species mass, respectively.
4.1.2 Coefficients of Transfer
Fig. 4.1-2 shows the transfer of z momentum, heat, and species A from an
interface, where they are more abundant, to an adjacent fluid, and from an adjacent
fluid, where they are more abundant, to an interface. The coefficients of transfer,
according to Eqs. [1.1-35], [2.1-14], and [3.1-21], are defined as follows:
3
C
'
f

 yz
y0
v z 0  v z

  (v z / y )
0  v
y0
(momentum
[4.1-10]
transfer coefficient)
4
h
km 
qy
y0

T0  T
j Ay
 k (T / y )
y0
( heat transfer coefficient )
T0  T
y0
 w A 0   w A

 D A (w A / y )
y0
(m ass transfer coefficie nt)
w A 0  w A
[4.1-11]
[4.1-12]
As mentioned in Sec. 3.1.6, Eq. [4.1-12] is for low solubility of species A in the fluid.
These coefficients share the same form listed as follows:
 C oefficient  

flux at the interface


 

 of transfer   difference in transport property 
[4.1-13]
or
k 
j
y0
0  

   ( / y )
y0
(m ass transfer coefficient)
0  
[4.1-14]
It is common to divide Cf by v/2 to make denominator appear in the form of the
kinetic energy v2∞/2. As shown in Eq. [1.1-36], the so-called friction coefficient is
defined by
'
Cf 
Cf
1
2
 

 yz
1
2
y0

[4.1-15]
2

5
4.1.3 The Chilton-Colburn Analogy
The analogous behavior of momentum, heat, and mass transfer is apparent from
Examples 1.4-6, 2.2-5, and 3.2-4, where laminar flow over a flat plate was considered.
From Eqs. [1.4-67], [2.2-71], and [3.2-56], at a distance z from the leading edge of the
plate,
C fz
 (1 2 )
 0.323 R e z
[4.1-16]
2
hz
 0.323 P r
1 3
k
km z
 0.323 Sc
DA
where
z  
Rez 
Pr 
Sc 

v
a

v
DA
C p
k


 DA
Rez
1 3
1 2
Rez
[4.1-17]
1 2
[4.1-18]
(local Reynolds number)
[4.1-19]
(Prandtl number)
[4.1-20]
(Schmidt number)
[4.1-21]
and υ∞ is the velocity of the fluid approaching the flat plate.
6
Equations [4.1-16] through [4.1-18] can be rearranged as follows
C fz
2
hz
 0.323 R e z
1
Pr
1
[4.1-22]
2 3
 0.323 R e z
Sc
2 3
k Pr Re
km z
 (1 2 )
 (1 2 )
 0.323 R e z
D A Sc R e z
[4.1-23]
1 2
[4.1-24]
Since these equations have the same RHS, we see
C fz
2

hz
1
Pr
2 3

k Pr R e
km z
1
Sc
2 3
D A Sc R e z
[4.1-25]
Substituting Eqs. [4.1-19] through [4.1-21] into Eq. [4.1-25], we obtain
C fz
2

h
 C p
Pr
2 3

km

Sc
2 3
[4.1-26]
7
This equation, known as the Chilton-Colburn analogy,1 is ofen written as follows
C fz
2
[4.1-27]
 jH  jD
where the j factor for heat transfer
jH 
h
v  C p
Pr
2/3
[4.1-28]
And the j factor for mass transfer
jD 
km
Sc
2/3
[4.1-29]
v
8
The Chilton –Colburn analogy for momentum, heat and mass transfer has been
derived here on the basis of laminar flow over a flat plate. However, it has been
observed to be a reasonable approximation in laminar and turbulent flow in
systems of other geometries provided no form drag is present . From drag, which
has no counterpart in heat and mass transfer, makes Cf/2 greater than jH and jD,
for example, in flow around (normal to) cylinders. However, when form drag is
present, the Chilton –Colburn analogy between heat and mass transfer can still be
valid, that is,
jH  jD
[4.1-30]
or
h
v  C p
Pr
2/3

km
Sc
2/3
[4.1-31]
v
These equations are considered valid for liquid and gases within the ranges
0.6 < Sc < 2500 and 0.6 < Pr < 100 . They have been observed to be a
reasonable approximation for various geometries, such as flow over flat plates,
flow around cylinders, and flow in pipes.
9
The Chilton –Colburn analogy is useful in that it allows one unknown transfer
coefficient to be evaluated from another transfer coefficient which is known or
measured in the same geometry. For example, by use Eq. [4.1-26] the mass transfer
coefficient km(for low solubility of species A in the fluid) can be estimated from a heat
transfer coefficient h already measured for the same geometry.
It is worth mentioning that for the limiting case of Pr=1, we see that from Eq.[4.1-26]
C fz
2

h
 C p
[4.1-32]
Which is known as the Reynolds analogy , in honor of Reynolds’ first
recognition of the analogous behavior of momentum and heat transfer in 1874.
10
4.1.4 Integral-Balance Equations
The integral-balance equations governing momentum, heat , and species
transfer, according to Eq. [1.4-3], [2.2-6], and [3.2-4], respectively, are as follows


t

 vd     (  vv  n ) dA     ndA 
A
A


( f b   p )d 
[4.1-33]
(momentum transfer)


t

 C v T d     (  vC v T )  ndA   q  ndA 
A
A


sd 
[4.1-34]
(heat transfer)


t

 w A     (  vw A )  ndA   j A  ndA 
A
A


rA d 
[4.1-35]
(species transfer)
11
In Eq. [4.1-33] the pressure term has been converted from a surface integral to
a volume integral using a Gauss divergence type theorem (i.e., Eq. [A.4-2]).
Furthermore, the body force fb and pressure gradient  p can be considered as
the rate of momentum generation due to these force. In Eq. [4.1-34] the kinetic
and potential energy, and the pressure, viscous, and shaft work are not included
since they are either negligible or irrelevant in most materials processing
problems. In Eq. [4.1-35] ρwA= ρA .
These integral balance equations share the same form as follows:
 rate of net
 R ate of
 

   inf low by
 accum ulation  
 convection
  rate of


 
  rate of
 other


 

generation


  net inf low 
 

[4.1-36]
or


t

     (  v )  ndA 
A

A
j  ndA 


s d 
[4.1-37]
12
These equations are summarized in Table 4.1-2. The following integral massbalance equation ,Eq.[1.2-4], is also included in the table:

t


 d Ω    (  v )  n d A
[4.1-38]
A
13
4.1.5 Overall Balance Equations
The overall balance equations for momentum, heat, and species transfer
according to Eqs.[1.4-9], [2.2-8], and [3.2-7], respectively, are as follows
dP
 ( m v ) in  ( m v ) out  Fv  ( F p  Fb )
dt
dE T
dt
dM
 ( m C v T ) in  ( m C v T ) out  Q  S
A
dt
 ( m w A ) in  ( m w A ) out  J A  R A
(momentum transfer)
[4.1.-39]
(heat transfer)
[4.1-40]
(species transfer)
[4.1-41]
These overall balance equations share the same form as follows
 R ate of
  rate of inflow   rate of outflow 




accum
ulation
by
convection

 
  by convection 
 rate of other net inflow
+
 from surroundings
or
  rate of

+
 

generation
 

[4.1-42]
14
d
dt
=(m  ) in -(m  ) out +J  +S 
[4.1-43]
Where the total momentum, thermal energy, or species A in the control
volume Ω is
 =    d 
[4.1-44]

In Eq.[4.1-39] the viscous force Fv at the wall can be considered as the rate
of momentum transfer through the wall by molecular diffusion. The pressure
force Fp and the body force Fb , on the other hand, can be considered as the
rate of momentum generation due to the action of these forces. In Eq.[4.1-40 ]
Q is by conduction, which is similar to diffusion.
The above equations are summarized in Table 4.1-3. The following overall
mass balance equation (i.e. Eq [1.2-6]), is also included in the table
dM
dt
= ( m ) in  ( m ) out
[4.1-45]
15
16
4.1.6 Differential Balance Equations
The differential balance equations governing momentum, heat, and species
transfer, according to Eqs. [1.5-6], [2.3-5] and [3.3-5], respectively, are as
follows:

(  v )=    (  vv )     + ( f b   p )
t

t

t
(momentum transfer)
[4.1-46]
(  C vT )     (  v C vT )    q  s
(heat transfer)
[4.1-47]
(  w A )     (  v w A )    j A  rA
(species transfer)
[4.1-48]
In Eq. [4.1-47] the viscous dissipation is neglected and in Eq. [4.1-48] ρwA =ρA
These differential balance equations share the same form as follows:
 R ate of
  rate of net inflow


accum
ulation

  by convection
or
  rate of other   rate of

+
+

net
inflow
generation
 
 

[4.1-49]
17

t
(  )    (  v  )    j  s
[4.1-50]
These equations are summarized in Table 4.1-4. The following equation of
continuity, Eq. [1.3-4], is also included in the table:

t
(  )    (  v )
[4.1-51]
Table 4.1-5 summarizes these equations for incompressible fluids.
18
19
Example
20