Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell
4.10 Analogy and Differences in Different
Transport Phenomenon Processes
 Most early work in predicting theoretically
the heat and/or mass transfer in both
laminar and turbulent flow cases were
done using the analogy between
moment m heat
momentum,
heat, and mass and predicting
the approximate results for heat and/or
mass transfer coefficient from momentum
transfer or friction coefficient.
4.10 Analogy and Differences in Different Transport Phenomenon
Processes
Chapter 4: External Convective Heat
and Mass Transfer
1
Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell


Clearly
Cl
l there
th
are severe limitations
li it ti
iin using
i thi
this
simple approach, however, it is beneficial to
understand the advantages and similarities for
physical and mathematical modeling as well as
the constraints involving this approach.
We present this analogy for the classical
problem of heat and mass transfer over a flat
plate in this section
section. It’s
It s applications to more
coupled geometries and boundary conditions as
well as turbulent flow is not proven and caution
should
h ld b
be ttaken
k iin applying
l i thi
this approach
h tto
other cases.
4.10 Analogy and Differences in Different Transport Phenomenon
Processes
Chapter 4: External Convective Heat
and Mass Transfer
2
Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell

As presented before a flat plate at constant wall temperature Tw is exposed to
free stream of constant velocity U∞, temperature T∞ and mass fraction ω1,∞, due
to binary diffusion can be presented with the following dimensionless
conservation boundary layer equations and boundary conditions corresponding
to Figure 4.31.
y, v
U∞
T∞
ω1, w
x,u
Tw and ω1, w
Figure 4.31 Mass, momentum and heat transfer in laminar boundary layer
4.10 Analogy and Differences in Different Transport Phenomenon
Processes
Chapter 4: External Convective Heat
and Mass Transfer
3
Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell
Continuity
u  v 

0
x  y 
Momentum
(4.348)

u 
 2u 
 u
u
v

x 
y 
y 2

Energy
(4 349)
(4.349)

  2
 
u
v



x
y
Pr y 2
  2
 
 
(4.350)

S
Species
i
u
x 
v
y 

(4.351)
((4.352))
(4.353)
(4.354)
Sc x 2
at y+ = 0
u+ = 0 θ = Φ = 0
v   vw
y+ = 0
y+ = ∞
u+ = 1 θ = Φ = 1
where the dimensionless variables are defined as:
T  Tw
u
 


,
,
,
u 
 
1
T  Tw
U
v 
v
U
,
y
y  ,
L
1, w
1,
x 
x
L
,
1, w
Re 
4.10 Analogy and Differences in Different Transport Phenomenon
Processes
U L
(4.355)

Chapter 4: External Convective Heat
and Mass Transfer
4
Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell
Let’s first consider the analogy between momentum and
heat transfer. Equations (4.349) and (4.350) and
appropriate
pp p
boundary
y conditions are the same if Pr = 1.
Therefore the solutions for u+ and θ are exactly the same
if Pr = 1 and one expects to have a relation between
friction coefficient Cf and heat transfer coefficient h.
h
Cf
2

w
U 
2

u

y
U
y 0
2

k
Nu 
T
y

u 
 
y
y  0
U L

u 
y 
y 0
((4.356)
356)
Re
L
hL

y 0

 
k
k T  T  y
4.10 Analogy and Differences in Different Transport Phenomenon
Processes
(4.357)
y 0
Chapter 4: External Convective Heat
and Mass Transfer
5
Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell
Si
Since
θ = u+ for
f Pr
P = 1,
1 one also
l concludes
l d th
thatt

y 

y 0
u 
 
y

(4.358)
y 0
therefore combining (4.356) and (4.357) and using (4.358)
C f Nu

(4.359)
2
Re
This relation between the friction coefficient and Nu is referred to as
Reynolds analogy and is appropriate for Pr = 1. If Pr ≠ 1 we already
concluded that
T
 Pr 1/ 3 for 0.5 ≤ Pr ≤ 10 from the similarity solution

presented in Section 4
4.6.
6 Using this information
information, one can generali
generalize
e
the result of Reynolds analogy to Pr ≠ 1 by
C f Nu 1/ 3
(4.360)

Pr
2
R
Re
4.10 Analogy and Differences in Different Transport Phenomenon
Processes
Chapter 4: External Convective Heat
and Mass Transfer
6
Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell
Now ffocus th
N
the attention
tt ti tto similarities
i il iti b
between
t
h
heatt and
d mass ttransfer
f
or comparison of equation (4.350) and (4.351) with their appropriate
boundary conditions. It is clear that the solution for differential equation
(4.349) and (5.350) for θ and Φ are same if Sc and Pr are
interchanged appropriately.
We already know the solution for equation (4.350) for vw = 0 from
similarity solution for 0.5 ≤ Pr ≤ 10
Nu x  0.332 Re x1/ 2 Pr1/ 3
(4.361)
Therefore it can also be assumed that the solution of equation (4.351)
for vw  0
Shx  0.332 Re x1/ 2 Sc1/ 3
(4.362)
Combining equations (4.361) and (4.362) gives
Nu Pr1/ 3
 1/ 3
Sh Sc
4.10 Analogy and Differences in Different Transport Phenomenon
Processes
(4.363)
Chapter 4: External Convective Heat
and Mass Transfer
7
Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell
It should
h ld b
be noted
t d th
thatt th
the convective
ti effect
ff t due
d to
t vertical
ti l
velocity at the surface in predicting h and hm are neglected
and therefore the analogy presented in (4.363) is for very
low mass transfer
f at the wall. This can be easily seen
since the contribution of vw in heat flux and mass flux were
primarily due to diffusion.
T
q "  Cp vw Tw  T   k
y

1

m "    1,1  vw  D12

y

4.10 Analogy and Differences in Different Transport Phenomenon
Processes
(4.364)
y 0



y 0 
((4.365))
Chapter 4: External Convective Heat
and Mass Transfer
8
Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell



Analogy in momentum, heat, and mass transfer
can also be applied to complex, coupled transport
phenomenon problems including phase change
and chemical reactions.
Obviously in these circumstances,
Obviously,
circumstances the simple
relation developed in equation (4.363) is not
applicable.
pp
To show the appropriate usefulness of this
analogy
gy of transport
p p
phenomenon we will apply
pp y it
to sublimation with chemical reaction for forced
convection over a flat plate.
4.10 Analogy and Differences in Different Transport Phenomenon
Processes
Chapter 4: External Convective Heat
and Mass Transfer
9
Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell

During combustion involving a solid fuel, the solid
fuel may burn directly or it may be sublimated
before combustion
combustion. In the latter case – which will
be discussed in this subsection – gaseous fuel
diffuses away from the solid-vapor surface.
M
Meanwhile,
hil th
the gaseous oxidant
id t diff
diffuses ttoward
d
the solid-vapor interface. Under the right
conditions, the mass flux of vapor fuel and the
gaseous oxidant meet and the chemical reaction
occurs at a certain zone known as the flame. The
fl
flame
iis usually
ll a very thi
thin region
i with
ith a color
l
dictated by the temperature of combustion.
4.10 Analogy and Differences in Different Transport Phenomenon
Processes
Chapter 4: External Convective Heat
and Mass Transfer
10
Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell
Figure 4.32 Sublimation with chemical reaction (Kaviany, 2001).

Figure 4.32 shows the physical model of the problem under
consideration The concentration of the fuel is highest at the solid
consideration.
fuel surface, and decreases as the location of the flame is
approached. The gaseous fuel diffuses away from the solid fuel
surface and meets the oxidant as it flows parallel to the solid fuel
surface Combustion occurs in a thin reaction zone where
surface.
temperature is the highest, and the latent heat of sublimation is
supplied by combustion. The combustion of solid fuel through
sublimation can be modeled as a steady-state boundary layer type
flow with sublimation and chemical reaction
reaction.
4.10 Analogy and Differences in Different Transport Phenomenon
Processes
Chapter 4: External Convective Heat
and Mass Transfer
11
Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell

To model the problem, the following
assumptions are made:



The fuel is supplied by sublimation at a
steady rate.
The Lewis number is unity, so the thermal
and concentration boundaryy layers
y
have the
same thickness.
The buoyancy
y
y force is negligible.
g g
4.10 Analogy and Differences in Different Transport Phenomenon
Processes
Chapter 4: External Convective Heat
and Mass Transfer
12
Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell

The conservations of mass
mass, momentum
momentum, energy and species
of mass in the boundary layer are
 (  u )  (  v)
(4.366)

0
x
u
y
u
u   u 
v
 

x
y y  y 


  T
(  c p uT )  (  c p vT )   k
x
y
y  y
(4.367)

  m ohc ,o




 
(  uo )  (  vo )    D o
x
y
y 
y
(4.368)

  m o

(4.369)
(4
369)
where is rate of oxidant consumption (kg/m3-s). is the heat
released by combustion per unit mass consumption of the
oxidant (J/kg)
(J/kg), which is different from the combustion heat
defined in Chapter 2. is mass fraction of the oxidant in the
gaseous mixture.
4.10 Analogy and Differences in Different Transport Phenomenon
Processes
Chapter 4: External Convective Heat
and Mass Transfer
13
Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell

The corresponding boundary conditions of eqs.
(4.366) – (4.369) are
u  U  , T  T , o  o, at y   (4.370)
u  0, v 
m f

,
o
0
y
at y  0 (4.371)
 f is the rate of solid fuel sublimation per
where m
unit area (kg/m2-s) and ρ is the density of the
mixture
mixture.
4.10 Analogy and Differences in Different Transport Phenomenon
Processes
Chapter 4: External Convective Heat
and Mass Transfer
14
Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell

The shear stress at the solid fuel surface is
u
w   ,
y

(4.372)
The heat flux at the solid fuel surface is
T
qw  k
,
y

y0
y0
(4 373)
(4.373)
The exact solution of the heat and mass problem described by eqs.
((4.366)) – ((4.369)) can be obtained using
g conventional numerical
simulation, which is very complex. However, it is useful here to
introduce the results obtained by Kaviany (2001) using analogy
between momentum and heat transfer. Multiplying eq. (4.369) by hc,0
and adding
g the result to eq.
q ((4.368)) , one obtains
 


  T
u(cpT  o hc,o )  v(cpT  o hc,o )  k   Dhc,o o  (4.374)
x
y
y  y
y 
4.10 Analogy and Differences in Different Transport Phenomenon
Processes
Chapter 4: External Convective Heat
and Mass Transfer
15
Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell

Considering the assumption that Lewis number is unity,
i.e. Le   / D  1, eq. (4.374) can be rewritten as


  u (c pT  o hc ,o )     v(c pT  o hc ,o ) 
x
y

(4.375)

 


(
c
T


h
)
p
o c ,o 
y 
y

which can be viewed as an energy equation with quantity
c pT  o hc ,o as a dependent variable.
 Since o / y  0 at y  0 , i.e., the solid fuel surface is not
permeable for the oxidant, eq. (4.373) can be rewritten as

(4.376)
qw    (c pT  o hc ,o ), y  0
y
4.10 Analogy and Differences in Different Transport Phenomenon
Processes
Chapter 4: External Convective Heat
and Mass Transfer
16
Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell

Analogy between surface shear stress and the surface
energy flux yields 
qw 


(c pT  o hc ,o ) w  (c pT  o hc ,o ) 
u
w
w
c p (Tw  T )  hc ,o (o , w  o , ) 
u
(4.377)
The energy balance at the surface of the solid fuel is
(4.378)
 qw  m f hsv  q
where the two terms on the right-hand side of eq. (4.378)
represent the latent heat of sublimation, and the sensible
heat required to raise the surface temperature of the
solid fuel to sublimation temperature and heat loss to the
solid fuel.
4.10 Analogy and Differences in Different Transport Phenomenon
Processes
Chapter 4: External Convective Heat
and Mass Transfer
17
Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell
Combining eqs. (4.377) and (4.378) yields the rate of
sublimation
sub
a o o
on the
e so
solid
d fuel
ue su
surface
ace
w


mf  Z
(4.379)
U
where Z is transfer driving force or transfer number defined
as
c p (T  Tw )  hc ,o (o ,  o , w )
Z
(4.380)


hsgg  q / m f

 Using the friction coefficient gives
q ((4.379)) becomes
eq
m f 
Cf
2
Cf 
U  Z
4.10 Analogy and Differences in Different Transport Phenomenon
Processes
w
U / 2
2

(4 381)
(4.381)
((4.382))
Chapter 4: External Convective Heat
and Mass Transfer
18
Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell

The surface blowing velocity of the gaseous fuel is then
vw 
m f


Cf
2
U Z
(4 383)
(4.383)
where the friction coefficient Cf can be obtained from the
solution of boundary layer flow over a flat plate with
blowing on the surface. The similarity solution of the
b
boundary
d
llayer flflow problem
bl
exists
i t only
l if bl
blowing
i
1/ 2
velocity satisfies vw  x . In this case, one can define a
g parameter as
blowing
(  v) w 1/ 2
B
Re x
(  u )
4.10 Analogy and Differences in Different Transport Phenomenon
Processes
(4.384)
Chapter 4: External Convective Heat
and Mass Transfer
19
Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell

Combination of eqs. (4.383) and (4.384)
yields
Z 1/ 2
(4.385)
B  Re x C f
2

Glassman (1987) recommended an
empirical form
f
off eq. (4.385)
(
) based on
numerical and experimental results:
ln(1  Z )
(4.386)
B
0.15
2.6Z
4.10 Analogy and Differences in Different Transport Phenomenon
Processes
Chapter 4: External Convective Heat
and Mass Transfer
20
Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell
Example 4.4

Air with a temperature of 27 °C flows at 1 m/s over a 1-m
long solid fuel surface with a temperature of 727 °C. The
concentration of the oxidant at the solid fuel surface is
0.1, and the heat released per unit mass of the oxidant
consumed is 12000 kJ/kg. The latent heat of sublimation
for the solid fuel is 1500 kJ/kg. Neglect the sensible heat
required to raise the surface temperature of the solid fuel
to sublimation temperature, and heat loss to the solid
fuel. Estimate the average blowing velocity due to
sublimation on the fuel surface.
4.10 Analogy and Differences in Different Transport Phenomenon
Processes
Chapter 4: External Convective Heat
and Mass Transfer
21
Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell
Solution

The mass fractions of the oxygen at the solid
fuel surface and in the incoming air are,
respectively, 0,0 w  0.1
0 21. The specific
0 1 and o ,   0.21
heat of gas, approximately taken as the specific
heat of air at Tave=(Tw+T∞)/2=377 °C, is cp=1.063
kJ/kg-K The combustion heat per unit oxidant
kJ/kg-K.
consumed is hc ,o  12000kJ / kg. The latent heat of
sublimation is hsv  1500kJ / kg. The density at the
wall and the incoming temperatures are
are,
respectively,  w  0.3482 kg / m 3 and    1.1614 kg / m 3
. The viscosity at Tave is =60.21×10-6m2/s
4.10 Analogy and Differences in Different Transport Phenomenon
Processes
Chapter 4: External Convective Heat
and Mass Transfer
22
Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell

Th transfer
The
t
f driving
d i i fforce can be
b obtained
bt i d ffrom eq. , ii.e.,
Z
c p (T  Tw )  hc , o (o,   o, w )
hsv
1 063  (27  727)  12000  (0
1.063
(0.21
21  00.1)
1)
1500
 0.3839

The blowing parameter obtained from eq
eq. (4
(4.386)
386) is
B
ln(1  Z ) ln(1  0.3839)

 0.1443
2.6 Z 0.15 2.6  0.38390.15
Th blowing
The
bl i velocity
l it att th
the surface
f
is
i obtained
bt i d ffrom eq. (4
(4.383)
383)
vw 


1/ 2
BU  Re x1/ 2   B U   x 1/ 2
w
w
which can be integrated to yield the average blowing velocity
4.10 Analogy and Differences in Different Transport Phenomenon
Processes
Chapter 4: External Convective Heat
and Mass Transfer
23
Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell
vw 
2 
w
B U  L 
1/ 2
1/ 2
2  1.1614
6

 0.1443  1 60.21 10  1
0.3482
 0.007469
0 007469 m/s
/
4.10 Analogy and Differences in Different Transport Phenomenon
Processes
Chapter 4: External Convective Heat
and Mass Transfer
24