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Thermal and Concentration Boundary Layers
Thermal and concentration boundary layers are very similar to the velocity boundary
layers discussed before, except we focus our attention to the temperature and
concentration profiles instead of velocity profiles. Instead of the growth of velocity
from zero to a free stream value, thermal or concentration boundary layers track
the changes in temperature decay or concentration decay. We may discuss all these
boundary layers by using similarity principles as introduced in the next section. Consider
a flow of polluted river water that is brought into a tank for purification. As the flow
enters the bed it may be considered a flow over a flat plate. We would pan heat into the
water to kill germs and apply other methods of pollutant control. The velocity profile,
temperature profile, and concentration profile on the flat plate are all shown in the figure
below:
y
vel(x
)
CA(y)
conc(x)
y
T(y)
u(y)
vel
u
Flat plate adds heat to flow
con
tem
temp(x)
c
p
x
TS
C AS
In this diagram the velocity boundary layer is shown to grow with distance x [marked by
vel(x)], where as the growth of the temperature and concentration profiles are marked as
temp(x) and conc(x). Depending on the transport characteristics of velocity, vorticity,
temperature, and concentration these boundary layers may grow at different rates. When
we speak about velocity changes in the boundary layer the fluid property that influences
them is viscosity, whereas for temperature and concentration boundary layers, the
corresponding properties are the convective heat transfer and mass transfer coefficients.
The governing equations for velocity boundary layers, thermal boundary layers, and
concentration boundary layers all follow similar patterns. Rather than deriving the
thermal and concentration boundary layer equation we simply present them below. For a
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fluid such as air that may be treated as an ideal, incompressible gas or, for an
incompressible liquid such as water,
T    T    T 
 T
 v    k    k     q
y  x  x  y  y 
 x
C  u
p
(A)
and,
u
C
C
C

v
 D
x
y x 
x
A
A
A
AB
C  
 
 N
 D
y 
 y 
(B)
A
AB
A
In the first equation k represents the thermal conductivity of a homogeneous solid,
q represents the rate of heat generation per unit volume and  represents the rate of
viscous dissipation per unit volume, given by:
2

 u 2  v  2  2  u v  2 
u v 
 

       2          
(C)
 y x 
y   3  x y  
 x 






Similarly in the equation (B), DAB represents the binary diffusion coefficients and
N A represents the rate of generation of the concentration CA. In deriving the above
relations some additional constitutive relations must be recalled. For example if the fluid
is an ideal gas, the gas law gives:
p  RT or, p  CA RT
R
where, R = Specific gas constant =
MA
R = Universal gas constant
MA = Molecular weight [kg/kmole] of gas, A.
[Q   M A  C A ]
Fourier’s law of heat conduction
T
Heat flux, q   k
in the y-direction where, k = Thermal conductivity of
y y  0
the wall. But heat convected into the fluid is given by the Newton’s law of cooling:
q  h(TS  T )
where, h = heat transfer coefficient (or, coefficient of heat convection)
TS = Surface temperature = T( y) y  0
T∞ = Temperature of the ambient fluid
Thus,
Q  qA =Rate of heat flow into fluid
S
  kA S
T
 h (TS  T )
y y  0
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where, AS = surface area through which heat flows. Therefore the heat transfer
coefficient may be expressed as
T
k
y y  0
h
(D)
TS  T
Similar to the heat transfer case the mass transfer constitutive relations are given by
 as
Fick’s law, which specifies molar flux, NA
C A
y y  0
where, DAB = Binary diffusion coefficient
   D AB
N A
But the molar flux coefficient may also be expressed as
  h m (C A , S  C A ,  )
N A
where hm = convective mass transfer coefficient
CA,S = Concentration of A at the surface
CA,∞ = Concentration of A in the ambient fluid
Therefore the convective mass transfer coefficient may be expressed as
C
 D AB A
y y  0
hm 
(E)
C A, S  C A, 
Remember that h and hm are variables defined by the above laws. For a finite size flat
plate we may define (similar to the overall skin friction coefficient, C f discussed in the
the velocity boundary layers)
1
1
h
and, h m 
 h m dA S
 h dA S
AS A
AS A
S
S
 may be related to the molar flux, NA
 yielding
The mean flux, n A
  M A N A
  h m ( A, S   A,  )
n A
where, MA = Molecular weight of A
and, A,S = M A  C A, S , etc
The above law shows striking similarity between the velocity boundary layer, thermal
boundary layer, and the concentration boundary layer.
Similarity Rules of Boundary Layers
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If you recall the work related to Prandtl’s analysis in the velocity boundary layer was
derived starting from a non-dimensionalization of the governing equations. The critical
parameter to analyze the velocity boundary layer was the Reynolds number. Similar
relations may be derived in cases of thermal boundary layer and concentration boundary
layer. We shall omit the derivations here. However the set of critical parameters resulting
from these operations must be noted carefully. For engineers, design solutions are
influenced by these numbers encountered everyday. A thorough understanding of these
numbers and their physical significance are essential. Only the non-dimensional numbers
relevant to this course are presented below.
Non-dimensional #
Expression
Physical Meaning
Reynolds No. (ReL)
UL

Inerta force
Viscous force
Prandtl No. (Pr)
Biot Number (Bi)
Mass Transfer Biot Number (Bim)
Schmidt Number (Sc)
Sherwood Number (Sh)
Nusselt Number (NuL)
C p
R
hL
R
Viscous Dissipatio n
Thermal Diffusion
Conductive Re sis tan ce
Convective Re sis tan ce
Conductive Re sis tan ce
hmL
Convective Re sis tan ce (in mass transfer )
R
Viscous Diffusion

Mean Diffusion
D AB
hmL
Convective Mass Transfer
Surface Mass Transfer
D AB
Fluid ' s Conductive Re sis tan ce
hL
Convective Re sis tan ce
Rf
By the use of the expressions (D) and (E) before, Sh and NuL may be also seen as the
non-dimensional concentration gradient and non-dimensional temperature gradient
respectively. The non-dimensional velocity, temperature, and concentration problems
may be summarized in functional forms as
u* = f1 (x*, y*, ReL, dp*/dx)
T* = f2 (x*, y*, ReL, Pr, dp*/dx)
and,
CA = f3 (x*, y*, ReL, Sc, dp*/dx)
These equations are solved to yield
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2 u *
|
Re y *
Nu = hL
= T *
|
k
y *
2/16/2016
 Cf = f4 (x*, ReL)
Cf =
y * 0
L
 Nu = f5 (x*, ReL, Pr) or, Nu = f6 (ReL, Pr)
y * 0
and,
= C
|
y
Sh = h L
D
 Sh = f7 (x*, ReL, Sc) or, Sh = f3 (ReL, Sc)
*
A
m
*
AB
y * 0
In other words, when we wish to solve the above problems in practice, we match the
corresponding non-dimensional numbers in parenthesis to solve for the desired physical
variables. In some problems of complex physics it is important to know the relationships
connecting the above non-dimensional parameters. For example, Stanton number (St),
defined as
St =
h
Vc
p
= Nu
Re Pr
Similarly, Stm = h
V
m
= Sh
Re Sc
The fact that Cf /2 = St = Stm is known as the Reynolds Analogy.
This can be applied only if Pr and Sc  1. For wider ranges of these parameters, we use
the modified Reynolds or, Chilton-Colburn analogies
Cf /2 = St . Pr 2/3 = jH ,
0.6 < Pr < 60, and,
Cf /2 = Stm . Pr 2/3 = jm , 0.6 < Pr < 3000,
where, jH and jm are known as the Colburn j – factors for heat and mass transfer.
The problems associated with these areas will now be illustrated.
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