thermal and concentration bl

7/27/2019 Thermal and Concentration BL

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Thermal and Concentration Boundary Layers

Thermal and concentration boundary layers are very similar to thevelocity boundary

layers discussed before, except we focus our attention to the temperature andconcentration 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 flowenters 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 figurebelow:

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

y

x

vel

tem

p

conc

vel

(x

)

temp

(x)

conc

(x)

u

y

u(y)T(y) C

A(y)

SAC

STFlat plate adds heat to flow
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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 thethermal and concentration boundary layer equation we simply present them below. For a

fluid such as air that may be treated as an ideal, incompressible gas or, for an

incompressible liquid such as water,q

y

Tk

yx

Tk

xy

Tv

x

TuC

p++

+

=

+

(A)

and,

A

A

AB

A

AB

AA Ny

CD

yx

CD

xy

Cv

x

Cu +

+

=

+

(B)

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:

+

++ +=

2222

yv

xu

32

yv

xu2

xv

yu (C)

Similarly in the equation (B), DAB represents the binary diffusion coefficients and AN

represents the rate of generation of the concentration CA. In deriving the above relationssome additional constitutive relations must be recalled. For example if the fluid is an

ideal gas, the gas law gives:TRp = or, RTCp A=

where, R = Specific gas constant =AM

R

R = Universal gas constant

MA = Molecular weight [kg/kmole] of gas, A. [ AA CM =Q ]

Fouriers law of heat conduction

Heat flux,0y

y

Tkq

=

= in the y-direction where, k = Thermal conductivity of

the wall. But heat convected into the fluid is given by the Newtons law of cooling:

)TT(hq S =

where, h = heat transfer coefficient (or, coefficient of heat convection)

TS = Surface temperature = 0y)y(T =

T = Temperature of the ambient fluid

Thus,S

AqQ = =Rate of heat flow into fluid

0y

Sy

TkA

=

= )TT(h S =

where, AS = surface area through which heat flows. Therefore the heat transfercoefficient may be expressed as


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=

=TT

y

Tk

hS

0y (D)

Similar to the heat transfer case the mass transfer constitutive relations are given by

Ficks law, which specifies molar flux, AN as

0y

AABA

y

CDN

=

=

where, DAB = Binary diffusion coefficient

But the molar flux coefficient may also be expressed as)CC(hN ,AS,AmA =

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

=

=,AS,A

0y

AAB

mCC

y

CD

h(E)

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, fC discussed in the

the velocity boundary layers)

S

SAS

dAhA

1h = and, S

SAm

Sm dAh

A

1h =

The mean flux, An may be related to the molar flux, AN yielding)(hNMn ,AS,AmAAA ==

where, MA = Molecular weight of A

and, A,S = S,AACM

, 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

If you recall the work related to Prandtls analysis in the velocity boundary layer wasderived 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 boundarylayer. We shall omit the derivations here. However the set of critical parameters resulting
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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 numbersrelevant to this course are presented below.

Non-dimensional # Expression Physical Meaning

Reynolds No. (ReL)

UL

forceViscous

forceInerta

Prandtl No. (Pr)R

Cp

DiffusionThermal

nDissipatioViscous

Biot Number (Bi)R

hL

cetansisReConvective

cetansisReConductive

Mass Transfer Biot Number (Bim) R

Lhm

)transfermassin(cetansisReConvective

cetansisReConductive

Schmidt Number (Sc)ABD

DiffusionMean

DiffusionViscous

Sherwood Number (Sh)AB

m

D

Lh

TransferMassSurface

TransferMassConvective

Nusselt Number (NuL)fR

hL

cetansisReConvective

cetansisReConductives'Fluid

By the use of the expressions (D) and (E) before, Sh and NuL may be also seen as thenon-dimensional concentration gradient and non-dimensional temperature gradientrespectively. 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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Cf =0*

|*

*

Re

2=

y

Ly

u Cf = f4 (x*, ReL)

Nu =k

hL=

0*|

**

=

y

yT Nu = f5 (x*, ReL, Pr) or, Nu = f6 (ReL, Pr)

and,

Sh =AB

m

D

Lh= 0**

*

|=

y

A

y

C Sh = f7 (x*, ReL, Sc) or, Sh = f3 (ReL, Sc)

In other words, when we wish to solve the above problems in practice, we match thecorresponding 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 =p

Vc

h

=

PrRe

Nu

Similarly, Stm =V

hm =

Sc

Sh

Re

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 . Pr2/3 = jH , 0.6 < Pr < 60, and,

Cf /2 = Stm . Pr2/3 = 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. Continue
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