1 DIFFUSIVE FLUX, HEAT & CONTAMINANT CONSERVATION Molecular diffusion is a process by which random...

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DIFFUSIVE FLUX, HEAT & CONTAMINANT CONSERVATION

Molecular diffusion is a process by which random molecular motion moves any quantity down the concentration gradient, i.e. from high concentration to low concentration. Diffusion does not require flow, but it operates in the presence of flow.

Consider the illustrated container of water. A dilute concentration of dye (molecules) is placed in the lower half of the container.

In time, molecular action cause the dye-free fluid to mix with the dye-laden fluid, so that the concentration eventually becomes uniform.

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3





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In the case below the dye is diffusing in the x3 direction. Let c denote the concentration of dye. Note that c is a decreasing function of x3, so that

The diffusive flux of dye in the vertical direction is from high concentration to low concentration, which happens to be upward in this case.

The simplest assumption we can make for diffusion is the linear Fickian form: where FD,con,3 denotes the diffusive flux of contaminant (in this case dye) in the x3 direction,

c

x3

0x

c

3

3c3,con,D x

cDF

where Dc denotes the kinematic molecular diffusivity of the contaminant.

FD,con,3


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The units of c are quantity/volume. For example, in the case of dissolved salt this would be kg/m3, and in the case of heat it would be joules/m3.

The units of Dc are thus

These units happen to be the same as those of the kinematic viscosity of the fluid, i.e. .

c

x3

In the case of heat, Dc is denoted as Dh and FD,con,3 is denoted as FD,heat, 3.

FD,con,3

The units of FD,con,3 should be quantity (crossing)/face area/time. In the case of dissolved salt, this would be kg/m2/s, and in the case of heat it would be joules/m2/s.

T

LL

Lquantity

TLquantity

]C[

L]F[]D[

2

3

2i,con,D

c


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The 3D generalization of the Fickian forms for diffusivity are

ici,con,D x

cDF

where c is the concentration of the contaminant (quantity/volume).

The concentration of heat per unit volume (Joules/m3) is given as cp. Thus

iiph

i

phi,heat,D x

kx

cDx

cDF

where k = cpDh denotes the thermal conductivity.

The dimensionless Prandtl number Pr and Schmidt number Sc are defined as

ch D,

D

ScPr

This comparison is particularly useful because we will later identify the kinematic viscosity with the kinematic diffusivity of momentum.


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Some numbers for heatHeat in air

kg/m3 J/kg/K N s/m2 m2/s J/s/m/K m2/s

C K cp k Dh Pr

-23.15 250 1.4131.005E+0

31.599E-

051.132E-

052.227E-

021.568E-

057.216E-

01

1.85 275 1.2351.006E+0

31.726E-

051.398E-

052.428E-

021.954E-

057.151E-

01

26.85 300 1.1171.005E+0

31.846E-

051.653E-

052.624E-

022.337E-

057.070E-

01

51.85 325 1.0861.008E+0

31.963E-

051.808E-

052.815E-

022.572E-

057.029E-

01Heat in water

kg/m3 J/kg/K N s/m2 m2/s J/s/m/K m2/s

C K cp k Dh Pr

0273.1

59.998E+0

24.209E+0

31.753E-

031.753E-

065.687E-

011.351E-

071.297E+0

1

10283.1

59.997E+0

24.194E+0

31.300E-

031.300E-

065.869E-

011.400E-

079.286E+0

0

20293.1

59.982E+0

24.184E+0

31.002E-

031.004E-

066.034E-

011.445E-

076.948E+0

0

40313.1

59.922E+0

24.177E+0

36.517E-

046.568E-

076.351E-

011.532E-

074.286E+0

0In the above relations denotes the dynamic viscosity of water.


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Some values of Dc and Dh are given as follows.

gases and vapors in air at 25C

= 1.54e-5 m2/s

m2/s

substance Dc Sc

H2 7.12E-05 0.216

CO2 1.64E-05 0.940

Ethyl alcohol 1.19E-05 1.290

Benzene 8.80E-06 1.750

dissolved solutes in water at 20C

= 1.004e-6 m2/s

m2/s

substance Dc Sc

H2

5.13E-09

1.957E+02

O2

1.80E-09

5.577E+02

CO2

1.77E-09

5.671E+02

N2

1.64E-09

6.121E+02

NaCl1.35E-

097.436E+0

2

Glycerol7.20E-

101.394E+0

3

Sucrose4.50E-

102.231E+0

3


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Consider a control volume that is fixed in space, through which fluid can freely flow in and out. In words, the equation of conservation of contaminant is:

/t(quantity of contaminant in control volume) = net inflow rate of contaminant in control volume + Net rate of production of contaminant in control volume

Contaminant concentration is denoted as c (quantity/volume). Contaminant can be produced internally by e.g. a chemical reaction (that produces heat or some some species of molecule). Let S denote the rate of production of contaminant per unit volume per unit time (quantity/m3/s). Where S is negative it represents a sink (loss rate) rather then source (gain rate) of contaminant.

The net inflow rate includes both convective and diffusive flux terms. Translating words into an equation,

dA

ni

VS

ii,con,Di,con,C

V

SdVdAnFFcdVt


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But by the divergence theorem

VS

ii,con,Di,con,C

V

SdVdAnFFcdVt

V

i,con,Di,con,CiS

ii,con,Di,con,C dVFFx

dAnFF

Thus the conservation equation becomes

0dVSFFxt

c

V

i,con,Di,con,Ci

or since the volume is arbitrary,

SFFxt

ci,con,Di,con,C

i


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SFFxt

ci,con,Di,con,C

i

Now

ici,con,Dii,con,C x

cDF,cuF

So the conservation equation reduces to a convection-diffusion equation with a source term:

Sxx

cD

x

cu

t

c

ii

2

ci

i

If the fluid is incompressible, i.e. ui/xi = 0, the relation reduces to

Sxx

cD

x

cu

t

c

ii

2

ci

i


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Special case of heat, for which c cp and Dc Dh, S Sh

hii

2

hpi

ip Sxx

Dcx

ut

c

or thus

p

h

ii

2

hi

i c

S

xxD

xu

t


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