Transport Phenomena: Mass Transfer
Procedure
1) Mass balance over a thin shell
2) Obtain 1st ODE
3) Insert the relation between mass flux and concentration gradient
4) Result in 2nd ODE
5) Integration
6) Apply BCs to determine the constants (from integration)
7) Obtain concentration distribution
The law of conservation of mass of species A in a binary system is written over the volume of the shell in the form
Rate of Rate of rate of production of
Mass of - Mass of - mass of A by =0
A in A out homogeneous reaction
Mass flux = the number of moles of A that go through a unit area in unit time.
Combined molecular convective
flux flux flux (Diffusive flow) (Due to Bulk motion of A)
Boundary conditions
1) The concentration at surface can be specified; for example .
2) The mass flux at surface can be specified; for example, .
3) Solid surface substance A is lost to a surrounding stream
4) Rate of chemical reaction at the surface can be specified; for example,
Example 1 Diffusivity through a stagnant gas (gas-air)
Air flow
B
S: Cross sectional area
z2
z1 A Loop A z
0
Figure 1 Steady-state diffusion of A through stagnant B with the liquid vapor interface maintained at a fixed position.
Assumption
1) B does not dissolve in liquid A
2) A does not react to B
3) Temperature and pressure is constant
Mole A balance around shell
Divided by and taking limit we obtain
Mass diffusion in binary system
From Fick’s law;
* is the molar flux of component A in the z direction due to molecular diffusion.
* is the molecular diffusivity of the molecule A in B
Mass convection in binary system
From Binary system:
From molar relative fixed co-ordinate:
Set equation (3) = equation (4) we get
Since B is stagnant,
We obtain the molar flux of A with
Substitute equation (5) into equation (1) we get
Divided through the equation by we obtain
Integration with respect to z gives
Second integration then gives
If we replace by and by , the above equation becomes
and may then be determined from the boundary conditions
BC1:
BC2:
Apply BC1,
Apply BC2,
Divided equation (7) by equation (8) gives
Substitute into equation (7) we obtain
Substitute and into the equation (6) we get
The profile of gas A are
The profile of gas B are obtained by using