MASS TRANSFER  CLASS

NOTES FOR ENGINEERING

Before discussing convective mass

transfer, it is necessary to compare

convective heat transfer and convective

mass transfer since there is a similarity

between these two. In heat transfer,

there is convective heat transfer

coefficient ‘ h’. Similarly there is mass

transfer coefficient in mass transfer.

Prandtl number (Pr = ν / α) is replaced

by Schmidt number (Sc = ν / DAB) and

the Nusselt number (Nu = hL/kf) is

replaced by Sherwood number

(Sh = kc L / DAB) and T_f is mean film

temperature. Mass transfer may occur in a gas mixture.

Further mass transfer may occur in a liquid

solution. Also, mass transfer may occur in a

solid. The basic mechanisms of mass

transfer are the same whether the phase is

a gas, or a liquid, or a solid. Driving force

for mass transfer is concentration gradient

of species. Mass transfer due to

concentration gradient is diffusion which is

very analogous to heat transfer by

conduction.

Table: Mass transfer coefficients in Forced Convection Mass Transfer

Sr.  No.	Empirical equation	Object	Type of flow	local or average  temp	Range of dimensionless number
1.	Shx= 0.332 Rex1/2 Sc 1/3	Flat plate	Laminar	Local	0.6≤Sc ≤50
2.	Shav= 0.664 Rex1/2 Sc 1/3	Flat plate	Laminar	Average	0.6 ≤Sc ≤50
3.	Sh= 0.0296 Rex4/5 Sc 1/3	Flat plate	Turbulent	local	Rex≤105, 0.6 ≤Sc ≤50
4.	Sh= (0.037 Rex4/5 −871) Sc 1/3	Flat plate	Transition	Local	5×105< Rex ≤107, 0.6≤Sc ≤50
5.	Sh = 0.3+ [0.62ReD1/2Sc1/3 ×[1 + (0.4/Sc)2/3]-1/4] ×[1 + (ReD/282,000)5/8]4/5	Cylinder	Cross flow	Average	ReD Sc > 0.2
6.	Sh= 2 + (0.4 ReD1/2+ 0.060 ReD2/3) Sc0.4 ×(μ/μs)1/2	Sphere	Cross flow	Average	3.5 < ReD< 7.6×104, 0.71 < Sc < 380 and 1.0 < (μ/μs) < 3.2

For an ideal gas

p_AV =n_A RT

C_A=n_A / V Normally in gases, there is equimolar counter diffusion which gives J*_AZ = — D_AB dC_A/dz = — J*_BZ = –(–) D_BA dC_B/dz = D_BA dC_B/dz D_AB = D_BA For a binary gas mixture of A and B, the diffusivity coefficient D_AB=D_BA in counter molar diffusion of gases. Other parameters of equimolar counter diffusion are J*_AZ = — J*_BZ or J* is constant in a steady state dc_A = –dC_B C= C_A + C_B D_AB = D_BA

Mass transfer Coefficients

These empirical correlations are valid for low mass transfer rate (or equi-molar mass transfer) where the mole fraction of species ‘A’ is less than about 0.05. For higher mass transfer rate, corrected mass transfer coefficients, using the log mean concentration difference, must be used. Hence k_c is replaced by k_c/(1 − y_A)lm

Where lm is logarithmic mean and is given by

(1 − y_A)lm = [(1–y_A) –(1–y_Ai)] /[ ln((1–y_A) / (1–y_Ai))] y_A is gas mole fraction of species A = p_A / p_t y_Ai is gas mole fraction of species A at the interface = p_Ai / p_t p_A is the partial pressure of species A p_Ai is the partial pressure of species A at the interface

Mass transfer coefficients at Macro level

When a fluid flowing outside a solid surface in forced convection motion, rate of convective molar flux is given by:

N_A = k_c (c_L1— c_Li)

N_A=molar flux of species A = kmol/s m² k_c = mass transfer coefficient (m/s) c_L1 = bulk fluid concentration, kmol/m³ c_Li = concentration of fluid near the solid surface, k mol/m³ k_c depends on the following factors: 1. System geometry—pipe or flat plate or channel 2. Flow velocity—laminar, transition or turbulent flow 3. Fluid properties—density, viscosity, specific heat etc. From the empirical equation of Sherwood number applicable, first mass transfer coefficient is calculated. Only then, mass flux can be known. These calculations are very similar to convective heat transfer calculations i.e.

Convective heat flux = h Δt

Where h is the convective heat transfer coefficient

Δt is the temperature difference

MASS TRANSFER

Four terms frequently used in mass transfer 

1. Molar concentration or Mole concentration or 3.5 < ReD< 7.6×104,

          0.71 < Sc < 380

          1.0 < (μ/μs) < 3.2

or volume concentration,  ’ C_A’

ii. Mass concentration, ‘ ρ_A’

iii. Mole fraction ‘ x_A’

iv. Mass fraction ‘m_A*’

Definition of Mole concentration’ C_A’ of Species A in a mixture

Number of molecules of species A present per unit volume of the mixture (atoms/m³ or molecules/m³ or moles/m³ or kmol/m³)

Definition of mass concentration, ‘ρ_A’_, of Species A in a mixture

Mass concentration of species A = Mass of species A per unit volume of the mixture ( kg of species A / m³ mixture)

Definition of Mole fraction’ x_A’ of Species A in a mixture

It is a ratio of molar concentration of A to the total molar concentration

x_A  = C_A/C = C_A/(C_A + C_B)

It has no units.

Definition of mass fraction ‘m_A*’ of Species A in a mixture

It is a ratio of mass concentration of A to the total mass concentration

m_A* = ρ_A/( ρ_A+ ρ_B)

It has no unit.

Diffusion

It is the movement of particles in a medium from an area of high concentration to an area of low concentration, resulting in the uniform distribution of the substance.

Diffusion in gases >> diffusion in liquids>>diffusion in solids.

Types of Diffusion

(a) Concentration diffusion: Concentration diffusion is due to concentration gradient. Fick’s law gives a linear relation between the rate of diffusion of chemical species and the concentration gradient of that species. (b) Thermal diffusion: Diffusion due to a temperature gradient. Usually negligible unless the temperature gradient is very large. (c) Pressure diffusion: Diffusion due to a pressure gradient. Usually negligible unless the pressure gradient is very large.

Diffusion in solids Knudsen diffusion: Diffusion phenomena in porous solids is called Knudsen diffusion.

Cause Of Diffusion Concentration difference—Whenever there is concentration difference in a medium, there is mass transfer by natural flow from the high concentration to the low concentration region.

Practical Examples Of Mass Transfer (Diffusion)

1. There is a tank which is divided into two equal parts by a partition. Initially, the left half of the tank contains oxygen O2 gas while the right half contains N2 gas at the same temperature and pressure. As soon as the partition is removed, the O2 molecules will start diffusing into the N2 while the N2 molecules will diffuse into O2. Over the passage of time, there will have a homogeneous mixture of N2 and O2 in the tank. It is mass transfer due to concentration gradients either way.

2. A drop of red liquid dye when added to a cup of water, the dye molecules will diffuse slowly by molecular diffusion in all directions in the remaining water till a homogeneous color is established. 3. Water evaporates from an open pan into air because of the difference in concentration of water vapor at the water surface and the surrounding air. 4. Evaporation of water from an open tray 5. Mass transfer is there in any open thermodynamic system i.e. suction, compression and discharge by a compressor is mass transfer.

6. Mass transfer in extraction

7. Mass transfer in distillation

8. Mass transfer in absorption

9. Mass transfer in separation

10. Mass transfer in purification