MASS TRANSFER – 4
CONVECTIVE MASS TRANSFER AND ITS EMPIRICAL CORRELATIONS
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.
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. | Sh_{x}= 0.332 Re_{x}^{1/2} Sc ^{1/3} | Flat plate | Laminar | Local | 0.6≤Sc ≤50 |
2. | Sh_{av}= 0.664 Re_{x}^{1/2} Sc ^{1/3} | Flat plate | Laminar | Average | 0.6 ≤Sc ≤50 |
3. | Sh= 0.0296 Re_{x}^{4/5} Sc ^{1/3} | Flat plate | Turbulent | local | Re_{x}≤105, 0.6 ≤Sc ≤50 |
4. | Sh= (0.037 Re_{x}^{4/5} −871) Sc ^{1/3} | Flat plate | Transition | Local | 5×10^{5}< Re_{x} ≤10^{7}, 0.6≤Sc ≤50 |
5. | Sh = 0.3+ [0.62Re_{D}^{1/2}Sc^{1/3} ×[1 + (0.4/Sc)^{2/3}]^{-1/4}] ×[1 + (Re_{D}/282,000)^{5/8}]^{4/5} | Cylinder | Cross flow | Average | Re_{D }Sc > 0.2 |
6. | Sh= 2 + (0.4 Re_{D}^{1/2}+ 0.060 Re_{D}^{2/3}) Sc^{0.4} ×(μ/μ_{s})^{1/2} | Sphere | Cross flow | Average |
3.5 < Re_{D}< 7.6×10^{4}, 0.71 < Sc < 380 and 1.0 < (μ/μs) < 3.2 |
For an ideal gas
p_{A}V =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 m^{2}
k_{c} = mass transfer coefficient (m/s)
c_{L1} = bulk fluid concentration, kmol/m^{3}
c_{Li} = concentration of fluid near the solid surface, k mol/m^{3}
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