r. shanthini 24 may 2010 content of lectures 7 to 9: mass transfer: concept and theory pm3125:...
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R. Shanthini 24 May 2010
Content of Lectures 7 to 9:
Mass transfer: concept and theory
PM3125: Lectures 7 to 9
R. Shanthini 24 May 2010
Mass transfer occurs when a component in a mixture goes from one point to another.
Mass Transfer
Mass transfer can occur by either diffusion or convection.
Diffusion is the mass transfer in a stationary solid or fluid under a concentration gradient.
Convection is the mass transfer between a boundary surface and a moving fluid or between
relatively immiscible moving fluids.
R. Shanthini 24 May 2010
Stirring the water with a spoon creates forced convection.
That helps the sugar molecules to transfer to the bulk water much faster.
Mass transfer can occur by either diffusion or by convection.
Diffusion(slower)
Example of Mass Transfer
R. Shanthini 24 May 2010
Stirring the water with a spoon creates forced convection.
That helps the sugar molecules to transfer to the bulk water much faster.
Mass transfer can occur by either diffusion or by convection.
Diffusion(slower)
Convection(faster)
Example of Mass Transfer
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Example of Mass TransferAt the surface of the lung:
Air Blood
Oxygen
Carbon dioxide
High oxygen concentrationLow carbon dioxide concentration
Low oxygen concentrationHigh carbon dioxide concentration
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Diffusion (also known as molecular diffusion)is a net transport of molecules
from a region of higher concentrationto a region of lower concentration
by random molecular motion.
Diffusion
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A B
A B
Liquids A and B are separated from each other.
Separation removed.
A goes from high concentration of A to low concentration of A. B goes from high concentration of B to low concentration of B.
Molecules of A and B are uniformly distributed everywhere in the vessel purely due to the DIFFUSION.
Diffusion
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• Scale of mixing: Mixing on a molecular scale relies on diffusion as the final step in mixing process because of the smallest eddy size
• Solid-phase reaction: The only mechanism for intra particle mass transfer is molecular diffusion
• Mass transfer across a phase boundary:Oxygen transfer from gas bubble to fermentation broth; Penicillin recovery from aqueous to organic liquid
Examples of Diffusion
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Fick’s Law of Diffusion
JA
CA
CA + ΔCA
Δx
JA = DAB
ΔCA
Δx
A & B
JA = -DAB
ΔCA
Δx
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JA = -DAB
ΔCA
Δx
Fick’s Law of Diffusion
diffusion coefficient (or diffusivity) of A in B
What is the unit of diffusivity?
concentration gradient (mass per volume per
distance)
diffusion flux of A in relation to the bulk motion in x-direction
(mass per area per time)
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= -kΔTΔx
Fourier’s Law of Heat Conduction
Thermal conductivity
Temperature gradient (temperature per
distance)
Heat flux (Energy per area per time)
Q.
A
Describe the similarities between
Fick’s Law and Fourier’s Law
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For ions (dissolved matter) in dilute aqueous solution at room temperature:
D ≈ 0.6 to 2 x10-9 m2/s
For biological molecules in dilute aqueous solution at room temperature:
D ≈ 10-11 to 10-10 m2/s
For gases in air at 1 atm and at room temperature:
D ≈ 10-6 to x10-5 m2/s
Diffusivity
Diffusivity depends on the type of solute, type of solvent, temperature, pressure, solution phase (gas, liquid or solid) and other characteristics.
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Prediction of Binary Gas Diffusivity
DAB - diffusivity in cm2/s
P - absolute pressure in atmMi - molecular weight
T - temperature in KVi - sum of the diffusion volume for component i
DAB is proportional to 1/P and T1.75
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Prediction of Diffusivity in Liquids
DAB - diffusivity in cm2/s
T - temperature in Kμ - viscosity of solution in kg/m sVA - solute molar volume at its normal boiling point
in m3/kmol
DAB is proportional to 1/μ and T
DAB =9.96 x 10-12 T
μ VA1/3
For very large spherical molecules (A) of 1000 molecular weight or greater diffusing in a liquid solvent (B) of small molecules:
applicable for biological
solutes such as proteins
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Prediction of Diffusivity in Liquids
DAB - diffusivity in cm2/s
MB - molecular weight of solvent B
T - temperature in Kμ - viscosity of solvent B in kg/m sVA - solute molar volume at its normal boiling point in m3/kmol
Φ - association parameter of the solvent, which 2.6 for water, 1.9 for methanol, 1.5 for ethanol, and so on
DAB is proportional to 1/μB and T
DAB =1.173 x 10-12 (Φ MB)1/2 T
μB VA0.6
For smaller molecules (A) diffusing in a dilute liquid solution of solvent (B):
applicable for biological solutes
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Prediction of Diffusivity of Electrolytes in Liquids
DoAB is diffusivity in cm2/s
n+ is the valence of cation
n- is the valence of anion
λ+ and λ- are the limiting ionic conductances in very dilute
solutions T is 298.2 when using the above at 25oC
DAB is proportional to T
DoAB =
8.928 x 10-10 T (1/n+ + 1/n-)
For smaller molecules (A) diffusing in a dilute liquid solution of solvent (B):
(1/λ+ + 1/ λ-)
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JA = - DAB
∆CA
∆x
Fick’s First Law of Diffusion (again)
JA is the diffusion flux of A in relation to the bulk motion in x-direction
where ED is the eddy diffusivity, and is dependent on the flow pattern
If circulating currents or eddies are present (which will always be present), then
NA = - (D + ED)∆CA
∆x
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Microscopic (or Fick’s Law) approach:
JA = - D∆CA
∆x
Macroscopic (or mass transfer coefficient) approach:
NA = - k ΔCA
where k is known as the mass transfer coefficient
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NA = - k ΔCA
is used when the mass transfer is caused by molecular diffusion plus other mechanisms such as convection.
Macroscopic (or mass transfer coefficient) approach:
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NA = - k ΔCA
mass transfer coefficient
concentration difference
(mass per volume)
net mass flux of A (mass per area per time)
What is the unit of k?
Macroscopic (or mass transfer coefficient) approach:
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Newton’s Law of Cooling in
Convective Heat Transfer
Q.
conv. = h (Tsurface – Tfluid)
Heat transfer coefficient
Heated surface at Tsurface
Flowing fluid at Tfluid
A
temperature difference
convective heat flux (energy per area per time)
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Describe the similarities between the convective heat transfer equation and the macroscopic approach to mass transfer.
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NA -k ΔCA=
NA
CA1
CA2
A & B
k (CA1 – C A2 )=
Macroscopic (or mass transfer coefficient) approach:
R. Shanthini 24 May 2010
NA -k ΔCA=
NA
CA1
CA2
A & B
k (CA1 – C A2 )=
CA1 = PA1 / RT
CA2 = PA2 / RT
Macroscopic (or mass transfer coefficient) approach:
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NA k (PA1 – P A2 ) / R T=
NA
PA1
PA2
A & B
Macroscopic (or mass transfer coefficient) approach:
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Other Driving Forces
Mass transfer is driven by concentration gradient as well as by pressure gradient as we have just seen.
In pharmaceutical sciences, we also must consider mass transfer driven by electric potential gradient (as in the transport of ions) and temperature gradient.
Transport Processes in Pharmaceutical Systems (Drugs and the Pharmaceutical Sciences, vol. 102),
edited by G.L. Amidon, P.I. Lee, and E.M. Topp(Nov 1999)
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Oxygen transfer from gas bubble to cell1. Transfer from the interior of the bubble to the gas-liquid interface
2. Movement across the gas film at the gas-liquid interface
3. Diffusion through the relatively stagnant liquid film surrounding the bubble
4. Transport through the bulk liquid
5. Diffusion through the relatively stagnant liquid film surrounding the cells
6. Movement across the liquid-cell interface
7. If the cells are in floc, clump or solid particle, diffusion through the solid of the individual cell
8. Transport through the cytoplasm to the site of reaction.
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1. Transfer through the bulk phase in the bubble is relatively fast2. The gas-liquid interface itself contributes negligible resistance3. The liquid film around the bubble is a major resistance to oxygen transfer4. In a well mixed fermenter, concentration gradients in the bulk liquid are
minimized and mass transfer resistance in this region is small, except for viscous liquid.
5. The size of single cell <<< gas bubble, thus the liquid film around cell is thinner than that around the bubble. The mass transfer resistance is negligible, except the cells form large clumps.
6. Resistance at the cell-liquid interface is generally neglected7. The mass transfer resistance is small, except the cells form large clumps or flocs.8. Intracellular oxygen transfer resistance is negligible because of the small
distance involved
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Interfacial Mass Transfer
air-waterinterface
air
water
Cw = concentration of solute in water
Ca = concentration of solute in airPa = partial pressure of solute in air
volatilization
absorption
Transport of a volatile chemical across the air/water interface.
Pa = Ca RT
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Interfacial Mass Transfer
air-waterinterface
air
water
δa and δw are boundary layer zones offering much resistance to mass transfer.
δa
δw
Cw,i
Pa,i
Pa = partial pressure of solute in air
Cw = concentration of solute in water
Pa,i vs Cw,i?
R. Shanthini 24 May 2010
Interfacial Mass Transfer
air-waterinterface
air
water
δa
δw
Cw,i
Pa,i
Pa
Cw
Henry’s Law:Pa,i = H Cw,i at equilibrium, where H is Henry’s constant
δa and δw are boundary layer zones offering much resistance to mass transfer.
R. Shanthini 24 May 2010
Henry’s Law
Pa,i = H Cw,i at equilibrium, where H is Henry’s constant
Unit of H = [Pressure]/[concentration]= bar / (kg.m3)
Pa,i = Ca,i RT is the ideal gas equation
Therefore, Ca,i = (H/RT) Cw,i at equilibrium,
where (H/RT) is known as the dimensionless Henry’s constant
H depends on the solute, solvent and temperature
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Gas-Liquid Equilibrium Partitioning Curve
Cw,i
Pa,i
Pa
Cw
P*a
C*w Cw
Pa
Pa,i = H Cw,i
P*a = H’ Cw
Pa = H’’ C*w
H = H’ = H’’if the partitioning
curve is linear
R. Shanthini 24 May 2010
Interfacial Mass Transfer
air-waterinterface
air
water
δa
δw
Cw,i
Pa,i
Pa
CwP*a
C*w
NA = KG (Pa – Pa,i)
KG = gas phase mass transfer coefficient
NA = KL (Cw,i – Cw)
KL = liquid phase mass transfer coefficient
R. Shanthini 24 May 2010
Interfacial Mass Transfer
air-waterinterface
air
water
δa
δw
Cw,i
Pa,i
Cw
C*w
KOG = overall gas phase mass transfer coefficient
KOL = overall liquid phase mass transfer coefficient
NA = KG (Pa – Pa,i)
NA = KL (Cw,i – Cw)
NA = KOG (Pa – P*a)
NA = KOL (C*w – Cw)P*
a
Pa
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Interfacial Mass Transfer
Cw,i
Pa,i
KOG = overall gas phase mass transfer coefficient
KOL = overall liquid phase mass transfer coefficient
NA = KG (Pa – Pa,i)
NA = KL (Cw,i – Cw)
= KOG (Pa – P*a)
= KOL (C*w – Cw)
KG = gas phase mass transfer coefficient
KL = liquid phase mass transfer coefficient
C*w
Pa
CwP*a
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Relating KOL to KL
Pa - Pa,i = H (C*w - Cw,i)
C*w – Cw = C*w – Cw,i + Cw,i – Cw
NA / KOL = C*w – Cw,i + NA /KL
If the equilibrium partitioning curve is linear over the concentration range C*w to Cw,i, then
NA / KG = H (C*w – Cw,i)
(1)
(2)
Combining (1) and (2), we get
1
KOL
1
H KG
= +1
KL
R. Shanthini 24 May 2010
Relating KOG to KG
Pa,i – P*a = H (Cw,i – Cw)
Pa - P*a = Pa – Pa,i + Pa,i – P*a
NA / KOG = NA /KG + Pa,i – P*a
If the equilibrium partitioning curve is linear over the concentration range Pa,i to P*a then
(3)
(4)
Combining (3) and (4), we get
1
KOG
1
KG
= +H
KL
Pa,i – P*a = H NA / KL
R. Shanthini 24 May 2010
Summary: Interfacial Mass Transfer
NA = KG (Pa – Pa,i)
NA = KL (Cw,i – Cw)
= KOG (Pa – P*a)
= KOL (C*w – Cw)
1
KOL
1
H KG
= +1
KL
1
KOG
1
KG
= +H
KL
H = P*a / Cw = Pa,i / Cw,i = Pa / C*w
Two-film Theory
1
KOG
=H
KOL
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Gas & Liquid-side Resistances in Interfacial Mass Transfer
1
KOL
1
H KG
= +1
KL
1
KOG
1
KG
= +H
KL
fG = fraction of gas-side resistance
=1/KOG
1/KG
1/KG
1/KG=+ H/KL KL
KL=+ H KG
fL = fraction of liquid-side resistance
=1/KOL
1/KL
1/HKG
1/KL=+ 1/KL + KL/H
KG=KG
R. Shanthini 24 May 2010
Gas & Liquid-side Resistances in Interfacial Mass Transfer
fG =1/KOG
1/KG
1/KG
1/KG=+ H/KL KL
KL=+ H KG
=1/KOL
1/KL
1/HKG
1/KL=+ 1/KL + KL/H
KG=KG
If fG > fL, use the overall gas-side mass transfer coefficient and the overall gas-side driving force.
If fL > fG use the overall liquid-side mass transfer coefficient and the overall liquid-side driving force.
fL
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For a very soluble gas
Gas-liquidinterface
gas
liquid
δa
δw
Cw,i
Pa,i
C*w
Cw ≈ Cw,i
Pa
NA = KG (Pa – Pa,i) = KOG (Pa – P*a)
P*a ≈ Pa,i
KOG ≈ KG
fG > fL
CwP*a
R. Shanthini 24 May 2010
Gas-liquidinterface
gas
liquid
δa
δw
Cw,i
Pa,i
Cw
C*wPa ≈ Pa,i
P*a
PaC*w ≈ Cw,i
KOL ≈ KL
NA = KL (Cw,i – Cw) = KOL (C*w – Cw)
For an almost insoluble gas fL > fG