Transcript C H A P T E R 13 The Transfer of Heat

PM3125: Lectures 7 to 9
Content of Lectures 7 to 9:
Mass transfer: concept and theory
R. Shanthini
24 May 2010
Mass Transfer
Mass transfer occurs when a component in a
mixture goes from one point to another.
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.
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Example of Mass Transfer
Mass transfer can occur
by either diffusion or
by convection.
Stirring the
water with a
spoon creates
forced
convection.
Diffusion
(slower)
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That helps the
sugar
molecules to
transfer to the
bulk water
much faster.
Example of Mass Transfer
Mass transfer can occur
by either diffusion or
by convection.
Stirring the
water with a
spoon creates
forced
convection.
Diffusion
(slower)
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Convection
(faster)
That helps the
sugar
molecules to
transfer to the
bulk water
much faster.
Example of Mass Transfer
At the surface of the lung:
Air
Blood
Oxygen
High oxygen concentration
Low carbon dioxide concentration
Low oxygen concentration
High carbon dioxide concentration
Carbon dioxide
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Diffusion
Diffusion (also known as molecular diffusion)
is a net transport of molecules
from a region of higher concentration
to a region of lower concentration
by random molecular motion.
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Diffusion
A
B
Liquids A and B are separated from each other.
Separation removed.
A
B
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.
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Examples of Diffusion
• 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
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Fick’s Law of Diffusion
JA = -D
DAB
AB
CA
ΔCA
Δx
A&B
JA
CA + ΔCA
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Δx
Fick’s Law of Diffusion
JA = -DAB
ΔCA
Δx
concentration
gradient
(mass per volume per
distance)
diffusion coefficient
(or diffusivity) of A in B
diffusion flux of A in relation to
the bulk motion in x-direction
(mass per area per time)
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What is the unit of diffusivity?
Fourier’s Law of Heat Conduction
.
Q
ΔT
= -k
A
Δx
Temperature gradient
(temperature per
distance)
Thermal conductivity
Heat flux
(Energy per area per time)
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Describe the
similarities between
Fick’s Law and
Fourier’s Law
Diffusivity
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
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Diffusivity depends on the type of solute, type of
solvent, temperature, pressure, solution phase
(gas, liquid or solid) and other characteristics.
Prediction of Binary Gas Diffusivity
DAB
P
Mi
T
Vi
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- diffusivity in cm2/s
- absolute pressure in atm
- molecular weight
- temperature in K
- sum of the diffusion volume for component i
DAB is proportional to 1/P and T1.75
Prediction of Binary Gas Diffusivity
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Prediction of Diffusivity in Liquids
For very large spherical molecules (A) of 1000 molecular
weight or greater diffusing in a liquid solvent (B) of small
molecules:
DAB =
DAB
T
μ
VA
9.96 x 10-12 T
μ VA1/3
applicable
for biological
solutes such
as proteins
- diffusivity in cm2/s
- temperature in K
- viscosity of solution in kg/m s
- solute molar volume at its normal boiling point
in m3/kmol
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DAB is proportional to 1/μ and T
Prediction of Diffusivity in Liquids
For smaller molecules (A) diffusing in a dilute liquid solution of
solvent (B):
DAB =
1.173 x 10-12 (Φ MB)1/2 T
μB VA0.6
applicable
for biological
solutes
DAB - diffusivity in cm2/s
MB - molecular weight of solvent B
T - temperature in K
μ - viscosity of solvent B in kg/m s
VA - 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
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DAB is proportional to 1/μB and T
Prediction of Diffusivity of Electrolytes in Liquids
For smaller molecules (A) diffusing in a dilute liquid solution of
solvent (B):
8.928 x 10-10 T (1/n+ + 1/n-)
DoAB =
(1/λ+ + 1/ λ-)
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
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DAB is proportional to T
Fick’s First Law of Diffusion (again)
JA = - DAB
∆CA
∆x
JA is the diffusion flux of A in
relation to the bulk motion in
x-direction
If circulating currents or eddies are present
(which will always be present), then
∆C A
NA = - (D + ED)
∆x
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where ED is the eddy diffusivity, and
is dependent on the flow pattern
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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Macroscopic (or mass transfer coefficient) approach:
NA = - k ΔCA
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is used when the mass
transfer is caused by
molecular diffusion plus
other mechanisms such
as convection.
Macroscopic (or mass transfer coefficient) approach:
NA = - k ΔCA
concentration
difference
(mass per volume)
mass transfer coefficient
net mass flux of A
(mass per area per time)
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What is the unit of k?
Newton’s Law of Cooling in
Convective Heat Transfer
Flowing fluid at Tfluid
Heated surface at Tsurface
.
Qconv.
A
= h (Tsurface – Tfluid)
convective heat flux
(energy per area per time)
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Heat transfer coefficient
temperature
difference
Describe the similarities between the
convective heat transfer equation and the
macroscopic approach to mass transfer.
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Macroscopic (or mass transfer coefficient) approach:
NA = -k ΔCA = k (CA1 – C A2 )
CA1
A&B
NA
CA2
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Macroscopic (or mass transfer coefficient) approach:
NA = -k ΔCA = k (CA1 – C A2 )
CA1 = PA1 / RT
CA2 = PA2 / RT
CA1
A&B
NA
CA2
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Macroscopic (or mass transfer coefficient) approach:
NA = k (PA1 – P A2 ) / R T
PA1
A&B
NA
PA2
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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)
R. Shanthini
24 May 2010
Oxygen transfer from gas bubble to cell
1.
2.
3.
4.
5.
6.
7.
8.
Transfer from the interior of the bubble to the gas-liquid
interface
Movement across the gas film at the gas-liquid interface
Diffusion through the relatively stagnant liquid film
surrounding the bubble
Transport through the bulk liquid
Diffusion through the relatively stagnant liquid film
surrounding the cells
Movement across the liquid-cell interface
If the cells are in floc, clump or solid particle, diffusion
through the solid of the individual cell
Transport through the cytoplasm to the site of reaction.
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1.
2.
3.
4.
5.
6.
7.
8.
Transfer through the bulk phase in the bubble is relatively fast
The gas-liquid interface itself contributes negligible resistance
The liquid film around the bubble is a major resistance to
oxygen transfer
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.
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.
Resistance at the cell-liquid interface is generally neglected
The mass transfer resistance is small, except the cells form large
clumps or flocs.
Intracellular oxygen transfer resistance is negligible because of
the small distance involved
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Interfacial Mass Transfer
Pa = partial pressure of solute in air
Ca = concentration of solute in air
air
volatilization
Pa = Ca RT
air-water
interface
water
absorption
Cw = concentration of solute in water
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Transport of a volatile chemical across the
air/water interface.
Interfacial Mass Transfer
Pa = partial pressure of solute in air
air
air-water
interface
water
δa
Pa,i
Cw,i
Pa,i vs Cw,i?
δw
Cw = concentration of solute in water
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δa and δw are boundary layer zones offering
much resistance to mass transfer.
Interfacial Mass Transfer
Pa
air
δa
air-water
interface
Cw,i
water
Cw
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Pa,i
δw
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.
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
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H depends on the solute,
solvent and temperature
Gas-Liquid Equilibrium Partitioning Curve
Pa
Pa = H’’ C*w
Pa
Pa,i = H Cw,i
Pa,i
P*
a
H = H’ = H’’
if the partitioning
curve is linear
P*a = H’ Cw
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Cw
Cw,i
C*w
Cw
Interfacial Mass Transfer
NA = KG (Pa – Pa,i)
air
δa
air-water
interface
Cw,i
Cw
C*w
Pa,i
water
P*a
Pa
δw
NA = KL (Cw,i – Cw)
KG = gas phase mass transfer coefficient
R. Shanthini
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KL = liquid phase mass transfer coefficient
Interfacial Mass Transfer
NA = KG (Pa – Pa,i)
Pa
C*w
NA = KOG (Pa – P*a)
air
δa
air-water
interface
Pa,i
Cw,i
water
P*a
Cw
δw
NA = KOL (C*w – Cw)
NA = KL (Cw,i – Cw)
KOG = overall gas phase mass transfer coefficient
K
= overall liquid phase mass transfer coefficient
R. Shanthini
OL2010
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Interfacial Mass Transfer
NA = KG (Pa – Pa,i) = KOG (Pa – P*a)
Pa
C*w
KG = gas phase mass transfer coefficient
KOG = overall gas phase mass transfer coefficient
Pa,i
Cw,i
P*a
Cw
NA = KL (Cw,i – Cw) = KOL (C*w – Cw)
KL = liquid phase mass transfer coefficient
KOL = overall liquid phase mass transfer coefficient
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Relating KOL to KL
C*w – Cw = C*w – Cw,i + Cw,i – Cw
NA / KOL = C*w – Cw,i + NA /KL
(1)
If the equilibrium partitioning curve is linear over
the concentration range C*w to Cw,i, then
Pa - Pa,i = H (C*w - Cw,i)
NA / KG = H (C*w – Cw,i)
(2)
Combining (1) and (2), we get
1
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KOL
=
1
H KG
+
1
KL
Relating KOG to KG
Pa - P*a = Pa – Pa,i + Pa,i – P*a
NA / KOG = NA /KG + Pa,i – P*a
(3)
If the equilibrium partitioning curve is linear over
the concentration range Pa,i to P*a then
Pa,i – P*a = H (Cw,i – Cw)
Pa,i – P*a = H NA / KL
(4)
Combining (3) and (4), we get
1
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KOG
=
1
KG
+
H
KL
Summary: Interfacial Mass Transfer
NA = KG (Pa – Pa,i) = KOG (Pa – P*a)
NA = KL (Cw,i – Cw) = KOL (C*w – Cw)
1
KOL
1
KOG
=
=
1
H KG
1
KG
+
+
1
KL
1
H
KOG
=
H
KOL
KL
H = P*a / Cw = Pa,i / Cw,i = Pa / C*w
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Two-film Theory
Gas & Liquid-side Resistances in Interfacial Mass Transfer
1
1
H
=
+
KOG
KG
KL
fG = fraction of gas-side resistance
1/KG
=
1/KOG
1
KOL
=
1
H KG
1/KG
=
1/KG + H/KL
+
=
KL
KL + H KG
1
KL
fL = fraction of liquid-side resistance
1/KL
=
1/KOL
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1/KL
=
=
1/HKG + 1/KL
KG
KG + KL/H
Gas & Liquid-side Resistances in Interfacial Mass Transfer
1/KG
fG =
1/KOG
1/KG
=
1/KG + H/KL
=
KL
KL + H KG
1/KL
fL =
1/KOL
1/KL
=
=
1/HKG + 1/KL
KG
KG + KL/H
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.
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For a very soluble gas
fG > fL
Pa
gas
Gas-liquid
interface
δa
C*w
Pa,i
Cw,i
δw
liquid
P*a
Cw
Cw ≈ Cw,i
P*a ≈ Pa,i
NA = KG (Pa – Pa,i) = KOG (Pa – P*a)
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KOG ≈ KG
For an almost insoluble gas
δa
Gas-liquid
interface
Pa
C*w ≈ Cw,i
Pa ≈ Pa,i
gas
fL > fG
Pa,i
Cw,i
liquid
P*a
C*w
δw
Cw
NA = KL (Cw,i – Cw) = KOL (C*w – Cw)
R. Shanthini
24 May 2010
KOL ≈ KL
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
Encyclopedia of Pharmaceutical Technology (Hardcover)
by James Swarbrick (Author)
R. Shanthini
24 May 2010