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Chapter 7 Mass Transfer
Mass transfer occurs in mixtures containing local concentration variation. For example, when dye is dropped into a cup of water, mass-transfer processes are responsible for the movement of dye molecules through the water until equilibrium is established and the concentration is uniform. Mass is transferred from one place to another under the influence of a concentration difference or concentration gradient in the system. Gas-liquid mass transfer is extremely important in bioprocessing because many processes are aerobic, oxygen must first be transferred from gas bulk through a series of steps onto the surfaces of cells before it can be utilized.
The solubility of oxygen within broth is very poor. Therefore, the enhancement of gas-liquid mass transfer during aerobic cultures and fermentations is always put into priority.
Direction of mass transfer Distance, y Fig. 7.1 Concentration gradient of component A inducing mass transfer Concentration of
A,
CA
C A a CA
Fick’s law of diffusion:
7.1.2 Role of Diffusion in Bioprocessing
(^) Mixing As discussed before, turbulence in fluids produces bulk mixing on a scale equal to the smallest eddy size. Within the smallest eddies, flow is largely streamline so that further mixing must occur by diffusion of fluid components. Mixing on a molecular scale therefore completely relies on diffusion as the final step in the mixing process. a dy J A = N (^) A D dCA AB
the overall reaction rate can be significantly reduced if diffusion is low. (^) Mass transfer across a phase boundary Mass transfer between phases occurs often in bioprocesses. Oxygen transfer from gas bubbles to fermentation broth, penicillin recovery from aqueous to organic liquid, and glucose transfer from liquid medium into mould pellets are typical examples. When different phases come into contact, fluid velocity near the phase interface is significantly decreased and diffusion becomes crucial for mass transfer across the phase interface.
7.1.3 Film Theory
Phase boundary Phase 2 Phase 1 Film 2 Film 1 Fig. 7.2 Two mass-transfer films formed within two phases CA CA 1 i CA 2 i CA 2 1
Mass transfer coupled with fluid flow is a more complicated process than diffusive mass transfer. The value of the mass-transfer coefficient reflects the contribution to mass transfer from all the processes in the system that affect the boundary layer, which depends on the combined effects of flow velocity, geometry of equipment, and fluid properties such as viscosity and diffusivity. Because the hydrodynamics of most practical systems are not easily characterized. k cannot be calculated reliably from theoretical equations. Instead, it is measured experimentally or estimated using correlations available from the literatures. In general, reducing the thickness of the boundary layer or improving the diffusion coefficient in the film will result in enhancement of k and improvement in the rate of mass transfer.
7.1.5 Liquid-Solid Mass Transfer
Fig. 7.3 Concentration gradient for liquid-solid mass transfer CAo CAi Solid-liquid interface Solid liquid film N A = k L a Δ C A = k L a ( C Ao C Ai)
At steady state, there is no accumulation of component A at the interface or anywhere else in the system, and component A transported through liquid 1 must be transported through phase 2, that is N A1 = N A2 = N A. If C A1i and C A2i are equilibrium concentrations, they can be related using the distribution coefficient m. C (^) A 2 i Therefore: m = CA 1 i or C A1i = mC A2i ( 7 .6) 1 A 1 A 2 L 1 L 2 A k a m k a
N ( ) C C (7.7)
and Here we define two overall mass-transfer coefficients: and Therefore: 1 1 A 2 L 1 L 2 A mk a k a m N ( ^ C ) CA 1 (7.8) m KL 1 a k (^) L 1 a k (^) L 2 a 1 1 ( 7 .9) 1 1 1 KL 2 a mkL 1 a kL 2 a (7.10)
7.1.7 Gas-Liquid Mass Transfer
Gas phase Liquid film (^) Gas film Fig 7.4 Concentration gradient for gas-liquid mass transfer Phase boundary Liquid phase CAG CAGi CALi CAL 2 1
of component A through the gas wide range concentration of concentration in the gas phase for some gases, equilibrium is a linear concentration. Therefore: function of liquid The rate of mass transfer boundary layer is: N AG = k G a ( C AG C AG i) (7.13) and the rate of mass transfer of component A through the liquid boundary layer is: N AL = k L a ( C ALi C AL) (7.14) If we assume that equilibrium exists at the interface, C AGi and C ALi can be related. For dilute concentration of most gases and for a
and the overall liquid-phase mass-transfer coefficient K L is defined as: Thus: KG a kG a k L a
1 1 m (7.18) KL a mkG a k L a
N A = K G a ( C AG mC AL) (7.20)
and Usually and
m
N A = K L a (
CA G
C
AL)^
N A = K G a ( C AG C
AG*) (7.2^2 )
N A = K L a ( C AL C AL)
