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Mass transfer coefficient
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When a fluid flows past a solid surface under condition such that turbulence generally prevails, there is a region immediately adjacent to the solid surface where the flow is predominantly laminar, followed by a transition zone and turbulent core. • Eddy movements is dominating in the turbulent core, whereas absence of eddy in laminar zone restricts the mixing only to molecular movement (i.e. molecular diffusion).
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Evaporation of water in to air
Heat transfer: flow of air past a heated plate
Mass transf t ransfer er coefficient •
Mechanism of the flow process involving involving movement of eddies in turbulent core is not fully developed, developed, whereas the mechanism in the laminar zone can be fully explained through kinetic theory, theory, at least for the gas phase.
•
Accordingly, it is customary to describe the transport equation for the turbulent core in the same manner as has been done for laminar zone.
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Mass transfer coefficient •
Steady-state molecular diffusion in fluid at rest and in laminar flow can be represented by the following equation: =
•
+
− + − +
is determined based on non-diffusional
consideration. Replacing , characteristic of molecular diffusion by F, the mass transfer coefficient: + − = + − +
F is known as F-type mas transfer coefficient
Mass transfer coefficient •
Since the surface through which mass transfer takes place may not be plane, N is defined as the flux at the interface or phase boundary. So, or - any one of these must be at the interface.
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F is a local mass transfer coefficient and it depends on the local nature of the fluid motion.
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For multi-component mixture, above equation is not fully correct, but it can be used with ∑ in place of + .
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k-type mass transfer coefficient •
For the two situation, which occur most frequently, transfer of A through stagnant B and equimolar counter-transfer, flux is usually expressed in the following form: flux = (coefficient) x (concentration difference)
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Since the concentration can be expressed in many forms, we have variety of coefficients.
Different k-type mass transfer coefficients
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•
Transfer of A through nontransferring B
k-type mass transfer coefficients correlations are analogous to the convective heat transfer coefficient relation, = ℎ − • Heat transfer equations are generally valid, but use of mass transfer correlations for transfer of A through non-transferring B is somewhat restricted. • For the above case, it is valid when the bulk flow term has small contribution and when the concentration difference is minimal, so that non-linear profile can be represented approximately by a linear profile. • k-type coefficient equations for equimolar counter-transfer are always applicable and correct.
Equimolar counter-transfer
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Relationship between F and k-type coefficients • Transfer of A through stagnant B:
=
− ⇒ =
But,
− + = ⇒= + − + So, =
Relationship between F and k-type coefficients • Equimolar counter-transfer:
= − = − = and = =
So, =
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Film Theory Total resistance to mass transfer is offered by laminar sub-layer, transition zone and turbulent core, all put together. This resistance can be replaced theoretically by fictitious laminar sub-layer.
− ⇒ ∝
So, =
Film Theory (Lewis)
− − = h∆
For mass transfer, = For heat transfer, =
= − = ∆
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Mass transfer coefficient correlations •
Experimentally, it has been observed that = = , , , ,
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Dimensional analysis by Buckingham pitheorem yields the following correlations:
= ′ , => Sh = ′(Re, Sc)
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Wetted-wall tower
Combining above two equations:
Colburn analogy: = = / 2
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Penetration theory HlGBlE Penetration Theory: •
The penetration theory was propounded in 1935 by HlGBlE who was investigating whether or not a resistance to transfer existed at the interface when a pure gas was absorbed in a liquid. In his experiments, a slug-like bubble of carbon dioxide was allowed to rise through a vertical column of water in a 3 mm diameter glass tube. As the bubble rose, the displaced liquid ran back as a thin film between the bubble and the tube, Higbie assumed that each element of surface in this liquid was exposed to the gas for the time taken for the gas bubble to pass it; that is for the time given by the quotient of the bubble length and its velocity.
Penetration theory
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Penetration theory The diffusion of solute A away from the interface (Y-direction) is thus given by
for conditions of equimolecular counter-diffusion, or when the concentrations of diffusing materials are sufficiently low for the bulk flow velocity to be negligible. Because concentrations of A are low, there is no objection to using molar concentration for calculation of mass transfer rates in the liquid phase
Solution of the above PDE with the mentioned conditions can be conveniently done by the method of Laplace Transform. The final solution is:
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flux
Penetration theory Mass transfer flux is defined as the flux prevailing at the interface, i.e. at y = 0
∝
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Regular surface renewal •
It is important to note that the mass transfer rate falls off progressively during the period of exposure, theoretically from infinity at t = 0 to zero at t = .
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Assuming that all the surface elements are exposed for the same time t e (Higbie's assumption), the moles of A (n A ) transferred at an area A in time t e is given by:
⇒ ∝
Regular surface renewal •
Thus, the shorter the time of exposure the greater is the rate of mass transfer. No precise value can be assigned to t e in any industrial equipment, although its value will clearly become less as the degree of agitation of the fluid is increased.
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Random surface renewal (DANCKWERTS) •
DANCKWERTS suggested that each element of surface would not be exposed for the same time, but that a random distribution of ages would exist. It was assumed that the probability of any element of surface becoming destroyed and mixed with the bulk of the fluid was independent of the age of the element and, on this basis, the age distribution function of the surface elements was calculated.
Random surface renewal (DANCKWERTS)
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Random surface renewal (DANCKWERTS)
The mass transfer rate at unit area of surface of age t is given by
Random surface renewal (DANCKWERTS) Thus, the overall rate of transfer per unit area when the surface is renewed in a random manner is
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Random surface renewal (DANCKWERTS) Above equation might be expected to slightly underestimate the mass transfer rate because, in any practical equipment, there will be a finite upper limit to the age of any surface element. The proportion of the surface in the older age group is, however, very small and the overall rate is largely unaffected. • It is be seen that the mass transfer rate is again proportional to the concentration difference and to the square root of the diffusivity. • The numerical value of s is difficult to estimate, although this will clearly increase as the fluid becomes more turbulent •
The film -penetration theory TOOR and MARCHELLO proposed this theory • The whole of the resistance to transfer is regarded as lying within a laminar film at the interface, as in the two-film theory, but the mass transfer is regarded as an unsteady state process. It is assumed that fresh surface is formed at intervals from fluid which is brought from the bulk of the fluid to the interface by the action of the eddy currents. • Mass transfer then takes place as in the penetration theory, except that the resistance is confined to the finite film, and material which traverses the film is immediately completely mixed with the bulk of the fluid. • The third boundary condition is applied at y = yb, the film thickness, and not at y = . •
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The film -penetration theory •
With DANKWERTS rate of renewal of surface elements, the final equation is,
, = . ℎ
The film -penetration theory
(B)
(A)
(B)
, = . ℎ
(A)
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The film -penetration theory
Inter-phase mass transfer •
Air-ammonia mixture in contact with water • Constant temperature and total pressure, maintained in a piston-cylinder assembly immersed in a constant temperature bath. • This curve results irrespective of relative amount of liquid & gas phases, depends only on T & P imposed on 3-comp system.
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Inter-phase mass transfer •
At a fixed set of conditions, referring to T & P, there exists a set of equilibrium relationships, which may be shown graphically in the form of equilibrium distribution curve. • For a system at equilibrium, there is no net diffusion of the component between the phases. • For a system not in equilibrium, diffusion of the components between the phases will occur in such a manner as to bring the system to the condition of equilibrium.
Whitman & Lewis two-film theory
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Whitman & Lewis two-film theory = − = − = ∆ = ∆ So,
− =− − Also, as per the equilibrium distribution relationship,
= f
Whitman & Lewis two-film theory Overall mass transfer coefficients
= − = − = − ∗ = ∗ −
Similarly,
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Special case - I •
Solute is highly soluble in the liquid (e.g. NH 3 + Air in contact with water) => m is very small =>small concentration of solute in gas phase give high concentration in the liquid phase.
•
Then
≈
⇒ ∆ ≈ ∆
Special case - I
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Special case - II •
Solute is sparingly soluble in the liquid (e.g. CO2 + Air in contact with water) => m is very high =>high concentration of solute in gas phase give low concentration in the liquid phase because of low solubility of solute gas in the liquid.
•
Then
≈
⇒ ∆ ≈ ∆
Relative resistance
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