Mass Transfer, Malik Nasir Wirali

Mass TransferMalik Nasir Wiarali

BSc. Chemical EngineeringUniversity of Gujrat, Pakistan

Mass TransferThe term mass transfer is used to denote the transference of a component in a mixture from a

region where its concentration is high to a region where the concentration is lower. Mass transfer process can take place in a gas or vapour or in a liquid, and it can result from the random velocities of the molecules (Molecules diffusion) or from the circulating or eddy currents present in a turbulent fluid (eddy diffusion).

In processing, it is frequently necessary to separate a mixture into its components and in a physical process, difference in a particular property are exploited as the basis for the separation process. Thus fractional distillation depends on differences on volatility, gas absorption on difference in solubility of gases in a selective absorbent, and similarly, liquid-liquid extraction is based on the selectivity of an immiscible liquid for one of the constituent. The rate at which process takes place is dependent both on driving force (concentration difference) and on the mass transfer resistance. In most of these applications, mass transfer takes place across a phase boundary where the concentrations on either side of the interface are related by phase equilibrium relationship. Where a chemical reaction takes place during the course of mass transfer process, the overall transfer rate depends on both the chemical kinetics of reaction and on the mass transfer resistance.

A simple example of a mass transfer process is that occurring in a box consisting of two components, each containing a different gas, initially separated by an impermeable partition. When the partition is removed the gases start to mix and mixing process continues at a constantly decreasing rate until eventually theoretically after the elapse of an infinite time the whole system requires a uniform composition. The process is one of molecular diffusion in which the mixing is attributable solely to the random motion of molecules. The rate of diffusion is governed by Fick’s Law, first proposed by Fick’s in 1855 which expresses the mass transfer rate as a function of the molar concentration gradient. In a mixture of two gases A and B, assumed ideal, Fick’s Law for steady state diffusion may be written as:

Where NA is the molar flux of A ( moles per unit area per unit time), CA is the concentration of A (moles of A per unit volume) DAB is known as diffusivity or diffusion coefficient for A in B, and y is the distance in the direction of transfer.

An equation of the same form may be written for B:

Where DBA is the diffusivity of B in A. For an ideal gas mixture, at constant pressure (CA + CB), is constant, and hence:

A simple example of a mass transfer process is that occurring in a box consisting of two compartments, each containing a different gas, initially separated by an impermeable partition. When the partition is removed the gases start to mix and mixing process continues at a constantly decreasing rate until eventually theoretically after the elapse of an infinite time the whole system requires a uniform composition. The process is one of molecular diffusion in which the mixing is attributable solely to the random motion of molecules. The rate of diffusion is governed by Fick’s Law, first proposed by Fick’s in 1855 which expresses the mass transfer rate as a function of the molar concentration gradient. In a mixture of two gases A and B, assumed ideal, Fick’s Law for steady state diffusion may be written as:

Where NA is the molar flux of A ( moles per unit area per unit time), CA is the concentration of A (moles of A per unit volume) DAB is known as diffusivity or diffusion coefficient for A in B, and y is the distance in the direction of transfer.

An equation of the same form may be written for B:

Where DBA is the diffusivity of B in A. For an ideal gas mixture, at constant pressure (CA + CB), is constant, and hence:

The condition for the pressure or molar concentration to remain constant in such a system is that there should be no net transfer of molecules. The process is then referred to as one of equimolecular diffusion, and NA + NB = 0

This relation is satisfied only if DBA = DAB and therefore the suffixes may be omitted and equation (1) becomes:

Equation (10.4) which describes the mass transfer rate arising solely from the random movement of molecules, is applicable to a stationary medium or a fluid in streamline flow. If circulating currents or eddies are present, then equimolecular mechanism will be reinforced and total mass transfer rate may be written as:

Where D is a physical property of the system and a function only of its composition, pressure and temperature, ED, which is known as the eddy diffusivity, is dependent on the flow pattern and varies with position.

The condition for the pressure or molar concentration to remain constant in such a system is that there should be no net transfer of molecules. The process is then referred to as one of equimolecular diffusion, and NA + NB = 0

This relation is satisfied only if DBA = DAB and therefore the suffixes may be omitted and equation (1) becomes:

Equation (10.4) which describes the mass transfer rate arising solely from the random movement of molecules, is applicable to a stationary medium or a fluid in streamline flow. If circulating currents or eddies are present, then equimolecular mechanism will be reinforced and total mass transfer rate may be written as:

Where D is a physical property of the system and a function only of its composition, pressure and temperature, ED, which is known as the eddy diffusivity, is dependent on the flow pattern and varies with position.

2. Diffusion in Binary Gas Mixtures2.1 Properties of Binary MixturesIf A and B are ideal gases in a mixture, the ideal law gas, may be applied to each gas separately

and to the mixturePA V = nA RT (10.9a)

PB V = nB RT (10.9b)

PV = n RT (10.9c)Where nA and nB are the number of moles of A and B and n is the total number in a volume V,

and PA, PB and P are the respective partial pressure and the total pressure

Thus

Where cA and cB are the mass concentrations and MA and MB molecular weights, and CA, CB, CT are, the molar concentrations of A and B respectively, and the total molar concentration of the mixture.

From Dalton’s law of partial pressures

Where xA and xB are the mole fractions of A and B

Thus for a system at constant pressure P and constant molar concentration CT

By substituting from equations (10.10a) and (10.10b) into equation (10.4), the mass transfer rates NA and NB can be expressed in terms of partial pressure gradients rather than concentration gradients. Furthermore, NA and NB can be expressed in terms of gradients of mole fraction.

Thus

Similarly

Equimolecular Counter DiffusionWhen the mass transfer rates of the two components are equal and opposite the process is said

to be one of equimolecular counter-diffusion. It occurs in distillation column when the molar latent heats of the two components are the same. At any point in the column a falling stream of liquid is brought into contact with a rising stream of vapour with which it is not in equilibrium. The less volatile component is transferred from the vapour to the liquid and the more volatile component is transferred to the opposite direction. If the molar latent heats of the components are equal, the condensation of a given amount of less volatile component releases exactly the amount of latent heat required to volatilize the same molar quantity of more volatile component. Thus at the interface, and consequently throughout the liquid and vapour phases, equimolecular counter-diffusion is taking place.

Under these conditions, the differential forms of equation fro NA (10.4, 10.18, and 10.19) may be simply integrated, for constant temperature and pressure, to give respectively:

Similar equations apply to NB which is equal to -NA, and suffixes 1 and 2 represent the values of quantities at positions y1 and y2 respectively.

Equation (10.22) may be written as NA = hd (CA1 – CA2) (10.25)

Where hd = D/(y2 – y1) is a mass transfer coefficient with driving force expressed as a difference in molar concentration, its dimension are those of velocity L T-1

Similarly, equation 10.23 may be written as:NA = kG’ (PA1 – PA2) --- (10.26)

Where kG’ = D/RT (y2 – y1) is a mass transfer coefficient with a driving force expressed as a difference in partial pressure

Mass Transfer through a Stationary Second component (Diffusion through a Stagnant gas)In several important processes, one component in gaseous mixture will be transported relative

to the fixed plane, such as liquid interface, for example, and the other will undergo no net movement. In gas absorption, a soluble gas A is transferred to the liquid surface where it dissolves, whereas the insoluble gas B undergoes no net movement with respect to the interface. Similarly, in evaporation from a free surface, the vapor moves away from the surface but the air has no net movement.

The concept of a stationary component may be envisaged by considering the effect of moving the box in the opposite direction to that in which B is diffusing at a velocity equal to diffusion velocity, so that to the external observer B appears to be stationary. The total velocity at which A is transferred will then be increased to its diffusion velocity plus the velocity of the box. The rate of diffusion is governed by Fick's law, first proposed by Fick in 1855 which expresses the mass transfer rate as a linear function of molar concentration gradient. In mixture of two gases A and B, assumed ideal, Fick’s Law for steady state diffusion may be written as:

The negative sign emphasizes that diffusion occurs in the direction of drop in concentration. For absorption of soluble gas A from a mixture with an insoluble gas B, the respective diffusion rates are given by:

Any movement of molecules B by diffusion away from the interface would lead to a decrease in concentration of B at the interface, since B is not liberated there. This would result in a total concentration difference being setup causing a bulk flow of A and B towards the interface in addition to transfer by diffusion. Since the total mass transfer rate of B is zero there must be a bulk flow of the system towards liquid interface exactly to counterbalance the diffusional flux away from the surface as shown in slide ( ), where:

The corresponding bulk flow of A must be CA/CB times that of B, since bulk flow implies that the gas moves in mass.

Therefore the total flux of A

Equation 10.30 is known as Stefan’s Law. Thus bulk flow enhances mass transfer rate by a factor CT/CB, known as drift factor. The fluxes of the components are given in slide ( ).

Writing equation (10.30) as:

On integration

By definition, CBm. The logarithmic mean of CB1 and CB2, is given by:

Thus substituting for in equation (10.31)

Or in terms of partial pressure:

Similarly in terms of mole fraction:

Equation 10.31 can be simplified when the concentration of the diffusing component A is small. Under these conditions CA is small compared with CT, and equation 10.31 becomes:

For small values of CA, CT – CA1 = CT, and only the first term is significant.

Thus

Equation (10.36) is identical to equation (10.22) for equimolecular counter diffusion. Thus, the effects of bulk flow can be neglected at low concentrations.

Equation (10.33) can be written in terms mass transfer coefficient hD to give:

It may be noted that all transfer coefficients are greater than those for equimolecular counter diffusion by factor , which is an integrated form of the drill factor.

Prediction of DiffusivitiesWhen the diffusivity D for transfer of one gas in another is not known and experimental

determination is not applicable, it is necessary to use one of the many predictive procedures. A commonly used method due to Gilliland is based on the “Stefan Maxwell” hard sphere model and this takes the form

Where D is the diffusivity in m2/s, T is the absolute temperature (K), MA, MB are the molecular masses of A and B. P is the total pressure in N/m2, and VA and VB are the molecular volume of A and B. The total molecular volume is the volume in m3 of one Kmol of the material in the form of liquid at its boiling point, and is a measure of the volume occupied by the molecules themselves. It may not always be known, although an approximate value can be obtained, for all but for simple molecules, by application of Kopp’s Law of additive volumes. Kop has presented a particular value for the equivalent atomic volume of each element and given in the literature.

Mass Transfer VelocitiesIt is convenient to express mass transfer rates in terms of velocities for the species under consideration where:Velocity = Flux / ConcentrationWhich, in the S.I System, has the units

For diffusion according to Fick’s law

As a result of the diffusional process, there is no net overall molecular flux arising from diffusion in a binary mixture, the two components being transferred at equal and opposite rates. In the process of equimolecular counter diffusion, which occurs, for example, in a distillation column, when the two components have equal molar latent heats, the diffusional velocities are the same as the velocities of molecular species relative to the walls of the equipment for phase boundary.

If the physical constraints placed upon the system result in a bulk flow, the velocities of the molecular species relative to one another remain the same, but in order to obtain the velocities relative to a fixed point in a equipment, it is necessary to add bulk flow velocity. As an example of a system in which there is a bulk flow velocity is that in which one of the component is transferred through a second component which is undergoing no net transfer, as for example in the absorption of a soluble gas A from a mixture with an insoluble gas. In this case, because there is no net flow of B, the sum of its diffusional velocity and the bulk flow velocity must be zero slides ( ).

General Case for Gas Phase Mass Transfer in a Binary MixtureWhatever the physical constraints placed on the system, the diffusion process causes the two

components to be transferred at equal and opposite rates and values of diffusion velocities UDA and UDB are always applicable.

It is bulk flow velocity uF which changes with imposed conditions and which gives rise to differences in overall mass transfer rates. In equimolecular counter diffusion, uF is zero. In the absorption of soluble gas A from a mixture the bulk velocity must be equal and opposite to the diffusion velocity of B as this latter component undergoes no net transfer.

In general for any component:Total transfer = Transfer by diffusion + transfer by bulk flow.For component ATotal transfer (Moles / unit area. time) = N’ADiffusional transfer according to Fick’s Law

Transfer by bulk flow = UF.CA

Thus for A N’A = NA + uF CA (10.46a)

And for B N’B = NB + uF CB (10.46b)

(10.47)

Substituting

Similarly for B:

For equimolecular diffusion N’A =- N’B and equation 10.48 reduces to Fick’s Law. For a system in which B undergoes no net transfer, N’B = 0 and equation 10.48 is identical to Stefan’s Law.

Diffusion as a Mass FluxFick’s law of diffusion is normally expressed in molar units or:

Where xA is the mole fraction of component A

The corresponding equation for component B indicates that there is an equal and opposite molar flux of that component. If each side of equation 10.4 is multiplied by the molecular weight of A, MA, then:

Where JA is the flux in mass per unit area and time (kg/m2.S in S.I units), and cA is a concentration in mass terms (Kg/m3 in S.I units).

Similarly for component B:

Although the sum of the molar concentration is constant in an ideal gas at constant pressure, the sum of mass concentrations is not constant, and are not equal and opposite.

Thus

Thus, the diffusional process does not give rise to equal and opposite mass fluxes.

Maxwell’s Law of DiffusionMaxwell postulated that the partial pressure gradient in the direction of diffusion for a constituent of two

components gaseous mixture was proportional toThe relative velocity of molecules in the direction of diffusion andThe product of molar concentrations of the componentsThus-dPA/dy = F CA CB (uA – uB) (10.77)

Where uA and uB are the mean molecular velocities of A and B respectively in the direction of mass transfer and F is a coefficient.

Noting that

And using PA = CA RT (for an ideal gas) (equation 10.10a)

On substitution in equation 10.77 gives-dCA/dy = F/RT (NA CB – NB CA) (10.80)

Equimolecular counter diffusionBy definition N’A = -N’B = NA

Substituting in equation 10.80

Then by comparison with Fick’s law (equation 10.4) ( 10.83 )

.

Transfer of A Through Stationary BBy definition N’B = 0

Thus

Substituting from equation 10.83

It may be noted that equation (10.86) is identical to equation 10.30 (Stefan’s Law) and, Stefan’s Law can therefore be derived from Maxwell’s Law of diffusion.

Diffusion in LiquidsThe diffusion of solution in a liquid is governed by the same equation as for the gas, the diffusion

coefficient is about two orders of magnitude smaller for liquid than for a gas.For an ideal gas, the total molar concentration CT is constant at a given total pressure P and

temperature T. This approximation holds quite well for real gases and vapours except at high pressure. For liquids however, CT may show considerable variations as the concentrations of the components change and, in practice, the total mass concentration (density of the mixture) is much more nearly constant. Thus for a mixture of ethanol and water for example, the mass density will range from 790 – 1000 Kg/m3 whereas the molar density will range from 17 to 56 K mol/m3. For this reason diffusion equations are frequently written in the form of a mass flux JA (mass/(Area) x (time) and the concentration gradients in terms of mass concentrations, such as cA.

Thus for component A, the mass flux is given by:

Where ρ is the mass density (now taken as constant), and ωA is the mass fraction of A in the liquid.

For component B:

Thus, the diffusional process in a liquid gives rise to a situation where the components are being transferred at approximately equal and opposite mass (rather than molar) rates.

Liquid phase diffusivities are strongly dependent on the concentration of the diffusing component which in strong contrast to gas phase diffusivities which are substantially independent of concentration. Values of liquid phase diffusivities which are normally quoted apply to very dilute concentrations of the diffusing component. For this reason, only dilute solutions are considered here, and in these circumstances no serious error is involved in using Fick’s first and second laws expressed in molar units.

Where D is now the liquid phase diffusivity and CA is the molar concentration in the liquid phase.

On integration, equation 10.4 becomes

And D/(y2 – y1) is the liquid phase mass transfer coefficient

Liquid Phase DiffusivitiesValues of the diffusivities of various materials in water are given in the literature. Where

experimental values are not available, it is necessary to use one of the predictive methods which are available. A useful equation for the calculation of the liquid phase diffusivities of dilute solutions of non-electrolytes has been given by Wilke and Chang. Using SI units.

Where D is the diffusivity of solute A in solvent B (m2/S) ФB is the association factor for the solvent (2.26 for water, 1.96 for methanol, 1.5 for ethanol and 1.0 for unassociated solvents such as hydrocarbons and ethers)

MB is the molecular weight of the solvent.

µ is the viscosity of the solution

T is the temperature (K) andVA is the molecular volume of the solute (m3/K mol).

It may be noted that for water a value of 0.0756 m3/K mol should be used.Equation 10.96 does not apply to either electrolytes or to the concentrated solutions.

Mass Transfer across a Phase BoundaryThe theoretical treatment which has been developed in the previous sections relates to mass transfer

within a single phase in which no discontinuities exist. In many important applications of mass transfer, however, material is transferred across a phase boundary. Thus, in distillation a vapour and liquid are brought into contact in the fractionating column and more volatile material is transferred from the liquid to vapour while the less volatile constituent is transferred in the opposite direction, this is an example of equimolecular counter diffusion. In gas absorption the soluble gas diffuses to the surface, dissolves in the liquid, and then passes into the bulk of liquid, and carrier gas is not transferred. In both of these examples, one phase in liquid and other a gas. In liquid-liquid extraction however, a solute is transferred from one liquid solvent to another across a phase boundary, and in the dissolution of crystal the solute is transferred from a solid to a liquid.

Each of these processes is characterized by transference of material across an interface. Because no material accumulates there, the rate of transfer on each side of the interface must be the same, and therefore the concentration gradients automatically adjust themselves so that they are proportional to the resistance to transfer at the interface, the concentrations on each side will be related to each other by the phase equilibrium relationship.

The mass transfer rate between two fluid phases will depend on the physical properties of the two phases, the concentration difference, the interfacial area, and the degree of turbulence. Mass transfer equipment is therefore designed to give a large area of contact between phases and to promote turbulence in each of the fluid.

A number of mechanisms have been suggested to represent conditions in the region of phase boundary. The earliest of these is the two-film theory proposed by Whitman is 1923 who suggested that resistance to mass transfer in each phase could be regarded in a thin film close to the interface. The transfer across these films is regarded as a steady state process of molecular diffusion.

Mass – Transfer ModelsFilm Theory (Whitman 1923) – First, and probably the simplest theory

proposed for mass transfer across a fluid. Detail of this model will be discussed later on.

Penetration Theory (Higbie 1935) – Assumes that the liquid surface in contact with the gas consists of small fluid elements. After contact with the gas phase, the fluid elements, return to the bulk of liquid and are replaced by another element from the bulk liquid phase. The time each element spends at the surface is assumed to be the same.

Surface Renewal Theory (Danckwertz 1951) – Improves on penetration theory by suggesting that the constant exposure time be replaced by an assumed time distribution.

Film Penetration Theory (Toor and Marchello 1958) – Combination of the film and penetration theories. Assumes that a laminar film exists at the fluid interface (as in film theory), but further assumes that the mass transfer is a non-steady process.

Two Film TheoryAccording to the two film theory, for a molecule of substance A to be absorbed,

it must proceed through a series of five steps. The molecule must:1. Migrate from the bulk gas phase to the gas film2. Diffuse across the interface3. Diffuse through the liquid film4. Mix into the bulk liquidThe theory assumes that complete mixing takes place in both gas and liquid

bulk phases and that the interface is at equilibrium with respect to the pollutant molecules transferring in and out of the interface. This implies that all resistance to movement occurs when the molecule is diffusing through the gas and liquid films to get to the interface area, hence the name double-resistance theory. The partial pressure (concentration) in the gas phase changes from PAG in bulk phase to PAi in the interface. Similarly, the concentration in changes from CAi at the interface to CAL in the bulk liquid phase as mass transfer occurs. The rate of mass transfer from one phase to the other then equals the amount of molecule A transferred multiplied by the resistance molecule A encounters in diffusing through the film.

Film TheoryA simple theoretical model for turbulent mass transfer to or from a fluid phase boundary

was suggested in 1904 by Nernest, who proposed that the entire resistance to mass transfer in a given turbulent phase is in a thin, stagnant region of that phase at the interface, called a film. This film is similar to laminar sub layer that forms when a fluid flows in the turbulent region parallel to a flat plate. This is shown schematically in Fig 3.18(a) for the case of gas liquid interface, where the gas is pure component A, which diffuses into non-volatile liquid B. Thus a process of absorption of A into liquid B takes place, without desorption of B into gaseous A. Because the gas is pure A at total pressure P = PA, there is resistance to mass transfer in the gas phase: At the gas-liquid interface, phase equilibrium is assumed so the concentration of A, CA, is related to the partial pressure of A, PA, by some form of Henry’s Law, for example CA = HA PA. In the thin stagnant liquid film of thickness δ, molecular diffusion only occurs with a driving force of CAi – CAb.

Since the film is assumed to be very thin all of the diffusing A passes through the film and into the bulk liquid. If, in addition, bulk flow of A is neglected, the concentration gradient is linear as shown in Fig. 3.18a. Accordingly, Flick’s First law, for the diffusion flux integrates to

If the liquid phase is dilute in A, the bulk flow effect can be neglected and (3.187) applies to the total flux

The Two Film TheoryThe two film theory of Whitman was the first serious attempt to represent conditions

accruing when material is transferred from one fluid stream to another.In this approach, it is assumed that turbulence dies out at the interface and that a laminar

layer exists in each of the two fluids. Outside the laminar layer, turbulent eddies supplement the action caused by random movement of the molecules, and resistance to transfer becomes progressively smaller. For equimolecular counter diffusion the concentration gradient is therefore linear close to the interface, and gradually becomes less at greater distances as shown in Fig. 10.5 by the full line ABC and DEf. The basis of this theory is the assumption that the zones in which the resistance to transfer lies can be replaced by two hypothetical layers, one on each side of the interface, in which the transfer is entirely by molecular diffusion. The concentration gradient is therefore linear in each of these layers and zero outside. The broken lines AGC and DHF indicate the hypothetical concentration distribution and the thicknesses of the two films are L1 and L2. Equilibrium is assumed to exist at the interface and therefore the relative positions of the points C and D are determined by the equilibrium relation between the phases.

From equation 10.22 the rate of transfer per unit area in terms of two-film theory for equimolecular diffusion is given for the first phase as:

Where L1 is the thickness of the film. CAo1 the molar concentration outside the film and CAi1 the molar concentration at the interface.

For the second phase, with same notation, the rate of transfer is:

Because material does not accumulate at the interface the two rates of transfer must be same and:

The relation between CAi1 and CAi2 is determined by the phase equilibrium relationship since the molecule layers on each side of the interface are assumed to be in equilibrium with one another.

The mass transfer coefficient is a diffusion rate constant that relates the mass transfer rate, mass transfer area and concentration gradient.

nA = is the mass transfer rate (mol/s)

A = is the effective mass transfer area (m2)ΔCA = is the driving force concentration difference (mol/m3)

The theory is equally applicable when bulk flow occurs. In gas absorption, for example where may be expressed the mass transfer rate in terms of the concentration gradient in the gas phase.

In this case, for steady state process, (dCA/dy) (CT/CB), as opposed to dCA/dy, will be constant through the film and dCA/dy will increase as CA decreases. Thus line GC and DH in Figure 10.5 will no longer be quite straight.





Penetration TheoryIn 1935 Higbie suggested that transfer process was largely attributable to fresh material being

brought by eddies to the interface, where a process of unsteady state took place for a fixed period at freshly exposed surface. This theory is generally known as penetration theory.

Higbie assumes that the liquid surface consists of elements , each of which is in contact with the gaseous phase for sometime and then penetrates in to the liquid phase (penetration theory ). It is then replaced by another element coming from the bulk of the liquid phase. The periods of contacts of the elements with the other phase are assumed to be equal . It was further supposed during this short period, which varied between 0.01 and 0.1 s, absorption took place as a result of unsteady state molecular diffusion into the liquid. In the case of turbulent flow this mechanism is undoubtedly more probable , but it is hardly acceptable for laminar flow.

From these assumptions Higbie derives the formula for the average flux over the time interval 0 to

The way in which the concentration gradient builds up as a result of exposing a liquid-initially pure-to

the action of soluble gas is shown in Figure 10.6 which is based on Higbie’s calculations.The percentage saturation of liquid is plotted against the distance from the surface for a number of

exposure times. Initially only the surface layer contains solute and the concentration changes abruptly from 100 percent to 0 percent at the surface. For progressively longer exposure times the concentration profile develops as shown, until after an infinite time the whole of liquid becomes saturated. The shape of profile is such that at any time the effective depth of liquid which contains an appreciable concentration of solute can be specified, and hence the theory is referred to as the penetration theory.

Higbie was the first to apply this equation to gas absorption in a liquid, showing that diffusion molecules will not reach the other side of a thin layer if the contact time is short. The depth of penetration, defined as the distance at which concentration change is 1 percent of the final value, is . For Dv = 10-5 cm2/s and tT = 10 S, the depth of penetration is only 0.036cm. In gas absorption equipment, drops and bubbles often have very short lifetimes because of coalescence, and penetration theory is likely to apply. An alternate form of penetration theory was developed by Danckwerts, who considered the case where elements of fluid at transfer surface are randomly replaced by fresh fluid from the bulk stream. An exponential distribution of ages or contact time results, and the average transfer coefficient is given by:

Where S is the fractional rate of surface renewal, in S-1

The equation 17.55) predicts that the coefficient varies with one half power of the diffusivity and give almost same value for a given average contact time.

The various forms of penetration theory can be classified as surface renewal models, implying either formation of a new surface at frequent intervals or replacement of fluid elements at the surface with fresh fluid from the bulk. The time tT and its reciprocal, the average rate of removal, are functions of the fluid velocity, the fluid property, and geometry of the system and can be predicted in only a few special cases.

Random Surface RenewalDanckwertz modifies the assumption of Higbie and assumes , instead , that the contact period of individual elements

with the gas phase is not the same but only a matter of chance .Hence he dervies the formula

Where s is the fraction of the surface , renewed in a unit of time relative to the total surface

In all these theories the quantity 1s proportional to . The basic difference lies , first of all , in a different relation of to the diffusion coefficient

Danckwertz showed how his theory could be useful when applied to the problem of absorption connected with a chemical reaction.

The Film Penetration TheoryA theory which incorporates some of the principles of both the two film theory and the penetration theory has been

proposed by Toor and Marchello. The whole of 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 fluid to the interface by the action of eddy currents. Mass transfer then takes place as in the penetration theory, except that resistance is confined to the finite film and the material which traverses the film is immediately completely mixed with the bulk of the fluid. For short times of exposure, when none of the diffusing material has reached the far side of the layer, the process is identical to that postulated in penetration theory. For prolonged period of exposure when a steady gradient has developed, conditions are similar to those considered in the two film theory.

The concentration profiles near an interface on the basis of:a) The film theory (steady-state)b) Penetration theoryc) The film penetration theoryAre shown is Figure 10.7

MASS TRANSFER COEFFICIENTSOn the basis of each of the theories discussed, the rate of mass transfer in the absence of bulk flow is

directly proportional to the driving force, expressed as molar concentration difference, and, therefore,NA = hD ( CAi - CAo ) (10.14a)

Where hd is a mass transfer coefficient. In the two film theory, hd is directly proportional to the diffusivity and inversely proportional to the film thickness. According to penetration theory it is proportional to the square root of the diffusivity and, when all surface elements are exposed for an equal time, it is inversely proportional to the square root of time of exposure, when random surface renewal is assumed, it is proportional to the square root of the rate of renewal. In the film penetration theory, the mass transfer coefficient is a complex function of the diffusivity, the film thickness, and either the time of exposure or rate of renewal of surface.

The penetration and film penetration theory have been developed for conditions of equimolecular diffusion only, the equations are too complex to solve explicitly for transfer through a stationary carrier gas. For gas absorption, therefore, they apply only when concentration of material undergoing mass transfer is low. On the other hand, in the two film theory the additional contribution to the mass transfer which is caused by bulk flow is easily calculated and hd is equal to (D/L) (CT/CBm) instead of D/L

When the film theory is applicable to each phase (the two-film theory), the process is steady state throughout and the interface composition does not then vary with time. For this case the two film coefficients can readily be combined. Because material does not accumulate at the interface, the mass transfer rate on each side of the phase boundary will be same and for two phases it follows that:

NA = hD1 ( CAo1 - CAi1 ) = hD2 ( CAi2 – CAo2 ) (10.150)

If there is no resistance to transfer at interface CAi1 and CAi2 will be corresponding values in the phase equilibrium relationship.

Usually, the values of the concentration at the interface are not known and the mass transfer coefficient is considered for the overall process. Overall transfer coefficients are then defined.

NA = K1 (CAo1 – CAe1) = K2 (CAe2 – CAo2) (10.15)

Where CAe1 is the concentration in phase 1 in equilibrium with CAo2 in phase 2, and CAe2 is the concentration in phase 2 in equilibrium with CAo1 in phase 1. If the equilibrium relationship is linear

Where H is a proportionality constant. The relationship between the various transfer coefficients are obtained as follows from equations 10.150 and 10.152

is the average slope of equilibrium curve and, when solution Obeys Henry Law,

Similarly

It follows, that when hD1 is larger compared with hD2, K2 and hd2 are approximately equal, and when hD2 is large compared with hD1, K1 and hD1 are almost equal. These relations between the various coefficients are valid provided that transfer rate is linearly related to the driving force and that equilibrium relationship is a straight line.

Counter-Current Mass Transfer and Transfer UnitsMass transfer processes involving two fluid streams are frequently carried out in a column,

counter-current flow is usually employed although co-current flow may be advantageous in some circumstances. There are two principal ways in which the two streams may be brought into contact in a continuous process so as to permit mass transfer to take place between them, and these are termed stage-wise processes and continuous differential contact processes.

Stage wise ProcessesIn a stage wise process the fluid streams are mixed together for a period long enough for them

come close to thermodynamic equilibrium, following which they are separated, and each phase is then passed counter currently to the next stage, where the process is repeated. An example of this type of process is the plate-type of distillation column in which a liquid stream flows down the column, overflowing from each plate to the one below as shown in Fig. 10.8. The vapour passes, up through the column, being dispersed into the liquid as it enters the plate, it rises through the liquid surface, disengaging from the liquid in the vapour space above the liquid , and then passes up wards to repeat the process on the next plate. Mass transfer takes place as the bubbles rise through the liquid, and as liquid droplets are thrown up into the vapour space above the liquid-vapour interface. On an ideal plate, the liquid and vapour stream leave in thermodynamic equilibrium with each another. In practice equilibrium may not be achieved and a plate efficiency, based on the composition in either the liquid or vapour stream is defined as the ratio of the actual change in composition to that which would have been achieved in an ideal stage. The use of stage wise processes for distillation, gas absorption and liquid-liquid extraction will be studied later on.

Continuous Differential contract processesIn this process, the two streams flow continuously through the column and undergo a continuous

change in composition. At any location are in dynamic rather than thermodynamic equilibrium. Such processes are frequently carried out in packed columns, in which the liquid (or one of the two liquids in the case of liquid-liquid extraction process) wets the surface of packing, thus increasing the interfacial area available for mass transfer and, in addition, promoting high film mass transfer coefficients within each phase.

In a packed distillation column, the vapour stream rises against the downward flow of a liquid reflux, and a state of dynamic equilibrium is setup in a steady state process. The more volatile constituent is transferred under the action of concentration gradient from the liquid to interface where it evaporates and then is transferred into the vapour stream. The less volatile component is transferred in the opposite direction and if the molar latent heats of the components are equal, equimolecular counter diffusion takes place.

In a packed absorption column, the flow pattern is similar to that in a packed distillation column but the vapour stream is replaced by a mixture of carrier gas and solute gas. The solute diffuses through the gas phases to the liquid phase where it dissolves and is then transferred to the bulk of liquid. In this case there is no mass transfer of the carrier fluid and transfer rate of solute is supplemented by the bulk flow.

In a liquid-liquid extraction column, the process is similar to that occurring in an absorption column except that both streams are liquids, and the lighter liquid rises through the denser one.

In distillation, equimolecular counter diffusion takes place if the molar latent heats of the components are equal and molar rate of flow of the two phases then remain approximately constant throughout the whole height of the column. In gas absorption, however, mass transfer rate is increased as a result of bulk flow and, at high concentrations of soluble gas, the molar rate of flow at the top of the column will be less than that at the bottom. At low concentrations, however, bulk flow will contribute very little to mass transfer and, in addition, flow rate will be approximately constant over the whole column.

The conditions existing in a column during the steady state operation of counter-current process are shown in Fig. 10.9. The molar rates of flow of the two streams are G1 and G2, which will be taken as constant over the whole column. Suffixes 1 and 2 denote the two phases, and suffixes t and b relate to the top and bottom of the column.

If the height of the column is z, its total cross-sectional area S, and a is interfacial area between the two phases per unit volume of the column, then the rate of transfer of a component in a height dZ of column is given by hD1:

Where G’1 is the molar rate of flow per unit cross-sectional of column. The exact interfacial area cannot normally be determined independently of transfer coefficient, and therefore values of the product hD1 a are usually quoted for any particular system.

The left side of equation 10.157 is the rate of change of concentration with height for unit driving force, and is therefore a measure of the efficiency of the column, and in this way, a high value of (hD1 a CT) / G’1 is associated with a high efficiency. The reciprocal of this quantity is G1’/ (hD1 a CT) which has linear dimensions and is known as the height of transfer unit H1 (HTU).

Rearranging equation (10.157) and integrating gives:

The right hand side of equation 10.158 is the height of the column divided by HTU and is known as the number of transfer units N1. It is obtained by evaluating the integral on left hand side of equation.

Therefore:In some cases, such as evaporation of liquid at an approximately constant temperature or the

dissolving of highly soluble gas in a liquid, the interface concentration may be either substantially constant or negligible with that of bulk. In such cases, the integral of left hand side of equation (10.158) may be evaluated directly to give:

Where b and t represent values at the bottom and the top as shown in Figure 10.9

Thus H1 is the height of the column over which driving force changes by a factor of e

Equation (10.156) can be written in terms of film coefficient for the second phase (hD2) or either of overall coefficients (K1, K2). Transfer units based on either film coefficient or overall coefficients can therefore be defined, and the following equations are analogues of equation (10.159)

Number of transfer unit based on phase 2N2 = Z/ H2 (10.162)

Number of overall transfer units based on phase 1.No1 = Z/Ho1 (10.163)

Number of overall units based on phase 2. No2 = Z/Ho2 (10.164)

Using this notation the introduction of a into the suffix indicates and overall transfer unit.The equation for H1, H2, Ho1, Ho2 are of the following form

If one phase is gas, as in gas absorption for example. It is often more convenient to express concentrations as partial pressures in which case:

Where in SI units, G’1 is expressed in K mole/m2 S, kg in

K mole/[m2.S (N/m2)], P in N/m2, and a in m2/m3

The overall values of HTU may be expressed in terms of film values by using equation 10.153, which gives the relation between the coefficients.

Substituting for coefficients in terms of the HTU:

The advantage of using the transfer unit in preference to the transfer coefficient is that the former remains much more nearly constant as flow conditions are altered. This particular important in problem of gas absorption where the concentration of the solute gas is high and flow of gas is high and flow pattern changes because of the change in the total rate of flow of gas at different sections. In most cases the coefficient is proportional to the flow rate slightly less than unity and therefore HTU is substantially constant.

Liquid Phase DiffusivitiesValues of the diffusivities of various materials in water are given in the literature. Where

experimental values are not available, it is necessary to use one of the predictive methods which are available. A useful equation for the calculation of the liquid phase diffusivities of dilute solutions of non-electrolytes has been given by Wilke and Chang. Using SI units.

Where D is the diffusivity of solute A in solvent B (m2/S) ФB is the association factor for the solvent (2.26 for water, 1.96 for methanol, 1.5 for ethanol and 1.0 for unassociated solvents such as hydrocarbons and ethers)

MB is the molecular weight of the solvent.

µ is the viscosity of the solution

T is the temperature (K) andVA is the molecular volume of the solute (m3/K mol).

It may be noted that for water a value of 0.0756 m3/K mol should be used.Equation 10.96 does not apply to either electrolytes or to the concentrated solutions.









If the bulk flow is not negligible, the from (3.30)

Which upon integration yields

Where c = total molar concentration or molar density (c = 1/v = ρ/M), v is molar volume and M is molecular weight).

In practice, the ratios DAB/δ in (3.188) and DAB/ δ (1-XA) lm in 3.189 are replaced by mass transfer coefficient kc and k’c respectively, because the film thickness, δ, which depends on the flow conditions, is not known and subscript c refers to a concentration driving force.

The film theory which is easy to understand and apply is often criticized because it appears to predict that the rate of mass transfer is directly proportional to the molecular diffusivity. This dependency is odd with experimental data which indicates a dependency of Dn, where n ranges from 0.5 to 0.75. Regardless of whether the criticism of the theory is valid, the theory has been and continuous to be widely used in the design of mass transfer separation equipment.

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Mass Transfer, Malik Nasir Wirali