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*Introduction *

In this chapter, we will study about mass transfer procedure unlike heat convey that has to do through temperature gradient, mass transfer is as a consequence of concentration gradient. In a system consisting of one or more components whose concentrations fluctuate from point to point, there is natural tendency for the transport of different species from the region of elevated to those of low concentrations. Therefore, the procedure of transfer of mass as an effect of the species concentration difference in a system/ mixture is said mass transfer.

*Mass Transfer *

Mass transfer is the net association of mass from one location, generally meaning a stream, phase, fraction or component, to another. Mass transfer happens in many chapter operation processes, these as absorption, evaporation, adsorption, drying, precipitation, membrane filtration, and distillation. Mass transfer is utilized via dissimilar scientific disciplines for different procedures and mechanisms. The phrase is usually utilized in engineering and applied sciences for physical procedures that engage diffusive and convective transport of chemical species inside physical systems.

Several ordinary instances of mass transfer procedures are the evaporation of water from a pond to the atmosphere, the sanitization of blood in the kidneys and liver, and the distillation of alcohol. In industrial procedures, mass transport process comprise division of chemical elements in distillation columns, absorbents these as scrubbers, adsorbents such as started carbon beds, and liquid-liquid extraction. Mass transfer is frequently coupled to extra transport processes, for instance in industrial chilling towers. These towers pair heat transfer to mass move via permitting hot water to flow in contact through hotter air and evaporate as it absorbs heat from the air.

*Modes of Mass Transfer *

The mechanism of mass transfer based significantly on the dynamics of the system in that it happens. Like those of heat transfer, there are dissimilar modes of mass transfer. Such are:

(i) Mass transfer via diffusion

(ii) Mass transfer via convection

(iii) Mass transfer via change of phase

Mass Transfer by Diffusion (Molecular or Eddy Diffusion)

The transport of water on a microscopic level as a consequence of diffusion from a region of elevated concentration to a region of low concentration in a system/mixture of liquids or gases is said molecular diffusion. It happens whenever a material disperses through a layer of stagnant fluid and might be due to concentration, temperature or pressure gradients. In a gaseous mixture, molecular diffusion happens due to random motion of the molecules.

When one of the diffusing fluids is in turbulent motion, the eddy diffusion takes place. Mass transfer is speedier through eddy diffusion than via molecular diffusion. An example of an eddy diffusion procedure is dissipation of smoke from a smoke stack. Turbulence reasons mixing and transfer of smoke to the ambient air.

Mass Transfer by Convection

Mass transfer through convection engages transfer between a moving fluid and a surface, or between 2 comparatively immiscible flowing fluids. The convective mass transfer based on the transport properties and on the dynamic (laminar or turbulent) traits of the flowing fluid. A good illustration is the evaporation of ether.

Mass Transfer by Change of Phase

Mass transfer happens whenever a transform from one place to another takes place. The mass transfer in these a case happens due to simultaneous action of convection and diffusion. Several instances are:

(i) Hot gases escaping from the chimney rise through convection and then spread into the air above the chimney.

(ii) Mixing of water vapour through air during evaporation of water from the lake surface (partly via convection and partly through diffusion).

(iii) Boiling of water in open air; there is 1^{st} transfer of mass from liquid to vapour state and then vapour mass from the liquid interface is transferred to the open air through convection in addition to diffusion.

*Mass transfer operations *

When a system encloses 2 or more components whose concentrations fluctuate from point to point, there is a normal tendency for mass to be transferred, minimizing the concentration dissimilarities inside a system.

The transport of one constituent from a region of elevated concentration to that of a lower concentration is said mass transfer. The transfer of mass inside a fluid mixture or across a phase boundary is a procedure, which plays a main role in many industrial processes. Instances of these processes are:

(i) Dispersion of gases from stacks.

(ii) Removal of pollutants from plant discharge streams by absorption or adsorption.

(iii) Stripping of gases from waste water.

(iv) Neutron diffusion within nuclear reactors.

(v) Air conditioning.

(vi) Gas absorption.

(vii) Distillation.

(viii) Extraction.

(ix) Adsorption.

(x) Drying.

Many of day-by-day air experiences also involve mass transfer, for example:

(i) A lump of sugar added to a cup of coffee eventually dissolves and then eventually diffuses to make the concentration uniform.

(ii) Water evaporates from ponds to increase the humidity of passing- air-stream.

(iii) Perfumes present a pleasant fragrance which is imparted throughout the surrounding atmosphere.

*Properties of Mixtures *

Mass transfer always engages mixtures. Consequently, we must account for the deviation of physical properties that normally exist in a specified system. Whenever a system encloses 3 or more components, as many industrial fluid streams do, the trouble becomes bulky very rapidly. The conventional engineering approach to problems of multi- component system is to attempt to decrease them to representative binary (2 components) systems.

In order to appreciate the future discussions, let us 1^{st} consider definitions and relations that are often utilized to clarify the role of components inside a mixture.

*Concentration of Species *

Concentration of species in multi-component mixture can be expressed in many ways. For species A, mass concentration signified through ρA is described as the mass of A, mA per unit volume of the mixture.

ρ_{A}= m_{A} /V

The total mass concentration density ρ is the sum of the total mass of the mixture in unit volume:

ρ = ∑ ρ_{i}

i

where ρ_{i }is the concentration of species i in the mixture.

Molar concentration of, A, C_{A} is described as the number of moles of A present per unit volume of the mixture.

By definition,

Number of moles = mass of A/ molecular weight of A

n_{A }= mA/M_{A}

Therefore from (1) & (2)

C_{A} = n_{A}/ V = ρ_{A} M_{A}

For ideal gas mixtures,

n_{A}= p_{A} V/ R T [from Ideal gas law PV = nRT]

C_{A}= n_{A} /V = p_{A} / R T

Where p_{A} is the partial pressure of species A in the mixture. V is the volume of gas, T is the absolute temperature, and R is the universal gas steady.

The total molar concentration or molar density of the mixture is given via

*Velocities *

C = ∑ C_{i} i

In a multi-component system the diverse species will usually shift at different velocities; and evaluation of velocity of mixture needs the averaging of the velocities of each species present.

If ν_{i} is the velocity of species i by respect to stationary fixed coordinates, then mass-average velocity for a multi-component mixture described in terms of mass concentration is,

ν = ∑ ρ ν ∑ ρν

i i i = i i i

∑ ρ_{i} ρ

i

Likewise, molar average velocity of the mixture ν* is explained as

C V

ν * = ∑i^{i}

i C

For most engineering problems, there will be little difference in ν * and ν and so the mass average velocity, ν, will be utilized in all additional discussions.

The velocity of a particular species relative to the mass-average or molar average velocity is termed as diffusion velocity.

Diffusion velocity = ν_{i} - ν

The mole fraction for liquid and solid mixtures, x_{A}, and for gaseous mixtures, y_{A}, are the molar concentration of species A divided by the molar density of the mixtures.

x_{A}= C_{A}/ C (liquids and solids).

y_{A}= C_{A}/ C (gases).

The sum of the mole fractions, through definition must equal 1;

(for example) = 1

∑ x_{i }i

i = 1

∑ y_{i}

i

Similarly, mass fraction of A in mixture is;

*Diffusion Flux *

w_{A}= ρ_{A }/ ρ

Just as momentum and energy (heat) transfers have two mechanisms for transport - molecular and convective, so does mass transfer. However, there are convective fluxes in mass transfer, even on a molecular level. The reason for this is that in mass transfer, whenever there is a driving force, there is always a net movement of the mass of a particular species that consequences in a bulk motion of molecules. Of course, there can also be convective mass transport due to macroscopic fluid motion.

The mass (or molar) flux of a given species is a vector quantity denoting the amount of the particular species, in either mass or molar units, that passes per given increment of time through a unit area normal to the vector.

Diffusivity

An empirical relation for the diffusional molar flux, first postulated via Fick, often termed to as Fick's first law, describes the diffusion of component A in an isothermal, isobaric system. For diffusion in only the Z-direction, the Fick's rate equation is

J_{A}= - D_{A}B dC_{A}

dZ

Where D_{AB} is diffusivity or diffusion coefficient for component A diffusing through component B, and dC_{A}/dZ is the concentration gradient in the Z-direction.

A more common flux relation that isn't restricted to isothermal, isobaric system could be written as

J_{A} = - C D_{A}B dy_{A} / dZ

Fick's Law proportionality, D_{AB}, is known as mass diffusivity (simply as diffusivity) or as the diffusion coefficient. DAB has the dimension of L^{2} / t, identical t the fundamental dimensions of the other transport properties: Kinematic viscosity, νη = (µ / ρ) in momentum transfer, and thermal diffusivity, α (= k / ρ C ρ) in heat transfer.

Diffusivity is normally reported in cm^{2}/sec; the SI unit being m^{2} / s or m^{2}s^{-1}.

Diffusivity based on pressure, temperature, and composition of the system.

In table, typical range of values of D_{AB }is specified for gas, liquid, and solid systems.

Diffusivities of gases at low density are almost composition independent, while they enhance through temperature and fluctuate inversely through pressure. Liquid and solid diffusivities are sturdily concentration dependent, while they enhance through temperature.

Table: Common Range of Values of Diffusivity

Type of System |
Diffusivities Range (m |

Gases |
5 x 10 |

Liquids |
10 |

Solids |
5 x 10 |

In the absence of experimental data, semi-theoretical expressions have been developed which give approximation, sometimes as valid as experimental values, due to the difficulties encountered in experimental measurements.

Diffusivity in Gases

Pressure dependence of diffusivity is given by

D_{AB}∝^{1 }(for moderate ranges of pressures, up to 25 atm).

P

And temperature dependency is according to

D_{AB}∝ T ^{3/2}

Diffusivity of a component in a mixture of components can be computed using the diffusivities for the diverse binary pairs included in the mixture. The relation specified via Wilke is 1

D_{1-mixture }= 1/′ y_{2} /D_{1-2} + ′ y_{3} /D_{1-3} +........... + y ′ /^{n} D_{1 -n }

Where D_{1-mixture }is the diffusivity for component 1 in the gas mixture; D_{1-n} is the diffusivity for the binary pair, component 1 diffusing through component n; and y_{n}′ is the mole fraction of component n in the gas mixture evaluated on a component -1 - free basis, that is y_{2} y ′ = 2 y_{2} + y_{3} + ....... y_{n}

Diffusivity in Liquids

Diffusivity in liquid is exemplified via the values specified in table. Most of these values are nearer to 10-5 cm2 / sec, and about ten thousand times lower than those in dilute gases. This characteristic of liquid diffusion often limits the overall rate of processes accruing in liquids (such as reaction between 2 components in liquids).

In chemistry, diffusivity limits the rate of acid-base reactions; in the chemical industry, diffusion is responsible for the rates of liquid-liquid extraction. Diffusion in liquids is important because it is slow.

Certain molecules diffuse as molecules, while others which are designated as electrolytes ionize in solutions and diffuse as ions. For instance, sodium chloride (NaCl), diffuses in water as Na^{+} and Cl^{-} ions.

Though each ion has a different mobility, the electrical neutrality of the solution indicates the ions must diffuse at the same rate; accordingly, it is possible to speak of a diffusion coefficient for molecular electrolytes such as NaCl. Though, if several ions are present, the diffusion rates of the individual cations and anions must be considered, and molecular diffusion coefficients have no meaning.

Diffusivity varies inversely with viscosity when the ratio of solute to solvent ratio exceeds five. In extremely high viscosity materials, diffusion becomes independent of viscosity.

Diffusivity in Solids

Typical values for diffusivity in solids are shown in table 1. One outstanding characteristic of these values is their small size, usually thousands of time less than those in a liquid, which are in turn 10,000 times less than those in a gas. Diffusion plays a major role in catalysis and is important to the chemical engineer and industrial chemist. For metallurgists, diffusion of atoms within the solids is of more importance.

Diffusion in Solids

In certain units operations of chemical engineering such as in drying or in absorption, mass transfer takes place between a solid and a fluid phase. If the transferred species is distributed uniformly in the solid phase and forms a homogeneous medium, the diffusion of the species in the solid phase is said to be structure-independent. In this case diffusivity or diffusion coefficient is direction - independent.

At steady state, and for mass diffusion which is independent of the solid matrix structure, the molar flux in the z direction according to Fick's

Law is specified via:

N_{A}= - D_{AB } dC_{A} /d z = constant,

Integrating the above equation,

N_{A }= D(C_{1} - C_{2}) /AB A z A

That is similar to the expression attained for diffusion in a stagnant fluid by no bulk motion (for example N = 0).

Instance: A steel rectangular container having walls 16mm thick is utilized to store hydrogen gas at elevated pressure. The molar concentrations of hydrogen in the steel at the inside and outside surfaces are 1.2kg mole/m^{3 }and zero correspondingly. Assuming the diffusion coefficient for hydrogen in steel as 0.248 x10^{-12}m^{2}/s, Compute the molar diffusion flux for hydrogen through the steel.

Solution: specified Z= 16mm= 0.016m, C_{A1}= 1.2kg mole/m3, CA2 =0, DA = 0.248 x10^{-12}m^{2}/s

Assuming 1 dimensional and steady state situation, the molar diffusion flux rate in the steel is specified by Fick's law of diffusion

Molar diffusion Flux = D (C_{1} - C_{2}) /AB A z A

N_{A} = 0.248 x10^{-12} (1.2- 0) 0.016

= 18.6 x 10^{-12}kg mole/s.m^{2}

Diffusion in Process Solids

In some chemical operations, such as heterogeneous catalysis, an important factor, affecting the rate of reaction is the diffusion of the gaseous component through a porous solid. The effective diffusivity in the solid is reduced below what it could be in a free fluid, for two reasons. First, the tortuous nature of the path increases the distance, which a molecule must travel to advance a given distance in the solid. Second, the free cross - sectional area is restricted. For many catalyst pellets, the effective diffusivity of a gaseous component is of the order of one tenth of its value in a free gas.

If the pressure is low enough and the pores are small enough, the gas molecules will collide with the walls more frequently than with each other. This is known as Knudsen flow or Knudsen diffusion. Upon hitting the wall, the molecules are momentarily absorbed and then given off in random directions. The gas flux is reduced by the wall collisions.

By the use of the kinetic flux, the concentration gradient is independent of pressure; whereas the proportionality constant for molecular diffusion in gases (diffusivity) is inversely proportional to pressure.

Knudsen diffusion occurs when the size of the pore is of the order of the mean free path of the diffusing molecule.

Transient Diffusion

Transient processes, in that the concentration at a specified point varies through time, are referred to as unsteady state procedures or time - dependent processes. This deviation in concentration is connected by a variation in the mass flux.

Such usually fall into 2 categories:

i) The procedure that is in an unsteady state only during its initial startup.

ii) The process which is in a batch operation throughout its operation.

In unsteady state procedures, there are 3 variables - concentration, time and position. Therefore the diffusion process must be described through partial rather than normal differential equations.

Even though the differential equations for unsteady state diffusion are simple to set up, most solutions to such equations have been bounded to situations involving simple geometries and boundary conditions, and a steady diffusion coefficient.

Many solutions are for one-directional mass transfer as defined by Fick's second law of diffusion:

∂ C_{A} / ∂ t = D_{AB} ∂^{2}C A /∂ z^{2}

This partial differential equation describes a physical condition in that there is no bulk-motion contribution, and there is no chemical reaction. This situation is encountered when the diffusion takes place in solids, in stationary liquids, or in system having equimolar counter diffusion. Due to the very slow rate of diffusion inside liquids, the bulk motion contribution of flux equation (for instance, y _{A}∑ N_{ i}) approaches the value of zero for dilute solutions; accordingly this system also satisfies Fick's 2^{nd} law of diffusion.

The solution to Fick's second law generally has one of the two standard forms. It might appear in the form of a trigonometric series which converges for large values of time, or it might engage series of error functions or related integrals that are most suitable for numerical evaluation at small values of time. Such solutions are commonly attained via using the mathematical methods of division of variables or Laplace transforms.

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