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 * =   ∑iⁱ

         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_AB   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_AB   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 L² / 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 cm²/sec; the SI unit being m² / s or m²s^-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 

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/′ y₂ /D_1-2 + ′ y₃ /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  y₂ y ′ = 2 y₂ +  y₃ +  ....... 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(C₁ - C₂) /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/m³ and zero correspondingly. Assuming the diffusion coefficient for hydrogen in steel as 0.248 x10^-12m²/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^-12m²/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 (C₁ - C₂) /AB A z A

   N_A                           =   0.248 x10^-12 (1.2- 0) 0.016   

                                  =   18.6 x 10^-12kg mole/s.m²

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 ∂²C A /∂ z²

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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