Mass transfer & diffusion hari

Basics of mass transfer operation and diffusion phenomenon

1. MASS TRANSFER & DIFFUSION PRESENTED BY:- HARERAM MISHRA GCT/1830155 EMAIL ID:[email protected] DEPARTMENT OF CHEMICAL ENGINEERING SANT LONGOWAL INSTITUTE OF ENGG. &TEC & TECHNOLOGY LONGOWAL,SUNGRUR,{PUB.},148106

2. CONTENTS:- 1.INTRODUCTION TO MASS TRANSFER 2.MECHANISM OF MASS TRASFER 3.CLASSIFICATION OF MASS TRANSFER 4.CONVECTIVE MASS TRANSFER 5.MASS TRANSFER COEFFICIENT 6.THEORIES OF MASS TRANSFER 7.MOMENTUM ,HEAT AND MASS TRANSFER ANALOGIES 8.INTERPHASE MASS TRANSFER 9.DIFFUSION & MOLECULAR DIFFUSION 10.KNUDSEN ,SURFACE&SELF DIFFUSION

3. INTRODUCTION TO MASS TRANSFER:- DEFINITION:- “ The process of transfer of mass as result of concentration difference of a component in a mixture or two phase in contact is called mass transfer”. Ex:- Evaporation of water from a pool of water In to a stream of air flowing over the water surface. Absorption of oxygen of air into blood occurs in the lungs of a animal in the process of respiration. In mass transfer operations, mass transfer may occur:- (a) In one direction e.g Gas Absorption. (b) In opposite direction e.g Distillation. (c) With simultaneous heat transfer e.g -Drying & crystallisation . (d ) With simultaneous chemical reaction e.g absorption of CO2 in an aqueous solution of KOH. (e) With a exchange of one or more components.

4. The phenomena those must exit in a mass transfer operation are:- (a) At least two phases must come in contact with each other (b) Materials must flow from one phase to other. (c) A part of total flow of material from one phase to other must occur by molecular diffusion. MECHANISM OF MASS TRANSFER:- (a) Equilibrium between the phases is attained after a sufficiently long time of phase contact between them. (b) Material transfer is caused by the combined effect of molecular diffusion and turbulence. (c) There is no resistance to mass transfer at the phase interphase (because of the existence of equilibrium at the interphase). (d) Rate of mass transfer is evaluated by deviation/departure from equilibrium concentration.

5. CLASSIFICATION OF MASS TRANSFER OPERATIOS:- ACCORDING TO THE PHASE IN CONTACT:- PHASES IN CONTACT MASS TRANS. OPERATIONS DRIVING FORCE DEGREE OF F-DOM F=C-P+2 (P-RULE) VARIALES OF OPERATIONS Liquid- vapour (gas) Distillation (fractionation) Vapour pressure diff. F=2-2+2=2 Liquid-gas Gas absorption Stripping Humidification Dehumidification Solubility diff. Solubility diff. Conc. diff. Conc. difference F =3-2+2=3 Liquid-solid Crystallisation Leaching Adsorption Liquid-liquid Extraction Diff. in solubil - Ity of solute Solid- vapour Sublimation Solid-gas Adsorption Solid (wet) gas (air) Drying F=3-2+2=3

6. CONVECTIVE MASS TRANSFER:- INTRODUCTION:- Mass transfer occurring under the the influence of motion in a fluid medium is called the ‘convective mass transfer’. Ex:- A simple mechanism of dissolution of sugar crystals in a stirred cup of water. The water just in contact with a crystals i.e. liquid at solid- liquid interface, get saturated with the sugar almost instantly. The dissolved sugar diffuses from the interface to the bulk of liquid through a thin layer or film of the solution adhering the crystals. More sugar dissolves in the liquid in the liquids simultaneously the interface. The thickness of the film decreases, if the stirring rate is more rapid, which results in fast or quicker rate of dissolution or rate of mass transfer. TYPES OF CONVECTIVE MASS TRANSFER:- Similar to the convective heat transfer, there are two types of convective mass transfer:- 1. forced convection mass transfer 2. free or natural convection mass transfer

7. 1.Free convection mass transfer:- In this mechanism of mass transfer,the motion or flow in the medium is caused by the diff. in density. 2. forced convection mass transfer:- This type of mass transfer involves that the motion in the medium is poroduced by an external agency such as pump, blower and agitator. Convective mass transfer is strongly influenced by the flow field,If the flow field is well defined,then mass transfer rate can be determined by mathematical analysis of mass and momentum balance sach as in the case of- Dissolution of the solid coated on a plate in a fluid flowing over it, the absorption of solute gas in a laminar liquid film falling down a wall. For more complex situation sach as the dissolution of solid in a mechanically stirred vessel,theor . calculation is difficult bcz flow field is complex.

8. MASS TRANSFER COEFFICIENT:- The mass transfer coefficient is defined as:- Rate of mass transfer conc. Driving force Rate of mass transfer ∞ area of contact between the phases W A ∞ a∆C A W A = k c a∆C A Where k c = mass transfer coefficient And,- W A = aN A = k C a∆C A So, N A = k c ∆C A Mass transfer coefficient,k C = N A / ∆C A Or k C = molar flux/ conc. DV For the purpose of comparison- Heat transfer coefficient, h=Heat flux/ temp.driving force. Mass transfer resistence ∞ 1/mass transfer coefficient Local flux/local driving force=local mass transfer coefficient. Avg.flux / Avg.driving force=Avg. mass transfer coefficient Units of k = unit of molar flux/ unit of driving force = kmol / m.m.s /unit of driving force If driving force is taken as conc. Diff.,then -unit of k = m.m /s(as unit of vel.) According to the units of D.F,units of mass trans.coff .(k ) will be changed.  

9. types of mass transfer coefficient:- Different types of mass transfer coefficients have been defined depending on:- (A) Whether mass transfer occurs in gas phase or in liquid phase (B) The choice of driving force ,and (C) Whether it is a case of diffusion of A through non- diffusing B or a case of counter-diffusion. the case of counter/concurrent diffusion Differents types of mass transfer coefficients are:- Diffusion of A through non-diffusing B Equimolar counter diffusion of A&B Units of mass Transfer coff . Flux,N A Mass transfer coefficient Flux,N A Mass transfer coefficient Gas -phase mass transfer coff . k G (p A1 -p A ) k G =D AB P/RT δ p BM K y (y A1 -y A2 ) k y =D AB P 2 /RT δ p BM K c (C A1 -C A2 ) k C =D AB P/ δp BM k’ G (p A1 -p A2 ) k’ G =D AB / δ R K’ y (y A1 -y A2 ) k’ y =D AB P/ δ RT k’ C (C A1 -C A2 ) k’ c =D AB / δ Mol /t(A)∆ p A Mol /t(A)∆ y A Mol /t(A)∆C A Liquid -phase mass transfer coff . k L (C A1 -C A2 ) k L =D AB / δ x BM k X (x A1 -x A2 ) k x =CD AB / δ x BM k’ L (C A1 -C A2 ) k’ L =D AB / δ k’ x (x A1 -x A2 ) k’ x =CD AB / δ Mol /t(A)∆C A Mol /t(A)∆ y A Conversion:- k G RT = RTk Y /P= k c ; k L = k x / C av k’ C = k’ G RT = RTk’ y /P; k’ L = k’ x / C av

10. DIMENSIONLESS GROUPS IN MASS TRANSFER:- There are most important dimensionless groups in mass transfer are:- Dimensionless groups and their physical significance in mass transfer Analogous groups in heat transfer and their physical significance Reynolds number,Re = interia /viscous force =lv ρ /µ=lv/ ν The same Schmidt number,Sc =momentum/mol. Diffu . =µ/ ρ D= ν /D Pr = c P /k=(µ/ ρ )/(k/ ρ c P )=v/ α =momentum diffusivity/thermal diffusivity Sherwood number,Sh =convective mass flux/diffusive flux across a layer of thick.l = k L l /D= k L ∆C /(D/l)∆C Nu=hl/k=convective heat flux/conduction heat flux across a thickness of l Stanton number,St M =convective mass flux/flux due to bulk flow of a medium = Sh /(Re)( Sc )= k L /v= k L ∆C / v∆C St H =Nu/(Re)( Pr )= h∆T / v∆T =convective heat flux/heat flux due to bulk flow Peclet number,Pe M =flux due to bulk flow of medium/diff. flux across a layer of l =(Re)( Sc )=lv/D= v∆C /(l/D)∆C Pe H =(Re)( Pr )=(v ρ c P )∆T/(k/)∆T=heat flux due to bulk flow/conduction heat flux across a thickness l Colburn factor,j D = St M ( Sc ) 2/3 = Sh /(Re)( Sc ) 1/3 The same Grashof number,Gr =l 3 ∆ ρ g/µ ν The same Lewis number,Le = Sc / Pr

11. THEORIES OF MASS TRANSFER :- There are no. of theories of mass transfer which aim at visualizing the mechanism and developing the expression for mass transfer coefficient theoretically.In fact,any such theory is based on a conceptual model for mass transfer. These important theories are:- 1. The Film theory (film model) 2. The Penetration Theory 3. The surface renewal theory THE FILM THEORY (FILM MODEL):- This theory was developed by Whitman,in 1923 . According to this theory,when the mass transfer occurred from a solid surface to a flowing fluid,even though the bulk liquid is in turbulent motion ,the flow near the wall may be considered to be laminar . The concentration of dissolved solid (A) will decreses from C ai, at the solid liquid interphase to C ab at the bulk of liquid. In reality the concentration profile will be very steep near the solid surface,where the effect of turbulence is practically absent. Molecular diffusion is responsible for mass transfer near the wall while convection dominates a little away from it. This theory is based on following assumptions such as:-

12. (a) Mass transfer occurs by purely molecular diffusion through a stagnant fluid layer at the phase boundary. Beyond this film,the fluid is well mixed having a concentration which is the same at that of bulk fluid. (b) Mass transfer through a film occurs at steady state. (c) The bulk flow term [(N A +N B )C A /C] is small. For the many practical situations, this assumption is satisfactory. According to the above figure,let us consider that an elementary volume of thickness ∆Z and of unit area normal to the z-direction. Making a steady-state mass balance over this element located at the position z.

13. Rate of input of solute at z = N A│Z Rate of output of solute at z+∆z = N A│z +∆z Rate of accumulation = 0 (at steady state) By the mass balance over the elementary volume- Input + accumulation = output N A │ z - N A │ Z+∆Z =0 Dividing by ∆z throughout and taking the limit ∆z→0, we get,- - dN A / dz = 0, and putting N A = - D AB dC A / dz D AB d 2 C A /dz 2 =0, or- d 2 C A /dz 2 = 0; integrating and using the following boundary conditions (1) & (2)- (1) at the z = 0 (i.e. the wall or the phase boundary or the interphase ), C A = C ai (2) at the z = δ (i.e. the other end of film of thickness, δ ), C A = C Ab , then,- C A = C Ai – ( C Ai – C Ab ) z/ δ , according to this- The theorectical concentration profile is linear as shown in figure.whereas the true concentration profile is curve type . At the steady state condition, the mass transfer flux through the film is constant . And, N A = -D AB [ dC A / dz ] Z=0 = D AB / δ ( C Ai – C Ab ) on and also, N A = k L (C A1 – C A2 ) comparing k L = D AB / δ Important points :- 1. The film theory has been extremely useful in the analysis of mass transfer accompanied by a chemical reaction . 2. The film thickness, is the thickness of the stagnant layer that offers a mass transfer resistence equal to the actual resistence of mass transfer offered the fluid

14. 3. The film theory predicts linear dependence of k L upon the diffusivity, i.e. D ∞ k L . 4. According to the experimental data on diverse system,- coefficient of mass transfer to a turbulent fluid varies as (D AB ) n , where n varies between 0 – 0.8. THE PENETRATION THEORY:- This theory was developed by Higbie.This theory can be explained by a simple phenomenon of mass transfer from a rising gas bubbles-for example, absorption of oxygen from an air bubble in a fermenter. According to the figure as shown- as the bubble rises,the liquid element from the bulk of fluid reach the top of bubble move along its spherical surface, reach its bottom and then deteached detached from it. The detached liquid element eventually get mixed up with the bulk liquid. Absorption of oxygen in a small liquid element occurs as long as the elements remains at gas-liquid interface (i.e. in contact with gas ). Thus the above theory concluded that- in the case of mass transfer at the phase boundary, an element of the liquid reaches the interface (by any mechanism whatsoever) and stay there for a short while when it receives some solute from other phase. At the end of its stay at the interface, the liquid element moves back into the bulk liquid carrying with it the solute it picked up during its brief stay at the interface. In this process, the liquid element makes room for another liquid element, fresh from bulk, on the surface of bubble.

15. BASIC ASSUMPTIONS OF PENETRATION THEORY:- 1. Unsteady state mass transfer occurs to a liquid element so long as it is in contact with the bubble (or other phase ) 2. equilibrium exists at the gas- liquid interface. 3. Each of the liquid element stays in contact with the gas(or other phase) for the same period of time. Consider the mass transfer of gas bubble of the diameter d b rising at velocity u b , the contact time of a liquid element during this period of time can be described by partial differential equation as- ∂C A / dt = D AB ∂ 2 C A /∂z 2 The appropriate boundary and initial conditions are:- Boundary condition 1:- t > 0, z =0; C A = C Ai Boundary condition 2:- t > 0, z = ∞; C A = C Ab Initial condition:- t = 0, z ≥ 0; C A = C ab The initial condition implies that in a fresh liquid element coming from the bulk, the concentration is uniform and is equal to the bulk concentration. The boundary condition 1 assumes that “interfacial equilibrium” exits at all time. The last condition implies that if the contact time of an element with the gas is small, the ‘ deapth of the penetration’ of the solute in the element should also be small and effectively the element can be considered to be of ‘infinite thickness’ in relative sense.

16. Due to subjected to initial and boundary conditions, the equation can be solved for the transient concentration distribution of the solute in the element by introducing a ‘similarity variable’ η :- C A - C Ab / C Ai -C Ab = 1 – erf η ; where η = z/2√D AB t The mass flux to the element at any time t is- N A (t) = -D AB [∂C A /∂z] z=0 = √D AB ( C Ai -C Ab )/ Π t The avg. mass flux over the contact time t C is- N A,avg = 1/ t C ∫ t N A (t) dt = 2√D AB ( C Ai -C ab )/ Π t c On comparing- Instantaneous mass transfer coefficient, k L =√D AB / Π t Average mass transfer coefficient, k L,avg =2√D AB / Π t c IMPORTANT POINTS:- 1. The flux decreases with the time because of a gradual build-up of solute concentration within the element and resulting decreases in the driving force. 2. At a large time, the element becomes nearly saturated and the flux becomes vanishingly small. 3. According to this theory, the mass transfer coefficient is proportional to the square root of diffusivity, i.e. k L ∞ D AB . 4. On comparing with film theory, the contact time t c is a model parameter like the film thickness δ in the film theory.

17. THE SURFACE RENEWAL THEORY:- One of the major drawback of penetration theory is the assumption that the contact time or ‘age’ of the liquid elements is same for all. In the turbulent medium, it is much more probable that some of the liquid elements are swept away, while still young, from the interface by eddies while some others, unaffected by eddies for the time being, may continue to be in contact with the gas for longer times. As a result, there will be a ‘distribution of age’ of the liquid element present at the interface at any moment. This is how danckwert (1951) visualized the phenomenon. This theory contains the following assumptions:- 1. The liquid element at the interface are being randomly displaced by the fresh elements from the bulk. 2. At any moment, each of the liquid elements at the surface has the same probability of being replaced by the fresh element. 3. Unsteady state mass transfer occurs to an element during its stay at the interface. The Danckwert’s theory is thus called the surface renewal theory. The model parameter is the fractional rate of surface renewal (s, the fraction of surface area renewed in unit time). The equation for the mass transfer coefficient is:- K L =√D AB s

18. IMPORTANT POINTS:- 1. Surface Renewal theory also predicts the square root dependence of mass transfer coefficient k L on the diffusivity, i.e. k L ∞√D AB. 2. It has also similar dependence on the fractional rate of surface renewal, s. 3. Increasing the turbulence in medium causes more brick surface renewal (i.e. larger s) thereby increasing the mass transfer coefficient. THE BOUNDARY LAYER THEORY:- Mass transfer in the boundary layer flows occurs in a way similar to that of heat tran . If the plate is located with a soluable subs. And the liquid (or gas ) flows over it, two boundary layers are formed- the ‘velocity boundary layer’ and the ‘concentration’ or ‘mass boundary layer’. It maybe recalled that in boundary layer flow over a heated plate, a thermal boundary layer is formed along with the velocity or moment- um boundary layer (thickness = δ ). The concentration distribution in the boundary layer is a function of position, C A = C A ( x,y ), and its thickness is a function of distance from the leading edge, δ = δ m (x). The relative thickness of the velocity and the concentration boundary layer depends on the value of Schmidt number, Sc.

19. Theoretical analysis of the mass transfer in laminar boundary layer is given by following correctional:- Sh x = k L,x x /D AB =0.332(Re x ) 1/2 ( Sc ) 1/3 where,- x =distance of a point from leading age of the plate k L,x = local mass transfer coefficient Re x =local R eynold number If ‘l’ is the length of the plate then ‘average Sherwood number’ is- Sh av = k L,av l /D AB =0.664( Re l ) 1/2 ( Sc ) 1/3 where,- k L,av = mass transfer coefficient averaged over the length of the plate, Re l = ρ V ∞ l /µ = ‘plate Reynold number’ based on the length of the plate,l . IMPORTANT POINTS:- 1. Boundary layer theory predicts that the mass transfer coefficient k L varies as (D AB ) 2/3 , which reasonably matches the experimental findings in many cases. 2. If the Schmidt number is greater than unity, the thickness of momentum boundary layer at any location on the plate is more than the concentration boundary layer,it can be shown as,-- δ / δ m = ( Sc ) 1/3

20. COMBINED STUDY OF MASS TRANSFER THEORY ALONG WITH MASS TRANSFER COEFFICIENT:- THEORY STEADY OR UNSTEADY STATE EXPRESSION FOR MASS TRANSFER COEFFICIENT,K L Dependence on diffusivity MODEL PARAMETER (UNIT) FILM THEORY Steady state k l = d/ δ K l ∞d δ (m) PENETRATION THEORY Unsteady state k l , ins =[d/ Π t] 1/2 K l,avg =2[D/ Π t c ] 1/2 K l ∞ d 1/2 K l ∞ d 1/2 t c (s) SURFACE RENEWAL THEORY Unsteady state K l,avg =[Ds] 1/2 K l ∞ d 1/2 s(s -1 ) BOUNDARY LAYER THEORY Steady state Boundary layer correlation K l ∞ d 2/3

21. CORRELATIONS FOR THE CONVECTIVE MASS TRANSFER COEFFICIENT:- A large number of mass transfer coefficient exist in literature,which is developed by researchers on the basis of large volume of experimental data:- DESCRIPTION RANGE OF APPLICATION CORRELATIONS Laminar flow through a circular tube Turbulent flow through a tube Boundary layer flow over a flat plate Flow through a wetted-wall tower Gas-phase flow through a packed bed Liquid flow through a packed bed d= tubediameter;Re l ,l =characteristics length V’’= Superfacial velocity of fluid ( i.e,velocity based on the bed cross-section ) Re≤2100 4,000≤Re≤60,000; 0.6≤Sc≤3000 Re l <80,000// Re l >5*10 5 3,000<Re’<40,000 0.5< Sc <3 10≤Re’’≤2500 Re’’<55 3<Re’’<10,000 Sh = k l d /D=1.62[(Re)( Sc ) (d/l)] 1/3 Sh =0.023(Re) 0.83 ( Sc ) 0.33 j D =0.664( Re l ) -0.5 // j D =0.037( Re l ) -0.2 j D =0.038(Re’) -0.3 1.17(Re’’) -0.415 j D =1.09(Re’’) -2/3 Sh =2+1.1(Re) 0.6 ( Sc ) 0.33

22. MOMENTUM,HEAT AND MASS TRANSFER ANALOGIES:- Transport of momentum, heat and mass in a medium in laminar motion are all diffusional processes and occurs by similar mechanisms. The three basic law in this connection— 1. Newton’s Law of viscosity 2. Fourier’s Law of Heat Conduction 3. Fick’s Law of Diffusion Newton’s Law:- Momentum flux, ς = -µdu x / dz ς = - vd ( ρ u x )/ dz Fourier’s Law:- Heat flux, q z = - kdT / dz q z = - α d( ρ c P T )/ dz Fick’s Law:- Mass flux, N A = - D AB dC A / dz v = µ/ ρ , momentum diffusivity and ρ u x =volumetric conc. Of momentum in x- direc . α = k/ ρ c P ,thermal diffusivity and ρ c p T = volumetric conc. Of thermal energy in x-di These three equations concluded that:- “Flux is proportional to the gradient of the quantity transported ( momentum,heat energy and mass), and proportionality constant is the corresponding diffusivity.” A negative sign is included in each equation to indicate that transport occurs in the direction of decreasing concentration (of momentum,heat and mass).

23. PRANDTL ANALOGY:- St H = (f/2)/1+5√f/2(Pr-1) and, St M = (f/2)/1+5√f/2(Sc-1) The prandtl analogy reduces to Reynold analogy if Pr = 1 or Sc = 1. COLBURN ANALOGY:- Colburn related the mass transfer coefficient to the friction factor by proposing the well known ‘ colburn analogy’.He introduced the colburn j-factor and suggested the following analogies for transport in pipe flow. j H = St H Pr 2/3 = Nu/Re.Pr 1/3 = 0.023Re -0.2 j D = St M Sc 2/3 = Sh /Re.Sc 1/3 = 0.023Re -0.2 The importance of the analogies lies in the fact that if the heat transfer coefficient is known at a particular hydrodynamic condition characterized by the Reynold Number, the mass transfer coefficient in a system having similar geometry and at a similar hydrodynamic condition can be determined:- j H = j D

24. In the turbulent flow,- The rate of transport except that an ‘eddy diffusivity’ is included to account for the contribution of eddy transport. Thus,-- Momentum Flux, ς = -(v + E v ) d( ρ µ X )/ dz Heat Flux, q z = -( α + E H )d( ρ c P T )/ dz Mass flux (at a low conc.) N A = -(D AB + E M ) dC A / dz Where E V , E H , E M , are the eddy diffusivities of momentum,heat and mass respectively. Since turbulence transport is a much faster process than the molecular transport, all the eddy diffusivities are larger the their molecular counterparts by about two order of magnitude. ANALOGY IN MASS TRANSFER ALONG WITH FLOW OF FLUID:- REYNOLD ANALOGY:- St H = Nu/ Re.Pr = f/2 and St M = Sh / Re.Sc = f/2 where, St H = Staton number for Heat Transfer St M = Staton number for Mass Transfer Nu = Nusselt Number involving the mass transfer coefficient f = friction factor for the fluid flowing through a pipe.

25. INTERPHASE MASS TRANSFER:- INTRODUCTION:- Mass transfer of the molucules from one point to another in a single phase or in a homogeneous medium occurs by Diffusion mechanism. However in practice, most of the mass transfer operations involve transport of one or more solute from one phase to another. Ex:-In the sulphuric acid plant, the air supplied to the sulphur burner should be moist-free, drying of air is done in contact with concentrated sulphuric acid in a ‘packed column’. Ammonia is removed from ammonia-air mixture by using H 2 O. Each of the above is a case of ‘Interphase mass transfer.’ How can we calculate the rate of mass transfer in two – phase system? In the case of interphase mass transfer, the driving force is not merely the difference of concentrations of the solute in two phases, but the driving force is rather measured by how far the phases are away from equilibrium. For the purpose of comparing- In the heat transfer, equilibrium means that the phases are at the same temperature ,so that driving force is zero. But in the case of mass transfer ,the equality of concentration of two phases does not necessarily ensure about the equilibrium.

26. EQUILIBRIUM BETWEEN THE PHASES IN MASS TRANSFER:- Equilibrium between two phases in contact means a state in which there is no net transfer of solute from one phase to to other. At the equilibrium, the chemical potentials of the solute in two phases are equal. Which can be explained by two phase system of absorption of SO 2 from air into water. Sulphur dioxide being soluabe in water, will be absorbed in it, when contacted with water. However some SO 2 molecules will simultaneously leave the water phase and re-enter the gas phase but necessa - rily at the same rate. The vessel is maintained at constant temperature and this process of absorption and desorption (stripping) continues. Eventually a time comes, at which the rate of absorption becomes equal to the rate of desorption. The parital pressure of SO 2 in air and its concentration in water will no longer change, thus the system is said to be at ‘equilibrium’. If some sulphur dioxide gas is fed into the vessel and sufficient time is allowed thereafter the system will reach a new equilibrium state and a new sets of equilibrium value ( x A * , p A ) is obtained. Sets of equilibrium values at constant temp. and at a constant total pressure is called the ‘Equilibrium Data’. A plot of these data is called the ‘Equilibrium line’ or ‘Equilibrium curve’.

27. LAW OF EQUILIBRIUM:- Two important law of equilibrium between the phases are Raoult’s Law and Henry’s Law. Raoult’s Law:- For the ideal gas-liquid or vapour -liquid system,the equilibrium relationship is given by Raoult’s Law- Statement:- “ The equilibrium partial pressure of a component in a solution at a given temp. is equal to the product of its vapour pressure in pure state and its mole fraction in liquid phase”. Mathmatically ,- p A = p o A x A Henry’s Law:- Equilibrium relationship for many non-ideal gas-liquid system at the low concentration is given br Henry’s Law- Statement:- “ The mole fraction of a solute gas dissolved in a liquid is directly proportional to the equilibrium partial pressure of the solute gas above the liquid surface”. Mathmatically ,- x A = 1/H A p A or,- p A = x A H A Where H A = Henry’s Law contact , which is strongly depends on Temperture (Increases). The temperature dependence of the Henry’s Law constant sometimes follows as- m = m exp(-E/RT).

28. Equilibrium diagram of SO 2 in water at different temp. is shown in figure .Equilibrium lines become steeper because the solubility of SO 2 in water decreases with increase with temperature. IMPORTANT POINTS:- 1. In a system at the equilibrium, the number of phases p P , the number of components C, and the number of variants F, also called ‘The Degree of Freedom’ and are related by ‘Phase Rule’- F = C – P + 2 In the above case,- F = 3 – 2 + 2 = 3 And, these three variants should be specified to define the system completely. 2. If two phase at equilibrium, there is no net transfer of solute from one phase to another. It rather means if a few molecules go from phase-1 to phase-2, the same number of molecules move from phase-2 to phase-1 in order to maintain the concentrations in the phases constant.This equilibrium is sometimes called ‘Dynamic Equilibrium’. 3. If two phase are not at equilibrium, mass transfer from phase-1 to phase-2 occurs so long as the concentration of solute in phase-2 is lower than the equilibrium concentration. “ The extent of Deviation from Equilibrium state is a measure of driving force”. And for the system at the equilibrium, the driving force is zero.

29. MASS TRANSFER BETWEEN TWO PHASES:- Concentration profile near the interface:- Mass transfer between two phases involves the following sequential steps:- 1. The solute (A) is transported from the bulk of the gas phase (G) to the gas-liquid interface 2. The solute (A) is picked-up or absorbed by the liquid phase (L) at the gas-liquid interface. 3. The absorbed solute (A) is transported from the interface to the bulk of liquid (L). The steps 1 & 3 are facilitated by the turbulence in the fluid medium. Interface means, geometrical plane or surface of contact between two phases. From the figure, p A and C A are the function of distance (z) along the mass transfer path and z = 0 means the interface. lim p A = p Ai (z → - ) and lim C A = C Ai (z → + ) These are called ‘Interfacial concentration’. In most it is assumed that equilibrium exits at the interface as - p Ai = Θ ( C Ai ) Where Θ represents the dependence of p A on C A at the equilibrium. This function is also called ‘Equilibrium relation’. If it so happens that the Henry’s Law is valid for solute-solvent pair, the equilibrium relation becomes linear as- p Ai = mC Ai

30. TWO FILM THEORY:- Lewis and Whitman (1924) visualized that in the case of mass transfer involving two phases in contact, there are two stagnant fluid film exits on either side of interface and mass transfer through these films , in sequence, purely by molecular diffusion. Beyond these films the concentration in a phase is equal to the bulk concentration.This is “Two film theory of lewis and Whitman”. The two films and concentration distributions are schematically shown in the figure. Determination of interfacial concentration:- Interfacial concentration can be determined either graphically or algebraically. If the mass transfer occurs from a gas phase to a liquid phase at the steady state, the mass flux of solute A from the bulk gas to the interface must be equal to that from the interface to the bulk liquid. This is because, at the steady state, there cannot be any accumulation of solute anywhere. In the term of gas and liquid phase coefficient- N A = k y ( y Ab – y Ai ) = k x ( x Ai -x Ab ) Gas-phase flux to the liquid –phase flux interface to interface And from the figure- y Ab – y Ai / x Ab – x Ai = - k x / k y slope of the line PM

31. DIFFUSION:- “ It is the movement of molecules or individual component through a mixture from a region of higher concentration to a region of lower concentration at a fixed temperature and pressure with or without the help of external force. Ex:- A drop of ink released in water in a glass gradually spreads to make the water uniformaly coloured . The fragrance of a bunch of rose kept on the centre table of drawing reaches out even to a remote corner of the room. Molecular Diffusion DIFFUSION Eddy or Turbulent Diffusion Molecular Diffusion:- Diffusion result from random movement of Molecules. Turbulent Diffusion:- When the movement of the molecule occurs with the help of an external force such as mechanical stirring and convective movement of fluid,then it is called ‘eddy or turbulent diffusion

32. IMPORTANT POINTS:- 1. Diffusion is a passive process. 2. The molecular diffusion is a slow process, whereas eddy diffusion is a fast process. 3. The molecular diffusion is a mechanism of stationary fluid i.e., a fluid at rest and flow is laminar whereas in the case of fluid in the turbulent flow ,the mechanism of mass transfer is by eddy molecular transfer eddy transfer. ROLE OF DIFFUSION IN MASS TRANSFER:- Diffusion may occur in one phase or in both phases in all the mass transfer operations. In the case of Distillation- The more volatile component diffuses through the liquid phase to the interface between the phases and away from interface into the vapour phase, the less volatile component diffuses in opposite direction. In the case of gas absorption - The solute gas diffuses through the gas phase to the interface and then through liquid phase from the interface between the phases. In the case of crystallization- The solid solute diffuses through a mother liquor to the crystals and deposit on the solid surface . In the case of drying operation- The liquid water (moisture) diffuses through a solid towards the surface of solid,evaporates and diffuses as a vapour into gas phase (Drying medium).

33. FICK’S LAW OF DIFFUSION:- The basic law of diffusion, called fick’s Law was enunciated by adolf Eugen Fick, a physiologist, in 1885. According to this law , the molar flux of a species relative to an observer moving with the molar average velocity is proportional to the concentration gradient of the species. Mathematicaly ,- J A ∞ dC A / dz J A = -D AB dC A / dz molar flux molecular diffusivity concentration gradient in or diffusion coefficient z-direction The diffusional flux J A is a positive quantity by the convection. Since Diffusion occurs spontaneously in the direction of decreasing concentration [( dC A / dz ) < 0],negative sign. Is involved in the mathematical representation of Fick’s law for diffusion. And, N A = N A + N B (C A / C) – D AB dC A / dz N A + N B (C A /C) Bulk Flow -D AB dC A / dz Molecular diffusion or N A = J A = -D AB dC A / dz (for small conc. Of A ) Diffusion velocity:- Diffusing velocity of species A, v A,d = u A - U = J A / C A = -D AB dC A / dz

34. For the gas phase diffusion- N A = (N A + N B ) p A /P – D AB dp A / RTdz In term of mole fraction- N A = (N A + N B ) y A –CD AB dy A / dz and J A = -CD AB dy A / dz Average total molar concentration of mixture If the gas mixture is ideal, the mutual diffusivities of A and B are equal, that is- D AB = D BA STEADY STATE MOLECULAR DIFFUSION IN A BINARY GAS MIXTURE- Diffusion of A through non- diffusing B- For the steady state diffusion of A through non-diffusing B,- N A = constant and N B = 0 And, Molar flux- N A = D AB P ln (p A0 – p Al )/ RTl * p BM Where p BM = log mean partial pressure of B or-N A = D AB P ln (P – p A )/ RTz *P-p A0 Equimolar counter current diffusion of A & B:- For the equimolar counter-current diffusion of A and B- N A = D AB (p A0 – p Al )/ RTl

35. KNUDSEN DIFFUSION,SURFACE DIFFUSION AND SELF-DIFFUSION:- (a) Knudsen Diffusion:- If the gas diffusion occurs in a very fine pore, particularly at a low pressure ,the mean free path of the molecules may be larger than the diameter of passage. Then collision with the wall becomes much more frequent than collision with other molecules. The rate of diffusional transport of a species is now governed by its molecular velocity, the dia. Of passage, and Of course, the gradient of concentration or partial pressure . This is called “Knudsen Diffusion” and becomes important if pore size is normally below 50 mm. Ex :- Intra-particle transport in a catalyst containing fine pores. (b) Surface diffusion:- Surface diffusion is the transport of adsorbed molecules on a surface in the presence of a conc. Gradient. Molecules adsorbed on a surface remain anchored to the active sites, if the fractional surface converage is less than unity,some of active sites remains vacant. The flux due to the surface diffusion is- J s = -D s dC s / dz (c) Self-diffusion:- The molecules of a gas or a liquid continuously move from one position to another as a result a mixture of uniform composition also diffuses but net rate is not changed.