 in this fourth and final lecture on mass transfer of module 6. Today, we consider the surface instability aspects of theories of mass transfer and certain confirmation of these theories from experiments on static and dynamic systems. We already have talked a little bit about the surface instability. That is if that is if at interfaces we have surface active solute at non-uniform concentration, it could create unstable surface tension gradients and therefore lead to movement within the surface on account of those surface concentration gradients. The experiment conducted by Langmuir was pertaining to evaporation of ether from a saturated solution in water. The example which I explained last time, if you want to visualize what happens in the surface, tulk particles could be sprinkled and by their movement one would be able to visualize the abrupt local movements caused by differences in surface tensions or surface pressures at different regions. Now, the mechanism here is that convection currents either originating in air or within water can cause more ether to be present at certain points in the surface than in the adjacent or neighboring regions and what that means is that on account of a higher concentration locally the surface pressure of ether at that point will be will be higher compared to the neighborhood. This would mean that this higher surface pressure will tend to take the surface active component from this point to the adjacent points. But that will not happen without dragging the liquid underneath and what it means is that if there is a movement within the surface here and it correspondingly drags the liquid underneath, we may be able to bring some of the undepleted ether solution from the bulk to this point. There by increasing the local concentration of ether even further and this will cascade. This will continue to happen until there is a visible movement present in the surface. So, one has to understand this that the origin might lie at a molecular scale, but ultimate cascading effect will be amplifying that initial fluctuation into observable dynamic movement within the surface and that is what will be visualized if we have this talc particles showing the abrupt motion which varies depending on these randomly varying surface concentration gradients. What it implies that what it implies is that along with the surface movement there is a movement induced in the liquid underneath and if there is a movement within the liquid clearly the resistance for solute transfer in the liquid phase that is R L will also be greatly reduced. So, surface instability can be a reason for observed reduction in the liquid phase resistance and this is especially important because R i and R g are likely to be small here. So, any decrease in the liquid phase resistance will manifest itself into observable rate of transfer and that is what I had tried to explain to you that ether could evaporate rather rapidly from such a surface and therefore, will be able to ignite the vapor in the gas phase. This ether itself is the surface active ingredient that we are talking of the surface pressure is caused by ether it is only ether and water. However, if you introduce a monolayer like of oleic acid or any other material with high enough surface viscosity then these unsteady movements in the surface could be constably reduced and therefore, the liquid phase resistance will also be kept at a higher value because effectively a larger liquid depth below the interface will be remaining stationary or unperturbed and this effect therefore, that of higher R L will reduce the evaporation rate of ether enough so that there is no burning visible. You can temporarily offset that effect by whipping the liquid surface covered with monolayer with a glass rod, but then as soon as you stop that disturbance of the surface the monolayer can take over and prevent burning. Similar effects have been observed by others in particular by Ruthfey and Crammers in 1955 for a system which was slightly different it was absorption of sulphur dioxide in a hydrocarbon like normal heften. In their experiments they found that the drop surface became violently agitated and therefore, that in turn also reduce the liquid phase resistance. So, turbulence of this kind would depend on the kind of solute which is transferring across the surface. Now, if eddies of fluid rich in solute coming from the bulk to the surface can locally increase the surface pressure pi, their redistribution could happen by spreading in the surface and this has to happen before the solute has a chance to transfer into another another phase. Only if that is guaranteed we will be able to see the cascading effect amplifying into the observed movement of tall particles in the surface. Now, thinking about factors what might be governing this effect one would expect one thing which is related to how the solute transport partitions between the two phase. In this case let us say ether between water and air. So, that distribution cohesion or partition cohesion is one thing which is which should be important. The second pertinent quantity should be how the surface tension depends on concentration. There is a if there is a change in concentration of the solute in the surface how sensitive is the surface tension. If the effect of change of unit concentration on the surface tension is negligible you do not expect the surface instability to become manifest. But on the other hand if it is a large and negative value then as you will see this could create surface instability. So, yeah. So, will it depend on the resistance of gas phase and interview? As we are saying the gas phase and liquid phase resistance otherwise are small. It is the interfacial resistance which is important. You are talking of evaporation from within from the liquid surface into the gas. So, it is the interfacial resistance which is now the dominant one and all the changes of miniscule changes in concentration if they create large changes in surface tension or surface pressure can you have this effect and it has to originate at the molecular level. Because of some fluctuation within air or within liquid. You are saying like the interfacial tension effect is related to the gamma wave. Gamma is your surface tension. Gamma is surface tension and C is the concentration like ether. In this case gamma will be the surface tension of ether solution in water and if there is a fluctuation in the concentration of ether there will be corresponding change in the surface tension. So, that surface tension is of a solution. It is not interfacial tension it is a surface tension of the solution of ether in water and depending on concentration that will change. So, now as opposed to ether you might be able to have other compounds which do not show a very sensitive variation of surface tension of the solution to concentration. Then you do not expect this kind of effect to be there. But we want to get a broader feel as to what factors might be governing the surface instability. And since we would not have time to go into details I will like to give you an overview of work done by a couple of people. There was a leading research done at University of Minnesota by Scriven in 1959 along with Sternley. So, they actually carried out a mathematical treatment of hydrodynamics of surface and interfacial turbulence and discussed the conditions under which a molecular fluctuation arising out of let us say some eddy random eddy movement within air or water how it could build up into an observable macroscopic eddy. That will in turn will be what you would see in in the movement of the talc particles. What their equations predict is that edding will be promoted if several of these factors are prevalent. First is the solute transfers from a phase of higher viscosity and lower diffusivity into a phase of lower viscosity and correspondingly higher diffusivity. So, our example here I will try to benchmark against these statements. It is now ether water solution that is a phase of higher viscosity. There the diffusivity of ether will be lower. In general in liquid phase the viscosity and diffusivity will be inversely related. So, edding is possible in system like ether water in contact with air and transfer of ether is taking place from the solution into air. Second if there are large differences in diffusivities and kinematic viscosity is between the phases. Once again a valid case the liquid against the gaseous phase. We expect the diffusivity is to be very different and kinematic viscosity is also because these are one is a dense condensed phase another is a rare gaseous phase. Third is if there are large changes large differences in concentration near the interface. Now there is a transfer of solute ether from solution into the bulk. So, where it is evaporating or vaporizing into air the concentration at the surface will become very low just underneath the concentration will be quite large. So, again that is favorable dou gamma by dou c the sensitivity of surface tension of the solution to changes in concentration should be large and negative. That is it should have something like a surface active impurity effect. There is a higher if there is higher concentration the surface tension should go down. And if surface active ingredients are absent we seeing that in absence of oleic acid we could see the burning of the vapor. So, but if you have surface active impurity like oleic acid then it is not possible to promote aiding and of course the interface should be large in extent. No, no these are all deductions from the theory. These are all from the mathematical model of the hydrodynamics of this instability for surface and interface. First point is you need you would not be able to see this unless you look at the coupling of the hydrodynamic equations with the mass transfer. And we are not going to details of that, but I am just trying to tell you that this is how it will promote. It is it does not mean please remember that these are all factors which will promote aiding. It does not mean that if this is not obeyed this first condition is not obeyed. It will not do the aiding otherwise it will be the that sulfur dioxide absorption in heptane would have been a counter example right. There it is possible to have the surface sensibility, but the transfer is from the phase of lower viscosity or higher diffusivity, but if these factors are there that will promote aiding that is all. Now, one question which may come to your mind is what kind of interfacial force will be produced if there is a heat of transfer coming into picture. We did talk about the cooling that occurs of the liquid due to evaporation, but how much of interfacial force can be created? Compared to concentration fluctuations this is negligible just about 0.1 percent of what you can achieve from concentration fluctuation. So, as a result of transfer of the solute whatever cooling might occur that is not going to be a very important factor. And the other group which was considering the same problem was Haydn's who also had a theory for spontaneous surface turbulence, but one generic comment could be made about both these theories is that neither of these analysis could be regarded as ultimate analysis of comprehensive analysis because there was no mention of what critical concentration of the solute would produce surface turbulence. One would have anticipated a theory to be able to give information as to what critical concentration will create interfacial turbulence nor was there any quantitative description of the distribution cohesion of the solute between the phases. How is the heat of transfer producing interfacial turbulence? If let us say ether evaporates then it will take the latent heat of cooling. So the interface will be cooled, temperature drops then the surface tension changes. If the surface tension changes with temperature and you know how sensitive surface tension to temperature is not that highly sensitive. Whereas a slight change in concentration or a concentration fluctuation can create enormous change in the surface tension or surface pressure. So, that way it should be understood. Now let us look at the theoretical values of rg. We could take an example like absorption of carbon dioxide in water. We could use the diffusion theory which gives you the mass transfer cohesion as equal to square root of d by pi t if the air is completely unstirred. That is the liquid phase mass transfer cohesion. Diffusivity for the gas is of the order of 0.2 cm square per second and at atmospheric pressure if we were to use a similar formula for the gas phase then the rg comes out to be just about 1% of liquid phase resistance. So the gas phase resistance is in general quite low and therefore the value comes out to be about 4 seconds per centimeter for a time of contact about or exposure about 1 second and then when you go for time like 100 seconds then there is a 10 fold increase in this gas phase resistance it jumps to about 40 seconds per centimeter. Usually however in practice the air would be in some kind of chaotic turbulent motion and therefore rg would depend on what is the thickness of layer which would contribute to the resistance. So typical range of values for rg is about 5 to 80 second per centimeter. If you take the other extreme example which I had used earlier if pure gas is absorbing into the liquid and the gas is not highly soluble in water then usually the liquid phase is the controlling resistance right. Pure gas and not that highly soluble gas in liquid will mean liquid phase resistance will be controlling. Thinking about the liquid phase resistance values we have kl equal to 1 by rl equal to root of d by pi t that will give you the value of rl they could be thermal convection currents or stirring which will reduce the rl much below what is calculated from here. So we come here to the spectrum of theories for mass transfer what we are going to talk about here in plane turbulent system. First is Lewis and bitman's model in 1924 is basically film theory. They propose that the stirring will maintain the composition uniform almost up to the surface except that delta x distance below the surface concentration is uniform up to there and within that distance delta x they postulated laminar flow which is parallel to the surface and through this laminar sub layer the transfer they said would be at steady state if the layer could be regarded as stagnant and then mass transquestion is related to diffusivity through a relation of that k equal to d by delta x linear relation between the mass transquestion k and diffusivity d or the rate of transfer will be dq by dt equal to ka delta c right. It will be presumed here that rl is a constant for a given delta x but if there is stirring then rl should be possible to be estimated from this provided we know how delta x will depend on stirring or the liquid movement. So there we could go for some kind of boundary layer kind of analysis which gives you delta x dependent on Reynolds number and delta x is proportional to this Reynolds to the power minus 0.67 where Reynolds number will be something like nl square by nu where n is the speed of stirring the rpm kind of thing and l is the length of tip to tip tip to tip length of the stirrer blades. So whatever is stirring that and nu is the kinetic viscosity you can see that nl is with the velocity units. So velocity length scale by kinetic viscosity that is a structure of Reynolds number and in addition if the concentration difference across this laminar sub layer does not vary much with time then we could integrate this easily and find the relation between q and time that q and q will be proportional to kt delta c. There should be area factor for equality and similarly between k and d there is a proportionality likewise between q and dt by delta c by delta x. You can see here that this delta c by delta x is the concentration gradient or you can look at d by delta x as the mass transformation from film theory delta c is the driving force that multiplied by area and time will give you total moles of q transfer or total moles of the solute q transferred across the interface. Now take a slightly different point of view this is one case let me illustrate this so that let us say our conditions are such that in one case we have the mixing occurring with the help of the stirrer with the rpmn and there is this delta x thickness which can be regarded as stagnant. Across this there is the concentration difference delta c which is causing the transfer of the solute either in this direction or in opposite direction gas to liquid or from liquid to gas. This is the film theory kind of picture and it is presumed that this portion is stagnant or it is in laminar flow but not to create too much difference from stagnancy. As opposed to this supposing that we increase the stirring way too much we create too rapid mixing over here then we might have an extreme situation where the turbulence created here could reach right up to the surface. So if this is now a turbulent system right up to the surface then you know that because of turbulent random movements the fluid particles will be moving in all directions. So one could visualize that the liquid actually comes from the bulk right up to the surface and then goes back it would happen quite randomly that is another extreme case. If that is the case we might on the average define some average perpendicular velocity to the interface in the liquid let us call it V n bar. This will be the velocity of the liquid which is reaching the interface and then because the volume of the liquid is conserved it will be returning back to the liquid right. So this is one extreme example that you can imagine in that extreme turbulence case the fresh solution will come in the form of eddies rapidly to the surface and then will be swept back into the bulk. Now we cannot have even that very thin laminar sub layer. We should forget about the stationary surface there is no stationary surface there is not even a laminar sub layer. Liquid then will be probably coming with that average velocity V n bar and if the difference in concentration between the interface and the bulk is delta c the rate of transfer should be d cube by d t equal to a times this V n bar which is like a flow rate of exchange between the bulk of liquid and the interface and that creates a transport for the two ends which differ in concentration by delta c. So a times V n bar can be looked at as a flow rate with which liquid is moving to the surface from bulk with the difference in concentration delta c. So that will be the molar flow rate from bulk to the surface if the liquid is transferring the solute to the gas phase. So this is one extreme example in practice this may not happen therefore the complexity of having to think of so many different models comes into picture. Why do we need the surface renewal or before that the penetration model and perhaps a combinations a combination of surface renewal and this kind of convection based model those questions will be the ones which will be coming. Sir this delta c is between the concentration difference just above and just below the surface no here if you are looking at the solute transfer from the bulk to the interface it is that delta c. So and it is understood that if the solute is being brought from the bulk to the interface thus the same amount will go from the interface to the bulk of air. It is not steady state it is it is this momentary transport of the solute from bulk of liquid to the interface and if that goes in the same proportion to the gas phase at that point you may say there is a balance of the two but steady state should not be brought in because this will not be steady state picture at another moment the v n bar may differ. So we only require that whatever is transported from bulk of liquid to the interface will be what will be going into the gas phase and this bar v n bar is the average. So that is indicating that is a turbulent situation you are doing the time averaging for the perpendicular component of velocity. So in other words in that extreme case we may say that our mass transfer equation kl becomes equal to v n bar okay. On one hand when we have stagnant liquid we know kl will be proportional to diffusivity the film model and in this extreme limit of turbulence diffusivity has no role to play. Your mass transfer equation or resistance for mass transfer in the liquid is entirely dependent on the turbulent state of the system and that characteristic perpendicular velocity which is time average v n bar okay. Obviously these are extremes and the Lewis-Wittman's film theory will not be applicable for short times of exposure. If the gas is in contact with turbulent liquid in all probability the liquid elements will pick up gas only for a short time. We are very unlikely to find the long times required for establishing steady state and we say in film theory. So film theory is much less credited today than it was in the beginning. Such conditions of short time exposure are actually important in practical situations like in gas absorption columns. For example in a gas absorption column you will have packing, stower packing. Each piece of packing will have liquid flowing over it and the gas going through the interstitial space. We may say that the gas and liquid are in contact and allow for mass transfer for very short exposure time which is the time taken by the liquid to reach from top of a packing to the beginning of the next packing. During that time we may regard the liquid to be in some sort laminar flow and the time of contact may be short but could be taken as constant. Those were the premises which led Higbie to postulate the penetration theory and that is what was done in 1935 for gas absorption in such columns. Higbie's proposition was that moderate liquid movement would have no effect on the diffusion rate for a very short time of exposure of liquid surface. I can paraphrase this this way that if there is a moderate liquid movement we may still say that diffusion is what is primarily governing the transport or you are saying what we been always given to understand that transport in stagnant stationary liquid or liquid in laminar flow slowly moving could be entirely described in terms of diffusion right that was the same assumption which Higbie makes and based on that constant time of exposure and unsteady state mass transport here I had at these relations d cube by dt is now a times the concentration difference C s minus infinity into a mass trans equation now which is dependent on time square root of d by pi t and by integrating that the amount absorbed in given time will come out as q equal to this 2 a C s minus infinity square root of dt by pi ok. And this theory was confirmed by absorption of carbon dioxide in water for a time of exposure about 0.12 seconds and k l and q were shown to vary very closely as square root of time as is required in this theory. In addition k l according to this theory should vary as square root of diffusivity which is bang in the middle of the absurd dependence of mass trans coefficient in the liquid on the diffusivity 0 to about 0.9. But again Higbie's theory can be valid only for very short exposure times and for such turbulence that the diffusing liquid is not able to penetrate deep within the liquid where the velocities may be appreciably different from the velocities in the surface. So, if longer times are coming into picture total time is still short to guarantee unsteady state conditions, but the depth of penetration is now sufficiently large such that the diffusion can actually be seen by that region of liquid deep within where the velocities are very different from surface velocities. If that happens then what one has to think about is an exchange of liquid from the bulk and the surface. At longer times we may say eddies could be pictured as continually exposing exposing fresh liquid surface to a gas and simultaneously sweeping into the bulk part of the surface which is already in contact with gas for some time. And then one may say that how long the liquid has been in contact with the gas, how long the liquid elements have been in contact with the gas that could be itself distributed need not be constant. So, that brings in what Kishinevsky and Danckwerts had proposed. Kishinevsky pointed out when this happens will have to use both the equations. There is a slight error here. The second equation should have left hand side reading as d cube by dt. So, the first equation is the standard unsteady state penetration model kind of equation d cube by dt is a C s minus infinity square root of dt d by pi t and the second equation is d cube by dt equal to a delta c v n bar. Both these equations should be simultaneously used. This is this was the suggestion by Kishinevsky. There is slightly different point of view which was proposed by Danckwerts who said that we can use these equations in series rather than in parallel. What he suggested was the elements of liquid had definite residence times in the region and after this time of residence they these elements are swept into the bulk and during this time the first equation should apply d cube by dt equal to this area concentration driving force and the square root of diffusivity by pi t. It is equivalent to putting these equations in series and if you do that then you need a new parameter s which is the fractional renewal rate. What surface fraction is renewed every second that is s. If you allow for that then a stationary age distribution could be derived as s e power minus s t and then the ultimate results give you the rate of absorption as d cube by dt equal to a times C s minus C infinity square root of d s where s is the fractional surface renewal rate and k l is square root of d s. Once again you find that k l is proportional to square root of diffusivity but s has to be estimated in some way right and if you integrate this you get the last equation q for amount absorbed which is k q equal to a delta c root of d s into t. How s could be precisely inferred will decide how we could discriminate between the surface renewal theory of dank words and the penetration theory of Higbie. It was Tour and Marshalo who consider very low rates of stirring and who showed that Lewis and Whitman model is valid for low rates of stirring. Although if the turbulence is moderate then the rate controlling state becomes the replacement of these liquid elements and therefore dank words equation the first equation of this set becomes applicable. We look at some experimental data now as evidence or in contradiction to observations and we need to look at two kinds of systems static systems and the dynamic systems. Let us first discuss the experiments on static systems. So for absorption of gases like oxygen and carbon dioxide into unstirred and chemically reactive solution. So we are talking of mass transfer and chemical reaction. It turns out that the monolayer like of sterile alcohol can reduce the uptake of gas by as much as 30%. So solute transfer from gas to liquid can be significantly reduced if you have certain monolayers like sterile alcohol. Rg can be made practically very low using pure gases that is the standard thing and Rl is fairly small because now reaction is accompanying diffusion because reaction is accompanying diffusion and if you carry out the experiments only for first few minutes of bringing the gas in contact with the liquid Rl can be designed to be a low value. Therefore the interfacial resistance Ri could be readily determined. It turns out that for such transfer oleic acid proteins and cholesterol are ineffective but if you look at oxygen diffusing through films of C16 and C18 alcohols then we get the resistances as 80 seconds per centimeter and 290 seconds per centimeter. For carbon dioxide the resistances are of similar magnitude. Expands of these kind systems of these kind have been studied by Harvey and Smith and they use interferometry for measurement of concentration gradient as close as 100th of a centimeter from the surface and for absorption of CO2 in water. They also use Sine camera so that the results could be obtained within first 5 seconds of exposure of liquid to CO2 and they claim was that by this combination of techniques the effects of convection could be eliminated and the resistances as low as quarter of a second per centimeter could be detected. Still today Harvey and Smith's 1959 paper remains as one of the important papers which deals with interfacial resistance. Their experiments reveal that if carbon dioxide were absorbed in distilled water there was not much practically detectable interfacial resistance. But if surface active agents or factors were dissolved then Ri of the order of 35 second per centimeter could be measured and these figures are higher than corresponding figures from evaporation studies which means now there is some kind of blocking effect of the head groups for the solute transfer from the gas to liquid. There is another thing which happens if there is a monolayer present then because of the dipoles of the movement head group there is a reorientation of water dipoles and that creates certain hydrogen bonded more viscous water which can retard the resist which can retard the transfer of the solute molecules from the gas phase into the liquid phase. So besides the head groups themselves providing the resistance the reorientation of the water dipoles leading to somewhat semi solid ice like water structure there is another factor which creates this additional barrier for transfer of solute molecules. We will wrap up the lecture here by talking about the dynamic experiments. Insturred vessels for steady state absorption of oxygen the liquid phase resistance could be anywhere between 120 to 12,000 second per centimeter. 120 is for the extreme level of mechanical agitation and 12,000 will be corresponding to the convection only or whatever transfer is occurring by virtue of convection only. Taking the diffusivity to be about 10 raise to minus 5 centimeter square per second and applying k equal to d by delta x, delta x works out to be about 1000 microns for water if it is not stirred. If you stir it to about let us say by about 50 rpm that is a rate of stirring this delta x reduces from 1000 to about 266 microns and at about 270 rpm it further reduces to about 58 microns. So you can see that as you increase the rate of stirring the thickness of that layer which could be identified as the film theories, film thickness that becomes smaller. But if you take the ratio of these thicknesses and the stirring powers then you find something interesting. The power is not expected power minus 0.67 as we had seen for the Reynolds number raise to minus 0.67 instead the power comes out to be about minus 0.9. This could be attributable to the surface or interface not being static because as you stir the interface itself moves whereas in the derivation of that delta x proportional to Reynolds to the power minus 0.67 you use the boundary layer analysis in which the solid surface is not movable. So that is the difference between the interface between a liquid and a gas or liquid and liquid and the interface that is between liquid and solid and therefore one could probably rationalize this higher power attributing it to the mobility of the interface between the fluid phases. At lower stirring speeds and with no convection in air a monolayer like of hexadecannol can retard the evaporation by 25 percent whereas there is not much reduction in the rate of oxygen transfer. This is because R i the interfacial resistance is now quite low about 10 second per centimeter and R L is unaffected by the presence of film. At higher stirring rates however the monolayer can contribute 5 fold increase of R L partially by damping out the eddies which are not now able to approach the surface as they would in case of a pure liquid gas system. And experimental measurements by Sherwood's group reveal scale proportionality to diffusivity to our 0.25 again at variance with the expectation from surface renewal or penetration models and that goes on to support what Kishinevsky has been advancing that in these experiments like carbon dioxide absorption into alkali the surface flow is probably much more complicated than has been envisaged in the Danckwert's theory. So with these closing words I think I will be able to convey to you this much that hydrodynamics plays a very important part in describing mass transfer in general and as yet we do not understand it adequately to be able to predict mass transfer rates or mass transfer equations absorption rates under all kinds of conditions from first principles. We have models which work reasonably well under the conditions which can be characterized and we have some kind of rationalization coming from experiments, but we certainly are far away as to be able to predict all mass transfer information under all conditions of hydrodynamics from first principles. We will stop here for today and in the last and final lecture for this particular course we will be talking about disperse phase systems. We just have one lecture for that. Thank you.