 Today, we resume the discussion on surface waves and ripples, the velocities and the effect of impurities through the surface tension and the surface compression modulus and effect of waves and ripples on the mass transfer basically rates of adsorption from liquid, bulk and absorption of a gas phase component from the bulk of gas and finally, we will talk about damping both at clean liquid surfaces and once containing mono layers. So before we begin this lecture proper, let us quickly see the velocity expression that we had seen earlier which combines the gravitational and surface tension effects in this expression where the small v is the velocity and we see the first part contributing to the gravitational nature of waves and the second term contributing to the capillary nature of waves. So, depending on which term dominates will have the respective kind of waves. So, capillary waves will be the short wavelength waves and gravity waves will be relatively having large wavelengths and those will be the individual expressions governing velocities for the capillary waves and gravity waves. We call those waves with wavelength shorter than about half a centimeter capillary waves because that is where the surface tension will be the dominating component and wherever surface tension is a predominant factor we call those phenomena capillary phenomena. So, we see here that for some reason if the surface tension is reduced then according to this equation 4 the velocity for the capillary waves will be correspondingly reduced. We will see later the effect of impurities on what happens to these waves through a factor other than the apparent effect through the surface tension. This is where we will begin today's discussion. If there is a surface film it would act also through the surface compressional modulus. I had explained to you earlier that the surface compressional modulus is the property which could be presumed to be attributing the rigidity to the surface and that should come into picture here because it is this C S inverse which would be major of the work done in resisting the distortions of the surface in the form of the contractions and extensions while a wave passes through. How does the wave or ripple affect the rate of adsorption from bulk of a liquid? We are talking here about surface active impurities which might be present in very low concentrations in the bulk of liquid and their adsorption at the interface. So, if we look at this picture if this is water in contact with air and there are surface active impurities in the bulk presuming the concentration of these surface active impurities or surface active surface active agents are low over here. How would their rate of adsorption at this interface get affected? If water is this solution is stagnant, there is no movement and air in contact is also stagnant. So, that there is no momentum transfer between air and water. The surface active agents molecules will be left to the process of diffusion. They will have to make their way from the bulk to the point on the plane of this water surface where presumably initially there is no concentration of any surface active agent whatsoever. So, the bulk will have a higher concentration compared to the surface concentration. So, if I represent by the bulk concentration of surfactant C b and the surface concentration to be C s in the beginning, we will have C s equal to 0 and then with time the C s will build up. There will be certain equilibrium distribution cohesion between the equilibrium concentration that can exist between the bulk and the surface, but till that equilibrium is reached, there would be movement of surface active molecules from the bulk to the surface. We expect this to be a relatively a slow process because first diffusion itself is quite slow and the surface active agents are relatively bulky molecules. They are not small molecules. It is a relatively long chain which will mean the diffusion cohesion itself will be quite low. So, rates of diffusion we expect to be quite small. Now, if the situation is one corresponding to a wave passing through or a ripple passing through the surface. If that is the case, then we understand that rest of this picture being similar, we would have appreciable components of velocity coming to picture. As the the surface wave passes through, we expect considerable velocity components or some kind of convection over here. At the beginning of the surface waves discussion, I had asked you to think about how the fluid particles, the liquid water particles would move in the surface while the wave is passing through and a direct consequence of incompressibility of water meant that you would have all the water particles going in circles. If one way to go towards the depth of the liquid, those circles will diminish in their diameter eventually vanishing. So, beyond certain depth, there would not be any circulation, but till such depth as they are going to be those circular moments, we will have appreciable components of velocity vectors upwards and downwards. What it means now is, we are coming to the classical situation where convection as soon as it comes into picture would by far exceed diffusion. This you would know from the basic understanding of convection against diffusion. Diffusion is a molecular process, very slow one. Convection happens over much larger length scales and slightest convection means, considerably more rapid rate of mass transfer here, it will contribute to the rate of adsorption. So, if the molecules are at low concentrations over here, they would tend to go to the surface a lot quicker now compared to a situation here. So, that is exactly what I am trying to show here. We will return to the slides now. These waves which are induced in the surface can increase the rate of adsorption by a factor as much as 7 fold. They can be increased in the rate of adsorption up to 7 fold. Let us return to the slides. No, eventually if the equilibrium is reached, then the concentration in the surface will be in equilibrium with the bulk concentration, but it need not be less than the bulk concentration. You have to remember that it is the equilibrium distribution cohesion which will determine what the final values are. All it means is that when the equilibrium is reached, the rate of adsorption and rate of desorption will be same. So, as we expect the surface active agents will have a lot more affinity for the interface. So, the surface concentration at equilibrium would expectedly be very much higher than the bulk value. That is not to mean that we have any violation of any first principle. We have this diffusion or convection occurring only until the surface cannot take anything more than what corresponds to the bulk value at equilibrium. Because the convection will be able to bring, but for the molecules to anchor into the surface, they will have to have an equilibrium level of state of occupancy. If the equilibrium already achieved and the surface is saturated, even though the convective movements might bring surfactant, the same amount will have to return because surface does not have capacity to hold on to any more surfactant. So, this is one part, how the induced waves would affect the rate of adsorption from the sub adjacent bulk of the liquid. But similarly, you could have increase in the rate of mass transfer when a solute is transferred from the gas phase into the liquid. So, if you are looking at the absorption of gases in liquids, even those are increased quite a lot because the same convective movements are now able to take the dissolved gas phase solute to the bulk a lot quicker. If it were only left to diffusion, it will be the diffusivity of the solute gas into the liquid times the gradient of concentration with a negative sign that would be the flux. And we know that diffusivity is even for small molecules are quite low about 10 raise to minus 5 centimeter square per second or 10 raise to minus 9 meter square per second. But if there is slight convection, then that convection by far dominates diffusion and will get a lot higher rate of mass transfer or absorption here. This is what was reported by Downing and Troustel in 1955. It is over here that I would like to tell you something more. So, let us take a surface which has already induced waves in the surface and we have a species solute A in some inert gas, let us say I over here and that is your water bulk. This solute A would dissolve in the surface and the interface being of very small thickness. And may conventionally take the concentration to be reaching the saturation concentration right away. So, the surface gets saturated with this solute A immediately. If the water is fewer in the beginning, then the concentration in the bulk of this solute A will be 0. And then under the influence of diffusion the solute will start moving into the bulk of water. If the wave or ripple is not there, then we have a picture which is corresponding to this flat interface from where the solute will enter the bulk. Once again the diffusivity is small and therefore the diffusional contribution is going to be very small. Whereas, if we have this wave over here, then there will be considerable circulation water going up and down and that would mean that compared to the diffusional component, these convective contributions will be very high. So, rate of absorption is expectedly a lot higher. The question is how do we quantify this or what kind of picture do we get? If you take a simple diffusion theory, then one will be able to find out the concentration profile corresponding to the stagnant liquid situation where this C A as a function of Z and T could be obtained by solving the diffusion equation with appropriate initial and boundary conditions. However, if these ripples are there or waves are there, then we will have in our equation a dominant convective contribution much larger than the diffusive contribution. And in that case when diffusion and convection both are present, one would be able to interpret the rate of mass transfer in terms of a mass transfer question. So, we go back to the conventional concept of mass transfer question of film transfer question or it could be according to the penetration model a transfer question K L which could be dependent on time. So, one could actually solve the diffusion equation and find out what is the predicted value for this mass transfer question and this could be done in absence of any convection for only diffusion. If on the other hand you actually measure the mass transfer question, you would find K L experimentally measured let us say on the average with this superscript A V. And what kind of setup would you require? Generally, you would use a falling film which means you take a tube on the external surface or internal surface, there is a liquid film flowing down. And then in contact with this film is gas containing this solute A and you find out what is the concentration on the average at the beginning and where the liquid leaves. Depending on that and the cross sectional area which gives you the thickness of the film and the internal surface area across which you have the transfer taking place, we should be able to find out what this K L average is experimentally. Now, what do you expect you would get? If you have this wetted wall column where in we have a flowing thin film of water in contact with the gas mixture A together with inner eye and let us say we start with very small flow rates. So, small that the flow could be guaranteed to be in the laminar region in which case at extremely small flow rates, we may expect the film to have a planar surface. Basically, if you have this representing the inner surface and we have a flowing film over here at very small flow rates, we may expect this surface of liquid film to be smooth and if you have this thickness very small effect of curvature could be neglected that is you could cut open this cylinder and look at this as a planar film of very small thickness. We expect the surface to be smooth and then you could calculate what is the concentration at the inlet and what is the concentration at the outlet. Knowing this interfacial area, you would be able to calculate experimental value for this mass transfer equation. Now, if things go as expected and if there is no complexity in the flow diffusion will be the determining factor and for an element of liquid moving from top to bottom, we can represent this picture in a unsteady state fashion with the time corresponding to 0 at the top and time corresponding to maximum at the bottom. So, and a liquid element may be regarded as sliding from top to bottom and during its travel from top to bottom, it will undergo transient diffusion. Based on that you can calculate what would be the theoretical value for mass transfer equation. If the picture is as simple as we presume here, then these two should be same because diffusion is a pretty well established phenomenon. If we can ensure conditions corresponding to simple diffusion, then we should be able to predict the experimental mass transfer equation. However, in this situation even in laminar flow at low enough Reynolds number of the order of about 18 may be up to 25 at the most, you find the surface developing ripples. The liquid film becomes wavy or ripples start appearing at this surface and now the rate of absorption of A as quantified by mass transfer equation based on the concentration here at inlet and in the exit you could say K L average experimental much larger than what you predict from theory for the same average mass transfer equation in absence of waves or ripples. And this can happen at Reynolds number of about 18 to 25 whereas, the turbulence would actually come into picture only at about Reynolds number of 1200. So, we are well within the laminar flow and yet we find the experimental value for mass transfer equation is lot larger than what you can expect from the theory if diffusion is the only factor. What it means is clearly it is these waves or ripples which are causing the experimental values of mass trans questions to by far exceed the theoretical predictions. So, if this is true if this argument is true it should be possible to ascertain whether this is really the cause. The question is simple if your experiment corresponds to picture different from theory force the experiment to agree with the assumptions in the theory or the model. If you can do that then we should find that the average mass transfer equation measured is same as the theoretical value. And what is the simplest way to do it one could possibly do it by incorporating a small amount of surface activity impurity surface active agent over here which would form a monolayer at the surface and suppress the waves or ripples. Once the waves or ripples are suppressed then we have conditions corresponding to what is presumed in the theory. There is only diffusion and convection, but no complexity of the oscillations arising out of ripples or waves. When we do this then it turns out that the K L average measured experimentally is approximately equal to the K L average from theory way up to the beginning of the turbulent flow. Once of course, turbulence begins then all our assumptions break down we do not expect the experiments to match the theory. This is exactly what happens. So, let us return to the slides to characterize the flow regime here in the case of a falling film. You would take a Reynolds number which is based on the flow rate flow rate the kinetic viscosity and the perimeter of the column. So, the Reynolds number is now 4 u by nu L where u is the volumetric flow rate of the liquid L is the perimeter of the column and nu is the kinematic viscosity. It is this Reynolds number whose values we have been addressing. So, at about Reynolds number of 18 the ripples appear and these become very distinct for Reynolds number of several hundreds and you could well be within the turbulent flow and the mass transfer data would start disagreeing with the theoretical predictions. Adding wetting agents or surface active agents makes the experiments match the theory until of course the complexity of turbulence creates differences again. In the context of what happens to the waves or ripples when surface active impurities or agents are introduced we could suspect that perhaps these surface active agents are preventing the formation of the waves or ripples. At this point mere agreement of the extent and theory does not make a case for one hypothesis or another. So, the possibilities are the surface active impurities when introduced into the liquid may prevent the formation formation of the waves or ripples what other possibility could exist. Perhaps the waves or ripples may still form but they may get damped out by action of the surface compression modulus or rigidity imparted by the monolayers of surface active agents at the interface. This has been discussed in literature by various researchers and the consensus today is that ripples will always form in such an experiment. However, they can be damped out rapidly if we have surface active material present. So, the idea is that waves will form but they will get damped out. So, that is the reason we need to consider damping. Before I go further I have to explain another experimental setup where the formation of waves could be studied and this is the conventional wind tunnel. This is what is commonly used in aerodynamic studies. You have a long channel tunnel through which air or whatever gas is under concentration is made to flow and the liquid will be in a trough kept at the bottom and the waves or ripples formed could be studied in this apparatus. The following film case of course is studied in the wetted wall columns. Now we want to generalize our discussion of the surface waves and the momentum transfer from gas bodies to the water having waves or ripples. So, the first question we should ask is if we have natural wind flowing over a reservoir of water, what would be the shear stress exerted by wind on water surface? For velocities of wind less than about 5 meters a second the shear stress is given by this expression where C H is the drag coefficient, rho g is the density of gas and V H is the velocity of gas at a height of h meters from the surface. So, V H is the velocity of wind at h meters from the surface that is the gas density or air density C H is the drag coefficient. What do we expect now if there is an increase in the wind velocity and if we had very clean water surface? The relative stress one would expect would increase. Why? Because as the wind starts flowing over the water surface at quicker rates, surface waves or ripples will form and if it is a clean water surface compared to the initial smooth surface we will have an equivalent roughness corresponding to the presence of ripples. So, now if the wind is seeing this rougher surface because of presence of the short wavelengths of ripples we expect with increase in the velocity they will be greater amount of shear stress exerted. Contrast this against the situation where we do not have a clean water surface. Let us say we have water again here and we have air flowing that is the action of wind and water is let us say to begin with pure or clean. In that case to begin with the surface is going to be very smooth as the wind velocity is increased we may have short wavelengths appearing. So, the surface becomes rougher. Therefore, the shear stress exerted by or momentum transferred from the wind to water will increase the shear stress will increase. Contrast this against water with surface active impurities. So, if you have now surface active impurities or agents over here then how does the surface behave that would be the question for you to think about. So, let us say we represent only a few of these adsorbed molecules here to begin with you have the surfactant molecules anchored in the surface giving you uniform surface concentration and now the action of wind begins. What do you expect will happen? You should be able to think because we have consider part of this earlier. If the monolayer is present we expect some kind of higher surface viscosity magnitude to exist at this new interface. In addition there will be corresponding surface compressional modulus or an elasticity or rigidity for this surface certain amount. As the wind velocity increases unlike here the surface will not readily split into a ripply surface or rough surface. So, the surface will remain more or less smooth. So, compared to this situation the increase in the relative stress here will be lesser there is something else which happens that is what I would like you to think about. Let us say for the sake of argument that this is the bank of the lake the shore. Let us say the liquid cannot move beyond this particular position that is a kind of wall it cannot go beyond that. The wind is in this direction follow the first principles step by step understand that if the wind is in this direction it will tend to drag the surface along with it and if the surface is a monolayer because there will be no slip between the monolayer and water underneath it will tend to drag water also underneath. So, successive layers of water underneath will be dragged, but lesser and lesser as you go to greater distances below the interface correct. So, the surface layer will be dragged by the action of wind, but this surface layer or a monolayer cannot cross this barrier. So, what it means is now these molecules will start moving in the surface, but they cannot go beyond this point. So, the picture will be that the once over here upstream will be taken downstream and therefore, this will vanish from here they will appear in greater concentration downstream right. There is a certain number of adsorb molecules the wind takes surface layer to the downstream end all the surface impurities will be swept towards that end. So, their concentration will increase downstream at this end and if the concentration of the surface active material here increases you know what happens greater the concentration of the surfactant lower will be surface tension. So, surface tension will be lower originally surface tension is gamma 0 for clean water. In this system surface tension is gamma and that will be lower than gamma 0 by action of the surfactant and greater the surfactant concentration lower will be gamma difference of gamma 0 and gamma is the surface pressure pi. So, in the downstream at the downstream end over here because the concentration of these adsorb molecules will be very high this would be lowest or the surface pressure will be highest which means in terms of surface pressure the surface pressure is high at the downstream end surface tension is lowest at the upstream end because the impurities and have been swept away surface tension will climb from initial gamma to nearly the value for pure water gamma 0. So, surface tension is high here surface tension is low here surface pressure is low here surface pressure is high here. So, in terms of the same arguments as we do for thinking about flow based on the pressure difference in the surface action will be for the monolayer to move from this end towards this end. So, the action of the surface pressure is to bring the monolayer in opposite direction to the direction in which wind is taking it. So, one could expect that in such a scenario there could be certain region near the shore or near the bank of this reservoir tank where the surface pressure is high enough to completely offset the action of shear exerted by wind on the monolayer. If that happens then this surface layer will be stationary if the shear stress is expressed by tau if shear stress is tau and this is exactly balanced by the gradient of surface pressure dou pi by dou L then we should have no movement in the surface. Does this happen can we visualize this you already know the answer. If we sprinkle on top of this monolayer ignited tall particles we should be able to see whether under the action of wind these particles are moving or not. If the monolayer has become stationary the particles will remain stationary indeed that happens you will have when dou pi by dou L balances tau the surface layer becoming stationary. What this means now is now there is no momentum transfer possible from the wind to the bulk of water. If there is no energy being transferred from wind to water that means these larger waves cannot form. So, thereby you are actually reducing the transfer of energy from wind to water through the action of the surface layer. However, when the energy that is in action in the waves comes from gravity there is very little you can do because those are gravitational waves with very long waves, but that is where one needs to think more about. Let us return to the slides to sum up what we have been talking about now. There is there will be a pressure gradient surface pressure gradient just enough to oppose the action of the wind. The wind is trying to drag the surface layer from upstream towards downstream. Surface pressure is trying to move the monolayer from downstream towards upstream. When the two actions balance the monolayer remains this is this is where in this end they will be action of the wind trying to move the surface over here. So, till such time as there are impurities here they will get kept they will keep they will keep getting drag to the downstream end. When there are no impurities left momentum will be transferred surface water will be transferred, but then movement of the surface water is opposed by the action of the monolayer. If the surface water is trying to move downstream and the monolayer is monolayer is trying to move upstream the two actions will balance somewhere. So, there will be region here where there would not be any movement possible. You would be able to see some movement here, but then it will be progressively less and less until this surface becomes stationary. So, returning back to our slides we see that if monolayer is present then the surface of the water would remain smoother and they would not be that much increase in the relative stress with the increase in the wind velocity. To get an idea about the magnitudes the drag coefficient here C h is about 0.9 into 10 raise to minus 3 when h is several meters, but if it is about 5 centimeters the drag coefficient is a lot higher 2.5 into 10 raise to minus 3. And these figures apply for all aerodynamically smooth surfaces whether covered with a film or not. And where the action of wind is counter balanced by the surface pressure gradient we would have the surface becoming stationary. So, to take an example if the shear stress is about 0.1 dynes per centimeter square and you have a monolayer which is about 400 centimeters long then it would mean that this monolayer will be compressed downstream to a surface pressure of 40 dynes per centimeter. And then the back stress will be able to counter balance the action of the shear stress and we should not have any movement possible in the surface layer. This has been demonstrated in large scale system in Van Don's experiments. He took readings on action of steady wind on a 260 meter yacht pond and using a commercial detergent added at the upwind end he eliminated ripples and the drag to what is expected for smooth surfaces. So, in this way you can reduce the effect of waves which have been formed. It is not that formation itself is prevented, but you actually overcome the effect of these waves or ripples by damping them out. Just to complete this lecture a couple of comments are required and that is about the wave damping at clean surfaces. Later on we will look at monolayer covered surfaces of water or any liquid. The wave damping you expect will be related to the viscosity of liquid, viscosity of water here. If the waves are of small amplitude, low amplitude ripples the amplitude will get damped according to this expression a equal to a naught exponential minus delta C l where delta C is damping coaction for clean water surface and l is the length from the source. So, if you create waves as they travel their amplitude will drop exponentially according to this expression a naught is amplitude at the source. If there is a line source creating these ripples or waves the magnitude of a naught is what is amplitude at the source and then as you go for greater distances away from this line source the wavelengths will decrease exponentially in this fashion and the damping cohesion is as expected dependent on viscosity, the group velocity of the wave train or you could express it in terms of the frequency of wavelength or you can bring it to depend on the surface tension. And since we are talking of low amplitude waves we are talking of capillary waves we expect surface tension to start affecting the amplitude damping through this delta C, but it suffices to note here that small levels of impurities which are able to change surface tension only v bit very little they might be able to alter the damping picture quite a lot because now we have an exponential factor here. This is actually only true for clean surface even for clean surfaces if you introduce such low level of impurity that the surface tension is not affected much you would still see the damping to be drastically altered. If you go for higher concentration of course the damping will be very pronounced, but that picture will have to be addressed separately. It is here that I would like to stop today and resume here, but let me plan that question in your mind what is it that will alter the damping if impurities surface active impurities are in such low concentrations that the surface tension is only change very little. Strictly speaking this expression may not be completely valid because delta C would differ right, but even if you go by this limited information that delta C is roughly given by this and any effect of surface active impurity on surface tension is still accounted for by this kind of dependence on surface tension. What is it physically that would govern this pronounced effect surface tension probably is not the factor. So, what is it that affects the damping so much and with that question we will end it here.