 In the last lecture, we continued our consideration of the effect of chemical reaction on mass transfer and completed a discussion of all the regimes. So, as the value of root M increases as we consider reactions of increasing severity, we first pass through the situation of slow reaction regime in which the reaction is so slow that it is unable to influence the concentration gradients within the film. And then we go to the fast reaction regime where quite in contrast to the slow reaction regime the reaction is virtually completed within the film itself. And then the extreme case of reaction severity arises when the reaction is instantaneous in relation to mass transfer in the sense that it is a situation where A and B cannot coexist within the film. So, in that situation we saw that the reaction actually takes place at a single plane which is located somewhere within the film. And the location of the film itself is determined by the condition that the fluxes of A and B should match at the film with respect to the reaction stoichiometry. So, towards the end of the lecture yesterday, we were considering the enhancement factor versus Hata number curve. And we saw that the curve generally has a shape of this kind. When the value of the Hata number is very small we are in the slow reaction regime the enhancement factor is equal to 1 that is to say that the mass transfer coefficient is equal to the physical mass transfer coefficient. And then as root M approaches a value of 1 the enhancement factor curve starts to lift off the floor and when the enhancement factor when the value of the Hata number is more than 3 we enter into what we called as the fast reaction regime in which the bulk of the liquid is not carrying out any reaction at all. So, the system proceeds along this asymptote where the enhancement factor is given by E equal to root M and then depending on the value of Q that is the relative abundance parameter that we have discussed earlier it settles down to one of these asymptotes as the value of the root M becomes much larger than Q. So, this is the situation and on this we can mark our regimes this is the slow reaction regime. This is the transition from slow to fast reaction in which the enhancement factor is given by that expression. This is the fast reaction regime and depending on the value of Q the system settles down an asymptotic enhancement factor equal to E infinity. So, that is the instantaneous reaction regime and this region here where the system has deviated from the fast reaction asymptote, but it has not quite reached the instantaneous reaction asymptote this we can call as the region of as the regime of transition from fast to instantaneous reaction. So, this is a name that we can give to that regime. So, as you have seen we have got expressions for enhancement factor in all the regimes analytical expressions within the framework of the film theory. So, here the enhancement factor is 1 here it is given by that expression here it is on this asymptote it is given by that expression and here it is equal to E infinity. This is the only regime the transition from fast to instantaneous where we do not have an expression, but knowing this asymptote and that asymptote we can do some kind of an interpolation in order to track the course of the enhancement factor as root M increases in the transition regime. We will in a moment derive an approximate solution to track that course as well, but for the moment although we do not have a rigorous solution here it is possible to see it is possible to do an approximate interpolation between the fast reaction asymptote and the instantaneous reaction asymptote. Just to complete the picture what we shall do is we shall erect an additional axis here which will track the concentration of A in the bulk and so this of course in the non-dimensional terms this has a maximum value equal to 1 and the concentration of AB is indeed equal to 1 in the kinetic sub regime or the slow reaction regime as we have seen. So it starts there and it comes down and then attains a value of 0 and then thereafter it remains 0. Now at what value of root M does the concentration decline to 0 depends on another parameter which we have called as the ratio of the film volume to the bulk volume. So this is the direction in which this parameter A hat delta increases. So depending on the value of A hat delta you have a kinetic sub regime and you have a diffusional sub regime before the reaction starts influencing the proceedings within the film. So this kind of summarizes this plot now summarizes all we have discussed in the last few lectures in terms of the effect of chemical reaction on mass transfer. Now before we leave this topic and go to see what the surface renewal theories have to say about the reaction various reaction regimes are there any similarities with what the film theory had to say and where are the differences and so on. We shall just address this transition regime where as we noted we do not have a rigorous analytical expression because that regime involves the simultaneous solution of two ordinary differential equations which are coupled. So obviously you know that is not going to be an easy task and numerical solutions would be needed but a very clever analytical solution has been approximate analytical solution has been derived in the literature and because of the physical insights it gives into the transition regime we shall consider that briefly. So we begin our discussion by noting that the equations that govern the transition regime are the second order differential equations d squared a upon d zeta squared equals m a b and d squared b by d zeta squared equals m divided by q a b with the boundary conditions zeta equal to 0 a equal to 1 and d b upon d zeta equal to 0 and zeta equal to 1 a is 0 and b is 1. So these are the boundary conditions that govern this these equations and we also noted that in general the profile would look something like this. So that is the film this is your zeta equal to 1 and zeta equal to 0 the gas liquid interface and that is your concentration equal to 1 non dimensional concentration 1 and we noted that in general the profiles would look something like this the concentration profile of a goes to a 0 gradient somewhere within the film and then the concentration of b starts with a 0 gradient at the interface and then it goes like that. So now what we note from this is that this is the region where a and b are both present and much of the reaction takes place in the part of this region which is close to the interface because that is where the concentration of a is high. Now we notice that because of this condition here the concentration profile of b is going to start with a 0 slope there which means for some small distance close to the gas liquid interface the concentration of b is not very different from its interfacial value which we shall call as bi. So we are saying that b equal to bi at zeta equal to 0 we of course do not know the value of bi but if we knew the value of bi a good approximation would be to consider the reaction to be first order once again because the concentration of b is approximately equal to bi. If we can make that approximation then we do not have to solve this equation because the after all what this equation gives is the variation of b with the distance which we now neglect and say that to a good approximation we can assume that in the region where we are interested in tracking the concentration profiles b is substantially equal to bi. So if that is the case then of course the equation becomes no different from the equation that we have already solved and if b is considered approximately equal to bi then we have seen that we have already solved this first order equation and that led to the situation that e is equal to root m divided by tan h root m except that here instead of m we have m times bi as the as the first order rate constant if you like that multiplies the concentration of a. So this is an expression that we can use in the transition regime first to instantaneous now that is all very fine except that how do we know the value of bi and unless we know the value of bi we cannot of course calculate the value of e. So in order to get the value of bi we go back to our original equations here and note that in general whether we assume b as equal to bi or not we can always write d squared a upon d zeta squared equals q times d squared b upon d zeta squared that is because if you look at these expressions the only difference between these two expressions is on the right hand side where on the in the second equation we are dividing by q. So if we multiply this equation by q the right hand sides become identical and we can then equate the left hand sides. So that is the equation that we have and we further note that the value of we know the gradient for b and we know the concentration for a. So what is the gradient for a? So if we ask that question then we see that d a or let us consider the definition of the enhancement factor the definition of the enhancement factor is nothing but minus d a d c a upon d x at x equal to 0 divided by minus d a c a star or plus d a c a star divided by delta which is the physical mass transfer rate. So the actual mass transfer rate in the presence of chemical reaction as given by the solution of these equations divided by the physical mass transfer rate. So that is the definition of e and so cancelling of d a and making use of the use of the dimensionless numbers that we have dimensionless concentration that we have defined earlier. We see that the concentration gradient a in dimensionless terms at zeta equal to 0 is nothing but equal to the enhancement factor. So we can use this and we shall of course calculate denote the concentration of b at the interface as b i. So we can say we have additional conditions available now at zeta equal to 0 we can say that d a upon d zeta at 0 is minus e that is this equation here and b equal to b i. Of course both of these are unknown at this stage but we shall see how we can relate these. So starting with this equation and integrating it once we get d a upon d psi equal to q d b upon d psi d zeta rather plus a integration constant which we shall call as d 1. Now if we implement this boundary condition there then we know that at zeta equal to 0 we have d a by d zeta is equal to minus e and d b by d zeta is 0. So I am implementing the boundary condition at zeta equal to 0 and I am saying that at this point e is or minus e is my concentration gradient of a and b is or rather the concentration gradient of b is 0. So this constant becomes equal to d 1 and therefore this equation becomes d a upon d zeta equals q d b upon d zeta minus e. So now we can integrate this once again and that gives us a equal to q times b minus e times zeta plus a second integration constant which we shall call as d 2. So now we apply the boundary condition that at zeta equal to 1 we have a as 0 and b is 1. So this gives we put a equal to 0 and q b is 1 minus e plus d 2 or d 2 is e minus q. So substituting this back in the expression for a that is in this expression here we have a as q b minus e zeta plus e minus q. So this is the concentration profile of a in terms of the concentration profile of b. So of course in this equation we do not know the value of e. So we bring in the additional condition that at zeta equal to 0 we have b equal to b i and a equal to 1. So that is 1 equals q b i minus this term goes plus e minus q or this is the equation that gives us the value of the concentration of b at the interface which is nothing but 1 plus q minus e divided by q. So we can write this in terms of since we are trying to interpolate between the instantaneous reaction regime and the fast reaction regime. We can note that e infinity is 1 plus q which implies that b i can be written as e infinity minus e divided by e infinity minus 1. So going back to this expression now for the enhancement factor e we can substitute the value of b i and finally conclude with this equation e. So this is the equation that governs the transition from fast to instantaneous. So e is given by square root of m b i is e infinity minus e divided by e infinity minus 1 divided by tan h of same quantity m e infinity minus e divided by e infinity minus 1. So this is an implicit equation that allows the calculation of e that is because e occurs in a transcendental form and however in most cases the tan h term in the transition from fast to instantaneous the tan h term goes to 1 and if that is the case you can take e equal to the numerator and square both sides end up in a quadratic which can be solved. So if you are trying to solve this equation by trial and error you can start with the solution that obtains by assuming that the denominator is equal to 1 with that as the starting or the initial guess you can do a Newton-Raphson or other methods of solving these kinds of equations. You can implement one of those methods with the initial guess given by the equating the numerator with e. So we now have an approximate way of calculating the enhancement factor in the transition regime as well. So no matter what our regime is we now have a way of calculating the absorption rate because once you know the enhancement factor you multiply the physical mass transfer rate with the enhancement factor and you have the absorption rate for that regime. So that completes the discussion of the various reaction regimes within the framework of the film theory. So now we address the question of whether you know we are justified at all in using the using the film theory and that we do by considering the same regimes or considering the same situation of mass transfer occurring with a chemical reaction with a second order chemical reaction of this stoichiometry A plus nu B going to C. A is in the gas phase, B is in the liquid phase and let us assume that C is in the liquid phase as well. The same reaction taking place in the in the liquid A is getting transported from the gas to the liquid, but now we assume that that transport is governed by a surface renewal type of mechanism. So in other words to recall the surface renewal mechanism there is the bulk that is in a state of churning in a state of agitation and because of this agitation the surface element the elements of liquid from the bulk are thrown on to the gas liquid interface. Each element spends a certain length of time at the gas liquid interface and during this time it is absorbing gas from the interface and while we did not consider the chemical reaction when we consider this picture in the context of physical mass transfer. Now we shall consider chemical reaction in other words the surface element has going on within it a process of unsteady state mass transfer accompanied by chemical reaction and so this process goes on and it leaves a certain concentration of A and a concentration of product in the element by the time it leaves the gas liquid interface and then these concentrations are evened out as the surface element gets completely mixed within the bulk of the liquid. Now we have seen that there are differences as to you know what is assumed for the distribution of residence times or distribution of surface ages you know as given by the Hickby theory and as given by the Danckwert's surface renewal theory, but in order to keep matters simple we shall take the attitude we shall take the distance that K L is given by square root of D by some quantity with dimensions of time which we shall call as tau. So, we note that if tau is pi T b divided by 4 we have the Hickby postulate in which every element of liquid is spending exactly the length of time given by T b at the interface or if tau is given by 1 over s the surface renewal rate or surface renewal frequency then we have the Danckwert's picture. So, both of these theories can be combined in this manner into one expression which gives that tau is this is dA incidentally. So, dA squared divided by K L. So, this sorry dA divided by K L squared. So, this is the definition of the time that is defined by this equation and if you substitute the orders of magnitude for diffusivity which is of the order of 10 to the power minus 9 meter square per second usually and K L is of the order of 10 to the power minus 4 squared and so that is meter per second squared. We note that this is of the order of 10 to the power minus 1 second. So, that is the order of magnitude of this characteristic time that we have defined by looking at the experimental mass transfer coefficient. So, that is one circumstance that we will make use of. The other one is that we have already made use of the expression for the Hata number in terms of the physical mass transfer coefficient rather than in terms of delta in terms of which we had first defined it. So, if you look at this expression now there is nothing in it to suggest that the film theory postulates are in any way involved because the film theory construct which is the film thickness has been eliminated in favor of the mass transfer coefficient which is an experimental parameter. So, we will keep these two things in the background that is the definition of root m in terms of the mass transfer coefficient and a characteristic time that is calculated by reference to the experimental mass transfer coefficient which is of the order of 10 to the power minus 1 second. So, the governing differential equations for the case of you know a second order reaction accompanying mass transfer would be can be written down in that manner d C A upon d t equals d A d squared C A upon d x squared minus k C A C B. So, what this is saying is that accumulation within a differential element within located within the surface element is equal to the input minus output minus consumption by chemical reaction. So, when we last considered the surface renewal theory in the context of physical mass transfer this reaction term was not there it now makes its appearance because we are now considering the effect of chemical reaction on mass transfer. Similarly, for B we have this equation d B d squared C B upon C upon d x squared minus nu times k C A C B which is the rate of consumption of B within the surface element. The initial and boundary conditions that are applicable are at t equal to 0 that is the initial condition for all values of x everywhere within the surface element. We have the concentration of A as equal to the concentration of bulk with where the element is coming from and the concentration of B is equal to the concentration of B in the bulk. In other words the surface element is taken out of the bulk liquid and therefore, when it just comes to the interface nothing has changed within the surface element. At t greater than 0 we have two conditions at x equal to 0 we have C A equals C A star in equilibrium with the prevailing partial pressure on the gas side and we have d C B upon d x equal to 0. And at large distances from the interface which we call as extending to infinity because the element is spending a fraction of a second at the interface there is not enough opportunity for the gas to diffuse very much deep into the surface element. Therefore, we can regard the surface element as infinitely deep from the point of view of the diffusing solute. So, we have C A bulk and C B bulk. In other words the conditions at x tending to infinity for large times for times larger than 0 are identical to the conditions at all x at t equal to 0. So, this as we have noted earlier suggests a combination of variables which we will come to in a minute. But before we do that as usual we want to minimize our work of having to solve these complicated partial differential equations which are coupled. Therefore, rather than go to the computer and start solving these by methods of brute force we shall do some analysis by first non-dimensionalizing these equations. So, in non-dimensionalization we have no difficulty in defining a non-dimensional concentration of A because we have already done that in the context of film theory. And we similarly can define a non-dimensional concentration of B again in identical terms to what we did in the context of film theory. But when it comes to a non-dimensional distance we note that there is a bit of a problem because the field of diffusion is semi infinite in this case it is bounded on one side by the gas liquid interface, but on the other side it goes to infinity. So, this problem has no characteristic length. So, however we note that there is a characteristic time for the process that is tau which we have defined by looking at the physical mass transfer coefficient. And in terms of tau a dimensionless time can always be defined and that is t divided by tau where tau we have seen is dA divided by kL square. And we note that a characteristic distance can be calculated from this characteristic time by reference to the equations of physical mass transfer. You recall the error function solution we derived for the concentration profile as a function of distance and time in the physical mass transfer case. And if you plug in the you know x is equal to square root of dt in that equation then it shows that this is the depth to which approximately the solute will penetrate in a time t. That is because if you plug this condition into the error function complement solution it shows that the a significant part of the concentration profile lies within this distance. So, this can be made use of as a characteristic distance that is the characteristic depth of penetration by substituting the characteristic time in this expression. So, x equal to square root of d tau will be our characteristic x with that we can define a dimensionless distance as x divided by square root of d tau. So, we are now ready to go through with the non dimensionalization of our equations. And if we do that for a we have the following equation we have this equation for a and this is the one that I am considering right now. So, we have C a star d a divided by d theta divided by tau here and d a C a star divided by the characteristic distance comes out here raised to the second power because we have a second derivative here. And therefore, that is d a tau minus k times C a star C b bulk a b alright. So, getting rid of all the variables from the left hand side we have d a upon d theta d a will cancel here and C a star upon tau will cancel there. So, we have d square a upon d zeta square minus k C b b tau multiplied by a b. Now, recalling that tau is nothing but d a divided by k l square. So, this term is 1 upon k l square d a k C b b and. So, this shows that this is nothing but square of the Hata number itself written in terms of the mass transfer coefficient. So, our equation now becomes d a upon d theta equals d square a upon d zeta square minus m a b which shows that the effect of any effect that the chemical reaction has on the concentration profiles. So, this term would be absent in physical mass transfer. So, any effect that the chemical reaction has on the concentration profile has to be understood in terms of the magnitude of this dimensionless parameter m. So, what is this m? m is of course, the same quantity that we have visited earlier, but if we want we can interpret m in terms of the postulates of the surface renewal theories. So, m is nothing but k C b b times tau. So, it is some kind of a Damkohler number in that sense, because k C b b is the first order rate constant and that is the time required for the reaction to proceed to a significant extent and tau is the time that is available for the reaction to occur, because after time tau typically, characteristically the surface element ceases to be at the surface. So, naturally if large values of m correspond to significant amounts of reaction taking place within the life of the surface element and small values of m correspond to a negligible amount of reaction taking place within the life of the surface element. So, that is as far as m is concerned. So, we can do a similar non dimensionalization as we did for the case of a for the case of b, I shall leave the details to you, but basically the equation that results is something like this m upon q a b. So, just as in the case of film theory you note that m upon q is making an appearance in the reaction term here, whereas m appeared in the equation for a. The other difference that we note from the case of film theory is that this ratio of diffusivity is a to b is appearing in this equation, whereas that was not a parameter that appeared anywhere in the in our considerations in the film theory. So, that will have some consequences as we go along, but for the moment we shall continue with this and write down the initial and boundary conditions in terms of the non dimensional parameters. The initial condition is at theta equal to 0 for all values of zeta, you have a is a b which is the ratio of the concentration of a in the bulk to c a star and b is 1, because the element is coming from the bulk. And for values of time greater than 0 that is once the element has landed at the interface at zeta equal to 0 then your a is 1 and d b upon d zeta is 0 and as zeta tends to infinity we have a going to a b and b going to 1 and these two conditions are similar. Now, so we have these equations we have the equation for we have the equation for a and the equation for b with the applicable initial and boundary conditions and let us consider these equations in terms of increasing severity of m. So, the q parameter that appears here is the relative abundance parameter that we have already seen in the in our considerations of film theory and we have noted that q is usually a number that is much larger than 1. So, that continues to be the case here. So, we shall first consider the case where our value of m is much smaller than q. So, we are considering very slow reactions for which the value of m is small anyway and considering that q is usually a large number we are approaching the matter of the effect of chemical reaction on mass transfer from a point of view from a point where the reaction is very, very slow. Now, that is the case then the equation for b becomes d a upon d b d b upon d theta equals d square b upon d zeta square because the other term is close to 0 the term that contains the ratio of m to q. Now, in the case of film theory the moment this assumption was made it fell out that b is equal to 1 everywhere within the film we did not have to do too much work. We shall persist with the equations of surface renewal theory for a bit. If only to demonstrate that there is a lot more work involved in arriving at similar conclusions as we arrived at in the case of film theory, but within the framework of surface renewal theory. Surface renewal theory is a transient theory the equations are partial differential equations the film theory is a steady state theory the equations are ordinary differential equations. So, we shall see how things work out. So, a conclusion that was self evident in the case of film theory will be made with considerable effort in this case. So, we need to it is not clear from this although in an intuitive sense since we are saying that the reaction is slow and the relative supply of b relative to the rate at which it is required by stoichiometry is much larger than 1. So, a combination of these two circumstances should mean that the concentration of b is uniform right up to the interface at all times in the case of the surface element, but while that is intuitively obvious the mathematics should also say the same thing and it is not obvious whether this equation is saying the same thing. So, we shall solve this equation once again by combining the variables in the manner in which we did it for the case of physical mass transfer and we can define a combination variable eta and I shall directly write the equation for eta. So, if you make the substitution here calculate these derivatives in terms of eta for example, you can say that dA upon d theta is dA upon d eta and d eta partial d eta upon d theta and so on. So, we can work out all these sorry this is I should have be there. So, we can work out all these derivatives on both sides by these kinds of application of chain rule of differentiation and if we did that and wrote down the final equation we see that theta and eta theta and zeta disappear from the equations leaving behind only eta. So, this is the equation that results and the corresponding two boundary conditions for this ordinary differential equation now is eta equal to 0 is dB upon d eta equal to 0 this is this comes from non volatility of B and as eta tends to infinity B equal to 1. This is the two conditions one for t equal to 0 and the other for extending to infinity which can be combined because under both of these conditions the concentrations are identical. So, we can integrate this in a straight forward manner because if we let u stand for dB upon d eta the first derivative then this equation is dU upon d eta equals minus 2 eta u which has a straight forward integral C e to the power minus eta squared and if we apply this boundary condition and noting that this is u at eta equal to 0 this term is 1 and we are saying u equal to 0. So, this application of the boundary condition this one shows that C equal to 0 which means u is 0 and if u is 0 dB upon d eta is 0 and that means that B is a constant for all eta and since B is equal to 1 at 1 value of eta that we know that we have this condition. So, this leads us to B equal to 1. So, we have to do so much work to realize that the concentration of B is uniform throughout. So, once we realize that we can now rewrite the equation for A as dA upon d theta equals d square A upon d zeta square minus mA because B is equal to 1 and this therefore, now becomes the pseudo first order case because the order with respect to B is degenerate and therefore, we have this equation here. Once again within the pseudo first order case we will first consider we are considering case 1 which is pseudo first order regime or pseudo first order case and we shall now consider under case 1 case 1 A which is m being far less than 1 it is not only far less than q it is additionally far less than 1. So, in this case we have dA upon d theta is equal to d square A upon d zeta square there is no appearance of the chemical reaction term anywhere in the works and we have the equation and initial and boundary conditions which are identical to the case of physical mass transfer. So, in other words what we are saying is that the reaction is sufficiently slow in this case that from the point of view of the surface element which is spending time at the gas liquid interface the reaction is unable to make any dent in the concentration profiles and therefore, the concentration profiles will work out to be exactly the same as in the case of physical mass transfer which means we have the error function complement solution applying and we arrive at the instantaneous flux which we average with respect to the weighted with respect to the surface H distribution and so on. So, everything works out in an identical manner to the case of physical mass transfer. So, the mass transfer rate or mass transfer coefficient is not influenced by chemical reaction this is equal to the case of physical mass transfer coefficient. So, this conclusion is identical to identical conclusion to the case of film theory. So, now we ask the question then what is the role of chemical reaction here and we come up with the answer that in order to look for the effect of chemical reaction. Now, we have to look at what is happening in the bulk because that is where the reaction is actually occurring. So, we make a bulk balance and again we have to anticipating that the volume of liquid in the surface elements is going to be much smaller than the volume of element volume of liquid in the bulk. We can write the bulk balance in a manner that is identical to the case of what we did when we were discussing film theory. So, this is the rate of mass transfer per unit liquid volume multiply that by the liquid volume and that is the we should have had the bulk volume here, but we are using the liquid volume here and so this is equal to V L into K into C AB times CBB. So, this is the same equation as we had written earlier and therefore, this will of course, lead to this equation that the bulk concentration of A is equal to 1 plus P where P is K CBB divided by K L A. We had shown this as equal to M divided by A delta here in the case of film theory and in this case it turns out if you do the manipulations that it is this quantity where the quantity within the in the denominator is nothing, but the volume of liquid within the penetration depth penetration depth divided by total volume of liquid. Since this is the same equation that same expression that we are dealing with this you can plug in the actual numbers tau is equal to minus 10 to the minus 1 dA is 10 to the power minus 9 and so on and so forth and this turns out of the of the order of 10 to the power minus 3. So, all these two just say that in the event of M being much less than 1 there is virtually no difference between what the film theory has to say and what the surface renewal theories have to say. So, these you know which variant of the surface renewal theory you are considering does not really matter because the mechanism of mass transfer itself is such that the reaction does not have an influence and therefore, whatever remarks we made for the surface the Higbee version of the surface renewal theory versus the Dankwoods version of the surface renewal theory in the case of physical mass transfer those remarks apply identically in this case. When we come to the considering the effect of chemical reaction it centers on the value of this parameter P and the value of parameter P can be either much greater than 1 or much less than 1 both of these conditions are possible because the volume of the liquid that is contained within the surface element within the penetration depth is much much smaller than the total volume of liquid there is in the tank and with that understanding both P much greater than 1 and P much less than 1 are possible. So, in one event we have the kinetic sub regime in the other event we have the diffusional sub regime. So, up to the end of the slow reaction regime we are solving I mean our solution of the surface renewal theory equations gives us nothing different from whatever we have concluded from our consideration of film theory. So, what happens for faster reactions is something we will take up in the next lecture.