 Welcome back. We have been considering the effect of chemical reaction on the mass transfer rate within the framework of the surface renewal theories and in the last lecture, we set up the equations in the framework of these theories that describe the instantaneous reaction regime. So, you recall that these were the equations. So, the instantaneous reaction regime is one in which A diffuses from the interface and B diffuses from the bulk and they meet and consume each other at a reaction plane which is located somewhere within the surface element. So, this reaction plane itself starts at the gas liquid interface at the instant the liquid element arrives at the gas liquid interface and as the element ages at the interface it is as it spends more and more time at the interface the reaction plane progressively moves inward into the liquid element. So, this becomes a moving boundary problem and if you look at the equations that were set up. So, there is a region between 0 and zeta 1 of theta where zeta 1 is the location in non-dimensional terms of the reaction plane. So, the region between 0 and the reaction plane the interface and the reaction plane is occupied by A and since there is no B there cannot be a reaction there. So, it is a case of pure diffusion. The region between the reaction plane and the rest of the element is occupied by B and once again since there is no A there there cannot be a reaction and this is the equation that governs the pure diffusion of B in that region. At the reaction plane itself because the reaction is assumed to be instantaneous the concentration of A and B both go to 0 and the rate of supply of B from the in the negative x direction if you like is in stoichiometric requirement as compared to the rate of supply of A in the positive x direction. So, that is why there is a sign difference here. So, now this is the formulation of the problem and in order to solve this we require a little more information and that is because this is a moving boundary problem we need to track the boundary movement. Now, this can be done in various ways and birds toward light foot in their in the first edition of their book transport phenomena suggest a very elegant way of doing this and we will simply state that before proceeding further. So, this approach consists in recognizing that at the reaction plane the following identity holds we have the concentration of A which is a function of zeta and theta and since we are considering the reaction plane it is zeta 1 which is itself a function of theta is identically equal to 0. So, A at the reaction plane is a function of the single variable theta and that is a constant value and it is 0. So, if we differentiate this we get d A by d theta equals partial d A by d zeta 1 d zeta 1 upon d theta plus partial d A by d theta and this is equal to 0 and this gives you the equation for the boundary movement which is d zeta 1 by d theta. So, this can be rearranged to get the rated which zeta 1 moves with theta and that equation can be used along with the previously shown equations in order to arrive at a solution to these equations. So, that solution is analytically possible and it is the details of the solution are available in the books that I mentioned in the previous lecture the book by Dan Quartz that is Gas Liquid Reactions and the book on Transport Phenomena by Burt Stewart and Lightfoot. We will not go into the details of the solution except to mention that the solutions you know looking at the nature of the equations the solutions can be expected to be of the error function type and. So, we formulate the error function solutions in terms of the variable x divided by or zeta divided by 2 root theta in non-dimensional terms and the constants are evaluated using the conditions that I have laid out earlier. So, now it turns out when you solve these equations that the rate of the instantaneous rate of absorption or the instantaneous flux which is given by negative of the diffusivity multiplied by the concentration gradient at the interface. So, this flux has the same kind of dependence on time as the flux in the case of physical mass transfer. In other words if you look at the absorption rate at every point during the life of the surface element the absorption flux is greater than the physical absorption flux by a constant factor. Since this factor which enhances the instantaneous mass transfer rate is a constant ultimately when you average these instantaneous rates weighted with the I of t dt function whether you assume the Higbee function or the Danckwerts function for the I of t dt that is the plug flow or the mixed flow assumption at the interface. It does not really matter as far as the effect of reaction on the mass transfer rate is concerned or in other words in so far as the calculation of enhancement factor is concerned. So, if you can define an instantaneous enhancement factor which is the rate by which is the factor by which the instantaneous flux is higher than the physical mass transfer flux at every time during the life of the surface element. This instantaneous enhancement factor is independent of time and therefore, the average enhancement factor over the entire life of the element whether you average the enhancement factors using the Higbee function or the Danckwerts function it gives you the same result. So, that result is what I will give you next. So, that is given by an expression of this kind error function of beta divided by square root of dA where this function beta where this parameter beta comes from the transcendental function exponential of beta squared divided by dB error function complement beta divided by square root of dA equals dB rather q square root of dA upon dB e to the power beta squared by dA error function of beta divided by square root of dA. So, this is the equation that is implicit in beta which has to be solved by a process of trial and error knowing the other quantities like q and dA and dB and once you get beta you substitute this in the enhancement factor expression and this is the same expression in both the surface renewal theories that we have considered that is Higbee and Danckwerts. So, the message that we are trying to convey here is that within the framework of these theories the surface renewal theories e infinity turns out to be or the maximum enhancement factor available to your system turns out to be a function of dA by dB and the parameter q. Recall that it was a function only of q in the film theory and here it there is an additional parameter which is the ratio of diffusivities. We have said that often this ratio is not very far from unity the rate of I mean the diffusivity of A ratioed with respect to the diffusivity of B is a number something like 1 in many cases and in that in those situations the e infinity becomes a function of q alone in the ambit of surface renewal theories as it does in the case of the film theory. Moreover there is also while these expressions look very different from the expression that we obtained for the film theory it turns out that if the value of the enhancement factor is large then there is a very simple approximation that obtains to this complicated expression here and that is e infinity equals square root of dA upon dB multiplied by 1 plus q and if you recall that without this diffusivity function what remains was the enhancement factor predicted in the instantaneous reaction regime by the film theory. So, the factor that distinguishes the surface renewal theory expression from the film theory expression is just this factor square root of the diffusivity ratio. Now, the this is as I said it is an approximation and the error that you make in calculating enhancement factor by this expression is of the order of 1 over 2 e infinity. So, the larger the value of the enhancement factor is the in other words the larger the value of q is the smaller the error that you make in using this expression. So, now, we are in a we have gone through the entire gamut of regimes in the surface renewal theory framework and we are now in a position to put the entire story down in terms of a plot of enhancement factor versus square root of m or also called as Hata number. So, we have Hata number of 1 somewhere there let us say Hata number of 3 somewhere here and so, the enhancement factor remains 1. So, this is the story according to surface renewal theories and so, this is a slow reaction regime where we said that there is absolutely no difference between the surface renewal theories and the film theory and about 1 the enhancement factors start to lift off the ground. So, this is 1 and we have a region here which I will show by dotted lines which is the transition slow to fast where there is a bit of a difference between the film theory and the surface renewal theory. So, this is the film theory asymptote would be here slightly below, but the difference is not much and as the Hata number becomes larger and larger you have the fast reaction asymptote obtaining E is equal to root m and depending on the value of the instantaneous enhancement factor you have got branches coming out of this asymptote and these are increasing values of E infinity and the E infinity itself is given by normally by this expression here with beta being calculated by that expression, but for larger values of E infinity we can use this approximation. So, we have the slow reaction regime where film theory is equal to the surface renewal theory is identical with the surface renewal theory. We have the transition regime where to a good approximation you can calculate the enhancement factor by either the expression given by the film theory or the expression given by the surface renewal theory. The two differ by a few percent at most and then in the fast reaction regime once again you have got a situation where the surface renewal theory and the film theory agree completely for reasons that we have elaborated on earlier and then in the instantaneous reaction regime there is a difference if the diffusivities of a and b are very different from each other. So, there is this region what we have normally called as the transition fast to instantaneous. So, this regime so before this regime we have the fast reaction regime E is equal to root m and beyond this we have got the instantaneous reaction regime E is equal to E infinity. So, how do we calculate the enhancement factor in this region? One can of course, graphically do an interpolation between this asymptote and the appropriate asymptote here, but it has been shown that a good approximation is to use the same expression that we derived with the film theory using making that is making a pseudo first order rate assumption except that the pseudo first order rate constant as is calculated as k C b i using the value of the concentration of b at the interface rather than as k C b b which uses the concentration of b in the bulk. So, the expression that we had derived then if you recall is square root of m E infinity minus E divided by E infinity minus 1 divided by tan h of the same quantity m E infinity minus E divided by E infinity minus 1. So, this equation obtains for the transition regime fast to instantaneous. So, the only difference between the way you calculate enhancement factor in the surface renewal theory and the way you do it in film theory is that while you are doing using the same expression the value of E infinity that you use comes from the penetration theory or the surface renewal theory expression which is different as we have seen when the diffusivities are different as compared to the film theory situation. So, the value of I mean the definition of E infinity is what changes in this expression depending on what theory you use. So, we have completed the discussion of the various reaction regimes in both the film theory and the surface renewal theory and an important point that it makes the entire discussion makes is that while the film theory is supposed to be less accurate as compared to the surface renewal theory in so far as the prediction of the mass transfer coefficient on the diffusivity is more correct in the surface renewal theories when it comes to predicting the effect of reaction on the mass transfer rate in the slow reaction regime in the transition slow to fast to a good approximation and certainly in the fast reaction regime that is in the entire pseudo first order situation. The film theory is nearly as good as the surface renewal theories for the purposes of predicting the enhancement factor. So, this is convenient because in more complex cases it is always simpler to use the film theory because there are ordinary differential equations and finite fields and so on as compared to the surface renewal theories. The only places where there is a difference is where the concentration of B starts to make a difference to the overall absorption rate that is in the second order regimes. So, having come so far now let us see whether we can relax some of the assumptions that we made at the very beginning. There are two important assumptions that we made number 1 was that there was no gas phase resistance and number 2 that the reaction is second order that is first order with respect to A and first order with respect to B. So, taking the first taking the second assumption first that is let us look at the effect of different reaction orders. So, we will see the effect of relaxing the assumptions. So, the first assumption is that the assumption of reaction order. So, we have assumed that A plus nu B going to C C is the reaction and we have assumed that the rate the intrinsic rate of this expression of this reaction is given by K C A C B that is the case for which we have developed the theory. So, far supposing the reaction is mth order in A and nth order in B and the rate constant is appropriately designated as K m n what difference does this make and how good are the theories that we have developed so far for the second order reaction in cases such as this. So, it turns out that all that needs to be done is to redefine the Herta number in the following manner. So, it is the physical mass transfer coefficient in the denominator and 2 divided by m plus 1 diffusivity of A the rate constant C A star raise to the power m minus 1 C B bulk raise to the power n. So, provided you calculate your root m in that manner this root m can be plugged into all expressions of the enhancement factor that we have derived and you get reasonable results. So, this is a very good way of extending the theory to reactions of other than second order. The second assumption that we want to examine is the assumption of negligible gas phase resistance and this turns out to be quite simple if the gas phase resistance is not negligible all that it means is that the value of C A star that we have been using in the theoretical expressions has now to be calculated as being in equilibrium with the interfacial partial pressure of the gas which in this case would be different from the bulk partial pressure of the gas. So, we do that by we calculate the interfacial partial pressure by equating the flux of the gaseous solute from the gas side to the flux on the liquid side. In other words we use the usual balance for two phase mass transfer if P stands for the partial pressure of A and P A B is the partial pressure of A in the bulk of the gas that is in the interior of the bubble if you like and P A I is the value of the partial pressure at the gas liquid interface then this would be equal to the rate at which the gas is taken away and that is K L into C A I minus C A B where C A is the concentration C A I is the concentration of A in the liquid at the gas liquid interface minus C A bulk is the concentration of A in the bulk. And usually as we have seen if the enhancement factor needs to be considered in if E is greater than 1 it usually implies that C A B is close to 0. So, either you have a situation where the rate is given by K L into C A I minus C A B or you have a situation where it is given by K L into A E into C A I C A B being equal to 0. And the C A I and P A I are in equilibrium with each other being the concentrations the partial pressure and the concentration on the two sides of the interface or we can write C A I is the C A star corresponding to P A I and this could be given by Henry's law or whatever a suitable thermodynamic expression. Therefore, the essential theory remains the same except that wherever we have had C A star being calculated as being in equilibrium with the bulk partial pressure of the gas we replace that by the quantity that is calculated as being in equilibrium with the partial pressure at the interface which itself is given by this expression here. So, that is all that needs to be done in order to take gas phase resistance into account. In order to see how these things work let us take an example and so we look at this example here for the instantaneous reaction regime and the transition to the instantaneous reaction. So, the example is about calculation of the maximum enhancement factor and the actual enhancement factor. So, let me read the problem carbon dioxide is being absorbed from a gas into a solution of sodium hydroxide at 20 degree centigrade in a packed tower. At a certain point in the tower the partial pressure of carbon dioxide is 1 bar and the concentration of sodium hydroxide is 0.5 kilo moles per meter cubed. Other data are as follows the physical mass transfer coefficient is 10 to the power minus 4 meters per second interfacial area per unit volume of packed space is 100 meter inverse that is 100 meter square per meter cubed of packed space. The concentration of A at the interface is given as 0.04 kilo moles per meter cube we assume that the gas phase resistance is negligible. The second order rate constant of the reaction is 10 to the power 4 meter cube per kilo mole second and the diffusivity of A and B are this should be diffusivity of B. A and B are given find the maximum enhancement possible and the actual enhancement and find also the actual absorption rate in units of kilo moles per second per unit volume of packed space. So, the reaction stoichiometry is given here. So, 1 mole of carbon dioxide reacts with 2 moles of sodium hydroxide. So, how do we attempt this example? The maximum enhancement factor we note first of all that the diffusivities of A and B are quite different from each other and in fact the ratio of the diffusivity of B to the diffusivity of A in this case turns out to be 1.7 if you calculate this ratio from the given values of the diffusivities. Therefore, because this is significantly different from 1 we have to use the surface renewal theories in order to calculate the maximum enhancement factor and rather than use that complex expression we will first see whether E infinity can be calculated by the approximate expression that we had which is square root of D A upon D B 1 plus Q. Q itself is given by D B C B B divided by nu D A C A star and putting in the values D B upon D A is 1.7 and C B B is 0.5 kilo moles per meter cube, nu the stoichiometric factor is 2 moles of NaOH per mole of carbon dioxide multiplied by C A star that is 0.04. This value turns out to be 10.625 not a particularly large number in this case we have said that Q is often of the order of 100 or more, but carbon dioxide has a relatively larger solubility as compared to gases such as oxygen, hydrogen etcetera. So, this value is a moderate value of about 10 or 11. If we substitute this value of Q in the expression for E infinity above we calculate the value of E infinity as 8.91. So, remember that this is an approximate value since we have used this expression and if we want to estimate the error we can do this as 1 over 2 E infinity and so approximately the error is going to be of the order of 1 over 18 or thereabouts. So, that is an acceptable value of the error because often the errors in the mass transfer coefficient itself are of a larger magnitude than this. So, let us proceed further and calculate the actual enhancement factor as the problem requires us to do. So, in order to calculate the actual enhancement factor we need to first estimate the regime. So, that we know which expression to use. So, this question can be answered by calculating the value of the Hata number or root m which for a second order reaction can be calculated in the following manner. This is the standard definition for a second order reaction and if you plug in the values turns out that 1 over m is 1 over k l which is 10 to the power minus 4 meters per second all units are in S i therefore, we do not have to do any conversions 1.8 multiplied by 10 raise to minus 9 is the value of diffusivity in square meters per second k is 10 to the power 4 second order rate constant multiplied by 0.5 is the value of the sodium hydroxide concentration. So, this gives you a value of 30. Now, so comparing this with the value of q we find that this is greater than 10.625. So, we have the situation root m greater than q. Now, in order to assume instantaneous reaction we would require that root m be far greater than q while 30 is more than 10.625 is the difference large enough that we can assume instantaneous reaction. So, we are not sure of that therefore, let us not assume that e is equal to e infinity and proceed to calculate the value of e using the transition regime expression which is this here square root of m into e infinity minus e divided by e infinity minus 1 this also under the square root divided by tan h of the same quantity right. In order to simplify our calculations and in order to put in place an iterative scheme for calculating the value of enhancement factor we will calculate a first approximation as e is equal to the numerator. In other words we assume that the value of this modified Hata number if you like m times e infinity minus e divided by e infinity minus 1 is large enough that is it is larger than 3. So, that we can assume the the hyperbolic tangent of this quantity to be nearly equal to 1. So, if we do that then we can calculate the first approximation as root m divided by e infinity minus 1 into square root of e infinity minus e where I have just separated out the values that we know from the value that we do not know. So, this is square root of well square root of m we know to be 30 divided by square root of e infinity is 8.91. So, we have 7.91 here multiplied by e infinity minus e under the square root and this gives you a value of this number here turns out to be 10.67 and therefore, we have the equation e equals 10.67 times square root of e infinity minus e and e infinity we know to be 8.91. So, we can square this and obtain the value of e as from this quadratic expression 10.67 square into e infinity minus e. This is a standard quadratic equation which we can solve and this will give you a value of e as 8.30 which is not too far from 8.91 which is the value that we had estimated for the instantaneous enhancement that is the maximum value of the enhancement factor. So, what we are saying is that even for root m a factor of 3 higher than q the reaction is almost totally in the diffusion control regime that is in the instantaneous reaction regime. So, now, since we made the approximation of the denominator that is the hyperbolic tangent term in the denominator being equal to 1 we can test out that assumption by calculating the second approximation where we calculate this quantity square root of m which is 30 square root of m is 30 e infinity minus 1 e infinity minus e divided by e infinity minus 1. So, this turns out to be a value that is sufficiently large that this is greater than 3. Therefore, the tan h of this quantity here is approximately equal to 1. So, the second approximation also turns out to be 8.30. So, we have converged on a value of the enhancement factor as 8.30 we accept that value and now we are ready to calculate the absorption rate. So, we do that in the following manner absorption rate is the absorption flux that is K L C star which is the physical absorption flux multiplied by the actual enhancement factor. This is the chemical absorption flux the actual absorption flux, but we want the rate in units of moles per centimeter cubed of packed space per second. Therefore, we can multiply this by the interfacial area ASP which is the square meters of area available per unit volume of packed space. So, if we can substitute this and calculate the value I will leave that to you. One point that we should note is that in our consideration of the various regimes and calculation of rates and so on. We have come across several definitions of this quantity which we have called the interfacial area. So, in our theoretical development we used a hat which we called as the interfacial area per unit volume of liquid. In one of the earlier examples we had the case of a sparse reactor in which the term interfacial area per unit volume of dispersion was introduced. Now, remember that the volume of dispersion includes the volume of liquid and the volume of the held up gas. So, this quantity of the this definition of the interfacial area is a little different from the interfacial area a hat that we have used in the theoretical development. And in this example we have encountered the interfacial area per unit volume of packed space. So, basically it does not make a difference to your calculation of the flux at all because that is given by this expression here. Depending on what units you want the rate to be in you have to multiply this by the appropriate type of interfacial area per unit volume. So, here the rate is required in moles per unit volume of packed space per second. So, use the interfacial area as square meters per meter cubed of packed space. So, that is a matter that is fairly trivial and does not need to engage our attention any further. So, now this is the rate expression that you would use in the design of the packed bed reactor itself. In other words in any design exercise you would do a mass balance on the flowing phase for a continuous equipment such as this in which you say that if the equipment is operating at steady state you would say that in any slice of the liquid in any slice of the packed bed there is a certain amount of solute that is entering and there is a certain amount of solute that is leaving and the balance is being absorbed within that slice of the packed bed. So, the absorption rate would be the rate of absorption per unit volume of the packed space multiplied by the volume of the slice. So, this rate of absorption per unit volume of the packed space is what we have calculated here. So, this is the local rate expression that would go into any of your macroscopic balances or reactor level balances. So, that completes our discussion of the gas liquid reactions. So, let us summarize what we have seen in the past 6 or 7 lectures. So, we have seen we started out by looking at how does mass transfer occur from a gas to an agitated liquid and we considered this in terms of two possible mechanisms. One is a steady state mechanism what we called as the film theory which assumes that the entire resistance to gas liquid gas liquid mass transfer is located in a thin film of thickness delta located at the gas liquid interface. Outside of delta the liquid is in a state of continuous mixing and because of the hydrodynamic forces there and because of that the concentration is uniform in that region. So, there is a concentration variation that goes from C A star to the bulk concentration that is prevalent in the rest of the liquid and this concentration drop occurs entirely in the diffusion film. So, that is the assumption on which the film theory is built and further we said that because this film is expected to be very very small and we by later on with reference to the available values of the mass transfer coefficient we estimated the volume of liquid in this film to be about 0.1 percent or so of the total volume of the liquid. So, because of the volume contained in this film is so small we are justified in treating this film to be always in a state of steady state. In other words any changes either on the gas side in terms of the changing partial pressure or on the liquid side in terms of the changing values of C A B the film is able to immediately adjust itself on an instantaneous basis to these changes therefore, the diffusing solute always proceeds as though the conditions were studied. So, we have a very simple equation to solve the steady state diffusion equation which is a second order ordinary differential equation with the constant boundary conditions. So, this theory predicts that the mass transfer coefficient is proportional to the linear power of the diffusivity and then we considered an alternative mechanism for the mass transfer of A from the gas side to the liquid side and this mechanism assumed that the action of turbulence is not to restrict the distance over which the concentration drop occurs as assumed in the case of the film theory, but it is to actually periodically throw elements of liquid from the bulk to the gas liquid interface and depending on the nature of the hydrodynamic field there is a certain time period during which individual elements of liquid stay at the gas liquid interface and then leave. Because these time periods are expected to be short the in general the processes assumed to be of an unsteady state nature. So, here that is number 1 and the second thing that happens is that because the time of exposure is small the depth of penetration is also small and therefore, the surface element can be assumed to be infinitely thick from the point of view of the diffusing solute. So, we get to solve a partial differential equation of the second order first order in time and second order in distance in order to calculate the absorption flux in a single surface element as a function of the time it has spent at the interface. In order to calculate the average absorption rate at that location we have to consider the unit gas liquid interface at that location which itself is a mosaic of several surface elements of various surface edges. So, the absorption rate in these different elements has to be averaged in order to get the overall absorption rate and we can do this averaging by assuming two types of surface age functions or age distribution functions if you like. And these give rise to two different theoretical pictures one which was originally proposed by Higbee where he said that every element of liquid spends exactly the same amount of time at the gas liquid interface as every other element. The second one due to Danckwerts who said that the picture at the gas liquid interface is more like in a well mixed vessel where elements of liquid are arriving randomly and departing randomly from the gas liquid interface. So, irrespective of which distribution function you use it turns out that the mass transfer coefficient is predicted to be predicted to have a square root dependence on the diffusivity. And if you compare these predictions of the film theory and the surface renewal theory with the experimental data it turns out that the surface renewal theories are closer to the actual picture than the film theory. But irrespective of that we should realize that neither of these theories is able to predict the physical mass transfer rate in any real sense because each of these theories has a parameter that usually in most realistic contacting situations cannot be calculated from first principles. Therefore, the theories are in some sense useless in their ability to predict mass transfer rate is concerned. But the usefulness of the theories is in their ability to predict the effect of chemical reaction on the mass transfer rate what is called as the enhancement factor. So, this is the business that we address next and we saw that whether you are a proponent of the film theory or whether you are a proponent of the surface renewal theories the effect of chemical reaction on mass transfer depends on a value of depends on the value of a parameter called as a Hata number which is the relative rate of reaction to the rate of diffusion. So, this there is a definition of the Hata number that arises on the non-dimensionalization of the relevant diffusion reaction equations. And as the Hata number increases the reactions are the reactions being considered are of ever increasing severity with respect to the mass transfer rate or the diffusion rate. So, we have the slow reaction regime and we have the fast reaction regime and in between we have the transition from slow to fast reaction. And all through this these three regimes the slow reaction the transition and the fast reaction the assumption of pseudo first order rate holds because the value of the Hata number or the value of m that is the square of the Hata number is much less than the relative abundance factor which governs whether concentration of B is going to be uniform right up to the interface or not. So, in all of these regimes it does not really matter as to whether you use the film theory to calculate the enhancement factor or the surface renewal theories to calculate the enhancement factor. Fair enough there is a bit of a difference in the transition regime, but that difference is of the order of a few percent and if you consider the uncertainties in the values of the physical mass transfer coefficient itself. So, this error is usually subsumed in the errors with which you can calculate the rate of absorption overall. So, the point is that as long as you have got a pseudo first order situation or as long as you have a situation in which the concentration of B does not play a role in the absorption rate expressions. The actual mechanism of mass transfer turns out to be unimportant the film theory and the surface renewal theory predict the much the same kind of values for the enhancement factor. So, then for larger values of root m that is for reactions which are of even higher severity than the ones that we have considered so far. The reaction is fast enough to deplete the concentration of B close to the interface and this is where the differences between the two sets of theories starts to surface and here by and large we should go with the more realistic theory that is the surface renewal theory. Fortunately, it turns out that for most practical situations the expressions from the surface renewal theories are not very different or not very difficult rather to evaluate the enhancement factors from and therefore, we can use the surface renewal theories with without too much difficulty. So, these are the various ways in which we can calculate the enhancement factor and once you have calculated the enhancement factor the local rate of absorption at any point within the equipment is given by the local mass transfer coefficient and multiplied by the driving force multiplied by this enhancement factor. So, you had the expression K L into C A star minus C A B for the physical mass transfer rate and all you got to do now is to put in this value of the enhancement factor as a multiplicator to this expression much as we use the effectiveness factor in the case of gas solid reactions to modify the intrinsic rate expression. So, there the basic case was the case of reaction and the effectiveness factor which had a value of less than 1 multiplied this intrinsic reaction rate to give you the actual reaction rate. Here, the base case is the base of mass transfer and this has to be multiplied by an enhancement factor which has a value greater than unity in order to give the actual rate of mass transfer. So, once you know the actual rate of mass transfer this is the rate expression that goes into various equipment level balances which you use either for analysis of process equipment or for design of process equipment. So, with that we have now completed setting up the requisite apparatus for doing an analysis or design of gas liquid reaction equipment where the enhancement factor calculations have to precede the writing the expression for the local rate. So, we shall close this set of lectures here and thank you for your attention.