 Friends, in this lecture we are going to look at how to estimate or how to identify whether a particular catalytic reaction, particular heterogeneous reaction is diffusion limited or reaction limited. In often in industry several experiments can be performed but it is important to understand under what situations, under what conditions the heterogeneous reaction will be diffusion limited or reaction limited and it is very useful to have certain tools which can be used to identify these regimes. So, what we are going to see today is to look at how to detect from the experimental data that is the observed reaction rates under different situations and different conditions, how to identify what is the, under what situation the heterogeneous reaction is diffusion limited and under what situation it is surface reaction limited. So, the diffusion limited situation can actually be of two types, one can be the external diffusion. So, diffusion limited experimental conditions can actually be diffusion limited reaction conditions can be either internal diffusion controlled or the external diffusion controlled. The other possibility is where it is the reaction limited, the conditions are reaction limited. So, as a first step we will look at the Weisprater criterion which is often used in industry in order to identify whether the reaction is internal diffusion limiting. So, we will start with Weisprater criterion and so the Weisprater criterion is actually used to answer this question is the, is internal diffusion controlling the, is internal diffusion limiting or the current conditions are such that it is internal diffusion limiting the actual performance of the heterogeneous catalytic reaction. So, suppose if the, if from experimental data we are able to obtain what is the observed reaction rate. So, if the observed reaction rate, the observed reaction rate is given by let us say minus r a prime of OBS is let us say it stands for the observed reaction rate and we will use this nomenclature for the rest of the lecture. And let us consider a first order reaction where a which is a reactant which undergoes a certain heterogeneous catalytic reaction and the results in forming a product and let us assume that it is a first order reaction. So, now for a first order reaction in the last few lectures you would have noticed that there is a relationship between the effectiveness factor and the Thiele modulus. So, for a first order reaction the relationship between the effectiveness and Thiele modulus is given by eta which is the effectiveness factor that is given by 3 divided by phi 1 square 1 stands for the 1 here refers to the first order reaction 3 by phi 1 square multiplied by phi 1 cot hyperbolic phi 1 minus 1. So, that is the relationship between the effectiveness factor and Thiele modulus for a first order reaction. Now, this equation can actually be modified as eta into phi 1 square that is equal to 3 into phi 1 cot hyperbolic phi 1 minus 1. So, all I have done is just multiply the whole equation with phi 1 square. So, once we do this we can now define a new parameter called the Weiss-Prater parameter. So, we define a parameter called the Weiss-Prater parameter and this parameter is you see WP see subscript WP represents the Weiss-Prater parameter and that is equal to eta into phi 1 square. So, we define a new parameter called CWP which is equal to eta into phi 1 square and that would be equal to 3 into phi 1 cot hyperbolic phi 1 minus 1. So, now what is this parameter? So, let us look at what is the meaning of this parameter CWP. So, we can look at the meaning of this parameter. So, CWP is eta into phi 1 square the effectiveness factor is essentially the ratio of the observed reaction rate to the reaction rate evaluated at the bulk concentrations A S. So, if the so that will be bulk observed reaction rate divided by the reaction rate at the bulk concentration if you assume that there is no mass transfer limitations that will be equal to the concentration at the surface of the catalyst and that multiplied by the Thiele modulus. So, phi 1 square is nothing but the reaction rate evaluated at the surface concentration if there is no mass transfer limitations divided by the diffusion rate of the species that is actually undergoing the reaction. So, there will be reaction rate at C A S divided by the diffusion rate of species A. So, now from here we can actually rewrite this expression. So, you can see that we can cancel off the denominator in the first expression and the numerator in the second expression and we can rewrite this vibrator parameter as observed as a ratio of observed reaction rate divided by the diffusion rate of species A. So, this parameter vibrator parameter provides or it provides a mechanism to compare the observed reaction rate divided by the diffusion rate of the species A. So, let us look at what we can do with this vibrator parameter remember that the objective is to find out whether the reaction is being conducted under internal diffusion limitations. So, now effectiveness factor eta if we put the expressions corresponding to the effectiveness factor in the Thiele modulus into the vibrator parameter what we see is that effectiveness factor is the observed rate divided by the reaction rate at the surface concentration and phi 1 square which is the Thiele modulus is given by the reaction rate per unit surface area of the catalyst multiplied by the surface area of the catalyst per gram of the catalyst multiplied by the density of the catalyst into R square where R is the radius of the catalyst pellet in which the reaction is being conducted divided by the diffusivity of the species multiplied by the concentration of the species at the surface and so that is equal to minus RAS into rho C into R square divided by CAS. So, now plugging in these two expressions into the vibrator parameter we will find that CWP is equal to eta phi 1 square and that is equal to minus R A observed rate multiplied by the density of the catalyst rho C multiplied by R square which is the square of the diameter of the pellet that is used in which the reaction is occurring heterogeneous reaction is occurring divided by the diffusivity divided by the concentration of the species at the surface of the catalyst. So, now if we observe this carefully if we observe this expression carefully we find that all the quantities which is represented here they are either measurable quantity or a known quantity for example they are all measurable or known quantities all information present in this particular expression they are all measurable or known quantities. So, the measurable quantity here is the reaction rate. So, the reaction rate is the measurable quantity in this expression here and the density radius and the diffusivity are again known quantities and the reaction rate and the concentration on the surface are all measurable quantities. So, once we know these quantities we know what the vibrator parameter is. So, the vibrator condition or the vibrator criterion is that vice-prater criterion vice-prater criterion is when the parameter CWP which is the vice-prater parameter if that is significantly smaller than 1 if that is significantly smaller than 1 which means that the ratio of the observed reaction rate to the rate of diffusion of the species into the catalyst in the catalyst if that is much smaller than 1 then it means that there are no diffusional limitations no diffusion limitation internal diffusion limitations. Now, this also means that the there is no concentration gradient within the catalyst pellet because the diffusion is very fast and so even before the reaction occurs all of the catalyst is now replenished with the reactant species. So, therefore there is no concentration gradient within the pellet within the catalyst pellet. So, that is an important parameter. So, once we know these parameter which can be estimated from the measurable and the known quantities one can actually decipher whether the reaction is being conducted at the diffusional limitations or not. So, when vice-prater criteria C parameter CWP if that is significantly larger than 1 then this implies that there is a strong internal diffusion internal diffusion limitations strong internal diffusion limitations and in fact it means that the strong internal diffusion limits the reaction severely. So, simply by using the observed quantities that is measurable quantities and some of the known properties of the system one can actually decipher using the vice-prater criterion whether the internal diffusion is limiting the reaction or not and that is an important aspect when it come in practice. It is also important to note that while the vice-prater criteria can be used for non-first order reactions the effectiveness factor in theory modulus for such reactions cannot be obtained using the method illustrated in this lecture. So let us look at a specific example. So, let us consider a first order reaction A going to B let us consider a first order reaction. Now an experiment has been performed with different catalysts and let us say that there is experiment one in which the catalyst of 10 millimeter diameter was used 10 millimeter diameter was used to conduct the heterogeneous catalytic reaction and let us assume that it was performed in an appropriate reactor such that there is no external mass transport diffusion limitations and so which means that the surfaces the concentration of the species at the surface is equal to the concentration of the species in bulk and the reaction rate that was observed and the reaction rate that was observed is about 0.18 into 10 power minus 2 moles per gram catalyst per second per minute that is the observed reaction rate for when the reaction was conducted using a catalyst pellet of 10 millimeter diameter. Now suppose the same reaction was conducted under same conditions except that as much smaller pellet was used so there is another experiment where the pellet that was used is 1 millimeter in diameter 1 millimeter spherical particles and the observed reaction rate is 0.9 into 10 power minus 2 moles per gram catalyst per minute. So now because the experiments were conducted such that with under appropriate conditions that there is no mass transfer resistance so here we need to estimate whether it is internal diffusionally controlled or not whether the internal diffusion are there any internal diffusion limitations so that is the first question. Then the next question is find the effectiveness factor eta and the effectiveness factor and the theory modulus for both these cases and what should be the radius of the pellet such that there is no diffusion limitations. So the question is find the effectiveness factor and theory modulus and what should be the radius of the catalyst such that there is no diffusional limitations. So how do we handle this question? So we have just now learned the vibrator parameter. So vibrator parameter is basically based on the observed quantities such as the reaction rate etc. So we can attempt to use the vibrator criteria in order to find out the effectiveness factor and the theory modulus from the observed reaction rate. So the theory modulus effectiveness factor relationship suggests that eta into phi 1 square which is the vibrator parameter that is equal to 3 times phi 1 cot hyperbolic phi 1 minus 1. Now so this is which is equal to the vibrator parameter Cwp and that is also equal to the observed reaction rate multiplied by the density of the catalyst multiplied by the square of the radius of the particle that is used divided by the diffusivity into the concentration at the surface because the mass transport limitations are not there. So the let us say that the surface of the catalyst concentration of the species in the surface is equal to the concentration of the species in bulk. So from here by looking at this relationship we find that minus rA into rho C into r square divided by diffusivity into CAS that should be equal to 3 times phi 1 cot hyperbolic phi 1 minus 1. So that brings a relationship between the observed reaction rate and the Thiele modulus. Now for each of these runs each of these experimental runs we could actually find out what this relationship is that is how the reaction rate at a given particular particle given particular radius of the pellet how is that related to the corresponding Thiele modulus. So for experiment 1 this is given by minus rA1 which is the observed reaction rate when the catalyst pellet is let us say 10 millimeters multiplied by rho C into r1 square divided by dE into CAS that should be equal to 3 into phi 1 1. Now phi 1 1 corresponds to the Thiele modulus of the first order reaction for the first experimental run and multiplied by cot hyperbolic phi 1 1 minus 1. And similarly for experiment 2 for the other particle other radius of the pellet for which the experimental data is available. So there will be minus A2 into rho C into r2 square divided by the diffusivity multiplied by the corresponding surface concentration. Remember that the experiment was conducted exactly under same conditions. So therefore the concentration of the species in at the surface of the reactor is same for both r1 and r2. So that is equal to 3 times phi 1 2 cot hyperbolic phi 1 2 minus 1. So now taking the ratio of these two expressions here we can now find a relationship between the observed reaction rates for both runs on the corresponding Thiele modulus. So by taking the ratio we find that rA2 into r2 square divided by 1 into r1 square. So that should be equal to phi 1 2 cot hyperbolic minus 1 divided by minus 1. So this relates the information that has been observed experimentally and the corresponding Thiele model line. So the left hand side of this expression is essentially the one which actually contains all the information that have been estimated experimentally and the parameters or the properties of the system that is being used and the right hand side is basically the Thiele model line, ratio of the Thiele model line, ratio of the function of Thiele model line. So now let us look at the expression for the Thiele modulus. So phi 1 1 which is the Thiele modulus for the first order reaction for the first experiment that is given by r1 which is the radius of the pellet multiplied by square root of minus rA s which is the reaction as if it were conducted on the surface concentration multiplied by rho c divided by the diffusivity into the concentration of the species at the surface and similarly phi 1 2 will be r2 into square root of minus rA s into rho c divided by dE into CAS. Now from here taking the ratio take dividing these two equations we will find that phi 1 1 by phi 1 2 that should be equal to r1 by r2. So the ratio of the Thiele modulus is given by the ratio of the radius of the pellet itself and from here one can find out the relationship between the Thiele modulus under different experimental conditions. So that will be phi 1 2 into r1 by r2. So now plugging in the numbers we will see that phi 1 1 is equal to 0.01 meter by 0.1 meter into phi 1 2 so that is equal to 10 times the phi 1 2. So the Thiele modulus under one experimental condition is about 10 times the Thiele modulus of the second experimental condition. So that is the relationship that we get from the Thiele modulus of these two experimental conditions. So now earlier we looked at we derived the relationship between the ratio of the reaction rates with the corresponding Thiele modulus. So now we have to we can plug in the relationship between the Thiele modulus under these two experimental conditions into that expression and that will be minus rA 2 into r2 square divided by minus rA 1 into r1 square and that should be equal to phi 1 2 which is the Thiele modulus for the second experimental run into cot hyperbolic phi 1 2 minus 1 divided by 10 times phi 1 2 into cot hyperbolic 10 times phi 1 2 minus 1. So all the information here is known the only variable is only unknown is is phi 1 2. So we need to estimate phi 1 2 using all the other information that we already know. So from here we can plug in the numbers. So the reaction rate for the second condition will be 0.9 into 10 power minus 2 multiplied by the square of the corresponding radius divided by 0.18 into 10 power minus 2 into 0.01 the whole square and that is equal to 0.05 and that will be equal to phi 1 2 cot hyperbolic phi 1 2 minus 1 divided by 10 times phi 1 2 into cot hyperbolic 10 times phi 1 2 minus 1. So now by solving this expression we can find out what is the Thiele modulus for the second experimental run. So solving we can find that phi 1 2 is about 1.65 this is for R 2 equal to 1 millimeter. We now know the we also know the relationship between the Thiele modulus of these two experimental conditions. So from there we can find out phi 1 1 which is 10 times phi 1 2 that is equal to 16.5 for R 1 equal to 10 millimeters. So we have now found the Thiele modulus. Now once we know the Thiele modulus we can actually estimate what is the effectiveness factor because effectiveness factor and the Thiele modulus under given experimental conditions are actually related. So for R 2 which is the second experimental run eta 2, eta 2 equal to 3 times phi 1 2 into cot hyperbolic phi 1 2 minus 1 divided by phi 1 2 square and so once we plug in the Thiele modulus data so we will find out that this effectiveness factor is about 0.856 and then similarly for the first experimental run the effectiveness factor eta 1 will be 0.182. Now the next question is can we find out what is the smallest radius at which the internal diffusion limitations does not exist that is there is no internal diffusion limitation that is limiting the reaction. So suppose if we assume that the effectiveness factor should be 0.95 suppose if the effectiveness factor should be 0.95 at which there is no internal diffusion limitations no internal diffusion limitations then we can actually use the relationship between the effectiveness factor and the Thiele modulus in order to find out what is the corresponding Thiele modulus. So we know that 0.95 that is equal to 3 times phi 1 3 cot hyperbolic phi 1 3 minus 1 divided by phi 1 3 square so by solving this we can find out that phi 1 3 is about 0.9. So that is the Thiele modulus that should be the Thiele modulus if the reaction has to be conductor under no internal diffusion limitations. So once we know the Thiele modulus we also know what is the relationship between the Thiele modulus ratio and the corresponding radii. So we can use that expression in order to find out what should be the particle radius in order for the internal diffusion limitations to not exist. So phi 1 3 divided by phi 1 1 so this is phi 1 1 is the Thiele modulus corresponding to one of the already conducted experiments and so that will be R3 by R1. So from here we can find out that R3 should be equal to about 0.55 millimeters. The smallest pellet at which the internal diffusion will not limit the reaction is about 0.55 for the given set of experimental conditions. So in this way using some trial experiments one can actually find out what should be the pellet size under which the reaction has to be conducted in order for the internal diffusion to not be present and not affect the reaction severely. Now let us come let us move on to the external diffusion we looked at the internal diffusion what is the what is the method to use the experimental observable data in order to find out whether the reaction is being conducted under internal diffusion limitations. So next let us look at the external diffusion limitations. So one has to use a criteria developed by Mears it is called the Mears criterion. So Mears criterion can be used to find out whether the catalytic reaction is being conducted under external diffusion limitation that is mass transport diffusion limitations that is the transport of species from the bulk to the surface of the catalyst is that limiting the reaction severely. So the expression is suppose if minus rA is the observed reaction rate it is the observed rate it is the observed rate then the criterion is that if minus rA into rho B into r into n divided by kC into CA bulk if that is smaller than 0.15. So if the ratio of if this expression on the left hand side that is basically contains the observed reaction rate. So the observed reaction rate multiplied by the corresponding density bulk density of the catalyst so the bulk density of the catalyst rho B is given by 1 minus phi into rho C where phi is the porosity of the of the bed that is used and rho C is the solid density of the catalyst. So if we know what is the bulk density then we can plug it in here and then we can find this expression and n is the order of reaction n stands for the order of reaction the reaction order and kC is the corresponding mass transfer coefficient kC is the mass transfer coefficient. So one needs to estimate what is the mass transport coefficient for transporting the species from bulk to the surface of the catalyst and then one needs to know what is the concentration of the species in bulk which is again a measurable quantity and one may also use different appropriate correlations. There are several correlations which are available to estimate the mass transport coefficient one may use the appropriate correlation in order to estimate what is the value of this mass transport coefficient for the given conditions that is for flow through packed beds. So once we know these numbers we will be able to estimate what is this quantity on the left hand side and if this is smaller than 0.15 then the external diffusional limitations are not playing any role and that can be ignored. If it is greater than 0.15 then certainly external diffusional limitations have to be taken into account. So now if the Mears criterion is satisfied if the Mears criterion is satisfied then this means that there is no concentration gradient between the bulk gas and the external surface of the catalyst pellet. So there is no concentration gradient between the so there is no concentration gradient between the bulk gas and the catalyst surface if the Mears criterion is satisfied. Now we looked at the concentration gradient now one may ask what is the what about the temperature gradient. So can we get an estimate can we find a criteria in order to know whether the temperature gradient external temperature gradient does it play a role that is the gradient of temperature between the temperature at the surface of the catalyst and the temperature of the bulk stream does it play any role in limiting the reaction. So Mears has come up with another criteria so it is a Mears criterion 2. Now this suggests that if the modulus of this expression where delta H is the heat of reaction into minus r A prime which is the observed reaction rate multiplied by rho B which is the bulk density of the catalyst into the radius of the pellet that is used multiplied by the activation energy E divided by a corresponding heat transport coefficient there are several correlations which are available in order to estimate the heat transport coefficient multiplied by the square of the bulk temperature of the fluid stream multiplied by the corresponding gas constant. So if this absolute value of this ratio if that is less than 0.15 then it means that the temperature gradient does not play any role which means that the bulk fluid temperature is approximately equal to approximately equal to the catalyst surface temperature. Note that intraparticle gradients need not necessarily be negligible however Weiss's platter criteria can be used to identify when these gradients may be neglected. So this is an important criterion so these two criterion can actually be used but these two criteria can actually be used in order to decipher whether the particular reaction is being conducted under diffusional limitations or not. So now let us summarize what we have learned so far. So we have looked at the Weiss's platter criterion Weiss's platter criterion and the first one is to decipher whether it is the Weiss's platter criterion is basically to decipher if the diffusional limitations exist and then we looked at the Meier's criterion. So the first criterion is basically to see if the external diffusional limitations exist to check whether the external diffusional limitations exist and then the second one is to check whether the temperature gradient is important external temperature gradient is important. So these three criteria vibrator criterion and the Meier's criterion they play a very very important role in real situations because the if the experimental data is available then this information can actually be used in order to decipher whether the reaction is being conducted under diffusional limitations or not. So next let us look at design of a packed bed reactor as to how we can incorporate the concept of effectiveness factor in order in the into the packed bed reactor design. Now let us consider a packed bed reactor packed bed reactor is essentially a tube I have filled with a catalyst and the fluid stream goes from one end of the tube into the other end of the tube. So let us consider a tube and there is a fluid stream which goes into this tube and it leaves the tube from this end. So if I mark this as z equal to 0 if z is my coordinate system so 0 to L so L is the length of the reactor and if AC is the cross sectional area of the tube and now I can write a mole balance in order to capture the dynamics or the behavior or the relationship of the conversion with all the other parameters. So now if I write a shell balance I now identify a small shell which is filled with catalyst and let us assume that the volume of this shell is delta v and the amount of catalyst which is packed inside is delta small w and if this the location where this element starts is z and the location where it ends is z plus delta z. So this immediately means that delta w which is the differential amount of catalyst which is present in that location should be equal to the bulk density of the catalyst multiplied by the cross sectional area multiplied by the delta z which is the thickness of that particular element that we have chosen. So now we can write a mole balance for this small element and we can find out what is the model equation which captures the relationship between the concentration of the species which is undergoing the catalytic reaction with respect to other parameters. So suppose if the flux of species that enters this small element is wAz that is the flux that enters at that location and the flux that leaves is wAz evaluated at z plus delta z. So that is the flux that leaves that location. Then we can write a mole balance and the mole balance is essentially the rate in the rate at molar rate at which the species enters that element minus the molar rate at which the species leaves the element plus whatever is generated that should be equal to 0 under steady state conditions. So if we assume steady state conditions then this is the balance that captures the process that is happening in this small element. So what is the rate at which the species is getting into the element that is given by the cross sectional area AC multiplied by the flux with which the species is actually entering into that element which is wAz evaluated at that particular location minus the rate at which the species leaves is given by the cross sectional area multiplied by the flux at which the species leaves that element that is wAz at z plus delta z. And the generation is given by the reaction rate per unit weight of the catalyst multiplied by the amount of catalyst which is packed inside that small element delta z. So now we know that the if the reactor volume is constant if the reactor volume is we know that we can actually rewrite this expression as delta w is equal to rho b AC into delta v delta z that is the differential amount of catalyst that is packed inside that element and so we can now incorporate this into the model and therefore the model will be AC into wAz minus AC Az at z plus delta z plus the rate multiplied by the bulk density into the cross sectional area into delta z equal to 0. So that is the model. Now we can divide this whole equation by cross sectional area into delta z so that will be 1 by AC into delta z that is multiplied by AC minus z plus delta z plus Ra prime into rho b into AC into delta z that is equal to 0. So now if we take limit delta z going to 0 that if the element is very very small infinitesimally small then the model equation becomes minus d w Az by dz plus Ra prime into rho b equal to 0. So this is the mole balance for species A which is undergoing a heterogeneous reaction in a packed bed reactor where w Az is essentially the flux with which the species is actually entering the flux of the species at a particular location z. Now if I assume that the total concentration is constant so if you assume that the total concentration of species all species is constant so total concentration of all species put together if it remains constant and if that value let us say is C. Now the flux with which the species crosses a particular location that can be written as minus d AB which is the diffusivity of the species equimolar counter diffusivity of that species A multiplied by the concentration gradient d AB divided by dz plus y AB which is the mole fraction of the species multiplied by the w Az which is the flux of the species at that location plus the flux of species B at that location. So now if we stare at this equation the first term here corresponds to the diffusion term corresponds to the species diffusion and the second term corresponds to the bulk flow. Second term corresponds to the bulk flow which is like a convective transport and the first term corresponds to the diffusive mode of transport. So now if we know what is the superficial velocity so suppose if the superficial velocity is u superficial velocity is u then the total flux that is crossing a particular location w Az plus w Bz that should be equal to the total concentration C multiplied by superficial velocity u. So now plugging in this expression we can rewrite the expression for the flux of species that is w Az and that should be equal to minus d AB into d C AB by dz so that is the bulk concentration of the species at that particular location plus u into C into y AB y AB small b. So now if you look C into y A bulk is nothing but the concentration of the species at that location and therefore it will be minus d AB into d C AB by dz plus u into C AB so that is the expression for the flux of species that is crossing any particular location z inside the reactor. So now plugging in the expression for the flux in the mole balance we can rewrite the mole balance as so now the mole balance can be rewritten as the mole balance can be rewritten as d AB which is the dispersion coefficient of the species d into d square C AB by dz square. So that is the second differential of this concentration of the species minus u which is the superficial velocity multiplied by d C AB by dz plus r A prime which is the rate at which the species is being consumed rate at which the species is being generated multiplied by the bulk density that should be equal to 0. So now this first term here which is the diffusion into the second derivative of the bulk concentration of the species so that is because that is due to diffusion and or it could be either because of diffusion the transport of species that is captured by this term could be either because of pure diffusion or because of axial dispersion because of dispersion in the axial direction. So therefore for rest of the rest of the design equation mole balances we would consider d AB which is the equimolar counter diffusivity should be approximately equal to the equal to d A which is basically the quantity that captures the extent of diffusion or the extent of dispersion or diffusion and dispersion together. So here after we will refer to it as d A in this lecture. So the modified mole balance using this representation is d A into d square C AB by dz square minus u into d C AB by dz plus r A prime into rho B equal to 0. So now using this expression now we need the next step we need to do is we need to relate r A to concentration we need to find out what is this relationship with the concentration of the species. So if suppose if omega is the overall effectiveness factor if this is the overall effectiveness factor so it is the overall effectiveness factor then the rate of reaction r A prime can actually be given by minus r A prime can actually be is given by minus r A bulk. So this is the reaction rate in the catalyst now that should be equal to the reaction rate if all the catalyst surface is now exposed and is present at the concentration of the species on the surface then that concentration of species in bulk then multiplied by the overall effectiveness factor will actually be the actual reaction rate inside the catalyst. So now if we assume that it is a first order reaction if it is a first order reaction then minus r A prime is given by the effectiveness factor multiplied by the specific reaction constant into the surface area of the catalyst per gram per unit weight of the catalyst multiplied by the bulk concentration C AB. So now plugging in this expression into the mole balance we can find that we can find that the mole balance is dA which is the dispersion coefficient multiplied by multiplied by d square C AB by dz square minus U which is the superficial velocity into dCA bulk by dz minus omega into the bulk density rho B into k prime which is the specific reaction constant multiplied by the surface area per gram catalyst into C AB equal to 0. So now if the flow through bed so we need to solve this equation in order to find out the relationship between C AB and the how C AB or the bulk concentration changes with location inside the reactor. So now suppose if the flow through the bed is large suppose if the flow through the bed is significantly large then one can actually neglect the axial dispersion one can neglect the axial dispersion and that is not true always only when these flow through bed is large and in fact one can actually quantify what is the situation under which the axial dispersion can be neglected. So the condition that has to be satisfied in order to assume that the axial dispersion is can be neglected is given by minus rA prime into rho B which is the bulk density multiplied by the diameter of the particle divided by the superficial velocity if that is much smaller than this quantity U0 dP by dA then it is acceptable to neglect the axial dispersion from the mole balance. So this also suggests that this suggests that the so when axial dispersion can be neglected it also suggests that the U into dCA by dz which is the term that corresponds to the convective transport that is significantly larger than the diffusive transport inside the packed bed reactor. So it suggests that the convective transport is larger than the diffusive transport and under these conditions one can actually neglect the dispersion inside the reactor and so the modified mole balance which accounts for these assumptions is dCA by dz that is equal to minus omega is the effectiveness factor into rho B into k is the reaction rate constant and SA is the area of the catalyst per gram of the catalyst surface area of the catalyst which is available divided by the superficial velocity into CAB. So if suppose at the entry of the reactor z equal to 0 if CAB is equal to the some constant value CA0 which is the concentration with which the species enters the reactor then we can integrate this expression and find that CAB is equal to CAB0 into exponential of minus omega rho B SA by superficial velocity U into z and or we can use this expression in order to find out what is the conversion of the species in the reactor and the conversion is given by and the conversion is given by this expression here where x is equal to 1 minus CAB by CAB0 which is equal to 1 minus exponential of minus omega into rho B into k into SA by U which is the superficial velocity into z. So this expression captures the relationship between the conversion as a function of the location inside the reactor and so this this is very useful in terms of estimating what is the reaction what is the conversion given the overall effectiveness factor and the system parameters and the configuration of the reactor that is what is the particle diameter etc. So what we have seen in today's lecture is we have looked at the criterion for when the internal diffusion is important and when the external diffusional limitations are important and that is simply based on the experimental data that has been obtained from the real reactor system and also we had attempted to see how to incorporate the effectiveness factor into the more balance of the packed bed reactor in order to find out what is the how the conversion changes as a function of position in the packed bed reactor. Thank you.