 Friends, let us look at how to obtain the limiting cases from the experimental data that is what whether the reaction is under diffusional limitations or under the surface reaction limiting conditions and how it depends upon various parameters of the system. So first let us consider the external mass transport limited case, external mass transport limited case. Now here under the external mass transport limitations the reaction rate is given by Kc Ac into Ca where Kc is the mass transport coefficient, Ac is the surface to volume ratio and Ca is the bulk concentration. Now Kc can be estimated using appropriate correlations such as the Thonus-Kramer correlation can be estimated using the Thonus-Kramer correlation. Now the correlation is as follows, it says that the Sherwood number should be equal to Reynolds number to the power of half multiplied by the Schmidt number to the power of 1 by 3. So the mass transport coefficient is embedded in Sherwood number while other properties like velocity etc. they are all embedded in Reynolds number and the Schmidt number. So now looking at the functional form of these three terms we could be able to discern how the mass transport coefficient depends upon various parameters such as the diameter of the particle or the temperature of the temperature at which the reaction is being conducted etc. And then find out what is the relationship or functional dependence of the overall reaction rate on various systems parameters such as the diameter of the particle and velocity with which the fluid is flowing into the reactor. So now if I look at the expression for Sherwood number you see that the Sherwood number is given by mass transport coefficient Kc multiplied by the diameter of the particle dp divided by the Aqmola counter diffusivity dAb multiplied by phi divided by 1 minus phi. Here phi refers to porosity of the bed and it does not represent the Thiele modulus. Here phi is the porosity of the bed and Reynolds number is given by u times dp, u is the superficial velocity, dp is the diameter of the particle, diameter of the catalyst pellet in which the catalytic reaction is being conducted divided by the 1 minus phi where phi is porosity into nu which is kinematic viscosity and similarly Schmitt number is given by kinematic viscosity divided by the Aqmola counter diffusivity of the species A that is of interest. So from here we can now write the all these plug in all these expression in the Toner-Skremer correlation. We find that Kc dp into phi divided by dAb into 1 minus phi that should be equal to u dp by 1 minus phi into nu to the power of half multiplied by nu by dAb to the power of one third. So that is the dependence of the mass transport coefficient where here Kc refers to the mass transport coefficient and this is the functional dependence of mass transport coefficient on the terminal velocity u and dp and other parameters. So from here we can discern that the mass transport coefficient K is now proportional to Kc is proportional to square root of u where u is the superficial velocity and also it is proportional to 1 by square root of dp. Kc is proportional to 1 by square root of dp. Now if I look at the reaction rate reaction rate is Kc into surface area per unit volume multiplied by the corresponding concentration Ca. So now the from here we can see that the area per unit volume is given by 1 by dp and therefore the reaction rate therefore the mass transport coefficient Kc is proportional to square root 1 by square root of dp into 1 by dp. So that will be given by 1 by dp to the power of 3 by 2, 1 by dp to the power of 3 by 2. So that is the functional form of the mass transport coefficient on the diameter of the particle. So it is dp to the power of 3 by 2. Now similarly one can see that the mass transport coefficient is typically proportional to the temperature at which the reaction is being conducted. So therefore from here we can see that the reaction rate is proportional to square root of u which is the superficial velocity with which the fluid is flowing into the reactor and also it is proportional to 1 by dp to the power of 3 by 2 and it is also proportional to the temperature at which the reactor is being operated. So that provides the functional form of the reaction rate or dependence of the reaction rate on various parameters of the system. So therefore now next step we look at the internal diffusional limitation. If you look at the internal diffusional limitations then the reaction rate is for any nth order reaction is given by Ra prime is equal to 3 by r that is the radius of the pellet, catalyst pellet that is being used into square root of 2 times the diffusivity, effective diffusivity of the species divided by n plus 1 multiplied by kn to the power of half where k is the nth order reaction rate constant multiplied by the surface concentration CaS to the power of n plus 1 by 2. So that is the rate law. Now the rate constant actually follows Arrhenius type dependence on the temperature. So therefore we can rewrite this expression as 3 by r into square root of 2 times De divided by n plus 1 multiplied by the frequency factor At to the power of half into exponential of minus e by RT to the power of half into CaS to the power of n plus 1 by 2. So that is the expression. So given that where e is the activation energy and t is the temperature at which the reactor is being operated. So from here one can discern that the rate of reaction Ra is proportional to 1 by diameter of the particle dp because of this functional dependence on because of this term 3 by r and also the rate of reaction is has an exponential dependence on the temperature on the temperature at which the reactor is being operated. It is important to also note that diffusivity is also a function of temperature. So this suggests that the unlike in the case of external mass transport control situation in the internal diffusion control situation the reaction rate does not depend upon the superficial velocity with which the fluid stream is being flown into the reactor. Now it should be noted that here the exponential dependence is exponential to the power of 1 by 2. So therefore the exponential dependence on temperature is not as strong. It is not as strong as in the case of in the case of surface reaction limited in the case of surface reaction limited situations. So therefore we can see that the reaction rate in the case of internal diffusion limitations is 1 by dp and depends exponentially on the temperature at which the reactor is being operated. So next let us look at the surface reaction limited case. So in the surface reaction limited case, the surface reaction limited case, the reaction rate is now is independent of the size of the particle. It is independent of the particle size and it is a strong function of the temperature. It is a strong function of the temperature at which the reactor is being operated. So now if you put them all together, if you put all of these 3 dependencies of the reaction rate on various systems parameters for all these 3 cases together, we can now form a table which captures the dependence of the reaction rate on various parameters. So now let us draw a table. So let us look at the external diffusion, the case of external diffusional limitations under the external diffusional limitations. What is its functional dependence on the velocity? It is u to the power of half. So that is the functional dependence on velocity and if I look at the particle size, it depends upon dp to the power of minus 3 by 2 and then on temperature, it is approximately linear. So the reaction rate is approximately linear with respect to temperature in the external diffusional limitation case. Next if we look at the internal diffusional limitations case, internal diffusional limitations case, the reaction rate is now independent of velocity. It is independent of the velocity with which the fluid is being pumped into the reactor and it is a function of dp to the power of minus 1 and it depends exponentially on the temperature. It is an exponential dependence on the temperature. Now if we look at the surface reaction case, surface reaction limited case, then it is independent of the velocity with which the fluid is being flown into the reactor. It is independent of that and it is also independent of the particle size and it has a strong exponential dependence on the temperature. So therefore, if reaction is being performed, the first step would be to perform the reaction at a certain velocity and different diameter of the particle. Now if we look at the different diameter of the particle, if the reaction rate by decreasing the particle size, it correspondingly uses this expression of dependent, this shows this dependence of 1 by dp, then it suggests that the particular reaction is under the internal diffusional limitations. Now if it has a functional dependence of dp to the power of minus 3 by 2, then it suggests that it is an exponentially external diffusion limited situation. Now if it is independent of the particle diameter, if it is completely independent of the particle diameter, then the current conditions suggest that the experimental conditions are essentially it is under the surface reaction limited situation. So as a rule of thumb, if the activation energy is of the order of 8 to 24 kilojoules per mole, if that is the order of magnitude, then it means that the reaction is actually strongly under strongly diffusional limitations, strongly external diffusion control regime that is external mass transfer controlling. It is under strong diffusion control regime. However, if it is about 200 kilojoules per mole, then more often than likely it will be a surface, it will be under surface reaction control regime. So the reaction will be surface reaction controlling. So this table summarizes the functional dependence of various parameters and how can an experiment be designed using these parameters in order to estimate whether the reaction is likely to be external diffusion limited or internal diffusion limited or the surface reaction limited. So now summarizing what we have learnt about various criteria is, we have looked at the Weisprater criterion. It essentially uses a parameter Cwb which is equal to the internal effectiveness factor multiplied by the corresponding Thiele modulus and this does not work and for all types of rate loss, it works for certain type of rate loss where the effectiveness factor Thiele modulus behaviour is monotonic. And then we looked at the Meier's criterion. Meier's criterion to discern from experimental data whether it is external diffusion controlled or not and then we looked at the generalized criterion where once again parameter capital Phi which is equal to the local effectiveness or internal effectiveness factor for any general reaction mechanism multiplied by the Thiele modulus of that particular general mechanism. So that particular product gives you a factor called capital Phi which is used in which is used as a general criterion. So if Phi is less than 1 then it is considered that the diffusional limitations does not exist and if it is greater than 1 then the diffusional limitation, strong diffusional limitations exist in the particular type of reaction. So now once we know how to identify whether the reaction is under the diffusion or the internal diffusional limitation or external diffusion limitation or surface reaction limited conditions then one may use these kind of information such as the effectiveness factor into the reactor design. So the general algorithm to perform a reactor design is first one writes a mole balance, a mole balance is written and then the rate law is identified and then based on the rate depending upon the experimental data one may be able to find out whether it is external diffusion controlled or internal diffusion controlled. So based on that we can find out what is the expression for the overall effectiveness factor which incorporates all types of limitations and then by solving the mole balance you can use this overall effectiveness factor and you can find out what is the rate law based on the observable quantity such as the bulk concentration and then we can find what is the relationship between the conversion x by taking stoichiometry into account versus the weight of the catalyst which may be used for conducting the reaction to attain a certain conversion or the position inside the reactor. Then following which can be obtained by integration of the mole balance so integrate the mole balance and one can get the relationship between x and the weight of the catalyst or the position inside the catalyst. Now the position at which the desired exit conversion can be achieved is what is the length of the reactor that needs to be designed or that is the required length of the reactor for attaining a certain conversion. Then the next step is to find the mass transport coefficient Kc this can be obtained using various correlations one may use the appropriate correlations one may use appropriate correlations. Now if it is not external mass transport control then the mass transport coefficient really does not play much role here so in that case this step can be ignored and the effectiveness factor will simply be equal to the internal effectiveness factor and so the overall effectiveness factor will simply be equal to the internal effectiveness factor. So then next step is to estimate based on the various properties and the parameters and next step is to estimate the overall effectiveness factor if mass transport if the mass transport limitations are negligible then the overall effectiveness factor is approximately equal to the internal effectiveness factor and then next step is to find the weight of the catalyst for a specified conversion for a specified conversion and then find the length of the reactor. So this is the general recipe for performing a reactor design in order to obtain the design parameters such as how much weight of catalyst is required to obtain a certain conversion what should be the length of the reactor to obtain the desired conversion. So now let us look at little bit more details of packed bed reactor and the design of packed bed reactors. The packed bed reactor hereafter will simply be referred to as PBR which stands for packed bed reactor. So packed bed reactor is essentially a tube which is filled with catalyst inside and the reactants are flown from one end and the gas stream the reactant species present in the gas stream they go into the catalyst and then get adsorbed onto the site of the catalyst and the reaction occurs and soon after the reaction occurs the product leaves the catalyst and then moves into the gas stream and the product leaves the reactor. So it's essentially a tube it's a tube it's a tube which is filled with catalyst filled with catalyst pellet. So these are catalyst pellets so the catalyst pellets may be spherical it may be small cylindricals there are various types of shapes that may be used depending upon the nature of the reaction that is being conducted and the typical size of the catalyst if it is spherical particle or the approximate hydraulic radius would be about one-eighth inch or one-sixth inch that's the typical size of the catalyst that is actually being used inside a commercial packed bed reactor. Now the fluid stream may be flowing from the bottom of the reactor let's say with a velocity superficial velocity of u so that's the superficial velocity and the product comes out from the top so this is where the product comes out the feed goes here the feed stream goes in from here and then the product comes out from the top so it may be that there may be different kinds of reactions that can be conducted inside such a packed bed reactor. So a packed bed reactor is actually the work horse of chemical petrochemical and pharmaceutical industries so it's the work horse of chemical petrochemical and pharmaceutical industries slash pharma industries it's used in many different processes conversion of many different chemicals and a good example of that will be steam reforming it's one of very good examples of situation where packed bed reactor is used and then it's used for catalytic reforming catalytic reforming process it's used in isomerization process it's used in polymerization process and it's used in ammonia synthesis these are some of the examples where packed bed reactor is routinely being used in industrial conditions industrial settings ammonia synthesis it's also used in SO3 synthesis so these are some of the examples of situations where packed bed reactor is being used in reality and the packed bed reactor can in principle be operated under two modes one can be can be operated under adiabatic mode or a non adiabatic mode so adiabatic mode is a situation where no heat that is actually lost from the reactor to the external surrounding so that's called an adiabatic situation another configuration is a non adiabatic situation where the reactor may not be conducted at an adiabatic condition the reactor maybe it's an maybe it's an exothermic reaction where there is a cooling fluid which is flowing in the jacket and it might be removing the heat that is being generated because of the exothermicity of the reaction now this is required this may be required under many situations because if there's an exothermic reaction then the local temperature inside the reactor can actually increase and that may lead to if it crosses the melting point of the catalyst then it may lead to melting of the catalyst or if it is present close to the walls and it's a if it's an undesirable temperature then it poses a very serious safety concern so therefore it's very important to manage the heat inside the reactor and so a fluid is being circulated around and constantly the heat is being removed if it is an exothermic reaction which generates heat due to the reaction so there are generally two modes of operating one is the adiabatic where there is no heat is removed from the reactor where the exterior of the tubular reactor is insulated and then another situation where it's a non adiabatic situation where the heat may be removed from the exterior walls of the tubular reactor now the packed bed reactor also is being operated under different kind of configuration there are several configurations that have been used in the in the industry and some of these are multi bed reactor so multi bed configuration in a multi bed configuration there are multiple beds which are present there are multiple tubular reactors multiple tubular reactors where the fluid stream goes into one reactor and then there may be some unit operation that may be present in between let's say to separate something or it may be to heat the fluid or any other unit operation and whichever is required and from there the input goes into the second reactor and then from the second reactor the fluid stream leaves and then there may be some of the unit operation if it is required otherwise it goes directly into the third stream so on and so forth so it's a multi bed reactor where the fluid stream constantly goes from reactor 1 to reactor 2, reactor 3 etc and while going through this the conversion increases and hopefully at the end of the multi bed the desired conversion is achieved and this is done for various reasons for example the nature of the catalyst might be required might be different if the temperature conditions at which the reaction is conducted is different so therefore one may actually pack different type different catalyst in each of these multiple bed and so that way a desired conversion can be achieved and as well the catalyst deactivation can be prevented so another very well known configuration is called the reverse flow reactor configuration so reverse flow reactor configuration so in the reverse flow reactor configuration what is done is suppose if this is a tubular packed bed reactor now the fluid is flown let's say this is at z equal to 0 so this is the entry point at certain time and this is z equal to L so at a certain time t the fluid enters the reactor at z equal to 0 and then it leaves the reactor at z equal to L at a later time t1 at a later time t1 let's say if this is t0 at a later time t1 which is greater than t0 the fluid is now entering from into the reactor from the from z equal to L and it flows in the reverse direction and then leaves the reactor at z equal to 0 so this is at time t equal to t1 so this kind of switching of the direction of flow is done at a certain frequency and this is very useful if the catalyst deactivation has to be prevented for example when an exothermic reaction is conducted inside the packed bed reactor there is a constant increase in the temperature because of the inside the reactor because of the heat that is being liberated by the reaction while it is happening and due to which the temperature at the other end of the reactor that is the downstream of the reactor the temperature is likely to increase significantly now in order to and that may actually deactivate the catalyst or it may it may be post some severe safety problems in order to prevent that at a certain preset frequencies if the fluid stream is now the direction of the fluid stream is reverted then the temperature front which is actually generated because of the exothermicity of the reaction is now preserved inside the reactor itself and which may also be used to preheat and as a byproduct there are most part of the reactor will not be exposed to a certain very high temperatures and this can actually prevent the deactivation of the catalyst and this provides a very useful mechanism to protect the catalyst which is very expensive so this is a very well well known configuration and the other configuration which is also fairly used particularly if you want if the reactant feed stream is of lean concentration if the concentration of the reactant is very small in the feed then counter current flow reactor is being used so a counter current flow reactor is essentially a two chamber reactor so it will be two concentric cylinders two concentric cylinders now in this concentric cylinder the feed stream is flown in one direction in one chamber and in the other direction in the other chamber so the fluid stream is now flown at a in a counter current fashion where the fluid stream in the inside tube leaves first enters the reactor at z equal to 0 and leaves the reactor at z equal to L while the fluid stream in the outside concentric cylinder external cylinder enters at z equal to N and leaves at z equal to 0 so this has a unique advantage that if the concentration is very lean then it can be this kind of a configuration can actually be used to preheat the fluid which increases the reaction rate remember that the reaction rate is increases if it is an exothermic reaction then the reaction rate actually increases as a function of temperature and therefore if the if we have a counter current flow then the feed stream which is present near the inlet of the reactor in the in the inside tube can actually obtain heat from the fluid stream which leaves the reactor from the outside chamber and because the conversion of the fluid stream near the exit is expected to be high and as a result the fluid stream also is expected to carry a lot of heat while it leaves the reactor so in order in order not to lose the heat that it is gained because of the reaction such a counter current flow reactor provides a mechanism by which the heat can be transferred from the outside chamber to the inside chamber and the inside chamber to the outside chamber wherever it is desirable so such kind of an operation where simultaneous heat transport from from two chambers into each other and and exothermic reaction is what is called as auto thermal operation is called the auto thermal operation of a packed bed reactor so such kind of a reactor is very useful if the reactant concentration is very lean and yet the desired conversion is required. At another configuration which is also being used in industry is called the radial flow reactor so in a radial flow reactor what happens is suppose if this is the tube suppose if this is a tube and and there is a small concentric channel which is present inside the reactor and the space between the concentric channel inside and the exterior of this tube is actually filled with catalyst. Now the the fluid stream which is fluid stream containing the reactant is made to flow through the inside chamber from both sides the fluid stream flows simultaneously from both sides inlet fluid stream now when it flows from both sides a sparger is kept along the periphery of the inside tube these are tiny holes which are placed at a certain pre-calculated distance and so the the fluid stream which enters through the reactor on either side they leave they go into the chamber which contains the catalyst particle through this sparger so the feed stream is sparse uniformly in all directions inside the radial flow reactor and the reaction happens in the catalyst and then the fluid stream leaves from the curved surface of this exterior tube so this is what is called as a radial flow reactor at a side view of this reactor it looks like this so if I take a cross section of this reactor and make a side view of it then I can see that the fluid stream actually leaves from the inside tube uniformly in all directions and then the catalyst is present here so that's the catalyst and the reaction happens in the in the region where the catalyst is packed and then the fluid stream leaves from the exterior of this of this chamber so this kind of an operation is what is called as a radial flow reactor configuration and that also is being used in many different types of situations the main advantage of this configuration is that it facilitates operation under low pressure drop so let us look at the design of packed bed reactor let's look at a packed bed reactor design now the in a tubular reactor a packed bed reactor is essentially a tube in which the catalyst are packed so therefore the temperature in the concentration gradient is expected to be present in the axial direction if this is the flow of the fluid then it's expected to be present in the direction of the flow of the fluid which is called the axial direction and then it is also expected the gradients are expected to be present in both the radial direction radial direction and the angular direction so the in principle the temperature and the concentration gradients are expected to be present in all three dimensions and it's only concentration gradient if the reactor's operator isothermal conditions so therefore the gradients concentration and temperature in all three dimensions is present in all three directions however for the purpose of design we will assume that the gradients are present only in the axial direction axial that is essentially flow direction we will assume that the gradients are present only in the flow direction so now for this situation we have already written the mole balance in one of the lectures before so if the if the reaction is a first-order reaction if the reaction is a first-order reaction then the mole balance essentially is axial dispersion coefficient DEA if A is the reaction which is leading to product formation then DEA is the axial dispersion coefficient multiplied by D square CAB which is the concentration of the species in bulk divided by DZ square that's the that captures the diffusion of species in the reactor in the axial direction minus U into DC AB by DZ plus RA prime into rho B equal to 0 now for so let us draw the coordinates so suppose if this is the tubular reactor suppose this is the tubular reactor then this is the z direction and this is the radial direction and then the angular direction is let's let's call it theta so theta is the angular direction so there are no gradients we assume that there are no gradients in the theta and the r direction and the gradients are present only in the z direction so this correspond the first term here corresponds to the diffusion in the z direction and the flow is happening from z equal to 0 to z equal to L so the gradients are the first term here corresponds to the diffusion in the in the direction of flow and the second term corresponds to the bulk convection in the direction of the flow and third term corresponds to the rate at which the species A is being consumed to form the necessary product so suppose if the if we know the overall effectiveness factor then we can write the reaction rate as minus RA minus RA prime equal to the overall effectiveness factor multiplied by the corresponding specific reaction constant into the surface area per unit weight of the catalyst multiplied by the bulk concentration CAB now for a first order reaction the overall effectiveness factor is given by the local effectiveness factor eta divided by 1 plus eta into the first order reaction constant k k double prime multiplied by the surface area of the catalyst per gram of catalyst multiplied by the bulk density of the catalyst divided by the mass transport coefficient into area per unit volume of the catalyst so that gives the reaction rate in terms of the measurable quantity such as the bulk concentration of the species which is which is reacting to form the necessary products and so now we can suppose if we assume that the flow through the bed is significant if the flow is large if the flow is large then we can assume that the diffusion as done before we can assume that the diffusion of species in the in the bed is significantly smaller compared to the bulk flow of the fluid so under this condition we can actually we can rewrite the mole balance as DCA by DCA bulk by dz that's equal to minus omega into rho B which is the bulk density of the catalyst into k double prime into SA into CAB divided by U now sometimes the design parameter that one needs to estimate may be the weight of the catalyst or in sometimes it may be the it may be the size of the reactor given given a certain conversion so therefore we can actually convert this equation in terms of the weight of the catalyst so the weight of the catalyst that is packed till a certain location in the axial direction is given by the bulk density of the catalyst multiplied by the cross-sectional area of the reactor multiplied by the corresponding position so if this is the tubular reactor and let's say we are looking at this particular ZZ and if the cross-sectional area is AC cross-sectional area of the reactor is AC then we want to know what is the total amount of weight of catalyst that is packed from Z equal to 0 to up to a some particular point Z so that's given by rho B which is the bulk density of the catalyst multiplied by the cross-sectional area into Z which gives the volume of the reactor till that location and therefore we can rewrite the model equation as DCAB by DW that's equal to minus overall effectiveness factor into rho B into K double prime SA CAB divided by U into rho B into AC so that's because of the dependence of this weight of the catalyst on the density and the cross-sectional area and that's why we get these two terms here so we can cancel out these two and U into AC if U is the superficial velocity with which the fluid is entering the reactor U into AC which is the cross-sectional area will give you the volumetric flow rate with which the fluid stream is entering the reactor so therefore this can be written as minus overall effectiveness factor into SA divided by the volumetric flow rate CAB so the volumetric flow rate at the inlet could be V0 and the volumetric flow rate at that particular location if there is volume expansion then the volumetric flow rate at that particular location could be different. So if you assume that there is no volumetric expansion then we can assume that V is equal to V0 and if and also if we assume that and if you assume that at if the weight of the catalyst 0 then the CAB which is the bulk concentration that's equal to the concentration with which the fluid actually is entering the reactor so CAB 0 is the concentration with which the fluid is actually entering the packed bed reactor in which the reaction is happening. This equation can now be integrated to find that the conversion conversion X is given by 1 minus CAB by CAB0 and that's equal to 1 minus exponential of minus omega which is the overall effectiveness factor multiplied by k double prime multiplied by the surface area of the catalyst available for the reaction per unit gram of catalyst multiplied by W divided by the volumetric flow rate V0 with which the fluid is actually entering the reactor or we can rewrite this as W equal to V0 divided by omega k double prime SA into ln of 1 by 1 minus X. So this provides a relationship between the weight of the catalyst that is required for a specified conversion. If we specify the conversion then we can find out how much weight of catalyst is required. So if we know the weight of the catalyst then we should be able to find out what should be the length of the reactor which is required for a given cross-section and that can be found simply by using this relationship the length of the reactor is given by weight of the catalyst divided by the cross-sectional area into density. So this method provides a mechanism this design provides a mechanism by which for a specified conversion if the assumption that the bulk flow is significantly larger than that of the diffusion then we can actually find out what is the weight of the catalyst which is required for a given conversion and what is the length of the reactor for a specified conversion. So suppose if we look at a second order reaction suppose if we consider a second order reaction and if A going to products is the reaction scheme and if minus rA prime is the rate at which the species A is being consumed if that is given by the overall effectiveness factor omega multiplied by the corresponding specific rate constant k double prime into the surface area of the catalyst per unit weight of the catalyst into CAB square so that is the rate law and the mole balance for the reactor will be DA which is the dispersion coefficient of the species into D square CAB by DZ square that's the diffusion term which with the rate at which the species is diffusing inside the reactor and minus U times DCAB by DZ minus omega into k prime into SA into rho B into CAB square so it should be a rho B here so the that's the mole balance which captures the dependence or the that captures the dependence of the concentration of the bulk concentration of the species as a function of position and other parameters of the system including the specific rate constant. So now we can solve this if you assume once again that the bulk flow is significantly larger than the corresponding diffusion rate then the DA square is significantly smaller to the corresponding bulk flow then the mole balance will simply be reduced to DCA bulk by DZ that's equal to minus overall effectiveness factor omega multiplied by k double prime into SA into rho B into CAB square divided by U. Now once again we can reformulate this in terms of the dependence of the concentration of bulk species concentration of species A in bulk as a function of the weight of the catalyst so we can rewrite this as rho B AC into DCAB by DW which is the where small w stands for the weight of the catalyst and that's equal to minus omega CAB square divided by U into AC into rho B that's the bulk density of the catalyst and so from here we can find that DCAB by DW that should be equal to minus omega k double prime SA divided by the volumetric flow rate V multiplied by CAB 0 into 1 minus x the whole square so this expression provides a relationship between the concentration of the species and other parameters and where CAB is equal to CAB 0 into 1 minus x where x is the corresponding conversion. So now we can also incorporate the relationship between the concentration and conversion on the left hand side of the expression and we can integrate the equation as by assuming that the volumetric flow rate at any location is equal to the inlet volumetric flow rate by assuming that the volume changes are negligible and also by assuming that at Z equal to 0 the weight of the catalyst is 0 that is there are no catalysts before the reactor starts and the reaction is actually happening inside the reactor. This differential equation cannot be integrated analytically as in the case of a first order reaction this is because the overall effectiveness factor for this case will be a function of conversion. So now such kind of a design can actually be used for various kinds of reactions it could be such kind of a design can be performed for rate law which is non-monotonic that is it's a non n-th order type reaction one could actually write a design equation for adsorption inhibition type reactions for Langmuir-Hinshelwood type reactions and such kind of a reaction can all design equation can also be written for situations where multiple species are actually participating in the reaction. So it's a good time to summarize that what we have seen so far is we have looked at various criterion for deciphering what is the what how deciphering from experiments experimental conditions whether the reaction is actually internal diffusion limited or external diffusion limited or surface reaction limited also we have looked at what is the recipe general recipe for performing the reactor design and also performed the looked at the reactor design equations for a first order and a second order reaction thank you.