 Friends, it is a good time to summarize what we have learned in the last lecture. We initiated discussion on the fluidized bed reactor and the objective is to design a fluidized bed reactor. What is a fluidized bed reactor? Fluidized bed reactor is essentially a tube which has a certain catalyst particles and then the gas is flown through this tube and as the velocity of the gas stream increases then the drag force that this fluid stream exerts on the gas particle on the solid catalyst particle that is equal to the weight that is actually gravitational force exerted by these particles because of its natural weight. So, when that equals then the catalyst particle starts raising and that is called the fluidization phenomena. Now, once the fluidization occurs then these gas bubbles are formed and when these gas bubbles are formed there is transport between the transport of the reactants from the gas bubble to the catalyst particle where the reaction occurs in the catalytic sites of the catalyst particle which is already fluidized and then the product which is formed in the sites is actually transported back into the bubbles which is the gas stream and then the bubbles carry these product and leave the reactor. So, this is the process that occurs in a fluidized bed reactor and then we looked at different regimes different fluidization regimes in the fluidized bed reactor. So, if FPR stands for the fluidized bed reactor we looked at different flow regimes. One is the fixed bed regime. In a fixed bed regime the fluid velocity is not significant enough to offset the gravitational force which is exerted by the catalyst particle and as a result the particles they remain packed at the bottom of the reactor where they are sitting on a perforated or a porous plate. Then the second regime is called the minimum fluidization regime, minimum fluidization regime. Now, in this regime the velocity of the fluid is just sufficient to offset the gravitational force that is the drag force that is that the particles experience because of the flow of this gas which is flowing at a certain superficial velocity which is which is being flown into the bed at a certain superficial velocity U0. Then the drag force offsets the or it is just equal to the gravitational force exerted by the catalyst particles then the particles starts raising or they get fluidized and that velocity minimum velocity is called the fluidization velocity and the regime is called the minimum fluidization regime. Typically, the bubbles which are present in the near the minimum fluidization regime they are bubbles are formed near the perforated plate and then the bubble starts travelling through this bed which is being fluidized and the third regime is called the aggressive bubbling regime. So when the velocity with which the gas stream is being flown inside the bubbling significantly increases and so there is aggressive bubbling lots of bubbles are formed which causes tremendous amount of recirculation of the fluid and also the catalyst particles and so that regime is called the aggressive bubbling regime and then the fourth one is called the slugging regime where these velocity is significantly higher that there are channels of these channels are actually created and through these streams the gas stream simply escapes the bed and that is what is called the slugging regime. And the last one is the lean phase where the all the particles are suspended and the density of the porosity is significantly higher and they are all suspended all through the reactor. So that is what is called the lean phase regime. So these are the five different regimes that we described in the last lecture and we initiated the discussion on Cuny-Levenspiel model Cuny-Levenspiel model. So the Cuny-Levenspiel model assumes that all particles are of same size and it also assumes that the solid flow in the emulsion the solid flow in the emulsion phase as though like it is a plug flow and then the emulsion phase exists at a minimum emulsion phase always exists at a minimum fluidization velocity and then it also assumes that the gas void fraction is equal to the void fraction as that of at the minimum fluidization velocity and then it assumes that the solid which is flowing down downwards is actually the concentration of the solids which is present in the emulsion phase is equal to the concentration of the solids which is present in the wakes which are actually formed just below the bubble. Just to recap the bubbles are formed as soon as these fluid actually is going through the bed the bubbles are formed and then the bubbles carry a certain amount of particles and then there is a wake which is formed below the bubble where a lot of particles of high concentration is actually carried along with the bubble around this around the bubbles there is a phase called the cloud phase there is a phase called the cloud phase and in this cloud there are some particles are present in this cloud and then around the cloud is the around the cloud is the phase called the emulsion phase which also contains lots of particles and the emulsion phase is essentially has the same porosity as that of the nearly the porosity of the resting bed and therefore the transport which occurs the mass transport process which occurs is the reactant species are transported from the bubble to the cloud phase. So, this is the cloud phase and this is the emulsion phase and this is the wakes so the mass transport occurs from the bubble phase into the cloud phase and the reactants are transported from the cloud phase into the emulsion phase where the particles are present and the reaction occurs in the particles and then the product is actually transported back into the cloud phase and back into the bubble phase. So, once this happens the bubbles carry the product and then the product leaves the fixed fluid as bed reactor. So, this is the process that occurs in a fluid as bed reactor. So, now we looked at some of the expressions for the there are several parameters that one needs to actually find out before we can model the fluid as bed reactor. So, the velocity with the velocity of gas in emulsion phase is given by u e equal to u m f which is the velocity of the minimum fluidization divided by the porosity at minimum fluidization minus u s where u s is the velocity of solids flowing downwards flowing downwards. Now, the bubble velocity so remember that the one of the important aspects that control the conversion or the performance of the fluid as bed reactor is the time that is spent by the bubble in the fluid as bed reactor because the reactants are carried inside the bubble and the reactants have to get in contact with the solids in order for the catalytic reaction to occur. So, therefore, the amount of time that is spent by the bubble inside the reactor significantly contributes to the performance of the fluid as bed reactor. So, the time that is spent by the bubble inside the reactor is controlled by the velocity with which the bubble is actually raising inside the fluid as bed reactor. So, let us look at how to estimate this velocity of the bubble which is raising inside the fluid as bed reactor. So, the bubble velocity is calculated for a single bubble there are correlations that exist for a single bubbles which is given by u b r and that is equal to 0.71 into gravity into diameter of the bubble to the power of half. So, one needs to know what is the diameter of the bubble in order to estimate what is the single bubble velocity. So now, if many bubbles are present which is typically the case in a fluid as bed reactor then the velocity with which a single bubble is going to raise together with many bubbles is going to be very different because of the presence of other bubbles. So, therefore, in a fluidized state in fluidized state the bubble velocity is given by the bubble there has to be some correction that is associated with the bubble velocity if there were to be just a single bubble. So, therefore, the correction is basically given by u naught minus u m f that is the correction to the single bubble velocity and that tells us what is the velocity of the bubble in a fluidized state where u naught is the superficial velocity with which the gas stream is being flown into the fluidized bed reactor and u m f is the corresponding velocity of at minimum fluidization point minimum fluidization at the minimum fluidization and so, therefore, plugging in the correlation for the for the velocity of a single bubble if we can find that the velocity of the bubble in a fluidized bed reactor in the fluidized state is given by u naught minus u m f plus 0.71 into gravity into diameter of the bubble to the power of half. So, now we need to find out what is the diameter of the bubble what in terms of the other properties of the reactor. So, there are correlations which are actually available. So, one needs to use correlations in order to estimate the diameter of the bubble. So, let us look at what these correlations are. So, the first correlation that you are going to look at is called the Mori-Wen correlation it is the Mori-Wen correlation the correlation is as it goes like this. So, dBm which is the maximum possible bubble diameter minus the diameter of a particular bubble divided by the maximum possible diameter minus the diameter of the bubble when the bubble is just formed that is the initial bubble diameter and that should be equal to exponential of minus 0.3 into h by dt where h is the height at which the particular bubble is being observed and dt is the bed diameter diameter of the bed. And so, now Dbo which is the initial bubble diameter initial bubble diameter that is given by 0.0037 into u naught minus u m f the whole square. If it is a perforated plate if it is a porous plate sorry if it is a porous plate then the correlation that gives an estimate of what is the initial bubble diameter is given by 0.0037 into superficial velocity minus the corresponding fluidization velocity and square of that difference. And that is equal to 0.347 multiplied by the area of cross section into u naught which is the superficial velocity minus the minimum fluidization velocity v m f divided by the number of perforations which is actually present in the perforated plate to the power of 0.4. So, this is the correlation for estimating the initial bubble diameter if the plate which actually holds these particles are actually a it is a perforated plate, if it is a perforated plate then this is the correlation that actually gives an estimate of what is the initial bubble diameter. So, next we need to know what is the maximum possible bubble diameter so the maximum bubble diameter is given by d B m is the maximum bubble diameter and that is given by the correlation 0.652 into the area of cross section of the bed into u naught minus u m f u naught is the superficial velocity with which the gas stream is let inside the fluidized bed reactor and u m f is the minimum fluidization velocity a whole to the power of 0.4. So, that provides an estimate of what is the maximum possible bubble diameter and it is known that the predictions are poor if the bed is very large. So, the d B m prediction by using this correlation is poor for large beds for a large fluidized bed reactor the correlation does not work very well the experimental observation suggests that this correlation does not give a good estimate if the bed is very large. The other correlation which is also available which seems to work over a wide range of fluidized bed reactor is called the Werther correlation the Werther correlation is that the diameter of the bubble is given by 0.853 into 1 plus 0.272 into u naught minus u m f whole to the power of 1 by 3 into 1 minus 0.684 into h which is the height of the bed at that particular instance it to the power of 1.21. So, this correlation is known to give a better prediction of the diameter of the bubble into the fluidized conditions. So, the next estimate that we need to make is the velocity of the solid when it is flowing in the emulsion phase. So, we need to find out what is the solids velocity u s that is what that is the next parameter that needs to be estimated. So, suppose in order to estimate the solids velocity we can perform a very simple material balance this is because of the fact that whatever solid particles which are actually flowing down in the emulsion phase should be equal to what are the solids which are actually being taken up in the wakes which is actually following the bubbles. So, wakes actually have the maximum concentration of the solid particles that are being lifted because of fluidization. So, therefore, the amount of solid particles which are actually flowing upwards in the wake should be equal to the amount of solid particles which are actually transported in the emulsion phase. So, therefore, if we make a material balance across the solids flowing in these 2 phases then we should be able to estimate what should be the relationship for finding the solids velocity. So, let us look at this material balance. So, the solids flowing in emulsion phase that should be equal to flowing down in emulsion phase should be equal to solids flowing solids actually taken upwards in the wakes. So, now putting the corresponding expression solids flowing down in the emulsion is given by area of cross section AC multiplied by the density of the catalyst rho C into 1-delta-alpha into delta into U s where delta is basically the fraction of the total bed that is actually bubbles. So, delta is the fraction of total bed that is bubbles. Now, this excludes the this excludes the wakes that are actually formed this is just the bubbles this is the fraction of the total bed which is basically the bubbles. So, that is delta and then alpha is essentially alpha is the fraction of or the volume of the wake per volume of the bubbles which are actually formed because of fluidization. This is the volume of wakes per volume of the bubbles which are actually formed by fluidization. So, therefore, this quantity 1-delta into alpha delta it actually it actually estimates as to how much what what fraction of the cross sectional area is actually filled with solids is given by 1-delta-alpha into delta. So, therefore, using the this relationship for the solids flowing in emulsion that can be equated to the fraction that contains fraction of volume in a given cross section that is actually filled by wakes is alpha into delta because alpha is the volume of wakes per volume of the bubbles and delta is the fraction of the bed that is actually bubbles multiplied by U b which is the velocity with which the bubble is raising up upwards into multiplied by rho c into ac. So, this material balance of the solids which is flowing down in emulsion and the solids which are actually taken upwards in the wakes can be used to estimate U s and note that the U b is the bubble rising velocity which can be obtained using the correlations that is Mori-Ven or the Werther correlation and alpha and delta are supposedly a known property for a for a particular fluidized bed reactor and we are going to see how to estimate them. So, based on this correlation we can find out that the velocity of the solids U s is actually given by alpha into delta into U b divided by 1-delta-alpha into delta. So, that is the expression for the that is the expression for the velocity with which the solid is flowing. So, the next step is to find out what is this value of delta which is basically the fraction of the of the bed which is actually filled with the bubbles. So, now let us write a simple material balance on the gas flow to find out what is this value of delta which is the fraction of bubbles fraction of the bed which is actually bubbles. So, this we can write a material balance on gas flow to estimate the value of delta. So, that is what we are going to do next. So, the material balance on gas flow basically is just to account for what is the how the gas is being split into different sections and what is the total material balance for the gas flow and that is given by Ac into U naught where U naught is the superficial velocity. So, that is the superficial velocity with which the fluid is actually being pumped into the fluidized bed reactor. So, the total gas that enters the reactor should be equal to the cross section area of the fluidized bed reactor multiplied by the corresponding superficial velocity. So, that should be equal to the gas that is actually carried by the bubbles. So, that will be the cross sectional area into delta which tells you what is the area fractional area that contains that is actually occupied by the bubbles multiplied by the velocity U b tells us what is the flow rate mass flow rate of the of the gas that is actually carried by the bubbles plus some of this gas is now going to go in the wakes. So, that can be estimated as Ac which is the cross sectional area multiplied by the porosity with at the minimum fluidization velocity epsilon mf into alpha into delta into U b. So, alpha is basically the volume of wakes per volume of the bubbles. And then there is another component which is basically the gas that may be present in the emulsion phase. So, that is actually Ac into epsilon mf into 1 minus alpha minus alpha 1 minus delta into alpha into delta into the velocity with which the emulsion is moving. So, this basically tells us what is the material balance for the gas flow. Now, we know that this is the this is the total gas flow, this is the gas flow in the bubbles and this is gas flow in wakes and this is in the emulsion phase that is in the emulsion phase. So, that tells us what is the total amount of material balance on gas flow and we know that the velocity of the emulsion phase is given by U mf divided by epsilon mf minus the velocity of the solids. So, now plugging in these expressions we will be able to find out what is delta. So, delta is now given by U naught which is the superficial velocity minus U mf which is the minimum fluidization velocity divided by U b minus U mf into 1 plus alpha. Now, if U b is significantly larger than U mf that is if the velocity with which the bubbles are actually raising is higher than the minimum fluidization velocity then we can further simplify the expression for the delta which is the fraction of the bed that is actually occupied by the bubbles that is given by U naught minus U mf divided by U b. So, now we have estimated what is delta. So, the next exercise is to characterize the mass transport in fluidized bed reactor characterize the mass transport in fluidized bed. Now, there are two forms of mass transport which actually occurs in the fluidized bed reactor one is there is mass transport between the gas and solids. So, that is very much like the gas solid transport and gas solid reactions. So, gas solid catalytic systems and then the second type of transport is basically between the bubble and the cloud phase between the bubble and the cloud phase and also between the cloud and the emulsion. So, remember that when for the reaction catalytic reaction to occur the reactants which is actually present in the gas stream is carried by the bubbles and this mass transport of the reactants from the bubble phase into the cloud phase and cloud phase into the emulsion phase which actually contains the particles where the reaction occurs and then the products are now transported back. So, therefore, the mass transport process in fluidized bed reactor is slightly different from the slightly in addition to the classical mass transport in the gas solid catalytic systems. So, let us look at what happens here. So, suppose if this is the bubble suppose the bubble is here and then if the cloud is present around here. So, now we may represent W if A is the species, W A B to C is essentially the transport from bubble to the cloud region and this is the emulsion region and then W A C to E is basically the flux of transport or rate of transport from the cloud to the emulsion phase and similarly after the product is formed the W B if B is the product that is formed that is the transport from the emulsion to the cloud phase and then W A W B C to B is the transport of the product from the cloud to the bubble phase. So, this is the in addition to the classical gas solid transport in the catalytic reactors this is an extra transport mechanism which is been observed in which is observed in the fluidized bed reactor. So, let us look at how to characterize the how to find the gas solid mass transport coefficient first and then we will look at the other mode of transport. So, now between gas and single particles there are correlations which is called the Frosling correlation that is the Frosling correlation and that is basically characterizes the mass transport between gas and single particle and that is given by Sherwood number that is equal to 2 plus 0.6 into Reynolds number to the power of half into Schmidt number to the power of 1 by 3 and for emulsion phase for emulsion phase the Cooney and Levenspiel have developed a correlation Cooney and Levenspiel have developed a correlation to estimate the mass transport coefficient in the emulsion phase and that is given by Sherwood number equal to 2 plus 1.5 into Schmidt number to the power of 1 by 3 into 1 minus epsilon into Reynolds number to the power of half and the validity of this correlation is basically between the Reynolds number of 5 and 120 and epsilon should be less than 0.8, 0.84. So, if the porosity is less than 0.84 then this correlation works in the emulsion phase. So, there are different mass transport coefficients and one need to actually combine all of these mass transport coefficient in order to estimate what is the overall mass transport coefficient because mass transport occurs through both these cases both these mechanisms then let us look at the mass transport between bubble and cloud. So, suppose if the so let us first look at the transport of the reactant from the bubble to the cloud. So, the suppose if A is the reactant so the flux at which the species A is being transported from the bubble to the cloud flux at which it is transported is actually given by the mass transport coefficient KBC multiplied by CAB minus CAC where CAB is basically the concentration. So, that is the concentration of the species in the bubble phase and this is the concentration in the cloud phase. So, this expression provides the flux at which the species is being transported from the bubble phase to the cloud phase and similarly the for transport from the cloud phase to the emulsion phase the expression can be written as KCB where KCB is the corresponding mass transport coefficient and the previous case the KBC is the corresponding mass transport coefficient multiplied by CAC minus CAC where CAC is the concentration of species A. So, CAC in the cloud phase and this is the concentration of species in the emulsion phase. So, this is for the mass transport between the cloud phase and the emulsion phase. So, similarly the one can actually write a similar transport for the product species which is being transported back from the emulsion phase back into the cloud phase and back into the bubble phase. So, Kuni-Levenspiel once again they have developed a correlation in order to find out what these mass transport coefficients are. So, they have developed a correlation for estimating these mass transport coefficient and so that is given by KBC which is the mass transport coefficient for reactant species to go from for species to go from the bubble phase into the cloud phase and that is given by 4.5 multiplied by the minimum fluidization velocity UMF divided by the diameter of the bubble plus 5.85 into the diffusivity DAB to the power of 1 by 2 into gravity to the power of 1 by 4 divided by the diameter of the bubble to the power of 5 by 4. So, that is the correlation which provides an estimate of what is the mass transport coefficient between the bubble phase and the cloud phase. Now because the mass transport actually occurs by the exchange of volume between the cloud phase and the bubble phase, one could assume that the mass transport between the mass transport coefficient for transport from the bubble to the cloud phase should be approximately equal to the mass transport coefficient from the cloud phase back into the bubble phase. And typically the order of magnitude of this mass transport coefficient is of order of 2 second minus 1. Remember that KCB is a mass transport coefficient. So, now let us look at the correlations for cloud to emulsion, mass transport from cloud to the emulsion and that is typically given by KCE equal to KEC. That is the KCE is the mass transport coefficient for transport from the cloud phase to the emulsion phase and KEC is the transport of products let us say from the emulsion phase back into the cloud phase. And so that is given by the correlation 6.77 into epsilon mf which is the porosity at the minimum fluidization velocity multiplied by the diffusivity DAB into the velocity of the raising bubble divided by the diameter of the bubble to the power of 3 cube of that to the power of 1 by 2 square root of the whole expression. And this is typically of the order of 1 second minus 1. So, that is the order of magnitude of the mass transport coefficient. So, let us next look at the reaction in the fluidized bed reactor. Remember there are 3 factors which are controlling. One is the mass transport of the reactant species from the bubble phase into the cloud and into the emulsion in order for it to get in contact with the catalytic particles. And then the next step is the reaction which is actually occurring inside the catalytic sites to form the product catalytic reaction which is happening inside the catalytic site to form the products. And once the products are formed they are transported back into the bubble phase. So, let us look at the reaction in the fluidized bed reactor. Let us look at the reaction. So, suppose if it is an nth order reaction, suppose if it is an nth order reaction, then the reaction rate in the bubble phase can actually be given as kB into CAB to the power of n. So, that is in the bubble phase. And similarly for the cloud phase it can be given as kC into CAC to the power of n where kB and kC are the corresponding rate constants. And then for the emulsion phase it can be written as kE into CAE to the power of n where the CAB is the concentration of the species in the bubble phase. And CAC is the concentration of the species in the cloud phase and CAE is the concentration of species in the emulsion phase. Now it is important to write these rate expressions in all three phases. Although the emulsion phase actually contains the maximum number of particles, the bubble phase and the cloud phase also will have some particles. And therefore, it is important to write these expressions because the reaction can in principle occur in the catalyst particles in each of these phases. So, now next we can write a mole balance. Once we know the reaction rates, we can now write a mole balance in the fluidized bed reactor in order to capture the behavior of the concentration of the reactant species in the reactor. So, now let us consider a small element. Let us consider a fluidized bed reactor and if let us say that the fluid is entering at a superficial velocity of U0 and then there is a small element between z and z plus delta z. And if you assume that the upward motion or the direction of the fluid flow is the positive direction, then we can now write a mole balance for the bubble phase. We can now write a mole balance for the bubble phase which is basically rate due to flow into the bubble phase minus the rate with which the fluid stream leaves because of flow plus the rate at which the fluid species is actually leaving the bubble phase because of mass transport plus whatever is being generated that should be equal to 0 under a steady state condition. So, if you assume a steady state conditions and this is the mole balance. So, now when we plug in all the corresponding expressions, rate due to flow that is into the bubble phase is given by the velocity of the bubble UB multiplied by the corresponding cross section AC into concentration of the species in the bubble phase multiplied by delta. So, delta is basically the fraction of the bed that is actually in the bubble phase and that at that particular location z that is the rate at which the species is entering this small element in the bubble phase and then the rate at which the species is leaving in the bubble phase at z plus delta z is given by UB AC CAB into delta at z plus delta z and mass transport is given by minus KBC into CAB that is the concentration of the species in the bubble phase minus the concentration of species in the cloud phase multiplied by cross section area into delta z. So, that is the rate at which the species is actually leaving the bubble phase and going into the cloud phase minus the corresponding reaction rate. So, that is KB if that KB is the reaction rate constant into CAB to the power of n. So, that is the CAB is the concentration of species in the bubble phase multiplied by the cross section into delta z into delta. So, that should be equal to 0. So, that is the mole balance for the species in the bubble phase. So, now we can rewrite this mole balance as by taking a limit that so, by taking limit that delta z goes to 0 we can rewrite this model as UB which is the velocity with which the bubble is racing inside the fluidized bed reactor into DCAB by DZ that should be equal to minus KB CAB to the power of n minus KBC which is the mass transport coefficient between the bubble and the cloud phase multiplied by CAB minus CAC. So, that is the model the mole balance for the bubble phase. So, similarly for the cloud phase the mole balance is given by UB into delta into 3 times UMF a very similar balance can be written and taking the limits of delta z going to 0. One would get that the expression for a mole balance for the concentration of the species in the cloud phase is basically given by this expression here into DCAC by DZ that should be equal to the mass transport coefficient of the species from the bubble to the cloud phase that is basically added into the cloud phase multiplied by CAB minus CAC minus KCE that is the mass transport coefficient for transport of the species from the cloud phase into the emulsion phase that is given by CAC minus CAE minus KC which is the rate at which is the rate constant for the reaction if the catalytic reaction is happening in the catalyst particles which may be present in the cloud phase. So, that is the mole balance for the cloud phase and then the mole balance for the emulsion phase is given by for the emulsion phase that is given by UE which is the velocity with which the emulsion phase is moving into 1 minus delta minus alpha into delta divided by delta into DCAE by DZ that should be equal to the mass transport coefficient between the cloud and the emulsion phase into CAC minus CAE minus the rate at which the species is actually being consumed because of the reaction that may be happening in the emulsion phase. So, if we need to find out what is the concentration of the species in the bubble phase in the cloud phase and the emulsion phase then these three equations have to be solved simultaneously. So, these things have to be solved simultaneously. So, once we solve them simultaneously then we can find out the expression for CAB, CAC and CAE what is its relationship how the profile changes with respect to the position inside the fluid as bed reactor. As this is a non-linear equation it cannot be solved analytically and one has to resort to numerical techniques to solve these set of equations. However, if we make an assumption that the reaction is a first order reaction, suppose if we assume that it is a first order reaction, suppose if the reaction the catalytic reaction which is happening is a first order reaction then we can actually write the expression as UB into DCAB by dz that is equal to minus KB into CAB to the power of n minus the mass transport coefficient KBC into n equal to 1 CAB minus CAC and suppose if we assume that DCAC by dz which is the rate of change of the concentration with respect to position in the cloud phase if this is very small and similarly if we assume that DCAE by dz is also very small then we can write the model equations for the these two concentrations as they basically become like this where the mass 0 equal to the mass transport coefficient KBC multiplied by CAB minus CAC minus KCE into CAC minus CAE minus KC into CAC and similarly for the emulsion phase the mole balance will become KCE into CAC minus CAE minus KE into CAE. So, that is the mole balance for the for the emulsion phase. Now, because it is a catalytic reaction, so the KB which is the corresponding rate constant suppose KB, KC and KE these are the corresponding rate constants for the reaction which is actually occurring in the bubble phase cloud phase and the emulsion phase respectively and emulsion phase respectively. So, now suppose if it is a catalytic reaction, if it is a catalytic reaction and if gamma B is basically the ratio of volume of solid catalyst in the bubble phase divided by the volume of the bubble. So, this provides an estimate of what fraction of the bubble volume is actually contained by the solid particles which are actually carried by the bubble phase. So, if we know this expression then we can actually rewrite the rate constant in terms of the intrinsic rates. So, that will be equal to KB is given by gamma B which is the fraction of the volume which inside the bubble which is basically which is occupied by the solid particles carried by the bubble into the corresponding reaction rate which is occurring in the catalyst surface of the particles and so now that can actually be rewritten as gamma B into rho C into K prime where K prime is the gram mole that is reactor per unit weight of the catalyst per unit time and rho C is the corresponding density of the catalyst. So, similarly we can actually write we can write KC is basically equal to gamma C which is the volume of solid catalyst which is present fraction of volume of the cloud phase which is occupied by the solid catalyst that multiplied by rho C into K prime. So, the and similarly K e equal to gamma e into rho C into K prime. So, notice that the K prime is basically same because it is the same catalytic reaction which is happening in the solid particles which are present in these three phases. The overall reaction rate is different in these three phases because the amount of catalyst particles which is present in each of these phases are different and therefore, the that is actually accounted for in the overall reaction rate constant. So, now we need to estimate what is this gamma B, gamma C and gamma E are. So, if we know that estimate then the mole balance can actually be solved. So, we need to find out what is gamma B, gamma C and gamma E. So, once we know this we can actually solve the model equation and so, gamma B is essentially given by 3 into U M F by epsilon M F divided by U B minus U M F by epsilon M F and similarly gamma E is given by 1 minus epsilon M F into 1 minus delta by delta minus gamma C minus gamma B. So, that is gamma E is the fractional volume in the emulsion phase which is occupied by the solid catalyst and gamma B is the corresponding fractional volume in the bubble phase and gamma C is the fractional volume in the cloud phase occupied by the solid particles which will be 1 minus epsilon M F into M F by epsilon M F divided by U B R minus U M F by epsilon M F plus alpha. So, these things can be estimated simply by estimating in terms of the properties such as the porosity and the minimum fluidization velocity etc. What is the volume of the bubble and what is the fraction of the bubble which contains the solid particles what is that volume. So, once we estimate these volume and take the ratio we can find out these expressions for the volume fractions in the in each of these respective phases that is contained by the solid particles. So, now if we know all these expressions then we can we can now solve for the solve the model equations in order to find out the design parameters of the fluidized bed reactor. So, the complete mole balance for the first order reaction is basically the set of mole balance equations are suppose if we assume that the time is equal to z by U B. So, remember that the time that is spent by the bubble inside the bed is actually an important parameter that controls the performance of the fluidized bed reactor. So, T here refers to the time that is actually spent by the bubble inside the fluidized bed reactor till this position z. So, U B where U B is basically the velocity with which the bubbles are actually rising inside the fluidized bed reactor. So, the mole balance will be dCA B by dt that is equal to minus gamma B into the specific rate constant for the catalytic reaction multiplied by C AB minus K BC into C AB minus C AC and the second equation is K BC which is the mass transport coefficient between the bubble and the cloud phase that multiplied by C AB minus C AC that should be equal to gamma C into the rate constant for the catalytic reaction into C AC plus the mass transport coefficient K CE into C AC minus CAE and then the third equation would be K CE into C AC minus CAE that should be equal to gamma E into the rate constant for the catalytic reaction into CAE. So, now if I look at the third equation from third equation I can actually rearrange third equation and find an expression for CAE and that CAE is equal to K CE divided by gamma E into K catalyst which is the reaction rate constant for the catalytic reaction plus K CE into C AC. So, further rearrangement of the expressions can actually be performed in order to estimate what is the concentration of the species in the cloud and that is from second equation we can find out. So, substituting the expression for the concentration of the species in the emulsion phase into the expression for the concentration into the mole balance expression for the concentration of the species in the cloud phase we can find out the expression for the concentration of the species in the cloud phase and that is given by K BC into CAB divided by gamma C into K catalyst plus K CE into gamma E K catalyst divided by gamma C into K catalyst plus K CE plus K BC. So, that is the corresponding mass transport coefficient now plugging in all these expressions into the into equation 1. So, plug in expression for C AC and CAE in equation 1 we will get. So, what we can find is that minus DC AB by DT that is equal to K cat into CAB into sum overall constant KR and this KR is essentially given by the overall constant KR is given by gamma B plus 1 divided by K catalyst by K BC plus 1 divided by gamma C plus 1 divided by 1 by gamma E plus K catalyst divided by K CE. So, that is the expression for the overall reaction rate and if I look at this expression gamma B gamma B is basically captures the 1 by gamma B is the resistance for resistance to reaction in the bubble and K cat by K BC is the resistance to mass transport from bubble to cloud and gamma C is resistance to 1 by gamma C is resistance to reaction in the cloud phase and this is the resistance to reaction in the emulsion phase and this is the resistance to mass transport from the cloud to the emulsion phase. So, this overall constant essentially captures the resistances that is all the resistances that are actually present in the in this system. So, now using the appropriate stoichiometry we can say that CAB is equal to CAB 0 into 1 minus x where CAB 0 is the concentration of the species at the inlet of the fluid as bed reactor and so we can now rewrite the mole balance as DX by DT equal to K cat into KR that is the overall rate constant which captures all the resistances which are involved into 1 minus x. So, now we can solve this equation and you can find that ln of 1 by 1 minus x that is equal to K cat into KR into T. So, this provides the relationship between the conversion as a function of the time that is actually spent by the raising bubble inside the fluid as bed reactor. So, from this we can find out what is the overall height that is required. So, that is equal to T into UB that is the height that is required for suppose if a specific conversion is set what should be the conversion supposing if for a desired conversion if T D is the time that is required from this expression for the desired conversion. So, this corresponds to the desired conversion if this corresponds to the time for the desired conversion then the height of the bed can actually be estimated by using this expression T D into UB and that is equal to UB divided by K cat into KR into ln of 1 by 1 minus x and from here we can find out what is the weight of the catalyst that is given by rho C AC into AH into 1 minus epsilon mf into 1 minus delta. So, that is given by rho AC UB into 1 minus epsilon mf into 1 minus delta divided by K cat into KR into ln of 1 by 1 minus x. So, that is the expression for the weight of the catalyst. So, let us quickly just rewrite the way to expression for the weight of the catalyst. So, that is given by rho AC UB into 1 minus epsilon mf into 1 minus delta divided by K cat into KR which captures the which is basically reflects the overall resistance into ln of 1 by 1 minus x and the height which is required for the bed which is an important design parameter is basically UB divided by K cat into KR into ln of 1 by 1 minus x. So, let us summarize what we have learnt in this lecture. So, what we have seen is we have actually designed a fluidized bed reactor. We started by looking at various parameters estimating the various parameters which is required to find to design a fluidized bed reactor. For example, what is the how to estimate the velocity of the racing bubble, how to estimate what is the fraction of the bed that is actually occupied by the bubbles etc. And then we found out what is the rate law that corresponds to the catalytic reaction that is happening in the particles which may be present in the bubble or cloud or the emulsion phase in any of these 3 phases. And then we looked at the mole balance for the species in each of these phases which incorporated the transport of species from the bubble phase to the cloud phase and cloud phase to the emulsion. And using this mole balance, we actually assume that it is a first order reaction and found out what are the important design parameters such as the height of the fluidized bed and weight of the catalyst which is required for a given conversion to be achieved using a fluidized bed reactor. Thank you.