 Friends, let us look at fluid solid non-catalytic systems. The next three lectures, we are going to look at what are the different aspects related to fluid solid reactions in non-catalytic systems and how to appreciate the reaction process which is occurring in the solid in non-catalytic systems and how to model these systems. So the fluid solid non-catalytic reactions, fluid solid non-catalytic systems, so these are essentially heterogeneous reactions which where the gas or the liquid, they actually come in contact with the solid and then they react with the solid and then they transform the reactants into a products. So these are a class of heterogeneous reactions. These are a class of heterogeneous reactions where the gas or a liquid which is essentially a reactant, gas or a liquid reactant, they come in contact with the solid, they contact the solid and then they react with the solid, with the solid and then form products. So this is very common in many industrial settings and we are going to see several examples of these today and also to look at how to model these systems. So suppose if the reaction is in general, if suppose there is a fluid A which is a fluid, it may react with a solid B to form various types of products. So there are various possibilities, one is the product might be a fluid, so it might be fluid products, so the fluid reacts with the solid B and then it forms a fluid product or it might form a solid product, it might form a solid product or it might form a combination of fluid and solid products together. So any of these three possibilities, any of these three can occur because of a fluid solid non-catalytic reaction. So now if we look at what happens exactly, suppose if there is a catalytic core, suppose if there is a solid and it is a non-catalytic solid, it is a non-catalytic solid core which is present, then suppose if this is B, B is that particular reactant and there might be A which is in the fluid stream which might be present around this particular solid, might be present around the solid, might be present around the solid and these, the species A now moves into the solid and then it reacts with the particular species B and then the product is now removed from this B, from this particular species, remember that it is a non-catalytic system. So now there are two possibilities, there are two possibilities or two different classes of such type of fluid solid non-catalytic systems. So I could call it, I call it two modes, there are two modes that are possible, two particular modes and this mode, this classification is based on the, it is based on the size of the, size of the solid core which is actually participating in the heterogeneous reaction. So one mode is where the size remains constant, one mode is where the size of the, size of the solid core which actually remains constant and another one where the size actually shrinks that is the particle actually shrinks. So there is a change in the size of the solid particle. So there are two modes based on the classification based on size, one is the, one is where the size remains constant, the other one is where the size actually shrinks. Here it is assumed that the density of the solid reactant and product are not very different. It should be noted that if the densities are different then the particle can also expand. So let us, let us look a little bit more deeply into these two modes. So the first mode is where the solid particles, solid particles remain unchanged in size, this is remain. So the size remains same which means that it is a same size all through. The same size all through the reaction is being conducted. So now suppose if I have a solid, suppose if I take a spherical particle. Now this is the initial, initial unreacted solid core. So let us say at time t equal to 0. Now as the reaction proceeds suppose there is a gas which is present outside and the gas actually gets into the solid and there is a certain reaction which is happening and after a certain time what happens is let us say that there is in this particular case, there is a, the size does not change and so there is some parts of the solid core which is actually undergone a reaction. So they might be, so it is basically partly reacted. So it is a partly reacted, remember that the size does not change. So if the radius of this core is r, the radius continues to remain as approximately the same size as the initial size of the particle. So this might be some time t1 which is greater than 0. Then as time further proceeds, as the reaction proceeds to further conversion as there is more conversion of the reactants then one can see that suppose if this is the size, this is the particle where remember that the size remains same, the size does not change in this particular mold. However there is more amount of reactant which gets converted. So this could be like a completely converted situation where t2 which is greater than t1. So as time proceeds, so this might be a complete conversion, this might be a complete conversion situation. So as time proceeds slowly the gases will go inside and then the reaction will start occurring from the outer periphery and then it will start going into the reactants will go inside and the reaction will proceed inside and the complete conversion will occur. However while the reaction occurs the size of the particle does not change. Now how is this possible? This particular mode is possible when the particles contain, particles contain lot of impurities. So these impurities are actually they are tightly bound to the solid core. So these impurities are tightly bound to the solid core and they remain as a non flaking per ash. So what happens is these impurities after the reaction is actually being conducted. So these impurities continue to stay inside the solid core and therefore it reserves the shape and size of the, size of the solid core and therefore these impurities they remain as a non flaking, they remain as a non flaking ash as the reaction proceeds. The other possibility how this particular mode can actually occur is where the products are actually solid products. If solid products are formed they are firmly bound to the solid core. So if the solid products form and they are very firmly bound to the core in which the heterogeneous reaction is occurring then the size does not change it actually preserves the size and pretty much the shape of the solid. So this particular, this aspect actually helps the solid core to retain its shape. So now let us look at a few examples of, few realistic examples of where under what kind of systems this mode has been observed. So the one good example is where the roasting of sulphide ores. So roasting of sulphide ores is basically a sulphide form for example zinc sulphide and roasting process is where the, there is a, there is an oxidation reaction. So the oxygen is, these zinc sulphide is exposed to oxygen under certain conditions and it lead to formation of the zinc oxides. So it is basically formation of zinc oxide from zinc sulphide and zinc oxide is a, zinc sulphide is actually a available in nature in solid form and the oxygen which is a fluid undergoes a non-catalytic reaction and it leads to the formation of the zinc oxide which also is a solid product. So therefore the nice example is where it is a zinc oxide preparation and the reaction scheme is two zinc sulphide and three oxygen molecules. They combine together to form two zinc oxide which is again in the solid form and then it leaves out sulphur dioxide. So that is the reaction scheme. So this particular example actually the reaction follows the first mode of fluid solid non-catalytic heterogeneous reaction type where the size of the zinc sulphide, particle sulphide ore that was used does not change much in the size. So the actual industrial process is the ore is taken and then it is crushed and then it is sized into a certain size and then it is put into a certain type of reactor where the oxygen, the air is being fed and then it reacts with zinc sulphide and then forms zinc oxide which is a solid product. Another example of that is the preparation of iron oxides and the way it is done is 4 times 4 ferric sulphide in the solid form that reacts with 11 molecules of oxygen and that gives 2 Fe2O3 solid and 8 sulphur dioxide. So that is another example of heterogeneous reaction which follows this first mode of fluid solid non-catalytic reaction. Another example is where we have magnetite which is Fe3O4 which is actually being reduced for the formation of the metal iron. So the second example is reducing oxides. So in the case of reducing oxides what happens is that the Fe3O4 which is actually present as in the solid form as a natural ore is actually, it is called the magnetite, what is done is it is actually crushed and then it is sized and then it is actually fed into a counter current fluidized bed reactor, continuous counter current fluidized bed reactor in the presence of hydrogen and this particular in the presence of hydrogen it gets reduced to forming the iron metal iron. So the reaction scheme is plus 4H2 which is again in the gas form that leads to the formation of 3 times Fe plus 4 times H2O. So that is the reaction scheme which is used for reducing the oxides. Another example is the formation of calcium cinamide where CaC2 which is basically the calcium carbide that is which is again for present in the solid form and that reacts with the nitrogen N2 and that leads to the formation of CaCN2 which is again in the solid form and then it also leaves amorphous carbon as deposits. So clearly in these examples there is a solid product which is formed which gets deposited onto the solid core in which the fluid solid non catalytic reaction actually occurs and this helps in preserving the size of the core. Another example is basically the plating of various metals. So for various purposes different metals are actually coated onto the surface using this method called electroplating and so in this plating where the original core does not change but then the reaction actually occurs in the fluid solid non catalytic reaction actually occurs without changing the size of the original solid core. So the second mode is basically where essentially the particles shrink. So there is a shrinking of the particles and in this particular case what happens is suppose if this is the initial unreacted core, if this is the initial unreacted core, so let us say at time t equal to 0, so as time proceeds the size of the particle changes because the solid is now being used and then it is converted into products and then it is being consumed. So there is a reduction in the size of the catalyst particle and that is at let us say time t1 greater than 0 and there is still unreacted core which is present inside. This is the unreacted core which is reduced in size and then as time further proceeds the size further reduces down. So this is let us say t2 greater than t1 that is a further time and then as time proceeds further it gets reduced into the solid core actually disappears. So this is because the solid which is present in the initial unreacted core is now almost completely consumed and so as time proceeds we can see that the core is slowly being consumed and then the core completely disappears. The particle can reduce in size even when there is a product layer around it. So there are several examples where this particular mode of heterogeneous reaction occurs and typically this occurs when there is reaction that involves carbonaceous materials. So typically reactions that involve carbonaceous material they generally follow this mode where the size of the solid core actually constantly decreases and then the core eventually disappears. So a nice example of that is the production of the producer gas and this is generally performed at air limiting conditions and so the reaction scheme is that carbon which is present in the solid form that reacts with oxygen which is in the gaseous form and that leaves to the formation of CO2 in the gas form. Now similarly another scheme which occurs simultaneously is carbon it reacts with oxygen to form carbon monoxide this again occurs simultaneously along with the first reaction and then another reaction which also occurs with is carbon when it reacts with CO2. So CO2 which is also formed by this reaction it again interacts with the unreacted carbon which is present in the solid core and that leads to formation of 2CO. So this is the reaction scheme for producer gas suppose if the reaction is conducted with in the presence of steam then it leads to the formation of CO2 and H2 which is the carbon gasification reaction and the scheme is the carbon and water. So steam is essentially the vapor form of water and that leads to CO and H2 and carbon in the presence of 2 water molecules that lead to formation of CO2 in the gas stream and 2 hydrogen molecules. So both these reactions typically occur in a situation where the solid core which is basically unreacted carbon when you start to start with it is slowly being consumed and then the carbon eventually disappears. So that is basically the mechanism that is been observed for in the producer gas production. So similar process actually occurs in the manufacture of carbon disulfide suppose if we look at carbon disulfide production we look at carbon disulfide production then the reaction scheme is carbon in its solid form reacts with 2 sulfur molecules which is again in the gas form and that leads to the formation of CS2 and this typically occurs at 750 degree to 1000 degree C. So that is at a very high temperature. So carbon in the solid form which is again present as a core solid core and sulfur which is the gas stream reacts to form the carbon disulfide and another example is the manufacture of sodium cyanide. So manufacture of sodium cyanide manufacture and the reaction scheme is sodium NaNH2 in the liquid form it reacts with the solid carbon solid core and that leads to the formation of NaCN in liquid and it liberates hydrogen. So that is the reaction scheme and it is typically conducted at 800 degree C. So that is another example where the core actually shrinks and eventually it disappears. So there are many other processes for which such kind of such mode of fluid solid non-catalytic reaction have been observed. So now so let us attempt to model this system, model the fluid solid heterogeneous reaction. So there are two types of models that have been proposed. So the first model is called the progressive conversion model, I call it PC model. So the first model is called the progressive conversion model and the second model is called the shrinking core model. So the shrinking core model is much more well known and more applicable to many systems than the progressive conversion model. However let us look at what, let us look into the details of what these two types of models are which to represent the fluid solid non-catalytic heterogeneous reactions. So let us start with PC model, the progressive conversion model. So progressive conversion model assumes that the reaction occurs everywhere, reaction occurs everywhere and not just that it occurs everywhere, it occurs everywhere at all times. So it assumes that all the time till complete conversion the reaction occurs everywhere in the solid core. So suppose if I, if we attempt to capture this assumption, what it suggests is that suppose if there is an initial core, suppose if this is the initial core that is at time t equal to 0, so that is the initial core which is filled with the solid reactant. Now as time proceeds reaction occurs in the solid core everywhere and let us say at time t1 which is greater than 0 and then at a further later time what happens is that the reaction occurs further and suppose if this is the solid core reaction occurs further and more reactant is being consumed, more solid is being consumed, again the reaction is actually occurring everywhere inside the reactor, however the rate of reaction that is occurring in different locations could in principle be different. So this might be at t2 which is greater than t1 and then at a further later time most of the reactants which are actually present in the solid core have actually been reacted. So suppose if the intensity of the red colour here suggests the extent of reaction at different times then at t3 which is greater than t2 one may assume that complete conversion has been occurred. So this might be the initials, initially when the solid is actually present everywhere in the core and this is basically when the complete conversion has occurred. Now suppose if we attempt to sketch the concentration profile in each of these time steps, so we can sketch it in the following way. Suppose if the radius of the particle is r and between 0 and r the concentration of the solid initially it reminds, it let us assume that it is uniform. So this is the concentration of solid, let us assume that it is uniform and then as time progresses at a later time, let us say that this is the initial concentration but because the reaction is occurring uniformly, reaction is occurring all through the particle, it may be non-uniform because the rates may be different in different locations. So we can sketch the profile in this way. So the concentration of the solids will attempt to decrease at every location inside the solid core and so let us say that at time t1 which is greater than 0, this is the kind of profile. Now at a later time what happens is that because the reaction is occurring everywhere inside the solid, so the concentration of the solid would actually decrease at every location inside the solid core. So therefore the kind of profile that one would expect is something like this where this is the end of the, this is basically r. At a much later time when the conversion is expected to be complete, so if this is the initial solid concentration and this is r, then the concentration of the solid inside the solid core will be almost 0. So clearly you can see that there is a progressive decrease in the concentration of the solid all through the solid core. So this is an important assumption that is actually made by the progressive conversion model. So next let us look at the shrinking core model. So the shrinking core model assumes that the reaction first occurs in the outer skin, reaction occurs in outer skin and then it assumes that the reaction zone actually moves inwards. So the first assumption also means that the reaction does not occur everywhere. The first assumption also means that the reaction does not occur everywhere in the solid core unlike what was assumed in the progressive conversion model and it also because of this proper this because the reaction first occurs in the outer skin the zone the reaction zone actually starts moving inward as the reaction proceeds and then once it moves as the reaction zone moves inside what it does is it leaves the leaves behind the completely converted material. So, it leaves behind the completely converted material and also some of the inert solids which may already be present in the solid core. So, the converted products are now assumed that it sticks back into the solid core and whatever inert solid which is unreacted they also will stick back in the solid core and so these two together is termed as ash. So, the ash in the solid core is basically a combination of the products which are actually completely converted material and also the inert solids which would have not reacted in the reaction. So, these two are called as ash products. Now, if you attempt to depict the concentration profiles if you attempt to include what is the concentration profile in the solid core in if you assume a shrinking core model is essentially it looks like this. Suppose if this is the solid core and let us say at some time as the reaction is started at some time if this is the unreacted core remember that the shrinking core model assumes the reaction first occurs at the outer skin and then the reaction front starts moving towards the inverse of the solid core. So, which means that at any time there will always be an small unreacted core until the conversion is complete which means that all the solid has been consumed. So, till all the solids have been consumed there will always be some location where the unreacted solids are present. So, let us say at some time t 1 which is greater than 0. So, this is the there is some amount of unreacted core which is present and the outside layer is called the ash layer. Now, as time proceeds as time proceeds then the scenario will be if this is the solid core which was started initially and the reaction has started this is the initial size of the solid core or is the initial size of the solid core. So, now the unreacted core will start shrinking in size because the reaction front has started moving towards the center of the solid particle and there will be again ash which is present outside this. So, the radius of this particle this unreacted core if I call this r 1. So, this r 1 will be smaller than the radius of the unreacted core which was present at an earlier time. So, this is r naught small r naught. So, if that radius will be smaller than this r naught which means that the reaction front has moved inside and the solid which is present between r naught and r 1 has now reacted to form products. So, this is at let us say time t 2 which is greater than t 1 and at a later time. So, this is the original size of the solid core which is which in which the reaction was started then the unreacted core will now reduce to a very small size until it completely did disappears. So, this is the unreacted core and everything outside is basically ash now where it is pretty much converted and this might be at a at a later time t 3 which is greater than t 2. So, now if we look at the concentration profile for this situation. So, if this is 0. So, at this time the the the solids are present only in this unreacted core. So, therefore, the concentration of the solids will essentially be the concentration of the solids which was present initially everywhere. However, it is present only. So, this is the initial concentration. So, this is the initial concentration of the solids then at this particular time t 1 the solids are present only in this location between 0 and r naught. However, the concentration will be as much as the initial concentration itself. And then at the next time step. So, the region where the unreacted solids are present is now going to shrink. However, the concentration in that location will continue to remain as that of the initial concentration. Then if I once again plot at a much later time this particular this region will start shrinking. So, this is at 0. So, 0 is the centre of the sphere and so the unreacted solids are now present only in the in this very small region which is around the centre of this particular sphere. If you assume that the solid core in which the reaction is started is actually a spherical particle. So, what this this model assumes that at any time there always exists at any time unreacted core always exists. So, this might be very small or very big, but at any time there is always unreacted core that exists. So, now there have been experiments that have been performed where the catalyst particle where the solid core after the reaction has occurred which is again a non catalytic reaction has actually been broke has been cut open to check the extent of carbon that has been consumed. So, if it is a carbonaceous reaction after some certain conversion at different times the carbon can actually be cut open and then looked at what is the extent of consumption of carbonaceous material towards forming the corresponding product. So, it has been observed that for that several systems actually follow this shrinking core model. So, now so let us look a little bit more deep into what exactly happens in the shrinking core model because it is actually very relevant that many systems actually follow a shrinking core model. Suppose if you assume that it is a spherical particle suppose if you assume spherical particle then if this is the initial core. So, let us say that there is a spherical particle and of radius r. So, that is the initial core that was actually started with and let us assume that the unreacted core is actually present somewhere here. So, that is again concentric with the let us assume that it is concentric with the periphery of the spherical particle and then if you assume that there is a gas which is present outside then there will be a small gas film. So, this gas film will exist around the spherical particle and the gas which is present in the bulk. So, let us say this is bulk gas stream this is bulk and this is the ash layer and this is the unreacted core. So, if species A let us say if this is the species A is present in the gas stream. So, the species A now first has to go into the gas film and then it has to go into the surface of solid core. So, this is the solid core surface. So, A has to now diffuse into the surface of this solid core and then it has to diffuse through the ash layer and it has to reach the unreacted core. So, it has to reach the location of the unreacted core. Remember that in this ash region there is no solid reactant which is present and therefore no reaction is happening in the ash layer. So, the reaction actually happens only at the periphery of this unreacted core. When the reaction occurs the product which is formed has now has to diffuse through the ash layer and then it has to diffuse from the ash layer into the gas stream gas film and then to the bulk gas. So, let us say that this is the product. So, the diffusion of the product. So, once the bulk species diffuses through the ash and then it reaches the surface of the unreacted core the product is formed and then the diffusion occurs and then the product leaves the spherical particle. So, this is the scheme that actually occurs in the this is how the this is the process that describes the transfer of reactants into the solid core and then the occurrence of the reaction in the unreacted core and the transport of the products back into the gas stream in the shrinking core type model. So, clearly there are three processes which are actually occurring. So, the first process is diffusion through gas film and then the second process is diffusion through ash and the third process is reaction at the interface of unreacted core and ash. So, therefore, the extent of reaction that is the overall conversion of the of the of the reactant of the solid depends upon which of these three processes a rate limiting step. So, it could be that the diffusion through the gas film is an extremely slow step. So, remember that the rate limiting step is essentially the one which is the slowest step and because it is the slowest step it controls the actual reaction which is occurring in the it control it controls the overall conversion. So, suppose if the diffusion of the gas from the bulk to the surface of the core through the gas film if that is an extremely slow process when can it be slow if the diffusivity if the effective diffusivity of the species through the gas stream to reach the surface of the solid core if that is that is extremely small if the if the diffusivity is very low then that can in principle serve as a as a rate controlling step. Now, particularly when the reaction is faster than the diffusion step. Now, the second possibility is where the diffusion through the ash. So, there is an unreacted core and there is ash which is surrounding the unreacted core. So, the reactants have to now diffuse from the surface of the core through the ash and it has to reach the location where the unreacted core is present. So, let me repeat the reactants have to actually diffuse from the periphery of the solid particle through the ash and it has to reach the surface of the unreacted core. So, if that diffusion of this reactant species through the ash layer if that is that is very slow then that can in principle be a limiting step. Now, the third possibility is where the reaction which is occurring at the periphery of the unreacted core. So, if that is an extremely slow if the net if the specific reaction constant is small then it can in principle actually be controlling and all the other processes can actually be faster. So, essentially there are 3 possibilities where so it could be diffusion controlling diffusion through ash layer controlling or reaction controlling. So, let us try to model each if model the system if the rate controlling step is any of these 3. So, let us start with the first one where we assume that the diffusion through gas film is the controlling step. Let us assume that diffusion through gas film is rate controlling. Now, what happens here? Suppose if this is the this is the original particle with which the reaction was initiated and if the radius of this particle is let us say r and if the unreacted core is let us say at. So, let us say that the initial radius is r naught and the unreacted core the radius of the unreacted core which is depicted in red color here if that is r. So, that is the at any time t at any time t. So, the radius of the unreacted core is now r and the original radius of the particle with in which the reaction was initiated is r naught and let us assume that there is a gas phase which is actually present around it. So, this is the gas phase and this is the unreacted core. Now, if we attempt to intuit what is going to be the concentration profile of let us say species A which is in the gaseous form and it now diffuses through to actually react with the solid in the in this particle. So, let us now try to intuit what is going to be the concentration of the species. So, suppose if this is the concentration of species A then because we assume that diffusion through the gas film is controlling the diffusion process in the gas film is going to be much slower than the other two processes which is actually occurring inside the solid core. So, therefore, we expect that there will be a concentration suppose if this is the this is the bulk concentration C A G is the bulk concentration is the bulk concentration of species A then if this we can expect that there will be a concentration gradient in the gas film. So, this is at the center of the that is the center of the core and this is r naught that is r naught and again this is r naught. So, we can expect that there will be concentration gradient outside the core and there will be no concentration gradient inside the core. And this is because as soon as the gas species actually enters the core the reaction the reactant A will be immediately consumed because it is a very fast process compared to the diffusion rate rate of diffusion of species A to the surface of the catalyst. So, the concentration of the species inside the core is expected to be 0. So, that is the concentration inside the of the surface and also the concentration of the species inside the core. So, now if we know this then how do we attempt to characterize this. So, we need to. So, if we need to if we need to understand the dynamics of the concentration change for this system we need to we need to start with a mole balance. So, suppose if the if the diffusion through the gas film is basically the limiting step then the driving force the driving force for the species to get transported from the bulk to catalyst surface the core surface surface of the solid core not catalyst that will be essentially be C A G which is the concentration of the gas concentration of the species in the bulk gas minus the concentration of the species in the surface. But we know that because the reaction and the diffusion through the ash layer is faster compared to the diffusion of the species through the gas phase C A gas is actually expected to be 0. So, therefore, the driving force is essentially the concentration of the species in the gas phase itself. Now, suppose if we assume that the gas phase bulk concentration is constant suppose if we assume that this is constant is reasonable because in a real system we may be able to feed the gas at a certain rate in order to maintain a constant concentration of that particular species in the gas phase. So, with this we can actually write a simple mole balance. Now, the rate at which rate of change of the number of moles of species A with respect to time that should be equal to the rate at which the species is coming in minus what at which the rate at which the moles are going out plus whatever is being generated. Now, the the only process which is actually occurring here is basically the transport of the species from the bulk on to the surface of the core. So, that is actually a mass transport process. So, suppose if we define a mass transport coefficient kg then we can write the rate as T N A. So, remember that A is the gas species which is actually moving from the bulk to the solid core where the reaction is occurring. So, D N A by D T where N A is the number of moles of A which is actually diffusing from the bulk into the surface to the surface of the core. And the reaction is occurring only inside the core and so there is no reaction in the gas phase. So, there is no reaction in gas filling. So, the only process which actually controls the rate of change of number of moles in the gas phase is essentially the transport of species from the bulk to the surface of the solid core because of diffusion. So, that should be equal to minus kg. Let us say that is the mass transport coefficient and typically the units are centimeters per second. And if S external is the surface area of the core external surface area of the core and that multiplied by the concentration of the species in the bulk phase. So, that is the mole balance which captures the rate of change of number of moles of species A in the gas phase. Now, because C A G is constant we can assume that this whole expression is basically a constant. The S A X is basically essentially the external surface area of the core and that is expected to be constant because we assume that the particle size is not changing significantly and then kg is the mass transport coefficient. There are several correlations which are actually available in order to estimate what are the what is the mass transport coefficient for transport of a particular species from a bulk phase to the surface of a solid core and C A G essentially can be maintained constant. So, therefore the right hand side of this balance is essentially a constant. So, suppose we assume that B is a solid which reacts with fluid A to form products. Then we can assume that there is an equimolar consumption. Suppose, if we assume that the B and A are basically under equimolar condition the reaction is occurring under equimolar conditions. Then the number of moles of A which is consumed because of the reaction should be equal to the number of moles of B which is consumed because of the reaction. Note that N A equal to N B is actually a special case and in general these two need not necessarily be equal. Now, why do we need this? Because we need to now monitor either the we can now monitor actually the amount of solid that is actually being consumed because of the reaction and that is what is of interest and this the mole balance that was written for the gas species in the gas film can actually be used in order to estimate what is the number of moles of the solid reactant that was actually being consumed because of the reaction and this is in the case of this when the diffusion in the gas film is actually controlling the overall reaction. So, therefore, if N A is equal to N B which is the number of moles of the solid that is being consumed in the reaction and from here we can rewrite the mole balance as D N B by D T equal to D N A by D T and that is equal to minus kg which is the mass transport coefficient into the external surface area multiplied by the concentration of the species in the gas phase. Now, we know that we know that the number of moles of the solid which is actually present in the in the solid is essentially given by the density of the species rho B multiplied by the volume and so we can rewrite this as D by D T rho B into volume that should be equal to so the number of moles of solid which is present in the core can actually be related to the volume of the core using this expression N B equal to density of the solid which is present in the core multiplied by the volume of the core at any instant time T. So, remember that the volume changes as a function of time because as the reaction occurs the core is now shrinking and so the volume of the unreacted or volume of the unreacted solid core changes with time. So, if volume of because we assume that it is a spherical particle the volume is given by 4 by 3 pi r cube. So, remember that this r is a function of time because as time progresses the core actually reduced shrinks in size and so the radius of the unreacted material is actually a function of time. So, now if we plug in this expression the mole balance we will find that rho B D by D T 4 by 3 pi r cube r is a function of time that should be equal to minus kg which is the mass transport coefficient into the surface area of the external surface area of the of the initial particle spherical particle that was chosen multiplied by the bulk concentration of this reactant species A. And so that can simply be rewritten as 4 by 3 pi 3 r square rho B D r by D T C A G that will be equal to r square D r by D T that is equal to minus kg into C A G divided by 4 pi into rho B. So, that is the balance. So, that describes how the radius of the unreacted core actually changes with time. Now, this is directly correlated to the number of moles or the amount of solid that is actually undergoing a reaction that is the amount of solid that is actually undergone the reaction at a certain time. So now, if at r equal to capital R so the ash is now present. So, if the we assume that the outer radius of the core is r naught and if the ash is actually present ash is present between r and r naught between r and r naught we can now integrate this expression between r naught and r D r by D T and that should be equal to minus kg divided by rho B into integral D D T. So, that is the so now this actually express this integration will now provide an expression for finding what is the radius of the unreacted core as a function of time. So, the integration suggests that it will be 1 by 3 that is equal to into rho B. So, this expression provides the relationship between the radius of the unreacted core as a function of time. So, if there is if the if the diffusion in the gas phase is actually controlling the overall reaction overall conversion then we can find out how the unreacted core actually decreases in size as the reaction proceeds. So, what we have seen in today's lecture is essentially introduction about the different what is the fluid solid non catalytic system and what are the different modes based on the size of the size based on the change in the size of the solid core that is participating in this fluid solid non catalytic heterogeneous reaction. And then we looked at two different models the progressive conversion and the shrinking core model a description of these two models and we have attempted to model the model the we have we have attempted to write a balance in order to find out what is the size of the unreacted solid core as a function of time for a shrinking core model when the diffusion in the gas phase is actually controlling. Thank you.