 we will start now right yeah so in the last class we have drawn the fluidized bed with marometer and all that right plus we also have the delta p graph right these two so now delta p graph we have to still draw this is second item pressure drop due to solids this is the graph what we have also drawn and I have given the diagram where how to measure the pressure drop right so the total pressure drop which you measure directly from the marometer is delta p t equal to delta p s plus delta p d plus walls usually walls are neglected but anyway we will keep that delta p s equal to delta p t minus delta p d plus delta p w okay the equation numbers we have so this is the first equation right yeah first one this is second equation okay so what we measure is this and before taking the actual delta p measurement with solids without solids you have to measure the pressure drop you know in the picture I do not want to draw it again in the picture before putting the solids you have only walls walls are there walls and distributor plate okay you just measure at various velocities what will be the delta p and then correspondingly you subtract from every pressure okay for every flow rate you can subtract these values and then you will get this and when you plot this only the pressure drop after fluidization will be constant otherwise as I mentioned in the last class also there will be slight increase in the pressure drop due to the distributor plate walls will not contribute much so okay but distributor plate will have tremendous pressure drop you know I hope you remember your Arianis equation not Arianis equation orifice equation orifice equation to measure pressure drop see we have single orifice where you measure the pressure drops okay and convert that into the flow rates okay so the same u square also will come here so that is why pressure drop will tremendously increase as the velocity is increasing that we have to subtract that one has to be very clear so once I have this then we have to find out is it I mean is there an equation which we can also derive for finding out the pressure drop due to solids okay is there an equation there is an equation very simple very good equation because weight per unit area is pressure drop and at minimum fluidization velocity the entire bed is supported by the drag force so the drag force and then weight of the solids if you are able to balance you are going to get the equation right so that equation you will check with the experimental data so that equation is at minimum fluidization velocity at umf that means at the point of fluidization you have the drag force exerted by fluid by solids exerted exerted by fluid on solids exerted by fluid on solids this must be equal to weight of the particles weight of the particles which is nothing but you know m into g mass into acceleration duty gravity okay that you just remember okay yeah so now this drag force how do you measure drag force yeah how do you measure it what you are telling is calculation Stokes law will give you calculated value right CD equal to 24 by N R E okay not N R I N R E N R I is different okay yeah R R E simply how do you measure is there any measurement possible for drag force how do you measure friction you have done it through pipes how do you measure friction Kavya always Kavya first attack just you are coming yeah take rest yeah shaker how do you measure velocity change I change the velocity what do you measure moody's charge you measure you read you do not measure measuring is doing experiment I say oh my god yeah but how do you measure energy loss right how do you measure total energy minus kinetic energy measurement I am asking where do you calculate yeah measure the pressure drop I say see this seems to be very simple questions are no they are not simple questions I think you do not have any idea at all you know how to measure the pressure drag force or friction factor or friction because you cannot directly put something there and then measure it no it is only indirect measurement which is used I mean which is measured by pressure okay even here we do the same thing drag force I think you know we do not really think beyond certain point I do not know what you know again if I go there I will time will go okay so the drag force is measured in terms of pressure drop okay so when I write the equation again of course in terms of words this will be pressure drop across the bed pressure drop across the bed in fact I gave you the clue how do you measure friction factor right friction factor all of you would have done the experiment may be Savitha would have not done okay and Kalpana what should I call you Kalpana or Sarojini okay yeah Kalpana would have not done but all others would have done it Rajithya would have not done it no seems like okay anyway so pressure drop across the bed and of course the cross sectional area that will be there and okay this is a length I do not know may be yeah cross sectional area equal to because drag force no weight of the bed okay it is nothing but of course weight of the bed mass into gravity how do you calculate mass of the bed volume of the bed I know yeah and then density okay so volume of bed and density and this density is the apparent density because already you have some other fluid and the solids would have lost some weight correct no because of the buoyancy okay into of course G is there okay so this is the one if I write the equation this will be delta of P and cross sectional area is A and volume of the bed is nothing but actually volume of the bed means here only what you are balancing is solids right so volume of the bed means we are talking about I think better write that means we feel the entire bed yeah solids so I have some the volume of the bed into 1 minus epsilon of that will be weight of the solids okay where epsilon is divided so that is what so that will be if I take what did I did not put anything there no okay let me put that one as HMF because we are talking about minimum fluidization velocity right into cross sectional area that is the volume into 1 minus epsilon MF because everything get minimum fluidization velocity right yeah this multiplied by density which is nothing but rho G or rho F to be general for liquid also that is possible and into G is very simple only yeah so then we have delta P by HMF is 1 minus epsilon MF rho S rho F into G so this equation if I write this also numbers 3 4 5 6 yeah this is is called the fluidization equation okay yeah so can I calculate from this the pressure drop almost equal to yes exactly so that is what I thought of asking you but you yourself asked and then answered that is right so at minimum fluidization velocity we know that the bed is almost packed so that is why except that a little bit of change and some people who would like to take that because there are lot of errors involved in these measurements because first of all your solids are not uniformly one size particles because always use sieves in reality right sieves will have some in passing through some retaining on so you will have always a little bit of distribution and that is starting point for all the errors in fluidized bed and 1 minus epsilon also many times we take as packed bed porosity or if there is already measured value available because some people in the literature some values are available that epsilon MF you can take so that is available rho S I know because solids which I am taking I know rho F I know whether I am taking liquid or gas and the densities we know and G we know we can calculate delta P provided we know the HF okay or HMF HMF is nothing but again I know same thing when you take that as packed bed porosity that also will be packed bed height so that will be the one good so you can calculate this in fact and we will take so this will give you an idea so I told you sometime back I think it was Anurag who was asking me the pressure drops are in a fluidized bed may be higher than the packed bed see the maximum pressure drop what you get is only this here due to solids right but distributor will have you know always increasing pressure drop as you go with that one but the same thing is also possible for packed bed packed bed also has the distributor plate right but there is slight difference between packed bed distributor plate and this distributor plate only in surface in free area which one will be more which one will be less free area for the distributor see you have to even you have to support the solids right so you have a perforated plate with holes okay so for packed bed we use also we have to use perforated plate otherwise fluid cannot enter now yes and for fluidized bed also you have to use free area in your opinion in which bed less free area in which bed more free area more free area means more amount of liquid can go through or the fluid can go through less free area means very less why answer is right exactly not that is the reason but I think can you think something it is not that because already bed has pressure drop so let me reduce the pressure drop here the same thing we can also do here right the same thing we can also do here but in packed bed plug flow will automatically come because of even bed presence because the bed presence will make the profile almost horizontal the presence of bed itself that is why that is a good thing even that is why we say for packed bed when do you get plug flow or when do you get turbulence you know almost flat velocity profile what is the Reynolds numbers I told you already this yeah and whereas for tubular flow single tube without anything inside it will be 40,000 50,000 to get almost flat velocity profile okay so that is why okay so that is not a problem but you know what Shaker set is right you will use more for packed bed okay not only to reduce the pressure drop to allow more and more throughput but if I use very large free area for fluidization fluidization itself will not take place it needs some pressure drop that means it needs some jet activity through the pores the fluid should come like jets okay so that will actually lift the entire bed for example you can take cloth you know filter cloths are there you can take cloth fix it and also you can fluidize you will never get fluidization you can get mesh what your mother is using you know to make flour and all that what we are also using sieves here so if you put just sieves you will not get fluidization you will get fluidization much much higher velocity okay so that is why actually the design next point is actually the pressure drop due to the perforated plate itself how much pressure drop you allow you should allow okay so that is why this is pressure drop yes I have been using yeah pressure drop due to solids so this equation which we have to use for measuring the I mean for calculating pressure drop and then it is very simple also particularly if I have gas phase this is almost zero when compared to this solids so it is actually delta P S you know that equation becomes even then it is not much complicated so one can easily calculate what will be delta P S and normally what we do is when you are doing this experiment we also draw this line where it is actually where that line is okay because I can calculate no delta P S I can calculate from that so delta that delta P S value I just draw there just for my reference right if this curve shows me something like this you know this is the pressure drop this is delta P S this is U let me say that I have like this what do you say about this graph is there anything wrong with the fluidized bed fluidization yeah that means not fluidization all the particles are not supported by by the fluid only some particles are fluidized some particles are not are not fluidizing that is very important point you can find out really you know that means the other part will be dead space there is no use of that right and sometimes you may get also some kind of you know like this like this that also may go up even if you plot for example exactly delta P S removing your pressure drop due to distributed plate even then you will get sometimes this may be difficult for you this may be due to this slightly increasing and in fact it is not constant it will be fluctuating up and down that pressure drop that is because of slugging because of slugging so this may be due to dead space that means all the particles are not fluidized and this is due to slugging what happens during slugging the entire cross sectional area is occupied by the gas and solids above they are just pushed up right so then the pressure drop may be more there because entire thing is just pushed up right and generally very narrow cross sectional area right so at that time when it goes up and then all the solids can be at one point they cannot be supported the slug will break and then entire bed will fall at that time pressure drop will be fluctuating up and down so these simple things one can see from the graphs and then find out what is the disease for this particular fluidized bed it is like doctors you know they will ask you to take some BP test or some some other test and then they see the results or graphs and then find out whether what disease you have similarly you are also doctors for the reactors or any equipment you can just find out by drawing or by looking at the you know some measurements and then find out whether something wrong with that or it is beautifully fluidizing so all these simple things will be there when you just plot this okay good so the next point is this is fine this is the equation this you do not please forget I think it is very for the people who have gone through fluidization you should remember this equation it is not very difficult to remember it is just weight of the bed okay supported by the gas okay so that is this in terms of pressure drop this is the equation so the next point is pressure drop due to perforated plate how much pressure drop we can give for the plate right so the next point if I write here 2 3 you know 3 pressure drop due to perforated plate or distributor perforated plate or distributor of course if you go deeper and deeper there will be lot of discussion about the distributor itself because that is the starting point for any fluidization and also solid particles so that is why solid particles have been categorized into the Geldatz ABCD groups then at least we will have an idea what kind of fluidization we may expect and here distributors various distributors have been used they use simplest one is perforated plate okay then they can also use bubble caps you know bubble caps probably bubble cap distributors how do they look look like it goes through that it won't open you know the cap is permanent so it will take the tortuous path like this and then again comes out why why do they require that not because of that it is not because of more contact time more contact time you can go to packet bytes think some more why do they put that bubble caps other one is see plate right even for distillation see plates they use yeah so why they don't prefer see plates but I think you know see plates will give you more much more bubble interaction because small small see is you know see means the perforations are smaller you generate large amount of bubbles vapor flowing through that so you will have more surface area more transfer but still we put sometimes bubble caps anyone I think any idea why do you need more pressure drop there actually we should have less pressure drop there no in distillation columns I am telling how can you operate I think in fact it that gives much more pressure drop at high velocities okay when you have bubble cap how this liquid is coming I am talking about distillation how the liquid is coming how the gas is going where is the gas going where is the liquid is liquid going because liquid has to come down from top plate to the bottom plate gas has to go from bottom plate to top plate yeah to the down comeer okay so there is always a way through the down comeer so liquid will come down there it comes through the down comeer that's not a problem for that see plate is better because see plate contains lot of small small holes right so when vapor is going through that it generates large number of small bubbles contact them is to some extent okay it's not mainly contact time you heard of dumping weeping and what is the problem with see plates to avoid meeping to make liquid happy you know because it should not weep no it should not weep so to make it happy you put bubble caps now what will happen happily it will go through the through the down comeers because you have to provide the down comeers when you put the bubble cap otherwise if you have see plates the liquid has to come through the same sieves gas also has to go through the same sieves so wonderful information is also there on that because sometimes even instabilities due to interaction between the bubbles and then liquid also may happen interaction I have seen not used for distillation for some other column turbulent bed contactor I have used and I also used perforated plates and then when liquid is coming from the top and under some conditions the entire liquid will be moving like this inside the bed my diameter was 10 inches a slightly bigger column okay it will be now if you imagine this kind of movement in the industry where they have 2 meters diameter okay it will have tremendous pressure the entire column also may be moving this way that way if you are not properly supporting it okay so that is because the interaction between the bubbles to liquid to avoid that for happy smooth flow bubble caps are better or see plates with down comeers also are there okay but see plates with down comeers still it is not good because some of the liquid is definitely coming through the sieves you cannot avoid that same problem here why do why we need why some people use distributor with bubble caps is to avoid solids flow here perforations perforated plate is the best one see plate is the best one the and sometimes what we do is over the sieve plate we will put mesh very fine mesh so that it won't create any pressure drop but it will not allow solids to fall through when they are falling through it may block even sieves so then you have that you know that whole totally not used for the flow then you have the dead space and like that more if you have more dead space it will create you see everything is important you know when you are designing a any equipment okay so that is why the general thumb ball is of course there are many many thumb rules for distributor design I am talking about this is one bubble caps can be used sieve plates can be used and also there is porous distributors which normally we never use for distillation or absorption we don't use because pressure drop will be tremendous okay porous you have seen the porous distributors it is like membranes porous distributor what they do is they take very fine powder and then compress it as a plate they use copper shots okay they can also use lead shots they would not use copper shots are very frequently used alumina glass you should have seen this I think you would have not noticed that when you are doing in physical chemistry lab they use for filtration you know sometimes not the filter paper but sometimes they use the glass frit they call that is porous distributor that is what exactly the same thing is used and that frit has given porosity you can ask for 10 percent 20 percent 30 percent that frits I mean what you have seen may be one inch very small one what you have done in the chemistry laboratory okay so that is what what we are I am telling you that also can be used that is very widely used because we need some kind of pressure drop across the distributor for fluidization that only gives sufficient energy because that will increase the you know the flow rate velocity okay velocity and what we are calling minimum fluidization velocity is the superficial velocity only we are not talking about each and every point what is the velocity in interstitial velocity we are not talking why because we cannot measure that very accurately same thing even with packet bed okay in packet beds we use whether a distillation column or any other use for packet beds when you are using packet beds for a certain process we never use interstitial velocity why because across any cross section the pressure the voidage itself varies right so that means somewhere I have slightly more velocity somewhere I have less velocity all this I can never take into consideration for the design even though we take another average average of all this voidage and sometimes we calculate and then tell people that this is the actual velocity going through the packet bed is superficial velocity divided by epsilon voidage that will give me the interstitial velocity okay that is only for some kind of you know satisfaction that okay I am I am also able to know what is now the interstitial velocity other than that if you look into many correlations most of the correlations will not give you this epsilon in the correlation the velocity Reynolds number they give for packet beds so most of the time they use only superficial velocity because which is easily measurable so that is the reason why superficial velocity cannot be because it is not simply manageable you cannot and that also varies from point to point everywhere in the bed unless you go for what are called structured packings I do not know whether you heard of them or not random pocket what normally we do in industry is random packing just take the solids and then just dump it but there are structured packings where I mean whatever you know like this like this like this also know like this and then you put the entire column with that you know may be 1 meter 1 meter 1 meter and then given diameter may be 1 meter diameter total may be 20 meters so for every 1 meter you put 1 piece and that is connected and you can see very beautiful structure in those packing those in the industrially these structured packings are also available what is the use of structured packing is that I know very clearly what is the surface area I am going to get that is all more clarity random packing I do not know unless you measure it and every time you change packing every time it changes right so that is the reason why random packings we use okay so the pressure drop due to perforated plate we will take just 2 plates commonly used one is the the sieve plate perforated plate and the other one is porous plate porous distributor and perforated plate distributor let me write that porous distributor and perforated distributor these 2 are widely used okay I think there are some thumb rules where the pressure drop should be around 0.2 to 0.4 of delta P S that is the general thumb rule delta P S by delta P B one of the oldest thumb rules is that 0.2 to 0.4 of delta okay that is all oh sorry the other one sorry this is delta P D yeah delta P D by delta P S okay this is what normally the thumb rule so this thumb rule was observed only by experiments so you get good good fluidization when you use this kind of values that means the pressure drop should be 20 percent of pressure drop due to solids which you can estimate from this you can calculate from this and I do not know whether you have designed who taught you process design equipment design oh yeah as a given the perforated plate design no that is simple one I think I will tell him I think he also may be semester is over at least for your juniors you know perforated plate design also he is a very very beautiful design okay how do you choose the diameter of the perforation itself how many perforations should be there how many perforations will be there automatically comes once you know the diameter of the perforation and once you know the total free area that fixes but what do you put I think he is the pitch whether it is triangular pitch or square pitch so what do you prefer generally why it is only to allow he remembers that for heat exchangers so trying to extend very good nothing wrong nothing wrong you can accommodate more number of perforations per unit area that is the reason why we go for that okay so the same thing even for heat exchangers but that is not good because when you for heat exchangers when you talk about cleaning square pitch is better because you will have at least some more place space in between so where you can clean it and then you know once in a while you have to clean it otherwise it will be falling at all that so heat exchange may not be there after some time so this is the general thumb rule this is general thumb rule but I want to take you slightly away from this thumb rule also because later people have used again the derivation is there but anyway I have also not seen the derivation but that equation that is used now later which also comes to this after some after under special conditions Delta P D by Delta P S should be greater than equal to H F height of the fluidizer bed divided by H M F at minimum fluidization velocity minus 1 this one is 1 minus U M F by U not to the power of N that is the equation 7 when N is a constant so this is the equation they try to use and then for porous plates for porous plates N equal to 1 and for perforated plates N equal to 2 again they do the experiments and then try to find out what is the exponent and all that okay so this is slightly some theory you know jet velocities and all that they will take and then they will try to derive this here of course one should know what is height H F H F is the height of the fluidizer bed there are some equations which relates H M F and H F okay beautiful relation is for particulate fluidization can you imagine that relation we have H P packet bed and the corresponding porosity is epsilon P right then it has fluidized to certain level where of course packet bed also you can take as M F minimum fluidization velocity right epsilon M F and at some other high velocities you have H F and epsilon F what should be the relation between these H F equal to what will happen to 1 minus epsilon F it is the volume balance it is the volume balance right so when you have the packet bed you have this height when you have almost minimum fluidization velocity yeah H F into 1 minus epsilon F equal to H M F into 1 minus epsilon M F that is simply volume balance you know height changes okay so that is why one can find out if it is very very good fluidization particulate fluidization that means volume is nicely expanding but in gas solid fluidization that will not work why the bubbles will destroy the height and you do not know where the bubbles form where the bubble break so that is why the overall height is not exactly like with this relationship but in the absence of anything we will also take that particular relation what I have told you sometimes we will use that so of course this we know M F why you not you know because you calculate M F let us say 1 meter per second you decide in the beginning itself that I want to use 5 meters per second that means 5 times 3 times okay or 20 times M F possible in industry particularly even 20 times M F also is used very easily in fact most of the time it is 22 or 10 to 40 times anywhere in between we use industrially okay because of large amount of solids are there distributor you cannot perfectly design even if you design perfectly the distributor gas flow is not uniform through all these perforations so to somehow you have to fluidize the entire bed so that is why try to use more and more gas okay so that is the reason why you know that kind of long large values are given that range 10 to 40 okay good so this is the one equation and yeah there is another thumb rule saying that you know if you are very close to M F you just use only 0.15 those are all details I think which we do not have to discuss that right this is what you can just remember for porous plates and perforated plates this is the equation which can be used to calculate delta P D because delta P S you know from this equation and using delta P D if it is a perforated distributor there is a specific A of calculating the perforated holes that you have to decide perforated holes okay then free area you have to decide normally free area we use here around 10 to 15 percent 10 to 15 percent okay free area good so I think this is fine yeah in fact you get this thumb rule if you take yeah this one if you take this ratio H F by H M F as 1.2 to 1.4 which is used I mean which is the height you know let me explain H F by H M F for many gas solid fluidized beds will be around 1.2 to 1.4 times L M F not able to follow sorry H F by H M F many times in the industry it will be 1.2 to 1.4 times for gas solid beds where you have bubbles and all that it cannot beautifully expand it will expand only when you are going for for the velocities beyond terminal velocities then only starts expanding and comes out of the bed okay so when you have this kind of values you substitute here for perforated plates and you know you will get this thumb rule because originally thumb rules are developed for perforated plates because easiest one is perforated plate so that is the reason and of course people saw that when it is a lot of leak is there two perforated plates they put slightly eccentric it is not the same holes but slightly eccentric then the pressure drop will be tremendous what is the problem there lot of powder will come and then stay in between these two plates okay industry is hell to run because I think everywhere you will have the problems good okay so that is the one and the next one is yeah Nila what is the next point this is 3 no that is 3 yeah 4 U M F minimum fluidization velocity how do you find out minimum fluidization velocity minimum fluidization velocity U M F nice ideas have come you know when you look at this graph okay I have written here fluidized bed fluidization before that what should I write here what is that region so at this point what is happening the pressure drop yeah the pressure drop at this point the packet bed pressure drop must be equal to fluidized bed pressure drop what is fluidized bed pressure drop equation 3 what is packet bed pressure drop remember name so at the point of minimum fluidization velocity simply we balance the pressure drop this equation and also Ergun equation Ergun equation also has delta P by L or LP or HP packet bed here H M F equal to HP because at the point of minimum fluidization velocity so simply we balance this and then that and then and Ergun equation has velocity terms you have 2 in fact quadratic equation yeah U and U square right so then it is a quadratic equation you solve it and then you will get the minimum fluidization velocity because you are balancing at point of minimum fluidization velocity that we will do quickly and before that no one asked me why this particular height and then falling and slightly above and then almost constant why no one asked me what will happen at that point why it is increasing and then again falling right you are on the correct track yeah so it needs a little bit of higher energy to break that inter particle forces and once they are broken then it is easy again so the particles will not have you know the gas need not have that much energy so that is the reason why it is a little bit falling and then comes down and if you do that if I do the experiment this is I have taken the bed put the solids and then slowly increasing the velocity in this direction from lower value to higher value but I can also do the experiment I can start at the highest value possible right not terminal velocity somewhere here terminal velocity means all these solids will go away right so somewhere here if I start and then slowly decrease I do not see this why abuse the inter particle that forces are not there because you have already broken them in the beginning itself okay so take it to higher velocities and then do it then you will get almost like this of course there will be slight hysteresis a little bit down okay but anyway I think you know you will get almost horizontal like this in the beginning that is when you are going in the 0 to smaller velocities to higher velocities okay that is why okay not lift the bed why it is why it has to lift the bed because all the particles are coming together if each particle is intuitively present then the gas will exert the drag force on each and every particle so uniformly it will fluidize but when you have normally we use fine powders so it has that surface energy where it will come together and then cling together so that you have to break before fluidization so that is the reason why it goes slightly above right and this kind of very beautiful points you cannot see particularly by the way this is generally for gel dot a and gel dot b particles okay if you go to cohesive powders then you cannot exactly find out that nice curve we cannot that is why we do not want to fluidize group C particles cohesive particles but if it is absolutely necessary we have to do it right so then use either vibrations or some people even use not only vibrations you have this sound ultrasound also breaks this you know the inter particle forces and one can also do that many many techniques can be used good okay so that is what I thought I will just inform you there liquid solid depends on what kind of not definitely this so much because but still it can have slight peak there also okay yeah for liquid solid also it is possible good so minimum fluidization velocity at the point of fluidization yeah so I think it is unfit may be to ask you to remember Ergun's equation so I will just draw here at umf delta p s equal to delta p okay this also I can write per unit length h m f h p packed bed yeah at a minimum fluidization velocity okay where of course we are taking h m f equal to h p at that point so now I have to write here the Ergun's equation okay here delta p s that familiar equation is 1 minus epsilon m f rho s rho g into g equal to the other equation is 150 1 minus epsilon m square m f epsilon m f cubed mu u m f because at at u m f already balanced this is mu viscosity and phi s by d p whole square okay that whole square comes and here I have yeah 1 point 75 again I have here 1 minus epsilon m f by epsilon m f cubed how you get this cubes and all that you would have done it know in the derivation Ergun's equation derivation you get here rho okay I will write here this term rho f yeah rho f u m f square by phi s d p where phi s is velocity okay yeah so yeah that is the equation and when you rearrange this I may give this derivation to you okay when I when I rearrange this what you what do I get is very nice equation 1 point 75 by epsilon m f cubed phi s r m f square plus 150 1 minus epsilon m f by epsilon m f cube phi s square into r e m f equal to accommodation number oh I have to give the number 7 this is 8 this is 9 this is 10 okay where I have to write where r e m f equal to d p u m f rho by mu rho f by mu and Archimedes is that is already g d p cubed rho f rho s minus rho f by mu square okay so this is the Reynolds number sorry Reynolds number and Archimedes number good I think I will remove this yeah I think this is first given by a person called when and you will also write here when and you that is why it is called when and you equation I think 1948 or so reference yeah okay I do not have here but sometime like that yeah so that is the one and they were very smart people you know they were all excellent engineers at that time they want to use the equation as simple as possible okay so that is why what they did was they found out epsilon m f and phi s square for large number of particles okay sand sand you know if you take river sand it will be different it will take if you take sea sand it will be different which one will be more round river sand or sea sand river sand because the river sea continuously flowing but there it is only back and forth action okay and also limited okay most of the time but there river in Vijiwada may bring sand to Nilur okay so during that it is only flowing like this you know at the bottom of the river or Ganga for example you take you do not know the sand from Himalayas may be coming to Bay of Bengal particles so that is why it is more round so like that they have taken various particles alumina copper you know not silver silver is costly zinc and many many materials and they have formulated this group as 1 by phi s epsilon m f cube is approximately 14 and the other group that is 1 minus epsilon 1 minus epsilon m f divided by what is that yeah phi s square and epsilon m f cube is approximately yeah 11 so then this constant will become how much this constant will become how much this will become yeah 24 point 5 and this becomes 1650 correct yeah so now that is the quadratic equation of which you can easily solve now this is a constant 24 point 5 multiplied by r m f square using these two right and 150 into 11 that is 1650 okay so then if you solve that the equation what you get for minimum fluidization velocity is r m f equal to 33 point 7 whole square plus 0.04 08 archimedes number this whole thing under square root into 33 point 7 this is called Wenland equation even now very widely used equation very widely used for small particles okay very widely used for small particles and you know the delta x people where you know they want to still see whether this is equation correct or can you modify a little bit delta x research normally what we do okay finding out a little bit of plus side negative side and all that what they did was that they take this as k1 constant this as k2 another constant and then solve the differential equation in terms of you know k1 and k2 right in terms of k1 and k2 what you get here is that yeah okay what I am trying to tell is that this k1 and k2 will be the okay let me write the corresponding differential equation in terms of k1 k2 k1 r m f square plus k2 r m f minus ar equal to 0 number is this is the level okay so this is 12 good so if they solve that what they get is r m f equal to yeah so root of k2 by 2 k1 whole square plus r comedies by k1 okay this one minus k2 by 2 k1 capital so this is equation number 3 you know 13 where now what they did was they took all the data that is available for minimum velocity because they know r comedies number right and then take this ratios k1 k2 you know to fix this 1 by k1 and k2 by k1 so they try to fit the data using this ratio one ratio is here I have yeah one ratio is k2 by 2 k1 other ratio is 1 by k1 that is this and anyway this is 2 okay yeah so yeah yeah so now this they try to fix the data like this we have 1, 2, 3, 4, 5, 6, 7 correlations 7 correlations so all these 7 correlations I think you know the popular correlations this is the first one one and you for generally small particles there is another one called cheat ester correlation cheat ester correlation this is in 1984 if you actually the cheat ester is the phd student where he has his work is only to find out minimum velocity in the thesis I think he is from France so for large particles his correlations seems to be better large particles I have not mentioned what is large what is small normally around 100 microns if you have 200 400 less than 500 microns this seems to be the better one when an equation above that cheat tester equation seems to be better okay so with that I think minimum fluidization velocity is over then I think I have to do tomorrow the remaining parameters okay