 We are looking at advanced reaction engineering. The objective here is to comprehend logic of decision making for design, operation, evaluation of reacting systems. So, it is a very large subject in which we look at reactions as a central way of understanding what is happening. Let us look at some examples where we might find this subject of much value. Process industry, for example, we make organic chemicals, we make biochemicals, various kinds of organic chemicals, fermentation products and then these are industries where they start with a certain raw material to produce certain products. Our raw materials could come from agriculture, it come from forestry, it come from mining, it could come from oceans in the form of various ocean products, all of which we use to produce various chemicals. Now, there could also be another way by which we can look at reaction engineering is we want to understand agriculture in a larger way. We want to understand what is happening in our environment or maybe we are trying to look at how a medicine is performing on a certain organism, maybe human, maybe an animal. Similarly, how a certain medicine can interact with the environment and so on. So, these are all situations where reactions are so central to what is happening in the system. So, to say that reaction engineering gives you some fundamental principles to be able to deal with small systems or large systems. What is our methodology? Our methodology in this course would be by a large problem solving. That means, we try to convey principles to you through solving small, but carefully selected problems. So, that whether it is a process industry problem or from agriculture or environment or biology or medicine, each of these situations we will be able to comprehend by looking at these small, but very carefully selected problems. I mean, we have as you go along this course, we will be formulating various models. Models as we will always remember models are to be seen carefully. So, we always remember that whatever we do science is only about models of reality. We like to understand reality, we build models and we see how these models describe reality and if it describes well we accept and as we understand more and more we find that these models are not so useful. So, better and better models are required. So, the base point is that science is only about models of reality and to the extent the model describe reality we will use them when they not describe reality well we look for better models. So, this is the philosophy with which we will look at this course. Chemical industry has been around for last more than 200 years now, may be much longer. So, the kinds of equipments that we will see in the chemical industry is tubular vessels, tubular reactor so common in the petroleum industry, stirred vessels very common in organic chemical industry. Then you have this fluid beds which you see in cat cracking industry or the lottery kill is very common in lime industry and of course, steel making is be known for a very long time, blast furnaces are such equipments. On other words what we try to do in this course is to see how we can use our understanding of the chemical reaction to build and design equipments and see how they operate, how they can be operated better, how best we can optimize its operations. So, that we produce a products safely, productively, economically so that we deliver useful value to society. So, whether it is tubular vessels, stirred vessels, fluid beds, rotary kills, blast furnaces each of these choices are based on an understanding of the reaction that we are trying to conduct. Now, the criteria that we will use the criteria of choice of these reaction equipment of course, depends on the quality of the product we want to produce, the reliability with which we want to make this equipments work, the safety feature that are associated with it, the productivity we desire and the cost at which we would like to produce. Or in other words our choice will be dependent on the importance that we attach to these 5 qualities. Of course, say I mean it is not that we would compromise any of these qualities, the fact is that all of them are equally important and therefore, the criteria should be such that every one of these qualities or criteria are satisfied to the extent that is required for the process. The task in front of us can be design, the task can be operation, the task can be development of a reactor design for a new application. Our principles must apply to all of these at all times. So, we need to develop principles and methods, so that we are able to address any of these situations that we would like to handle. So, to put this in the perspective, this course on advanced reaction engineering, essentially we want to look at the design equations that are required to understand the equipment that we are going to design, to operate the equipment to performance function that we are looking for. So, let us get on with this task of reaction engineering design equations. Now, this is a conceptual representation of what might be a reaction equipment, please do not think that this is how a reaction equipment looks like, this is not the idea. So, we have an enclosure of an arbitrary shape deliberately chosen, in which a certain amount of fluid F j at 0 is entering and it is leaving. Fluid enters and fluid leaves the equipment. Now, we can write a material balance, we call it as mole balance or any of the components that enter the system. F j 0 is the moles per time, which is entering the system and F j is the moles per time that leave the system. Well, G j is the moles per unit time, which is produced in the system or produced by means it can be consumed, in case it has a negative value. And this difference would be the rate at which the material is accumulating in the system. On other words, input minus of output plus generation equal to accumulation is a fundamental statement of material balance. Whatever comes in, whatever goes out, whatever gets generated, this combination the rest should accumulate. Generation can be positive, in which case this function is positive, if it is consumed it takes a negative value. This is general statement of material balance. Now, we recognize that this function G j, this function G j that means the rate at which the component j is produced or consumed at every point in the equipment. And this could vary from point to point, there could be some variation. On other words, the rate of generation of this component can be different at different points in the equipment. So, you must take that into account. So, keeping that in mind, what you have said is that at different points 1, 2, 3, 4, 5 up to n, r j 1 is the rate of generation of component j, delta v 1 is the volume of that elemental volume. Similarly, r j 2 is the rate of generation elemental volume delta t 2 up to n. On other words, rate at which component j is generated is an integral of r j multiplied by d v. So, r j d v is a rate of generation of component j. So, if you substitute this into our general statement here, material balance, replace this G j by r j times d v. So, then we have a more general statement of the conservation equation f j 0 minus of f j integral r j d v equal to d by d t of n j. Now, it could so happen that the equipment that we are dealing with is an equipment where there is a stirrer. So, that the composition, temperature and other intensive variables are essentially the same at different points in the equipment in which case we call it a stirred vessel. So, if you have a stirred vessel into which you have continuous input of component j and continuous output of component j, then we have so we have continuous input of component j and continuous output the stirred vessel in which case we can say that r j is uniform at every point inside the equipment. So, that we can take r j outside the integral and therefore, this equation can be written as f j 0 minus of f j plus r j times v integral d v is v equal d by d t of n j. On other words our statement of material balance for the case of a stirred vessel can be written as f j 0 minus of f j plus r j times v equal to d by d t of n j. Let us look at this in some more detail. Suppose, instead of a stirred vessel instead of a stirred vessel we have a different type of vessel which is often seen in the process industry which is called as a tubular reactor. What is a tubular reactor? We have a pipe of a chosen diameter perhaps we have a catalyst inside and your fluids come this way let us say and goes out this way a tubular reactor. Now, this is trying to represent a general situation where you have a vessel of an arbitrary shape through which component j is entering and component j is leaving component j is entering and leaving f j 0 is entering f j is leaving and we want to write a material balance on an elemental volume delta v. So, this is an elemental volume delta v over which we want to write our material balance. So, we notice here that f j at v at any time t f j at v plus delta v at any time t plus r j times v at any time t equal to this is the total amount of multiplied by delta v delta v. So, this is the statement of material balance for a flow through what we call as a plug flow vessel by plug flow what we mean is that every fluid element that enters here it moves through the equipment without recognizing the existence of other fluid elements. On other words as the fluid element moves through this equipment it does not mix with other fluid elements. For example, what is mean said here suppose there is a fluid element comes here it moves through and gets out. There is another fluid element which enters here it moves through and gets out. On other words as this fluid element moves through and as this element flows they do not recognize the existence of each other as a result this called as a plug flow. That means there is no intermixing between these two fluids and as such situations when we write a material balance for an element f j at v f j at v plus delta v rate of generation of the component j this is the difference that accumulates. So, if we take the as delta v tends to 0 and so on this how the material balance looks like for under the unsteady state when it is steady state where this term disappears the equation looks like this. On other words what we are trying to say here is there are two types of equipment ideal equipment that you will see in the process industry. Equipment number one we might pointed out is a stirred vessel this stirred vessel could be operated such that it could be a batch in the sense there is no continuous input and continuous output. It is still a stirred vessel there is a stirrer where the compositions are maintained. The other kind of equipment is what we have here is a tubular equipment where materials move through and then exit without mixing with each other it is called a plug flow vessel. The equations that described the variation of various properties for a stirred vessel is given by this and for a plug flow vessel is given by this. For a plug flow vessel at steady state your del by del v of f j is given by rate of generation of j and for stirred vessel the equation we already shown. Now, these two types of equipment what we call as plug flow this is for a place of a plug flow plug flow equipment. So, you find that plug flow equipments are very common particularly in the process industry dealing with catalyst because this pipe holds a catalyst and the fluids can enter and leave. Therefore, the catalyst that is performing work for you can be changed as and when the activity goes down and that is the kind of equipment most popularly used in the process industry. When you are dealing with fluids and removal of heat etcetera particularly in smaller scales you will find stirred vessels in batch and continuous mode. On other words what I am trying to get across to you is that you have the stirred vessels stirred vessels which are commonly used for smaller scales of operations where you have either in batch operation or continuous operation. You have this plug flow vessels particularly for catalytic reactions mostly in very large scales where you have a catalyst which is continuously used for processing a certain material to a certain final product. So, batch vessels continuous stirred vessels as well as plug flow vessels are the most common thing that you will see in the process industry. We have set up equations both for batch and stirred vessels here and then we have set up equations for steady state and unsteady state for the plug flow vessels. Our job now is to see how reactions perform in these ideal vessels. How do reactions perform and what we can learn from the performance of these reactions? Let us just to illustrate let us just assume that any reaction of this form A A plus B B giving you C C plus D D takes place in our reaction equipment. Now, if you choose A as our reference species then you can divide throughout by A. So, that our reaction the reaction now looks like A plus B by A B plus C by A C plus D by A D. So, this is assuming that this is component A is our reference. Now, we know that a reaction takes place at a certain rate if I call that rate of chemical reaction as R A if I say that rate of reaction of component A is R A we know from our understanding of basic chemistry that R A divided by minus of A equal to R B minus of minus of B equal to R C divided by plus C equal to R D divided by plus D or in other words this ratio of rate of reaction of A divided by its stoichiometric coefficient with appropriate sign for a given reaction they are all equal and therefore, I have turned it as R 1 sometimes called as the intensive rate of reaction. Now, how do you understand conversion? We understand conversions at steady state conversions are typically we start with so many moles of A at the end of the reactions. So, many moles of A is unreacted. So, we expressed the amount of component A converted with respect to what you started with that is gives you the conversion. On other words the whole meaning of conversion requires you to recognize a certain starting material or reference. So, our reference for defining conversion is N A 0. If instead of a batch process you have a continuous process this N A 0 has a units of moles where if it is a continuous process X A is defined as with respect to the reference F A 0 where F A 0 has the units of moles per time. So, whether it is batch or continuous we define conversion with respect to a certain reference. For continuous process we have taken F A 0 as the reference species and for batch process we have taken N A 0 as the reference. Choice of reference is always your choice in the sense you can define with respect to your reference. Generally a reference component with the limiting substrate or limiting reagent that is in the reaction that is how it is normally done, but it is not important any reference is. Let us say we have a batch system where the reaction A A plus B B let me just write it down once again our reaction is A A plus B B equal to C C plus D D or if you divide throughout by A it looks like this. So, for this reaction if you look at species A B C D and I is inert typically every system comes with some amount of inert. So, you start with so many moles of A N A 0 N B 0 N C 0 N B 0 and so on. Therefore, as per stoichiometry what we have written is if X is the extent of reaction or conversion. So, the unreacted A is given by N A 0 times 1 minus of X A this definition and similarly N B is N B 0 minus of N A 0 X A multiplied by B by A. So, this term B by A this term comes because of the stoichiometry as we have represented there. So, as per the way we have defined conversion and the way we have taken the chemical reaction we can write what is the unreacted amounts of A B C D and I in the system at any given conversion. This follows directly from stoichiometry we can add up all this when you add up all this we find it is the number of moles we start with is N T 0 when you add up all these moles this is. So, many moles have coming in and so many moles are going out when you add up all this we notice here this is N T 0 and all this terms common N A 0 X is common I have forgotten the X A here and then this term inside D by A plus C by A minus B by A minus 1 this term is the effect of the chemical reaction. So, what we get here from stoichiometry number of moles we start with the number of moles we get when there is a certain amount of reaction X A. Now, we can write the number of moles at the end of certain extent of reaction certain conversion X A is given by N T 0 plus this term is the effect of chemical reaction as expressed with respect to component A which is what is called as delta A. So, delta A is the meaning of delta A is change in moles change in moles due to reaction with respect to the reference component A. On other words delta A tells you what is the number of moles change that comes because of the reaction D by A plus C by A minus B by A minus 1 and this is the known quantity for every reaction. So, what we get from stoichiometry is that given a conversion X A given conversion X A N T by N T 0 is given by this relationship where Y A 0 is the mole fraction of component A where we start with and then N T 0 is the total number of moles which we start and delta A is the change in moles due to reaction expressed with respect to component A. So, in other words this representation tells us what is the number of moles at any conversion at which the reactor is working. Now, what is important is what we have done here with respect to component A we can do it with any other component B C T it does not matter. So, just to put it in the perspective what I have said here is that if you chose B as the reference species then your equation will look like this where instead of Y A 0 instead of Y A 0 here you will get Y B 0 instead of X A you will get X B and what is X B? What is X B? X B is generally just like X B equal to N B 0 minus of N B divided by N B 0 or in other words X B is the conversion as measured with respect to component B and we recognize that in a reaction this could be different when you measure conversion with respect to component A and component B they could be different. Now, if instead of a batch system see we talked about this stoichiometry all this stoichiometry we have talked about is considering our system is a batch system by batch what we mean is we have an equipment which is a vessel which is got no outlet there is no input inlet certain amount of fluid is inside the equipment and then these are the changes that we are going to observe. But you could also have a system where there is a continuous input there is a continuous input and continuous output like a tubular reactor there is a continuous input there is a continuous output. So, how do we deal with systems where there is a continuous input and continuous output. So, when we have a flow system our equations now look exactly the same there is no difference of stoichiometry still does not change I notice here this here instead of writing F N A 0 I have written F T 0 F B 0 F C 0 F D 0 F I 0 and F T 0. In other words as far as stoichiometry is concerned it does not matter whether you are talking about a flow system or a batch system. Therefore, whatever relationship that we get for N T is the same we will get when it is for F T because stoichiometry does not get affected because of flow etcetera. So, we can say now F T the total number of moles leaving the system is F T 0 which is total number of moles entering the system multiplied by F A 0 X A multiplied by delta A same like here N T equal to N T 0 N A 0 this is delta X A. So, essentially what we are saying is that if you have a flow system then total moles leaving is related to the system by the same kind of equation that we have done for a batch system. So, that F T divided by F T 0 is given by 1 plus Y A 0 X A delta A previously we had N T divided by N T 0 given by this. So, whether it is N T by N T 0 or F T by F T 0 the right hand side does not change because that is coming from stoichiometry. Now, having said all these things now let us go a little bit forward and see how we can look at various kinds of design equations for our systems. First systems we are taking is a batch system let me just remind you what is a batch system or batch system looks like this you have a vessel the vessel is closed it is well stirred which means that compositions are the same at different points. There is an inlet, but this is closed there is an outlet which is closed the idea of this inlet or valve and outlet valve. So, that we can charge and discharge. So, as far as the reaction is concerned during the reaction there is no inlet there is no outlet. So, whatever is you take inside the equipment that is what brings about the reaction. So, for the case of a batch system where there is no input where there is no output there is only generation and accumulation therefore, first term and second term disappears therefore, you have R J times V the rate of generation of component A equal to D by D T of N J. So, this is statement of material balance for a batch system that rate at which the reaction takes place equal to the rate of accumulation in the system. Now, our reaction is given by A A plus B B equal to C C plus D D or in terms of component reference it is looking like A plus B by A B equal to C by A C and D by A D. Therefore, now we can say if A is the reference species R A times V just replacing J by A we have R A times V equal to D by D T of N A. So, statement of material balance for a batch system is simply R A times V equal to D by D T of N A. Now, we can go forward we have been talking about stoichiometric table. Now, I mean something that we know that conversion is defined as N A 0 minus of N A divided by N A 0. Point to be noted here is that N A 0 is a reference. This reference is our choice in this particular case we have taken A as the reference. Now, as a result of this representation you know that the moles of N A is given by N A 0 times 1 minus of x. On other words if you look at a batch reactor we find minus of N A 0 D x A by D T is given by R A times V. This is the statement of material balance. So, that we can now say D T is simply equal to N A 0 x A divided by minus of R A V. When you integrate this we get the reaction time is N A 0 D x A divided by minus of R A times V. Notice here that the volume volume volume effect appears here. On other words if volume of the reactor changes during the process of reaction that effect should have to be accounted here. If volume is constant it can be taken out of the integral. So, that this becomes C A 0. So, that T R is C A 0 times D x A by minus of R A. A familiar representation in many textbooks, but it is so provided V is constant. This is an important issue that we must bear in mind. Now, let us take another very common instance that you might see in the process industry what is called as continuous stirred tank reactor. Now, stirred tanks are examples where the fluid comes in continuously fluid leaves continuously. The batch equipment please recall batch equipment we said the process works the process works and during the process there is no input. So, we said this if a batch equipment is an instance where there is no addition or removal of material as the reaction takes place that is a batch equipment. We charge all the raw materials close all the valves conduct the reaction at the end of the reaction time you open this valve to discharge the product. Now, a continuous a continuous reactor is where you have continuous input of fluid continuous removal of fluid. A reaction takes place inside this equipment continuously and therefore, we are able to continuously withdraw the product at a rate that we desire. So, continuous reaction what is called continuous stirred tank reactor is one example of a process equipment wherein you will produce your product continuously. On other words you do not have to worry about charging time, discharging time and so on which could be very significant in a batch process. So, all those charging and discharging of material is eliminated only the process time is what is of concern to you and this way you are able to produce large quantities of material. One of the advantages of this continuous processing is that we do not lose time during charging and discharging. Now, let us see how the equation looks like our statement of material balance does not change input minus of output plus generation equal to accumulation is a fundamental statement of conservation of mass. Our input is what if our reaction once again let us recognize our reaction let me write down once again a a plus b b equal to c c plus d d and with a as reference it becomes b by a b plus equal to c by a c plus d b. So, we are still considering the same reaction with a as the reference. So, we have input is f a 0 we can see here input is f a 0 output is f a as you can see here rate of generation of component a and d by d t of n a and we are operating at steady state. We will look at unsteady state situations later on for the moment let us assume that this process is running at steady state. What is meant by steady state? By steady state we mean that if you look at the composition of the system at any time and look at the composition of the output at any time. We find the composition inside the equipment and composition of the exit are the same. That means it does not change with time steady state is an instance of a process operation where the compositions that you measure at the exit does not change with time. The composition that you measure inside the equipment with time does not change this time that means it is steady with respect to time. Under situations where it is steady this equation f a 0 minus f a plus r a v equal to d by d t of n a the right hand side which represents accumulation. Because it is steady there is no change with respect to time therefore, there is d by d t of this will go to 0 therefore, the right hand side is 0. So, steady state representation of material balance gives you f a 0 times x a equal to minus of r a v because right hand side is 0. Please recognize f a 0 minus of f a equal to r a v this comes from the fundamental definition of x a. We define x a as f a 0 minus of f a divided by f a 0 that is how x is defined that is why we write this term in this form. So, that this representation gives us volume of the continuous third tank reactor is given by f a 0 times x a divided by minus of r a. So, a continuous reaction in which you have a CSTR to bring about this reaction the volume of the equipment is given by f a 0 x a by minus of r a. f a 0 is something you know x a you can measure and r a is the chemical kinetics in way at which the reaction is occurring. So, right hand side is all a known quantity therefore, given the chemical kinetics given the extent to which you must react given the product production that you want all this can be taken together to determine what is the size of the equipment which we must design for. Now, why is it that we have moved away from batch equipment to continuous equipment like CSTR there are many reasons for this batch to continuous is a decision that an entrepreneur will take depending upon the scale of production. Generally, you will find continuous productions are preferred if the scales are very large the scales are small of course, batch productions are preferred, but that is not saying enough. Let us also recognize that when you have some products which are very expensive where you require very careful control of the process during the process of reaction the general preference is to do it in batch. So, that you can address all the issues that is required to be addressed during the course of reaction. So, there are certain advantages in doing batch processing and there are advantages using continuous processing and that decision will be a decision that that is taken based on technical economic and commercial issues. What we are now saying is that if our stirred tank continuous CSTR design equation is this recognizing that residence time is actually volume of the equipment divided by the flow at the inlet. Then we can write our design equation in terms of v we can also write the design equation in terms of residence time tau which looks like this. On other words this is in terms of volume this is in terms of residence time they are stating the same thing the residence time and react volume or related depending upon the volume flow at which we process the fluids. So, this is the fundamental statement of design equation for CSTR or this might be another way of saying that this is an equation for CSTR. Where if you know the chemical kinetics if you know the extent to which you want to react and if you know this fluid then you can say what is the size of the equipment that is required for your process. What has happened in the process industry is that there are situations where you may need not one stirred tank, but many stirred tanks to operate in sequence. There are situations which require many tanks to be operated in series, but more importantly this sequence of stirred tanks gives us certain insights into what happens in a process. Let us just look at what this sequence tells us this sequence tells us that if you look at tank 1 input output generation equal to 0 at steady state. We are studying all these are at steady state that means we are writing the material balance at steady state. We will come to unsteady state at a later date first of now we are looking at steady state. Therefore, our statement of material balance leads us to size of the equipment given by F A 0 x 1 by minus of r a or residence time given by C A 0 x 1 by minus of r a that means the residence time in tank 1 is given by this equation C A 0 x 1 by minus of r a 1. Now, if you write the material balance for tank 2 which is F A 1 minus of F A 2 input output generation equal to 0 where this is the rate at which reaction occurs in tank 2 r a 1 is the rate at which reaction occurs in tank 1. Therefore, this equation is a way of understanding what is the size of V 2 that is required. So, we can simplify this write our material balance please our material balance this is our material balance for tank 2 is what is written once again on the next page F A input output generation equal to 0. We find that the size of reactor 2 equipment 2 can now be given as F A 0 times x 2 minus of x 1 x 2 is what is x 2 x 1 is conversion at the at the outlet tank here x 2 is conversion here. So, what are we saying here what we are saying here is that the size of tank 2 depends upon F A 0 times x 2 minus of x 1 what is x 2 minus of x 1 it is the moles of component a that is under gone chemical reaction in tank 2 what is r a 2 r a 2 is the rate at which chemical reaction occurs in tank 2. So, the total moles converted divided by the rate of conversion is the size of the equipment. So, it is stating the obvious is nothing new being said the obvious is being said, but said in a form which is useful for doing calculations. So, what do we get the size of equipment V 2 if you want to express in terms of residence time our residence time is defined as equal to V 2 divided by V 0. So, we have size of equipment 2 I mean reactor 2 is given by C A 0 times the conversion change divided by the rate of chemical reaction. Now, we can continue this whole process and then say for tank 3 tank 4 tank 5 and so on. We can say for tank n we are looking at its F A 0 term x n minus x n minus 1 divided by r a n where what is F A 0 F A 0 is please recognize we have n tanks here tank 1 and tank 2 and then tank n this is tank n 1 2. Now, inputs this is F A 0 this is this is coming this is what is coming out is x n. So, what we are saying is that the size of tank V n this is V n is F A 0 what comes here multiplied by what happen this is x n minus 1 what is coming in here is x n minus 1 you see. So, x n minus 1 time x this is the difference change in conversion in this divided by the rate of chemical reaction. So, it is stating the obvious, but in a forms that is very useful to us for doing calculations. So, what have we said what we have said is that if you have if you have tanks in sequence instead of a single tank tanks in sequence. Then we find that we can use the same statement of material balance to find out what is the size of equipment 1 tank 1 what is the residence time in tank 1 then we can see what is the size of equipment 2 what is the residence time in tank 2. Similarly, we can say what is the size of equipment n what is the residence time in tank n. So, our statement of material balance helps us to calculate all this because it is put in that form. Suppose as we said when we started this we said we have two types of equipment that we will see in the process industry. One is what is called as stirred vessel where it is mechanically stirred the other equipment you have a plug flow vessel where there is no mechanical stirring inside the equipment. That means, the plug flow equipment you have let us say a pipe may be inside the pipe there is a catalyst and then fluid is coming in fluid is going out. There is no mechanical equipment inside here to do stirring. So, and as a result you might expect that there is plug flow plug flow what we mean is that they move inside without mixing with each other. So, this is a plug flow kind of equipment. Now, we have already derived what is our design equation for a plug flow equipment for a reaction like this we have already said this d by d by d by d v of f a is r a what are we saying here what we are saying is that d f a d v is what it is rate of change of moles of a per unit volume per unit time. So, rate at which moles of a change per unit volume per unit time that must be equal to the rate of chemical reaction it is stating the obvious. What I am trying to put across to you is that every equation that is written here it is stating the obvious it is in a form that we can understand from first principles nothing new is being said. Now, we can represent this f a from our stoichiometry what we have said in our stoichiometry we said in our stoichiometry f a is equal to f a 0 multiplied by 1 minus of x a this is how it is defined this conversion itself is defined like this. Therefore, we are able to substitute for f a here and simplify and we get volume of this of this equipment is f a 0 what is the entering here this is this is f a 0 what is the entering and what is leaving. So, volume of the equipment is f a 0 multiplied by this integral d x a by minus of r a please recognize that when we had third tank it was c a 0 x 1 x 1 minus of r a on other words here we had an algebraic equation here we have a integral. So, there is a difference see this is a differential equation while here this is an algebraic equation. So, when we have stirred tanks we deal with algebraic equations when we have plug flow reactors we deal with differential equations. That means, the equation that relates various variables only change as we change the equipment from one to the other. Now, that we know the size of the equipment you can now talk in terms of what is called as residence time that is defined as volume of this equipment divided by this flow rate flow rate at the inlet. So, the way it is defined is that the residence time is defined as volume divided by the flow rate at the inlet what in other words why I am saying this to you is that this volume flow can change as it flows through the equipment. But the way it is defined it is defined with respect to inlet flow on other words residence time also involves certain reference flows at which we do our calculations it is convenient it is convenient to do it that way. So, if you define residence time as v by v 0 then this equation becomes c a 0 this is a known quantity this integral d x a by minus of r a what is r a rate of chemical reaction this number comes from understanding of the chemical kinetics and we will at a later date look at how we can determine this function that determines this function r a now having said this let us just quickly understand these functions. Recognize that this volume depends on x and r a suppose we plot having said this suppose I make a plot of 1 by r a please see this function x and minus. So, if I plot x versus 1 by r a you notice that this area this area is what we are talking about if you look at this if I make a plot of 1 by r a versus x that is what I have done here 1 by r a versus x. Therefore, this term x 1 by minus of r a is simply the area under the under this curve this rectangle is the area. So, if this area of this rectangle multiplied by f a 0 becomes the volume of the equipment as you can see here if I plot x versus r a as I have done here x a r a you can see here this area multiplied by f a 0 becomes the volume of the equipment. Now, for a plug flow reactor we said 0 to x d x a by minus how do you this when you plot 1 by r a versus x a this integral is simply under this curve area under the curve. So, on other words what we notice here is that the way we have formulated the problem of single reaction taking place in ideal vessels whether it is stirred tank whether it is plug flow the plots of 1 by r a versus x actually gives you a way of determining the size of the equipment or for example, in many cases it may be easier for you to determine this function 1 by r a versus x from your experiments. The functionality r a may not be known, but data may be available that means you may have experimental data here you may have experimental data you may have experimental data. So, that you can use the experimental data and determine what is the size of your equipment in many cases particularly reactions which involve many complex reactions etcetera you will find it easier to actually make a plot of 1 by r a versus x a. So, that you can determine the area under the curve and hence size of the equipment is very convenient that way and it is preferred most cases you will find this might be much faster way of determining the size of the equipment that might be of use to you. So, what we have tried to say so far is that if you have a batch if you have a continuous equipment like CSTR or a continuous equipment like PFR you can plot this 1 by r a versus x and then find out the size of these equipments. Recognize that suppose instead of one tank you have many tanks and other words if you have several tanks in sequence where v 3 is f a 0 times x 3 minus of x 2 divided by r a 2 with a minus sign. On other words if you have a plot of minus of 1 by r a our reaction once again please recognize that our reaction is still this there is no change we are still looking at a single reaction where a plus b by a plus b equal to c by a c plus d by a d we are still looking at this. So, if you have a sequence of tanks 1 so this is tank 1 which is this area this is for tank 1 this is the area for tank 2 this is tank 3. Therefore, size of tank 1 2 and 3 can be calculated by multiplying the area under the curve by the flow f a 0. So, this graphical procedure gives you a way of determining size of the equipment. So, this equipment simply by experimental data and that is the great advantage of this procedure because you do not have to look for a functionality that will determine the rate of chemical reaction. It is a huge advantage that is there particularly for reactions where you have great difficulty in understanding what is going on experiment may be easier another point that frequently that we might want to to know is suppose you have a continuous equipment a continuous equipment. Let us say let us say catalytic for example, and due to chemical reaction as you can see here due to chemical reaction there could be some volume change and how do you account for this volume change we account for this volume change by recognizing that our equation d f a by d v equal to r a it is the fundamental statement of a material balance for a plug flow vessel. And we also know that the time of residence is given by d v by v on other words if you have a plug flow equipment then the residence time at every point is given by d v by v. And this is the general statement of material balance by combining these two now we can tell that what is that actual time of residence of fluid elements inside this equipment. So, what we are trying to put across here is that if you have a plug flow equipment in which there is a chemical reaction taking place as a result of which there is some volume change. And we can take that effect into account through this equation. So, we not only can find out residence time using our residence time can be on the basis of inlet flow we can do residence time this way or we can do residence time actual which is given by what is this equation. So, both firms both approaches are available and whatever is appropriate to you you would use in a given application. I will stop here with this we will take up this at a in the next lecture the implications of what I have said for its applications to various problems that we would encounter. Thank you.