 It is a good time to summarize what we have learnt in residence time distribution so far. So we have looked at what is a non-ideal reactor and what is the residence time distribution function, what are its definitions and we had looked at what are the ways to measure it experimentally that is looking at the pulse and the step input and we have also came we also looked at what are the RTD or the residence time distribution functions E curve and the cumulative distribution function F curve in the last lecture. So today let us start with this lecture, let us start with looking at the properties of different functions and also proceed further. So suppose if I look at the an important property of the residence time distribution is the mean residence time. So the mean residence time is actually given by the first moment. So if I Tm is the symbol that I will use for mean residence time it is actually given by the first moment of the of E of t that is the RTD function. So E of t is actually a distribution and that distribution can actually be used to decipher some of the properties of the distribution itself and some of the properties of the reactor system. For example mean residence time is an important property that is actually used to control various things in the system. When there is no dispersion across boundaries that is between the point of injection and the entrance of the reactor. Here in these situations the space time that is tau which is equal to V by the volumetric flow rate with which the fluid is actually flowing through the reactor that is equal to the mean residence time itself. Now this is independent of any RTD function that is actually representing the non-ideal behavior of the reactor under no dispersion conditions irrespective of the RTD function the mean residence time that we obtained would be exactly equal to the space time of the reactor itself. So this is true for all RTDs this is true for all RTD all residence time distributions irrespective of what type of reactor as long as the dispersion is actually absent. So now let us look at how to calculate the mean residence time from the residence time distribution function E of t. So tm which is the mean residence time is actually given by the first moment as we observed as I mentioned in the previous note a few moments ago that is 0 to infinity t into E of t dt divided by integral 0 to infinity E of t dt. So that is the residence time distribution and because the integral of the E curve which is the RTD function between 0 to infinity that is equal to 1 this expression can further be simplified as integral between 0 to infinity t into E of t dt. So that is the expression for the mean residence time if the RTD function E of t is known. So if the residence time distribution function is known one can simply plug it in in this expression and find out what is the mean residence time. Now suppose let us look at suppose let us consider a reactor and let us assume that it is filled with species A and let us say that at time t equal to 0 a tracer molecule tracer species B is injected into the reactor let us say it is a dye and then in some time dt. So let us say that the amount of tracer which is actually leaving the reactor in this time delta t whose age is actually lies between that time is actually given by V times dt where V is the volumetric flow rate with which the fluid actually leaves the reactor and that is equal to the volume of the tracer which is actually leaving the that is actually the volume of the effluent stream which is actually leaving the reactor not the tracer. So now suppose if we want to know that the species has been there for a long time suppose see so species A has been in the reactor for a long time. So remember V dt is the volume of the effluent which is actually leaving the reactor in this time dt and if you want to know what is the volume of species A which is actually leaving in that time delta t so then that will be given by dV which is equal to the total volume of the fluid that is actually leaving the reactor multiplied by 1 minus f of t. So f of t is basically the fraction that has been in the reactor for time which is greater than t. This is the fraction which is actually so that is the fraction in the reactor residing for time larger than t. So 1 minus f t multiplied by the volume of the effluent stream will actually tell us what is the amount of species A which is actually leaving the reactor in that small time dt. So now if we sum this over all the molecules of A then that will tell us what is the net volume of the species which is actually leaving the reactor. So if we sum over all A molecules see the total volume that is leaving is given by 0 to infinity V dt into 1 minus f of t. So from here if we assume that the volumetric flow rate with which the fluid stream leaves the reactor if that remains constant and this is generally not true for gas stream but it is normally true for liquid streams that is actually leaving the reactor. If it is a gas stream suppose if it is operated under constant pressure and under isothermal conditions that is constant temperature and if the number of molecules or number of moles does not change because of the reaction then one may also assume that the volumetric flow rate with which the fluid leaves the reactor the effluent stream volumetric flow rate is probably perhaps remains constant. So by using this we can say that V equal to V0 into integral 1 minus f t dt. So now we can integrate this by parts. So if we integrate we will find that V by V0 that is equal to t into 1 minus f of t limits 0 to infinity plus integral 0 to 1 tdf. So that is the integral this is basically when we do an integration by parts we can see that we can split the integral into two sections this t into 1 minus f t evaluated between 0 and infinity and 0 to 1 t times df. Now if I look at the f curve the f curve typically looks like this. So this is with respect to time and this is 1. So at time t equal to 0 f of t is 0 and when t goes to infinity 1 minus f of t is 0. So that can actually be easily seen from the f t curve or the f curve. So now substituting these expressions we will find that V by V0 that is equal to tau which is the space time of the reactor and that is 0 to 1 t times df. So what is df? df is nothing but the residence time distribution itself dt into dt gives the first differential of the f curve and therefore V by V0 that is equal to tau and that is equal to integral 0 to 1 t times E of t dt and that is nothing but the mean residence time itself. So this shows that for any RTD if there is no dispersion between the point of injection and the entrance of the reactor one can show that the mean residence time is actually equal to the space time of the reactor itself irrespective of what is the RTD function E of t. So clearly for V equal to V0 for constant volumetric flow rate then tau equal to tm if no dispersion. And remember that this V equal to V0 is true for gases only if the reactor is operated under constant pressure drop and the temperature is maintained constant that is isothermal conditions and if the number of moles does not change because of the reaction only under those conditions the effluent stream volumetric flow rate may be assumed as a constant. So therefore the exact volume of the reactor, exact volume of the reactor if there is no dispersion is actually given by V0 multiplied by the average residence time. So if the average residence time is known then we can actually calculate what is the exact volume of the reactor in which the fluid is actually flowing. So are there other properties so we looked at mean residence time and we also showed that the mean residence time should be equal to the space time irrespective of the RTD function as long as the dispersion is negligible or 0 and also if the volumetric flow rate with which the fluid stream leaves remains nearly constant. So are there other properties and the answer is yes there are other properties so the other properties is we can also estimate what is the variance of the distribution and that can be obtained using the second moment obtained using the second moment. So the sigma square which is the variance is given by 0 to infinity t minus tm square into E of t dt and so now if we expand this square this product here so we can expand this as 0 to infinity t square plus tm square minus 2 into t t into tm into E of t dt. So that is the integral and this is nothing but 0 to infinity t square E of t dt minus tm square. So this essentially variance essentially quantifies the spread in the distribution of the RTD function. So that is another property that is actually very commonly used in the real systems and the third property which is not very commonly used is the skewness property it is called the skewness and that is obtained using the third moment of the distribution and that is given by if s q s is the skewness parameter there will be 1 by sigma to the power of 3 by 2 where sigma is the standard deviation that is square root of the variance 0 to infinity t minus tm the whole cube into E of t dt. So that is the skewness and this is basically reflects the extent to which the distribution residence time distribution function is skewn. So remember that it may be skewed in either directions so for example if the residence time distribution looks like this then it is sort of skewed to the right hand side of the mean. So the s cube essentially says how skewed is the distribution with respect to the mean of the distribution itself. So now once we know these properties next the question is from real reactor data suppose if there is a tracer that goes inside and from the real data is it possible to estimate some of these parameters and what are the steps that is involved. So let us look at how to calculate the mean residence time and sigma square from the actual data. So normally the actual data that one would get is basically the measurement of concentration as a function of time. So let us say that there are several concentrations that has been measured let us say from time 1 to 10 and there has been concentration CT that has been measured. So then one needs to create a table where as a first step one calculates E of t. So we know the formula for E of t which is essentially given by CT divided by the integral of C over the whole time domain and then the next thing one needs to estimate is t into E of t. So this column provides an estimate of the first moment which is the mean residence time can be used to find the mean residence time and the next step is to estimate t minus tm the whole square and then find out t minus tm the whole square into E of t and then from here one can actually find out what is tm square into E of t. So one can make such a table moment the experimental data of time versus concentration is available of the tracer is available then one can actually fill up this table and from this express from this column one can estimate the mean residence time and from this column one can actually estimate what is the sigma square. So and one needs to use an appropriate numerical integration scheme remember that the concentration is actually discrete values at different time points and so one has to use appropriate numerical integration appropriate numerical integration in order to complete this table. Once this table is complete we will actually be able to estimate what is the mean residence time and the variance for the distribution that represents the RTD function for the reactor. Now the suppose if we change the suppose if there is a reactor and we know the RTD function suppose we know the RTD function suppose we know the E curve for a given volumetric flow rate V1. Now if we want to find out what is the E curve or the RTD function for a different volumetric flow rate. So now let us consider the situation where we are actually feeding the reactor with a fluid of volumetric flow rate which is less than V1. So then the amount of time that the fluid stream spends inside the reactor is going to be larger because the volumetric flow rate is actually lesser than V1 and as a result the E curve would actually look like this the slope of the E curve will correspondingly change. So now because of this problem so this corresponds to volumetric flow rate V2 and because of this issue it is very difficult to now compare the E curves at different conditions because the E curve is now going to be dependent on the volume of the reactor and also on the volumetric flow rate with which the fluid is actually being fed into the reactor. Even for a fixed volume the E curve is now going to be a function of the volumetric flow rate because the volumetric flow rate decides the residence time of the fluid stream inside the reactor. So therefore the tau 1 which is the space time when the volumetric flow rate is V1 is given by V by V1 and tau 2 is given by V by V2. So clearly the amount of time that is spent by the second in the second case that is when when the fluid is being fed at a volumetric flow rate of V2 that is going to clearly be larger than that of the time that is actually spent by the fluid elements inside the reactor when the volumetric flow rate is V1 because V2 is actually smaller than V1. So because ET depends on properties such as volumetric flow rate it is difficult to compare. So as a result it is useful to actually define a normalized RTD function in order to facilitate the ability to compare different RTD curves. So let us look at what is the normalized RTD function. So suppose if we define theta as the ratio of T divided by tau where tau is the space time of the reactor. If we define theta as the ratio of time versus the space time of the reactor then we can now rewrite the RTD function E theta as basically tau multiplied by ET. So that is tau is the space time multiplied by the corresponding RTD function gives the normalized RTD function E of theta and so now theta here which is the ratio of time to tau essentially represents the number of reactor volumes of fluid based on the entrance condition that have actually flowed through the reactor in that particular time t. So now this normalized RTD function E theta provides a facilitates a way by which the performance of the reactor or the RTD function itself can be compared when the sizes are different. So therefore if we look at the RTD curve of the normalized RTD function then the curve looks like this where so irrespective of whatever is the volumetric flow rate for a given reactor volume the RTD function essentially looks like this. So now there is another definition that one needs to know is the internal age distribution and the symbol that is commonly used is I of alpha where I of alpha d alpha that essentially represents the fraction of the material that is present inside the reactor in a time span of alpha in a time span for a period between that is between alpha and alpha plus d alpha. So that represents the fraction of the material that is actually residing inside the reactor whose period of residing inside lies between this lies between alpha and alpha plus d alpha in that small interval. So E alpha essentially represents the age of the fluid that actually is leaving the reactor and I alpha represents the age of the fluid that is actually present inside the reactor. So these two have its own utility and particularly the internal the age of the fluid elements that is actually present inside the reactor has a significant importance when one looks at when one wants to study the unsteady state behavior unsteady state behavior of a particular reactor. In particular a good example of that would be that suppose if there is a catalytic reaction and the catalyst is actually decaying with time then it is important to know what is the internal age distribution and it is important to actually consider the age distribution in modeling the performance of such kind of a reactor. So I alpha the internal age distribution is essentially given by 1 by tau into 1 minus f of alpha and E of alpha as we know is actually given by minus d by d alpha tau into alpha because of the connection between the E curve and the f curve. So the relationship between the E curve and the I curve is nothing but E of alpha is minus d by d alpha into tau into I alpha. Now for a CSTR for an ideal CSTR I alpha is essentially given by 1 by tau into exponential of minus alpha by tau. So that is the internal age distribution for a CSTR. After all these definitions that we have seen that is the E curve, f curve and the I curve and the mean residence time variance and skewness let us look at the residence time distribution in ideal reactors. So particularly we will consider two cases one is plug flow and ideal batch reactor and second one is we will look at the single CSTR case. So these two we will look at and we will attempt to find out how to get the RTD for RTD curves for these two types of ideal reactors. So let us first consider the plug flow reactor. Let us consider the plug flow reactor. So what is the property of the plug flow reactor? All atoms or all molecules of the material which is actually entering the reactor will spend exactly the same amount of time before they leave the reactor which means that all elements or all molecules of the material will have exactly the same residence time. So same residence time for all fluid elements that is actually entering and leaving the reactor. So therefore the RTD function must have the following properties. So first thing is it must have a spike of infinite height because all of them will have same residence time therefore they will all leave like a plug. So therefore the E curve must have a spike of infinite height and also it must have 0 width and not just that the area under the curve should be equal to 1. The spike will be exactly at the mean residence time and that is very important because that is the property which actually captures the nature of the plug flow reactor. So therefore the spike will be exactly at t equal to v by v0 that is equal to tau which is the space time of the reactor and because there is no dispersion the space time of the reactor will also be equal to the mean residence time of the reactor or in the non-dimensional terms theta equal to t by tau that is equal to 1. So therefore the corresponding E curve because of these properties of the RTD function for the plug flow reactor the E curve should simply be represented by the direct delta function centered at the space time of the reactor. So this is the direct delta function that is the direct delta function and it is defined as follows. So direct delta function delta x that is equal to 0 if x is not equal to 0 and it is equal to infinity when x is exactly equal to 0 and the property of this E curve is actually given by minus infinity to plus infinity delta x dx should be equal to 1 that is the property of the direct delta function in addition to that the another important property is by that the satisfies the convolution integral that is equal to g of tau. So integral of gx if gx is some function of x multiplied by the delta function to dx that is equal to g evaluated at that value of tau itself where x minus tau is actually equal to 0 that is where the spike is actually present. So now let us calculate the mean residence time for this RTD curve. So the mean residence time tm is actually given by integral 0 to infinity t into E of t dt that is the that is the definition for the mean residence time in terms of the RTD function. So that is equal to plugging in the E curve for plug flow reactor we will find that 0 to infinity t into delta of t minus tau into dt and that is nothing but tau itself. So therefore the mean residence time is exactly equal to the space time and this actually one would easily guess because we said that the an important property of the plug flow reactor is that all material that is actually entering the reactor and leaving the reactor will actually have exactly the same residence time and that the E curve is actually going to be centered at the space time. So therefore the mean residence time must be exactly equal to the space time of the plug flow reactor itself which is which one would actually guess and it is also clearly shown by the RTD function also. So now let us look at the second moment that is the variance of the of the distribution. So that is given by 0 to infinity t minus tm the whole square into E of t into dt. So that is equal to tm square into delta function into dt and so now we open up this the t minus tm whole square and then if one integrates we will find that this essentially reduces to t square into delta of t minus tau into dt 0 to infinity plus integral 0 to infinity tm square delta of x minus tau dt minus 2 integral t into t m into delta of x minus tau dt and that is essentially so the first term here because of the property of the delta function is it will simply be equal to tau square and the second property will simply be equal to plus tm square. So that will be the second one and the third one will simply be 2 into t into tm t into delta function integral tm is constant so that will come out of the integral and t into the delta function will essentially be equal to the mean residence time. So that will be equal to 2 tm square and that is equal to 0 because the mean residence time and the space time are exactly equal. So therefore the variance is actually equal to 0 and that reflects the property of the RTD function that actually we intuitively guessed that is the there has to be a spike at exactly t equal to tau with an area under the curve is equal to 1 and the height of the spike is equal to infinity which means that the variance should be equal to 0 for the distribution. So let us look at the f curve for the plug flow reactor. So for a plug flow reactor the f curve f of t is essentially given by 0 to t e of t by dt that is by definition and so that is equal to integral 0 to t delta of t minus tau dt that is equal to 1 by we know that this integral is equal to 1 and therefore the f of t curve is nothing but 1 and so as a result the properties or the RTD function for the plug flow reactor is essentially given by e of t to summarize is equal to delta function of t minus tau. So that is the summary for plug flow reactor where the residence time distribution function is essentially given by delta t minus tau and the mean residence time is equal to the space time of the reactor which is the volume divided by the volumetric flow rate and the sigma square is essentially 0 the variance is actually 0 and the f t is essentially equal to 1. So therefore if we actually attempt to sketch the e curve and the f curve we will find that so that is time and suppose if this is tau here at t equal to 0 if there is a spike tracer that is actually put into the plug flow reactor so that is the spike then exactly after a delay of tau time which is the space time of the plug flow reactor the tracer will actually come out and the same amount same quantity of tracer will actually come out of the reactor so that is the out stream and the height will be infinity. Now suppose if I look at the f curve so this is the e curve and suppose if I look at the f curve of the reactor suppose if I look at the f curve of the reactor then exactly at tau equal to exactly at t equal to tau that is the space time or the mean residence time of the plug flow reactor the f of f value will be exactly equal to 1. So that is the e curve and the f curve for a plug flow reactor. Now let us look at the CSTR case what is the RTD function for a single CSTR. Now suppose if here is a CSTR and this is the inlet stream and this is the outlet stream of the CSTR and the CSTR is well mixed and it is assumed that it is a ideal CSTR and therefore it is a completely well mixed system and let us now because it is completely well mixed system the concentration of the species which inside the reactor should be equal to the concentration of the species and the effluent stream as well. So which means that the outlet concentration is equal to the concentration of the species in the reactor and let us now write a material balance on an inert tracer suppose there is an inert tracer which is actually fed into the reactor. So let us say that an inert tracer is fed into the reactor and if the concentration of the inert tracer is actually C0 at t equal to 0. So time t equal to 0 some C0 quantity of tracer is actually fed into the reactor and now we can write a material balance in order to find out what is the RTD function. So for any time greater than 0 whatever fluid is actually whatever tracer is entering the reactor that should minus whatever is actually leaving that should be equal to the accumulation of the tracer inside the reactor. Now if we assume that it is a pulse tracer if it is actually a pulse tracer which means that the time at which the tracer is actually fed into the CSTR is exactly t equal to 0 and nothing before and nothing after t equal to 0. So therefore at any time greater than 0 no tracer is actually entering the reactor. So therefore the inlet is 0 minus what leaves is the volumetric flow rate V of the effluent stream multiplied by the concentration of the tracer C and that should be equal to V into dC by dt which is the accumulation of the tracer in the CSTR. Now because the concentration of the species inside the reactor is equal to the concentration at which the species is actually leaving the reactor the C here essentially represents the outlet concentration of the species from the reactor that reflects the concentration of the species with which it actually leaves the reactor in the effluent stream. So now one can actually integrate this expression to find out that C of t is equal to C naught into exponential of minus t by tau where T C naught is the initial tracer concentration initial pulse tracer concentration concentration of the initial tracer that is actually fed as a pulse tracer and from this we can find out that E of t is given by C t divided by integral 0 to infinity C of t dt. So now we know the expression for C t the dependence of C on time and other properties so we can plug that in here we will see that is exponential of minus t by tau divided by integral 0 to infinity minus t by tau. So performing the integration we will find that because C naught is constant one can actually cancel out C naught from the numerator and denominator and so we will find that this will be equal to 1 by tau into exponential of minus t by tau. So that will be the residence time distribution function for a single CSTR. Now in terms of the dimensionless in terms of the normalized RTD function E of theta is essentially given by exponential of minus theta where theta is actually t by tau and E of theta is nothing but tau into E of t. So that is the normalized residence time distribution function and now we can actually find out what is the f curve so f of theta is nothing but integral 0 to theta E theta into d theta that is actually 1 minus exponential of minus theta. So that is the f curve that is the expression for f curve which is 1 minus exponential of minus theta where theta is t by tau and tau is the space time of the reactor where tau is nothing but v by v naught that is the space time of the reactor. So let us attempt to sketch the E curve and the f curve. So the normalized RTD function so the E curve essentially looks like this it is an exponential dk and then the corresponding f curve is actually looks like this. So this is 1 and this is theta. So it actually essentially looks like this and so the mean can actually be estimated as tm that is equal to integral 0 to infinity. These are different properties of the distribution t into E of t dt that should be equal to integral 0 to infinity t by tau into exponential of minus t by tau and that should be equal to tau. So that is exactly what we observed before if there is no dispersion then irrespective of whatever is the RTD then the mean distribution time should be equal to the space time of the reactor itself and now the next the variance sigma square is given by 0 to infinity t minus tm square into E of t dt and that should be equal to tau square integral 0 to infinity x minus 1 the whole square into exponential of minus x dx. So where the change of variable is done by by setting x equal to t by alpha and so integrating this is a standard expression so while integrating this expression one can find that is equal to tau square which means that the standard deviation of the distribution is actually equal to the space time of the reactor itself. So for a single CSTR for a single CSTR the mean residence time is equal to the space time of the reactor and the standard deviation of the residence time function is also equal to the mean residence time of the reactor itself. So now if we compare the various compare the RTD function and the various properties of CSTR we can find that so suppose if you make a comparison we can summarize the function and the properties that we have found so far for a plug flow reactor and a CSTR. So the residence time distribution function E of t is essentially the delta function for a plug flow reactor which means that there is just a delay and whatever is fed into the reactor is going to come out of the reactor exactly after a certain delay and the delay is given by the space time of the reactor and here delta x is actually defined as 0 for x not equal to 0 and infinity for x equal to 0 and the corresponding RTD function for CSTR is 1 by tau exponential of minus t by tau where tau is given by the v by v tau is the space time which is given by volume of the reactor divided by the corresponding volumetric flow rate and then the mean residence time for a plug flow reactor is given by tau and it is the same for the CSTR because there is no dispersion and so the mean residence time should be equal to the space time of the reactor itself and the variance for a plug flow reactor is 0 while for a CSTR it is actually equal to the space time of the reactor itself and then the f curve actually is 1 for a plug flow reactor and it is 1 minus exponential of minus t by tau for a CSTR. So that summarizes the various properties of the RTD that summarizes the RTD function and the various properties of the function for the plug flow reactor and a CSTR. So next let us look at the another reactor laminar flow reactor. Let us try to estimate the RTD function for the laminar flow reactor LFR will be referred to as LFR hereafter. So suppose if there is a tank and there is a fluid stream which is actually entering at 0 and leaving at L so that is the length of the reactor that is L and the fluid is actually entering under laminar conditions and it is expected that there will be a parabolic velocity profile, there will be a parabolic velocity profile with maximum at the centre and 0 near the walls, maximum at the centre and 0 near the walls. So suppose if the centre of the reactor is so that is the centre of the reactor and that is r equal to 0. So if I label this coordinates as r and at r equal to 0 it will be maximum velocity and at r equal to r which is the periphery the velocity will be 0. So that is a parabolic velocity profile that is the parabolic velocity with which the velocity profile with which the fluid is actually flowing through the reactor. Now clearly this suggests that the fluid particles which are actually fluid elements which are actually at the centre they will actually have the shortest residence time because they have the maximum velocity and so they will leave the reactor much faster than the they will leave the reactor faster than the other fluid elements which are actually present in other radial locations other than 0. So now so therefore the velocity profile u is actually given by u max which is the maximum velocity at the centre multiplied by 1 minus r by r the whole square. Now often this maximum velocity may not be known so instead what may be known is the average velocity that is the velocity of the fluid stream averaged across the whole cross section and that can actually be estimated from the velocity profile from the local velocity expression. So u average which is the average velocity at a given cross section is given by volumetric flow rate divided by the area of the reactor at that cross section and that is given by 1 by pi r square into integral 0 to r u max into 1 minus r by r the whole square into 2 pi r dr. So here I have assumed that if this is the cross section of the reactor if that is the cross section of the reactor then let us assume that there is a small element which is present here from the centre and that is located at a distance r and the thickness of this is actually dr. So therefore the volumetric flow rate of the fluid and at any cross section is given by the local velocity multiplied by 2 pi r dr into integrated over the whole integrated between 0 and r. So that gives the volumetric flow rate at that cross section and pi r square is the corresponding area at that cross section. So from this integrating this expression we will find that will be equal to u max by pi r square multiplied by 2 pi r square by 2 minus 2 pi by r square into r power 4 by 4 into and the limits are 0 to r and that is equal to u max by 2. So the maximum velocity is simply twice the average velocity that is the average over the cross section of a of the reactor and in fact the average velocity is also called as the cup mixing average and so u max is equal to 2 times u average. So substituting this in the expression for the velocity we can actually rewrite the velocity expression as u equal to 2 times u average multiplied by 1 minus r by r the whole square and that is equal to 2 v naught by pi r square v naught is the volumetric flow rate with which the fluid is actually flowing at that cross section into 1 minus r by r the whole square. So that is the expression for the volumetric velocity with which the fluid is actually flowing as a function of the radial position. Now we can now estimate what is the time that is actually spent by the fluid particles at a that is entering at a given location r. So that is actually given by the length of the reactor L divided by the velocity with which the fluid is actually flowing in that radial location r which is actually u r and that is given by pi r square by v naught into L into 1 by 2 times 1 minus r by r the whole square. So that is the time that is taken by different fluid elements that is actually entering the reactor at any r location. So that is actually equal to tau divided by 2 into 1 minus r by r the whole square where tau is given tau is the space time of the reactor which is given by v divided by v naught. So now we need to relate the we need to now relate what is the we need to find out what is the RTD function ET. So in order to find that we need to know what is the fraction of this fluid that is leaving and what is the age of that particular fluid. So now the volume of the fluid the volumetric flow rate between r and r plus dr. So that is the volumetric flow rate of the fluid which is actually flowing between r and r plus dr and that is given by dv that is equal to u r into 2 pi r dr. So that is the volumetric flow rate of the fluid which is actually flowing in this element dr that is between r and r plus dr. So now the fraction of the total that is actually flowing through this small element dr is actually given by dv divided by v naught where v naught is the total volumetric flow rate dv by v naught gives the fraction of the fluid that is actually flowing through this element dr. So that is given by u r divided by v naught into 2 pi r dr. So that is the fraction of the fluid that is actually flowing through this element dr and in fact that is nothing but the E of t into dt because the fraction of the fluid that is actually flowing through this small element dr and also the fluid which is actually between v and dv which is spending the time t and t plus delta t is what is given by this RTD function E of t dt and that should be equal to dv by v naught which is actually the fluid which is flowing between v and v plus dv whose residence time is actually between t and t plus delta t. So what we have seen so far in this lecture is essentially different properties of the residence time distribution which is the mean we have looked at the variance and we looked at this queerness and then we went on moved on to the residence time distribution of the ideal reactors particularly we considered the plug flow reactor and then we found out what is the residence time distribution for this particular reactor and what are the properties of the residence time distribution and specifically we found out what is the E curve and the F curve as related to the time as a function of time and next we looked at the residence time distribution function for a single CSTR we found the E curve and the F curve and the corresponding properties and then initiated discussion on the laminar flow reactor. Thank you.