 Welcome to the lecture series on process integration. This is module 4, lecture 4. The topic of the lecture is problem table algorithm second part. In this lecture, we will see a new problem table algorithm which is used to find out the hot and cold utilities. Based on the conventional problem algorithm, one can determine hot and cold pinch temperatures and can also target hot and cold utility demands. However, the hot and cold utilities thus computed are at highest and lowest temperatures. The above algorithm is not useful if one needs to accommodate multiple level hot and cold utilities. For this purpose, grand composite curve a graphical method is employed in pinch technology to accommodate multiple level hot and cold utilities. In this present lecture, an extension to the problem table algorithm based on the work of Andre et al. 2009 is suggested to directly deal with multi level hot and cold utilities. To demonstrate this algorithm that is extension to the problem table algorithm, a four stream problem for modified problem table algorithm for delta T minimum equal to 20 degree is taken. There are two hot streams and two cold streams whose source temperature and target temperatures are given and in capacity flow rate are given. If you do the PTA analysis of this stream table, then we find that hot utility requirement is 105 kilowatt, cold utility demand is 30 kilowatt and pinch temperature is at 80 degree centigrade. Now, this is the grid diagram of the problem. We see there are three streams in the above pinch area and four streams in the below pinch area. This is the composite curve. It very clear shows that multiple hot utilities can be used. We can use a hot utility here. If you want, you can use a hot utility here. You can use a hot utility here or you can use a hot utility here based on the delta T minimum criteria. This is the cascading and these are the temperature levels and it shows the amount of heat cascading down. Now, let us see some nomenclature which is being used for the present algorithm. In this procedure, the number of utilities and the temperature regions must be noted at the beginning of the procedure. The exact temperature level of each utility are determined later from the algorithm, from these ranges. The selection of utilities are based on transition temperature between different utilities. This is very important line in which we will keep different utilities based on that transition temperature. This will be clear. For the illustration considered here, table below shows the corresponding values of two hot utilities and two cold utilities. Hot utilities are numbered from hotter to cooler. This is very important. That means, if I say hot utility number one, then it will be hotter than the hot utility number two and hot utility number two will be hotter than hot utility number three. So, hot utilities are from one to NHU and the cold utilities are from one to NCU number and cold utility number one will be cooler than the cold utility number two and cold utility number two will be cooler than cold utility number three. So, hot utility one being hottest and cold utility one being coldest in this range. So, for this problem, we are taking two hot utilities and two cold utilities. Hot utility number one and hot utility number two and that transition temperature is 130. That means, 130 and above we are using hot utility number one and under 30 below we are using hot utility number two. Cold utility number one and cold utility number two. That means, two cold utilities we are using. So, the cold utility number one is below 65 and cold utility number two is equal to 65 or above 65. So, in this algorithm, the transition temperature of hot utilities and cold utilities are given. Now, if we analyze what is the range of hot utility or the bounds of hot utility, then we will see that the maximum bound of the hot utility is this and the minimum bound of hot utility is the pinch temperature. So, the all the hot utilities will range from here to here whether I am using two hot utilities or two cold three hot utilities or four hot utilities. So, the maximum bound will be within these two lines. Similarly, the minimum bound for the cold utility will be this and the maximum bound for the cold utility will be the pinch temperature level. So, all the cold utilities will fall within this range. Now, here in this diagram which is a grand composite curve and this is shifted temperature versus delta H. Here we have put the transition temperature of the hot utilities. So, this line is the transition temperature of the hot utilities. Above this is hot utility one is used and below the hot utility two is used. Similarly, the cold utility transition temperature is given by this line. So, cold utility two is used above it and cold utility one is used below this and these are the different temperature levels of the system. So, between this and this temperature this is level interval one, this is interval two, this is interval three and this is interval four, this is interval five, this is interval six, this is seven and this is eight. So, there are eight intervals and these are the heat input and output to a interval. So, to the first interval heat input is 105 and heat output from this interval is 117.5. So, for these values these interval values these are the heat input and heat output and in this algorithm we will calculate this value and based on these values all decisions will be taken. In this approach the transition temperature between utilities are added to the set of source and target temperatures and the problem table algorithm is then applied. This is a very important factor that what we have done the transition temperatures of the utilities that means hot utilities as well as cold utilities are added to the source and target temperatures of different streams and the problem table algorithm is then applied. So, now the transient temperatures are a integral part of the problem table algorithm. Now, if you see this the actual and shifted temperatures of hot and cold utilities we know that delta T minimum is 20 degree. So, here if you take the actual temperatures of the hot streams the highest temperature is 150 degree, then 60 degree, then 90 degree, then 60 degree and this T H U is the transition temperature between the two hot utilities hot one and hot two. Now, if you find out the shifted temperature of this the delta T minimum is 20 degree centigrade. So, we deduct 10 degree centigrade to this because the shifted temperatures are actual temperatures minus delta T minimum by 2 and delta T minimum by 2 in this case is 10. So, 150 minus 10 is 140 degree. Now, for this 140 degree centigrade if I compute this actual temperature of the hot and cold. So, if it is a hot shifted temperature I have to add 10 degree that becomes 150 and if this is a cold shifted temperature then I have to deduct 10 degree from this 140. So, it becomes 130. So, my range is 150 to 130 that means, if it is a hot temperature it is 150 actual temperature if it is a cold 130. So, two actual temperatures of hot and colds are written for this shifted temperatures here. Similarly, here 60. So, I will deduct 10. So, it becomes 50. So, for hot it is 60 and for cold it is 40. Similarly, this is 90 I deduct 10 this is 80, 90 to 70. So, I write down these temperatures like this and for this T H O T is the transition temperature of hot utility 1 to 2 I also write the same thing. Then for cold streams I repeat this this is 20, 125, 2500 and this is the transition temperature of cold 1 to cold 2 which is 65 degree centigrade in this case. So, here we will add 10 degree to this to convert it to the shifted temperatures. So, this is 30. For this 30 shifted temperature if I convert it into a hot temperature then it becomes 40 and for sorry cold temperature it become hot temperature it will be added 10. So, 40 and cold temperature it is 20. Similarly, hot and cold temperature for all temperature levels that is shifted temperature levels we have computed. This thing that is mass at a strict mark the temperature inside this bracket shows the actual hot stream temperature and actual cold stream temperatures separated by a dash for the given shifted temperature. This is used for computing the effective temperature range of multiple utilities. Why we have done so because we have we want to calculate the actual temperatures of the multiple utilities that means actual temperature range of the multiple utilities and that is why we have preserved this data and with the temperature levels. Now, here all temperature levels shifted temperature levels have been written that is all shifted temperature levels of the hot as well as all the shifted temperature levels of the cold and these are the temperature intervals 1 to 8. These are the actual temperature range that is hot to cold for this shifted temperature intervals. Similarly, for all the shifted intervals I have written the hot to cold temperatures and this we name T 1, T 2, T 3, T 4 to T 9 these temperatures. Then if you see the range of streams that means the streams are working number one stream is working from this shifted temperature level to this shifted temperature level. The C p value is 2.5 2 is working from this shifted temperature level to this shifted temperature level C p is 8. The 3 is working from this temperature level to this temperature level and this is C p is 3 and 4 is working from this temperature level to this C p is 3. Now, we will use this information for making other tables. So, here the shifted temperature levels these are the temperature intervals and this is delta T T i minus T i minus T i plus 1. So, here 140 and 135 the difference is 5 here 135, 120 difference is 15. So, we have written the difference of two temperature levels in the interval. Then summation C p c minus C p h value. So, based on the stream availability of stream at different temperature levels which has been shown you in the last slide we have computed these values 2.5, minus 0.5, minus 0.5, minus 3.5, 4.5, 4.5, minus 6 and 3. Then we have multiplied this with this that is delta T with the summation C p c minus summation C p h values and this is computed. So, this is minus 12.5, this is minus 7.5, this is 5, this is 105, this is minus 22.5, this is minus 112.5, 90 and 15. Now, a negative quantity shows the surplus that means in this temperature interval heat is available. And in this temperature level which is below this two number temperature it is deficit because we have a positive sign here. So, negative sign shows surplus and positive sign shows deficit. So, this is a deficit temperature interval, this is also a deficit temperature interval, this is also a deficit temperature interval, this is a surplus interval, this is a surplus interval, this is a deficit interval, this is a deficit interval. So, we also know that heat will flow from higher temperature to lower temperature naturally. So, the heat available here can be given to this to compensate this deficit, but heat from here to here it cannot go, but when the deficit is more then the heat has to come from top that means from the hot utility. So, let us see that now suppose 0 heat is coming from outside to this temperature interval. So, the heat requirement is 0 minus 12.5 this is 12.5 the heat requirement here is 5, heat requirement is 0 here, heat requirement is here minus 105, 82.5, 30, 60 and 75. So, maximum negative quantity is 105 and this is the requirement at this temperature level. Now even so to make it 0 what we have to do we have to add the heat from the top. So, if you add 105 heat input to this interval then we see that in this interval this is 117.5, this is 1110, this is 105 and this becomes 0, this is 22.5, this is 135, this is 45 and this is 30. So, this is the heat flow in different temperature intervals and we see that all these values are positive in nature and that is why this is a feasible table that means if it works like this the system will work or hen will work. Now in this algorithm which is little bit different than the earlier PTI algorithm the heat input to different levels and heat output from different levels are recorded simultaneously and this two values or columns which will be generated now will be used for the computation of the requirement of different hot utilities as well the requirement of different old utilities plus this information will be used to compute the temperature range of hot utility 1, temperature range of cold utility 1 and temperature range of cold utility 2 and the hot utility 2. So, here let us try to understand to this temperature this interval the input is 105 and 170 is the heat output from interval i equal to 1, but it is input to i equal to 2. Similarly 110 is the heat output from i equal to 2 and heat input to i equal to 3. So, 105 is heat output from i equal to 3 and input to 4 and this is 0 is heat output from i equal to 4 and input to i equal to 5. So, this 0 shows this is the pinch heat out from this 22.5 shows heat output from i equal to 5 and input to i equal to 6. Similarly we can write down for 135, 45 and 30 which is the heat output from i equal to 8 and heat input to none. So, this is our cold utility requirement total cold utility requirement and this is our total hot utility requirement. Now, this shows the cascade heat input to different intervals and this shows the cascade heat output to different from different intervals. So, heat output to interval heat input to interval 2 is 117 and heat output from interval 1 is 117.5 this is input to second interval. So, we have similar values here because one is input other is output this shows our cold utility requirement that is 30 and this shows our hot utility requirement which is 105. Now, based on the last two columns this is cumulative Q in and this is Q out cumulative Q in and cumulative Q out and these are the temperatures that is hot and cold temperatures of a temperature level. Now, we divide this into sub tables now this is the transition temperature 130. So, above this hot utility 1 is used. So, this is the table this value are in this table which we called sub table 1 and this is for hot utility 1 this is hot utility 1 sub table 1. So, these are the data which is available here. Similarly, here up to this is the pinch temperature up to this the hot utility 2 works in this region. So, this is hot utility 2 sub table 2. So, from this temperature level to this temperature level the data is this will be used for computation of hot utilities and cold utilities for different temperature levels. This is for cold utility 2 and similarly this is for cold utility 1. So, this is table number 3 and this is cold utility 1 table number 4. I will see that this is cold utility 1 this is cold utility 2 because cold utility 1 will be the coolest 1. So, it will be in lower temperature than cold utility number 2, but here this is reverse hot utility 1 will be the highest temperature and hot utility 2 will be at the lower temperature. So, computation for hot utility will from this direction and computation for cold utility will be from the bottom side. So, this shows cold utility 1 range this is cold utility 2 this is hot utility 2 and hot utility 1. So, now this has been divided into sub tables 1, 2, 3 and 4 and then operations will take place in this sub table. So, if I am interested in computing the amount of hot utility 1. So, I will work with this sub table. If I am interested in hot utility number 2 will work in this table if cold utility 2 will work in this table and cold utility 1 will work on this table. So, these 3 columns will be used for calculating the amount of hot utility 1, hot utility 2, cold utility 1, cold utility 2 and also the temperature ranges of this hot utilities and cold utilities this we are going to see. Now, let us see the algorithm now how to compute the amount of hot utilities for each hot utility sub table h 1 to n h u the consumption is evaluated by the following algorithm. So, first job will be to compute hot utility 1. So, this is the table we will be using table number 1 for the hot utility. Evaluate the difference delta h u between the total minimum hot utility consumption and the sum of the utility consumption values computed in the previous sub tables. For hot utility number 1 the total minimum hot utility is 105 this is the total minimum hot utility consumption and consumption above this is 0 because there is no table above this hot utility consumption utility 1. So, its consumption is 0 then identify the smallest heat load r h u of this sub table right column. So, for hot utility computation we will go for this right column and for the cold utility consumption we will go for this left column. So, in this table the right column is this and there are two entries in the right column 117.5 and 110. So, we have to calculate the identify the smallest heat load of the sub table right column. So, this is 117.5 and 1110. So, obviously smallest load will be 1110 the hot utility consumption h u correspond to the result indicated by equation 1 below. So, this is the equation we will be using to compute the amount of hot utility 1 and 2. Now, below this if it is positive otherwise it means that there is no utility consumption at this line. That means, if the value of u h h u comes out to be negative we will take it to be 0 and if it is positive then it has got a meaning and it will show the hot utility consumption. Similarly, when we go for the table 2 that means, hot utility 2 we will take these two values to compute. Now, let us see for the cold utilities now for the cold utilities for each cold utilities sub table C 1 to N C u will be using now the cold utility 1 is this area. So, there are three entries in the left column of the sub table and we will use this and this 30 shows the total minimum C u consumption that is cold utility consumption. Now, evaluate the difference delta C u between the total minimum cold utility consumption and the sum of the utility consumption values calculated in the previous sub table. Now, there is no previous table below this cold utility 1. So, total minimum consumption is 30 minus 0 because there is no sub table here below this. Then identify the smallest heat load L C u of the sub table left column so, in the left column these are three entries in the sub table. So, I have to calculate the minimum of these three then the cold utility consumption U C C u correspond to the results indicated by equation 2 this is the result I will be using and here also that if it is positive then all right if it is negative then this will be taken as 0. These evaluations can be conducted over the sub table or problem table directly without any complex mathematical manipulation as we will see now. So, when we calculate a hot utility or cold utility we will remain in its table. So, data of this table will be used. Now, let us go for the computation. So, we go for the hot utility 1. So, here it shows the tables this is table 1. So, these 4 values are there this shows the marking. So, we will compute the hot utility from here we will come here and cold utility from this direction. So, hot utility estimation now hot utility number 1 sum of the total minimum of the utility consumption value in the previous sub table is 0, because there is no previous sub table. So, energy consumption or hot utility consumption in the previous table is taken to be 0. So, delta H u is calculated by this 105 this is the hot utility requirement minus the hot utility consumption value in the previous table what is which is 0. So, this is 105 minus 0. Now, the R H u value is the minimum of this 2. So, 101, 17.5 and 110 the minimum is 110. So, U H H u can be calculated from this equation it becomes 105 minus 110 which is minus 5. As we have already told that if U H H u value is negative then it will be taken as 0 and if it is positive then only we will take the value of the hot utility. So, as the value of U H H u is negative it is taken to be 0. So, U H H u is 0. So, we can conclude here that the hot utility demand or hot utility 1 demand for this problem is 0. Now, we go for the hot utility number 2 how to compute this. Now, some of the utility consumption values in previous sub table is 0, because we know that the hot utility requirement of hot utility 1 is 0 and above the hot utility 1 also the requirement is 0. So, some of the utility consumption values in previous sub table is 0. So, this is very clear. Now, delta H u will be the hot utility requirement minus the sum of the utility consumption values in the previous table which is 0. So, here also it is 105 minus 0, but this 105 comes from here that is hot utility requirement. So, delta H u is 105 minus 0, R H u will be calculated from this table this is utility number 2 table. So, minimum of 105 and 0. So, this is minimum of 105 and 0 is 0. So, delta H u value is 105 R H u value is 0. So, I can put it here U H H u equal to 105 minus 0 and this comes out to be 105 sorry this is not negative this is 105. So, U H H u is 105 kilo watt this is 105 minus 0 is 105 here is not negative. So, this is 105 kilo watt. So, my hot utility consumption of the utility number 2 is 105. If I see the hot utility requirement is 1 is 0, hot utility requirement 2 is 105 and if you remember that the hot utility requirement of the total problem is 105 also that is hot utility number 1 plus hot utility number 2 is 105 which is the requirement of total hot utility for this problem. Now, cold utility estimation now from here in this direction we will estimate the cold utility because cold utility number 1 is the coolest cold utility. So, here if you see the sum of the utility consumption value in the previous sub table it is 0 because there is no cold utility requirement below this table. So, it is taken to be 0 now cold utility requirement is 30. So, delta C u is 30 minus 0 which comes from here. Now for cold utility we will go for this left hand side column for hot utility we are using this, but cold utility for L C u will go for this computation. So, L C u will be minimum of this three quantities. So, minimum of 135, 45 and 22.5 this comes out to be 22.5. So, now delta C u is known to us and L C u is known to us. So, you see C u is delta C u minus L C u is 30 minus 22.5 this is 7.5. So, the cold utility consumption for utility number cold utility number 1 is 7.5 kilowatt this is a positive value. So, we accept this. So, this is 7.5 now let us go to the cold utility number 2. So, sum of the utility consumption values in previous sum table is 7.5 because the cold utility requirement was 7.5 and hence sum of the utility consumption values in previous sub table is 7.5 and below this was 0. So, obviously this is 7.5. So, delta C u will be 30 which is the cold utility requirement minus the sum of the utility consumption values in previous sub table which is 7.5. So, it comes 7.5 here. So, when I deduct 7.5 from 30 it is 22.5 kilowatt. Now, L C u is 0 as this is only one entry here and minimum of 0 is 0. So, U C C u for the utility number 2 is delta C u minus L C u this is 22.5 minus 0 is equal to 22.5. So, the cold utility demand for the utility number 2 is 22.5. Now, if I add these two values that means cold utility demand 1 and cold utility demand 2 then this is 22.5 plus 7.5 which is 30 kilowatt and we know that for this problem the total cold utility demand is 30 kilowatt. Once we have computed the hot utility 1 demand, hot utility 2 demand, cold utility 1 demand and cold utility 2 demand then we will try to compute the temperature ranges of these utilities. Now, you see the algorithm for the determination of feasible utility temperature ranges. The analysis of the sub table of problem table can also identify the thermodynamic constraints that can reduce the temperature range of the utilities. These reduction include the need for a lower bound on a hot utility temperature higher than the original one and a upper bound on a cold utility temperature lower than the original one. This can be done as follows. For hot utilities for each hot utility sub table 1 to n h u the temperature range is analyzed through the following algorithm. If there is no utility demand the original temperature range is not modified because the introduction of energy could be done in any level inside of the temperature range since it would cascade down entirely. Otherwise we have to take the step 2 which is step 2 here. The temperature upper bound of the utility correspond to the original one and the lower bound is determined by interpolation. So, for this interpolation along the right column this is the right column along the right column that is q cumulation out of the current sub table. So, if I am working in sub table 2 this is the values which I will work for. So, corresponding to identity of the position t star which correspond to the first t in q cumulative out column from top to bottom such that q t 1 to h u is less than or equal to delta h u. In our case delta h u is 105. So, in this column I will come from top to bottom. So, at the top I get 105 here. So, this is the point t star because this condition is satisfied this condition is satisfied q 1 to h u is 105 delta h u is 105. So, this condition is satisfied and hence this is the t star point. Here subscript 2 represent the column corresponding to q cumulative out. This is the column which is in question and this is represented by subscript 2. Thus for sub table corresponding to heat utility 2 t star corresponding to the value of q cumulative out in the first row of the column and hence this is the value q t star 2. This is the value 105 is equal to delta h u which is the hot utility requirement in this case because for the utility number 1 the hot utility requirement is 0 and that is why delta h u is 105. Now, the lower bound can be evaluated by this expression. This is the expression to compute the lower bound of the hot utility where t ceiling h u and t floor h u are the hot stream temperatures levels above and below the position t star. Now, this is the t star position if I draw a line here this is above and this is below. So, this is the hot temperatures we are dealing with and when we will compute the cold temperature we will go for this. So, in this case 130 is the t ceiling and 120 is the t floor h u. So, this is the line and above is this and below is this. So, t ceiling is 130 t floor is 120. The remark is that if the position t star is located at the first line of the sub table the interpolation will involve the last position of the previous sub table. That means, if it is in the first line then the interpolation can use a data from the above sub table that is here 110. In this case if the current sub table is the first one it means that the temperature lower bound corresponding to original upper bound. Similarly, we can have for the cold utilities if there is no utility demand the original temperature range is not modified otherwise we have to go to step 2. The temperature lower bound of the utility corresponds to the original one and the upper bound is determined by the interpolation. Here we are through interpolation we are finding out the upper bound of the cold utilities. Along the left column now you will use this column which is q cumulative in. So, remember for hot utility we have used this column for cold utility we will use this column and for cold utility we will use this value which are cold stream actual temperature values. Along the left column q cumulative minimum of the current sub table identify the position t star which correspond to the first position t in q cumulative in column from top to bottom. So, here this is the top to bottom we will go. So, this is the temperature this is the value which satisfy because delta t c u value is 30 because there is no consumption of cold utility below this. So, this is 30 and this is 30 is greater than 22.5 and that is why this is the value denoted by t star and this is q t star 1 c u 1 stands for this column. So, this condition is satisfied and hence this is the t star value here subscript 1 represents the column corresponding to q cumulative in. Thus for sub table corresponding to cold utility 1 t star correspond to the first position of the column q cumulative in of that sub table. So, this is q t comma 1 c u is equal to t star comma 1 c u is equal to 22.5 is greater than 30 that is greater than this and that is the value of delta c u. So, this is my t star value. Now, again as we have done in the case of hot utility to find out the bounds here we will use this by this expression where t ceiling and t floor are the cold steam temperature levels above and below the position t star. For example, in the sub table corresponding to cold utility 1 t star is equal to 22.5 this we have seen. So, if we I see the table above is 65 and below is 40. So, t ceiling for cold utility is 65, t ceiling for t floor for cold utility is 40 it is wrongly written t h u this is should be c u this should be c u. So, this is 65 and 40. Now, we can compute the t star value temperature ranges. Now, for the computation of hot utility 1 temperature no utility you says in this range because we have computed that no hot utility is required in this range that is 132, 150. So, the range is 132, 150 because 150 is the highest temperature and 130 is the transition temperature of hot utility to 1 to hot utility 2. So, h u 1's range is 130, 250 it remains and there is no consumption of hot utility in this range. So, we will calculate the for hot utility 2. So, for hot utility 2 temperature range hot delta h u is the transition temperature is 105 this we have already computed q t 2 h u q t star 2 h u is 105 this we all know and computed q t minus 1 2 h u is 110. So, this is the a t star now t minus t star minus 1 is this value it is the in the first entry. So, minus 1 entry is taken from the above table this is 110. So, this is taken as 110 now t ceiling is 130, t floor is 120. So, this is 130, 120 this is our equation if I put this values are all known to me if I put this values all these values here then I find that this is 120 degree centigrade. So, the feasible range for h u 2 which is hot utility 2 is 120 to 130 degree centigrade. Similarly, we will compute for cold utilities now if I compute for the cold utility 1 temperature range delta c u is 30 minus 0 q star 1 c u is 22.5 this we have already computed in the earlier slides t star plus 1 i is 135 if you see here that is t star and this is t star plus 1 and the above is t star minus 1. So, t star minus 1 we have used for hot utility computation and for cold utility computation it is t star plus 1 which is this value t ceiling is 65 this is t floor is 40 this value. So, all the data is available to compute this expression and when we compute this it comes out to be 63.33 degree centigrade. So, the feasible range for cold utility 1 is 20 to 63.3 degree centigrade 20 is the lowest temperature. So, from the lowest temperature to the 63.3 degree centigrade is the feasible range of cold utility 1 and we can use any temperature within this range if some cold water is available within this range of temperature we can use it as cold utility 1. Now for cold utility 2 let us compute the computation of cold utility 2 temperature range. So, in this case delta c u is equal to 30 which is the cold utility requirement and consumption of cold utility in the previous subtable is 7.5. So, delta c u comes out to be 22.5 this is q t star c u is equal to 0 here this is 0 and q t star plus 1 c u is the next lower value here this is 22.5 and if I go here this is t ceiling c u is 70 and t floor c u is 65 these are two temperature which are one away and one below this. So, this is the value. So, when these values are known to me I can use this equation. So, when I calculate this become 65 degree centigrade. So, the feasible range of this is c u 2 this is c u 2 is 65 degree centigrade. So, this is my results. So, hot utility number 1 the consumption is 0 range is 130 to 150 hot utility number 2 the consumption is 105 kilowatt. The range is 120 230 cold utility number 1 the consumption is 7.5 the range is 20 to 63.3 and cold utility 2 the consumption is 22.5 and the it can be given only at one temperature that is 65 degree centigrade. Now, we can place this in a grand composite curve because grand composite curve is a curve in which multiple utilities are placed and this has been created for that. So, let us place it. So, this is my grand composite curve I can use the hot utility at this temperature the maximum temperature the requirement is 105 and cold utility as the lowest temperature my requirement is 30 kilowatt and this is a shifted temperature. Now, this shows the bound of hot utility to the highest temperature is 130 degree centigrade and the lowest temperature is 120 degree centigrade. So, we can use the hot utility in any temperature between these two temperatures. Similarly, this is cold utility 1 that is a large range for this. So, this is cold utility 1 which range is 20 to 63.3 degree centigrade this is cold utility 1 not 2 and this is cold utility number 2 this is not 1 this is 2 which is 65 degree centigrade. The amount of cold utility number 1 is 7.5 kilowatt and the amount of cold utility 2 is 22.5 kilowatt and it will be delivered at 65 degree centigrade and cold utility 1 can be delivered from 20 to 63.3 degree centigrade. So, at any temperature we can deliver this. So, what we have understood that a the same PTA can be used to find out different hot utilities and different cold utilities. If we know the transition temperatures of this hot utilities and cold utilities which is easier to find in a industry once this transition temperatures are known we can use this modified PTA to find out the amount of these utilities and their temperature ranges which is very beneficial for us. Otherwise we have to use the GCC to find out this hot and cold utilities. Thank you.