 Welcome to the lecture series on process integration. Today, we will see how the number of units target is created. So, the topic of this lecture is number of unit target is module number 4, lecture number 5. We have already seen how to do targeting of hot utility and cold utility and the unit this is the second lecture on targeting. Here, we will target the number of units in the heat exchanger network. This is a known fact that the capital cost of chemical processes is dominated by the number of items on the flow sheet and hence there is a strong incentive to decrease the number of items on the flow sheet. This logic is also true for heat exchanger networks. There is a strong incentive to reduce the number of matches between hot and cold streams. This units targeting gives prior to the design the minimum number of units in the heat exchanger network. Suppose, I have designed two heat exchanger networks N 1 and N 2. If N 1 has got 10 number of units and N 2 has 15 number units, obviously I should go for N 1 because the cost of N 1 will be less than the N 2 and this logic is based on the fact that one big exchanger cost less than the two small exchangers which has equal area of that big heat exchanger. Now, let us demonstrate this numbers of unit targeting with an example. We see a stream table which has seven streams, four hot streams and three cold streams having delta T minimum is equal to 10 degree centigrade. The PTA analysis of this problem shows that it is a threshold problem and needs only cooling and no heating. That means, hot utility stream is not present and only cold utility stream is present. So, if we count number of streams including the utility streams, it will be seven process streams and one cold utility streams making total number of stream count to be 8. The minimum cooling load required for the above system is computed using PTA as 2126.89 units. Now, if I translate that stream table into a picture like this, we have four hot streams, hot one is 373.66 kilowatt, hot two is 604 kilowatt, hot three is 2100.8 kilowatt and hot four is 721.6 kilowatt, cold three is 819.36 kilowatt, cold two is 297.8 kilowatt, cold one is 556.01 kilowatt and cold utility requirement using the PTA is 2126.89 kilowatt. Let us remember again that this problem does not need a hot utility and that is why hot utility stream is not present here. Now, our aim is to design a heat exchanger network which will satisfy the need of cold utilities, cold streams from the heat of the hot streams and excess heat which is available with the hot stream will be pushed to the cold utility. So, total number of streams including the cold utility stream is 8 in this case. So, we start designing from the right to left. So, the cold three stream requires 819.36 kilowatt. Now, I can push the total heat available with the hot four stream that is 721.6 kilowatt at once to the cold stream. I cannot also push by dividing this heat into two three parts, but if I do so the number of heat exchangers will increase. Now, in a unit's target we always take of the streams to bring down the number of heat exchanger to a bare minimum. So, 721.6 kilowatt heat of hot stream number four is pushed to the cold stream number three. After pushing this heat the cold stream number three requires 97.76 kilowatt of heat to satisfy it. This heat comes from hot stream number three to cold stream number three and hence now the cold stream number three is satisfied, but there is some heat available with hot stream number three. Similarly, we see that cold stream number two requires 297.8 kilowatt heat and hot two has 604 kilowatt of heat. So, it can directly satisfy the cold two by giving its heat. So, it pushes 297.8 kilowatt of heat directly to the cold stream number two and ticks it up. That means this amount of heat satisfy the cold stream. The remaining heat with the hot two which is 306.2 is now pushed to cold stream number one which needs 556.01 kilowatt. Hence, it partially satisfy the cold one and another 294.81 kilowatt is required by the cold stream to be completely satisfied. This heat comes from hot one who has which has got a value of 373.6 kilowatt with it. So, the remaining 123.85 kilowatt with hot one will now be pushed to the cold utility because it is the extra heat which is available with hot one. Now, we see that some heat is available with hot three which is 2003.01 kilowatt extra and some extra heat is available with hot one. These two extra heats are now pushed to the cold utility which is 2126.89 kilowatt and which satisfies the cold utility. So, what we see now the all the heat which is available with the hot streams are pushed to the cold streams as well as cold utility. So, the whole system is in thermal balance. Now, to do so, we see that we are using seven number of heat exchangers H X 1, H X 2, H X 3, H X 4, 5, 6 and 7. So, can we develop a method to calculate that how much number of heat exchangers will be required if number of streams including hot utility and cold utilities are known to us. So, we see that 8 minus 1 is 7 that means we have 8 number of streams minus 1 gives us the number of heat exchangers required for this design. Let us take another example in this process only hot utility is required, but no cold utility is required. So, hot utility as steam is available with 1068 units of heat, one hot steam is available as 2570 units of heat. There are 3 cold streams cold stream number 1 requires 2233 units of heat, cold stream number 2 requires 413 units of heat and cold 3 requires 992 units of heat. Now, we have to develop a heat exchanger network for this so that the heat available with steam as well as hot stream number 1 is passed to this cold streams numbering 3. So, the total amount of heat available with steam which is 1068 is now pushed to cold 1 which requires 2233 units of heat. So, obviously this steam with its 1068 units of heat is not able to satisfy completely the cold 1. So, some heat from the hot stream number 1 has to go to the cold stream number 1 to satisfy it completely. So, we pass on now the remaining 1165 units of heat from hot stream to satisfy the cold stream. Now, the remaining 2 cold streams are then completely satisfied by transferring heat from the hot stream which are to satisfy cold 2 we are transferring 413 units of heat and to satisfy cold 3 we are transferring 992 units of heat. Now, this way all the heat available with the hot stream and this steam is now pushed to cold 1, 2, 3 streams and they are in thermal balance. Now, the heat exchangers which are required for this purpose is 4 that means 4 number of heat exchangers are required. Capacities are 992, 413, 1165 and 1068. Now, for the earlier problem the number of heat exchangers were given by 8 minus 1 is equal to 7 where 8 is the number of streams available. In this case the number of streams available where 5, so 5 minus 1 is 4 which is the number of heat exchangers for this hen last hen. So, following the principle of maximizing the load which is called ticking of streams or utility roads or residuals leads to a design with a total number of 4 matches why we have done so because if you do not do the maximization of load by ticking of stream or utility loads or residuals we cannot reach to a minimum number of units and our aim is to calculate the minimum number of heat exchangers in the network which will satisfy the design or the requirement. So, a small correlation or formula can be generated which says that u minimum that is minimum number of units is equal to n minus 1 where u minimum is the minimum number of units including heaters and coolers and n is equal to total number of streams including utilities. So, if I develop this equation and apply to the earlier two problems I see that they predict the number of heat exchangers accurately. Now, the question is whether this equation is fool proof we will see that in some of the cases it fails and hence this equation has to be modified or enlarged. Now, let us see a problem with two hot streams two cold streams a hot utility and a cold utility hot utility is given by S T which is a steam and cold utility C W cold water and hot streams are H 1 H 2 cold streams are C 1 and C 2 this steam which is given by S T as 30 units of heating H 1 steam hot stream has got 70 units of heating H 2 is 90 unit of heating cold water that is C W is 50 units it can take heat C 2 can take 100 units of heat and C 1 can take 40 units of heat. This can be satisfied with this arrangement that means H 2 can pass on 50 units of heat to C W and take it off the remaining 40 can be passed on to the C 2, but it will not satisfy the C 2. So, the remaining 60 units to satisfy C 2 will come from H 1. So, this way H 2 and H 1 will satisfy C 2 by giving them giving it 100 units of heat the remaining 10 units of heat with H 1 will now pass on to C 1 and the remaining 30 units of heat which is required by C 1 will pass from S T that is steam. Now, we see here the all the heat available with the hot streams and hot utility are able to satisfy the cold utility and cold streams. So, they are in thermal balance and we are using 5 heat exchangers in this case having capacities 50, 40, 60, 10 and 30 units and the number of streams with us including the hot utility and cold utility is 6. So, if I apply this equation it satisfies N minus 1 because N minus 1 is 5 where N is 6 and here we have 5 heat exchangers. Now, the same problem can be done in this way that means H 2 gives 50 units of heat to cold water and remaining 40 of units of heat is passed to C 1 then H 1 gives 70 units of heat to C 2 and remaining 30 is passed on by the steam to C 2. If I arrange this way the heat exchangers I find that I am using one less heat exchangers that is 5 minus 1 equal to 4 and with 4 heat exchangers I am able to solve the problem a special case we see that H 2 satisfies C 1 and C W whereas, S T and H 1 satisfies C 2. So, there are two subsets inside it if there is a single subset obviously, there will be a second subset. So, we here we see that two subsets are available one subset is C 1 C W H 2 and other subset is S T H 1 and C 2. So, it appears as if it is a two problems because two separate components are available which are thermally satisfying each other or which are thermally satisfied internally. So, if I apply my rule to both the subset then 3 minus 1 or subset S T H 1 and C 2 is 2 and for the second H 2 C 1 and C W 3 minus 1 is 2. So, 2 plus 2 is 4. So, this way we can predict the number of heat exchanger in this arrangement. So, we see that the subset equality also plays a role in the determination of number of heat exchangers. So, in a heat exchanger network we should always search for subset equality and we by chance if we get subset equalities then we will be able to decrease the number of heat exchangers. Now, we see a second case here I put a extra line between steam to C 2 which transfers x amount of heat. If it transfers x amount of heat to C 2 then 30 minus x amount of heat will be transferred to C 1 and 10 plus x amount of heat from H 1 will be transferred to C 1 and when we add 30 minus x plus 10 plus x it is 40. So, it is C 1 is satisfied similarly 60 minus x will be transferred to C 2 and 40 units of heat will come from H 2 to C 2 and 50 units of heat goes to C W. So, this is also a arrangement which satisfies the need, but requires 6 heat exchangers. Now, let us analyze why it requires 6 heat exchangers we see that S T C 1 C 2 and H 1 forms a loop in many a times loops are necessary because in a loop the heat can be transferred from one unit to another unit and the heat loads become flexible, but for that flexibility we have to pay one extra unit. So, we see that if there is a loop in the heat exchanger network it will increase the number of heat exchangers. So, for each loop one heat exchanger will be added as in this arrangement there is only a single loop we are paying one more heat exchanger for this loop. So, though a loop gives flexibility in operation, but we have to give a tax in terms of a additional heat exchanger per loop. So, breaking loop will decrease the number of heat exchangers in a heat exchanger network. This we will see when we will develop non-MER designs and in a MER design we will find that always loops will exist and if you want to decrease the number of heat exchanger in the end then these loops are to be broken. So, loop also contributes to the number of heat exchangers and hence our equation should contain this. So, we saw that number of separate component contributes, number of loops contribute and number of streams including the hot utility and cold utility also contributes. So, u minimum is equal to n plus l minus s where u minimum is the number of units including heaters and coolers, n is the total number of streams including utilities that is hot utility and cold utility, l is the number of loops present in the hand and s is the number of separate components present in the hand. This equation is basically from graph theory in mathematics known as Euler's general network theorem. This theorem translates into the terminology of n and states that u minimum is equal to n plus l minus s. So, this is the full-fledged equation and we will always use this equation to target number of units in a hand. Normally we will like to avoid extra units and hence we will design for l is equal to 0 that means we will break the loop if there is a loop present in the hand and we will decrease the number of units and hence our aim will be to design hence for l equal to 0 and if we are lucky enough then there will be subset equality otherwise there will be no subset equality and hence the s value will be equal to 1. If I keep this values then the e q u minimum becomes n minus 1. Since the pinch divides the problem into two thermodynamically independent regions the targeting formula must be applied to each separately. When I am using pinch analysis for the design of heat exchanger network then it breaks the problem into two thermodynamically independent regions as through the pinch no heat transfer takes place. So, as far as thermal independence is concerned they are independent from each other but physically they look as if one unit one heat exchanger network but thermodynamically they are divided into two heat exchanger networks and hence I should apply this equation in the upper part of the pinch as well as to the lower part of the pinch and then the results should be added to find out what should be the minimum number of units in a design which uses the pinch analysis. So, the u minimum m e r that is maximum energy recovery design is equal to u minimum of the hot end which is the upper part of the pinch plus the u minimum of the cold end which is the lower part of the pinch. Let us take an example of a five stream problem when we use PTA we find that the hot utility requirement is 822.61 kilowatt cold utility requirement is 5 4 5 0.95 kilowatt hot pinch is 60 cold pinch is 50 because delta T minimum is 10. Now here we see that if the heat load of the cold stream 3 which is 4 4 7 9 can be brought to 4 4 7 8 there is a chance for subset equality and S will have a value of 2 and thereby decrease of number of units will be 1 for this case n is equal to 7 including h u and c u that is hot utility and cold utility l is equal to 0 and S equal to 2. So, n plus l minus S is equal to 5. So, if somehow a unit is cut down from the load of cold 3 7 7 7 4 4 7 9 and make it 4 4 7 8 we can drastically cut down the number of units by 1. So, in this case process modifications we should go for process modification and should try to bring down the colds 3 stream to 4 4 7 8 units. If subset equality is not created then n is equal to 7 l equal to 0 and S equal to 1 then the number of units minimum number of units is 6. Now if we apply this formula for the hot end as well as cold end then above the pinch this is hot end the number of streams are 4 including hot utility and S is equal to 1. So, 4 minus 1 is equal to 3 and for the below pinch this is 5 minus 1 is equal to 4. So, overall u minimum for the M e r design is 3 plus 4 equal to 7. So, what we conclude out of this if you do the targeting for the minimum number of units for M e r design then what we find that u minimum for the network is 3 plus 4 is equal to 7 units. If pinch division is not considered then number of streams including hot and cold utility is 7 S is equal to 1 l equal to 0. So, if pinch division is not considered then we call it a non M e r design and in a non M e r design u minimum is equal to 6 that means a non M e r design gives less number uses less number of heat exchangers than a M e r design. So, this conclusion we can find out here that u minimum M e r is always greater than the u minimum non M e r design. The number of units obtained in targeting of M e r design is more than the u minimum due to the fact that streams that cross the pinch are counted twice in a M e r design. This conclusion is that there is a trade up between the energy recovery and number of units employed in a M e r design. What does it mean if I am going for a M e r design then I am recovering maximum energy, but I am and my hot utility and cold utilities are minimum, but I am paying in terms of more number of units. Now, if I go for a non M e r design my hot utility requirement and cold utility requirement will increase, but I will able to decrease the number of units. So, decision is very simple if gain is there in a non M e r design then I should prefer a non M e r design. If there are losses in the non M e r design I should not prefer a non M e r design I should go for a M e r design and this decision will be clear when I do the cost targeting of a M e r design or I do the costing I find out the tax total annual cost of a M e r design and the total annual cost of a non M e r design. So, this I should remember there is always a trade up between energy recovery and number of units. Thank you.