 Welcome to the lecture series on process integration, this is module 4 lecture number 6, the topic of the lecture is number of cell targeting. Let us first try to know while why cell targeting is required, we have already done number of units targeting. So, why specially this cell targeting is required, now to know this we should concentrate that what type of heat exchangers are used in a heat exchanger network. If we question this we will find that in most of the cases cell and tube heat exchangers are common heat transfer equipment used in a process industry or in a hand. The area and energy targeting algorithm used for heat exchanger network are based on the assumption that heat exchange matches are pure counter current flow. So, the targeted values will be close to real one only if heat exchanger employed in hen are 1 1 that means 1 cell pass and 1 tube pass design. If we use such type of heat exchangers in the hen then it will offer lowest surface for the heat exchanger network. However, exchangers employed in industries are not 1 1 cell and tube heat exchangers instead they may be 1 2 that means 1 cell 2 tube pass, 1 4 that is 1 cell 4 tube pass and even 2 4 that is 2 cell 4 tube pass heat exchanger designs which involve cross flow, counter flow, co-current flow and partially mixed flow. Thus in such type of heat exchangers the effective temperature difference of heat exchanger is reduced as compared to a pure counter current device. So, to account for this reduce in effective temperature difference a factor F t is used in the basic heat exchanger design equation. Further it has been seen both through experimental and numerical computation that the temperature crossover can be achieved in cell and tube heat exchangers with length to width ratio greater than 4.62 and cannot be achieved anymore in cell and tube heat exchangers with length by width ratio less than 3.08. The results also indicate that heat transfer performance decreases with increasing L by W ratio that is length to width ratio. Further it should be noted that the optimum heat exchangers are those heat exchangers whose L by W ratios are high. Thus for a long silent tube heat exchangers where the temperature range of both hot and cold streams through the exchanger is large compared to the temperature driving force. This may exhibit temperature cross whereas, design through a temperature approach or small temperature cross can be accommodated in a single 1, 2, cell and tube heat exchanger whereas, the design with large temperature cross becomes infeasible. To accommodate the effect of temperature cross or to nullify the effect of temperature cross in a single silent tube heat exchanger it has to be distributed into multi cell increasing the total cost of silent tube heat exchanger meaning that if there is a temperature cross considerable amount of temperature cross then the design tells that we should go for multi cell heat exchangers to accommodate this and if you go for a multi cell heat exchanger then the cost of the silent tube heat exchanger increases considerably. Thus targeting the minimum number of cells are useful and realistic in comparison with number of units target as it provides better opportunity to compare two hand designs. Let us try to understand what is a F T factor which is a LMTD log mean temperature difference correction factor. In the design practice a correction factor F T is introduced into the basic heat exchanger design equation to account for counter flow as well as parallel flow in 1, 2, 7 tube heat exchangers and the equation written is q is equal to UA LMTD into F T where the value of F T can vary from 0 to 1 where q is the heat exchanger duty in kilowatt, u is the overall heat transfer coefficient, A is the heat exchanger area and LMTD is the log mean temperature difference. The F T factor can be represented as the ratio of actual mean temperature difference in a 1, 2 exchanger to counter flow LMTD for the same terminal temperatures. The F T correction factor is usually correlated in terms of two main factors which are dimensionless in nature. The thermal effective of the heat exchanger called P and the ratio of two heat capacity flow rates called R. So, this shows the F T, the presence right shows the F T as a function of R and P where P is equal to T h i minus T h o divided by T h i minus T c i and R is C p h minus C p c is equal to T c o minus T c i divided by T h i minus T h o where T h i is the hot stream inlet temperature, T h o hot stream outlet temperature, T c i cold stream inlet temperature and T c o cold stream outlet temperature. Let us now try to understand what is a temperature approach and temperature cross. The figure clearly shows that the hot stream the inlet temperature is this and the outlet temperature is this whereas the cold stream inlet temperature is this and outlet temperature is this. This axis shows the enthalpy or length of the heat exchanger and this is the temperature axis. From here we see that the outlet temperature of the hot stream is little bit more than the outlet temperature of the cold stream. So, there is a gap and this gap is called temperature approach. Now, if it is the case and we want to correlate a heat exchanger with some temperature approach then this design falls in the upper part of this f t versus p graph where R is a parameter. So, it is a feasible design and here the f t is around 0.9. So, in this plot the red portion shows n feasible design and the green portion shows feasible design. Let us see the same graph but where here the outlet temperature of the hot is below the outlet temperature of the cold. So, there is considered amount of temperature cross here though the temperature cross is not very high. The outlet temperature of the cold stream is slightly lower than the outlet temperature of the sorry outlet temperature of the hot stream is slightly lower than the outlet temperature of the cold stream. This is called temperature cross. This situation is usually straight forward to design provided the temperature cross is small as it can be accommodated in a single cell. However, it decreases the f t value of the heat exchanger that means it will require significantly higher amount of area. So, if I want to map this type of heat exchanger into the f t p diagram then this exchanger can be given by this point which is very close to infeasible region. This line which separates the feasible and infeasible region the value of f t is 0.75 that means if I compute the value of f t to be 0.75 then it is almost in the borderline case of feasible and infeasible design. If the f t is lower than the 0.75 then it is not advisable to design the heat exchanger with a single cell and some design modification has to be done to increase this f t value and the general design modification is to increase the number of cells to improve the f t factor. A third type of situation will occur when there is a large temperature cross. Now, if I map this type of heat exchangers where there is a large temperature cross then it will map into the infeasible region of the f t versus p curve and here the value of f t will be far low than 0.75. In such a situation either the heat exchanger is not designed or multi cell heat exchanger has to be designed to improve the f t till improve the f t factor is more than 0.75. Infeasible exchanger design returns f t very low value of f t or f t less than 0. Having f t greater than 0 however is not enough to make a design practical. A commonly used rule of thumb requires f t greater than 0.75 for the design to be considered practical. It is well known fact that for multi pass heat exchangers the heat recovery is limited by LMTD correction factor f t. As temperature approach decreases f t decreases rapidly if f t is less than 0.75 one should increase the number of cells till f t becomes greater than 0.75. For a 1 2 exchanger if f t falls sharply with increasing temperature cross the ability to accommodate a temperature cross increases rapidly as the number of cell pass is increased. Meaning that if the f t is less than 0.5 the designer has to opt a multi cell heat exchanger or the number of cells should be increased such that f t becomes more than 0.75. Now, this gives the basis for cell targeting. Traditionally the designer would approach a problem requiring multiple cells by trial and error till the f t factor becomes more than 0.75. The design begins by assuming a number of cells usually one in the first instance and the f t is evaluated. If the f t is not acceptable then the number of cells in series is progressively increased until a satisfactory value of f t is obtained for each cell. Now, having known that why cells are necessary multiple cells are necessary in the design and with the increase in the cells the cost of the heat exchanger increases. That is why if we compare the number of units target with vis-a-vis number of cells targeting then we find that the number of cell targeting is far accurate than number of units targeting and this gives us a method to evaluate the hence. That means if there are two hence n 1 and n 2 n 1 has got 8 cells and n 2 has got 10 cells probably we would like to take the n 1 because it has got less cells and hence its fixed cost will be less than the n 2. Now, let us find out how to proceed for cell targeting f t can be evaluated from f t charts and p r values. If p and r values are known we can find out f t from f t charts. Ahma Dattel in 1985 gave an analytical expression for calculating number of cells directly based on the fact that for any value of r a maximum asymptotic value of p exists which gives f t value which tends to minus infinite. So, p x p maximum is equal to 2 divided by r plus 1 plus root over r square plus 1. A 1 2 exchanger design for p equal to p max will not be feasible because at this point p max point f t will tends to minus infinite and we generally do not design if f t falls below 0.75. They defined a practical design to be limited to some fraction of x p of the p max. The fraction is x p and a practical design will occur when I multiply p max with a fraction x p where the value of x p can be 0 to 1. It has been seen that if the x p value is equal to 0.9 it is sufficient to satisfy a design where f t is greater or equal to 0.7 while also avoiding reasons of steep slope and therefore, assuring a more reliable design. Now, based on this we can have two equations to find out the number of cells for r is not equal to 1 s is equal to log 1 minus r p divided by 1 minus p and then whole divided by l n in brackets 1 minus r p 12 r into p 12 brackets close divided by in bracket 1 minus p 12 where p 12 is or p 12 is equal to x p into p max for r equal to 1 the number of cells are p divided by in brackets 1 minus p whole divided by p 12 divided by in brackets 1 minus p 12 the whole bracket close. The number of cells predicted by above equation is a real quantity that means, it is a fractional non-integer quantity and thus when we predict cells it would be obviously, to taken into the next largest integer value. Now, the question is how to calculate number of cells for a hen before the design as I had already told that all targets are achieved before the design. So, this is a exclusively a very good technique of pinch technology where before the design the targets are fixed. To calculate the number of cells in the hen the composite curves that is hot and cold composite curves are divided into a number of enthalpy intervals and here also we it obeys the vertical temperature difference. Now, for this purpose we create a balanced composite curve which is called B C C. We have seen up till now the composite curves hot and cold composite curves when in a hot and cold composite curves the hot utility as well as cold utilities are plotted then it converts into a balanced composite curve because in such sense stage the heat available with the hot utility plus the hot streams is exactly equal to the heat required for the cold streams and cold utility and that is why such a plot is called balanced composite curve. And then this balanced composite curve B C C is divided into vertical enthalpy intervals as in the case of area targeting. In the area targeting we have taken it and a details of constructing a B C C is given. If each match enthalpy interval i requires n i number of streams using temperatures of interval i then the maximum cells count for the interval is s i into n i minus 1. The foundation of this equation is from units target we know that the number of units is equal to the number of streams in a particular interval minus 1. So, once number of streams is known and I know that in each number of streams there is a certain amount of cells then when this is multiplied with number of units I get total number of cells in that enthalpy interval which is given by s i into n i minus 1 n i minus 1 gives you the number of units and s i gives that how what is the number of cells per number of unit. In fact the temperature defining s i are achieved by the minimum of n minus 1 matches. The real or the non-intention number of cells target is then simply the sum of the real number of cells from all the enthalpy intervals. Suppose I have m number of enthalpy intervals then in the each interval I will find out what is the value of number of cells and then I will multiply this with the number of streams including hot cold utilities minus 1 in that interval. So, it will give me the number of cells in that enthalpy interval and then I will sum up this for m intervals that will give me the number of cells in the whole BCC curve. Furthermore actual design will normally observe the pinch division if I go for a MER design I have a pinch division. It has been told that through pinch no heat flows when I go for a maximum energy recovery design and hence pinch point divides the whole problem into two different parts one is called above pinch and the another is called below pinch as they are thermally balanced. So, for a pinch design we have to find out number of cells above the pinch and then we have to find out number of the cells below the pinch. Then for each number has to be converted into a whole number and then when we add it up then we find out the what is the total number of cells in that design. To see that how the cells targeting is done we will show this through an example. Here we see a stream data table which has two hot streams and two cold streams one hot utilities steam and another cold utility which is cold water. So, in fact we have six streams here whose supply and target temperatures are given and MCP values are given. Now if we plot this or we do the PTA of this stream data then we find that hot pinch is at 125 degree centigrade and cold pinch is at 105 degree centigrade and this is for delta T minimum equal to 20 degree centigrade and the hot utility demand is 605 kilowatt and the cold utility demand is 525 kilowatt. If I find out the number of units the total number of units defying the pinch division then it is four streams hot and cold streams plus two utility streams minus one is five. So, if I ignore pinch division then five number of units can do the heat exchange that means the heat exchanger network will required only five number of heat exchangers and if I go for MER design that means pinch division I consider the pinch division then in the upper part there are four streams including the utility four minus one and the lower part there are five streams including utility. So, five minus one. So, this is three plus four is equal to seven units. So, in the MER design the number of units target gives seven units if I consider the pinch division and number of units target gives five units if I do not consider the pinch division. So, this will remember and will compare with the sales target data. Now, this shows a balanced composite curve because here we have used the cold utility and here we have used the hot utility and the total heat available with the hot stream as well as hot utility is given to the cold stream with cold utility and that is why it is called a balanced composite curve. Now, here we have divided it where there is a slope change from that point we have divided this into many enthalpy intervals this is 1, 2, 3, 4, 5 like this. Now, here we see that this temperature is known to us, but this temperature is not known to us, this temperature is known to us, but this temperature is not known to us this temperature is known to us, but this temperature is not known to us. This temperature is known to us, this temperature is not known to us, this temperature is known to us, this is not known to us, so on so forth. So, our first job will be to find out this unknown temperatures. So, if I find out these unknown temperatures, so both end are known, that means this temperature is known, this temperature is known of the hot, number temperature is known and this temperature is known for the cold. So, this looks like a temperature profile of a counter current heat exchanger and knowing this temperatures I can find out the value of P and R and once P and R values are known I can find out what will be the value of S i because S i computation only needs P and R values and we have taken X P is equal to 0.09. So, once I know the S i value for this enthalpy interval and if I know the number of cold streams and hot streams and including the cold utility then I can find out what will be the number of units here using number of units target and then I can multiply this with the number of cells per unit and can find out the number of cells required for this enthalpy interval. Similarly, I will do for this enthalpy interval this enthalpy interval this enthalpy interval this enthalpy interval this enthalpy interval this enthalpy interval this enthalpy interval and then I will add up all the you know the cells and then I will find out the total number of cells which are required for this problem. Here also we will see that we will divide this problem into two parts above the pinch and below the pinch and we will add up the number of cells above the pinch and below the pinch and then we will convert them to integers and then we will add up to find out the total number of cells for this problem. So, how to do this has been clearly demonstrated in area targeting. So, I will request the readers to see the area targeting lecture and find out how for each intervals the T H I and T C I have been computed it requires a certain amount of efforts to do this and these are you done by interpolation method. So, let us see the calculation procedure for P I have told you that for calculation of number of cells we have to first calculate the value of P and R this is required and P values and R values can be computed through the values known that is T H I and T C I. So, to have 0 interval 1 2 3 4 5 6 7 8 9 intervals. So, to calculate P in the ith interval. So, the formula is P I T H I minus T H I minus 1 divided by T H I minus T S I minus 1. So, for the first interval if I calculate this is T H I is 65 T H I minus 1 is equal to 45 T H I is 65 T C I minus 1 is 15. So, for I equal to 1 this is the value and my P I is 0.4. So, this 0.4 goes here similarly I can calculate the P I values of other intervals that is this enthalpy intervals and we can fill it up. So, we are able to calculate the P values for all the temperature intervals which are enthalpy intervals in this case. Then the next stage is to compute the R values. So, here the R value is equal to T C I minus T C I minus 1 T H I minus T H I minus 1. So, for I equal to 1 we will be using this 4 set of tata here. So, here T C I for I equal to 1 is this value 18.81 and T C I minus 1 is this value which is 15. So, 18.81 minus 15 divided by T H I is this value 65 and T H I minus 1 is this value 45. So, 65 minus 45. So, it comes out to be 0.1905. So, for I equal to 1 we can fill this. Then subsequently you can calculate for other enthalpy intervals and we can fill it up. Maybe for this interval 2 we will use this 2 data and this 2 data and for this third interval we will use this 2 data and this 2 data and fourth interval we will use this 4 data. That means, 4th row data and third row data and for fifth we will use fifth row data this 2 data and this 2 data and so on so forth for ninth we will use this 2 data and this 2 data. In this manner we can find out the value of R. Now, once P and R are known the next stage will be to find out what will be the value of number of cells. Now, here we are calculating P 12 this P 12 is nothing but x p into P max. So, P 12 is equal to x p into P max and the formula of P max is this 2 divided by R plus 1 root over R square plus 1 where x p value is 0.9. So, I have the value of R. So, I can calculate P max I know the value of x p which is 0.9. So, I can multiply the value of x p with P max and I can calculate the value of P 12. So, if I do this for I equal to 1 this P 12 comes out to be 0.815. This uses the R value here that is 0.105 and this is the value of x p which we have used this is 0.9. Now, I can fill up the P 12 values for all the enthalpy intervals starting from 1 to 9. Once P R and P 12 is known we can calculate the S values that is number of cell values and the formula for this is given. And here we can see that the S is a function of R P and P 12 when R is not equal to 1 and when R is equal to 1 it is a function of P and P 12. Because for R equal to 1 you can put the value of R as 1 and obviously the functional relationship with R vanishes because it becomes a constant. So, there are two equations are available if R is equal to 1 and when R is not equal to 1. So, we can see here then none of the intervals R is equal to 1. So, we will use this formula which tells to calculate which gives us the method to calculate the value of S when R is not equal to 1. When S will be calculate for a interval this will be called S i. So, for i equal to 1 we can use these three quantities to compute the S i value here this is 1 minus R R value is 0.1905 into 0.4 this is the value of P this is 0.4 this is 0.4 then 1 minus P 1 minus the value of P 0.4 bracket closed and we are going for logarithm of this value whole value and then logarithmic of this value. So, we get a computed value as 0.2841. So, this is 0.2841 here basically there is a division line here I think it is divided here also this is divided. So, this is 0.2841 and I can fill up the values here. So, once S i number is calculated then I have to find out the value of N i minus 1 that means, for each enthalpy interval we have to compute how many number of streams are there and then we have to find out N i minus 1 value and then we multiply S i value to that and for each interval we will find out what will be the value of number of cells. So, for this purpose we have to plot the stream population here if I see in interval number 1 I have only 2 streams operating in the interval number of 2 I have 3 streams operating whereas, in interval number of 3 I have 1 2 3 4 streams are operating. So, from here I can calculate what is the N i value in each interval and then with N i minus 1 I can multiply this S i value this is S i S i value and then I can find out the S cells for each interval. So, this is filled up I can check for interval number 3 there are 1 2 3 4 streams are present. So, N i is equal to 4 and for interval 2 there are 1 2 3 streams present. So, it is N i is equal to 3. So, for i equal to 2 S i N i minus 1 is S i is 0.0237 into this is 3. So, 3 minus 1 this is 0.474. So, for this i equal to 2 the number of cells is 0.474. Similarly, for all other intervals I can compute the value of cells. So, we fill it up. So, here we find 0 because there is a temperature jump at this point and for we are joining 2 streams with a fictitious hot stream which will not require any heat transfer and hence the number of cells for that part of fictitious fictitious stream is 0 that is why this has come 0. Now, we have this table that intervals are that T h i is that T c i is that and number of cells are there for each enthalpy interval. Now, we see that this is not a integer value. So, now we have to find out where is the pinch division and then we will divide the problem into 2 parts upper pinch area and lower pinch area and then we will count the number of cells and then convert it into the next integer and for the upper pinch section and lower pinch section we will do this and then we will add it them up to find out the number of cells. So, the pinch point is this I have already told you that the pinch of this problem is 125 the hot pinch and 105 is the cold pinch and at this point interval which is the fifth interval it falls the number of cells is 5.1912. So, now we will compute the number of cells available above the pinch. So, these are 0.2841 this calls here 0.0474, 0.3481, 0.4305 and 5.1912 when we add this it becomes 6.3013 and if we round off it it becomes 7. Now, cell above the pinch we can calculate like that and it becomes to 2. So, cells have a pinch 2 cells below the pinch 7 and then when add those numbers it becomes 9. So, number of cells target gives of a value that for this problem number of cells will be 9. Now, if we compare this with the units target if I ignore the pinch then number of units target will be 5 and number of units for a MEA design when I consider pinch it will be 7. So, the number of cells target gives a better figure than number of units target and we are more close to the actual scenario if we compare the cells target and units target cell target gives you a better picture which is more close to the real scenario and hence cells target is better than the units target and by knowing the cells target we compare different hand designs which can be generated for a single problem utilizing the pinch rules. Thank you.