Minimum number of units (MNU) and minimum total surface area (MTA) targets 525 Contribution Ns i brought in by stream i to the total number of shells is equal to: Ns i = ent k K i Ns k (16.51) Notice that contributions are rounded-off to integer numbers for each stream. The total number of shells can then be calculated from: K Ns min = i I Ns i - k=1 Ns k (16.52) The equation was presented by Ahmad and Smith (1989) without rigorous derivation. Proof that Equation (16.52) can be derived from Equation (16.50) is presented in Je owski and Je owska (2002). However, this proof is only valid z z if we do not account for rounding-off operations. The rounding-off operations are more frequent in the case of the method based on stream contributions since rounding-off is performed for each stream contribution. One can expect that the results from both methods can be different and that those from Equation (16.52) will be higher. This was shown by Je owski and Je owska (2002) for some exam- z z ples. In all of them the results from the stream contribution based approach were systematically higher. Moreover, those authors observed that Equation (16.50) could yield unreliable results, such that the number of shells is smaller than the number of counter-current 1-1 heat exchangers, that is, matches. Je owski and z Je owska (2002) noticed that such results could be expected for high HRAT z values. Concluding, it can be stated that only Equation (16.52) applying stream con- tributions should be applied. It is expected to give rather conservative results. Moreover, it is difficult to estimate its accuracy since results from more rigor- ous approaches are not available. The Nmin-Transs model could be applied to determine the MNS target after inserting MTD parameters. However, there is no such extension in the literature. 16.5. MINIMUM TOTAL AREA FOR SHELLS (MTA-S) TARGET Here, we explain approaches for the minimum surface area target for a multi- shell apparatus (MTA-s). We begin with the approach proposed by Je owski et al. z (2003e). They focused on developing a linear model. To reach the goal they pro- posed creating sufficiently small temperature intervals such that temperatures in "mini-matches" among the mini-streams in intervals can be approximated by temperatures of intervals. Thus, for known temperatures of "mini-matches" the logarithmic mean temperature differences and Ft correction factors are also known. The result is that only heat loads of "mini-matches" are variables and the problem becomes linear. To model heat transfer Je owski et al. (2003e) z applied the transportation model. Though it requires more variables than the