**6. Case Study**

The ammonia synthesis column with diameter 0.8 m is considered. The limiting WPHE plate diameter to fit in this column is 0.6 m, corresponding to an effective plate length of *Lpl* = 0.54 m. The operating conditions required by the process are given in Table 4. In the process of design, the height of corrugations *b* was changed from 0.3 to 0.5 mm and, correspondingly, the cross-section area of the channel, *fch*. It is done for consecutive numbers of passes *n* in the heat exchanger. The fixed design parameters are given in Table 5. The average angle of corrugation is β<sup>1</sup> = 40◦ for the hot side and β<sup>2</sup> = 50◦ for the cold side. The coefficients ζ*DZ1* = 11 for the hot side, and ζ*DZ2* = 17 for the cold side are taken according to laboratory tests. The design is made for bypass 20% on the cold stream to enable process control with the aging of the catalyser.

Attempts to calculate one-pass WPHE have shown that, at given temperatures, programming one-pass WPHE is not feasible at any increase of heat transfer area. The calculations with not fixed L*pl* for different pass numbers show the existence of local optimums with certain minimal values of WPHE heat transfer in area F. That value is lower for a smaller number of passes. However, for its fulfilment, an unacceptable plate length can be required; e.g., for a two-pass WPHE with a corrugation height of 3 mm, the minimal heat transfer area is 95 m2 and required plate length 0.9 m. In this case, the loss of mean temperature difference due to crossflow is too big. The calculations for WPHE with a number of passes, *n* = 3 and higher, produce the heat transfer area from 64 to 90 m2 with smaller required plate lengths. However, there is also an adverse influence of crossflow in one pass that is more emphasised at smaller passes numbers. In these cases, WPHE construction features and imposed constraints must be accounted for.


**Table 4.** The required operating conditions for case study.

**Table 5.** The fixed values in the WPHE optimal design.


As is discussed in [13] (pp. 54–65), for compact heat exchangers, the decrease of the equivalent diameter of channels leads to a smaller heat transfer area and improved compactness. However, then the problem of adjusting the plate length into its required limits arises. In the considered case, to fit into the high-pressure shell, the specified plate length is *LplS* = 0.54 m. The locally optimal value for the given passes number can be selected only if they fit values of corrugation height *b* at which *Lpl* = *LplS.* In the case under consideration, these values are *b* = 3.3 mm at *n* = 3 with F = 68.3 m<sup>2</sup> (that is considered as global optimum) and *b* = 4.7 mm at *n* = 4 with F = 72.9 m<sup>2</sup> that is shown in Figure 8. Line (1) in Figure 8 shows the minimal values of the heat transfer area that could be achieved without constraint for plate length. By line (2), there is also a designated heat transfer area that is needed to satisfy the required process conditions with inequality constraints for heat load *Q*◦ or pressure drop Δ*P*◦ at *Lpl* = 0.54 m. In that case, only one constraint is satisfied strictly, but another with some margin. At *b*, lower than the locally optimal value, the optimal plate length is smaller than *LplS.* The number of plates in WPHE and its heat transfer area must be increased to fulfil pressure drop requirement ΔP = ΔP◦ and WPHE can transfer more heat *Q* > *Q*◦. At *b*, higher than the locally optimal value, the optimal plate length is bigger than *LplS.* The number of plates in WPHE and its heat transfer area has to also be increased to make it shorter, but to satisfy constraint strictly on the heat load, *Q* = *Q*◦*.* With this, the pressure drop became smaller than required, ΔP < ΔP◦.

When a designer has already manufactured plate, as the example in Figure 1 investigated here, the required number of plates in WPHE and heat transfer area can be estimated by calculations for specified corrugations height *b* = 4 mm. The results of such calculations are shown at graphs in Figure 8 for the number of passes *n* = 3 and for *n* = 4. For *n* = 3, the heat load is strictly satisfied at F = 132.94 m<sup>2</sup> with pressure drop 4430 Pa, that on 82% lower than specified. For *n* = 4, strictly satisfied is a pressure drop at F = 85.12 m2 while heat load has 3.1% of margin. This second option is preferable not only for a smaller heat transfer area but also for better utilisation of pressure drop which is leading to high velocities and wall shear stress in channels that can mitigate possible fouling with catalyst

dust. While this heat transfer area is on 24.6% bigger than an optimal global solution, it is 25.4% smaller than of prototype WPHE with unsymmetrical passes arrangement described above in Section 4. The heat transfer effectiveness with symmetrical passes arrangement *n* = 4 is 0.842 compared to 0.862 at pure countercurrent flow or the loss is only 2.3% compared to 14.6%–17.2% for the asymmetrical arrangement in Table 3.

**Figure 8.** The calculated results for the set of corrugations height b and symmetrical flow arrangement: line 1—the WPHE area for *Lpl* not fixed; line 2—the WPHE area for *Lpl* = 0.54 m.
