4. Example

One example, which was adapted from Chen and Hung [10–12], has been used to illustrate how the newly presented methodology works. The example was first presented by Floudas and Grossmann [3] and has also been solved by other authors [14, 15]. It has two hot streams, two cold streams, one hot utility (steam) and one cold utility (cooling water). Table 1 shows the stream parameters and other costing details.

In the example, for hot stream 1, the heat capacity flow rate fluctuates around a nominal value of 1.4 kW/K by a magnitude of ±0.4, while its supply temperature fluctuates around a nominal value of 583 K by a magnitude of ±10. For cold stream 2, the heat capacity flow rate fluctuates around a nominal value of 2.0 kW/K by a magnitude of ±0.4, while its supply temperature fluctuates around a nominal value of 388 K by a magnitude of ±5. The objective in this example is to develop a flexible HEN that can feasibly transfer heat for the specified disturbance range in a minimum TAC network. In solving this problem, Chen and Hung [10] identified four extreme operating points within the uncertain process parameter range which include those for nominal conditions, maximum area, maximum cooling load and maximum heating load. The second, third and fourth sets of operating points, that is, the maximum area, maximum cooling load and maximum heating load, respectively, were appended one after the other, to the candidate network generated using the nominal operating conditions.

In solving this problem using the new method developed in this chapter, the first step entails identifying 10 sets of operating points (termed periods) that would be used to generate the base candidate multi-period network. The parameters for the 10 periods are shown in Table 2. The parameters listed in Table 2 are then solved in Step 2 as a multi-period problem having 10 periods of operations with equal-period durations and unequal-period durations using Eqs. (3)


Operating hours = 8600 (hours/year), heat exchanger capital cost function = 4333A0.6 (\$/year), capital annualisation factor (AF) = 0.2, A in m<sup>2</sup> , overall heat transfer coefficient (U) for all matches = 0.08 kW/(m<sup>2</sup> <sup>K</sup>−<sup>1</sup> ), ΔTmin = 10 K.

Table 1. Stream, cost and capital equipment data for the example.


Table 2. Periods of operations used to generate candidate network.


Table 3. Total annual cost for each dominant period.

in Step 3. The model at this stage is solved with further minor adjustments to exchanger

One example, which was adapted from Chen and Hung [10–12], has been used to illustrate how the newly presented methodology works. The example was first presented by Floudas and Grossmann [3] and has also been solved by other authors [14, 15]. It has two hot streams, two cold streams, one hot utility (steam) and one cold utility (cooling water). Table 1 shows

In the example, for hot stream 1, the heat capacity flow rate fluctuates around a nominal value of 1.4 kW/K by a magnitude of ±0.4, while its supply temperature fluctuates around a nominal value of 583 K by a magnitude of ±10. For cold stream 2, the heat capacity flow rate fluctuates around a nominal value of 2.0 kW/K by a magnitude of ±0.4, while its supply temperature fluctuates around a nominal value of 388 K by a magnitude of ±5. The objective in this example is to develop a flexible HEN that can feasibly transfer heat for the specified disturbance range in a minimum TAC network. In solving this problem, Chen and Hung [10] identified four extreme operating points within the uncertain process parameter range which include those for nominal conditions, maximum area, maximum cooling load and maximum heating load. The second, third and fourth sets of operating points, that is, the maximum area, maximum cooling load and maximum heating load, respectively, were appended one after the other, to

In solving this problem using the new method developed in this chapter, the first step entails identifying 10 sets of operating points (termed periods) that would be used to generate the base candidate multi-period network. The parameters for the 10 periods are shown in Table 2. The parameters listed in Table 2 are then solved in Step 2 as a multi-period problem having 10 periods of operations with equal-period durations and unequal-period durations using Eqs. (3)

> TS (K)

Hot utility, HU1 - 573 573 171.428 · 10−<sup>4</sup> Cold utility, CU1 - 303 323 60.576 · 10−<sup>4</sup>

Operating hours = 8600 (hours/year), heat exchanger capital cost function = 4333A0.6 (\$/year), capital annualisation factor

Hot-process stream 1, H1 1.4 ± 0.4 583±10 323 - Hot-process stream 2, H2 2.0 723 553 - Cold-process stream 1, C1 3.0 313 393 - Cold-process stream 2, C2 2.0 ± 0.4 388 ± 5 553 -

, overall heat transfer coefficient (U) for all matches = 0.08 kW/(m<sup>2</sup>

Supply temperature

Target temperature

Cost (\$/kWh)

Tt (K)

<sup>K</sup>−<sup>1</sup>

), ΔTmin = 10 K.

the candidate network generated using the nominal operating conditions.

rate FCp (kW/K)

Table 1. Stream, cost and capital equipment data for the example.

areas, if needed, so as to accommodate as many operating points as possible.

4. Example

the stream parameters and other costing details.

100 Heat Exchangers– Design, Experiment and Simulation

Stream Heat capacity flow

(AF) = 0.2, A in m<sup>2</sup>

and (4) as well as Eq. (A1) in the appendix. For the unequal-period duration scenario, each of the selected periods of operations is made to dominate the total length of periods for all the 10 periods by a significant amount. The purpose of this is to ensure that the final flexible network is able to cater for all possible scenarios including the worst-case scenarios in terms of heatexchanger area and utility requirement, irrespective of the duration of period for each of the parameter points lying within the full disturbance range. The resulting solution for each scenario is shown in Table 3. The average solution generation time for each of the solutions in Table 3 is 5 S of CPU time. The TAC shown in the first column of Table 3 is for a case where all periods have equal duration, that is, each period contributes 10% of the total period duration. The second column is for a case where the parameter points of period 1 dominates the total period durations by 99.1%. The same scenario applies for the rest of the columns, that is, each period concerned dominates the total period duration by 99.1%.

Step 3 requires that the selected matches, as well as their areas, for the equal-period duration solution network (TAC = 38,133 \$ in Table 3) and the network obtained for the most dominant period (i.e. period 4, TAC = 46,573 \$ in Table 3) be identified from Table 3. Table 4 shows the selected matches, as well as their areas, for the two cases. It should be known that the solution network for the equal-period scenario has more units compared with that of a case where the dominant period (i.e. period 4) is considered. This implies that the solution for the dominant period will result in a simpler network with fewer units, but with higher operating cost. In this chapter, the solution of the equal-period scenario is used in subsequent steps so as to get a TAC


Table 4. Selected matches for equal-period duration and most dominant period.


Table 5. TAC for equal period and one dominating period at a time in the flexible network tested for 10 periods.

that competes with those presented in the literature. However, the resulting network of the equal-period scenario is still tested for flexibility to feasibly transfer heat in scenarios of unequal-period durations.

Step 3 further requires that the matches selected (including their areas) in the equal-period case be used to initialise the multi-period MINLP model of the 10-period problem data shown in Table 2, by fixing the areas of the matches to those of the equal-period scenario shown in Table 4. Note that the model is still solved as an MINLP at this stage, despite the fact that the matches and areas are fixed, because in getting compromise solutions for a flexible network, not all matches in Table 4 (for the equal-period case) may be selected, in fact matches which do not exist on the table may even be added to the network. Solving the 10-period candidate multi-period network of Table 2, using the fixed areas of the matches shown in Table 4 for the equal-period case, gives a range of TACs for each period dominating one at a time as shown in Table 5.

In Step 4, the flexibility of the network obtained in Step 3 was then further tested for more randomly generated parameter points lying within the full disturbance range by solving a large multi-period model with equal-period durations. The model at this stage is initialised using the matches, as well as their areas, shown for the equal-period case in Table 4. The network was not feasible for a case having 100 periods of equal durations, so the areas of the network were adjusted to obtain new set of areas as shown in Table 6.

After the adjustments, the total network area increased from 132.96 m2 in Table 4 to 135.4 m<sup>2</sup> in Table 6. The resulting network was then feasible to transfer heat in a cost-efficient manner


Table 6. Heat-exchanger areas for final flexible network for the example.

Figure 2. Final flexible structure of this study.

that competes with those presented in the literature. However, the resulting network of the equal-period scenario is still tested for flexibility to feasibly transfer heat in scenarios of

Equal-period scenario Period 4 dominating

Table 4. Selected matches for equal-period duration and most dominant period.

HU1,C1,1 4.71 HU1,C2,1 43.04 H1,C1,5 32.74 H1,C1,5 14.08 H1,C2,4 6.04 H1,CU1,6 57.34 H1,CU1,6 57.34 H2,C1,5 4.77 H2,C1,5 4.77 H2,C2,3 17.26

Total area 132.96 136.49

Dominant period Equal 1 2 3 4 5 6 7 8 9 10 Period duration 10% 99.1 99.1 99.1 99.1 99.1 99.1 99.1 99.1 99.1 99.1 TAC (\$/yr) 38,134 39,396 35,064 37,127 47,235 41,479 43,419 31,247 33,669 40.943 31,763

Table 5. TAC for equal period and one dominating period at a time in the flexible network tested for 10 periods.

) Match Area (m2

)

Step 3 further requires that the matches selected (including their areas) in the equal-period case be used to initialise the multi-period MINLP model of the 10-period problem data shown in Table 2, by fixing the areas of the matches to those of the equal-period scenario shown in Table 4. Note that the model is still solved as an MINLP at this stage, despite the fact that the matches and areas are fixed, because in getting compromise solutions for a flexible network, not all matches in Table 4 (for the equal-period case) may be selected, in fact matches which do not exist on the table may even be added to the network. Solving the 10-period candidate multi-period network of Table 2, using the fixed areas of the matches shown in Table 4 for the equal-period case, gives a range of TACs for each period dominating one at a time as shown in

In Step 4, the flexibility of the network obtained in Step 3 was then further tested for more randomly generated parameter points lying within the full disturbance range by solving a large multi-period model with equal-period durations. The model at this stage is initialised using the matches, as well as their areas, shown for the equal-period case in Table 4. The network was not feasible for a case having 100 periods of equal durations, so the areas of the

After the adjustments, the total network area increased from 132.96 m2 in Table 4 to 135.4 m<sup>2</sup> in Table 6. The resulting network was then feasible to transfer heat in a cost-efficient manner

network were adjusted to obtain new set of areas as shown in Table 6.

unequal-period durations.

Match Area (m<sup>2</sup>

102 Heat Exchangers– Design, Experiment and Simulation

H2,C2,2 27.36

Table 5.

for all of the 100 possible periods of operations. The TAC of the 100-period scenario, which was obtained in 20.94 S of CPU time, is 38,992 \$. The flexible network, which is shown in Figure 2, is deemed flexible at this stage; hence, it is selected as the final flexible network. In this figure, the areas of each of the heat exchangers are indicated on the units. It should be known that only the final flexible network obtained in Step 4 is shown because the network structure remains unchanged in each of the steps. Table 7 shows a comparison of the final solution obtained in this example with those of other papers. It is worth mentioning that the solutions of other works, which are shown in Table 7, are presented for a case where the periods have equal duration. However, in


Table 7. Comparison of solutions for the example.


Table 8. Total annual cost for each dominant period in final flexible network for the example.

reality, the period durations may not be equal; hence, there is a need to also test or demonstrate the flexibility of the final network for unequal-period durations as done in this chapter. The final network obtained in this study has been tested for a 10-period scenario where period durations may be unequal and the TAC that should be anticipated for cases where each of the periods in the set of 10 periods dominates by 99.1% of the time is shown in Table 8. It is expected that the TAC that would be obtained when any of the 100 possible periods of operations dominates significantly by 99.1% of the time, or less, will not be too different from what is shown in Table 8.

Figures 3 and 4 show the final flexible networks obtained by other authors as found in Refs. [12, 14]. What is common to these two structures is that they both have six units, out of

Figure 3. Final flexible HEN structure of Li et al. [14].

A Multi-Period Synthesis Approach to Designing Flexible Heat-Exchanger Networks http://dx.doi.org/10.5772/66694 105

Figure 4. Final flexible HEN structure of Chen and Hung [12].

which two are coolers. This is unlike the structure obtained in this chapter which has only one cooler.

#### 5. Conclusions

reality, the period durations may not be equal; hence, there is a need to also test or demonstrate the flexibility of the final network for unequal-period durations as done in this chapter. The final network obtained in this study has been tested for a 10-period scenario where period durations may be unequal and the TAC that should be anticipated for cases where each of the periods in the set of 10 periods dominates by 99.1% of the time is shown in Table 8. It is expected that the TAC that would be obtained when any of the 100 possible periods of operations dominates significantly by 99.1% of

Dominant period Equal 1 2 3 4 5 6 7 8 9 10 Period duration 10% 99.1 99.1 99.1 99.1 99.1 99.1 99.1 99.1 99.1 99.1 TAC (\$/yr) 38,429 39,698 35,365 37,428 47,537 41,759 43,720 31,549 33,926 41,245 32,065

Costs Floudas and Grossmann [3] Chen and Hung [10] Li et al. [14] This work Annual operating cost (\$) 10,499 11,772 11,866 8554 Annual capital cost (\$) 39,380 30,104 28,626 30,438 Total annual cost (\$) 49,879 41,876 40,492 38,992

Figures 3 and 4 show the final flexible networks obtained by other authors as found in Refs. [12, 14]. What is common to these two structures is that they both have six units, out of

the time, or less, will not be too different from what is shown in Table 8.

Table 8. Total annual cost for each dominant period in final flexible network for the example.

Table 7. Comparison of solutions for the example.

104 Heat Exchangers– Design, Experiment and Simulation

Figure 3. Final flexible HEN structure of Li et al. [14].

This chapter has presented a new simplified synthesis method for designing small- to medium-sized flexible heat-exchanger networks using a mixed integer non-linear programming multi-period synthesis approach. The new method improves on currently available methods in the literature based on the fact that the final flexible network is selected considering the possibility of one or more sets of process parameter points dominating the total period durations more than others. This is essential so as to effectively plan for utility management. Key limitations of the new method are its tediousness of application, especially in large-scale problems, due to the fact that the impact on the overall network TAC of the possibility of each set of selected critical points dominating the total period duration needs to be investigated in a sequential manner. Also, during the flexibility tests, heat-exchange areas need to be manually adjusted, and there is no specific criterion to consider for the adjustments.

#### Appendix

The set of equations shown in Eq. (A1) was used in the multi-period model of this study. The details of each of these equations can be found in Verheyen and Zhang [5] and Isafiade et al. [7].

ðTs i,p−T<sup>t</sup> <sup>i</sup>,pÞFi,<sup>p</sup> <sup>¼</sup> ∑ k∈K ∑ j∈C qi,j, <sup>k</sup>,pi∈Hp∈P ðTt j,p−T<sup>s</sup> <sup>j</sup>,pÞFj,<sup>p</sup> <sup>¼</sup> ∑ k∈K ∑ i∈H qi,j, <sup>k</sup>,pj∈Cp∈P 8 >>< >>: 9 >>= >>; overal energy balances <sup>ð</sup>ti, <sup>k</sup>,p−ti, <sup>k</sup>þ1,pÞFi,<sup>p</sup> <sup>¼</sup> ∑ j∈C qi,j, <sup>k</sup>,pk∈Ki∈Hp∈P <sup>ð</sup>tj, <sup>k</sup>,p−tj, <sup>k</sup>þ1,pÞFj,<sup>p</sup> <sup>¼</sup> ∑ i∈H qi,j, <sup>k</sup>,pk∈Kj∈Cp∈P 8 >>< >>: 9 >>= >>; stage energy balances ti, <sup>k</sup>¼2,<sup>p</sup> <sup>¼</sup> <sup>T</sup><sup>s</sup> <sup>i</sup>,pi∈Hp∈P tj,K−1,<sup>p</sup> <sup>¼</sup> <sup>T</sup><sup>s</sup> <sup>j</sup>,pj∈Cp∈P ( ) assignment of superstructure inlet temperatures ti, <sup>k</sup>,p≥ti, <sup>k</sup>þ1,pi∈Hp∈P tj, <sup>k</sup>,p≥tj, <sup>k</sup>þ1,pj∈Cp∈P ( )temperature feasibility qi,j, <sup>k</sup>,p−Ωpzi,j, <sup>k</sup> ≤ 0 n ological constraint for heat load dti,j, <sup>k</sup>,<sup>p</sup> ≤ ti, <sup>k</sup>,p−tci,j, <sup>k</sup>,<sup>p</sup> þ φð1−zi,j, <sup>k</sup>Þk ∈ Ki∈ Hj ∈ Cp ∈ P dti,j, <sup>k</sup>þ1,<sup>p</sup> ≤ thi,j, <sup>k</sup>þ1,p−tj, <sup>k</sup>þ1,<sup>p</sup> þ φ ð1−zi,j, <sup>k</sup>Þ k ∈ Ki ∈ Hj ∈ Cp ∈ P ( ) exchanger approach temperatures dti,j, <sup>k</sup>,<sup>p</sup> ≥ ε k ∈ Ki ∈ Hj ∈ Cp ∈ P n obound for approach temperature LMTDi,j, <sup>k</sup>,<sup>p</sup> <sup>¼</sup> <sup>2</sup> 3 � ðdti,j, <sup>k</sup>,pÞðdti,j, <sup>k</sup>þ1,pÞ �1=2 þ 1 3 ðdti,j, <sup>k</sup>,pÞþðdti,j, <sup>k</sup>þ1,pÞ 2 � � � � logarithmic mean temperature difference zi,j, <sup>k</sup>∈f0, 1g ti, <sup>k</sup>,p, tj, <sup>k</sup>,p, qi,j, <sup>k</sup>,p, dti,j, <sup>k</sup>,p≥0 8 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>< >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>: 9 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>= >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>; (A1)
