2. Problem statement

minTAC <sup>¼</sup> AF � ∑

96 Heat Exchangers– Design, Experiment and Simulation

minTAC <sup>¼</sup> <sup>1</sup>

<sup>þ</sup>∑ iϵH ∑ jϵC ∑ kϵK

<sup>þ</sup> ∑ pϵP

> <sup>þ</sup>∑ iϵH ∑ jϵC ∑ kϵK

same interval of the superstructure at different periods of operations.

iϵH ∑ jϵC ∑ kϵK

DOPp NOP <sup>∑</sup> jϵC

NOP <sup>þ</sup> <sup>1</sup> <sup>∑</sup>

(

CFzi,j, <sup>k</sup> <sup>þ</sup> ∑

<sup>i</sup>,j, <sup>k</sup> <sup>þ</sup> ∑ jϵC

CUHqj,hu,<sup>p</sup> <sup>þ</sup> ∑

<sup>i</sup>,j, <sup>k</sup> <sup>þ</sup> ∑ jϵC

ACi,j, kAAEi,<sup>j</sup>

pϵP ∑ iϵH

ACi,j, kAAEi,<sup>j</sup>

where AF is the annualisation factor, CF is the fixed charge for heat exchangers, AC is the area costs for heat exchangers, AE is the area cost exponent for heat exchangers, CUH and CUC are the costs of hot and cold utilities, respectively, DOP is the duration of period p, NOP is the number of periods/operating conditions, Ai,j,k is the area of heat exchanger for hot and coldprocess stream pairs i,j in interval k. Aj,hu and Ai,cu are the area of heat exchangers exchanging heat between hot utility and cold-process streams and cold utility and hot-process streams, respectively. H,C,HU,CU are the set of hot streams, cold streams, hot utilities and cold utilities, respectively. It should be known that the area Ai,j,k is the representative heat exchange area which, as explained previously, are used by the same pair of streams exchanging heat in the

Eqs. (1) and (2) are the objective functions used in determining the TAC for the first-step initial candidate multi-period network that is later tested for flexibility using various kinds of approaches in some of the existing methods [5, 9–14]. It can be seen that the utility cost calculation component of these equations will result in allotting equal contributions, in terms of utility usage durations, for each of the periods of operations present in the first-step candidate multi-period network. This implies that the candidate multi-period network that is designed at the initial step, and later tested for flexibility, may be limited based on the fact that it is designed with the assumption that these initial candidate critical points have equal-period durations. Since TAC is being solved for at the first step, the objective functions in Eqs. (1) and (2) will aim to simultaneously minimise both utility consumption and investment costs. The investment cost is influenced by the size of heat-exchanger areas and the number of units. Allowing this limitation at the first step of the flexible network synthesis process means that the flexibility analysis step needs to be sophisticated so as to compensate for this limitation. This is because some candidate networks that lie in the uncertain process parameter range that are tested in the flexibility step may be disqualified from being included in the flexible network feasible space due to the fact that Eqs. (1) and (2) were used as the objective functions for

iϵH ∑ CU

CFzi, cu <sup>þ</sup> ∑

ACj, huAAEj,hu

DOPp NOP <sup>∑</sup> iϵH

pϵP ∑ jϵC

ACj, huAAEj,hu

pϵP

" #

" #

CUCqi, cu, <sup>p</sup> <sup>þ</sup> ∑

jϵC ∑ HU

<sup>j</sup>,hu <sup>þ</sup> ∑ iϵH

CFzj, hu

CUCqi, cu,<sup>p</sup>

CUHqj, hu, <sup>p</sup>

<sup>j</sup>,hu <sup>þ</sup> ∑ iϵH

ACi, cuAAEi, cu i, cu

> ACi, cuAAEi, cu i, cu

)

(1)

(2)

Given a set of hot-process streams and a set of cold-process streams, which have to be cooled and heated, respectively. Given also are the supply and target temperatures and the flow rates of these streams. Hot and cold utilities are also available. The task is to synthesise a flexible heatexchanger network which is optimally operable (i.e. featuring a minimum TAC network) for any unforeseen process-operating parameter point lying within an uncertain operating range.
