*2.4. Optimization Problem Defenition*

The objective of the optimization study was to minimize the energy consumption of the MR pre-cooling process whilst satisfying a minimum temperature approach constraint. The objective function was formulated as described in Equation (1):

$$\min \{ \text{SEC}\_{\text{MR}} \} \_{\text{\\_such}} \text{ that} \begin{cases} \text{ lb}\_i < \text{OP}\_i < \text{lb}\_i\\ \Delta T\_{\text{min}} - \Delta T\_{\text{acc}} > 0\\ \dot{m}\_{\text{MR}} > 0 \end{cases} . \tag{1}$$

In Equation (1), SECMR is the specific energy consumption of the MR process, OP*<sup>i</sup>* are the set of *i* optimization parameters (see Table 5), lb*<sup>i</sup>* and ub*<sup>i</sup>* are a set of lower and upper bounds for each parameter, Δ*T*min is the minimum approach temperature in HX-1 (Δ*T*min <sup>=</sup> min{Δ*Tn*}), <sup>Δ</sup>*T*acc is the minimum acceptable approach temperature in HX-1 and . *m*MR is the mass flowrate of the MR. SECMR was calculated from the sum the compression stage energy consumptions, *W*MR, which are, in turn, a function of OP*i*, MP*<sup>i</sup>* (see Table 1) and *T*c is described by Equation (2):

$$\text{SEC}\_{\text{MR}} = \sum W\_{\text{MR}}(\text{OP}\_{i\prime} \text{MP}\_{i\prime} T\_c) / \dot{m}\_{\text{F2}}.\tag{2}$$

**Table 5.** Summary of Optimization Parameters with Initial (OP*i*,0) and Constraint Values.


In Equation (2), . *m*H2 is the mass flowrate of hydrogen in the pre-cooling process.

The set of optimization parameters, OP*i*, used in the study are summarized in Table 5 along with the initial values used (OP*i*,0) and initial values of the boundary constraints (lb*<sup>i</sup>* and ub*i*).

Although the ultimate purpose of the boundary constraints shown in Table 5 was to limit the optimization process to physically meaningful solutions—e.g., component mole fractions greater than zero—the initial boundary constraints were also used to limit the search area around the likely optimum values. This was done to reduce optimization time. The initial values of lb and ub shown in Table 5 were set based on results from the reference case, but where the optimization solution was found close to the initial limits, the bounds were extended to ensure that the overall optimum solution was not missed.

In addition to the optimization parameters listed in Table 5, the MR compressor inter-stage pressure, MR compressor discharge pressure and HX-1 warm-end approach temperature could be considered as optimization parameters. However, in this work these have been excluded to limit complexity. The MR compressor discharge pressure is, therefore, fixed at the value used in the reference study, the MR inter-stage pressure set in each case to maintain equal stage pressure ratios, and the HX-1 warm-end approach set to 5 ◦C. The MR mole fraction for butane is also not identified as an optimization parameter because it is calculated from the sum of the other components.

### *2.5. Optimization Algorithm*

In a phase of initial testing the *Fmincon* (FMC) algorithm with the SQP option was found to provide fast and generally accurate optimization results, although in some cases local minima were found. In all subsequent cases, FMC was used with the solution tolerance set to 0.001 kWh/kg and all other options left as default.

To help identify the global minimum solutions for each *T*c, the boundary constraints shown in Table 5 were evaluated in a manual, stepwise, process: after the initial results had been gathered, new initial guesses were specified when the original initial guess was found to be a long way from the solution. When a stable set of bounds enclosing the global solution had been found, the *MultiStart,* MS, and *GlobalSearch*, GS, algorithms were used to help test the quality of the results. In both cases the MS and GS runs were again based on the FMC algorithm with the parameters as before.

The quality each optimization result was assessed qualitatively using the results from other *T*c cases. The basis of this assessment was the assumption that a simple, monotonic, relationship was likely between each of the optimization parameters and *T*c. In addition to this assessment, the temperature profiles in HX-1 for each case were reviewed qualitatively to determine if Δ*T*acc was consistently approached throughout the heat exchanger.
