*2.6. Performance Variation with Cooling Temperature*

Performance variation with cooling temperature was studied for the MR pre-cooling process by finding the optimum operating parameters, OP*i*, for each cooling temperature, *T*<sup>c</sup> case. The fixed modelling parameters shown in Table 2 were used as the basis in all cases. The cooling temperature range studied was 5 to 50 ◦C.

In the model developed for the cryogenic-cooling step, process parameters were not optimized: flowrates and pressure levels in the cryogenic cycle were held constant at the values shown in Table 3. The variation of the energy consumption of the cryogenic cycle compressor with *Tc* was modelled using the more simplistic assumption that, since the composite cooling curves in HX-2 are straight and parallel, a constant warm-end approach temperature exists across the range of cooling temperatures studied. The energy consumption of the cryogenic cycle compressor was calculated using the same basis as that of the MR pre-cooling process. A 2 ◦C warm-end approach temperature was assumed across the cooling temperature range 5 to 50 ◦C.

The overall SEC for the hydrogen liquefaction process was calculated as the sum of the energy consumption for the MR pre-cooling step, SECMR, and the cryogenic-cooling step, SECCY, which was—in turn—calculated as the sum of the cycle compressor stage energy consumptions minus the energy recovered in the cryogenic-cooling step expanders as described in Equations (3) and (4):

$$\text{SEC} = \text{SEC}\_{\text{MR}} + \text{SEC}\_{\text{CY}} \tag{3}$$

$$\text{SEC}\_{\text{CY}} = \sum W\_{\text{H2}} (\text{MP}\_{\text{H2}}, T\_{\text{c}}) / \dot{m}\_{\text{F2}} - \text{SEC}\_{\text{Ex}} \tag{4}$$

In Equation (3), *W*H2 is the energy consumption of the cycle compressors shown in Figure 2, and in Equation (4), MPH2 is the set of fixed modelling parameters for the cryogenic-cooling cycle compressor (see Table 2).

To provide an independent means of reviewing the trends shown in the results, the SEC for an ideal process that cooled the hydrogen from *T*<sup>c</sup> to a final temperature of −259 ◦C was also calculated. This ideal energy consumption, SECID, was then used to calculate a second law efficiency, *η*id = SEC/ SECID, for the overall process. The method used to calculate SECID was to summate the ideal Carnot cycle energy consumption for a set of very small temperature steps along temperature–enthalpy data for hydrogen as explained previously by Jackson et al. [29].

#### **3. Results and Discussion**

### *3.1. Process Modelling and Validation*

Table 6 shows the results from the model validation work. In addition to the results from the reference study, three sets of results are presented in Table 6: Case A uses the TREND implementation of the Peng Robinson (PR) equation of state; Case B the TREND/SRK equation of state; and Case C the TREND/ Helmholtz free energy properties method.


**Table 6.** Summary of modelling parameters for the model validation work.

\* Given only as an approximate value in the reference case. \*\* Adjusted to give a positive value for min. approach.

Of the three cases compared in Table 6, Case C—using the TREND/ Helmholtz free energy properties method—is considered to represent the closest match to the reference case, but since Case B also offers good agreement and significantly reduced calculation time, SRK is selected as the basis for further work.

Figure 3 presents the composite temperature profile data for Case B and C in Table 6. The results show that, although the shape of the curves differs between the two cases, the results from both cases show a very good fit between the warm and cold curves throughout the heat exchanger. These results, therefore, add confidence to the validation work and the selection of Case B as the modelling basis.

**Figure 3.** Composite Curves and Key Performance Parameters for HX-1, MR Pre-cooling Process: (**a**) Case B; (**b**) Case C.

While Table 6 and Figure 3 show that the selection of a good modelling basis is important to the determination of the optimum operating parameters for this process, no claim is made here that the modelling basis selected is the one that is most accurate for the modelling of this process, just that it provided a good match with the reference case in the validation work presented.

A limitation of the present study is that the heat generated during ortho-para hydrogen conversion is omitted form the model. This is a simplification that limits the extent to which this modelling work reflects the performance of a hydrogen liquefaction process operating in the real world. The main claim made here regarding the modelling basis is that it provides provide a consistent basis to study performance across the operating cases considered. The implication of this for further work is that the study of the variation in energy consumption with cooling temperature made here is valid and can provide some insight into how the performance real hydrogen liquefaction processes can be expected to vary when designed for utility cooling at different temperatures.

#### *3.2. Performance Variation with Cooling Temperature*

Figures 4 and 5 show how the five optimization parameters vary with *T*c, and Figures 6 and 7 provide two examples of the optimized cooling curves resulting from these runs. In Figures 4 and 5 all of the data collected over the final set of optimization runs (two using GS and two using MS) are presented as points and the overall optimum datasets are connected by dotted lines.

**Figure 4.** Variation in OP for the MR pre-cooling step with cooling temperature.

**Figure 5.** Variation in SEC for the MR pre-cooling step with cooling temperature.

The results presented in Figure 4 for MR composition show quite clear trends with the component mole fraction of each component a monotonic function of cooling temperature in the majority of cases. The impact on butane is largest, which is due to the steadily increasing heat duty at the warm end of HX-1 as the cooling temperature increases. The impact on the optimum nitrogen content in the MR is affected least by cooling temperature, reflecting the relatively static conditions at the cold end of HX-1.

The data presented in Figure 4 that represents optimum MR pressure solutions is less consistent with a slight upward trend visible across the range of cooling temperatures considered. This indicates that the optimum combination of MR composition and MR operating pressure is more difficult to determine and that the overall minimum may not have been found in all cases. However, Figure 5 shows a very consistent trend in how the SEC for the MR pre-cooling process varies with *T*c, which provides confidence that a solution close to the overall minimum was found in all cases.

Figures 6 and 7 present the hot and cold composite cooling curves for the overall minimum SEC solutions found for *T*<sup>c</sup> = 5 ◦C and *T*<sup>c</sup> = 50 ◦C. Generally, the results in Figures 6 and 7 show that the optimization algorithm has found a good fit for the cooling curves, with the 2 ◦C pinch temperature approached in multiple locations within HX-1 in both cases. The cooling curves for each of the temperature points studied between *T*<sup>c</sup> = 5 ◦C and *T*<sup>c</sup> = 50 ◦C are presented in Figures A2–A9, which are contained in the Appendix A.

**Figure 6.** Composite curves in HX-1 and key performance parameters, 5 ◦C cooling temperature case.

**Figure 7.** Composite curves in HX-1 and key performance parameters, 50 ◦C cooling temperature case.

Comparing the variation in minimum approach temperature data presented in Figure 6 for *T*<sup>c</sup> = 5 ◦C with that presented in Figure 7 for *T*<sup>c</sup> = 50 ◦C, it can also be observed that the optimization process has found a set of parameters that better minimize the temperature approach in HX-1 for the *T*<sup>c</sup> = 5 ◦C case. Looking at the *T*<sup>c</sup> = 50 ◦C case, we see that it becomes more difficult to maintain a close approach at the warm end of the heat exchanger suggesting that SEC could be reduced further through the addition of heavier components to the MR.

Figure 8 presents the SEC for the pre-cooling step, the cryogenic-cooling step, and the overall process. Figure 9 presents the same data in terms of the % change relative to the 25 ◦C case. Moreover, presented in Figure 9 are the corresponding second law efficiencies expressed as a percentage.

**Figure 8.** Variation in SEC with cooling temperature for the pre-cooling, cryogenic-cooling and overall cooling processes.

**Figure 9.** Percentage variation in overall SEC and second law efficiency for the pre-cooling, cryogeniccooling and overall cooling processes.

Figure 8 shows that the contribution of the pre-cooling process to overall SEC across the range of cooling temperatures investigated is approximately 10%. In addition, Figure 8 shows non-linear variation in SEC with cooling temperature for the pre-cooling part of the overall process that contrasts with the linear relationship between SEC and cooling temperature for the cryogenic process. This non-linear relationship for the pre-cooling process reflects the fact that lower cooling temperatures both reduced cooling duty and increase efficiency, whereas the close to linear impact on the cryogenic process is a result of only reduced increased efficiency. Further insight into this is provided by the results presented in Figure 9.

The results presented in Figure 9 show that energy consumption for the overall liquefaction process increases by around 20% across the cooling temperature range 5 to 50 ◦C and 5% over the range 20 to 30 ◦C. For the pre-cooling process the increase is close to 80% over the full temperature range. Figure 9 also shows that while the second law efficiency of the cryogenic-cooling process increases slightly across the range of temperatures considered, the efficiency of the pre-cooling process drops above 25 ◦C. The cause of this drop in efficiency as the cooling temperature increases can be seen in Figures 6 and 7, which show that the mean temperature approach for the higher temperature cases is higher than that of the lower temperature cases. It is this reduced level of optimization as cooling temperature increases above 25 ◦C that accentuates the non-linear behavior notable in Figure 8.

The implication of the results presented in Figure 9 is the same as discussed earlier: that design changes in the MR process could help to improve performance for the cases where the cooling temperature is higher than 25 ◦C. Both the addition of heavier components to the MR mixture could provide a more optimized design or the division of the MR loop into additional pressure levels. Both of these design alternatives could form the basis of further study work.

#### **4. Conclusions**

A model for a hydrogen liquefaction process has been developed and validated against results from an independent study. Although the validation process highlighted the significant impact that different properties models can have on model predictions, the validation results also indicate that the present model is suitable for the study of the impact of ambient temperature on process performance.

A set of optimization parameters were selected, and an optimization method developed that was shown to be suitable for the study of process performance across a range of process cooling temperatures through the consistency of the results obtained. The MR studied is limited to a mixture of five components. It is indicated in the results presented that the addition of heavier components could be used to improve efficiency for cooling temperatures above 25 ◦C, although the available gains would be small.

The results of the optimization work show that the specific energy consumption, SEC, of the MR pre-cooling process increases by around 80%, from approximately 0.57 to 1.0 kWh/kg, across the cooling temperature range 5 to 50 ◦C. These results, combined with the calculated process performance for the cryogenic-cooling step (not optimized here), show that total energy consumption for the hydrogen liquefaction process increases by around 20%, from 5.8 to 7.1 kWh/kg, across the same temperature range. Considering just the range 20 to 30 ◦C, there is a 5% increase, which illustrates the significant impact ambient temperature can have on energy consumption.

The variation in energy consumption with cooling temperature implies a significant benefit for liquefaction processes operating in low ambient temperature locations, especially given that the hydrogen liquefaction process represents a very energy intensive step in the supply of liquid hydrogen. The aim of further work is to combine these results into a larger system model that considers the impact of ambient temperature on the supply of low-carbon energy from natural gas.

**Author Contributions:** Conceptualization, S.J.; methodology, S.J. and E.B.; validation, S.J.; formal analysis, S.J.; investigation, S.J.; data curation, S.J.; writing—original draft preparation, S.J.; writing review and editing, S.J. and E.B.; supervision, E.B. Both authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Appendix A**

**Figure A1.** Block flow diagram of the overall liquefaction process.

**Figure A2.** Optimized composite cooling curves for 10 ◦C cooling temperature.

**Figure A3.** Optimized composite cooling curves for 15 ◦C cooling temperature.

**Figure A4.** Optimized composite cooling curves for 20 ◦C cooling temperature.

**Figure A5.** Optimized composite cooling curves for 25 ◦C cooling temperature.

**Figure A6.** Optimized composite cooling curves for 30 ◦C cooling temperature.

**Figure A7.** Optimized composite cooling curves for 35 ◦C cooling temperature.

**Figure A8.** Optimized composite cooling curves for 40 ◦C cooling temperature.

**Figure A9.** Optimized composite cooling curves for 45 ◦C cooling temperature.
