Pressure ratio profile

reason of this minimum is due to two different and opposite effects: the approximation to the ideal isothermal work, described previously, and the relationship between the polytropic efficiency of the compressor and its diameter.

As the CO2becomes compressed, its volume flowrate decreases, making the compressor diame- ter decrease as well so the diameter of the compressor decreases along the train. Since the efficiency penalty decreases with the diameter (see section A.1.5), the last compressors of the train are the ones with the lowest efficiency, illustrated in figure 7.4(b). Having these two influences, the final number of compressors from the optimization results in a balance between the approximation to the isothermal ideal work and the decreasing of efficiency.

In spite of having the problem solved by solving many small problems for fixed values of number of compressors, as a proof of concept, the problem was also solved with the original formulation from chapter 6. In other words, the problem was solved as a MINLP optimization problem by optimizing both number of compressors and pressure ratio.

Table 7.1:Critical output results from initial point optimization.

Output variable Variable Ns Fixed Ns Deviation (%)

Pressure ratio 2.3073 2.3068 0.02

Number of compressors 4 4 -

Total cost (M$/yr) 6.515 6.513 0.03

CAPEX (M$/yr) 0.856 0.855 0.12

OPEX (M$/yr) 5.659 5.658 0.02

Average MINLP iterations 2 0 -

Average NLP iterations 21 10 -

Average total CPU time (s) 12.1 7.5 -

The results from table 7.1 were obtained using 3 as an initial guess for the number of compressors and using 3 and 5 as lower and upper bounds, knowing that the solution is already inside that range.

As for the pressure ratio, the initial guess and bounds were the same used in the previous optimization (initial guess of 2.67, lower bound of 1.2 and upper bound of 4).

Even thought the complexity of the problem is higher than the previous one, since the number of DOF increased from 1 to 4, the optimal solution can be considered the same, since the deviation is smaller then 0.5% for all critical variables of the problem. Moreover, both problems are still relatively simple, since the number of DOF is reasonably small, so it’s normal that the difference between both methods is not a problem in terms of optimality and optimization time.

Note that, during this optimization, the compressor duty (around 9200 Hp) and cooler area (around 220 m²) were inside the range of applicability of the cost functions from section 6.2.2.

Based on that, the equation assumed in section 6.4 (equation 6.13) was removed from the previ- ous formulation and a new optimization was performed in order to determine how is it going to affect both cost and optimal solution.

Using the same methodology from the previous section, i.e., optimizing the pressure ratio fixing the number of compressors and using the same initial guess and bounds (except for the 3 compressor case where the upper bound used for the pressure ratio was 4.5 instead of 4), the NLP problems were solved and the results were compared against the previous formulation. The decision variables are now the pressure ratio of each individual compressor, having the following number of DOF, depending on the value of the number of compressor sections:

DOF=Nsections (7.1)

Figure 7.5: Total cost in function of number of stages for pressure ratio profile formulation (a), Comparison of total cost between constant pressure ratio and profile formulations (b).

By analyzing figure 7.5(a), it is possible to observe that the optimal number of compressors for the new formulation is 5 instead of the 4 compressors train given by the previous formulation. This is probably related to the effect of a better balanced compression work along the train and is going to be analyzed in the next figures.

Figure 7.5(b) shows that, by optimizing the pressure ratio of each individual compressor, the total cost is smaller for this case than for the constant pressure ratio case and that difference increases with the number of compressors. This is coherent with the fact that, the more compressors the train has, the more DOF it has to change in order to get a better compression balance when compared with the constant pressure ratio formulation.

Also, it can be observed that the variation of total cost with the number of compressors is much smaller for the pressure ratio profile formulation than for the constant pressure ratio formulation. Be- tween the optimal solution, the difference in cost between 4, 5 and 6 compressors is smaller than 0.5%, leading to two different conclusions:

The first was that it is more important to determine the right pressure ratio distribution than to precisely determine the optimal number of compressors, since the last one is highly related to CAPEX and the the other one to OPEX.

The second was that, as a chef engineer that is designing a compression train, it is much more

useful to have three different possible solutions to choose from, allowing him to check which one is better in off-design conditions. This is something that wouldn’t be detected if only an MINLP opti- mization problem was performed instead of the discretization to NLP problems for different values of number of compressors.

Figure 7.6: Pressure ratio profile for optimal solution of five compressors (a), Pressure ratio profile for different number of compressors (b).

Figures 7.6(a) and 7.6(b) confirm that the pressure ratio follows the path of the efficiency along the train, i.e., it is higher in the first compressor, where the efficiency is bigger, and gradually decreases as it moves towards the last compressor.

The only exception to this is the last compressor, that always has a bigger pressure ratio than the previous one and the reason for this comes from the way the superstructure was formulated. When the CO2stream reaches the last compressor and is by-passed to the end of the train, it doesn’t flow to a last cooler. So, since the CO2 doesn’t have to be cooled after the last compressor, the last compressor has a pressure ratio slightly higher than the previous one. Although it is just a slight change, it doesn’t necessary mean that is incorrect since intercooling can or cannot be necessary before the dehydrator, depending on the water content in CO2stream from the capture plant and the discharge temperature of the last compressor.

Figure 7.7:CAPEX in function of number of stages (a), OPEX in function of number of stages (b).

and OPEX, where the difference between both formulation is much bigger in OPEX than in CAPEX.

It is also possible to see that the difference in total cost verified in figure 7.5(b) comes mainly from the difference of OPEX from figure 7.7(b).

The reason for both things is that OPEX is highly related with the pressure ratio so, due to the power consumption reduction caused by the compression work distribution, it has a bigger impact in OPEX than in CAPEX.

Like in the previous section, the problem was solved with the original formulation from chapter 6 but where it was optimized the pressure ratio of each compressor section. In other words, the problem was solved as a MINLP optimization problem by optimizing both number of compressors and pressure ratio of each compressor.

Table 7.2:Critical output results from optimization with pressure ratio profile.

Output variable Variable Ns Fixed Ns Deviation (%)

Number of compressors 5 5 -

Pressure ratio (1) 3.4777 3.5089 -0.9

Pressure ratio (2) 1.9268 1.9206 0.3

Pressure ratio (3) 1.6680 1.6438 1.5

Pressure ratio (4) 1.4926 1.5009 -0.6

Pressure ratio (5) 1.6607 1.6657 -0.3

Total cost (M$/yr) 6.4083 6.513 0.02

CAPEX (M$/yr) 0.8658 0.855 -0.002

OPEX (M$/yr) 5.5425 5.658 0.002

Average MINLP iterations 3 0 -

Average NLP iterations 32 20 -

Average total CPU time (s) 15.7 9.2 -

The results from table 7.2 were obtained using 4 as an initial guess for the number of compressors and using 3 and 6 as lower and upper bounds, knowing that the solution is already inside that range.

As for the pressure ratio, the initial guess and bounds were the same used in the previous optimization (initial guess of 2.67, lower bound of 1.2 and upper bound of 4).

The optimal solution can be considered the same, since most of the deviation values are smaller then 1% for all critical variables of the problem. Also, by the number of iterations and CPU time that both problems took, it maybe worth spending more optimization time by doing many NLP optimizations for fixed number of compressors, because the final result will have a format of a total cost versus number of compressors curve, where the user can choose himself what configurations he wants to test in off-design conditions.

Having analyzed the effect of the optimization of the pressure ratio of each individual compressor section, it is possible to conclude that this formulation tends to lead to more compressors, since a better compression work distribution reduces OPEX in a way that surpasses the increasing in CAPEX by adding an extra compressor. So, the designed train obtained by this formulation has a smaller total cost than the one from the previous formulation.

Due to this, the solving methodology used for the next sections of this thesis is going to be this last formulation.