Thermodynamic Principle-Based Hydrogen Partial Pressure Correction for Optimization of Refinery Hydrogen Networks
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Abstract
The cost of hydrogen consumption takes up a high percentage of the total crude oil refining cost. Fresh hydrogen and compression work are main operating cost to be minimized in hydrogen networks. In hydroprocessing reactions, hydrogen partial pressure is a key index to hydrogen demand and compression work. However, such a decisive factor is estimated by Dalton’s law which is not suitable to real gas especially in higher pressure case. To cope with such an issue, thermodynamic principles including the theorem of corresponding states and compressibility factor are employed to quantify hydrogen partial pressure accurately. Based on relative concentration and hydrogen partial pressure properties for hydrogen streams, this paper establishes a nonlinear programming (NLP) model for the minimization of total exergy including fresh hydrogen and compression work. Case studies show that hydrogen partial pressure calculated by Dolton’s law is larger than the requirement and thermodynamic principles are not only more accurate but are also able to save compression work.
Keywords
Hydrogen networks Thermodynamic principle Relative concentration Hydrogen partial pressure Mathematical programmingNomenclature
- ch
A constant for the exergy calculation of fresh hydrogen (kW h Nm^{−3})
- Cp
Specific heat capacity of gas at constant temperature
- d
Connection matrix of hydrogen network structure
Exergy useful work (kW)
- F
Flow rate (Nm^{3} h^{−1})
- P
Pressure (MPa)
- R
Gas constant (J mol^{−1} K^{−1})
- SUM
Summation
- T
Temperature (K)
- V
Volume (m^{3})
- W
Compressor power consumption (kW)
- y
Absolute concentration (%)
- Z
Compressibility factor
Subscript
- H
H_{2}
- i
Number of hydrogen source
- j
Number of hydrogen sink
- k
Three impurities H_{2}S, NH_{3}, CH_{4}
Superscript
- c
Critical properties
- lo
The lower limit
- r
Reduced properties
- RC
Relative concentration
- up
The upper limit
Greek letter
- γ
The ratio of specific heat of gas at constant pressure to that at constant volume
- η
Efficiency of compressors
Abbreviations
- AC
Absolute concentration
- RC
Relative concentration
- sk
Hydrogen sink
- sr
Hydrogen source
Introduction
As the demand for petroleum products increases (Wang and Zhang 2005), hydrodesulfurization, hydrocracking, and hydroforming operations consume more and more hydrogen. Therefore, hydrogen resources are so expensive that the way to improve efficiency utilization and conservation of fresh hydrogen is the key issue to increase profit for refineries. It has been proved that hydrogen network integration is an effective method which can minimize the waste of hydrogen. Usually, the streams that can supply hydrogen is named hydrogen source (sr) while that consuming hydrogen is defined as hydrogen sink (sk). The advantage of hydrogen network integration is taking all hydrogen consumption units into comprehensive consideration instead of optimizing one by one.
Analysis methods of hydrogen network nowadays mainly include pinch technology and mathematical programming approaches. Alves and Towler (2002) used the diagram of surplus hydrogen to target minimum fresh hydrogen consumption, which was an earlier pinch technology used to solve integration problems of hydrogen network. El-Halwagi et al. (2003) proposed a rigorous graphical targeting approach to minimize fresh resource demand by segregating, mixing, and direct reuse of all sources. Zhao et al. (2006) proposed a material recovery pinch diagram through hydrogen load versus flow rate coordinate to target minimum fresh hydrogen demand. Foo et al. (2006) put forward an algebraic calculation and ultimately achieved the purpose of maximum resource recovery. Bandyopadhyay (2006) proposed a source composite curve to design networks that achieve the minimum fresh resource target. Limiting the composite curve for targeting both hydrogen and water networks was presented by Agrawal and Shenoy (2006), and the nearest neighbor algorithm was also proposed to design the corresponding networks. Afterwards, Deng et al. (2015) used an improved problem table to determine the optimal performance of the purifier and consequently minimize the hydrogen utility. Zhang et al. (2011) proposed an improved graphical method coupling material recovery pinch diagram and mass balance triangle rule for the hydrogen networks with purification reuse. Relative concentration based on pinch analysis (Zhang et al. 2016a) was used to release the total flow rate and concentration normalization constraints so that fresh hydrogen can further be reduced. Pinch-based conceptual methods take advantage of clear concepts and visual targeting and design process. However, the disadvantage of pinch technology is not suitable to complex problems.
For mathematical programming methodologies, Hallale and Liu (2001) first established a superstructure and nonlinear programming (NLP) model to optimize the hydrogen network with basic constraints. Afterwards, mathematical models used to synthesize hydrogen networks encompass much more necessary and complex constraints so that it can be more practical and reasonable. Liu and Zhang (2004) made a trade-off of operation and capital cost among different purifier placement scenarios. Liao et al. (2010) proposed a sensitivity analysis to make the trade-off between hydrogen network cost and oil processing network for improvement of refinery profit. Zhou et al. (2012) introduced a desulfurization unit into the hydrogen network to minimize total annual cost through a MINLP model. Jiao et al. (2013) took the fierce market competition and stringent environmental legislation into consideration and established a mixed-integer nonlinear programming (MINLP) which can deal with high-dimensional and complex hydrogen system problems to gain good economic improvements. Deng et al. (2014) set hydrogen utility headers considering the requirement of hydrogen consumers to simplify the network configuration and ultimately enhanced the expandability and flexibility of hydrogen networks. An algebraic targeting method methodology to minimize compression work was proposed by Bandyopadhyay et al. (2014) to address flow, concentration, and pressure-constrained hydrogen allocation network (HAN). Similarly, Jagannath and Almansoori (2017) used stream-dependent properties and realistic compressor cost correlations to determine the compression duty and costs for synthesizing hydrogen networks with minimum compression cost. In 2016, the relative concentration property is employed by mixed linear programming model (MILP) (Zhang et al. 2016b) to analyze not only multi-impurity but also the fluctuant hydrogen networks. Liang et al. (2017) proposed a subperiod partitioning method based on clustering of uncertain operating parameters, to minimize the total annual cost and improve the flexibility of the multi-period hydrogen networks. To reduce the number and the capital cost of compressors, Liang and Liu (2017) proposed two strategies for further merging compressors. Recently, a linearization algorithm (Mahmoud et al. 2017) is used to simplify power equations and reduce an MINLP to an MILP model for operating cost minimization. Mathematical programming methods take the advantage of being able to deal with complex network integration problems. However, the pressure constraint has not been dealt with accurately enough.
Hydrogen partial pressure is an important index for hydroprocessing operations, but its current determination of total pressure times hydrogen mole fraction is defective. Ideal gas means that its equation of state is able to ignore the molecular volume and intermolecular forces. However, the conditions of hydrogenation reaction are generally 5~20 MPa and 550~650 K, so under such temperature and pressure, hydrogen stream cannot be considered as an ideal gas. As a result, it is obvious that the hydrogen partial pressure calculated by Dalton’s law adopted by existing methods is not accurate. Therefore, relative concentration and thermodynamic principles are introduced to correct the defect of traditional methods and a nonlinear programming (NLP) model is formulated for optimization of hydrogen network. Results show that hydrogen partial pressures identified by real gas state equation are more accurate and compression work can also be reduced.
Theory
Superstructure of Hydrogen Networks with Hydrogen Partial Pressure Constraint
Thermodynamic Principle
There are several state equations for gases. Among them, the van der Waals equation was the first one that has been proposed for real gas and afterwards, the cubic equation of state (Valderrama 2003) was put forward. The following equations like Redlich-Kwong equation of state (Redlich and Kwong 1949), Soave modification of Redlich-Kwong (Soave 1972), Peng-Robinson equation of state (Peng and Robinson 1976), Peng-Robinson-Stryjek-Vera equations of state (Stryjek and And 1986), Elliott-Suresh-Donohue equation of state (Elliott et al. 1990), which were developed based on van der Waals equation. Equations that also can be used for real gases are Virial equation of state (Gómeznieto and Robinson 1979) (also called the Kamerlingh Onnes equation) and multi-parameter equations of state (Span 2000). Van der Waals and RK equations were rarely used in engineering calculations because of obvious error. SRK and PR equations have good-enough accuracy; however, with terms like V^{3}, the iteration calculation is more complex and furthermore they are good at non-polar systems. By comparison, Virial equation of state and the principle of corresponding states, its accuracy is able to meet requirements and it has faster calculation speed, what is more, there is a strong applicability to different gases.
The thermodynamic principle of this paper is mainly based on Virial equation of state and the principle of corresponding states (Guggenheim 1945). Z is calculated by fitting the generalized compressibility factor map based on these principles.
Mathematical Programming
The mathematical model of this paper takes exergy as the objective function to solve the problem that fresh hydrogen consumption (FSR1) and the sum of compression work (SUMW) are with different units. And this model can be divided into two parts for a clearer description. The first part is the accurate calculation of hydrogen partial pressure based on the principle of thermodynamics; the second part is composed of the common constraints of the hydrogen network.
Thermodynamic Method for Accurate Calculation of Hydrogen Partial Pressure
Generally, the constraint of hydrogenation reactions with hydrogen flow is based on the constraint of hydrogen partial pressure. The simple calculation is based on Dalton’s law of partial pressure to set the operating conditions of the compressor, that is, the total pressure of the compressor outlet got through the lower limit of hydrogen partial pressure in the reactor divided by hydrogen mole fraction of the flow. While thermodynamic method is, based on Virial equation of state and the principle of corresponding states, using the known hydrogen partial pressure calculates the operation setting of the compressor. Following is specific description.
The above mathematical model is nonlinear, because the relationship between reduced properties, T^{ r }, and P^{ r }, and the compressibility factor Z is nonlinear.
Owing to the total pressure of mixed gas P we got, data calculation on the compressor can be carried out. Furthermore, the following is used in hydrogen network integration.
Mathematical Model of Hydrogen Network Integration
Objective Function
Relative Concentration Property
Constraint Equations of Sources
Constraint Equations of Hydrogen Sinks
This mathematical model consists of Eqs. (1)~(25). Among them, equations used to calculate hydrogen partial pressure and the compressor work consumption are non-linear, so it is a nonlinear programming (NLP) model.
Case Study
A hydrogen network case
F (Nm^{3} h^{−1}) | y_{ H } (mol%) | P (MPa) | y_{ k }(%) | |||
---|---|---|---|---|---|---|
H_{2}S | NH_{3} | CH_{4} | ||||
sr _{ 1 } | 35,000.00 | 99.99 | 2.15 | 0 | 0.01 | 0 |
sr _{ 2 } | 14,192.64 | 87.2 | 4 | 1 | 2.3 | 9.5 |
sr _{ 3 } | 34,513.92 | 95.4 | 8.8 | 1.3 | 1.5 | 1.8 |
sr _{ 4 } | 46,448.64 | 97.2 | 2.1 | 1.7 | 0.5 | 0.6 |
sr _{ 5 } | 27,388.80 | 93.22 | 2.2 | 0.6 | 5.14 | 1.04 |
sr _{ 6 } | 8386.56 | 85.4 | 18.5 | 2.26 | 7.23 | 5.11 |
sr _{ 7 } | 21,853.44 | 89.4 | 6 | 4.5 | 2.6 | 3.5 |
sk _{ 1 } | 17,660.16 | 82.3 | 4.5 | 0 | 7.8 | 9.9 |
sk _{ 2 } | 10,241.28 | 91.33 | 9.2 | 1.2 | 5.4 | 2.07 |
sk _{ 3 } | 15,402.24 | 87.1 | 2.2 | 3.5 | 3.9 | 5.5 |
sk _{ 4 } | 22,176.00 | 98.2 | 2.3 | 0.5 | 0.6 | 0.7 |
sk _{ 5 } | 25,401.60 | 84.7 | 19.1 | 3 | 1.2 | 11.1 |
sk _{ 6 } | 31,530.24 | 96.8 | 6.2 | 1.5 | 0.7 | 1 |
Hydrogen network structure of the case
d _{ i,j } | sk _{ 1 } | sk _{ 2 } | sk _{ 3 } | sk _{ 4 } | sk _{ 5 } | sk _{ 6 } |
---|---|---|---|---|---|---|
sr _{ 1 } | 1 | 1 | 1 | |||
sr _{ 2 } | 1 | |||||
sr _{ 3 } | 1 | 1 | 1 | |||
sr _{ 4 } | 1 | 1 | 1 | 1 | ||
sr _{ 5 } | 1 | 1 | ||||
sr _{ 6 } | 1 | 1 | ||||
sr _{ 7 } | 1 | 1 |
Optimization results of pressure constraint using total pressure
Allocations | Total pressure basis | ||
---|---|---|---|
F_{ j,i } (Nm^{3} h^{−1}) | W_{ j,i } (kW) | P_{ j,i } (MPa) | |
sk _{ 1 } , sr _{ 1 } | 14,536 | 316.1 | 4.50 |
sk _{ 2 } , sr _{ 4 } | 1445 | 70.2 | 9.20 |
sk _{ 2 } , sr _{ 5 } | 6000 | 280.4 | 9.20 |
sk _{ 2 } , sr _{ 6 } | 2758 | 0 | 9.20 |
sk _{ 3 } , sr _{ 5 } | 3327 | 0 | 2.20 |
sk _{ 3 } , sr _{ 7 } | 11,536 | 0 | 2.20 |
sk _{ 4 } , sr _{ 1 } | 13,641 | 24.6 | 2.30 |
sk _{ 4 } , sr _{ 3 } | 8529 | 0 | 2.30 |
sk _{ 5 } , sr _{ 3 } | 18,700 | 428.5 | 19.10 |
sk _{ 5 } , sr _{ 4 } | 3548 | 288.7 | 19.10 |
sk _{ 5 } , sr _{ 7 } | 253 | 9.2 | 19.10 |
sk _{ 6 } , sr _{ 1 } | 1945 | 63.6 | 6.20 |
sk _{ 6 } , sr _{ 3 } | 7284 | 0 | 6.20 |
sk _{ 6 } , sr _{ 4 } | 22,250 | 746.5 | 6.20 |
FSR1(Nm^{3} h^{−1}) | 30,121.87 | ||
SUMW (kW) | 2227.82 | ||
Exergy (kW) | 89,638.40 |
Optimization results of pressure constraint using the hydrogen partial pressure calculated by Dalton’s law
Allocations | Dalton’s law basis | ||
---|---|---|---|
F_{ j,i } (Nm^{3} h^{−1}) | W_{ j,i } (kW) | P_{ j,i } (MPa) | |
sk _{ 1 } , sr _{ 1 } | 14,536 | 226.2 | 3.70 |
sk _{ 2 } , sr _{ 4 } | 1445 | 66.6 | 8.64 |
sk _{ 2 } , sr _{ 5 } | 6000 | 275.6 | 9.01 |
sk _{ 2 } , sr _{ 6 } | 2758 | 0 | 9.20 |
sk _{ 3 } , sr _{ 5 } | 3327 | 0 | 2.20 |
sk _{ 3 } , sr _{ 7 } | 11,536 | 0 | 2.20 |
sk _{ 4 } , sr _{ 1 } | 13,641 | 17.9 | 2.26 |
sk _{ 4 } , sr _{ 3 } | 8529 | 0 | 2.30 |
sk _{ 5 } , sr _{ 3 } | 18,700 | 356.8 | 16.96 |
sk _{ 5 } , sr _{ 4 } | 3548 | 264.9 | 16.64 |
sk _{ 5 } , sr _{ 7 } | 253 | 8.7 | 18.10 |
sk _{ 6 } , sr _{ 1 } | 1945 | 61.4 | 6.00 |
sk _{ 6 } , sr _{ 3 } | 7284 | 0 | 6.20 |
sk _{ 6 } , sr _{ 4 } | 22,250 | 743.2 | 6.17 |
FSR1(Nm^{3} h^{−1}) | 30,121.87 | ||
SUMW (kW) | 2021.62 | ||
Exergy (kW) | 89,432.20 |
Optimization results of pressure constraint using hydrogen partial pressure calculated by thermodynamic principles
Allocations | Thermodynamic principle basis | ||
---|---|---|---|
F_{ j,i } (Nm^{3} h^{−1}) | W_{ j,i } (kW) | P_{ j,i } (MPa) | |
sk _{ 1 } , sr _{ 1 } | 14,536 | 226.2 | 3.70 |
sk _{ 2 } , sr _{ 4 } | 1445 | 66.6 | 8.64 |
sk _{ 2 } , sr _{ 5 } | 6000 | 275.0 | 9.00 |
sk _{ 2 } , sr _{ 6 } | 2758 | 0 | 9.20 |
sk _{ 3 } , sr _{ 5 } | 3327 | 0 | 2.20 |
sk _{ 3 } , sr _{ 7 } | 11,536 | 0 | 2.20 |
sk _{ 4 } , sr _{ 1 } | 13,641 | 17.9 | 2.26 |
sk _{ 4 } , sr _{ 3 } | 8529 | 0 | 2.30 |
sk _{ 5 } , sr _{ 3 } | 18,700 | 355.0 | 16.91 |
sk _{ 5 } , sr _{ 4 } | 3548 | 264.6 | 16.61 |
sk _{ 5 } , sr _{ 7 } | 253 | 8.6 | 17.97 |
sk _{ 6 } , sr _{ 1 } | 1945 | 61.4 | 6.00 |
sk _{ 6 } , sr _{ 3 } | 7284 | 0 | 6.20 |
sk _{ 6 } , sr _{ 4 } | 22,250 | 742.7 | 6.17 |
FSR1(Nm^{3} h^{−1}) | 30,121.87 | ||
SUMW (kW) | 2018.01 | ||
Exergy (kW) | 89,428.59 |
By result comparison, the minimum fresh hydrogen consumptions determined by three scenarios are all the same, 30,121.87 Nm^{3} h^{−1}. However, the compression works consumed in three cases are 2227.82, 2021.62, and 2018.01 kW, respectively. Therefore, the difference of total exergy consumption is caused by compression work. Such results indicate that the thermodynamic principle gives not only the accurate hydrogen partial pressure but also reduces the compression work consumption.
As for the reason, Dalton’s law of partial pressure has certain errors for nonideal gases, so based on the same set value of hydrogen partial pressure in hydrogen sink, different gas flows correspond to different total pressure compared with thermodynamic principle that can calculate the P-V-T properties of nonideal gases accurately.
Comparison of all outlet pressure on three bases
Allocations | P_{ j,i } (MPa) HPP: hydrogen partial pressure | ||
---|---|---|---|
Total pressure | HPP (Dalton’s law basis) | HPP (thermodynamic principle) basis | |
sk _{ 1 } , sr _{ 1 } | 4.50 | 3.70 | 3.70 |
sk _{ 2 } , sr _{ 4 } | 9.20 | 8.64 | 8.64 |
sk _{ 2 } , sr _{ 5 } | 9.20 | 9.01 | 9.00 |
sk _{ 2 } , sr _{ 6 } | 9.20 | 9.20 | 9.20 |
sk _{ 3 } , sr _{ 5 } | 2.20 | 2.20 | 2.20 |
sk _{ 3 } , sr _{ 7 } | 2.20 | 2.20 | 2.20 |
sk _{ 4 } , sr _{ 1 } | 2.30 | 2.26 | 2.26 |
sk _{ 4 } , sr _{ 3 } | 2.30 | 2.30 | 2.30 |
sk _{ 5 } , sr _{ 3 } | 19.10 | 16.96 | 16.91 |
sk _{ 5 } , sr _{ 4 } | 19.10 | 16.64 | 16.61 |
sk _{ 5 } , sr _{ 7 } | 19.10 | 18.10 | 17.97 |
sk _{ 6 } , sr _{ 1 } | 6.20 | 6.00 | 6.00 |
sk _{ 6 } , sr _{ 3 } | 6.20 | 6.20 | 6.20 |
sk _{ 6 } , sr _{ 4 } | 6.20 | 6.17 | 6.17 |
In Figs. 2 and 3, in addition to the data of hydrogen supply, we can see the pressure of every compressor outlet stream, and the lower pressure limit in sink j is based on constraint of the total pressure. For the two hydrogen networks with the same hydrogen supply and network structure, the numbers in red reflect more accurate information of hydrogen pressure compared with the blue ones. The pressure of every compressor outlet seems too small, while the partial pressure of hydrogen is adequate. It can also show us that the saving potential of the power consumption of compression based on the constraint of the total pressure of flow. Compared with the constraint of the total pressure, the hydrogen partial pressure constraint corrected based on thermodynamic principles has saved 209.81 kW compression work, about 9.42%. And it has saved 3.61 kW compression work, about 0.2%, in contrast with the pressure constraint used in the hydrogen partial pressure calculated by Dalton’s law.
Above all is that, changing the pressure constraint of hydrogen network, results are mainly about SUMW. It means that if a hydrogen network used the total pressure of hydrogen flow to optimize the system, there is superfluous SUMW. While hydrogen partial pressure constraint based on Dalton’s law of partial pressures is the same but more slight, however, the worse is that the redundant SUMW must influence hydrogen partial pressure, which would result in a bad quality of the oil product. Therefore, paying more attention to such an issue is necessary.
Conclusions
In this paper, thermodynamic principles are introduced to correct hydrogen partial pressure constraint and a NLP model is formulated to address optimization of hydrogen networks. The results prove that the Dalton’s law will lead to greater hydrogen partial pressures while thermodynamic principles can give accurate levels of them. Meanwhile, the thermodynamic principle based the mathematical method proposed in this paper can further reduce the compression work. Since the case is from a practical refinery, the revision of hydrogen partial pressure based on thermodynamic principle is indispensable and the method proposed is contributory not only theoretically but also applicably.
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