Stable and high-precision liquefied natural gas density calculation method and system

Abstract

The invention discloses a stable and high-precision liquefied natural gas density calculation method and system. According to the method, aiming at the problem that a GERG-2008 equation possibly generates a plurality of density solutions at low temperature, the solutions with physical significance can be accurately identified and calculated. According to the characteristics of a solution with physical significance, a method of combining one-dimensional search and Newton iteration is provided for calculation, and finally, accurate judgment is carried out by utilizing a Gibbs free energy principle, and a correct solution is obtained. Meanwhile, the invention further provides a corresponding calculation system in a matched mode, and the system can accurately and rapidly calculate the density of the liquefied natural gas. Compared with a traditional method, the method has the advantages that the calculation precision and stability are remarkably improved, and a reliable tool is provided for technicians. The method is particularly suitable for calculation of the density of the liquefied natural gas, effectively solves the problems caused by multiple solutions of the GERG-2008 equation, and ensures the accuracy and practicability of the calculation result of the density of the liquefied natural gas.

Description

Technical Field

[0001] The present invention relates to the fields of cryogenic physics and thermodynamics, and in particular to a stable and high-precision method and system for calculating the density of liquefied natural gas. Background Art

[0002] As a clean energy, the global trade volume of liquefied natural gas continues to grow. In the production, transportation and trade handover of liquefied natural gas, density, as a core physical parameter, plays a vital role in ensuring measurement accuracy and optimizing storage and transportation efficiency. However, the traditional method of directly measuring the density of liquefied natural gas by relying on experimental instruments is not only time-consuming and labor-intensive, but also costly, and it is difficult to adapt to the fast pace and cost control requirements of large-scale industrial applications. To solve this problem, high-precision mathematical prediction models based on empirical data have become the mainstream method for calculating the density of liquefied natural gas. Among them, the GERG-2008 equation is recommended as the standard equation for calculating the density of liquefied natural gas in the International Organization for Standardization (ISO) due to its excellent accuracy and wide applicability. However, under the low temperature conditions unique to liquefied natural gas, the relationship between pressure and molar density described by the GERG-2008 equation is complex and nonlinear, which brings challenges to the solution of the equation. Specifically, under the temperature conditions of liquefied natural gas, there may be multiple density solutions in the GERG-2008 equation, but only one of them is the true density solution. Traditional numerical solution methods, such as Newton iteration method and bisection method, have limitations when dealing with such complex nonlinear problems, and it is often difficult to ensure stable convergence to the true density solution. The selection of the initial value of the iteration, the control of the iteration process, and the screening of solutions in the case of multiple solutions are all key factors that restrict the traditional methods from accurately solving the density. Therefore, given the importance of LNG density calculation in practical applications and the limitations of existing solution methods, it is particularly urgent to develop a method and system that can stably, accurately and efficiently calculate the density of LNG. This new method needs to be able to overcome the shortcomings of traditional solution methods in solving the GERG-2008 equation, especially under the low temperature conditions of LNG, to ensure stable convergence to the only physical actual solution, thereby providing solid technical support for the measurement, storage and transportation of the LNG industry. Summary of the invention

[0003] The present invention proposes a new calculation method and system to solve the complex nonlinear relationship between pressure and molar density in the GERG-2008 equation under liquefied natural gas temperature conditions, and the problem that traditional calculation methods (such as Newton iteration method or dichotomy method) are difficult to stably obtain the correct density during calculation. The method combines one-dimensional search with Newton iteration, and introduces the Gibbs free energy criterion to ensure the authenticity of the solution, so that the density of liquefied natural gas can be stably and accurately calculated.

[0004] According to the first aspect of the present application, a stable and high-precision calculation method for the density of liquefied natural gas is provided. The method includes: 1) Inputting the molar composition x of liquefied natural gas i (%), temperature T (K), and pressure P (Pa), where the range of T is 60 - 700 K and the range of P is 0 - 70 MPa.

[0005] 2) Obtaining the first molar density solution that meets the requirements through a one-dimensional search calculation method. The specific steps are as follows:

[0006] S21, Given an initial value a for calculating the molar density (the value range is 0 < a ≤ 0.01 mol / L), according to the molar composition (x i ), temperature (T), and pressure (P) input in 1), calculate the corresponding pressure f(a) when the molar density is a according to Equation (1 - 6). If f(a) > 0, then perform the calculation; otherwise, set d1 = 0 and end the calculation;

[0007]

[0008] Where: f(d) is the pressure corresponding to the molar density d, in Pa; d is the molar density, in mol / L; T is the temperature, in the unit of K; where The calculation formula is as shown in Equations (2 - 4):

[0009]

[0010] Where: d i,k , n i,k , t i,k , c i,k , d ij,k , n ij,k , η ij,k , t ij,k , ε ij,k , β ij,k , γ ij,k —— are the coefficients and exponents corresponding to different components, obtained from the standard ISO 20765 - 2. The calculation formulas for the parameters δ and τ are as shown in Equations (5) and (6):

[0011]

[0012] Where: d c,i , d c,j are the critical molar densities, in mol / L; T c,i , T c,j are the critical temperatures, in K; β V,ij , γ V,ij , β T,ij , γ T,ij are the binary interaction parameters;

[0013] S22. Set the step size to 1. The increased molar density \(x = a + 1\). Calculate the partial derivative \(f'(x)\) of pressure with respect to molar density \(x\) according to Equation (7). If \(f'(x)>0\), then proceed to step S23. If \(f'(x)\leq0\), proceed to step S26;

[0014]

[0015] S23. If \(f(x)<P\), then let \(a = x\) and return to step S22. Otherwise, execute step S24;

[0016] S24. Let \(b=(a + x) / 2\). Calculate the pressure \(f(b)\) when the molar density is \(b\) according to Equation (1 - 7). If \(|f(b)-P|>10\) -6 then execute step S25. Otherwise, let \(d1 = b\) and the calculation ends;

[0017] S25. If \(f(b)-P>0\), then let \(x = b\) and return to step S24. Otherwise, let \(a = b\) and return to step S24;

[0018] S26. Let \(b=(a + x) / 2\). If \(|a + x|>0.01\), then execute step S27. Otherwise, let \(d1 = 0\) and the calculation ends;

[0019] S27. If \(f'(x)\times f'(b)<0\), then let \(x = b\) and return to S26. Otherwise, execute step S28;

[0020] S28. If \(f(b)<P\), then let \(a = b\) and return to step S26. Otherwise, let \(x = b\) and execute step S24;

[0021] 3) Given a relatively large initial value of Newton iteration molar density (\(d>25\ mol / L\)), obtain the second molar density solution that meets the requirements through Newton iteration. The specific solution steps are as follows:

[0022] S31. According to the temperature, pressure, and liquefied natural gas components input in 1), set the initial molar density \(d = 30\ mol / L\);

[0023] S32. Obtain the pressure \(f(d)\) corresponding to the molar density \(d\) and the partial derivative \(f'(d)\) of \(f(d)\) with respect to the molar density \(d\).

[0024] S33. Let \(d\) ite \(=d-(f(d)-P) / f'(d)\). Calculate the pressure \(f(d\) ite ) corresponding to the molar density dite.

[0025] S34. If \(|f(d\) ite )-P|>10 -6 , then let \(d = d\) ite, return to step S32, otherwise d2 = d ite The calculation is finished.

[0026] 4) Based on the minimum Gibbs free energy principle, determine the Gibbs free energy corresponding to the two molar densities. The specific solution steps are as follows:

[0027] S41, if d1≠0, proceed to step S42, otherwise the true molar density d=d2, and the calculation ends;

[0028] S42, if |d1-d2|>10 -6 , proceed to step S43, otherwise the true molar density d = (d1 + d2) / 2, and the calculation ends;

[0029] S43, calculate the Gibbs free energy g(d1) and g(d2) according to formula (8-12). If g(d1)≤g(d2), then d=d1, otherwise d=d2, and the calculation ends.

[0030]

[0031] Where: ci , T ci , c oi,k ——The coefficients and indices corresponding to the different components are obtained in standard ISO 20765-2.

[0032] 5) Input the molar component x according to 1) i The density of liquefied natural gas is calculated according to formula (13) using the true molar density d calculated by 2), 3) and 4).

[0033] ρ=x i M i d (13)

[0034] Where: M i is the molar mass corresponding to component i, mol / kg.

[0035] According to the second aspect of the present invention, a stable and high-precision liquefied natural gas density calculation system is provided. The system mainly includes two components: a memory and a solver. The memory is used to store computer program instructions and can temporarily store input and output data. The solver is responsible for executing these computer program instructions to implement a stable and high-precision liquefied natural gas density calculation method described in the first aspect of the present invention. Specifically, the system includes the following main modules:

[0036] 1) Data input and output management module, responsible for receiving and storing the necessary parameters such as liquefied natural gas composition information, temperature value, pressure value, etc. input by the user. Provide a data output interface for saving and displaying the calculated liquefied natural gas density and other results.

[0037] 2) Density calculation module, which has built-in specific implementation logic of the calculation method provided by the first aspect of the present invention. According to the data input by the user, the module sequentially performs steps such as one-dimensional search, Newton iteration and Gibbs free energy criterion to stably and accurately calculate the density of liquefied natural gas.

[0038] The present invention has the following advantages and beneficial effects:

[0039] (1) The present invention calculates 56 groups of liquefied natural gas densities with a temperature range of 100 to 160 K and a pressure range of 0 to 8.5 MPa, and compares the calculation results obtained by the solution method of the present invention with the experimental values ​​for verification. The average relative deviation between the calculation results and the experimental values ​​using the solution method provided by the present invention is in the range of 0.01% to 0.12%;

[0040] (2) The present invention can be applied not only to density calculation under low temperature conditions, but also to density calculation under normal and high temperature conditions. BRIEF DESCRIPTION OF THE DRAWINGS

[0041] Figure 1 A flow chart of the one-dimensional search for solving the GERG-2008 equation provided by the present invention;

[0042] Figure 2 A flow chart of the Newton iteration method for solving the GERG-2008 equation provided by the present invention;

[0043] Figure 3 A flow chart of a solution for determining true molar density based on the Gibbs free energy principle provided by the present invention. DETAILED DESCRIPTION

[0044] The present invention is further described below in conjunction with the embodiments, but is not intended to be limiting of the present invention. The present invention provides a stable and high-precision method for calculating the density of liquefied natural gas, which comprises the following steps:

[0045] 1) Input liquefied natural gas molar composition x i (%), temperature T (K), pressure P (Pa), where T ranges from 60 to 700 K and P ranges from 0 to 70 MPa.

[0046] 2) Obtain the first molar density solution that meets the requirements through a one-dimensional search calculation method. The specific steps are as follows:

[0047] S21. Given the initial value a of the molar density (where 0 < a ≤ 0.01 mol / L), according to the molar composition (x i ), temperature (T), and pressure (P) input in 1), calculate the corresponding pressure f(a) when the molar density is a according to equations (1 - 6). If f(a) > 0, then perform the calculation; otherwise, set d1 = 0 and end the calculation;

[0048]

[0049] where: f(d) is the corresponding pressure when the molar density is d, in Pa; d is the molar density, in mol / L; T is the temperature, in K; The calculation formulas are as shown in equations (2 - 4):

[0050]

[0051] where: d i,k , n i,k , t i,k , c i,k , d ij,k , n ij,k , η ij,k , t ij,k , ε ij,k , β ij,k , γ ij,k are the coefficients and exponents corresponding to different components, obtained from the standard ISO 20765 - 2. The calculation formulas for the parameters δ and τ are as shown in equations (5) and (6):

[0052]

[0053]

[0054] where: d c,i , d c,j are the critical molar densities, in mol / L; T c,i , T c,j are the critical temperatures, in K; β V,ij , γ V,ij , β T,ij , γ T,ij are the binary interaction parameters;

[0055] S22. Set the step size to 1. The increased molar density x = a + 1. Calculate the partial derivative f'(x) of the pressure with respect to the molar density x according to equation (7). If f'(x) > 0, then perform step S23; if f'(x) ≤ 0, perform step S26;

[0056]

[0057] S23, if f(x) < P, then let a = x and return to step S22; otherwise, execute step S24;

[0058] S24, let b = (a + x) / 2, and calculate the pressure f(b) at the molar density of b according to equations (1 - 7). If |f(b) - P| > 10 -6 then execute step S25; otherwise, let d1 = b and the calculation ends;

[0059] S25, if f(b) - P > 0, then let x = b and return to step S24; otherwise, let a = b and return to step S24;

[0060] S26, let b = (a + x) / 2. If |a + x| > 0.01, then execute step S27; otherwise, let d1 = 0 and the calculation ends;

[0061] S27, if f'(x) × f'(b) < 0, then let x = b and return to S26; otherwise, execute step S28;

[0062] S28, if f(b) < P, then let a = b and return to step S26; otherwise, let x = b and execute step S24.

[0063] 3) Given a relatively large initial value of the Newton iteration molar density (d > 25 mol / L), obtain the second molar density solution that meets the requirements through Newton iteration. The specific solution steps are as follows:

[0064] S31, according to the temperature, pressure, and liquefied natural gas components input in 1), set the initial molar density d = 30 mol / L;

[0065] S32, obtain the pressure f(d) corresponding to the molar density d and the partial derivative f'(d) of f(d) with respect to the molar density d;

[0066] S33, let d ite = d - (f(d) - P) / f'(d), and calculate the pressure f(d ite ) corresponding to the molar density dite;

[0067] S34, if |f(d ite ) - P| > 10 -6 , then let d = d ite , and return to step S32; otherwise, d2 = d ite and the calculation ends.

[0068] 4) Based on the principle of the lowest Gibbs free energy, judge the Gibbs free energy corresponding to the two molar densities. The specific solution steps are as follows:

[0069] S41, if d1≠0, proceed to step S42, otherwise the true molar density d=d2, and the calculation ends;

[0070] S42, if |d1-d2|>10 -6 , proceed to step S43, otherwise the true molar density d = (d1 + d2) / 2, and the calculation ends;

[0071] S43, calculate the Gibbs free energy g(d1) and g(d2) according to formula (8-12). If g(d1)≤g(d2), then d=d1, otherwise d=d2, and the calculation ends;

[0072]

[0073]

[0074] Where: ci , T c,i , c oi,k The coefficients and indices corresponding to the different components are obtained in standard ISO 20765-2.

[0075] 5) According to the molar component x input in 1) i The molar density d calculated by 2), 3) and 4) is used to calculate the density of liquefied natural gas according to formula (13):

[0076] ρ=x i M i d (13)

[0077] Where: M i is the molar mass corresponding to component i, mol / kg.

[0078] The above is an introduction to the implementation steps of the method. The following further describes the calculation effect of the present invention through a stable and high-precision liquefied natural gas density calculation system proposed by the present invention.

[0079] 1. Liquefied Natural Gas Density Experiment Information

[0080] The experimental data of liquefied natural gas density were collected, and its molar composition is shown in Table 1. Its experimental temperature and pressure range and the number of experimental data points are shown in Table 2.

[0081] Table 1 Sample molar composition

[0082] composition <![CDATA[CH4]]> <![CDATA[C2H6]]> <![CDATA[C3H8]]> <![CDATA[n-C4H 10 ]]> <![CDATA[i-C4H 10 ]]> <![CDATA[n-C5H 12 ]]> <![CDATA[i-C5H 12 ]]> <![CDATA[N2]]> M1 87.8854 7.2738 2.9257 1.5647 0 0 0 0.3504 M2 81.5626 13.3744 3.6793 0.6884 0 0 0 0.6953 M3 84.6362 12.8 1.499 0.2093 0.2199 0.0301 0.0201 0.5854 M4 97.8898 0.999 0.4971 0.2092 0.1771 0.0162 0.0183 0.1933 M5 98.9977 0 0 0 0 0 0 1.0023

[0083] Table 2 Liquefied natural gas experimental conditions

[0084] sample Temperature / K Pressure / MPa Data Points M1 105～135 0～8.9 35 M2 105～135 0～8.9 32 M3 105～135 0～8.9 28 M4 105～135 0～8.9 28 M5 100～160 0～8.5 28

[0085] 2. Calculation of liquefied natural gas density

[0086] According to the temperature and pressure conditions given in the table, taking the calculation of M6 at 100K and 0-8.1MPa as an example, the density is calculated by a stable and high-precision liquefied natural gas density calculation method and system proposed by the present invention. And the initial value of Newton iteration is 20mol / m 3 , the binary interval is 0~30mol / m 3 , the density of liquefied natural gas is calculated by Newton iteration and dichotomy method respectively. The calculation results are shown in Table 3:

[0087] Table 3 Relative deviation between calculated and experimental values ​​of liquefied natural gas M6 density

[0088]

[0089] As shown in Table 3, the real density solution is obtained by combining one-dimensional search with Newton iteration based on the Gibbs free energy principle. The average relative deviation between the density calculation results of this method and the experimental value is 0.01%. The relative deviation ranges of the binary method and Newton iteration are 62.93% to 63.40% and 21.37% to 22.99%, respectively.

[0090] III. Comparison of the calculation method of the present invention with the dichotomy method and Newton iteration

[0091] Table 4 Average relative deviation between calculated and experimental values ​​of liquefied natural gas density

[0092]

[0093] As shown in Table 4, the average relative deviation of the results of the liquefied natural gas density calculation of the present invention is in the range of 0.01% to 0.12%, while the average relative deviations of the dichotomy method and Newton iteration are 45.28% to 62.88% and 25.29% to 46.09%, respectively. The calculation method proposed in the present invention effectively solves the problem that the dichotomy method and Newton iteration are overly dependent on the Newton iteration initial value and the dichotomy interval when solving the GERG-2008 equation, making it difficult to judge and obtain the true density solution.

[0094] The above descriptions are merely embodiments of the present invention and are not intended to limit the present invention. Any modifications, equivalent substitutions and improvements made within the spirit and principles of the present invention should be included in the protection scope of the present invention.

Claims

1. A stable and high-precision method and system for calculating the density of liquefied natural gas, characterized by: The calculation method takes into account the complex non - linear relationship between pressure and molar density in the GERG - 2008 equation, as well as the problem that traditional calculation methods (such as Newton's iterative method or bisection method) are difficult to stably obtain the correct density during calculation. A new calculation method and system are proposed. This method combines one - dimensional search with Newton's iteration and introduces the Gibbs free energy criterion to ensure the authenticity of the solution, so as to be able to stably and accurately calculate the density of liquefied natural gas. The calculation process includes the following steps: 1) Input liquefied natural gas molar composition x i (%), temperature T (K), pressure P (Pa), where T ranges from 60 to 700 K, and P ranges from 0 to 70 MPa; 2) Obtain the first molar density solution that meets the requirements through the one - dimensional search solution method. The specific steps are as follows: S21. Given the initial value a of the molar density (the value range is 0 < a ≤ 0.01 mol / L), according to the molar composition (x i ), temperature (T), and pressure (P) input in 1), calculate the corresponding pressure f(a) when the molar density is a according to equations (1 - 6). If f(a) > 0, perform the calculation; otherwise, let d1 = 0 and end the calculation; Where: f(d) is the pressure corresponding to the molar density d, Pa; d is the molar density, mol / L; T is the temperature, unit K; The calculation formula is shown in formula (2-4): Where: d i,k 、n i,k ,t i,k 、c i,k d ij,k 、n ij,k , η ij,k ,t ij,k , ε ij,k , β ij,k , γ ij,k —— are the coefficients and indexes corresponding to different components, obtained in standard ISO 20765-2. The calculation formulas of parameters δ and τ are shown in formula (5) and formula (6): Where: d c,i ,d c,j is the critical molar density, mol / L; T c,i , T c,j is the critical temperature, K; β V,ij , γ V,ij , β T,ij , γ T,ij is the binary interaction parameter; S22. Set the step size to 1, and the increased molar density \(x=a + 1\). Calculate the partial derivative \(f'(x)\) of pressure with respect to molar density \(x\) according to Equation (7). If \(f'(x)>0\), then proceed to step S23; if \(f'(x)\leq0\), proceed to step S26; S23. If \(f(x)<P\), then let \(a = x\) and return to step S22; otherwise, execute step S24; S24, let b = (a + x) / 2, and calculate the pressure f (b) when the molar density is b according to equations (1 to 7). If |f (b) - P|> 10 -6 Then execute step S25, otherwise set d1 = b and the calculation ends; S25. If \(f(b)-P>0\), then let \(x = b\) and return to step S24; otherwise, let \(a = b\) and return to step S24; S26. Let \(b=(a + x) / 2\). If \(|a + x|>0.01\), then execute step S27; otherwise, let \(d1 = 0\) and the calculation ends; S27. If \(f'(x)\times f'(b)<0\), then let \(x = b\) and return to S26; otherwise, execute step S28; S28. If \(f(b)<P\), then let \(a = b\) and return to step S26; otherwise, let \(x = b\) and execute step S24; 3) Given a relatively large initial value of molar density for Newton's iteration (\(d>25mol / L\)), obtain the second molar density solution that meets the requirements through Newton's iteration. The specific solution steps are as follows: S31. According to the temperature, pressure, and liquefied natural gas components input in 1), set the initial molar density \(d = 30mol / L\); S32. Obtain the pressure \(f(d)\) corresponding to the molar density \(d\) and the partial derivative \(f'(d)\) of \(f(d)\) with respect to the molar density \(d\); S33, let d ite =d-(f(d)-P) / f'(d), calculate the molar density d ite The corresponding pressure f(d ite ); S34, if |f(d ite )-P|>10 -6 , then let d = d ite , return to step S32, otherwise d2 = d ite The calculation is finished; 4) Based on the principle of the lowest Gibbs free energy, judge the Gibbs free energy corresponding to the two molar densities. The specific solution steps are as follows: S41. If \(d1\neq0\), proceed to step S42; otherwise, the true molar density \(d = d2\) and the calculation ends; S42, if |d1-d2|>10 -6 , proceed to step S43, otherwise the true molar density d = (d1 + d2) / 2, and the calculation ends; S43. Calculate the Gibbs free energies \(g(d1)\) and \(g(d2)\) according to Equations (8 - 12). If \(g(d1)\leq g(d2)\), then \(d = d1\); otherwise, \(d = d2\) and the calculation ends; Where: ci , T ci , c oi,k The coefficients and indices corresponding to the different components are obtained in standard ISO 20765-2; 5) According to the molar component x input in 1) i The density of liquefied natural gas is calculated according to formula (13) using the molar density d calculated by 2), 3) and 4); p=x i M i d (13) Where: M i is the molar mass corresponding to component i, mol / kg.

2. A stable and high-precision liquefied natural gas density calculation system, the system comprising: A memory and a solver. The memory stores computer program instructions and can temporarily store input and output data. The computer program instructions are executed by the solver to implement the steps of a stable and high - precision liquefied natural gas density calculation method provided in the first aspect of the present application.

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