Method for calculating thermophysical properties of high-temperature non-equilibrium triatomic gas and method for generating database

Abstract

The application discloses a high-temperature non-equilibrium model three-atom gas thermal property calculation method and a database generation method, and first, according to the fact that the energy level distribution probability on the electronic, vibration and rotation modes respectively meets the Boltzmann distribution hypothesis, a high-temperature non-equilibrium gas model is established; then, according to the rule that the vibration and rotation energy is less than the minimum dissociation energy of a molecule, the vibration and rotation energy level range of the molecule is determined; finally, according to the high-temperature non-equilibrium gas model and the vibration energy level and rotation energy level range of the molecule, a non-equilibrium three-atom gas specific heat calculation formula containing a single temperature variable is derived. Based on the non-equilibrium three-atom gas specific heat calculation formula, combined with the piecewise linear interpolation and piecewise quadratic interpolation method, a high-temperature non-equilibrium three-atom gas thermal property database is constructed. The application can support the engineering application of three and four temperature models, and has important significance for the scientific research and engineering application in the field of high-temperature non-equilibrium calculation aerodynamics.

Description

Technical Field

[0001] This invention relates to the field of computational aerodynamics, specifically to a method for calculating the thermophysical properties of high-temperature non-equilibrium triatomic gases and generating a database. Background Technology

[0002] High-precision simulation of high-altitude, high-speed flow fields is of great significance for the design of new high-speed aircraft, the prediction of communication disruptions, and the development of future manned spacecraft. Real gas effects and non-equilibrium are two major characteristics of high-altitude, high-speed flow. To improve the accuracy of flow field simulation, both a physical model that effectively describes the non-equilibrium process and reliable high-temperature gas data are indispensable.

[0003] In the field of nonequilibrium physics models, after nearly 40 years of development, finite-rate chemical reaction models describing chemical nonequilibrium processes, such as the Park model and the Gupta model, have been established; multi-temperature models describing thermal nonequilibrium processes, such as two-temperature models, three-temperature models, and four-temperature models, have also been established. Although existing research shows that considering the nonequilibrium effect of electron energy can improve the accuracy of plasma flow field prediction, and that rotational nonequilibrium phenomena become prominent after the translational temperature exceeds approximately 12000 K, the lack of reliable high-temperature nonequilibrium gas thermophysical property data, especially high-temperature nonequilibrium triatomic gas thermophysical property data, means that currently only two-temperature models are widely used in engineering practice.

[0004] Unlike cryogenic gases, which can only excite translational and rotational energy modes, high-temperature triatomic gases can excite up to four energy modes: translational, electronic, vibrational, and rotational. The electronic, vibrational, and rotational modes are collectively referred to as internal modes, which are independent of the translational modes and exhibit a series of complex characteristics at high temperatures, such as vibrational anharmonicity and rotational non-rigidity. Furthermore, the distribution of internal degrees of freedom among the modes varies depending on the molecular spatial structure. It is precisely because of these characteristics that modeling and solving the thermophysical properties of high-temperature nonequilibrium gases are extremely difficult. Summary of the Invention

[0005] The purpose of this invention is to overcome the aforementioned deficiencies and provide a method for calculating the thermal properties of triatomic gases applicable to thermal nonequilibrium models with temperatures above two temperatures. Based on this method, a database of thermal properties of high-temperature nonequilibrium triatomic gases is established to support the engineering application of thermal nonequilibrium models with temperatures above two temperatures. This invention first establishes a high-temperature nonequilibrium gas model based on the partition function, assuming that the energy level distribution probabilities of triatomic molecules in electronic, vibrational, and rotational modes conform to the Boltzmann distribution hypothesis. Then, based on the criterion that the vibrational and rotational energies of triatomic molecules are less than the minimum dissociation energy of the molecules, the ranges of the vibrational and rotational energy levels of triatomic molecules are determined. Finally, under the modal decoupling assumption T... ex ≈T vib ≈T rotUnder these conditions, based on the high-temperature non-equilibrium gas model and the range of vibrational and rotational energy levels of triatomic molecules, formulas for calculating the specific heat of electronic, vibrational, and rotational modes of non-equilibrium triatomic gases, containing only a single temperature variable, are derived. In the database generation method, based on the calculation formulas for the specific heat of non-equilibrium triatomic gases, a high-temperature non-equilibrium triatomic gas thermophysical property database is constructed by combining piecewise linear interpolation and piecewise quadratic interpolation methods.

[0006] The technical solution of this invention is:

[0007] A method for calculating the thermophysical properties of a non-equilibrium triatomic gas at high temperatures includes the following steps:

[0008] A high-temperature nonequilibrium gas model based on the partition function is established;

[0009] Based on the high-temperature nonequilibrium gas model based on the partition function, the maximum vibrational energy level of each vibrational mode and the maximum energy level of the rotational mode of a triatomic molecule are determined.

[0010] Based on the high-temperature non-equilibrium gas model and the range of vibrational and rotational energy levels of triatomic molecules, a formula for calculating the specific heat of non-equilibrium triatomic gas electronic, vibrational, and rotational modes containing only a single temperature variable is derived, and specific heat data of non-equilibrium triatomic molecules under arbitrary temperature conditions are obtained.

[0011] Preferably, the high-temperature non-equilibrium gas model based on the partition function is as follows:

[0012]

[0013] Among them, Q int It is the molecular internal energy partition function, σ n It is the molecular symmetry constant, n is the electron quantum number, and v is the molecular symmetry constant. s1 v b v s2 J is the vibrational quantum number, and J and k are the rotational quantum numbers; and These are electronic energy, vibrational energy, linear triatomic molecule rotational energy, and nonlinear triatomic molecule rotational energy; g n It is the electron level degeneracy, k B n is the Boltzmann constant, ε0 is the zero-point energy; max For the maximum electronic energy level, v max,n,s1 v max,n,b v max,n,s2 J represents the maximum vibrational energy level for each vibration mode. max,n It represents the maximum rotational energy level of the triatomic molecule's rotational mode.

[0014] Preferably, the electron energy of a triatomic molecule Vibration energy linear form of rotational energy and nonlinear form of rotational energy Calculate using the following formulas respectively:

[0015]

[0016]

[0017]

[0018]

[0019] Among them, h P Let T be Planck's constant, c be the speed of light, and T be the speed of light. e,n ω e,n,s1 ω e,n,b ω e,n,s2 A e,n B e,n C e,n These are the spectral constants of a triatomic molecule determined through experimental measurement or theoretical calculation.

[0020] Preferably:

[0021] The maximum vibrational energy level v of each vibrational mode of the molecule max,n,s1 v max,n,b v max,n,s2 Based on the fact that the vibrational energies of each mode of a triatomic molecule are less than the molecular dissociation energy D, n diss Assumptions are established;

[0022] The maximum energy level of the rotational mode of a triatomic molecule J max,n According to the fact that the rotational energy of a triatomic molecule is less than the minimum dissociation energy D n diss Assumptions are established.

[0023] Preferably, the quantum numbers (v) of each vibrational mode energy level of the triatomic molecule s1 v b v s2 The quantum number J of the rotational mode energy level must satisfy the following inequality:

[0024] For the first stretching vibration mode s1 and the second stretching vibration mode s2 of a triatomic molecule, we have:

[0025]

[0026] For triatomic molecules with different spatial structures, the bending vibrational mode b is as follows:

[0027]

[0028] For triatomic molecules with different spatial structures, the rotational modes are as follows:

[0029]

[0030] v max,n,s1 v max,n,b v max,n,s2 J max,n v s1 v b v s2 The largest integer that satisfies the above inequality with J.

[0031] Preferably, the formula for calculating the specific heat of non-equilibrium triatomic gases with only a single temperature variable, including electronic, vibrational, and rotational modes, is as follows:

[0032]

[0033]

[0034]

[0035] Where R is the gas constant, Q int It is the partition function of the internal energy of the molecule; T ex T vib and T rot These are the electronic temperature, vibrational temperature, and rotational temperature of a triatomic molecule, respectively. These are the electronic mode specific heat, vibration mode specific heat, and rotation mode specific heat of the gas, respectively.

[0036] Secondly,

[0037] A method for generating a high-temperature non-equilibrium triatomic gas thermophysical property database is proposed. Based on the specific heat calculation formulas for the electronic, vibrational, and rotational modes of non-equilibrium triatomic gases obtained in the first aspect, three specific heat distribution curves of high-temperature non-equilibrium triatomic gases are obtained by piecewise linear interpolation, and three enthalpy distribution curves of high-temperature non-equilibrium triatomic gases are obtained by piecewise quadratic interpolation. A gas thermophysical property database is then established based on the curves.

[0038] Preferably, in T min ~T max The specific heat c of triatomic gas was obtained using a piecewise linear interpolation method within a temperature range. p Distribution curve F cp , i (T), the distribution curves of the enthalpy h of the triatomic gas at various temperatures were obtained using piecewise quadratic interpolation. h,i (T);

[0039] The specific steps are as follows:

[0040] Establish T min ~Tmax The interpolation formula for enthalpy and specific heat over the temperature range is F. h,i (T) and F cp,i (T), using n points to divide T min ~T max The interval is divided into n-1 segments of equal length. Let ΔT be the interval between adjacent interpolation points, and let T be the temperature of the i-th interpolation point. i =iΔT, where the interval between interpolation point i and i+1 is the i-th interpolation region, i = 1, 2, 3…n-1. The temperature range is 50K to 50000K, and the value of ΔT is no greater than 150K.

[0041] Interpolation interval T i ~T i+1 Within, the interpolation formula F for the enthalpy corresponding to the piecewise quadratic interpolation curve. h,i (T) and the interpolation formula for the specific heat corresponding to the piecewise linear interpolation curve. as follows:

[0042]

[0043] Among them, A i,0 A i,1 A i,2 Here, T represents the interpolation coefficients, and T represents the temperature.

[0044] The interpolation formula F is determined based on the following conditions. h,i (T) and Interpolation coefficients in:

[0045] Condition 1: Interpolation F h,i (T) and In T∈[T min ,T max ] Continuous within, At the interpolation point i, the specific heat value c is equal to that obtained through the specific heat calculation formula of the non-equilibrium triatomic gas electron, vibrational, or rotational modes. p (T i );

[0046] Condition 2: F h,i The first derivative of (T) is equal to

[0047] Condition 3: When i = 1,

[0048] Based on conditions 1 to 3, according to the interpolation formula F h,i (T) and The equations established to determine the interpolation coefficients are as follows:

[0049]

[0050] Preferably, it further includes:

[0051] In T∈(T max Within the range of (, +∞), the following definition applies:

[0052] The extrapolated expression for enthalpy is h = F h,n (T), extrapolated curve expression for specific heat

[0053] And F h,n (T n ) = F h,n-1 (T n ),

[0054] F h,n (T) and The expression is as follows:

[0055] F h,n (T)=-ABexp[-B(TT n )]+C

[0056]

[0057] The constants A, B, and C in the expression are:

[0058]

[0059]

[0060] C = F h,n-1 (T n )+ABexp[-B(TT n )];

[0061] In T∈(0,T) min Within the range, define the extrapolation curve expression for enthalpy. Extrapolation curve expression of specific heat

[0062] Compared with the prior art, the present invention has the following advantages:

[0063] (1) In the high-temperature non-equilibrium triatomic gas thermophysical property calculation method of the present invention, the effects of high-temperature real gas effects such as high-order electron energy excitation, oscillator nonharmonicity, rotor nonrigidity, and dissociation energy constraint are comprehensively considered, which can better reflect the properties of high-temperature gas. The accuracy of the enthalpy and specific heat value predicted by the present invention is significantly improved.

[0064] (2) The high-temperature non-equilibrium triatomic gas thermophysical property calculation method of the present invention, based on reasonable assumptions, completely decouples the electronic, vibration and rotation modes of triatomic gas to obtain the approximate values ​​of enthalpy and specific heat, which can support the engineering application exploration of three- and four-temperature models and has strong applicability;

[0065] (3) In the method for generating a high-temperature non-equilibrium triatomic gas thermophysical property database of the present invention, based on the calculation formula of the specific heat of non-equilibrium triatomic gas, a triatomic gas enthalpy and specific heat database is established by using piecewise linear interpolation and piecewise quadratic interpolation methods, avoiding the Runge phenomenon caused by high-order interpolation, with good continuity and effectively improving the calculation efficiency. Attached Figure Description

[0066] Figure 1 This is a step diagram illustrating the method for calculating the thermophysical properties of a triatomic gas and generating a database using a high-temperature non-equilibrium model according to the present invention.

[0067] Figure 2 The diagram shows two different spatial construction methods for a triatomic molecule, and the corresponding vibrational and rotational modes of motion.

[0068] Figure 3(a) shows a comparison of the specific heat of CO2 molecules with temperature.

[0069] Figure 3(b) shows a comparison of the specific heat of NO2 molecules with temperature. Detailed Implementation

[0070] The features and advantages of the present invention will become clearer and more apparent from the following detailed description.

[0071] The term “exemplary” as used herein means “serving as an example, embodiment, or illustration.” Any embodiment illustrated herein as “exemplary” is not necessarily to be construed as superior to or better than other embodiments. Although various aspects of embodiments are shown in the accompanying drawings, the drawings are not necessarily drawn to scale unless specifically indicated otherwise.

[0072] like Figure 2 As shown, the spatial configurations of triatomic molecules can be divided into two types: linear and nonlinear. A linear triatomic molecule contains four vibrational modes and two rotational modes, of which the two bending vibrational modes and the two rotational modes can be degenerate; a nonlinear triatomic molecule contains three vibrational modes and three rotational modes.

[0073] Currently, in simulations of thermal nonequilibrium flow fields at temperatures above two temperatures, the low-temperature assumption simplification method is the only way to obtain enthalpy and specific heat data for triatomic gases. The advantages of this method are: the calculation formula has an analytical form, thorough decoupling of internal degrees of freedom modes, and applicability to three- and four-temperature models. The disadvantages are: it considers a limited number of energy levels, does not consider modal coupling, and has large prediction deviations for the thermal properties of high-temperature gases. The low-temperature assumption simplification method makes the following assumptions: 1) Electrons can be excited to at most one order; 2) Molecular vibrations are harmonic; 3) Molecules are rigid rotors. Based on these assumptions, the enthalpy and specific heat calculation formulas are as follows:

[0074]

[0075]

[0076]

[0077]

[0078]

[0079]

[0080] in, The energy degeneracy of the electronic ground state and the first excited state. The characteristic temperature of the first excited state. Here, represents the characteristic temperatures of the first stretching vibration mode, the second stretching vibration mode, and the bending vibration mode of the electronic ground state, and R is the gas constant. This method can reflect the properties of low-temperature triatomic gases relatively well, but due to simplification assumptions, its prediction of the thermophysical properties of high-temperature triatomic gases has a large deviation. h represents enthalpy, and c... p The superscripts ex, rot, and vib represent the specific heat, respectively, and represent the electronic, rotational, and vibrational modes of a triatomic molecule. The temperatures corresponding to each electronic, vibrational, and rotational mode are T. ex T vib T rot .

[0081] To address this, considering the effects of high-temperature real gas phenomena such as higher-order electron energy excitation, oscillator anharmonicity, rotor non-rigidity, and dissociation energy constraints, a high-temperature non-equilibrium gas model was completed. An internal degree-of-freedom modal decoupling assumption was proposed, and a method for calculating the thermal properties of high-temperature non-equilibrium triatomic gases was established. Based on this, a method for generating a database of high-temperature non-equilibrium triatomic gas thermal properties was designed. The main steps in determining the range of vibrational and rotational energy levels of triatomic molecules include:

[0082] Based on the spatial configuration of the triatomic molecule, the vibrational energies of the first stretching vibration mode s1, the second stretching vibration mode s2, the bending vibration mode b, and the rotational mode are all determined to be less than D. ndiss Given the constraints, take the maximum vibrational quantum number of a triatomic molecule that satisfies the above constraints, v. s1 v b v s2 By combining the rotational quantum number J, we obtain the ranges of the vibrational and rotational energy levels.

[0083] like Figure 1 As shown, the present invention includes the following steps:

[0084] S1 establishes a high-temperature non-equilibrium gas model based on the partition function, based on the assumption that the energy level distribution probabilities of triatomic molecules in electronic, vibrational and rotational modes respectively conform to the Boltzmann distribution.

[0085] S2 determines the maximum energy levels of each vibrational and rotational mode of a triatomic molecule based on the criterion that the vibrational and rotational energies of the triatomic molecule are less than the minimum dissociation energy of the molecule.

[0086] S3 T ex T vib and T rot These are the electronic temperature, vibrational temperature, and rotational temperature of a triatomic molecule, respectively, under the modal decoupling assumption T. ex ≈T vib ≈T rot Under these conditions, based on the high-temperature non-equilibrium gas model and the range of vibrational and rotational energy levels of triatomic molecules, a formula for calculating the specific heat of non-equilibrium triatomic gas electronic, vibrational, and rotational modes containing only a single temperature variable is derived, and specific heat data of non-equilibrium triatomic molecules under arbitrary temperature conditions are obtained.

[0087] The high-temperature non-equilibrium gas model based on the partition function established in step S1 is as follows:

[0088]

[0089] Among them, Q int It is the molecular internal energy partition function, σ n It is the molecular symmetry constant, n is the electron quantum number, and v is the molecular symmetry constant. s1 v b v s2 J is the vibrational quantum number, and J and k are the rotational quantum numbers. and These are electronic energy, vibrational energy, linear triatomic molecule rotational energy, and nonlinear triatomic molecule rotational energy, respectively, g n It is the electron level degeneracy, k B n is the Boltzmann constant, ε0 is the zero-point energy; max For the maximum electronic energy level, v max,n,s1 v max,n,b v max,n,s2 J represents the maximum vibrational energy level for each vibration mode.max,n This is the maximum rotational energy level.

[0090] Electron energy of triatomic molecular spectra Vibration energy linear form of rotational energy and nonlinear form of rotational energy Calculate using the following formulas respectively:

[0091]

[0092]

[0093]

[0094]

[0095] Among them, h P Let T be Planck's constant, c be the speed of light, and T be the speed of light. e,n ω e,n,s1 ω e,n,b ω e,n,s2 A e,n B e,n C e,n These are the spectral constants of a triatomic molecule determined through experimental measurement or theoretical calculation.

[0096] The vibrational and rotational energy levels are constrained by the minimum dissociation energy. The specific steps of step S2 are as follows:

[0097] S2.1 Based on the fact that the vibrational energies of each mode of a triatomic molecule are less than the molecular dissociation energy D n diss Assuming we determine the maximum vibrational energy level v of each vibrational mode of the molecule. max,n,s1 v max,n,b v max,n,s2 .

[0098] S2.2 Based on the fact that the rotational energy of a triatomic molecule is less than the minimum dissociation energy Assuming the maximum energy level J of the molecular rotational mode is determined... max,n .

[0099] In steps S2.1 and S2.2, based on the assumption that the vibrational energies of each vibrational mode of the triatomic molecule are less than their dissociation energies, the quantum numbers (v) of each vibrational mode energy level of the triatomic molecule are... s1 v b v s2 The quantum number J of the rotational mode energy level must satisfy the following inequality:

[0100] For the first stretching vibration mode s1 and the second stretching vibration mode s2 of the triatomic molecule, we have

[0101]

[0102] For triatomic molecules with different spatial structures, the bending vibrational mode b is as follows:

[0103]

[0104] For triatomic molecules with different spatial structures, the rotational modes are as follows:

[0105]

[0106] Among them, v max,n,s1 v max,n,b v max,n,s2 J max,n v s1 v b v s2 The largest integer that satisfies the above inequality with J.

[0107] Using T ex ≈T vib ≈T rot Assuming the decoupling of electronic energy, vibrational energy, and rotational energy is achieved, the formulas for calculating the specific heat of non-equilibrium triatomic gases in electronic, vibrational, and rotational modes are as follows:

[0108]

[0109]

[0110]

[0111] Where R is the gas constant, Q int It is the partition function of the internal energy of the molecule. ex T vib and T rot These are the electronic temperature, vibrational temperature, and rotational temperature of a triatomic molecule, respectively. These are the electronic mode specific heat, vibration mode specific heat, and rotation mode specific heat of the gas, respectively.

[0112] A method for generating a high-temperature non-equilibrium triatomic gas thermophysical property database is proposed. Based on the above-obtained specific heat calculation formulas for the electronic, vibrational, and rotational modes of non-equilibrium triatomic gases, three specific heat distribution curves for high-temperature non-equilibrium triatomic gases are obtained by piecewise linear interpolation, and three enthalpy distribution curves for high-temperature non-equilibrium triatomic gases are obtained by piecewise quadratic interpolation. A gas thermophysical property database is then established based on the curves.

[0113] In T min ~T max The specific heat c of triatomic gas was obtained using a piecewise linear interpolation method within a temperature range. pDistribution curve The distribution curves of the enthalpy h of the triatomic gas at various temperatures were obtained using piecewise quadratic interpolation. h,i (T).

[0114] The specific steps are as follows:

[0115] (1) Establish T min ~T max The interpolation formula for enthalpy and specific heat over the temperature range is F. h,i (T) and Using n points to divide T min ~T max The interval is divided into n-1 segments of equal length. Let ΔT be the interval between adjacent interpolation points, and let T be the temperature of the i-th interpolation point. i =iΔT, where the interval between interpolation point i and i+1 is the i-th interpolation region, i = 1, 2, 3…n-1. Based on engineering application background and experience, T was selected. min =50K,T max =50000K, ΔT=50K. Interpolation interval T i ~T i+1 Within, the interpolation formula F for the enthalpy corresponding to the piecewise quadratic interpolation curve. h,i (T) and the interpolation formula for the specific heat corresponding to the piecewise linear interpolation curve. as follows:

[0116]

[0117] Among them, A i,0 A i,1 A i,2 Here, T represents the interpolation coefficients, and T represents the temperature.

[0118] (2) Determine the interpolation formula F based on the following conditions h,i (T) and Interpolation coefficients in:

[0119] Condition 1: Interpolation F h,i (T) and F cp,i (T) in T∈[T] min ,T max ] Continuous within, At the interpolation point i, the specific heat value c is equal to that obtained through the specific heat calculation formula of the non-equilibrium triatomic gas electron, vibrational, or rotational modes. p (T i );

[0120] Condition 2: F h,i The first derivative of (T) is equal to

[0121] Condition 3: When i = 1,

[0122] Based on conditions 1 to 3, according to the interpolation formula F h,i (T) and The equations established to determine the interpolation coefficients are as follows:

[0123]

[0124] (3) In T∈(T max Within the range of (, +∞), the following definition applies:

[0125] The extrapolated expression for enthalpy is h = F h,n (T), extrapolated curve expression for specific heat

[0126] And F h,n (T n ) = F h,n-1 (T n ), F determined based on research experience h,n (T) and The expression is as follows:

[0127] F h,n (T)=-A / Bexp[-B(TT n )]+C

[0128]

[0129] The constants A, B, and C in the expression are:

[0130]

[0131]

[0132] C = F h,n-1 (T n )+A / Bexp[-B(TT n )).

[0133] (4) In T∈(0,T) min Within the range, define the extrapolation curve expression for enthalpy. Extrapolation curve expression of specific heat

[0134] Example

[0135] Step S1: High-temperature non-equilibrium gas modeling;

[0136] There is strong coupling between the vibrational and rotational modes of triatomic molecules. To establish a non-equilibrium triatomic gas model, the internal energy of triatomic molecules is first divided into three parts: electronic energy, vibrational energy, and rotational energy. The following formulas are used to calculate these components according to different molecular spatial configurations:

[0137]

[0138]

[0139]

[0140]

[0141] Among them, h P Let T be Planck's constant, c be the speed of light, and T be the speed of light. e,n ω e,n,s1 ω e,n,b ω e,n,s2 A e,n B e,n C e,n These are the spectral constants of a triatomic molecule determined through experimental measurement or theoretical calculation.

[0142] Assume that the triatomic molecule independently follows a Boltzmann distribution at each energy level of its electronic, vibrational, and rotational modes, and let T be the temperatures corresponding to each electronic, vibrational, and rotational mode. ex T vib T rot This yields the non-equilibrium form of the partition function, i.e., the high-temperature non-equilibrium gas model based on the partition function:

[0143]

[0144] Among them, Q int It is the molecular internal energy partition function, σ n It is the molecular symmetry constant, n is the electron quantum number, and v is the molecular symmetry constant. s1 v b v s2 J is the vibrational quantum number, and J and k are the rotational quantum numbers. and These are electronic energy, vibrational energy, linear triatomic molecule rotational energy, and nonlinear triatomic molecule rotational energy, respectively, g n It is the electron level degeneracy, k B n is the Boltzmann constant, ε0 is the zero-point energy; max For the maximum electronic energy level, v max,n,s1 v max,n,b v max,n,s2 J represents the maximum vibrational energy level for each vibration mode. max,n This is the maximum rotational energy level.

[0145] Step S2: Calculation of vibrational and rotational energy level ranges

[0146] Constrained by the fact that rotational and vibrational energies are less than the minimum molecular energy of decomposition, there are a finite number of combinations of rotational and vibrational quantum numbers. The following will take the ground state electronic energy of carbon dioxide (CO2) and nitrogen dioxide (NO2) as examples to illustrate the specific steps for determining the range of vibrational and rotational energy levels of linear and nonlinear molecules, respectively.

[0147] Table 1. Main spectral data of ground state electronic energy of CO2 and NO2 molecules.

[0148]

[0149] *Note: Units in the table are in cm. -1

[0150] Table 1 presents the main spectral data of nitrogen gas in its electronic ground state. The minimum energy required for the dissociation of an electronically ground-state molecule is given. The CO2 molecule has a linear configuration, while the NO2 molecule has a nonlinear configuration.

[0151] 1) Calculation of the range of vibrational and rotational energy levels of linear triatomic molecules

[0152] Based on the assumption that the vibrational energy of each vibrational mode is less than the dissociation energy, the constraint equations for the first stretching vibrational mode s1 and the second stretching vibrational mode s2 of a linear triatomic molecule are as follows:

[0153]

[0154] Therefore, the quantum number of the maximum energy level is determined to be:

[0155]

[0156] Further, the constraint equations for the two degenerate bending vibration modes b are:

[0157]

[0158] Therefore, the quantum number of the maximum energy level is determined to be:

[0159]

[0160] For two degenerate rotational modes, we have:

[0161]

[0162] Therefore, the quantum number of the maximum energy level is determined to be:

[0163]

[0164] 2) Calculation of the range of vibrational and rotational energy levels of nonlinear triatomic molecules

[0165] Based on the assumption that the vibrational energy of each vibrational mode is less than the dissociation energy, constraint equations exist for the first stretching vibrational mode s1, the second stretching vibrational mode s2, and the bending vibrational mode b of the nonlinear triatomic molecule:

[0166]

[0167] Therefore, the quantum number of the maximum energy level is determined to be:

[0168]

[0169] For a nonlinear triatomic molecule, there are three independent rotational modes:

[0170]

[0171] Therefore, the quantum number of the maximum energy level is determined to be:

[0172]

[0173] It should be noted that the above method for solving the energy level range can be extended to all electronic energy levels of all triatomic molecules.

[0174] Step S3: Derivation of the specific heat calculation formula for a single temperature variable

[0175] Based on theoretical derivation, the formula for calculating the specific heat of a gas derived from the non-equilibrium partition function is as follows:

[0176]

[0177]

[0178]

[0179] In the above formula, the specific heat of the gas is a function of the electron, vibration, and rotation temperatures. Introducing T... ex ≈T vib ≈T rot Assuming modal decoupling, we obtain a formula for calculating the specific heat of a nonequilibrium gas that depends only on a single temperature variable:

[0180]

[0181]

[0182]

[0183] Methods for establishing a database of thermophysical properties of high-temperature nonequilibrium gases:

[0184] To avoid the Runge phenomenon caused by high-order interpolation, this invention employs piecewise linear interpolation to establish a non-equilibrium triatomic gas specific heat database and piecewise quadratic interpolation to establish a non-equilibrium triatomic gas enthalpy database. The database covers a temperature range of 50K to 50000K, with adjacent interpolation points spaced 50K apart. Outside the database coverage areas of 0K–50K and >50000K, the enthalpy and specific heat of triatomic gases are calculated using two forms of extrapolation.

[0185] In the temperature range of 50K to 50000K, the piecewise interpolation formulas for enthalpy and specific heat are F1 and F2, respectively. h (T) and To ensure interpolation accuracy, the interval between adjacent interpolation points is fixed at 50K, and the temperature T of the i-th (i = 1, 2, 3…n-1, n = 1000) interpolation point is... i =50i, let the area between interpolation point i and i+1 be the i-th interpolation region, and let F be the interpolation formula for enthalpy and specific heat within the region. h,i (T) and The format is as follows:

[0186] F h,i (T)=A i,0 +A i,1 T+0.5A i,2 T 2

[0187]

[0188] A i,0 A i,1 A i,2 These are the interpolation coefficients. The process of determining the interpolation coefficients will be explained below. Interpolation formula F h (T) and Within the range of 50K to 50000K, the following conditions must be met: 1) Continuous within the range of 50K to 50000K, and at interpolation point i, equal to the specific heat value given by the non-equilibrium gas specific heat calculation formula with a single temperature variable; 2) F h (T) is differentiable in the range of 50K to 50000K, and its first derivative is equal to Based on this, the following equations are established:

[0189]

[0190] The above equation contains 3n-2 terms and 3(n-1) unknowns. Assume that when i = 1... We obtain supplementary equations that provide a definite solution to the system of equations.

[0191] A 1,0 +A 1,1 T1+0.5A 1,2 T1 2=(A 1,1 +A 1,2 T1)T1

[0192] For cases outside the database coverage area, the present invention adopts the following processing method:

[0193] 0K~50K: (This likely refers to a specific value or process, possibly related to the concept of "c") p,1 Extrapolating to the specific heat of this region, the enthalpy is continuous at interpolation point 1, thus in this region:

[0194] h = c p,1 T,c p ≡c p,1

[0195] >50000K: Within this region, assume h is of the form -ABexp[-B(T-50000)]+C, c p The form is Aexp[-B(T-50000)], where h equals F at the interpolation point n. h,999 (T 1000 ), c p equal c p The first derivative equals Therefore, we can conclude that:

[0196]

[0197] This application example selects two typical triatomic molecular gases, CO2 and NO2, to verify the effectiveness of the high-temperature non-equilibrium gas thermophysical property calculation method of the present invention (hereinafter referred to as the "new algorithm") in predicting the specific heat of high-temperature triatomic gases. Taking the calculation of gas enthalpy and specific heat as an example, the efficiency improvement of the lower temperature assumption simplification method in the prior art (hereinafter referred to as the "old algorithm") is tested.

[0198] 1) Calculation and testing of specific heat of triatomic molecular gases

[0199] The verification test selected two typical triatomic molecules, CO2 and NO2, and set the temperature range to 50K to 50000K.

[0200] Figures 3(a) and (b) show the temperature-dependent curves of the internal degrees of freedom specific heat of two typical triatomic molecular gases, CO2 and NO2, predicted using the method of this invention, the low-temperature assumption simplification method, and theoretical predictions. They illustrate the prediction results of the new and old algorithms for the specific heat of four triatomic gases, as well as the total specific heat data calculated by molecular physics theory. Throughout the entire test temperature range, the new algorithm's prediction of the total specific heat is consistent with the theoretical data, while the old algorithm's prediction of the total specific heat deviates significantly from the theoretical data after the temperature exceeds approximately 7000 K. Further comparison of the prediction results of the new and old algorithms for the specific heat of each internal energy mode is as follows: First, a comparison... The prediction results are different because the new algorithm considers more electronic energy levels. Relatively high; then compare The prediction results of the two algorithms are as follows: when the temperature does not exceed approximately 2000K. This aligns with the fact that, as temperature increases, the new algorithm, constrained by the minimum dissociation energy, provides... It tends to 0, while the old algorithm predicts... Linear molecules tend to be 4 times R, while nonlinear molecules tend to be 3 times R; finally, a comparison is made. The prediction results are based on the rigid rotor assumption. The linear molecule is always equal to R, and the nonlinear molecule is very close to 1.5R. After considering the rotor's non-rigidity and the minimum dissociation energy constraint, the new method predicts... The temperature gradually approaches 0 after reaching approximately 15000 K. These results demonstrate that the new algorithm better reflects the characteristics of high-temperature gases and is superior to the old algorithm.

[0201] 2) Computational efficiency test

[0202] To demonstrate the computational efficiency advantages of the new algorithm, a comparative test was designed based on the enthalpy and specific heat calculations of CO2 and NO2 gases. The test consisted of five rounds, each round including calculations of the enthalpy and specific heat of three internal energy modes using both the old and new algorithms, totaling 12 cases. Each case was set up with a calculation point every 0.001 K within a temperature range of 50 K to 50000 K, resulting in 49,950,001 calculation points per case.

[0203] Tables 2 to 5 record the results of five rounds of testing, as well as the average time taken to solve each parameter. The test results show that the new algorithm takes significantly less time to calculate the enthalpy and specific heat of electronic and vibrational modes than the old algorithm, while it takes slightly more time to calculate the enthalpy and specific heat of rotational modes. In particular, the new algorithm takes only one-twentieth the time to calculate the specific heat of vibrational modes.

[0204] Table 2 Time consumed in calculating the enthalpy of CO2 gas

[0205]

[0206] Table 3 Time consumed in calculating the specific heat of CO2 gas

[0207]

[0208]

[0209] Table 4. Time consumed in calculating the enthalpy of NO2 gas.

[0210]

[0211] Table 5. Time consumed in calculating the specific heat of NO2 gas

[0212]

[0213] This invention comprehensively considers the effects of high-temperature real gas effects such as high-order electron energy excitation, oscillator anharmonicity, rotor non-rigidity, and dissociation energy constraint. Based on reasonable assumptions, it completely decouples the electronic, vibrational, and rotational modes of a triatomic gas to obtain approximate calculation methods for enthalpy and specific heat. This method can support the engineering application exploration of three- and four-temperature models and has important guiding significance for research in the field of computational aerodynamics.

[0214] The present invention has been described in detail above with reference to specific embodiments and exemplary examples; however, these descriptions should not be construed as limiting the present invention. Those skilled in the art will understand that various equivalent substitutions, modifications, or improvements can be made to the technical solutions and embodiments of the present invention without departing from the spirit and scope of the invention, and all such modifications and improvements fall within the scope of the present invention. The scope of protection of the present invention is defined by the appended claims.

[0215] The contents not described in detail in this specification are common knowledge to those skilled in the art.

Claims

1. A method for calculating the thermophysical properties of a high-temperature non-equilibrium triatomic gas, characterized in that, include: A high-temperature nonequilibrium gas model based on the partition function is established; Based on the high-temperature nonequilibrium gas model based on the partition function, the maximum vibrational energy level of each vibrational mode and the maximum energy level of the rotational mode of a triatomic molecule are determined. Based on the high-temperature non-equilibrium gas model and the range of vibrational and rotational energy levels of triatomic molecules, a formula for calculating the specific heat of electronic, vibrational, and rotational modes of non-equilibrium triatomic gases containing only a single temperature variable is derived, and specific heat data of non-equilibrium triatomic molecules under arbitrary temperature conditions are obtained. The high-temperature nonequilibrium gas model based on the partition function is as follows: in, It is the molecular internal energy partition function. It is the molecular symmetry constant. n Electron quantum number v s1 , v b , v s2 It is a vibrational quantum number. J and k It is the rotation quantum number; , , and These are electronic energy, vibrational energy, linear triatomic molecule rotational energy, and nonlinear triatomic molecule rotational energy, respectively. It is the electron level degeneracy. k B It is Boltzmann's constant. It is zero-point energy; The highest electronic energy level, , , The maximum vibrational energy level for each vibration mode. It represents the maximum rotational energy level of the triatomic molecule's rotational mode.

2. The method for calculating the thermophysical properties of a high-temperature non-equilibrium triatomic gas according to claim 1, characterized in that, Electron energy of triatomic molecules Vibrational energy linear form of rotational energy and nonlinear forms of rotational energy Calculate using the following formulas respectively: in, h P is Planck's constant. c At the speed of light, , , , , , , These are the spectral constants of a triatomic molecule determined through experimental measurement or theoretical calculation.

3. The method for calculating the thermophysical properties of a high-temperature non-equilibrium triatomic gas according to claim 1, characterized in that: Maximum vibrational energy levels of each vibrational mode of the molecule , , Based on the fact that the vibrational energies of each mode of a triatomic molecule are less than the molecular dissociation energy... Assumptions are established; Maximum energy level of triatomic molecule rotational mode According to the fact that the rotational energy of a triatomic molecule is less than its minimum dissociation energy Assumptions are established.

4. The method for calculating the thermophysical properties of a high-temperature non-equilibrium triatomic gas according to claim 3, characterized in that, Quantum numbers of each vibrational mode energy level of a triatomic molecule ( v s1 , v b , v s2 ) and rotational mode energy level quantum number J The following inequalities must be satisfied: For the first stretching vibration mode s1 and the second stretching vibration mode s2 of a triatomic molecule, we have: For triatomic molecules with different spatial structures, the bending vibrational mode b is as follows: For triatomic molecules with different spatial structures, the rotational modes are as follows: , , , They are respectively v s1 , v b , v s2 and J The largest integer that satisfies the above inequality.

5. The method for calculating the thermophysical properties of a high-temperature non-equilibrium triatomic gas according to claim 1, characterized in that, The formulas for calculating the specific heat of nonequilibrium triatomic gases with only a single temperature variable, including electronic, vibrational, and rotational modes, are as follows: in, R The gas constant is... It is the partition function of the internal energy of the molecule; , and These are the electronic temperature, vibrational temperature, and rotational temperature of a triatomic molecule, respectively. , , These are the electronic mode specific heat, vibration mode specific heat, and rotation mode specific heat of the gas, respectively.

6. A method for generating a high-temperature non-equilibrium triatomic gas thermophysical property database, characterized in that, According to any one of claims 1-5, the specific heat calculation formulas for the electronic, vibrational, and rotational modes of non-equilibrium triatomic gases are obtained by piecewise linear interpolation to obtain three high-temperature non-equilibrium triatomic gas specific heat distribution curves, and by piecewise quadratic interpolation to obtain three high-temperature non-equilibrium triatomic gas enthalpy distribution curves. A gas thermophysical property database is then established based on the curves.

7. The method for generating a high-temperature non-equilibrium triatomic gas thermophysical property database according to claim 6, characterized in that, exist The specific heat of triatomic gases was obtained using a piecewise linear interpolation method within a temperature range. c p Distribution curve The enthalpy of the triatomic gas at various temperatures was obtained using a piecewise quadratic interpolation method. h Distribution curve ; Specifically as follows: Establish The interpolation formulas for enthalpy and specific heat over the temperature range are as follows: and ,use n Each point will The intervals are divided into those of equal length. Segment, denoted as the interval between adjacent interpolation points. , No. i Temperature at each interpolation point ， interpolation point i arrive i +1 is the first i Interpolation region, i =1, 2, 3… , +1; the temperature range is 50K to 50000K. The value range is no greater than 150K; interpolation interval The interpolation formula for the enthalpy corresponding to the piecewise quadratic interpolation curve. Interpolation formula for specific heat corresponding to piecewise linear interpolation curves as follows: ， in, , , These are the interpolation coefficients. T For temperature; The interpolation formula is determined based on the following conditions. and Interpolation coefficients in: Condition 1: Interpolation and exist Internal continuity, At the interpolation point i The specific heat value is equal to that obtained by using the specific heat calculation formula of non-equilibrium triatomic gas electronic, vibrational or rotational modes. ; Condition 2: The first derivative equals ; Condition 3: i When =1, ; Based on conditions 1-3, according to the interpolation formula and The equations established to determine the interpolation coefficients are as follows: 。 8. The method for generating a high-temperature non-equilibrium triatomic gas thermophysical property database according to claim 7, characterized in that, Also includes: exist Within the scope, defined: Extrapolation curve expression of enthalpy Extrapolated curve expression for specific heat ; and , , ; and The expression is as follows: Wherein, the constant in the expression A , B , C for: ； ； ； exist Within the defined range, the extrapolation curve expression for enthalpy. h = Extrapolated curve expression for specific heat c p = .

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