CN114491962A - High-temperature non-equilibrium diatomic gas thermophysical property calculation method and database generation method - Google Patents
High-temperature non-equilibrium diatomic gas thermophysical property calculation method and database generation method Download PDFInfo
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Abstract
The invention discloses a diatomic gas thermophysical property calculation method and a database generation method of a high-temperature non-equilibrium model. Firstly, according to the assumption that vibration-rotation coupling energy is distributed between rotation and vibration, the energy level distribution probability on the electron, vibration and rotation modes respectively accords with the Boltzmann distribution assumption, and a high-temperature non-equilibrium gas model is established; then determining the molecular vibration and rotation energy level range according to the criterion that the sum of the vibration and rotation energy is less than the minimum dissociation energy of the molecules; and finally, deriving a non-equilibrium diatomic gas specific heat calculation formula containing a single temperature variable according to the high-temperature non-equilibrium gas model, the molecular vibration energy level and the rotation energy level range. And constructing a high-temperature non-equilibrium diatomic gas thermophysical property database based on a non-equilibrium diatomic gas specific heat calculation formula and by combining a piecewise linear interpolation method and a piecewise quadratic interpolation method. The method can support engineering application exploration of three-temperature and four-temperature models, and has important guiding significance for research in the technical field of computational aerodynamics.
Description
Technical Field
The invention relates to the technical field of computational aerodynamics, in particular to a method for calculating thermophysical properties of high-temperature non-equilibrium diatomic gas and generating a database.
Background
The high-precision simulation of the high-altitude high-speed flow field has important significance for the design of a novel high-speed aircraft, communication interruption prediction and the development of future manned spacecrafts. The real gas effect and the non-equilibrium are two main characteristics of high-altitude high-speed flow, and are used for improving the flow field simulation precision and effectively describing a physical model of the non-equilibrium process and reliable high-temperature gas data.
In the aspect of an unbalanced physical model, through the development of more than 40 years, a finite rate chemical reaction model describing a chemical unbalanced process is established at present, such as a Park model, a Gupta model and the like; and describing a multi-temperature model of the thermal non-equilibrium process, such as a double-temperature model, a three-temperature model, a four-temperature model and the like. Although the existing research shows that the prediction precision of the plasma flow field can be improved by considering the electron energy non-equilibrium effect, and the rotation non-equilibrium phenomenon is prominent after the translation temperature is more than 12000K, the method is limited by the lack of reliable high-temperature non-equilibrium gas thermophysical property data, particularly high-temperature non-equilibrium diatom gas thermophysical property data, and only a double-temperature model is widely applied in the current engineering practice.
The high-temperature diatomic gas can excite four energy modes of translation, electron, vibration and rotation at most, wherein the four energy modes of electron, vibration and rotation are jointly called as Internal degree of freedom (Internal mode), and the four energy modes are mutually independent from the translation mode, and have a series of complex characteristics such as vibration non-simple harmony, rotation non-rigidity, modal coupling and the like at high temperature. Because of these characteristics, high temperature non-equilibrium gas modeling and thermophysical solution are very difficult.
Disclosure of Invention
The invention aims to overcome the defects and provide a diatomic gas thermophysical property calculation method suitable for a thermal nonequilibrium model with more than two temperatures, and a high-temperature nonequilibrium diatomic gas thermophysical property database is established according to the diatomic gas thermophysical property calculation method so as to support engineering application of the thermal nonequilibrium model with more than two temperatures. The invention firstly establishes the assumption that the energy level distribution probability of the diatomic molecules on the electron, vibration and rotation modes respectively accords with the Boltzmann distribution assumption according to the assumption that the diatomic molecule vibration-rotation coupling energy is distributed between the rotation mode and the vibration mode in a certain proportionA high-temperature non-equilibrium gas model based on a partition function; then determining the vibration energy level and the rotation energy level range of the diatomic molecule according to the criterion that the sum of the vibration energy and the rotation energy of the diatomic molecule is smaller than the minimum dissociation energy of the molecule; finally decoupling hypothesis T at modeel≈Tvib≈TrotUnder the condition of (1), according to a high-temperature non-equilibrium gas model and the range of the vibration energy level and the rotation energy level of the diatomic molecule, a non-equilibrium diatomic gas electron, vibration and rotation mode specific heat calculation formula only containing a single temperature variable is derived. In the generation method of the database, a high-temperature non-equilibrium diatomic gas thermophysical property database is constructed based on a non-equilibrium diatomic gas specific heat calculation formula and by combining a piecewise linear interpolation method and a piecewise quadratic interpolation method.
In the preferred mode of determining the range of the vibrational energy level and the rotational energy level of the diatomic molecule, the main steps comprise: determining upper bound J of rotation quantum number according to potential energy function in diatom moleculelim.n(ii) a At J<Jlim,nWithin the range, according to the property that the first derivative of the potential energy function at the maximum value point is zero and the second derivative is less than zero, solving r corresponding to the maximum value point of the potential energy curvemax,n,J；rmax,n,JSubstituting into potential energy function to obtain minimum energy required by molecular dissociation
Solving the condition that the sum of all diatomic molecular vibrations and rotational energy meeting the constraint condition is less than
The number v of the vibration quanta and the number J of the rotation quanta, and the vibration energy level and the rotation energy level range are obtained.
The method comprehensively considers the high-temperature real gas effect influences such as high-order electronic energy excitation, vibrator non-syntactical property, rotor non-rigidity, rotation-vibration coupling, dissociation energy constraint and the like, completely decouples the diatomic gas electronic, vibration and rotation modes on the basis of reasonable assumption to obtain an enthalpy and specific heat approximate value calculation method, can support engineering application exploration of three-temperature and four-temperature models, and has important guiding significance for research in the technical field of computational aerodynamics.
In order to achieve the above purpose, the invention provides the following technical scheme:
a method for calculating the thermophysical properties of high-temperature non-equilibrium diatomic gas comprises the following steps:
s1, establishing a high-temperature non-equilibrium gas model based on a partition function according to the assumption that diatomic molecule vibration-rotation coupling energy is distributed between a rotation mode and a vibration mode, and the energy level distribution probability of diatomic molecules on the electron, vibration and rotation modes respectively accords with the Boltzmann distribution assumption;
s2, determining the vibration energy level and the rotation energy level range of the diatomic molecule according to the criterion that the sum of the vibration energy and the rotation energy of the diatomic molecule is smaller than the minimum dissociation energy of the molecule;
S3 Tel、Tviband TrotThe electron temperature, the vibration temperature and the rotation temperature of diatomic molecules are respectively adopted, and the hypothesis T is assumed in modal decouplingel≈Tvib≈TrotUnder the condition of (1), according to a high-temperature non-equilibrium gas model and the range of the vibration energy level and the rotation energy level of the diatomic molecule, a non-equilibrium diatomic gas electron, vibration and rotation mode specific heat calculation formula only containing a single temperature variable is derived.
Further, the high-temperature non-equilibrium gas model based on the partition function established in step S1 is as follows:
wherein QintIs the function of molecular internal energy distribution, sigma is the molecular symmetric constant, n, v, J are the numbers of electron, vibration and rotation quanta,
and
respectively, electronic energy, vibration energy, rotational energy and vibration-rotational coupling energy, gnIs degree of degeneracy of energy level, kBolBoltzmann constant, f is a coupling energy distribution control parameter,from vibration temperature TvibRotational temperature TrotAnd vibration characteristic temperature
Determining that the value range is 0-1:
ε0is zero energy;
nmaxis the maximum electron energy level, vmax,nAnd Jmax,n,vRespectively, a maximum vibrational energy level and a maximum rotational energy level determined according to the diatomic molecular vibrational energy level range and the rotational energy level range.
Further, the specific steps of step S1 are as follows:
s1.1 the diatomic molecular vibration-rotation energy is divided into three parts: only the part affected by the mode of vibration, i.e. vibration energy
Part effected only by the mode of rotation, i.e. the energy of rotation
And portions influenced by both the rotary mode and the vibrational mode, i.e. vibration-rotary coupling energy
Wherein h isPlkIs Planck constant, c is speed of light，ωe,n、ωe,nxe,n、ωe,nye,n、ωe,nze,n、αe,n、βe,n、Be,n、De,nIs a diatomic molecular spectral constant determined via experimental measurements or theoretical calculations;
s1.2, introducing a coupling energy distribution control parameter f with a value range of 0-1 according to the assumption that diatom molecular vibration-rotation coupling energy is distributed between a rotation mode and a vibration mode, and carrying out vibration mode selection on the coupled energy distribution control parameter f
Is divided into
And
two parts are respectively distributed into a vibration mode and a rotation mode to obtain new vibration energy
With new rotational energy
Wherein the energy distribution control parameter is coupled
S1.3 respectively conforming to Boltzmann distribution hypothesis according to energy level distribution probabilities of diatomic molecules on electron, vibration and rotation modes, and obtaining new vibration energy in step S1.2
With new rotational energy
And establishing a high-temperature non-equilibrium gas model based on a distribution function.
Further, the step S2 includes the following specific steps:
s2.1 determining the upper bound J of the number of rotating quanta according to the potential energy function in the diatomic moleculelim.n；
S2.2 according to the potential energy function in the diatomic molecule, J is more than or equal to 0<JlimDetermination of minimum dissociation energy of molecules within range
S2.3 solving the sum of all the vibration and rotation energies of diatomic molecules
Less than the minimum dissociation energy of the molecule
The number of vibration quanta v and the number of rotation quanta J of the criterion are combined, i.e. the vibration level and the rotation level range.
Further, in the step S2.1, according to the property that the first derivative of the diatomic intramolecular potential energy function at the extreme point is zero, the following equation is obtained, and the quantum number upper bound J is rotatedlim.nThe smallest integer for which the following equation is unsolved:
wherein the content of the first and second substances,
to a ground state of rotational energy J of 0 minMinimum energy required for dissociation of the atoms, μ is the reduced mass of the diatomic molecule, r is the interatomic distance between the atoms constituting the diatomic molecule, re,nJ is 0 and corresponds to r, h at the lowest point of the potential energy curvePlkIs Planck constant, c is speed of light, omegae,nJ is the number of rotational quanta for the diatomic molecular spectral constant determined via experimental measurements or theoretical calculations;
in the step 2.2, J is more than or equal to 0<JlimWithin the range, solving a potential energy maximum value r which satisfies that the first derivative of the potential energy function in the diatomic molecule is zero and the second derivative is less than zeromax,n,JR is tomax,n,JSubstituting into a diatomic intramolecular potential function to obtain
Further, the calculation formula of the specific heat of the non-equilibrium diatomic gas electron, vibration and rotation modes obtained in the step S3 is as follows:
wherein R is a gas constant, QintIs the intramolecular energy partitioning function.
A high-temperature nonequilibrium diatomic gas thermophysical property database generation method adopts a piecewise linear interpolation and piecewise quadratic interpolation method according to the nonequilibrium diatomic gas electron, vibration and rotation mode specific heat calculation formula to establish a high-temperature nonequilibrium diatomic gas thermophysical property database containing diatomic gas specific heat and enthalpy.
Further, in a method for generating a high-temperature non-equilibrium diatomic gas thermophysical database, at Tmin～TmaxWithin the temperature range, the specific heat c of the diatomic gas at each temperature is obtained by adopting a piecewise linear interpolation methodpObtaining the diatomic gas enthalpy h at each temperature by adopting a segmented quadratic interpolation method;
the method comprises the following specific steps:
(1) establishing Tmin～TmaxThe interpolation formula of enthalpy and specific heat in the temperature range is Fh,i(T) and
using n points to calculate Tmin～TmaxIs divided into n-1 segments with the same length, the interval between adjacent interpolation points is recorded as delta T, and the temperature T of the ith interpolation point is recorded asiI Δ T, the ith interpolation region is defined between the interpolation points i and i +1, i is 1, 2, 3 … n-1,
interpolation interval Ti～Ti+1Inner Fh,i(T) and
the following were used:
(2) determining interpolation formula F based on the following conditionsh,i(T) and
the interpolation coefficient of (1):
condition 1: interpolation formula Fh,i(T) and
in T e [ T ∈ [ ]min,Tmax]The inner part is continuous with the outer part,
equaling by non-equilibrium diatoms at the interpolation point iSpecific heat value c obtained by calculation formula of specific heat of gas electron, vibration or rotation modep(Ti)；
Condition 2: fh,iThe first derivative of (T) being equal to
Condition 3: when the value of i is 1, the reaction condition is shown,
based on the conditions 1-3, according to an interpolation formula Fh,i(T) and
the equation established for determining the interpolation coefficients is as follows:
further, the method for generating the high-temperature non-equilibrium diatomic gas thermophysical property database further comprises the following steps:
(3) in T e (T)maxWithin a range of + ∞), define:
h＝Fh,n(T)，
and Fh,n(Tn)＝Fh,n-1(Tn)，
Wherein, Fh,n-1(Tn) Represents Tmin～TmaxInterpolation of temperature range Fh,i(T) temperature T of nth interpolation point in nth-1 th interpolation regionnThe value of time; t isn＝Tmax；
(4) In T epsilon (0, T)min) Within the scope, define
T1＝Tmin。
Further, in the step (3) of the high-temperature nonequilibrium diatomic gas thermophysical property database generation method, in T epsilon (T ∈ (T)) (max, + ∞) temperature range Fh,n(T) and
the expression is as follows:
Fh,n(T)＝-A/Bexp[-B(T-Tn)]+C
wherein, the parameter A, B, C in the expression is:
C＝Fh,n-1(Tn)+A/Bexp[-B(T-Tn)]。
further, in a method for generating a high-temperature non-equilibrium diatomic gas thermophysical database, the Tmin＝50K，Tmax＝50000K，ΔT＝50K。
Compared with the prior art, the invention has at least one of the following beneficial effects:
(1) in the calculation method for the thermophysical properties of the high-temperature non-equilibrium diatomic gas, the influence of high-temperature real gas effects such as high-order electronic energy excitation, vibrator non-simple harmonic property, rotor non-rigidity, rotation-vibration coupling, dissociation energy constraint and the like is comprehensively considered, the properties of the high-temperature gas can be better reflected, and the accuracy of the predicted enthalpy value and the specific heat value is obviously improved;
(2) according to the calculation method for the thermophysical properties of the high-temperature non-equilibrium diatomic gas, the electronic, vibration and rotation modes of the diatomic gas are completely decoupled on the basis of reasonable assumption, so that an approximate calculation method for enthalpy and specific heat is obtained, engineering application exploration of a three-temperature model and a four-temperature model can be supported, and the applicability is strong;
(3) according to the method for generating the high-temperature non-equilibrium diatomic gas thermophysical property database, the diatomic gas enthalpy and specific heat database is established by adopting the piecewise linear interpolation and the piecewise quadratic interpolation method according to the non-equilibrium diatomic gas specific heat calculation formula, so that the Runge phenomenon caused by high-order interpolation is avoided, the continuity is good, and the calculation efficiency is effectively improved.
Drawings
FIG. 1 is a step diagram of a high temperature non-equilibrium gas thermophysical property calculation method and a database generation method according to the invention;
FIG. 2 shows nitrogen gas (N)2) A molecular potential energy curve under different rotation quantum numbers J of an electron energy ground state (n is 0);
FIG. 3 is (a) N predicted using the method of the present invention, the low temperature hypothesis reduction method and the theory in example 12、(b)CO、(c)O2And (d) a curve of the change of the degree-of-freedom specific heat with the temperature of four typical diatomic molecular gases of NO.
Detailed Description
The features and advantages of the present invention will become more apparent and appreciated from the following detailed description of the invention.
The word "exemplary" is used exclusively herein to mean "serving as an example, embodiment, or illustration. Any embodiment described herein as "exemplary" is not necessarily to be construed as preferred or advantageous over other embodiments. While the various aspects of the embodiments are presented in drawings, the drawings are not necessarily drawn to scale unless specifically indicated.
In the current thermal non-equilibrium flow field simulation of more than two temperatures, a low-temperature hypothesis simplification method is the only way for obtaining the enthalpy and specific heat data of the diatomic gas, and the method has the advantages that: the calculation formula has an analytic form, has thorough decoupling on the internal freedom degree mode, and is suitable for three-temperature and four-temperature models; the disadvantages are that: considering that the energy level is limited, not considering modal coupling, the prediction deviation of the thermophysical property of the high-temperature gas is large, and the hypothesis of the low-temperature hypothesis simplification method is as follows: 1) the electrons can be excited to one order at most; 2) molecular vibration simple resonance; 3) the molecule is a rigid rotor. The enthalpy and specific heat based on this are calculated as:
hrot＝RTrot 
wherein the content of the first and second substances,
the degeneracy of the energy levels of the electronic ground state and the first excited state,
is the characteristic temperature of the first excited state,
r is a gas constant, which is a vibration characteristic temperature of an electron ground state. The method can better reflect the properties of the low-temperature diatomic gas, but is limited by simplifying hypothesis, the method has larger deviation on the thermophysical property prediction of the high-temperature diatomic gas, h represents enthalpy, c represents enthalpypRepresenting specific heat, the superscripts el, rot and vib respectively represent electron, rotation and vibration modes of diatomic molecules, and the corresponding temperatures of the electron, vibration and rotation modes are Tex、Tvib、Trot。
In contrast, high-temperature real gas effect influences such as high-order electronic energy excitation, vibrator non-simple harmonic, rotor non-rigidity, rotation-vibration coupling, dissociation energy constraint and the like are comprehensively considered, high-temperature non-equilibrium gas modeling is completed, an internal degree of freedom modal decoupling hypothesis is provided, a high-temperature non-equilibrium diatom gas thermophysical property calculation method is established, and a high-temperature non-equilibrium diatom gas thermophysical property database generation method is designed according to the high-temperature non-equilibrium diatom gas thermophysical property calculation method.
As shown in fig. 1, the method for calculating the thermophysical properties of the high-temperature non-equilibrium diatomic gas comprises the following steps:
step S1: modeling high-temperature non-equilibrium gas;
strong coupling exists between diatomic molecular vibration and rotation modes, and in order to establish an unbalanced diatomic gas model, rotation-vibration energy is divided into three parts: part influenced only by vibration mode
Part influenced only by rotary mode
Part influenced by both rotation mode and vibration mode
Wherein h isPlkIs Planck constant, c is speed of light, n, v, J are numbers of electronic, vibration and rotation quanta,
is the corresponding electronic, vibration and rotational energy, ωe,n、ωe,nxe,n、ωe,nye,n、ωe,nze,n、αe,n、βe,n、Be,n、De,nIs a diatomic molecular spectral constant determined via experimental measurements or theoretical calculations. Introducing a coupling energy distribution control parameter f with a value range of 0-1, and
is divided into
And
the two parts are respectively distributed to a vibration mode and a rotation mode to obtain new vibration energy and rotation energy:
the invention is designed in the form of f, which is measured by the vibration temperature TvibRotational temperature TrotAnd vibration characteristic temperature
Determining:
assuming that the probability of diatomic molecules on each energy level of the electronic, vibration and rotation modes independently satisfies the Boltzmann distribution, and recording the corresponding temperature of each electronic, vibration and rotation mode as Tex、Tvib、TrotObtaining a distribution function in an unbalanced form, namely a high-temperature unbalanced gas model based on the distribution function:
wherein Q isintIs the intramolecular energy partition function, σ is the molecular symmetry constant, kBolIs the boltzmann constant;
nmax，vmax,nand Jmax,n,vMaximum electron energy level, maximum vibration energy level and maximum rotation energy level, respectively, when the molecular species is determined, the maximum electron energy level nmaxIs a known quantity, vmax,nAnd Jmax,n,vDetermined by step S2, vmax,nMaximum vibration level, J, obtained for a fixed diatomic molecular electron quantum number, nmax,n,vMaximum rotation level J obtained for fixed vibration quantum number v and electron quantum number nmax,n,v。
Step S2: vibration level and rotation level range calculation
Limited combinations of rotational and vibrational quanta exist, constrained by the sum of rotational and vibrational energy being less than the minimum understood molecular energy, and nitrogen (N) will be used below2) The electronic energy ground state is taken as an example to explain the specific steps of vibration energy level and rotation energy level range determination. Taking into account the influence of rotation, N2The intramolecular potential energy function and the first and second derivatives thereof are:
TABLE 1N2Gas electronic energy ground state main spectrum data
Note that in the table, r is removede,0Has the unit of
In addition, the rest units are cm-1
In the above formula, the first and second carbon atoms are,
minimum energy required for molecular dissociation when the rotational energy is in the ground state, re,nCorresponding to the distance between N atoms at the lowest point of the potential energy curve of the ground state of the rotation energyN2＝mNmN/(mN+mN) In order to reduce the molecular mass, where mNTable 1 gives the main spectral data of nitrogen in the ground state of electron energy for the monatomic molecular mass.
FIG. 2 shows the intramolecular potential energy curve under different rotation quantum numbers J under the electron ground state, wherein rmin,n,JAnd rmax,n,JRespectively corresponding to a minimum value point and a maximum value point r of the potential energy curvemax,n,JCorresponding U (r)max,n,J) For a given quantum number n, J, the minimum dissociation energy of the molecule, i.e.
The sum of the rotation and vibration energy needs to be satisfied
The invention determines the energy level range by referring to the following steps:
1) determining the upper bound J of the number of rotational quantalim.n. According to the property that the first derivative of the function at the extreme point is zero, the equation is obtained:
gradually increasing J from J to 0, and defining the smallest integer J which makes the equation have no solution as Jmin,n。
Specifically, in this step, the diatomic molecular electron quantum number n is fixed, a molecular potential energy curve is drawn according to diatomic molecular spectral data, the rotational quantum number J is gradually increased until no extreme point exists in the potential energy curve, and the rotational quantum number at this time is defined as Jlim,n。
2) J is more than or equal to 0<Jlim,nDetermination of minimum energy required for dissociation within range
According to the property that the first derivative of the function at the maximum point is zero and the second derivative is less than zero:
solving to obtain r satisfying the above formulamax,n,JSubstituting into potential energy function to obtain
3) Solving all the satisfies
The number of vibration quanta v and the number of rotation quanta J of the constraint conditions are combined, i.e. the vibration level and the rotation level range.
Recording v as the maximum potential energy vmax,0V is not less than 0 and not more than vmax,0In the range, the maximum J corresponding to each v is recorded as Jmax,n,vTable 2 shows the J corresponding to the partial v in the ground state of nitrogen electron energymax,n,v。
TABLE 2N2Gas electron energy ground state portion v corresponds to Jmax
It is noted that the above energy level range solution method can be generalized to all electron energy levels of all diatomic molecules.
Step S3: specific heat calculation formula of single temperature variable is derived
According to theoretical derivation, the gas specific heat calculation formula derived from the non-equilibrium partition function is as follows:
in the above formula, the specific heat of gas is a function of three temperatures of electrons, vibration and rotation, and T is introducedel≈Tvib≈TrotAnd (3) assuming modal decoupling, obtaining a non-equilibrium gas specific heat calculation formula only with a single temperature variable:
the method for establishing the thermophysical database of the high-temperature non-equilibrium gas comprises the following steps:
in order to avoid Runge phenomenon caused by high-order interpolation, the invention adopts piecewise linear interpolation to establish an unbalanced diatomic gas specific heat database, and adopts piecewise quadratic interpolation to establish an unbalanced diatomic gas enthalpy database, wherein the temperature range of the database is 50K-50000K, and the interval between adjacent interpolation points is 50K. In the area outside the coverage range of the database of 0K-50K and >50000K, the enthalpy and specific heat of the diatomic gas are calculated by adopting two forms of extrapolation formulas.
In the temperature range of 50K-50000K, the segmented interpolation expressions of the enthalpy and the specific heat are respectively Fh(T) and
in order to ensure the interpolation precision, the interval between adjacent interpolation points is fixed to be 50K, and the ith (i is equal to 1, 2, 3 … n-1, n is equal to 1000) interpolation point temperature TiWhen the value is 50i, the ith interpolation region is recorded between interpolation points i and i +1, and the interpolation formula of the enthalpy and the specific heat in the region is Fh,i(T) and
the form is as follows:
Fh,i(T)＝Ai,0+Ai,1T+0.5Ai,2T2
Ai,0、Ai,1、Ai,2is an interpolation coefficient, and the determination process of the interpolation coefficient will be described below. Interpolation formula Fh(T) and
satisfies the following condition 1 in the range of 50K to 50000K
Continuously in the range of 50K-50000K, and at an interpolation point i, the specific heat value is equal to the specific heat value given by the non-equilibrium gas specific heat calculation formula of the single temperature variable; 2) fh(T) is derivable over a range of 50K to 50000K with a first derivative equal to
Based on this, the following equation is established:
the above equation contains 3n-2 terms, 3(n-1) unknowns, assuming that i is 1
A complementary equation is obtained that solves the system of equations.
A1,0+A1,1T1+0.5A1,2T1 2＝(A1,1+A1,2T1)T1
For the case outside the database coverage, the invention adopts the following processing mode:
0K-50K: c is top,1Extrapolating as the specific heat of the zone, the enthalpy value continues at the interpolation point 1, thus within the zone:
h＝cp,1T,cp≡cp,1
>50000K: within this region, the h form is assumed to be-A/Bexp [ -B (T-50000)]+C，cpIn the form of Aexp [ -B (T-50000)]At the interpolation point n, h equals Fh,999(T1000)、cpIs equal to
cpIs equal to the first derivative of
This gives:
example 1:
n is selected from the application example2、CO、O2And four typical diatomic molecular gases of NO, the prediction effectiveness of the high-temperature non-equilibrium gas thermophysical property calculation method (hereinafter referred to as a new algorithm) on the specific heat of the high-temperature diatomic gas is verified, and N is used2Enthalpy and specific heat of the gas are calculated as an example, and the efficiency improvement condition of a lower temperature hypothesis simplification method (hereinafter referred to as an 'old algorithm') in the prior art is tested.
1) Calculation test for specific heat of diatomic molecular gas
Verification test selection N2、CO、O2And four typical diatomic molecules of NO, and the set temperature range is 50K-50000K.
As shown in FIG. 3, the predicted results of the new algorithm and the old algorithm on the specific heat of the four diatomic gases and the total specific heat data calculated by molecular physics theory are shown. The total specific heat predicted by the new algorithm is consistent with theoretical data in the whole test temperature range, and the total specific heat predicted by the old algorithm is obviously deviated from the theoretical data after the temperature is more than about 7000K. And further comparing the prediction conditions of the new algorithm and the old algorithm on the specific heat of each internal energy modality: first, compare
Because the new algorithm considers a large number of electronic energy levels, the predicted situation is
Is relatively high; re-comparison
Predicted by both algorithms when the temperature does not exceed about 2000K
Coincidence, is influenced by vibration non-simple harmony along with temperature rise and is predicted by a new algorithm
Slightly higher than the old algorithm, further rise with temperature, and are influenced by the constraint of minimum dissociation energy
Tend to 0, predicted by the old algorithm
A trend toward a gas constant R; final comparison
Based on rigid rotor assumptions
Constant equal to R, predicted by the new method, taking into account the rotor non-rigidity and minimum dissociation energy constraints
And gradually approaches 0 after a temperature greater than about 15000K. The above results all show that the new algorithm is significantly better than the old algorithm.
2) Computational efficiency testing
For illustrating the advantage of the calculation efficiency based on the new algorithm, the method based on N is designed2And calculating and comparing and testing gas enthalpy and specific heat. The test is carried out for five rounds in total, each round comprises calculation tests of enthalpy and specific heat of three internal energy modes by a new algorithm and an old algorithm, 12 calculation examples are totally included, a single calculation example is arranged in the temperature range of 50K-50000K at an interval of 0.001K, and each calculation example comprises 49950001 calculation points.
The results of five rounds of tests and the average time spent on solving each parameter are recorded in tables 3 and 4. The test result shows that the average time consumption of the new algorithm for solving the enthalpy of the electron and the vibration modes is about one fourth of that of the old algorithm, the average time consumption of the new algorithm for solving the specific heat of the electron and the vibration modes is about one sixth of that of the old algorithm, and the average time consumption of the new algorithm for solving the enthalpy and the specific heat of the rotation modes is slightly lower than that of the old algorithm.
TABLE 3N2Gas enthalpy calculation time
TABLE 4N2Gas specific heat calculation is time consuming
The invention has been described in detail with reference to specific embodiments and illustrative examples, but the description is not intended to be construed in a limiting sense. Those skilled in the art will appreciate that various equivalent substitutions, modifications or improvements may be made to the technical solution of the present invention and its embodiments without departing from the spirit and scope of the present invention, which fall within the scope of the present invention. The scope of the invention is defined by the appended claims.
Those skilled in the art will appreciate that those matters not described in detail in the present specification are well known in the art.
Claims (11)
1. A method for calculating the thermophysical properties of high-temperature non-equilibrium diatomic gas is characterized by comprising the following steps:
s1, establishing a high-temperature non-equilibrium gas model based on a partition function according to the assumption that diatomic molecule vibration-rotation coupling energy is distributed between a rotation mode and a vibration mode, and the energy level distribution probability of diatomic molecules on the electron, vibration and rotation modes respectively accords with the Boltzmann distribution assumption;
s2, determining the vibration energy level and the rotation energy level range of the diatomic molecule according to the criterion that the sum of the vibration energy and the rotation energy of the diatomic molecule is smaller than the minimum dissociation energy of the molecule;
S3 Tel、Tviband TrotThe electron temperature, the vibration temperature and the rotation temperature of diatomic molecules are respectively adopted, and the hypothesis T is assumed in modal decouplingel≈Tvib≈TrotUnder the condition of (1), according to a high-temperature non-equilibrium gas model and the range of the vibration energy level and the rotation energy level of the diatomic molecule, a non-equilibrium diatomic gas electron, vibration and rotation mode specific heat calculation formula only containing a single temperature variable is derived.
2. The method according to claim 1, wherein the distribution function-based high-temperature non-equilibrium gas model established in step S1 is as follows:
wherein QintIs the function of molecular internal energy distribution, sigma is the molecular symmetric constant, n, v, J are the numbers of electron, vibration and rotation quanta,
and
respectively, electronic energy, vibration energy, rotational energy and vibration-rotational coupling energy, gnIs degree of degeneracy of energy level, kBolIs Boltzmann constant, f is a coupled energy distribution control parameter, and is determined by the vibration temperature TvibRotational temperature TrotAnd vibration characteristic temperature
Determining that the value range is 0-1:
ε0is zero energy;
nmax，vmax,nand Jmax,n,vThe maximum electron energy level, the maximum vibration energy level and the maximum rotation energy level, respectively.
3. The method for calculating the thermophysical property of the high-temperature non-equilibrium diatomic gas of claim 2, wherein the step S1 comprises the following steps:
s1.1 the diatomic molecular vibration-rotation energy is divided into three parts: only the part affected by the mode of vibration, i.e. vibration energy
Part only influenced by the mode of rotation, i.e. rotational energy
And portions influenced by both the rotary mode and the vibrational mode, i.e. vibration-rotary coupling energy
Wherein h isPlkIs Planck constant, c is speed of light, omegae,n、ωe,nxe,n、ωe,nye,n、ωe,nze,n、αe,n、βe,n、Be,n、De,nIs a diatomic molecular spectral constant determined via experimental measurements or theoretical calculations;
s1.2, introducing a coupling energy distribution control parameter f with a value range of 0-1 according to the assumption that diatom molecular vibration-rotation coupling energy is distributed between a rotation mode and a vibration mode, and carrying out vibration mode selection on the coupled energy distribution control parameter f
Is divided into
And
two parts are respectively distributed into a vibration mode and a rotation mode to obtain new vibration energy
With new rotational energy
Wherein the energy distribution control parameter is coupled
S1.3 respectively conforming to Boltzmann distribution hypothesis according to the energy level distribution probability of diatomic molecules on electron, vibration and rotation modes, and obtaining new vibration energy in the step S1.2
With new rotational energy
And establishing a high-temperature non-equilibrium gas model based on a distribution function.
4. The method for calculating the thermophysical property of the high-temperature non-equilibrium diatomic gas of claim 1, wherein the step S2 comprises the following steps:
s2.1 determining the upper bound J of the number of rotating quanta according to the potential energy function in the diatomic moleculelim.n；
S2.2 according to the potential energy function in the diatomic molecule, J is more than or equal to 0<JlimDetermination of minimum dissociation energy of molecules within range
S2.3 solving the sum of all the vibration and rotation energies of diatomic molecules
Less than the minimum dissociation energy of the molecule
The combination of the number of vibration quanta v and the number of rotation quanta J of the criterion, i.e. the vibration level and the range of rotation levels, i.e. the maximum vibration level vmax,nAnd maximum rotational energy level Jmax,n,v。
5. The method as claimed in claim 4, wherein in step S2.1, the equation for rotating the upper bound J of quantum number is obtained according to the property that the first derivative of the potential energy function in diatomic molecule at the extreme point is zerolim.nThe smallest integer for which the following equation is unsolved:
wherein the content of the first and second substances,
the ground state of the rotational energy, J, is 0, the minimum energy required for the dissociation of a molecule, μ is the reduced mass of a diatomic molecule, r is the interatomic distance that makes up the diatomic molecule, re,nJ is 0 and corresponds to r, h at the lowest point of the potential energy curvePlkIs Planck constant, c is speed of light, omegae,nJ is the number of rotational quanta for the diatomic molecular spectral constant determined via experimental measurements or theoretical calculations;
in the step 2.2, J is more than or equal to 0<JlimWithin the range, solving a potential energy maximum value r which satisfies that the first derivative of the potential energy function in the diatomic molecule is zero and the second derivative is less than zeromax,n,JR is tomax,n,JSubstituting into a diatomic intramolecular potential function to obtain
6. The method for calculating the thermal properties of the high-temperature non-equilibrium diatomic gas of claim 1, wherein said calculation formula of the electron, vibration and rotation modal specific heats of the non-equilibrium diatomic gas obtained in step S3 is as follows:
wherein R is a gas constant, QintIs the intramolecular energy partitioning function.
7. A high-temperature nonequilibrium diatomic gas thermophysical property database generation method is characterized in that a high-temperature nonequilibrium diatomic gas thermophysical property database containing the specific heat and the enthalpy of diatomic gas is established by adopting a piecewise linear interpolation and piecewise quadratic interpolation method according to the nonequilibrium diatomic gas electronic, vibration and rotation mode specific heat calculation formula obtained by any one of claims 1-6.
8. The method of claim 7, wherein T is the temperature at which the diatomic gases are not in equilibriummin～TmaxWithin the temperature range, the specific heat c of the diatomic gas at each temperature is obtained by adopting a piecewise linear interpolation methodpObtaining the diatomic gas enthalpy h at each temperature by adopting a segmented quadratic interpolation method;
the method comprises the following specific steps:
(1) establishing Tmin～TmaxThe interpolation formula of enthalpy and specific heat in the temperature range is Fh,i(T) and
using n points to calculate Tmin～TmaxIs divided into n-1 segments with the same length, the interval between adjacent interpolation points is recorded as delta T, and the temperature T of the ith interpolation point is recorded asiI Δ T, the ith interpolation region is defined between the interpolation points i and i +1, i is 1, 2, 3 … n-1,
interpolation interval Ti～Ti+1Inner Fh,i(T) and
the following were used:
(2) determining interpolation formula F based on the following conditionsh,i(T) and
the interpolation coefficient of (1):
condition 1: interpolation formula Fh,i(T) and
in T e [ T ∈ [ ]min,Tmax]The inner part is continuous with the outer part,
at the interpolation point i is equal to the specific heat value c obtained by calculation of the specific heat of the non-equilibrium diatomic gas electron, vibration or rotation modep(Ti)；
Condition 2: fh,iThe first derivative of (T) being equal to
Condition 3: when the value of i is 1, the reaction condition is shown,
based on the conditions 1-3, according to an interpolation formula Fh,i(T) and
the equation established for determining the interpolation coefficients is as follows:
9. the method for generating the thermophysical property database of the high-temperature non-equilibrium diatomic gas of claim 8, further comprising the steps of:
(3) in T e (T)maxWithin a range of + ∞), define:
h＝Fh,n(T)，
and Fh,n(Tn)＝Fh,n-1(Tn)，
Tn＝Tmax；
(4) In T epsilon (0, T)min) Within the scope, define
T1＝Tmin。
10. The method for generating the high-temperature nonequilibrium diatomic gas thermophysical property database of claim 9, wherein in step (3), in T e (T ∈)max, + ∞) temperature range Fh,n(T) and
the expression is as follows:
Fh,n(T)＝-A/Bexp[-B(T-Tn)]+C
wherein, the parameter A, B, C in the expression is:
C＝Fh,n-1(Tn)+A/Bexp[-B(T-Tn)]。
11. the method as claimed in claim 8, wherein T is the value of Tmin＝50K，Tmax＝50000K，ΔT＝50K。
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