A method for calculating the characteristic time of gas vibration relaxation using the state-to-state transition rate

Abstract

The present invention discloses a method for calculating the characteristic time of gas vibration relaxation by using the "state-to-state" transition rate, which includes the following processes: setting the molecular vibration energy at energy level i as ev(i); when an inelastic collision occurs from i to j once, the molecular vibration energy will increase by ev(j) - ev(i); calculating the change in the gas vibration energy Ev by using the "state-to-state" transition rate between any two vibration energy levels i and j, combining with formula (1), and the gas state equation p = nkBT (kB is the Boltzmann constant), to obtain a method for calculating the vibration relaxation characteristic time from the "state-to-state" transition rate. The present invention adopts a new method to calculate the vibration relaxation time τ v , which can give the vibration relaxation time τ v when the translational temperature T and the vibration temperature T v are not equal; the calculation range of the calculated vibration relaxation time τ v is relatively large, and can cover the high temperature region above 10,000 K; only some basic physical and chemical data are needed to calculate the vibration relaxation time τ v , without the need to conduct a wind tunnel experiment, which greatly reduces the cost.

Description

Technical Field

[0001] The invention belongs to the technical field of hypersonic aircraft, and in particular relates to a method for calculating gas vibration relaxation characteristic time by using "state-to-state" transition rate. Background Art

[0002] Hypersonic vehicles have a wide range of military and commercial applications and are a hot research topic in the field of aerospace. When a vehicle flies at an extremely high speed, a strong shock wave will form in front of the vehicle. The temperature of the incoming gas will increase sharply after passing through the shock wave, which will then produce a series of high-temperature real gas effects, which will have an important impact on key issues such as the aerodynamic force and aerodynamic heat of the vehicle.

[0003] Vibrational excitation is an important high-temperature real gas effect that has a significant impact on the thermodynamic properties of the gas and the thermochemical coupling process. After vibrational excitation, the vibration temperature T v There is a difference between the translation temperature T and the vibration temperature T. At this time, the non-equilibrium evolution of vibration energy can be described by the Laudau-Teller equation, that is,

[0004]

[0005] Where Ev is the gas vibration energy when the vibration temperature is Tv, E v,eq is the gas vibration energy when the vibration temperature reaches T, τ v is the vibration relaxation characteristic time. τ v The Millikan-White formula is usually used for calculation, that is,

[0006] pτ v =exp(aT -13 -b) atm·s (2)

[0007] Where p is the pressure; a and b are constants, which can be fitted using the data measured in the shock tube experiment. However, recent studies have shown that the Millikan-White formula has some obvious shortcomings, including: (1) The Millikan-White formula only contains the effect of the translational temperature T on τ v The influence of vibration temperature Tv on τ is not considered. v (2) Most of the experimental data in the existing literature cannot cover the high temperature range above 10000K; (3) Regarding the above-mentioned vibration relaxation characteristic time τ v The curve comes from a wind tunnel test. A set of equipment for a wind tunnel test costs at least several million dollars, and it is very expensive to conduct an experiment. In addition, non-ideal factors such as shock wave boundary layer interference in wind tunnel shock tube experiments will also introduce additional errors. Summary of the invention

[0008] The present invention aims to solve the problems existing in the prior art by proposing a method for calculating the characteristic time of gas vibration relaxation using the "state-to-state" transition rate. The first purpose is to solve the problem that the Millikan-White formula in the prior art only contains the effect of the translation temperature T on τ v The influence of vibration temperature Tv on τ is not considered. v The second purpose is to solve the problem that most of the experimental data in the existing literature cannot cover the high temperature zone above 10000K; the third purpose is to solve the problem of the vibration relaxation characteristic time τ in the prior art v The curve comes from a wind tunnel test. A set of equipment for a wind tunnel test costs at least several million dollars, and it is very expensive to conduct an experiment. In addition, non-ideal factors such as shock wave boundary layer interference in the shock tube experiment of the wind tunnel test will also introduce additional errors.

[0009] The present invention proposes the following technical solutions to solve the technical problems:

[0010] A method for calculating the characteristic time of gas vibration relaxation using the state-to-state transition rate is characterized by comprising the following processes:

[0011] 1) Set the vibration energy of the molecule at energy level i to ev(i);

[0012] 2) When passing through an inelastic collision from i to j, the vibration energy of the molecule will increase by ev(j)-ev(i);

[0013] 3) The change of gas vibration energy Ev is calculated using the "state-to-state" transition rate between any two vibration energy levels i and j. The formula is:

[0014]

[0015] The left side of formula (3) is the change in gas vibration energy Ev calculated using the "state-to-state" transition rate between any two vibration energy levels i and j, where Ev is the gas vibration energy when the vibration temperature is Tv;

[0016] On the right side of formula (3), n is the molecular number density; i and j are vibrational energy levels; P v (i) is e v (i) probability of occurrence; k v is the state-to-state rate; T is the translational temperature; i and j are vibrational energy levels; ev(j)-ev(i) is the increase in molecular vibrational energy when the vibrational energy level changes from i to j;

[0017] 4) Combine formula (1) and the gas state equation p = nk B T (kB is the Boltzmann constant), we get

[0018]

[0019] The left side of formula (4) is the pressure p multiplied by the vibration relaxation characteristic time τ v , formula (4) gives a method for calculating the vibrational relaxation characteristic time from the state-to-state transition rate.

[0020] The numerator k on the right side of formula (4) B is the Boltzmann constant; T is the translation temperature; E v,eq is the gas vibration energy when the vibration temperature reaches the translation temperature T; Ev is the gas vibration energy when the vibration temperature is Tv; E v,eq -E v is the increase of gas vibration energy when the temperature of gas vibration increases from the vibration temperature Tv to the translation temperature T;

[0021] The denominator P on the right side of formula (4) v (i) is the probability of vibration level i; k v is the state-to-state rate; T is the translation temperature; T,i→j is k v The variables include the time variable T and the vibration energy level variables i and j; ev(j)-ev(i) is the increase in molecular vibration energy when the vibration energy level changes from i to j.

[0022] Furthermore, the combined formula (1) of step 4 and the gas state equation p=nk B T, we get formula (4), and the derivation process is as follows:

[0023] By taking the equal sign on the right side of Formula 1 and Formula 3, we can get

[0024]

[0025] Multiply both sides of the above equation by k B T, can be obtained

[0026]

[0027] Finally, using p = nk B T, we can get formula 4

[0028]

[0029] Furthermore, formula (3) calculates the vibration relaxation time τ v It can be obtained with only some basic physical and chemical data, specifically:

[0030] ① In formula (3), P v It is a quantity related to temperature. As long as the temperature and the molecule are given, P vIt is determined and can be directly calculated using known physical and chemical data. For example, if the composition of a gas is given as oxygen or nitrogen, the probability P of each energy level at a certain temperature is v It is determined;

[0031] ② In formula (3), the e corresponding to each energy level is v (i) is also known;

[0032] ③In formula (3), the additional calculation is k v , k v It can also be calculated through quantum chemistry methods and obtained through computer simulation.

[0033] Advantages and effects of the present invention

[0034] 1. The present invention adopts a new method to calculate the vibration relaxation time τv. The new method calculates the vibration relaxation time τ v The calculation range is relatively large, and it can give the translation temperature T and vibration temperature T v Unequal vibration relaxation time τ v , while the traditional method is only related to the translation temperature T, and cannot give T and T v Unequal vibration relaxation periods.

[0035] 2. The present invention adopts a new method to calculate the vibration relaxation time τ v , the new method to calculate the vibration relaxation time τ v The calculation range is relatively large and can cover high temperature areas above 10,000 K, for example, it can cover high temperature areas of 15,000 K and 20,000 K.

[0036] 3. The present invention adopts a new method to calculate the vibration relaxation time τ v , the new method to calculate the vibration relaxation time τ v Only some basic physical and chemical data are needed to calculate the vibration relaxation time τ v , there is no need to do wind dynamic experiments. Compared with wind dynamic experiments that require equipment worth millions, the cost is greatly reduced. BRIEF DESCRIPTION OF THE DRAWINGS

[0037] Figure 1 This is a flow chart of the present invention using the "state-to-state" transition rate to calculate the gas vibration relaxation characteristic time;

[0038] Figure 2 This is a graph of the "state-to-state" transition rate kv(T,i→j) (nitrogen) of the present invention;

[0039] Figure 3 When the translation temperature is given in the present invention, the vibration temperature has an effect on pτ vThe influence of (nitrogen) curve;

[0040] Figure 4 pτ calculated by the "state-state" database and the Millikan-White formula of the present invention v Compare (nitrogen).

[0041] Figure 5 pτ calculated by the "state-state" database and the Millikan-White formula of the present invention v Contrast (oxygen). DETAILED DESCRIPTION

[0042] Design principle of the present invention

[0043] 1. Innovation of the present invention: The innovation lies in using a new method to calculate the vibration relaxation time τ v ,

[0044] First, the new method calculates the vibration relaxation time τ v Only some basic physical and chemical data are needed to calculate the vibration relaxation time τ v , and there is no need to do a wind experiment, the wind experiment is a shock tube wind experiment;

[0045] Second, the new method calculates the vibration relaxation time τ v The calculation range is relatively large, and it can give the vibration relaxation time τ when the translation temperature T and the vibration temperature Tv are not equal. v However, the traditional method is only related to the translation temperature T, and cannot give the vibration relaxation period when T and Tv are not equal.

[0046] Third, the new method is used to calculate the vibration relaxation time τ v Only some basic physical and chemical data are needed to calculate the vibration relaxation time τ v , specifically:

[0047] ① In formula (3), P v It is a quantity related to temperature. As long as the temperature and the molecule are given, P v It is determined and can be directly calculated using known physical and chemical data. For example, if the composition of a gas is given as oxygen or nitrogen, the probability P of each energy level at a certain temperature is v It is determined;

[0048] ② In formula (3), the e corresponding to each energy level is v (i) is also known;

[0049] ③In formula (3), the additional calculation is k v , k vIt can also be calculated through quantum chemistry methods and obtained through computer simulation.

[0050] Based on the above invention principle, the present invention designs a method for calculating the characteristic time of gas vibration relaxation using the "state-to-state" transition rate, which is characterized by comprising the following processes:

[0051] 1) Set the vibration energy of the molecule at energy level i to ev(i);

[0052] 2) When passing through an inelastic collision from i to j, the vibration energy of the molecule will increase by ev(j)-ev(i);

[0053] Supplementary Note 1:

[0054] The above statement "when passing through an inelastic collision from i→j, the vibration energy of the molecule will increase by ev(j)-ev(i)" can be explained as follows: the vibration energy of the molecule is in a series of discrete vibration energy levels (i=0,1,2,…,imax), and the vibration energy of the molecule at energy level i is ev(i). During a molecular collision, the vibration energy level may change, and this collision is called an inelastic collision. The "state-to-state" transition rate kv(T,i→j) is defined as the rate at which the molecular vibration energy level changes from i to j through an inelastic collision at a translation temperature of T.

[0055] 3) The change of gas vibration energy Ev is calculated using the "state-to-state" transition rate between any two vibration energy levels i and j. The formula is:

[0056]

[0057] The left side of formula (3) is the change in gas vibration energy Ev calculated using the "state-to-state" transition rate between any two vibration energy levels i and j, where Ev is the gas vibration energy when the vibration temperature is Tv;

[0058] On the right side of formula (3), n is the molecular number density; i and j are vibrational energy levels; P v (i) is e v (i) probability of occurrence; k v is the state-to-state rate; T is the translational temperature; i and j are vibrational energy levels; ev(j)-ev(i) is the increase in molecular vibrational energy when the vibrational energy level changes from i to j;

[0059] Supplementary Note 2:

[0060] The above formula (3) shows that when an inelastic collision from i to j occurs, the molecular vibration energy will increase by ev(j)-ev(i). Therefore, the change in gas vibration energy Ev can be calculated by using the "state-to-state" transition rate between any two vibration energy levels i and j.

[0061] 4) Combine formula (1) and the gas state equation p = nk B T (kB is the Boltzmann constant), we get

[0062]

[0063] The left side of formula (4) is the pressure p multiplied by the vibration relaxation characteristic time τ v , formula (4) gives a method for calculating the vibrational relaxation characteristic time from the state-to-state transition rate.

[0064] The numerator k on the right side of formula (4) B is the Boltzmann constant; T is the translation temperature; E v,eq is the gas vibration energy when the vibration temperature reaches the translation temperature T; Ev is the gas vibration energy when the vibration temperature is Tv; E v,eq -E v is the increase of gas vibration energy when the temperature of gas vibration increases from the vibration temperature Tv to the translation temperature T;

[0065] The denominator P on the right side of formula (4) v (i) is the probability of vibration level i; k v is the state-to-state rate; T is the translation temperature; T,i→j is k v The variables include the time variable T and the vibration energy level variables i and j; ev(j)-ev(i) is the increase in molecular vibration energy when the vibration energy level changes from i to j.

[0066] Supplementary Note 3:

[0067] Figure 3 is the calculation result of formula (4). The five curves represent the vibration temperature τ when the temperature is 1000K, 2000K, 5000K, 15000K, and 20000K. v For pτ v As can be seen from the figure, as the temperature approaches 20000K, the vibration temperature τ v Close to the translation temperature T, at this time, the vibration temperature has a great influence on pτ v The influence of is the smallest, and when the temperature is relatively low, the vibration temperature τ v When the vibration temperature is relatively different from the translation temperature T, the vibration temperature has a great influence on pτ. v has the greatest impact.

[0068] Figure 4 , Figure 5 When the translation temperature and vibration temperature tend to be equal, the pτ calculated by the "state-state" database of the present invention and the Millikan-White formula is vBy comparing the results, it can be proved that the results obtained by the present invention based on the "state-state" database of physical and chemical data are roughly the same as the results calculated by the Millikan-White formula based on wind tunnel tests in the prior art, and the calculation results of the present invention take into account the effect of vibration temperature on pτ v , and is therefore more accurate.

[0069] Furthermore, the combined formula (1) of step 4 and the gas state equation p=nk B T, we get formula (4), and the derivation process is as follows:

[0070] By taking the equal sign on the right side of Formula 1 and Formula 3, we can get

[0071]

[0072] Multiply both sides of the above equation by k B T, can be obtained

[0073]

[0074] Finally, using p = nk B T, we can get formula 4

[0075]

[0076] Furthermore, formula (3) calculates the vibration relaxation time τ v It can be obtained with only some basic physical and chemical data, specifically:

[0077] ① In formula (3), P v It is a quantity related to temperature. As long as the temperature and the molecule are given, P v It is determined and can be directly calculated using known physical and chemical data. For example, if the composition of a gas is given as oxygen or nitrogen, the probability P of each energy level at a certain temperature is v It is determined;

[0078] ② In formula (3), the e corresponding to each energy level is v (i) is also known;

[0079] ③In formula (3), the additional calculation is k v , k v It can also be calculated through quantum chemistry methods and obtained through computer simulation.

[0080] Embodiment 1

[0081] A specific example is given below. This example is calculated based on the STELLER "state-to-state" database (Silva et al., 2012) published in the literature. The STELLER database provides the "state-to-state" transition rates of air components such as nitrogen and oxygen calculated using the FHO and QCT methods, with a temperature range of 100-100000K. This example uses data in the range of 200-20000K.

[0082] like Figure 2 As shown in Figure 2, a part of the state-to-state transition rate in the STELLER database is given. Using the calculation formula given in this patent, the vibration relaxation characteristic time τ at any translation temperature T and vibration temperature Tv can be obtained. v The calculation results are shown in Figure 2 It can be seen that both the translation temperature and the vibration temperature have a significant effect on the vibration relaxation characteristic time.

[0083] like Figure 3 As shown, the vibration relaxation characteristic time calculated by the "state-to-state" transition rate in the special case of Tv=T is given, and compared with the calculation result of the Millikan-White formula. It can be seen that the "state-to-state" results of the present invention are basically consistent with the Millikan-White formula results, indicating that the vibration relaxation characteristic time given by the Millikan-White formula is actually an approximate result when Tv is close to T.

[0084] Embodiment 2

[0085] In order to calculate the vibration relaxation characteristic time more accurately, this patent proposes a method for calculating the vibration relaxation characteristic time using the "state-to-state" transition rate. Among them, the "state-to-state" transition rate can be given by theoretical models such as FHO or computational chemistry methods such as QCT. The "state-to-state" transition rates given by these theories or calculation methods can cover a higher temperature range and are not affected by experimental errors. Therefore, these data can be used to calculate a more accurate vibration relaxation characteristic time. The specific method is as follows.

[0086] Molecular vibration energy is in a series of discrete vibration energy levels (i = 0, 1, 2, ..., imax), and the molecular vibration energy at energy level i is ev(i). During molecular collisions, the vibration energy level may change, and this collision is called an inelastic collision. The "state-to-state" transition rate kv(T, i→j) is defined as the rate at which the molecular vibration energy level changes from i to j through an inelastic collision at a translation temperature of T.

[0087] When an inelastic collision from i to j occurs, the molecular vibration energy will increase ev(j)-ev(i). Therefore, using the state-to-state transition rate between any two vibration energy levels i and j, the change in gas vibration energy Ev can be calculated using the formula:

[0088]

[0089] Where n is the molecular number density, Pv(i) is the probability of vibration level i. Combined with the Laudau-Teller equation and the gas state equation p = nk B T (kB is the Boltzmann constant), we can get

[0090]

[0091] This formula gives a method for calculating the vibrational relaxation characteristic time from the state-to-state transition rate.

[0092] The above contents are merely examples and explanations of the concept of the present invention. The technicians in this technical field may make various modifications or additions to the specific embodiments described or replace them in a similar manner. As long as they do not deviate from the concept of the invention or exceed the scope defined by the claims, they should all fall within the protection scope of the present invention.

Claims

1. A method for calculating the characteristic time of gas vibration relaxation using the "state-to-state" transition rate, which is used to generate high-temperature real gas effects during the flight of hypersonic aircraft, and is characterized by: The process includes: Step 1: Set the vibration energy of the molecule at energy level i to e v (i); Step 2: When an inelastic collision from i to j occurs, the vibration energy of the molecule will increase by e v (j)-e v (i); Step 3: Calculate the gas vibration energy E using the "state-to-state" transition rate between any two vibration energy levels i and j v The change of The left side of formula (3) is the calculation of the gas vibration energy E using the "state-state" transition rate between any two vibration energy levels i and j. v The change of E v is the gas vibration energy when the vibration temperature is Tv; On the right side of formula (3), n is the molecular number density; i and j are vibrational energy levels; P v (i) is e v (i) probability of occurrence; k v is the state-to-state rate; T is the translational temperature; i and j are vibrational energy levels respectively; e v (j)-e v (i) is the increase in molecular vibration energy when the vibration energy level changes from i to j; Step 4: Combine the Laudau-Teller equation and the gas state equation p=nk B T, we get formula (4): The left side of formula (4) is the pressure p multiplied by the vibration relaxation characteristic time τ v , formula (4) gives a method for calculating the vibration relaxation characteristic time from the "state-to-state" transition rate; The numerator k on the right side of formula (4) B is the Boltzmann constant; T is the translation temperature; E v,eq E is the gas vibration energy when the vibration temperature reaches the translation temperature T; v,eq -E v is the increase of gas vibration energy when the temperature of gas vibration increases from the vibration temperature Tv to the translation temperature T; The denominator P on the right side of formula (4) v (i) is the probability of vibration level i; k v is the state-to-state rate; T is the translation temperature; T,i→j is k v The variables include time variable T and vibration level variables i, j; e v (j)-e v (i) is the increase in molecular vibration energy when the vibration energy level changes from i to j.

2. According to claim 1, a method for calculating the characteristic time of gas vibration relaxation using the "state-to-state" transition rate is characterized in that: The combined Laudau-Teller equation of step 4 and the gas state equation p=nk B T, we get formula (4), and the derivation process is as follows: By equating the Laudau-Teller equation and the right side of formula (3), we can obtain: Multiply both sides of the above equation by k B T, we can get: Finally, using p = nk B T, we can get formula (4) 3. The method for calculating the characteristic time of gas vibration relaxation using the "state-to-state" transition rate according to claim 1, characterized in that: Formula (3) calculates the vibration relaxation time τ v It can be obtained with only some basic physical and chemical data, specifically: ① In formula (3), P v It is a quantity related to temperature. As long as the temperature and the molecule are given, P v It is determined, that is, the probability P of each energy level at a certain temperature v It is determined; ② In formula (3), the e corresponding to each energy level is v (i) is known; ③In formula (3), the additional calculation is k v , k v It can also be calculated through quantum chemistry methods and obtained through computer simulation:

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