An experimental data fitting method for the fluorescence intensity of pressure-sensitive paint

Abstract

An experimental data fitting method for the fluorescence intensity of pressure-sensitive paint, measuring the oxygen diffusion coefficient D of the pressure-sensitive paint and the thickness H of the pressure-sensitive paint coating; establishing a partial differential equation set for the temporal and spatial variations of the oxygen molecule concentration and the excited-state fluorescence molecule concentration in the pressure-sensitive paint; establishing a partial differential equation set for the temporal and spatial variations of the relative oxygen molecule concentration and the relative fluorescence intensity in the pressure-sensitive paint; experimentally measuring the dynamic change data of the relative fluorescence intensity of the pressure-sensitive paint under a certain pressure input, establishing the partial differential equation set, obtaining the temporal and spatial distribution function of the relative fluorescence intensity and the kinetic mathematical model of the relative fluorescence intensity of the pressure-sensitive paint; fitting the kinetic mathematical model of the relative fluorescence intensity of the pressure-sensitive paint with the test measurement results to obtain the final kinetic model of the relative fluorescence intensity of the pressure-sensitive paint, and completing the experimental data fitting.

Description

Technical Field

[0001] The present invention relates to an experimental data fitting method, and specifically, to an experimental data fitting method for the fluorescence intensity of pressure-sensitive paint. Background Art

[0002] The pressure-sensitive paint (PSP) technology is a gas pressure measurement technology based on the principle of photoluminescence-oxygen quenching. Spraying the pressure-sensitive paint on the surface of an aerodynamic model such as an aircraft or a wing can completely display the surface pressure distribution of the model during wind tunnel testing. The pressure-sensitive paint mainly consists of two components, a polymer matrix and fluorescent molecules dispersed therein. The pressure-sensitive paint measurement technology is based on two characteristics of the fluorescent molecular group: photoluminescence and oxygen quenching effect. Illuminating the fluorescent molecules with a short-wavelength excitation light source raises their energy levels, making them excited states. The excited-state fluorescent molecules can emit fluorescence with a longer wavelength, and this process is called photoluminescence. When the excited-state fluorescent molecules encounter ground-state oxygen molecules, they will be converted into ground states by the oxygen molecules and no longer emit fluorescence, and this process is fluorescence oxygen quenching. Therefore, spraying the pressure-sensitive paint on the surface of the aerodynamic model, the intensity of the pressure-sensitive paint light is inversely proportional to the oxygen partial pressure on the model surface. Assuming that the proportion of each component in the ambient atmosphere remains unchanged, the oxygen partial pressure on the model surface is proportional to the static pressure. Therefore, by measuring the light intensity distribution of the pressure-sensitive paint on the model surface, the corresponding static pressure distribution can be calculated.

[0003] Since the light intensity of the pressure-sensitive paint is inversely proportional to the oxygen partial pressure, the oxygen content in the pressure-sensitive paint layer directly determines the fluorescence light intensity of the pressure-sensitive paint. The greater the oxygen content in the pressure-sensitive paint layer, the smaller the fluorescence light intensity. Oxygen diffuses from the environment into the polymer matrix by passive diffusion and quenches the excited-state fluorescent molecules. That is, after a certain response time, the oxygen content in the polymer matrix can reach equilibrium with the oxygen content in the environment, and at this time, the light intensity of the pressure-sensitive paint can reflect the static pressure of the environment. Establishing a kinetic model of the fluorescence light intensity of the pressure-sensitive paint and fitting the experimental data of the fluorescence light intensity of the pressure-sensitive paint according to the model helps to analyze the kinetic process of the fluorescence light intensity response of the pressure-sensitive paint, predict the light intensity response of the pressure-sensitive paint at different pressures, and perform parameter analysis on the pressure-sensitive paint coating, providing guidance for its performance optimization. At present, the research on pressure-sensitive paint at home and abroad focuses on the improvement of its response performance, and there are few related studies on the fitting of experimental data of fluorescence light intensity. The commonly used fitting method for the fluorescence light intensity of pressure-sensitive paint is the first-order kinetic model based on Fick's second law. The modeling and fitting process is to assume that the diffusion of oxygen in the pressure-sensitive paint follows Fick's second law, solve the partial differential equation according to the boundary conditions, and obtain the spatio-temporal distribution of oxygen in the pressure-sensitive paint. According to the relationship between the oxygen concentration and the fluorescence light intensity in the pressure-sensitive paint, a mathematical model of the fluorescence light intensity changing with time is obtained. This model can fit the trend of the fluorescence light intensity changing with time, but there is a large deviation between the model and the experimental results in the initial stage. The literature (DOI: 10.2514 / 3.13099) discloses an improved double-exponential kinetic model. When this model is used to fit the experimental data of the fluorescence light intensity of the pressure-sensitive paint, it can correct the deviation of the first-order kinetic model in the initial stage to a certain extent, but it does not explain the physical mechanism of the model and is only a mathematical fitting. Summary of the Invention

[0004] Aiming at the deficiencies in the prior art, the purpose of the present invention is to provide a method for fitting experimental data of the fluorescence light intensity of pressure-sensitive paint.

[0005] To achieve the above purpose, the present invention adopts the following technical solutions:

[0006] A method for fitting experimental data of the fluorescence light intensity of pressure-sensitive paint, where the pressure-sensitive paint is coated on the surface of the test piece, and the specific steps include:

[0007] Step 1: Measure the oxygen diffusion coefficient D of the pressure-sensitive paint and the thickness H of the pressure-sensitive paint coating;

[0008] Step 2: Based on the oxygen diffusion coefficient and the assumed fluorescence oxygen quenching rate coefficient K0, establish a system of partial differential equations for the spatio-temporal changes of the oxygen molecule concentration and the excited-state fluorescent molecule concentration in the pressure-sensitive paint respectively;

[0009] Step 3: Establish a system of partial differential equations for the spatio-temporal changes of the relative oxygen molecule concentration and the relative fluorescence light intensity in the pressure-sensitive paint;

[0010] Step 4: Experimentally measure the dynamic change data of the relative fluorescence intensity of the pressure-sensitive paint under a certain pressure input;

[0011] Step 5: Based on the oxygen diffusion coefficient D and the assumed fluorescence oxygen quenching rate coefficient K0, combined with the boundary conditions, solve the partial differential equation system established in Step 3 to obtain the spatio-temporal distribution function of the relative fluorescence intensity;

[0012] Step 6: Accumulate and normalize the spatio-temporal distribution function of the relative fluorescence intensity obtained in Step 5 in the thickness direction of the pressure-sensitive paint to obtain a mathematical model of the change of the relative fluorescence intensity of the pressure-sensitive paint with time, that is, the kinetic model of the relative fluorescence intensity of the pressure-sensitive paint;

[0013] Step 7: Fit the kinetic model of the relative fluorescence intensity of the pressure-sensitive paint with the test measurement results, calculate the actual fluorescence oxygen quenching rate coefficient K, substitute K for K0 into Steps 5 and 6, and obtain the final kinetic model of the relative fluorescence intensity of the pressure-sensitive paint to complete the fitting of the experimental data of the fluorescence intensity of the pressure-sensitive paint.

[0014] Preferably, the assumed fluorescence oxygen quenching rate coefficient K0 in Step 2 should be a positive value.

[0015] Preferably, the specific form of the partial differential equation system of the concentration of oxygen molecules and the concentration of excited-state fluorescent molecules changing with space and time in the pressure-sensitive paint in Step 2 is:

[0016]

[0017]

[0018] where x is the distance from a certain point in the pressure-sensitive paint coating to the surface of the pressure-sensitive paint coating; t is the time; u e0 is the concentration of excited-state fluorescent molecules when the pressure is the lowest and the oxygen concentration is the lowest, that is, the concentration of fluorescent molecules in the pressure-sensitive paint coating; u O0 is the oxygen concentration corresponding to the highest pressure and the lowest concentration of excited-state fluorescent molecules; u O (x, t) is the oxygen concentration at a distance x from the surface at time t in the pressure-sensitive paint coating; u e (x, t) is the concentration of excited-state fluorescent molecules at a distance x from the surface at time t in the pressure-sensitive paint coating. The right side of equation (2) represents the concentration of excited-state fluorescent molecules when the oxygen concentration at a distance x from the surface of the pressure-sensitive paint is u O (x, t) and reaches equilibrium. The right side of equation (2) represents the oxygen quenching rate of excited-state fluorescent molecules, that is, the concentration of excited-state fluorescent molecules at a distance x from the surface of the pressure-sensitive paint at time t changes from u e (x, t) and reaches the equilibrium concentration The fluorescence oxygen quenching rate coefficient determines the length of the time delay.

[0019] Preferably, the partial differential equations of the relative concentration of oxygen molecules and the relative fluorescence intensity varying with time and space in the pressure-sensitive paint in step 3 are in the following specific forms:

[0020]

[0021]

[0022] Among them, u RO (x, t) is the relative concentration of oxygen at a distance x from the surface at time t in the pressure-sensitive paint coating; RI(x, t) is the relative fluorescence intensity of the excited-state fluorescent molecules at a distance x from the surface at time t in the pressure-sensitive paint coating. The derivation process of equations (3) and (4) is as follows: Substituting u O (x, t) = u O0 ·u RO (x, t) into equation (1) can obtain equation (3); substituting u O (x, t) = u O0 ·u RO (x, t) and u e (x, t) = u e0 ·RI(x, t) into equation (2) can obtain equation (4). For the convenience of software solution, equations (3) and (4) are rewritten as:

[0023]

[0024] Preferably, the oxygen content in the gas of the measurement environment in step 4 is a fixed value The pressure on the surface of the pressure-sensitive paint is P(t).

[0025] Preferably, in step 5, the boundary conditions are: at x = 0, that is, on the surface of the pressure-sensitive paint, at x = H, that is, on the bottom surface of the pressure-sensitive paint, Among them, P0 is the maximum pressure on the surface of the pressure-sensitive paint.

[0026] Preferably, in step 6, the expression of the kinetic model of the relative fluorescence intensity of the pressure-sensitive paint is:

[0027]

[0028] This fitting method and the kinetic model used in the fitting method not only consider the influence of the oxygen diffusion process on the kinetic model, but also consider the oxygen quenching process. Therefore, compared with the existing fitting methods, the method of the present invention can more accurately fit the experimental results and has a smaller fitting deviation. Description of the Drawings

[0029] Figure 1 Comparison of the relative fluorescence intensity fitted by different fitting methods with the experimental results. Among them, Exp.: experimental data in the literature (DOI: 10.2514 / 3.13099); DM: data fitted by the existing first-order kinetic model; DeM: data fitted by the double-exponential kinetic model proposed in the literature (DOI: 10.2514 / 3.13099); DQM: data fitted by the kinetic model considering diffusion and oxygen quenching processes in the present invention.

[0030] Figure 2 The deviation of the fitting results of the three kinetic models of DM, DeM, and DQM. It can be seen that the kinetic model (DQM) used in the fitting method of the present invention can control the maximum fitting deviation within 5%, which is the highest fitting accuracy among the three models. Specific Embodiments

[0031] An experimental data fitting method for the fluorescence intensity of pressure-sensitive paint provided by the present invention. The pressure-sensitive paint is coated on the surface of the test piece. The specific steps include:

[0032] Step 1: Measure the oxygen diffusion coefficient D = 900 μm 2 / s and the thickness H = 35 μm of the pressure-sensitive paint coating;

[0033] Step 2: Based on the oxygen diffusion coefficient D and the assumed fluorescence oxygen quenching rate coefficient K0 = 10, respectively establish partial differential equations for the concentration of oxygen molecules and the concentration of excited-state fluorescence molecules in the pressure-sensitive paint changing with time and space:

[0034]

[0035]

[0036] Step 3: Establish partial differential equations for the relative concentration of oxygen molecules and the relative fluorescence intensity in the pressure-sensitive paint changing with time and space:

[0037]

[0038]

[0039] Step 4: Experimentally measure the dynamic change data of the relative fluorescence intensity of the pressure-sensitive paint under a certain pressure input, as shown in the Exp. data in Figure 1 ;

[0040] Step 5: Divide the pressure-sensitive paint into 100 equal-spacing units along the thickness direction. The distance of the i-th unit from the surface of the pressure-sensitive paint is x i. Based on the oxygen diffusion coefficient D and the assumed fluorescence oxygen quenching rate coefficient K0 = 10, combined with the boundary conditions, solve the partial differential equations established in step 3 to obtain the spatio-temporal distribution function RI(x i ,t); where the boundary conditions are: at x = 0, that is, on the surface of the pressure-sensitive paint, u RO (0,t) = 1, at x = 35 μm, that is, on the bottom surface of the pressure-sensitive paint,

[0041] Step 6: Accumulate and normalize RI(x,t) obtained in step 5 in the thickness direction of the pressure-sensitive paint to obtain a mathematical model of the relative fluorescence intensity of the pressure-sensitive paint varying with time:

[0042]

[0043] Step 7: Fit the relative fluorescence intensity kinetic model of the pressure-sensitive paint with the experimental measurement results, calculate the actual fluorescence oxygen quenching rate coefficient K = 14.6, substitute K = 14.6 for K0 = 10 into steps 5 and 6 to obtain the final relative fluorescence intensity kinetic model of the pressure-sensitive paint, and complete the fitting of the experimental data of the fluorescence intensity of the pressure-sensitive paint. The trend of the relative fluorescence intensity varying with time obtained by fitting is as Figure 1 shown in the DQM data.

[0044] The specific embodiments of the present invention have been described above. It should be noted that for those of ordinary skill in the art, without departing from the principle of the present invention, several improvements and refinements can be made. These improvements and refinements should also fall within the protection scope of the present invention.

Claims

1. An experimental data fitting method for the fluorescence intensity of pressure - sensitive paint, where the pressure - sensitive paint is coated on the surface of the test piece, characterized in that, The specific steps include: Step 1: Measure the oxygen diffusion coefficient D and the thickness H of the pressure-sensitive paint coating. Step 2: Based on the oxygen diffusion coefficient and the assumed fluorescence oxygen quenching rate coefficient K0, establish partial differential equations for the temporal and spatial variations of the oxygen molecule concentration and the excited-state fluorescence molecule concentration in the pressure-sensitive paint, respectively. Step 3: Establish partial differential equations for the temporal and spatial variations of the relative oxygen molecule concentration and the relative fluorescence intensity in the pressure-sensitive paint. Step 4: Experimentally measure the dynamic change data of the relative fluorescence intensity of the pressure-sensitive paint under a certain pressure input. Step 5: Based on the oxygen diffusion coefficient D and the assumed fluorescence oxygen quenching rate coefficient K0, and combined with the boundary conditions, solve the partial differential equations established in Step 3 to obtain the temporal and spatial distribution function of the relative fluorescence intensity. Step 6: Accumulate and normalize the temporal and spatial distribution function of the relative fluorescence intensity obtained in Step 5 in the thickness direction of the pressure-sensitive paint to obtain a mathematical model of the relative fluorescence intensity of the pressure-sensitive paint varying with time, that is, the kinetic model of the relative fluorescence intensity of the pressure-sensitive paint. Step 7: Fit the kinetic model of the relative fluorescence intensity of the pressure-sensitive paint with the test measurement results, calculate the actual fluorescence oxygen quenching rate coefficient K, substitute K for K0 into Steps 5 and 6 to obtain the final kinetic model of the relative fluorescence intensity of the pressure-sensitive paint, and complete the fitting of the experimental data of the fluorescence intensity of the pressure-sensitive paint.

2. The experimental data fitting method for the fluorescence intensity of pressure - sensitive paint according to claim 1, characterized in that, The assumed fluorescence oxygen quenching rate coefficient K0 in Step 2 should be a positive value.

3. The experimental data fitting method for the fluorescence intensity of pressure - sensitive paint according to claim 1, characterized in that, The specific form of the partial differential equations for the temporal and spatial variations of the oxygen molecule concentration and the excited-state fluorescence molecule concentration in the pressure-sensitive paint in Step 2 is as follows: where x is the distance from a certain point in the PSP coating to the surface of the PSP coating; t is the time; u e0 is the concentration of excited-state fluorescent molecules at the minimum pressure and the lowest oxygen concentration, that is, the concentration of fluorescent molecules in the PSP coating; u O0 is the oxygen concentration corresponding to the maximum pressure and the lowest concentration of excited-state fluorescent molecules; u O u(x, t) is the oxygen concentration at a distance x from the surface at time t in the PSP coating; u e u*(x, t) is the concentration of excited-state fluorescent molecules at a distance x from the surface at time t in the PSP coating; the right side of Equation (2) represents the concentration of excited-state fluorescent molecules when the oxygen concentration at a distance x from the surface of the PSP coating is u O (x, t) reaches equilibrium; the right side of Equation (2) represents the oxygen quenching rate of excited-state fluorescent molecules, that is, the concentration of excited-state fluorescent molecules at a distance x from the surface of the PSP coating at time t changes from u e (x, t) and reaches the equilibrium concentration after a certain time delay. The fluorescence oxygen quenching rate coefficient determines the length of the time delay.

4. The experimental data fitting method for the fluorescence intensity of pressure - sensitive paint according to claim 1, characterized in that, The specific form of the partial differential equations for the temporal and spatial variations of the relative oxygen molecule concentration and the relative fluorescence intensity in the pressure-sensitive paint in Step 3 is as follows: Among them, u RO (x,t) is the relative oxygen concentration at a distance x from the surface at time t in the PSP coating; RI(x,t) is the relative fluorescence intensity of the excited fluorescent molecules at a distance x from the surface at time t in the PSP coating; The derivation process of equations (3) and (4) is as follows: Substituting u O (x, t) = u O0 ·u RO (x, t) into equation (1) gives equation (3); substituting u O (x, t) = u O0 ·u RO (x, t) and u e (x, t) = ue 0·RI(x, t) into equation (2) gives equation (4); for the convenience of software solution, equations (3) and (4) are rewritten as: 。 5. The experimental data fitting method for the fluorescence intensity of pressure - sensitive paint according to claim 1, characterized in that, The oxygen content in the gas of the measurement environment in Step 4 is a fixed value The surface pressure of the PSP is P(t).

6. The experimental data fitting method for the fluorescence intensity of pressure - sensitive paint according to claim 1, characterized in that, In step 5, the boundary conditions are: at x = 0, i.e., on the surface of the PSP, at x = H, i.e., on the bottom surface of the PSP, where P0 is the maximum pressure on the surface of the PSP.

7. The experimental data fitting method for the fluorescence intensity of pressure - sensitive paint according to claim 1, characterized in that, The expression of the kinetic model of the relative fluorescence intensity of the pressure-sensitive paint in Step 6 is: 。

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