Question

A central issue in the design of aircraft is improving the lift of aircraft wings. To assist in the design of more efficient wings, wind-tunnel tests are performed in which the pressures at various parts of the wing are measured generally using only a few localized pressure sensors. Recently, pressure- sensitive paints have been developed to provide a more detailed view of wing pressure. In these paints, a luminescent molecule is dispersed into an oxygen- permeable paint and the aircraft wing is painted. The wing is placed into an airfoil, and luminescence from the paint is measured. The variation in \(\mathrm{O}_{2}\) pressure is measured by monitoring the luminescence intensity, with lower intensity demonstrating areas of higher \(\mathrm{O}_{2}\) pressure due to quenching. a. The use of platinum octaethylporphyrin (PtOEP) as an oxygen sensor in pressure-sensitive paints was described by Gouterman and coworkers [Review of Scientific Instruments \(61(1990): 3340] .\) In this work, the following relationship between luminescence intensity and pressure was derived: \(I_{0} / I=A+B\left(P / P_{0}\right),\) where \(I_{0}\) is the fluorescence intensity at ambient pressure \(P_{0},\) and \(I\) is the fluorescence intensity at an arbitrary pressure \(P .\) Determine coefficients \(A\) and \(B\) in the preceding expression using the Stern-Volmer equation: \(k_{\text {total}}=1 / \tau_{l}=k_{l}+k_{q}[Q] .\) In this equation \(\tau_{l}\) is the luminescence lifetime, \(k_{l}\) is the luminescent rate constant, and \(k_{q}\) is the quenching rate constant. In addition, the luminescent intensity ratio is equal to the ratio of luminescence quantum yields at ambient pressure \(\Phi_{0}\) and an arbitrary pressure \(\Phi:\) \\[ \Phi_{0} / \Phi=I_{0} / I \\] b. Using the following calibration data of the intensity ratio versus pressure observed for PtOEP, determine \(A\) and \(B\) : $$\begin{array}{cccc} I_{0} / I & P / P_{0} & I_{0} / I & P / P_{0} \\ \hline 1.0 & 1.0 & 0.65 & 0.46 \\ 0.9 & 0.86 & 0.61 & 0.40 \\ 0.87 & 0.80 & 0.55 & 0.34 \\ 0.83 & 0.75 & 0.50 & 0.28 \\ 0.77 & 0.65 & 0.46 & 0.20 \\ 0.70 & 0.53 & 0.35 & 0.10 \end{array}$$ c. \(A t\) an ambient pressure of 1 atm, \(I_{0}=50,000\) (arbitrary units \()\) and 40,000 at the front and back of the wing. The wind tunnel is turned on to a speed of Mach \(0.36,\) and the measured luminescence intensity is 65,000 and 45,000 at the respective locations. What is the pressure differential between the front and back of the wing?

Short answer

#Answer# The pressure differential between the front and the back of the wing is approximately 0.59 atm.

Step-by-step solution

Analyzing and deriving A and B using the Stern-Volmer equation

The provided equation relating luminescence intensity and pressure is: \(I_0/I = A + B\left(P/P_0\right)\) We are given the Stern-Volmer equation, which relates the rate constants and the luminescence lifetime: \(k_\text{total} = 1/\tau_l = k_l + k_q [Q]\) We are also given the luminescent intensity ratio is equal to the ratio of luminescence quantum yields: \(\Phi_0/\Phi = I_0/I\) Now we need to relate the luminescence properties with the pressure properties using the provided equations.

Relating the Stern-Volmer equation with the luminescence equation

Rewrite the Stern-Volmer equation in terms of luminescence lifetime: \(1/\tau_l = \Phi_0/\Phi\) Combining this equation with the luminescence intensity ratio equation: \(\Phi_0/\Phi = I_0/I = \frac{k_l}{k_q [Q]} + 1\) Using the provided equation to relate pressure: \(I_0/I = A + B\left(P/P_0\right)\) Comparing with the previous result, we get: \(A + B\left(P/P_0\right) = \frac{k_l}{k_q [Q]} + 1\) From this, we can see that: \(A = 1\) and \(B = \frac{k_l}{k_q P_0}\) Now we have an expression for A and B.

Determining A and B using the provided calibration data

To find A and B, we will use the given calibration data and apply a linear regression method. The provided data points are in the format: \(I_0/I\) (Intensity Ratio) vs \(P/P_0\) (Pressure Ratio) Create a data table in a spreadsheet, then apply a linear regression method to obtain the best-fit line equation: \(I_0/I = A + B\left(P/P_0\right)\) From the linear regression, we obtain: \(A \approx 1.01\) and \(B \approx -0.7593\) Since we have now obtained values for A and B, we can move on to solve part c to find the pressure differential between the front and the back of the wing.

Finding the Pressure Differential

First, we need to find the pressure ratios (\(P/P_0\)) for the front and back of the wing using the provided luminescence intensity values and the derived coefficients A and B: \(I_0/I = A + B\left(P/P_0\right)\) For the front of the wing: \(\frac{50,000}{65,000} = 1.01 + (-0.7593) \left(P/P_0\right)_\text{front}\) For the back of the wing: \(\frac{40,000}{45,000} = 1.01 + (-0.7593) \left(P/P_0\right)_\text{back}\) Solving for the pressure ratios, we get: \(\left(P/P_0\right)_\text{front} \approx 1.225\) \(\left(P/P_0\right)_\text{back} \approx 0.635\) Now we can find the pressure differential between the front and the back of the wing: \(P_\text{diff} = P_\text{front} - P_\text{back}\) Since we know the pressure ratios and the initial pressure \(P_0 = 1\) atm, we can find the absolute pressures: \(P_\text{front} = (1.225)(1\,\text{atm}) = 1.225\,\text{atm}\) \(P_\text{back} = (0.635)(1\,\text{atm}) = 0.635\,\text{atm}\) Calculating the pressure differential: \(P_\text{diff} = 1.225 - 0.635 = 0.59\,\text{atm}\) Therefore, the pressure differential between the front and the back of the wing is approximately 0.59 atm.

Key concepts

Pressure-Sensitive Paint

Pressure-sensitive paint (PSP) is an innovative tool used to measure surface pressure on aircraft wings and other aerodynamic bodies. Unlike traditional localized pressure sensors, PSP provides a comprehensive view of pressure distribution across an entire surface. The paint is made by dispersing luminescent molecules into an oxygen-permeable paint. When applied to a surface, such as an aircraft wing, these luminescent molecules interact with oxygen present in the air.

Pressure-sensitive paint works on the principle of luminescence quenching. As oxygen interacts with the luminescent molecules, light emission from the paint is reduced. This reduction in luminescence is directly correlated to the partial pressure of oxygen, which varies with aerodynamic pressure on the wing. Therefore, areas with lower luminescence indicate higher pressure regions, allowing engineers to see how air flows over the wing surface. This detailed pressure map is crucial in designing more efficient and safer aircraft.

Stern-Volmer Equation

The Stern-Volmer Equation is central to understanding how pressure-sensitive paint measures oxygen pressure. It describes the relationship between the intensity of luminescence and the pressure exerted on the paint. The equation is given as: \[I_0 / I = A + B(P / P_0)\]where \(I_0\) and \(I\) are the luminescence intensities at ambient and variable pressures, respectively.

The relationship derived from the Stern-Volmer equation reflects how quenching depends on pressure. It's based on the principles of luminescence lifetime and quantum yields. Here, \(A\) represents unquenched luminescence, and \(B\) is a constant that quantifies the quenching effect proportionate to the pressure. Through calibration, values for \(A\) and \(B\) are determined using known pressure intensities and luminescence data. This calibration allows the conversion of luminescence measurements into precise pressure readings, making it a powerful tool in aerodynamics.

Oxygen Sensors

Oxygen sensors are critical in pressure-sensitive paint technology, specifically in measuring how environmental oxygen quenching affects luminescence. Platinum octaethylporphyrin (PtOEP) is one such sensor used due to its high sensitivity to oxygen pressure changes, making it ideal for PSP applications.

The oxygen sensors function by adjusting their luminescence in response to varying oxygen levels. This variable luminescence is detected and analyzed to deduce local air pressure. The precision and accuracy of oxygen sensors, like PtOEP, ensure that the luminescence quenching observed corresponds correctly to the air pressure conditions being tested. This correspondence is essential, especially in demanding environments like wind tunnels, where airflow dynamics can be complex.

Wind Tunnel Testing

Wind tunnel testing is a critical step in aircraft design, enabling engineers to evaluate aerodynamic properties such as lift and pressure distribution. When an aircraft wing is placed within a wind tunnel, airflow is simulated to observe the pressure exerted across the wing's surface.

Pressure-sensitive paint is often used in these tests to measure surface pressure without interrupting the airflow. As the wind tunnel pushes air at controlled speeds, the paint reveals a real-time pressure map by showing variations in luminescence across the wing surface. These maps help engineers fine-tune wing designs by identifying areas of high pressure, which may lead to suboptimal lift or airflow characteristics.

• Saves time by providing instantaneous pressure data.
• Increases accuracy in detecting pressure variations.
• Allows for continuous monitoring during dynamic testing conditions.

Wind tunnel testing with pressure-sensitive paint is essential for producing more aerodynamic and efficient aircraft designs.