Understanding Pitot Tubes and Standard Atmosphere in AAE 251

Name Team Number Arya Patel R303 AAE 251: Introduction to Aerospace Design Homework 1—Pitot Tubes and The Standard Atmosphere Due Tuesday, September 5t, 11:59 PM on Gradescope Instructions This assignment has three sections. Start now, or you will run out of time! Section A: We will apply Bernoulli’s equation to understand how a Pitot tube works. You may find Anderson Chapter 4 very helpful, especially the examples. Section B: We will write a MATLAB code to reproduce the standard atmosphere. You will find the code useful in future Homework and your projects. Section C: We will apply relations between pressure, temperature, density, and altitude we derived in class to solve numerical problems. Show all the work. You will not receive full credit for a correct final answer if you don’t show your work. You must use the template provided. Anything thatyou want the graders to grade must . be within the boxes provided to you. There are multiple steps involved to complete your submission on Gradescope. You can follow the step-by-step guide posted on Brightspace. The steps are listed under the section “Gradescope” in Table of Contents. Select the relevant pages of your final answer for each question in your Gradescope submission. Scanned with CamScanner

SECTION A In this section, you will learn how pitot tubes, which are used to measure airspeed on aircraft, work. Step 0: Watching old TV (from a 16 mm film reel) Here is a clear, yet old video discussing static and dynamic pressure, and, right at the end, pitot tubes. Take a look and marvel at how TV has improved, and the laws of physics have remained true: WWW. ] 'watch?v=B X Step 1: Define Stagnation Pressure [10 points] When a stream of uniform velocity flows into a blunt body like an airfoil, the streamlines are as shown in Figure 1. Y Figure 1: Stagnation Point Some of the streamlines move over the airfoil, and some of them move under it. But one streamline goes to the leading edge of the airfoil and then stops. At this stagnation point, the velocity is zero. Let’s use the simplified Bernoulli equation we derived in class to calculate the pressure at this point, or the stagnation pressure. First, indicate your velocities and pressures on Figure 1: Designate a point upstream where the flow is undisturbed as having velocity and pressure (u,p;), and the stagnation point as (4, p3). a) What four assumptions do we have to make about the fluid to apply our simplified form of Bernoulli’s equation? [4] Scanned with CamScanner

Answer (Ala): 1. Fluid is incompressible 2. The density and velocity of the fluid is steady throughout 3. Fluid has inviscid flow: no internal friction or viscosity 4. Streamline flow b) Write down our simplified form of Bernoulli’s equation to relate these two pairs of values, and solve for the stagnation pressure: [2] Answer (Alb): A v e U %/sﬂ'man;&bﬂ fg‘”d UZ:O 2 Scanned with CamScanner

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

¢) Lookatyour equation. Mathematically, is p, larger or smaller than p,;? Why do you say that? [2] Answer (Alc): Mo\filew\c{hmﬂzg, P v aA)t%e)( Jhran P IW (’2 " eq/ua/ To #ht valwe o p, (:!ZAAA a ?W vaﬂua(ipu.l), d) Now consider the physical point of view. Why might the stagnation pressure be larger or smaller than p;? [2] Answer (A1d): p2 is larger that p1. This difference is called the dynamic pressure and it is positive in this situation because the fluid is going from positive velocity at location 1 to stopping at location 2. Step 2: Create a simple velocity measurement device [5 points] Your answer in Step 1(b) should contain a velocity term and therefore suggest a way of measuring airspeed—if we have the static pressure, stagnation pressure, and air density. For now, let's assume that we know our altitude and can therefore use the standard atmosphere model to find the air density. What about the static and stagnation pressures? Figure 2 shows the beginnings of a simple device to measure airspeed. This device has two ports, labelled “1” and “2”. -0 ] T Figure 2: A static port (1) and stagnation port (2) combination exposed to fluid flow (horizontal arrow) Scanned with CamScanner

Let’s start with Port 1. Note how the tube is at 90° to the fluid flow as indicated by the arrow. This placement ensures that the tube only sees the static pressure, not the dynamic pressure. This configuration is also the basic form of a piezometer. A piezometer (“piezo” from the Ancient Greek “to squeeze or press”) measures static pressure by relating the height of a fluid in a tube to the pressure that causes the fluid to rise against gravity. Port 2 faces directly into the flow. The port is small enough that the air flow stagnates. Like with the static port, we can relate the pressure the stagnation port sees to the height of fluid in the tube above it. The stagnation port and tube combination are called a “Pitot tube” in honor of its French inventor (Henri Pitot, 1695-1771). Assuming that we have such a system mounted on an aircraft, correctly oriented to gravity, shade in the fluid on Figure 2, and indicate its relative height in the two tubes. In which tube is the fluid higher, and why? [2] Answer (A2a): The fluid in the second tube is higher because in that tube, the stagnation pressure is greater. In an aircraft, there’s no guarantee that the tubes will always be parallel to gravity, or that gravity will be the only acceleration the fluid sees, so, rather than using fluid height, we use different types of pressure gauges that do not rely on fluid height. b) Complete your Pitot tube derivation by rewriting your equation to create an expression for u; in terms of the static pressure, stagnation pressure, and air density [3]: Scanned with CamScanner

R g AR A u.l 1 \02'93 P Scanned with CamScanner

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

SECTION B MATLAB Assignment [25 points] Write a MATLAB program to reproduce the standard atmosphere in Appendix B of Anderson. Recreate this data, and also calculate the speed of sound at each altitude. For better bookkeeping, include the following details at the start of all your code submissions for AAE251: %AAE 251 Fall 2023 %Homework <Enter Homework #> %Title of the code %Authors: <Enter your and your collaborators’ names> 120 100 80 4 = g 60 2 3 < 40 Stratosphere Z Z ZZ Gl 20 Nearly all manned aircraft fly below this altitude Troposphere \ 0 160 180 200 220 240 260 280 300 Temperature, T, Kelvin Figure 3: Standard Atmosphere temperature model (Brandt, 2004) Use equations from Anderson Ch 3.4. No credit will be given for simply entering the data from the tables. Program your code to produce correctly labeled plots for altitude vs. temperature, pressure, density, and the speed of sound similar to Figure 3 (up to 100,000 ft or 30.48 km). Paste your commented code in the appropriate answer box. Create a set of graphs showing how pressure, temperature, density, and the speed of sound vary with altitude, plotted in SI and English units. Make sure your graphs are properly labeled. Paste your graphs in the appropriate answer box. Scanned with CamScanner

MATLAB Code; % AAE 251 Fall 2023 % Homework <1> % Title of the code: Standard Atmosphere clear; clc; close all; % Constants PO_SI = 101325; % Pressure [Pa] rho0_SI = 1.225; % Density [kg/m*3] TO_SI = 288.16; % Temperature [K] g0_SI = 9.81; %Gravitational acceleration [m/s"2] R_SI = 287.05; % Specific gas constant for dry air [J/kg"K] gamma_SI = 1.4; % Heat capacity ratio altitude_m = 0:10:30480; altitude_ft = altitude_m * 3.28084; temperature_S| = zeros(length(altitude_m), 2); pressure_S| = zeros(length(altitude_m), 2); density_S| = zeros(length(altitude_m), 2); speed_of_sound_S| = zeros(length(altitude_m), 2); %Atmospheric Parameter Calculations for i = 1:length(altitude_m) h_m = altitude_m(i); % Temperature if h_m <= 11000 % Troposphere T_SI=T0_SI-0.0065 * h_m; elseif h_m <= 25000 % Tropopause to Stratosphere T_SI =216.65; else % Stratosphere T_SI=T0_SI +0.001 * (h_m - 25000); end % Pressure and density P_SI = P0O_SI * exp(-g0_S! * (h_m - altitude_m(1)) / (R_SI * T_SI)); rho_SI = rho0_SI * exp(-g0_S! * (h_m - altitude_m(1)) / (R_SI * T_SI)); % Speed of sound a_S| = sqrt(gamma_SI * R_S| * T_SI); % Store data Scanned with CamScanner

temperature_SI(i,:) = [h_m, T_SIJ; pressure_Sl(i,:) = [h_m, P_SI]; density_SI(i,:) = [h_m, rho_SlI]; speed_of_sound_SI(i,:) = [h_m, a_SI]; end % Plot results in Sl units figure(1) subplot(2, 2, 1) plot(temperature_SI(:,2), temperature_SI(:,1), 'LineWidth', 2) xlabel('Temperature, T, Kelvin', 'FontSize', 15) ylabel(‘Altitude, h, m', 'FontSize', 15) grid on subplot(2, 2, 2) plot(pressure_SI(:,2), pressure_SI(:,1), 'LineWidth', 2) xlabel('Pressure, P, Pa', 'FontSize', 15) ylabel(‘Altitude, h, m', 'FontSize', 15) grid on subplot(2, 2, 3) plot(density_SI(:,2), density_SI(:,1), 'LineWidth', 2) xlabel('Density, rho, kg/m*3', 'FontSize', 15) ylabel(‘Altitude, h, m', ‘FontSize', 15) grid on subplot(2, 2, 4) plot(speed_of_sound_SI(;,2), speed_of_sound_SI(:,1), 'LineWidth', 2) xlabel('Speed of Sound, a, m/s', 'FontSize', 15) ylabel('Altitude, h, m', ‘FontSize', 15) grid on xlim([280 360]) % Convert properties temperature_Eng = 1.8 * temperature_SI(:,2); pressure_Eng = 0.020885438 * pressure_SI(:,2); density_Eng = density_SI(:,2) / 515.3788184; speed_of_sound_Eng = 3.28084 * speed_of_sound_SI(:,2); % Altitude in feet altitude_Eng_ft = altitude_ft; % English units figure(2) subplot(2, 2, 1) plot(temperature_Eng, altitude_Eng_ft, 'LineWidth', 2) Scanned with CamScanner

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

xlabel('Temperature, T, °R’, 'FontSize', 15) ylabel('Altitude, h, ft', 'FontSize', 15) grid on subplot(2, 2, 2) plot(pressure_Eng, altitude_Eng_ft, ‘LineWidth', 2) xlabel('Pressure, P, Ib/ftA2', ‘FontSize', 15) ylabel(‘Altitude, h, ft', 'FontSize', 15) grid on subplot(2, 2, 3) plot(density_Eng, altitude_Eng_ft, 'LineWidth', 2) xlabel('Density, rho, slug/ftr3', 'FontSize', 15) ylabel(‘Altitude, h, ft', 'FontSize', 15) grid on subplot(2, 2, 4) plot(speed_of_sound_Eng, altitude_Eng_ft, 'LineWidth', 2) xlabel("Speed of Sound, a, ft/s', 'FontSize', 15) ylabel('Altitude, h, ft', 'FontSize', 15) grid on xlim([900 1200]) figure(1) sgtitle("Standard Atmosphere (S| Units)', 'FontSize', 18) figure(2) sgtitle('Standard Atmosphere (English Units)', ‘FontSize', 18) 10 Scanned with CamScanner

Graphs: Standard Atmosphere (S| Units) x10% x10% g° ‘ g3 = Bl ol ) ) he) © Sl 21 < 3 | < . 200 250 300 0 5 10 Temperature, T, Kelvin Pressure, P, Pa:10* 3 x10% 5 x10% S ‘ S L3 < [)] Q ko) he) 21 S = < g | < 5 0 0.5 1 15 280 300 320 340 360 Density, rho, kg/m® Speed of Sound, a, m/s 11 Scanned with CamScanner

Altitude, h, ft Altitude, h, ft = o [ 10 Standard Atmosphere (English Units) x10* x10* / [l | 400 450 500 550 Temperature, T, °R x10% Altitude, h, ft 10 (&) 0 0 500 1000 1500 2000 Pressure, P, Ib/ft? i x10% 0 0.5 1 15 2 Density, rho, slugfil‘81 Altitude, h, ft 0 900 1000 1100 1200 Speed of Sound, a, ft/s 12 Scanned with CamScanner

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

SECTION C Question 1 [10 points] An airplane flying at an altitude of 8 km has a malfunctioning Pitot tube with only the stagnation pressure probe providing data. Somehow the static pressure probe suddenly stopped providing information. Knowing that the measured stagnation pressure is 38.2 kPa with a 1% error, find the indicated (measured) airplane speed and its uncertainty in percent. You should use your standard atmosphere code and assume a 3% error for the density and pressure of air at 8 km altitude. 11 Scanned with CamScanner

A!!_Swer 1‘,_ p ol Skw=0.5uC6 Voir A Thia= 3.064x13 P A*a?r.cdim P W = 392 kVa 2%.2 \f\’,\ = 3.031;10”Pq x ); (0.5uLs) \12 26 $20.3U3 % wg £ \ Vo= 163,709 ws 12 Scanned with CamScanner

Question 2 [5 points] As you may have seen as you enter WALC, a common classroom location for AAE 251, this site used to be a power plant for our West Lafayette campus (see: ttps://www.purdue.e wsroom/purduetod eleases/2 4 /then-and- now-wilmeth.html if you are curious). That power plant most likely used an oil-based thermal management system to keep its components operating at the right temperatures. A simple way to check the velocity of the oil in that system is a pitot tube! Consider the setup shown below and that, under optimal conditions, the power plant operators wanted to keep the left-hand tube oil level at 48.2 cm (from the top to the oil pipe) and at 62.4 cm on the right-hand tube (again, from the top of the oil pipe). Consider the density of oil to be 873 kg/m? and Earth’s gravitational acceleration as 9.81 ms2. Figure 4. Simplified schematic of an oil system most likely used in the old Purdue power plant. Flow is from left to right. From this information, you are asked to calculate: 1. The pressure difference between points A and B as shown on Figure 4. (in Pascals) 2. The velocity of the oil at point B. (in m/s) The velocity of the oil at point A. (in m/s) 4. Keeping in mind that oil is quite viscous (compared to water for example), is the velocity of the oil at the extreme right of the pipe the same, smaller, or higher than at point A? Briefly justify why that is. w 13 Scanned with CamScanner

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

JauuedSWED YlIM pauuess 1 oy o vad My oo g oy oy . o of Cuoly o7 fops e g o g s/ 1,991 = ¥ o ezin + #n (l1ess) ¢ . ¢nncg L ;ndlzf - 9) ¢ o = W 2 w9z = Y4 - 94 = AV Y4 1o takh = o220 ) (S g)) (Q‘M‘C\ <ED) - q6d - v) A 20'hh&eg = 29 (1) (464 269) = u6d = ¥l 7 TSy

AAE_251_HW_1

Related Documents