EAE 130A Project 2 Report: Airplane Drag and Performance Analysis

To calculate the individual drag contributions of the external aircraft components, equivalent flat plate area method can be used where parasitic drag is: 𝐷𝑝𝑎𝑟𝑎𝑠𝑖𝑡𝑖𝑐 = 𝑞∞ ∗ ∑ 𝐶𝐷𝑗 𝑆𝑗 𝑛 𝑗=1 = 𝑞∞ ∗ 𝑓 = 𝑒𝑞𝑢𝑖𝑣𝑎𝑙𝑒𝑛𝑡 𝑓𝑙𝑎𝑡 𝑝𝑙𝑎𝑡𝑒 𝑎𝑟𝑒𝑎 Where Sj is the reference area and CD,j is the drag coefficient of the part. Certain components of the aircraft require detailed calculation of the flat plate area, which includes the wing, fuselage, and the empennage. For the wing, the zero-lift drag coefficient, 𝐶𝐷0,𝑤, is (Roskam, 148): 𝐶𝐷0,𝑤 = 𝑅𝑤𝑓𝑅𝐿𝑆𝐶𝑓𝑤 {1 + 𝐿′(𝑡 𝑐 ⁄ ) + 100(𝑡 𝑐 ⁄ )4} 𝑆𝑤𝑒𝑡𝑤 𝑆 [3] Rwf is the wing-fuselage interference factor and can be obtained from Figure 5.11 in Airplane Aerodynamics and Performance by Roskam. RLS is the lifting surface correction factor that is dependent on the sweep angle of the aircraft and can be obtained from Figure 5.12. Cfw is the turbulent flat plate friction coefficient of the wing and can be obtained from Figure 5.13 after calculating the wing Reynolds number, ReNw. L’ is the airfoil thickness parameter from Figure 5.15, (𝑡 𝑐 ⁄ ) is the thickness ratio of the wing at mean geometric chord. 𝑆𝑤𝑒𝑡𝑤 is the wetted area of the wing and 𝑆 is the wing reference area: 𝑆𝑤𝑒𝑡𝑤 = 𝐾𝑤𝑆𝑒𝑥𝑝𝑜𝑠𝑒𝑑𝑤 𝐾𝑤 = 1.9767 + 0.5333 (𝑡 𝑐) 𝑓𝑜𝑟 𝑡 ≥ 0.05 Same steps can be taken to determine the zero-lift drag coefficient for the horizontal and the vertical tail. For the fuselage, the zero-lift coefficient can be found by another formula: 𝐶𝐷𝑜𝑓 = 𝑅𝑤𝑓𝐶𝑓𝑓 { 1 + 60 ( 𝑙𝑓 𝑑𝑓 ) 3 + 0.0025 ( 𝑙𝑓 𝑑𝑓 ) } 𝑆𝑤𝑒𝑡𝑓 𝑆 9 In this formula, 𝐶𝑓𝑓 is the turbulent flat plate friction coefficient from Figure 5.13, (𝑙𝑓 𝑑𝑓 ) is the fuselage fineness ratio from Table 5.1, and 𝑆𝑤𝑒𝑡𝑓 is the wetted area of the fuselage. To use Table 5.1, the values for Cessna 185 is used as a substitute for 152, due to lack of data for the latter. Using engineering judgement, the fuselage of the 185 is determined to be most similar to the 152. After adding up the equivalent flat plate areas of all the components, 𝐶𝐷0 can be found by dividing the f value by the total wetted area of the aircraft. Once all the 𝐶𝐷0 values are combined, drag force can be found by combining 𝐶𝐷0 and 𝐶𝐷𝑖 along with the dynamic pressure and aircraft reference area. After calculating the total drag force, the power required for various aircraft speed can be found using: 𝑃𝑟𝑒𝑞 = 𝐷𝑉∞ In addition to the power required, the power available can be calculated using the given performance parameters of Cessna 152: 𝑃𝑎𝑣 = 𝜂𝑝𝐵𝐻𝑃 These two power parameters can be plotted against aircraft speed, which yields the desired parameters to be investigated in this report: maximum rate of climb and maximum speed. Maximum speed is determined by the intersection of the two curves. Maximum rate of climb is determined by the following equation: 𝑅𝐶𝑚𝑎𝑥 = 𝑀𝑎𝑥𝑖𝑚𝑢𝑚 𝐸𝑥𝑐𝑒𝑠𝑠 𝑃𝑜𝑤𝑒𝑟 𝑊 = (𝑃𝑎𝑣−𝑃𝑟𝑒𝑞)𝑚𝑎𝑥 𝑊 Using this methodology, a detailed drag analysis can be performed on the selected aircraft. 10 Results For the analysis, some initial parameters were set to easily calculate the drag coefficients of the individual aircraft components: 1978 Cessna Model 152 and Freestream Parameters Airspeed at Cruise: 107 kts = 180.6 ft/s Air Dynamic Viscosity at SL: 3.737 * 10-7 lb*sec/ft2 Air Density at SL: 2.377 * 10-3 slugs/ft3 Re/c: 1.149*106 The summary of the individual drag contributions of the aircraft external components is given in Table 2 below. The detailed calculations are included in Appendix A. Component Area [ft2] CD,0 f = CD * S [ft2] Reference Wings 326.51 0.0088 1.4121 Roskam, J. Fuselage 176.00 0.0098 1.7255 Roskam, J. Nose Cone 0.7162 0.4672 0.3346 White, Frank M. Nose Wheel 0.4219 0.6267 0.2588 White, Frank M. Step-up Handle (x2) 0.0035 0.7300 0.0025 White, Frank M. Exhaust 0.0174 0.6800 0.0118 White, Frank M. Nose Landing Gear Cylinder 0.0208 0.7217 0.0150 White, Frank M. Nose Landing Gear Wheel Holder 0.0069 1.1950 0.0083 White, Frank M. Wing Strut (x2) 1.9722 0.1225 0.2416 White, Frank M. Step (x2) 0.0039 1.4600 0.0057 White, Frank M. Landing Gear Strut (x2) 0.6667 0.1175 0.0783 White, Frank M. Landing Gear Wheel (x2) 1.2917 0.6155 0.2588 White, Frank M. Pitot Tube 0.0137 0.7300 0.0100 White, Frank M. Table 1. [4] 1978 Cessna Model 152 and Freestream Parameters. Table 2. 1978 Cessna Model 152 Aircraft Component Drag Contributions. 11 Using this chart, it is seen that the total flat plate area of the aircraft is 5.9432. Combined with the total wet area of the aircraft, 𝐶𝐷0 is found by: 𝐶𝐷0 = 𝑓 𝑆𝑤𝑒𝑡,𝑡𝑜𝑡𝑎𝑙 = 5.9432 𝑓𝑡2 599.01𝑓𝑡2 = 0.0099217 Then, induced drag can be calculated by equating lift to the weight of the aircraft, which is 1670 lbs max. First, the lift coefficient for the wing and fuselage is calculated: 𝐶𝐿𝑤 = 1670 1 2 (2.377 ∗ 10−3) (180.6𝑓𝑡 𝑠 ) 2 ∗ 160 = 0.2693 𝐶𝐿𝑓 = 1670 1 2 (2.377 ∗ 10−3) (180.6𝑓𝑡 𝑠 ) 2 ∗ 176 = 0.2448 𝐶𝐿 = (𝐶𝐿𝑤 𝑆𝑤 + 𝐶𝐿𝑓 𝑆𝑓) 1 (𝑆𝑡𝑜𝑡𝑎𝑙) = 0.2564 Then, the induced drag can be calculated using the following: Landing Strut Step (x2) 0.0104 1.2067 0.0126 White, Frank M. Fuel Sump 0.0069 0.7640 0.0053 White, Frank M. Horizontal Tail 61.22 0.0097 0.2919 Roskam, J. Static Port 0.0052 1.1900 0.0062 White, Frank M. Door Lock (x2) 0.0278 1.1850 0.0329 White, Frank M. Vertical Tail 30.00 0.0097 0.1427 Roskam, J. Beacon 0.0347 0.6400 0.0222 White, Frank M. Wing Lights (x2) 0.0208 0.6533 0.0136 White, Frank M. ADS-B 0.0347 0.6120 0.0213 White, Frank M. Subtotal f1 = 4.9117 Cooling drag 10% of f1 = 0.4912 van Dam, C.P. Interference drag 10% of f1 = 0.5403 Total ftot = 5.9432 12 𝐶𝐷𝑖 = 𝐶𝐿 2 𝜋 ∗ 𝐴𝑅 ∗ 𝑒 = 0.25642 𝜋 (33.332 160 ) (0.7) = 0.004308 In this calculation, the Oswald’s efficiency factor is assumed to be e = 0.7, as it is in the drag analysis notes (van Dam, 6). Now, the total drag coefficient can be calculated by adding 𝐶𝐷𝑖 and 𝐶𝐷0: 𝐶𝐷 = 𝐶𝐷𝑖 + 𝐶𝐷0 = 0.004308 + 0.0099217 = 0.0142285 As drag is a function of airspeed, the power required, 𝑃𝑟𝑒𝑞, can be found as a function of 𝑉∞: 𝑃𝑟𝑒𝑞 = 𝐷𝑉∞ = 𝐶𝐷 ∗ 1 2 ∗ 𝜌 ∗ 𝑉∞ 3 ∗ 𝑆 = (1.356/(7.457 ∗ 102)(0.0142285) (1 2) (2.377 ∗ 10−3)(599.01)𝑉∞ 3 In this formula, the highlighted part is the conversion factor from 𝑠𝑙𝑢𝑔𝑠 ∗ 𝑓𝑡2/𝑠3 to hp (Torenbeek, 515). In addition, the power available for the aircraft is dependent on the engine and the propeller. For this analysis, the propeller efficiency, 𝜂𝑝, is estimated to be 75%. Then, 𝑃𝑎𝑣 = 𝜂𝑝𝐵𝐻𝑃 = (0.75)(110) = 82.5 ℎ𝑝 Using these two power values, a power curve can be plotted: 13 Looking at Figure 3, it can be seen that the maximum speed of the aircraft is around 97 knots. Also, the maximum excess power is the maximum difference between these two curves, which is 72.69 hp. So, the maximum rate of climb is: 𝑅𝐶𝑚𝑎𝑥 = 𝑀𝑎𝑥𝑖𝑚𝑢𝑚 𝐸𝑥𝑐𝑒𝑠𝑠 𝑃𝑜𝑤𝑒𝑟 𝑊 = 550 ∗ 72.69 1670 = 23.94 𝑓𝑡 𝑠 = 1436 𝑓𝑡/𝑚𝑖𝑛 These values will be further discussed in the next section. 40 50 60 70 80 90 100 110 Airspeed [knots] 0 20 40 60 80 100 120 Horsepower [hp] Power Diagram Power Required Power Available Figure 3. Calculated Power Curve of Cessna 152. Maximum Speed Maximum Excess Power 14 Discussion In the project, the drag analysis was performed on a Cessna 152 in order to compare the performance values supplied by the manufacturer to the estimated performance values. Observing the Pilot’s Operating Manual, it is stated that the maximum speed of the aircraft is 110 knots while the rate of climb is rated as 715 ft/min. As drag force on an aircraft ultimately determines the limits of the aircraft in terms of vertical and horizontal speed where the power available from the propulsion systems set the limit on the power that can be used to reach to a higher speed. Therefore, a detailed drag analysis with consideration of all the significant external aircraft components is key to evaluating the performance statistics of an airplane. This project is also important for the future RFP work that requires evaluating the team’s own design of a short range high capacity transport aircraft. The results found from the drag analysis show that the maximum speed of Cessna 152 is 97 knots with a maximum rate of climb of 1436 ft/min. The speed estimated from drag analysis shows a lower value for maximum speed, which is 13% slower. This result makes sense in the way where the drag analysis performed was conservative as a result of the assumptions made during the drag coefficient and equivalent flat plate area calculations. The areas used for the drag calculations simplified the protrusions on the aircraft as simple shapes, such as cylinders, flat plates, and 3D ellipses, instead of streamlined shapes that significantly reduce the drag. Because measuring the exact wetted area of these components were not viable by the means of using a measuring tape, a caliper, and a ruler, it was decided that the use of simple shapes would help the processing time of the raw data significantly. In addition, the use of the simple shapes enabled the author to reference to the textbook, Fluid Mechanics by Frank M. White, which includes drag coefficient charts for certain defined shapes. In addition, the fuselage areas, wetted and reference, were not included in Table 5.1 in Airplane Aerodynamics and Performance, by Jan Roskam. Therefore, the areas were used were in fact of a Cessna 185, which was considered to be the most similar airplane structure to a Cessna 152. For future work, these assumptions can be replaced by more precise measurements and modeling of the aircraft that would yield more accurate results, bringing the overall drag of the airplane down, and increasing the maximum speed of the aircraft. The rate of climb found through the analysis is 1436 ft/min, which is 100% higher than the rated rate of climb. This value is significantly higher, therefore indicates to an error in calculation of the 15 maximum excess power. While calculating the induced drag of the aircraft, the induced drag coefficient was found for cruise parameters, including the speed of the aircraft. In a more accurate calculation, the induced drag would depend on the speed of the aircraft, which would yield a different power available curve. If the current model for calculations is substituted by a varying induced drag coefficient, it is expected that the maximum excess power would occur somewhere in between the stall velocity and the maximum velocity of the aircraft. This way, the rate of climb can be calculated more accurately and account for variations in freestream parameters in the power curve functions. Conclusions The drag analysis and performance characteristics of the Cessna 152, which was the main focus of the project, showed the effect of assumptions made during the modeling of the aircraft. The key ideas used in the analysis included utilizing the power curves for an aircraft, which indicate the maximum available power and the required power for various speeds. From the two curves plotted, it was seen that the maximum speed of the aircraft was 97 knots while the operating manual indicated 110 knots. These two values were considered close and the discrepancy was attributed to the shape approximation of the external aircraft components in order to calculate the drag coefficients. On the other hand, the rate of climb of the aircraft was found via calculating the maximum excess power of the aircraft and dividing the value by the maximum weight of the aircraft. The value found was 1436 ft/min, which was significantly different than the listed value in Pilot’s Operating Manual, which was 715 ft/min. The difference between these two values indicated a major source of error in the calculation of the minimum power required for equilibrium flight across various values of airspeed. The suspected error is due to the calculation of lift-induced drag, which was calculated to be a single value assuming cruise conditions, while it should have been a function of airspeed. For future drag analyses for this aircraft, it is recommended that the aircraft is modeled in a more detailed fashion where the actual shapes of the aircraft is incorporated for drag coefficient calculations. One possible method would include using CFD in order to capture a more accurate drag force estimation. 16 References [1] [4] Pilot’s Operating Handbook Cessna 152. 2nd ed. Cessna Aircraft Company. Wichita, KS, USA. 1977. Ch. 1-2. [2] van Dam, C.P. EAE 130A – Aircraft Performance & Design - Aircraft Drag Buildup and Analysis. 20 January 2020. [3] Roskam, J. and Lan, C.T. Airplane Aerodynamics and Performance. DARcorporation. 1997. [5] White, Frank M. Fluid Mechanics. McGraw-Hill Education, 2016. Torenbeek, Egbert, and H. Wittenberg. Flight Physics: Essentials of Aeronautical Disciplines and Technology, with Historical Notes. Springer, 2009. 17 Appendices Appendix A1. Example Aircraft Component Drag Calculations – Wings 𝐶𝐷0,𝑤 = 𝑅𝑤𝑓𝑅𝐿𝑆𝐶𝑓𝑤 {1 + 𝐿′(𝑡 𝑐 ⁄ ) + 100(𝑡 𝑐 ⁄ )4} 𝑆𝑤𝑒𝑡𝑤 𝑆 𝑅𝑒𝑁𝑓 = 𝜌𝑉∞𝑙𝑓 𝜇 = 2.77 ∗ 107 𝑀𝑎 = 𝑎 𝑉 = 0.16 Mach number used for the charts is Ma = 0.25, due to lack of data for Ma<0.25. 𝑅𝑤𝑓 = 1.06 (𝐹𝑟𝑜𝑚 𝐹𝑖𝑔𝑢𝑟𝑒 5.11) 𝑐𝑜𝑠Λ = 1 → 𝑅𝐿𝑆 = 1.08(𝐹𝑟𝑜𝑚 𝐹𝑖𝑔𝑢𝑟𝑒 5.12) 𝑆𝑒𝑥𝑝𝑜𝑠𝑒𝑑𝑤 = 160𝑓𝑡2 𝑆𝑤𝑒𝑡𝑤 = 𝐾𝑤𝑆𝑒𝑥𝑝𝑜𝑠𝑒𝑑𝑤 𝐾𝑤 = 1.9767 + 0.5333 (𝑡 𝑐) = 2.04 𝑓𝑜𝑟 𝑡 ≥ 0.05 𝑆𝑤𝑒𝑡𝑤 = 𝐾𝑤𝑆𝑒𝑥𝑝𝑜𝑠𝑒𝑑𝑤 = (2.04)(160) = 326.51𝑓𝑡2 𝑐̅𝑤𝑒 = 𝑤𝑖𝑛𝑔 𝑎𝑟𝑒𝑎 𝑤𝑖𝑛𝑔𝑠𝑝𝑎𝑛 = ( 160 29.88) = 5.36𝑓𝑡 𝑅𝑒𝑁𝑤 = 𝜌𝑉∞𝑐̅𝑤𝑒 𝜇 = 6.15 ∗ 106 𝐶𝑓𝑤 = 0.455 (log10 𝑅𝑁)2.58(1 + 0.144𝑀𝑎2)0.58 = 3.24 ∗ 10−3 (𝐹𝑟𝑜𝑚 𝐹𝑖𝑔𝑢𝑟𝑒 5.13) NACA 2412 Wings → (𝑡 𝑐) max @ 0.3𝑐 → 𝐿′ = 1.2 (𝐹𝑟𝑜𝑚 𝐹𝑖𝑔𝑢𝑟𝑒 5.15) (𝑡 𝑐) = 0.12 Using all these coefficients, 18 𝑓𝑤𝑖𝑛𝑔 = 𝐶𝐷0,𝑤𝑆 = 1.4121 Wing drag due to compressibility is negligible due to a low Mach Number. Same calculations can be applied to the vertical and the horizontal wing by changing the airfoil used in the equations to NACA 0012. Appendix A2. Example Aircraft Component Drag Calculations – Fuselage 𝐶𝐷𝑜𝑓 = 𝑅𝑤𝑓𝐶𝑓𝑓 { 1 + 60 ( 𝑙𝑓 𝑑𝑓 ) 3 + 0.0025 ( 𝑙𝑓 𝑑𝑓 ) } 𝑆𝑤𝑒𝑡𝑓 𝑆 𝑅𝑤𝑓 = 1.06 (𝐹𝑟𝑜𝑚 𝐹𝑖𝑔𝑢𝑟𝑒 5.11, 𝑠𝑎𝑚𝑒 𝑎𝑠 𝑡ℎ𝑒 𝑤𝑖𝑛𝑔) 𝑅𝑒𝑁𝑓 = 𝜌𝑉∞𝑙𝑓 𝜇 = 2.77 ∗ 107 𝐶𝑓𝑓 = 0.455 (log10 𝑅𝑁)2.58(1 + 0.144𝑀𝑎2)0.58 = 2.56 ∗ 10−3 (𝐹𝑟𝑜𝑚 𝐹𝑖𝑔𝑢𝑟𝑒 5.13) To use Table 5.1, the values for Cessna 185 is used as a substitute for 152, due to lack of data for the latter. Using engineering judgement, the fuselage of the 185 is determined to be most similar to the 152. Therefore, 𝑙𝑓 𝑑𝑓 = 5.15 𝑆 = 176 𝑓𝑡2 𝑆𝑤𝑒𝑡𝑓 = 292𝑓𝑡2 Using the found coefficients, 𝐶𝐷𝑜𝑓 = 0.0065 50% is added to the drag coefficient of the fuselage to account for the canopy, as it was done in the Drag Analysis Notes (van Dam, 6) Then, the drag coefficient is: 19 𝐶𝐷𝑜𝑓 = 0.0098 Appendix A3. Example Aircraft Component Drag Calculations – Nose Cone Referencing to Frank M. White’s Fluid Mechanics 8th Edition Textbook, Table 7.3, pg. 483: The Reynolds number is calculated to check if the CD can be approximated from this graph, which states that, to use this table 𝑅𝑒 ≥ 104. Using the Re/c parameter in Table 1, Reynolds number is calculated to be Re = 1.097 * 106. Drag coefficient for the nose cone of Cessna 152 is determined to be between 0.40 and 0.55, which can be interpolated and found to be CD = 0.4672. For the other components, same method is applied by looking up the CD charts of various shapes that the components were approximated. θ 1 ft. 0.95 ft. θ = 25.52° Figure 4. Nose Cone Drawing of Cessna 152. Figure 5. Drag Coefficient Chart for a Cone. [5]