HW5_soln

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Jan 9, 2024

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Name: HW5 Page 1 of 11 ENAE283 Homework #5 Due Saturday, 7/1/22 by 11:59PM For all homework assignments in this course, you are required to submit fully- explained solutions, indicating the sources for any numbers and equations used. Boxed areas are provided for your final answers. If you cannot use this document directly as your worksheets, please work neatly on your own paper. Include a header on each page like the ones shown, write the problem statement at the top and box your final answer at the lower right corner. Pay attention to appropriate significant figures in final answers.
Name: HW5 Page 2 of 11 (1-11) You have a choice for this portion of the HW: you can study the performance characteristics of a propeller driven airplane (Beechcraft Bonanza) or a twin jet attack aircraft (Fairchild Republic A-10). Pick one and then follow through the whole series of problems relating to that airplane. Each of these problems is most easily accomplished by employing a spreadsheet tool like Excel, since all of the calculations are repeated at different velocities and altitudes. Your Excel spreadsheet or MATLAB arrays should contain the following calculations. 𝑉𝑉 𝑓𝑓𝑓𝑓 𝑠𝑠 𝐶𝐶 𝐿𝐿 = 𝑊𝑊 𝑞𝑞 𝑆𝑆 𝐶𝐶 𝐷𝐷 = 𝐶𝐶 𝐷𝐷 , 0 + 𝐶𝐶 𝐷𝐷 , 𝑖𝑖 = 𝐶𝐶 𝐷𝐷 , 0 + 𝐶𝐶 𝐿𝐿 2 / 𝜋𝜋𝜋𝜋𝜋𝜋𝜋𝜋 𝐶𝐶 𝐿𝐿 / 𝐶𝐶 𝐷𝐷 𝑇𝑇 𝑅𝑅 = 𝑊𝑊 𝐶𝐶 𝐿𝐿 / 𝐶𝐶 𝐷𝐷 [ 𝑙𝑙𝑙𝑙 ] 𝑃𝑃 𝑅𝑅 = 𝑇𝑇 𝑅𝑅 𝑉𝑉 �𝑓𝑓𝑓𝑓 𝑙𝑙𝑙𝑙 𝑠𝑠 𝑃𝑃 𝐴𝐴 �𝑓𝑓𝑓𝑓 𝑙𝑙𝑙𝑙 𝑠𝑠 𝑇𝑇 𝐴𝐴 [ 𝑙𝑙𝑙𝑙 ] 𝜋𝜋𝐶𝐶 = 𝑃𝑃 𝐴𝐴 − 𝑃𝑃 𝑅𝑅 𝑊𝑊 𝑓𝑓𝑓𝑓 𝑠𝑠 The only variable that changes is the density between Sea Level and altitude. Besides changing the thrust/power required curves via the 𝑞𝑞 term, this also affects the power available from the engine which is assumed to be proportional to density, so 𝑃𝑃 𝐴𝐴 , 𝑆𝑆𝐿𝐿 𝜌𝜌 𝑆𝑆𝐿𝐿 = 𝑃𝑃 𝐴𝐴 , 𝑎𝑎𝑎𝑎𝑎𝑎 𝜌𝜌 𝑎𝑎𝑎𝑎𝑎𝑎 𝑃𝑃 𝐴𝐴 , 𝑎𝑎𝑎𝑎𝑎𝑎 = 𝑃𝑃 𝐴𝐴 , 𝑆𝑆𝐿𝐿 𝜌𝜌 𝑎𝑎𝑎𝑎𝑎𝑎 𝜌𝜌 𝑆𝑆𝐿𝐿 For PR at altitude, you can either rerun the above calculations with a new density value, or multiply both 𝑉𝑉 and 𝑃𝑃𝜋𝜋 𝑆𝑆𝐿𝐿 by the altitude correction factor �𝜌𝜌 𝑆𝑆𝐿𝐿 𝜌𝜌 𝑎𝑎𝑎𝑎𝑎𝑎 . Note: If you don’t multiply 𝑉𝑉 by the correction factor, your plot will be incorrect. See below.
Name: HW5 Page 3 of 11 Can Prop Plane - Beechcraft Bonanza V-tailed, Single Engine Private Plane The characteristics of this airplane are as follows: wing area = 180 ft 2 ; aspect ratio = 6.2; Oswald efficiency factor = 0.92; weight = 3000 lb; zero-lift drag coefficient = 0.027; single piston engine power = 350 hp (at sea level); propeller efficiency = 0.81; specific fuel consumption = 0.42 lb fuel/hp.hr; fuel capacity = 45 gallons; maximum gross weight = 3400 lb; C Lmax = 1.2 on take-off; wings are 4 ft off the ground during take-off roll; C Lmax = 1.8 on landing (with flaps). 1. Find a picture of this airplane online, copy and paste it into your homework and cite. 2. Plot the power required versus velocity curve at sea level.
Name: HW5 Page 4 of 11 3. Determine the maximum velocity at sea level. From PR=PA max , 𝑉𝑉 𝑚𝑚𝑎𝑎𝑚𝑚 = 293 𝑓𝑓𝑓𝑓 / 𝑠𝑠 4. Draw the power required versus velocity curve at 14,000 ft altitude. See above 5. Determine the maximum velocity at 14,000 ft altitude. 𝑉𝑉 𝑚𝑚𝑎𝑎𝑚𝑚 = 283 𝑓𝑓𝑓𝑓 / 𝑠𝑠 6. Calculate the maximum rate of climb at sea level and at 14,000 ft altitude. 𝜋𝜋 . 𝐶𝐶 . 𝑚𝑚𝑎𝑎𝑚𝑚 = ( 𝑃𝑃 𝐴𝐴 − 𝑃𝑃 𝑅𝑅 ) 𝑚𝑚𝑎𝑎𝑚𝑚 / 𝑊𝑊 𝜋𝜋 . 𝐶𝐶 . 𝑆𝑆𝐿𝐿 = 42.3 𝑓𝑓𝑓𝑓 𝑠𝑠 = 2,538 𝑓𝑓𝑓𝑓 𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑚𝑚𝑎𝑎 𝜋𝜋 . 𝐶𝐶 . 14𝑘𝑘 = 21.8 𝑓𝑓𝑓𝑓 𝑠𝑠 = 1,308 𝑓𝑓𝑓𝑓 / 𝑚𝑚𝑚𝑚𝑚𝑚 7. Estimate the absolute ceiling of this airplane (assuming rate of climb varies linearly with altitude). From the R.C. calculations above, 𝜋𝜋 . 𝐶𝐶 . ( ) = 42 . 3−21 . 8 0−14 , 000 + 42.3 [ 𝑓𝑓𝑎𝑎 𝑠𝑠 ] Extrapolating the 𝜋𝜋 . 𝐶𝐶 . trend from above to find when 𝜋𝜋 . 𝐶𝐶 . = 0, 𝑎𝑎𝑎𝑎𝑠𝑠 28,888 𝑓𝑓𝑓𝑓 8. Calculate the range and endurance of this aircraft at sea level. For propeller-driven aircraft, 𝜋𝜋 𝑚𝑚𝑎𝑎𝑚𝑚 = 𝜂𝜂 𝑐𝑐 𝐶𝐶 𝐿𝐿 𝐶𝐶 𝐷𝐷 𝑚𝑚𝑎𝑎𝑚𝑚 ln 𝑊𝑊 0 𝑊𝑊 1 𝐶𝐶 𝐿𝐿 𝐶𝐶 𝐷𝐷 𝑚𝑚𝑎𝑎𝑚𝑚 𝑜𝑜𝑐𝑐𝑐𝑐𝑜𝑜𝑜𝑜𝑠𝑠 𝑤𝑤ℎ𝜋𝜋𝑚𝑚 𝐶𝐶 𝐷𝐷 , 0 = 𝐶𝐶 𝐷𝐷 , 𝑖𝑖 = 𝐶𝐶 𝐿𝐿 2 𝜋𝜋𝜋𝜋𝜋𝜋𝜋𝜋 → 𝐶𝐶 𝐿𝐿 = 0.696 𝑎𝑎𝑚𝑚𝑎𝑎 𝐶𝐶 𝐷𝐷 = 𝐶𝐶 𝐷𝐷 , 0 + 𝐶𝐶 𝐷𝐷 , 𝑖𝑖 = 0.054 → � 𝐶𝐶 𝐿𝐿 𝐶𝐶 𝐷𝐷 𝑚𝑚𝑎𝑎𝑚𝑚 = 12.88 𝜋𝜋 𝑚𝑚𝑎𝑎𝑚𝑚 = 3.81 × 10 6 𝑓𝑓𝑓𝑓 = 822.6 𝑚𝑚𝑚𝑚 𝐶𝐶 𝐿𝐿 3 2 / 𝐶𝐶 𝐷𝐷 𝑚𝑚𝑎𝑎𝑚𝑚 = 12.167 𝐸𝐸 𝑚𝑚𝑎𝑎𝑚𝑚 = 𝜂𝜂 𝑐𝑐 𝐶𝐶 𝐿𝐿 3 2 𝐶𝐶 𝐷𝐷 𝑚𝑚𝑎𝑎𝑚𝑚 2 𝜌𝜌𝑆𝑆 1 �𝑊𝑊 1 1 �𝑊𝑊 0 = 2.93 × 10 4 𝑠𝑠 = 8.15 ℎ𝑜𝑜 9. Estimate the sea level take-off distance on a paved runway. 𝑆𝑆 𝐿𝐿𝐿𝐿 = 1.44 𝑊𝑊 2 /( 𝑔𝑔𝜌𝜌𝑆𝑆𝐶𝐶 𝐿𝐿 , 𝑚𝑚𝑎𝑎𝑚𝑚 { 𝑇𝑇 − [ 𝐷𝐷 + 𝜇𝜇 𝑅𝑅 ( 𝑊𝑊 − 𝐿𝐿 )]} 𝑉𝑉 𝑎𝑎𝑎𝑎 where 𝑉𝑉 𝑎𝑎𝑎𝑎 = 0.7( 𝑉𝑉 𝐿𝐿𝐿𝐿 ) = 0.7 1.2 2 𝑊𝑊 𝑚𝑚𝑎𝑎𝑚𝑚 𝜌𝜌𝑆𝑆𝐶𝐶 𝐿𝐿 , 𝑚𝑚𝑎𝑎𝑚𝑚 = 115.1 𝑓𝑓𝑓𝑓 / 𝑠𝑠 so 𝑆𝑆 𝐿𝐿𝐿𝐿 = 543.1 𝑓𝑓𝑓𝑓 (670 ft w/o ground effect) 10. Estimate the sea level landing roll distance assuming the plane is landing with a weight of 3000 lb and that lift is zero after touchdown. 𝑆𝑆 𝐿𝐿 = 1.69 𝑊𝑊 2 /( 𝑔𝑔𝜌𝜌𝑆𝑆𝐶𝐶 𝐿𝐿 , 𝑚𝑚𝑎𝑎𝑚𝑚 [ 𝐷𝐷 + 𝜇𝜇 𝑟𝑟 ( 𝑊𝑊 − 𝐿𝐿 )] 0 . 7𝑉𝑉 𝑇𝑇 where 0.7 𝑉𝑉 𝑇𝑇 = 0.7(1.3 𝑉𝑉 𝑠𝑠𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 ) = 0.7 1.3 2 𝑊𝑊 𝜌𝜌𝑆𝑆𝐶𝐶 𝐿𝐿 , 𝑚𝑚𝑎𝑎𝑚𝑚 = 80.3 𝑓𝑓𝑓𝑓 𝑠𝑠 so 𝑆𝑆 𝐿𝐿 = 495.7 𝑓𝑓𝑓𝑓 11. Look up some interesting fact about this airplane and include that in your homework.
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