COMPRESSIBLE FLUID FLO
COMPRESSIBLE FLUID FLO MAE 3293
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This 0 page Class Notes was uploaded by Felipe McLaughlin on Sunday November 1, 2015. The Class Notes belongs to MAE 3293 at Oklahoma State University taught by David Lilley in Fall. Since its upload, it has received 12 views. For similar materials see /class/232792/mae-3293-oklahoma-state-university in Mechanical and Aerospace Engineering at Oklahoma State University.
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Date Created: 11/01/15
Table of Contents Table of Contents 3 Objective 4 Appmarh 4 Results and niltrllltltinn 5 r 10 n 10 17 Introduction Martin s nia KUiVu due Due to these hihn d h394 H thebodyof extreme temperatures to occur The airfoii design though was one used for many 39 39 e design 39 a triangie 39 asa deita wing This concept of supersonic airfoi39is ieads towards the focus ofthe second proiect caicuiating the ii39ft and drag ofa supersonic airfoi39i The cornpiexities are irnportant to consider since ii39ft and drag characteristics iead to determining the proper aircrattdesign required to fi39H its niche Objective The objective of this second project as set forth by Dr Lilley was to calculate the lift and drag characteristics of a supersonic airfoil The airfoil which is shown below by using Professional Engineer CAD program with an angle of attack of six degrees and the leading edge has an angle of two degrees while the trailing edge has an angle of three degrees The dimensions ofthe airfoil can be found in the appendices The defining parameters of the project were as such Design Mach number M 3 N50 where N33 and will be for all subsequent calculations and yield a design Mach number of 234 The pressure will be taken at an altitude of 5km from the surface ofthe Earth and has a value of 5405 KPa In addition the class textbook refers to Gas Dynamics 3 John amp Keith Approach The approach that was taken towards this project was to simply analyze the pressures and conditions associated with the upper and lower surfaces The solution to part a ofthe project was very methodical and the steps taken can be found in the appendices The approach taken towards the coefficients and their graphs was simply by choosing a valid range of angle of attacks in which the shock didn t become detached Once that range was determined the lift and drag forces were written into Excel and then multiplied by the span width of 03 meters Results amp Discussion To begin the project part a is stated as such a What are the Lift and Drag in Newtons and Lift and Drag coef cients CL and CD for one example angle of attack situation Explain the development equations words etc don t just give the answers Show the calculations choosing an angle of attack to be six degrees the following calculations were used to develop the lift and drag For the lower pressure of the airfoil Appendices D of the class textbook was utilized Because the airfoil is traveling a speed greater than Mach 1 a shock wave is going to develop around the leading edge of the wing For further explanation of this phenomena consult the class textbook but this shock causes changes to occur to the isentropic flow around the airfoil Since the angle of attack of the leading edge was two degrees and the angle of attack was six degrees a 5 of 4 degrees was used At a Mach speed of 234 and a 5 of 4 degrees a shock angle 0 was found to be 284 Now since the normal Mach speed needs to be found by simply multiplying the Mach speed by the sine of the shock angle as such 234sin284 1113 Taking this value into the normal shock tables or Appendix C of the class textbook a normal Mach speed after the shock is found to be 0902 Taking this value into the lsentropic Flow Tables or Appendix B of the class textbook results in a pressure ratio amp of 1279 this process will be taken further but next it is now required to calculate the upper pressure acting on the wing The upper wing surface has yet again another phenomenon associated with it that being the expansion fan angle associated with PrantlMeyer fluid flow Again for a further explanation of the intricacies associated with PrantlMeyer expansion consult the class textbook In short the expansion that the fluid on the upper surface of the airfoil experiences causes a pressure decrease resulting in a velocity increase To calculate the upper wing pressure Appendix D was again utilized Taking 5 to be 5 6 and at a Mach speed of 234 a shock angle was found to be 301 As mentioned earlier since the conditions after the shock change a normal Mach speed needed to be found after the shock Multiplying the Mach speed 234 by the sine of 301 yields a value of 1174 This speed is valid for segment AB See diagram in Appendices But since the angle changes after segment AB the fluid is going to experience an expansion through the angle change To calculate the speed after the angle change the following was done since M2 1568 gt U2 29 U3 2 3 29 6 from Appendices E in the class textbook The addition of 2 and 3 are the two interior angles of the airfoil and since the two interior angles of a triangle sum to the exterior angle of a triangle the preceding was performed The 6 was added due it being the angle of attack Since U3 139 by using Appendix E the Mach speed M3 can be found at this fan angle which is 1568 Since M2 and M3 values are now known the pressure ratios can now be found by going to the isentropic flow tables With M2 and M3 equal to their respective values amp 00751 and amp 0246 and multiplying them by their respective terms results in upper surface having a total pressure distribution of 18 KPa To calculate the lift and drag simple geometry was incorporated in order to solve In addition upper pressures were subtracted from lower pressures to solve for lift To calculate the coefficients of lift and drag the dynamic pressure was first found Dynamic pressure is equal to val where A is equal to the planform area or cspan which is equal to 06 At 5 km p is equal to 0736 k ga Finally the v2 term is equal to m 234 14 287 062 The coefficients of lift and drag are there by calculated by simply dividing them by the previous constant Part b of the project asks to do the following Now use COMPROP to make calculations for other angles of attack This is computer experimentation Explain compute tabulate and draw gures to illustrate the graphs CLvsoc CDvsoc CLCD vsoc CL vs CD with points for different on this is called the quotlift drag polar This graph simply shows the Coefficient of Lift verses the Angle of Attackand goes further to demonstrate that this relationship is linear It can also be deduced that as angle increases the coefficient of lift becomes greater meaning that a higher angle of attack yields to better lift Ao A Degrees This graph shows further that Drag and the Angle of Attack are not linearly related It also shows that there is an angle where the drag will become a minimum meaning that is where there is the least amount of overall drag and that is an optimal design point since increasing the angle of attack will only increase drag as well after the minimum CLCd This graph is indicative of the ability of an aircraft to lift heavy payloads without much thrust Hence it explains why the graph is at one extreme when there is a large coefficient of drag and on the other with a large coefficient of lift LiftDrag Polar E c B 2 2 t a B U Li yDlag 01000 0 1500 02000 02 500 Coeffitiem of mag This graph tends to decrease for the drag up until zero lift and then begin to increase after that decrease This liftdrag polar graph is used to find the optimized point at a certain angle between the lift and drag forces For part c c What is the imponant angle of attack for zero li Is there a maximum angle of attack for which the oblique shock is still attached Would any of the infoimation in Pans b and c change for different ight spee d5 The important angle of attack for zero lift would be found by taking the lift force calculation and finding where it goes to zero and perform simple interpolation as such From Appendices AoA L 30000 O897 40000 3509 interpolating results in ADA 5 09 where AoA is 4 3 3619 equal to 319quot The calculations that were made in part b will definitely change due to the fact that the lift coefficient was based on the dynamic pressure which as it was mentioned earlier contains a velocity squared term The AoA and zero lift condition will not change due to the fact that neither relied on the speed of the aircraft to calculate the value Conclusions It can be concluded that the shock drag and lift calculations can be somewhat long or tedious which makes the importance of an executable file very convenient and timesaving when it comes to fan angle and shock and shock angle calculation The results that were just unto here given were variable by classmates Brad Bolt and Manny Cortez Recommendations One of the recommendations that could be made to the computer program is the option of more drawing features on the airfoil palette The option that exists already in the program only allows for cambered airfoils or triangular supersonic airfoils These triangular airfoils while they are a model are not entirely accurate of what s going within the real world supersonic wings Aggendlces o2 2o kNm Pvessuve We 87 s 73 n 58 6 M 2 so a 3 a P56D5kPa292 ms nu Dan 2873mm I Ang eafauack s degvee Pvessuve a ang uppev smlace ar am Pvessuve a ang my smlace ar am Span q m 7Kpaj Dylan Sirbaugh MAE 3293 Project 3 Due 112108 Problem a 10 N b 10 N2 pumps C quot S quot Nozzle chamber 0 Schematic of a Liquid Fuel Rocket Take the fuel to be a composite liquid fuel with pentane heptane and decane in mass percentage proportions a b and c given as above Deduce from these the molar percentages that are needed in the AFTC CP Part 1 7 CombustionPropulsion Consider aquot quot 39 t fuel as given 39 39 O2 Throat area 150 N cm7 Take combustion chamber pressure 2 N20 MPa Here N your code number 1 For stoichiometric combustion compute the combustion chamber temperature and product species composition using the AFTC computer code Heat losses andor dissociation will lower Tc 2 Consider the design AA etc find exit velocity exit temperature exit pressure etc for a design altitude of5 km Use an appropriate specific heat ratio y for your particular combustion products at say 3000K You also need to generate and use an isentropic ow table that you generate for your specific heat ratio Lhi with Linc y m LplujCLL 3 Find thrust at design altitude at sea level and in space with this same convdiv nozzle as the designedfor altitude Part 2 7 Rocket Flight Based on the rocket performance class material and possibly elsewhere calculate and discuss the performance of your rocket in vertical upward ight You will calculate velocity vs time from the Rocket equation probably on an Excel spreadsheet and hence determine velocity vs time and distance vs time Include things like initial total mass propellant loading blurring rate time of burn velocity at burnout altitude at burnout and at maximum height reached when the mass remaining continues in vertical ight etc Given the mass percentages of each individual fuel component I calculated the mole percentage using the formula mass M l l M M0l 08w ar ass Totalmoles Using these numbers the AFTC program provided by Dr Lilley was used to calculate the flame temperature of the components burning in 100 oxygen as well as the molar composition of the resulting product of the reaction It should be noted that the case that was investigated did not consider disassociation effect and if such effect were taken into account the flame temperature would drop from the 8000 K to 6468 K Next several values had to be calculated for the gas that was exiting the rocket The Gas constant R was calculated using gt3 Elm Where M Z xiM i From this the specific heat capacity of the reaction was calculated using the formula C 10 CP M Where C79 Z 139 With M the same as used for the gas constant calculation The values of the heat capacities of each gas were taken from the provided Fuels handout For the value of specific heat constant for volume the following equation was used With the values of cp and R as calculated previously From these values the specific heat ratio could be calculated as shown below Using this value the program that was developed for the first project was used to produce and sentropic flow table for the reactants exiting the rocket through the convergingdiverging nozzle This table was then used to find the properties of the flow after experiencing the nozzle Assuming that since the nozzle was designed for an altitude of five kilometers the gas would be designed to expand to a pressure of 5405 kPa Using this as the outside pressure and the provided chamber pressure the ppl ratio was calculated The sentropic Flow table generated from the specific heat ratio as calculated before was then used to find the Mach number temperature at the exit and the area ratio at the location where ppl was found The area ratio found gave the required area of the final nozzle shape which was found to be 1095 cm2 This number gives a minimum diameter that the rocket must have in order to accommodate these conditions Next the velocity was calculated from the data found on the table using the following equation V MyRT The value for the mass flow rate limited by the throat conditions was found by x P A RT m R7 V This allowed for the calculation of thrust at design conditions by the equation Thrust rille This value was then corrected for rocket use at Sea Level and in outer space assume the pressure in space to be zero with the equation Thrust thrustdesign 19 paAe Since the atmospheric pressure drops approximately half its previous value with ever five kilometers of altitude the thrust corrections showed that the difference in thrust at sea level was noticeable different than at the design altitude compared to the thrust achieved at altitudes higher than the design Part 2 The second part of the assignment took a look at the performance that a rocket could be expected to show given the parameters as calculated before First several assumptions had to be made about the rocket in question The value of initial mass was made to be 1000 kg and the propellant loading made to be 06 The value of burn rate was assumed to be the same as the mass flow rate through the nozzle as found before The time of the burn was found using the equation mart burn rate This gave the time to be only nineteen seconds a fairly quick time to go through 600 kg of fuel The velocity of the rocket at burnout was then found using the equation m V i ln1 Pr m0 The height achieved was the found using the simplified equation below The acceleration was assumed to be constant for this flight because the thrust values change very little over altitude and the effects of air resistance were neglected Alt at2 With the acceleration assumed to be Vbumout a tbumout This produced a result of forty three kilometers of altitude after the initial burn From this point the rocket is purely coasting upwards During this stage of flight the gravitational pull of the Earth was assumed to be a constant 981 msz Thus the total altitude achieved by this rocket was then calculated using and assuming the velocity at max height to be zero Vbumout 2 Alt 293981 Altbmm The total rime of the flight was then found using Vbumout H t Part3 For the last part of this investigation the results of the rocket flight were plotted against each other to investigate the trends of the flight These graphs are attached to the end of this report The first graph is the change in thrust with respect to altitude This graphs shows that since the thrust achieved is so high the changes due to atmospheric pressure have very little effect on the performance after the first fifteen or so kilometers of flight The second graph emphasizes the interesting part of the curve only graphing the first seventy five kilometers of flight The Third graph shows the altitude of the rocket with respect to time The initial part of this curve shows the accelerated flight of the rocket due to the burn and the rest shows the deceleration of the rocket due to gravity The fourth graph is a closer view of the curve during the burn and the transition to coasting flight during the first minute of flight From this graph it s interesting to see how little altitude that rocket actually achieves during the trusting portion of flight and how much the momentum of the rocket carries it to higher altitudes The fifth graph shows the relationship between the velocity of the rocket and time of the flight The last graph also shows the velocity during just the first minute of flight There is an error in the graph due to rounding errors Because the equations of motion change as soon as the rocket burn is over a new equation had to be applied to the curve at 1862 seconds Because a step size of one second was used the new equation had to be applied at nineteen seconds because the top of the graph to have a bump that in reality doesn t exist These graphs show how fast the velocity increases during the thrusting stage as compared to the coast attributing to the high altitudes achieved Summary The effects of different chemical reactions to produce thrust produce very interesting rocket motion In this investigation a 1000 kilogram rocket with 600 kilograms of a mixture of Pentane Heptane and Decane combusting in pure oxygen allowed the rocket to achieve a total altitude of 1177 kilometers with less than nineteen seconds of burning The results of this burn can be seen graphed at the end of this report This investigation did leave out several parameters the will greatly affect the actual flight of the rocket These include the effect of air resistance on the rocket as it travels and the decreasing effect of gravity on the rocket as it moves further from the planet earth Also random events such as gusts causing the rocket to not achieve perfect vertical flight would also lower the actual altitude achieved The effects of disassociation on the fuel products were also neglected This effect would significantly lower the effectiveness of the combustion reaction