Homework 5b

# Homework 5b - 4262 Rockets and Mission Analysis Homework#5b...

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4262: Rockets and Mission Analysis Assigned: October 14, 2010 Homework #5b Due: November 2, 2010 Question 1: A communication satellite is launched from low earth orbit (LEO) 200 km high above the earth, in order to place it in geosynchronous earth orbit (GEO) that has a period of one sidereal day (1 sidereal day = 23 hours, 56 minutes, 4.09 seconds). a. At GEO, how high above the earth’s surface is the satellite transmitting? b. What is the Hohmann (or free-space) mission velocity requirement for the LEO-GEO launch of the satellite? c. How long would it take for the LEO-GEO transfer? d. What is the eccentricity of the transfer ellipse? Question 2: In Section 11.3, an expression for the divergence loss of a conical nozzle is developed and shown to be: 2 cos 1 α λ + = Consider a linear (as opposed to axisymmetric) rocket nozzle, with a wedge-shaped exit profile, as shown below. Assuming all properties are constant on the cylindrical surface (shown by a dashed line), calculate the divergence loss factor λ , due to the finite angle α . Figure 1: Linear Rocket Nozzle Schematic Plot the divergence loss for a conical and linear nozzle for divergence angles from 0 to 45 degrees. For how larger of a range of half angles would you believe these results? What physical phenomenon may begin to occur as the half angle is continually increased? α R θ 1

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The schematic below shows an ideally expanded 2-D spike nozzle which expands uniform sonic flow (with Cp/Cv = 1.2) to a uniform Me = 3 exit condition. The sonic flow is directed at an angle θ o (to be determined) to the final flow direction. Figure 1: 2-D Spike Nozzle Schematic The whole expansion area is a “simple flow region” because it is adjacent to uniform regions, from which one of the two families of characteristics will propagate a constant invariant into it. The expansion is centered at the cowl lip (point O), and it is a Prandtl- Meyer flow. All of the expansion characteristics pass through the point O. The flow is choked at Mach 1 at the throat location, where the height is ho. a. Based on the above statement and schematic, calculate θ o. b. Using continuity, calculate the ratio he/ho c. Calculate the angle α made by the last expansion characteristic with the final flow direction. d.
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Homework 5b - 4262 Rockets and Mission Analysis Homework#5b...

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