performance of the c-5a galaxy and the c-5m super...
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Performance of the C-5A Galaxy and the C-5M Super Galaxy
Tanveer Singh Chandok (Point Performance)
In collaboration with: Ju S Lee (Dimensions, Weights, Missions, Aerodynamics and Propulsion)
and John Mueller (Integral Performance)
AE 3310 – Spring 2012
Georgia Institute of Technology
Atlanta, GA
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Contents Executive Summary ...................................................................................................................................... 3
Dimensions, Weights, and Missions ............................................................................................................. 3
Task 1-3 .................................................................................................................................................... 3
Task 4 ........................................................................................................................................................ 4
Aerodynamics ............................................................................................................................................... 4
Task 5 ........................................................................................................................................................ 4
Task 6 ........................................................................................................................................................ 5
Task 7 ........................................................................................................................................................ 7
Propulsion ..................................................................................................................................................... 7
Task 8 ........................................................................................................................................................ 7
Task 9 ........................................................................................................................................................ 8
Task 10 ...................................................................................................................................................... 9
Point Performance ....................................................................................................................................... 10
Task 11 .................................................................................................................................................... 10
Max Rate of Climb Algorithm ............................................................................................................. 10
Max Speed Algorithm .......................................................................................................................... 11
Takeoff Ground Roll Algorithm .......................................................................................................... 12
Task 12 .................................................................................................................................................... 13
Task 13 .................................................................................................................................................... 20
Task 14 .................................................................................................................................................... 21
Integral Performance ................................................................................................................................... 23
Task 15 .................................................................................................................................................... 23
Task 16 .................................................................................................................................................... 24
Task 17 .................................................................................................................................................... 25
Appendix A ................................................................................................................................................. 27
MATLAB Code used in Task 13 ............................................................................................................ 27
MATLAB Code used in Task 14 .............................................................................................................. 28
References ................................................................................................................................................... 30
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Executive Summary
This paper provides a full analysis of the C-5A Galaxy and C-5M. The C-5A was a heavy
cargo aircraft built in Marietta, GA by the Lockheed Martin Company in the late 1960s to the
early 1980s. The C-5M Super Galaxies are the fleets that began to undergo the Reliability
Enhancement and Re-Engining Program (RERP). The RERP program replaced the GE TF-39
engine with GE CF6-80C2 engine and redesigned the pylons for the new engine. This paper will
analyze how that RERP program changed the dimensions, weights, aerodynamics, propulsion,
and the performance of the aircraft. The empty weight of C-5M is greater than C-5A due to
heavier power plants and additional structural weight. The C-5M is more efficient in
aerodynamics. Due to shorter nacelle and new pylon design, C-5M has less drag than C-5A. In
terms of propulsion, the new engine produces more thrust and has lower thrust specific fuel
consumption. C-5M’s performance has also improved from C-5A.
Dimensions, Weights, and Missions
Task 1-3
Span (ft) 222.7
Area (sq. ft) 6200
Sweep (degrees) 25
Aspect Ratio 7.75
Table 1.1: Basic Dimensions
C-5A C-5M
Empty Weight (OEW) (lbs) 320,086 332,986
Maximum Takeoff Gross
Weight (MTOGW) (lbs)
769,000 769,000
Maximum Payload Weight (lbs) 265,000 252,400
Maximum Fuel Weight (lbs) 177,038 177,038
Table 1.2: Weight of C-5A and C-5M
Compared to C-5A, C-5M has greater empty weight and less payload weight. C-5M
empty weight is calculated by adding the additional structural weight and the power plant weight
to the C-5A empty weight. 320086 + 4*(9400-7500) + 5000 = 332986 lbs. Now the payload for
C-5M is the payload of the C-5A minus the additional weight. 265000 – 4*(9400-7500) – 5000 =
252,400 lbs. TOGWmax does not equal to OEW + Wpayloadmax + Wfuelmax for both C-5A and C-
5M, but rather greater than the sum of the three. This is because the TOGWmax is defined as the
maximum weight ceiling that the aircraft can take. Due to the structural limit, the airplane would
not carry the maximum weight. Also, aircrafts do not usually carry maximum payload weight
every flight. According to the payload-distance graph, the range of the aircraft increases as the
payload decreases at any condition. Therefore, to maximize the range, they minimize the payload
weight.
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Task 4
C-5A C-5M
Takeoff Gross Weight (TOGW)
(lbs)
732,500 732,500
Maximum Payload Weight (lbs) 220,000 225,000
Maximum Fuel Weight (lbs) 185,538 185,538
Table 4.1: Mission III
C-5A C-5M
Takeoff Gross Weight (TOGW)
(lbs)
732,500 732,500
Maximum Payload Weight (lbs) 87,038 92,038
Maximum Fuel Weight (lbs) 318,500 318,500
Table 4.2: Mission V
C-5A C-5M
Takeoff Gross Weight (TOGW)
(lbs)
646,462 646,462
Maximum Payload Weight (lbs) None 5,000
Maximum Fuel Weight (lbs) 318,500 318,500
Table 5: Mission IX
Aerodynamics
Task 5
Using the Matlab code c5polar.m, the drag polar of C-5A had been calculated at 6
different conditions, as shown in Figure 5.1.
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Figure 5.1: Drag Polar for C-5A
According to Figure 5.1, the drag coefficient increases as the Mach number increases.
This is due to the shock waves that form on the upper surface of the airfoil, which can induce
flow separation and adverse pressure gradients on the aft portion of the wing.
Task 6
CDo calculation:
The equation for calculating the zero-lift drag coefficient:
CDo = CD – CDi
The equation for calculating the induced drag:
CDi = CL2/(π*eo*AR)
Approximate estimation of Oswald-efficiency:
eo = 1.78(1 – 0.045AR0.68
) – 0.64
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CL CD CDi CD0
0.2 0.0182 0.002009 0.0162
0.3 0.0193 0.004521 0.0148
0.4 0.0217 0.008038 0.0137
0.5 0.0267 0.012559 0.0141
0.6 0.035 0.018085 0.0169
0.7 0.0476 0.024616 0.0230
Table 6.1: Lift and Drag Coefficient
As the lift coefficient increases, the drag coefficient also increases and the induced drag
increases significantly. The zero-lift drag coefficient, however, decreases until about 0.4 Mach
and bounces back up and starts increasing.
(L/D)max Calculation:
The equation for calculating the maximum lift to drag ratio:
(L/D)max =
The equation for the induced drag factor:
K = 1/(πe0AR)
CL CD CDi CD0 (L/D)max
0.2 0.0182 0.002009 0.0162 17.532
0.3 0.0193 0.004521 0.0148 18.35031
0.4 0.0217 0.008038 0.0137 19.08541
0.5 0.0267 0.012559 0.0141 18.75955
0.6 0.035 0.018085 0.0169 17.15244
0.7 0.0476 0.024616 0.0230 14.71453
Table 6.2: Maximum Lift to Drag Ratio
The lift to drag ratio behaves exactly the opposite of the zero-lift drag coefficient. It
increases until 0.4 Mach and starts decreasing after 0.4 Mach.
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Task 7
Figure 7.2: Drag Polar for C-5M
Figure 7.2 represents the drag polar for the new installation. C-5M has very slightly lower
drag than C-5A due to the new nacelle and pylon designs. In order to install the new engines on
the C-5, a new pylon design was desired. The new aerodynamically efficient pylon minimized
the interference drag. Also, the new CF6-80C2 engine has a shorter nacelle than that of the TF-
39 engine. According to the wind tunnel test, the new engine had 5 drag counts less than the old
engine and according to the CFD, the new engine had 7 drag counts less than the old engine.
Propulsion
Task 8
Propulsion had been analyzed by using the Matlab code thrustmaxTF39.m. This
calculates the maximum thrust at a certain conditon. This code analyzed the lapse rate at four
different altitudes: sea level, 15,000 feet, 30,000 feet, and 35,000 feet.
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Figure 8.1: TF-39 Maximum Thrust
Task 9
Figure 9.1: CF6 Maximum Thrust
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Figure 8.1 and Figure 9.1 shows that the maximum thrust decreases as the Mach number
increases at any altitude. Also, the altitude and Mach number lapse rate is lower at higher
altitude.
Task 10
Now using the powerhookTF39.m and powerhookCF6.m files, the thrust specific fuel
consumption has been analyzed as shown below.
Figure 10.1: Thrust Specific Fuel Consumption vs. Thrust for TF39 and CF6
In reality, the curves would not look similar at all, but rather, TF39 will have higher
TSFC at low thrusts (< 10000 lbs) and CF6 will have higher TSFC at high thrusts (>10000 lbs).
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Point Performance
Task 11
There are three algorithms present in the c5pointperf.m file. These are used to calculate
the Max Rate of Climb, the Max Speed, and the Takeoff Ground Roll. However, before any of
these calculations can be made, the configuration of the aircraft must be defined in terms of the
‘Wing Area’ (S) and ‘Takeoff Gross Weight (TOGW)’ (W).
Max Rate of Climb Algorithm
Variables used
h Altitude
VRCmax Matrix to store value of the maximum vertical rate of climb
RCmax Matrix to store value of maximum rate of climb
CL Coefficient of lift matrix
Minf Freestream mach number
Tmax Maximum thrust
D Drag force matrix
S Wing area
W Takeoff gross weight (TOGW)
ρ Density
Methodology
The first step of this algorithm is to find the atmospheric properties of the flying
condition. Next, a root finding method (based on the actual drag polar and thrust lapse model) is
used since the maximum thrust depends on the velocity nonlinearly. The root find is
implemented using the ‘fzero’ function in MATLAB. This function finds the zero of an inputted
function at an inputted location. A detailed explanation of the function can be found on the
MathWorks websitei. Furthermore, in order to obtain VRCmax, the VRCmaxfcn function must
be used as the input in ‘fzero’. VRCmaxfcn outputs RCmax and takes velocity, height, TOGW
and wing area as inputs.
After generating the VRCmax matrix, the Mach numbers at specific heights can be
calculated using the speed of sound. The relationship used here is Equation 11.1, where V is
velocity and a is the speed of sound. Since all these methods are being run within a for loop, the
Minf matrix is thus generated.
Moving on, the coefficient of lift (CL) and drag (D) must be calculated. CL is calculated
in order to obtain D, as D is in turn required to obtain RCmax. CL can be calculated using
Equation 11.2. Drag (D) can then be found using Equation 11.3. Velocity used in both of the
previous relations refers to VRCmax. To calculate the drag coefficient, the function c5polar.m
was used. This function outputs the drag coefficient (CD) and the lift-to-drag ratio (L/D), while
taking CL, Minf and h as inputs.
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Finally, the matrix of RCmax can be calculated using Tmax, D, VRCmax and W. The
relationship used is Equation 11.4. Since this is within a for loop, the entire matrix of RCmax
over different altitudes is generated. The last step is to plot the graph of Max Rate of Climb in
ft/min to Altitude in ft. A comparison between the calculated values and the SAC chart data is
made by plotting two separate lines on the graph.
Equations used
Equation 11.1
Equation 11.2
Equation 11.3
Equation 11.4
Max Speed Algorithm
Variables used
Vmax Matrix of maximum speeds
VlimitCAS Velocity limit of calibrated air speed in m/s
Vlimit Velocity limit of correctairspeed function
Methodology
As was done in the algorithm to calculate Max Rate of Climb, the atmospheric properties
at the flying conditions must be defined. Next, a root finding method (based on the actual drag
polar and thrust lapse model) is used since the maximum thrust depends on the velocity
nonlinearly. The root find is implemented using the ‘fzero’ function in MATLAB. Furthermore,
in order to obtain Vmax, the Vmaxfcn function must be used as the input in ‘fzero’. Vmaxfcn
outputs Vmax and takes velocity, height, TOGW and wing area as inputs. This process is the
same as the one detailed in the previous algorithms explanation, except with different input
functions.
The max speed is limited to 350 KCAS (knots calibrated airspeed). As an error catch,
Vlimit is calculated using MATLAB’s inbuilt ‘correctairspeed’ function. This function computes
the conversion factor from specified input airspeed to specified output airspeed using speed of
sound and static pressure. Detailed information about the function can be found on the
MathWorks websiteii. Lastly, a check must be made as an error catch to make sure that the max
speed is at or under the speed limit.
Finally, the graph of maximum speed (in knots) to the altitude (in feet) can be plotted.
This plot also contains the comparison between the calculated data and the SAC chart data.
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Takeoff Ground Roll Algorithm
Variables used
Wto Weight at takeoff
CLmaxto Max coefficient of life at takeoff
Vstallto Stalling velocity of takeoff
Vlo Liftoff speed
Mlo Mach number at liftoff
Tmaxto Max thrust at takeoff
sg Ground roll
g Acceleration due to gravity
Methodology
Since the method used is an approximate analysis approach, there are a few estimations
and assumptions made. CLmaxto is estimated to be 1.8 and Tmaxto is ‘averaged’ at 0.7Mloiii
. To
calculate Vstallto, Equation 11.5 is used. Another estimate is used to calculate Vlo. This
relationship is Equation 11.6. Equation 11.1 can then be used again to calculate Mlo. To
calculate Tmaxto, the function thrustmaxTF39 is used. This function returns the maximum thrust
by taking the height and freestream Mach number as inputs. Finally sg can be calculated using
Equation 11.7. This equation is modeled as a combination of Equation 6.94 and 6.95 in Chapter
6 of Anderson (Aircraft Performance and Design). Equation 6.94 and 6.95 (from the Anderson
textbook) are:
In Equation 11.7, the part to the R.H.S of the plus sign is Vlo times 3 (N). This represents
an approximation made for the ground distance covered during the rotation periodiv
.
Equations Used
Equation 11.5
Equation 11.6
Equation 11.7
ρ
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Task 12
Analysis for TOGW of 719,886 lbs for C-5A
Figure 12.1: Max Rate of Climb vs. Altitude Figure 12.2: Maximum Velocity vs. Altitude
Figure 12.3: Gross Weight vs. Ground Roll Distance
Here a TOGW of 719,886 lbs was used. Fuel consumption was neglected. This is just a
demonstration of the c5pointperf.m file’s standard output. The data here is not very important to
point performance evaluations.
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Analysis for Mission III (C-5A)
Mission III: Design Cargo
Initial weight used (Empty weight + payload weight + fuel weight) = 725,624 lbs
Payload weight = 220,000 lbs
Fuel Weight = 185,538 lbs
Figure 12.4: Max Rate of Climb vs. Altitude Figure 12.5: Max velocity vs. Altitude
Figure 12.6: Gross Weight vs. Ground roll distance
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Analysis for Mission III (C-5M)
Initial weight used (Empty weight + payload weight + fuel weight) = 743,524 lbs
Payload weight = 225,000 lbs
Fuel Weight = 185,538 lbs
Figure 12.7: Max Rate of Climb vs. Altitude Figure 12.8: Max velocity vs. Altitude
Figure 12.9: Gross Weight vs. Ground roll distance
Comparing Figures 12.4 to Figures 12.9 (for Mission III), it is seen that there are some
performance differences between the two aircraft models. Comparing the Max rate of climb vs.
Altitude graphs of the two aircraft, it is seen that there is variation as to how the aircrafts behave.
The C-5M Has an overall slightly better rate of climb as compared to the C-5A. There is very
slight difference between the Max velocity vs. Altitude of both aircrafts. This is justifiably due to
the different distributions in the weight compared to how the fuel is consumed. The C-5A has a
greater TOGW, however it has lesser fuel carrying capacity (the C-5M is vice versa). There is a
change in the Gross Weight vs. Ground roll distance due to the difference in the different initial
weights of the aircrafts.
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Analysis for Mission V (C-5A)
Mission V: Max fuel volume
Initial weight used (Empty weight + payload weight + fuel weight) = 725,624 lbs
Payload weight = 87,038 lbs
Fuel Weight = 318,500 lbs
Figure 12.10: Max Rate of Climb vs. Altitude Figure 12.11: Max velocity vs. Altitude
Figure 12.12: Gross Weight vs. Ground roll distance
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Analysis for Mission V for C-5M
Initial weight used (Empty weight + payload weight + fuel weight) = 743,524 lbs
Payload weight = 92,038 lbs
Fuel Weight = 318,500 lbs
Figure 12.13: Max Rate of Climb vs. Altitude Figure 12.14: Max velocity vs. Altitude
Figure 12.15: Gross Weight vs. Ground roll distance
Comparing Figures 12.10 through 12.15, we can analyze the difference in performance
between the two aircrafts during mission V. The Max rate of climb graph is better adapted by the
C5-M. As in the previous mission, the Max velocity graph is quite similar in both cases (for
reasons discussed above in previous mission). Lastly, we can see that the Gross weight vs.
Ground roll distance graph’s are not exact, but quite similar. This is mainly due to the difference
in initial weights used for both aircrafts.
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Analysis for Mission IX for C-5A
Mission IX: Ferry Range
Initial weight used (Empty weight + payload weight + fuel weight) = 638,586 lbs
Payload weight = 0 lbs
Fuel Weight = 318,500 lbs
Figure 12.16: Max Rate of Climb vs. Altitude Figure 12.17: Max velocity vs. Altitude
Figure 12.18: Gross Weight vs. Ground roll distance
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Analysis for Mission IX for C-5M
Initial weight used (Empty weight + payload weight + fuel weight) = 656,486 lbs
Payload weight = 5,000 lbs
Fuel Weight = 318,500 lbs
Figure 12.19: Max Rate of Climb vs. Altitude Figure 12.20: Max velocity vs. Altitude
Figure 12.21: Gross Weight vs. Ground roll distance
Comparing Figures 12.16 through 12.21, it is seen that the model is now breaking down
and it not approximating values close to the SAC data from specific conditions such as Max Rate
of Climb vs. Altitude. Still, the C-5M has better performance while compared to the C-5A. The
Gross Weight vs. Ground roll distance graphs are different. This can be attributed to the fact that
the C-5A (in this configuration) has no payload, but carries the same amount of fuel.
Looking at all the curves plotted above, the following deductions can be made:
a) The C-5A’s data sometimes matches the SAC chart data. For the Max velocity vs.
Altitude plots, the data is very similar to the SAC charts, and only breaks away beyond
altitudes of approximately 23,000 ft. This holds true for all 3 missions. However, it is
seen that moving through mission III, V and IX respectively, the data calculated starts to
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differ to a greater degree from the SAC chart results, this is in the case of Max Rate of
Climb vs. Altitude. It is seen that in mission IX, the calculated data is vastly different
from the SAC charts. Lastly, the Gross Weight vs. Ground roll distance graphs are not
very realistic. Due to the method that was used to approximate the values, a lot of
information is lost and hence a good model is not developed. Figures 12.5, 12.11, and
12.17 had good matching regions while figures 12.16 had poor matching. Figures 12.4
and 12.10 had a few well matched regions, but also a lot of poorly matched ones.
b) The C-5M’s data matches the SAC charts better than the C-5A’s data matches (in most
cases). This goes to show that the C-5M would perform better as well. The Max velocity
vs. Altitude graphs for the C-5A and C-5M are quite similar, and start disintegrating after
approximately the 23,000 ft mark. In regards to the Max Rate of Climb vs. Altitude
graphs, the C-5M performs better across mission III and V, but also displays poorly for
mission IX. Same as mentioned in sub clause ‘a’, the Gross Weight vs. Ground roll
distance graphs could have been much better, had a better approximation method been
used. In regards to max rate of climb for mission III and V, there is a 60% improvement
and 50% improvement in how well matched the charts are. It can be seen that there are a
lot more points (calculated for C-5M) that match well with the SAC chart data.
Task 13
A time-to-climb calculation algorithm plots a graph of time (x-axis) vs. the altitude (y-
axis). As taken from the source Aviation Weekv the table below displays what the altitude to time
relations are for a few values (that were considered world records). These values can be
manipulated to find the relationship between the altitude and the time taken. One major
constraint however, is to keep in mind the rate of climb (how high can an airplane go in 1
second). This is a limiting factor. The TOGW was taken as 649,680 lbs.
Using the relationship shown in Equation 13.1, and the programming code from
c5pointperf.m, the following graph (13.1) was obtained.
Equation 13.1
Table 13.1: Relationship between Altitude and Time
Altitude (feet) Time taken (seconds)
9843 4min 13sec = 253 seconds
19685 7min 27sec = 447 seconds
29528 13min 8sec = 788 seconds
39371 23min 59sec = 1439 seconds
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Figure 13.1: Time vs. Altitude relationship while keeping rate of climb as a limiting factor
The code used can be found in Appendix A.
Task 14
As mentioned in previous sections, the takeoff ground analysis was weak as it is based on
considerable approximations. Please refer to TASK 11 for a deeper explanation into the
assumptions made and estimates used. One way of overcoming these shortcomings is to use
equations (such as equations 6.94 from the Anderson textbook). Please refer to page 12 to
explore equation 6.94. One problem with using an equation such as this one is that some of the
values are difficult to find with the limited data provided. Instead, further approximations are
used to get a better model. At first, this may seem counterproductive; however, the
approximations made will decrease the error margin that was created in the previous analysis.
The new method used estimates CLmaxto to be 1.8 (greater than 1.5) and multiplies all
Tmax values at 0.7 Mlo by the ratio of thrusts 50.6/41. This hence causes the function to
integrate to a more realistic measure of takeoff ground distance analysis.
A demonstration of how this new method works better can be seen from the result of
Figure 14.1. In this configuration, a C-5A was performing mission III.
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Figure 14.1: Gross weight vs. Ground roll distance using the new method
As can be seen, the calculated graph corresponds much closer to the SAC data (when
compared with the old method used to generate Figure 12.6). Another example is shown in
Figure 14.2. This is the graph plotted considering a C-5M performing mission V.
Figure 14.2: Gross weight vs. Ground roll distance using the new method
Again, it can be seen that the graph obtained is closer to the SAC data than the one
obtained using the previous method (Figure 12.15). In both the cases shown above, there is
approximately a 5% to 10% decrease in error when the new method was used. This shows us
how lax the approximations were before, and how small approximations to counter those, can
easily affect the new model.
The code used can be found in Appendix A.
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Integral Performance
Task 15
The algorithm in c5intperf.m uses C-5A specifications and various aerospace formulas to
recreate the SAC payload-range diagram. As the first set of comments note, Mach number and
lift coefficient are fixed during cruise, and maximum Mach number multiplied by lift-to-drag
ratio are used during climb. Wing area is also constant. There are three different configurations
that need to be calculated: maximum cargo configuration, maximum fuel configuration, and
maximum range configuration.
Takeoff gross weight, fuel weight, and payload weight are specified in the Mission 3
design parameters. The fuel weight is minimized and the payload weight and takeoff gross
weight are maximized. A vector W is formed using these parameters, with the first value being
the weight at takeoff and the last being the estimated weight at landing. Cruise height and
velocity are then specified, with the velocity corresponding to climb on maximum continuous
thrust, per the mission requirements on page 6 of the SAC chart. Using atmosphere.m,
temperature, pressure, density, and the speed of sound are found for the specified cruising
altitude. Mach number and lift coefficient are calculated based on these values. Finally,
rangecalc.m takes the Mach number, lift coefficient, weight vector, cruising altitude, and wing
area to return the maximum range the C-5A can fly under the Mission 3 design parameters. In
this function, range is calculated based on thrust, thrust-specific fuel consumption, and weight.
Thrust is found as a function of the drag coefficient, calculated from the C-5A drag polar
modeled in c5polar.m, and TSFC is found using linearly-extrapolated values from data in
powerhookTF39.m. At the end of the Mission 3 algorithm, a single R3 value has been
calculated, which is the farthest the plane can fly.
Missions 5 and 9 are calculated much the same way, with variations only in the initial
weight parameters. In Mission 5, payload weight is minimized and fuel weight and takeoff gross
weight are maximized, more than doubling the maximum range. In Mission 9, payload weight is
set to zero, decreasing the takeoff gross weight and adding an additional 1000 nautical miles to
the C-5A’s range.
The SAC chart presents one set of mission rules for Range Missions 3, 5, and 9. Takeoff
and climb are performed under maximum continuous thrust, as was shown in the formulas. Fuel
considerations are made for “five minutes at maximum continuous thrust at sea level for warm-
up and takeoff,” which corresponds to the initial weight value 0.98*TOGW shown in the Matlab
algorithm. Cruise should last until only reserve fuel remains, which consists of “fuel for 30
minutes loiter at sea level at recommended endurance speeds plus 5% of initial fuel load for
reserves.” This statement corresponds to the final weight value 0.9*Wfuel shown in the
algorithm.
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Task 16
From estimates presented in Dimensions, Weights, and Missions, the RERP program will
add 12,600 lbs to the OEW, decreasing the maximum payload correspondingly in each mission.
According to Ronald Segall, the new C-5M drag coefficient was measured to be approximately
0.00104 more than that of the C-5A while in cruise, a 4% increase.
Changes made to the C-5A algorithm:
For all missions:
A new file rangecalcM.m was created, identical to rangecalc.m except for the
changes noted below.
Because of the lack of new data which could approximate the C-5M drag polar in a
similar way to c5polar.m, the C-5M drag coefficient values were assumed to be 4%
higher than the C-5A ones at all times, not just in cruise. Consequently, the drag
coefficient value returned by c5polar.m was multiplied by 1.04.
A new file powerhookCF6.m was created, identical to powerhookTF39.m except for
data changes. The thrust-specific fuel consumption for the C-5M’s CF6-80C2
engines was approximated using the propulsion characteristics summary of the CF6-
50E, at 25,000 ft. and 301 knots. This new lookup function was substituted into the
thrust-specific fuel consumption equation.
For Missions 3 and 5:
Payload weight was decreased by 12,600 lbs and a new range was calculated using
rangecalcM.m.
For Mission 9:
Takeoff gross weight was increased by 12,600 lbs. An additional W vector and CL
value was created using the new TOGW, and a new range was calculated using
rangecalcM.m.
The results are displayed in Figure 6, found on the following page.
Item A
There is very little difference between the SAC chart data (red) and the C-5A calculation
(blue). Since the slopes of both lines are nearly identical, it can be reasoned that the difference
between them is not caused by an error in the TSFC estimation, but rather a discrepancy between
the reserve fuel levels used in each plot. Most likely, the C-5A calculation provides for a slightly
larger reserve fuel level, resulting in a decreased range. Increasing the reserve fuel level from
TOGW – 0.9*Wfuel to TOGW - .092*Wfuel more closely approximates the SAC chart data.
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Figure 16.1: Unified Payload-Range Plot
Item B
The C-5M has an average range increase of 500 nautical miles over the C-5A. This range
benefit slightly increases for longer range missions, an expected result caused by the decreased
thrust-specific fuel consumption of the C-5M’s new CF6-80C2 engines. However, this increased
range comes at a cost of reduced payload; C-5Ms provide no noticeable benefit over C-5As at
ranges of less than 3000 nautical miles when carrying identical payloads. Their effectiveness is
only apparent on long-haul missions of greater than 6000 nautical miles.
Task 17
Instead of first finding the maximum endurance of the C-5A, the code was modified to
calculate the farthest a plane could fly based on the Mission 7 parameters. As such, Item B was
calculated before Item A.
To find the range of a C-5A flying under Mission 7 constraints, a new block of code was
written in c5intperf.m, conforming to the specifications listed on page four of the 1975 SAC
chart:
Takeoff weight: 670,000 lbs.
Fuel weight: 318,500 lbs.
Payload weight: 34,538 lbs.
Average speed: 231 kn.
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Initial cruising altitude: 7,500 ft.
Additionally, new files powerhookTF39B.m and rangecalcB.m were created. In the
former file, the thrust-specific fuel consumption data was modified to more accurately reflect the
lower speed that the plane would be traveling at (originally at 452 knots, modified to 300 knots).
The latter file is identical to rangecalc.m except for the different TSFC function call.
Item B
Coincidentally, with unmodified initial and final weights (0.98*TOGW and TOGW -
0.9*Wfuel), the algorithm returned a range of 4105.7 nautical miles, extremely close to the SAC
chart value of 4037 nautical miles, a 1.7% error. Consequently, the weights were left as they
were in the other range calculations.
Using the range found in Item B, an additional line of code was added to c5intperf.m to
find the travel time. After multiplying and dividing by the correct conversion factors, the flight
time was found to be approximately 15.5 hours, a fair amount shorter than the expected 17.5
hours. This discrepancy is likely the result of an inaccurate TSFC model; the model used is
based on the 30,000 ft., 300 knot curve, where the mission specifies that the plane travel only
231 knots between 7,500 and 15,000 ft. Because of the extensive length of the flight, even a
slight change in the TSFC values could cover for the two-hour mismatch.
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Appendix A
MATLAB Code used in Task 13
% Time-to-climb calculation algorithm that integrates incremental % altitude steps using the maximum rate of climb algorithm (provided % in c5pointperf.m % Tanveer Singh Chandok % Georgia Tech % 17 April 2012
clear all S = 6200; % Area in ft^2 W=649680; %TOGW (lbs) for point performance calcs known_alt = [9843 , 19685 , 29528 , 39371]; % in ft known_time = [253 , 447 , 788 , 1439]; % in seconds
h=linspace(0,33000,51); time_rate = zeros(size(h)); inv = zeros(size(h)); VRCmax=zeros(size(h)); RCmax=zeros(size(h)); CL=zeros(size(h)); Minf=zeros(size(h)); [temp0,p0,rho0,a0]=atmosphere(0, 1); for i=1:length(h) [temp,p,rho,a]=atmosphere(h(i), 1); if i==1 VRCmax(i)=fzero(@(V) VRCmaxfcn(V,h(i),W,S),450); % Tmax depends on V
nonlinearly, so need a root find else VRCmax(i)=fzero(@(V) VRCmaxfcn(V,h(i),W,S),VRCmax(i-1)); end Minf(i)=VRCmax(i)/a; Tmax(i)=thrustmaxTF39(h(i),Minf(i)); CL(i)=2*W/(rho*VRCmax(i)^2*S); D(i)=0.5*rho*VRCmax(i)^2*S*c5polar(CL(i),Minf(i),h(i)); RCmax(i)=(Tmax(i)-D(i))*VRCmax(i)/W*60; %ft/min inv(i) = 1./RCmax(i); end time_rate = integral(inv,0,33000,'ArrayValued',true); plot(inv,h/100),xlabel('Time (seconds)'),ylabel('Altitude (100 ft)')
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MATLAB Code used in Task 14
% Program to estimate C-5 point performance and compare to SAC chart data % This method uses the new approximations as detailed in TASK 14 % B. German % Georgia Tech % 23 October 2009
% Edited by Tanveer S Chandok % Georgia Tech % 17 April 2012
clc clear all load SACdata
% Configuration % ------------- S=6200; %ft^2 TOGW=743524; % Takeoff gross weight (lbs) Wfuel=318500; %lbs
W=(0.98*TOGW:-500:TOGW-0.9*Wfuel); % Max rate of climb % ----------------- h=linspace(0,33000,51); VRCmax=zeros(size(h)); RCmax=zeros(size(h)); CL=zeros(size(h)); Minf=zeros(size(h)); [temp0,p0,rho0,a0]=atmosphere(0, 1); for i=1:length(h) [temp,p,rho,a]=atmosphere(h(i), 1); if i==1 VRCmax(i)=fzero(@(V) VRCmaxfcn(V,h(i),W(i),S),450); % Tmax depends on
V nonlinearly, so need a root find else VRCmax(i)=fzero(@(V) VRCmaxfcn(V,h(i),W(i),S),VRCmax(i-1)); end Minf(i)= 1.23414634 .* (VRCmax(i)/a); Tmax(i)=thrustmaxTF39(h(i),Minf(i)); CL(i)=2*W(i)/(rho*VRCmax(i)^2*S); D(i)=0.5*rho*VRCmax(i)^2*S*c5polar(CL(i),Minf(i),h(i)); RCmax(i)=(Tmax(i)-D(i))*VRCmax(i)/W(i)*60; %ft/min end %figure(1),plot(RCmax,h/1000,RCmaxdata,hdata1/1000,'r-
'),legend('Calculation','SAC chart'),xlabel('Max Rate of Climb (ft/min)'),
ylabel('Altitude (1000 ft)')
% Max speed % --------- Vmax=zeros(size(h)); VlimitCAS=350*0.5144; % Vlimit CAS in m/s for i=1:length(h) [temp,p,rho,a]=atmosphere(h(i), 1);
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if i==1 Vmax(i)=fzero(@(V) Vmaxfcn(V,h(i),W(i),S),850); % Tmax depends on V
nonlinearly, so need a root find else Vmax(i)=fzero(@(V) Vmaxfcn(V,h(i),W(i),S),Vmax(i-1)); end % Limit max speed to 350 KCAS (knots calibrated airspeed) ams=a*0.3048; % Speed of sound in m/s ppascal=p*47.880259; % Pressure in pascals Vlimit=correctairspeed(VlimitCAS,ams,ppascal,'CAS','TAS')*3.2808399; %
ft/s if Vmax(i)>Vlimit Vmax(i)=Vlimit; end end %figure(2), plot(Vmax*0.5925,h/1000,Vmaxdata,hdata2,'r-'), xlabel('Vmax
(knots)'), ylabel('Altitude (1000 ft)'),legend('Calculation','SAC
chart','Location','SouthEast')
% Takeoff ground roll % ------------------- % Approximate analytical approach from Anderson, Chp. 6, Eqn. 6.95 Wto=W; CLmaxto=1.8; % Estimate Vstallto=sqrt(2/rho0.*Wto/S/CLmaxto); Vlo=1.1*Vstallto; Mlo= 1.23414634.*(Vlo/a0); Tmaxto=thrustmaxTF39(0,0.7*Mlo); %Thrust at 0.7*Mlo as "average" (Anderson) sg=1.21*(Wto/S)./(32.2*rho0*CLmaxto*Tmaxto./Wto)+1.1*3*Vstallto; figure(3),plot(Wto/1000,sg/1000,TOGWdata,TOdistdata/1000,'r-'),xlabel('Gross
Weight (1000 lbs)') ylabel('Ground Roll Distance (1000 ft)'),legend('Calculation','SAC
chart','Location','Northwest')
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References
NOTE: All references shown below are from the Point Performance section only
i MathWorks (‘fzero’), mathworks.com/help/techdoc/ref/fzero.html ii MathWorks (‘correctairspeedfunction’), mathworks.com/help/toolbox/aerotbx/ug/correctairspeed.html
iii Anderson (Aircraft Performance and Design), Chapter 6, Page 361
iv Anderson (Aircraft Performance and Design), Chapter 6, Page 360
v Aviation Week and Space Technology, September 21
st, 2009 edition (AviationWeek.com/awst)