performance of the c-5a galaxy and the c-5m super...

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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AE 3310 Spring 2012 Project Tanveer Singh Chandok

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



Table 4.1: Mission III

C-5A C-5M


(lbs)

732,500 732,500



Table 4.2: Mission V

C-5A C-5M


(lbs)

646,462 646,462

Maximum Payload Weight (lbs) None 5,000


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)






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






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Analysis for Mission V for C-5M






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


Payload weight = 0 lbs




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Analysis for Mission IX for C-5M






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




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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)

performance of the c-5a galaxy and the c-5m super...

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