Embry-Riddle Aeronautical University
Department of Aeronautical Science
AS-310 Aircraft Performance Course Plan (Section 2, 3 & 4)
Spring Semester 2015
MWF Section 2: 1030-1130 COA 259; Section 3: 1300-1400 COA-255 Section 4: MWF 1530-1639 COA-259
Credit Hours: 3
Instructor: Dr. R. Rogers
Phone: 386-226-6436
Prerequisite: AS309 (implies PS104 and MA112)
Office: COA-233
email: rogers@erau.edu
Course Web Site: webfac.db.erau.edu/~rogers/as310
Office Hours: available at
webfac.db.erau.edu/~rogers//Schedule.xls
Dr. Rogers Home Page: webfac.db.erau.edu/~rogers
Course Text: Online at Course Web Site (webfac.db.erau.edu/~rogers/as310)
Required Course Materials: 1) Online Course Text (download from course web site links); 2) Supplemental
Materials for AS310 Aircraft Performance (Dr. R. Rogers' Sections Only), available in ERAU bookstore; 3)
Online Lecture Notes (download from course web site via Course Schedule / Links to Lecture Notes); 4)
Current copy of course outline (download from course web site link); 5) Draftsman triangle, available in
bookstore
Course Description: This course explores the performance of airplanes powered by reciprocating, turboprop,
and/or jet turbine and turbofan engines. Topics studied include stability and control, weight and balance, and
performance charts and graphs.
Goal: This course of study will provide the student with an understanding of the performance characteristics of
modern reciprocating, turbo-prop, and/or jet-powered airplanes. Students will acquire a knowledge of weight
and balance; takeoff and cruise control; and airplane performance charts and curves, from which they will
extract data that maximize performance.
Learning Outcomes:
1. Analyze the variable factors affecting an aircraft’s performance.
2. Interpret basic performance curves and analyze performance data.
3. Perform weight and balance calculations and relate aircraft weight and weight distribution to aircraft
performance and operational limitations.
4. Utilizing aircraft performance theory, explain the safe and efficient approaches to obtaining
maximum performance from an airplane.
Grading: 3 tests (EA: 75%); 9 quizzes (QA: 25%). A: 90 and above; B: 80-89; C: 70-79; D: 60-69; F: 59 and
below. Each quiz grade is an average of the Quiz Worksheet score submitted as homework and the Quiz Score
resulting from in-class group work; however, if you fail to submit a substantially complete Quiz Worksheet,
your Quiz Grade will be recorded as 0 because you came to class unprepared to participate in group work.
Three bonus points for attending class (AB) are added to your course average. Initially AB = +3. For each
absence from class, AB is decremented by 1 point, with negative values of AB resulting upon the 4th and
subsequent absences. Two late arrivals to class are equivalent to an absence. . Your final numeric average is
computed as 0.75*EA + 0.25*QA + AB. It follows that the highest possible numeric average is 103
(achievable only with 100% on time attendance). Note that absences in excess of 3 result in reducing the
highest possible numeric average below 100.
Course Policies: available at http:// webfac.db.erau.edu /as310/AS310CoursePolicies.doc. Students are
responsible for reading and understanding the contents of the Course Policies document and for abiding by its
provisions.
Class Schedule: available at http://webfac.db.erau.edu/~rogers/as310/as310LectureNotes.htm
Rogers’s AS310 Notes: Part I, Page 1
Class Schedule Including Reading Assignments and Work Due Dates
Unforeseen eventualities may require modifications to the following schedule.
http://pages.erau.edu/~rogers/as310/as310LectureNotes.htm
Note: the file found at the link above has links to online course lecture notes for each lecture!!!
Rogers’s AS310 Notes: Part I, Page 2
Showing Work on Quizzes and Exams. To receive full credit for a quiz or exam answer that involves a
calculation, you must show how you obtained the answer. A correct answer without substantiating work is
awarded only half credit. (This requirement doesn’t apply for objective questions, i.e. questions where you
pick the answer from a list.)
1. When an equation is used. If the answer involves using a formula or equation, you must
a. Write the equation with the unknown on the left hand side
b. Substitute full accuracy numeric values in the equation for each known in the equation
c. Use your calculator to determine and record the unknown value to the desired accuracy.
This is the minimum sufficient work required to receive full credit; additional work, if required, need not be
written down. Steps a and b completed correctly receive half credit even if the answer in step c is incorrect.
Example: Suppose you are asked to calculate the height of a right cylinder of known volume V, where you
also know the radius r of the cylinder’s base. The equation for the volume V of a cylinder with height h and
radius of the base r is given by the equation
V =  r2 h ( r2 gives the area of the circular base of the cylinder).
If the volume of the cylinder V = 143.875 in3 and the radius of the base of the cylinder r = 2.12543 in, then the
problem should be solved as follows. Note that we have solved algebraically for h in the volume formula and
written the unknown h on the left hand side of the resulting equation.
h = V / ( r2)
h = 143.875 / (3.141592654 * 2.125432)
h = 10.13775640 in
2. When no equation applies. If the answer involves a calculation that does not derive from a formula, there
is no equation to write. In that case, just show the numbers you used to calculate your answer, followed by the
answer.
Example: suppose a performance table lookup results in two numeric values 143.75 and 21.2, where the
smaller number must be subtracted from the larger number to produce the answer you are asked for. Then
write
143.75 – 21.2 = 122.55
3. Inaccuracy due to rounding. You should always use values to the full accuracy of your calculator when
making calculations. Otherwise, rounding may result in inaccuracy in your final answer, causing it to receive
reduced credit even though your calculation method is correct.
Example: Suppose the desired accuracy in the problem in 1 above is 6 significant figures (i.e. accuracy to 4
decimal places). Then the desired answer is 10.1378 inches. Suppose we rounded  to two decimal places.
Then the answer will be calculated as
h = V / ( r2)
h = 143.875 / (3.14 * 2.125432)
h = 10.14289842 in, which rounds to 10.1429, as opposed to 10.1378
If we rounded the radius to one decimal place with  at full accuracy, an even more serious inaccuracy would
result.
h = V / ( r2)
h = 143.875 / (3.141592654 * 2.12)
h =10.38476981 in, which rounds to 10.3848 as opposed to 10.1378
Rogers’s AS310 Notes: Part I, Page 3
PROPERTIES OF THE ATMOSPHERE
Standard Atmosphere
To facilitate the study of aircraft performance, atmospheric scientists have developed the notion of a standard
atmosphere. There are several similar models; the differences between them minimal. Shown below and on
the next page are the ICAO and NACA Standard Atmosphere Tables for altitudes between sea level and
100,000’. What follows explains the information in the tables.
ICAO Standard Atmosphere Table
Rogers’s AS310 Notes: Part I, Page 4
Figure 2.1. NACA Standard Atmosphere Table
In AS30, always use the NACA table to obtain standard atmosphere values used in calculations.
Atmospheric Measurements: Pressure, Temperature, Density Ratios
Pressure. Always distinguish between static pressure and dynamic pressure. Naively, static pressure P results
from the “weight”of air. Dynamic pressure q is created by relative motion between air and a body immersed in
it and is a function of air velocity V and air density:
q =  V2/2 (where  is measured in (#-sec2)/ft4and V in ft/sec, so q is measured in #/ft2)
Rogers’s AS310 Notes: Part I, Page 5
Total pressure is the algebraic sum of static and dynamic pressure i.e., T = P + q.
Static Pressure P is the force per unit area a column of air exerts on a surface At sea level on a standard day
(15o C or 59o F),
P0 = 2116 #/ft2 = (2116 #/ft2) / (144 in2/ft2) = 14.69444 #/in2.
An alternative measure of static pressure is inches of mercury. Standard pressure at sea level is 29.92 inches
of mercury (Hg). The millibar is a unit of static pressure used by meteorologists.
The pressure ratio  (small delta) at a given altitude with ambient pressure P is defined to be
 = P / P0.
For example, 18000’ = 0.49938 in a standard atmosphere, reflecting the fact that the standard pressure at
18,000 feet is about ½ the standard pressure at sea level.
In a standard atmosphere, pressure decreases as altitude increases, so 0    1 at sea level and above.
Approximate Lapse Rate for Pressure. When less than precise atmospheric measurements are required,
pilots sometimes use the rule of thumb that pressure decreases in the atmosphere at 1” Hg / 1000 feet.
It is easy to verify that this approximate decrease is somewhat accurate in the lower standard atmosphere, up to
say 10,000 feet, and that the rule of thumb in quite inaccurate as altitude decreases, For example, the rule
predicts that 10,000’ = (29.92 – 10.0) / 29.92 – 19.92 / 29.29 ≈ 0.66578, whereas the value in the table is
0.68770, a small difference3. At 20,000 feet the approximated value is 9.92 / 29.92 ≈ 0.33155, whereas the
“correct” value is 0.45954, a large difference.
If a lapse rate of 1” Hg /1000 ft of altitude were accurate, the atmosphere would end at 29,920 feet MSL.
Temperature: Absolute temperature (absolute zero = -273oC) must be use performance calculations. Since F
= (9/5) C + 32, -273o C = 9/5(-273) + 32 = -459.4  -460o F is absolute zero in Fahrenheit.
Add 273 (460) to centigrade (Fahrenheit) to get Kelvin (Rankine). K is the symbol for Kelvin, R for Rankine.
Sea level standard temperature:
T0 = 59o F = 59+460 = 519o R.
T0 = 15o C = 15 + 273 = 288o K.
Air temperature decreases with altitude until the tropopause (36, 089’), then stays constant at –69.7o F until
about 85,000’.
Temperature ratio  (theta) at a given altitude where the temperature is T is defined as
 = T / T0.
For example, at the tropopause,  = T / To = (-69.7+460) / (59 + 460) = ((5/9) (-69.7 – 32) + 273) / (15+273) =
0.752.
A common mistake in calculations involving  = T / To is forgetting to use absolute T.
In a standard atmosphere, temperature decreases as altitude increases, so 0    1at sea level and above.
Rogers’s AS310 Notes: Part I, Page 6
Standard Lapse Rate for Temperature. The standard lapse rate for temperature in the earth’s atmosphere is
2o C (3.8o F) / 1000 feet. This lapse rate applies quite precisely up to the tropopause (the point where
atmosphere temperature stops decreasing—between 36,000’ and 37,000’ feet), at which point the temperature
stabilizes (remains the same) for many additional thousands of feet.
If  is known, T in a standard atmosphere can be calculated. Suppose  = 0.93125. Then
T =  To = 0.93125 (15+273) = 268.2 K = (268.2 – 273)o C = -4.8o C.
This is a drop of approximately 20o C from standard temperature at SL. At a lapse rate of 2 o C / 1000 feet, the
altitude in a standard atmosphere should be about 10,000 feet. You can confirm this is the case from the
NACA Standard Atmosphere Temperature, where T 10,000’ = -0.4812
Density: Density  (rho) is mass per unit volume. Mass is
 = mass / unit volume = slugs/ft3 = (#-sec2/ft) / ft3 = (#-sec2)/ft4
The density at sea level on a standard day is 0 = 0.002377 slugs/ft3.
Density Ratio  (sigma) at an altitude where the density is  is defined to be
=/0
where 0 is as above. For example, 22000’ = 0.001183 slugs/ft3 in a standard atmosphere, so
22000’ = 22000 /  0 = 0.001183 /0.002377 = 0.497686159  0.5.
From this we may conclude only that, in a standard atmosphere, density at 22,000’ is ½ the density at sea level.
(Compare this to 18,000’ for static pressure.)
In a standard atmosphere, density decreases as altitude increases, so 0    1 at sea level and above.
SMOE. The ICAO Standard Atmosphere Table has a  column. The corresponding NACA Standard
Atmosphere column gives the reciprocal of this value, i.e., 1/, referred to as SMOE, an acronym for
standard means of evaluation.
In a standard atmosphere at sea level and above,     1.0, so SMOE  1.0.
Relationship between Density, Pressure, and Temperature Ratios
From the universal gas law, density of a fixed volume of gas is proportional to P/T; i.e.  = k P / T for some k
> 0, where the value of k depends on the properties of the gas. Thus
 =  /  0 = (kP/T) / (kP0/T0) = (P/P0) / (T/T0) =  / .
This is a very important relationship, one that you will use frequently in calculations. In English, the equation
says: “Density ratio is equal to pressure ratio divided by temperature ratio.” You should confirm this fact by
performing the required calculation for several arbitrarily selected rows in the standard atmosphere table.
Note that the equation also reflects the fact that both temperature and pressure affect air density. As
temperature increases (decreases), air density decreases (increases). As pressure increases (decreases), air
density increases (decreases). That is, density is inversely proportional to temperature and directly
proportional to pressure.
Speed of Sound in Air
Rogers’s AS310 Notes: Part I, Page 7
As air temperature decreases, the speed of sound (a) in air decreases, from 661.74 KTS at sea level to 573.80
KTS at the tropopause. In other words, the speed of sound in air is a function of temperature only. This
result is non-intuitive. An incorrect but common guess is that the speed of sound in air is a function of air
density (i.e., of both pressure and temperature).
It is possible to shows that the speed of sound varies in air as the square root of the temperature ratio. Let a 0 be
the speed of sound at sea level in a standard atmosphere, i.e. 661.74 KTS. Then the speed of sound a at an
altitude where the temperature ratio is θ is given by the equation
a = a0 √ θ.
For example, if θ = 0.79374 at a given altitude, then the speed of sound at this altitude is
a = a0 √ θ = 661.74 √0.79374 = 589.56 KTS.
From the Standard Atmosphere Table, this is the speed of sound at 30,000 feet, where θ = 0.79374.
Mach Number
Mach number (after the German physicist Ernst Mach) is defined as the ratio of TAS to the local speed of
sound, i.e., the speed of sound at operating altitude. More formally, if M is Mach number, and a is the local
speed of sound, then
TAS
M = TAS / a =
.
a0 
If Mach Number is known, one may calculate the corresponding temperature ratio. For example, if M =
614.56, then
√ θ = a /a0, or θ = (a / a0)2 = (614.56 / 661.74)2 = 0.86249.
In a standard atmosphere, the temperature ratio at 20,000 feet is 0.86249 and the speed of sound is 614.56.
Variations of Atmospheric Parameters with Altitude
We have observed that temperature, pressure, density, and the speed of sound in a standard atmosphere all
decrease, while SMOE increases, with an increase in altitude. That is, the following inequalities hold in a
standard atmosphere at sea level and above:
1.
2.
3.
4.
5.
0    1.0
0    1.0
so 0    1.0
SMOE  1.0
a  661.74 KTS
Figures 2.2 – 2.4 on the next page reveal the decrease is linear only for temperature and the speed of sound.
This reflects the 2o C per thousand feet lapse rate for temperature and the fact that the speed of sound in air is a
function of temperature only. Note however that the decrease in pressure in the lower atmosphere is nearly
linear, confirming the approximate accuracy of the rule of thumb about pressure lapse rate near the earth’s
surface —1” Hg /1000 feet.
Rogers’s AS310 Notes: Part I, Page 8
Figures 2.2, 2.4, and 2.3 Respectively
Rogers’s AS310 Notes: Part I, Page 9
ALTITUDE MEASUREMENT
Different measures of altitudes are of interest to aviators and others who must understand properties of the
atmosphere. What follows assumes we are talking about a barometric altimeter, not a radar altimeter.
Indicated, Pressure, and Density Altitude
Indicated Altitude is the altitude indicated by an altimeter. It is understood that the current altimeter setting
corrected for instrument error is set in the altimeter pressure window. Recall that the pressure window is
calibrated in inches of mercury.
Pressure altitude is altitude corresponding to a given static pressure in a standard atmosphere; i.e., is
indicated altitude corrected for non-standard pressure, assuming no altimeter error. For example, if pressure is
972 #/ft2, then  = 972 /2116 = 0.4594, which corresponds to approximately 20,000’ pressure altitude in a
standard atmosphere. The observed indicated altitude might be somewhat different, depending on whether or
not the atmosphere where the measurement is taken actually corresponds to the standard atmosphere model.
To read pressure altitude directly, set 29.92 plus or minus instrument correction in the altimeter pressure
window. From this we may conclude that flight levels are pressure altitudes.
Recall that in the lower atmosphere a 1000’ altitude differential corresponds approximately to a 1.0” Hg
pressure differential. Thus, a change of .01 in the altimeter setting reflects a change in pressure altitude of 10’.
You can use this correspondence to estimate the pressure altitude of your aircraft when flying at lower
altitudes. Remember that higher pressures correspond to lower altitudes. Thus if the current altimeter setting
is higher than 29.92, your pressure altitude is lower than your indicated altitude. The converse also holds, of
course.
As an example, suppose you are on a ramp 775’ above sea level, and the field altimeter setting is 30.17. With
30.17 set in the altimeter window, assuming no instrument error, your indicated altitude will be 775’.
However, your pressure altitude is lower than 775’, corresponding to ambient pressure higher than standard.
More precisely, the pressure altitude is 775 - (30.17 – 29.92) (1000) = 775 – (.25)(1000) = 775 – 250 = 525’.
We subtracted 250 from indicated altitude because pressure is higher than standard, indicating a pressure
altitude lower than standard. If the reported pressure was lower than the standard 29.92, we would have added
the correction.
As a second example, suppose you were taking off from a seal level airport during a hurricane where the
ambient pressure is 27.42” Hg. Your pressure altitude is 0 + (29.92 – 27.42) (1000) = 2.5 (1000) = 2500’. We
added 2500’ to indicated altitude (0’) because lower pressures correspond to higher pressure altitudes.
Density altitude is pressure altitude corrected for non-standard temperature. (Recall that  = k P / T, so that
density is directly proportional to pressure and inversely proportional to temperature.) Density altitude is
typically used to construct aircraft performance charts for various altitudes, since air density most directly
relates to performance parameters such as thrust, lift, and drag.
Finding Density Altitude from Indicated Altitude and Temperature
If you know indicated altitude and temperature in C, you can find density altitude and SMOE corresponding to
it from the density altitude chart (Figure 2.5) shown on the next page.
1.
2.
3.
4.
5.
Determine pressure altitude as above.
Find the temperature along the bottom of the chart.
Move vertically upward until intersecting the appropriate curved pressure altitude line.
From the intersection, move horizontally left and read density altitude.
To find SMOE corresponding to this density altitude, proceed horizontally to the right until
intersecting the right hand axis.
In Figure 2.5, each block on the horizontal scale represents one degree C. However, the vertical scale is a little
Rogers’s AS310 Notes: Part I, Page 10
confusing: since there are 20 blocks per 5000’, each block represents 250’. It is not uncommon for people to
misread values on this chart because of the unusual y-axis scale.
Example: suppose in the hurricane mentioned above the actual air temperature is 20 o C. Since the pressure
altitude is 2500’ at sea level, we follow the 20o line upward and stop halfway between the 2000’ and 3000’
pressure altitude line. The corresponding density altitude on the left vertical scale is about 3700’. The
corresponding value of SMOE, read at the right vertical scale, is approximately 1.06. Your aircraft will
perform as if it were flying at 3700’ density altitude in a standard atmosphere. The decrease in performance
might significantly increase takeoff roll and decrease climb-rate capability.
Note also that Figure 2.5 allows finding the standard temperature corresponding to a pressure altitude:
1.
2.
Find the intersection of the (straight) standard atmosphere temperature line with the appropriate
(curved) pressure altitude line.
Proceed vertically downward and read the corresponding standard temperature on the horizontal
scale.
Confirm your understanding of this process by determining that the standard temperatures at pressure altitudes
of SL, 5000’, 1000’, and 20,000’ are respectively 15 o C, 5o C, and -25o C. Note that these figures are precisely
what would be calculated using the 2o C lapse rate per 1000’ of altitude.
Figure 2.5. Density Altitude Chart
Rogers’s AS310 Notes: Part I, Page 11
TEMPERATURE RAM RISE
Many flight performance calculations require accurate temperature data. As a consequence, most aircraft have
a system to measure air temperature and display it in the cockpit. Unfortunately, the temperature indicated in
the cockpit is often higher than the actual outside air temperature. What follows briefly explains the cause of
this discrepancy and how to deal with it.
Definitions
Ambient Air Temperature (AAT). Consider an aircraft moving through an air mass. Ambient air
temperature is the temperature of the air slightly ahead of the aircraft, before the aircraft’s interaction with the
air mass can exert an effect on the temperature of the air. Performance calculations require ambient air
temperature.
AAT is also sometimes called




Static Air Temperature (SAT)
Outside Air Temperature (OAT)
True Outside Air Temperature (TOAT)
Free Air Temperature (FAT)
You probably want to be able to recognize these alternative terms, since they are often used in discussions
about aviation. However, for testing purposes in this course, we will settle on the term ambient air
temperature and ignore its synonyms (except when a synonym is used in a figure required for a test).
Indicated Outside Air Temperature (IOAT). Indicated outside air temperature (or indicated temperature,
for short) is the temperature reading displayed in the cockpit. We desire that IOAT accurately reflect AAT.
However, as previously indicated, ordinarily the former is higher than the latter, sometimes very significantly
higher.
Ram Air Effects on Temperature. A temperature indicating system consists of a sensor, located outside the
aircraft, connected to a display in the cockpit. There are two related reasons why the display shows IOATs
that exceed corresponding AATs:
1.
2.
Even at low airspeeds, ram air friction with the aircraft and related phenomena can cause air
temperatures to rise modestly in the vicinity of an aircraft. At higher airspeeds / Mach numbers,
shock waves caused by compression of airflow across the wings and other surfaces generate heat
which can significantly raise air temperature.
Similar air friction on the sensor creates heat that can raise the temperature at the sensor above
the temperature of nearby air.
We will refer to these phenomena as ram air effects. Temperatures in the air around the aircraft may differ
significantly from point to point. Of course, we are not directly interested in any of these temperatures.
Rather, we wish to determine AAT; i.e., temperature in the surrounding air unaffected by motion of the
aircraft.
Temperature Ram Rise (TRR). Temperature ram rise (also known as ideal adiabatic temperature rise) is
just the expected temperature rise due to ram air effects for a given airspeed. Figure 11.1 on the next page is
based on wind tunnel data for a typical temperature sensor. It plots TRR as a function of Mach number; i.e.,
the figure shows how much IOAT theoretically is expected to exceed AAT for a given airspeed.
The figure reveals that TRR can be many hundreds of degrees at high Mach. Even at or below Mach 1,
temperature rises up to 50 degrees can occur, and for cruising speeds characteristic of modern transport
aircraft, TRRs in the range of 25-35o C are expected.
Note that the curve in figure is parabolic (TRR = f(M2)), reflecting the fact that temperature ram rise increases
Rogers’s AS310 Notes: Part I, Page 12
as a function of the square of air velocity. For example, at Mach 1.25, the rise is 80 oC, and at 2*Mach 1.25 =
Mach 2.5, the rise is 320oC = 22 (80) = 4 (80). As a second example, at Mach 1 the expected rise is 50 o C, and
at Mach 2 it is 200o C, which is 22(50) = 4 (50). This also reflects the fact that dynamic air pressure is a
function on V2 (q =  V2 / 2) for constant density altitude, since of course dynamic pressure (ram air) is what
influences the heat rise.
Figure 11.1. Temperature Ram Rise as a Function of Mach Number.
Relationships Between IOAT and AAT
Discrepancies in Cockpit Temperature Readings. If every aircraft’s temperature indicator obeyed the TRR
of Figure 11.1, the data could be applied directly to correct indicated outside air temperature readings.
Unfortunately, this is not the case, so a pilot must know the characteristics of his/her temperature measuring
system to make the correction. Such systems may be grouped into two categories. Note that there are two
related systems discussed in the first category below.
1.
Systems where IOAT ≠ AAT. In such systems, some proportion of the theoretical TRR must
subtracted from IOAT to obtain AAT. The proportion (no greater than one) of TRR to be
applied is known as the recovery coefficient (RC), and it ordinarily varies between 0.75 and 1.0.
For older aircraft, a typical figure is 0.80. If RC < 1.0, IOAT is called ram air temperature
(RAT):
AAT = IOAT – RC (TRR) = RAT – RC (TRR)
For systems where the entire TRR is subtracted from IOAT to get the AAT, a recovery
coefficient of 1.0 applies, and IOAT is also called total air temperature (TAT):
AAT = IOAT – RC (TRR) = IOAC – 1.0 (TRR) = IOAT – TRR = TAT – TRR
2.
Systems where IOAT = AAT, i.e. where IOAT and the AAT are the same. Obviously, this kind
of system is what every pilot wants. However, they exist primarily on expensive, sophisticated
aircraft which only a few actually get to fly.
Rogers’s AS310 Notes: Part I, Page 13
Pilot-Friendly Approaches to Correcting IOAT
In a glass cockpit, a computer will correct IOAT so that IOAT = AAT. Absent a computer-based flight
director, pilots use some kind of table or graph to find AAT from IOAT. As an example, the table below
(Figure 11.3) allows correcting IOAs to AAT for a B737-300.
To use the table
1.
2.
3.
4.
Locate the column corresponding to indicated Mach number.
Move down that column until observed indicated outside air temperature is reached.
Move left across that row to the number in column 2 (labeled 0 Mach) of this row. This is the
ambient air temperature.
The number in column one of the same row is the pressure altitude corresponding to this ambient
air temperature in a standard atmosphere.
For example, suppose indicated Mach number is 0.86 and indicated outside air temperature (IOAT)is –17oC.
Enter the .86 Mach number column and proceed down to the temperature, -17. Now go left to the second
column, and find the ambient air temperature –50oC. Just to the left of –50 is 33,000, the pressure altitude
where –50oC is the standard temperature. Finally, since the chart presents Total Temperature (TAT), you may
conclude that the recovery coefficient for this system is 1.0, and that TRR = IOAT – AAT = TAT – AAT = -17
– (-50) = 33oC, which agrees with Figure 11.1. Your OAT gauge is reading 33o too high, and if atmospheric
conditions are near standard, you are operating at FL330.
Figure 11.3. Temperature Correction Table for Boeing 737-300.
Rogers’s AS310 Notes: Part I, Page 14
AIRSPEED MEASUREMENT
Pitot-Static Systems
A pitot static system compares total pressure T = P + q) to static pressure P, using the difference to compute q.
Since q =  V2/2, it is an indirect measure of airspeed V. Note that a pitot-static system does not correct for
density variations in the atmosphere. For example, V = 600 KTS will produce a very different q and
corresponding IAS at sea level than at 40,000 feet, where air density is a little less than half sea level density in
a standard atmosphere. A schematic of a pitot-static system is shown below.
Stagnation point of airflow at tip of pitot tube
(thus total pressure T is measured inside pitot tube)
static port (static
pressure here is P)
Pitot Tube
pressure here is P,
static pressure, which
remains constant
as q increases
pressure here is T = P+q,
with q increasing as airspeed
increases
degree of diaphragm displacement
measures H – P = q, the dynamic pressure
due to airflow velocity (airspeed)
Schematic of a Pitot-Static System
IAS, CAS, EAS, and TAS
Indicated Airspeed (IAS). IAS is the airspeed shown on the airspeed indicator.
Calibrated Airspeed (CAS). CAS is IAS corrected for pitot/static system errors. These are of two types:
1.
2.
Errors in the a/s indicator itself.
Position error due to interference with airflow at the static port.
Typical errors are small. . Assuming CAS = IAS results in negligible errors for most calculations.
Figure 3.1 Typical IAS to CAS Correction Chart
Rogers’s AS310 Notes: Part I, Page 15
Equivalent Airspeed (EAS). Equivalent airspeed is CAS corrected for compressibility effects. Such effects
occur when ram air impacting the pitot tube causes the air entering it to be compressed. This error always
results in an airspeed reading that is too high.
This problem causes A/S indicators to read as much as 28 kts fast in the subsonic operating regime. Problems
are worst at altitudes 20-40 M and airspeeds in the range 300-500 KTS.
Figure 3.2. CAS to EAS Compressibility Correction Chart
Other equivalent formats exist for presenting compressibility information. Shown below are low speed
compressibility corrections in tabular format. The table contains the same information as Figure 3.2. The
lower right hand entry is absent because 350 KTS CAS exceeds Mach 1 at 36,000’.
Pres. Alt.
SL
2000
4000
6000
8000
10000
12000
14000
16000
18000
20000
22000
24000
26000
28000
30000
32000
34000
36000
150 kts CAS
0
0
0
0.5
0.5
0.5
0.5
1.0
1.0
1.0
1.0
1.5
1.5
1.5
2.0
2.0
2.5
3.0
3.0
200 kts CAS
0
0
0.5
0.5
1.0
1.0
1.5
1.5
2.0
2.5
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.5
7.0
250 kts CAS
0
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
5.0
5.5
6.5
7.5
8.0
9.0
10.0
11.5
12.5
300 kts CAS
0
0.5
1.0
1.5
2.5
3.5
4.0
5.0
6.0
7.0
8.0
9.0
10.5
12.0
13.5
15.0
17.0
18.5
20.5
350 kts CAS
0
0.5
1.5
2.5
4.0
5.0
6.0
7.5
9.0
10.5
12.5
14.0
16.0
18.0
20.5
22.5
25.0
28.0
Figure 3.4. Airspeed Compressibility Correction Table (Corrections in Knots)
Rogers’s AS310 Notes: Part I, Page 16
EAS (and not TAS) is the airspeed that determines how an aircraft “feels” when you control it, and how it
performs. Thus EAS has been termed perfect indicated airspeed; i.e., it is the airspeed a pitot-static system
would register if corrections for instrument error and compressibility were already applied. Glass cockpit
aircraft can be designed to present an EAS readout. However, most pilots don’t fly glass, and must make the
corrections themselves.
True Airspeed (TAS). True airspeed is EAS corrected for non-standard density (i.e., for density that differs
from standard density at sea level). Useful primarily for navigation, TAS has little or nothing to do with the
way the aircraft responds to control input from the pilot’s point of view. However, the increasing difference
between TAS and EAS as density altitude increases means that landing rollout and stop distance, for example,
will increase for the same EAS approach speed as the landing elevation increases.
The following derivation shows the formulaic relationship between EAS and TAS. Recall that q is dynamic
pressure:
q =  V2 / 2 and  =  / 0, so  =  0, and q =  0 V2 / 2 = (0/2) ( V2).
That is, dynamic pressure is directly proportional to (V2, which is EAS2), since (0/2) is a constant. It follows
that
(TAS)2  = (EAS)2 0 = (EAS)2, or (TAS)2 = (EAS)2 / , and TAS = EAS
1 EAS
= EAS  SMOE.



Mach Number Corresponding to Airspeed
Computing Mach Number from IAS by First Finding TAS. The steps to compute Mach number from IAS
are as follows:
1.
2.
3.
4.
Convert IAS to CAS by applying pitot-static corrections for your specific aircraft.
Convert CAS to EAS using a compressibility chart or graph, or a comparable computer.
Convert EAS to TAS using the TAS = EAS SMOE.
Determine the speed of sound a at your pressure altitude, and compute M = TAS / a. This is
TAS
equivalent to computing Mach number using the formula M =
.
a0 
Computing Mach Number from EAS without First Finding TAS. It is possible to compute Mach number
for EAS:
TAS
M = TAS / a =
.
a0 
Then since TAS = EAS /  ,  =   , and a = a0  :
M=
TAS EAS EAS 
EAS 
EAS




a
a 
a 
a0   a0 
This equation is very useful for computing Mach number quickly when EAS is known without first computing
TAS, so we will write it one more time for emphasis:
M = EAS / (ao √)
Rogers’s AS310 Notes: Part I, Page 17
Using a Graph/Chart to Find Mach Number from CAS. Ordinarily, algebraic computation of Mach
Number is not required of line pilots, since the information is typically available from graphs or charts. An
example of such a chart is Figure 3.6, shown below. To use the figure:
1.
2.
3.
4.
5.
6.
7.
8.
Observe IAS, IOAT, and pressure altitude.
Computer CAS from observed IAS using an aircraft specific chart or graph.
On the horizontal axis, locate CAS.
Move upward vertically to intersect the pressure altitude line.
Move horizontally left to read Mach number on the left vertical axis.
Move horizontally right to intersect the 0 pressure altitude reference line.
Move downward vertically to intersect the IOAT line.
Move horizontally right to read TAS on the right vertical axis.
In the example traced in the figure, 250 kts at 35,000’ and -40o C IOAT corresponds to 0.735 Mach and 430
kts TAS. Note that the figure automatically applies the correction to get AAT from IOAT, for this particular
(F14) aircraft. Figures 3.7 and 3.8 in the text are similar charts for higher airspeed/Mach values.
Note: there are three such charts in the supplemental material (from the Bookstore) required for the course.
The figure below is only the first of these three.
Figure 3.6. Navy F14 Tomcat Airspeed Conversion Chart.
Rogers’s AS310 Notes: Part I, Page 18
AIRSPEED ENVELOP FOR SUBSONIC JET
Airspeed Envelop for Subsonic Jet Aircraft. Shock waves form on an aircraft which reaches the critical
Mach Number Mcrit, the speed where airflow on the aircraft first becomes supersonic. As speed increases
beyond Mcrit, these shock waves can adversely affect control and lead to a transonic phenomenon known as
high-speed buffet. Because of control and structural integrity considerations, subsonic jet transport aircraft
must at all cost avoid entering the transonic flight region. There are also other aerodynamic issues which limit
allowable airspeed/Mach, specifically stall speed VS, and maximum allowable EAS, which is dictated by
structural integrity considerations. Transient speeds below VS produce low-speed buffet, as opposed to highspeed buffet that occurs in the transonic region.
Assume some fixed operating gross weight. Airspeeds / Mach numbers relevant to subsonic flight include:
1.
2.
3.
4.
5.
VS – minimum 1 G EAS; i.e., stall speed (invariant with altitude)
VMO – maximum operating EAS due to structural integrity (invariant with altitude)
MMO – maximum operating Mach due to high speed buffet (invariant with altitude)
VMMO – EAS associated with MMO (decreases as altitude increases)
MDF – maximum demonstrated flight Mach; it has been demonstrated by flight tests that the
aircraft is controllable at speeds up to and including this Mach (invariant with altitude)
It should be clear that MDF > MMO. Exceeding MDF conceivably could lead to loss of control, together with the
inability to recover. No one officially knows, because no test pilot has flown the aircraft faster and formally
reported the results as published data.
As already indicated, VS and VMO measured as EAS remain constant with altitude change for a given gross
weight. Also, since EAS = M a0  , and since  decreases as altitude increases, the EAS associated with
MMO increases as altitude decreases, and vice versa.
We may conclude that top speed is sometimes limited by VMO (structural integrity) and sometimes by the
VMMO associated with MMO (control considerations due to high-speed buffet). Remember that while MMO is
invariant with altitude, VMMO is not. In addition, it is impossible to sustain level flight below VS.
These ideas are reflected in the Airspeed Envelop for a transport aircraft, also called the Allowable Airspeed
Range Chart, which results when VS, VMO, and MMO (VMMO) are plotted as a function of altitude for fixed
gross weight. An example for a B767 follows. Airspeeds plotted for MMO correspond to Mach 0.88.
Figure 3.11. Airspeed Envelop for Boeing 767 at 300,000# Gross.
Rogers’s AS310 Notes: Part I, Page 19
In the figure on the previous page, the three curves together with the vertical axis enclose an area which
indicates the permissible airspeeds at various altitudes. To reiterate, the airspeed is measured in KEAS, which
is not grossly different from what the pilot observes on the A/S Indicator, since EAS is just IAS corrected for
pitot-static and compressibility errors.
Recall that VMMO is the airspeed at a given altitude corresponding to M MO. The notation min (x, y) denotes the
smaller of values x and y. In the present context, the allowable airspeed range of an aircraft operating at that
altitude may be formally defined as:
min (VMMO, VMO) - VS
Be sure you understand what the above equation means. For a given altitude



the minimum allowable airspeed is a) the stall speed (V S) at that altitude.
the maximum allowable airspeed is b) the smaller of the maximum operating airspeed (V MO) and
the airspeed (VMMO) at that altitude corresponding the maximum operating mach number.
the allowable airspeed range is just b) – a).
From Figure 3.11, some conclusions we may draw are as follows:
1.
2.
3.
4.
5.
6.
7.
8.
VMO  390 KEAS is the limiting airspeed below 20,000’ (value given in text).
At 20,000’, the airspeed corresponding to MMO = 390 KEAS = VMO.
VS  156 (this is value given in the text; I can’t read the scale that accurately).
At 20,000’, the allowable airspeed range is 390 – 156 = 234 KEAS.
At 30,000’, the allowable airspeed range is 325 – 156 = 169 KEAS.
At 40,000’, the allowable airspeed range is 260 – 156 = 104 KEAS
The airspeed corresponding to MMO equals VS at 60,000’.
At 60,000’, the allowable airspeed range is 156 – 156 = 0.
(You may read the vertical axis values slightly differently that I did for items 5 –7. And it is certain that the
aircraft is capable of operating at 60,000’.)
The altitude where VMMO (MMO) = VS is aptly called the coffin corner altitude. At this altitude, the stall speed
and the max allowable airspeed are the same, so the allowable airspeed range is zero. You need to have the
right stuff to fly at this altitude. Suffice it to say that only test pilots take airliners to their operating limits, and
then only with good reason.
Note that as the allowable airspeed range decreases, stall speed and max speed due to high-speed buffet lie
increasingly close together. There have been accidents caused by a mistaken analysis of the meaning of buffet
when flying in the airspeed envelop near the coffin corner: Does the buffet you feel on the aircraft result from
a stall or from exceeding MMO and entering shock wave stall? Increased power and forward yoke effect low
speed stall recovery. But think what would happen if you applied this technique when you were already at
Mcrit. The aircraft could enter a regime of transonic uncontrollability. A number of fatal accident have
developed from such a situation. Other airliners have lost tens of thousands of feet of altitude before
recovering, with serious injuries and even fatalities in the cabin.
The altitude where VMMO (MMO) = VMO is called the crossover altitude. In figure 3.11, the crossover altitude is
20,000’.
Figure 3.12 below is the airspeed envelop for the B767 redrawn to reflect a 2G (60o bank angle) safety margin,
to guard against effects of turbulence and resulting unanticipated angles of bank. A 60o bank increases stall
speed by a factor of 2 = 1.414, so now VS = 156 (1.414) = 221 KEAS. The coffin corner altitude has
decreased from 60, 000’ to 45,000’, and the allowable airspeed range at 40,000’ is only about 20 KEAS (240 –
221). It requires skill and vigilance (or an autopilot) to cruise safely at this altitude.
Rogers’s AS310 Notes: Part I, Page 20
Figure 3.12. Airspeed Envelop with 2G Safety Margin for Boeing 767 at 300,000# Gross
Airspeed envelopes are nice learning devices, but aren’t very useful for quick reference in the cockpit. Instead,
similar, more practically oriented information is put into a table (or in the memory of a computer) which shows
permissible maximum and minimum airspeeds at various altitudes and gross weights.
Figure 3.13 on the next page is for the B737. A 1.3 G safety margin is built into the table, and airspeeds are
IAS as opposed to CAS, for the convenience of the flight crew. It appears that missing values (for high gross
weights and high altitudes) reflect the fact that that an allowable airspeed range smaller than 13 kts IAS is
deemed intolerable.
To use the table:
1.
2.
3.
Determine gross weight and cruising altitude
At the intersection of the gross weight column and the cruising altitude row, read the maximum
and minimum allowable IAS values to stay within the airspeed envelop.
If no entry exists at the intersection, cruise at this gross weight and altitude places the airplane
outside the envelop.
Rogers’s AS310 Notes: Part I, Page 21
Figure 3.13. Maximum and Minimum B737 Airspeeds by Gross Weight
Rogers’s AS310 Notes: Part I, Page 22
SOME AERODYNAMIC PRINCIPLES
AOA, Lift, Drag, Stall, CL vs. α
Angle of Attack. AOA  (alpha).is the angle between the relative wind and the wing chord.
L
AF

D
relative wind

Lift. When air is accelerated over the upper wing surface more rapidly than over the lower surface, static
pressure on the upper wing surface is lower than static pressure on the lower wing surface in accordance with
Bernoulli’s principle. The resulting unbalanced force is called the aerodynamic force (AF). The component of
the aerodynamic force perpendicular to the flight path / relative wind is called lift. The component of the
aerodynamic force parallel the flight path is drag, and consists of both parasite and induced drag. See the
following item an explanation of drag.
Drag. Total drag (DT) consists of two components: parasite drag (DP) and induced drag (DI). Parasite drag
results from ram air pressure q =  V2 / 2. Induced drag is created when high static pressure at the lower wing
surface forces air around the wingtips to the lower pressure area at the upper wing surface. This constant flow
from lower to upper wing surface produces wingtip vortices and an associated downwash angle. AOA must be
increased to compensate for the loss of lift caused by wingtip vortices. The increased AOA creates additional
drag, called induced drag.
It is true that induced drag is drag inevitably caused when lift is increased. It is not true that all drag due to lift
is induced drag; some of it is parasite drag. That is, the component of the aerodynamic force parallel the flight
path is composed of both induced and parasite drag. The fuselage and other aircraft surfaces also produce
parasite drag, but only the wing creates induced drag.
Parameters Affecting Lift and Drag. Lift and associated drag are influenced primarily by







Dynamic pressure q in #/ft2
Air density ratio  (dimensionless)
Planform wing area S in ft2
Shape of airfoil section, especially camber
Air viscosity
Compressibility effect
AOA
Lift Equation. Dynamic pressure q =  V2 / 2 is a function of airspeed and air density. Air density is
measured in slugs/ft3, or #-sec2/ft. Since  =  0,
q =  V2 / 2 = 0  V2 / 2, with 0 a constant, and V in ft/sec.
The lift equation then is:
Rogers’s AS310 Notes: Part I, Page 23
L = CL q S = C L
0
V 2S ,
2
where CL and q are as previously defined, and S is planform wing area in ft2. V is in ft/sec. If V is to be
expressed in kts TAS, then (since there are 6076 feet in a nautical mile and 0 = 0.002377 slugs/ft3 = 0.002377
#-sec2/ft4)
 0  6076 
C V 2 S C L EAS 2 S
0.002377  6076 
.

V  S  CL
V S  L


2
L = CL
2
 3600
2

2
The unit of lift is the pound, since 0 V2 S has units
 3600

295 .37
# sec2 ft 2
ft
4
sec
2
295 .37
ft 2  # (CL and  are dimensionless).
Drag Equation. A similar derivation produces the drag equation (V in kts TAS):
D = CD q S =
C DV 2 S C D EAS 2 S

295 .37
295 .37
EAS vs. TAS. Regardless of altitude, lift depends on EAS and the coefficient of life only, regardless of
altitude. This helps explain why—as we have seen previously—that EAS is sometimes called perfect
indicated airspeed. EAS is of interest to pilots; TAS (with a few exceptions) is of interest to navigators.
Similarly, regardless of altitude, drag depends on EAS and the coefficient of drag only.
Stall. When AOA α increases beyond a certain point, air cannot negotiate the curved path it must follow along
the upper wing surface. The air flow—called the boundary layer—begins to detach from the upper wing,
causing a loss of lift referred to as stall.
Relationship between CL and α. As shown in the plot of CL vs. AOA below, the coefficient of lift CL for an
wing varies directly with AOA, When the coefficient of lift is (CL)max, the wing is producing maximum lift for
any fixed airspeed. The AOA associated with (CL)max is called αCRIT. Note that at αCRIT, also called the stall
angle of attack, the wing is not stalled; stall commences only when αCRIT is exceeded. Important point: For a
given configuration (e.g., clean or dirty), stall always occurs at the same AOA. Also, for a given configuration
and weight, 1 G stall always occurs at the same EAS regardless of bank.
Typical Plot of CL vs. 
Rogers’s AS310 Notes: Part I, Page 24
Effect of Changes in Weight, CL, and Density Altitude, &c. on Velocity
If we assume L = W in 1G steady state level flight, then V is related to W, CL, and  as follows:
V
295 .37 L

CL S
295 .37 W
(this results from solving the lift equation for V).
CL S
From this, we see that the speed required for 1G steady state level flight in a given configuration varies directly
with the square root of gross weight, and inversely with the square root of density ratio and coefficient of lift.
Let V1 and V2 be any two true airspeeds for a given aircraft. From the lift equation, again assuming that L =
W, we see that when only lift (weight) varies
V2

V1
295 .37 L 2
CLS
295 .37 W2
CLS
L2

L1

295 .37 L1
CLS

295 .37 W1
CLS
W2
.
W1
In a similar manner, if only density varies
V2

V1
295 .37 L
CL2 S
295 .37 L
CL 1 S

1
.
2
Finally, if only CL varies (e.g., because of a configuration change),
V2

V1
295 .37 L
C L 2  S
295 .37 L
C L 1  S

C L 1
C L 2
In addition, recall from AS 309 that, for steady state level turning flight, if only bank angle  changes,
V2

V1
L2
W / cos  2


L1
W / cos  1
cos  1

cos  2
G2
,since G = 1/cos).
G1
To sum up:
V2

V1
{
W2  1
,
,
W1  2
C L 1
,
C L 2
cos 1 G2
,
cos  2
G1
}
The above says a given TAS value, e.g., stall speed or the speed for maximum endurance, increases if gross
weight, density altitude, bank angle, or G force increases, and decreases if coefficient of lift increases.
Important: since EAS = TAS , all the above identities except one also hold for EAS.
Rogers’s AS310 Notes: Part I, Page 25
EAS 2

EAS1
{
W2
,
W1
C L 1
,
C L 2
cos 1 G2
,
cos  2 G1
}
Since EAS is ordinarily much closer to IAS than is TAS, this observation is important for a pilot.
Example application: suppose an airplane stalls at 155 kts TAS wings level at SL in the clean configuration
(CL = 1.5) at 100,000# gross weight. What is the clean stall speed if 1) gross weight increases to 140,000; 2)
density altitude increases to 20,000; 3) bank angle increases to 35 o? 4) What is the wings level dirty stall
speed at SL (CL = 2.6)? 5) What is the wings level clean stall speed EAS at 100,000# gross and 20,000’?
1.
V2 = V 1
W2 W1  155 140,000 100,000  183.4 KTAS .
2.
V2 = V 1
SL 20000  155 1.0 0.53281  212.3 KTAS .
3.
V2 = V 1
cos 0 cos 35  155 1.0 0.81915  171.3 KTAS .
4.
V2 = V 1
CL clean CL dirty  155
5.
155 KEAS, assuming a standard atmosphere, since EAS = TAS / SMOE, and SMOE = 1.0 at SL.
1.5 2.6  117.7 KTAS .
Note that calculations 1, 3, and 4 above apply to EAS as well as TAS, but that calculation 2 does not.
Rogers’s AS310 Notes: Part I, Page 26
BODY ATTITUDE AS A MEASURE OF AOA
An AOA indicator is one of the most important instruments in an aircraft cockpit. For example, you don’t
need to worry about stall in the dirty configuration if you just fly the prescribed angle of attack for approach:
this value is constant, while approach speed IAS varies with gross weight and bank angle, among other
parameters. As another example, the AOA for maximum engine out glide distance is invariant with gross
weight. Certainly the AOA is the critical airspeed control instrument for landing on aircraft carriers, regardless
of whether the landing is being controlled by the pilot or by ACLS (automatic carrier landing system). Due to
kinetic energy limitations of the arresting gear, approach speed is close to the stall, and must be maintained
within a range of approximately 3 knots. Too fast gets a waveoff from the landing safety officer; too slow
also gets a waveoff, to avoid settling a few feet in close and hitting the ramp.
Angle of
Attack
Pitch Angle =
Body Attitude
Angle of
Incidence
Body AOA
Velocity
Flight Path Angle
Surprisingly, man contemporary airliners don’t have AOA indicators. Instead, the cockpit crew sets body
attitude or deck angle to achieve a desired AOA. Body attitude is just the angle between the longitudinal axis
of the aircraft and the horizon, i.e., the pitch angle. For steady state flight at a given configuration, gross
weight, airspeed, throttle setting, and rate of descent or climb, body attitude can always be used together with
aircraft design geometry to determine the angle of attack.
B737-300 body angles corresponding to various flight regime conditions are given in the table on the next
page. N1 is the speed of the fan (forward blade set) in a high by-pass turbofan engine, measured in percentage
of allowable RPM. A non-linear measure of engine thrust, it is set for each engine using a cockpit gauge
resembling a tachometer, but graduated in units between 0 and 100. (More information on jet engine design
and instrumentation is presented later.)
The table shown in Figure 4.1 is useful for conducting flight with an unreliable airspeed indicator. However,
for a given flight condition, information in the table serves as a pilot reminder about the pitch angle (body
attitude), configuration, and thrust setting required to sustain that flight condition, and the associated rate of
climb or descent.
To use the table:
1.
2.
3.
4.
5.
Determine the flight regime (climb, cruise, descent, holding, final approach). Enter this major
row in the table.
Determine altitude/flight level. Enter the corresponding minor row within the major row.
Read the desired airspeed at the intersection of the airspeed column with the minor row.
Determine gross weight. Follow right from gross weight sub-row to intersect target body attitude,
climb/descent rate, and thrust setting columns in order.
At each intersection in order, read target body attitude, expected climb/descent rate, and required
thrust setting.
Rogers’s AS310 Notes: Part I, Page 27
Figure 4.1. B737-300 Body Attitude Table
Rogers’s AS310 Notes: Part I, Page 28
TOTAL DRAG (DT) AND THRUST REQUIRED (TR) CURVES
Recall that DT = DI + DP. Induced drag, resulting from wingtip vortices, is high at high angles of attach and
low at low angles of attack. Parasite drag, due to dynamic pressure q = ρV2/2, is high at high velocity and low
at low velocity. The total drag curve for an airplane is just the scalar sum of its induced and parasite drag
curves. An example is shown below.
For any total, induced, or parasite drag curves, the following constraints apply:
1.
2.
3.
4.
5.
For a given type airplane (in this case the T38 Talon)
In a fixed configuration (in this case clean)
At a fixed gross weight (in this case 10,000 # gross)
At a fixed density altitude (in this case SL)
Steady-state straight and level flight
T38 Induced, Parasite, and Total Drag Curves (SL Clean at 10,000 Gross)
Rogers’s AS310 Notes: Part I, Page 29
If we assume TR (thrust required) = D (drag) in straight and level flight, then the D T curve is also the TR curve.
Thrust Required Curve for T38 (SL Clean at 10,000 Gross)
Minimum Point. L/D is called the lift-drag ratio. The minimum point on the Thrust Required Curve occurs at
the airspeed for (L/D)max. Thrust at this airspeed is the minimum possible for maintaining straight and level
flight. At lower thrust settings, the aircraft cannot maintain altitude. We see in the next section that the best
power-off glide airspeed occurs at (L/D) max, i.e. where the lift-drag ratio is maximum.
Back Side of the Thrust Curve. The part of the curve corresponding to airspeeds below the airspeed for
(L/D)max, so called because in this airspeed regime it is necessary to increase thrust to maintain level flight as
airspeed decreases. The reason this happens is that induced drag in this airspeed range is increasing faster than
parasite drag is decreasing. More drag means more thrust required to overcome it.
Region of Positive Control. The part of the curve corresponding to airspeeds above the airspeed for (L/D) max.
In this region, more power is required to fly faster. Induced drag is decreasing more slowly than parasite drag
is increasing.
Region of Neutral Airspeed Stability. This is the airspeed region near the (L/D) max airspeed, i.e. around the
bottom portion of the curve. Here the slope of the curve is near 0. A small increase or decrease of thrust
results in a significant steady-state airspeed change. As a result, it is challenging to maintain airspeed
precisely while flying in this region.
Shock Wave (Transonic) Drag Increase. The sharp increase in total drag around 625 KTS (Mach 0.95)
results as the T38 reaches the critical Mach number and shock waves begin to form on the wing.
Rogers’s AS310 Notes: Part I, Page 30
POWER OFF GLIDING FLIGHT
Fundamentals
From the vector diagram for power off gliding flight shown below:
L = W cos a
D = W sin a
Since sin a / cos a = tan a, cos a / sin a = 1 / tan a, and it follows that L/D = 1 / tan a
AA
a (glide angle)
L
D
W cos a a
W
W sin a
glide path
GD
a
GD is the glide distance, AA is the absolute altitude (altitude above ground level), and a is the glide angle. W
is the weight of the aircraft. D is drag. Because the aircraft is in steady state power off gliding flight, L = W
cos a, and W sin a = D.
The glide ratio GR is defined as GR = GD/AA, i.e. the ratio of glide distance to absolute altitude.
Because the small and large triangles in the above figure are similar (both are right triangles with the same
acute angle a, the ratio of any two sides in one triangle is equal to the ratio of the two corresponding sides in
the other. It follows that
GR =
GD W cos a
1
L


 .
AA W sin a tan a D
Note that the glide ratio is not a function of the weight of the aircraft. Also, the best glide ratio is obtained
when flying at the airspeed corresponding to (L/D) mx. This airspeed, which varies with weight, is always
obtained at the same angle of attack. When the ratio L/D is maximum, the glide angle a is minimum.
Thus we may calculate glide distance using any one of the following identities.
GD = GR (AA) = (L/D) (AA) = (1/tan a) (AA)
Low Point on TR Curve. We noted earlier that the airspeed corresponding to the lowest point on the DT / TR
curve corresponds to the airspeed for (L/D)max, i.e. the best glide speed. Now we prove this claim. The
Rogers’s AS310 Notes: Part I, Page 31
identity of the two curves depends on the assumption that T = D in steady-state straight and level flight. If this
is so, then if follows that L = W. Then
L/D = W/D, or D = W / (L/D)
That is, if L/D is maximum for a given weight, then drag must be minimum. That is,
Dmin =
W
L D max
This identity applies for any arbitrary gross weight, i.e. L/D is maximum when D is minimum.
The associated power off glide velocity V will vary with weight, but the AOA for (L/D) max is constant for a
given aircraft and configuration.
Suppose weight is 20,000# and (L/D)max = 11.0. Then
Dmin = W / (L/D)max = 20,000# / 11.0 = 1818.18#.
Example Glide Distance Calculations
Example 1. Suppose (L/D)max for an airplane is 12.0 in the clean configuration and 5.0 in the dirty
configuration. Then the maximum glide distance from 10,000’ AGL in the clean configuration is
GD = AA (L/D)max = 10,000feet (12.0) = 120,000’ = 120,000feet / 6076 feet/nm = 19.75 nm.
In the dirty configuration, the maximum glide distance from 10,000’ is (10,000feet (5.0) / 6076 ft/nm = 8.23
nm.
Example 2. Suppose an airplane can glide 50 nm from 20,000 feet above the ground. To two decimal places,
the glide ratio is
GR = GD / AA = (50 nm) (6076 ft/nm) / 20,000 ft = 15.19
Example 3. A plane glides at an angle of 8o. From 15,000 feet, how far can the plane glide in feet? What is
the glide ratio? The answers should be accurate to two decimal places.
GD = GR (AA) = (1/tan a) (AA) = (1/tan 8) (15,000 feet) = 106,739,5458 (106.739.55 to two decimal places.
GR = 1 / tan 8 = 7.115369723 =GD / AA = 106,739,5458 / 15,000 = 7.115369720 = 7.12 to two decimal
places. (Note that GR has been calculated in two distinct ways, one using the glide angle and the other using
the glide distance.)
Rogers’s AS310 Notes: Part I, Page 32
DRAG POLARS
The drag polar for a given airplane in a given fixed configuration plots CL vs. CD for various airspeeds/AOAs.
From a drag polar, it is possible to find (L/D) max for that configuration.
Given a drag polar plot, find (L/D) max as follows:
1.
2.
3.
4.
5.
Locate the origin of the coordinate system.
From the origin, draw a tangent line to the curve on the graph
Locate 3 to 5 points on the tangent line, and record CD (x-axis) and CL (y-axis) values
corresponding to each point.
For each point located in step 3, calculate the quotient C L / CD
Average the quotients calculated in step 4, omitting any quotient that is obviously inaccurate.
(All quotients theoretically should be the same. If one or more are outliers, omit such value(s)
from the average. If two or more out of five points do not lie close together, you have made an
error. Start over.
The value you have calculated in step 5 above is (L/D)max. Why is this true? For any fixed airspeed in steady
state level flight, the ratio CL/CD = L/D. It follows that any line drawn from the origin that intersects the curve
has slope L/D corresponding to the points on the curve intersected by the line. Of all such lines intersecting
the curve, the tangent line has the steepest slope, i.e. the largest value for L/D.. Thus it’s slope is (L/D)max.
Note that while it is possible to find (L/D)max from the drag polar, it is not possible to find the associated
airspeed. That airspeed can be found from the corresponding total drag or thrust required curve. To sum up
this distinction:

From the drag polar one can find (L/D) max but not the associated airspeed.

From the total drag / thrust required curve, one can find the airspeed corresponding to (L/D) max but
not the value of (L/D)max,
Example. Using the drag polar in Figure 4.5 (next page), find (L/D)max for the B767 in the clean
configuration.
CD (x-axis)
0.01
0.02
0.03
0.04
0.05
CL (y-axis)
0.195
0.4
0.58
0.78
0.975
CL / CD
19.5
20.0
19.3
19.5
19.5
The average of the quotients (with 20.0 omitted) rounds to 19.5, so (L/D) max = 19.5 for this model of the
Boeing 767 in a clean configuration.
The airplane has a twin engine failure during a trans-Atlantic flight. A suitable landing field in Iceland is 110
NM away. Can the plane glide there no thrust to land?
GD = GR (AA) = 19.5 (40,000 feet) / (6076 feet/NM) = 128.4 NM.
110 NM is long glide for a big airplane, but it’s possible to make Iceland. By contrast, the glide ratio of a
C172 is something like 11 to 12. Can you explain why the B767 is a better glider than the Chicken Hawk?
Rogers’s AS310 Notes: Part I, Page 33
CD (x-axis)
0.01
0.02
0.03
0.04
0.05
CL (y-axis)
0.195
0.4
0.58
0.78
0.975
CL / CD
19.5
20.0
19.3
19.5
19.5
Black values are chosen; red values are looked up (column 2) or calculated (column 3).
Figure 4.5. B767 Low Speed Drag Polar, Clean Configuration
Rogers’s AS310 Notes: Part I, Page 34
Example 2. Sometimes the origin of the coordinate system must be extrapolated from the intersection of the
nominal x and y axes.
CD (x-axis)
0.04
0.06
0.08
0.10
0.11
CL (y-axis)
0.565
0.87
1.15
1.45
1.61
CL / CD
14.1
14.5
14.4
14.5
14.6
Black values are chosen; red values are looked up (column 2) or calculated (column 3).
The average of the quotients rounds to 14.4, so (L/D) max = 14.4 for this model of the Boeing 767 at flaps 15.
From 40,000 feet the B767 at flaps 15 can glide GD = GR (AA) = 14.4 (40,000 feet) / (6076 feet/NM) = 94.8
NM.
Lowering flaps increases camber and hence increases lift. Can you glide farther by extending your flaps when
you lose thrust? Explain the change in B767 glide distance from 40,000 feet when flaps are extended.
Figure 4.14. B767 Drag Polar at with Flaps-15, Gear Up
Rogers’s AS310 Notes: Part I, Page 35
The family of high speed B767 drag polar shown below reveals that L/D decreases as airspeed increases. One
can also infer from this graph that glide ratio goes down as airspeed increases above the airspeed for (L/D)max.
This is seen in the decreasing slopes of tangents from the origin to each curve as airspeed (Mach number) goes
up. It’s also true that glide ratio goes down as airspeed is decreased below the airspeed for (L/D) max.
Figure 4.6. Clean Configuration High Speed Cruise B767 Drag Polars
Rogers’s AS310 Notes: Part I, Page 36
JET ENGINES
Thrust Production. A jet engine produces force (thrust) by accelerating air, in accordance with Newton’s 2 nd
law (action and reaction). Recall that acceleration is rate of change of velocity with respect to time, so that
constant acceleration a is just the velocity change V divided by the elapsed time t.
F (force) = T (thrust) = m (mass) a (acceleration) = m (V/t) = (m/t) V.
Since m is measured in slugs (#-sec2/ft), t in sec, and V in ft/sec, the unit of thrust is the pound
 # sec2 1 ft


 #  . The term (m/t), which measures the rate of flow of the air mass, is known as mass
 ft

sec sec


flow. V here is the velocity change of the air mass flow, not aircraft velocity. Note that high mass flow and/or
high velocity change reflect high thrust..
The density of air varies indirectly (and very significantly) with pressure altitude and temperature, affecting the
mass of air accelerated by a jet engine to produce thrust. As a consequence, the same engine will produce
much more thrust at Daytona Beach on a cold day than, say, at Denver International (approximately 5000’
elevation) when the temperature is 40o C. Accordingly, takeoff rolls (and other performance parameters) will
vary drastically with changes in ambient conditions. In my own experience, actual takeoff rolls (using military
thrust) for the same aircraft at the same gross weight have varied from as little as 2500’ as SL on a very cold
day to as much as 10,600’ at 4000’ elevation on a very hot day.
Pure Jet vs. Turbofan. A pure jet engine consists of a compressor section; a combustion section where fuel is
injected and burned; and a turbine section, where the force of exhaust gasses is captured to power the
compressor. See below. N or NC is the symbol usually chosen to represent compressor RPM. As shown, pure
jet engines with dual stage compressors exist. In this case, N1 and N2 are used respectively to denote the
forward (low speed) and aft (high speed) compressor RPM. The low speed (high speed) compressor is
powered by the low pressure (high pressure) turbine. Air flow in both figures is from left to right. As
discussed shortly, N, NC, N1, and N2 are all potential measures of thrust being developed by a pure jet engine.
Figures 6.10 and 6.11. Schematics of Pure Jet Engine with Single Stage and Dual Stage Compressor
Rogers’s AS310 Notes: Part I, Page 37
A more recent innovation in jet engines is the turbofan bypass engine, where a significant proportion of the
intake air is compressed and then caused to bypass the hot section of the engine. In this circumstance,
additional thrust results from accelerating the compressed air, in accordance with Newton’s second law. The
figure below depicts a three-stage high by-pass turbine engine. The first stage is the bypass compressor, while
the second and third constitute a two stage compressor similar to the figure at the bottom of the previous page.
Here there are three different RMP measurements which potentially can be used to measure performance: N 1,
N2, and N3. In the figure, airflow is again from left to right.
Figure 6.13. Schematic of Turbofan Engine with Multi-Stage Compressor.
Instruments to Measure Thrust. In AS310, we need to be aware of the various cockpit instruments which
allow us to determine how much thrust a jet aircraft engine is generating. Failure to understand and utilize
these instruments correctly has resulted in a number of fatal aircraft accidents in the past, and will probably
continue to do so in the future. Instruments to measure thrust include
1.
Engine Pressure Ratio (EPR). This is the ratio of air pressure leaving the engine to air pressure
entering the engine. Higher EPR means higher thrust.
2.
Exhaust Gas Temperature (EGT). This is the temperature of air leaving the engine. Higher EGT
means higher thrust, assuming the engine is operating correctly. However, high EGT is also
associated with abnormalities such as compressor or turbine failures, or engine fires. In such a
circumstance, EPR will usually be abnormally low, reflecting the fact that reduced thrust is being
produced.
3.
Engine Revolutions per Minute (RPM). Depending on the type of jet engine being discussed,
this is the rotational speed of the engine at some designated location, ordinarily some stage of the
compressor, or the turbine. Instead of being measured in terms of absolute revolutions, it is
measured in terms of percent of allowable revolutions, so the gauge reading RMP is graduated
from 0% to 100% (with a small region above 100% available to show overspeeds.) High RMP is
associated with high thrust. The relationship between RPM and thrust is not linear. In fact, for
many jet engines, half of the available thrust is produced in the last 10% - 15-% of engine RPM,
i.e., at or above 85% RPM. Jet engine idle RMP is quite high, perhaps in the region of 70%, and
very little thrust is produced at idle.
4.
Fuel Flow, measured in # of fuel / hour. If the engine is working correctly, high fuel flow is
associated with high thrust. For a given throttle setting, the fuel control unit on a jet engine is
designed to meter fuel to the engine based on the density of air entering the intake. Thus a climb
with the throttle set at 100% RPM might begin with 10,000 #/hr of fuel flow at SL, and terminate
at 40,000’ at 3000 #/hr.
Rogers’s AS310 Notes: Part I, Page 38
Figure 6.9 indicates where these measurements might be taken in a two-stage turbine engine.
Figure 6.9/ Engine Performance Measurement in Typical Two-Stage Turbine Engine
Figure 6.8. Cockpit Instruments to Measure Turbine Engine Performance
Parameters that Affect Thrust. RPM, density altitude, and airspeed, among other parameters, affect thrust
Rogers’s AS310 Notes: Part I, Page 39
developed by a jet engine. While developing quantitative models to predict thrust under different conditions is
a challenge, empirical data reflects how thrust characteristically changes as a function of these variables.
Variation of Thrust with Airspeed. Figure 6.1 illustrates the non-intuitive fact that, for constant RPM and
density altitude, thrust is fairly constant as velocity changes at across the airspeed operating range. Air
compression as airspeed increases causes an increase in air density and hence in thrust.. However, this effect
is offset by the fact that exhaust exit velocity is limited, so that air mass acceleration—the difference between
intake and exhaust velocity—decreases with an increases of velocity. Since power is work / time = force x
distance / time = thrust x distance / time = thrust x velocity, the power curve in the figure is also close to linear
Figure 6.1. Thrust vs. Velocity for Jet Turbine Engine in Mid Velocity Range
Variation of Thrust with RPM. Figure 6.2 illustrates the non-linear relationship between RPM and. thrust.
The curve is exponential of the form Nx, where x ≈ 3.5 for this engine.
Figure 6.2. Thrust vs. Compressor Speed (N) for Typical Non-Bypass Turbine Engine
Thrust Specific Fuel Consumption (TSFC). TSFC is the pounds of fuel used per hour for each pound of
thrust produced by a jet engine.
Rogers’s AS310 Notes: Part I, Page 40
TSFC = (fuel flow in # /hour) / (# of thrust produced) = FF / T.
TSFC varies with altitude, airspeed, and engine RPM, among other factors. Counter-intuitively, low RPM
does not result in efficient use of fuel. In fact, fuel efficiency increases as RPM increases. Figure 6.4 graphs
TSCF as a function of engine RPM. The curve reveals that TSFC decreases as RPM increases; i.e., higher the
RPM, the more thrust is obtained for a given expenditure of fuel.
As an example of this principle, a Navy anti-submarine plane commander was able to get 45 minutes more onstation time in a twin-engine jet airplane than other pilots using the same amount of fuel. What he did was to
run one engine at idle, and the other at high power, thus reducing overall TSFC (fuel flow). In other words, to
get total thrust T, instead of operating each engine at thrust T/2, he operated one engine at very low thrust and
the other at high thrust.
Figure 6.4. Specific Fuel Consumption vs. Engine RPM for Typical Turbine Engine
Figure 6.3 graphs (for a turbojet engine) thrust, TSFC, and density ratio as a function of altitude. Since density
decreases with altitude, the mass of air entering a jet intake decreases as altitude increases. Hence power at a
fixed airspeed also decreases. The decrease is not quite proportional to the density ratio  =  / 0. A value of
0.8 is more accurate for the data given in Figure 6.3. Note that the density altitude and thrust ratio curves have
approximately the same shape.
As illustrated in the figure, TSFC for a turbojet at fixed RPM decreases pretty much linearly as a function of
altitude up to the tropopause. That is, at higher altitudes up to the tropopause, one gets more thrust for the
same amount of fuel flow. This is one reason why jet airplanes achieve more range at high altitudes than at
low. The other reason is that the TAS/EAS ratio for fixed EAS—which (along with wind and fuel available)
determines range—grows increasingly large as altitude increases.
However, the question of how high to climb to achieve maximize rang—as in a divert, for example—is not
easily addressed, since climbing uses much more fuel than cruising, and cruising than descending.. A longer
climb implies a shorter cruise distance and a longer glide. Thus to optimize total fuel consumption, one must
weigh high fuel flow in the climb against lower fuel flow during cruise and very low fuel flow during descent.
Without computer assistance, this is an art, not a science, but an art based on principles. No doubt computers
in the newest airplanes suggest solutions to such problems to grateful pilots. But many airplanes still in use
today do not. You should be prepared to solve such problems yourself.
Note: TSFC for a high bypass turbofan does not necessarily decrease as altitude increases. We have so far
been unsuccessful in obtaining specific data for a high bypass turbofan TSFC variance with altitude, but
Rogers’s AS310 Notes: Part I, Page 41
anecdotal information from both engines and pilots indicates that TSFC for such an engine does NOT decrease
with altitude. In fact, it may in fact increase slightly. This result is not necessarily non-intuitive. Most of the
thrust from a high bypass turbofan comes from the fan, which is essentially a shrouded propeller, and
propellers are more efficient in dense air at low altitude than in at high altitude.
One implication of this observation is that increasing cruise altitude in a high bypass turbofan airliner gives
more TAS for the same EAS, but perhaps slightly less excellent fuel specifics. Thus cruise altitude for a given
gross weight is quite specific. As a friend Bill Roberson, Boeing Chief Test Pilot for Safety, has observed in
an email he sent me: “The contrast I have seen from the turbojet to the turbo fan is how critical it is to be at the
optimal altitude. Two thousand feet high is just as bad, if not worse than being 2000 feet low for fuel
consumption. In the turbojet, or even the low bypass ratio turbofan, we always were better off going higher.”
Figure 6.3. Thrust vs. Density Altitude for Typical Turbine Engine
Rogers’s AS310 Notes: Part I, Page 42