Aeronautical math issue?

slowjoe · Jan 3, 2010

I was fooling around with my electron E6B this morning and found an interesting inconstancy between two formulas:

If I go into the P/D Alt calculator and enter an Indicated Alt of 1462ft, Baro of 30.13 and a TRUE temp of 4C you get the following:

PALT: 1252ft
DALT: 204ft

But, if you enter those numbers in Actual True Airspeed: PALT: 1252ft, INDICATED temp of 4C and a CAS of say 100 you get the following:

TAS: 100.2
Mach: 0.15
DALT: 36.9ft

A DALT difference of 167ft? I know that True Temp and Indicated Temp are different, but everything I've found on the subject indicate that below True Airspeeds of around 0.2 Mach the difference is negligible. Any Aerospace engineers out there know why the difference between the two formulas?

--Joe

gismo · Jan 4, 2010

slowjoe said:
I was fooling around with my electron E6B this morning and found an interesting inconstancy between two formulas:

If I go into the P/D Alt calculator and enter an Indicated Alt of 1462ft, Baro of 30.13 and a TRUE temp of 4C you get the following:

PALT: 1252ft
DALT: 204ft

But, if you enter those numbers in Actual True Airspeed: PALT: 1252ft, INDICATED temp of 4C and a CAS of say 100 you get the following:

TAS: 100.2
Mach: 0.15
DALT: 36.9ft

A DALT difference of 167ft? I know that True Temp and Indicated Temp are different, but everything I've found on the subject indicate that below True Airspeeds of around 0.2 Mach the difference is negligible. Any Aerospace engineers out there know why the difference between the two formulas?

--Joe

The effects of compressibility on temp readings varies with installation effects so there's no exact formula for general use.

Here's more info:

Mach Number (M) = TAS/CS
CS = sound speed= 38.967854*sqrt(T+273.15) where T is the OAT in celsius.
TAS is true airspeed in knots. Because of compressibility, the measured IAT (indicated air temperature) is higher than the actual true OAT. Approximately:
IAT=OAT+K*TAS^2/7592 The recovery factor K, depends on installation, and is usually in the range 0.95 to 1.0, but can be as low as 0.7. Temperatures are Celsius, TAS in knots.
Also:
OAT = (IAT + 273.15) / (1 + 0.2*K*M^2) - 273.15 The airspeed indicator measures the differential pressure, DP, between the pitot tube and the static port, the resulting indicated airspeed (IAS), when corrected for calibration and installation error is called "calibrated airspeed" (CAS).
For low-speed (M<0.3) airplanes the true airspeed can be obtained from CAS and the density altitude, DA.
TAS = CAS*(rho_0/rho)^0.5=CAS/(1-6.8755856*10^-6 * DA)^2.127940 (DA<36,089.24ft) Roughly, TAS increases by 1.5% per 1000ft.
When compressibility is taken into account, the calculation of the TAS is more elaborate:
DP=P_0*((1 + 0.2*(IAS/CS_0)^2)^3.5 -1)
M=(5*( (DP/P + 1)^(2/7) -1) )^0.5 (*)
TAS= M*CS [(*) If this results in M>1 - ie supersonic flight, we have to account for the shock wave ahead of the pitot tube, using Rayliegh's Supersonic Pitot equation.
Using the M from above as the first guess on the RHS, iterate:
M=0.881285 sqrt((DP/P + 1)(1 - 1/(7*M^2))^(5/2))
to convergence.] P_0 is is (standard) sea-level pressure, CS_0 is the speed of sound at sea-level, CS is the speed of sound at altitude, and P is the pressure at altitude.
These are given by earlier formulae:
P_0= 29.92126 "Hg = 1013.25 mB = 2116.2166 lbs/ft^2
P= P_0*(1-6.8755856*10^-6*PA)^5.2558797, pressure altitude, PA<36,089.24ft
CS= 38.967854*sqrt(T+273.15) where T is the (static/true) OAT in Celsius.
CS_0=38.967854*sqrt(15+273.15)=661.4786 knots

[Example: CAS=250 knots, PA=10000ft, IAT=2°C, recovery factor=0.8
DP=29.92126*((1+0.2*(250/661.4786)^2)^3.5 -1)= 3.1001 "
P=29.92126*(1-6.8755856*10^-6 *10000)^5.2558797= 20.577 "
M= (5*( (3.1001/20.577 +1)^(2/7) -1) )^0.5= 0.4523 Mach
OAT=(2+273.15)/(1 + 0.2*0.8*0.4523^2) - 273.15= -6.72C
CS= 38.967854*sqrt(-6.7+273.15)=636.08 knots
TAS=636.08*0.4523=287.7 knots] In the reverse direction, given Mach number M and pressure altitude PA, we can find the IAS with:
x=(1-6.8755856e-6*PA)^5.2558797
ias=661.4786*(5*((1 + x*((1 + M^2/5)^3.5 - 1))^(2/7.) - 1))^0.5 (for M <=1)
Some notes on the origins of some of the "magic" number constants in the preceeding section:
6.8755856*10^-6 = T'/T_0, where T' is the standard temperature lapse rate and T_0 is the standard sea-level temperature.
5.2558797 = Mg/RT', where M is the (average) molecular weight of air, g is the acceleration of gravity and R is the gas constant.
0.2233609 = ratio of the pressure at the tropopause to sea-level pressure.
4.806346*10^-5 = Mg/RT_tr, where T_tr is the temperature at the tropopause.
4.2558797 = Mg/RT' -1
0.2970756 = ratio of the density at the tropopause to the density at SL (rho_0)
145442 = T_0/T'
38.967854 = sqrt(gamma R/M) (in knots/Kelvin^0.5), where gamma is the ratio of the specific heats of air

BillTIZ · Jan 4, 2010

gismo said:
The effects of compressibility on temp readings varies with installation effects so there's no exact formula for general use.

Here's more info:

Mach Number (M) = TAS/CS
CS = sound speed= 38.967854*sqrt(T+273.15) where T is the OAT in celsius.
TAS is true airspeed in knots. Because of compressibility, the measured IAT (indicated air temperature) is higher than the actual true OAT. Approximately:
IAT=OAT+K*TAS^2/7592 The recovery factor K, depends on installation, and is usually in the range 0.95 to 1.0, but can be as low as 0.7. Temperatures are Celsius, TAS in knots.
Also:
OAT = (IAT + 273.15) / (1 + 0.2*K*M^2) - 273.15 The airspeed indicator measures the differential pressure, DP, between the pitot tube and the static port, the resulting indicated airspeed (IAS), when corrected for calibration and installation error is called "calibrated airspeed" (CAS).
For low-speed (M<0.3) airplanes the true airspeed can be obtained from CAS and the density altitude, DA.
TAS = CAS*(rho_0/rho)^0.5=CAS/(1-6.8755856*10^-6 * DA)^2.127940 (DA<36,089.24ft) Roughly, TAS increases by 1.5% per 1000ft.
When compressibility is taken into account, the calculation of the TAS is more elaborate:
DP=P_0*((1 + 0.2*(IAS/CS_0)^2)^3.5 -1)
M=(5*( (DP/P + 1)^(2/7) -1) )^0.5 (*)
TAS= M*CS [(*) If this results in M>1 - ie supersonic flight, we have to account for the shock wave ahead of the pitot tube, using Rayliegh's Supersonic Pitot equation.
Using the M from above as the first guess on the RHS, iterate:
M=0.881285 sqrt((DP/P + 1)(1 - 1/(7*M^2))^(5/2))
to convergence.] P_0 is is (standard) sea-level pressure, CS_0 is the speed of sound at sea-level, CS is the speed of sound at altitude, and P is the pressure at altitude.
These are given by earlier formulae:
P_0= 29.92126 "Hg = 1013.25 mB = 2116.2166 lbs/ft^2
P= P_0*(1-6.8755856*10^-6*PA)^5.2558797, pressure altitude, PA<36,089.24ft
CS= 38.967854*sqrt(T+273.15) where T is the (static/true) OAT in Celsius.
CS_0=38.967854*sqrt(15+273.15)=661.4786 knots

[Example: CAS=250 knots, PA=10000ft, IAT=2°C, recovery factor=0.8
DP=29.92126*((1+0.2*(250/661.4786)^2)^3.5 -1)= 3.1001 "
P=29.92126*(1-6.8755856*10^-6 *10000)^5.2558797= 20.577 "
M= (5*( (3.1001/20.577 +1)^(2/7) -1) )^0.5= 0.4523 Mach
OAT=(2+273.15)/(1 + 0.2*0.8*0.4523^2) - 273.15= -6.72C
CS= 38.967854*sqrt(-6.7+273.15)=636.08 knots
TAS=636.08*0.4523=287.7 knots] In the reverse direction, given Mach number M and pressure altitude PA, we can find the IAS with:
x=(1-6.8755856e-6*PA)^5.2558797
ias=661.4786*(5*((1 + x*((1 + M^2/5)^3.5 - 1))^(2/7.) - 1))^0.5 (for M <=1)
Some notes on the origins of some of the "magic" number constants in the preceeding section:
6.8755856*10^-6 = T'/T_0, where T' is the standard temperature lapse rate and T_0 is the standard sea-level temperature.
5.2558797 = Mg/RT', where M is the (average) molecular weight of air, g is the acceleration of gravity and R is the gas constant.
0.2233609 = ratio of the pressure at the tropopause to sea-level pressure.
4.806346*10^-5 = Mg/RT_tr, where T_tr is the temperature at the tropopause.
4.2558797 = Mg/RT' -1
0.2970756 = ratio of the density at the tropopause to the density at SL (rho_0)
145442 = T_0/T'
38.967854 = sqrt(gamma R/M) (in knots/Kelvin^0.5), where gamma is the ratio of the specific heats of air

That should teach you a lesson... NEVER NEVER EVER ask an Aero Engineer a simple question and expect a simple answer.. :mad2:

BillTIZ · Jan 4, 2010

slowjoe said:
I was fooling around with my electron E6B this morning and found an interesting inconstancy between two formulas:

If I go into the P/D Alt calculator and enter an Indicated Alt of 1462ft, Baro of 30.13 and a TRUE temp of 4C you get the following:

PALT: 1252ft
DALT: 204ft

But, if you enter those numbers in Actual True Airspeed: PALT: 1252ft, INDICATED temp of 4C and a CAS of say 100 you get the following:

TAS: 100.2
Mach: 0.15
DALT: 36.9ft

A DALT difference of 167ft? I know that True Temp and Indicated Temp are different, but everything I've found on the subject indicate that below True Airspeeds of around 0.2 Mach the difference is negligible. Any Aerospace engineers out there know why the difference between the two formulas?

--Joe

When you are computing "Act TAS", they don't care about the Baro setting, they are computing DA based on PAlt and Temp. So the "effective DA" based on the PAlt and temp can vary by your 167 ft.

In some math formula's.. that 100ft can happen with a decimal rounding issue.

But I'm not an Aero Engineer... :rofl:

:smile:

RJM62 · Jan 4, 2010

gismo said:
The effects of compressibility on temp readings varies with installation effects so there's no exact formula for general use.

Here's more info:

Mach Number (M) = TAS/CS
CS = sound speed= 38.967854*sqrt(T+273.15) where T is the OAT in celsius.
TAS is true airspeed in knots. Because of compressibility, the measured IAT (indicated air temperature) is higher than the actual true OAT. Approximately:
IAT=OAT+K*TAS^2/7592 The recovery factor K, depends on installation, and is usually in the range 0.95 to 1.0, but can be as low as 0.7. Temperatures are Celsius, TAS in knots.
Also:
OAT = (IAT + 273.15) / (1 + 0.2*K*M^2) - 273.15 The airspeed indicator measures the differential pressure, DP, between the pitot tube and the static port, the resulting indicated airspeed (IAS), when corrected for calibration and installation error is called "calibrated airspeed" (CAS).
For low-speed (M<0.3) airplanes the true airspeed can be obtained from CAS and the density altitude, DA.
TAS = CAS*(rho_0/rho)^0.5=CAS/(1-6.8755856*10^-6 * DA)^2.127940 (DA<36,089.24ft) Roughly, TAS increases by 1.5% per 1000ft.
When compressibility is taken into account, the calculation of the TAS is more elaborate:
DP=P_0*((1 + 0.2*(IAS/CS_0)^2)^3.5 -1)
M=(5*( (DP/P + 1)^(2/7) -1) )^0.5 (*)
TAS= M*CS [(*) If this results in M>1 - ie supersonic flight, we have to account for the shock wave ahead of the pitot tube, using Rayliegh's Supersonic Pitot equation.
Using the M from above as the first guess on the RHS, iterate:
M=0.881285 sqrt((DP/P + 1)(1 - 1/(7*M^2))^(5/2))
to convergence.] P_0 is is (standard) sea-level pressure, CS_0 is the speed of sound at sea-level, CS is the speed of sound at altitude, and P is the pressure at altitude.
These are given by earlier formulae:
P_0= 29.92126 "Hg = 1013.25 mB = 2116.2166 lbs/ft^2
P= P_0*(1-6.8755856*10^-6*PA)^5.2558797, pressure altitude, PA<36,089.24ft
CS= 38.967854*sqrt(T+273.15) where T is the (static/true) OAT in Celsius.
CS_0=38.967854*sqrt(15+273.15)=661.4786 knots

[Example: CAS=250 knots, PA=10000ft, IAT=2°C, recovery factor=0.8
DP=29.92126*((1+0.2*(250/661.4786)^2)^3.5 -1)= 3.1001 "
P=29.92126*(1-6.8755856*10^-6 *10000)^5.2558797= 20.577 "
M= (5*( (3.1001/20.577 +1)^(2/7) -1) )^0.5= 0.4523 Mach
OAT=(2+273.15)/(1 + 0.2*0.8*0.4523^2) - 273.15= -6.72C
CS= 38.967854*sqrt(-6.7+273.15)=636.08 knots
TAS=636.08*0.4523=287.7 knots] In the reverse direction, given Mach number M and pressure altitude PA, we can find the IAS with:
x=(1-6.8755856e-6*PA)^5.2558797
ias=661.4786*(5*((1 + x*((1 + M^2/5)^3.5 - 1))^(2/7.) - 1))^0.5 (for M <=1)
Some notes on the origins of some of the "magic" number constants in the preceeding section:
6.8755856*10^-6 = T'/T_0, where T' is the standard temperature lapse rate and T_0 is the standard sea-level temperature.
5.2558797 = Mg/RT', where M is the (average) molecular weight of air, g is the acceleration of gravity and R is the gas constant.
0.2233609 = ratio of the pressure at the tropopause to sea-level pressure.
4.806346*10^-5 = Mg/RT_tr, where T_tr is the temperature at the tropopause.
4.2558797 = Mg/RT' -1
0.2970756 = ratio of the density at the tropopause to the density at SL (rho_0)
145442 = T_0/T'
38.967854 = sqrt(gamma R/M) (in knots/Kelvin^0.5), where gamma is the ratio of the specific heats of air

That's exactly what I was thinking...

JeffDG · Jan 5, 2010

gismo said:
The effects of compressibility on temp readings varies with installation effects so there's no exact formula for general use.

Here's more info:

Mach Number (M) = TAS/CS
CS = sound speed= 38.967854*sqrt(T+273.15) where T is the OAT in celsius.
TAS is true airspeed in knots. Because of compressibility, the measured IAT (indicated air temperature) is higher than the actual true OAT. Approximately:
IAT=OAT+K*TAS^2/7592 The recovery factor K, depends on installation, and is usually in the range 0.95 to 1.0, but can be as low as 0.7. Temperatures are Celsius, TAS in knots.
Also:
OAT = (IAT + 273.15) / (1 + 0.2*K*M^2) - 273.15 The airspeed indicator measures the differential pressure, DP, between the pitot tube and the static port, the resulting indicated airspeed (IAS), when corrected for calibration and installation error is called "calibrated airspeed" (CAS).
For low-speed (M<0.3) airplanes the true airspeed can be obtained from CAS and the density altitude, DA.
TAS = CAS*(rho_0/rho)^0.5=CAS/(1-6.8755856*10^-6 * DA)^2.127940 (DA<36,089.24ft) Roughly, TAS increases by 1.5% per 1000ft.
When compressibility is taken into account, the calculation of the TAS is more elaborate:
DP=P_0*((1 + 0.2*(IAS/CS_0)^2)^3.5 -1)
M=(5*( (DP/P + 1)^(2/7) -1) )^0.5 (*)
TAS= M*CS [(*) If this results in M>1 - ie supersonic flight, we have to account for the shock wave ahead of the pitot tube, using Rayliegh's Supersonic Pitot equation.
Using the M from above as the first guess on the RHS, iterate:
M=0.881285 sqrt((DP/P + 1)(1 - 1/(7*M^2))^(5/2))
to convergence.] P_0 is is (standard) sea-level pressure, CS_0 is the speed of sound at sea-level, CS is the speed of sound at altitude, and P is the pressure at altitude.
These are given by earlier formulae:
P_0= 29.92126 "Hg = 1013.25 mB = 2116.2166 lbs/ft^2
P= P_0*(1-6.8755856*10^-6*PA)^5.2558797, pressure altitude, PA<36,089.24ft
CS= 38.967854*sqrt(T+273.15) where T is the (static/true) OAT in Celsius.
CS_0=38.967854*sqrt(15+273.15)=661.4786 knots

[Example: CAS=250 knots, PA=10000ft, IAT=2°C, recovery factor=0.8
DP=29.92126*((1+0.2*(250/661.4786)^2)^3.5 -1)= 3.1001 "
P=29.92126*(1-6.8755856*10^-6 *10000)^5.2558797= 20.577 "
M= (5*( (3.1001/20.577 +1)^(2/7) -1) )^0.5= 0.4523 Mach
OAT=(2+273.15)/(1 + 0.2*0.8*0.4523^2) - 273.15= -6.72C
CS= 38.967854*sqrt(-6.7+273.15)=636.08 knots
TAS=636.08*0.4523=287.7 knots] In the reverse direction, given Mach number M and pressure altitude PA, we can find the IAS with:
x=(1-6.8755856e-6*PA)^5.2558797
ias=661.4786*(5*((1 + x*((1 + M^2/5)^3.5 - 1))^(2/7.) - 1))^0.5 (for M <=1)
Some notes on the origins of some of the "magic" number constants in the preceeding section:
6.8755856*10^-6 = T'/T_0, where T' is the standard temperature lapse rate and T_0 is the standard sea-level temperature.
5.2558797 = Mg/RT', where M is the (average) molecular weight of air, g is the acceleration of gravity and R is the gas constant.
0.2233609 = ratio of the pressure at the tropopause to sea-level pressure.
4.806346*10^-5 = Mg/RT_tr, where T_tr is the temperature at the tropopause.
4.2558797 = Mg/RT' -1
0.2970756 = ratio of the density at the tropopause to the density at SL (rho_0)
145442 = T_0/T'
38.967854 = sqrt(gamma R/M) (in knots/Kelvin^0.5), where gamma is the ratio of the specific heats of air

It's a sad commentary on the state of teaching pilots today that you have to explain this to some people who hold certificates.

If your CFI didn't make all of this perfectly clear to you, you were seriously dis-served. If you haven't finished yet, you should fire your CFI if he can't explain this to you!

Richard · Jan 5, 2010

Great Scott!, Lance.

Now that I think about it, maybe that was Lance's birthday present to himself: to prove he still has the mettle to confound and and befuddle.

Dr. O · Jan 5, 2010

Holy density, Batman... High to low, look out below...

denny-o

Areeda · Jan 5, 2010

Wow Lance, it's going to take a while to digest that but thanks for the reference.

The one formula that I get that relates to the original post is
(mach < 0.3)
TAS = CAS*(rho_0/rho)^0.5=CAS/(1-6.8755856*10^-6 * DA)^2.127940 (DA<36,089.24ft) Roughly, TAS increases by 1.5% per 1000ft.

Given that the E6B he's using said DA was 37ft, it seems his TAS is right on. (Rho_0/rho) = 1.005 so TAS is 100.23

My favorite DA calculator is http://wahiduddin.net/calc/calc_da.htm

It calculates DA (assuming dew point of 0°F) at 156ft.

I think what's going on is which rule of thumb they are using for temperature compensation.

I use 60'/°F or 110'/°C.

So 1400' MSL should have a standard temperature of 15-2*1400/1000 = 12.2 °C so we're 8.2 °C below standard. So 1250-8.2*110 = 350, I'm even further off, but good enough for what I need.

My personal feeling is that there is no relevant difference between 200' DA and 36' DA. Especially since we're not considering humidity.

Joe

gismo · Jan 5, 2010

BillTIZ said:
That should teach you a lesson... NEVER NEVER EVER ask an Aero Engineer a simple question and expect a simple answer..

Well, I did give a "simple answer" above the detail. Did you miss that?

Aeronautical math issue?

slowjoe

Pre-Flight

gismo

Touchdown! Greaser!

BillTIZ

Final Approach

BillTIZ

Final Approach

RJM62

Touchdown! Greaser!

JeffDG

Touchdown! Greaser!

Richard

Final Approach

Dr. O

Pattern Altitude

Areeda

Pattern Altitude

gismo

Touchdown! Greaser!