24-04-2012, 03:40 PM
TOTAL TEMPERATURE SENSORS TECHNICAL REPORT
[attachment=20718]
1. INTRODUCTION
Modern jet power aircraft require very accurate
measurements of Outside Air Temperature (OAT) for inputs
to the air data computer and other airborne systems. What
does OAT mean? There are actually four temperatures of
concern. These are defined as follows:
Static Air Temperature (SAT or Ts)
The temperature of the undisturbed air through which the
aircraft is about to fly;
Total Air Temperature (TAT or Tt)
The maximum air temperature which can be attained by
100% conversion of the kinetic energy of the flight;
Recovery Temperature (Tr)
The adiabatic value of local air temperature on each portion
of the aircraft surface due to incomplete recovery of the
kinetic energy; and
Measured Temperature ™
The actual temperature as measured, which differs from Tr,
because of heat transfer effects due to imposed
environments.
For flight conditions in clear air at low altitudes and low
airspeeds the four temperatures are practically the same and
OAT can apply to any of these four temperatures. As the
airspeed and altitude increase the four temperatures will
differ, and the term OAT becomes meaningless.
Figure 1 illustrates the general relationship between the four
temperatures. The first three can be related by equations to
flight speed, as will be discussed in the next section. The
measured temperature, on the other hand, must be defined
at a particular flight condition. Its value may be higher or
lower than TAT or Tr, due to sensor design, location or
imposed heat transfer environment. If a severe environment
is imposed, the value of Tm can fall below that of SAT (e.g.,
vortex-producing designs or for severe weather).
The static air temperature is difficult to measure accurately,
and total air temperature, by definition, can never be
measured exactly (100% energy conversion). We are then left
with a design objective for total air temperature sensors to
impose an environment which will make Tm approximately
equal to the ideal TAT value for all flight conditions.
As will be shown later, the ratio of TAT to SAT (Tt/Ts) is
known for each flight condition. Thus, if Tm is close enough
to TAT, SAT can be calculated with greater accuracy than it
can be measured. In general, our total air temperature
sensors exhibit a Tm V 0.995 Tt.
2. PERFORMANCE PARAMETERS
A thorough understanding of the application of total air
temperature sensors involves an acquaintance with a
number of performance parameters. Many of the
parameters depend on Reynolds number and Mach number
as described in the referenced literature. In turn, these two
parameters are dependent upon the flight condition; that is,
the speed and altitude of the aircraft.
In the past, our technical literature introduced the usage of
total pressure behind the normal shock, which was valid but
somewhat difficult to work with. This bulletin utilizes
terminology familiar to all who work with aircraft: Mach
number and air density. To maintain the non-dimensional
concept, density ratio (ρ/ρ0) will be used. The density ratio
variation with altitude for the 1962 U.S. Standard
Atmosphere is given in Table 1.
Total air temperature sensor performance, given the
aircraft’s flight envelope in terms of Mach number and
altitude (converted to ρ/ρ0 from Table 1), now is directly
obtainable. Figure 2 shows how the parameters relate in
subsonic flight. The free stream values of M and (ρ/ρ0)
apply in subsonic flight for a properly located total air
temperature sensor; the product being a parameter upon
which the correction of measured temperature to recovery
temperature depends.
TOTAL TEMPERATURE SENSORS TECHNICAL REPORT 5755
1
Figure 1: Temperature Relationships in Flight
In supersonic flow, a correction must be made for the normal
shock wave which forms just upstream of the sensor inlet. The
shock wave causes the Mach number to drop to a subsonic
value at the inlet, but the local air density behind the shock
wave increases. Table 2 lists values of the correction factor for
twenty-two values of flight Mach number. Using a subscript to
denote conditions downstream of a normal shock, the
correction factor is the product of M1/M and ρ1/ρ. To
determine the performance of a total air temperature sensor for
a supersonic flight application select specific values of Mach
number M and multiply these by the ρ/ρ0 values from Table 1
for the altitude appropriate application. Then correct the M
(ρ/ρ0) product to M1 (ρ1/ρ0) using Table 2. Parameter Z (zeta)
will be used for brevity: Z=M1(ρ1/ρ0).
For a particular flight condition or set of M and (ρ/ρ0) values,
sensor design variations (even seemingly insignificant
variations in the sensor geometry) yield detectable variations in
sensor performance. These design-dependent variations are
categorized by a number of performance parameters which
have been used extensively in various technical reports and
design specifications. The important parameters are defined
and discussed in the following paragraphs.
2.1 THERMAL RECOVERY
The relationship between total and static temperature, absolute
units (°K or °R), is:
TOTAL TEMPERATURE SENSORS TECHNICAL REPORT 5755
2
Table 1: U.S. Standard Atmosphere: ρ/ρ0 Density Ratio Versus Geopotential Altitude, 1962 Values
Figure 2: Subsonic Flight in U.S. Standard Atmosphere
Equation 1
tT γ - 1
tS 2
= 1 + m2
Table 2: Multiplier Values for Correcting M (ρ/ρ0)
to M1 (ρ1/ρ0) = Z (Normal Shock Theory)
1.0 1.0000
1.1 .9690
1.2 .9419
1.3 .9166
1.4 .8929
1.5 .8703
1.6 .8489
1.7 .8281
1.8 .8080
1.9 .7887
2.0 .7700
2.1 .7516
2.2 .7339
2.3 .7168
2.4 .7001
2.5 .6840
2.6 .6684
2.7 .6533
2.8 .6388
2.9 .6247
3.0 .6109
3.5 .5493
FREE sTREAM
mACH nUMBER mULTIPLIER
Where the γ is the ratio of specific heats.
This is the equation programmed into air data computers to
calculate SAT from Tm values which are very close to TAT.
If the total temperature sensor is designed such that there
are no significant heat sources or thermal paths to heat
sinks and such that flow over the sensing element is
uniform and continuous, the sensor will indicate the
adiabatic value Tr very closely.
One parameter which relates Tr to TAT and SAT is called the
recovery factor, defined as follows:
Some geometric shapes have constant recovery factors
(e.g., the classic flat plate with airflow parallel to its
surface). For sensors with a constant adiabatic recovery
factor, Equation 1 and 2 can be combined.
However, most sensors exhibit a variable recovery factor,
and it is more convenient to use a recovery correction
defined as:
Our total temperature sensors exhibit a variable η for M
values below 1.0 but constant η for M values above 1.0.
Once Tr and η are known, Tt is calculated by:
The relationships between η and r are given by:
For non-adiabatic conditions, additional parameters are
involved in calculating TAT and SAT.
2.2 THERMAL CONDUCTION
A condition error may occur when the fuselage of the
aircraft is at a different temperature than the sensor. This
usually occurs on the ground during taxi conditions on hot
summer days or in the winter. Our total air temperature
sensor is designed to have its air inlet located in the free
stream beyond the aircraft boundary layer. Thus, during
flight, this design provides sufficient internal airflow over
the sensing element to negate the conduction mode of heat
transfer, when the Mach number is below 2, the altitude is
below 50,000 feet and Z (zeta) is above 0.15.
2.3 THERMAL RADIATION
When the total temperature being measured is relatively
high, heat is radiated from the sensing element, resulting in
a reduced indication of temperature. This effect is increased
at very high altitude for a sensor of simple design, where
the low air density decreases the ability of the internal air
flow to make up for this radiation heat loss. Radiation error
is negligible for our multiple shielded total air temperature
sensors when the Mach number is below 2, the altitude is
below 50,000 feet, and Z (zeta) is above 0.15.
2.4 TIME CONSTANT
An instantaneous response by a sensor to a temperature
change is impossible due to the heat capacity of the sensor
parts and surrounding structure. This results in an
indicated temperature/time transient. The time constant is
a performance parameter typically used to describe this
temperature/time transient.
The time constant is classically defined as the time required
for the sensor to respond to 63.2% of an instantaneous
(step) change in temperature, Figure 3-A. Thus time
constant is an index of how rapidly the temperature sensing
element can follow a changing temperature. Something
close to a step change in temperature may occur as the
aircraft emerges from a cloud bank into clear air.
A more common type of temperature change in flight is the
gradual change or ramp change, reference Figure 3-B. This
type of change occurs when the aircraft changes altitude or
speed. In this case the sensor begins its response to the
TOTAL TEMPERATURE SENSORS TECHNICAL REPORT 5755
3
Equation 2
tR - tS
tT - tS
R =
Equation 3
tR γ - 1
tS 2
= 1 + R ( ) m2
Equation 4
tT - tR
tT
η =
Equation 5
tR
1 - η
tT =
Equation 6
2
(γ - 1) m2
R = 1 - η ( 1 + )
Equation 7
η =
(1 - R) ( γ - 1 ) m2
2
1 + ( γ - 1 ) m2
2
temperature change slowly and approaches a straight line
parallel to the ramp change in temperature. The transient is
displaced by one time constant as in Figure 3-B after a time
period equal to approximately 4 time constants.
When the temperature is not changing or changing very
slowly, the error due to the time constant is insignificant.
When the rate of change of temperature with time becomes
greater, the error caused by a sensor time lag can be
approximated by the product of the time constant times the
rate of change of temperature with time. This method of
computing the time response error is only an
approximation because most temperature changes are not
characteristic of a true ramp change as in Figure 3-B and all
temperature sensors have some small second-order effects.
2.5 AIRFLOW DIRECTION
When the airstream approaches the inlet of a total air
temperature sensor from a direction other than directly
forward, errors may be introduced. Our total temperature
sensors are designed to be insensitive to a wide range of
angle of attack in powered flights.
2.6 SELF-HEATING
Total air temperature sensors which use resistance
elements require that a small electrical current pass through
the sensing element. This current causes a self-heating
effect (I2R - Joule heating) which results in a small increase
in the measured temperature. The magnitude of the
temperature increase can be kept below 0.10°C if care is
taken to keep the power low. The self-heating effect is
controlled by convective cooling in flight and is minimized
by keeping the applied electrical current to a practical
minimum.
2.7 DEICING HEAT ERROR
For total temperature sensors with deicing heaters,
application of the deicing heat can cause Tm to increase at
low airspeeds. Basically, this effect is a conduction error,
internal to the sensor, caused by the close proximity of
heated portions of the sensor housing to the sensing
element. Our deiced total air temperature sensors are
designed to provide maximum thermal resistance between
the heated housing and the element, thus minimizing the
deicing heater effect.
2.8 AERODYNAMIC DRAG
Although the drag is not involved with the accuracy of total
air temperature sensors, it can be important in trade-offs
with other performance parameters. For example, deiced
total air temperature sensors usually exhibit a higher drag
than non-deiced sensors. The drag is influenced by the
shape and size of the sensor and varies with the aircraft
speed and altitude.
[attachment=20718]
1. INTRODUCTION
Modern jet power aircraft require very accurate
measurements of Outside Air Temperature (OAT) for inputs
to the air data computer and other airborne systems. What
does OAT mean? There are actually four temperatures of
concern. These are defined as follows:
Static Air Temperature (SAT or Ts)
The temperature of the undisturbed air through which the
aircraft is about to fly;
Total Air Temperature (TAT or Tt)
The maximum air temperature which can be attained by
100% conversion of the kinetic energy of the flight;
Recovery Temperature (Tr)
The adiabatic value of local air temperature on each portion
of the aircraft surface due to incomplete recovery of the
kinetic energy; and
Measured Temperature ™
The actual temperature as measured, which differs from Tr,
because of heat transfer effects due to imposed
environments.
For flight conditions in clear air at low altitudes and low
airspeeds the four temperatures are practically the same and
OAT can apply to any of these four temperatures. As the
airspeed and altitude increase the four temperatures will
differ, and the term OAT becomes meaningless.
Figure 1 illustrates the general relationship between the four
temperatures. The first three can be related by equations to
flight speed, as will be discussed in the next section. The
measured temperature, on the other hand, must be defined
at a particular flight condition. Its value may be higher or
lower than TAT or Tr, due to sensor design, location or
imposed heat transfer environment. If a severe environment
is imposed, the value of Tm can fall below that of SAT (e.g.,
vortex-producing designs or for severe weather).
The static air temperature is difficult to measure accurately,
and total air temperature, by definition, can never be
measured exactly (100% energy conversion). We are then left
with a design objective for total air temperature sensors to
impose an environment which will make Tm approximately
equal to the ideal TAT value for all flight conditions.
As will be shown later, the ratio of TAT to SAT (Tt/Ts) is
known for each flight condition. Thus, if Tm is close enough
to TAT, SAT can be calculated with greater accuracy than it
can be measured. In general, our total air temperature
sensors exhibit a Tm V 0.995 Tt.
2. PERFORMANCE PARAMETERS
A thorough understanding of the application of total air
temperature sensors involves an acquaintance with a
number of performance parameters. Many of the
parameters depend on Reynolds number and Mach number
as described in the referenced literature. In turn, these two
parameters are dependent upon the flight condition; that is,
the speed and altitude of the aircraft.
In the past, our technical literature introduced the usage of
total pressure behind the normal shock, which was valid but
somewhat difficult to work with. This bulletin utilizes
terminology familiar to all who work with aircraft: Mach
number and air density. To maintain the non-dimensional
concept, density ratio (ρ/ρ0) will be used. The density ratio
variation with altitude for the 1962 U.S. Standard
Atmosphere is given in Table 1.
Total air temperature sensor performance, given the
aircraft’s flight envelope in terms of Mach number and
altitude (converted to ρ/ρ0 from Table 1), now is directly
obtainable. Figure 2 shows how the parameters relate in
subsonic flight. The free stream values of M and (ρ/ρ0)
apply in subsonic flight for a properly located total air
temperature sensor; the product being a parameter upon
which the correction of measured temperature to recovery
temperature depends.
TOTAL TEMPERATURE SENSORS TECHNICAL REPORT 5755
1
Figure 1: Temperature Relationships in Flight
In supersonic flow, a correction must be made for the normal
shock wave which forms just upstream of the sensor inlet. The
shock wave causes the Mach number to drop to a subsonic
value at the inlet, but the local air density behind the shock
wave increases. Table 2 lists values of the correction factor for
twenty-two values of flight Mach number. Using a subscript to
denote conditions downstream of a normal shock, the
correction factor is the product of M1/M and ρ1/ρ. To
determine the performance of a total air temperature sensor for
a supersonic flight application select specific values of Mach
number M and multiply these by the ρ/ρ0 values from Table 1
for the altitude appropriate application. Then correct the M
(ρ/ρ0) product to M1 (ρ1/ρ0) using Table 2. Parameter Z (zeta)
will be used for brevity: Z=M1(ρ1/ρ0).
For a particular flight condition or set of M and (ρ/ρ0) values,
sensor design variations (even seemingly insignificant
variations in the sensor geometry) yield detectable variations in
sensor performance. These design-dependent variations are
categorized by a number of performance parameters which
have been used extensively in various technical reports and
design specifications. The important parameters are defined
and discussed in the following paragraphs.
2.1 THERMAL RECOVERY
The relationship between total and static temperature, absolute
units (°K or °R), is:
TOTAL TEMPERATURE SENSORS TECHNICAL REPORT 5755
2
Table 1: U.S. Standard Atmosphere: ρ/ρ0 Density Ratio Versus Geopotential Altitude, 1962 Values
Figure 2: Subsonic Flight in U.S. Standard Atmosphere
Equation 1
tT γ - 1
tS 2
= 1 + m2
Table 2: Multiplier Values for Correcting M (ρ/ρ0)
to M1 (ρ1/ρ0) = Z (Normal Shock Theory)
1.0 1.0000
1.1 .9690
1.2 .9419
1.3 .9166
1.4 .8929
1.5 .8703
1.6 .8489
1.7 .8281
1.8 .8080
1.9 .7887
2.0 .7700
2.1 .7516
2.2 .7339
2.3 .7168
2.4 .7001
2.5 .6840
2.6 .6684
2.7 .6533
2.8 .6388
2.9 .6247
3.0 .6109
3.5 .5493
FREE sTREAM
mACH nUMBER mULTIPLIER
Where the γ is the ratio of specific heats.
This is the equation programmed into air data computers to
calculate SAT from Tm values which are very close to TAT.
If the total temperature sensor is designed such that there
are no significant heat sources or thermal paths to heat
sinks and such that flow over the sensing element is
uniform and continuous, the sensor will indicate the
adiabatic value Tr very closely.
One parameter which relates Tr to TAT and SAT is called the
recovery factor, defined as follows:
Some geometric shapes have constant recovery factors
(e.g., the classic flat plate with airflow parallel to its
surface). For sensors with a constant adiabatic recovery
factor, Equation 1 and 2 can be combined.
However, most sensors exhibit a variable recovery factor,
and it is more convenient to use a recovery correction
defined as:
Our total temperature sensors exhibit a variable η for M
values below 1.0 but constant η for M values above 1.0.
Once Tr and η are known, Tt is calculated by:
The relationships between η and r are given by:
For non-adiabatic conditions, additional parameters are
involved in calculating TAT and SAT.
2.2 THERMAL CONDUCTION
A condition error may occur when the fuselage of the
aircraft is at a different temperature than the sensor. This
usually occurs on the ground during taxi conditions on hot
summer days or in the winter. Our total air temperature
sensor is designed to have its air inlet located in the free
stream beyond the aircraft boundary layer. Thus, during
flight, this design provides sufficient internal airflow over
the sensing element to negate the conduction mode of heat
transfer, when the Mach number is below 2, the altitude is
below 50,000 feet and Z (zeta) is above 0.15.
2.3 THERMAL RADIATION
When the total temperature being measured is relatively
high, heat is radiated from the sensing element, resulting in
a reduced indication of temperature. This effect is increased
at very high altitude for a sensor of simple design, where
the low air density decreases the ability of the internal air
flow to make up for this radiation heat loss. Radiation error
is negligible for our multiple shielded total air temperature
sensors when the Mach number is below 2, the altitude is
below 50,000 feet, and Z (zeta) is above 0.15.
2.4 TIME CONSTANT
An instantaneous response by a sensor to a temperature
change is impossible due to the heat capacity of the sensor
parts and surrounding structure. This results in an
indicated temperature/time transient. The time constant is
a performance parameter typically used to describe this
temperature/time transient.
The time constant is classically defined as the time required
for the sensor to respond to 63.2% of an instantaneous
(step) change in temperature, Figure 3-A. Thus time
constant is an index of how rapidly the temperature sensing
element can follow a changing temperature. Something
close to a step change in temperature may occur as the
aircraft emerges from a cloud bank into clear air.
A more common type of temperature change in flight is the
gradual change or ramp change, reference Figure 3-B. This
type of change occurs when the aircraft changes altitude or
speed. In this case the sensor begins its response to the
TOTAL TEMPERATURE SENSORS TECHNICAL REPORT 5755
3
Equation 2
tR - tS
tT - tS
R =
Equation 3
tR γ - 1
tS 2
= 1 + R ( ) m2
Equation 4
tT - tR
tT
η =
Equation 5
tR
1 - η
tT =
Equation 6
2
(γ - 1) m2
R = 1 - η ( 1 + )
Equation 7
η =
(1 - R) ( γ - 1 ) m2
2
1 + ( γ - 1 ) m2
2
temperature change slowly and approaches a straight line
parallel to the ramp change in temperature. The transient is
displaced by one time constant as in Figure 3-B after a time
period equal to approximately 4 time constants.
When the temperature is not changing or changing very
slowly, the error due to the time constant is insignificant.
When the rate of change of temperature with time becomes
greater, the error caused by a sensor time lag can be
approximated by the product of the time constant times the
rate of change of temperature with time. This method of
computing the time response error is only an
approximation because most temperature changes are not
characteristic of a true ramp change as in Figure 3-B and all
temperature sensors have some small second-order effects.
2.5 AIRFLOW DIRECTION
When the airstream approaches the inlet of a total air
temperature sensor from a direction other than directly
forward, errors may be introduced. Our total temperature
sensors are designed to be insensitive to a wide range of
angle of attack in powered flights.
2.6 SELF-HEATING
Total air temperature sensors which use resistance
elements require that a small electrical current pass through
the sensing element. This current causes a self-heating
effect (I2R - Joule heating) which results in a small increase
in the measured temperature. The magnitude of the
temperature increase can be kept below 0.10°C if care is
taken to keep the power low. The self-heating effect is
controlled by convective cooling in flight and is minimized
by keeping the applied electrical current to a practical
minimum.
2.7 DEICING HEAT ERROR
For total temperature sensors with deicing heaters,
application of the deicing heat can cause Tm to increase at
low airspeeds. Basically, this effect is a conduction error,
internal to the sensor, caused by the close proximity of
heated portions of the sensor housing to the sensing
element. Our deiced total air temperature sensors are
designed to provide maximum thermal resistance between
the heated housing and the element, thus minimizing the
deicing heater effect.
2.8 AERODYNAMIC DRAG
Although the drag is not involved with the accuracy of total
air temperature sensors, it can be important in trade-offs
with other performance parameters. For example, deiced
total air temperature sensors usually exhibit a higher drag
than non-deiced sensors. The drag is influenced by the
shape and size of the sensor and varies with the aircraft
speed and altitude.