E1C08 09/14/2010
14:54:2 Page 364
where T p represents the equilibrium temperature of the stationary (with respect to the flow) real
temperature probe. In general, r may be a function of the velocity of the flow, or more precisely, the
Mach number and Reynolds number of the flow, and the shape and orientation of the temperature
probe. For thermocouple junctions of round wire, Moffat (15) reports values of
r ¼ 0:68 Æ 0:07 95%
ð
Þ for wires normal to the flow
r ¼ 0:86 Æ 0:09 95%
ð
Þ for wires parallel to the flow
These recovery factor values tend to be constant at velocities for which temperature errors are
significant, usually flows where the Mach number is greater than 0.1. For thermocouples having a
welded junction, a spherical weld bead significantly larger than the wire diameter tends to a value of
the recovery factor of 0.75, for the wires parallel or normal to the flow. The relationships between
temperature and velocity for temperature probes with known recovery factors are
T p ¼ T 1 þ
rU
2
2c p
ð8:36Þ
or in terms of the recovery error, e U ,
e U ¼ T p À T 1 ¼
rU
2
2c p
ð8:37Þ
The probe temperature is related to the stagnation temperature by
T p ¼ T t À
1 À r
ð
ÞU
2
2c p
ð8:38Þ
Fundamentally, in liquids the stagnation and static temperatures are essentially equal (5), and the
recovery error may generally be taken as zero for liquid flows. In any case, high-velocity flows are
rarely encountered in liquids.
Normally, the uncertainty assigned to the recovery error is set at u ¼ e U . The uncertainty
interval is often not symmetrical.
Example 8.15
A temperature probe having a recovery factor of 0.86 is to be used to measure a flow of air at
velocities up to the sonic velocity, at a pressure of 1 atm and a static temperature of 30
C. Calculate
the value of the recovery error in the temperature measurement as the velocity of the air flow
increases, from 0 to the speed of sound, using Equation 8.37.
KNOWN r ¼ 0:86 p 1 ¼ 1 atm abs ¼ 101 kPa abs
M 1
T 1 ¼ 30
C ¼ 303 K
FIND The recovery error as a function of air velocity.
ASSUMPTION Air behaves as an ideal gas.
SOLUTION Assuming that air behaves as an ideal gas, the speed of sound is expressed as
a ¼
ffiffiffiffiffiffiffiffiffi
kRT
p
ð8:39Þ
For air at 101 kPa and 303 K, with R ¼ 0.287 kJ/kg K, the speed of sound is approximately 349 m/s.
364 Chapter 8 Temperature Measurements
14:54:2 Page 364
where T p represents the equilibrium temperature of the stationary (with respect to the flow) real
temperature probe. In general, r may be a function of the velocity of the flow, or more precisely, the
Mach number and Reynolds number of the flow, and the shape and orientation of the temperature
probe. For thermocouple junctions of round wire, Moffat (15) reports values of
r ¼ 0:68 Æ 0:07 95%
ð
Þ for wires normal to the flow
r ¼ 0:86 Æ 0:09 95%
ð
Þ for wires parallel to the flow
These recovery factor values tend to be constant at velocities for which temperature errors are
significant, usually flows where the Mach number is greater than 0.1. For thermocouples having a
welded junction, a spherical weld bead significantly larger than the wire diameter tends to a value of
the recovery factor of 0.75, for the wires parallel or normal to the flow. The relationships between
temperature and velocity for temperature probes with known recovery factors are
T p ¼ T 1 þ
rU
2
2c p
ð8:36Þ
or in terms of the recovery error, e U ,
e U ¼ T p À T 1 ¼
rU
2
2c p
ð8:37Þ
The probe temperature is related to the stagnation temperature by
T p ¼ T t À
1 À r
ð
ÞU
2
2c p
ð8:38Þ
Fundamentally, in liquids the stagnation and static temperatures are essentially equal (5), and the
recovery error may generally be taken as zero for liquid flows. In any case, high-velocity flows are
rarely encountered in liquids.
Normally, the uncertainty assigned to the recovery error is set at u ¼ e U . The uncertainty
interval is often not symmetrical.
Example 8.15
A temperature probe having a recovery factor of 0.86 is to be used to measure a flow of air at
velocities up to the sonic velocity, at a pressure of 1 atm and a static temperature of 30
C. Calculate
the value of the recovery error in the temperature measurement as the velocity of the air flow
increases, from 0 to the speed of sound, using Equation 8.37.
KNOWN r ¼ 0:86 p 1 ¼ 1 atm abs ¼ 101 kPa abs
M 1
T 1 ¼ 30
C ¼ 303 K
FIND The recovery error as a function of air velocity.
ASSUMPTION Air behaves as an ideal gas.
SOLUTION Assuming that air behaves as an ideal gas, the speed of sound is expressed as
a ¼
ffiffiffiffiffiffiffiffiffi
kRT
p
ð8:39Þ
For air at 101 kPa and 303 K, with R ¼ 0.287 kJ/kg K, the speed of sound is approximately 349 m/s.
364 Chapter 8 Temperature Measurements