Error Estimate in Temperature Measurements
Physical Errors in Temperature Sensors: As mentioned previously, there are errors due to the thermocouple insertion and junction heating or cooling. Insertion errors are due to conduction, convection, radiation, and recovery effects. Junction point heating or cooling errors are due to non-isothermal connections and the reference junction. In addition, errors can be due to aging/annealing/work hardening, magnetic field effects, and reference junction errors. Radiation errors need to be considered for high temperature measurements (T > 500°C) and recovery effects if Ma > 0.2.
The following lead wire model (Figure 3.14) can be used to estimate thermocouple reading errors for measuring temperatures in fluids and solids. The model is based on the concept of heat conduction through a fin with proper boundary conditions [1].
Lead Wire Model
Apply the concept of heat conduction through a fin.
Axial heat conduction,
if kA = k_{w}A_{w} + kiAj (overestimation)
FIGURE 3.14 Concept of single lead wire with insulation, where cIQ, = h2nrdr(T_{iulf} -7})
‘"Г—)
^{where R} = 17^ ^{+} -ШГ
Similarly, use the concept of heat conduction through a fin for the double wire with inner and outer insulations (Figure 3.15).
Examples
1. A thermocouple temperature sensor is inserted in a gas flow (Figure 3.16).
What is an estimate of the temperature error? [1]
From fin equation solution:
Temperature error: The temperature difference between the thermocouple and fluid(7]_{c} — T_{r}) decreases with increasing the values of L, and L
yfkAR
If M < 0.2, 7, = T_{f}
If M > 0.2, T, = T, (1 + r^-M^{2})
where:
T, is the recovery temperature,
M is the Mach number,
r is the Recovery Factor, it depends on the following conditions:
VpF |
Laminar |
Pr'^{!}‘ |
Turbulent |
0.68 ±0.07 |
Wire normal to flow |
0.86 ±0.09 |
Wire parallel to flow |
FIGURE 3.16 A thermocouple temperature sensor is inserted in a gas flow.
FIGURE 3.17 A thermocouple temperature sensor is embedded in a massive solid.
2. A thermocouple temperature sensor is embedded in a massive solid (Figure 3.17). What is an estimate of the temperature error? [1]
From the solution of previous case, one gets the similar temperature results as shown in below equation (3.1).
In— In— where /?, = —^-+—
2nkj btkj
where r_{3} = adhesive radius
Fin theory: Consider a fin in solid 1 and extended into fluid 2.
From (3.1) and (3.2):
Temperature error: The temperature difference between the thermocouple and solid (T_{tc} -7_{s}) decreases with increasing the values of L, and Rn,
3. In a surface-mounted thermocouple temperature sensor (surface temperature measurement of a massive solid, Figure 3.18), what is an estimate of the temperature error? [1]
From the fin equation solution:
Temperature error:
The temperature difference between the thermocouple and solid (7_{su[}f - T_{lc}) will reduce, if the Biot number increases and if the thermocouple con- ductance/solid conductance decreases.
High Velocity Gas Temperature Measurements
Stagnation Temperature, T_{0}, or Total Temperature, T,: When a high-speed flow approaches a sphere, or thermocouple bead, flow dynamic energy converts into thermal energy which increases its temperature. This is referred to as the stagnation temperature or total temperature (Figure 3.19). If a gas flow is brought to rest adiabatically, the stagnation temperature can be determined as shown below. The stagnation temperature is the same as the static temperature for low-speed flows and in liquids [6]. The total temperature, 7), (or stagnation temperature, Г_{0}) = static temperature T_{s} (or T,,) + dynamic temperature j:
IfM= 1. deviate 20%
= 0.5, deviate 5% where:
у = — = 1.4 for air M is the Mach number = -r= yjyRT is the speed of sound ~ 1000 fps at room temperature
Adiabatic Wall Temperature, T_{m}., or Recovery Temperature, T_{r}: When a high-speed flow passes over an adiabatic surface, frictional heat (~v(ff)^{2}) is generated inside the boundary layer due to viscosity and the velocity gradient. This heat increases the surface temperature, and this elevated temperature is referred to as the adiabatic wall temperature, T_{m}., or recovery temperature, T_{r} (Figure 3.20). Note that the recovery temperature (T_{r}) is the stagnation temperature (7~_{0}), if the recovery factor, r = 1 [6]. The recovery temperature is strongly dependent upon the probe design. Adiabatic wall temperature T_{aw} (or recovery temperature, T_{r}) = static temperature T_{s} (or T,) + dynamic temperature Ui. ■
2c
where:
r_{c} = recovery factor = Pr'^{12} Laminar flow = Pi^{jp} turbulent flow
= 0.75-0.99
= 1 if Pr = 1 (assumption, ideal)
FIGURE 3.21 Concept of high velocity and high temperature probe.
In general T_{m}, < T_{0}, T_{r} < T,
High Velocity/High Temperature Probe for Gas Temperature Measurements: Temperature measurements in high-speed gas flows, such as a gas turbine combustor, ramjet, or scramjet are difficult and require special measurement probes. The gas velocity must be reduced to zero in order to measure the true temperature (Figure 3.21). High-velocity gas temperature probes (HVGT) must be calibrated.
Example
The space shuttle Columbia broke apart upon re-entry into the Earth's atmosphere at approximately 8:00 a.m. on Saturday, February 1, 2003. As the space shuttle broke apart, debris fell to the ground across East Texas. During this tragedy, all seven crew members on Columbia were killed.
The space shuttle broke apart at approximately 200,000 feet ~ 39 miles above the state of Texas. The shuttle was traveling at approximately 12,500 mph (or 18 times the speed of sound). The shuttle had just reached the hottest point of its descent into the Earth's atmosphere. NASA officials estimated the external temperature generated by friction (so-called aerodynamic heating) as the shuttle travelled into the compressed atmosphere (so-called the stagnation temperature) was 3,000°F.
The heat shield is comprised of approximately 20,000 tiles, each 6" x 6". A section of tile was damaged during the shuttle launch as a block of foam insulation (20" x 16" x 5") fell from the fuel tank and struck the heat shield beneath the left wing of the shuttle. The damaged tiles on the shuttle were not detected before re-entry of the shuttle. Note that 1 mile = 5128 feet, speed of sound = {y.R.T_{a})^{m} = 1000 fps, у = 1.4, M, = Mach number = shuttle velocity/speed of sound = 18, T,, = air temperature = 45°F = 280°K at 39 miles sky. Based on the compressible flow equations with a normal shock wave [7], the estimated Mach number and air temperature after the shock M_{2} = 0.38 and T_{a2} = 63.8
7,i = 17,885°K, respectively. The following is the estimated shuttle leading edge temperature (the hottest temperature) T_{0 }