The time constant of a temperature sensor (or any other sensor) is defined as the time required for the sensor to respond to 63.2% of its total output signal when subjected to a step-change in temperature. Sensor response time is not the same as the time constant, but is a measure of how long it takes for the sensor to reach its full output. Thus, response time can be defined as five time constants (Fig. 1), which is based on the fact that the temperature sensor is subjected to a step change in temperature.
A step change in temperature occurs very rarely in the physical world (some examples are shock waves, explosions, gun barrel tests, internal combustion engines, and injection molding). However, these are not truly step changes when measured using a fast responding sensor. Thus, a different term is needed to define more accurately what occurs in the physical world. Nanmac uses "rise time" to define the time required for the test event to reach its full output. For example, a particular temperature event might take one second to reach its full output even though the time constant of the temperature sensor may be only one microsecond. A temperature sensor cannot reach its full output before the rise time of the event reaches its full output.
Response times
Response in static air. Air response tests can be performed in static and moving air. The response time of a thermocouple suddenly immersed into hot air depends on the diameter of the thermocouple wires if all other variables are held constant. Figure 2 illustrates the typical response times of exposed bare wire thermocouples.
Bath response time. Most temperature sensors have a protection sheath (such as stainless steel, Inconel, ceramic and molybdenum) surrounding the sensing elements and the required electrical insulation. The time constant of an assembled temperature sensor depends on the overall response time of all its components. If all but one variable are held constant, the effect of that variable on sensor response times can be determined graphically (e.g., effect of sensor diameter graphically illustrated in Fig. 3). Similar relationships can be determined for other variables such as sheath types, type and amount of insulation, wire size and properties, size of measuring junction, physical shape of the sensor (round, elliptical, or flat) etc.
The sensing tip must absorb heat faster than heat is conducted away to increase in temperature. Also, the heat capacity of the medium must be large compared with the heat conducted away by the sensor so the step change can be maintained during the test. In air tests, the temperature of the air surrounding the probe decreases as soon as heat is applied to the sensor until more heat flows into this region from the undisturbed air region, which takes a considerable length of time in still air. The time decreases as the air mass flow rate increases as shown in the table.
In liquid tests, the heat capacity of the liquid is much greater than air, so the heat input into the sensor is much greater. Thus, the superimposed temperature change more closely approaches a step function.
In liquid metal (or contacting a hot plate), the heat capacity of the metal is much greater than those of gas or liquid, so the temperature change approaches a step change even more. Thus, these tests produce the fastest response times for a given temperature sensor.
Time constants
The time constant of a temperature sensor affects the accuracy of temperature measurements in transient applications as illustrated by the following examples.
Example 1. Measure the temperature of a hot plate, bearing wall, etc. by contacting the sensor to the hot wall for three seconds duration. From the sensors point of view, this is a three-second transient. Because it takes five time constants to reach 99% of its output, the time constant of this sensor can be no slower than 600 milliseconds.
Example 2. Measure the temperature of a cyclical application such as in injection molding. In a typical molding cycle of ten seconds, it takes about 1.5 seconds to fill the mold with hot plastic. The remainder of the cycle time is used for curing the part, opening the mold, ejecting the part and then closing the mold to begin the next cycle. Thus, the transient portion of the cycle is only 1.5 seconds, and a sensor having a time constant of 300 milliseconds is required.
It is not sufficient to immerse a temperature sensor in a container of hot water to determine its time constant. Laminar layers adjacent to the cool probe, the speed at which the probe is immersed into the water and the heat transfer coefficient of the water mask the response time. Such a test gives a rise time, which for a sensor having a fast response time, simply changes proportionally with the probe's velocity through the water. A more accurate measure of response time can be obtained by slowly heating the sensing tip with a small torch and quickly removing the heat source while its output is increasing. The time lag between the instant the heat source is removed and the output of the sensor begins to decrease is a good indication of the sensor's response time.
Two types of errors produced when using a temperature sensor with inadequate re-sponse times in transient applications are: 1) the peak temperature is either missed completely or greatly reduced; and 2) peak temperature is delayed, inaccurately indicating when the peak temperature occurred. In many industrial processes, true peak temperature and/or when it occurs are critical to control process efficiently.
Ribbon thermocouples in hot-water tests. Figure 4 shows temperature versus time data obtained by manually immersing a right-angle thermocouple from ice water to boiling water then back to ice water.
Response tests in moving air. The effect of moving air on temperature sensor response time also must be considered. When sensor response time in air is specified, it is necessary to indicate the air mass flow rate under which the test will be conducted, or the data is meaningless.
Other factors affecting response times
Sensor response times can be greatly affected by two variables even if the diameter of the two elements are the same, and the orientation of the probe with respect to the isotherms is the same, then the thermocouple with the lowest combined thermal conductivity will produce the fastest response time. Thus, a Chromel-Alumel (Type K) thermocouple has a much faster response time than a Cu-Constantan (Type T) thermocouple of the same diameter and construction. Type T is much faster if the probe is installed parallel to the plane of heat flow.
Conduction problems and isotherms
A thermocouple installed in a furnace is subject to large errors caused by conduction along the stem. Because heat is always conducted from a hotter medium to a cooler medium, it is very important to eliminate (or minimize) the error caused by conduction along the sensor stem, which usually contains a metallic sheath, electrical insulation and wire. The sensor measures the temperature of its sensing tip after the tip reaches an equilibrium temperature; that is, when heat input to the sensing tip offsets the heat losses (conduction along the stem through the wall and thus to the outside environment). This stem effect must be eliminated to obtain the true temperature of the interior of the furnace.
Thus, a 0.25 in. (~6 mm) diameter sensor must be installed parallel to the isotherms for a distance of 20 times its diameter, or 5 in. (127 mm) to offset conduction error. Ideally, sensor orientation should be considered during the design stage of the furnace, not after. Generally, accurate temperature measurement is achieved by installing the sensor parallel to the longitudinal axis of the workload for a distance equal to 20 times the sensor diameter. It is easy to determine if there is a significant error in the installation by temporarily installing a smaller diameter thermocouple into the same location under the same conditions. If there is a conduction error caused by stem effect, the smaller diameter sensor will indicate a higher temperature. Errors as high as 300 F (165 C) have been observed in heat-treating applications. Figures 5 and 6 illustrate correct and incorrect temperature-sensor installations in furnaces.
Thermal properties of the thermowell
In an experiment designed to determine the effect of thermal properties of the thermowell on the recorded temperature, several identical thermocouples were constructed using various materials (combinations of phenolic, stainless steel and molybdenum) in the thermowell, keeping all other conditions the same. The thermocouples were installed flush with the inner surface of a rocket motor nozzle containing phenolic insulation.
Figure 7 shows the results of one test in which a phenolic probe and molybdenum probe were used in the same test. The sensors differed by about 2000 F (1080 C) seven seconds after the test started, and still differed by 1000 F (535 C) after 18 seconds. Repeated tests produced similar results. Other materials such as stainless steel, tantalum, graphite, etc. also were used. Generally, recorded surface temperatures were inversely proportional to the thermal properties of the thermowells. The true surface temperature of the nozzle was obtained using a thermocouple with a phenolic thermowell.
Using only a Mo-sheathed Type C thermocouple would show that the highest surface temperature of the phenolic wall was only 3090 F (1700 C), and specifications based on this data would produce catastrophic results.
Recording-system errors
In addition to sensor errors, there are other errors created by the recording system including connectors, extension wires, reference junctions, amplifiers, controllers, etc. To illustrate, the errors possible using a high-temperature Type C thermocouple (W5%Re vs. W26%Re) are itemized.
The ANSI standard limits of error for a Type C thermocouple is +/-1% between 800 and 4200 F (430 and 2320 C), thus Type C elements can be "off" by a maximum of +/-42 F (+/-23 C) and still be acceptable for use at 4200 F. ANSI established the acceptable maximum deviation (limits of error) for all thermocouple types from NIST calibrations. This is the first source of error in temperature measurements.
Additional sources of errors are introduced when calibrated ANSI Type C wires are assembled into a thermocouple including the welded junction, two-hole insulators and protection sheath. Several more sources of error are introduced when the assembled thermocouple is installed into a heated chamber or furnace including conduction errors due to temperature gradients within the chamber wall, response time errors if transients are occurring and radiation errors due to cold walls. Finally, a few more sources of error are introduced when the output of the assembled thermocouple is recorded including reference-junction, transmitter, amplifier and other errors; compensated lead-wire errors and recorder or controller errors.
Because all of the above errors are included in every installation, minimizing them requires paying very careful attention to details. If each source of error cannot be isolated and measured, the measuring system is not capable of detecting them. Errors are either electronic or thermal.
Electronic errors at 2730 F (1500 C) include:
- Wire calibration error: +/-1% = +/-15 C
- Extension wire error: +/- 0.5% = +/-7.5 C
- Typical ref. junction error: +/-1 C
- Typical recorder error: +/-0.5% or +/-7.5 C
The maximum possible electronic error is +/-31 C at 1500 C (+/-56 F at 2730 F). All of these errors fortunately are not always accumulative, because some errors cancel out others.
Thermal errors include:
- Stem-effect error caused by conduction from the tip of the sensor to the wall. This error is most pronounced when there are large temperature differences between the wall and the tip of the sensor.
- Radiation error caused by temperature differences between the tip of the thermocouple (including its protection sheath) and the inside walls and/or the workpiece or load in the furnace.
