Thermal time constant

A temperature detector does not react immediately to a change in temperature. The reaction time of the sensor or thermal time constant is a measure of the time it takes for the sensor to stabilize internally to the external temperature change, and is determined by the thermal mass and thermal conduction resistance of the device. Thermometer bulb size, probe size, or protection well can affect the

Figure 8.7 Thermocouple emf versus temperature for various types. 70 60 50 40 30 20 10 0 0 1200 1600 Temperature °C EMF Output (mV) E Type J Type K Type R Type S Type T Type 800 400

response time of the reading, i.e., a large bulb contains more liquid for better sensitivity, but this will also increase the time constant taking longer to fully respond to a temperature change.

The thermal time constant is related to the thermal parameters by the fol- lowing equation:

(8.16) where tc=thermal time constant

m= mass c= specific heat

k= heat transfer coefficient A= area of thermal contact

t mc

kA

c=

TABLE 8.8 Summary of Sensor Characteristics

Type Linearity Advantages Disadvantages Bimetalic Good Low cost, rugged, and Local measurement or for

wide range ON/OFF switching only Pressure Medium Accurate and wide range Needs temperature

compensation and vapor is nonlinear

Resistance Very good Stable, wide range, Slow response, low sensitivity, and accurate expensive, self heating, and

limited range Thermistor Poor Low cost, small, Nonlinear, range, and

high sensitivity, and self heating fast response

Thermocouple Good Low cost, rugged, and Low sensitivity and very wide range reference needed

Semiconductor Excellent Low cost, sensitive and Self heating, Slow response, easy to interface range, and power source TABLE 8.7 Temperature Range and Accuracy of Temperature Sensors

Sensor type Range (degree Celsius) Accuracy (FSD) Expansion Mercury in glass −35 to 430 ±1%

Liquid in glass −180 to 500 ±1%

Bimetallic −180 to 600 ±20%

Pressure–spring Liquid filled −180 to 550 ±0.5% Vapor pressure −180 to 550 ±2.0% Gas filled ₋180 to 550 _±0.5% Resistance Metal resistors −200 to 800 ±5%

Platinum ₋180 to 650 _±0.5% Nickel −180 to 320 ±1% Copper −180 to 320 ±0.2% Thermistor 0 to 500 ±25% Thermocouple ₋60 to 540 _±1% −180 to 2500 ±10% Semiconductor IC −40 to 150 ±1%

When the temperature changes rapidly, the temperature output reading of a thermal sensor is given by

T−T2=(T1− T2)e−

t/tc

(8.17) where T=temperature reading

T1=initial temperature T2=true system temperature

t=time from when the change occurred

The time constant of a system tc is considered as the time it takes for the

system to reach 63.2 percent of its final temperature value after a temperature change, i.e., a copper block is held in an ice–water bath until its temperature has stabilized at 0°C, it is then removed and placed in a 100°C steam bath, the temperature of the copper block will not immediately go to 100°C, but its tem- perature will rise on an exponential curve as it absorbs energy from the steam, until after some time period (its time constant) it will reach 63.2°C, aiming to eventually reach 100°C. This is shown in the graph (line A) in Fig. 8.8. During the second time constant the copper will rise another 63.2 percent of the remain- ing temperature to get to equilibrium, i.e., (100−63.2) 63.2 percent =23.3°C, or at the end of 2 time constant periods, the temperature of the copper will be 86.5°C. At the end of 3 periods the temperature will be 95°and so on. Also shown in Fig. 8.8 is a second line B for the copper, the time constants are the same but the final aiming temperature is 50°C. The time to stabilize is the same in both cases. Where a fast response time is required, thermal time con- stants can be a serious problem as in some cases they can be of several seconds duration. Correction may have to be applied to the output reading electronically to correct for the thermal time constant to obtain a faster response. This can be

100 90 80 70 60 50 40 30 20 10 0 0 1 2 3 4 Time constants 5 Line B Line A 6 7 95 °C 86.5 °C 63.2 °C 31.6 °C 43.2 °C Temperature ° C

done by measuring the rate of rise of the temperature indicated by the sensor and extrapolating the actual aiming temperature.

The thermal time constant of a body is similar to an electrical time constant which is discussed in the chapter on electricity under electrical time constants. 8.5.4 Installation

Care must be taken in locating the sensing portion of the temperature sensor, it should be fully encompassed by the medium whose temperature is being measured, and not be in contact with the walls of the container. The sensor should be screened from reflected heat and radiant heat if necessary. The sensor should also be placed downstream from the fluids being mixed, to ensure that the temperature has stabilized, but as close as possible to the point of mixing, to give as fast as possible temperature measurement for good control. A low ther- mal time constant in the sensor is necessary for a quick response.

Compensation and calibration may be necessary when using pressure-spring devices with long tubes especially when accurate readings are required. 8.5.5 Calibration

Temperature calibration can be performed on most temperature sensing devices by immersing them in known temperature standards which are the equilibrium points of solid/liquid or liquid/gas mixtures, which is also known as the triple point. Some of these are given in Table 8.9. Most temperature sensing devices are rugged and reliable, but can go out of calibration due to leakage during use or contamination during manufacture and should therefore be checked on a regular basis.