When two dissimilar metals are joined together at two points to form a closed loop and a temperature difference exists between the junctions, an electrical potential is set up between the junctions. Such an arrangement is known as thermocouple and is frequently used for the measurement of temperature. Measurement of temperature by a thermocouple is an important application of the lumped parameter analysis.
The response of a thermocouple is defined as the time required for the thermocouple to reach the source temperature when it is exposed to it.
Referring to the lumped-parameter solution for transient heat conduction;
It is evident that larger the parameter hA/ρVc, the faster the exponential term will reach zero or more rapid will be the response of the thermocouple. A larger value of hA/ρVc can be obtained either by increasing the value of convective coefficient, or by decreasing the wire diameter, density and specific heat.
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The sensitivity of the thermocouple is defined as the time required by the thermocouple to reach 63.2% of its steady state value. When such a condition is attained, equation 6.2 can be written as-
The parameter ρVc/hA has units of time and is called time constant of the system and is denoted by τ*.
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Thus-
Using time constant, the temperature distribution in the solids can be expressed as-
The time constant is indicative of the speed of response, i.e., how fast the thermocouple tends to reach the steady state value. A large time constant corresponds to a slow system response, and a small constant represents a fast response.
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A low value of time constant can be achieved for a thermocouple by:
(i) Decreasing the wire diameter
(ii) Using light metals of low density and low specific heat
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(iii) Increasing the heat transfer coefficient
Depending upon the type of fluid used, the response times for different sizes and materials of thermocouple wires usually lie between 0.04 to 2.5 seconds.
Example 1:
A mercury thermometer with bulb idealized as a sphere of 1 mm radius is used for measuring the temperature of fluid whose temperature is varying at a fast rate.
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For mercury:
k = 10 W/mK; α = 0.00005 m2/s and h = 10 W/m2K
If the time for the temperature change of the fluid is 3 s, specify whether or not the thermometer is able to read the temperature faithfully? If not, what should be the radius of a thermocouple (k = 100 W/mK; α = 0.0012 m2/s and h = 8 W/m2K) to read the temperature of the fluid?
Solution:
The characteristic linear dimension defined as the ratio of the volume of bulb to its surface area works out as:
Since the time constant (6.67 s) is more than the time for the temperature change of the fluid (3 s), the thermometer will not give a faithful record of the time varying temperature of the fluid.
(b) The diameter of the thermocouple with the given properties can be worked out from the correlation for time constant.
That is-
Example 2:
A thermocouple junction of spherical is to use to measure the temperature of a gas stream. The junction is initially at 20°C and is placed in gas stream which is at 200°C. Make calculations for (a) junction diameter needed for the thermocouple to have thermal time constant of one second and (b) time required for the thermocouple to reach 197°C temperature.
Assume the thermo-physical properties as given below:
k (thermocouple junction) = 20 W/m-deg;
h = 350 W/m2-deg
c = 0.4 kJ/kg K and ρ = 8000 kg/m3
Solution:
The time constant for a thermocouple is given by:
