The temperature of two junctions of a thermocouple are maintained at 0°C and [latex]\theta^oC[/latex]. The thermo-emf generated is given by the relation, [latex]E = 10^{-5}\theta – 10^{-7}\theta^2[/latex]. Calculate the neutral temperature of the thermocouple and maximum value of thermoemf.

Solution

Here,

[latex]E = 10^{-5}\theta – 10^{-7}\theta^2[/latex]

Neutral temperature [latex](\theta_n)[/latex] =?

We know that, neutral temperature is that temperature at which thermo emf is maximum. For maximum thermo emf, we have,

[latex]\dfrac{dE}{d\theta} = 0[/latex]

[latex]Or,\ \dfrac{d}{d\theta}(10^{-5}\theta-10^{-7}\theta^2) = 0[/latex]

[latex]Or,\ 10^{-5} – 10^{-7}\times2\theta = 0[/latex]

For neutral temperature, [latex]\theta = \theta_n[/latex]

So, [latex]2\times10^{-7}\theta_n = 10^{-5}[/latex]

Or, [latex]\theta_n = \dfrac{10^{-5}}{2\times10^{-7}} = 50^oC[/latex]

Now, for the maximum value of thermoemf, put, [latex]\theta = \theta_n = 50^oC[/latex], we get,

[latex]E = 10^{-5}\times50 – 10^{-7}\times50^2[/latex]

[latex]= 2.5\times10^{-4}\ V[/latex]

In a given thermo-couple the temperature of cold junction is 25°C while the neutral temperature is 270°C. What is the value of temperature of inversion?

Solution

Temperature of cold junction [latex](\theta_c) = 25^oC[/latex]

Neutral temperature [latex](\theta_n) = 270^oC[/latex]

Now, temperature of inversion [latex](\theta_i)[/latex] =?

We know that,

[latex]\theta_n = \dfrac{\theta_i+\theta_c}{2}[/latex]

Or, [latex]270 = \dfrac{\theta_i+25}{2}[/latex]

Or, [latex]\theta_i = 540-25 = 515^oC[/latex]

The thermo – emf E and the temperature of hot junction [latex]\theta[/latex] satisfy a relation [latex]E = a\theta + b\theta^2[/latex] where a = [latex]4.1\times10^{-5}\ V(^oC)^{-1}[/latex] and b = [latex]-4.1\times10^{-8}\ V(^oC)^{-2}[/latex]. If the cold junction temperature is 0°C, find the neutral temperature.

Solution

Given,

[latex]E = a\theta + b\theta^2[/latex]

a = [latex]4.1\times10^{-5}\ V(^oC)^{-1}[/latex]

b = [latex]-4.1\times10^{-8}\ V(^oC)^{-2}[/latex]

Cold junction temperature [latex](\theta_c) = 0^oC[/latex]

Neutral temperature [latex](\theta_n)[/latex] =?

We know that, neutral temperature is that temperature at which thermo emf is maximum. For maximum thermo emf, we have,

[latex]\dfrac{dE}{d\theta} = 0[/latex]

[latex]Or,\ \dfrac{d}{d\theta}(a\theta+b\theta^2) = 0[/latex]

[latex]Or,\ a + 2b\theta = 0[/latex]

For neutral temperature, [latex]\theta = \theta_n[/latex]

So, [latex]2b\theta_n = -a[/latex]

Or, [latex]\theta_n = -\dfrac{a}{2b} = -\dfrac{4.1\times10^{-5}}{-2\times4.1\times10^{-8}}[/latex]

∴ [latex]\theta_n = 500^oC[/latex]