Answer:
The inversion temperature of a thermocouple is the temperature at which the electromotive force (EMF) produced by the thermocouple changes sign as the temperature changes. This temperature can be calculated using the Seebeck coefficients (a and b) of the two metals used in the thermocouple. The formula for the inversion temperature is:
T_inv = b1 / (b1 - b2) * (a2 / a1) * T1
where b1 and a1 are the Seebeck coefficients of the first metal, b2 and a2 are the Seebeck coefficients of the second metal, and T1 is the temperature at which the EMF is zero.
In this case, the first metal is copper, with Seebeck coefficients a1 = 18.5 µV/K and b1 = -0.2 µV/K, and the second metal is iron, with Seebeck coefficients a2 = 4 µV/K and b2 = 0.09 µV/K.
To calculate the inversion temperature, we need to find the temperature at which the EMF is zero. This is the temperature at which the two junctions of the thermocouple are at the same temperature, and the Seebeck effect does not produce a voltage difference between them. In other words, it is the temperature at which the thermocouple is not sensitive to temperature changes.
Assuming that the temperature of the copper-iron thermocouple is initially at room temperature (20°C or 293 K), we can calculate the inversion temperature by setting the EMF to zero and solving for T1:
0 = b1 / (b1 - b2) * (a2 / a1) * (T1 - 293 K)
Solving for T1, we get:
T1 = 293 K * b1 / ((b1 - b2) * (a2 / a1))
Substituting the values of a and b for copper and iron, we get:
T1 = 293 K * (-0.2 µV/K) / ((-0.2 µV/K - 0.09 µV/K) * (4 µV/K / 18.5 µV/K))
Simplifying this expression, we get:
T1 = 1,189 K
Therefore, the inversion temperature of the copper-iron thermocouple is approximately 1,189 K (916°C or 1681°F).