Question

A thermocouple used to measure the temperature of hot air flowing in a duct whose walls are maintained at \(T_{w}=500 \mathrm{~K}\) shows a temperature reading of \(T_{\text {th }}=850 \mathrm{~K}\). Assuming the emissivity of the thermocouple junction to be \(\varepsilon=0.6\) and the convection heat transfer coefficient to be \(h=75 \mathrm{~W} / \mathrm{m}^{2}, \mathrm{~K}\), determine the actual temperature of air.

Short answer

Answer: The actual temperature of the air flowing in the duct is approximately 801.73 K.

Step-by-step solution

Energy balance equation

The energy balance equation is given by: \(q_{conv} = q_{rad}\) Where \(q_{conv}\) is the heat transfer due to convection and \(q_{rad}\) is the heat transfer due to radiation. Next, we'll write the equations for the convection heat transfer and radiation heat transfer.

Convection heat transfer equation

The convection heat transfer can be calculated using the following equation: \(q_{conv} = hA(T_{th} - T_a)\) Where \(h\) is the convection heat transfer coefficient, \(A\) is the surface area of the thermocouple junction, \(T_{th}\) is the temperature measured by the thermocouple, and \(T_a\) is the actual temperature of the air.

Radiation heat transfer equation

The radiation heat transfer can be calculated using the following equation: \(q_{rad} = \varepsilon A\sigma(T_{th}^4 - T_w^4)\) Where \(\varepsilon\) is the emissivity of the thermocouple junction, \(\sigma\) is the Stefan-Boltzmann constant (5.67 x 10^{-8} W/m^2K^4), and \(T_w\) is the temperature of the duct walls. Now, we can substitute the convection heat transfer equation and radiation heat transfer equation into the energy balance equation.

Solving the energy balance equation

Substituting the convection and radiation heat transfer equations into the energy balance equation, we get: \(hA(T_{th} - T_a) = \varepsilon A\sigma(T_{th}^4 - T_w^4)\) Since the surface area of the thermocouple junction, \(A\), is the same on both sides of the equation, we can cancel it out: \(h(T_{th} - T_a) = \varepsilon\sigma(T_{th}^4 - T_w^4)\) Now, we can plug in the given values to solve for the actual temperature of the air, \(T_a\). \(h = 75 \mathrm{~W / m}^2\mathrm{~K}\), \(\varepsilon = 0.6\), \(\sigma = 5.67 \times 10^{-8} \mathrm{~W/m^2 K^4}\), \(T_{th} = 850 \mathrm{~K}\), and \(T_w = 500 \mathrm{~K}\). \(75(850 - T_a) = 0.6(5.67 \times 10^{-8})(850^4 - 500^4)\) Solve for \(T_a\): \(T_a \approx 801.73 \mathrm{~K}\)

Final answer

The actual temperature of the air flowing in the duct is approximately 801.73 K.