Question

A thermocouple with a spherical junction diameter of \(1 \mathrm{~mm}\) is used for measuring the temperature of a hydrogen gas stream. The properties of the thermocouple junction are $k=35 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \rho=8500 \mathrm{~kg} / \mathrm{m}^{3}\(, and \)c_{p}=320 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$. The hydrogen gas, behaving as an ideal gas at \(1 \mathrm{~atm}\), has a free-stream temperature of \(200^{\circ} \mathrm{C}\). If the initial temperature of the thermocouple junction is $10^{\circ} \mathrm{C}$, evaluate the time for the thermocouple to register 99 percent of the initial temperature difference at different free-stream velocities of the hydrogen gas. Using appropriate software, perform the evaluation by varying the free-stream velocity from 1 to \(100 \mathrm{~m} / \mathrm{s}\). Then, plot the thermocouple response time and the convection heat transfer coefficient as a function of free-stream velocity. Hint: Use the lumped system analysis to determine the time required for the thermocouple to register 99 percent of the initial temperature difference (verify the application of this method to this problem).

Short answer

Question: Compute the time required for a spherical thermocouple junction to register 99% of the initial temperature difference between the thermocouple and free-stream velocity in hydrogen gas. Plot the thermocouple response time and the convection heat transfer coefficient as a function of free-stream velocity. Note: You should use the lumped system analysis method to solve this problem. Use the provided characteristic length, \(L_c\), thermal conductivity, \(k\), and verify that the Biot number is less than 0.1 for each case.

Step-by-step solution

1. Compute the Biot number

To ensure that the lumped system analysis is applicable to this problem, we must calculate the Biot number. The Biot number is given by the formula: \[Bi = \dfrac{hL_c}{k}\] Here, \(h\): the convection heat transfer coefficient \(L_c\): the characteristic length (radius of the spherical thermocouple junction, \(r\)) \(k\): the thermal conductivity of the junction However, we don't have the value for \(h\). To calculate \(h\), we can use the Nusselt number: \[Nu = \dfrac{hD}{k}\] Here, \(D\) is the diameter of the spherical thermocouple junction. Since we know that the diameter is \(1 \mathrm{~mm} = 0.001 \mathrm{~m}\), the radius, \(r\), equals \(0.0005 \mathrm{~m}\). The Nusselt number is a function of the Reynolds and Prandtl numbers, so we must first find these values. We will have to calculate Reynolds number: \[Re = \dfrac{uD}{\nu}\] Prandtl number: \[Pr = \dfrac{\nu}{\alpha}\] Nusselt number as a function of Reynolds and Prandtl numbers: \[Nu = f(Re,Pr)\] Finally, we can calculate the Biot number for each free-stream velocity: \[Bi = \dfrac{hL_c}{k}\] where \(L_c=r=\dfrac{D}{2}\).

2. Find the time required for the thermocouple to register 99% of the initial temperature difference

Once we verify that the lumped system analysis can be applied in this problem by having Biot number less than 0.1, we have to determine the time required for the thermocouple to register 99% of the initial temperature difference as a function of different free-stream velocities. Lumped system analysis has a solution for spherical objects: \[ \dfrac{T - T_\infty }{T_{i} - T_\infty} = e^{-\dfrac{3h}{\rho c_{p}r}t} \] We want to evaluate the time for the thermocouple to register 99 percent of the initial temperature difference: \[\dfrac{T - T_\infty}{T_{i} - T_\infty} = e^{-\dfrac{3h}{\rho c_{p}r}t} = 0.01\] Here, we have \(T_\infty= 200^{\circ}C\), \(T_i=10^{\circ}C\), and using the material properties, we find \(\dfrac{3}{\rho c_{p}r}=\dfrac{3}{8500 \cdot 320 \cdot 0.0005} \approx 0.4084 \cdot 10^{-5}\). Now, we need to calculate the time \(t\) for various free-stream velocities and with the corresponding \(h\) value.

3. Plot the thermocouple response time and the convection heat transfer coefficient as a function of free-stream velocity

Finally, we have to create two plots: one for thermocouple response time as a function of free-stream velocity, and the other for the convection heat transfer coefficient as a function of free-stream velocity. Using a suitable software like Microsoft Excel or Python, perform calculations for different free-stream velocities of hydrogen gas from 1 to \(100 \mathrm{~m}/\mathrm{s}\) and plot the response time (\(t\)) and convection heat transfer coefficient (\(h\)) as a function of free-stream velocity.