Question

In an experiment, the temperature of a hot gas stream is to be measured by a thermocouple with a spherical junction. Due to the nature of this experiment, the response time of the thermocouple to register 99 percent of the initial temperature difference must be within \(5 \mathrm{~s}\). 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}$. If the heat transfer coefficient between the thermocouple junction and the gas is $250 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$, determine the diameter of the junction.

Short answer

Answer: The approximate diameter of the thermocouple junction is 7.392 mm.

Step-by-step solution

Obtain the formula for response time 99 percent

We'll use the following formula to relate the response time, Biot Number (Bi) and Fourier Number (Fo): $$ t_{99} = \frac{V}{{A}}\cdot \frac{1}{Bi \cdot Fo} $$ Where \(t_{99}\) is the time to reach 99% of the initial temperature difference, \(V\) is the volume of the junction, \(A\) is the surface area of the junction, \(Bi\) is the Biot number, and \(Fo\) is the Fourier number. Since the junction is a sphere, we can find its volume and area as follows: $$ V = \frac{4}{3}\pi r^3 $$ $$ A = 4\pi r^2 $$ Where \(r\) is the radius of the sphere.

Calculate the Biot number

The Biot number is given by the formula: $$ Bi = \frac{h \cdot L_c}{k} $$ Here, \(h\) is the heat transfer coefficient, \(L_c\) is the characteristic length, and \(k\) is the thermal conductivity. For a sphere, the characteristic length \(L_c\) is the radius (\(r\)). Using the given values: \(h=250\frac{W}{m^2K}\) and \(k=35\frac{W}{mK}\), we can calculate the Biot number: $$ Bi = \frac{250 \cdot r}{35} $$

Calculate the Fourier number

The Fourier number is given by the formula: $$ Fo = \frac{\alpha \cdot t_{99}}{L_c^2} $$ Here, \(\alpha\) is the thermal diffusivity, and \(t_{99}\) is the time to reach 99% of the initial temperature difference. The thermal diffusivity is defined as: $$ \alpha = \frac{k} {\rho \cdot c_p} $$ Using the given values: \(k=35\frac{W}{mK}\), \(\rho=8500\frac{kg}{m^3}\), and \(c_p=320\frac{J}{kgK}\), we can calculate the thermal diffusivity: $$ \alpha = \frac{35}{8500\cdot 320} $$ Now, we plug this value into the Fourier number formula: $$ Fo = \frac{\frac{35}{8500\cdot 320}\cdot 5}{r^2} $$

Solve for the radius

Combine the equations from Steps 1-3 to solve for the radius: $$ 5 = \frac{\frac{4}{3}\pi r^3}{4\pi r^2} \cdot \frac{1}{\frac{250 \cdot r}{35} \cdot \frac{\frac{35}{8500\cdot 320}\cdot 5}{r^2}} $$ $$ 5 = \frac{4}{3}\cdot \frac{r}{250 \cdot \frac{35}{8500\cdot 320}} $$ Solve for the radius, \(r\): $$ r = 3.696\times 10^{-3} m $$

Determine the diameter of the junction

The diameter \(d\) of the junction needs to be calculated since it has been found that the radius is \(3.696\times 10^{-3} m\). To do so, use the formula: $$ d = 2r $$ Plug the value of \(r\): $$ d = 2(3.696\times 10^{-3} m) $$ $$ d = 7.392\times 10^{-3} m $$ The diameter of the junction is approximately \(7.392 mm\).