Question

The temperature of a gas stream is to be measured by a thermocouple whose junction can be approximated as a \(1.2\)-mm-diameter sphere. The properties of the 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}$, and the heat transfer coefficient between the junction and the gas is \(h=110 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Determine how long it will take for the thermocouple to read 99 percent of the initial temperature difference.

Short answer

Question: Determine the time it takes for a thermocouple to read 99 percent of the initial temperature difference. Given: - Diameter of the thermocouple junction sphere (d) = 1.2 mm - Density (ρ) = 8500 kg/m³ - Specific heat capacity (cp) = 320 J/(kg·K) - Heat transfer coefficient (h) = 110 W/(m²·K) Following the steps in the provided solution: 1. Calculate the volume (V) and surface area (A) of the junction sphere. 2. Calculate the time constant (τ) using the lumped capacitance method. 3. Calculate the time (t) for the thermocouple to reach 99 percent of the initial temperature difference. Answer: The time it takes for the thermocouple to read 99 percent of the initial temperature difference is the calculated value for t.

Step-by-step solution

Calculate Volume and Area of the Junction Sphere

We can first find the volume and surface area of the junction sphere: $$ V=\frac{4}{3} \pi\left(\frac{d}{2}\right)^{3} \quad \text { and } \quad A=4 \pi\left(\frac{d}{2}\right)^{2} $$ where, \(d\) is the diameter of the thermocouple junction sphere (\(1.2\) mm).

Calculate Lumped Capacitance Time Constant

Now we can calculate the time constant in the lumped capacitance method: $$ \tau=\frac{\rho V c_{p}}{h A} $$ Using the given values for density (\(\rho = 8500 \frac{kg}{m^3}\)), specific heat capacity (\(c_p = 320 \frac{J}{kg \cdot K}\)), and heat transfer coefficient (\(h = 110 \frac{W}{m^2\cdot K}\)).

Calculate the Time for 99 Percent of Temperature Difference

Given that we want to find the time it takes to reach 99 percent of the initial temperature difference, we can use the following formula: $$ t = -\tau \ln (1 - 0.99) $$ Calculating the time using the obtained value for the time constant, \(\tau\).

Result

The time it takes for the thermocouple to read 99 percent of the initial temperature difference is the calculated value for \(t\).