# The junction of a thermocouple can be approximated as a sphere with a diameter of (1 mathrm{~mm}).

## Question:

The junction of a thermocouple can be approximated as a sphere with a diameter of $$1 \mathrm{~mm}$$. As shown in Figure 7.32, the thermocouple is used to measure the temperature of a gas stream. For the junction, the density is $$ho=8500 \mathrm{~kg} / \mathrm{m}^{3}$$, the specific heat capacity is $$c=320 \mathrm{~J} /\left(\mathrm{kg} \cdot{ }^{\circ} \mathrm{C}\right)$$, and the thermal conductivity is $$k=40 \mathrm{~W} /$$ $$\left(\mathrm{m} \cdot{ }^{\circ} \mathrm{C}\right)$$. The temperature of the gas $$T_{\mathrm{f}}$$ is $$120^{\circ} \mathrm{C}$$ and the initial temperature of the sphere $$T_{0}$$ is $$25^{\circ} \mathrm{C}$$. The heat transfer coefficient between the gas and the junction is $$h=70 \mathrm{~W} /\left(\mathrm{m}^{2 . \circ} \mathrm{C}\right)$$.

a. Determine if the junction's temperature can be considered uniform.

b. Derive the differential equation relating the junction's temperature $$T(t)$$ and the gas's temperature $$T_{\mathrm{f}}$$.

c. Using the differential equation obtained in Part (b), construct a Simulink block diagram to find out how long it will take the thermocouple to read $$99 \%$$ of the initial temperature difference.

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