Subtopic: Transient heat transfer in simple geometry
Question Statement:
A spherical metal temperature probe of diameter $D = 0.012\text{ m}$ (characteristic length $L_c = R/3 = 0.002\text{ m}$) has density $\rho = 8000\text{ kg/m}^3$ and specific heat $c_p = 500\text{ J/(kg}\cdot\text{K)}$. It is suddenly plunged into an agitated liquid where the convective heat transfer coefficient is $h = 200\text{ W/(m}^2\cdot\text{K)}$. The thermal time constant $\tau = \frac{\rho c_p L_c}{h}$ of the probe in seconds is ________ (answer in integer).
Answer Key:
40
Step-by-Step Solution & Explanation
The thermal time constant of a lumped capacitance system is:
$\tau = \frac{\rho V c_p}{h A_s} = \frac{\rho c_p L_c}{h}$
Substitute values:
$\tau = \frac{8000\text{ kg/m}^3 \times 500\text{ J/(kg}\cdot\text{K)} \times 0.002\text{ m}}{200\text{ W/(m}^2\cdot\text{K)}} = \frac{8000}{200} = 40.0\text{ seconds}$