Let's solve the question step by step: ### Given Data: - Diameter, \( d = 1 \text{ mm} = 0.001 \text{ m} \) - Thermal conductivity, \( k = 35 \text{ W/m·K} \) - Density, \( \rho = 8500 \text{ kg/m}^3 \) - Specific heat, \( c_p = 320 \text{ J/kg·K} \) - Convection heat transfer coefficient, \( h = 210 \text{ W/m}^2\text{·K} \) ### Time Constant for a Sphere (Lumped Parameter Analysis): The time constant (\( \tau \)) for a lumped system is: \[ \tau = \frac{\rho V c_p}{hA} \] For a sphere: - Volume, \( V = \frac{\pi}{6} d^3 \) - Surface area, \( A = \pi d^2 \) Now, substitute these: \[ \tau = \frac{\rho \left(\frac{\pi}{6} d^3\right) c_p}{h (\pi d^2)} \] \[ \tau = \frac{\rho d c_p}{6h} \] ### Substitute the Values: \[ \tau = \frac{8500 \times 0.001 \times 320}{6 \times 210} \] Calculate numerator: \[ 8500 \times 0.001 = 8.5 \] \[ 8.5 \times 320 = 2720 \] Calculate denominator: \[ 6 \times 210 = 1260 \] Now, \[ \tau = \frac{2720}{1260} \approx 2.158 \text{ s} \] --- ## **Final Answer:** **The time constant is** \( \boxed{2.158 \text{ s}} \). So, the correct option is **2.158 s**.