A long (i.e., L >> D) stainless steel rod 6.4 mm in diameter is initially at a uniform temperature of 25°C and is suddenly immersed in a liquid at 150°C with h = 120 W/m2 °C. Calculate the time necessary for the center of the rod to reach 120°C.

To calculate the time necessary for the center of the rod to reach 120°C, we would typically make use of transient heat conduction equations. However, since this requires integral calculus and typically the use of tables or software for transient conduction into infinitely long cylinders, it's not straightforward to solve without more advanced mathematical tools. A key simplification in this scenario is the fact that the rod is "long," which allows us to treat it as a semi-infinite solid for the purposes of our model.

But even with this simplification, we'll need to make use of the concept of the Biot number (Bi) to ensure that we can treat the problem as a lumped system (where temperature throughout the rod is assumed uniform) or whether we must consider the temperature gradient along the rod.

The Biot number is calculated as: \[ Bi = \frac{hL_c}{k} \] where \( h \) is the convective heat transfer coefficient, \( L_c \) is the characteristic length (for a cylinder, \( L_c = \frac{V}{A_s} = \frac{{\text{volume}}}{{\text{surface area}}} \)), and \( k \) is the thermal conductivity of stainless steel.

For this problem, however, we need to know that the Biot number is much less than 0.1 to simplify the situation into a lumped system analysis. If so, we could apply the lumped system analysis formula to solve for time (\( t \)): \[ T(t) - T_\infty = (T_i - T_\infty)e^{-\frac{hA_s}{\rho V C_p} t} \]

where: - \( T(t) \) is the temperature at time \( t \) (120°C in your case) - \( T_\infty \) is the surrounding temperature (150°C) - \( T_i \) is the initial temperature (25°C) - \( A_s \) is the surface area of the rod - \( \rho \) is the density of stainless steel - \( V \) is the volume of the rod - \( C_p \) is the specific heat of stainless steel

Nevertheless, this formula is only applicable if we are truly in the lumped capacity regime. If not, we must solve a more complex partial differential equation that describes heat transfer within the rod over time, often using numerical methods or tables/charts derived from the analytical solution to the heat equation.

Given that we are dealing with a "long rod" and thus, length is not a factor, we really are thinking about radial heat transfer. Without thermal properties of stainless steel and more information about the rod's specific geometrical constraints, it is not possible to provide an exact numerical answer. Typically, one would use either analytical methods to solve the transient heat conduction equation (e.g., using Bessel functions for cylindrical coordinates) or numerical methods like finite element analysis for an exact solution.

For schooling or pedagogical purposes, you would likely be provided with tables, charts, or past solved examples that correspond specifically to the thermal properties of the rod material, allowing you to estimate the time to reach \( T(t) \) without fully resolving the equation analytically or numerically.