Question

A spherical metal ball of radius \(r_{o}\) is heated in an oven to a temperature of \(T_{i}\) throughout and is then taken out of the oven and dropped into a large body of water at \(T_{\infty}\), where it is cooled by convection with an average convection heat transfer coefficient of \(h\). Assuming constant thermal conductivity and transient one-dimensional heat transfer, express the mathematical formulation (the differential equation and the boundary and initial conditions) of this heat conduction problem. Do not solve.

Short answer

In a cooling process of a spherical metal ball of radius \(r_o\) with an initial temperature \(T_i\) dropped into water with a temperature \(T_\infty\), the transient heat conduction equation in spherical coordinates is given by: $$\frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2 k\frac{\partial T}{\partial r}\right) = \rho c_p \frac{\partial T}{\partial t}$$ The initial condition is \(T(r, 0) = T_i\), and the boundary conditions are: 1. At the center of the sphere: \(\left. \frac{\partial T}{\partial r} \right|_{r=0} = 0\) 2. At the surface of the sphere: \(-k \left. \frac{\partial T}{\partial r} \right|_{r=r_{o}} = h \left[T(r_{o}, t) - T_{\infty}\right]\)

Step-by-step solution

Write down the given conditions and variables

We are given the following variables and parameters: - Radius of the spherical metal ball: \(r_{o}\) - Initial temperature of the metal ball: \(T_{i}\) - Temperature of the water: \(T_{\infty}\) - Convection heat transfer coefficient: \(h\) - Thermal conductivity of the metal: \(k\) - Density of the metal: \(\rho\) - Specific heat at constant pressure: \(c_p\)

Write the general heat conduction equation

In spherical coordinates, the general transient heat conduction equation is: $$\frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2 k\frac{\partial T}{\partial r}\right) = \rho c_p \frac{\partial T}{\partial t}$$

Write the initial condition

At time \(t=0\), the whole sphere is at uniform initial temperature \(T_{i}\). Mathematically, this can be expressed as: $$T(r, 0) = T_{i}$$ This serves as the initial condition for the problem.

Write the boundary conditions

There are two boundary conditions for this problem, one at the center (no heat flux) and one at the surface (convection cooling). 1. At the center of the sphere (\(r=0\)), there is no heat flux. Therefore, the temperature gradient is zero: $$\left. \frac{\partial T}{\partial r} \right|_{r=0} = 0$$ 2. At the surface of the sphere (\(r=r_{o}\)), the heat conduction in the solid is equal to the heat transfer by convection in the water. Mathematically, we can express this as: $$-k \left. \frac{\partial T}{\partial r} \right|_{r=r_{o}} = h \left[T(r_{o}, t) - T_{\infty}\right]$$ These two boundary conditions complete the mathematical formulation of the problem.