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

Based on the heat conduction problem for a transient one-dimensional spherical metal ball of radius \(r_{o}\) with initial temperature \(T_{i}\), which is dropped into a large body of water at temperature \(T_{\infty}\), provide the differential equation, boundary conditions, and initial condition of the problem. Differential equation: \[ \frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2 \frac{\partial T}{\partial r}\right) = \frac{1}{\alpha}\frac{\partial T}{\partial t} \] Boundary conditions: \[ \frac{\partial T}{\partial r}\bigg|_{r=0} = 0 \] \[ -k \frac{\partial T}{\partial r}\bigg|_{r=r_{o}} = h(T - T_{\infty}) \] Initial condition: \[ T(r, 0) = T_{i} \]

Step-by-step solution

Heat Conduction Differential Equation

We will begin by writing the general heat conduction equation for a sphere in a transient state. This equation is the general form of any heat conduction problem and can be expressed as follows in spherical coordinates: \[ \frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2 \frac{\partial T}{\partial r}\right) = \frac{1}{\alpha}\frac{\partial T}{\partial t} \] Here, T is the temperature as a function of radial distance r and time t, and \(\alpha\) is the thermal diffusivity.

Boundary Condition at r = 0

Now, we need to define the boundary conditions for this problem. For heat conduction in spherical coordinates, there will be a boundary condition at r = 0 (the center of the sphere). Since no heat is being generated at the center, we can impose symmetry at r = 0: \[ \frac{\partial T}{\partial r}\bigg|_{r=0} = 0 \]

Boundary Condition at r = \(r_{o}\)

Next, we need to define the boundary condition at r = \(r_{o}\), the outer surface of the sphere. The sphere is cooled by convection with a heat transfer coefficient of h. The convective heat transfer is given by: \[ q_{conv} = h(T - T_{\infty}) \] where \(T_{\infty}\) is the temperature of the water and q is the heat flux. We can relate the heat flux to temperature using Fourier's law in spherical coordinates: \[ q_{cond} = -k \frac{\partial T}{\partial r} \] At the boundary, the heat flux due to conduction must equal the heat flux due to convection: \[ -k \frac{\partial T}{\partial r}\bigg|_{r=r_{o}} = h(T - T_{\infty}) \]

Initial Condition

Finally, we need to define the initial condition for this problem. The initial condition is given as the sphere being at a uniform temperature of \(T_{i}\): \[ T(r, 0) = T_{i} \] To summarize, the mathematical formulation of this heat conduction problem consists of the heat conduction differential equation, boundary conditions, and initial condition: Differential equation: \[ \frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2 \frac{\partial T}{\partial r}\right) = \frac{1}{\alpha}\frac{\partial T}{\partial t} \] Boundary conditions: \[ \frac{\partial T}{\partial r}\bigg|_{r=0} = 0 \] \[ -k \frac{\partial T}{\partial r}\bigg|_{r=r_{o}} = h(T - T_{\infty}) \] Initial condition: \[ T(r, 0) = T_{i} \]

Key concepts

Spherical Coordinates Heat Transfer

Heat transfer analysis in spherical geometries is vital for objects like spheres, where temperature may vary from the center to the surface. Unlike Cartesian coordinates where the heat conduction is typically linear, spherical coordinates consider a radial dimension, requiring us to account for the curvature in the thermal analysis.

When dealing with transient heat conduction in a sphere, it's important to remember that the temperature changes not only over time but also varies with the radial position. The governing differential equation reflects this complex behavior. The rate of temperature change is proportional to the Laplacian of the temperature field, taking on a form that emphasizes the role of radius in the sphere's geometry. By considering such factors, we can model scenarios such as a heated metal ball cooling in water, capturing the nuances of radial heat flow.

Initial and Boundary Conditions

The success of mathematically modeling heat conduction problems hinges on accurately defining initial and boundary conditions. These conditions frame the problem and provide constraints for solving the differential equations.

The initial condition typically reflects the temperature distribution at the initial moment—in our case, the uniform temperature of the spherical object before it starts cooling. Without this information, predicting the future state of the system becomes nearly impossible.

Boundary conditions, on the other hand, describe how the system interacts with its surroundings. The boundary at the sphere's center is straightforward, given the symmetry of the problem. However, the outer boundary condition combines conduction and convection phenomena, which dictate the rate at which heat is transferred from the sphere to the surrounding water.

Convection Heat Transfer Coefficient

The convection heat transfer coefficient (\(h\)), a critical parameter in convective cooling scenarios, measures the effectiveness of convection in transferring heat between a solid surface and the fluid it is in contact with. This coefficient is influenced by various factors, including the type of fluid, its velocity, and the surface characteristics.

In practice, a higher value of the convection heat transfer coefficient indicates that the fluid is efficient at removing heat, which results in quicker cooling. In an educational setting, it's crucial to emphasize that while students might encounter average values to simplify analyses, in the real world, the coefficient can fluctuate based on local conditions. Understanding its role helps explain how rapidly the spherical object will cool down when immersed in water.

Fourier's Law of Thermal Conductivity

Central to heat transfer analysis is Fourier's law of thermal conductivity, which provides a linear relationship between the heat flux and the temperature gradient within a material. The law, represented by \( q_{cond} = -k \frac{\partial T}{\partial r} \), where \(k\) is the thermal conductivity, conveys the intuitive concept that heat flows from regions of higher temperature to lower temperature.

For students, Fourier's Law is a cornerstone in understanding how materials conduct heat and how quickly they will respond to temperature differences. In our example with the spherical metal ball, the law allows us to connect the rate of heat loss due to conduction at the surface of the sphere to the cooling effect provided by the water, facilitating a comprehensive analysis of the cooling process.