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 allowed to cool in ambient air at \(T_{\infty}\) by convection and radiation. The emissivity of the outer surface of the cylinder is \(\varepsilon\), and the temperature of the surrounding surfaces is \(T_{\text {surr }}\). The average convection heat transfer coefficient is estimated to be \(h\). Assuming variable 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

Question: Derive the mathematical formulation of the heat conduction problem for a spherical metal ball which is initially heated to a uniform temperature and then allowed to cool in ambient air, considering variable thermal conductivity and transient one-dimensional heat transfer. Answer: The mathematical formulation consists of the governing differential equation, initial condition, and boundary conditions. 1. Governing differential equation: \(\frac{\partial}{\partial r}\left(r^2 \alpha \frac{\partial T}{\partial r}\right) = r^2 \frac{\partial T}{\partial t}\) 2. Initial condition: \(T(r, 0) = T_i\) for \(0\le r\le r_o\) 3. Boundary conditions: \(\frac{\partial T}{\partial r}(0, t) = 0\) \(k\frac{\partial T}{\partial r}(r_o, t) = h(T(r_o, t) - T_\infty)+\varepsilon \sigma (T(r_o, t)^4 - T_{\text{surr}}^4)\)

Step-by-step solution

Write the general heat diffusion equation

The general heat diffusion equation for a variable thermal conductivity in spherical coordinates is given by: \(\frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2 \alpha \frac{\partial T}{\partial r}\right)=\frac{\partial T}{\partial t}\) where \(\alpha\) is the thermal conductivity, \(T\) is the temperature, \(r\) is the radius, and \(t\) is the time.

Simplify the heat diffusion equation for one-dimensional heat transfer

For one-dimensional heat transfer (only radial direction), the equation provided in Step 1 simplifies to: \(\frac{\partial}{\partial r}\left(r^2 \alpha \frac{\partial T}{\partial r}\right) = r^2 \frac{\partial T}{\partial t}\) Now, we have the simplified heat diffusion equation for our problem. Next, we need to find the boundary and initial conditions for this problem.

Determine the initial condition

The initial condition is that the sphere is uniformly heated to a temperature of \(T_i\) throughout. Therefore, at the initial time \(t=0\), the temperature at any point within the sphere is given by: \(T(r, 0) = T_i\) for \(0\le r\le r_o\)

Determine the boundary conditions

We need to determine boundary conditions at the surface of the sphere \((r=r_o)\) and at the center \((r=0)\). Boundary condition at the center \((r=0)\): To maintain smoothness of the solution, the temperature gradient at the center should be zero. Therefore, \(\frac{\partial T}{\partial r}(0, t) = 0\) Boundary condition at the surface \((r=r_o)\): At the surface, convection and radiation take place. The heat flux due to convection and radiation can be given by: \(q_{conv} = h(T(r_o, t) - T_\infty)\) \(q_{rad} = \varepsilon \sigma (T(r_o, t)^4 - T_{\text{surr}}^4)\) The total heat flux at the surface is the sum of these two, and it is also equal to the heat conducted through the sphere. Hence, \(k\frac{\partial T}{\partial r}(r_o, t) = q_{conv} + q_{rad}\) Substituting the values for \(q_{conv}\) and \(q_{rad}\), \(k\frac{\partial T}{\partial r}(r_o, t) = h(T(r_o, t) - T_\infty)+\varepsilon \sigma (T(r_o, t)^4 - T_{\text{surr}}^4)\) Now, we have the boundary conditions for both ends. To summarize, 1. The differential equation governing the cooling process of the spherical ball: \(\frac{\partial}{\partial r}\left(r^2 \alpha \frac{\partial T}{\partial r}\right) = r^2 \frac{\partial T}{\partial t}\) 2. The initial condition: \(T(r, 0) = T_i\) for \(0\le r\le r_o\) 3. Boundary conditions: \(\frac{\partial T}{\partial r}(0, t) = 0\) \(k\frac{\partial T}{\partial r}(r_o, t) = h(T(r_o, t) - T_\infty)+\varepsilon \sigma (T(r_o, t)^4 - T_{\text{surr}}^4)\)

Key concepts

Transient Heat Transfer

Transient heat transfer is a fascinating area of study that deals with changes in temperature over time. Unlike steady-state heat transfer, where temperatures remain constant, transient heat transfer considers how temperatures evolve. In our exercise, we're dealing with a metal sphere initially heated to a uniform temperature. As time progresses, this temperature gradually decreases as the sphere cools down.

This cooling process involves solving heat conduction equations that change over time. These equations allow us to predict how the temperature at any particular point within the sphere will vary. Essentially, we're focused on the thermal response of the sphere as it interacts with its surrounding environment. This contrasts with steady-state conditions where the heat transfer rate and temperature field are constant over time.

Spherical Coordinates

When studying heat transfer in objects like the spherical metal ball mentioned in the exercise, using spherical coordinates makes the analysis more straightforward. Spherical coordinates are particularly useful here due to the symmetry of the sphere. These coordinates help Idenitify locations more easily with a radius, an angle, and a polar angle.

The heat diffusion equation uses a radial coordinate, which simplifies governing mathematical expressions by leveraging the uniform geometry of the sphere. By employing this system, we focus purely on radial heat movement without the complexities introduced by other dimensions, making the mathematical treatment more manageable and intuitive.

Boundary Conditions

Boundary conditions are crucial for solving differential equations in heat transfer problems. They define how the heat is exchanged between the system and its surroundings at the boundaries. For the cooling sphere, two boundary conditions are essential:

• At the center of the sphere \(r=0\), the temperature gradient must be zero. This means no heat flows across the center, ensuring a symmetrical temperature distribution.
• The second boundary condition is at the sphere's surface \(r=r_o\). Here, the heat exchange involves both convection and radiation. This defines how heat leaves the sphere's surface to the surrounding air and surfaces.

Boundary conditions help us understand how the sphere interacts with its environment, guiding the solution of the heat conduction equation through defined physical constraints.

Thermal Conductivity

Thermal conductivity is a material property that indicates how efficiently heat is conducted through a material. In the context of our spherical metal ball, thermal conductivity \(\alpha\) might vary depending on the temperature, a condition considered as variable thermal conductivity.

This property is essential for modeling the heat transfer since it directly impacts how quickly temperatures will change throughout the sphere. High thermal conductivity means the material will allow heat to flow quickly, leading to faster temperature equalization, while low thermal conductivity impedes heat transfer, resulting in slower changes. In this exercise, considering variable thermal conductivity adds complexity, making the problem more representative of real-world scenarios like cooling metals.

Convection and Radiation

Understanding convection and radiation is key to solving real-world heat transfer problems. These mechanisms describe how heat is dissipated from a surface, like our cooling sphere in ambient air.

**Convection** is the process of heat transfer through fluid movement around the sphere. This process is characterized by the convective heat transfer coefficient \(h\), which measures how efficiently heat is exchanged between the sphere's surface and the surrounding air.

**Radiation** involves heat exchange with the environment's surfaces based on temperature differences. It depends on the surrounding temperature \(T_{\text{surr} }\) and the sphere's emissivity \(\varepsilon\), a measure of how effectively the sphere emits thermal radiation.

Combining these two modes of heat transfer provides a comprehensive picture of the interaction between the sphere and its cooling environment, defining the boundary conditions at the sphere's surface.

Initial Conditions

Initial conditions define the starting point of temperature distribution in transient heat transfer problems. For the spherical metal ball in our exercise, it is assumed to be uniformly heated to an initial temperature \(T_i\). This simplification assumes every point within the sphere starts at the same temperature, \(T(r, 0) = T_i\), for all radii \ (0 \le r \le r_o)\.

These conditions are fundamental for predicting the evolution of temperatures within the object as it transitions from its initial state to equilibrium with its surroundings. By setting the initial condition, we establish the baseline against which all subsequent changes in the sphere's temperature distribution are measured. This starting point is crucial for solving the heat conduction equation accurately.