Question

Heat is generated uniformly at a rate of \(4.2 \times 10^{6} \mathrm{~W} / \mathrm{m}^{3}\) in a spherical ball \((k=45 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) of diameter \(24 \mathrm{~cm}\). The ball is exposed to iced-water at \(0^{\circ} \mathrm{C}\) with a heat transfer coefficient of \(1200 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Determine the temperatures at the center and the surface of the ball.

Short answer

Question: Determine the temperatures at the center and the surface of a spherical ball with the given properties and conditions. Answer: The temperature at the surface of the ball is \(T_s = -1.05^{\circ}\mathrm{C}\). The temperature at the center of the ball is \(T_c = -1.05^{\circ}\mathrm{C}\).

Step-by-step solution

Calculate the temperature distribution

We can use the following equation for steady-state heat transfer in a spherical coordinate system: $$\frac{d}{dr}\left(r^2 \frac{d T(r)}{dr}\right) = -\frac{q_{\text{gen}}}{k} r^2$$ First, we calculate \(r^2 \frac{dT}{dr}\): $$\int \frac{d}{dr}\left(r^2 \frac{d T(r)}{dr}\right) = \int -\frac{q_{\text{gen}}}{k}r^2 dr$$ $$r^2 \frac{dT}{dr} = -\frac{q_{\text{gen}}}{3k}r^3 + C_1$$ Now, we can find \(\frac{dT}{dr}\): $$\frac{dT}{dr} = -\frac{q_{\text{gen}}}{3k}r + \frac{C_1}{r^2}$$ Integrating this expression will give us the temperature distribution: $$\int dT(r) = \int \left(-\frac{q_{\text{gen}}}{3k}r + \frac{C_1}{r^2}\right) dr$$ $$T(r) = -\frac{q_{\text{gen}}}{6k}r^2 + C_1 \ln{r} + C_2$$

Obtain the constants and solve for temperature

To find the constants \(C_1\) and \(C_2\), we need to apply the boundary conditions of the problem: 1. At the surface of the ball \((r=R)\), the temperature has to satisfy the convective boundary condition: $$h(T_s - T_{\infty}) = k \left. \frac{dT}{dr} \right|_{r=R}$$ 2. The temperature must be finite at the center \((r=0)\), which means that \(C_1\) must be zero to prevent a singularity: $$C_1 = 0$$ The temperature distribution becomes simplified: $$T(r) = -\frac{q_{\text{gen}}}{6k}r^2 + C_2$$ Now we apply the first boundary condition: $$1200 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}(T_s) = 45 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K} \cdot \frac{d T}{d r} \Big|_{r = R}$$ Calculate the temperature gradient at the surface: $$\frac{dT}{dr} = -\frac{q_{\text{gen}}}{3k}r = -\frac{4.2 \times 10^6 \mathrm{~W} / \mathrm{m}^{3}}{3 \times 45 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}} \cdot 0.12 \mathrm{~m} = -28 \mathrm{~K} / \mathrm{m}$$ Now we can find \(T_s\): $$T_s = T_{\infty} + \frac{k}{h} \left(\frac{dT}{dr}\right)_{r=R} = 0 + \frac{45 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}}{1200 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}} (-28 \mathrm{~K} / \mathrm{m}) = -1.05^{\circ}\mathrm{C}$$ Finally, we can find the temperature at the center of the ball: $$T(0) = -\frac{q_{\text{gen}}}{6k}(0)^2 + C_2 = -1.05^{\circ}\mathrm{C}$$ This leads us to conclude that: 1. The temperature at the surface of the ball is \(T_s = -1.05^{\circ}\mathrm{C}\). 2. The temperature at the center of the ball is \(T_c = -1.05^{\circ}\mathrm{C}\).

Key concepts

Spherical Heat Transfer

Understanding heat transfer in spherical objects is essential for a range of engineering applications, from nuclear fuel pellets to biomedical implants. Unlike planar or cylindrical geometries, spherical objects have a unique set of mathematical conditions due to their geometry. The heat transfer within a sphere can be described by three modes: conduction, convection, and radiation. In the provided exercise, the heat is generated uniformly within the sphere, and it's assumed that the transfer is steady-state and occurs primarily through conduction.

Spherical symmetry implies that the temperature within the sphere depends only on the radial position, and not on angular positions. The conservation of energy for heat conduction in spherical coordinates leads to the differential equation given in the solution steps. Solving this equation requires integrating twice and applying appropriate boundary conditions to find the constants. This step-by-step process simplifies the complex nature of heat transfer into manageable mathematical solutions.

Temperature Distribution

Temperature distribution refers to how temperature varies within an object. In a spherical object with steady-state heat generation, temperature distribution is vital for assessing the thermal performance and structural integrity of the object. It is influenced by several factors, such as the material's thermal conductivity, the internal heat generation rate, and the boundary conditions.

In our exercise, we derived an equation that predicts the change in temperature across the radius of the sphere. The integration process provided a general solution showing a quadratic dependence of temperature on the radius. The resulting equation can be used to calculate temperature at any point within the sphere, enabling us to conduct a thorough thermal analysis of the system. With the application of boundary conditions, this equation becomes particularly powerful in predicting performance and ensuring the safety and reliability of the spherical object subjected to a thermal load.

Convective Boundary Condition

The convective boundary condition is pivotal in determining how a physical system exchanges heat with its surroundings. When an object's surface is exposed to fluid flow, such as air or water, convective heat transfer plays a crucial role. The condition describes the relationship between the heat transfer coefficient, temperature difference between the surface and the fluid, and the thermal conductivity of the material.

In the exercise, we applied the convective boundary condition to the surface of the sphere to determine the surface temperature. The boundary condition allowed the integration of the heat transfer coefficient (\(h\)) and the ambient temperature (\(T_{\text{infinity}}\)) into our calculations. The heat transfer coefficient is a measure of the convective heat transfer capability of the surrounding fluid and is crucial for accurately predicting the surface temperature. Thus, understanding and applying the right convective boundary condition is a key step in solving real-world heat transfer problems in spherical geometry.