Question

Consider uniform heat generation in a cylinder and a sphere of equal radius made of the same material in the same environment. Which geometry will have a higher temperature at its center? Why?

Short answer

Answer: The sphere will have a higher temperature at its center compared to the cylinder.

Step-by-step solution

Understand the problem

We have a cylinder and a sphere, both have equal radii and the same material properties. They are also in the same environment, meaning they have the same ambient temperature. The goal is to find out which one has a higher temperature at their respective centers.

Use heat transfer equation

We will use the heat transfer equation to determine the temperature distribution in both geometries. The heat transfer equation for 3D heat generation can be written in cylindrical coordinates for the cylinder and in spherical coordinates for the sphere. For both cases, we can simplify the heat transfer equation by assuming steady-state conditions, constant thermal conductivity, and no heat convection. The simplified heat transfer equation in cylindrical coordinates (for the cylinder) is: $$\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial T}{\partial r}\right) = -\frac{q}{k}$$ And in spherical coordinates (for the sphere) is: $$\frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\frac{\partial T}{\partial r}\right) = -\frac{q}{k}$$ Where: - r is the radial coordinate - T is the temperature - q is the volumetric heat generation rate (assumed to be uniform) - k is the thermal conductivity of the material

Solve the heat transfer equation

Now, we will solve the heat transfer equation for both geometries. Since both materials have the same thermal conductivity and volumetric heat generation rate, we will focus on the radial part of the heat transfer equation. For the cylinder: Integrate the heat transfer equation for the cylindrical coordinate system twice with respect to r, then use the boundary conditions to find the constants of integration. The solution of the temperature for the cylinder can be determined by the following function: $$T_{cylinder} = -\frac{q}{4k}(R^2 - r^2) + T_{cylinder, ambient}$$ For the sphere: Similarly, integrate the heat transfer equation for the spherical coordinate system twice and use the boundary conditions to find the constants of integration. The final temperature distribution for the sphere is given by the following function: $$T_{sphere} = -\frac{q}{6k}(R^2 - r^2) + T_{sphere, ambient}$$ Where R is the radius of both geometries, and T_ambient is the ambient temperature for both cases.

Compare the central temperatures

Now, compare the temperature at the center of both geometries, which corresponds to r = 0. For the cylinder, the temperature at the center (r = 0) is: $$T_{cylinder, center} = -\frac{q}{4k}R^2 + T_{cylinder, ambient}$$ For the sphere, the temperature at the center (r = 0) is: $$T_{sphere, center} = -\frac{q}{6k}R^2 + T_{sphere, ambient}$$ We can now observe that the temperature at the center of the sphere is higher than that of the cylinder. Therefore, the sphere will have a higher temperature at its center. The reason by this difference can be explained by how efficiently the heat is transported within the geometries— the sphere's shape conducts and distributes heat more effectively than the cylinder, leading to a higher temperature at its center.

Key concepts

Uniform Heat Generation

Uniform heat generation refers to a scenario where heat is produced at a consistent rate throughout a material's volume. In our exercise, both the cylinder and the sphere generate heat in such a fashion. The uniformity in heat generation simplifies the heat transfer equations, as the heat source term, denoted by the symbol \( q\), remains constant.

Understanding uniform heat generation is crucial in determining temperature distribution because it directly influences the internal energy change within an object. In practical applications, this could represent materials that generate heat due to chemical reactions, electrical resistance, or other energy transformations.

Cylindrical Coordinates

Cylindrical coordinates are a three-dimensional coordinate system that extends polar coordinates with an extra dimension for height, making them suitable for solving problems with cylindrical symmetry. The variables used are radial distance \( r\), angular coordinate \( \theta\), and height \( z\).

In heat transfer problems, when we consider a cylinder, we often focus on the radial component \( r\) because heat flows radially outward or inward. This system simplifies the calculation of temperature distribution in cylindrical objects, like the one described in our exercise.

Spherical Coordinates

Spherical coordinates are another three-dimensional coordinate system, perfect for analyzing phenomena with radial symmetry from a central point, like a sphere. This system uses a radial distance \( r\), an inclination angle \( \theta\), and an azimuthal angle \( \phi\).

For example, when studying the temperature distribution of a sphere undergoing uniform heat generation, spherical coordinates make the problem more approachable by aligning with the object's natural symmetry. This allows us to simplify the heat transfer equation as we did in the given exercise for finding the temperature distribution within a sphere.

Temperature Distribution

Temperature distribution refers to how temperature varies within an object. It's a critical concept in heat transfer as it influences the thermal stresses and the physical properties of the material. The distribution depends on various factors like the shape of the object, the material's properties, the boundary conditions, and the presence of uniform or non-uniform heat generation.

In our solved exercise, we calculated the temperature distribution across a cylinder and a sphere with uniform heat generation. The differences in the geometry affected how heat diffused through each object, resulting in different temperature profiles, and ultimately, different central temperatures for the cylinder and sphere.

Thermal Conductivity

Thermal conductivity, denoted by \( k\), is a material's ability to conduct heat. Materials with high thermal conductivity can transport heat more efficiently than those with low thermal conductivity. In the context of our exercise, both the cylinder and sphere are made from the same material, meaning they have the same thermal conductivity.

This property is fundamental in the heat transfer equation as it determines how easily heat flows through the material. The rate of heat transfer through a material directly impacts the temperature distribution within it. In equations, the thermal conductivity serves as a scaling factor for the relationship between heat generation and temperature rise.