Question

Consider a short cylinder of radius \(r_{o}\) and height \(H\) in which heat is generated at a constant rate of \(\dot{e}_{\text {gen. }}\). Heat is lost from the cylindrical surface at \(r=r_{o}\) by convection to the surrounding medium at temperature \(T_{\infty}\) with a heat transfer coefficient of \(h\). The bottom surface of the cylinder at \(z=0\) is insulated, while the top surface at \(z=H\) is subjected to uniform heat flux \(\dot{q}_{H}\). Assuming constant thermal conductivity and steady two-dimensional heat transfer, express the mathematical formulation (the differential equation and the boundary conditions) of this heat conduction problem. Do not solve.

Short answer

Based on the given information, the mathematical formulation for the heat conduction problem in a short cylinder can be derived as the following: 1. Governing equation in cylindrical coordinates (steady state, 2D, constant thermal conductivity, and heat generation): \(\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial T}{\partial r}\right) + \frac{\partial^2 T}{\partial z^2} = \frac{\dot{e}_{\text{gen}}}{k}\) 2. Boundary conditions: a. Heat loss from the cylindrical surface: \(\left.-k\frac{\partial T}{\partial r}\right|_{r=r_o} = h(T(r_o, z) - T_\infty)\) b. Insulated bottom surface: \(\left.\frac{\partial T}{\partial z}\right|_{z=0} = 0\) c. Uniform heat flux from the top surface: \(\left.-k\frac{\partial T}{\partial z}\right|_{z=H} = \dot{q}_H\) d. Symmetry condition at the centerline: \(\left.\frac{\partial T}{\partial r}\right|_{r=0} = 0\) These equations and boundary conditions together define the mathematical model for the heat transfer problem in the short cylinder.

Step-by-step solution

Write down the equation governing heat transfer in cylindrical coordinates

Since the problem is related to heat transfer in a cylindrical body, we should use the heat conduction equation in cylindrical coordinates. For a steady state, two-dimensional problem with constant thermal conductivity \(k\) and heat generation \(\dot{e}_{\text{gen}}\), the equation is: \(\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial T}{\partial r}\right) + \frac{\partial^2 T}{\partial z^2} = \frac{\dot{e}_{\text{gen}}}{k}\)

Write down the boundary conditions

We have four boundary conditions for this problem: 1. Heat is lost from the cylindrical surface at \(r=r_o\) by convection. We can write the boundary condition for this as: \(\left.-k\frac{\partial T}{\partial r}\right|_{r=r_o} = h(T(r_o, z) - T_\infty)\) 2. The bottom surface at \(z=0\) is insulated, which means no heat is conducted through it. The boundary condition for this is: \(\left.\frac{\partial T}{\partial z}\right|_{z=0} = 0\) 3. The top surface at \(z=H\) has a uniform heat flux \(\dot{q}_H\). The boundary condition for this is: \(\left.-k\frac{\partial T}{\partial z}\right|_{z=H} = \dot{q}_H\) 4. Symmetry condition: Since the problem is axisymmetric, there should be no heat flux along the radial direction at the centerline \(r=0\). The boundary condition for this is: \(\left.\frac{\partial T}{\partial r}\right|_{r=0} = 0\) Now we have the mathematical formulation of the problem, consisting of the heat conduction equation and the boundary conditions.

Key concepts

Heat Generation

Heat generation refers to the process by which energy is produced internally within a material, leading to a temperature rise. In our cylindrical coordinate system, heat generation is denoted by a constant rate \(\dot{e}_{\text{gen}}\). This is crucial as it defines an internal source of heat.

In the context of the cylindrical body, imagine the cylinder generates heat uniformly across its volume. This uniform distribution simplifies our calculations and helps to form the foundational heat conduction equation in cylindrical coordinates.

• Uniform heat generation ensures that the equation remains manageable, maintaining consistency through the cylinder's structure.
• The intensity of heat generation heavily impacts thermal management strategies applied to the system.

Recognizing uniform heat generation aids in understanding and predicting the steady-state conditions of the system. It serves as a significant factor influencing the overall temperature distribution.

Boundary Conditions

Boundary conditions are constraints that describe how the system interacts with its surroundings. They play a crucial role in solving heat conduction problems by simplifying the mathematical models to accurately reflect physical settings.

For our cylindrical body, the problem defines four specific boundary conditions:

• **Convection at \(r=r_o\):** The heat is lost through convection to external surroundings, influenced by the heat transfer coefficient \((h)\).
• **Insulated bottom surface at \(z=0\):** No heat can escape or enter through this surface, represented by zero heat flux.
• **Uniform heat flux at \(z=H\):** Implies a consistent amount of heat transfers across this top surface.
• **Symmetry condition at \(r=0\):** Indicates no radial heat flux at the cylinder’s centerline.

These conditions guide us in shaping the thermal profile across the cylinder, establishing how our system evolves toward equilibrium.

Convection Heat Transfer

Convection is the process of heat transfer through fluid (such as air or liquid) movement. It directly impacts the heat loss from the surface of our cylindrical system.

When we state that heat is lost from the cylindrical surface via convection, it interacts with a surrounding medium at temperature \(T_{\infty}\). This process is quantitatively described using the convection heat transfer coefficient \(h\).

• The convection term in our boundary condition accounts for the heat transfer from the cylinder to its environment.
• It is critical in determining the total heat loss from the surface, especially for materials where surface interaction is significant.

This aspect of heat transfer is pivotal for accurately modeling how the cylindrical body dissipates its generated heat into the surroundings.

Thermal Conductivity

Thermal conductivity \(k\) defines a material's ability to conduct heat. It forms the backbone of the heat conduction equation in cylindrical coordinates.

In our problem, we assume constant thermal conductivity, which simplifies the calculations, allowing us to assume uniform material properties. This assumption affects heat transfer through the material:

• High thermal conductivity denotes the material can efficiently distribute generated heat.
• Low thermal conductivity could indicate localized heating, potential hotspots, leading to material stresses.

Given these considerations, constant thermal conductivity is an essential simplification aiding the development of precise thermal models.

Steady-State Heat Transfer

Steady-state signifies a condition where temperatures within the system remain constant over time, despite the presence of heat generation or transfer processes.

In the context of this problem, reaching a steady state implies that:

• Heat inflow consistently equals heat outflow, balancing the system's energy equation.
• It provides a simpler snapshot of system behavior, eliminating the complexity introduced by transient temperature changes.

Understanding the steady-state conditions helps us predict that the temperatures will no longer change with time under the provided constraints, thereby making the mathematical solution more direct and interpretable.