Question

Consider a spherical container of inner radius \(r_{1}\), outer radius \(r_{2}\), and thermal conductivity \(k\). Express the boundary condition on the inner surface of the container for steady onedimensional conduction for the following cases: \((a)\) specified temperature of \(50^{\circ} \mathrm{C},(b)\) specified heat flux of \(45 \mathrm{~W} / \mathrm{m}^{2}\) toward the center, (c) convection to a medium at \(T_{\infty}\) with a heat transfer coefficient of \(h\).

Short answer

Question: Determine the boundary conditions for steady one-dimensional conduction on the inner surface of a spherical container in the given cases: (a) specified temperature of \(50^{\circ}\mathrm{C}\), (b) specified heat flux of \(45 \mathrm{W}/\mathrm{m}^2\) toward the center, and (c) convection to a medium at a temperature \(T_{\infty}\) with a heat transfer coefficient \(h\). Answer: (a) \(T(r_1) = 50^{\circ}\mathrm{C}\) (b) \(-k \frac{dT}{dr}\bigg|_{r=r_1} = 45 \mathrm{W}/\mathrm{m^2}\) (c) \(-k \frac{dT}{dr}\bigg|_{r=r_1} = h\big(T(r_1) - T_{\infty}\big)

Step-by-step solution

Case (a): Specified temperature of \(50^{\circ}\mathrm{C}\)

In this case, the boundary condition on the inner surface of the container is that the temperature is a fixed \(50^{\circ}\mathrm{C}\). Since the temperature is provided in degree Celsius, we can just state the boundary condition as follows: $$ T(r_1) = 50^{\circ}\mathrm{C} $$

Case (b): Specified heat flux of \(45 \mathrm{~W} / \mathrm{m}^{2}\) toward the center

In this case, the inner surface has a heat flux of \(45 \mathrm{W}/\mathrm{m}^2\) towards the center of the container. The heat flux \(q\) is related to the temperature gradient by Fourier's Law: $$ q = -k \frac{dT}{dr} $$ Since we know the heat flux \(q = 45 \mathrm{W}/\mathrm{m}^2\) toward the center, we must take into account the negative sign in Fourier's Law as the temperature gradient will be negative. The boundary condition on the inner surface can be expressed as: $$ -k \frac{dT}{dr}\bigg|_{r=r_1} = 45 \mathrm{W}/\mathrm{m^2} $$

Case (c): Convection to a medium at \(T_{\infty}\) with a heat transfer coefficient of \(h\)

In the last case, we have convection to a medium at a temperature \(T_{\infty}\) with a heat transfer coefficient \(h\). The heat being transferred from the inner surface to the medium can be expressed by Newton's Law of Cooling: $$ q = h\big(T(r_1) - T_{\infty}\big) $$ We can also relate the heat flux \(q\) to the temperature gradient using Fourier's Law, as shown in the previous case: $$ q = -k \frac{dT}{dr}\bigg|_{r=r_1} $$ Combining both these equations, we can express the boundary condition on the inner surface as: $$ -k \frac{dT}{dr}\bigg|_{r=r_1} = h\big(T(r_1) - T_{\infty}\big) $$

Key concepts

Boundary Conditions

Boundary conditions are vital in solving heat transfer problems as they describe the thermal behavior at the surface where heat transfer processes are occurring. They define how external or internal influences affect the inner surface temperature. In our exercise, we have three types of boundary conditions to consider:

• Specified Temperature: This condition fixes the temperature at a specific boundary. In the case, it's simplified to stating that the temperature at the inner radius of the sphere is set to 50°C. It means the surface must maintain this temperature, influencing the heat conduction within the sphere.
• Specified Heat Flux: This is when the rate of heat flow per unit area is defined at the boundary. In our scenario, the inner surface of the sphere has a heat flux of 45 W/m² directed towards the center. Here, "heat flux" specifies how fast heat travels through the surface.
• Convection: Convection occurs when heat is transferred between the surface and a fluid like air or water surrounding it. The boundary condition involves the medium’s temperature and a heat transfer coefficient, which indicates the efficiency of heat transfer between the surface and the fluid.

Understanding these conditions helps define how temperature and heat will change over time or remain steady within a system.

Heat Flux

Heat flux refers to the flow rate of thermal energy through a given surface in a specified direction. It is typically measured in watts per square meter (W/m²).
Heat flux is crucial as it tells how much thermal energy is entering or leaving a system, providing insight into temperature distribution and the thermal balance of objects.
In the case of the sphere with a specified heat flux, it's directed towards the center, meaning heat is being transferred from the surface inward.
The heat flux boundary condition can be mathematically represented via Fourier's Law. It's calculated as:\[q = -k \frac{dT}{dr}\]where:

• \(q\) is the heat flux;
• \(k\) is the thermal conductivity of the material;
• \(\frac{dT}{dr}\) is the temperature gradient in the radial direction.

The negative sign signifies that heat moves from high to low temperature areas, opposing the temperature gradient. The heat flux representation provides clarity on conduction and helps design systems for efficient thermal management.

Fourier's Law

Fourier’s Law establishes the relationship between heat flux and temperature gradient through materials, illustrating how heat conduction occurs within solids and fluids.
It is foundational for determining heat flow in conductive mediums and is expressed as:\[q = -k \frac{dT}{dx}\]This law calculates the rate of heat transfer within a material based on its thermal conductivity and the gradient of temperature change across it.

• \(q\) is the heat flux;
• \(k\) is the material's thermal conductivity, indicating how well it conducts heat;
• \(\frac{dT}{dx}\) represents the temperature change across a small distance.

The negative sign denotes heat naturally flows from warmer to cooler regions. Fourier's Law is essential for engineering applications, helping designers create efficient thermal systems. By understanding how conductive heat transfer operates, engineers can better predict temperature changes and ensure the systems function safely and effectively.

Newton's Law of Cooling

Newton's Law of Cooling describes the rate of heat transfer from a solid surface to a fluid (like air or water) when they are at different temperatures. It gives a clear representation of convective heat transfer processes with the following equation:

• \[q = h (T_s - T_∞)\]

where:

• \(q\) is the convective heat flux;
• \(h\) is the convective heat transfer coefficient, measuring efficiency;
• \(T_s\) is the surface temperature of the object;
• \(T_∞\) is the fluid's temperature far from the surface.

The law indicates that the heat transfer rate depends greatly on the temperature difference between the surface and its surroundings and the transfer coefficient's magnitude. This concept is important for designing systems that rely on effective heat exchange, like radiators, cooling systems, and various industrial processes.
By understanding Newton’s Law of Cooling, engineers can predict cooling rates and manage temperatures, ensuring systems perform effectively under varying environmental conditions.