Question

A spherical vessel has an inner radius \(r_{1}\) and an outer radius \(r_{2}\). The inner surface \(\left(r=r_{1}\right)\) of the vessel is subjected to a uniform heat flux \(\dot{q}_{1}\). The outer surface \(\left(r=r_{2}\right)\) is exposed to convection and radiation heat transfer in a surrounding temperature of \(T_{\infty}\). The emissivity and the convection heat transfer coefficient on the outer surface are \(\varepsilon\) and \(h\), respectively. Express the boundary conditions and the differential equation of this heat conduction problem during steady operation.

Short answer

Answer: 1. Steady-state heat conduction equation: \(\frac{1}{r^2}\frac{\partial}{\partial r}\left( k r^2 \frac{\partial T}{\partial r}\right) = 0\) 2. Boundary conditions: (a) At \(r = r_1\): \(-k \frac{\partial T}{\partial r}\Big|_{r=r_1} = \dot{q}_1\) (b) At \(r = r_2\): \(-k \frac{\partial T}{\partial r}\Big|_{r=r_2} = h \left(T\left(r=r_2\right) - T_\infty\right) + \varepsilon \sigma \left(T^4\left(r=r_2\right) - T_\infty^4\right)\)

Step-by-step solution

Heat Conduction equation in spherical coordinates

The heat conduction equation in spherical coordinates is given by: \(\frac{1}{r^2}\frac{\partial}{\partial r}\left( k r^2 \frac{\partial T}{\partial r}\right) = 0\) Since we are asked for steady-state operation, we can ignore the time-dependent part of the equation. Now, we need to find the boundary conditions at the inner (\(r = r_1\)) and outer (\(r = r_2\)) surfaces of the vessel.

Boundary Condition at Inner Surface \((r = r_1)\)

At the inner surface of the vessel, we are given a uniform heat flux \(\dot{q}_1\). Heat flux is related to the temperature gradient by Fourier's law: \(\dot{q}_1 = -k \frac{\partial T}{\partial r}\Big|_{r=r_1}\) Thus, the boundary condition at \(r = r_1\) is: \(-k \frac{\partial T}{\partial r}\Big|_{r=r_1} = \dot{q}_1\)

Boundary Condition at Outer Surface \((r = r_2)\)

At the outer surface, the vessel loses heat through convection heat transfer to the surrounding fluid and radiation heat transfer. The total heat transfer can be expressed as: \(\dot{q}_2 = -k \frac{\partial T}{\partial r}\Big|_{r=r_2}\) Heat transfer due to convection can be calculated using Newton's law of cooling: \(\dot{q}_{conv} = h \left(T\left(r=r_2\right) - T_\infty\right)\) Heat transfer due to radiation is given by the Stefan-Boltzmann law: \(\dot{q}_{rad} = \varepsilon \sigma \left(T^4\left(r=r_2\right) - T_\infty^4\right)\) Since the total heat transfer should be equal to the sum of convection and radiation heat transfer, we can write the boundary condition at \(r = r_2\) as: \(-k \frac{\partial T}{\partial r}\Big|_{r=r_2} = h \left(T\left(r=r_2\right) - T_\infty\right) + \varepsilon \sigma \left(T^4\left(r=r_2\right) - T_\infty^4\right)\) Now we have the steady-state heat conduction equation and the boundary conditions at both the inner and outer surfaces:

Differential Equation and Boundary Conditions

1. Differential equation: \(\frac{1}{r^2}\frac{\partial}{\partial r}\left( k r^2 \frac{\partial T}{\partial r}\right) = 0\) 2. Boundary conditions: (a) At \(r = r_1\): \(-k \frac{\partial T}{\partial r}\Big|_{r=r_1} = \dot{q}_1\) (b) At \(r = r_2\): \(-k \frac{\partial T}{\partial r}\Big|_{r=r_2} = h \left(T\left(r=r_2\right) - T_\infty\right) + \varepsilon \sigma \left(T^4\left(r=r_2\right) - T_\infty^4\right)\)

Key concepts

Spherical Coordinates

When dealing with heat conduction in shapes like spheres, we use spherical coordinates to describe the system mathematically. This is because spherical coordinates are naturally suited to problems with spherical symmetry, simplifying the complex mathematics involved in heat transfer. In spherical coordinates, a point in space is represented by three variables: radial distance \(r\), polar angle \(\theta\), and azimuthal angle \(\phi\). For the heat conduction problem, the radial direction \(r\) is of primary interest.
The heat conduction equation in spherical coordinates for steady-state problems is given by:

• \[\frac{1}{r^2}\frac{\partial}{\partial r}\left( k r^2 \frac{\partial T}{\partial r}\right) = 0\]

Here, \(k\) is the thermal conductivity, and \(T\) is the temperature. This equation helps determine how temperature changes as you move through the sphere. It assumes that there are no temperature changes over time, focusing instead on the steady-state condition where time plays no role.

Boundary Conditions

Boundary conditions are essential in solving differential equations as they specify the behavior of the system at its boundaries. In spherical heat conduction problems, boundary conditions define how the sphere interacts with its surroundings. For our problem, we have two boundary conditions: one at the inner surface \((r = r_1)\) and one at the outer surface \((r = r_2)\).
At the inner surface, there is a uniform heat flux \(\dot{q}_1\). According to Fourier's law, this relationship can be expressed as:

• \[-k \frac{\partial T}{\partial r}\Big|_{r=r_1} = \dot{q}_1\]

This equation says that the rate at which heat leaves the inner surface is equal to the uniform heat flux.
At the outer surface, heat loss occurs through both convection and radiation. This is represented by the equation:

• \[-k \frac{\partial T}{\partial r}\Big|_{r=r_2} = h \left(T\left(r=r_2\right) - T_\infty\right) + \varepsilon \sigma \left(T^4\left(r=r_2\right) - T_\infty^4\right)\]

Here:

• \(h\) is the convection heat transfer coefficient.
• \(\varepsilon\) is the emissivity of the outer surface.
• \(\sigma\) is the Stefan-Boltzmann constant.
• \(T_\infty\) is the surrounding temperature.

This equation captures the complex interaction of heat being lost to the environment through both air (convection) and electromagnetic waves (radiation).

Steady-State Heat Transfer

Steady-state heat transfer occurs when the temperature distribution in a system does not change over time. This simplifies the analysis because we don't have to consider any transient effects or time-dependent terms in the equations. In the spherical vessel problem, once the system reaches steady state, the heat entering the system at the inner surface equals the heat leaving at the outer surface.
Mathematically, the steady-state heat conduction equation in spherical coordinates is:

• \[\frac{1}{r^2}\frac{\partial}{\partial r}\left( k r^2 \frac{\partial T}{\partial r}\right) = 0\]

This equation focuses solely on the radial part \((r)\) because it assumes the material is isotropic and homogeneous, and thermal properties are consistent throughout. As a result,

• The heat balance becomes much easier because any internal thermal storage is zero.
• The equation becomes a function solely of spatial variables.

In practical terms, for engineers, this implies designing for scenarios where the heat source and environmental conditions remain constant, ensuring efficient and predictable thermal performance.