Question

Consider a cylindrical shell of length $L$, inner radius $r_1$, and outer radius $r_2$ whose thermal conductivity varies in a specified temperature range as $k(T)=k_0(1+\beta T^2)$ where $k_0$ and $\beta$ are two specified constants. The inner surface of the shell is maintained at a constant temperature of $T_1$ while the outer surface is maintained at $T_2$. Assuming steady one-dimensional heat transfer, obtain a relation for the heat transfer rate through the shell.

Short answer

Answer: The final relation for the heat transfer rate is given by: $$\frac{dQ}{dt}=\frac{2\pi k_{0}L}{\ln \left(\frac{r_2}{r_1}\right)}\left[\frac{\arctan \left(\sqrt{\beta }{T}_2\right)-\arctan \left(\sqrt{\beta }{T}_1\right)}{\sqrt{\beta }}\right]$$

Step-by-step solution

Write down Fourier's Law for heat conduction

Fourier's Law of heat conduction for a cylindrical shell can be written as: $$\frac{dQ}{dt}=-k(T)A\frac{dT}{dr}$$ where $\frac{dQ}{dt}$ is the heat transfer rate, $k(T)$ is the thermal conductivity, $A$ is the surface area of the cylindrical shell, and $\frac{dT}{dr}$ is the temperature gradient with the radius $r$.

Calculate surface area A

The surface area of a cylindrical shell of length $L$ and radius $r$ can be calculated as: $$A=2\pi rL$$

Substitute the given conductivity variation

Now we need to substitute the given conductivity variation $k(T)=k_0(1+\beta T^2)$ in the Fourier's Law: $$\frac{dQ}{dt}=-k_0(1+\beta T^2)\cdot 2\pi rL\frac{dT}{dr}$$

Solve the differential equation

The equation we derived in step 3 is an ordinary differential equation, to solve it, we can separate the variables as follows: $${\int }_{T_1}^{T_2}\frac{1}{1+\beta T^2}dT=-\frac{k_{0}L}{\frac{dQ}{dt}}{\int }_{r_1}^{r_2}\frac{1}{r}dr$$

Evaluate the integrals

Evaluate the left-hand side (LHS) and the right-hand side (RHS) of the equation obtained in step 4: $${\int }_{T_1}^{T_2}\frac{1}{1+\beta T^2}dT=\frac{1}{\sqrt{\beta }}\arctan \left(\sqrt{\beta }T\right){|}_{T_1}^{T_2}$$ and $$-\frac{k_{0}L}{\frac{dQ}{dt}}{\int }_{r_1}^{r_2}\frac{1}{r}dr=-\frac{k_{0}L}{\frac{dQ}{dt}}\ln (r){|}_{r_1}^{r_2}$$

Obtain the final relation for heat transfer rate

Equate the LHS and RHS evaluation and simplify the equation obtained in step 5 to get the final relation for the heat transfer rate $\frac{dQ}{dt}$: $$\frac{dQ}{dt}=\frac{2\pi k_{0}L}{\ln \left(\frac{r_2}{r_1}\right)}\left[\frac{\arctan \left(\sqrt{\beta }{T}_2\right)-\arctan \left(\sqrt{\beta }{T}_1\right)}{\sqrt{\beta }}\right]$$

Key concepts

Fourier's Law

Fourier's Law is fundamental to understanding heat conduction. It describes how heat energy is transferred through a material. In its basic form, the law states that the rate of heat transfer through a material is proportional to the negative gradient of temperature and the area through which it is being transferred.

In the context of a cylindrical shell, which is our focus here, Fourier's Law can be expressed specifically for radial heat conduction. This formulation considers the shape and nature of the material. For cylindrical geometry, the heat transfer rate along the radius is determined by the formula:

• $\frac{dQ}{dt}=-k(T)A\frac{dT}{dr}$

where:

• $\frac{dQ}{dt}$ is the heat transfer rate.
• $k(T)$ represents temperature-dependent thermal conductivity.
• $A$ is the surface area of the cylindrical shell.
• $\frac{dT}{dr}$ is the radial temperature gradient.

Understanding this law helps us predict how quickly heat flows through structures like pipes or tubes, which commonly have cylindrical geometries.

cylindrical shell

A cylindrical shell is a hollow cylinder with a specific thickness, characterized by two radii: the inner radius $r_1$ and the outer radius $r_2$. Understanding heat transfer in cylindrical shells is crucial for many practical applications, such as insulation of pipes in plumbing, chemical processes, and mechanical engineering.
The critical aspect of a cylindrical shell involves its surface area, which is calculated as:

• $A=2\pi rL$

where:

• $r$ is the radius at which we're considering the surface area.
• $L$ is the length of the cylinder.

This surface area plays a crucial role in heat transfer calculations using Fourier's Law. When heat flows radially, the surface area changes with radius, impacting the rate of heat conduction.
By understanding these properties, engineers can design systems to control heat flow effectively, ensuring efficiency and safety in a variety of applications.

thermal conductivity

Thermal conductivity is a material property that measures a material's ability to conduct heat. It tells us how well heat moves through a substance. In the context of our problem, it varies with temperature, modeled by the equation:

• $k(T)=k_0(1+\beta T^2)$

where:

• $k_0$ is the base thermal conductivity at a reference temperature.
• $\beta$ is a constant that adjusts how much the thermal conductivity changes with temperature squared.
• $T$ is the temperature of the material.

This dependence highlights that as temperature changes, so does the material's ability to transfer heat. For example, in our cylindrical shell, the conductivity increases as the temperature rises, due to the positive $\beta$ term.
Engineers often need to consider these variations in thermal properties to predict and control heat transfer in devices effectively. It plays a vital role in the design of materials used for thermal management in engineering systems.

ordinary differential equation

An ordinary differential equation (ODE) is a mathematical equation that relates a function with its derivatives. In our context, it is a powerful tool to model the relationship between temperature and heat flow in systems like the cylindrical shell.
We arrive at an ODE while solving for heat transfer across the shell:

• $\frac{dQ}{dt}=-k_0(1+\beta T^2)\cdot 2\pi rL\frac{dT}{dr}$

To find the heat transfer rate, we need to solve this equation by integrating it over the given boundaries. This involves separating variables and considering both temperature and radius as factors.
The integration gives us a function that links the temperature difference across the shell with the heat transfer rate, which is calculated explicitly in specific boundary conditions. By evaluating these integrals, we obtain a comprehensive relationship for designing and analyzing heat transfer systems.