Question

A sphere of radius $R$ is at temperature $T=0$ throughout. At time $t=0$, it is immersed in a liquid bath at temperature $T_0$. Find the subsequent temperature distribution $T(r,t)$ inside the sphere. [Let $\kappa =$ thermal conductivity $/($ density $\times$ specific heat).]

Short answer

The temperature distribution is: $$T(r,t)=T_0(1-\frac{2}{r}\sum _{n=1}^{\mathrm{\infty }}\frac{(1-(-1{)}^n)}{n\pi }{e}^{-\kappa (n\pi /R{)}^{2}t}\sin \left(\frac{n\pi r}{R}\right))$$

Step-by-step solution

Understand the Heat Equation

The heat equation in spherical coordinates for radial symmetry is given by $$\frac{\partial T}{\partial t}=\kappa \left(\frac{1}{r^2}\frac{\partial }{\partial r}\left(r^2\frac{\partial T}{\partial r}\right)\right)$$where $T$ is the temperature, $\kappa$ is the thermal diffusivity, $r$ is the radial distance, and $t$ is time.

Apply Initial and Boundary Conditions

The initial condition is $T(r,0)=0$ for all $0\le r\le R$. The boundary condition is $T(R,t)=T_0$ for time $t>0$. Additionally, due to the symmetry, the derivative at the center must be zero: $\frac{\partial T}{\partial r}{|}_{r=0,t}=0$.

Assume a Solution Form

Assume a solution of the form $T(r,t)=T_0(1-f(r,t))$, where $f(r,t)$ is a function to be determined. Substitute this into the heat equation and simplify.

Solve for the Function $f(r,t)$

Substituting $T(r,t)=T_0(1-f(r,t))$ into the heat equation reduces the problem to solving for the equation of $f(r,t)$. This leads to a separation of variables solution: $f(r,t)=\sum _{n=1}^{\mathrm{\infty }}{A}_n{e}^{-\kappa {\alpha }_n^{2}t}\frac{\sin ({\alpha }_{n}r)}{r}$, where ${\alpha }_n$ are the roots of ${\alpha }_{n}R=n\pi$.

Apply Boundary Conditions

Using boundary conditions, express $A_n$ terms. Evaluating the Fourier coefficients, we find: $$A_n=\frac{2(1-(-1{)}^n)}{n\pi }$$

Write the Final Solution

Combine all terms to express the final temperature distribution. The result is: $$T(r,t)=T_0(1-\frac{2}{r}\sum _{n=1}^{\mathrm{\infty }}\frac{(1-(-1{)}^n)}{n\pi }{e}^{-\kappa (n\pi /R{)}^{2}t}\sin \left(\frac{n\pi r}{R}\right))$$

Key concepts

thermal conductivity

Thermal conductivity measures how easily heat flows through a material. It's crucial for determining temperature distribution in objects like our sphere. When a sphere with initial temperature zero is immersed in a liquid bath at temperature $T_0$, the temperature throughout the sphere changes over time depending on the thermal conductivity. In our equation, $\text{/}(...$ denotes thermal diffusivity, combining thermal conductivity with density and specific heat. This tells us how fast heat spreads. In simpler terms, higher thermal conductivity means quicker heat transfer, affecting how soon the sphere reaches equilibrium.

temperature distribution

Temperature distribution inside the sphere shows how temperature varies at different points within the sphere over time. Initially, the temperature is zero throughout. When the sphere is placed in the liquid bath, heat transfers from the surface to the center. We use the heat equation in spherical coordinates to model this: $$\frac{\partial T}{\partial t}=\kappa \left(\frac{1}{r^2}\frac{\partial }{\partial r}\left(r^2\frac{\partial T}{\partial r}\right)\right)$$. This equation helps predict how the temperature changes with time and space. The goal is to find the function $T(r,t)$ showing temperature $T$ at distance $r$ from the center, at time $t$.

boundary conditions

Boundary conditions are necessary to solve the heat equation fully. In our case, we have:

• The initial condition: At $t=0$, the temperature $T(r,0)=0$ for $0\le r\le R$.
• The boundary condition at the surface: $T(R,t)=T_0$ for time $t>0$.
• Symmetry condition: The derivative of temperature at the center is zero, $\frac{\partial T}{\partial r}{|}_{r=0}=0$. This reflects that there's no heat flux escaping the center. These conditions help us find the unique solution for the temperature distribution.

separation of variables

Separation of variables is a method to solve partial differential equations like our heat equation. We assume a solution of the form $(T(r,t)=T_0(1-f(r,t))$. Substituting this into the heat equation helps simplify it. We then separate the equation into independent functions of $r$ and $t$. This approach reduces the problem to solving ordinary differential equations. In our solution for $(f(r,t)$, the final form involves a series that satisfies all boundary conditions: $$f(r,t)=\sum _{n=1}^{\mathrm{\infty }}{A}_n{e}^{-\kappa {\alpha }_n^{2}t}\frac{\sin ({\alpha }_{n}r)}{r}$$. Here, ${\alpha }_n$ are the roots derived from boundary conditions, and $A_n$ are coefficients found using Fourier series analysis.