Heat conduction problem: sphere

In summary, the conversation discusses the problem of finding the temperature within a solid sphere immersed in a vat of fluid at a fixed boundary temperature T_0 and initial temperature T_1. The reasoning involves using a solution of the form T=X(t)R(r) and plugging it into the heat conduction equation. The solution for X(t) is found to be C*[exp(-t/(D*k^2))], and for R(r) it is Acos(kr)+Bsin(kr). The problem arises when trying to set boundary conditions, as only one variable can be eliminated. The use of a Fourier series is suggested, but the Laplacian in spherical coordinates is corrected to be \frac{1}{r}\
  • #1
genius2687
12
0
A solid sphere of radius a is immersed in a vat of fluid at a temperature T_0. Heat is conducted into the sphere according to

dT/dt = D(d^2T/dr^2)
(d-> partial derivative btw)

If the temperature at the boundary is fixed at T_0 and the initial temperature of the sphere is T_1, find the temperature within the sphere as a function of time.

My reasoning

Ok. Here's my reasoning. Use a solution of the form T=X(t)R(r), and plug into the above equation to get

R''/R=X/(DX')=-k^2.

I get X(t) = C*[exp(-t/(D*k^2))]. (k^2>0 for convergence)

Then I have R'' +k^2*R = 0

so R= Acos(kr) + Bsin(kr), since k^2>0.

The Problem

Assuming that the above steps are right, we could set the boundary conditions and have Acos(ka)+Bsin(ka)=T_0. That only eliminates one variable, either A or B. I don't know where to go from there however. Should this be like a Fourier series or something?
 
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  • #2
The Laplacian in spherical coords is not d^2T/dr^2.
It is [itex]\frac{1}{r}\left[\partial_r^2(rT)\right][/itex]
 

1. What is heat conduction?

Heat conduction is the transfer of thermal energy between two bodies that are in direct contact with each other. This transfer occurs from the hotter body to the colder body until both reach equilibrium.

2. What is the heat conduction problem in a sphere?

The heat conduction problem in a sphere refers to the mathematical modeling and analysis of heat transfer in a spherical object. It involves understanding the flow of heat through the sphere, taking into account factors such as temperature, material properties, and boundary conditions.

3. How is heat conduction in a sphere calculated?

The heat conduction in a sphere can be calculated using the Fourier's Law of Heat Conduction, which states that the heat flux (amount of heat transferred per unit area) is proportional to the temperature gradient (change in temperature over distance) and thermal conductivity of the material. This can be expressed mathematically as q = -kA(dT/dr), where q is the heat flux, k is the thermal conductivity, A is the surface area, and dT/dr is the temperature gradient.

4. What factors affect heat conduction in a sphere?

The rate of heat conduction in a sphere is influenced by several factors, including the thermal conductivity of the material, the temperature difference between the two bodies, the surface area of the sphere, and the thickness of the sphere. Additionally, the presence of any insulating materials or external heat sources can also affect heat conduction.

5. What are some real-life applications of the heat conduction problem in a sphere?

The heat conduction problem in a sphere has many practical applications, such as in the design of thermal insulation materials, heat exchangers, and refrigeration systems. It is also important in fields such as geology, where it is used to understand heat transfer within the Earth's interior, and in medicine, where it is utilized in treatments such as hyperthermia therapy for cancer. Understanding heat conduction in a sphere can also help in predicting and preventing heat-related accidents in industries such as manufacturing and transportation.

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