Question about initial and boundary conditions with the heat equation

In summary, when analyzing boundary conditions in a transient heat conduction differential equation, the temperature at the surface of a sphere does not depend on time. This is due to the fact that the temperature on the surface is not constant in time and is affected by convection and radiation. The boundary condition is written with the convection coefficient and the difference between the surface temperature and the surrounding temperature.
  • #1
patricio ramos
8
0
I am seeing the heat conduction differential equation, and I was wondering about a boundary condition when the equation is of transient (unsteady) nature.

When analyzing boundary conditions at the surface of say, a sphere, the temperature does not depend on time. For example, if you have conduction, but at the surface you have convection, the boundary condition is written like this:

$$-k* dT(r,t)/dx = h(T(r)-Tsurrounding)$$

r is the radius of the sphere, t is time and h is the convection coefficient. I notice that T is independent on time when writing radiation and convection boundary conditions. Why is this? Is it because the temperature at the surface is constant even if the problem is transient?

Thanks
 
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  • #2
The temperature on the surface is not constant in time, and your equation should read:$$-k* dT(r,t)/dx = h(T(r,t)-Tsurrounding)$$
 

FAQ: Question about initial and boundary conditions with the heat equation

What is the heat equation?

The heat equation is a mathematical representation of how heat flows through a material over time. It describes the relationship between temperature, time, and the material's thermal properties.

What are initial conditions in the heat equation?

Initial conditions in the heat equation refer to the temperature distribution within a material at the beginning of the analysis. These conditions are typically specified as a function of position and can be used to determine the temperature distribution at any given time.

What are boundary conditions in the heat equation?

Boundary conditions in the heat equation refer to the constraints placed on the temperature distribution at the boundaries of the material being analyzed. These conditions can be either prescribed temperatures or heat fluxes and are used to solve for the temperature distribution within the material.

How do initial and boundary conditions affect the solution of the heat equation?

The initial and boundary conditions play a crucial role in determining the solution of the heat equation. They define the starting point and constraints for the temperature distribution, which ultimately determines how heat will flow through the material over time.

What are some common methods for solving the heat equation with initial and boundary conditions?

Some common methods for solving the heat equation with initial and boundary conditions include the finite difference method, finite element method, and the method of separation of variables. These methods involve discretizing the material and solving a system of equations to determine the temperature distribution at different points in time.

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