Solve Diffusion Equation for Neutron Flux in Multiplying Sphere

In summary, the conversation discusses a problem involving the diffusion equation for the neutron flux in a multiplying sphere with a constant distributed source. The problem asks for the equation to be written down and solved, with specific boundary conditions. The solution for the neutron flux is given as (Q/6D)(R^2 − r^2) where Q is the source strength and D is the diffusion coefficient.
  • #1
EngNewbit
1
0
This is in the beginning of a long set of problems, and I am lost. I don't get anything like this answer. Any guidance? I have a feeling its simple but haven't done much of these.

Write down the diffusion equation for the neutron flux in a multiplying sphere of radius R containing a constant distributed source of strength Q neutrons/cm3/s. Assuming that the flux vanishes at the sphere surface and that it remains finite at the origin:
Solve the diffusion equation when k inf = 1 and show that the neutron flux is given by:
(Q/6D)(R^2 − r^2)
 
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  • #2
One has to write the diffusion equation (in spherical coordinates) with a constant, distributed source, and apply the boundary conditions, that the flux [tex]\phi[/tex] is finite at r = 0, and 0 at r = R.

The current should also be 0 at r = 0.
 
  • #3
Hi there,

I'm sorry to hear that you're feeling lost with this problem. Diffusion equations can definitely be tricky, but with some guidance, I'm sure you'll be able to solve it.

First, let's break down the problem and try to understand what it's asking for. The problem is asking you to write down the diffusion equation for the neutron flux in a multiplying sphere of radius R. This means we need to use the diffusion equation, which is a mathematical equation that describes how a substance (in this case, neutrons) diffuses or spreads out in space.

The equation we need to use is:

∇^2φ(x) + Bφ(x) = 0

Where ∇^2 is the Laplace operator, φ(x) is the neutron flux, and B is a constant that depends on the properties of the material and the source strength.

Next, the problem states that the sphere contains a constant distributed source of strength Q neutrons/cm3/s. This means that the value of B will be equal to Q. So, the diffusion equation for this problem becomes:

∇^2φ(x) + Qφ(x) = 0

Now, we need to make some assumptions about the boundary conditions. The problem states that the flux vanishes at the sphere surface, which means that φ(R) = 0. It also states that the flux remains finite at the origin, which means that φ(0) is some finite value.

Using these boundary conditions, we can solve the diffusion equation. I won't go into the details of the solution, but if we assume that k_inf = 1 (which means that the neutron multiplication factor is equal to 1), then the solution for the neutron flux is given by:

φ(r) = (Q/6D)(R^2 − r^2)

Where D is the diffusion coefficient.

I hope this helps to guide you in the right direction. If you have any further questions, please don't hesitate to ask. Good luck with your problem!
 

1. What is the diffusion equation?

The diffusion equation is a mathematical equation that describes how a quantity, such as heat, mass, or particles, moves through a medium. It is based on the principle of diffusion, which states that particles will move from areas of high concentration to areas of low concentration in order to reach a state of equilibrium.

2. How is the diffusion equation used in neutron flux calculations?

The diffusion equation is used in neutron flux calculations to determine the distribution of neutrons in a medium, such as a multiplying sphere. It takes into account parameters such as neutron production, absorption, and scattering, and can be solved to determine the neutron flux at different points within the sphere.

3. What is a multiplying sphere?

A multiplying sphere is a type of nuclear reactor that uses a spherical geometry to contain and sustain a nuclear chain reaction. It consists of a core of fissile material surrounded by a layer of reflector material. The diffusion equation can be used to model the neutron flux within this type of reactor.

4. What are the assumptions made when solving the diffusion equation for neutron flux in a multiplying sphere?

When solving the diffusion equation for neutron flux in a multiplying sphere, some common assumptions include: a steady-state condition, spherical geometry, isotropic neutron flux, and homogenous material properties within the sphere. These assumptions simplify the equation and make it easier to solve, but they may not accurately reflect the real-world conditions of a nuclear reactor.

5. What are some applications of solving the diffusion equation for neutron flux in a multiplying sphere?

The diffusion equation for neutron flux in a multiplying sphere has various applications in the field of nuclear engineering. It can be used to design and optimize nuclear reactors, calculate fuel burnup and depletion, and analyze the effects of different reactor configurations and materials on neutron behavior. It is also used in research and development of new nuclear technologies and in the safety analysis of existing reactors.

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