Deriving the critical radius of Uranium using diffusion equation

In summary, the student attempted to solve the equation for the neutron density as a function of position and time, but ran into a problem because the coefficient of sin(kr)/r + cos(kr)/r in the equation includes m, which is an integer. The student then attempted to solve for the critical radius using boundary conditions, but ran into another problem because the expression for the critical radius contains m.
  • #1
vengeance123
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Homework Statement



I have solved the equation for the neutron density as a function of position and time. I need the boundary conditions to change my infinite number of solutions (the varying separation constant) into one value so that my answer for the critical radius does not contain a sum!

Homework Equations



del squared (n) - A(dn/dt) = -Bn (which i have solved)

The Attempt at a Solution



i assumed spherically symmetric solutions so using spherical polar coordinates n varies only (spatially) from the distance to the centre of the sphere. The r dependence is of form cos(kr)/r + sin(kr)/r. So the coefficients of cos term must all be 0 (as the density at the centre of any given sphere cannot be infinite). I thought that at the surface, the density is 0 as neutrons do not diffuse back into the sphere once they are out. Is this right?
 
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  • #2
maybe i should add a little more to this. The time dependence part of the neutron density is found to be exp(B-K)t where K=k*k (k is the separation constant). now at the critical radius i know that the neutron density cannot be increasing or decreasing with time (it must be stable). So i reasoned that B-K = 0. If also we reason that the density of neutrons on the surface of a sphere R must be zero, then we have that sin(kR)=0 or kR=m*pi where m is an integer. So k=m*pi/R and subbing this to B-K=0, i have an expression for the critical radius in terms of B (which are the product of some intrinsic properties of uranium like mean free path etc). The only problem I have now is that the expression for R, the critical radius, contains m! so I have infinitely many expressions! How do I fix the value of m using the boundary conditions?! this is driving me insane..
 
  • #3
Help Please!
 

1. What is the diffusion equation used for?

The diffusion equation is a mathematical tool used to describe the process of diffusion, which is the movement of particles from an area of high concentration to an area of low concentration.

2. How does the diffusion equation relate to uranium?

The diffusion equation can be used to model the diffusion of uranium atoms in a medium, such as in a nuclear reactor. It can help us understand how the concentration of uranium changes over time and determine the critical radius at which the concentration becomes critical for a nuclear reaction.

3. What is the critical radius of uranium?

The critical radius of uranium is the minimum distance between uranium atoms that is required for a self-sustaining nuclear reaction to occur. It is an important factor in the design and operation of nuclear reactors.

4. How is the critical radius of uranium derived using the diffusion equation?

The critical radius of uranium can be derived by solving the diffusion equation with certain boundary conditions, such as the concentration of uranium at the center of the reactor and the flux of uranium at the outer boundary. This calculation allows us to determine the critical radius at which the concentration of uranium reaches the critical level for a nuclear reaction to occur.

5. What other factors can affect the critical radius of uranium?

The critical radius of uranium can also be influenced by other factors, such as the temperature and pressure of the reactor, the presence of other materials, and the shape and size of the reactor vessel. These factors can affect the rate of diffusion and the concentration of uranium, ultimately impacting the critical radius.

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