18 Equation of continuity
An equation of continuity generally describes the rate of change of a selected parameter in a defined environment. It is a mass balance of the parameter in time through a control volume. This could be temperature in a medium, fluid flow in a channel, chemical compounds through soil, etc.
Here, the parameter of interest is of course neutrons.
In words this is –
[rate of change of neutrons] = [production rate] - [absorption rate] - [leakage rate]
Typically, a fixed volume is assumed. Depending on the phenomenon to be modeled, there could be other terms. For example, a chemical reaction rate could be added for chemical compounds or a decay rate for radionuclides.
Equation of continuity
[latex]\frac{d}{dt}\int_V n dV \; = \; \int_V s dV \; - \; \int_V \Sigma_A \phi dV \; - \; \int_V \underline{J} \cdot \underline{n} dA[/latex]
- [latex]\int_V n dV \equiv \;[/latex] total number of neutrons
- [latex]\frac{d}{dt}\int_V n dV \equiv \;[/latex] rate of change of the total number of neutrons
- [latex]\int_V s dV \equiv \;[/latex] production rate
- [latex]\int_V \Sigma_A \phi dV \equiv \;[/latex] absorption rate
- [latex]\int_A \underline{J} \cdot \underline{n} dA \equiv \;[/latex] leakage rate
[latex]\phi[/latex] is the neutron flux in the reactor. Note that current ([latex]\underline{J}[/latex]) appears in the leakage term. The ([latex]\underline{n}[/latex]) is just the unit normal vector. Essentially, this term describes the flux of neutrons leaking out of a ‘face’ of the control volume.
With applying the following known, mathematical relationships –
[latex]\frac{d}{dt}\int_V n dV = \int_V \frac{\partial{n}}{\partial{t}} dV[/latex]
and
[latex]\int_A \underline{J} \cdot \underline{n} dA = \int_V \nabla \underline{J} dV[/latex]
where the first is a form of the Leibniz integral rule, and the second is a form of Gauss’ divergence theorem.
Then –
[latex]\int_V \frac{\partial{n}}{\partial{t}} dV \; = \; \int_V s dV \; - \; \int_V \Sigma_A \phi dV \; - \; \int_V \nabla \underline{J} dV[/latex]
The integrals essentially cancel because integration is performed over the same control volume.
Therefore –
[latex]\frac{\partial{n}}{\partial{t}} \; = \; s \; - \; \Sigma_A \phi \; - \; \nabla \underline{J}[/latex]
This is the equation used to describe reactor operations and other neutron transport phenomenon.