## Inhour Equation

If the reactivity is constant, the model of point kinetics equations contains a set (**1 + 6**) of linear ordinary **differential equations** with constant coefficient and can be solved analytically. Solution of six-group point kinetics equations with Laplace transformation leads to the relation between the **reactivity** and the **reactor period**. This relation is known as the **inhour equation** (which comes from **inverse hour**, when it was used as a unit of reactivity that corresponded to e-fold neutron density change during one hour) may be derived.

**General Form:**

The **point kinetics equations** may be solved for the case of an initially critical reactor without external source in which the properties are changed at t = 0 in such a way as to introduce a **step reactivity ρ _{0}** which is then constant over time. The system of coupled first-order differential equations can be solved with Laplace transformation or by trying the solution

**n(t) = A.exp(s.t)**(equation for the neutron flux) and

**C**(equations for the density of precursors).

_{i}(t) = C_{i,0}.exp(s.t)Substitution of these assumed exponential solutions in the **equation for precursors** gives the relation between the coefficients of the neutron density and the precursors.

The subsequent substitution in the equation for neutron density yields an equation for **s**, which after some manipulation can be written as:

This equation is known as the **inhour equation**, since the constants of** s _{0 – 6}** was originally determined in inverse hours. For a given value of the reactivity

**ρ**the associated values of

**s**are determined with this equation. The following figure shows the relation between

_{0 – 6}**ρ**and roots

**s**graphically. From this figure it can be seen that for a given value of ρ seven solutions exist for s. The figure indicates that for positive reactivity

**only s**. The remaining terms rapidly die away, yielding an asymptotic solution in the form:

_{0}is positivewhere **s _{0} = 1/τ_{e}** is the

**stable reactor period**or

**asymptotic period of reactor**. This root,

**s**, is

_{0}**positive for ρ > 0**and

**negative for ρ < 0**, therefore this root describes the reactor response, which is lasting after the transition phenomena have died out. The figure also shows that a negative reactivity leads to a negative period: All of the s

_{i}are negative, but the root s

_{0}will die away more slowly than the others. Thus the solution

**n(t) = A**is valid for positive as well as negative reactivity insertions.

_{0}exp(s_{0}t)To determine the reactivity required to produce a given period a plot of ρ vs. τ_{e} must be constructed using the delayed neutron data for a particular fissionable isotope or mix of isotopes, and for a given prompt generation time. To determine the stable reactor period, which results from a given reactivity insertion, it is convenient to use the following form of inhour equation.