## 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.

where:

**β**** _{eff}** = effective delayed neutron fraction

**λ**** _{eff}** = effective delayed neutron precursor decay constant

**τ**** _{e}** = reactor period

**ρ** = reactivity

The first term in this formula is the **prompt term** and it causes that the positive reactivity insertion is followed immediately by a immediate power increase called the **prompt jump**. This power increase occurs because the rate of production of prompt neutrons changes immediately as the reactivity is inserted. After the **prompt jump**, the rate of change of power cannot increase any more rapidly than the built-in time delay the precursor half-lives allow. Therefore the **second term** in this formula is called the **delayed term**. The presence of delayed neutrons causes the power rise to be controllable and the reactor can be controlled by control rods or another reactivity control mechanism.

**Reactivity**is not directly measurable and therefore most power reactors procedures do not refer to it and most technical specifications do not limit it. Instead, they specify a limiting rate of neutron power rise (measured by excore detectors), commonly called a

**startup rate**(especially in case of PWRs).

**The reactor startup rate** is defined as the number of factors of ten that power changes in one minute. Therefore the units of **SUR** are powers of ten per minute, or **decades per minute** (**dpm**). The relationship between reactor power and startup rate is given by following equation:

**n(t) = n(0).10 ^{SUR.t}**

where:

**SUR = reactor startup rate [dpm – decades per minute]**

**t = time during reactor transient [minute]**

The higher the value of SUR, the more rapid the change in reactor power. The startup rate may be positive or negative. If SUR is positive, reactor power is increasing. If SUR is negative, reactor power is decreasing. The relationship between reactor period and startup rate is given by following equations:

Example:

Suppose **k _{eff} = 1.0005** in a reactor with a generation time

**l**. For this state calculate the reactor period –

_{d}= 0.01s**τ**, doubling time –

_{e}**DT**and the startup rate (

**SUR**).

ρ = 1.0005 – 1 / 1.0005 =

**50 pcm**

τ_{e} = l_{d} / k-1 = 0.1 / 0.0005 = **200 s**

DT = τ_{e} . ln2 = **139 s**

SUR = 26.06 / 200 = **0.13 dpm**

## Special Cases of Inhour Equation

**ρ is very small.**This causes the first root, s

_{0}, of the reactivity to be also very small. In this case,

**s**

_{0}**≪**

**λ**

**and therefore, we can ignore the term**

_{i}**s**

**in the denominator of the inhour equation. The**

_{0}**inhour equation**then takes form:

As can be seen, this special case results in the same period as in the case of **simple point kinetics equation**, which also uses the **mean generation time with delayed neutrons**** (l**_{d}**)**:

Thus for small reactivities—positive or negative—the reactor period is governed almost completely by the delayed neutron properties. Despite the fact the **amount of delayed neutrons** is only on the order of **tenths of percent** of the total amount, **the timescale in seconds (**τ_{i}**) plays the extremely important role.**

The assumption is **s**_{0}**≪** **λ**_{i}**. **The largest value of **τ**_{i}** = 1/λ**_{i}** = 80s, **therefore the this formula is valid for periods, τ_{e}, higher than **80s**. On the other hand, this formula is useful and accurate enough for most purposes for reactivities up to about **ρ = 0.0005 = 50pcm.**

Note that:

**Mean generation time with delayed neutrons**** (l**_{d}**)**:

**l**_{d}** = (1 – β).l**_{p}** + ∑l**_{i}** . β**_{i}** => l**_{d}** = (1 – β).l**_{p}** + ∑τ**_{i}** . β**_{i}

where

**(1 – β)**is the fraction of all neutrons emitted as prompt neutrons**l**is the prompt neutron lifetime_{p}**τ**is the mean precursor lifetime, the inverse value of the decay constant_{i }**τ**_{i}**= 1/λ**_{i}- The weighted delayed generation time is given by
**τ = ∑τ**_{i}**. β**_{i}**/ β = 13.05 s** - Therefore the weighted decay constant
**λ = 1 / τ ≈ 0.08 s**^{-1}

The number, **0.08 s**** ^{-1}**, is relatively high and have

**a dominating effect of reactor time response**, although delayed neutrons are a small fraction of all neutrons in the core. This is best illustrated by calculating a weighted mean generation time with delayed neutrons:

**l**_{d}** = (1 – β).l**_{p}** + ∑τ**_{i}** . β**_{i}** = (1 – 0.0065). 2 x 10**^{-5}** + 0.085 = 0.00001987 + 0.085 ≈ 0.085**

In short, **the mean generation time with delayed neutrons** is about **~0.1 s**, rather than ~**10**** ^{-5}** as in section Prompt Neutron Lifetime, where the delayed neutrons were omitted.

Example:

Let us consider that **the mean generation time with delayed neutrons is ~0.085** and k (k_{∞} – neutron multiplication factor) will be step increased **by only 0.01%** (**i.e. 10pcm or ~1.5 cents**), that is k_{∞}=1.0000 will increase to k_{∞}=1.0001.

It must be noted such reactivity insertion (10pcm)** is very small** in case of LWRs (e.g. one step by control rods). The reactivity insertions **of the order of one pcm** are for LWRs **practically unrealizable**. In this case the reactor period will be:

**T = l**_{d}** / (k**_{∞}**-1) = 0.085 / (1.0001-1) = 850s**

This is a very long period. In ~14 minutes the neutron flux (and power) in the reactor would increase by a factor of e = 2.718. This is completely different dimension of the response on reactivity insertion in comparison with the case without presence of delayed neutrons, where the reactor period was 1 second.

**ρ is higher than β.**In this case, the production of prompt neutrons alone is enough to balance neutron losses and increase the neutron population. The number of neutrons is

**exponentially increasing**in time (as rapidly as the prompt neutron generation lifetime ~

**10**

^{-5}**s**).

This causes the first root, **s _{0}**, of the reactivity becomes large compared to each

**λ**

**. In this case,**

_{i}**λ**

_{i}**≪**

**s**

**and therefore, we can ignore the term**

_{0}**λ**

**in the denominator of the**

_{i}**inhour equation**. The inhour equation then takes form:

As can be seen, this special case results in the same period as in the case of** simple point kinetics equation without delayed neutrons**.

Obviously, reaching the reactivity of **β (e.g. 650pcm) **completely changes the response of a reactor, therefore this situation can only be accidental. As **prompt criticality** is reached, the distinction between the **prompt jump** and the** reactor period** vanishes, for now the prompt neutron lifetime rather than the delayed neutron half-lives largely determines the rate of exponential increase. Reactors with such a kinetics would be very difficult to control by mechanical means such as the movement of control rods. In normal operation, the reactivity of a reactor must remain far below the prompt criticality threshold with sufficient margin.

In **design basis accidents **(DBA), such a kinetics can occur. For example under RIA conditions (**Reactivity-Initiated Accidents**) reactors should withstand a jump-like insertion of relatively large (~1 $ or even more) positive reactivity (e.g. in case of control rod ejection) and the **PNL** (prompt neutron lifetime) plays here the key role. The longer prompt neutron lifetimes can substantially improve kinetic response of reactor (**the longer prompt neutron lifetime gives simply slower power increase**). Therefore the PNL should be verified in a reload safety evaluation (RSE) process. Management of this accident is based on reactivity feedbacks, especially on the **doppler temperature coefficient (DTC)**. This coefficient is of the highest importance in the **reactor stability**. The **doppler temperature coefficient** is generally considered to be even **more important** than the **moderator temperature coefficient**** (MTC)**. Especially in case of a control rod ejection the doppler temperature coefficient will be **the first** and the most important feedback, that will compensate the inserted positive reactivity. The time for heat to be transferred to the moderator is usually measured in seconds, while the fuel temperature coefficient is effective almost **instantaneously**. Therefore this coefficient is also called the **prompt temperature coefficient** because it causes an **immediate response** on changes in fuel temperature.

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