NUKEFACT #68

last update May 29, 2004

A set of seven balance equations are the foundation for defining all reactor behavior. The set consists of a neutron balance equation and six precursor balance equations. These equations define the difference between production and loss rates of neutrons and of precursor atoms in the reactor. The seven equations are "coupled", meaning that one cannot be solved independently of the others. This is quite similar to the algebraic exercise of solving of two or more equations simultaneously. The solution must satisfy all equations. The balance equations model the reactor in nuclear plant simulators.

By coalescing the six precursor equations into a single effective precursor group, the set of seven reduces to just two equations, which are amenable to substantial analytic manipulation. The results are useful toward understanding the physical process underlying observed behavior. The balance equations themselves, and the relationships derived therefrom, can be used to generate realistic numeric results. However, because of the complexity of reactor behavior, as evidenced by the number of equations that must be solved simultaneously, it is not practical to manually calculate the entirety of even a simple transient. Modeling approximations are possible that, although not producing exact behavior, constitute a powerful learning tool for understanding the inner workings of reactor behavior.

In applying the balance equations to a simple transient, certain conditions and minor assumptions are made. These are:

1. Initial condition: steady-state, equilibrium source multiplication or criticality, k_eff constant
   
2. Final condition: off-critical, exponential power change, T and k_eff constant
   
3. A step change in k_eff is used to establish the off-critical condition: to separate rapid prompt neutron response from the sluggish delayed neutron response. Phase-1 of the transient is the prompt neutron response. Phase-2 is the delayed neutron response.

THE BALANCE EQUATIONS

The neutron balance can be expressed as:

68.1

• The neutron balance equation applies to all possible values of k_eff in the steady-state and the transient state, for both the Sub-Critical region and the Delayed-Critical region.
• The balance is on thermal neutrons causing fission, to allow direct conversion to reactor power.
• The first [bracketed] term on the right-hand-side represents the difference between the number of prompt neutrons causing thermal fission in the second lifetime and total number of neutrons lost in the first lifetime by causing thermal fission.
• The second [bracketed] term on the right-hand-side represents the number of delayed neutrons causing thermal fission in the second lifetime.
• The third [bracketed] term on the right-hand-side represents the number of non-fission neutrons causing thermal fission in the second lifetime.

68.2

delta-C = the change in precursor inventory in successive prompt neutron lifetimes, atoms

• The precursor balance applies to all possible values of k_eff in the steady-state and the transient state, for both the Sub-Critical region and the Delayed-Critical region.
  
• The first (bracketed) term on the right-hand-side represents the number of precursor atoms produced in the first lifetime.
  
• The second (bracketed) term on the right-hand-side represents the number of precursor atoms lost by decay in the first lifetime.
  

Solving the neutron balance equation for n(t), the number of thermal neutrons causing fission at time "t", gives:

68.3

• This expression defines the population of thermal neutrons causing fission at any instant in time, given the total source strength and k_eff value at that time.
• The neutron population, n(t) is the dependent variable in that a change in k_eff is required to initiate a transient from the steady-state condition and a changing precursor inventory is required to maintain the transient.
• The right-hand-side describes the process as one of source multiplication; the numerator is the total source strength and (1/denominator) is the multiplier. In fact, the underlying physical process in a nuclear reactor is always source multiplication.
• The numerator represents the total neutron source strength at time "t", indicating that delayed neutrons act as source neutrons.
• A unique feature of Equation 68.3 is that it reveals that source multiplication exists in the transient state.
• A second unique feature is that this equation is valid for all possible values of k_eff and is not restricted to k_eff values less than 1.0.
• The equation form is shorthand notation that sums the contributions of the multitude of ongoing chain reactions to yield the total neutron population at time "t".

68.4

• It can be shown analytically that Equation 68.4 is equivalent to Equation 68.3.

Solving the precursor balance equation for C(t), the precursor inventory at time "t" is:

68.5

• This expression applies to all possible values of reactor period in the steady-state and the transient state, for both the Sub-Critical region and the Delayed-Critical region.

Assuming a negligible non-fission source strength in Equation 68.3 and rearranging terms gives:

68.6

• The right-hand-side of this expression is the ratio of the delayed neutrons causing thermal fission to the total number of neutrons causing thermal fission in the Delayed-Critical region.
• This makes the left-hand-side of Equation 68.6 the exact expression for the fraction of the total neutron population that consists of delayed neutrons.
• A unique feature of this definition of the delayed neutron population fraction is that no matter how large k-effective, the fraction never goes to zero. It is impossible for delayed neutrons to be insignificant in reactors where the underlying physical process is one of source multiplication.
• The (l_p/T) term represents the change in precursor inventory directly related to the introduction of delta-k.
• The delayed neutron population fraction is important because on inversion it becomes the source multiplication factor for both the Delayed-Critical region and the Sub-Critical region (see next item).

Equation 68.3 can be written as:

68.7

68.8

• "M" represents the source multiplication factor, being a measure of how many times prompt neutrons multiply the neutron source strength.
• A unique feature of this expression is that it applies to all possible values of k_eff , and particularly to the transient state, in both the Sub-Critical region and the Delayed-Critical region. Conventional treatments of the source multiplication factor are for the steady-state in the Sub-Critical region, namely for the condition of equilibrium multiplication.
• The source multiplication factor is the inverse of the delayed neutron population fraction.
• As the delayed neutron population fraction becomes extremely small, as it does for very rapid excursions, the source multiplication factor becomes extremely large.

Having established the source multiplication factor "M", it follows that:

68.9

k_p = the prompt neutron multiplication factor

• This expression applies to all possible values of k_eff in the steady-state and the transient state, for both the Sub-Critical region and the Delayed-Critical region.
• k-prompt is the factor propagating the chain reactions over successive prompt neutron lifetimes.
• Since k-effective itself applies to a model that does not properly account for delayed neutrons, modifications to k_eff are required; k_p incorporates the necessary modifications.
• The unique feature of this expression is that k_p is always less than 1.0. For extreme supercriticality, i.e. for k_eff values approaching, and greater than, 1.0065, the value of k_p asymptotically approaches 1.0, but remains less than 1.0.
• A reactor is always subcritical on prompt neutrons; the reactor always acts as a subcritical source multiplier.
• Chain reactions are always finite in length.
• Physically, k-prompt represents the prompt neutron population fraction.

A step change in k_eff from an initial condition of criticality causes the neutron population to "jump" or "drop" by a factor of:

68.10

PJF = the prompt jump factor

• The unique feature of this expression, which applies to the Delayed-Critical region, is that it defines the magnitude of the prompt jump for all possible values of k_eff , including those greater than 1.0065. Conventional representations of prompt jump break down as k_eff approaches 1.0065.
• A step change in k_eff causes a near instantaneous change in the source multiplication factor "M"
• Prompt neutrons, with an extremely short lifetime, respond almost instantaneously to a step change in k_eff, which accounts for the "jump" in neutron population.
• The prompt jump in neutron population constitutes phase-1 of the two phase transient.
• The prompt neutron response terminates in quasi steady-state condition because the reactor is always subcritical on prompt neutrons, i.e. k_p is always less than one.
• The phase-1 prompt jump factor (PJF) is a ratio of the source multiplication factor after a step change in k_eff to the initial source multiplication factor at criticality.
• The prompt jump creates an imbalance between precursor production and precursor loss, setting the stage for phase-2 of the transient.

Following phase-1 establishment of an off-critical condition, the balance equations define the rate of change in the neutron population as:

68.11

• This expression defines the stable reactor period in the Delayed-Critical region for all possible values of k_eff, including values greater than 1.0065.
• A stable reactor period, T, represents the time required for both the precursor inventory and the neutron population to change by a factor-of-e. A constant reactor period exists for exponential change with time.
• Phase-1, the prompt jump, creates a mismatch between the rate of precursor production and precursor loss, which leads to phase-2 of the transient. The precursor inventory begins to change with time, as does the delayed neutron source strength. The fractional change in each prompt neutron lifetime is l_p/T.
• Ongoing change in the delayed neutron source strength is the prime mover of the transient following the prompt jump.
• If delta-k is large and positive (> +0.0040), the (l_p/T) term on the right-hand-side becomes significant and an iterative solution is required, with the aid of a lambda_eff diagram.

68.12

• It can be shown analytically that Equation 68.12 is equivalent to Equation 68.11.
• Here also, if delta-k is large and positive (> +0.0040), the (l_p/T) term on the right-hand-side becomes significant and an iterative solution is required.

The classic in-hr equation is commonly expressed as:

68.13

rho = k_eff /( k_eff - 1) = reactivity
l^*_p = l_p /k_eff

• The in-hr equation applies to the Delayed-Critical region for all possible values of k_eff, including values greater than 1.0065.
• A unique feature of this form of the in-hr equation is that reactivity is equated to two terms, both of which relate to the prompt jump in phase-1, but both of which define delayed neutron behavior and importance in phase-2.
• The first term on the right-hand-side, l_p/T, represents incremental change in the precursor inventory associated with the prompt jump of phase-1. In phase-2 (l_p/T) accounts for the fractional change in the delayed neutron source strength over each prompt neutron lifetime.
• [1/(1 + lambda_eff × T)] represents the factor of change in the initial steady-state delayed neutron population fraction. The change results from the promt jump in phase-1.
• A potential problem in using Equation 68.13 is that k_eff appears on the right-hand-side of the equation, forcing an iterative solution for very short reactor periods.
• The in-hr equation is equivalent to the reactor period equations, except that it solves for k_eff when given the reactor period, T.

68.14

• The unique feature of this form of the in-hr equation is that iteration is never required for any value of the reactor period.

1. The balance equations provide the basis for defining all reactor behavior.

2. The underlying physical process determining the neutron population is always source multiplication, for both the steady-state and the transient-state.

3. The delayed neutron population fraction is always significant; delayed neutrons are never negligible, even for k_eff greater than 1.0065.

4. The source multiplication factor "M" is the inverse of the defined delayed neutron population fraction of the Delayed-Critical region.

5. The propagating factor in the chain reaction, "k_p", incorporates modification of k_eff so as to correctly account for delayed neutrons.

6. k-prompt is always less than 1.0 because an incremental change in the precursor inventory occurs simultaneously with the prompt jump in neutron population ; chain reactions are never self-sustaining.

7. k-prompt represents the prompt neutron population fraction.

8. The phase-2 stable period "T" results from an ongoing change in the delayed neutron source strength caused by a mismatch between the rate of precursor production and the rate of precursor loss.

9. The phase-2 period equation solves for T, given k_eff; the phase-2 in-hr equation solves for k_eff, given T; the two equations are equivalent.

10. For operational application, delta-k < +0.0040, terms containing l_p are insignificant and can be omitted from each derived equation. This operational approximation in no way lessens the importance of the l_p/T term to the understanding of the physical process of reactor behavior. In fact, it is the absence of the l_p/T term that causes the operational approximations to break down for values of k_eff approaching and exceeding 1.0065.