description: The life cycle model is typically represented as a closed loop block diagram involving six specific neutron interactions. In Figure 31.1 the sixth interaction is displayed in two parts.
 |
where:
n1 = fast fission neutrons in generation-1, starting the first life cycle
epsilon = the fast fission factor
Lf = the fast neutron non-leakage probability
p = the resonance escape probability
Lth = the thermal neutron non-leakage probability
f = the thermal utilization factor
eta = the fuel reproduction factor
n2 = fast fission neutrons produced at end of generation-1, starting the second life cycle
keff = the effective multiplication factor
assumptions: the only assumptions listed for each model are those having direct impact on the models ability to represent reactor behavior, of which the student should be cognizant.
- average fission neutrons - no prompt/delayed neutron distinction
- no non-fission neutrons
- duration of life cycle = generation time (weighted avg of prompt/delayed neutron lifetimes)
- constant k-effective
- fission neutrons traverse life cycle as a group - interactions occur in a sequence
- none of generation-1 neutrons survive into generation-2
- The principal result from the life cycle model is the six-factor formula. Each of the six factors are defined by specific parameters of the system, i.e. by compositions, cross sections, neutron diffusion lengths, and system size.
31.1 THE SIX FACTOR FORMULA - From interaction 6.b. in Figure 31.1, a second important result of the life cycle model is a definition of k-effective based on the fission neutron population in successive life cycles:
31.2 K-EFFECTIVE AS THE RATIO OF FISSION NEUTRONS
- The fact that the life cycle model does not account explicitly for delayed neutrons is a major deficiency. Without delayed neutrons the entire time response is compromised and several important reactor behavioral characteristics cannot be explained, including reactor rate behavior during reactivity ramp, prompt jump in reactor power, and power turning.
- Equation 31.2, is used to establish the following relationships between the value of k-effective and the neutron population response:
k-effective > 1.0000 neutron population increases k-effective = 1.0000 neutron population constant k-effective < 1.0000 neutron population decreases - Another significant retained misconception that arises from Equation 31.2 is that at keff = 1.0000 chain reactions are self-sustaining. Again, this is true for the crude life cycle model but is not true in actuality. Since delayed neutrons act as source neutrons to initiate chain reactions, they do not participate in the propagation of each chain. Prompt neutrons propagate chain reactions and, operationally, the reactor is always subcritical on prompt neutrons. As a result, chain reactions are always finite in length. The terminating chains are replaced by new chains initiated by ongoing delayed neutron emissions (see NUKEFACT #4).
1. LifeCycle | 2. Eq-Diff | 3. Transient | 4. SourceMult | 5. Non-EqDiff | 6. SingleGrp | Summary
description: Neutron diffusion theory, combined with Fermi Age theory for neutron slowing down and considerable mathematical manipulation, quantifies the two non-leakage terms.
assumptions: are as for the life cycle model, the neutron population is at a steady-state condition, with additional assumptions as to neutron spatial symmetry and the slowing down process.
results: The principal result from the neutron diffusion model is that the six-factor formula is more fully defined as:
 |
31.3 |
where:
k-infinity = eta x epsilon x p x f (the four factor formula, no leakage)
e-B2 x TAU = the fast neutron non-leakage probability
1/(1 + B2 x L2) = the thermal neutron non-leakage probability
comments: With the two non-leakage terms defined, the six-factor formula is complete and a better understanding of neutron leakage is provided. Otherwise, the six-factor formula retains the limitations of the life cycle model.
1. LifeCycle | 2. Eq-Diff | 3. Transient | 4. SourceMult | 5. Non-EqDiff | 6. SingleGrp | Summary
The transient model demonstrates the underlying nature of neutron change with time.
description: the generation time model applies a constant k-effective from Equation 31.2 to a series of life cycles having a life time equivalent to the average fission neutron. Repetitive multiplication by k-effective in successive life cycles results in exponential change with time.
assumptions:
- average fission neutrons
- no non-fission neutrons
- duration of life cycle - generation time (weighted avg of prompt/delayed neutron lifetimes)
- constant off-critical k-effective - exists prior to time zero
results: in standard form, using base-e, the exponential change in neutron population is expressed as:
 |
31.4 |
where:
N(0) = the initial neutron population at time zero
N(t) = the neutron population at time "t", seconds
delta-k = keff - 1
lg = the neutron generation time, seconds
t = elapsed time in seconds
comments: The underlying character of off-critical reactor behavior with constant k-effective, is, in fact, exponential with time.
- As for the life cycle model, the fact that this model does not account for delayed neutrons is a major deficiency. The Transient Model cannot represent actual reactor behavior.
- This model can track exponential change after an off-critical k-effective is established and the transient is underway. However, it cannot generate the change in neutron level that occurs during introduction of the off-critical k-effective that initiates a transient from an initial steady state condition, for example from criticality to a supercritical condition.
- In some texts, this model is used to demonstrate that, without delayed neutrons, the neutron population changes too rapidly to be controllable. The prompt neutron life time, lp is substituted for generation time in the exponent of Equation 31.4. The common misconception drawn, is that it is generation time that makes the reactor controllable. This is false. There are no "average" fission neutrons. Average fission neutrons are a fiction, as is their weighted generation time. It is the decay characteristics of the precursor atoms that emit delayed (source) neutrons that make the reactor controllable.
Unfortunately, this all prompt neutron model is also used as a basis for including the term l*/rho in the standard stable rate equation to represent rate behavior at, and beyond, prompt criticality (see Equation 31.14). This is a grievous mistake.
1. LifeCycle | 2. Eq-Diff | 3. Transient | 4. SourceMult | 5. Non-EqDiff | 6. SingleGrp | Summary
The subcritical multiplication model is based on the creation of chain reactions from a continuously emitting non-fission neutron source in a subcritical multiplying medium.
description: The source multiplication model incorporates non-fission neutrons into the life cycle, implements the concept of a chain reaction by linking a series of life cycles, and illustrates that in the Sub-Critical region with constant k-effective, the neutron level seeks an equilibrium condition.
assumptions:
- non-fission source emits fast neutrons into each life cycle
- non-fission neutrons initiate chain reactions
- average fission neutrons propagate chain reactions
- duration of life cycle = generation time (weighted avg of prompt/delayed neutron lifetimes)
- k-effective is constant and less than 1.0000
- k-effective applies to fast non-fission neutrons as well as fission neutrons; non-fission neutrons are emitted at same energy as fission neutrons
- non-fission and fission neutrons traverse the life cycle as a group
- neutrons in one generation do not survive into the next generation
results: The presence of a non-fission neutron source in a subcritical multiplying medium creates an equilibrium neutron population, where the fission neutron losses are balanced by the non-fission neutron production:
 |
31.5 |
where:
N = the equilibrium fast neutrons entering each life cycle
S = the number of fast non-fission neutrons emitted into each life cycle
keff = the effective multiplication factor (< 1.0000)
To apply k-effective to non-fission neutrons the definition of k-effective given in Equation 31.2 must be modified to:
 |
31.6 |
This modified definition of k-effective appears to differ little from that of Equation 31.2. However, the denominator refers to fast neutrons starting the generation, not just to the fission neutrons starting the generation. The denominator now includes non-fission neutrons emitted into the life cycle. As it turns out, the neutrons starting a generation need not be limited to fission neutrons. However, neutrons produced into the next life cycle, the numerator of Equation 31.6, are fission neutrons alone. Any non-fission source neutrons emitted into the next life cycle must be added to the fission neutron population starting the next life cycle. In any case, applying k-effective to non-fission source neutrons for several life cycles gives the following series:
 |
31.7 |
where:
N = the number of fast neutrons in the chain
S = the number of fast non-fission neutrons emitted into a life cycle
keff = the effective multiplication factor
The first term on the right-hand-side of Equation 31.7 represents non-fission neutrons. All subsequent terms represent fission neutrons. The total neutron population in a life cycle consists of both non-fission and fission neutrons. For a typical reactor, even in a shutdown condition, the fission neutrons make up the greater part of the neutron population. Allowing the number of generations, "i", to approach infinity, and creating a model of parallel chains, leads to the equilibrium condition represented by Equation 31.5.
comments: The subcritical source multiplication model provides an excellent means for introducing the concept of the chain reaction and for illustrating how a non-fission source will ensure a minimum neutron level in the reactor. Because this process is developed step-by-step, i.e. life cycle to life cycle, students generally grasp this model readily. However, the following points should be noted:
- By using average fission neutrons this model overlooks the most important neutron source in the reactor, namely the delayed neutrons. The net result is that source multiplication is limited to the Sub-Critical region, i.e. to situations where the reactor is subcritical and the non-fission neutron source is significant. As discussed in NUKEFACT #3, source multiplication applies to all modes of reactor operation, from shutdown to full power.
- The multiplication process of Equation 31.7 is frequently used to illustrate transient behavior in the Sub-Critical region. Since this model does not incorporate delayed neutrons, such transients do not provide the correct time behavior.
- This model creates a second definition of k-effective. Generally, this change occurs without note, leaving the original definition in place. Even though the difference in the altered definition is subtle, it can unsettle student understanding. In fact, the INPO Exam Bank contains test items that require each of these definitions for credit. In addition, the INPO Exam Bank contains at least one test item that requires an incorrect definition of k-effective (see NUKEFACT #13). This means that the student must be prepared to give one of three definitions of k-effective, depending on the test item encountered. COOL !
1. LifeCycle | 2. Eq-Diff | 3. Transient | 4. SourceMult | 5. Non-EqDiff | 6. SingleGrp | Summary
The Non-Equilibrium Diffusion Theory model refines the equilibrium model by allowing for transient behavior and by explicit treatment of the standard precursor groups.
description: Neutron diffusion theory, Fermi Age theory slowing down, imbalance between neutron production and loss, precursor contributions, and considerable mathematical manipulation, yields the classic in-hour equation.
assumptions: are as for the equilibrium diffusion model, except with incorporation of transient capability and precursor groups.
- six precursor groups - yield and decay characteristics
- prompt neutrons
- duration of life cycle = prompt neutron lifetime
- constant k-effective
- no non-fission source
results: an important result from this model is the in-hour equation, which provides the definitive link between reactivity and reactor rate.
 |
31.8 |
where:
rho = reactivity
l* = lp/keff
T = reactor period, seconds
beta-i = the yield fraction of the ith precursor group
lambda-i = the decay constant of the ith precursor group
comments: setting the duration of the lifecycle equal to the prompt neutron lifetime is a major feature of this model. It is the first model, discussed herein, that can provide a realistic representation of actual reactor behavior. The in-hour equation defines the stable reactor rate for any off-critical reactivity condition, from far subcritical to beyond prompt criticality.
1. LifeCycle | 2. Eq-Diff | 3. Transient | 4. SourceMult | 5. Non-EqDiff | 6. SingleGrp | Summary
For purposes of understanding the physical process of reactor behavior, it is helpful to coalesce the six precursor groups of the Diffusion and In-hour equations into a single effective group, so as to simplify the resultant formulations.
description: The precursor atoms are important to reactor transient behavior because the precursors act as a variable neutron source which governs the rate of power change with time. In defining a single effective group it is not possible to chose both a fixed precursor yield fraction and a fixed precursor decay constant that will provide the same transient behavior as the combination of six precursor groups. Convention is to employ a constant precursor yield fraction, beta, and a variable precursor decay rate, lambdaeff. The single group precursor decay constant is, in reality, not a constant, but rather a dynamic property that depends on the mix of precursor atoms resulting from the rate and direction of power change. This effective group provides the same stable rate characteristics as the in-hour equation.
In addition, with some further mathematical manipulation of the diffusion equation, it allows for the definition of reactor rate for transient reactivity conditions, as well as with a non-fission source present. Importantly, the duration of the life cycle remains equal to the prompt neutron lifetime.
assumptions:
- single precursor group
- prompt neutrons
- duration of life cycle = prompt neutron lifetime
- constant beta
- lambda-effective - a function of reactivity
- constant reactivity or reactivity ramp
results: solving the non-equilibrium diffusion equation, sometimes called the neutron balance equation, first for the steady-state condition gives:
 |
31.9 |
where:
P = equilibrium, or steady-state, reactor power, watts
S-bar = S/(3.1x1010 x 2.5)
S = the core non-fission source strength, neutrons/second
lambda = the effective precursor decay constant
C-bar = C/(3.1x1010 x 2.5)
C = the core precursor inventory, equilibrium concentration
beta = the total yield fraction of precursors
rho = reactivity (constant)
The steady-state condition may be either equilibrium subcritical multiplication or criticality. Though not found in standard treatments, Equation 31.9 is crucial to understanding reactor behavior because it identifies delayed neutrons as source neutrons and because it indicates that source multiplication occurs from shutdown to full power.
The rate of power change can be obtained from the in-hour equation, modified to incorporate a single precursor group:
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31.10 |
Rearranging terms to solve for the stable reactor rate gives:
 |
31.11 |
As with the in-hour equation, Equation 31.11, when used with the appropriate value for lambda-effective (see NUKEFACT #30), provides the correct stable rate for all constant reactivity conditions. Since "T" appears on both sides of the equation, the solution involves iteration.
By deleting the two terms containing l*, i.e. l-star, which are only important for large reactivities approaching prompt critical and beyond, and incorporating terms to account for reactivity change with time and the presence of non-fission source neutrons, the general expression for operational reactor rate becomes:
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31.12 |
where:
rho-dot = the rate of reactivity change with time, delta-rho/second
With the proper selection of terms and values, Equation 31.12 gives realistic reactor rate for all operational reactivity conditions, up to about rho = +0.0040. This rate equation is used by many Training Centers. The use of a rule-of-thumb lambda-effective is common and adequate for most operational situations.
Then, applying the reactor rate to power change with time, we have:
 |
31.13 |
where:
P(t) = reactor power at time = t, watts
P(o) = reactor power at time = o, watts
t = elapsed time of transient, seconds
T = average reactor period over time interval t, seconds
Equation 31.13 can be applied to situations where the reactor rate is changing with time, by using appropriate average rates over short time intervals.
comments: Equations 31.9, 31.12, and 31.13 represent the substance of the single precursor group model, which is the only model that provides realistic representation of reactor behavior from shutdown to full power, for both the steady-state and the transient state.
Be aware that an incorrect form of the reactor rate equation is in common usage at many U.S. Training Centers. It is:
 |
31.14 |
The second term on the right-hand-side of Equation 31.14 supposedly makes it applicable to prompt criticality, and beyond. This is not true.
1. LifeCycle | 2. Eq-Diff | 3. Transient | 4. SourceMult | 5. Non-EqDiff | 6. SingleGrp | Summary
SUMMARY
After progressing through the subject of reactor behavior by employing a series of models, one might expect that the concepts to be emphasized for student retention would be those of the most refined and realistic model developed, namely those of model-6. Unfortunately, in Reactor Operator training, this is not the case.
Of the six models listed, the first four do not explicitly treat delayed neutrons. Therefore, these models cannot provide realistic representation of reactor behavior with time. Yet, from these four models come the following entrenched misconceptions:
- Direction of power change: is determined by k-effective, e.g. with k-effective > 1.0000 ... neutron level increases with time.
- Self-sustaining chain reactions: with k-effective = 1.0000 chain reactions are self-sustaining
- Controllability: generation time makes the reactor controllable
- Source neutrons: non-fission neutrons are the only source neutrons in the reactor
- Source multiplication: applies only in the Sub-Critical region
- Power behavior with time: transient behavior in the Sub-Critical region can be described using the subcritical multiplication model
These false concepts, and others, persist as conventional wisdom because they are erroneous test items in the INPO Exam Catalog. Contrast these myths with comparable valid concepts of model-6.
- Direction of power change: Equation 31.12 states that the direction of ongoing power change is determined by net of three contributors, namely reactivity ramp rate, the precursor inventory change, and the relative non-fission source effect. K-effective, by itself, does not determine the direction of power change (see NUKEFACTs #5 and #9)
- Self-sustaining chain reactions: chain reactions are initiated by source neutrons, which are non-fission and delayed neutrons, and are propagated by prompt neutrons. Operationally, the reactor is always subcritical on prompt neutrons. Hence, chain reactions are not self-sustaining, they are finite in length (see NUKEFACT #4)
- Controllability: The decay characteristics of the precursors in emitting delay (source) neutrons renders the reactor controllable. Weighted neutron generation time is a fiction.
- Source neutrons: from Equation 31.9, it is evident that delayed neutrons are the most important neutron source in the reactor (see NUKEFACT #2)
- Source multiplication: from Equation 31.9, source multiplication is the underlying process of all reactor behavior, from shutdown to full power (see NUKEFACT #3); not just in the Sub-Critical region.
- Power behavior with time: all operational transient power behavior can be reasonably approximated using Equations 31.12 and 31.13. The subcritical multiplication model fails because it does not explicitly treat delayed neutrons.
The misconceptions are compounded by:
- Equation error: any educational program that purports to teach the basics of reactor behavior, and yet cannot get the reactor rate equation right, has a serious problem. Even when given the widely accepted in-hour equation for the stable rate, simple algebraic manipulation is apparently not an option. Equation 31.14 is in error by 1300% at prompt critical (see NUKEFACT #29), and, it is promoted by both INPO and the NRC as acceptable.
- Multiple definitions of k-effective: one definition would suffice, and be of great benefit to the students, but, as noted in describing the Subcritical Multiplication Model, this isn't the case.
- Erroneous definition of beta: error in defining the fundamental parameters of reactor behavior is fatal to everything that follows (see NUKEFACT #1).
- Other error: exclusive of the conceptual errors identified, fully 25% of the INPO test items on Reactor Theory contain technical error.
To supplement the brief description of Model-6 given herein, you may find it helpful to refer to NUKEFACT #22, How a Nuclear Reactor Works.