In this Nukefact we address the questions under the subsection Reactor Kinetics and Neutron Sources. This subsection is an extremely important topic to the prospective reactor operator because it relates directly to reactor behavior and what the operator will be dealing with in the control room. Of the seventy-five questions in the INPO Catalog pertaining to the Reactor Kinetics and Neutron Sources we find thirty-six to be technically incorrect, that's 48% wrong folks. And from the existing technical error it becomes evident that INPO hasn't a clue about the following:
1. the difference between reactivity and reactivity change
2. that reactivity change itself causes power to change
3. the meaning of betaeff
4. that delayed neutrons act as source neutrons
5. that the lifecycle model and the six-factor formula are not capable of representing actual reactor behavior
6. that initial conditions in a question are important and must be defined for the student
7. that a stable period cannot exist in the power range
The sloppiness in structuring the questions and in terminology is truly disturbing ... and appalling.
This is the only subsection of Reactor Theory that provides a set of definitions prior to the question list ... for the stated purpose of consistency. Unfortunately, one result is that certain questions are consistently in technical error. But, you know, whatever's important.
The introductory statement reads: "It is recognized that various symbols are defined differently in different sources. For consistency within this catalog, the following symbols and terms are used."
Five terms are then defined, namely: beta, betaeff, beta-bareff, lambda, and lambda-bar. We will address just the first three of these, since that is where the problem lies.
"beta - delayed neutron fraction - the fraction of neutrons born delayed from fission of a particular nuclide"
"betaeff - effective delayed neutron fraction - the fraction of neutron-induced fissions caused by delayed neutrons for a particular nuclide"
"beta-bareff - the average effective delayed neutron fraction - the weighted average of the betaeff for each fissionable nuclide in the core"
The definition of beta is correct ... although we prefer to identify beta as "the precursor yield fraction", which represents the fraction of fission produced precursor atoms plus prompt neutrons that is precursor atoms (not neutrons). Using the more correct terminology "precursor yield fraction" avoids confusion with "the delayed neutron population fraction", which is neither beta, betaeff, nor beta-bareff. The definition of betaeff is completely incorrect. In modifying beta to betaeff adjustment is made to the magnitude of beta to account for the fact that delayed neutrons are born at lower energy than prompt neutrons, which results in fewer fast fissions (a reduction in beta) and in reduced fast leakage (an increase in beta). This adjustment in beta allows all fission neutrons, prompt and delayed, to be treated mathematically as if born at the same energy ... and in no way changes the definition, or character, of beta from a yield term to one that quantifies the fraction of fissions caused by delayed neutrons. Beta-bareff is nothing more than an isotopic weighted average of the incorrectly defined betaeff. NUKEFACT #1 addresses this misinterpretation error. It now appears in several of the questions that follow ... along with many other errors.
Question 2 : Which of the following statements is true concerning subcritical multiplication?
a. Time to reach equilibrium count rate increases as Keff approaches one.*
b. Source range count rate is directly proportional to Keff.
c. Source strength increases as Keff approaches one.
d. Adding additional neutron sources increases Keff.
Comment: The asterisk indicates the intended correct answer is choice "a". There are two difficulties with this question, namely:
1. Choice "a" is too general, and therefore not necessarily true. Whether or not a longer time is required to reach the equilibrim count rate depends on the magnitudes of the changes in reactivity. For example, if a positive delta-rho of +0.0005 follows a positive delta-rho of +0.0050, the time required to reach equilibrium is not likely to be longer for the smaller delta-rho. The question should provide some measure of the reactivity increments involved, as do several other similar questions.
2. There is a second correct answer. Choice "c" is also correct because delayed neutrons act as source neutrons. The delayed neutron source strength increases as criticality is approached, and exceeds the non-fission source strength for negative reactivities smaller than -0.0065.
The question is technically incorrect. Restate the question and choices as follows:
Question 2: (revised) Which of the following statements is true concerning subcritical multiplication during a power doubling startup?
a. Time to reach equilibrium count rate after a reactivity change increases as Keff approaches one.*
b. Source range count rate is directly proportional to Keff.
c. Non-fission source strength increases as Keff approaches one.
d. Adding additional neutron sources increases Keff.
Question 3: Explain how the neutron population in a subcritical reactor can be made to increase.
Answer: In a subcritical reactor, the fission process is not self-sustaining and would tend to cause the neutron population to decrease. However, source neutrons, which are produced independent of fission, can "make up" for the net loss of neutrons from fission, resulting in a constant neutron population. If positive reactivity is then inserted, the fraction of neutrons lost in the fission process from one generation to the next is reduced. This reduction in neutron "losses" from fission, accompanied by a constant source neutron input, results in an increase in neutron population in the subcritical reactor. (Reference 73, pages 8-83 and 8-4)
Comment: The context in which the reference provides this description is unknown, but appears to be for the Fission Neutron Lifecycle Model, where prompt and delayed neutrons are combined into a single "average" neutron (see NUKEFACT #31: The Many Models of Reactor Behavior). For an actual BWR, the answer is incorrect because there are two reasons that the subcritical neutron population increases in response to a positive reactivity change. The fractional neutron losses in the chains are certainly reduced, over successive prompt neutron lifetimes, as the subcritical keff is increased. But, this increase in prompt neutrons does not persist because the reactor is subcritical on prompt neutrons. There is another more important response occurring when prompt and delayed neutrons are treated separately, as in The Single Precusor Group Model. Here, as in actuality, a positive reactivity increment immediately increases the prompt neutron population, because of reduced losses in the chain. The increased fissions caused by these prompt neutrons, while precursor decay changes very slowly, creates a mismatch between precursor production and precursor loss, which results in an increasing delayed neutron source strength. As the subcritical keff approaches 1.0000, the delayed neutron source strength approaches, and exceeds, that of the non-fission source. It is this increasing delayed neutron source strength as criticality is approached that accounts for the lengthening time to attain equilibrium subcritical multiplication. After reaching initial criticality, it is the increase in delayed neutron source strength that accounts for all power increase.
In using "generation time", the given answer to this question illustrates a major fault in the INPO Catalog section on Reactor Theory. It uses a primitive, and inappropriate, model to explain reactor behavior. This is the so called Fission Neutron Lifecycle Model. This model is incapable of explaining either the subcritical behavior in this question or any behavior in the Delayed-Critical region where the non-fission source is negligible. The model that should be applied, and used exclusively, is the Single Precursor Group Model, which provides a reasonable representation of all reactor behavior. The net result is that the student is burdened, and probably unknowningly confused, by the apparent lack of a single, or standard, model for explaining reactor behavior. Since there is, in fact, a single model that fully explains all reactor behavior, namely The Single Precursor Group Model, the use of two models, one of which is inadequate, is unnecessary and inexcusable.
Question 4: Describe and explain the response of neutron population as equal insertions of positive reactivity are made in a subcritical reactor. Assume the reactor remains subcritical, and that the reactor reaches a stable condition before each reactivity insertion.
Answer: When positive reactivity is inserted, the net "loss" of neutrons from the fission process in each generation is reduced. Therefore, neutron population will increase until the net number of neutrons lost per generation from fission equals the number of source neutrons produced.
The same process occurs with the next reactivity insertion. However, the fraction of neutrons "lost" from the fission process is already smaller than before the first insertion, causing the effect of the second equal-size insertion to be greater. Thus, neutron population will increase by a greater amount and therefore take longer to reach steady state.
The larger increases in population and longer time to equilibrium become more pronounced with each subsequent reactivity insertion (Reference 73, pages 8-48 through 8-55)
Comment: The question is technically incorrect for several reasons, which are as follows:
1. The initial condition is not defined. If the non-fission source is negligible, the initial neutron population will be decreasing with time. The answer makes the assumption that the inital neutron population is at equilibrium subcritical multiplication, but the question never stated this to be the case.
2. Positive reactivity (rho) is never inserted into a reactor. A positive change in reactivity (delta-rho) is introduced. In moving from an initial reactivity condition (rho1) to a final reactivity condition (rho2), the reactivity change (delta-rho) can be expressed as:
|
38.1 |
and on rearranging terms, we have:
 |
38.2 |
Notice that in moving to a new reactivity condition, rho2, reactivity change, delta-rho, is added to the initial reactivity, rho1. Or, conversely, notice that reactivity, rho, is definitely not added to the initial reactivity. Although use of terminology to indicate reactivity change is preferred, to say that an "increment" of reactivity is added is certainly better than saying reactivity is added. Now for those who may consider this nothing more than nitpicking semantics, not so. In later test items where numeric choices for reactivity change are given, they are indicated to represent reactivity. Mathematically, according to equation 38.2, this is flat out WRONG.
3. The wording is ambiguous. What is ... "until the reactor reaches a stable condition" ... supposed to mean? The word stable is totally and completely overused in this Catalog. In one test item or another everything is eventually described as being stable. The word "stable" should be reserved for describing the period associated with an exponential rise, or decline, in power.
4. The comments given for question #3 about using an inappropriate model apply.
The explanation for a greater increase between equilibrium levels for equal positive reactivity increments is rather feeble and, in fact, is difficult to make without resorting to the equation for subcritical multiplication.
Question 6: Which of the following is not a characteristic of subcritical multiplication?
a. Doubling the indicated count rate by reactivity additions will reduce the margin to criticality by approximately one half.
b. For equal reactivity additions, it takes longer for the equilibrium subcritical neutron population level to be reached as Keff approaches 1.
c. If an addition of 1x10-3 percent delta-k/k positive reactivity to a subcritical reactor causes the count rate to increase by 10 CPS, then an addition of 2x10-3 percent delta-k/k positive reactivity will cause count rate to increase by 20 CPS.*
d. A constant neutron population is achieved when the total number of neutrons produced in one generation is equal to the number of source neutrons plus the number of fission neutrons in the next generation.
Comment: The asterisk indicates the intended correct answer is choice "c". The question is technically incorrect because both choice "b" and choice "c" indentify a reactivity change, or a reactivity increment, as reactivity itself (delta-k/k). Reactivity change is expressed as delta-rho (see comment #2 to question #4). Choice "d" is not appropriate because it is based upon the Fission Neutron Life Cycle Model instead of the Single Precusor Group Model (see comment to question #3). Restate the question as follows:
Question 6: (revised)Which of the following is not a characteristic of subcritical multiplication?
a. Doubling the indicated count rate by incremental reactivity additions will reduce the margin to criticality by approximately one half.
b. For equal incremental reactivity additions, it takes longer for the equilibrium subcritical neutron population level to be reached as Keff approaches 1.
c. If an addition of +1x10-3 percent delta-rho to a subcritical reactor causes the count rate to increase by 10 CPS, then an addition of +2x10-3 percent delta-rho will cause count rate to increase by 20 CPS.*
d. A constant neutron population is achieved when the total number of neutrons produced in one prompt neutron lifetime is equal to the number of source neutrons plus the number of fission neutrons in the next prompt neutron lifetime.
Question 7: A reactor startup is in progress. Which one of the following statements describes the response to control-rod withdrawal when taking the reactor critical?
a. The nuclear instrumentation will take longer to stabilize at each new subcritical level.*
b. The reactor will be critical when the period and power level remain constant, with no further rod withdrawal.
c. Each complete control-rod withdrawal will result in the same amount of change in subcritical power level.
d. Each control-rod withdrawal results in an initial negative period followed by a strong positive period.
Comment: The asterisk indicates the intended correct answer is choice "a". The question is technically incorrect because the size of the reacitivity increments is not specified (see comment to question #2). In addition, the correct choice "a" has the nuclear instrumentation stabilizing ... whatever that means. Restate the question as follows:
Question 7: (revised) A power doubling reactor startup is in progress. Which one of the following statements describes the power response to the series of control-rod withdrawals:
a. The nuclear instrumentation indication of neutron level will take longer to reach each new equilibrium subcritical level.*
b. The reactor will be critical when the period and power level remain constant, with no further rod withdrawal.
c. Each complete control-rod withdrawal will result in the same amount of change in subcritical power level.
d. Each control-rod withdrawal results in an initial negative period followed by a strong positive period.
Question 8: Which of the following statements best describes subcritical multiplication during a reactor startup?
a. Subcritical multiplication is the process of using source neutrons to maintain a self-sustaining reaction when Keff is less than 1.*
b. As keff approaches unity, a smaller change in neutron level occurs for a given change in keff.
c. The equilibrium subcritical neutron level is dependent on the source strength and the time between successive reactivity insertions.
d. As keff approaches unity, less time is required to reach the equilibrium neutron level for a given change in keff.
Comment: The asterisk indicates the intended correct answer is choice "a". The question is poorly worded in using "best" ... there should be only one correct answer. Subcritical multiplication is a process that occurs whether a reactor startup is in progress or not. The response given by choice "a" has nothing to do with a reactor startup. In addition, subcritical chains are finite in length ... they are not self-sustaining. Choice "c" incorrectly refers to reactivity insertion rather than incremental reactivity insertion. Restate the question as follows:
Question 8: (revised) Which of the following statements describes subcritical multiplication during a reactor startup?
a. The equilibrium subcritical neutron level is dependent on the non-fission source strength and the subcritical reactivity following each rod withdrawal increment.*
b. As keff approaches unity, a smaller change in neutron level occurs for a given change in keff.
c. The equilibrium subcritical neutron level is dependent on the source strength and the time between successive incremental reactivity insertions.
d. As keff approaches unity, less time is required to reach the equilibrium neutron level for a given change in keff.
Question 9 : Of the following conditions, which group is necessary for subcritical multiplication to occur and be detected?
a. neutron source, moderator, and fissionable material*
b. keff less than one, moderator, and control rods
c. moderator, neutron source, and keff greater than one
d. fissionable material, moderator, and keff less than one
Comment: The asterisk indicates the intended correct answer is choice "a". The question is technically incorrect because it refers to "conditions" and then provides choices that are not conditions, but primarily components of the reactor. The only condition listed is keff. The indicated correct answer includes no conditions. Restate the question and choices as follows:
Question 9: (revised) Which group of reactor constituents is necessary for subcritical multiplication to occur and be detectable on BWR nuclear instruments?
a. neutron source, moderator, and fissionable material*
b. fissionable material, moderator, and control rods
c. control rods, neutron source, and primary coolant
d. fissionable material, moderator, and burnable poison
Question 13: Assume your reactor is being taken critical by periodically withdrawing equal reactivity control-rod increments. Which one of the following statements is correct as Keff approches unity?
a. The neutron level change for successive rod increment pulls becomes smaller.
b. A longer period of time is required to reach the equilibrium neutron level after each rod withdrawal.*
c. A rod withdrawal will result in the reactor becoming slightly supercritical, due to a "prompt jump," and then return to a subcritical level.
d. If the rod withdrawal is stopped for several hours, the neutron level will decrease to source level.
Comment: The asterisk indicates the intended correct answer is choice "b". This question is included only to illustrate that some questions of this nature do correctly include the stipulation as to equal reactivity increments. Note particularly the use of "increments".
Question 15: A subcritical reactor has an initial Keff of 0.8000. Positive reactivity is added until the subcritical count rate is doubled. What reactivity addition caused the count rate to double?
a. .0139 delta-K/K
b. .0361 delta-K/K
c. .1389 delta-K/K*
d. .3611 delta-K/K
Comment: The asterisk indicates the intended correct answer is choice "c". The question is technically incorrect because it states that "positive reactivity is added" and because all choices are indicated to represent reactivity ... instead of reactivity change (see comment #2 to question #4). Restate the question as follows:
Question 15: (revised) A subcritical reactor has an initial Keff of 0.8000. Positive reactivity change is introduced until the subcritical count rate is doubled. What incremental reactivity addition caused the count rate to double?
a. +0.0139 delta-rho
b. +0.0361 delta-rho
c. +0.1389 delta-rho*
d. +0.3611 delta-rho
Question 18: The average effective delayed neutron fraction (beta-bareff) is defined as:
a. (no. of neutrons born delayed)/(no. of neutrons born prompt)
b. (no. of fissions caused by delayed neutrons)/(total no. of fissions caused by fission neutrons)*
c. (no. of neutrons born delayed)/(total no. neutrons born from fission)
d. (no. of delayed neutrons that reach thermal energy)/(no. of prompt neutrons that reach thermal energy)
Comment: The asterisk indicates the intended correct answer is choice "b". The question is technically incorrect for the reason discussed in the introductory remarks to this subsection. Restate the question as follows:
Question 18: (revised) The average effective delayed neutron fraction (beta-bareff) is defined as:
a. (no. of neutrons born delayed)/(no. of neutrons born prompt) corrected for core life and power level
b. (no. of fissions caused by delayed neutrons)/(total no. of fissions caused by fission neutrons) correct for fast fission and fast leakage
c. (no. of precursors produced)/(total no. of precursors plus prompt neutrons produced) corrected for neutron energy and isotopic core compostion*
d. (no. of delayed neutrons that reach thermal energy)/(no. of prompt neutrons that reach thermal energy)
Question 20: Define and contrast the delayed neutron fraction (beta), the effective delayed neutron fraction (betaeff), and the average effective delayed neutron fraction (beta-bareff).
Answer: The delayed neutron fraction (beta) is the fraction of neutrons born delayed from fission of a particular nuclide. The value of beta for each fissionable nuclide is a constant.
The effective delayed neutron fraction (betaeff) is the fraction of neutron-induced fissions caused by delayed neutrons for a particular nuclide. It differs from beta in that it recognizes the lower birth energy of delayed neutrons compared to prompt neutrons. This lower birth energy means delayed neutrons are less likely to cause fast fission or to leak out while slowing down. Depending on core size and fuel loading, betaeff might be greater or smaller than beta. In a typical large BWR betaeff < beta.
The average effective delayed neutron fraction (beta-bareff) is a weighted average of the beta for each fuel isotope in a given core. Because betaeff considers the effect of delayed neutrons on fission in the entire core, it is the term of most use in discussing and predicting reactor response to reactivity changes. (Reference 73, pages 7-28 through 7-38).
Comment: In the third paragraph of the answer, beta is incorrectly used in place of betaeff and betaeff is incorrectly use in place of beta-bareff. The answer is technically incorrect for the reason discussed in the introductory remarks to this subsection. Note the use of ... "response due to reactivity change" ... in the answer. Restate the answer as follows:
Answer (revised): The delayed neutron fraction (beta) is the fraction precursor atoms and prompt neutrons produced that are precusor atoms (not neutrons)for a particular nuclide. The value of beta for each fissionable nuclide is a constant.
The effective delayed neutron fraction (betaeff) is beta corrected for energy. It differs from beta in that it recognizes the lower birth energy of delayed neutrons compared to prompt neutrons. This lower birth energy means delayed neutrons are less likely to cause fast fission or to leak out while slowing down. Depending on core size and fuel loading, betaeff might be greater or smaller than beta. In a typical large BWR betaeff < beta.
The average effective delayed neutron fraction (beta-bareff) is a weighted average of the betaeff for each fuel isotope in a given core. Because beta-bareff considers isotopic makeup of fissile nuclides in the core, it is the term of most use in discussing and predicting reactor response to reactivity changes.
Question 21: Explain the difference between beta (delayed neutron fraction) and betaeff (effective delayed neutron fraction).
Answer: The delayed neutron fraction, beta, is a constant for any specific fissionable nuclide. It is the fraction of neutrons from fission of that nuclide that are born delayed.
The effective delayed neutron fraction, betaeff, represents the relative contribution of delayed neutrons in producing fission. Because delayed neutron birth energies are less than those of prompt neutrons, delayed neutrons are less likely to cause fast fissions (tending to make betaeff < beta) but more likely to remain in the core while slowing down (tending to make betaeff > beta). (Reference 73, pages 7-28 through 7-38).
Comment: The answer is technically incorrect for the reason discussed in the introductory remarks to this subsection. Restate the answer as follows:
Answer (revised): The delayed neutron fraction, beta, is a constant for any specific fissionable nuclide. It is the fraction of neutrons from fission of that nuclide that are born delayed.
The effective delayed neutron fraction, betaeff is an energy corrected form of beta. Because delayed neutron birth energies are less than those of prompt neutrons, delayed neutrons are less likely to cause fast fissions (tending to make betaeff < beta) but more likely to remain in the core while slowing down (tending to make betaeff > beta).
Question 22 : The difference between delayed neutron fraction and effective delayed neutron fraction is that the
a. delayed neutron fraction is based on a finite-sized reactor and the effective delayed neutron fraction is based on an infinite-sized reactor
b. effective delayed neutron fraction will remain constant over core life but the delayed neutron fraction changes due to fuel changes that occur as the core ages
c. delayed neutron fraction considers neutrons at their birth while the effective delayed neutron fraction considers neutrons causing fission*
d. delayed neutron fraction is a weighted average of various fission products and the effective delayed neutron fraction is not
Comment: The asterisk indicates the intended correct answer is choice "c". The question is technically incorrect for the reason stated in the introductory remarks to this subsection. Restate the question and choices as follows:
Question 22: (revised) The difference between delayed neutron fraction and effective delayed neutron fraction is that the
a. delayed neutron fraction is based on a finite-sized reactor and the effective delayed neutron fraction is based on an infinite-sized reactor
b. effective delayed neutron fraction will remain constant over core life but the delayed neutron fraction changes due to fuel changes that occur as the core ages
c. the effective delayed neutron fraction is corrected for the lower emission energy of the delayed neutrons*
d. delayed neutron fraction is a weighted average of various fission products and the effective delayed neutron fraction is not
Question 26: Which of the following terms defines reactor period?
a. the time for reactor power to change by a factor of "e" (or 2.71873)*
b. the rate of change of reactor power expressed in decades per minute (DPM)
c. the time for reactor power to change by a factor of 10
d. the rate of change of reactor power expressed in decades per second (DPS)
Comment: The asterisk indicates the intended correct answer is choice "a". The question is flawed because reactor period is expressed in specific time units, namely seconds. All other questions related to reactor period use units of seconds. Restate the question as follows:
Question 26: (revised) Which of the following terms defines reactor period?
a. the number of seconds for reactor power to change by a factor of "e" (or 2.71873)*
b. the rate of change of reactor power expressed in decades per minute (DPM)
c. the number of minutes for reactor power to change by a factor of 10
d. the rate of change of reactor power expressed in decades per second (DPS)